GIFT OF Yellow Yellow and Red Yellow, Red and Blue Finished Result Yellow, Red, Blue and Black FOUR COLOR PRINTING PRACTICAL PHYSICS BY HENRY S. CARHART, Sc.D., LL.D. \\ FORMERLY PROFESSOR OF PHYSICS, UNIVERSITY OF MICHIGAN AND HORATIO N. CHUTE, M.S. INSTRUCTOR IN PHYSICS IN THE ANN ARBOR HIGH SCHOOL ALLYN AND BA.CON BOSTON NEW YORK CHICAGO ATLANTA SAN FRANCISCO COPYRIGHT, 1920, BY HENRY S. CARHART AND HORATIO N. CHUTE XortoonD J. S. Gushing Co. Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE Practical Physics aims above all things to justify its title. In introducing an exceptionally large number of applications, the authors have not lost sight of the fact that the most prac- tical book is the one from which the pupil can most easily learn. To secure this practical quality, the material is presented in the simplest and clearest language : short sentences and paragraphs, terse statements, careful explanations, and an in- ductive development of each principle. The subject matter is logically arranged in the order followed by most secondary schools. Each principle is illustrated not only by diagrams, problems, and questions, but by its most striking applications. While the number of these practical applications is un- usually large, nothing has been included merely because it is sensational ; every application illustrates some principle treated in the book. The utmost care has been taken to keep the whole work within easy range of the average pupil's ability, and to provide material that can be easily mastered in a school year. The World War has emphasized once more the universality of physics. This universality is brought out in Practical Physics from the very outset, always, however, with care that the pupil understand the basic physical principles which under- lie each application. July 4, 1920. m 459995 CONTENTS Chapter I. Introduction PAGE I. Matter and Energy . , 1 II. Properties of Matter . . ' $ III. Physical Measurements ...... 16 Chapter II. Molecular Physics I. Molecular Motion 25 II. Surface Phenomena . . ' . . . . .... 29 III. Molecular Forces in Solids . .. . ... 34 Chapter III. Mechanics of Fluids I. Pressure of Fluids ,39 II. Bodies Immersed in Liquids . . . , . 53 III. Density and Specific Gravity . . ._ ., .58 IV. Pressure of the Atmosphere . .- . '. VV 65 V. Compression and Expansion of Gases . , . , 73 VI. Pneumatic Appliances . . . . . . 83 Chapter IV. Motion I. Motion in Straight Lines 91 II. Curvilinear Motion . . . . . 99 III. Simple Harmonic Motion . . - . . . . 101 Chapter V. Mechanics of Solids I. Measurement of Force * . . . 104 II. Composition of Forces and of Velocities . . .107 III. Newton's Laws of Motion . . . ; H6 IV. Gravitation. . . , ^ .122 V. Falling Bodies 128 VI. Centripetal and Centrifugal Force . . . .133 VII. The Pendulum .136 v VI CONTENTS Chapter VI. Mechanical Work PAG , I. Work and Energy . . r . . . . 143 II. Machines . . 155 Chapter VII. Sound I. Wave Motion 176 II. Sound and its Transmission 181 III. Velocity of Sound 184 IV. Reflection of Sound 186 V. Resonance . . . . . . . . .188 VI. Characteristics of Musical Sounds .... 191 VII. Interference and Beats 194 VIII. Musical Scales . . . . . . . .196 IX. Vibration of Strings . 201 X. Vibration of Air in Pipes 205 XI. Graphic and Optical Methods 208 Chapter VIII. Light I. Nature and Transmission of Light . . . . 214 II. Photometry . .- . . . . . .219 III. Reflection of Light . . . . . . .223 IV. Refraction of Light . . . . . . . 238 V. Lenses 246 VI. Optical Instruments ....... 255 VII. Dispersion 263 VIII. Color 271 IX. Interference and Diffraction . . 276 Chapter IX. Heat I. Heat and Temperature ...... 280 II. The Thermometer 282 III. Expansion .287 IV. Measurement of Heat 297 V. Change of State 299 VI. Transmission of Heat 309 VII. Heat and Work. 319 CONTENTS Vii Chapter X. Magnetism PA . .. . . 354 VI. Atmospheric Electricity . . . ... 358 Chapter XII. Electric Currents I. Voltaic Cells . . 361 II. Electrolysis ...... . . .373 III. Ohm's Law and its Applications . . . . 378 IV. Heating Effects of a Current . . , . .385 V. Magnetic Properties of a Current . - . . . 387 VI. Electromagnets . . * ... . .392 VII. Measuring Instruments ... . . . 394 Chapter XIII. Electromagnetic Induction I. Faraday's Discoveries 401 II. Self-Induction 405 III. The Induction Coil 406 IV. Radioactivity and Electrons . . '. . .416 Chapter XIV. Dynamo-Electric Machinery I. Direct Current Machines 421 II. Alternators and Transformers 431 III. Electric Lighting 442 IV. The Electric Telegraph 447 V. The Telephone . . . . '. . . .451 VI. Wireless Telegraphy 453 yiii CONTENTS Chapter XV. The Motor Car PAGE I. H. in. IV. The Engine . . . e . . The Storage Battery . The Chassis and Running Gear . The Brake 460 466 467 468 V. The Clutch 469 VI. Transmission and Differential 470 VII. VIII. The Steering Device The Starter -. 471 471 IX. On the Road .- 472 X. The Pedestrian 473 Appendix I. II. 475 479 Conversion Tables . . . . . in. Mensuration Rules ....... 481 IV. Table of Densities 482 V. Index Geometrical Construction for Refraction of Light . 483 1 FULL PAGE ILLUSTRATIONS Four Color Process Printing Frontispiece FACING PAGE Electric Welding 16 Bureau of Standards, Washington .... Common Crystals ... 35 Galileo Galilei 42 Blaise Pascal Elephant Butte Dam . . . . . . . .'-_- .50 Dry Dock " Dewey " 58 fifi Hydro-airplanes Motion and Force Sir Isaac Newton 124 Yosemite Fall Centrifugal Force Pisa Cathedral United States 16-inch Gun 150 Lord Kelvin 154 LordRayleigh Photographs of Sound Waves I 83 Echo Bridge ' - Hermann von Helmholtz . 193 Niagara Falls Power Plant . . . . Parabolic Mirror at Mount Wilson Moving Picture Film ... 258 Various Spectra Bridge over the Firth of Forth .... James Watt James Prescott Joule .... A Row of Corliss Engines .... qOO Four-valve Engine Section and Rotor of Steam Turbine 323 Front and Rear Views of Airplane Benjamin Franklin 358 tx X FULL PAGE ILLUSTRATIONS FACING PAGE Hans Christian Oersted . . . . . . . 366 Alessandro Volta . . . . 382 Georg Simon Ohm . . . . . . . . . 382 James Clerk-Maxwell ......... 392 Joseph Henry . 400 Michael Faraday 401 Sir William Crookes . - 412 Wilhelm Konrad Roentgen ........ 412 Madame Curie . . . . . . . . 418 Sir Joseph John Thomson 419 Field Magnet and Drum Armature of D. C. Generator . . 426 Electric Engine Crossing the Rockies 430 Armature Core and Field Magnet of A. C. Generator . . 431 Dam and Power House, Great Falls, Montana .... 438 Transformers and Switches 439 Stator and Field of A. C. Generator 440 Stator of Three-phase Motor and Motor Complete . . . 441 Alexander Graham Bell 449 Samuel F. B. Morse 449 Field Wireless of the United States Army 452 Wireless Room in a Transatlantic Liner ..... 453 Heinrich Rudolf Hertz 458 Thomas Alva Edison 459 Guglielmo Marconi 459 PRACTICAL PHYSICS CHAPTER I INTRODUCTION I. MATTER AND ENERGY 1. Physics Defined. Physics is the science which treats of the related phenomena of matter and energy. It includes mechanics, sound, light, heat, magnetism, and electricity. A MOTOR CAR. Probably the automobile is the best general application of the principles of physics. Mechanics of solids is illustrated by its springs, bolts, and most of its moving parts ; mechanics of liquids by the circulation of its water- cooling system ; sound by its horn ; light by its headlights ; heat by the explosions of its engine ; and magnetism and electricity by its electrical battery and its starting and lighting systems. : INTRODUCTION r and fwwrgy scarcely admit of definition except by means of their properties. Matter is everything we can see, taste, or touch, such as earth, water, wood, iron, gas in short, everything that occupies space. Energy is whatever produces a change in the motion or condition of matter, especially against resistance opposing BRITISH "TANK" CROSSING A SHELL-HOLE. The tank is a land battleship, carrying guns and running on its own track which it carries with it. In this way it can cross holes and trenches which would stop a vehicle with wheels. the change ; that is, energy is the universal agency by means of which work is done. Water in an elevated reservoir, steam under pressure in a boiler, a flying shell with its content of explosives, all these may do work, may over- come resistance, or change the position or motion of other bodies. They possess energy which is transferred from them to the bodies on which work is done. 2. The Universal Science. Since everything which we recognize by the senses is matter, and every change in THE UNIVERSAL SCIENCE 3 matter involves energy, it is plain that physics is a uni- versal science, touching our life at every point. Count- less physical phenomena are taking place about us every day ; a girl cooking, a boy playing ball, the fire-whistle blowing, the sun giving light and heat, a flag flapping in the wind, an airplane soaring aloft, an apple falling from A WRIGHT-MARTIN " BOMBER." An airplane which can cross the United States from coast to coast with only one stop. a tree, a train or motor car whizzing by, a British " tank " crossing a shell-hole, all are examples of matter and associated energy. Physics is not so much concerned with matter alone or with energy alone as with the relations of the two. A baseball is of little interest in itself ; it becomes interesting only in connection with a bat and the energy of the player's arm. The engine driver's inter- est is not so much in the engine itself as in the engine with steam up ready to drive it. No one would care to buy an automobile to 4 INTRODUCTION stand in a garage ; its attractiveness lies in the fact that it becomes a thing of life when its motor is vitalized by the heat of combustion of gasoline vapor. 3. Applications of the Principles of Physics. The appli- cations of the principles of physics in the household and in the familiar arts are very numerous and affect us constantly in daily life. Water under pressure is deliv- ered for domestic use, and fuel is used in the liquid or in the gaseous form as well as in the solid. Electricity lights our houses, toasts our bread, and even cooks our daily food. The electric motor runs our vacuum cleaners and our sewing machines. The applications of physics in modern life are so nu- merous and they are changing so rapidly that we cannot expect to learn about all of them in a year's study ; but physical principles remain the same ; and if we acquire a knowledge of these principles and of their familiar appli- cations, we shall be prepared to understand and to ex- plain other applications that have been made possible by the science of physics. So this book lays emphasis on the underlying princi- ples of physics, illustrating them by some of their inter- esting applications, leaving it to the enthusiasm and ingenuity of both teacher and pupils to supplement the applications with others drawn from life and from scien- tific journals. 4. States of Matter. Matter exists in three distinct states, exemplified by water, which may assume either the solid, the liquid, or the gaseous form, as ice, water, or water vapor. Briefly described, Solids have definite size and shape, and offer resistance to any change of these. FORCE Liquids have definite size, but they take the shape of the container and have a free surface. G-ases have neither definite size nor shape, both depending on the container. These are not all the differences between solids, liquids, and gases, but they serve to distinguish between them. Some substances are neither wholly in the one state nor in the other. Sealing wax softens by heat and passes gradually from the solid to the liquid state. Shoemaker's wax breaks into fragments like a solid under the blow of a hammer, but under long-continued pres- sure it flows like a liquid, though slowly, and it may be molded at will. 5. Force. Our primitive idea of force is that of a push or a pull ; it is derived from experience in making muscular exertion to move bodies or to stop their motion. Pushing a chair, throwing a stone, pulling a cart, row- ing a boat,' stretch- ing a rubber band, bending a bow, catching a ball, lift- ing a book, all re- quire muscular ef- fort in the nature of a push or a pull. Thus force implies A TRIP HAMMER - a push or a pull This wei hs several tons and will exert an , , ' * enormous force on the red-hot iron below it. though not neces- sarily muscular; and the effect of the action of a force on a body free to move is to give it motion or to change 6 INTRODUCTION its motion. For the present we shall make use of the units of force familiar to us, such as the pound of force and the gram of force, meaning thereby the forces equal to that required to lift the mass of a pound and that of a gram respectively. II. PROPERTIES OF MATTER 6. The Properties of Matter are those qualities that serve to define it, as well as to distinguish one substance from another. All matter has extension or occupies space, and so extension is a general property of matter. On the other hand common window glass lets light pass through it, or is transparent, while a piece of sheet iron does not transmit light, or is opaque. A watch spring recovers its shape after bending, or is elastic, while a strip of lead possesses this property in so slight a degree that it is classed as in- elastic. So we see that transparency and elasticity are special properties of matter. 7. Extension. All bodies have three dimensions, length, breadth, and thickness. A sheet of tissue paper or of gold leaf, at first thought, appears to have but two dimensions, length and breadth; but while its third dimension is rel- atively small, if its thickness should actually become zero, it would cea,se to be either a sheet of paper or a piece of gold leaf. Extension is the property of occupying space or having dimensions. 8. Impenetrability. While matter occupies space, no two portions of matter can occupy the same space at the same time. The volume or bulk of an irregular solid, such as a lump of coal, may be measured by noting the volume of liquid displaced when the solid is completely immersed in it. The general property of matter that no two bodies can occupy the same space at the same time is known as impenetrability. INERTIA Put a lump of coal into a tall graduate partly filled with water, as in Fig. 1. Note the reading at the surface of the water be- fore and after putting in the coal ; the differ- ence is the volume of water displaced, or the volume of the piece of coal. 9. Inertia. The most conspicuous and characteristic general property of matter is inertia. Inertia is the prop- erty which all matter possesses of resist- ing any attempt to start it if at rest, to stop it if in motion, or to change either FIGURE 1. MEASURING the direction or the amount of its motion. VOLUME BY DlSPLACEMENT - If a moving body stops, its arrest is always owing to something outside of itself ; and if a body at rest is set moving, motion must be given to it by some other body. It is a familiar fact that no body of any sort will either start or stop moving of itself. 10. Illustrations of Inertia- Many familiar facts are due to inertia. When a street car stops suddenly, a person standing con~ tinues by inertia to move forward, or is apparently thrown toward the front of the car ; the driver of a racing motor car is apparently thrown with violence when the rapidly moving car collides with a post or a tree ; the fact is the car is violently stopped, while the driver continues to move forward as be- FIGURE 2. STATUE TWISTED AROUND BY EARTHQUAKE. 8 INTRODUCTION fore the collision. When a fireman shovels coal into a furnace, he suddenly arrests the motion of the shovel and leaves the coal to move forward by inertia. A smooth cloth may be snatched from under a heavy dish with- out disturbing it. The violent jar to a water pipe when a faucet is quickly closed is accounted for by the inertia of the stream. Tall columns, chimneys, and monuments are sometimes twisted around by violent earthquake move- ments (Fig. 2). The sudden circular motion of the earth under a column leaves it standing still, while the slower return motion carries it around. The persistence with which a spinning top maintains its axis of rotation in the same direction is due to its inertia. If it is spun on a smooth surface, like a mirror, and is tossed into the air, it will not tumble over and over, but will keep upright (Fig. 3) and may be caught on the mirror, still spin- ning on its point. The gyrostat wheel acts on the same principle, and so does Sperry's gyrostatic compass and his stabilizer for ships and aeroplanes. If a round flat biscuit is pitched into the air, there is no certainty as to how it will come down ; but if it is given a spin before it leaves the hand, the axis of spin- ning keeps parallel to itself (Fig. 4). If one wants to throw a hoop or a hat to some one to catch on a stick, one gives to the hoop or the hat a spin. So also if one wants to throw a quoit and be FIGURE 3. SPINNING MAINTAINS ITS Axis OF TATION. TOP Ro- FIGURE 4. SPINNING BISCUIT. MA88 9 certain how it will alight, one gives it a spin. Its inertia keeps it spinning around the same axis in space. Tie a piece of twine to a heavy weight, such as a flatiron. By pull- ing slowly the flatiron may be lifted, but a sudden jerk on the twine will break it because of the inertia of the weight. Suspend a heavy weight by a cotton string, as in Fig. 5, and tie a piece of the same string to the under side of the weight. A steady downward pull at B will break the upper string because it carries the greater load. A sudden downward pull on B will break the lower string before the pull reaches the upper one on account of the inertia of the weight. 11. Mass. We are all familiar with the fact that the less matter there is in a body, the more easily it is moved, and the more FIGURE 5. - ., ., . 11- / ^ INERTIA EXPERI- easily it is stopped when in motion. One MENT. can tell an empty barrel from a full one by a kick, a block of wood from a brick by shoving it with the foot, and a tennis ball from a baseball by catching it. The mass of a body is the quantity of matter it contains ; but since the inertia of a body is proportional to the quantity of matter in it, it is not difficult to see that the mass of a body is the measure of its inertia. While mass is most easily measured by means of weigh- ing, it must not be confused with weight ( 132), because mass is independent of the earth-pull or gravity. The mass of a meteoric body is the same when flying through space as when it strikes the earth and embeds itself in the ground. If it could reach the center of the earth, its weight would become zero; at the surface of the sun it would weigh nearly twenty-eight times as much as at the earth's surface; but its mass would be the same every- where. For this reason, and others which will appear later, in discussing the laws of physics we prefer to speak 10 INTRODUCTION of mass when a student thinks the term weight might be used as well. 12. Cohesion and Adhesion. All bodies are made up of very minute particles, which are separately invisible, and are called mole- cules. Cohesion is the force of at- traction between molecules, and it binds together the molecules of a substance so as to form a larger mass than a mol- ecule. Adhe- sion is the force uniting bodies by their adjacent surfaces. When two clean sur- faces of white- hot wrought iron are brought into close con- tact by hammer- ing, they cohere and become a single body. If a clean glass rod be dipped into water and then withdrawn, a drop will ad- here to it. Glue, adhesive plaster, and postage stamps stick by adhesion. Mortar adheres to bricks and nickel plating to iron. COHESION. Mr. Lambirth, dean of blacksmiths in America, has spent 35 years at the head of the forge work at the Massachusetts Institute of Technology. POROSITY 11 Suspend from one of the arras of a beam balance a clean glass disk by means of threads cemented to it (Fig. 6). After counterpoising the disk, place below it a vessel of water, and adjust so that the disk just touches the surface of the water when the beam of the balance is horizontal. Now add weights to the opposite pan until the disk is pulled away from the water. Note that the under surface of the disk is wet. The adhesion of the water to the glass is greater than the cohesion between the molecules of the water. If lycopodium powder be carefully sifted on the sur- face of the water, the water will not wet the disk and there will be no adhesion. If mercury be substituted for water, a much greater force will be necessary to separate the disk from the mercury, but no mercury will adhere to it. The force of cohesion between the molecules of the mercury is greater than the adhesion between it and the glass. Cut a fresh, smooth surface on each of two lead bullets and hold these surfaces gently together. They will not stick. Now press them tightly together with a slight twisting motion. They will adhere quite firmly. This fact shows that molecular forces act only through insensible distances. It has been shown that they vanish in water at a range of about one five-hundred-thousandth of an inch. An interesting example of selective adhesion occurs in the winning of diamonds in south Africa. The mixed pebbles and other worthless stones, with an occasionaUdiamond, are washed down an inclined shaking surface covered with grease. Only the diamonds and a few other precious stones stick to the grease ; the rest are washed away. 13. Porosity. Sandstone, unglazed pottery, and similar bodies absorb water without change in volume. The water fills the small spaces called pores, which are visible either to the naked eye or under a microscope. All matter is FIGURE 6. GLASS ADHERES TO WATER. 12 INTRODUCTION probably porous, though the pores are invisible, and the corresponding property is called porosity. In a famous experiment in Florence many years ago, a hollow sphere of heavily gilded silver was filled with water and put under pressure. The water came through the pores of the silver and gold and stood in beads on the surface. Francis Bacon observed a similar phenomenon with a lead sphere. Oil penetrates into marble and spreads through it. Even so dense a substance as agate is porous, for it is artificially colored by the ab- sorption, first of one liquid and then of another which acts chemically on the first ; the result is a deposit of coloring matter in the pores of the agate. 14. Tenacity and Tensile Strength. Tenacity is the resist- ance which a body offers to being torn apart. The tensile strength of wires is tested by hanging them vertically and loading with successive weights until they break (Fig. 7). The breaking weights for wires of different materials but of the same cross section differ greatly. A knowl- edge of tensile strength is essential in the de- sign of telegraph wires and cables, suspension bridges, and the tension members of all steel structures. Tenacity diminishes with the duration of the pull, so that wires sometimes break with a load which they have supported for a long time. Lead has the least tenacit 7 of a11 solid metals, STRENGTH and cast steel the greatest. Even the latter is OF WIRE. excee d e d by fibers of silk and cotton. Single fibers of cotton can support millions of times their own weight. 15. Ductility. Ductility is the property of a substance which permits it to be drawn into wires or filaments. Gold,' TENACITY AND TENSILE STRENGTH 13 copper, silver, and platinum are highly ductile. The last is the most ductile of all. It has been drawn into wire only 0.00003 inch in diameter. A mile of this wire would weigh only 1.25 grains. AERIAL TRAMWAY OVER THE WHIRLPOOL RAPIDS. The cables have great tensile strength to support the car. Other substances are highly ductile only at high tem- peratures. Glass has been spun into such fine threads that a mile of it would weigh only one third of a grain. Melted quartz has been drawn into threads not more than 0.00001 inch in diameter. Such threads have nearly as great tenacity as steel. 14 INTRODUCTION 16. Malleability. Malleability is a property which per- mits of hammering or rolling some metals into thin sheets. Gold leaf, made by hammering between skins, is so thin that it is partially trans- parent and trans- mits green light. Zinc is malleable when heated to a temperature of from 100 to 150 C. (centigrade scale). It can then be rolled into sheets. Nickel at red heat can be worked like wrought iron. Mal- leable iron is made from cast iron by heating it for sev- eral days in contact with a substance which removes some of the carbon from the cast iron. 17. Hardness and Brittleness. Hardness is the resistance offered by a body to scratching by other bodies. The relative hardness of two bodies is ascertained by finding which will scratch the other. Diamond is the hardest of all bodies because it scratches all others. Sir William Crookes has shown that diamonds under great hydraulic pressure be- tween mild steel plates completely embed themselves in the metal. Carborundum, an artificial material used for grinding metals, is nearly as hard as diamond. Brittleness is aptness to break under a blow. It must be POURING MOLTEN IRON INTO MOULDS. HARDNESS AND BRITTLENESS 15 distinguished from hardness. Steel is hard and tough, while glass is hard and brittle. Tool steel becomes glass-hard and brittle when suddenly cooled from a high temperature. The tempering of steel is the process of giving the degree of hardness required for various purposes. It consists usually in first plunging the article at red heat into cold water or other liquid to give it an excess of hardness ; then reheating gradually until the hardness is reduced, or " drawn down," to the required degree. The indication of the hardness is the color appearing on a polished portion, such as straw- yellow, brown-yellow, purple, or blue. The process of annealing as applied to iron and glass is used to render them less brittle. It is done by cooling very slowly and uni- formly from a high temperature. Soft iron is thus made more ductile, while glass is relieved from the molecular stresses set up in rapid cooling, and it thus becomes tougher and more uniform. The best lamp chimneys are annealed by the manufacturer. Disks of glass for telescope lenses and mirrors must be carefully annealed to prevent fracture and warping during the process of grinding and polishing. Prince Rupert drops (Fig. 8) are made by dropping melted glass into cold water. The outside is suddenly chilled and solidified, while the interior is still fused, and when it cools it must ac- commodate itself to the dimensions of the outer skin. The drop is thus under great tension. With a pair of pliers break off the tip of the drop under water in a tumbler, or scratch with a file ; the whole drop will fly to powder with almost explosive violence. A large tall jar on foot is usually thick at the bot- : torn, and imperfectly annealed. Such jars have not infrequently been broken by a scratch inside, made, for example, by stirring emery powder in water by means of a long wooden stick. A scratch inside is usually fatal to a lamp chimney. A large glass tube may be cut in two by scratching it around on the inside by means of an appropriate tool, and then carefully heating it in a small gas flame. 16 INTRODUCTION Exercises 1. Given a large crystal of rock candy. Can its volume be deter- mined by the method outlined under impenetrability ? How ? 2. The volume of a bar of lead can be reduced by pounding it. Explain. 3. A small quantity of sugar can be dissolved in a cup of water without increasing the volume. Explain. 4. A quick blow with a heavy knife will often remove smoothly the neck of a glass bottle, while a less vigorous blow will shatter the bottle. Explain. 5. Why can an athlete jump farther in a running jump than in a standing jump ? 6. By striking the end of the handle it can be driven into a heavy ax much better than by pounding the ax. Why ? 7. A man standing on a flat-bottomed car that is moving jumps 'vertically upward. Will he come down on the spot from which he jumped ? Explain. 8. If a top be set spinning it stands up ; if not spinning it topples over. Explain. 9. A bullet fired from a rifle will pass through a pane of glass, cutting a fairly smooth hole ; a stone thrown by the hand on striking a pane of glass will shatter it. Explain. 10. Name three properties of matter that are characteristic of it. 11. A rolling wheel does not fall over, but one not rolling topples over. Why ? 12. Why hold a heavy hammer against a spring board when driv- ing a nail into it? 13. In Jules Verne's Trip to the Moon the incident is told that when a few days on the way the dog died and was thrown overboard. To their surprise the dead dog followed along after them. Is Verne's Physics correct ? Explain. II. PHYSICAL MEASUREMENTS 18. Units. To measure any physical quantity a certain definite amount of the same kind of quantity is used as the unit. For example, to measure the length of a body, some MEASURES OF LENGTH 17 arbitrary length, as a foot, is chosen as the unit of length; the length of a body is the number of times this unit is con- tained in the longest dimension of the body. The unit is always expressed in giving the magnitude of any physical quantity; the other part of the expression is the numerical value. For example, 60 feet, 500 pounds, 45 seconds. In like manner, to measure a surface, the unit, or stand- ard surface, must be given, such as a square foot; and to measure a volume, the unit must be a given volume, such, for example, as a cubic inch, a quart, or a gallon. 19. Systems of Measurement. Commercial transactions in most civilized countries are carried on by a decimal system of money, in which all the multiples are ten. It has the advantage of great convenience, for all numerical operations in it are the same as those for abstract numbers in the dec- imal system. The system of weights and measures in use in the British Isles and in the United States is not a dec- imal system, and is neither rational nor convenient. On the other hand most of the other civilized nations of the world within the last fifty years have adopted the metric system, in which the relations are all expressed by some power of ten. The metric system is in well-nigh universal use for scientific purposes. It furnishes a common numer- ical language and greatly reduces the labor of computation. 20. Measures of Length. In the metric system the unit of length is the meter. In the United States it is the dis- tance between two transverse lines on each of two bars of platinum-iridium at the temperature of melting ice. These bars, which are called "national prototypes," were made by an international commission and were selected by lot after two others had been chosen as the " international pro- totypes " for preservation in the international laboratory on neutral ground at Sevres near Paris. Our national 18 INTRODUCTION prototypes are preserved at the Bureau of Standards in Washington. Figure 9 shows the two ends of one of them. The only multiple of the meter in general use is the kilometer, equal to 1000 meters. It is used to measure such distances as are expressed in miles in the English system. FIGURE 9. ENDS OF METER BAR. The Common Units in the Metric System are : 1 kilometer (km.) = 1000 meters (m.) 1 meter = 100 centimeters (cm.) 1 centimeter =10 millimeters (mm.) The Common Units in the English System are: 1 mile (mi.) = 5280 feet (ft.) 1 yard (yd.) = 3 feet 1 foot = 12 inches (in.) By Act of Congress in 1866 the legal value of the yard is fffy meter; conversely the meter is equal to 39.37 inches. The inch is, therefore, equal to 2.540 centimeters. 100 MILLIMETERS = 10 CENTIMETERS = 1 DECIMETER = 3. 937 INCHES. INCHES AND TENTHS FIGURE 10. CENTIMETER AND INCH SCALES. The unit of length in the English system for the United States is the yard, defined as above. The relation between the centimeter scale and the inch scale is shown in Fig. 10. CUBIC MEASURE 19 Square Inch 21. Measures of Surface. In the metric system the unit of area used in the laboratory is the square centimeter (cm. 2 ). It is the area of a square, the edge of which is one centimeter. The square meter (m. 2 ) is often em- ployed as a larger unit of area. In the English system both the square inch and the square foot are in common use. Small areas are measured in square inches, while the area of a floor and that of a house lot are given in square feet; larger land areas are in acres, 640 of which are contained in a square mile. The square inch contains 2.54 x 2.54 = 6.4516 square centimeters. The relative sizes of the two are shown in Fig. 11. The area of regular geometric figures is obtained by computation from their linear dimensions. Thus the area of a rectangle or of a parallelogram is equal to the product of its base and its altitude (A = b x h) ; the area of a triangle is half the product of its base and its altitude (A =^b x h) ; the area of a circle is the product of 3.1416 (very nearly %p) and the square of the radius (A irr 2 ) ; the surface of a sphere is four times the area of a circle through its center (A = 4ur 2 ). For other surfaces, see Appendix III. FIGURE 11. SQUARE CENTIMETER AND SQUARE INCH. FIGURE 12. CUBIC CENTIMETER AND CUBIC INCH. 22. Cubic Measure. The smaller unit of volume in the metric system is the cubic centimeter. It is the vol- ume of a cube, the edges of which are one centimeter long. The cubic inch equals (2.54) 8 or 16.387 cubic centimeters. The relative' sizes of the two units are shown in Fig. 12. In the English system the 20 INTRODUCTION Cm .3 15C. -500 FIGURE 13. CYLINDRI- CA L GLASS GRADUATE. cubic foot and cubic yard are employed foi larger volumes. The cubical capacity of a room or of a freight car would be expressed in cubic feet ; the volume of building sand and gravel or of earth embankments, cuts, or fills would be in cubic yards. The unit of capacity for liquids in the metric system is the liter. It is a decimeter cube, that is, 1000 cubic centimeters. The imperial gallon of Great Britain contains about 277.3 cubic inches, and holds 10 pounds of water at a temperature of 62 Fahrenheit. The United States gallon has the capacity of 231 cubic inches. Common Units in the Metric Sys- tem : 1 cubic meter (m. 3 ) = 1000 liters (1.) 1 liter = 1000 cubic centimeters (crn. 3 ) Common Units in the English System : 1 cubic yard (cu.yd.)= 27 cubic feet (cu. ft.) 1 cubic foot = 1728 cubic inches (cu. in.) 1 U. S. gallon (gal.) = 4 quarts (qt.) = 231 cubic inches 1 quart = 2 pints (pt.) The volume of a regular solid, or of a solid geometrical figure, may be calculated from its linear dimensions. Thus, the number of cubic feet in a room or in a rectangular block of marble is found by get- ting the continued product of its length, its breadth, and its height, all measured in feet. The volume of a cylinder is equal to the product of the area of its base (Tir 2 ) and its height, both measured in the same system of units. Liquids are measured by means of graduated vessels of metal or of glass. Thus, tin vessels holding a gallon, a quart, or a pint are used UNITS OF MASS 21 FIGURE 14. VOLUMETRIC FLASK. for measuring gasoline, sirup, etc. Bottles for acids usually hold either a gallon or a half gallon, and milk bottles contain a quart, a pint, or a half pint. Glass cylindrical graduates (Fig. 13) and volumetric flasks (Fig. 14) are used by phar- A macists, chemists, and physicists to measure liquids. In the metric system these are graduated in cubic cen- timeters. 23. Units of Mass. The unit of mass in the metric system is the kilogram. The United States has two prototype kilograms made of platinum-iridium and preserved at the Bureau of Standards in Washington (Fig. 15). The gram is one thousandth of the kilogram. The latter was originally de- signed to represent the mass of a liter of pure water at 4 C. (centigrade scale). For practical purposes this is the kilogram. The gram is therefore equal to the mass of a cubic centimeter of water at the same temperature. The mass of a given body of water can thus be immediately inferred from its vol- ume. The unit of mass in the English system is the avoirdupois pound. The ton of 2000 pounds is its chief multiple ; its submultiples are the ounce and the grain. The avoirdupois pound is equal to 16 ounces FIGURE 15. STANDARD KILOGRAM. and to 7000 grains. 22 INTRODUCTION The " troy pound of the mint " contains 5760 grains. In 1866 the mass of the 5-cent nickel piece was legally fixed at 5 grams ; and in 1873 that of the silver half dollar at 12.5 grams. One. gram is equal approximately to 15.432 grains. A kilogram is very nearly 2.2 pounds. More exactly, one kilogram equals 2.20462 pounds. All mail matter transported between the United States and the fifty or more nations signing the International Postal Convention, including Great Britain, is weighed and paid for entirely by metric weight. The single rate upon international letters is applied to the standard weight of 15 grams or fractional part of it. The International Parcels Post limits packages to 5 kilograms; hence the equivalent limit of 11 pounds. Common Units in the English System : 1 ton (T.) = 2000 pounds (Ib.) 1 pound = 16 ounces (oz.) 1 ounce = 437.5 grains (gr.) Common Units in the Metric System : 1 kilogram (kg.) = 1000 grams (g.) 1 gram = 1000 milligrams (mg.) 24. The Unit of Time. The unit of time in universal use in physics and by the people is the second. It is .g-g-J^ of a mean solar day. The number of seconds be- tween the instant when the sun's center crosses the me- ridian of any place and the instant of its next passage over the same meridian is not uniform, chiefly because the motion of the earth in its orbit about the sun varies from day to day. The mean solar day is the average length of all the variable solar days throughout the year. It is divided into 24 x 60 x 60 = 86,400 seconds of mean solar time, the time recorded by clocks and watches. PROBLEMS 23 The sidereal day used in astronomy is nearly four minutes shorter than the mean solar day. 25. The Three Fundamental Units. Just as the meas- urement of areas and of volumes reduces simply to the measurement of length, so it has been found that the measurement of most other physical quantities, such as the speed of a ship, the pressure of water in the mains, the energy consumed by an electric lamp, and the horse power of an engine, may be made in terms of the units of length, mass, and time. For this reason these three are considered fun dam ental units to distinguish them from all others, which are called derived units. The system now in general use in the physical sciences employs the centimeter as the unit of length, the gram, as the unit of mass, and the second as the unit of time. It is accordingly known as the c. g. s. (centimeter-gram- second) system. Problems In solving these problems the student should use the relations and values given in 20, 22, and 23. 1. Reduce 76 cm. to its equivalent in inches. 2. Express in feet the height of Eiffel Tower, 355 m. 3. The metric ton is 1000 kg. Find the difference between it and an American ton. 4. If milk is 15 cents a quart, what would be the price per liter? 5. On the basis that one liter of water weighs a kilogram, what would a gallon of water weigh in pounds ? 6. What per cent larger than a pound avoirdupois is half a kilogram ? 7. If the speed limit on a state road is 25 mi. per hour, what would that be expressed in kilometers per hour ? 8. How many liters in a cubic foot of water? 24 IN TR OD UCTION 9. What is the equivalent in the metric system of a velocity of 1090 ft. per second ? 10. If a cylindrical jar is 4 in. in diameter and one foot deep, how many liters will it hold ? 11. Express the velocity of light, 186,000 miles per second, in kilometers per second. 12. What would be the error made if in measuring 12 ft. a bar 30 cm. long is used as a foot ? 13. If a cubic foot of water weighs 62.4 lb., what would a pint of water weigh? 14. If coal sells at $12 per ton, what would 20,000 kilograms cost? CHAPTER II MOLECULAR PHYSICS I. MOLECULAR MOTION 26. Diffusion of Gases. If two gases are placed in free communication with each other and are left undisturbed, they will mix rather rapidly. Even though they differ in density and the heavier gas is at the bottom, the mixing goes on. This process of the spontaneous mixing of gases is called diffusion. The rapidity with which gases diffuse may be illus- trated by allowing illuminating gas to escape into a room, or by exposing ammonia in an open dish. The odor quickly reveals the presence of either gas in all parts of the room, even when air currents are suppressed as far as possible. A more agreeable illustration is furnished by a bottle of smelling salts. If it is left open, the perfume soon pervades the whole room. Fill one of a pair of jars (Fig. 16) with the fumes of strong hydrochloric acid, and the other with gaseous ammonia, and place over them the glass covers. Bring the jars together as shown, and after a few seconds slip out the cover glasses. In a few minutes both jars will be filled with a white cloud of the chloride p IGURE 1 6. DIFFUSION of ammonia. Instead of these vapors, air and OF GASES illuminating gas may be used, and after dif- fusion, the presence of an explosive mixture in both jars may be shown by applying a flame. to the mouth of each separately. 25 MOLECULAR PHYSICS 27. Effusion through Porous Walls. The passage of a gas through the pores of a solid is known as effusion. The rate of effusion for different gases is nearly inversely proportional to the square root of their relative densities. Hydrogen, for ex- ample, which is one sixteenth as heavy as oxygen, passes through very small open- ings four times as fast as oxygen. Cement a small unglazed battery cup to a funnel tube, and connect the latter to a flask nearly filled with water and fitted with a jet tube, as shown in Fig. 17. Invert over the porous cup a large glass beaker or bell jar, and pass into it a stream of hy- drogen or illuminating gas. If all the joints are air-tight, a small water jet will issue from the fine tube. The hydrogen passes freely through the in- visible pores in the walls of the porous cup and produces gas pressure in the flask. If the beaker is now removed, the jet subsides and the pressure in the flask quickly falls to that of the air outside by the passage of hydrogen outward through the pores of the cup. FIGURE 17 _ EFFUSION OF HY- DROGEN. 28. Molecular Motion in Gases. The simple facts of the diffusion and effusion of gases lead to the conclusion that their molecules (12) are not at rest, but are in constant and rapid motion. The property of indefinite expansibility is a further evidence of molecular motion in gases. No matter how far the exhaustion is carried by an air pump, the gas remaining in a closed vessel expands and fills it. This is not due to repulsion between the molecules, but to their motions. Gases move into a good vacuum much more quickly than they diffuse through one another. In diffusion their motion is frequently arrested by molecular collisions, and hence diffusion is impeded. The property of rapid expansion into a free space is a THE VELOCITY OF MOLECULES 27 highly important one. The operation of a gasoline engine, in which the inlet valve presents only a narrow opening for a small fraction of a second, is an excellent illustration ; and yet this brief period suffices for the explosive mixture to enter and fill the cylinder 29. Pressure Produced by Molecular Bombard- ment. It would be possible to keep an iron plate suspended hori- zontally in the air by the impact of a great many bullets fired up against its under sur- face. The clatter of an indefinitely large num- ber of hailstones on a roof forms a continuous sound, and their fall beats down a field of grain flat to the ground. So the rapidly moving molecules of a gas strike CROSS SECTION OF AUTOMOBILE MOTOR. The valve V is open only a fraction of a second, but the gas fills the cylinder C completely. innumerable minute blows against the walls of the containing vessel, and these blows compose a con- tinuous pressure. This, in brief, is the kinetic theory of the pressure of a gas. 30. The Velocity of Molecules. It has been found pos- sible to calculate the velocity which the molecules of air must have under standard conditions to produce by their impact against the walls of a vessel the pressure of one 28 MOLECULAR PHYSICS atmosphere, or 1033 g. per square centimeter. It is about 450 m. per second. For the same pressure of hydrogen, which is only one fourteenth as heavy as air, the velocity has the enormous value of 1850 m. per second. The high speed of the hydrogen molecules accounts for their rela- tively rapid progress through porous walls. 31. Diffusion of Liquids. Liquids diffuse into one an- other in a manner similar to that of gases, but the process is indefinitely slower. Diffusion in liquids, as in gases, shows that the molecules have independent motion because they move more or less freely among one another. Let a tall jar be nearly filled with water colored with blue litmus, and let a little strong sulphuric acid be intro- duced into the jar at the bottom by means of a thistle tube (Fig. 18). The density of the acid is 1.8 times that of the litmus solution, and the acid therefore remains at the bot- tom with a well-defined surface of separation, which turns red on the litmus side because acid reddens litmus. But if the jar be left undisturbed for a few hours, the line of separation will lose its sharpness and the red color will move gradually upward, showing that the acid molecules have made their way toward the top. 32. Diffusion of Solids. The diffusion of solids is much less pronounced than the diffusion of gases and liquids, but it is known to occur. Thus, if gold be overlaid with lead, the presence of gold throughout the lead may in time be detected. Mercury appears to diffuse through lead at ordinary temperatures ; in electroplating the deposited metal diffuses slightly into the baser metal ; at higher temperatures metals diffuse into one another to a marked degree, so that there is evidence of molecular motion in solids also. MOLECULAR FORCES IN LIQUIDS 29 II. SURFACE PHENOMENA 33. Molecular Forces in Liquids. By an easy transition of ideas we carry the primitive conception of force derived from the sense of muscular exertion over to forces other than those exerted by men and animals, such as those be- tween the molecules of a body. Molecular forces act only through insensible distances, such as the distances separat- ing the molecules of solids and liquids. A clean glass rod does not attract water until there is actual contact between the two. If the rod touches the water, the latter clings to the glass, and when the rod is with- drawn, a drop adheres to it. If the drop is large enough, its weight tears it away, and it falls as a little sphere. By means of a pipette a large globule of olive oil may be introduced below the surface of a mixture of water and alcohol, the mixture having been adjusted to the same density ( 69) as that of the oil by varying the proportions. The globule then assumes a truly spherical form and floats anywhere in the mixture (Fig. 19). Cover a smooth board with fine dust, such as lycopodium powder or powdered charcoal. If a little water be dropped upon it from a height of about two feet, it will scatter and take the form of little spheres (Fig. 20). FIGURE 1 9. SPHERICAL GLOBULE OF OIL. FIGURE 20. SPHERICAL DROPS OF WATER. In all these illus- trations the spheri- cal form is ac- counted for by the forces between the molecules of the liquid. They produce uniform molecular pressure and form little spheres, because a spherical surface is the smallest that will inclose the given volume. MOLECULAR PHYSICS FIGURE 21 . NEEDLE FLOATING ON WATER. 34. Condition at the Surface of a Liquid. Bubbles of gas re- leased in the interior of a cold liquid and rising to the surface often show some difficulty in breaking through. A sewing needle carefully placed on the surface of water floats. The water around the needle is depressed and the needle rests in a little hollow (Fig. 21). Let two bits of wood float on water a few mil- limeters apart. If a drop of alcohol is let fall on the water between them, they suddenly fly apart. A thin film of water may be spread evenly over a chemically clean glass plate; but if the film is touched with a drop of alcohol on a thin glass rod, the film will break, the water retiring and leaving a dry area around the alcohol. The sewing needle indents the surface of the water as if the surface were a tense membrane or skin, and tough enough to support the needle. This surface skin is weaker in alcohol than in water; hence the bits of wood are pulled apart and the water is withdrawn from the spot weakened with alcohol. 35. Surface Tension. The molecules composing the surface of a liquid are not under the same conditions of equilibrium as those within the liquid. The latter are ''attracted equally in all directions by the surrounding molecules, while those at the surface are attracted downward and laterally, but not up- ward (Fig. 22). The result is an unbalanced molecular force toward the interior of the liquid, so that the surface layer is compressed and tends to contract. The contraction means that the surface acts like a stretched membrane, which molds the liquid into a volume with as small a surface as possible. Liquids in small masses, therefore, always tend to become spherical. FIGURE 22. MOLECULAR ATTRACTIONS. ILLUSTRATIONS 31 FIGURE 23. CIRCLE IN LIQUID FILM. 36. Illustrations. Tears, dewdrops, and drops of rain are spherical because of the tension in the surface film. Surface tension rounds the end of a glass rod or stick of sealing wax when softened in a flame. It breaks up a small stream of molten lead into little sections, and molds them into spheres which cool as they fall and form shot. Small globules of mer- cury on a clean glass plate are slightly flattened by their weight, but the smaller the globules the more nearly spherical they are. Of stout wire make a ring three or four inches in diameter with a handle (Fig. 23) . Tie to it a loop of soft thread so that the loop may hang near the middle of the ring. Dip the ring into a soap solu- tion containing glycerine, and get a plane film. The thread will float in it. Break the film inside the loop with a warm pointed wire, and the loop will spring out into a circle. The tension of the film attached to the thread pulls it out equally in all directions. Interesting surfaces may be obtained by dipping skeleton frames made of stout wire into a soap solu- tion. The films in Fig. 24 are all plane, and the angles where three surfaces meet along a line are necessarily 120 for equilibrium. A bit of gum camphor on warm water, quite free from an oily film, will spin around in a most erratic The camphor manner. dissolves unequally at different points, and thus pro- duces unequal weaken- ing of [the surface ten sion in different direc- tions. Make a tiny wooden boat and cut a notch in the stern ; in this notch FIGURE 24. PLANE FILMS. FIGURE 25. BOAT DRAWN BY SURFACE TENSION. 32 MOLECULAR PHYSICS FIGURE 26. CONTRACTION OF SOAP BUBBLE. put a piece of camphor gum (Fig. 25). The cam- phor will weaken the tension astern, while the ten- sion at the bow will draw the boat forward. Surface tension makes a soap bubble contract. Blow a bubble on a small funnel and hold the open tube near a candle flame (Fig. 26). The expelled air will blow the flame aside, and the smaller the bubble the more energetically will it expel the air. A small cylinder of fine wire gauze with solid ends, if completely immersed in water and partly filled, may be lifted out horizontally and still hold the water. A film fills the meshes of the gauze and makes the cylinder air-tight; if the film is broken by blowing sharply on it, the water will quickly run out. 37. Capillary Elevation and Depression. If a fine glass tube, commonly called a capillary or hairlike tube, is partly immersed vertically in water, the water will rise higher in the tube than the level outside ; on the other hand, mercury is depressed below the outside level. The top of the little column of water is con- cave, while that of the column of mercury is convex upward (Fig. 27). Familiar examples of capillary action are numer- ous. Blotting paper absorbs ink in its fine pores, and oil rises in a wick by capillary action. A sponge absorbs water for the same reason ; so also does a lump of sugar. A cotton or a hemp rope absorbs water, increases in diameter, and shortens. A liquid may be carried over the top of a vessel by capillary action in a large loose cord. Many salt solution c construct their own capillary high- way up over the top of the open glass vessel in which they stand. They first rise by capillary action along the surface of the glass, then the water evaporates, leaving the salt in fine crystals, through which the solution rises still higher by capillary action. This process may continue until the liquid flows over the top and down the outside of the vessel. FIGURE 27. CONCAVE AND CONVEX SURFACES IN TUBES. CAPILLARY ACTION IN SOILS 33 38. Laws of Capillary Action. Support vertically several clean glass tubes of small internal diameter in a vessel of pure water (Fig. 28). The water will rise in these tubes, highest in the one of smallest di- ameter, and least in the one of greatest. With mercury in place of water, the de- pression will be the greatest in the smallest tube. If two chemically clean glass plates, in- clined at a very small angle, be supported with their lower edges in water, the height to which the water will rise at different points will be inversely as the distance be- tween the plates, and the water line will FIGURE 28. CAPILLARY be curved as in Fig. 29. ELEVATIONS. These experiments illustrate the following laws : I. Liquids ascend in tubes when they wet them, that is, when the surface is concave; and they are depressed when they do not wet them, that is, when the surface is convex. H- -^ or tubes of small diameter, the elevation or depression is inversely as the diameter of the tube. 39. Capillary Action in Soils. The distribution of mois- ture in the soil is greatly affected by capillarity. Water spreads through compact porous soil as tea spreads through a lump of loaf sugar. As the moisture evaporates at the surface, more of it rises by capillary action from the sup- ply below. To conserve the moisture in dry weather and in " dry farming," the surface of the soil is loosened by cultivation, so that the interstices are too large for free FIGURE 29. CAPILLARY ELEVATION BETWEEN PLATES. 34 MOLECULAR PHYSICS FIGURE 30. ELEVATION BY SURFACE TEN- SION. capillary action. The moisture then remains at a lower level, where it is needed for the growth of plants. 40. Capillarity Related to Surface Tension. The attrac- tion of water for glass is greater than the attraction of water for itself ( 12). When a liquid is thus attracted by a solid, the liquid wets it and rises with a concave surface upward (Fig. 30). The surface tension in a curved film makes the film contract and produces a pressure toward its center of curvature, as shown in the case of the soap bubble ( 36). When the surface of the liquid in the tube is concave, the result of this pressure toward the center of curvature is a force upward ; the downward pressure of the liquid under the film is thus reduced, and the liquid rises until the weight of the column AE down- ward just equals the amount of the upward force. When the liquid is of a sort like mercury, which does not wet the tube, the top of the column is convex, the pressure of the film toward its center of curvature is downward, and the column sinks until the downward pressure is counter balanced by the upward pressure of the liquid outside. III. MOLECULAR FORCES IN SOLIDS 41. Solution of Solids. The solution of certain solids in liquids has become familiar by the use of salt and sugar in liquid foods. The solubility of solids is limited, for it depends on the nature of both the solid and the solvent, the liquid in which it dissolves. At room temperatures, table salt dissolves about three times as freely in water as in alcohol ; while grease, which is practically insoluble in water, dissolves readily in benzine or gasoline. ^^^p COMMON CRYSTALS. Quartz (ideal). Quartz (actual). Galena or Lead Sulphide. Garnet. Alum. CRYSTALLIZATION 35 Solution in a small degree takes place in many unsuspected cases. Thus, certain kinds of glass dissolve to an appreciable extent in hot water. Many rocks are slightly soluble in water, and the familiar adage that the " constant dropping of water wears away a stone " is accounted for, in part at least, by the solution of the stone. Flint glass, out of which cut glass vessels are made, dissolves to some extent in aqua ammonia ; this liquid should not be kept in cut glass bottles, nor should cut glass be washed in water containing ammonia. There is a definite limit to the quantity of a solid which will dissolve at any temperature in a given volume of a liquid. For example, 360 g. of table salt will dissolve in a liter of water at ordinary temperatures; this is equiva- lent to three quarters of a pound to the quart. When the solution will dissolve no more of the solid, it is said to be saturated. As a general rule, though it is not without exceptions, the higher the temperature, the larger the quantity of a solid dissolved by a liquid. A liquid which is saturated at a higher temperature is supersaturated when cooled to a lower one. 42. Crystallization. When a saturated solution evapo- rates, the liquid only passes off as a vapor; the dissolved substance remains behind as a solid. When the solid thus separates slowly from the liquid and the solution remains undisturbed, the conditions are favorable for the molecules to unite under the influence of their mutual attractions, and they assume regular geometric forms called crystals. Similar conditions exist when a saturated solution cools and becomes supersaturated. The presence of a minute crystal of the solid then insures the formation of more. The process of the separation of a solid in the form of crystals is known as crystallization. Dissolve 100 gm. of common alum in a liter of hot water. Hang some strings in the solution and set aside in a quiet place for several hours. The strings will be covered with beautiful transparent octa- 36 MOLECULAR PHYSICS hedral crystals. Copper sulphate may be used in place of the alum; large blue crystals will then collect on the strings. Filter a saturated solution of common salt and set aside for twen- ty-four hours. An examination of the surface will reveal groups of crystals floating about. Each one of these, when viewed through a magnifying glass, will be found to be a little cube. Ice is a compact mass of crystals, and snow consists of crystals formed from the vapor of water. They are of various forms but all hexagonal in outline (Fig. 31 ). 1 FIGURE 31. SNOW CRYSTALS. 43. Elasticity. Apply pressure to a tennis ball, stretch a rubber band, bend a piece of watch spring, twist a strip of whalebone. In each case the form or the volume has been changed, and the body has been strained. A strain means either a change in size or a change in shape. As soon as the distorting force, or stress, has been withdrawn, these bodies recover their initial shape and dimensions. The word stress is applied to the forces acting, while the word strain is applied to the effect produced. The property of recovery from a strain when the stress is removed is called elasticity. It is called elasticity of form when a body re- covers its form after distortion ; and elasticity of volume when the temporary distortion is one of volume. Gases and liquids have perfect elasticity of volume, because 1 These figures were made from microphotographs taken by Mr. W. A. Bentley, Jericho, Vermont. HOOKERS LAW 37 they recover their former volume when the original pres- sure is restored. They have no elasticity of form. Some solids, such as shoemaker's wax, lead, putty, and dough, when long-continued force is applied, yield slowly and never recover. The elasticity of a body may be called forth by pressure, by stretching, by bending, or by twisting. The bound- ing ball and the popgun are illustrations of the first ; rubber bands are familiar examples of the second ; bows and springs of the third ; and the stretched spiral spring exemplifies the fourth. 44. Hooke's Law. Solids have a limit to their distor- tion, called the elastic limit, beyond which they yield and are incapable of re- covering their form or volume. The elastic limit of steel is very high <, steel breaks before there is much permanent FlGURE 32. -BENDING PROPORTIONAL TO WEIGHT. distortion. On the other hand, lead does not recover completely from any distortion. When the strain in an elastic body does not exceed the elastic limit, in general the distortion is proportional to the distorting force, or the strain is proportional to the stress, This relation is known as Hookes law. Clamp a meter stick to a suitable support (Fig. 32), and load the free end with some convenient' weight in a light scale pan ; observe the bending of the stick by means of the vertical scale and the pointer. Then double the weight and note the new deflection. It should be double the first. The amount of bending or distortion of the bar is proportional to the weight. 38 MOLECULAR PHYSICS Generally, for all elastic displacements within the elastic limit, the distortions of any kind, due to bending, stretching, or twisting, are proportional to the forces pro- ducing them. Questions and Exercises 1. When a glass tube or rod is cut off its edges are sharp. Why do they become rounded by softening in a blowpipe flame ? 2. Why does a small vertical stream of water break into drops? 3. Why does a dish with a sharp lip pour better than one with- out it? 4. A soap bubble is filled with air. Is the air inside denser or rarer than the air outside ? 5. Explain the action of gasoline in removing grease spots. How should it be applied so as to avoid the dark ring which often remains after its use ? 6. The hairs of a camel's-hair brush separate when placed in water, but gather to a point when the brush is removed from the water. Explain. 7. Are the divisions on the scale of a spring balance equal ? What law is illustrated ? 8. In the stone quarries of ancient Egypt it is said that large blocks of stone were loosened by drilling a series of holes in the rock, driving in wooden plugs, and then thoroughly wetting them. Ex- plain. 9. Why is it difficult to write on clean glass with a pen? 10. Analysis of the air in a closed room shows little or no difference in its composition in different parts of the room. Explain. 11. If a capillary tube is supported vertically in a vessel of water and the tube is shorter than the distance to which water would rise in it, will the water flow out of the top ? Why ? 12. If water rises 15 mm. in a capillary tube of 1.9 mm. diameter, what must be the diameter of a tube in which water will rise 45 mm. ? CHAPTER III MECHANICS OF FLUIDS I. PRESSURE OF FLUIDS 45. Characteristics of Fluids. A fluid has no shape of its own, but takes the shape of the containing vessel. It cannot resist a stress unless it is supported on all sides. The molecules of a fluid at rest are displaced by the slight- est force; that is, a fluid yields to the continued applica- tion of a force tending to change its shape. -But fluids exhibit wide differences in mobility, or readiness in yield- ing to a stress. Alcohol, gasoline, and sulphuric ether are examples of very mobile liquids; glycerine is very much less mobile, and tar still less so. In fact, liquids shade off gradually into solids. A stick of sealing wax supported at its ends yields continuously to its own weight; in warm weather paraffin candles do not maintain an upright position in a candlestick, but curve over or bend double; a cake of shoemaker's wax on water, with bullets on it and corks under it, yields to both and is traversed by them in opposite directions. At the same time, sealing wax and shoemaker's wax when cold break readily under the blow of a hammer. 46. Viscosity. The resistance of a fluid to flowing under stress is called viscosity. It is due to molecular friction. The slowness with which a tine precipitate, thrown down by chemical action, settles in water is owing to the vis- cosity of the liquid; and the slow descent of a cloud is 39 40 MECHANICS OF FLUIDS accounted for by the viscosity of the air. Viscosity varies between wide limits. It is less in gases than in liquids; hot water is less viscous than cold water; hence the rela- tive ease with which a hot solution filters. THE MOBILITY OF GASOLINE VAPOR. In this six-cylinder automobile engine, gasoline from the tank at the right is vaporized in the carburetor at the center. The mobility of the vapor is so great that it passes readily through the pipe to the cylinders. 47. Liquids and Gases. Fluids are divided into liquids and gases. Liquids, such as water and mercury, are but slightly compressible, while gases, such as air and hydro- gen, are highly compressible. A liquid offers great resist- ance to forces tending to diminish its volume, while a gas offers relatively small resistance. Water is reduced only PASCAL'S PRINCIPLE 41 0.00005 of its volume by a pressure equal to that of the atmosphere (practically 15 Ib. to the square inch), while air is reduced to one half its volume by the same additional pressure. Pressure means force per unit of surface. Then, too, gases are distinguished from liquids by the fact that any mass of gas when introduced into a closed vessel always completely fills it, whatever its volume. A liquid has a bulk of its own, but a gas has not, since a gas expands indefinitely as the pressure on it decreases. 48. Pressure Transmitted by a Fluid. Fit a perforated stopper to an ounce bottle, preferably with flat sides, and mounted in a suitable frame (Fig. 33). Fill the bottle with water and then force a metal plunger through the hole in the stopper. If the plunger fits the stopper water-tight, the force applied to the plunger will be transmitted to the water as a bursting force ; and the whole force transmitted to the inner surface of the bottle will be as many times greater than the force applied as the area of this surface is greater than that of the end of the plunger. Figure 34 is a form of syringe made of glass ; the hollow sphere at the end has several small openings. Fill with water and apply force to the piston. The water will escape in a series of jets of apparently equal velocities, although only one of them is directly in line with the piston. Fit a glass tube to the stem of a small rubber balloon; blow into the tube; the balloon will ex- pand equally in all directions, forming a sphere and showing equal pressures in all directions. A large soap bubble shows the same thing. 49. Pascal's Principle. A solid transmits pressure only in the direction in which the force acts ; but a fluid trans- A 77 r RESSURE IN /\LU DIRECTIONS. 42 MECHANICS OF FLUIDS mits pressure in every direction. Hence the law first announced by Pascal in 1653 : Pressure applied to an inclosed fluid is transmitted equally in all directions and without diminution to every part of the fluid and of the interior of the contain- ing vessel. This is the fundamental law of the mechanics of fluids. It is a direct consequence of their mobility, and it applies to both liquids and gases. 50. The Hydraulic Press. An important application of Pascal's principle is the hydraulic press. Figure 35 is a section showing the principal parts. A heavy piston P works water-tight in the larger cylin- der A, while in the smaller one the piston p is moved up and down as a force pump; it pumps water or oil from the reservoir D and forces it through the tube into the cylinder A. When the piston p of the pump is forced down, the liquid transmits the pressure to the base of the larger piston, on which the force R is as many times the force E applied to p as the area of the large piston is greater than the area of the small one. If the cross-sectional area of the small piston is represented by a, and that of the large one by A, the ratio between the forces acting on the two pistons is S = JL-O 2 E a cP' FIGURE 35. HYDRAULIC PRESS. where D and d are the diameters of the large and small pistons respectively. Galileo Galilei (1566- 1642) was born at Pisa, Italy. He was a man of great gen- ius, and an experimental philosopher of the first rank. He was educated as a phys- ician, but devoted his life to mathematics and physics. He discovered the properties of the pendulum, invented the telescope bearing his name, and was ardent in his support of the doctrine that the earth revolves around the sun. Besides his original work in physics, he made interesting discoveries in astronomy* Blaise Pascal (1623-1662) was born at Clermont in Au- vergne. He was both a math- ematician and a physicist. Even as a youth he showed remarkable learning, and at the age of seventeen achieved renown with a treatise on conic sections. He is best known for his announcement in 1653 of the important law of fluid pressure bearing his name. He distinguished him- self by his researches in conic sections, in the properties of the cycloid, and the pressure of the atmosphere APPLICATION OF THE HYDRAULIC PRESS 48 Thus, if the area A is 100 times the area a, a force of 10 pounds on the piston p becomes 1000 pounds on P. The hydraulic press is a device which permits of the exer- tion of enormous forces. 51. Application of the Hydraulic Press. This machine is used in the industries for lifting very heavy weights and for compressing materials into small volumes. Instances of the former use are the lifting of large crucibles filled FIGURE i3o. COMMERCIAL HYDRAULIC PRESS. with molten steel, and of locomotives to replace them on the track. The enormous force of the hydraulic press is applied also to the baling of cotton and paper, to punching holes through steel plates, to making dies, embossing metal, and forcing lead through a die in the manufacture of lead pipe. A small white pine board one inch thick, compressed in an hydraulic press to a thickness of three-eighths inch, becomes capable of a high polish and has many of the properties of hard wood. The commercial press (Fig. 36) is the same in principle as Fig. 35, with the addition of some auxiliary parts to 44 MECHANICS OF FLUIDS make a working machine. The piston s of the force pump may be worked by any convenient power. It has a check valve d which closes when rises and prevents the return of the water from the large working cylinder. The piston P is surrounded by a peculiar leather collar, without which the press is a failure. The larger the pressure in P, the closer the leather collar presses against the piston and prevents leakage. The upper portion of the machine, cut away in the figure, differs according to the use to which the press is put. If the ratio between the cross-sections of the two pistons is 500, then when 8 is pressed down with a force of 100 Ib. the piston P is forced up with a force of 50,000 Ib. In the hydraulic press it is evident that the small piston travels as many times farther than the large one as the force exerted by the large piston is greater than the effort applied to the small one. 52. The Hydraulic Elevator. A mod- ern application of Pascal's principle is the hydraulic elevator. A simple form is shown in Fig. 37. A long piston P carries the cage A, which runs up and down between guides and is partly coun- terbalanced by a weight W. The piston runs in a tube sunk in a pit to a depth equal to the height to which the cage is designed to rise. Water under pressure enters the pit from the pipe m through the valve v. Turned in one FIGURE 37. HY- DRAULIC ELEVATOR. direction the valve admits^ water to the sunken cylinder, DOWNWARD PRESSUEE OF A LIQUID 45 and the pressure forces the piston up ; when the operator turns it in the other direction by pulling a cord, it allows the water to escape into the sewer, and the elevator de- scends by its own weight. When greater speed is required, the cage is connected to the piston indirectly by a system of pulleys. The cage then usually runs four times as fast and four times as far as the piston. 53. Downward Pressure of a Liquid. Pascal's principle relates to the transmission of pressure applied to a liquid in a closed vessel. But a liquid in an open vessel, such as water in a bucket, produces pressure because it is heavy ; and the pressure of any layer is transmitted to every other layer at a lower level. Since each layer adds t its pressure, there must be in- creasing pressure as the depth ' increases. A glass cylinder, 5, is cemented into a metal ferule, T, which screws into a short cylinder, D (Fig. 38). This short cylinder is closed at the bottom by an elastic diaphragm of thin metal, any motion of which caused by water in B is communicated by a rack and pinion device to the hand on the dial, FIGURE 38. DOWNWARD PRES- E. As the tank, A, filled with water SURE PROPORTIONAL TO DEPTH is moved up the supporting rod water flows through the tube, F, into the cylinder, B, causing the hand to move over the dial. The reading of the hand divided by the depth of the water in B at any moment will be practically constant. Hence, The downward pressure is proportional to the depth. Repeat the experiment with a saturated solution of common salt, which is heavier than water. Every pointer reading will be greater 46 MECHANICS OF FLUIDS than the corresponding ones with water, but the same relation will exist between them. Hence, The downward pressure of a liquid is proportional to its density ( 69). 54. Upward Pressure. Let a glass cylinder A (Fig. 39), such as a straight lamp chimney, have its bottom edge ground off so as to be closed water tight by a thin piece of glass 0. Holding this against the bottom of the cylinder by means of a thread C, immerse the cylinder in water. The thread may then be released and the bottom will stay on be- cause the water presses up against it. To release the bottom we shall FIGURE 39. UPWARD PRESSURE. have to pour water into the cylinder until the levels inside and outside are the same, The upward pressure on the bottom of the cylinder is then the same as the downward pressure inside at the same depth. Or, In liquids the pressure upward is equal to the pressure downward at any depth. 55. Pressure at a Point The three glass tubes of Fig. 40 have short arms of the same length, measured from the bend to the mouth. They open in different directions, upward, downward, and sidewise. Place mercury to the same depth in all the tubes, and lower them into a tall jar filled with water. When the open ends of the short arms are kept at the same level, the change in the level of the mercury is the same in all of them. Hence, The pressure at a point in a liquid is the same in all directions. FIGURE 40. PRESSURE SAME IN ALL DIRECTIONS. TOTAL FORCE ON ANY SURFACE 47 The equality of pressure in all directions may also be inferred from the absence of currents in a vessel of liquid, since an unbalanced pressure would produce motion of the liquid. 56. Bottom Pressure Inde- pendent of the Shape of the Vessel. Proceeding as in 53, use in succession the three vessels shown in Fig. 41. They have equal bases, but differ in shape and vol- ume. Thev are known as Pascal's _, T J FIGURE 41. PRESSURE INDEPEND- vases. Fill each in succession to ENT OF SHAPE. the same height, and note the read- ing of the pointer. It will be the same for all, notwithstanding the great difference in .the amount of water. Hence, The downward pressure in a liquid is independent of the shape of the vessel. The apparent contradiction of unequal masses of a liquid producing equal pressures is known as the hydro- static paradox. Thus, suppose the circular bottom of a tin pail has an area of 200 cm. 2 It would be about 16 cm. in diameter. Suppose the pail filled with water to a depth of 25 cm. Then the pressure on the bottom would be the weight of a prism of water 1 cm. 2 in section and 25 cm. high, or 25 g., since a cm. 8 of water weighs one gram. The whole force on the bottom would be 200 x 25 = 5000 g., or 5 kg. If the pail flares, it would contain more than 5000 cm. 8 of water and would require more than 5 kg. of force to lift it, but the pressure on the bottom would be the same. 57. Total Force on Any Surface. It will be seen from the example in the last section that the pressure on any area is equal to the product of its depth h below the sur- face of the liquid and the weight d of a unit volume of the liquid, or p = lid. If the depth is in centimeters and 48 MECHANICS OF FLUIDS the weight in grams, the pressure p in water is equal to the depth A, since a cubic centimeter of water weighs one gram. The pressure is then in grams per square centi- meter. But if h is in feet and d in pounds per cubic foot, then jt? = Ax62.4 pounds per square foot, since a cubic foot of water weighs 62.4 pounds. To get the pressure in pounds per square inch, divide by 144, because there are 144 square inches in a square foot. The force on any horizontal area A is then P A x h x d . . (Equation 1) If the given surface is inclined, then the pressure in- creases from its value at the highest point submerged to its value at the lowest point. In this case h means the mean depth of the area, or the depth of its center of figure. The total force on any given plane area is always normal, that is, perpendicular to it. Equation 1 still applies, Examples. To calculate the force on the bottom and sides of a cubical box 30 centimeters on each edge, filled with water, and stand- ing on a horizontal plane : The area of each face is 30 x 30 = 900 cm. 2 Then the force on the bottom at a depth of 30 cm. is 900 x 30 = 27,000 g. On the sides the pressure varies from zero to 30 g. per square centimeter. The average pressure is halfway down at a point 15 cm. deep and is 15 g. per square centimeter. Hence the force tending to push out each FIGURE 42. FORCE AGAINST DAM. . . & * Knr . side is 900 x 15 = 13,500 g. The upstream face of a dam measures 20 ft. from top to bottom, but it slopes so that its center of figure is only 7 ft. from the surface of the "water when the dam is full (Fig. 42). Find the perpendicular force against the dam for every foot of length. The area of the face of the dam per foot in length is 20 sq. ft. Hence the weight of the column of water to represent the force is 20 x 7 x 62,4 = 8736 Ib. LEVEL OF LIQUID IN CONNECTED VESSELS 49 FIGURE 43. SAME LEVEL IN ALL BRANCHES. 58. Surface of a Liquid at Rest. The free surface of a liquid under the influence of gravity alone is horizontal. Even viscous liquids assume a hori- zontal surface in course of time. The sea, or any other large ex- panse of water, is a part of the spheroidal surface of the earth. When one looks with a field glass at a long straight stretch of the Suez Canal near Port Said, the water and the retaining wall as contrasting bodies appear dis- tinctly curved as a portion of the rounded surface of the earth. 59. Level of Liquid in Connected Vessels. The water in the apparatus of Fig. 43 rises to the same level in all the branches. (Why should the spout of a teakettle be as high as the lid ?) There is equi- librium because the pressures on opposite sides of any cross section of the liquid in the connecting tube are equal, since they are due to liquid columns of the same height. The glass water gauge, used to show the height of the water in a steam boiler, is an important appli- cation of this principle. A thick- FIGURE 44. - WATER GAUGE. wal]ed glagg ^ ^ (pig> ^ . g connected at the top with the steam and at the bottom with the water in the boiler. The pressure of steam is then the same on the water in the boiler and in the gauge tube, and the water level is the same in 50 MECHANICS OF FLUIDS the two. The stopcocks C and D are kept open except when it be comes necessary to replace the glass tube. Another stopcock E serves to clean out the tube by running steam through it. Another application is the water level, consisting of two glass tubes, joined by a long rubber tube, and employed by builders for leveling foundations. 60. Artesian Wells. Artesian or flowing wells illustrate on a grand scale the tendency of water to " seek its level." In geology an artesian basin is one composed of long strata one above the other. One of these permits the passage of water, and lies .between two layers of clay or other material through which water does not pass (Fig. 45). FIGURE 45. ARTESIAN WELL. This stratum K crops out at some higher level and here the water finds entrance. When a well I is bored through the overlying strata in the valley, water issues on account of the pressure transmitted from higher points at a distance. There are 8000 or 10,000 artesian wells in the western part of the United States ; some notable ones are at Chicago, St. Louis, New Orleans, Charleston, and Denver. In Europe there are very deep flowing wells in Paris (2360 ft.), Berlin (4194 ft.), and near Leipzig (5740 ft.). 61. City Water Supply. In some cases, where a supply of water for city purposes is available at an elevation higher than the points of distribution, as in San Francisco, Los Angeles, Denver, and New York, the water from the source, or from a storage reservoir, is conducted to the city in open channels, or in pipes or " mains," and the pressure causing it to flow is due to gravity alone. Arriving at the ELEPHANT BUTTE DAM. Largest mass of masonry in the world. The lake formed by the dam is 45 miles long and has a capacity four times that of the Assouan Dam in Rcrvnt f.nmicrh tn rnver the state of Delaware to B. death of two feet. CITY WATER SUPPLY 51 city, it is distributed through the streets, the pipes ter- minating at fire hydrants in the streets, and at plugs and faucets in buildings. The water is under pressure ade- quate to carry it to the highest desired points. In the absence of a water supply at an elevation, it is necessary to pump the water into a reservoir on a high FIGURE 46. CITY WATER SUPPLY. point, or into a " staridpipe" or water tower as a part of the distributing system. The water rises in the water tower to a height corresponding to the pressure maintained by the pump. This device serves to equalize the pressure throughout the system, and in the smaller systems it may take the place of a reservoir ; it may exert pressure for domestic pur- poses and for fire protec- tion even when the pump is not running (Fig. 46). For limited domestic supply the hydraulic ram (Fig. 47) is some- times used. Its action depends on the inertia of a stream of water in a pipe. The valve at B is normally open and the other valve open- ing upward into the air dome is closed. The flow of water through the pipe A closes the ball valve B, and the shock of the sudden arrest of the flow opens the valve into the air dome ; the water enters to re- lieve the sudden pressure. Valve B then opens again and the other one closes. The flow thus takes place by a succession of pulses. FIGURE 47. HYDRAULIC RAM. 52 MECHANICS OF FLUIDS Questions and Problems 1. If a pressure gauge be attached to the water pipe on the top floor of a tall building and a second one be attached in the basement, will the readings be the same ? Why ? 2. Why is there danger of bursting a thermos bottle by forcing in the stopper when the bottle is full of liquid ? 3. In a city supplied with water from a reservoir, to what height will the water rise in a vertical pipe connected with the system ? 4. Why does a coiled garden hose tend to straighten out when the water is turned on ? 5. Is the pressure against a dam that backs up the water for a mile greater than on one that backs up the water for a half mile, the depth of water at the dam being the same in both cases ? 6. The cylinders of a hydraulic press are respectively 6 in. and 1 in. in diameter. If a force of 100 Ib. is applied to the piston of the smaller cylinder, what force will the larger piston exert ? 7. A tank 10 ft. square and 10 ft. deep, full of water, will -exert how much force on the bottom ? How much on one side ? 8. A glass tube 76 cm. long is full of mercury. What is the pressure in grams per cm. 2 on the bottom ? (1 cm. 3 of mercury weighs 13.6 g.) 9. A glass cylinder 6 in. in diameter and 12 in. deep is full of water. What is the force of the water against its cylindrical surface ? 10. If the pressure gauge of a water system registers 50 Ib., how high will water rise in a vertical pipe attached thereto ? 11. What weight can be supported on the platform of a hydraulic elevator, if the piston is 10 in. in diameter and the pressure gauge register 50 Ib. to the square inch ? 12. Sea water weighs 64 Ib. to the cubic foot. What force will be exerted on a board 10 ft. long and 1 ft. wide sunk horizontally in the sea to a depth of a mile ? 13. The pressure gauge of a water system registers 60 Ib. to the sq. in. on the ground floor and 30 Ib. to the sq. in. on the top floor. What is the difference of level? 14. A kerosene tank is 10 ft. in diameter and 10 ft. deep. When THE MEASURE OF BUOYANCY 53 full of kerosene what force will there be against the cylindrical sur- face ? (One cubic foot of kerosene weighs 54 Ib.) 15. A wooden box one foot square has fitted into its top a vertical tube 40 ft. long and 1 in. in diameter. When both the tube and box are full of water what bursting force is exerted on the inner surface of the box? II. BODIES IMMERSED IN LIQUIDS 62. Buoyancy. A marble sinks in water and floats in mercury ; a fresh egg sinks in water and floats in a satu- rated solution of common salt; a piece of oak floats in water and the dense wood lignum-vitae sinks ; a swim- mer in the sea . is nearly lifted off his feet by the heavy salt water. Suspend a pound or two of iron from the hook of a draw scale, and note its weight. Now bring a beaker of water up under the iron and partly immerse it ; note that its weight is diminished; immerse farther and the loss of weight increases ; after it is fully submerged, the loss of weight does not increase with the depth of immersion. If salt water is used, the apparent loss of weight will be greater ; if kerosene, it will be less. In popular language the body immersed is said to have lost weight. Its real weight has not changed in the least; bat an upward force has been brought to bear on it. The lifting force of a liquid on a body immersed in it is called buoyancy. 63. The Measure of Buoyancy. The law of buoyancy was discovered by a Greek philosopher Archimedes about 240 B.C. while engaged in determining the composition of the golden crown of Hiero, king of Syracuse, who sus- pected that the goldsmith had mixed silver with the gold. The law is as follows : JL body immersed in a liquid is buoyed up by a force equal to the weight of the liquid displaced by it. 54 MECHANICS OF FLUIDS The following experiments illustrate the principle of Archimedes, which is the basis of the theory of floating bodies : The hollow brass cylinder A (Fig. 48) and the solid brass cylinder B, which exactly fits into A, are suspended from one arm of a balance and carefully counterpoised. If now the cylinder A be filled with water, the equilibrium will be disturbed ; but if at the same time cylinder B is immersed in water, as in the figure, the equilibrium will be restored. The upward force on the solid cylinder is therefore equal to the weight of the water in A, and this is equal in volume to that of the immersed cylinder. If the experiment is tried with any other liquid which does not attack brass, the result will be the same. A metal cylinder 5.1 cm. long, and 2.5 cm. in di- ameter has a volume of almost exactly 25 cm. 8 Suspend it by a fine thread from one arm of a balance (Fig. 49) and counterpoise. Then submerge it in water as in the figure. The equilibrium will be re- stored by placing 25 g. in the pan above the cylinder. The cylinder displaces 25 cm. 8 of water weighing 25 g., and its ap- parent loss of weight is 25 g. The temperature of the water should be near freezing. 64. Explanation of Archimedes' Principle. If a cube be immersed in water (Fig. 50), the pressures on the vertical sides a and b are equal and in opposite direc- tions. The same is true of the other pair of vertical faces. There is therefore no resultant horizontal force. On d there is a downward force equal to the weight of the column of water having the face d as a base, and the height dn. On c there is an upward force equal to the weight of a column of water whose base is the area of c, and whose height is en. FIGURE 48. ILLUSTRATING PRINCIPLE OF ARCHIMEDES. FIGURE 49. Ai - PARENT LOSS OF WEIGHT. FLOATING BODIES 55 The upward force therefore exceeds the downward force by the weight of the prism of water whose base is the face c of the cube, and whose height is the difference between en and dn, or cd. This is the weight of the liquid displaced by the cube. In general if a cube of any material be immersed in water, the water pressure at every point will be independent of the sub- stance of the cube. Suppose then it is a cube of the water itself. Its weight will be a Fioure 50. - EXPLANATION OF PRINCIPLE. vertical force downward. But it is in equilibrium, for it does not move. Hence its own weight downward is offset by an equal force acting vertically upward. But this upward force of the water is the same, whatever the material of the cube. Hence, there is an upward force on any submerged cube equal to the weight of the water displaced by it. A similar argument applies to a body of any shape submerged in any liquid. 65. Floating Bodies. If a body be immersed in a liquid, it may displace a weight of the liquid less than, equal to, or greater than its own weight. In the first case, the upward force is less than the weight of the body, and the body sinks. In the second case, the upward force is equal to the weight of the body, and the body is in equilibrium. In the third case, the upward force ex- ceeds the weight of the body, and the body rises until enough of it is out of the liquid so that these forces be- come equal. The buoyancy is independent of the depth so long as the body is wholly immersed, but it decreases 56 MECHANICS OF FLUIDS as soon as the body begins to emerge from the liquid. Hence, When a body floats on a liquid it sinks to such a depth that the weight of the liquid displaced equals its own weight. 66. Experimental Proof. Make a wooden bar 20 cm. long and 1 cm, square (Fig. 51). Drill a hole in one end and fill with enough shot to give the bar a vertical position when float- ing with nearly its whole length in water. Gradu- ate the bar in millimeters along one edge^ beginning at the weighted end, and coat with hot paraffin. Weigh the bar and float it in water, noting the vol- ume in cubic centimeters immersed. This volume is equal to the volume of water displaced; and since 1 cm. 3 of water weighs 1 g., the weight of the water displaced is numerically equal to the volume of the bar immersed. This will be found also very nearly, if not quite, equal to the weight of the loaded bar. 67. The Cartesian Diver. Descartes, a French scientist, illustrated the principle of flota- tion by means of an hydrostatic toy, since called the Cartesian diver. It is made of glass, is hollow, and has a small opening near the bottom. The figure is partly filled with water so that it just floats in a jar of water (Fig. 52). Pressure applied to the sheet rubber tied over the top of the jar is transmitted to the water, more water enters the floating figure, and the air is compressed. The figure then displaces less water and sinks. When the pressure is relieved, the air in the diver expands and forces water out again. The actual displacement of water is then increased, and the figure rises to the surface. The water in the diver may be so nicely adjusted that the little figure will sink in cold water, but will rise again when the water has reached the tempera- ture of the room, and the air in the figure has expanded. FIGURE 51. EXPERI MENTAL PROOF. FIGURE 52. CAR- TESIAN DIVER. THE FLOATING DRY DOCK 57 FIGURE 53. A UNITED STATES SUBMARINE. A good substitute for the diver is a small inverted homeopathic vial in a flat 16-oz. prescription bottle, filled with water and closed with a rubber stopper. When pressed, the sides yield, and the vial sinks. A submarine boat is a modern Cartesian diver on a large scale. It is provided with tight compartments, into which water may be ad- mitted to make it sink. It may be made to rise to the surface by ex- pelling some of the water by powerful pumps. 68. The Floating Dry Dock. The floating dock re- sembles the submarine in principle. It is made buoyant THE SAME SUBMARINE SUBMERGING. 58 MECHANICS OF FLUIDS FIGURE 54. DRY DOCK. by pumping water out of water-tight compartments, and by floating it lifts a vessel out of the water. In Fig. 54 J., A are compartments full of air. When they are filled with water, the dock sinks to the dotted position j5, B and the vessel is floated into it. When the water is pumped out, the dock takes the position indi- cated by the full lines and the vessel is lifted out of water. III. DENSITY AND SPECIFIC GRAVITY 69. Density. We are familiar with the fact that bodies of different kinds may have the same size or bulk and yet differ greatly in weight, that is, in mass. A block of steel, for example, is nearly forty times as heavy as a block of cork of the same dimensions, that is, its mass is nearly forty times as great. This difference is expressed as a difference in density. The density of a substance is the number of units of mass of it contained in a unit of volume. In the e.g. s. system density is the number of grams per cubic centimeter. For example, the density of steel is 7.816 grams per cubic centimeter (expressed as 7.816 g. /cm. 3 ), while that of cork has a mean value of about 0.2 g./cm. 3 , and that of mercury 13.596 g./cm. 3 So mass or in symbols, j , //e-M/ort density = , volume m d = ; whence m = dv, and v = v d (Equation 2) To illustrate, a slab of marble 20 x 50 x 2 cm. has a volume of 2000 cm. 8 and weighs 5.4 kg. or 5400 g. Hence its density is 5400/2000 = 2.7 g./cm. 8 DENSITY AND SPECIFIC GRAVITY COMPARED 59 70. Specific Gravity. The specific gravity of a body is the ratio of its weight to the weight of an equal volume of water. If, for example, a cubic inch of lead weighs 11.36 times as much as a cubic inch of water, the specific gravity of lead is 11.36. The principle of Archimedes furnishes a simple method of finding specific gravity, since the loss of weight of a heavy body suspended in water is equal to the weight of the water displaced, or the weight of a vol- ume of water equal to that of the suspended body. Hence ./. ., weight of body specific gravity = * .-' , y loss of weight in water For example, a piece of copper weighs 880 g. in air and 780 in water. Its loss of weight is then 100 g., and this is the weight of the water displaced. Hence the specific gravity of copper is 880/100 = 8.8. 71. Density and Specific Gravity Compared. Specific gravity and density have not quite the same meaning. For example, the specific gravity of lead is the abstract num- ber 11.36, while the density of lead is 11.36 g. /cm. 3 , or 62.4x11.36 = 708.9 Ib./cu. ft., both of them concrete numbers. Specific gravity is only a ratio between two masses or weights, and is therefore independent of the units em- ployed in determining it ; while density depends on the units used to express it. In the e.g. s. system density and specific gravity are numerically the same, because the density of water is one gram per cubic centimeter, or density (g./cm. 3 ) = specific gravity. But in the English system density (Ib./cu. ft.) = 62.4 x specific gravity. 60 MECHANICS OF FLUIDS It is worth remembering that if the density of any sub- stance is expressed in c. g. 8. units, its numerical value is always that of the specific gravity. Table IV in the Appendix of this book gives the densities in grams per cubic centimeter. 72. Density of Solids. The density of a solid body is its mass divided by its volume. Its mass may always be obtained by weighing, but the volume of an irregular solid cannot be obtained from a measurement of its dimensions. In the c. g. 8. system, however, the principle of Archimedes furnishes a simple method of finding the volume of a solid, however irregular it may be ; for in this system the volume of an immersed solid is numerically equal to its loss of weight in water ( 63). Then the equa- tion which defines density ( 69), mass becomes density volume mass of body loss of weight in water FIGURE 55. SOLIDS HEAVIER THAN WATER. ' 73. Solids Heavier than Water. Find the mass of the body in air in terms of grams; if it is insoluble in water, find its apparent loss of weight by suspending it in water (Fig. 55). This loss of weight is equal to the weight of the volume of water dis- placed by the solid ( 63). But the volume of a body in cubic centimeters is the same as the mass in grams of an equal volume of water. The mass divided by this volume is the density. 74. Solids Lighter than Water. If the body floats, its volume may still be obtained by tying to it a sinker heavy SOLIDS LIGHTER THAN WATER 61 enough to force it beneath the surface. Let w l denote the weight in grams required to counterbalance when the body is in the air, and the attached sinker in the water ; and let w 2 denote the weight to counterbalance when both body and sinker are under water (Fig. 56). Then obviously w 1 w 2 is equal to the upward force on the body alone, and is therefore numerically equal to the volume of the body. The mass divided by this volume is the density. If the solid is soluble in water, a liquid of known density, in which the body is not soluble, must be used in place of water. Then the loss of weight is equal to the weight of the liquid displaced, and if this FIGURE 56. is divided by the density of the liquid SoLIDS LlGHTER (Equation 2), the volume of the body will be obtained. Then the mass of the body divided by this volume will be the density sought. EXAMPLES. First, for a body heavier than water. Weight of body in air . t , . ; 10.5 g. Weight of body in water 6.3 g. Weight of water displaced 4.2 g. Since the density of water is 1 g. per cubic centimeter, the volume of the water displaced is 4.2 cm. 8 . This is also the volume of the body. Therefore, 10.5 -H 4.2 = 2.5 g. per cubic centimeter is the density. Second, for a body lighter than water. Weight of body in air 4.8 g. Weight of sinker in water 10.2 g. Weight of body and sinker in water ... 8.4 g. The combined weight of the body in air and the sinker in water is, then, 4.8 + 10.2 = 15 g. But when the body is attached to the 62 MECHANICS OF FLUIDS sinker, their apparent combined weight is only 8.4 g. Therefore the buoyant effort on the body is 15 8.4 = 6.6 g., and this is the weight of the water displaced by the body, and hence its volume is 6.6 cm. 8 . The density is, then, 4.8 -4- 6.6 = 0.73 g. per cubic centimeter. Third, for a body soluble in water. Suppose it is insoluble in alcohol, the density of which is 0.8 g. per cubic centimeter. Weight of body in air 4.8 g. Weight of body in alcohol 3.2 g. Weight of alcohol displaced 1.6 g. The volume of alcohol displaced is 1.6 -s- 0.8 = 2 cm. 8 . This is also the volume of the body. Therefore, the density of the body is 4.8 -4- 2 = 2.4 g. per cubic centimeter. 75. Density of Liquids. (a) By the specific gravity bottle. A specific gravity bottle (Fig. 57) is usually made to hold a definite mass of distilled water at a specified temperature, for example, 25, 50, or 100 g. Its volume is therefore 25, 50, or 100 cm. 3 . To use the bottle, weigh it empty, and filled with the liquid, the density of which is to be determined. The weight of the liquid divided by the capacity of the bottle in cubic centimeters (the number of grams) is equal to the density of the liquid. (b) By the density bulb. The density bulb is a small glass globe loaded with shot, and having a hook for suspension (Fig. 58). To use it, suspend from the arm of a balance with a fine platinum wire, and weigh first in air and then in water. The apparent loss of weight is the weight of the water displaced by the bulb. Then weigh it again when suspended in the liquid. The loss of weight is this time the weight of a volume of the FIGURE 57. SPECIFIC GRAVITY BOTTLE. FIGURE 58. DENSITY BULB. DENSITY OF LIQUIDS 63 liquid equal to that of the bulb. Divide this loss of weight by the loss in water, and the quotient will be the specific gravity of the liquid, or its density in grams per cubic centimeter ( 71). (e) By the hydrometer. The common hydrometer is usually made of glass, and consists of a cylindrical stem and a bulb weighted with mercury or shot to make it sink to the required level (Fig. 59). The stem is graduated, or has a scale inside, so that readings can be taken at the surface of the liquid in which the hydrometer floats. These readings give the densities directly, or they may be reduced to densities by means of an ac- companying table. Hydrom- eters sometimes have a ther- mometer in the stem to indicate the temperature of the liquid at the time of taking the read- ing. Specially graduated in- struments of this class are used to test milk, alcohol, acids, etc. For liquids lighter than water, in which the hydrometer sinks to a greater depth, it is cus- tomary to use a separate instrument to avoid so long a stem and scale. FIGURE 60 -^ or testing the acid of a storage battery, the ACID HY- hydrometer is inclosed in a large glass tube (Fig. DROMETER. 60 ^ By means o f ^ e rubber bulb at the top of the large tube enough acid may be drawn in to make the hydrometer float. The hydrometer is then read as usual and the acid is returned to the cell by squeezing the bulb. FIGURE 59. HY- DROMETER. 64 MECHANICS OF FLUIDS Questions and Problems 1. Why does an ocean steamer draw more water after entering fresh water ? 2. If the Cartesian diver should sink in the jar, why will the addition of salt cause it to rise ? 3. What is the density of a body weighing 15 g. in air and 10 g in water ? What is its specific gravity ? 4. A hollow brass ball weighs 1 kg. What must be its volume so that it will just float in water? 5. What is the density of a body weighing 20 g. in air and 16 g. in alcohol whose density is 0.8 g. per cubic centimeter? 6. A bottle filled with water weighed 60 g. and when empty 20 g. When filled with olive oil it weighed 56.6 g. What is the density of olive oil ? 7. A density bulb weighed 75 g. in air, 45 g. in water, and 21 g. in sulphuric acid. Calculate the density of the sulphuric acid. 8. A piece of wood weighs 96 g. in air, 172 g. in water with sinker attached. The sinker alone in water weighs 220 g. Find the density of the wood. 9. A piece of zinc weighs 70 g. in air, and 60 g. in water. What will it weigh in alcohol of density 0.8 g. per cubic centimeter ? 10. The mark to which a certain hydrometer weighing 90 g. sinks in alcohol is noted. To make it sink to the same mark in water it must be weighted with 22.5 g. What is the density of the alcohol ? 11. A body floats half submerged in water. What is its specific gravity? What part of it will be submerged in alcohol, specific gravity 0.8? 12. If an iron ball weighs 100.4 Ib. in air, what will it weigh in water if its specific gravity is 7.8? 13. What is the specific gravity of a wooden ball that floats two thirds under water ? 14. A ferry boat weighs 700 tons. What will be the displacement of water if it takes on board a train weighing 600 tons? 15. A liter flask weighing 75 g. is half filled with water and half with glycerine. The flask and liquids weigh 1205 g. What is the density of the glycerine? What is its specific gravity? PRESSURE PRODUCED BY THE AIR 65 IV. PRESSURE OF THE ATMOSPHERE 76. Weight of Air. It is only a little more than 250 years since it became definitely known that air has any weight at all. Even now we scarcely appreciate its weight. Place a globe holding about a liter (Fig. 61) on the pan of a bal- ance and counterpoise; the stopcock should be open. Remove the globe and force in more air with a bicycle pump, clos- ing the stopcock to retain the air under the increased pressure; the balance will show that the globe is heavier than before. Remove it again and exhaust the air with an air pump ; the balance will now show that the globe has lost weight. A large incandescent lamp bulb may be used in place of the globe by first counterbalancing and then admitting air by punctur- ing with the very pointed flame of a blowpipe. Thus air, though invisible, may be put into a vessel or re- FIGURE 61. moved like any other fluid ; and, like any other fluid, GLOBE FOR it has weight. WEIGHING AIR. The weight of a body of air is surprisingly large. A cubic yard of air at atmospheric pressure weighs more than 2 Ib. The air in a hall 50 ft. long, 30 ft. wide, and 18 ft. high weighs more than a ton. Precise measure- ments have shown that air at the temperature of freezing and under a pressure equal to that of a column of mer- cury 76 cm. high weighs 1.293 g. per liter, or 0.001293 g. per cubic centimeter. 77. Pressure Produced by the Air. Since the air sur- rounding the earth has weight, it must exert pressure on any surface equal to the weight of a column of air above it, just as in the case of a liquid. Many experiments prove this to be true. We are not aware of this pressure because it is equalized in all directions, and we are built to sustain it, just as deep-sea fishes sustain the much greater pressure of water above them. 66 MECHANICS OF FLUIDS Stretch a piece of sheet rubber, and tie tightly over the mouth of a glass vessel, as shown in Fig. 62. If the air is gradually exhausted from the vessel, the rubber will be forced down more and more by the pressure of the air above it, and it may break. The depression will be the same in whatever direction the rubber membrane may be turned. Fill a common tumbler full of water, cover with a sheet of paper so as to ex- FIGURE 62. DOWNWARD clude the air > and holdin S the hand against PRESSURE OF THE AIR. the paper, invert the tumbler (Fig. 63). When the hand is removed, the paper is held against the mouth of the glass with sufficient force to keep the water from run- ning out. Cut about 20 cm. from a piece of glass tubing of 3 or 4 mm. bore. Dip it vertically into a vessel of water, and close the upper end with the finger. The tube may now be lifted out, and the water will remain in it. Figure 64 illustrates a pi- pette ; it is useful for convey- ing a small quantity of liquid from one vessel to another. FIGURE 63. UPWARD PRESSURE OF THE AIR. 78. The Rise of Liquids in Exhausted Tubes. Near the close of Galileo's life his patron, the Duke of Tuscany, dug a deep well near Florence, and was surprised to find that he could get no pump in which water would rise more than about 32 feet above the level in the well. He appealed to Galileo for an explanation ; but Galileo appears to have been equally surprised, for up to that time every- body supposed that water rose in tubes exhausted by suction because "nature abhors a vacuum." Galileo sug- FIGURE 64. PIPETTE. HYDRO -AIRPLANES. When in the air these are sustained by the air-pressure against their planes; when on the water, by the water-pressure against their pontoons. PASCAL'S EXPERIMENTS 67 gested experiments to find out to what limit nature abhors a vacuum, but he was too old and enfeebled in health to perform them himself and died before the problem was solved by others. 79. Torricelli's Experiment. Torricelli, a friend and pupil of Galileo, hit upon the idea of measuring the resist- ance nature offers to a vacuum by a column of mercury in a glass tube instead of a column of water in the Duke of Tuscany 's pump. The experiment was performed in 1643 by Viviani under Torricelli's direction. A stout glass tube about a yard long, sealed at one end and filled with clean mercury, is closed at the open end with the finger, and in- verted in a vessel of mercury in a vertical po- sition (Fig. 65). When the finger is removed, the column falls to a height of about 30 inches. The space above the mercury is known as a Torricellian vacuum. The column of mercury in the tube is counterbalanced by the pressure of the atmosphere on the mercury in the larger vessel at the bottom. 80. Pascal's Experiments. To Pascal is due the credit of completing the demonstration that the weight of the column of mercury in the Tor- ricellian experiment measures the pressure of the atmos- phere. He reasoned that if the mercury is held up simply by the pressure of the air, the column should be shorter at higher altitudes because there is then less air above it. Put to the test by carrying the apparatus to the top of the Tour St. Jacques (Fig. 66), 150 feet high, at that time the bell tower of a church in Paris, his theory was confirmed. A statue of Pascal now stands at the FIGURE 65. TORRICEL- LI'S TUBE. 68 MECHANICS OF FLUIDS base of the old tower. Desiring to carry the test still further, he wrote to his brother-in-law to try the experi- ment on the Puy de Dome, a mountain nearly 1000 m. high, in southern France. The result was that the column of mercury was found to be nearly 8 cm. shorter than in Paris. Pascal repeated the experiment with red wine instead of mercury, and with glass tubes forty -six feet long ; and he found that the lighter the fluid, the higher the column sustained by the pressure of the air. Further, a balloon, half filled with air, appeared fully inflated when car- ried up a high mountain, and collapsed again grad- ually during the descent. Thus the question of FIGURE 66. TOUR ST. JACQUES. ,, -^ -, m the Duke of Tuscany was fully answered; liquids rise in exhausted tubes be- cause of the pressure of the atmosphere on the surface of the liquid outside. 81. Pressure of One Atmosphere. The height of the column of mercury supported by atmospheric pressure varies from hour to hour and with the altitude above the sea. Its height is independent of the cross section of the tube, but to find the pressure, or force per unit area, a tube of unit cross section must be assumed. Suppose an THE HAHOMETEH 69 internal cross-sectional area of 1 cm. 2 . The standard height chosen is 76 cm. of mercury at the temperature of melting ice (0 C.), and at sea level in lati- tude 45. The density of mercury at this tem- perature is 13.596 grams per cubic centimeter. Hence, standard atmospheric pressure, which is the weight of this column of mercury, is 76 x 13.596 = 1033.3 g. per square cen- timeter, or roughly 1 kg. per square centimeter, equivalent to 14.7 Ib. per square inch. The height of a column of water to produce ti pressure of one atmosphere is 76 x 13.596 = 1033.3 cm. = 33.9 ft. 82. The Barometer. The barometer is an instrument based on Torricelli's experiment, and is designed to measure the varying pres- sure of the atmosphere. In its simplest form it consists of a J-shaped glass tube about 86 cm. (34 in.) long, and attached to a support- ing board (Fig. 67). The short arm has a piuhole near the top for the admission of air. A scale is fastened by the side of the tube, and the difference of readings at the top of the mercury in the long arm and the short one gives the height of the mercury column sustained by atmospheric pressure. This varies from about 73 to 76.5 cm. for places near sea level. When accuracy is required, the barometer reading ] must be corrected for temperature. A good barometer must contain pure mercury, and the mercury must be boiled in the glass tube to expel air and moisture. FIGURE 67. THE BA- 70 MECHANICS OF FLUIDS 83. The Aneroid Barometer. A more convenient barom- eter to carry about is the aneroid barometer, which contains no liquid. It consists essentially of a shallow cylindrical box (Fig. 68), from which the air is partially exhausted. It has a thin cover corrugated in circular ridges to give it greater flexibility. The cover is prevented from col- lapsing under atmos- pheric pressure by a stiff spring attached to the center of the cover (shown in the figure under the pointer). This flexible coyer rises and falls as the pres- sure of the atmosphere varies, and its motion is transmitted to the pointer by means of delicate levers and a chain. A scale graduated by comparison with a mer- curial barometer is fixed under the pointer. These instru- ments are so sensitive that they readily indicate the change of pressure when carried from one floor of a building to the next, or even when moved no farther than from a table to the floor. 84. Utility of the Barometer. The barometer is a faithful indicator of all changes in the pressure of the atmosphere. These may be due to fluctuations in the atmosphere itself, or to changes in the elevation of the observer. The barometer is constantly used by the Weather Bureau in forecasting changes in the weather. Experi- ence has shown that barometric readings indicate weather changes as follows : FIGURE 68. ANEROID BAROMETER. CYCLONIC STORMS 71 I. A rising barometer indicates the approach of fair weather. II. A sudden fall of the barometer precedes a storm. III. An unchanging high barometer indicates settled fair weather. The difference in the altitude of two stations may be computed from barometer readings taken at the two places simultaneously. Various complex rules have been pro- posed to express the relation between the difference in barometer readings and the difference in altitude ; a sim- ple rule for small elevations is to allow 0.1 in. for every 90 ft. of ascent. 85. Cyclonic Storms. Weather maps are drawn from observations made at many places at the same time and telegraphed to a central station. In this way cy- clonic storms are discov- ered and followed. At the center of the storm is the lowest reading of the ba- rometer. Curves called isobars are traced through points of equal pressure around this center (Fig. 69) . The wind blows from areas of higher pressure toward those of lower, but in the northern hemisphere the inflowing winds are de- flected toward the right on account of the rotation of the earth. This gives to the storm a counter-clockwise rota- tion, as indicated by the arrows in a weather map. Cy- FIGURE 69. ISOBARS. 72 MECHANICS OF FLUIDS clonic storms usually cross the northwest boundary of the United States from British Columbia, travel in a south- easterly direction until they cross the Rocky Mountain range, and then turn northeasterly toward the Atlantic coast. Storms coming from the Gulf of Mexico usually travel along the Atlantic coast toward the northeast. Questions and Problems 1. What are the objections to the use of water as the liquid for barometers ? 2. Why must the air be completely removed from the barometer tube? 3. Why is mercury the best liquid to use in a barometer tube ? 4. Point out some of the good points as well as some of the objec- tionable features of an aneroid barometer. 5. Must a barometer be suspended out of doors in order to get the air pressure ? Why ? 6. Does the diameter of the bore of a barometer tube affect the height to which the mercury rises ? 7. How can you make water run in a regular stream from a 8. The barometer reading is 75.2 cm. Calculate the atmospheric pressure per square centimeter. 9. The barometer reading is 29 in. Calculate the atmospheric pressure per square inch. 10. Calculate the buoyancy of the air for a ball 10 cm. in diameter if a liter of air weighs 1.29 g. II- The density of glycerine is 1.26 g. per cubic centimeter. If a barometer were constructed for glycerine what would be its reading when the mercurial barometer reads 73 cm. ? 12, When the density of the air is 0.0013 g. per cubic centi- meter, how much less will 200 cm. 3 of cork weigh in air than in a vacuum ? COMPRESSIBILITY OF AIR 73 12. If a barometer at the foot of a tower reads 29.5 in., while one at the top reads 29.2 in., what is the height of the tower ? 13. A bottle is fitted air-tight with a rubber stopper and a tube as in Fig. 70. If water be sucked out by the tube, what will happen when the tube is released? If air is blown in through the tube, what will happen when the tube is re- leased? 14. Figure 71 represents a pneu- matic inkstand, nearly full of ink. Why does the ink not run out? FIGURE 70. FIGURE 71. V. COMPRESSION AND EXPANSION OF GASES 86. Compressibility of Air. The inflation of a toy bal- loon, an air cushion, and a pneumatic tire illustrates the ready compressibility of the air. Push a long test tube under water with its open end down. The deeper the tube is sunk, the higher the water rises in it and the smaller becomes the volume of the in- closed air ; also the reaction tending to lift the tube in- creases. The expansibility of air, or its tendency to increase in volume whenever the pressure is reduced, is shown by its escape from any vessel under pressure, such as the rush of compressed air from a popgun, an air gun, or a punctured pneumatic tire. The air in a building shows the same tendency to expand. When the pressure outside is sud- denly reduced, as in the passage of a wave due to an ex- plosion, the force of expansion of the air within often bursts the windows outward. Blow air into the bottle (Fig. 70) through the open tube. The air forced in bubbles up through the water and is compressed within. As soon as the tube is released and the pressure in it falls to that of 74 MECHANICS OF FLUIDS Air Cushion the atmosphere, the expansive force of the imprisoned air forces water out through the tube with great velocity. This principle is applied in many forms of devices for spraying plants and shrubbery. The compression and the expansion of air are both illustrated by the com- mon pneumatic door check for light doors ; also by the air dome on a force pump ; and the air cushion on a water pipe (Fig. 72), which is usually carried a few inches higher than the faucet so that the air confined in the closed FIGURE 72. AIR CUSHION WATER PIPE. ON -100 -80 -60 end may act as a cushion to take up any sudden shock due to the inertia of the water when the stream is sud- denly checked. The " pounding " of the pipes when the water is turned off quickly is owing to the absence of this air cushion. 87. Boyle's Law. In dealing with air in a state of compression or expansion, the question at once arises, how does a given volume of air change when the pressure on the air changes? The answer is contained in the dis- covery by Robert Boyle at Oxford, England, in 1662. The principle discovered by Boyle (and later in France by Mariotte) is known as Boyle's law ; it applies to all gases at a con- stant temperature. Boyle in his experiments used a J-tube with the short arm closed ; both arms were pro- vided with a scale (Fig. 73). In his experi- ments the pressures extended only from -fa of an atmosphere to 4 atmospheres. Mercury was poured in until it stood at the same level in both arms of the tube. The air in the short arm was then under the same pres- -110 -no o FIGURE 73. BOYLE'S EXPERIMENT. THE LAW APPROXIMATE 75 sure as the atmosphere outside. Its volume was noted by means ot the attached scale, and more mercury was then poured into the tube, The difference in the level of the mercury in the two arms of the tube gave the excess of pressure on the inclosed air above that of the at- mosphere. When this difference amounted to 76 cm., the pressure on the gas in the short tube was 2 atmospheres, and its volume was re- duced to one half. When the difference became twice 76 cm., the pressure on the inclosed air was 3 atmospheres and its volume became one third ; and so on. This is the law of the compressibility of gases ; it may be expressed as follows : At a constant temperature the volume of a given mass of gas varies inversely as tJw pressure sustained by it. If the volume of gas v under a pressure p becomes volume v' when the pressure is changed to p 1 ', then by the law: , ^- = 2^-; whence pv =p'v f . (Equation 3) (Notice the inverse proportion.) In other words, the product of the volume of the gas and the corresponding pressure remains constant for the same temperature. 88. The Law Approximate. Extended investigations have shown that Boyle's law is not rigorously exact for any gas. In general, gases are more compressible than the law requires, and this is especially true for gases which are easily liquefied, such as carbon dioxide (CO 2 ), sulphur dioxide (SO 2 ), and chlorine. Within moderate limits of pressure, however, Boyle's law is exceedingly useful in dealing with the volume and pressure of gases. An example will illustrate its use : If a mass of gas under a pres- sure of 72 cm. 8 of mercury has a volume of 1900 cm. 8 , what would its volume be if the pressure were 76 cm. 8 ? By Equation 3, pv = p'v' ] hence, 72 x 1900 = 76 x v'. From this equation v' = 1800 cm. 8 . 76 MECHANICS OF FLUIDS FIGURE 74. SECTION OF AIR COMPRESSOR. 89. The Air Compressor. A pump designed to compress air or other gases under a pressure of several atmospheres is shown in section in Fig. 74, and complete in Fig. 75. The piston is solid, and there are two metal valves at the bottom. Air or other gas is admitted through the left-hand tube when the piston rises ; when it de- scends, it compresses the inclosed air, the pressure closes the left-hand valve, and opens the outlet valve on the right, and the compressed air is discharged into the compression tank. A bicycle pump (Fig. 76) is an air compressor of a very simple type. The piston has a cup- shaped leather collar e, which permits the air to pass by into the cylinder when the piston is withdrawn, but closes when the piston is forced in. The collar thus serves as a valve, allowing the air to flow one way but not the other. The compressed air is forced through the tube form- ing the piston rod, and the check valve in the tire inlet prevents its return. 90. The Air Pump. The air pump for re- moving air or any gas from a closed vessel de- pends for its action on the expansive or elastic force of the gas. The first air pump was invented by Otto von Guericke, burgomaster of Magdeburg, about 1650. FIGURE 75. AIR COMPRESSOR. FIGURE 76. -BICY- CLE PUMP. THE AIR PUMP 77 FIGURE 77. SIMPLE PUMP. AIR In the very simplest form the two valves, corresponding with those of the air compressor, are worked by the pressure of the air. But though they may be made of oiled silk and very light, the pressure in the vessel to be exhausted soon reaches a lower limit below which it is too small to open the valve between it and the cylinder of the pump. On this account automatic valves, operated mechanically, are in use in the better class of pumps. The modern pump in its sim- plest form is shown in Fig. 77. The two valves are oper- ated by the pressure of the air ; they are of oiled silk so as to be as light as possible. When the piston descends, valve Fin the piston opens and V at the bottom of the cylinder closes ; the reverse is true when the piston^ ascends. The limit of exhaustion is reached when the elastic force of the rare- fied air is not sufficient to open the valves. Figure 78 shows in section the cylinder of an air pump iu which the valves are automatic. A piston P, with a valve at S, works in a cylindrical barrel, commu- nicating with the outer air by a valve V at its upper end, and with the receiver FIGURE 78. AIR PUMP. to be exhausted by the horizontal tube at the bottom. The valve S' is carried by a rod passing through the piston, and fitting tightly enough to be lifted when the upstroke begins. The ascent of the rod is almost immediately 78 MECHANICS OF FLUIDS arrested by a stop near its upper end, and the piston then slides on the rod during the remainder of the upstroke. The open valve S' allows the air to flow from the vessel to be exhausted into the space below the piston. At the end of the upstroke the valve S' is closed by the lever shown in dotted lines. During the downward movement the valve S is open, and the inclosed air passes through it into the upper part of the cylinder. The ascent of the piston again closes S ; and as soon as the air is sufficiently compressed, it opens the valve V and escapes. Each complete double stroke re- moves a cylinder full of air; but as it be- comes rarer with FIGURE 79. FOOTBALL RECEIVER. FlGURE 80 - ~ AFTER AIR IN ~ P HAUSTED each stroke, the mass removed each time is less. 91. Experiments with the Air Pump. 1. Expansibility of air. (a) Football. Fill a small rubber foot- ball half full of air, and place under a big bell jar on the table of the air pump (Fig. 79). When the air is exhausted from the jar, the football expands until it is free from wrinkles (Fig. 80). A toy balloon may be substituted. (&) Bolthead. A glass tube with a large bulb blown on one end (Fig. 81) is known as a bolthead. The stem passes air-tight through the cap of the bell jar, and dips below the surface of the water in the inner vessel. When the air is exhausted from the jar, the air in the bolthead FIGURE expands and escapes in bubbles through the water. 81. BOLT- Readmission of air into the jar restores the pressure, HEAD ' and drives water into the bolthead. Why? 2. Air pressure, (a) Downward. Wet a piece of parchment paper, and tie it tightly over the mouth of a glass cylinder (Fig. 82). A EXPERIMENTS WITH THE AIR PUMP' 79 FIGURE 82. BURSTING PARCHMENT PAPER. sheet of stout paper may be pasted over the cylinder instead. When the air is exhausted, the paper will break with a loud report. (6) The vacuum fountain. A tall glass vessel has an inner jet tube which may be closed on the outside with a stop- cock. Exhaust the air, place the open- ing into the jet tube in water, and open the stopcock. The water is forced by atmospheric pres- sure into the exhausted tube like a fountain (Fig. 83). (c) Upward pressure. A strong glass cyl- inder supported on a tripod is fitted with a piston (Fig. 84). The brass cover of the FIGURE 83. VACUUM cylinder i s connected with the air pump by FOUNTAIN. v,. , , , a thick rubber tube. When the air is exhausted, the piston is lifted by atmospheric pressure, and carries the heavy attached weight. (of) The Magdeburg hemispheres. This historical piece of apparatus was designed by Otto von Gue- ricke to exhibit the great pressure of the atmosphere ( Fig. 85) . The lips of the two parts are accurately ground to make an air-tight joint when greased. When they are , i .1 FIGURE 84. LIFTING brought together and the air is WEIQHT BY PRESSURE QF exhausted, it requires consider- A TMO SPHERE. able force to pull them apart. M AG^E- The ori g inal hemispheres of von Guericke were about 22 BURG HEMI- in - in diameter, and the atmospheric pressure holding SPHERES. them together was about 5600 Ib. 80 MECHANICS OF FLUIDS FIGURE 86. THE BAROSCOPE. 92. Buoyancy of the Air. A small beam balance has attached to one arm a hollow closed brass globe ; it is counterbalanced in air by a solid brass weight on the other arm. When the balance is placed under a bell jar, and the air is exhausted, the globe^ overbalances the solid weight (Fig. 86). The apparatus just described is called a baroscope. It shows that the atmos- phere exerts an upward or buoyant force on bodies immersed in it; that is, the principle of Archimedes applies to gases as well as to liquids. The buoy- ancy or lifting effect of the atmosphere is equal to the weight of the air dis- placed by a body. Whenever a body is less dense than the weights, it weighs more in a vacuum than in the air. 93. Balloons and airships also illustrate the buoyancy of the air. A soap bubble and a toy balloon filled with air fall because they are heavier than the air displaced; but if filled with hydrogen or coal gas, they rise in the air. Their buoyancy is greater than their weight, including the inclosed gas. The weight of a balloon with its car and contents must be less than that of the air displaced by it. The essential part of a balloon is a silk bag, var- nished to make it air-tight ; it is filled either with hydro- gen or with illuminating gas. A cubic meter of hydrogen weighs about 0.09 kg., a cubic meter of illuminating gas, 0.75 kg., while a cubic meter of air weighs 1.29 kg. With hydrogen the buoyancy is 1.29 0.09 = 1.2 kg. per cubic meter; with illuminating gas it is 1.29 0.75 = 0.54 kg. per cubic meter. The latter is more commonly used because it is much cheaper. BALLOONS 81 A balloon is not fully inflated to start with, but it expands as it rises- because the pressure of the air on the outside diminishes. The buoyancy then decreases slowly as the balloon ascends into a rarer atmosphere. If it were fully inflated at the start, the inside pressure of THE BRITISH AIRSHIP R 34. This was the first airship to cross the Atlantic. It is shown here at its moorings on Long Island, the gas at a high altitude would be greater than the out- side atmospheric pressure, and the bag would burst. Airships are balloons with steering and propelling de- vices attached. They are made of large volume so as to give them considerable lifting force. Huge Zeppelins have been made 775 feet long, and holding 32,000 cubic feet of gas. They are driven by several gasoline engines aggregating from 4000 to 5000 horsepower. Figure 87 82 MECHANICS OF FLUIDS is a picture of a Zeppelin with the outer rubberized cotton cloth D partly cut away, to show the gas balloons inside. G-Q- are propellers, shown also in the front view in the corner of the picture. The balancing planes and the rudder may be seen at the rear end. FIGURE 87. A ZEPPELIN. Problems and Questions 1. What limits the height to which a balloon will ascend? 2. A pound of feathers exactly counterpoises a pound of shot on the scale pans of a balance. Do they represent equal masses of mat- ter ? Explain. 3. What force will be required to separate a pair of Magdeburg hemispheres, assuming the air to be entirely removed from the inside, the diameter of the hemispheres being 4 in. and the height of the barometer 30 in. ? 4. The volume of hydrogen collected over mercury in a graduated cylinder was 50 crn. 8 , the mercury standing 15 cm. higher in the cylinder than outside of it. The reading of the barometer was 75 cm. How many cubic centimeters of hydrogen would there be at a pressure of 76 cm. ? SUGGESTION. The height of the mercury in the cylinder above the surface of the mercury outside must be subtracted from the barometer reading to get the pressure of the gas in the cylinder. THE SIPHON 83 5. A test tube is forced down into water with its open end down, until the air in it is compressed into the upper half of the tube. How deep down is the tube if the barometer stands at 30 in. ? (The specific gravity of mercury may be taken as 13.6.) 6. With what volume of illuminating gas must a balloon be filled in order to rise, if the empty balloon and its contents weigh 540 kg. ? 7. A mass of iron, density 7.8, weighs 2 kg. in air. How much will it weigh in a vacuum ? VL PNEUMATIC APPLIANCES 94. The Siphon. The siphon is a U-shaped tube em- ployed to transfer liquids from one vessel over an inter- vening elevation to another at a lower, level by means of atmospheric pressure. If the tube is filled and is placed in the position shown in Fig. 88, the liquid will flow out of the vessel and be discharged at the lower level D. If the liquid flows outward past the highest point of the tube in the direction BO, it is because the pressure on the liquid outward is greater than the pres- sure in the other direction. Now the outward pressure at the top is the pres- sure of the atmosphere transmitted by the liquid to the top minus the weight FlGUI ~ THE of the column of liquid AB ; while the pressure inward is the atmospheric pressure transmitted to the top by the liquid in BD minus the weight of the column DO. Hence, the pressure inward is less than the pressure outward by the weight of a column of the liquid equal in height to the difference between AB and DO. AB and D(7are the lengths of the arms of the siphon. If the outer arm dips into the liquid in the receiving vessel, MECHANICS OF FLUIDS the arm terminates at the surface of the liquid. To in- crease the length CD is to increase the rate of flow. As AB and D C approach equality the rate of flow decreases and the flow ceases when this difference is zero. The siphon fails to work also when B is about 33 feet above A. Why ? On a small scale siphons are used to empty bottles and carboys, which cannot be tilted to pour out a liquid; also to draw off a liquid from a vessel without dis- turbing the sediment at the bottom. On a large scale engineers have used siphons for drain- ing lakes and marshes; also for lifting water from the ocean or other large body of water through a pipe leading to a steam condenser in a power plant, whence it flows back through the return pipe to the level of the water supply. The pipes are con- tinuous and air-tight, and the pump has no work to do ex- cept to keep the water run- ning against friction in the pipes. There is also a slight back pressure because the water on the discharge side is warmer and therefore lighted than on the intake side. When the mains of a water supply run over hills to a lower level, they constitute in reality siphons. Air is carried along with the water and collects in the bends at, the tops. If there are several of these siphons one after another, the back pressure may actually stop SIPHON OVER A MOUNTAIN. On the far side of the mountain the water is lifted by gravity pressure to within 32 feet of the top. THE LIFT PUMP 85 the flow of water, unless the air is removed by air pumps, or is allowed to escape under pressure through relief valves. An intermittent siphon (Fig. 89) has its short arm inside a vase and its long arm passing through the bottom. The vase will hold water until its level reaches the top of the bend of the siphon. It then discharges and empties the vessel, if it discharges faster than it is filled. Again the water rises in the vase, and the siphon again emp- ties it. Intermittent springs are supposed to operate on the same principle. A siphon fountain may be made * * ' ^ K w with a Florence flask and glass *~ tubing (Fig. 90). The flask is partly filled with water, and the apparatus is then inverted as shown. The water enters the flask as a jet. If a piece of rubber tubing is attached to the longer arm, the jet will rise as the end of the tubing is lowered. A portion of the water runs out at first, producing a partial vacuum inside. A siphon in a vacuum made of glass tubing about 2 mm. in diameter may be set up with mercury as the liquid. If it is set in action under a tall bell jar on the air pump, it will stop working when the air is exhausted from the jar, but will be- gin again when the air is admitted. The water in an S-trap, in common use under sinks and washbowls, may be siphoned off when the discharge pipe is filled with water for a short distance below the trap, unless the trap is ventilated at the top of the S. Fig. 91 shows the method of ventilating such traps. 95. The Lift Pump. -The common FrouRE 91 ._ VENTILA . lift or suction pump acts by the pres- TION OF S-TRAP. -Vent Pipe to Roof FIGURE 90. SIPHON FOUNTAIN. First Floor 86 MECHANICS OF FLUIDS sure of the air ; it is, in fact, a simple form of air pump ; but it was in use 2000 years before the air pump was invented. The first few strokes serve merely to draw out air from the pipe below the valve F (Fig. 92) ; the pressure of the air on the water in the well or cistern W, then forces it up the pipe $, and finally through the valve V. After that, when the piston descends, the valve V closes and checks the return of the water, and water passes through the valve V above the piston. The next upstroke lifts the water to the level of the spout. Since the pressure of the air lifts the water to the highest point to which the piston ascends, it is obvious that this point cannot be more than the limit of about 33 ft. above the water in the well. Practically it is less on account of leakage through the imper- fect valves. The priming of a pump by pour- ing in a little water to start it serves to wet the valves and make them air-tight. For deep wells the piston rod is lengthened and the valves v and v' are placed far down the well ; the long pump rod serves to lift the FIGURE 93. water from the piston to the spout (Fig. 93). -LIFT PUMP. 96. The Force Pump. The force pump (Fig. 94) is used to deliver water under pressure, either at a point FIGURE 92. SUCTION PUMP. THE AIR BEAKS 87 FIGURE 94. FORCE higher than the pump into pipes, as in the fire engine, into boilers against steam pressure, or into the cylinder of the hydraulic press. The air dome D is added to secure a continuous flow through the delivery pipe d. Water flows out through v f only while the piston is descending; without the air dome, therefore, water would flow through the pipe d only during the downstroke of the piston; but the water under pressure from the piston enters the dome and compresses the air. The elastic force of the air drives the water out again as soon as v' closes. Thus the flow is practically continuous. The pump of a steam fire engine is double acting, that is, it forces water out while the piston is moving in either direction (Fig. 95) ; so also are pumps for waterworks and mines. 97. The Air Brake. The well- known Westinghouse air brake is oper- ated by compressed air. In Fig. 96 P is the train pipe leading to a large reservoir at the engine in which an air compressor maintains a pressure of about 75 Ib. per square inch. So long as this pressure is applied through P, the automatic valve V maintains communication between P and an auxiliary reservoir R under each car, and at the same time shuts off air from the brake cylinder C. But as soon as the pressure in P falls, either by the movement of a lever in the engineer's cab or by the accidental parting of the hose coupling k, the valve V cuts off P and connects the reservoir R with the cylinder C. The pressure on the piston in C drives it powerfully FIGURE 95. FIRE ENGINE PUMP. 88 MECHANICS OF FLUIDS to the left and sets the brake shoes against the wheels. As soon as air from the main reservoir is again admitted to the pipe P, the valve V reestablishes com- munication between P and R, and the confined air in C escapes. The brakes are released by the action of the spring S in forcing the piston back to the right. 98. Other Applications of the Air Pump and the Air Compressor. The air pump and the air com- pressor are extensively used F.GURE 96.-A.R BRAKE. in industry. Sugar refiners employ the air pump to re- duce the boiling point of the sirup by lowering the pressure on its sur- face in the evaporating pan ; manufacturers of soda water use a com- pressor to charge the water with carbon dioxide ; in pneumatic dispatch FIGURE 97. RIVETING HAMMER. tubes, now extensively used for carrying small packages, both the air pump and the compressor are used, one to exhaust the air from the tube in front of the closely fitting carriage, and the other to compress air in the tube behind it, so as to propel the carriage with great velocity. The air compressor is employed to make a forced draft for QUESTIONS AND PROBLEMS 89 steam boilers, to ventilate buildings, and to operate machinery in places difficult of access, as in mines, where it furnishes fresh air as well as power. It is employed also in the pneumatic caisson for making excavations and laying foundations under water. The cais son is a large heavy air chamber which sinks as the soft earth is removed from within. When its bottom is below water level, air is forced in under sufficient pressure to prevent the entrance of water. Access to it is gained by air-tight locks. Compressed air is frequently used for operating railway signals, and to control automatic heating and ventilating appliances. Pneu- matic tools are used for calking seams and joints, for stone cutting, chipping iron, and riveting. Figure 97 shows a riveting hammer ; A is the air pipe, B the trigger for controlling the air, and C the hammer. The vacuum cleaner is essentially a fan driven by an electric motor. The fan pushes the air away from one face and atmospheric pressure forces air through the mouthpiece of a tube leading to the fan to fill the partial vacuum. This stream of air carries with it the dust of the rug o carpet. Questions and Problems 1. What will happen if the tip of an incandescent lamp bulb be broken off under water? 2. How can a tumbler of water be inverted (with the aid of a card) without spilling the water? 3. Explain why the " priming " of a dry suction pump restores it to working condition. 4. What sort of rubber tube must be used to connect a receiver to be exhausted by an air pump ? 5. What is the limit of pressure to which a large suction water pump can subject the intake pipe? Will the pipe collapse if the pump "sucks" hard enough? 6. When the barometer stands at 29 in., what is the limiting height over which a siphon can carry water ? 7. A vessel 36 in. deep is filled with mercury; can it be com- pletely emptied by means of a siphon ? 90 MECHANICS OF FLUIDS 8. A diver works in 35 feet of sea water, specific gravity 1.025 What pressure must the compression pump supply to counterbalance the water pressure ? 9. When the barometer reading is 73 cm., what is the greatest possible length of the short arm of a siphon when used for sulphuric acid, density 1.84 g. per cubic centimeter? 10. If the pressure against the 8 in. piston of an air brake is 80 Ib. per square inch, what is the force driving the piston forward ? CHAPTER IV MOTION I. MOTION IN STRAIGHT LINES 99. All Motion Relative. Rest and motion are relative terms only. A body is at rest when its relative position with respect to some point, line, or surface remains un- changed ; but when that relative position is changing, the body is in motion. The moving about of a person on a ship is relative to the vessel; the movement of the ship across the ocean is relative to the earth's surface ; the daily motion of the earth's surface is relative to its axis of rotation ; the motion of the earth as a whole is relative to the sun ; while the sun itself is drifting with other stars through space. 100. Types of Motion. Many familiar motions are irreg- ular in every way, both as to direction and speed. The flight of a bird, the running of a boy at play, and even the motion of a man riding a horse, are illustrations. We shall study only those motions that can be classified and reduced to simple terms. The line described by a moving body is its path. When this path is straight, like that of a falling body, the motion is rectilinear ; when it is a curved line, like that of a rocket, the motion is curvilinear. Then there is also simple harmonic motion, exemplified by the to-and-fro swing of a pendulum ; and rotary motion about an axis, such as the rotation of the earth on its axis, and that of the pulley and armature of a stationary elec- 91 92 MOTION trie motor. The motion of a carriage wheel along a level road, and that of a ball along the floor of a bowling alley combine motion of rotation with rectilinear motion. 101. Speed or Velocity. If an automobile runs thirty miles in an hour and a half, its average speed is 20 miles per hour. Speed or velocity is the rate of motion, that is, it is the distance traversed per unit of time. In express- ing a speed or a velocity the time unit must be given as THE TWENTIETH CENTURY LIMITED AT SIXTY MILES AN HOUR. The railway train is one of our most familiar examples of motion. well as the numerical value. Thus, 60 miles per hour, 5280 feet per minute, and 26.82 meters per second are all expressions for the same speed. There is but little distinction between speed and ve- locity. Both express the rate of motion, but velocity is generally used to express the rate of motion in a definite direction, while speed is rate of motion without reference to direction. 102. Uniform Motion. If the motion is over equal dis- tances in equal and successive units of time, the motion is uniform and the velocity is constant. In uniform motion ACCELERATION 98 the whole distance traversed is found by multiplying the speed by the time, or distance = speed x time. In symbols this is written, s = v x t ; from which v = - and t = - . . (Equation 4) t v EXAMPLE. A railway train runs uniformly covering 660 ft. in 10 min. Then the speed v = 6 T 6 < 66 ft. per minute, or mi. per hour. The distance s = 66 x 10 = 660 ft. The time t = 6 g 6 ^ = 10 min. The average speed in variable motion is found in the same way as in uniform motion, namely, by dividing the space traveled by the time. 103. Velocity at any Instant. When the motion is vari- able, the velocity of a body at any instant is the distance it would travel in the next unit of time if at that instant its motion were to become uniform. For example : The velocity of a falling body at any moment is the distance it would fall during the following second, if the attraction of the earth and the resistance of the air were both to be withdrawn. The velocity of a ball as it leaves the muzzle of a gun is the distance it would pass over in the second following if from that instant it should continue to move for a second without any change in speed. Actually the motion of the body and the ball for the suc- ceeding second is variable ; the question is, what would be the velocity if the motion were invariable ? 104. Acceleration. When a train runs a mile a minute for several minutes, it moves with uniform velocity ; but when it is starting or slowing down, it is said to be accel- erated. If the velocity increases, the acceleration is posi- tive; if it decreases, it is negative. A falling body goes 94 MOTION faster and faster ; it has a positive acceleration. A body thrown upward goes more and more slowly ; it has a negative acceleration. A loaded sled starts from rest at the top of a long hill ; it gains in velocity as it descends the hill ; it has a positive acceleration. When it reaches the bottom, it loses velocity and is retarded, or has a negative acceleration, until it stops. Acceleration is the rate of change of speed. Acceleration = change in speed per unit time. Acceleration is always expressed as so many units of speed per unit of time. If, for example, a street car start- ing from rest gains uniformly in speed, so that at the end of ten seconds it has a speed of 10 miles per hour, its ac- celeration is its gain in speed-per-hour acquired in one second, or 1 mile-per-hour per second. 105. Uniform Acceleration. If the change in velocity is the same from second to second, the motion is uniformly accelerated. The best example we have of uniformly ac- celerated motion is that of a falling body, such as a stone or an apple. Neglecting the resistance of the air, its gain in velocity is 9.8 m.-per-second for every second it falls. Its acceleration is therefore 9.8 m.-per-second per second ; in other words, it gains in velocity 9.8 m.-per-second for every second of time. This is equivalent to an increase in velocity of 588 m.-per-second acquired in a minute of time. The unit of time enters twice into every expression for acceleration, the first to express the change in velocity, and the second to denote the interval during which this change takes place. If an automobile starts from rest and increases its speed one foot a second for a whole minute, its velocity at the end of the minute is 60 ft. per second. Since it gains in one second a velocity of one foot DISTANCE TRAVERSED 95 a second, and in one minute a velocity of 60 ft. a second, its accelera- tion may be expressed either as one foot-per-second per second, or as 60 ft.-per-second per minute. Its velocity is constantly changing ; its acceleration is constant. 106. Velocity in Uniformly Accelerated Motion. Suppose a body to move from rest in any given direction with a constant acceleration of 5 ft.-per-second per second. Its velocity at the end of the first second will be 5 ft. per second ; at the end of two seconds, 2 x 5 ft. ; at the end of three seconds, 3 x 5 ft. ; and at the end of t seconds, t x 5 ft. per second ; that is, final velocity = time x acceleration, or in symbols, . v = ta ; whence a = -. . (Equation 5) Hence, In uniformly accelerated motion the speed acquired in any given time is proportional to the time. 107. Distance traversed in Uniformly Accelerated Motion. If we can find the mean or average velocity for any period of t seconds, the distance s traversed in t seconds may be found precisely as in the case of uniform motion ( 102). For a body starting from rest with an accelera- tion of 5 feet-per-second per second, for example, its ve- locity at the end of four seconds is 4 x 5 ft. per second, and the average velocity for the four seconds is the mean be- tween and 4 x 5, or 2 x 5 ft. per second, the velocity at the middle of the period. So at the end of t seconds the average velocity is \ ta ft. per second. Then we have distance = average velocity x time, 96 MOTION or in symbols, s = %taxt = \at\ . (Equation 6) Hence, In uniformly accelerated motion the distance traversed from rest is proportional to the square of the time. 108. Uniformly Accelerated Motion Illustrated. The old- est method of demonstrating uniformly accelerated motion was devised by Galileo. It con- sists of an inclined plane two or three meters long (Fig. 98), made of a straight board with a shallow groove, down which a mar- FIGURE 98. GALILEO'S INCLINED PLANE. ble or a steel ball may roll slowly enough to permit the distances to be noted. For measuring time, a clock beating seconds, or a metro- nome, may be used. Assume a metronome as shown in the figure adjusted to beat seconds. One end of the board should be elevated until the ball will roll from a point near the top to the bottom in three seconds. Hold the ball in the groove against a straightedge in such a way that it may be quickly released at a click of the metronome. Find the exact position of the straight- edge near the top of the plane from which the ball will roll to the bottom and strike the block there so that the blow will coincide with the third click of the metronome after the release of the ball. Measure exactly the dis- tance between the upper edge of the straightedge and the block at the bottom and call it 9 d. Next, since distances are proportional to the square of the times, let the straight- UNIFORMLY ACCELERATED MOTION 97 edge be placed at a distance of 4d from the block; the ball released at this point should reach the block at the second click of the metronome after it starts. Finally, start the ball against the straightedge at a distance d from the block; the interval this time should be that of one beat of the metronome. TABULAR EXHIBIT NUMBER OF BEATS, t WHOLE DISTANCE FALLEN, * DISTANCE IN SUCCES- SIVE INTERVALS VELOCITIES ATTAINED, * I d d 2d 2 d 3d 4d 3 9d 5d Qd 4 IQd Id 8d The third column is derived by subtracting the successive numbers of the second. To get the fourth column, we notice that if t is one second in Equation 6, then = \ a\ that is, the distance traversed in the first second is one half the accel- eration. But the acceleration is the same as the velocity acquired the first second. Hence s = J v and d = J v. Therefore the veloc- ity at the end of the first second on* the inclined plane is 2 d. Since by Equation 5 the velocities are proportional to the time, the suc- ceeding velocities are 4 d, 6 df, etc. ^. =8d 7"; The numbers in the second 3d 4.d FIGURE 99. LAWS OF FALL- ING BODIES. 98 MOTION column show that the distances traversed are proportional to the squares of the time [compare Equation 6] ; those of column three show that the distances in successive seconds are as the odd numbers 1, 3, 5, etc. The results are shown graphically in Fig. 99. Problems NOTE. For the relation between the circumference of a circle and its diameter, see the Mensuration Table in the Appendix. 1. An aviator drives his aeroplane through the air a distance of 500 km. in 8 hr. 20 min. What was his average speed per minute? 2. The engine drives a boat downstream at the rate of 15 mi. an hour, while the current runs 3 ft. a second. How long will it take to go 50 mi.? 3. A man runs a quarter of a mile in 48.4 seconds. At that speed, what was his time for 100 yd.? 4. If a man can run 100 yd. in 10 sec., what would be his time for a mile, if it were possible to maintain the same speed? 5. A procession 100 yd. long, moving at the rate of 3 mi. an hour, passes over a bridge 120 yd. long. How long does it take the proces- sion to pass entirely over the bridge ? 6. An express train is running 60 mi. an hour. If the train is 500 ft. long, how many seconds will it be in passing completely over a viaduct 160 ft. in length ? 7. A locomotive driving wheel is 2 m. in diameter. If it makes 200 revolutions per minute, what is the speed of the locomotive in kilometers per hour, assuming no slipping of the wheel on the track? 8. An automobile running at a uniform speed of 25 mi. per hour is 10 mi. behind another one on the same highway running 20 mi. per hour. How long will it take the former to overtake the latter, and how far will each machine have gone during this time ? 9. If the acceleration of a marble rolling down an inclined plane is 40 cm.-per-second per second, what will be its velocity after 3 sec. from rest ? 10. How far will a marble travel down an inclined plane in 3 sec. if the acceleration is 40 cm.-per-second per second? DIRECTION OF MOTION ON A CURVE 99 11. A body starts from rest, and moving with uniformly acceler- ated motion acquires in 10 sec. a velocity of 3600 m. per minute. What is the acceleration per-second per second. How far does the body go in 10 sec. ? 12. What acceleration per-minute per minute does a body have if it starts from rest and moves a distance of a mile in 5 min.? What will be its velocity at the end of 4 minutes ? 13. If a train acquires in 2 min. a velocity of 60 mi. an hour, what is its acceleration per-minute per minute, assuming uniformly accel- erated motion ? 14. An electric car starting from rest has uniformly accelerated motion for 3 rnin. At the end of that time its velocity is 27 km. an hour. What is its acceleration per-minute per minute ? 15. A sled is pushed along smooth ice until it has a velocity of 4 m. per second. It is then released and goes 100 m. before it stops. If its motion is uniformly retarded, what is the retardation in centi- meters-per -second per second ? 16. To acquire a speed of 60 mi. an hour in 10 min., how far would an express train have to run, provided it started from rest and its motion were uniformly accelerated ? II. CURVILINEAR MOTION 109. Direction of Motion on a Curve. Curvilinear motion, or motion along a curved line, occurs more frequently in nature than motion in a straight line. The motion of a point on the earth's surface and about its axis is in a circle; the motion of the earth in its path around the sun is along a curve only approximately circular; the motion of a rocket or of a stream of water directed obliquely upward is along a parabolic curve. So also is the motion of a baseball when batted high in air. The thrown " curved ball," too, illustrates curvi- FIGURE 100. MOTION linear motion. ALONG A CURVE. 100 MOTION When the motion is along a curved line, the direction of motion at any point, as at B (Fig. 100), is that of the line CD, tangent to the curve at the point. This is the same as the direction of the curve at the point. 110. Uniform Circular Motion. In uniform circular mo- tion the velocity of the moving body, measured along the circle, is constant. There is then no acceleration in the direction in which the body is going at any point. But while the velocity remains unchanged in value, it varies in direction. If a body is moving with constant velocity in a straight line, its acceleration is zero in every direction; but if the direction of its motion changes continuously, then there is an acceleration at right angles to its path and its motion becomes curvilinear. If this ac- celeration is constant, the motion is uni- form in a circle. Hence, in uniform circular motion there is a constant acceler- ation directed toward the center of the circle. It is called centripetal accelera- tion. FIGURE 101. CEN- Uniform circular motion consists of a TRIPETAL ACC^LERA- uniform motion in the circumference of the circle and a uniformly accelerated motion along the radius. If v is the uniform velocity around the circle whose radius is r, the value of the cen- tripetal acceleration is v 2 * a= , .... (Equation 7) . . . 7 7 ^ square of velocity in circle or centripetal acceleration = -^ *- J t radius of circle * Let ABC (Fig. 101) be the circle in which the body revolves, and AB the minute portion of the circular path described in a very small interval of time t. Denote the length of the arc AB by s. Then, since MOTION AND FORCE. Above : White Star Liner " Britannic." Below : Part of Boston Elevated Company's Power Plant. SIMPLE HARMONIC MOTION > 101 III. SIMPLE HARMONIC MOTION 111. Periodic Motion. The motion of a body is said to be periodic when it goes through the same series of move- ments in successive equal periods of time. It is vibratory if it is periodic and reverses its direction of motion at the end of each period. The motion of the earth around the sun is periodic, but not vibratory. A hammock swinging in the wind, the pendulum of a clock, a bowed violin string, and the prong of a sounding tuning fork illustrate both periodic and vibratory motion. 112. Simple Harmonic Motion. Suspend a ball by a long thread and set it swinging in a hori- zontal circle (Fig. 102). Place a white screen back of the ball and a strong light, such as an arc light or a Welsbach gas light, at T-r a distance of JL twelve or fif- teen feet in front and on a level with the ball. Viewed in darkened room, a shadow of the ball will be seen FIGURE 102. SIMPLE on the screen, moving to and fro in a straight HARMONIC MOTION. line. This motion is very nearly simple harmonic and would be perfectly so if the projection could be made with sun- light, so that the projecting rays were perpendicular to the screen. the motion along the arc is uniform, s = vt. AB is the diagonal of a very small parallelogram with sides AD and AE. The latter is the distance through which the revolving body is deflected toward the center while traversing the very small arc AB. Since the acceleration is constant, AE = \ at 2 by Equation 6. The two triangles ABE and ABC are simi- lar. Hence AB 2 AE x AC. Calling the radius of the circle r and substituting for AB, AE, and AC their values, vW = |a 2 x 2r = at*r. Then a = - 102 MOTION Simple harmonic motion is the projection of a uniform cir- cular motion on a straight line in the plane of the circle. All pendular motions of small arc are simple harmonic. The name appears to be due to the fact that simple musical sounds are caused by bodies vibrating in this manner. The graph of a simple harmonic motion is obtained as follows : Draw the circle adgk (Fig. 103) representing the path of the ball, and the straight line ADG its projection on the screen. Divide the circumference into any even number of equal parts, as twelve. Through the points of division let fall per- pendiculars on AG, as aA, bB, cC, etc. Now as the ball moves along the arc adg, its shadow appears to the observer to move from A through B, C, etc., to- G, where it momentarily comes to rest. It then starts back toward A, at first slowly, but with increasing velocity until it passes D. Its velocity then decreases, and at A it is again zero, and its motion reverses. c D FIGURE 103. GRAPH SIMPLE HARMONIC MOTION. The radius of the circle, or the distance AD, is the amplitude of the vibration. The period of the motion is the time taken by the ball to go once around the circle ; it is the same as the time of a double oscillation of the projected motion. The frequency of the vibration is the reciprocal of the period. For example, if the period is J a second, the frequency is 2, that is, two complete vibrations per second. This relation finds frequent illustration in musi- cal sounds, where pitch depends on the frequency ; in light, where frequency determines the color ; and in alter- nating currents of electricity, where a frequency of 50, for example, means that a complete wave is produced every fiftieth of a second, and that the current reverses 100 times per second. PROBLEMS 103 Two simple harmonic motions of the same period are said to differ in phase when they pass through their maximum or minimum velocities at a different time. Thus, if one has its maximum velocity at the same instant that the other has its minimum, the two motions differ in phase by a quarter of a period. Problems 1. At what speed must an automobile be driven to go four times around a circular track one mile in diameter in thirty minutes? 2. The equatorial diameter of the earth is about 8000 miles. What is the speed in miles per minute of a point on the equator, owing to the earth's rotation on its axis ? 3. A conical pendulum swinging in a circle whose diameter is 50 cm. makes 5 complete revolutions in 15 seconds. What is the centrip- etal acceleration of the bob ? 4. The radius of the moon's orbit is 240,000 miles, and the moon revolves around the earth in 27 days, 8 hours. What is its centripetal acceleration with respect to the earth in f eet-per-second per second ? 5. A balance wheel on a stationary engine is 10 ft. in diameter and makes 100 revolutions per minute. A point on its circumference has what centripetal acceleration per-second per second ? 6. The earth's equatorial radius is 20,926,000 feet, and the period of the earth's rotation on its axis is 23 h., 56 min., 4 sec. Calculate the the centripetal acceleration per-second per second at the equator. CHAPTER V MECHANICS OF SOLIDS I. MEASUREMENT OF FORCE 113. Force. A preliminary definition of force as a push or a pull has already been given. The effects of force in producing motion are among our commonest observations. A BRITISH "TANK" GOING INTO ACTION. The " tank " exerts enough force to break down trees and walls. A brick loosened from a chimney or pushed from a scaffold falls by the force of gravity ; a mountain stream rushes down by reason of the same force in nature; the leaves of a tree rustle in the breeze, the branches sway violently in the wind, and their trunks are even twisted off by the 104 UNITS OF FORCE 105 force of the tornado; powder explodes in a rifle and the bullet speeds toward its mark; loud thunder makes the earth tremble and vivid lightning rends a tree or shatters a flagstaff. From many such familiar facts is derived the conception that force is anything that produces motion or change of motion in material bodies. It remains now to explain how force is measured. 114. Units of Force. Two systems of measuring force in common use are the gravitational and the absolute. The gravitational unit of force is the weight of a standard mass, such as the pound of force, the gram of force, or the kilo- gram of force. A pound of force means one equal to the force required to lift the mass of a pound against the down- ward pull of gravity. The same is true of the metric units with the difference in the mass lifted. Gravitational units of force are not strictly constant because the weight of the same mass varies from point to point on the earth's surface, and at different elevations. The actual force necessary to lift the mass of a pound at the poles of the earth is greater than at the equator; it is less on the top of a high mountain than in the neighboring valleys, and still less than at the level of the sea. Gravi- tational units of force are convenient for the common pur- poses of life and for the work of the engineer, but they are not suitable for precise measurements, especially in the domain of electricity. The so-called " absolute " unit of force in the c.g.s. system is the dyne (from the Greek word meaning force). The dyne is the force which imparts to a gram mass an acceler- ation equal to one centimeter -per-second per second. This unit is invariable in value, for it is independent of the vari- able force of gravitation. It is indispensable in framing the definitions of modern electrical and magnetic units. 106 MECHANICS OF SOLIDS 115. Relation between the Gram of Force and the Dyne. The gram of force is the pull of the earth on a mass of one gram, definitely at sea level and latitude 45. Since the attraction of the earth in New York imparts to a gram mass an acceleration of 980 cm.-per-second per second, while the dyne produces an acceleration of only 1 cm.-per- second per second, it follows that the gram of force in New York is equal to 980 dynes, or the dyne is -g| 7 of the gram of force. The pull of gravity on a gram mass in other latitudes is not exactly the same as in New York, but for the purposes of this book it will be sufficiently accurate to say that a gram of force is equal to 980 dynes. It will be seen, therefore, that the value of any force expressed in dynes is approximately 980 times as great as in grams of force. Conversely, to convert dynes into grams of force, divide by 980. 116. How a Force is Measured Mechanically. The simplest device for measuring a force is the spring balance (Fig. 104). The common draw scale is a spring balance graduated in pounds and fractions of a pound. If a weight of 15 lb., for example, be hung on the spring and the position of the pointer be marked, then any other 15 lb. of force will stretch the spring to the same extent in any direction. If a man by pulling in any direction 104. stretches a spring 3 in., and if a weight of 150 SPRING pounds also stretches the spring 3 in., the force RAT ATMPP " exerted by the man is 150 pounds of force. The spring balance may be graduated in pounds of force, kilograms or grams of force, or in dynes. If correctly graduated in dynes, it will give right readings at any latitude or elevation. Why are the divisions of the scale equal? COMPOSITION OF FORCES 107 117. Graphic Representation of a Force. A force has not only magnitude but also direction; in addition, it is often necessary to know its point of application. These three particulars may be represented by a straight line drawn through the point of application of the force in the direction in which the force acts, and as many units in length as there are units of force, or some multiple or submultiple of that number. If a line 1 cm. long stands for a force of 15 dynes, a line 4 cm. long, in the direction -* ' B AB (Fig. 105), will represent 5 ' 7 ^ REPRESENT A a force of 60 dynes acting in the direction from A to B. Any point on the line AB may be used to indicate the point at which the force is applied. If it is desired to represent graphically the, fact that two forces act on a body at the same time, for example, B 4 kg. of force horizontally and 2 kg. tof force vertically, two lines are drawn from the point of application A (Fig. 2 ~* *c 106), one 2 cm. long to the right, and FIGURE 106. Two the other 1 cm. long toward the top FORCES AT RIGHT o f the page. The lines AB and AC represent the forces in point of appli- cation, direction, and magnitude, on a scale of 2 kg. of force to the centimeter. II. COMPOSITION OF FORCES AND OF VELOCITIES 118. Composition of Forces. The resultant of two or more forces is a single force which will produce the same effect on the motion of a body as the several forces acting together. (Note the exception in the case of a couple, 121.) The process of finding the resultant of two or more 108 MECHANICS OF SOLIDS forces is known as the composition of forces. It will be con- venient to consider first the composition of parallel forces, and then that of forces acting at an angle. The several forces are called components. 119. The Resultant of Parallel Forces. Suspend two draw scales, A and B (Fig. 107), from a suitable support by cords. Attach to them a graduated bar and adjust the draw scales and the attached cords so that they are vertical. Read the scales, then attach the weight W and again read the scales. Note the dis- tances CE and ED. Correct each drav: scale reading by sub- tracting from it the reading before the weight W was added. Compare W with the sum of these corrected readings, and also the ratio of the corrected readings of A and B to that of ED and EC. Change the position of E and repeat the observations. It will be A ~t? Ty found in each case that = . Hence the following principle : B EC The resultant of two parallel forces in the same direc- tion is equal to their sum; its point of application divides the line joining the points of application of the two forces into two parts which are inversely as the forces. 120. Equilibrium. If two or more forces act on a body and no motion results, the forces are said to be in equi- FIGURE 107. PARALLEL FORCES. RESULTANT OF TWO FORCES AT AN ANGLE 109 librium. In Fig. 107 the weight W is equal and opposite to the resultant of the two forces measured bythe draw scales A and B. The three forces A, B, and W are in equilibrium. Further, each force is equal and opposite to the resultant of the other two and is called their equi- librant. The equilibrium of a body does not mean that its velocity is zero, but that its acceleration is zero. Rest means zero velocity ; equilibrium, zero acceleration. 121. Parallel Forces in Opposite Directions. If two par- allel forces act in opposite directions, their resultant is their difference, and it acts in the direction of the larger force. In Fig. 107 the resultant of A and W is equal and opposite to B. When the two parallel forces acting in opposite direc- tions are equal, they form a couple. The resultant of a couple is zero ; that is, no single force can be substituted for it and produce the same effect. A couple produces motion of rotation only, in which all the particles of the body to which it is applied rotate in circles about a com- mon axis. For example, a magnetized sewing needle floated on water is acted on by a couple when it is dis- placed from a north-and-south position. One end of the needle is attracted toward the north, and the other toward the south, with equal and parallel forces. The effect is to rotate the needle about a vertical axis until it returns to a north-and-south position. The common auger, as a carpenter employs it to bore a hole, illustrates a couple in the equal and opposite parallel forces applied by the two hands. 122. The Resultant of Two Forces Acting at an Angle. Tie together three cords at D (Fig. 108) and fasten the three ends to the hooks of the draw scales A, B, C. Pass their rings over pegs set in a board at such distances apart that the draw scales will all be 110 MECHANICS OF SOLIDS stretched. Record the readings of the scales, and by means of a protractor (see Appendix I) measure the angles formed at D by the cords. Draw on a sheet of paper three lines meeting at a point Z>, and forming with one another these angles. Lay off on the three lines on some convenient scale, distances to represent the readings of the draw scales, DF for A, DE for B, and DC for C. With DF and DE as adjacent sides, complete the parallelogram DFGE and draw the diagonal DG. DG is the resultant of the forces A and jB, and its Scale : 50 gm. to 1 OB. \j FIGURE 108 RESULTANT OF Two FORCES AT AN ANGLE. length on the scale chosen will be found equal to that of DC, their equilibrant. Here again, each force is equal and opposite to the resultant of the other two. When two forces act together on a body at an angle, the resultant lies between the two ; its position and value may be found by apply- ing the following principle, known as the parallelogram offerees : If two forces are represented by two adjacent sides and DE) of a parallelogram, their resultant is represented by the diagonal (D 6r) of the parallelogram drawn through their common point of application COMPONENT IN A GIVEN DIRECTION 111 When the two forces are equal, their resultant lies mid- way between them. If the two forces are at right angles (Fig. 109) the parallelogram becomes a rectangle and the two forces and their resultant are represented by the three sides of a right triangle, AB, ED, AD. The value of the resultant in this case may be found by com- FIGURE 109. FORCES puting the hypotenuse of the triangle. AT RlGHT ANGLES> For example, if the forces at right angles are 6 kg. of force and 8 kg. of force, their resultant is V6 2 + 8 2 = 10 kg. of force. 123. Component of a Force in a Given Direction. It frequently occurs that if a force produces any motion, it must be in a direction other than that of the force itself. For example, suppose the force AB (Fig. 110) applied to cause FIGURE 1 10. COMPONENT IN a car to move along the rails mn. The force AB evidently produces two effects ; it tends to move the car along the rails, and it increases the pressure on them. The two effects are produced by the two forces OB and DB re- spectively. They are therefore the equivalent of AB. The force CB is called the component of AB in the direc- tion of the rails mn< and DB is the component perpen- dicular to them. The component of a force in a given direction is its effective value in this direction. To find the component in a given direction, construct on the line representing the force, as the diagonal, a rectangle, the sides of which are respectively parallel and perpendicular to the direction of the required component ; the length of the 112 MECHANICS OF SOLIDS side parallel to the given direction represents the component sought. EXAMPLE. Let a force of 200 Ib. be applied to a truck, as AB in Fig. 110 ; and let it act at an angle of 30 with the horizontal. Find the horizontal component pushing the truck forward. Construct a parallelogram on some convenient scale (Appendix I) with the angle ABC equal to 30 and AB representing 200 Ib. Measure the side CB and obtain by the scale used its equivalent in pounds of force. CB may be calculated since A CB is a right triangle. Since ABC is an angle of 30, AC is one half of AB. Then, since A C denotes 100 Ib. of force, CB = = V200 2 - 100 2 = 173.2 Ib. of force. 124. Illustrations of the Resolution of a Force. The kite, the sailboat, and the aeroplane are familiar illustrations of the resolution of the force of the wind. In the case of the kite, the forces acting are the weight of the kite AB (Fig. Ill), the pull of the string A C, and the force of the wind LA. AD is the resultant of AB and A C. Re- solve the force of the wind into two components, one perpendicular to HK, the face of the kite, and the other parallel to HK. If HK sets itself at such an angle that the component of LA perpendicular to HK coincides with AD and is equal to it, the kite will be in equi- librium ; if it is greater than AD, the kite will move upward ; if less, it will descend. In the case of the sailboat, the sail is set at such an angle that the wind strikes the rear face. In Fig. 112 BS represents the sail, and AB the direc- FIGURE 112 _ tion and force of the wind. This force may be re- FORCES ON SAIL- solved into two rectangular components, CB and BOAT. FIGURE 111. FORCES ACTING ON KITE. COMPOSITION AND RESOLUTION OF VELOCITIES 113 DB, of which CB represents the intensity of the force that drives the boat forward. In the case of the aeroplane (Fig. 113), if a large flat surface, placed obliquely to the ground, be moved along rapidly, it will be lifted upward by the vertical component of the reaction of the air against it, equivalent to a wind, just as the kite is lifted. In both the monoplane and the biplane, large bent surfaces attached to a strong light frame are forced through the air by a rapidly rotating propeller driven by a powerful gasoline engine ( 380). By means of suitable FIGURE 113. THE FRENCHMAN VEDRINES AND HIS MONOPLANE. levers under control of the driver, these planes, or certain auxiliary planes, can be set at an angle to the stream of air against which they are propelled. Then, as with the kite, they rise through the air. Vertical planes are attached to the frame to serve as rudders in steer- ing either to the right or the left. 125. Composition and Resolution of Velocities. At the Paris exposition in 1900 a continuous moving sidewalk carried visitors around the grounds. A person walking on this platform had a velocity with respect to the ground made up of the velocity of the sidewalk relative to the ground and the velocity of the person relative to the mov- 114 MECHANICS OF SOLIDS ing walk. The several velocities entering the result are the component velocities. Velocities may be combined and resolved by the same methods as those applying to forces. When several motions are given to a body at the same time, its actual motion is a compromise between them, and the compromise path is the resultant. The following is an example of the composition of two velocities at right angles : A boat can be rowed in still water at the rate of 5 mi. an hour ; what will be its actual velocity if it be rowed 5 mi. an hour across a stream running 3 mi. an hour ? Let AB (Fig. 114) represent in length and direction the velocity of 5 mi. an hour across the stream, and AC, at right angles to AB, the velocity of the current, 3 mi. an hour, both on the same scale. Complete the parallelo- gram ABDC, and draw the diagonal AD through the point A common to the two component velocities. AD represents the actual velocity of the boat ; its length on the same scale as that of the other lines is 5.83. The resultant velocity is therefore 5.83 miles an hour in the direction AD. When the angle between the components is a right angle, as in the present case, the diagonal AD is the hypotenuse of the right triangle ABD. Its square is therefore the sum of the squares of 5 and 3, or AD = V5 2 + 3 2 = 5.83. When the angle at A is not a right angle, the approximate resultant may be found by a graphic process of measurement. A velocity, like a force, has both direction and magni- tude, and a component of it in any given direction may be found in precisely the same way as in the case of a force, ( 123). The most common case is the resolution into components at right angles to each other. In most cases it suffices to find the component in the direction in which the attention for the time being is directed. The other one at right angles is without effect in this particular direction. FIGURE 1 14. BOAT RUN- NING ACROSS STREAM. PROBLEMS 115 Problems NOTE. Solve graphically the problems involving forces and velocities at an angle. Where possible, verify by calculation. Consult Appendix I for methods of drawing. 1. Plot a force of 25 g. on a scale of 4 cm. to the gram. 2. Represent a force of 50 g. by a straight line on a scale of 10 cm. to the gram. 3. Represent by a figure two forces acting at a common point, the forces being 15 g. and 20 g. respectively, and the angle between their directions being 60. 4. A body is acted on by two parallel forces, 20 and 30 Ib. respec- tively. These forces act in the same direction and have their points of application 60 in. apart. Find the magnitude of the resultant and the distance of its point of application from the less force. SUGGESTION. Let x be the distance of the point of application of the resultant from the force 20. Then 60 x will be the distance from the force 30. Then by Art. 119, ^ = ^=^ . 5. A weight of 200 Ib. is fastened to the middle of a bar four feet long. A boy and a man take hold of the bar to carry it. The boy takes hold at one end of the bar. Find where the man must take hold so that he will carry two-thirds of the load. 6. A horse and a colt are hitched side by side in the usual manner to a loaded wagon. A force of 300 Ib. will just move the wagon. At what point of the double-tree must it be attached to the tongue of the wagon so that the colt will pull two pounds to the horse's three, the double-tree being 40 in. long. 7. A stiff bar firmly fastened at one end sticks out horizontally over a cliff for 10 ft., and will just support without breaking a weight of 100 Ib. at the outer end. How far out on the bar may a weight of 150 Ib. be placed with safety? 8. Resolve a force of 50 dynes into two parallel forces, with their points of application 20 and 30 cm. respectively from the given force. 9. Two forces, 30 and 40 grams, act on a body at an angle of 60. Find the resultant. 10. A ball is given an eastward direction by the action of a force of 20 dynes. At the same time a force of 30 dynes acts on the ball 116 MECHANICS OF SOLIDS to give it a northward direction. In what direction does it go, and what single force will produce the same effect ? 11. In towing a boat along a stream two ropes were used, the angle between them when taut being 45. A force of 100 Ib. was acting on one rope and 150 Ib. on the other. What resistance did the boat offer to being moved ? 12. A sailboat is going eastward, the wind is from the northwest, and the sail is set at an angle of 30 with the. direction of the wind. If the wind's velocity is 12 miles an hour, what is the component velocity at right angles to the sail ? III. NEWTON'S LAWS OF MOTION 126. Momentum. So far we have considered different kinds of motion, or how bodies move, without reference to the mass moved, and without considering the relation between force on the one hand and the moving mass and its velocity on the other, or why bodies move. Before taking up the laws of motion, which outline the relations between force and motion, it is necessary to define two terms intimately associated with these laws. The first of these is momentum. Momentum is the product of the mass and the linear velocity of a moving body. Momentum mass x velocity, or M = mv. (Equation 8) In the c.g.s. system, the unit of momentum is the mo- mentum of a mass of 1 g. moving with a velocity of 1 cm. per second. It has no recognized name. In the English system, the unit of momentum is the momentum of a mass of 1 Ib. moving with a velocity of 1 ft. per second. 127. Impulse. Suppose a ball of 10 g. mass to be fired from a rifle with a velocity of 50,000 cm. per second. Its momentum would be 500,000 units. If a truck weighing 50 kg. moves at the rate of 10 cm. per secpnd, its momen- tum is also 500,000 units. But the ball has acquired its FIRST LAW OF MOTION 117 momentum in a fraction of a second, while a minute or more may have been spent in giving to the truck the same momentum. In some sense the effort required to set the ball in motion is the same as that required to give the equivalent amount of motion to the truck, be- cause the momenta of the two are equal. This equality is expressed by saying that the impulse is the same in the two cases. Since the effect is doubled if the value of the force is doubled, or if the time during which the force continues to act is doubled, it follows that impulse is the product of the force and the time it continues to act. In estimating the effect of a force, the time ele- ment and the magnitude of the force are equally impor- tant. The term impulse takes both into account. 128. Newton's Laws of Motion. The laws of motion, formulated by Sir Isaac Newton (1642-1727), are to be regarded as physical axioms, incapable of rigorous experi- mental proof. They must be considered as resting on convictions drawn from observation and experiment in the domain of physics and astronomy. The results de- rived from their application have so far been found to be invariably true. They form the basis of many of the important principles of mechanics. 129. First Law of Motion. Every body continues in its state of rest or of uniform motion in a straight line, unless compelled by applied force to change that state. This is known as the law of inertia ( 9), because it asserts that a body persists in a condition of rest or of uniform motion, unless it is compelled to change that state by the action of an external force. It is further true that a body offers resistance to any such change in 118 MECHANICS OF SOLIDS proportion to its mass. Hence the term mass is now often used to denote the measure of a body's inertia ( 11). From this law is also derived the Newtonian definition of force, for the law asserts that force is the sole cause of change of motion. 130. Second Law of Motion. Change of momentum is proportional to the impressed force which produces it, and takes place in the direction in which the force acts. The second law points out two things : First. What the measure is of a. force which produces change of motion. Maxwell restated the second law in modern terms as follows : " The change of momentum of a body is numerically equal to the impulse which produces it, and is in the same direction " ; or in other words, momentum (mass x velocity) = impulse (force x time). Expressed in symbols, mv=ft. . . . (Equation 9) TT /. mv Hence, / = . t The initial velocity of the mass m before the force / acted on it is here assumed to be zero, and v is the veloc- ity attained in t seconds. Then the total momentum im- parted in the time t is mv, and therefore - - is the rate of t change of momentum. Force is therefore measured by the rate of change of momentum. Since - is the rate of change t of velocity, or the acceleration a (see Equation 5), we may write /= ma. . . . (Equation 10) THIRD LAW OF MOTION 119 We see from this that force may also be measured by the product of the mass moved and the acceleration imparted to it. Therefore when the mass m is unity, the force is numerically equal to the acceleration it produces. Hence the definition of the* dyne ( 114). Second. This law also points out that the change of momentum is always in the direction in which the force acts. Hence, when two or more forces act together, each pro- duces its change of momentum independently of the others and in its own direction. This principle lies at the founda- tion of the method of finding the resultant effect of two forces acting on a body in different directions ( 118). On a horizontal shelf about two meters above the floor are placed two marbles, one on each side of a straight spring fixed vertically over a hole in the shelf. One marble rests on the shelf and the other is held over the hole between the spring and a block fixed to the shelf (Fig. 115). When the hammer falls and strikes the spring, it pro- jects the one marble horizontally and lets the other one fall vertically. The two reach the floor at the same instant. Both marbles have the same vertical acceleration. FIGURE 1 15. ILLUSTRATING 131 . Third Law of Motion. To every action SECOND LAW OF there is always an equal and contrary re- action ,* or the mutual actions of two bodies are always equal and oppositely directed. The essence of this law is that all action between two bodies is mutual. Such action is known as a stress and a stress is always a two-sided phenomenon, including both action and reaction. The third law teaches that these two aspects of a stress are always equal and in opposite direc- tions. The stress in a stretched elastic cord pulls the two 120 MECHANICS OF SOLIDS bodies to which it is attached equally in opposite directions; the stress in a compressed rubber buffer or spring exerts an equal push both ways ; the former is called a tension and the latter a pressure. . '<\ *-" ILLUSTRATIONS. The tension in a rope supporting a weight is a stress tending to part it by pulling adjacent portions in opposite directions. The same is obviously true if two men pull at the ends of AMERICAN AIRPLANE SQUADRON IN FORMATION. Note the perfect alinement. the rope. An ocean steamship is pushed along by the reaction of the water against the blades of the propeller. The same is true of an aeroplane, only in this case the reaction against the blades is by the air, and the blades are longer and revolve much faster than in water in order to move enough air to furnish the necessary reaction. When a man jumps from a rowboat to the shore, he thrusts the boat back- wards. An athlete would not make a record standing jump from a feather bed or a spring board. When a ball is shot from a gun, the gun recoils or " kicks." All attraction, such as that between a mag- net and a piece of iron, is a stress, the magnet attracting the iron and the iron the magnet with the same force. PROBLEMS 121 Practical use is made of reaction to turn the oscillating electric fan from side to side so as to blow the air in different directions. A rec- tangular sheet of brass is bent lengthwise at right angles and is pivoted so as to turn 90 about a vertical axis (Fig. 116). When one half of this bent sheet is ex- posed to the air current, the reaction sustained by the blades of the fan on this side is in part balanced by the reaction of the bent sheet; but on the opposite half of the fan the reaction of the blades is not balanced. Hence the whole fan turns about a vertical axis on the- standard until a lever touches a stop and shifts the bent strip so as to expose the other half of it to the air current from the opposite half of the fan. The fan then reverses its slow motion and turns to the other side. FIGURE 116. OSCIL- LATING FAN. Since force is measured by the rate at which momentum changes, the third law of motion is equivalent to the fol- lowing: In every action between two bodies, the momentum gained by the one is equal to that lost by the other, or the momenta in opposite directions are the same. Problems 1. What relative velocities will equal impulses impart to the masses 5 Ib. and 8 Ib. respectively ? 2. A body of 50 g. is moving with a velocity of 20 cm. per second. What is its momentum ? 3. Find the ratio of the momentum of a body whose mass is 10 Ib., moving with a uniform velocity of 50 ft. per second to that of a body whose mass is 25 Ib. and whose velocity is 20 ft. per second. 4. Two bodies have equal momenta. One has a mass of 2 Ib. and a velocity of 1500 ft. per second, the other a mass of 100 Ib. What is the velocity of the second body ? 5. What is the velocity of recoil of a gun whose mass is 5 kg., the mass of the ball being 25 g. and its velocity 600 m. per second? 122 MECHANICS OF SOLIDS 6. An unbalanced force of 500 dynes acts for 5 sec. on a mass of 50 g. "What will be the velocity produced ? 7. A force of 980 dynes acts on a mass of 1 g. What is the acceleration ? How far will the body go in 10 sec. ? 8. A force of 400 dynes acts on a body for 10 sec. What will be the momentum at the end of this period? 9. A body is acted on by a force of 100 dynes for 20 sec. and acquires a velocity of 200 cm. per second. What is its mass? 10. A force of 10 g. acts for 5 sec. on a body whose mass is 15 g. What velocity is imparted ? 11. What force in grams of force can impart to a mass of 50 g. an acceleration of 980 cm.-per-second per second? 12. A force of 50 g. acts for 5 sec. on a mass of 50 g. How far will the body have gone in that time, starting from rest ? IV. GRAVITATION 132. Weight. The attraction of the earth for all bodies is called gravity. The weight of a body is the measure of this attraction. It is a pull on the body and therefore a force. It makes a body fall with uniform acceleration called the acceleration of gravity and denoted by g. If we represent the weight of a body by w and its mass by w, by Equation 10, w = mg. From this it appears that the weight of a body is proportional to its mass, and that the ratio of the weights of two bodies at anyplace is the same as that of their masses. Hence, in the process of weighing with a beam balance, the mass of the body weighed is compared with that of a standard mass. When a beam balance shows equality of weights, it shows also equality of masses. 133. Direction of Gravity. The direction in which the force of gravity acts at any point is very nearly toward the earth's center. It may be determined by suspending a weight by a cord passing through the point. The cord LAW OF UNIVERSAL GRAVITATION 123 is called a plumb line (Fig. 117), and its direction is a ver- tical line. A plane or line perpendicular to a plumb line is said to be horizontal. Vertical lines drawn through neighboring points may be considered parallel without sensible error. 134. Center of Gravity. In Physics a body is thought of as composed of an indefinitely large number of parts, each of which is acted on by gravity. For bodies of ordinary size, these forces of gravity are parallel and proportional to the masses of the several small parts. The point of application of their resultant is the center of gravity of the body. If the body is uniform throughout, the position of its center of gravity depends on its geometri- cal figure only. Thus, the center of gravity (1) of a straight rod is its middle point ; (2) of a circle or ring, its center ; (3) of a sphere or a spherical shell, its center ; (4) of a parallelo- gram, the intersection uf its diagonals ; (5) of a cylinder or a cylindrical pipe, the middle point of its axis. i i 7 e It is necessary to guard against the idea that P L u M F the force of gravity on a body acts at its center of INE ' gravity. Gravity acts on all the particles composing the body, but its effect is generally the same as if the resultant, that is, the weight of the body, acted at its center of gravity. It will be seen from the examples of the ring and the cylin- drical pipe that the center of gravity may lie entirely out- side the body. 135. Law of Universal Gravitation. It had occurred to Galileo and the other early philosophers that the attrac- tion of gravity extends beyond the earth's surface, but it 124 MECHANICS OF SOLIDS remained for Sir Isaac Newton to discover the law of uni- versal gravitation. He derived this great generalization from a study of the planetary motions discovered by Kep* ler. .The law may be expressed as follows : Every portion of matter in the universe attracts every other portion, and the stress between them is directly pro- portional to the product of their masses and inversely proportional to the square of the distance between their centers of mass. For spherical bodies, like the sun, the earth, and the planets, the attraction of gravitation is the same as if all the matter in them were concentrated at their centers; hence, in applying to them the law of gravitation, the distance between them is the distance between their cen- ters. Calculations made to find the centripetal accelera- tion of the moon in its orbit show that it is attracted to the earth with a force which follows the law of universal gravitation. The law of universal gravitation does not refer in any way to weight but to mass. It would be entirely meaningless to speak of the weight of the earth, or of the moon, or of the sun, but their masses are very definite quantities, the ratios of which are well known in as- tronomy. Thus the mass of the earth is about 80 times that of the moon, and the mass of the sun is about 332,000 times that of the earth. The weight of a pound mass at the distance of the moon is only jgVrr the weight of a pound mass at the surface of the earth. 136. Variation of Weight. Since the earth is not a sphere but is flattened at the poles, it follows from the law of gravitation that the acceleration of gravity, and the weight of any body, increase in going from the equa- tor toward either pole. If the earth were a uniform sphere and stationary, the value of g would be the same Sir Isaac Newton (1642-1727) is celebrated for his discoveries in mathematics and physics. He was a Fellow of Trinity Col- lege, Cambridge. He discovered the binomial theorem in alge- bra and laid the foundation of the calculus. His greatest work is the Principia, a treatise on motion and the laws governing it. His greatest discoveries are the laws of gravitation and the composi- tion of white light. From Kepler's laws of the planetary orbits Newton proved that the attraction of the sun on the planets varies inversely as the squares of their distances. He was also distinguished in public life. He sat in Parliament for the University of Cambridge, was at one time Master of the Mint, and the reformation of the English coinage was largely his work. EQUILIBRIUM UNDER GRAVITY 125 all over its surface. But the value of g varies from point to point on the earth's surface, even at sea level, both because the earth is not a sphere and because it rotates on its axis. The centripetal acceleration of a point at the equator, owing to the earth's rotation on its axis, is ^Q the acceleration of gravity g. Since 289 is the square of 17, and the centripetal acceleration varies as the square of the velocity ( 110), it follows that if the earth were to rotate in one seventeenth of a day, that is, ' 17 times as fast as it now rotates, the apparent value of g at the equator would become zero, and bodies there would lose all their weight. The value of g at the equator is 978.1 and at the poles 983.1, both in centimeters-per-second per second. At New York it is 980.15 centimeters-per-second per second, or 32.16 feet-per-second per second. 137. Equilibrium under Gravity. When a body rests on a horizontal plane, its weight is equal and opposite to the reaction of the plane. The vertical line through its cen- ter of gravity must therefore fall within its base of sup- port. If this vertical line falls outside the base, the weight of the body and the reaction of the plane form a couple ( 121), and the body overturns. The three kinds of equilibrium are >(!) stable, for any displacement which causes the center of gravity to rise ; (2) unstable, for any displacement which causes the cen- ter of gravity to fall ; (3) neutral, for any displacement which does not change the height of the center of gravity. Fill a round-bottomed Florence flask one quarter full of shot and cover them with melted paraffin to keep them in place (Fig. 118). Tip the flask over ; after a few oscillations it will return to an up- right position. Repeat the experiment with a similar empty flask; it will not stand up, but will rest in any position on its side and with 126 MECHANICS OF SOLIDS the top on the table. The loaded flask cannot be tilted over without raising its center of gravity; in a vertical position it is therefore stable and when tipped over, unstable, for it returns to a vertical position. For the empty flask, its center of grav- ity is lower when it lies on its side than when it is erect. Rolling it around does not change the height of its center FIGURE 1 18. - STABILITY OF FLASKS. of S ravit y and its equilibrium is thus neutral. The three funnels of Fig. 119 illustrate the three kinds of equi- librium on a plane. A rocking horse, a rocking chair, and a half sphere resting on its convex side are examples of stable equilibrium. An egg lying on its side is in neutral equilibrium for rolling and stable equilibrium for rocking ; it is unstable on either end. A lead pencil supported on its point is in unstable equilib- Any such body may become stable by attaching weights to it in such a manner as to lower the center of gravity below the supporting point (Fig. 120). 138. Stability. Stability is the state of being firm or stable. The higher the center of gravity of a body must be lifted to put the body in unstable equilibrium or to overturn it, the greater is its stability. This condition is met by a relatively large base and a low center of gravity. A pyramid is a very stable form. On account of the large area lying within the four feet of a quadruped, its stability is greater than that of a biped. A child is therefore able FIGURE 119. STABILITY OF FUNNELS. num. FIGURE 120. CENTER OF GRAVITY BELOW SUPPORT. QUESTIONS AND PROBLEMS 127 to creep " on all fours " before it learns to maintain stable equilibrium in walking. A boy on stilts has smaller sta- bility than on his feet because his support is smaller and his center of gravity higher. Stability may be well illustrated by means of a brick. It has greater stability when lying on its -narrow side (2" x 8") than when standing on end; and on its broad side (4" x 8") its sta- ^-^ bility is still greater. Let Fig. /' \ A \ \^-\d 121 represent a brick lying on its narrow side in A and stand- ing on end in B. In both case", to overturn it its center of gravity c is lifted to the same FIGURE 121. DEGREES OF STABILITY. height, but the vertical dis- tance bd through which the center of gravity must be lifted is greater in A than in B. A tall chimney or tower has no great stability because its base is relatively small and its center of gravity high. A high brick wall is able to support a great crushing weight, but its stability is small unless it is held by lateral walls and floor beams. Questions and Problems 1. If one jumps off the top of an empty barrel standing on end, why is one likely to get a fall? 2. Where is the center of gravity of a knife supported as in Fig. 122 ? 3. Given a triangle cut from a uni- form sheet of cardboard or thin wood. Describe two methods of finding its center of gravity. How can you tell when the right center has been found ? 4. Represent a hill by the hypotenuse of a right triangle, and a ball on the hill by a circle, the circumference of the circle just touching the hypote- nuse of the triangle. How would you represent the weight of the ball ? By resolving this force into two components, find the force that rolls FIGURE 122. 128 MECHANICS OF SOLIDS the ball down the hill and the force with which the ball presses against it. 5. Which is less likely to " turn turtle " in rounding a sharp curve, an underslung or an overslung automobile ? Why? 6. A body weighing 150 Ib. on a spring balance on the earth would weigh how much on the moon, the radius of the moon being that of the earth and its mass ? FIGURE 123. CATHEDRAL OF PISA AND LEANING TOWER. 7. If the acceleration of gravity is 32.2 ft.-per-second per second on the earth, what must it be on the sun, the radius of the sun being taken as 110 times that of the earth and its mass as 330,000 times ? 8. With what force will a man weighing 160 Ib. press on the floor of an elevator when it starts with an acceleration of 4 ft.-per-second per second, first going up, and then going down? V. FALLING BODIES 139, Bate at which Different Bodies Fall. It is a familiar fact that heavy bodies, such as a stone or a piece of iron, RESISTANCE OF THE AIR 129 fall much faster than such light bodies as feathers, bits of paper, and snow crystals. Before the time of Galileo it was supposed that different bodies fall with velocities pro- portional to their weights. This erroneous notion was corrected by Galileo by means of his famous experiment of dropping various bodies from the top of the leaning tower of Pisa (Fig. 123) in the presence of professors and students of the university in that city. He showed that bodies of different materials fell from the top of the tower to the ground, a height of 180 feet, in practically the same time; also that light bodies, such as paper, fell with ve- locities approaching more and more nearly those of heavy bodies the more compactly they were rolled together in a ball. The slight differences in the velocities observed he rightly ascribed to the resistance of the air, which is relatively greater for light bodies than for heavy compact ones. This inference Galileo could not completely verify because the air pump had not yet been invented. 140. Resistance of the Air. Place a small coin and a feather, or a shot and a bit of tissue paper, in a glass tube from 4 to 6 feet long. It is closed at one end and fitted with a stopcock at the other (Fig. 124). Hold the tube in a vertical position and suddenly in- vert it ; the coin or the shot will fall to the bottom first. Now exhaust the air as perfectly as possible ; again invert the tube quickly ; the lighter body will now fall as fast as the heavier one. This experiment is known as the " Guinea and Feather Tube." It demon- strates that if the resistance of the air were wholly re- moved, all bodies at the same place would fall with eration. An interesting modification of the experiment is Cut a round piece of paper slightly smaller than a the cent and the paper side by side; the cent will FIGURE 124. GUINEA AND FEATHER TUBE. the same accel- the following: cent and drop reach the floor 130 MECHANICS OF SOLIDS first. Then lay the paper on the cent and drop them in that position ; the paper will now fall as fast as the cent. Explain. The friction of the air against the surface of bodies moving through it limits their velocity. A cloud floats, not because it is lighter than the atmosphere, for it is actually heavier, but because the surface fric- tion is so large in comparison with the weight of the minute drops of water, that the limiting velocity of fall is very small. When a stream of water flows over a high precipice, it is broken into fine spray and falls slowly. At the Yosemite Fall (Fig. 125) a large stream is broken by the resistance of the air until at the bottom of its 1400 foot drop it becomes fine spray. 141. Laws of Falling Bodies. Galileo verified the fal- lowing laws of falling bodies: I. The velocity attained by a falling body is propor- tional to the time of falling. II. The distance fallen is proportional to the square of the time of descent. III. The acceleration is twice the distance a body falls in the first second. These laws will be recognized as identical with those derived for uniformly accelerated motion, 106 and 107. If the inclined plane in Galileo's experiment be tilted up steeper, the effect will be to increase the acceleration down the plane ; and if the board be raised to a vertical position, the ball will fall freely under gravity and the acceleration will become # ( 136). Since the acceleration g is sensibly constant for small distances above the earth's surface, the equations already obtained for uniformly accelerated motion may be applied directly to falling bodies, by substituting g for a in Equa- tions 5 and 6. Thus we have v = gt^ . . . (Equation 11) and * = \ yt 2 . . . (Equation 12) LAWS OF FALLING BODIES 131 FIGURE 125. YOSEMITE FALL. 132 MECHANICS OF SOLIDS If in Equation 12 t is one second, s= \g\ or the dis- tance a body falls from rest in the first second is half the acceleration of gravity. A body falls 490 cm. or 16.08 ft. the first second ; and the velocity attained is 980 cm. or 32.16 ft. per second. 142. Projection Upward. When a body is thrown verti- cally upward, the acceleration is negative, and it loses each second g units of velocity (980 cm. or 32.16 ft.). Hence, the time of ascent to the highest point is the time taken to bring the body to rest. If the velocity lost is g units a second, the time required to lose v units of velocity will be the quotient of v by #, or , . /i velocity of projection upward time of ascent y . r acceleration oj~ gravity In symbols t = - . . . . (Equation 13) 9 For example, if the velocity of projection upward were 1470 cm. per second, the time of ascent, neglecting the frictional resistance of the air, would be -Vg 7 ^, or 1.5 sec- onds. This is the same as the time of descent again to the starting point ; hence, the body will return to the start- ing point with a velocity equal to the velocity of projection but in the opposite direction. In this discussion of projection upward, the resistance of the air is neglected. Problems Unless otherwise stated in the problem, g is to be taken as 980 cm.- or 32 ft.-per-second per second. 1. The tower of Pisa is 180 ft. high. In what time would a ball dropped from the top reach the ground? With what velocity would it strike? CENTRIPETAL AND CENTRIFUGAL FORCE 133 2. From what height must a ball fall to acquire a velocity of 1 km. per second ? 3. With what velocity in a vertical direction must a shell be fired just to reach an aeroplane flying at an elevation of one mile? 4. A ball is fired vertically with an initial velocity of 500 m. per second. Neglecting the resistance of the air, to what height will it rise, and in what time will it return to the earth ? 5. An aeroplane flying westward with a velocity of 60 mi. per hour and at an elevation of one mile, dropped a bomb while vertically over a cathedral. How far from the cathedral did the bomb strike the ground and in which direction ? 6. A ball fired horizontally reaches the ground in 4 sec. What was the height of the point from which it was fired? 7. A cannon ball is fired horizontally from a fort at an elevation of 122.5 m. above the neighboring sea. How many seconds before it will strike the water ? 8. The Washington monument is 555 ft. high. Two balls are dropped from its top one second apart. How far apart will the balls be when the first one strikes the ground ? 9. An iron ball was dropped from an aeroplane moving eastward at the rate of 45 mi. per hour. It reached the ground 528 ft. east of the vertical line through the point from which it was dropped. What was the elevation of the aeroplane ? 10. A body slides without friction down an inclined plane 300 cm. long and 24.5 cm. high. If it moves 40 cm. during the first second, what is the computed value of g ? VI. CENTRIPETAL AND CENTRIFUGAL FORCE 143. Definition of Centripetal and Centrifugal Force. Attach a ball to a cord and whirl it around by the hand. The ball pulls on the cord, the pull increasing with the velocity of the ball. If the ball is replaced by a heavier one, with the same velocity the pull is greater. If 'a longer cord is used, the pull is less for the same velocity in the .circle. 134 MECHANICS OF SOLIDS The constant putt which deflects the body from a rectilinear path and compels it to move in a curvilinear one is the cen- tripetal force. The resistance which a body offers on account of its inertia, to deflection from a straight line is the centrifugal force. When the motion is uniform and circular, the force is at right angles to the path of the body around the circle and constant. These two forces are the two aspects of the stress in the cord (third law of motion), the action of the hand on the ball, and the reaction of the ball on the hand. 144. Value of Either Force The centripetal acceleration Q for uniform circular motion ( 110) is a = , where v is T the uniform velocity in the circle, and r is the radius. Further, in 130 the relation between force and accelera- tion was found to be as follows: force equals the product of the mass and the acceleration imparted to it by the force. Hence we have centripetal force = mass x centripetal acceleration, or /= .' . . . (Equation 14) This relation gives the value of either the centripetal or the equal centrifugal force in the absolute system of measurement, because it is derived from the laws of motion and is independent of gravity. In the metric system m must be in grams, v in centimeters per second, and r in centimeters ; f is then in dynes. To obtain f in grams of force, divide by 980 ( 115). In the English system, m must be in pounds, v in feet per second, and r in feet ; dividing by 32.2, the result will be in pounds of force. CENTRIFUGAL FORCE. Above : Auto Race on a Circular Raised Track. Below : Sled in Swiss Winter Sports being thrown over the embank- ment by centrifugal force. ILLUSTRATIONS OF CENTRIFUGAL FORCE 135 For example : If a mass of 200 g. is attached to a cord 1 m. long and is made to revolve with a velocity of 140 cm. per second, the ten- sion in the cord is 20 X 14 2 = 39,200 dynes = = 40 grams of f 100 980 force. Again if a body having a mass of 10 Ib. 1 oz. move in a circle of 5 ft. radius with a velocity of 20 ft. per second, then the centripetal force is/= 10 rV * 2 <> 2 = 25 pounds of force. 5 x 32.2 145. Illustrations of Centrifugal Force. Water adhering to the surface of a grindstone leaves the stone as soon as the centrifugal force, increasing with the velocity, is greater than the adhesion of the water to the stone. Grindstones and flywheels occasionally burst when run at too high a speed, the latter when the engine runs away after a heavy load is suddenly thrown off. When the centripetal force ceases to deflect the body from the tangent to the circle, the body flies off along the tangent line. A stone is thrown by whirling it in a sling and releasing one of the strings. A carriage or an automobile rounding a curve at high speed is sub- ject to strong centrifugal forces, which act through the tires. The centripetal force consists solely of the friction between the tires and the ground. If the friction is insuffi- cient, " skidding " takes place. When a spherical vessel containing some mercury and water is rapidly whirled on its axis (Fig. 126), both the mercury and the water rise and form separate bands as far as possible from the axis of rotation, the mercury out- FIGURE 126. WHIRLING LIQUIDS. Centrifugal machines are used in sugar refineries to separate sugar crystals from the sirup, and in dye- works and laundries to dry yarn and wet clothes by whirling them rapidly in a large cylinder with openings in the side. Honey is ex- tracted from the comb in a similar way. When light and heavy par- ticles are whirled together, the heavier ones tend toward the outside. New milk is an emulsion of fat and a liquid, and the fat globules are lighter than the liquid of the emulsion. Hence, when fresh milt is whirled in a dairy separator, the cream and the milk form distinct layers and collect in separate chambers. 136 MECHANICS OF SOLIDS VII. THE PENDULUM 146. Simple Pendulum. Any body suspended so as to swing about a horizontal axis is a pendulum. A simple pendulum is an ideal one. It may be denned as a material particle without size suspended by a cord without weight. A small lead ball suspended by a long thread without sen- sible mass represents very nearly a simple pendulum. When at rest the thread hangs vertically like a plumb line; but if the ball be drawn aside and released, it will oscillate about its position of rest. Its oscillations become gradually smaller ; but if the arc described be small, the period of its swing will remain unchanged. This feature of pendular motion first attracted the attention of Galileo while watching the slow oscillations of a " lamp " or bronze chandelier, suspended by a long rope from the roof of the cathedral in Pisa. Galileo noticed the even time of the oscillations as the path of the swinging chandelier became shorter and . shorter. Such a motion, which repeats itself over and over in equal time intervals, is said to be periodic. 147. The Motion of a Pendulum. A N in Fig. 127 is a nearly simple pendulum with the ball at N. When the ball is drawn aside to the position B, its weight, represented by BG, may be resolved into two components, BD in the direc- tion of the thread, and BC at right angles to it and tangent to the arc BNE. The latter is the force which produces motion of the ball toward N. As the ball moves from B toward N the component BC becomes smaller and smaller and vanishes at N, where the whole weight of the batt is in the direction of the thread. In falling from B to N, the ball moves in the arc of a circle under the influence of a force which is greatest at B and becomes zero at TV. The motion is therefore FIGURE 127. FORCES ACTING ON A PENDULUM. THE MOTION OF A PENDULUM 137 INTERIOR OF PISA CATHEDRAL. The bronze chandelier which Galileo observed hangs just in front of the altar. 138 MECHANICS OF SOLIDS accelerated all the way from B to N, but not uniformly. The velocity increases continuously from B to N, but at a decreasing rate. The ball passes N with its greatest velocity and continues on toward E. From N to E the component of the weight along the tangent which is always directed toward N, opposes the motion and brings the pendulum to rest at E. It then retraces its path and continues to oscillate with a periodic and pendular motion. 148. Definition of Terms. The center of suspension is the point or axis about which the pendulum swings. A single vibration is the motion comprised between two successive passages of the pendulum through the lowest point of its path, as the motion from Nio B (Fig. 128) and back to N again. A complete or double vibration is the motion between two successive pas- sages of the pendulum through the same point and going in the same direction. A complete vibration is double that of a single one. The period of vibration is the FIGURE 128. time consumed in making a complete or SIMPLE PENDU- double vibration. The amplitude is the LUM arc BN OT the angle BAN. 149. Laws of the Pendulum. The following are the laws of a simple pendulum which are independent of both the material and the weight: I. For small amplitudes, the period of vibration is independent of the amplitude. ' II. The period of vibration is proportional to the square root of the length of the pendulum. III. The period of vibration is inversely proportional to the square root of the acceleration of gravity. One of the earliest and most important discoveries by Galileo was that of the experimental laws of the motion of CENTER OF OSCILLATION 139 a pendulum, made when he was about twenty years of age. This was long before their theoretical investigation. If the period of a single vibration of a simple pendulum is denoted by f, the length by Z, and the acceleration of gravity by g, it can be shown that t = TTV/-. . . . (Equation 15) To illustrate Law I. It is only necessary to count the vibrations of a pendulum which take place in some convenient time with different amplitudes. Their number will be found to be the same. This result will hold even when the ampli- tudes are so small that the vibrations can only be ob- served with a telescope. To illustrate Law II. Mount three pendulums (Fig. 129), making the lengths 1 m., \ m., and ^ m. re- spectively. Observe the period of a single vibra- tion for each. They will be 1 sec., sec., and % sec. nearly, or in periods proportional to the square root of the lengths. In accordance with Law III a pendulum oscillates more slowly on the top of a high mountain than at sea level, and more slowly at the equator than at the poles. Place a strong magnet just under the bob of the longest pendulum, which must be iron. It will then be found to vibrate in a slightly shorter period than before. The downward magnetic pull on the bob is equivalent to an increased value of g. 150. Center of Oscillation. Insert a small staple in one end of a meter stick, and suspend it so as to swing as a pendulum about a horizontal axis through the staple (Fig. 130). With a ball and a thread make a simple pendulum that will vi- brate in the same period as the meter stick. Beginning at the staple, lay oft' on the meter stick the length of this FIGURE 129. PENDULUMS OF DIFFERENT LENGTHS. FIGURE 1 30. - C ENTER OF OS- CILLATION. 140 MECHANICS OF SOLIDS pendulum. It will extend two thirds of a meter down. Bore a hole through the meter stick at the point thus found, and suspend it as a pendulum by means of a pin through this hole. Its period of vibration will be the same as before. The bar is a compound pendulum, and the new axis of vibration is called the center of oscillation. The distance between the center of suspension and the center of oscilla- tion is the length of the equivalent simple pendulum that vibrates in the same period as the compound pendulum. The centers of suspension and of oscillation are inter- changeable without change of period. 151. Center of Percussion. Suspend the meter bar by the staple at the end and strike it with a soft mallet at the center of oscillation. It will be set swinging smoothly and without perceptible jar. Hold a thin strip of wood a meter long and four or five centimeters wide by the thumb and forefinger near one end. Strike the flat side with a soft mallet at different points. A point may be found where the blow will not throw the wood strip into shivers, but will only set it swinging like a pendulum. The center of oscillation is also called the center of per- cussion; if the suspended body be struck at this point at right angles to the axis of suspension, it will be set swing- ing without jar. A baseball club or a cricket bat has a center of percussion, and it should strike the ball at this point to avoid breaking the bat and " stinging " the hands. 152. Application of the Pendulum. Galileo's discovery suggested the use of the pendulum as a timekeeper. In the common clock the oscillations of the pendulum regulate the motion of the hands. The wheels are kept in motion by a weight or a spring, and the regulation is effected by means of the escapement (Fig. 131). The pendulum rod, passing between the prongs of a fork a, communicates its motion to an axis carrying the escapement, which ter- QUESTIONS AND PROBLEMS 141 minates in two pallets n and m. These pallets engage alternately with the teeth of the escapement wheel 72, one tooth of the wheel escaping from a pallet every double vibration of the pendulum. The escapement wheel is a part of the train of the clock ; and as the pendulum controls the escapement, it also controls the motion of the hands. 153. Seconds Pendulum. A seconds pen- dulum is one making a single vibration in a second. Its length in New York is 99.31 cm. This is the length of the equivalent simple pendulum vibrating sec- onds. The value of gravity g increases from the equator to the poles, and the length of the seconds pendulum increases in the same proportion, s Questions and Problems 1. Why can a heavy shot be thrown much farther by swinging it from the end of a short wire or cord than by hurling it from the shoulder as in " putting the shot " ? 2. Why is the outer rail on a railway curve elevated above the inner one ? 3. A ball weighing 10 Ib. is attached to a cord 2 ft. long and is whirled about the hand at the rate of ten revolutions in three seconds. What is the tension in the cord ? 4. A ball swings as a conical pendulum; its mass is 2 kg., its distance from the center of its circular path is 30 cm., and it makes ten revolutions in 35 seconds. What horizontal force in grams would be necessary to hold the ball at any point in its path if it were not revolving ? 5. Find the period of vibration of a pendulum 70 cm. long, the value of g being 980 cm.-per-second per second. FIGURE 131. Es- CAPEMENT. 142 MECHANICS OF SOLIDS 6. Calculate the length of a seconds pendulum at a place where the value of g is 980 cm.-per-second per second. 7. At a place where g is 32 ft.-per-second per second what is the length of a pendulum that vibrates in sec. ? 8. What would be the acceleration of gravity if a pendulum one meter long had a period of vibration of one second ? 9. If a simple pendulum 90 cm. long makes 64 single vibrations per minute, what is the value of (Equation 16) Since force is equal to the product of mass and accelera- tion ( 130), w = ma x 8. . . . (Equation 17) 156. Units of Work. Before use can be made of these expressions for work, it is necessary to define the units employed in measuring work.' Three or four such units are in common use : 1. The foot pound (ft. lb.), or the work done by a pound of force working through a space of one foot. If a pound weight is lifted a foot high, or if a body is moved a distance of one foot by a force of one pound, a foot pound of work is done. This unit is in common use among English-speaking engineers. It is open to the objection that it is variable, since a pound of force varies with the latitude and with the elevation above sea level. 2. The kilogram meter (kg. m.), or the work done by a kilogram of force working through a space of one meter. It is the gravitational unit of work in the metric system, and varies in the same manner as the foot pound. The gram-centimeter is also used as a smaller gravitational POWER 145 unit of work. The kilogram meter is equal to 100,000 gram-centimeters. 3. The erg, 1 or the work done by a dyne working through a distance of one centimeter. The erg is the absolute unit in the c. g. 8. system and is invariable. Since a gram of force is equal to 980 dynes ( 115), if a gram mass be lifted vertically one centimeter, the work done against gravity is 980 ergs. Hence one kilogram meter is equal to 980 x 1000 x 100 = 98,000,000 ergs. The mass of a "nickel" is 5 g. The work done in lifting it through a vertical distance of 5 m. is the continued product of 5, 500, and 980, or 2,450,000 ergs. The erg is therefore a very small unit and not suitable for measuring large quantities of work. For such purposes it is more convenient to use a multiple of the erg, called the joule. 2 Its value is 1 joule = 10 7 ergs = 10,000,000 ergs. Expressed in this larger unit, the work done in lifting the " nickel " is 0.245 joule. 8 157. Power. While it takes time to do work, it is plain that time is not an element in the amount of work done. To illustrate : Suppose a ton of marble is lifted by a steam engine out of a marble quarry 300 ft. deep. The work is done by means of a wire rope, which the engine winds on a drum. If now the drum be replaced by another of twice the diameter, and running at the same rate of rotation, the ton of marble will be lifted in half the time ; but the total work done against gravity remains the same, namely, 600,000 ft. Ib. In an important sense the engine as an agent for doing work is twice as effective in the second instance as in the 1 The erg is from the Greek word meaning work. 2 From the noted English investigator Joule. 8 The joule is equal to about f of a foot pound. MECHANICAL WORK first. Time is an essential element in comparing the capacities of agents to do work. Such a comparison is made by measuring the power of an agent. Power tells us not how much work is done, but how fast it is done. A MARBLE QUARRY. Power is the time rate of doing work, Or power = =*J. time t (Equation 18) This expression may be used directly to measure power, due regard being paid to the units employed. The result will be in foot pounds per second, kilogram meters per second, gram-centimeters per second, or ergs per second, according to the consistent units used. The units of power universally used by engineers are GIANT ORE CRANE. When these jaws close, as shown in the picture on page 153, the bucket holds 12 tons of iron ore. either the horse power or the watt and its multiple the kilowatt. The horse power (H.P.) is the rate of working equal to 33,000 ft. Ib. per minute, or to 550 ft. Ib. per second. 148 MECHANICAL WORK Hence in which /is in pounds of force, 8 in feet, and t in seconds. In the c. g. s. system the watt 1 is the rate of working equal to one joule per second. A kilowatt (K.W.) is 1000 watts. Hence watts = xs ^ ; K.W. = * * (Equation 20) In Equation 20 / is in dynes, s in centimeters, and t in seconds. One horse power equals 746 watts, or 0.746 kilowatt (nearly -| K. W.). To convert kilowatts into horse powers approximately, add one third ; to convert horse powers into kilowatts, subtract one fourth. For example, 60 K.W. are equal to 80 H.P., and 100 H.P. are equal to 75 K.W. The power capacity of direct current dynamo electric generators is now universally expressed in kilowatts ; the steam engines and water turbines used to drive these generators are commonly rated in the same unit of power; so, too, the capacity of electric motors is more often given in kilowatts than in horse powers. A kilowatt hour means power at the rate of a kilowatt expended for one hour. Thus, 20 kilowatt hours mean 20 K.W. for one hour, or 5 K.W. for four hours, etc. 158. Energy. Experience teaches that under certain conditions bodies possess the capacity for doing work. Thus, a body of water at a high level, gas under pressure in a tank, steam confined in a steam boiler, and the air moving as a wind, are all able to do work by means of appropriate motors. In general, a body or system on which work has been done acquires increased capacity for doing work. It is then said to possess more energy than before. 1 From the noted English engineer James Watt. POTENTIAL ENERGY 149 " Work may be considered as the transference of energy from one body or system to another." " Energy we know only as that which in all natural phenomena is constantly passing from one portion of matter to another." Since the work done on a body^is the measure of its increase of energy, work and energy are measured in the same units. 159. Potential Energy. A mass of compressed air in an air gun tends to expand ; it possesses energy and may expend it in propelling a bullet. Energy is stored also in the lifted weight of the pile driver (Fig. 132), the coiled spring of the clock, the bent bow of the archer, the impounded waters behind a dam, the chemical changes in a charged storage battery, and the mixed charge of gasoline vapor and air in the cylinder of a gas engine. In all such cases of the storage of energy a stress ( 43) is pres- ent. The compressed air pushes outward in the air gun ; gravity pulls on the lifted weight; the FIGURE 132. PILE DRIVER. spring tends to uncoil in the clock ; the bent bow tries to unbend ; the water presses against the dam ; the electric pressure is ready to produce a current ; and the explosive gas mixture awaits only a spark to set free its energy. The energy thus stored, which is associated with a stress or with a position with respect to some othe body, is energy of stress, or, more commonly, potential energy. The energy of an elevated body, of bending, twisting, of chemical sep- 150 MECHANICAL WORK aration, and of air, steam, or water under pressure, are all examples of potential energy. 160. Kinetic Energy. A body also possesses energy in consequence of its motion ; the energy of a moving body is known as kinetic energy. The. descending hammer forces the nail into the wood, the rushing torrent carries away bridges and overturns buildings ; the swiftly moving can- non ball, by virtue of its high speed, demolishes fortifica- tions or pierces the steel armor of a battleship ; the energy stored in the massive rotating flywheel keeps the engine running and may do work after the steam is shut off. When the engine is speeding up, it pushes and pulls on the shaft to increase the speed of the flywheel ; in other words, the engine does work on the flywheel. After normal speed has been reached, all the work done by the engine goes into the driven machinery; but if an extra load comes on the engine, its speed does not drop sud- denly, because it is sustained by the flywheel giving out some of its stored energy to help along the engine. The engine tends to stop the flywheel, and this now does work instead of absorbing energy. When a meteoric body, or " shooting star," enters the earth's atmosphere, its energy of motion is converted into heat by friction with the air ; the heat generated raises the temperature of the meteor (at least on its surface) until it glows like a star. If it is small, it may even burn up or become fine powder. The energy of the invisible molecular motions of bodies constituting heat is included under kinetic energy no less than that of their visible motion. Heat is a form of kinetic energy. Kinetic energy must not be confused with force. A mass of moving matter carries witli it kinetic energy, but KINETIC ENERGY 151 it exerts no force until it encounters resistance. Energy is then transferred to the opposing body, and force is exerted only during the transfer. 161. Measure of Energy. Energy is measured in the same terms as those used in measuring work. In general, potential energy is the measure of the mechanical work done in storing the energy, or P.E.=fxs. . . (Equation 21) If / is in pounds of force and 8 in feet, the result is in foot pounds. Similarly, if / is in grams of force and in centimeters, the potential energy is expressed in gram- centimeters. Since a gram of force is equal to 980 dynes, expressed 1 Tl f^T^O^ P.E. = 980 x grams x centimeters. 162. Kinetic Energy in Terms of Mass and Velocity. The work fs done by the force / on the mass m to give it the velocity v, while working through the distance , measures the kinetic energy acquired, or, K.E. =f x s. But it is highly desirable to express kinetic energy in terms of the mass m and the acquired velocity t>, instead of / and s. By the second law of motion ( 130) / = ma. Hence K.E. =maxs. But s = \at^. Therefore K.E. = ma?t 2 . Also v = at ( 106); therefore K.E. = \mv*. . . (Equation 22) Both m and v are magnitudes independent of gravitation ; it follows that the results calculated from Equation 22 can- not be in gravitational units. If m is expressed in grams and v in centimeters per second, the kinetic energy is in 152 MECHANICAL WORK ergs. Since the gram -centimeter is equal to 980 ergs, to reduce the result to gram-centimeters, divide by the value of g in this system, or 980. In precisely the same way, if m is in pounds and v in feet per second, to obtain the energy in foot pounds, divide by the value of g in the English system, 32.2. To illustrate : If an automobile, weighing 3000 lb., is running at a speed of 30 miles per hour, find its kinetic energy. A mile a minute is 88 ft. per second, and 30 miles an hour or half a mile a minute is 44 ft. per second. Hence the kinetic energy of the m VingCari8 3000 x 44 ml8fi ,,, 2x32.2 = 9 ' 186 ft lb> This energy represents very nearly the work required to lift the car 30 ft. high against gravity, for this work is 3000 x 30 = 90,000 ft. lb. A large ship, moving toward a wharf with a motion scarcely per- ceptible, will crush with great force small intervening craft. The moving energy of the large vessel is great because of its enormous mass, even though its velocity is small. Its weight is supported by the water and has nothing to do with its crushing force. 163. Transformations of Energy. When a bullet is shot vertically upward, it gradually loses its motion and its kinetic energy, but gains energy of position or potential energy. When it reaches the highest point of its flight, its energy is all potential. It then descends, and gains energy of motion at the expense of energy of position. The one form of energy is, therefore, convertible into the other. The pendulum illustrates the same principle. While the bob is moving from the lowest point of its path toward either extremity, its kinetic energy is converted into potential energy ; the reverse transformation sets in DISSIPATION OF ENERGY 153 when the pendulum reverses its motion. All physical processes involve energy changes, and such changes are in ceaseless progress. 164. Conservation of Energy. Whenever a body gains energy as the result of work done on it, it is always at the expense of energy in some other body or system. The agent, or body, which does work always loses energy ; the body which has work done on it gains energy equal to the work done. On the whole there is neither gain nor loss of energy, but only its transfer from one body to an- other. Innumerable facts and observa- tions show that it is as inlpossible to cre- ate energy as it is to create matter. So the law of conserva- tion of energy means that no energy is created and none destroyed by the action of forces we know anything about. 165. Dissipation of Energy. Potential energy is the more highly available or useful form of energy. It always tends to go over into the kinetic type, but in such a way that only a portion of the kinetic energy is available to effect useful changes in nature or in the mechanic arts. CLOSED JAWS OF ORE BUCKET. This crane makes one trip per minute from the hold of the vessel to the ore train on the dock. 154 MECHANICAL WORK The remainder is dissipated as heat. This running down of energy by passing into an unavailable form is known as the dissipation of energy. It was first recognized and dis- tinctly stated by Lord Kelvin in 1859. The capacity which a body possesses for doing work does not depend on the total quantity of energy which it may possess, but only on that portion which is available, or is capable of being transferred to other bodies. In the problems of physics our chief concern is with the varia- tions of energy in a body and not with its total value. Questions and Problems 1. A cord that will just support an iron ball will generally break if the attached ball is lifted and allowed to drop. Explain. , 2. In what form is the energy of a coiled spring ? Of a bomb ? Of a pile driver? 3. Lake Tahoe in the Sierra Nevadas is at an elevation of 6225 ft. above the sea. Account for the energy of position stored there in the water. 4. Why has the ball in leaving the gun so much more energy of motion than the gun has in the recoil ? 5. Why is " perpetual motion " impossible ? 6. Is not the case of the earth going around the sun a case of perpetual motion? How does this differ from what is commonly meant by " perpetual motion " ? 7. A man weighing 200 Ib. climbs to the top of a hill 900 ft. high. How much work does he do ? 8. A man carries a ton of coal up a flight of stairs 14 ft. high. How much work does he do ? 9. A force of 200 dynes moves a mass of 100 g. through a dis- tance of 50 cm. How much work is done ? 10. A load of two tons was drawn up a hill half a mile long by a traction engine. The hill was 100 ft. high. How much work was done ? What force did the engine exert ? SUGGESTION. Notice that the work done by the engine in pulling the load half a mile is the same as lifting it vertically 100 ft. Lord Kelvin (Sir William Thomson), 1824-1907, was born at Belfast. He graduated at Cambridge in 1845 and in the same year received^the appointment of professor of natural philosophy in the University of Glasgow, a position which he held for fifty- three years. He was one of the greatest mathematical physicists of his day. His invention of the astatic mirror galvanometer and the siphon recorder has made successful marine cables a reality. His laboratory for the use of students was the first of the kind to be established. His most noteworthy investigations were in heat, energy, and electricity, yet there is scarcely any portion of physi- cal science that has not been greatly enriched by his genius. WHAT A MACHINE IS 155 11. How much work can a 40 H.P. engine do in an hour? How many tons of coal can it raise out of a mine 400 ft. deep in 10 hours ? 12. Express in joules the work done by a force of 100 kg. in mov- ing 100 kg. through a distance of 100 km. 13. An electric motor rated at 100 K.W. is used to operate a pump. The water has to be raised 100 m. How many liters will it be pos- sible to pump per hour ? 14. The mass of a railroad train is 250 tons, and the resistan.ee to its motion on a level track is 15 Ib. per ton. What H.P. must the loco- motive develop to maintain a speed of 40 miles per hour on the level ? 15. What is the potential energy of a stone weighing 100 Ib. as it rests on the top of a column 50 ft. high ? What will be its kinetic energy at the moment of reaching the ground if it should fall ? How much work would be done in placing the stone back on the column ? 16. A ball with a mass of 100 g. is given a velocity of 100 m. per sec. by being struck with a club. What was the energy of the blow ? 17. An automobile weighing 2500 Ib. when running at the rate of 30 mi. an hour strikes a telephone pole. Calculate the energy of the blow. 18. A force of 100 g. moves a mass 1000 g. through a distance of 100 m. in 10 sec. Express the activity of the agent in watts. 19. The mass of the ram of a certain pile driver is 2000 Ib. It falls from a height of 20 ft. upon the head of a pile and drives it 2 ft. into the ground. What is the energy of the blow delivered to the pile ? What is the resistance offered by the ground ? 20. A cannon ball weighing 10 Ib. is fired from a cannon whose barrel is 5 ft. long with a velocity of 1500 ft. per sec. Calculate the momentum of the ball ; also the energy of the ball ; also the average force acting on the ball in the barrel. II. MACHINES 166. What a Machine is. A machine is a device designed to change the direction or the value of a force required to do useful work, or one to transform and transfer energy. Simple machines enable us to do many things that would be impos- sible for us to do without them. A boy can draw a nail with a claw 156 MECHANICAL WORK hammer (Fig. 133) ; without it and with his fingers alone he could not start it in the least. By the use of a single pulley, the direction of the force applied may be changed, so as to lift a weight, for example, while the force acts in any convenient direction. Two men can easily lift a piano up to a second story window with a rope and. tackle. Perhaps the mosfimportant use of machines is for the purpose of utilizing the forces exerted by animals, and by wind, water, steam, or electricity. A water wheel transforms the potential and kinetic FIGURE 133 energy of falling water into mechanical energy rep- HAMMER AS LEVER, resented by the energy of the rotating wheel. A dynamo electric machine transforms mechanical energy into the energy of an electric current, and an electric motor at a distance transforms the electric energy back again into useful mechanical work. 167. General Law of Machines. Every machine must conform to the principle of the conservation of energy ; that is, the work done by the applied force equals the work done in overcoming the resistance, except that some of the applied energy may be dissipated as heat or may not appear in mechanical form. A machine can never produce an increase of energy so as to give out more than it receives. Denote the applied force, or effort, by E and the resist- ance by R, and let D and d denote the distances respectively through which they work. Then from the law of conser- vation of energy, the effort multiplied by the distance through which it acts is equal to the resistance multiplied by its displacement, or ED = Rd. . . . (Equation 23) 168. Friction. Friction is the resistance which opposes an effort to slide or roll one body over another. It is called into action whenever a force is applied to make one surface FRICTION 157 move over another. Friction arises from irregularities in the surfaces in contact and from the force of adhesion. It is diminished by polishing and by the use of lubricants. Experiments show that friction (a) is proportional to the pressure between the surfaces in contact, (5) is inde- MACHINE FOR MEASURING FRICTION AT MASS. INST. OF TECHNOLOGY. pendent of the area of the surfaces in contact within cer- tain limits, and (c) has its greatest value just before motion 'takes place. The friction of a solid rolling on a smooth surface is less than when it slides. Advantage is taken of this fact to reduce the fric- tion of bearings. A ball-bearing (Fig. 134) substitutes the rolling friction between balls and rings for the sliding FlGURE 134 _ B friction between a shaft and its journal. BEARING. 158 MECHANICAL WORK FIGURE 135. ROLLER BEARING. Roller bearings (Fig. 135) are also used with similar advantages. 169. Advantages and Disadvantages of Friction. Friction has innumer- able uses in preventing motion between surfaces in contact. Screws and nails hold entirely by friction ; we are able to walk because of friction between the shoe and the pavement ; shoes with nails in the heels are dangerous on cast-iron plates because the friction between smooth iron surfaces is small. Friction is useful in the brake to stop a motor car or railway train, in holding the driving wheels of a locomo- tive to the rails, and in enabling a gaso- line engine to drive an automobile by friction between the tires and the street. On the other hand, friction is also a re- sistance opposing useful motion, and Whenever motion CATERPILLAR TRACTOR. takes place, work The chain be i t around the wheels greatly in- must be done against creases the friction with the ground. SIMPLE MACHINES 159 this frictional resistance. The energy thus consumed is converted into heat and is no longer available for useful work. 170. Efficiency of Machines. On account of the impos- sibility of avoiding friction, every machine wastes energy. The work done is, therefore, partly useful and partly waste- ful. The efficiency of a machine is the ratio of the useful work done by it to the total work done by the acting force, or efficiency = ^ work done total energy applied For example, an effort of 100 pounds of force applied to a machine produces a displacement of 40 ft. and raises a weight of 180 Ib. 20 ft. high. Then 100 x 40 = 4000 ft. Ib. of energy are put into the ma- chine, and the work done is 180 x 20 = 3600 ft. Ib. Hence efficiency = = Q.9 = 90 per cent. Ten per cent of the energy is wasted and ninety per cent recovered. Since every machine wastes energy, a machine which will do either useful or useless work continuously without a supply of energy from without, a so-called "perpetual motion machine," is thus clearly impossible. Let e denote the efficiency of a machine; then from the relations just explained, Equation 23 becomes eED = Ed. . . . (Equation 24) This relation is the strictly correct one to apply to all machines; but in most problems dealing with simple machines, friction is neglected. 171. Simple Machines. All machines can be reduced to six mechanical powers or simple machines: the lever, the pulley, the inclined plane, the wheel and axle, the wedge, and the screw. Since the wheel and axle is only a modi- 160 MECHANICAL WORK fied lever, and the wedge and the screw are modifications of the inclined plane, the mechanical powers may be re- duced to three. In solving problems relating to simple machines in ele- mentary physics it is customary to neglect friction and to consider that the parts of machines are rigid and without weight. With these limitations, the law expressed by Equation 23 holds good. 172. Mechanical Advantage. A man working a pump handle and pumping water is an agent applying energy ; the pump and the water compose a system receiving energy. In a simple machine the force exerted by the agent ap- plying energy, and the opposing force of the system re- ceiving energy, may be denoted by the two terms, effort, E, and resistance, R. The problem in simple machines consists in finding the ratio of the resistance to the effort. The ratio of the resisting force R to the applied force E is called the mechanical advantage of the machine. This ratio may always be expressed in terms of certain parts of simple machines. 173. Moment of *, Force. In the application of the lever, the pulley, or the wheel and axle there is motion about an axis. The application of a single force to a body with a fixed axis produces rotation only. Examples are a door swinging on its hinges and the flywheel of an engine. The effect of a force in producing rotation depends, not only on the value of the force, but on the distance of its line of application from the axis of rotation. A smaller force is required to close a door when it is applied at right angles to the door at the knob than when it is applied near the hinge. Also, an increase in the speed of rota- tion of a flywheel may be secured either by increasing the THE LEVER 161 F FIGURE 136 MO- MENT OF FORCE. applied force or by lengthening the crank. Both these elements of effectiveness are included in what is known as the moment of a force. The moment of a force is the product of the force and the perpendicular distance between its line of action and the axis of rotation. Let M be a body which may rotate about an axis through (Fig. 136). The moment of the force F ap- plied at B in the direction CB is F x OB ; applied in the direction AB, its moment is Fx OA. The point is called the center of moments. A moment is considered positive if it produces rotation in a clockwise direction, and negative if in the other. If the sum of the positive moments equals that of the negative moments, there is equilibrium. The principle of moments is a very useful one in solv- ing a great variety of problems. c 174. The Lever. The lever ^| I is more frequently used than any other simple machine. In E* I *B its simplest form the lever is a rigid bar turning about a fixed axis called the fulcrum. It is convenient to divide levers into three classes, distinguished by the relative position of the ful- crum with respect to the two forces. In the first class the fulcrum is between the effort E and the resistance R (Fig. 137) ; in the second class the resistance is between the effort and the fulcrum ; in FIGURE 137. LEVERS. 162 MECHANICAL WORK FIGURE 138. LEVER, FIRST CLASS. the third class the effort is between the resistance and the fulcrum. FIGURE 139. -LEVER, SECOND CLASS. FIGURE 140. SCIS- SORS. 175. Examples of Levers. A crowbar used as a pry (Fig. 138) is a lever of the first class, but when used to lift a weight with one end on the ground (Fig. 139), it is a lever of the second class. Scissors (Fig. 140) are double levers of the first class. So also are the tongs of a blacksmith, and those used in chemi- cal laboratories for lifting crucibles (Fig. 141). The forearm when it supports a weight in the extended hand (Fig. 142), and the door when it is closed by pushing it near the hinge, are examples of levers of the third class. Nut - crack- ers (Fig. 143) and FIGURE 141. TONGS. FIGURE 142. FOREARM AS LEVER. FIGURE 143. NUT CRACKER. lemon squeezers are double levers of the second class. The steelyard (Fig. 144) is a lever of the first class with unequal arms. The common balance (Fig. 145) is a lever of the first class with equal arms. The two weights are thus also equal. The conditions for a sensitive balance, to show a small excess of weight in one pan over ^^^ that in the other, are small friction at the fulcrum, a light beam, and the center of gravity only slightly lower than the " knife-edge " forming the f ul- 176. Mechanical Advantage of the Lever. In STEELYARD. Fig. 146 U is the effort, R the resistance or MECHANICAL ADVANTAGE OF THE LEVER 163 weight lifted, O the fulcrum, and AC and BO the lever arms. Consider the lever to be weightless and to rotate about O without fric- tion; then the moment of the force E about the fulcrum ( 173) is E x A 0, and that of the force R is R x BO. These two forces tend to produce rotation in op- posite directions ; for equilibrium their mo- ments are therefore FlGURE 145. -COMMON BALANCE. equal, that is, ExAO=RxBO', from which E BO (Equation 25) Hence, the mechanical advantage of the lever equals the inverse ratio FIGURE 146. MECHANICAL ADVANTAGE OF of its arins. LEVER - If the weight of the lever has to be taken into account, it is to be treated as a force acting at the center of gravity of the lever, and CZ its moment must be added to that of the force turning the lever in the same direc- tion as its own weight. EXAMPLE. The weights W^ and W 2 are placed at distances 5 and 8 units respectively from (Fig. 147). If W l is 20 lb., what mustW 2 be for equilibrium? By the principle of moments about O, [ U Jb rrr FIGURE 147. 164 MECHANICAL WORK 20 x 5 = W z x 8 ; whence W 2 = 12.5 Ib. If the lever is uniform, it is balanced about the fulcrum and its moment is zero. Suppose the weight of the bar to be 1 Ib. and its center of gravity 4 units to the left of 0. The equation for equi- librium would then be 20 x 5 + 1 x 4 = W 2 x 8. Whence ^ = 13 lb> 177. The Wheel and Axle consists of a cylinder and a wheel of larger diameter usually turning together on the same axis. In Fig. 148 the axle passes through (7, the radius of the cylinder is BO, and that of the wheel is AC. The weights P and W are sus- pended by ropes wrapped around the cir- cumference of the two wheels; their moments about the axis are P x AO and Wx BO respectively. For equilib- rium these moments are equal, that is, PxAC= WxBO. Hence, FIGURE 148.- J?=^=:il. (Equation 26) WHEEL AND AXLE. P BO r R and r are the radii of the wheel and the axle respec- tively. The weight P represents the effort applied at the circumference of the wheel, and the weight W the resist- ance at the circumference of the axle. Therefore, the mechanical advantage of the wheel and axle is the ratio of the radius of the wheel to that of the axle. 178. Applications. The old well wind- lass for drawing water from deep wells (Fig. FIGURE 149. WELL 149) by means of a rope and bucket is an ap- WINDLASS. TEE PULLEY 165 plication of the principle of the wheel and axle. In the windlass a crank takes the place of a wheel and the length of the crank is the radius of the wheel. In the capstan (Fig. 150) the axle is vertical, and the effort is applied by means of handspikes inserted in holes in the top. The derrick (Fig. 151) is a form of wheel and axle much used for raising FIGURE 150. CAPSTAN FIGURE 151. DERRICK. heavy weights. In the form shown it is essentially a double wheel and axle. The axle of the first sys- tem works upon the wheel of the second by means of the spur gear. The mechanical advantage of such a compound machine is the ratio of the product of the radii of the wheels to the product of the radii of the axles. In the case of gearing, the number of teeth is substituted for the ra- dius. 179. The Pulley consists of a wheel, called a sheave, free to turn about an axle in a frame, called a block (Fig. 152). The effort and the resist- ance are attached to a rope BLOCK AND which moves in a groove cut SHEAVE - in the circumference of the wheel. A simple fixed pulley is one whose axis does not change its position ; it is used to change the direction of the applied force (Fig. 153). If friction and the FIGURE 153.- SINGLE ri g idit y of the r P e are neglected, the PULLEY. tension in the rope is everywhere the FIGURE 152. w 166 MECHANICAL WORK PRACTICAL USE OF DERRICKS. These enormous derricks are used for raising the huge blocks of marble from the quarry. same ; the effort and the resistance are then equal to each other and the mechanical advantage is unity. In the movable pulley (Fig. 154) it is evident that the weight W is sup- ported by two parts of the cord, one half of it by means of the hook fixed in the beam above and the other half by the effort E applied at the free end of the cord. If the weight is lifted, it rises only half as fast as the cord travels. 180. Systems of Fixed and Movable W FIGURE 154. MOV- ABLE PULLEY. Pulleys. Fixed and movable pulleys MECHANICAL ADVANTAGE OF SIMPLE PULLEY 167 are combined in a great variety of ways. The most com- mon is the one employing a continuous cord with one free end and the other attached to a rigid support or to one of the blocks. Figure 155 represents a combination of one fixed and one movable pulley. Figure 156 il- lustrates the common " block and tackle," where each block has more than one sheave. 181. Mechanical Advantage of the Simple Pulley. In Fig. 157 the cord passes in succession around each pulley. It is evi- dent that if the movable pulley and the resistance W are moved toward the FIGURE 155 fixed pulley a distance #, FIXED AND Mov- each cord passing between ABLE PULLEYS - the two blocks must be shortened by a units. The effort E therefore travels through a distance of na units, n being the number of parts to the cord between the two pulleys. Then by the general law of machines ( 167), x na = W X a; whence W (Equation 27) Hence, when a continuous cord is. used^ the mechanical advantage of the pulley is equal FIGURE 156. to the number of times the cord passes to BLOCKANDTACKLE. ^ from the movable block. It should be noticed that n is equal to the entire num- ber of sheaves in the fixed and movable blocks, or to that 168 MECHANICAL WORK number plus one. If the upper block in Fig. 157 were the movable one, that is, if the system were inverted, so that the effort E is upward, n would be equal to one more than the number of sheaves. 182. The Differential Pulley. The differ- ential pulley (Fig. 158) is much used for lifting heavy machinery by means of a rela- tively small force. In the upper block are two sheaves of different diameters turning rigidly to- gether. The lower block has only one sheave. An end- less chain runs over the three sheaves in succession. It is kept from slip- ping by projections on the sheaves, which fit between the links of the chain. A practical advantage of the differential pulley is that there is alwaj 7 s enough friction to keep the weight from dropping when there is no force applied to the chain. FIGURE 157. MULTIPLE PUL- LEYS. The mechanical advantage of the dif- ferential pulley may be found as follows : In Fig. 159, which is an outline drawing of this pulley, let the radius A C of the FIGURE larger sheave be denoted by R, and that \J 158. DIFFERENTIAL PULLEY. TEE INCLINED PLANE 169 of the smaller one A B by r. Suppose a force E to move the chain some convenient distance as R ; then a length r winds off the smaller sheave at B and a length R winds on the larger sheave at D. The length of chain between the two blocks is thus shortened by a length R r, and the weight W D is lifted a distance %(R r). The work done by the effort E is E x R and the work dojie on W is W x \(R r). Neglecting friction, these expres- sions may be placed equal to each other, or Whence W E R-r . (Equation 28) Since the difference R r may be made small, it _ is obvious that the mechanical advantage of the dif- OUTLINE OF DIF- ferential pulley is large, and it is larger the nearer r FERENTIAL PUL- approaches R in length. LEY. 183. The Inclined Plane. Any plane surface making an angle with the horizontal is an inclined plane. Planks or FIGURE 160. HUGE FLOATING CRANE. 170 MECHANICAL WOKK skids used to roll casks and barrels up to a higher level are examples of inclined planes. Every road, street, or railway not on a level is an inclined plane. The steeper the incline, the greater the push required to force the load up the grade. If a body rests on an inclined plane without friction, the weight of the body acts vertically downward, while the reaction of the plane is perpendicular to its surface, and therefore a third force must be applied to maintain the body in equilibrium on the incline. 184. Mechanical Advantage of the Inclined Plane. Con- sider only the case in which the force applied to maintain equilibrium is parallel to the face of the plane (Fig. 161). The most convenient way to find the relation between the force E and the weight JTof the body D is to apply the principle of work ( 167). Suppose D to be moved by the force E from A to O. Then the work done by E is E x AC. Since the body D is lifted through a vertical dis- tance BO, the work done on it against gravity is Wx EG. Therefore, ExAC= Wx BO, and (Equation 29) FIGURE 161. INCLINED PLANE. E BC or the mechanical advantqge, when the effort is applied parallel to the face of the plane, is the ratio of the length of the plane to its height. 185. Grades. The grade of an inclined roadway is ex- pressed as the number of feet rise per hundred feet along the incline. If the rise, for example, is 3 feet for every 100 feet measured along the roadway, the road has a three GRADES 171 per cent grade. The grade of railways seldom exceeds 2 per cent, but county roads and state highways may have 8 or 10 per cent grades. Various expedients are adopted for the purpose of lengthening the incline on roads and railways so as to keep the grades within practical limits. THE GREAT PYRAMID. The huge stones of which the pyramids are made were probably raised to their great height by inclined planes. Zigzags and " switchbacks " are common expedients for the purpose. A most remarkable inclined railway track is on the northern approach to the St. Gotthard tunnel in Switzer- land. This tunnel reaches a culminating elevation of 3786 feet. In at least one instance the railway forms three turns of a screw, one above the other, each turn lying partly on the face of the mountain and partly in a tunnel cut 172 MECHANICAL WORK FIGURE 162. THE WEDGE. through the rock. This novel grade enables the road to surmount a precipice by means of an inclined plane, the necessary length of which was secured along the thread of a mammoth screw. 186. The Wedge is a double inclined plane with the effort applied parallel to the base of the plane, and usually by a blow with a heavy body (Fig. 162). Although the principle of the wedge is the same as that of the inclined plane, yet no ex- act statement of its me- chanical advantage is pos- sible, because the resistance has no definite relation to the faces of the planes, and the friction cannot be neglected. Many cutting instruments, such as the ax and the chisel, act on the principle of the wedge ; also nails, pins, and needles. 187. The Screw is a cylinder, on the outer surface of which is a uniform spiral projection, called the thread. The faces of this thread are inclined planes. If a long triangular strip of paper be wrapped around a pencil (Fig. 163), with the base of the triangle perpendicular to the axis of the cylindrical pencil, the hypotenuse of the triangle will trace a spiral like the thread of a screw. The screw (Fig. 164) works in a block called a nut, on the inner surface of which is a groove, the THE NUT. exact counterpart of the thread. FIGURE 163. THE SCREW. FIGURE 164, APPLICATIONS OF THE SCREW 173 FIGURE 165. PITCH OF SCREW. The effort is applied at the end of a lever or wrench, fitted either to the screw or to the nut. When either makes a complete turn, the screw or the nut moves through a distance equal to that between two adjacent threads, measured parallel to the axis of the screw cylinder. This distance, s in Figure 165, is called the pitch of the screw. It is usually expressed as the number of threads to the inch or to the centimeter. 188. Mechanical Advantage of the Screw. Since the screw is usually combined with the lever, the simplest method of finding the mechanical advantage is to apply the principle of work, as expressed in the general law of machines ( 167). If the pitch be denoted by s and the resistance overcome by R, then, ignoring friction, the work done against R in one revolu- tion of the screw is R x s. If the length of the lever is , the work done by the effort JE in one revolution is E x 2 irl. Whence E x 27rl=R x s, or FIGURE 166. JACK- SCREW. : = . (Equation 30) E s Hence, the mechanical advantage of the screw equals the ratio of the dis- tance traversed by the effort in one revolution of the screw to the pitch of FIGURE 167. LETTER the screw. PRESS. 189. Applications of the Screw. The jackscrew (Fig. 166), the letter press (Fig. 167), the vise (Fig. 168), the two blade propeller of a 174 MECHANICAL WORK FIGURE 168. THE VISE. flying machine, and the two, three, or four blade propeller of a ship are familiar examples of the use of a screw. The rapid rotation cf the propeller blades tends to push backward the air in the one case and the water in the other, but the inertia of the fluid medium produces a reaction against the propeller and forces the vessel forward. The screw propeller pushes against the fluid and so forces itself and the vessel to which it is attached in the other direction. An important application of the screw, though not as a machine, is that for measuring small dimensions. The wire micrometer (Fig. 169) and the spherometer (Fig. 170) are instruments for this pur- pose. In both, an accurate screw has a head divided into a number of equal parts, 100 for example, so as to register any portion of a revolution. If the pitch of the screw is 1 mm., then turning the head through one of its divisions causes the screw to move parallel to its axis 0.01 mm. All wood screws, augers, gimlets, and most machine screws and bolts are right-handed, that is, they screw in or away from the observer by turn- ing around in the direction of watch hands. An example of a left-handed screw is the turnbuckle (Fig. 171). This has a right- handed screw at one end and a left-handed screw at the other. It is used for tightening tie rods, stays, etc. One turn of the buckle brings the rods together a distance equal to twice the pitch of the screws. FIGURE 169. MICROMETER. FIGURE 170. SPHEROME- TER. Questions and Problems 1. What are the relative positions of the effort, the resistance, and the fulcrum in the following : the lever as applied to the jack- screw, the oar of a boat in row- ing, the claw hammer in pull- ing a nail, and a bar applied to a car wheel to move the car? FIGURE 171. TURNBUCKLE. QUESTIONS AND PROBLEMS 175 2. In which direction does friction on the rails act on the wheels of a locomotive? On those of a freight car? Does it act in the same direction on the front and rear wheels of an automobile ? 3. Calculate the efficiency of a machine that lifts a weight of 1000 Ib. a distance of 8 ft. by the action of a force of 100 Ib. through 100 ft. 4. A motor whose efficiency is 90 % delivers 10 H.P. What must be the input ? 5. In a system of pulleys a tension of 100 Ib. is applied to the rope and the rope is drawn 60 ft., while a weight of 500 Ib. is lifted 10 ft. What is the efficiency of the system ? 6. A weight of 100 Ib. is lifted by a lever of the second kind. The weight is placed 2 ft. from the fulcrum and the lever is 12 ft. long. What force is necessary ? 7. A bar 4 m. long is of uniform size and weighs 1 kg. to the meter. A weight of 10 kg. is placed at one end, and the fulcrum is 1 m. from that end. What weight at the other end will produce a balance ? 8. In order to lift a weight of 500 Ib. at one end of a bar 15 ft. long, a weight of 100 Ib. is used at the other end. The bar is of uni- form size and weighs 25 Ib. Where must the fulcrum be placed ? 9. The axle on which the rope wound in a windlass was 8 in. in diameter. The crank was 12 in. long and the weight lifted was 200 Ib. What force was applied ? 10. The diameter of a ship's capstan is 16 in. What force must be applied to each of two handspikes at an effective distance of 6 ft. to turn the capstan and lift an anchor weighing 2400 Ib. if the efficiency of the machine is 80 per cent ? 11. In a system of six pulleys, three of which are movable, how many kilograms can a force of 25 kg. support ? 12. A jackscrew was used to lift a weight of 200 Ib. The lever was 2 ft. long and the screw had 4 threads to the inch. Assuming an efficiency of 100 96, what force was applied at the end of the handle? 13. The radii of a wheel and the axle are 5 ft. and 5 in. respec- tively. It was found that a force of 100 Ib. could lift a weight of 960 Ib. What weight would 100 Ib. of force lift if there were no friction ? What is the efficiency of the machine ? 14. If the front sprocket wheel of a bicycle contains 24 sprockets and the rear one 8, how far will one complete turn of the pedals drive a 28 in. wheel? CHAPTER VII SOUND I. WAVE MOTION 190. Vibrations. A vibrating body is one which re- peats its limited motion at regular short intervals of time. A complete or double vibration is the motion between two successive passages of the moving body through any point of its path in the same direction. If we suspend a ball by a long thread and set it swinging like a common pendulum, it will return at regular intervals to the starting point. If we set the ball moving in a circle, the string will describe a conical surface and the ball will again return at the same inter- vals to the starting point. 191. Kinds of Vibration. Clamp one end of a thin steel strip in a vise (Fig. 172) ; draw the free end aside and release it. It will move repeatedly from D' to D" and back again. The shorter or thicker the strip, the quicker its vibration ; when it becomes like the prong of a tuning fork, it emits a musical sound. D' D Vibrations like these are transverse. FIGURE 172. VIBRATION A body vibrates transversely when the OF STEEL STRIP. direction of the motion is at right angles to its length. The strings of a violin, the reeds of a cabinet organ, and the wires of a piano are familiar ex- amples. 176 TRANSVERSE WAVES 177 Fasten the ends of a long spiral spring securely to fixed supports with the spring slightly stretched. Crowd together a few turns of the spiral at one end and release them. A vibratory movement will travel from one end of the spiral to the FlGURE 173. VIBRATORY MOTION IN SPRING. other, and each turn of wire will swing backward and forward in the direction of the length of the spiral (Fig. 173), The vibrations of the spiral are longitudinal. A body vibrates longitudinally when its parts move backward and forward in the direction of its length. The vibrations set up in a long glass tube by stroking it lengthwise with a damp cloth are longitudinal ; so are those of the air in a trumpet and the air in an organ pipe. 192. Wave Motion. Tie one end of a soft cotton rope, such as a clothesline, to a fixed support ; grasp the other end and stretch the rope horizontally. Start a disturbance by an up-and-down motion of the hand. Each point of the rope will vibrate with simple har- monic motion ( 112), while the disturbance will travel along the rope toward the fixed end. This progressive change of form due to the periodic vibra- tion of the particles of the medium is a wave. The particles are not all in the same phase ( 112) or stage of vibration, but they pass through corresponding positions in suc- cession. 193. Transverse Waves. A small camel's-hair brush is at- tached to the end of a long slender strip of clear wood, mounted as FIGURE 174. INSCRIBING TRANSVERSE WAVE. shown in Fig. 174, which was made from a photograph giving an oblique view of the apparatus. The brush should touch lightly the 178 SOUND paper attached to the narrow board, which may be moved in a straight line against the guiding strip. Ink the brush and while it is at rest push the paper along under it. The brush will mark the straight middle line running through the curve shown in the figure. Replace the board in the starting position; then pull the strip aside and release it. Again draw the board under the brush with uniform motion. This time the brush traces the curved line. The strip of wood vibrates at right angles to the direc- tion of motion of the paper with a simple harmonic motion (112); the board moves with a uniform rectilinear motion; the curve is a simple harmonic curve. It is the resultant of the two motions, and illustrates a transverse wave. A transverse wave is one in which the vibration of the particles in the wave is at right angles to the direction in which the wave is traveling. 194. To Construct a Transverse Wave. Suppose a series of particles, originally equidistant in a horizontal straight line, to 1 % c T T H a ] ] I t ! ! It y r m , | 1 i rrti 1 ! i i i j i i i i * i FIGURE 175. POSITION OF PARTICLES IN WAVE. Vibrate transversely with simple harmonic motion. Let Fig. 175 represent the position of the particles at some particular instant, the displacement of each one from the straight horizontal line being found by means of an auxiliary circle as in 112. They will out- line a transverse wave. At g the particle has reached its extreme displacement in the positive direction and is momentarily at rest; the particle at s has reached its maximum negative displacement, and is also at rest. The particle at m is moving in the positive direction LONGITUDINAL WAVE 179 with maximum velocity, and the particles a and y with maximum velocity in the negative direction. If the wave is traveling to the right, then an instant later the displacement of g will have diminished and that of i will have increased to a maximum, the crest having moved forward from g to i in the short interval. The successive particles of the wave all differ in phase by the same amount. 195. Longitudinal Wave. Place a lighted candle at the conical end of the long tin tube of Fig. 176. Over the other end stretch a m, FIGURE 176. WAVE OF COMPRESSION IN TUBE. piece of parchment paper. Tap the paper lightly with a cork mallet; the transmitted impulse will cause the flame to duck, and it may easily be blown out by a sharper blow. The air in the tube is agitated by a vibratory motion, and a wave, consisting of a compression followed by a rarefaction, traverses the tube. The dipping of the flame indicates the arrival of the compression. Each particle of air vibrates longitudinally in the tube, the disturbance being similar to that of the vibrating spiral. E A C E G FIGURE 177. PARTICLES IN WAVE OF COMPRESSION. Figure 177 illustrates the distribution of the air particles when disturbed by such a longitudinal wave of com- pressions and rarefactions. B^ D, F, etc., are regions of compressions ; A, C, E, etc., those of rarefaction. The 180 SOUND distances of the different points of the curve from the straight line denote the relative velocities of the air particles. The greatest velocity forward is at the middle of the condensation, as at B, and the greatest velocity backward is at the middle of the rarefaction, as at A. A and (7, or B and 2>, are in the same phase, that is, in corresponding positions in their path. A longitudinal wave is one in which the vibrations are backward and forward in the same direction as the wave is traveling. 196. Wave Length. The length of a wave is the distance from any particle to the next one in the same phase, as from a to y (Fig. 175), or from A to or B to D (Fig. 177). Since the wave form travels from a to y, or from A to (7, during the time of one complete vibration of a particle, it follows that the wave length is also the distance traversed by the wave during one vibration period. 197. Water Waves. One of the most familiar examples of transverse waves are those on the surface of water. For deep water FIGURE 178. WATER WAVE. the particles describe circles, all in the same vertical plane containing the direction in which the wave is traveling, as illustrated in Fig. 178. The circles in the diagram are divided into eight equal arcs, and the water particles are supposed to describe these circles in the direction of watch hands and all at the same rate ; but in any two consecutive circles their phase of motion differs by one eighth of a period, that is, the water particles are taken at such a distance apart that each one begins to move just as the preceding one has completed one eighth part of its orbit. When a has completed one revolution, b is one eighth of a revolution behind it, c two eighths or one quarter, etc. SOURCE OF SOUND 181 A smooth curve drawn through the positions of the particles in the several circles at the same instant is the outline or contour of a wave. When a particle is at the crest of a wave, it is moving in the same direction as the wave ; when it is in the trough, its motion is opposite to that of the wave. The crests and troughs are not of the same size, and the larger the circles (or amplitude), the smaller are the crests in comparison with the troughs. Hence the crests of high waves tend to become sharp or looped, and they break into foam or white caps. II. SOUND AND ITS TRANSMISSION 198. Sound may be defined as that form of vibratory mo- tion in elastic matter which affects the auditory nerves, and produces the sensation of hearing. All the external phenom- ena of sound may be present without any ear to hear. Sound should therefore be distinguished from hearing. 199. Source of Sound. If we suspend a small elastic ball by a thread so that it just touches the edge of an inverted bell jar, and strike the edge of the jar with a felted or cork mallet, the ball will be repeatedly thrown away from the jar as long as the sound is heard. This shows that the jar is vibrating energetically. Stretch a piano wire over the table and a little above it. Draw a violin bow across the wire, and then touch it with the suspended ball of the previous paragraph. So long as the wire emits soun^. the ball will be thrown away from it again and again. If a mounted tuning fork (Fig. 179) is sounded, and a light ball of pith or ivory, suspended by a thread, is brought in contact with one of the prongs at the back, FIGURE 179. it will be briskly thrown away by the energetic vibra- -VIBRATION OF tionsofthefork. Partly fill a glass goblet with water, and produce a musical note by drawing a bow across its edge. The tremors of the glass will throw the surface of the water into violent agitation in four sectors, with intermediate regions of relative repose. This agitation disappears when the sound ceases. 182 SOUND A glass tube, four or five feet long, may be made to emit a musical sound by .grasping it by the middle and briskly rubbing one end with a cloth moistened with water. The vibrations are longitudinal, and may be so energetic as to break the tube into many narrow rings. Experiments like these show that the sources of sound are bodies in a state of vibration. Sound and vibratory movement are so related that one is strong when the other is strong, and they diminish and cease together. 200. Media for Transmitting Sound. Suspend a small electric bell in a bell jar on the air pump table (Fig. 180). When the air is exhausted, the bell is nearly inau- dible. Sound does not travel through a vacuum. Fasten the stem of a tuning fork to the middle of a thin disk of wood. Set the fork vibrating, and hold it with the disk resting on the surface of water in a tumbler, standing on a table. The sound, which is scarcely audible FIGURE 180. BELL IN VACUUM. . ... . J . J when there is no disk attached to the fork, is now distinctly heard as if coming from the table. Hold one end of a long, slender wooden rod against a door, and rest the stem of a vibrating fork against the other end. The sound will be greatly intensified, and will come from the door as the apparent source. Press down on a table a handful of putty or dough, and insert in it the stem of a vibrating fork ; the vibrations will not be conveyed to the table to an appreciable extent. Only elastic matter transmits sound, and some kinds transmit it better than others. 201. Transmission of Sound to the Ear. Any uninter- rupted series of elastic bodies will transmit sound to the ear, be they solid, liquid, or gaseous. A bell struck under water sounds painfully loud if the ear of the listener is also under water. A diver under water can hear voices in the air. By placing the ear against the steel rail of a railway, two sounds may be heard, if the rail is struck some distance away: a Lord Rayleigh (John William Strutt) was born at Essex in 1842, and graduated from Cambridge University in 1865. In 1884 he was appointed professor of experimental physics in that institution, and three years later he was elected professor of natu- ral philosophy at the Royal Institution of Great Britain. His work is remarkable for its extreme accuracy. The discovery of argon in the atmosphere, while attempting to determine the density of nitrogen, was the result of a very minute difference between the result obtained by using nitrogen from the air and that from another source. Nearly every department of physics has been enriched by his genius. His treatise on Sound is one of the finest pieces of scientific writing ever produced. His determination of the electrochemical equivalent of silver and the electromotive force of the Clark standard cell are important contributions to modern electrical measurements. He died in 1919. Photographs of Sound- Waves produced by an Electric Spark behind a Black Disk. (Taken by Professor Foley of Indiana University.) 1. A spherical sound-wave. 2. The same wave a fraction of a second later. 3. Spherical sound-wave reflected from a plate of plane glass. 4. The same wave a moment later. The broken line near the black disk shows the effect of the puff of hot air from the spark. 5. Sound-wave reflected from a parabolic reflector. The source is at the focus and the reflected wave is plane. 6. The same wave a moment later, showing its central portion advanced by the puff of hot air from the spark. MOTION OF THE PARTICLES OF A WAVE 183 louder one through the rails and then another through the air. The faint scratching of a pin on the end of a long stick of timber, or the ticking of a watch held against it, may be heard very distinctly if the ear is applied to the other end. The earth conducts sound so well that the stepping of a horse may be heard by applying the ear to the ground. This is understood by the Indians. The firing of a cannon at least 200 miles away may be heard in the same way. The report of a mine blast reaches a listener sooner through the earth than through the air. The great eruption of Krakatoa in 1883 gave rise to gigantic sound waves, which produced at a distance of 2000 miles a report like the firing of heavy guns. 202. Sound Waves. When a tuning fork or similar body is set vibrating, the disturbances produced in the air about it are known as sound waves. They consist of a series of condensations and rarefactions succeeding each other at regular intervals. Each particle of air vibrates in a short path in the direction of the sound transmission. Its vibrations are longitudinal as distinguished from the transverse vibrations in water waves. 203. Motion of the Particles of a Wave. The motion of the particles of the medium conveying sound is distinct from the motion of the sound wave. A sound wave is composed of a condensation followed by a rarefaction. In the former the particles have a forward motion in the direction in which the sound is traveling ; in the latter they have a backward motion, while at the same time both condensation and rarefaction travel steadily forward. The independence of the two motions is aptly illustrated by a field of grain across which waves excited by the wind are coursing. Each stalk of grain is securely anchored to the ground, while the wave sweeps onward. The heads of grain in front of the crest are rising, while all those behind the crest and extending to the bottom of the trough are falling. They all sweep forward and backward, not simultaneously, but in succession, while the wave itself travels continuously forward. 184 SOUND III. VELOCITY OF SOUND 204. Velocity in Air. In 1822 a scientific commission in France made experiments to ascertain the velocity of sound in air. Their method was to divide into two par- ties at stations a measured distance apart, and to determine the interval between the observed flash and the report of VIEW OF LAKE GENEVA. a cannon fired alternately at the two stations. The mean of an even number of measurements eliminated very nearly the effect of the wind. The final result was 331 m. per second at C. The defect of the method is that the per- ception of sound and of light are not equally quick, and they vary with different persons. Subsequent observers, employing improved methods, and correcting for all sources of error, have obtained as the most probable velocity 332.4 m., or 1090.5 ft., per second at C. At higher temperatures sound travels QUESTIONS AND PROBLEMS 185 faster, the correction being 0.6 m., or nearly 2 ft., per de- gree Centigrade. At 20 C. (68 F.) the velocity is very nearly 1130 ft. per second. 205. Velocity in Water. In 1827 Colladon and Sturm, 'by a series of measurements in Lake Geneva, found that sound travels in water at the rate of 1435 m. per second at a mean temperature of 8.1 C. They measured with much care the time required for the sound of a bell struck under water to travel through the lake between two boats anchored at a distance apart of 13,487 m. It was 9.4 sec- onds. A system of transmitting signals through water by means of sub- merged bells is in use by vessels at sea and for offshore stations. Spe- cial telephone receivers have been devised to operate under water and to respond to these sound signals. Indeed, the vessel itself acts as a sounding board and as a very good receiver. 206. Velocity in Solids. The velocity of sound in solids is in general greater than in liquids on account of their high elasticity as compared with their density. The ve- locity in iron is 5127 m. per second ; in glass 5026 m. per second ; but in lead it is only 1228 m. per second, at a temperature in each case of C. Questions and Problems 1. Why do the timers in a 200-yd. dash start their stopwatches by the flash of the pistol rather than by the report ? 2. If the flash of a gun is seen 3, sec. before the report is heard, how far is the gun from the observer, the temperature being 20 C. ? 3. The interval between seeing a flash of lightning and hearing the thunder was 5 sec. ; the temperature was 25 C. How far away was the lightning discharge ? 4. Signals given by a gun 2 mi. away would be how much in error when the temperature is 20 C. and the wind is blowing 10 mi. an hour in the direction from the listener to the gun ? 186 ' SOUND 5. A man sets his watch by a steam whistle which blows at 12 o'clock. The whistle is 1.5 mi. away and the temperature ' 15 C. How many seconds will the watch be in error? 6. A ball fired at a target was heard to strike after an interval of 8 sec. The distance of the target was 1 mi. and the temperature of the air 20 C. What was the mean velocity of the ball ? 7. The distance between two points on a straight stretch of rail- way is 2565 m. An observer listens at one of these points and a blow is struck on the rails at the other. If the temperature is C., what is the interval between the arrival of the two sounds, one through the rails and the other through the air? 8. A man watching for the report of a signal gun saw the flash 2 sec. before he heard the report. If the temperature was C. and the distance of the signal gun was 2225 ft., what was the velocity of the wind ? 9. A shell fired at a target, distance half a mile, was heard to strike it 5 sec. after leaving the gun. What was the average speed of the bullet, the temperature of the air being 20 C. ? IV. REFLECTION OF SOUND 207. Echoes. An echo is the repetition of a sound by re- flection from some distant surface. A clear echo requires a vertical reflecting surface, the dimensions of which are large compared to the wave length of the sound. A cliff, a wooded hill, or the broad side of a large building may serve 'as the reflecting surface. Its inequalities must be small compared to the length of the incident sound waves ; otherwise, the sound is diffused in all directions. A loud sound in front of a tall cliff an eighth of a mile away will be returned distinctly after about a second and a sixth. If the reflecting surface is nearer than about fifty feet, the reflected sound tends to strengthen the original one, as illustrated by the greater distinctness of sounds indoors than in the open air. In large rooms where the echoes produce a confusion of sounds the trouble may be I I 2 b) O It AERIAL ECHOES 187 diminished by adopting some method to prevent regular reflection, such as the hanging of draperies, or covering the walls with absorbing materials. 208. Multiple Echoes. Parallel reflecting surfaces at a suitable distance produce multiple echoes, as parallel mir- rors produce multiple images ( 261). The circular baptistery at Pisa and its spherical dome prolong a sound for ten or more seconds by successive reflec- tions; the effect is made more conspicuous by the good reflecting surface of polished marble. Ex- traordinary echoes some- times occur between the parallel walls of deep canons. 209. Aerial Echoes. Whenever the medium transmitting sound changes suddenly in density, a part of the energy is transmitted and a part reflected. The in- tensity of the reflected system is the greater the greater the difference in the densities of the two media. A dry sail reflects a part of the sound and transmits a part ; when wet it becomes a better reflector and is almost im- pervious to sound. Aerial echoes are accounted for by sudden changes of density in the air. Air, almost perfectly transparent to light, may be very opaque to sound. When for any rea- son the atmosphere becomes unstable, vertical currents THE BAPTISTERY AT PISA. 188 SOUND and vertical banks of air of different densities are formed. The sound transmitted by one bank is in part reflected by the next, the successive reflections giving rise to a curious prolonging of a short sound. Thus, the sound of a gun or of a whistle is then heard apparently rolling away to a great distance with decreasing loudness. 210. Whispering Gallery. Let a watch be hung a few inches in front of a large concave reflector (Fig. 181). A place may be found for the ear at some distance in front, as at E, where the ticking of the watch may be heard with great distinctness. The sound waves, after reflection from the concave surface, converge to a point at E. The action of the ear trumpet FIGURE 181. REFLECTOR FOR -, ,, a ,. ,. j _ depends on the reflection of sound from curved surfaces; the sides of the bell-shaped mouth reflect the sound into the tube which conveys it to the ear. An interesting case of the reflection of sound occurs in the whispering gallery, where a faint sound produced at one point of a very large room is distinctly heard at some distant point, but is inaudible at points between. It re- quires curved walls which act as reflectors to concentrate the waves at a point. Low whispers on one side of the dome of St. Paul's in London (see page 81) are distinctly audible on the opposite side. V. RESONANCE 211. Forced Vibrations. A body is often compelled to surrender its natural period of vibration, and to vibrate with more or less accuracy in a manner imposed on it by an external periodic force. Its vibrations are then said to be forced. SYMPATHETIC VIBRATIONS 189 Huyghens discovered that two clocks, adjusted to slightly differ' ent rates, kept time together when they stood on the same shelf. The two prongs of a tuning fork, with slightly different natural periods on account of unavoidable differences, mutually compel each other to adopt a common frequency. These two cases are examples of mutual control, and the vibrations of both members of each pair are forced. The sounding board of a piano and the membrane of a banjo are forced into vibration by the strings stretched over them. The top of a wooden table may be forced into vibration by pressing against it the stem of a vibrating tuning fork. The vibrations of the table are forced and it will respond to a fork of any period. 212. Sympathetic Vibrations. Place two mounted tuning forks, tuned to exact unison, near each other on a table. Keep one of them in vibration for a few seconds and then stop it; the other one will be heard to sound. In the case of these forks, the pulses in the air reafch the second fork at intervals corresponding to its natural vibration period and the effect is cumulative. The ex- periment illustrates sympathetic vibrations in bodies hav- ing the same natural period. If the forks differ in period, the impulses from the first do not produce cumulative effects on the second, and it will fail to respond. Suspend a heavy weight by a rope and tie to it a thread. The weight may be set swinging by pulling gently on the thread, releas- ing it, and pulling again repeatedly when the weight is moving in the direction of the pull. Suspend two heavy pendulums on knife-edges on the same stand, and carefully adjust them to swing in the same period. If then one is set swinging, it will cause the other one to swing, and will give up to it nearly all its own motion. When the wires of a piano are released by pressing the loud pedal, a note sung near it will be echoed by the wire which gives a tone of the same pitch. A number of years ago a suspension bridge of Manchester in Eng- land was destroyed by its vibrations reaching an amplitude beyond 190 SOUND IP css^3\ ^Bi ^^^PII the limit of safety. The cause was the regular tread of troops keep- ing time with what proved to be the natural rate of vibration of the bridge. Since then the custom has always been observed of breaking step when bodies of troops cross a ^ bridge. 213. Resonance. Reso- nance is the reenf or cement of sound by the union of direct and reflected sound waves. Hold a vibrating tuning fork over the mouth of a cylindrical jar (Fig. 182). , Change the length of the air column by pouring in water slowly. The sound will increase in loudness until a certain length is reached, after which it becomes weaker. A fork of different pitch OF will require a different length of air column to reenforce its sound. The " sound of the sea" heard when a sea shell is held to the ear is a case of resonance. The mass of air in the shell has a vibration rate of its own, and it amplifies any faint sound of the same period. A vase with a long neck, or even a tea- cup, will also exhibit resonance. The box on which a tuning fork is mounted (Fig. 183) is a resonator, designed to increase the volume of sound. The air within the body of a violin and all instruments of like character acts as a resonator. The air in the mouth, the larynx, and the nasal passages is a resonator ; the -'- FIGURE 182. REENFORCEMENT SOUND. FIGURE 183. MOUNTED TUNING length and volume of this body of air FORK. can be changed at pleasure so as to reenforce sounds of different pitch. PITCH 191 214. The Helmholtz Resonator. The resonator devised by Helmholtz is spherical in form, with two short tubes on opposite sides (Fig. 184). The larger opening A is the mouth of the resonator ; the smaller one B fits in the ear. These resonators are made of thin brass or of glass, and their pitch is determined by their size. When one of them is held to the ear, it strongly reenforces any sound of its own rate of vibration, but is silent to others. FIGURE 1 84. HELMHOLTZ RESONATOR. VI. CHARACTERISTICS OF MUSICAL SOUNDS 215. Musical Sounds. Sounds are said to be musical when they are pleasant to the ear. They are caused by regular periodic vibrations. A noise is a disagreeable sound, either because the vibrations producing it are not periodic, or because it is a mixture of dis- cordant elements, like the clapping of the hands. Musical sounds have three distinguish- ing characteristics : pitch, loudness, and quality. 216. Fitch. Mount on the axle of a whirling machine (Fig. 185), or on the armature of a small electric motor, a cardboard or metal disk D with a series of equidistant holes in a circle near its edge. While the disk is rotating rapidly, blow a stream of air through a small tube against the circle of holes. A distinct musical tone will be produced. If the experiment be repeated with the disk rotating more slowly, or with a circle of a smaller number of holes, the tone will be lower; if the disk is rotated more rapidly, the tone will be higher. FIGURE 185. SIREN. 192 SOUND The air passes through the holes in a succession of puffs producing waves in the air. These waves follow one another with definite rapidity, giving rise to the characteristic of sound called pitch. We conclude that the pitch of a musical sound depends only upon the number of pulses which reach the ear per second. To Galileo belongs the credit of first pointing out the relation of pitch to frequency of vibration. He illustrated it by drawing the edge of a card over ths milled edge of a coin. 217. Relation between Pitch, Wave Length, and Velocity. If a tuning fork makes 256 vibrations per second, and in that time a sound travels in air, at 20 C., a distance of 344m., then the first wave will be 344m. from the fork when it completes its 256th vibration. Hence, in 344 m., there will be 256 waves, and the length of each will be m., or 1.344 m. In general, then, wave length = velocit V , frequency or in symbols, I = -, v = nl, and n -. . (Equation 31) n I 218. Loudness. The loudness of a sound depends on the intensity of the vibrations transmitted to the ear. The energy of the vibrations is proportional to the square of their amplitude ; but since it is obviously impracticable to express a sensation in terms of a mathematical formula, it is sufficient to say that the loudness of a sound increases with the amplitude of vibration. As regards distance, geometrical considerations would go to show that the energy of sound waves in the open decreases as the square of the distance increases, but the actual decrease in the intensity of sound is even greater than this. The energy of sound waves is gradually dis- sipated by conversion into heat through friction and viscosity. Hermann von Helmholtz (1821-1894) was born at Potsdam. He received a medical education at Berlin and planned to be a specialist in diseases of the eye, ear, and throat. His studies soon revealed to him the need of a knowledge of physics and mathe- matics. To these subjects he gave his earnest attention and soon became one of the greatest physicists and mathematicians of the nineteenth century. He made important contributions to- all de- partments of physical science. He is the author of an important work on acoustics and is celebrated for his discoveries in this field. But perhaps his most useful contribution is that of the ophthalmoscope, an instrument of inestimable value to the oculist in examining the interior of the eye. QUALITY 193 The area of the vibrating body affects the loudness. This is illustrated in the piano, where strings of different diameters produce sounds differing in loudness. The thicker vibrating string sets more air in motion, and the wave has in consequence more energy. The less dense the. medium in which the vibration is set up, the feebler the sound. On a mountain top the report of a gun is comparable in loudness with that produced by the breaking of a stick at the base. The electric bell in a partially exhausted receiver ( 200) is nearly inaudible. Fill three large battery jars with coal gas, air, and carbonic acid respectively. Ring in them successively a small bell. There will be a marked difference in loudness. 219. Quality. Two notes of the same pitch and loud- ness, such as those of a piano and a violin, are yet clearly distinguishable by the ear. This distinction is expressed by the term quality or timbre. Helmholtz demonstrated that the quality of a note is determined by the presence of tones of higher pitch, whose frequencies are simple mul- tiples of that of the fundamental or lowest tone. These are known as overtones. The quality of sounds differs because of the series of overtones present in each case. Voices differ for this reason. Violins differ in sweetness of tone because the sounding boards of some bring out overtones different from those of others. Even the untrained ear can readily appreciate differences in the character of the music pro- duced by a flute and a cornet. Voice culture consists in training and developing the vocal organs and resonance cavities, to the end that purer overtones may be secured, and greater richness may by this means be imparted to the voice. 194 SOUND VII. INTERFERENCE AND BEATS 220. Interference. Hold a vibrating tuning fork over a cylin- drical jar adjusted as a resonator, and turn the fork on its axis until a position of minimum loudness is found. In this position cover one prong with a pasteboard tube without touching (Fig. 186). The sound will be restored to nearly maxi- mum loudness, because the paper cylinder cuts off the set of waves from the covered prong. It is well known that the loudness of the sound of a vibrat- ing fork held freely in the hand near the ear, and turned on its stem, exhibits marked variations. In four positions the sound is nearly inaudible. Let A, B (Fig. 187) be the ends of the two prongs. They vibrate with the same fre- quency, but in opposite direc- tions, as indicated by the arrows. \ When the two approach each other, a condensation is pro- duced between them, and at the same time rarefactions start from the backs at c and d. The condensations and rarefactions meet along the dotted lines of /' \ equilibrium, where partial ex- ' FIQURE 187 ._ lNTERPERENOE N tinction occurs, because a rare- FROM PRONGS OF TUNING FORK. FIGURE 186. INTERFERENCE. \ BEATS 195 faction nearly annuls a condensation. When the fork is held over the resonance jar so that one of these lines of interference runs into the jar, the paper cylinder cuts off one set of waves, and leaves the other to be reenforced by the air in the jar. Interference is the superposition of two similar sets of waves traversing the medium at the same time. One of the two sets of similar waves may be direct and the other reflected. If two sets of sound waves of equal length and amplitude meet in opposite phases, the condensation of one corresponding with the rarefaction of the other, the sound at the place of meeting is extinguished by interference. 221. Beats. Place near each other two large tuning forks of the same pitch and mounted on resonance boxes. When both are set vibrating, the sound is smooth, as if only one fork were sounding. Stick a small piece *of wax to a prong of one fork: this load increases its periodic time of vibration, and the sound given by the two is now pulsating or throbbing. Mount two organ pipes of the same pitch on a bellows, and sound them together. If they are open pipes, a card gradually slipped over the open end of one of them will change its pitch enough to bring out strong pulsations. With glass tubes and jet tubes set up the apparatus of Fig. 188. One tube is fitted with a paper slider so that its length may be varied. When the gas flame is turned down to the proper size, the tube gives a continu- ous sound known as a "singing flame." By making the tubes the same length, they may be made to yield the same note, the com- bined sound being smooth and steady. Now change the position FIGURE 188. INTERFERENCE WITH SINGING FLAMES. 196 SOUND of the slider, and the sound will throb and pulsate in a disagreeable manner. These experiments illustrate the interference of two sets of sound waves of slightly different period. The outbursts of sound, followed by short intervals of comparative silence, are called beats. Figure 189 illustrates the composition of two transverse waves of slightly different length. The addition of the FIGURE 189. INTERFERENCE OF Two TRANSVERSE WAVES. ordinates of the two waves ABO gives the wave A' B'C', with a minimum amplitude at B* . 222. Number of Beats. If two sounds are produced by forks, for example, making 100 and 110 vibrations per second respectively, then in each second the latter fork gains ten vibrations on the former. There must be ten times during each second when they are vibrating in the same phase, and ten times in opposite phase. Hence, in- terference of sound must occur ten times a second, and ten beats are produced. Therefore, the number of beats per second is equal to the difference of the vibration rates (frequencies) of the two sounds. VIII. MUSICAL SCALES 223. Musical Intervals. A musical interval is the rela- tion between two notes expressed as the ratio of their frequencies of vibration. Many of these intervals have THE MAJOR DIATONIC SCALE 197 names in music. When the ratio is 1, the interval is called unison ; 2, an octave ; |, a fifth ; %, a fourth ; etc. Any three notes whose frequencies are as 4:5:6 form a major triad, and alone or together with the octave of the lowest note, a major chord. Any three notes whose fre- quencies are as 10 : 12: 15 form a minor triad, and alone or with the octave of the lowest, a minor chord. Mount the disk of Fig. 190 on the whirling table of Fig. 185. The disk is perforated with four circles of equidistant holes, numbering 24, 30, 36, and 48 respectively. These are in the relation of 4, 5, 6, 8. Rotate with uniform speed, and beginning with the inner circle, blow a stream of air against each row of holes in succession. The tones produced will be recognized as do, mi, sol, do', forming a major chord. If now the speed of rotation be increased, each note will rise in pitch, but the musical sequence will remain the same. FlGURE 190. DISK FOR MAJOR CHORD. It will be seen from the fore- going relations that harmonious musical intervals consist of very simple vibration ratios. 224. The Major Diatonic Scale. A musical scale is a succession of notes by which musical composition ascends from one note, called the keynote, to its octave. This last note in one scale is regarded as the keynote of another series of eight notes with the same succession of intervals. In this way the series is extended until the limit of pitch established in music is reached. The common succession of eight notes, called the major diatonic scale, was adopted about three hundred and fifty years ago. The octave beginning with middle is written : / 1 g' : & : d"[::4:5:6 /' : ' : *"J The frequency universally assigned to c' in physics is 256. It is convenient because it is a power of 2, and it is practically that of the "middle <7" of the piano If c' is due to 256, or m, vibrations per second, the frequency of the other notes of the diatonic scale may be found by proportion from the three triads above; they are as follows : 256 288 -320 341 384 426f 480 512 c 1 d' e' f g' a f b f c" do re mi fa sol la si do m f m 4m 4m \m & m -^ m 2m O t o A o o If the fractions representing the relative frequencies be reduced to a common denominator, the numerators may be taken to denote the relative frequencies of the eight notes of the scale. They are 24 27 30 32 36 40 45 48 An examination of these numbers will show that there are only three intervals from any note to the next higher. They are |, a major tone ; - a g-, a minor tone ; and if, a half tone. The order is f , J, If, f , ^-, f , if. 225. The Tempered Scale. If were always the key- note, the diatonic scale would be sufficient for all purposes except for minor chords ; but if some other note be chosen for the keynote, in order to maintain the same order of intervals, new and intermediate notes will have to be in- troduced. For example, let D be chosen for the key- LIMITS OF PITCH 199 note, then the next note will be 288 x f = 324 vibrations, a number differing slightly from E. Again, 324 x *- = 360, a note differing widely from any note in the series. In like manner, if other notes are taken as keynotes, and a scale is built up with the order of intervals of the dia- tonic scale, many more new notes will be needed. This interpolation of notes for both the major and minor scales would increase the number in the octave to seventy-two. In instruments with fixed keys such a number is un- manageable, and it becomes necessary to reduce the num- ber by changing the value of the intervals. Such a modi- fication of the notes is called tempering. Of the several methods proposed by musicians, that of equal temperament is the one generally adopted. It makes all the intervals from note to note equal, interpolates one note in each whole tone of the diatonic scale, and thus reduces the number of intervals in the octave to twelve. BE IKS The only accurately ^^ tuned interval in this scale is the octave ; all the others are more or less modified. The fol - F]OURE m _ SCALE op c lowing table shows the differences between the diatonic and the equally tempered scales : c' d' e' f g r a' V c" Diatonic . . . 256 288 320 341.3 384 426.7 480 512 Tempered. . .256 287.3 322.5 341.7 383.6 430.5 483.3 512 Figure 191 illustrates the scale of on the staff and the keyboard. 226. Limits of Pitch. The international pitch, now in general use in Europe and America, assigns to a r the vi- uTe'l/Tg"T 200 SOUND bration frequency of 435. But some orchestras have adopted 440 vibrations for a' '. In the modern piano of seven octaves the bass A has a frequency of about 27.5, the highest A, 3480. The lowest note of the organ is the O of 16 vibrations per second ; the highest note is the same as the highest note of the piano, the third octave above a 1 ', with a frequency of 3480. The limits of hearing far exceed those of music. The range of audible sounds is about eleven octaves, or from the O of 16 vibrations to that of 32,768, though many persons of good hearing perceive nothing above a fre- quency of 16,384, an octave lower. Questions and Problems 1. Why is the pitch of the sounds given by a phonograph raised by increasing the speed of the cylinder or the disk containing the record? 2. A megaphone or a speaking tube makes a sound louder at a distance. Explain why. 3. The teeth of a circular saw give a note of high pitch when they first strike a plank. Why does the pitch fall when the plank is pushed further against the saw ? 4. Miners entombed by a fall of rock or by an explosion have signaled by taps on a pipe or by pounding on the rock. How does the sound reach the surface ? 5. Two Rookwood vases in the form of pitchers with slender necks give musical sounds when one blows across their mouth. Why does the larger one give a note of lower pitch than the smaller? 6. What note is made by three times as many vibrations as c f (middle C) ? 7. If c' is due to 256 vibrations per second, what is the frequency of g" in the next octave ? 8. What is the wave length of g' when sound travels 1130 feet per second? 9. If c' has 264 vibrations per second, how many has a'? 10. When sound travels 1120 ft. per second, the wave length of the note given by a fork was 3.5 ft. What was the pitch of the fork ? LAWS OF STRINGS 201 IX. VIBRATION OF STRINGS 227. Manner of Vibration. When strings are used to produce sound, they are fastened at their ends, stretched to the proper tension, and are made to vibrate transversely by drawing a bow across them, striking with a light ham- mer as in the piano, or plucking with the fingers as in the banjo, guitar, or harp. 228. The Sonometer. The sonometer is an instrument for the study of the laws governing the vibration of FIGURE 192. SONOMETER. strings. It consists of a thin wooden box, across which are stretched violin strings or thin piano wires (Fig. 192). The wires pass over fixed bridges, A and B, near the ends, and are stretched by tension balances at one end. They may be shortened by movable bridges (7, sliding along scales under the wires. 229. Laws of Strings.. Stretch two similar wires on the so- nometer and tune to unison by varying the tension. Shorten one of them by moving the bridge C to f, f , f , f , etc. The successive inter- vals between the notes given by the two wires will be f, f, $, f, etc. The notes given by the wire of variable length are those of the major diatonic scale. Hence, The frequency of vibration for a given tension varies inversely as the length. Starting with a given tension and the strings or wires in unison, increase the stretching force on one of them four times ; it will now give the octave of the other with twice the frequency. Increase the 202 SOUND tension nine times ; it will give the octave plus the fifth, or the twelfth, above the other with three times the frequency. These statements may be verified by dividing the comparison wire by a bridge into halves and thirds, so as to put it in unison with the wire of variable tension. Hence, When the length is constant, the frequency varies as the square root of the tension. Stretch equally two wires differing in diameter and material, that is, in mass per unit length. Bring them to unison with the movable bridge. The ratio of their lengths will be inversely as that of the square roots of the masses per unit length. Hence, The length and tension being constant, the frequency varies inversely as the square root of the mass per unit length. 230. Applications. In the piano, violin, harp, and other stringed instruments, the pitch of each string is determined partly by its length, partly by its tension, and partly by its size or the mass of fine wire wrapped around it. The tuning is done by varying the tension. 231. Fundamental Tone. Fasten one end of a silk cord about a meter long to one prong of a large tuning fork, arid wrap the other end around a wooden pin in- serted in an upright bar in such a way tha* tension can be applied to the cord by turning the pin. Set the fork vibrating, and adjust FIGURE 193. FUNDAMENTAL OF A STRING, the tension until the cord vibrates as a whole (Fig. 193). Arranged in this way, the frequency of the fork is double that of the cord. The experiment shows the way a string or wire vibrates when giving its lowest or fundamental tone. A body NODES AND SEGMENTS 203 yields its fundamental tone when vibrating as a whole, or in the smallest number of segments possible 232. Nodes and Segments. With a silk cord about 2 m. long, and mounted as in the last experiment, adjust the tension until the cord vibrates in a number of parts, giving the appearance of a succession of spindles of equal length (Fig. 194). The frequency of the fork is twice that of each spindle. Stretch a wire on a sonom- eter with a thin slip of cork strung on it. Place the cork at one third, one fourth, one fifth, or one sixth part of the wire from one end ; touch it lightly, and bow the shorter portion of the wire. The wire will vibrate in equal segments (Fig. 195). The division into segments may be made more conspicu- ous by placing on the wire, before bowing it, narrow V-shaped pieces of paper, or riders. If, for example, the cork is placed at one fourth FIGURE 194. STRING VIBRATING IN SEG- MENTS. FIGURE 195. WIRE VIBRATING IN SEGMENTS. the length of the wire, the paper riders should be in the middle, and at one fourth the length from the other end, and at points midway be- tween these. When the wire is deftly bowed, the riders at the fourths will remain seated, and the intermediate ones will be thrown off. The latter mark points of maximum, and the former those of mini- mum vibration. The ends of a wire and the intermediate points of least motion are called nodes ; the vibrating portions between 204 SOUND the nodes are loops or segments; and the middle points of the loops are called antinodes. The last two experiments illustrate what are known as stationary waves. They result from the interference of the direct system of waves and those reflected from the fixed end of the wire. At the nodes the two meet in opposite phase; at the anti- nodes in the same phase. At the former the motion is re- duced to a minimum ; at the latter it rises to a maximum. 233. Overtones in Strings. Stretch two similar wires on the sonometer and tune to unison; then place a movable bridge at the middle of one of them. Set the longer wire in vibration by pluck- ing or bowing it near one end. FIGURE 196. - FUNDAMENTAL AND Oc- The tone mogt distinctl heard is TAVE TOGETHER. J its fundamental. Touch the wire lightly at its middle point ; instead of stopping the sound, a tone is now heard in unison with that given by the shorter wire, that is, an octave higher than the fundamental and caused by the longer wire vibrating in halves (Fig. 196). If the wire be again plucked, both the funda- mental and the octave may be heard together. Touching the wire one third from the end brings out a tone in FlGURE 19 ~ FUNDAMENTAL AND Oc- ... ,, J . , ,, TAVE PLUS FIFTH TOGETHER. unison with that given by the second wire reduced to one third its length by the movable bridge, that is, it yields a tone of three times the frequency, or an octave and a fifth higher than the fundamental. Figure 197 illustrates the man- ner in which the wire is vibrating. The experiment shows that a wire may vibrate not only as a whole but at the same time in parts, yielding a com- plex note. The tones produced by a body vibrating in parts are called overtones or partial tones. 234. Harmonics. If the frequency of vibration of the overtone is an exact multiple of the fundamental, it is called an harmonic partial or simply an harmonic. In AIR AH A SOURCE OF SOUND 205 strings the overtones are usually harmonics, but in vibrat- ing plates and membranes they are not. The harmonics are named first, second, third, etc., in the order of their vibration frequency. The frequency of any particular harmonic is found by multiplying that of the fundamental by a number one greater than the number of the harmonic. For example, the frequency of the first harmonic of c 1 of 256 vibrations per second is 256 X 2 = 512 ; that of the second is 256 x 3 = 768, etc. X. VIBRATION OF AIR IN PIPES 235. Air as a Source of Sound. In the use of the res- onator we saw that air may be thrown into vibration when FIGURE 198. CLARINET. it is confined in tubes or globes, and that it thus becomes the source of sound. Such a body of air may be set FIGURE 199. FLUTE. vibrating in two ways; by a vibrating tongue or reed, as in the clarinet (Fig. 198), the fish horn, etc., or by a FIGURE 200. TROMBONE. stream of air striking against the edge of an opening in the tube, as in the whistle, the flute (Fig. 199), the organ pipe, etc. In several pipe or wind instruments the lips 206 SOUND of the player act as reeds, as in the trumpet, trombone (Fig. 200), the French horn, and the cornet. Wind instruments may be classed as open or stopped pipes, according as the end remote from the mouthpiece is open or closed. 236. Fundamental of a Closed Pipe. Let the tall jar of Fig. 201 be slowly filled with water until it responds strongly to a c' fork, for example. The length of the column of air will be about 13 in. or one fourth of the wave length of the note. When the prong at a moves to b, it makes half a vibration, and generates half a sound wave. It sends a condensed pulse down the tube AB, and this pulse is reflected from the water at the bot- tom. Now, if AB is one fourth a wave length, the distance down and back is one half a wave length, and the pulse will return to A at the instant when the prong begins to move from b back to a, and to send a rarefaction down AB. This in turn will FIGURE ^201. - run d own the tube and back, as the prong com- pletes its vibration ; the co- vibration is then re- peated indefinitely, the tube responds to the fork, and its length is one quarter of the wave length. Hence, The fundamental of a closed pipe is a note whose wave length is four times the length of the pipe. 237. Laws for Columns of Air. Set vertically in a wooden base eight glass tubes each about 25 cm. long and 2 cm. in diameter (Fig. 202). Pour in them melted paraffin to close the bottom. A musical note may be produced by blowing a stream of air across the top of each tube. From the confused flutter made by the air striking the edge of the tube, the column of air selects for reenforcement the frequency corresponding to its own rate. Hence the pitch may be varied by pouring in water. Adjust all the tubes with water until they give the eight notes of the major diatonic scale . The measured lengths of the columns of air will be found to be nearly as 1, f, |, $, f, f, &, . STATE OF THE AIR IN A SOUNDING PIPE 207 The notes emitted have the frequencies 1, f , f , f, f, f, V> 2 ( 224). Hence, The frequency of a vi- brating column of air is inversely as its length. This is the principle employed in playing the trombone. FIGURE 202. PIPES FOR NOTES OF MAJOR DIATONIC SCALE. Blow gently across the end of an open tube 30 cm. long and about 2 cm. in diameter arid note the pitch. Take another tube of the same diameter and 15 cm. long ; stop one end by pressing it against the palm of the hand, and sound it by blowing across the open end. The pitch of the closed pipe will be the same as that of the open one. The experiment may be varied by comparing the notes ob- tained by the shorter pipe when open and when closed at one end; the former will be an octave higher than the latter. Hence, For the same frequency, the open pipe is twice the length of the stopped one. The length of the open pipe is, therefore, half the wave length of the fundamental note in air. 238. State of the Air in a Sounding Pipe. Employing an open organ pipe, preferably with one glass side (Fig. 203), lower into it a miniature tambourine about 3 cm. in diameter and covered with fine sand, while the pipe is sounding its fundamental note. The sand will be agitated most at the ends of the pipe and very little IGIJ ^ at the middle. There is, therefore, a node at the middle NODE AT ^ an P en Pi? 6 - -A. node is a place of least motion and MIDDLE OF greatest change of density; an antinode is a place of PIPE. greatest motion and least change of density. The closed *208 SOUND end of a pipe is necessarily a node, and the open end an antinode Hence, In an open pipe, for the fundamental tone, there is a node at the middle and an antinode at each end; in the stopped pipe, there is a node at the closed end and an antinode at the other end. 239. Overtones in Pipes. Blow across the open end of a glass tube about 75 cm. long and 2 cm. in diameter. A variety of tones of higher pitch than the fundamental may be obtained by varying the force of the stream of air. These tones of higher pitch than the fundamental are overtones ; they are caused by the column of air vibrating in parts or segments with intervening nodes. Open pipes give the complete series of overtones, with fre- quencies 2, 3, 4> 5, etc. times that of the fundamental. In stopped pipes only those overtones are possible whose frequencies are 3, 5, 7, etc. times that of the fundamental. Briefly, the reason is that with a node at one end and an antinode at the other, the column of air can divide into an odd number of equal half segments only. It follows that the notes given by open pipes differ in quality from those of closed pipes. XI. GRAPHIC AND OPTICAL METHODS 240. Record of Vibrations. Graphic methods of study- ing sound are of service in determining the frequency of vibration. Figure 204 shows a practical device for this purpose. A sheet of paper is wrapped around a metal cylinder, and is then smoked with lampblack. A large fork is securely mounted, so that. a light style attached to one prong touches the paper lightly. The cylinder is MANOMETRIC FLAMES 209 mounted on an axis, one end of which has a screw thread cut in it, so that when the cylinder turns it also moves in the direction of its axis. The beats of a seconds pendulum may be marked on the paper by means of electric sparks between the style and the cylinder. The number of waves be- tween successive marks made by the spark is equal to the frequency FIGURE 204. INSCRIBING THE VIBRATIONS of the fork. OF A FORK - 241. Manometric Flames. A square box with mirror faces is mounted so as to turn around a vertical axis (Fig. 205). In front of the revolving mirrors is sup- ported a short cylinder A., which is divided into two shallow chambers by a partition of gold- beater's skin or thin rub- ber. Illuminating gas is admitted to the com- partment on the right through the tube with a stop-cock, and burns at the small gas jet on the little tube running into FIGURE 205. MANOMETRIC FLAME AP- this same compartment. PARATUS. rr(1 The speaking tube is connected to the compartment on the other side of the flexible partition. 210 SOUND FIGURE 206. MANOMETRIC FLAMES. ,, now of gas to the burner. The flame changes shape and flickers, but its vibrations are too rapid to be seen directly. But if it is examined by reflection from the rotating mirrors, its image is a serrated band (Fig. 206). Koenig fitted three of these little cap- sules with jets to the side of an open organ pipe (Fig. 207), the membrane on the inner side of the gas chamber forming part of the wall of the pipe. When the pipe is blown so as to sound its fundamental tone, the middle point is a node with the great- est variations of pressure in the pipe, and the flame at that point is. more violently agitated than at the other two, giving in the mirrors the top band of Fig. 206. By increasing the air blast, the fundamental is made to give way to the first overtone ; the two outside jets then vibrate most strongly, and give the second band in the figure, with twice as many tongues of flame When the mirrors are turned, the image of the gas jet is drawn out into a smooth band of light. Any pure tone at the mouthpiece produces alter- nate compressions and rare- factions in both chambers separated by the mem- brane, and these aid and retard the FIGURE 207. ORGAN PIPE WITH GAS FLAMES. THE PHONODEIK 211 as in the image for the fundamental. The third band may be obtained by adjusting the air pressure so that both the fundamental and the first overtone are produced at the same time. This same figure may be obtained by singing into the mouthpiece or funnel of Fig. 205 the vowel sound o on the note B, showing that this vowel sound is composed of a funda- FlGURE 2 08. THE PHONODEIK. mental and its octave. 242. The Phonodeik is an instrument devised by Professor Dayton C. Miller to exhibit sound waves. It consists of a very small and thin glass mirror mounted on a minute steel spindle resting in jeweled bearings. On this spindle is a little pulley around which wraps a fine thread. One end of the thread is attached to a very thin glass diaphragm closing the small end of a resonator horn ; the other is connected to a delicate tension spring (Fig. 208). A small pencil of light is focused on the mirror by a lens and is reflected by the mirror to a sensitized film moving at right angles to it. Any vibration of the diaphragm traces on the film a wave form marked with all the perculiari- FlGURE 210. WAVE FORM OF . , _ j . VIOLIN TONE. ties of the sound producing the vibrations of the diaphragm. These photographs are af- terwards enlarged. Fig. 209 shows the wave form caused by a heavy tuning fork. Fig. 210 represents the wave of FIGURE 209. WAVE FORM FROM TUNING FORK. 212 SOUND a violin tone, the irregulari- ties marking the overtones. Fig. 211 is the wave form of the sound of the human voice FIGURE 211. WAVE FORM OF . VOICE. saying " ah. Questions and Problems 1. Name three ways in which musical sounds may differ. 2. Pianos are made so that the hammers strike the wires near one end and not in the middle. Why? 3. Why does the pitdh of the sound made by pouring water into a tall cylindrical jar rise as the jar fills? 4. What effect does a rise of temperature have on the pitch of a given organ pipe ? Explain. 5. If the pipes of an organ are correctly tuned at a temperature of 40 F., will they still be in tune at 90 F. ? Explain. 6. The tones of three bells form a major triad. One of them gives a note a of 220 vibrations per second, and its pitch is between those of the other two. What are the frequencies of three bells, and what is the note given by the highest? 7. How much must the tension of a violin string be increased to raise its pitch a fifth ( 223) ? 8. If the E string of a violin is 40 cm. long, how long must a similar one be to give G ? 9. The vibration frequency of two similar wires 100 cm. long is 297. How many beats per second will be given by the two wires when one of them is shortened one centimeter ? 10. Two c' forks gave 5 beats per second when one of them was weighted with bits of sealing wax. Find the frequency of the weighted fork. 11. What will be the length of a stopped organ pipe to give c' of 256 vibrations per second when the temperature of the air is 20 C.? 12. Calculate the length of an open organ pipe whose fundamental tone is one of 32 vibrations per second, and the temperature of the air is20C. QUESTIONS AND PROBLEMS 213 13. An open organ pipe sounds c' (256) ; what notes are its two lowest overtones? 14. What is the frequency of an 8-foot stopped pipe when the velocity of sound is 1120 ft. per second? 15. Two open organ pipes 2 ft. in length are blown with air at a temperature of 15 and 20 C., respectively. How many beats do they give per second? 16. When the temperature of the air is such that the velocity of sound is 1105 ft. per second, what will be the frequency of the funda- mental note produced by blowing across one end of a tube 12.75 in. long, the other end being closed? What will be the frequency of its first overtone? CHAPTER VIII LIGHT I. NATURE AND TRANSMISSION OF LIGHT 243. The Ether. Exhaust the air as far as possible from a glass bell jar. Place a candle on the far side of the jar ; it will be seen as clearly before the air has been let into the bell jar as after. It is obvious that the medium conveying light is not the air and it must be something that exists even in a vacuum. This medium is vaguely known as the ether. It exists everywhere, even penetrating between the molecules of ordinary matter. 244. Light. The prevailing view about the nature of light is that it is a transverse wave motion in the ether. Huyghens, a Dutch physicist, in 1678 proposed the theory that light is a wave motion; later, Fresnel, a French physicist, showed that the disturbance must be transverse ; finally Maxwell modified the theory to the effect that these disturbances are probably not transverse physical movements of the ether, but transverse alterations in its electrical and magnetic conditions. 245. Transparent and Opaque Bodies. When light falls on a body, in general, a part of it is reflected, a part passes through or is transmitted, and the rest is absorbed. A body is transparent when it allows light to pass through it with so little loss that objects can be easily distinguished through it, as in the case of clear glass, air, pure water. Translucent bodies transmit light, but so imperfectly that 214 NIAGARA FALLS POWER PLANT. Used for light and power in several cities of New York State. SPEED OF LIGHT 215 objects cannot be seen distinctly through them, as horn, oiled paper, very thin sheets of metal or wood. Other bodies, such as blocks of wood or iron, transmit no light, and these are opaque. No sharp line of separation between these classes can be drawn. The degree of transparency or opacity depends on the nature of the body, its thick- ness, and the wave length ( 310) of the light. Water when deep enough cuts off all light; the bottom of the deep ocean is dark. Stars invisible at the foot of a moun- tain are often visible at the top ; bodies opaque to light of one wave length are often transparent to light of a differ- ent wave length. 246. Speed of Light. Previous to the year 1676 it was believed that light traveled infinitely fast, because no one had found a way to measure so great a velocity. But in that year Roemer,ayoung Danish astron- omer, made the very important discovery that light travels with finite speed. Roemer was en- gaged at the Paris Observa- tory in observing the eclipses of the inner moon of the planet Jupiter. At each revolution of the moon M (Fig. 212) in its orbit around the planet J, it passes into the shadow of the planet and becomes invisible from the earth at E, or is eclipsed. By comparing his observations with FIGURE 212. SPEED OF LIGHT FROM JUPITER'S INNER MOON. 216 LIGHT much earlier recorded ones, Roemer found that the mean in- terval of time between two successive eclipses was 42.5 hours. From this it was easy to calculate in advance the time at which succeeding eclipses would occur. But when the earth was going directly away from Jupiter, as at U v the eclipse interval was found to be longer than anywhere else ; and at 27 2 , across the earth's orbit from Jupiter, each eclipse occurred about 1000 sec. later than the predicted time. To account for this difference Roemer advanced the theory that this interval of 1000 sec. is the time taken by light to pass across the diameter of the earth's orbit. This gave for the speed of light 309 million meters, or 192,000 miles per second. Later determinations in our own country by Michelson and Newcomb show that the speed of light is 299,877 km., or 186,337 mi. per second. 247. Direction of Propagation. Place a sheet-iron cylinder over a strong light, such as a Welsbach gas lamp, in a darkened room. The cylinder should have a small hole opposite the light. Stretch a heavy white thread in the light streaming through the aperture. When the thread is taut it is visible throughout its entire length, but if permitted to sag it becomes invisible. The experiment shows that light travels in straight lines. It will appear later that this is true only when the medium through which light passes has the same physical proper- ties in all directions. 248. Bay, Beam, Pencil. Light is propagated outward from the luminous source in concentric spherical waves, as sound waves in air from a sonorous body. Rays are the radii of these spherical waves, and they are, therefore, normal (perpendicular) to them. They mark the direc- tion of propagation. When the source of light is at a great distance, the rays SHADOWS 217 incident on any surface are sensibly parallel. A number of parallel rays form a beam of light. For example, in the case of light from the sun or stars, the distance is so great that the rays are sensibly parallel. Rays of light pro- ceeding outward from a point form a diverging pencil; rays proceeding toward a point, a converging pencil. 249. Shadows. Place a ball between a lighted lamp and a white screen. From a part of this screen the light will be wholly cut off, and surrounding this area is one from which the light is excluded in part. If three small holes be made in the screen, one where it is darkest, one in the part where it is less dark, and one in the lightest part, it will be found when one looks through them that the flame of the lamp is wholly invisible through the first, a part of it is visible through the second, and the whole flame through the third. The space behind the opaque object from which the light is excluded is called the shadow. The figure on the screen is a section of the shadow. The darkest part of the shadow, called the umbra, is caused by the total exclusion of the light by the opaque object ; the lighter part, caused by its partial exclusion, is called the penumbra. p FIGURE 213. SOURCE OF LIGHT A POINT. When the source of light is a point L (Fig. 213), the shadow will be bounded by a cone of rays, ALB, tangent to the object, and will have only one part, the umbra. When the source of light is an area, such as LL (Fig. 214), the space ABDO behind the opaque body receives no light, and the parts between AC and AQ' , and between BD and 218 LIGHT BD { r , receive some light, the amount increasing as AC' and BD 1 are approached. From these figures the cases when the luminous body is larger than the opaque body, FIGURE 214. SOURCE OF LIGHT AN AREA. and when it is of the same size, may be understood and illustrated by the student. 250. Images by Small Openings. Support two sheets of card- board (Fig. 215) in vertical and parallel planes. In the center of one cut a hole H about 2mm. square and in front of it place a lighted candle or lamp. An inverted image of the flame will appear on the other sheet if the room is dark. The area of the image will vary with any FIGURE 215. IMAGE BY SMALL OPENING. chan e in the Position of the screen or candle, the bright- ness with the size of the aperture, but no change in the shape of the aperture, affects the image. With a larger aperture the image gains in brightness but loses in definition. Every point of the candle flame is the vertex of a cone of rays, or a diverging pencil, passing through the opening and forming an image of it on the screen. These numer- ous pictures of the opening overlap and form a picture of the flame, and the number at any one place determines the brightness. The edge of the image will therefore be less bright than other portions. In the case of a large LAW OF INTENSITY 219 opening, the overlapping of the images of the aperture destroys all resemblance between the image and the object, the resulting image having the shape of the aperture. 251. Illustrations. The pinhole camera is an applica- tion of the foregoing principle. It consists of a small box, blackened within, and provided with a small opening in one face (Fig. 21*6) ; the light passes through this and forms an image on the sensitized plate placed on the oppo- FIGURE 216. PINHOLE CAMERA. site side. When the sun shines through the small chinks in the foliage of a tree, a number of round or oval spots of light may be seen on the ground. These are images of the sun. During a partial solar eclipse such figures as- sume a crescent shape. II. PHOTOMETRY 252. Law of Intensity. The intensity of illumination is the quantity of light received on a unit of surface. Every- day experience shows that it varies, not only with the source of the light, but also with the distance at which the source is placed. Cut three cardboard squares, 4, 8, and 12 cm. on a side respectively, and mount them on supports (Fig. 217). The centers of these screens should be at the same distance above the table as the source of light. 220 LIGHT Use a Welsbach gas lamp with an opaque chimney having a small opening opposite the center of the light, and set it 99 cm. from the largest screen. Place the medium-sized screen so that it exactly cuts off the light from the edges of the largest. In like manner place the smallest screen with respect to the intermediate one. If these screens are placed with care, it will be found that their distances from the light are 33, 66, and 99 cm. respectively, or as 1:2:3. Now as each screen exactly cuts off the light from the one next farther away, it FIGURE 217. LAW OF INTENSITY OF ILLUMINATION. follows that each receives the same amount of light from the source when the light is not intercepted. The surfaces of the screens are as 1:4:9, and hence the quantity of light per unit of surface must be inversely as 1 : 4 : 9, the square of 1, 2, and 3 respectively. This experiment shows that the intensity of illumination varies inversely as the square of the distance from the source of light. If the medium is such as to absorb some of the light, the decrease in intensity is greater than that ex- pressed by the law of inverse squares. This law of illumination assumes that the source of light is a point, and that the receiving surface is at right angles to the direction of the rays. When the surface on which the light (and heat) falls is inclined, the intensity is still less. In northern latitudes the earth is nearer the sun in winter than in summer, but the intensity of the radiation received is less than in summer, because the alti- THE BUN SEN PHOTOMETER 221 tude of the sun at noon is less, that is, because the earth's surface is more inclined to the direction of the radiations. 253. The Bunsen Photometer. A photometer is an instru- ment for comparing the intensity of one light with that of another. The principle applied is a consequence of the law of the intensity of illumination ; it is that the ratio of the intensities of two lights is equal to the square of the ratio of the distances at which they give equal illumination. In the Bunsen photometer a screen of paper A (Fig. 218), having a translucent spot made by applying a little FIGURE 218. BUNSEN PHOTOMETER. hot paraffin, is supported on a graduated bar between a standard candle B and the light to be compared with it. An old but imperfect standard candle is the light emitted by the sperm candle of the size known as " sixes," when burning 120 grains per hour. The photometer screen is usually inclosed in a box open toward the two lights, and back of it are two mirrors placed with their reflecting sides toward each other in the form of a V, so that the observer standing by the side of A can see both sides of the screen by reflection in the mirrors. The 222 LIGHT position of A or of B may then be adjusted until both sides of the screen look alike. Then the intensity of is to the intensity of B as AC 2 is to A?. In the Joly photometer two rectangular blocks of paraffin, separated by a sheet of tinfoil, take the place of the sheet of paper. When the lights are balanced the edges of the paraffin blocks are equally lighted. Questions and Problems 1. What is the cause of an eclipse of the sun ? Explain by diagram 2. What is the cause of an eclipse of the moon ? Explain by dia- gram. 3. Why does a small aperture in the camera give a more sharply denned image than a large one? 4. Why is a larger aperture in the camera necessary for a snapshot than for a time exposure? 5. In an attempt to determine the height of a tree the following data were obtained : Length of the tree's shadow, 50 ft. ; length of the shadow of a vertical 10-f t. pole, 4 ft. What is the height of the tree 4 6. Two lights, 25 and 100 c.p. respectively, are placed 60 ft. apart. Where must a screen be placed between them and on the line joining them so as to be equally illuminated on its two sides? 7. In measuring the candle power of a lamp the following data were obtained : Distance of the standard lamp from the photometer disk, 20 cm. ; distance of lamp, 120 cm. What is the candle power? 8. If a book can be read at a distance of 1 ft. from a 20 c.p. electric lamp, at what distance from a 60 c.p. lamp can it be read with equal clearness ? 9. The picture of a tree taken with a pinhole camera was 10 cm. long. The aperture was 20 cm. from the sensitive plate and 30 m. from the tree. What is the height of the tree ? 10. Two Mazda lamps are to be used to give equal illumination to the two sides of a screen. One of them is 20 c.p. and distant 8 ft. from the screen ; the other is 40 c.p. How far from the screen must the second lamp be placed to secure the desired illumination ? LAW OF REFLECTION 223 11. What is the length of the umbra of the earth's shadow, the diameter of the earth arid sun being 8000 and 880,000 miles respec- tively, and the distance from the center of the earth to that of the sun being 93,000,000 miles ? III. EEFLECTION OF LIGHT 254. Regular Reflection. When a beam of light falls on a polished plane surface, the greater part of it is re- flected in a definite di- rection. This reflec- tion is known as regu- lar reflection. In Fig. 219 a beam of light IB is incident on the plane mirror B and is re- flected as BE. IB is the incident beam, BR is the reflected beam, the angle IBP between the incident beam and the normal (perpendicular) to the reflecting surface is the angle of incidence, and the angle PBR between the reflected beam and the normal is the angle of reflection. 255. Law of Reflection. On a semicircular board are mounted two arms, pivoted at the center of the arc (Fig. 220). One arm carries a vertical rod P, and the other a paper tube T with parallel threads stretched across a diameter at each end. A plane mirror Mis mounted at the center of the semicircle, with its reflecting surface parallel to the di- ameter at the ends FIGURE 220. LAW OF REFLECTION. of the arc. On the FIGURE 219. INCIDENCE AND REFLECTION. 224 LIGHT edge of the semicircle is a scale of equal parts with the zero on the normal to the mirror. Place the arm P in any desired position and move the arm T until the image of the rod in the mirror is exactly in line with the two threads. The scale readings will show that the two arms make equal angles with the normal to the mirror. Hence, The angle of reflection is equal to the angle of inci- dence; and the incident ray, the normal, and the reflected ray all lie in the same plane. 256. Diffused Reflection. Cover a large glass jar with a piece of cardboard, in which is a hole about 1 cm. in diameter. Fill the jar with smoke, and reflect into it through the hole in the cover a beam of sunlight. The whole of the interior of the jar will be illuminated. The small particles of smoke floating in the jar furnish a great many reflecting surfaces ; the light falling on them is reflected in as many directions. The scattering of light by uneven or irregular surfaces is diffused reflection. To a greater or less extent all reflecting surfaces scat- ter light in the same way as the smoke particles. Figure 221 illustrates in an exaggerated way the difference between a perfectly smooth surface and one FIGURE 221. REGULAR AND DIFFUSED REFLEC- somewhat uneven. TION - It is by diffused re- flection that objects become visible to us. Perfect reflec- tors would be invisible ; it is almost impossible to see the glass of a very perfectly polished mirror. The trees, the ground, the grass, and particles floating in the air reflect the light from the sun in every direction, and thus fill the space about us with light. If the air were free from all floating particles and gases, the sky would be dark in all IMAGE IN A PLANE MIRROR 225 directions, except in the direction of the sun and the stars. This conclusion is confirmed by aeronauts who have reached very high altitudes, where there was almost a complete ab- sence of floating particles. 257. Image in a Plane Mirror. Any smooth reflecting surface is called a mirror. A plane mirror is one whose reflecting surface is a plane. A spherical mirror is one whose reflecting surface is a portion of a sphere. Support a pane of clear window glass in a vertical position, and place a red-colored lighted candle back of it. Place a white un- lighted candle in front. Move the unlighted candle until its image in the glass as a mirror coincides exactly with the lighted candle seen through the glass. The distance of the two candles from the mirror will be the same. Let A be a luminous point in front of a plane mirror 222). The group of waves included between the rays AB and AC after reflection proceed as if from A r , situated on the normal AK and as far behind the reflecting surface as A is in front of it. An eye placed at DE receives these waves as if they came directly from a source A 1 . The point A 1 is called the image of A in the mirror MN. It is known as a virtual image, because the light only appears to come from it. Therefore, the image of a point in a plane mirror is virtual, and is as far back of the mirror as the point is in front. The image may be found by drawing A FIGURE 222. POSITION OF IMAGE OF A POINT. 226 LIGHT FIGURE 223. CONSTRUCTION IMAGE. from the point a perpendicular to the mirror, and pro- ducing the perpendicular until its length is doubled. 258. Construction for an Image in a Plane Mirror. As the image of an object is composed of the images of its points, the image may be located by finding those of its points. Let AB (Fig. 223) represent an object in front of the plane mirror MN. Draw perpendiculars from A and B to the mirror and produce them until their length is doubled. AB' is the image of AB. It is virtual, erect, and of the same size as the object. An image in a plane mirror is reversed from right to left. This is clearly seen when a printed page is held in front of a mirror, the letters all being reversed, or perverted, as it is termed. Otherwise the image is so like the object that illusions are produced, because a well-polished mirror itself is invisible. In general, the image in a plane mirror is the same size as the object, is virtual, and is as far back of the mirror as the object is in front. 259. Path of the Rays to the Eye. It is important to notice that the image of any fixed object is fixed in space, and is entirely independent of the position of the observer. The paths of the rays for the image for one observer are not the same as those for another. Let AB (Fig. 224) represent an object in front of the plane mirror MN. Drop perpendiculars from points of the object to the mirror, and produce them until their length is doubled. In this manner the image of AB is found at A' B' . Let E and E 1 be the position of the eye USES OF A PLANE MIRROR 227 FIGURE 224. PATH OF RAYS TO THE EYE. for two observers. To find the path of the rays entering the eye at E, draw lines from A 1 and B' to E. These lines are the directions in which the light enters the eye from A 1 and B 1 respectively. But no light comes from be- hind the mirror, and so the in- tersections of these lines with the mirror are the points where the rays from A and B are re- flected to E. In a similar man- ner the path of the rays may be traced for the position of the eye at E'. The full lines in front of the mirror are the paths of the rays from A and B, which give the images at A and B'. 260. Uses of a Plane Mirror. The employment of the plane mirror as a " looking glass " dates from a period of great antiquity. The process of covering a glass surface with an amalgam of tin and mercury came into use in Venice about three centuries ago. The process of cover- ing glass with a film of silver was invented during the last century. The fact that the image in a plane mirror is virtual has been used to produce many optical illusions, such as the stage ghost, the magic cabinet, the decapitated head, etc. To produce the illusion of a ghost, a large sheet of un- silvered plate glass, with its edges hidden by curtains, is so placed that the audience has to look obliquely through it to see the actors on the stage. Other actors, hidden from direct view, and strongly illuminated, are seen by reflection in the glass as ghostly images on the stage. 228 LIGHT 261. Multiple Reflection. Place two mirrors so that their re- flecting surfaces form an angle (Fig. 225). If a lighted candle be placed between them, several images may be seen in the mirrors; three when they are at right angles, more when the angle is less than a right angle. When the mirrors are parallel, all the images are in a straight line perpendicular to the mirrors. FIGURE 225. MULTIPLE REFLECTION. The i mage i n one m i rror serves as an object for the second mirror, and the image in the second becomes in turn an object for the first mirror. In Fig. 226 the two mirrors are at right angles. O f is the image of in AB, and is found as in 258. 0'" is the image of 0' in A C, and is found by the line 0' O flf drawn perpendicular to AC produced. 0" is the image of in A@, and since the mirrors are at right angles, 0'" is also the image of 0" in AB. 0'" is situated behind the plane of both mirrors, and no im- ages of it can be formed. All the images are situated in the circumference of a circle whose center is A and radius A 0. If E is the po- sition of the eye, then 0' FIGURE 226. and 0" are each seen by one reflection, and O rff by two reflections, and for this reason it is less bright. To trace the path of a ray for the image O' ri ', draw 0" f E, cutting AB at 6, and from the intersec- tion "b draw 10", cutting A at a. Join aO ; the path of the ray is OabE. It is interesting to find the images when the mirrors are at various angles. MIRRORS AT .RIGHT ANGLES. SPHERICAL MIRRORS 229 262. Illustrations. The double image of a bright star and the several images of a gas jet in a thick mirror (Fig. 227) are examples of multiple reflection, the front surface of the mirror and the metallic surface at the back serving as parallel reflectors. Geometrically the number of images is infinite ; but on account of their faint- ness only a limited number is visible. The kaleidoscope, a toy invented by Sir David Brewster, is an interesting application of the same principle. It consists of a tube containing three mir- rors extending its entire length, the angle between any two ol them being 60. One end of the tube is closed by ground glass, and the other by a cap with a round hole in it. Pieces of colored glass are placed loosely between the ground glass and a plate of clear glass parallel to it. On looking through the hole at any source of light, multiple images of these pieces of glass are seen, symmetrically arranged around the center, and form- ing beautiful figures, which vary in pattern with every change in the position of the pieces of glass. 263. Spherical Mirrors. A mirror is spherical when its reflecting surface is a portion of the surface of a sphere. If the inner surface is polished for reflection, the mirror is concave ; if the outer surface, it is convex. Only a small portion of a spherical surface is used as a mirror. In Fig. 228 the center of the mirror MN is the center of curvature of the FIGURE 228. SPHERICAL S P here of which the ^fleeting surface MIRROR. is a part. The middle point A of the FIGURE 227. MULTIPLE IMAGES. 230 LIGHT reflecting surface MN is the pole or vertex of the mirror, and the straight line AB passing through the center of curvature Q and the pole A of the mirror is its principal axis. Any other straight line through the center and in- tersecting the mirror is a secondary axis. The figures of spherical mirrors in this chapter are sections of a sphere made by passing a plane through the principal axis. The difference between a plane mirror and a spherical one is that the normals to a plane mirror are all parallel lines, while those of a spherical mirror are the radii of the surface, and all pass through the center of curvature. 264. Principal Focus of Spherical Mirrors. A focus is the point common to the paths of all the rays after inci- dence. It is a real focus if the rays of light actually pass through the point, and virtual if they only appear to do so. Let the rays of the sun fall on a concave spherical mirror. Hold a graduated ruler in the position of its principal axis, and slide along it a small strip of cardboard. Find the point where the image of the sun is small- est. This will mark the principal focus, and it is a real one. If a convex spherical mirror be used the light will be reflected as a broad pencil diverging from a point back of the mirror. The focus is then a virtual one. FIGURE 229. PRINCIPAL Focus, CONCAVE MIRROR. a pencil of rays parallel to the principal axis falls on a concave spherical mirror, the point to which the rays converge after reflection is called the principal focus of the mirror (Fig. 229). In the case of a convex spherical mirror, the prin- cipal focus is the point on the axis behind the mirror from which the reflected rays diverge (Fig. 230). The dis- POSITION OF THE PRINCIPAL FOCUS 231 FIG.URE 230. PRINCIPAL Fo- cus, CONVEX MIRROR. tance of the principal focus from the mirror is its principal focal length. 265. Position of the Principal Focus. Let MN (Fig. 231) be a concave mirror whose center is at C and principal axis is AB. Let ED be a ray parallel to BA. Then CD is the normal at D; and CDF, the angle of reflection, must equal EDC, the angle of incidence. Since the ray BA is normal to the mirror, it will be reflected back along AB. The reflected rays DF 1 and AB have a common point F, which is the principal focus. The triangle CFD is isosceles with the sides CF and FD equal. (Why ?) But when the point D is near A, FD is equal to FA ; F is therefore the middle point of the radius CA. Other rays parallel to BA will pass after reflection nearly through F. Hence, the principal focus of a concave spheri- cal mirror is real and is halfway between the center of curva- ture and the vertex. Let MN (Fig. 232) be a convex spherical mirror. ED and BA are rays parallel to the principal axis. When produced back of the mirror, after reflection, their common point F is back of the mirror and half way bet ween ^4. and C. ~ OQO n TT FIGURE 232. PRINCIPAL Focus VIR- .(Why >) Hence, the pmnci- TUAL FOR CONVEX MIRROR. FIGURE 231. - POSITION OF PRINCIPAL Focus. u _ F c 232 LIGHT FIGURE 233. CONJUGATE Foci, CONCAVE MIRROR. pal focus of a convex spherical mirror is virtual and halfway between the center of curvature and the mirror. 266. Conjugate Foci of Mirrors. When a diverging pen- cil of light AED (Fig. 233) falls on the spherical mirror MN, it is focused after reflection at a point B' on the axis AB which passes through the ra- diant point or source of light ; after reflec- tion the rays diverge from this focus B 1 as a new radiant point. When rays diverging from one point converge to another, the two points are called conjugate foci. In Fig. 234, the rays BA and BD diverge from B as the radiant point ; after reflection they diverge as if they came from B' behind the reflecting surface ; B 1 is a virtual focus and B and B' are conjugate foci. In the first case the source of light is farther from the mirror than the center of curvature, and the focus is real ; in the second case it is nearer the mirror than the principal focus, and the focus is virtual. 1 FIGURE 234. CONJUGATE Foci, ONE Focus VIRTUAL. 1 In Fig. 233, CD bisects the angle BDH. Hence, - = ^-. If D B' D B' C is close to A, we may, without sensible error, place BD = BA and B'D = B'A. Put BA = p, B'A = g, CA=r = 2/. Then BC = p - r, B' C = r - g, and - =^37^, from which - + - = - = j. By measuring j) IMAGES IN SPHERICAL MIRRORS 233 267. Images in Spherical Mirrors. In a darkened room sup- port on the table a concave spherical mirror, a candle, and a small white screen. Place the candle anywhere beyond the focus, and move the screen until a clear image of the flame is formed on it (Fig. 235). Notice the size and position of the image, and whether it is erect or inverted. When the can- dle is between the focus and the mirror, an image of it cannot be obtained on the screen, but it can be seen by looking into the mirror. FlGURE 235 - ~ IMAGE BY CONCAVE The same is true for the convex MIRROR. mirror, whatever be the position of the candle ; in these last cases the image is a virtual one. The experiment shows the relative positions of the object and its image for a concave mirror, all depending on the position of the object with re- spect to the mirror. If these positions are carefully noted it will be seen that there are six FIGURE 236. OBJECT BEYOND CENTER OF CURVA- j nc cases as fol- TURE. lows : First. When the object (AB, Fig. 236) is at a finite distance beyond the center of curvature, the image is real, inverted, smaller than the object, and between the center of curvature and the principal focus. Second. When a small object is at the center of curva^ ture, the image is real, inverted, of the same size as the and g, we may compute r and /. For the convex mirror, q and r are m 234 LIGHT FIGURE 237. OB- JECT AT CENTER OF CURVATURE. object, and at the center of curvature (Fig. 237). Third. When the object is between the center and the principal focus, the image is real, inverted, larger than the object, and is beyond the center (Fig. 238). This is the converse of Case I. Fourth. When the object is at the principal focus, the rays are reflected parallel and no distinct image is formed (Fig. 239). Fifth. When the ob- ject is between the prin- cipal focus and the mir- ror, the image is virtual, erect, and larger than the object (Fig. 240). Sixth. When the mirror is convex, the image is always virtual, erect, and smaller than the object (Fig. 241). 268. Construction for Images. To find images in spherical mirrors by geometrical construction, it is only necessary to find conjugate focal points. To do this trace two rays for each point for the object, one along the secondary axis through it, and the other parallel to the principal axis. The first ray is reflected back on itself, FIGURE 239. OBJECT AT PRINCI- PAL Focus. and the second through the FIGURE 238. OBJECT BETWEEN CENTER AND PRINCIPAL Focus. SPHERICAL ABERRATION IN MIRRORS 235 FIGURE 240. OBJECT BETWEEN PRINCIPAL Focus AND MIRROR. principal focus. The intersection of the two reflected rays from the same point of the object locates the image of that point. For instance : In Fig. 236, AC is the path of both the incident and the reflected ray, while the ray AD is reflected through the principal focus F. Their intersection is at a. The rays B and BE are reflected similarly through b. Hence, ab is the image of AB. In Fig. 240, the ray AC along the secondary axis, and AD reflected back through F as DF, must be produced to meet back of the mirror at the virtual focus a. A and a are conjugate foci ; also B and , and ab is a virtual image. For the convex mir- ror (Fig. 241) the con- struction is the same. From the point A draw A C along the normal or secondary axis, and AD parallel to the principal axis. The latter is re- flected so that its direc- tion passes through F. The intersection of these two lines is at a. The image ab is virtual and erect. v 269. Spherical Aberration in Mirrors. Bend a strip of bright tin into as true a semicircle as possible and fasten it to a vertical board as in Fig. 242. At right angles to the board at one end place N FIGURE 241. IMAGE ALWAYS VIRTUAL IN CONVEX MIRROR. 236 LIGHT a vertical sheet of cardboard containing three parallel slots. Send a strong beam of light through each of these slots ; the three beams will be reflected by the curved tin through different points, the beam nearest the straight rim of the mirror crossing the axis nearest the mir- FIGURE 242. SPHERICAL ABERRATION. The experiment shows that rays in- cident near the mar- gin of a spherical mirror cross the axis after reflection be- tween the principal focus and the mirror. This spreading out of the focus is known as spherical aberration by reflec- tion. It causes a lack of sharpness in the outline of images formed by spherical mirrors. It is reduced by decreasing the aperture of the mirror by means of a dia- phragm to cut off marginal rays, or by decreasing the curvature of .f < the mirror from the vertex out- ward. The result then is a para- bolic mirror (Fig. 243), which finds use in searchlights, light- houses, headlights of locomotives and automobiles, and in reflecting telescopes. 270. Caustics by Reflection. Use the tin reflector of the last experi- ment as shown in Fig. 244. The light from a candle or a lamp is focused on a curved line. The curve formed by the rays reflected from a spherical mirror is called the caustic by reflection. It may be seen \ \ FIGURE 243. PARABOLIC MIRROR. QUESTIONS AND PROBLEMS 237 by letting sunlight fall on a tin milk pail partly full of milk, or on a plain gold ring on a white surface. FIGURE 244. CAUSTIC BY REFLECTION. Questions and Problems 1. Why is the image of an object seen in the bowl of a silver spoon distorted ? 2. Show by an arrangement of plane mirrors how to see around an obstruction. 3. How can a concave, a convex, and a plane mirror be distinguished from one another, even when their outer surfaces are flat, as is often the case? 4. Construct all the images that would be formed of a luminous point placed between two mirrors forming an angle of 60. 5. Show by a diagram that a person can see his whole length in a short plane mirror placed on a vertical wall by tipping the top of the mirror forward and standing close to the mirror. 6. A candle foot is the intensity of illumination of a 1 c.p. light at a distance of one foot from the illuminated surface. What will be the illumination in foot candles of a surface 10 ft. away from a 50 c.p. lamp? 238 LIGHT 7. How far must a surface be from a 40 c.p. lamp to receive the same illumination as it would receive from a 4 c.p. lamp two feet distant? 8. If a person can just see to read a book when 10 ft. away from a 16 c.p. lamp, how far away from a 1600 c.p. arc light can he see to read the book ? 9. Where must a 16 c.p. lamp be placed between two parallel walls of a room 20 ft. apart in order that one wall may be four times as strongly illuminated as the other? 10. A gas burner consuming 5 cu. ft. per hour gives a flame of 16 c.p. A 16 c.p. electric bulb consumes 44 watts per hour. With gas at $ 1 per 1000 cu. ft. and electricity at 12 cents per K. W. hour, which is the cheaper ? 11. If an object is 18 ft. distant from a concave spherical mirror and the image formed of it is 2 ft. from the mirror, what is its focal length ? 12. Find by a diagram what effect it has on the image of an object in a convex spherical mirror to vary the distance of the object from the mirror. 13. The mirror formula applies equally well to the convex spheri- cal mirror if q, r, and /are made negative. Find the position of the image of an object as given by a convex spherical mirror when the radius of curvature is 20 inches, the object being 10 ft. from the mirror. 14. If a plane mirror is moved parallel to itself directly away from an object in front of it, show that the image moves twice as fast as the mirror. IV. REFRACTION OF LIGHT 271. Refraction. Fasten a FIGURE 245. -REFRACTION OF LIGHT. P a ? er P rofcractor sc * le centrally on one face of a rectangular bat- tery jar (Fig. 245), and fill the jar with water to the horizontal di- ameter of the scale. Place a slotted cardboard over the top. With CAUSE OF REFRACTION 239 FIGURE 246. CUP OF WATER AND COIN. a plane mirror reflect a beam of light through the slit into the jar, at such an angle that the beam is incident on the water exactly back of the center of the scale. The path of this ribbon of light may be traced ; its direction is changed at the surface of the water. The change in the course of light in passing from one transparent medium into another is called refraction. Place a coin at the bottom of an empty cup standing on a table, and let an observer move back until the coin just passes out of sight below the edge of the cup; now pour water into the cup, and the coin will come into view (Fig. 246). The changes in the apparent depth of a pond or a stream, as the observer moves away from it, are caused by re- fraction. The broken appearance of a straight pole thrust obliquely into water is accounted for by the change in direction which the rays coming from the part under water suffer as they emerge into the air. 272. Cause of Refraction. Foucault in France and Michelson in America have measured the veloc- ity of light in water, and have found that it is only three-fourths as great as in air. The velocity of light in all transparent liquids and solids is less than in air, while the velocity in air is practi- cally the same as in a vacuum. If now a beam of light FIGURE 247. REFRACTION EXPLAINED. 240 LIGHT is incident obliquely on the surface MN of water (Fig. 247), all parts of a light wave do not enter the water at the same time. Let the parallel lines perpendicular to AB represent short portions of plane waves. Then one part of a wave, as/, will reach the water before the other part, as e, and will travel less rapidly in the water than in the air. The result is that each wave is swung around, that is, the direction of propagation BO, which is perpendicular to the wave fronts, is changed; in other words, the beam is refracted. The refraction of light is, therefore, due to its change in velocity in passing from one transparent medium to another. 273. The Index of Refraction. Let a beam of light pass obliquely from air to water or glass, and let AB (Fig. 248) be the incident wave front. From A as a center and with a radius AD equal to the distance the light travels in the second medium while it is going from B to in air, draw the dotted arc. This limits the distance to which the disturbance spreads in the second medium. Then from (7 draw CD tangent to this arc and draw AD to the point of tangency. CD is the new wave front. The distances BC and AD are traversed by the light in the same time. They are therefore proportional to the velocities of light in the two media. Then the T j / / * speed of light in air v 1 Index of refraction = r . ^- = -: . speed in second medium v FIGURE 248. INDEX OF RE- FRACTION. lr The older mathematical definition of the index of refraction is the ratio of the sine of the angle of incidence to the sine of the angle of re- LAWS OF REFRACTION 241 The angle NOB is the angle of incidence. It is equal to the angle BAG between the incident wave front and the surface of separation of the two media. The angle of refraction is the angle N 1 AD. It is equal to the angle A CD between the wave front in the second medium and the surface of separation. The angle at (7, between the direction of the incident ray and the refracted ray, is the angle of deviation. The following are the indices of refraction for a few substances: Water .... 1.33 Crown glass . . 1.51 Alcohol .... 1.36 Flint glass 1.54 to 1.71 Carbon bisulphide 1.64 Diamond .... 2.47 For most purposes the index of refraction for water may be taken as , for crown glass , for flint glass J^-, and for diamond J. 274. Laws of Refraction. The following laws, which summarize the facts relative to single refraction, were discovered by Snell, a Dutch physicist, in 1621: I. When a pencil of light passes obliquely from a less highly to a more highly refractive medium, it is bent toward the normal ; when it passes in the reverse direc- tion, it is bent from the normal. II. Whatever the angle of incidence, the index of re- fraction is a constant for the same two media. fraction. Now the sine of an angle in a right triangle is the quotient of the side opposite by the hypotenuse. Thus, the sine of angle BAG is , and the sine of ACD is 45- Dividing one by the other, the common BO v term AC cancels out, and the index of refraction equals -^ = -j , as before. The two definitions are therefore equivalent to each other. For the con- struction to find the refracted ray, see the Appendix. 242 LIGHT III. The planes of the angles of incidence and refrac- tion coincide. 275. Refraction through Plate Glass. Draw a heavy black line on a sheet of paper, and place over it a thick plate of glass, cover- ing a part of the line. Look obliquely through the glass ; the line will appear broken at the edge of the plate, the part under the glass appearing laterally displaced (Fig. 249). To explain this, let MN (Fig. 250) represent a thick plate of glass, and AB a ray of light incident obliquely upon it. If the path of the ray be determined, the emergent ray will be parallel to the inci- dent ray. Hence, the apparent position of an object viewed through a plate of glass is at one side of its true position. 276. A Prism. Let AB (Fig. 251) represent a section of a glass prism made by a plane perpendicular to the refracting edge A. Also, let LI be a FIGURE 249. IMAGE OF LINE DIS- PLACED. FIGURE 250. I NCI- DENT AND EMERGENT RAYS PARALLEL. ray incident on the face BA. This ray will be refracted along IE, and entering the air at the point E will be refracted again, taking the di- rection EO. Reflect across the table a strong beam of light and intercept it with a sheet of green glass. Let this ribbon of green light be incident on a prism of small re- fracting angle in such a manner that only part of the beam passes through the prism. Two lines of light may be traced through the .Jf B C FIGURE 251. PATH OF LIGHT THROUGH PRISM. TOTAL INTERNAL REFLECTION 243 Horizon dust of the room or by means of smoke. By turning the prism about its axis, the angle between these lines of light can be varied in size. It is the angle of deviation, represented by the angle D in the figure. The angle of deviation is least when the angles of incidence and emergence are equal ; this occurs when the path of the ray through the prism is equally inclined to the two faces. 277. Atmospheric Refraction. Light coming to the eye from any heavenly body, as a star, unless it is directly overhead, is gradually s^ bent as it passes through \^ the air on account of the increasing density of the atmosphere near the earth's surface. Thus, if S in Fig. 252 is the real position of a star, its ap- parent position will be S r to an observer at E. Such an object appears higher above the horizon than its real altitude. The sun rises earlier on account of atmos- pheric refraction than it otherwise would, and for the same reason it sets later. Twilight, the mirage of the desert, and the looming of distant objects are phenom- ena of atmospheric refraction. 278. Total Internal Reflec- tion. Take the apparatus of 271 and place the cardboard against the end of the jar so that the slit is near the bottom (Fig. 253). Reflect a strong beam of light up through the water and incident on its under surface just back of the center of the protractor scale. Adjust the slit so that the beam shall be incident at an angle a little greater than 50. It will be reflected back into the water as from a plane mirror. FIGURE 252. ATMOSPHERIC REFRACTION. FIGURE 253. TOTAL INTERNAL REFLECTION. 244 LIGHT CRITICAL ANGLE. As the angle of refraction is always greater than the angle of incidence when the light passes from water into air, it is evident that there is an incident angle of such a value that the corresponding angle of refraction is 90, that is, the refracted light is parallel to the surface. If the angle of incidence is still further increased, the light no longer passes out into the air, but suffers total internal reflection. 279. The Critical Angle. The critical angle is the angle of incidence corre- sponding to an angle of refraction of 90. This angle varies with the in- dex of refraction of the substance. It is about 49 for water, 42 for crown glass, 38 for flint glass, and 24 for diamond. Of all the rays diverging from a point at the bottom of a pond and incident on the surface, only those within a cone whose semi-angle is 49 pass into the air. All those incident at a larger angle un- dergo total internal reflection (Fig. 254). Hence, an observer under water sees all objects outside as if they were crowded into this cone ; beyond this he sees by reflection ob- jects on the bottom of the pond. Total reflection in glass is shown by means of a prism whose cross sec- tion is a right-angled isosceles tri- angle (Fig. 255). A ray incident normally on either face about the right angle enters the prism without refraction, FIGURE 255. TOTAL RE- FLECTION BY PRISM, QUESTIONS AND PEOBLEMS 245 and is incident on the hypotenuse at an angle of 45, which is greater than the critical angle. The ray there- fore suffers total in- ternal reflection and leaves the prism at right angles to the in- cident ray. A simi- ______ lar prism is sometimes ^ 256. - E R ECT, N C PK.SM. used in a projecting lantern for making the image erect (Fig. 256). It would otherwise be inverted with respect to the object. Questions and Problems 1. Why are reflectors back of wall lamps frequently made concave at the outer edge and convex in the central part ? 2. Show that atmospheric refraction increases the length of daylight. 3. A plane mirror is revolved through an angle of 20. Show by diagram that a ray of light incident on the mirror will be displaced 40. 4. Show that the deviation of a ray of light by a glass prism is in- creased by increasing the angle of the prism. 5. Show that the deviation of a ray of light by a prism is increased by increasing the index of refraction. 6. Why does the full moon when seen near the horizon appear just a little elliptical, the longer axis being horizontal? 7. Why does a stream of water, to one standing on its bank, appear less than its true depth ? 8. A genuine diamond is distinctly visible in carbon disulphide, a paste or false diamond is nearly invisible. Explain. (The paste diamond is flint glass.) 9. What peculiarity will the image of an object have if the mirror is convex cylindrical ? 10. In spearing a fish from a boat would you strike directly at the apparent position of the fish ? Explain. 11. Show by diagram the apparent displacement of a body as seen by looking obliquely at it through a plate glass window. 246 LIGHT 12. Why is powdered glass opaque ? 13. Show by diagram that a triangular prism of air within water has the opposite effect on the direction of a ray of light passing through it that a prism of water in air has. V. LENSES 280. Kinds of Lenses. A lens is a portion of a transpar- ent substance bounded by two surfaces, one or both being FIGURE 257. FORMS OF LENSES. curved. The curved surfaces are usually spherical (Fig. 257). Lenses are classified as follows: 1. Double-convex, both surfaces convex . . 2. Plano-convex, one surface convex, one plane 3. Concavo-convex, one surface convex, one concave 4. Double-concave, both surfaces concave . . 5. Plano-concave, one surface concave, one plane 6. Convexo-concave, one surface concave, one convex . Converging lenses, thicker at the middle than at the edges. Diverging lenses, thinner at the middle than at the edges. TRACING BAYS. THROUGH LENSES 247 The concavo-convex and the convexo-concave lenses are frequently called meniscus lenses. The double-con- vex lens may be regarded as the type of the converging class of lenses, and the double-concave lens of the diverg- ing class. 281. Definition of Terms relating to Lenses. The centers of the spherical surfaces bounding a lens are the centers of curvature. The optical center is a point such that any ray passing through it and the lens suffers no change of di- rection. In lenses whose surfaces are of equal curv- ^x^ x \/'' ature, the optical center is their center of volume, as 0, in Fig. 258. In piano-lenses, the optical center is the middle point of the curved face. The straight line, <7C V , through the centers of curvature, is the principal axis, and any other straight line through the op- tical center, as EH, is a secondary axis. The normal at any point of the surface is the radius of the sphere drawn FIGURE 238. OPTICAL CENTER OF LENS. FIGURE 259. TRACING RAY THROUGH CONVERGING LENS. to that point ; thus CD is the normal to the surface AnB at D. 282. Tracing Rays through Lenses. A study of Figs. 259 and 260 shows that the action of lenses on rays of 248 LIGHT light traversing them is similar to that of prisms, and conforms to the principle illustrated in 276. A ray is always refracted toward the perpendicular on entering a denser medium (glass) and away from it on entering a medium of less optical density. Thus we see that the convex lens bends a ray toward the principal axis, while the concave lens (Fig. 260) bends it away from this axis. N FIGURE 260. TRACING RAY THROUGH DIVERGING LENS. (For tracing the path of a ray geometrically, consult Appendix V.) 283. The Principal FOCUS. Hold a converging lens so that the rays of the sun fall on it parallel to its principal axis. Beyond the lens hold a sheet of white paper, moving it until the round spot of light is smallest and brightest. If held steadily, a hole may be burned through the paper. This spot marks the principal focus of the lens, and its distance from the optical center is the principal focal length. For double-convex lenses, the two faces having the same radius of curvature, the principal focus is at the center of curvature when the index of refraction is 1.5. If the index is greater than 1.5, the focal length is less than the radius of curvature ; if less than 1.5, it is greater than this radius. Converging lenses are sometimes called burning glasses because of their power to focus the heat rays, as shown in the experiment. CONJUGATE FOCI OF LENSES 249 Figure 261 shows that parallel rays are made to con- verge toward the principal focus .Fby a converging lens, and the focus is real; on the other hand, Fig. 262 illus- A trates the diverging effect of a concave lens on parallel rays; the focus F is now virtual because the FIGURE 261. PRINCIPAL Focus OF A rays after passing through the lens only apparently come from F. In gen- eral, converging lenses increase the convergence of light, while diverging lenses decrease it. FIGURE 262. PRINCIPAL Focus OF A DIVERGING LENS. 284. Conjugate Foci of Lenses. If a pencil of light di- verges from a point and is incident on the lens, it is focused at a point on the axis through the radiant point. FIGURE 263. CONJUGATE Foci, CONVERGING LENS. 250 LIGHT These points are called conjugate foci, for the same reason as in mirrors. In Fig. 263 a pencil of rays BAE diverges from A and is focused by the lens at the point H. It is evident that if the rays diverge from H, they would be brought to a focus at A. Hence A and H are conjugate foci. 285. Images by Lenses. Place in a line on the table in a dark- ened room a lamp, a converging lens of known focal length, and a white screen. If, for example, the focal length of the lens is 30 cm., place the lamp about 70 cm. from it, or more than twice the focal length, and move the screen until a clearly denned image of the lamp appears on it. This image will be inverted, smaller than the object, and situated between 30 cm. and 60 cm. from the lens. By placing the lamp successively at 60 cm., 50 cm., 30 cm., and 20 cm., the images will differ in position and size, and in the last case will not be received on the screen, but may be seen by looking through the lens toward the lamp. If a diverging lens be used, no image can be received on the screen because they are all virtual. The results of such an experiment may be summarized as follows : I. When the object is at a finite distance from a con- verging lens, and farther than twice the focal length, the N FIGURE 264. OBJECT FARTHER THAN TWICE FOCAL LENGTH FROM LENS. image is real, inverted, at a distance from the lens of more than once and less than twice the focal length, and smaller thanrthe object (Fig. 264).. IMAGES BY LENSES 251 IT. When the object is at a distance of twice the focal length from a converging lens, the image is real, inverted, FIGURE 265. OBJECT TWICE FOCAL LENGTH FROM LENS at the same distance from the lens as the object, and of the same size (Fig. 265). III. When the object is at a distance from a con- verging lens of less than twice and more than once its FIGURE 266. OBJECT LESS THAN TWICE FOCAL LENGTH FROM LENS. focal length, the image is real, inverted, at a distance of more than twice the focal length, and larger than the object (Fig. 266). IV. When the object is at the principal focus of a con verging lens, no distinct image is formed (Fig. 267). FIGURE 267. OBJECT AT PRINCIPAL Focus. 252 LIGHT V. When the object is between a converging lens and its principal focus, the image is virtual, erect, and en- larged (Fig. 268). FIGURE 268. OBJECT LESS THAN FOCAL LENGTH FROM LENS. VI. With a diverging lens, the image is always virtual, erect, and smaller than the object (Fig. 269). 286. Graphic Construction of Images by Lenses. The im- age of an object by a lens consists of the images of its points. If the object is represented by an arrow, it is FIGURE 269. IMAGE VIRTUAL IN DIVERGING LENS. This necessary to find only the images of its extremities, is readily done by following two general directions: First. Draw secondary axes through the ends of the arrow. These represent rays that suffer no change in direction because they pass through the optical center ( 281). Second. Through the ends of the arrow draw rays parallel to the principal axis. After leaving the lens, these pass through the principal focus ( 283). SPHERICAL ABERRATION IN LENSES 253 The intersection of the two refracted rays from each extremity will be its image. To illustrate. Let AS be the object and MN the lens (Figs. 264-269). Rays along secondary axes through pass through the lens without any change in direction. The rays AD and BH, parallel to the principal axis, are refracted in the lens along DE and HI respectively, and emerge from the lens in a direction which passes through the principal focus F. The intersection of Aa with Ea is the image of A, and that of Bb with Ib is the image of B. Other rays from A and B also pass through a and b respectively, and therefore ab is the image of AB. The image is virtual when the intersection of the refracted rays is on the same side of the lens as the object. The relative size of object and image is the same as their rela- tive distance from the lens. 287. Spherical Aberration in Lenses. If rays from any point be drawn to different parts of a lens, and their FIGURE 270. SPHERICAL ABERRATION. directions be determined after refraction, it will be found that those incident near the edge of the lens cross the principal axis, after emerging, nearer the lens than those incident near the middle (Fig. 270). The principal focal length for the marginal rays is therefore less than for central rays. This indefiniteness of focus is called spherical aberration by refraction, the effect of which is to lessen the 254 LIGHT distinctness of images formed by the lens. In practice a round screen, called a diaphragm, is used to cut off the marginal rays ; this renders the image sharper in outline, but less bright. In the large lenses used in telescopes the curvature of the lens is made less toward the edge, so that all parallel rays are brought to the same focus. 288. Formula for Lenses. The triangles AOK and aOL in Fig. 271 are similar. Hence, j- = -jr If the lens is thin, a straight line connecting D and H will pass very nearly through the optical A FIGURE 271. RELATION BETWEEN OBJECT AND IMAGE. center 0. Then DFO is a triangle similar to aFL, and = . a,L LF Since DO is equal to AK, the first members of the two equations above are equal to each other, and therefore LO - q, and OF =/. Then LF = q -/, and K^O OF above are equal to each other, and therefore = -- Put KO =p, LO LF Clearing of fractions and dividing through by pqf, we have 1 = 1 + 1 ...... (Equation 32) / P q By measuring p and q we may compute /. For diverging lenses /and q are negative. Questions and Problems 1. How can a convex lens be distinguished from a concave one ? 2. Why does common window glass often give distorted images of objects viewed through it? THE MAGNIFYING GLASS 255 3. How can the principal focal length of a concave spherical mirror be found ? 4. Given a collection of spectacle lenses ; select the concave from the convex. 5. Why do so many cheap mirrors give distorted images ? 6. If an oarsman sticks his oar into the water obliquely, why does it appear broken at the point of entrance ? 7. Concave spherical mirrors are often mounted in frames to be used as hand glasses. Such mirrors are usually made by silvering one face of a lens. Why can several images be seen in such a mirror? 8. When is the distance between the object and its real image as formed by a converging lens the least possible? 9. The focal length of a camera lens is two inches. How far must the sensitized plate be from the lens, when the object is distant 100 ft. ? 10. If a reading glass has a focal length of 16 in. and in its use is held 10 in. from the book, what is the position of the virtual image? 11. An object 100 cm. in front of a converging lens gives an image 25 cm. back of the lens. What is the focal length of the lens? 12. Show by diagram what effect it has on the image of an object by a diverging lens to move it farther away from the lens. 13. Why is a convex mirror used on an automobile to view objects back of the driver, instead of a plane mirror ? 14. In a diverging lens, show that a pencil of light that converges to a point beyond the focus of the lens issues as a diverging pencil. 15. Where must a diverging lens be placed to render parallel a converging pencil of light? VI. OPTICAL INSTRUMENTS 289. The Magnifying Glass, or simple microscope, is a double-convex lens, usually of short focal length. The object must be placed nearer the lens than its principal focus. The image is then virtual, erect, and enlarged. If AB is the object in Fig. 272, the virtual image is ab ; and if the eye be placed near the lens on the side opposite 256 LIGHT the object the virtual image will be seen in the position of the intersection of the rays produced, as at ab. FIGURE 272. MAGNIFYING GLASS. 290. The Compound Microscope (Fig. 273) is an instru- ment designed to obtain a greatly enlarged image of very small objects. In its simplest form it consists of a con- verging lens JOT (Fig. 274), called the object glass or ob- jective, and another con- verging lens RS, called the eye-piece. The two lenses are mounted in the ends of the tube of Fig. 273. The object is placed on the stage just under the objective, and a little beyond its principal focus. A real image ab (Fig. 274) is formed slightly nearer the eye-piece than its focal length. This image formed by the objective is FIGURE 273. - MICROSCOPE. viewed b ? the eyepiece, and the latter gives an enlarged virtual image. (Why?) Both the objective and the eyepiece produce magnification. THE ASTRONOMICAL TELESCOPE 257 291. The Astronomical Telescope. The system of lenses in the refracting astronomical telescope (Fig. 275) is simi- lar to that of the compound microscope. Since it is in- tended to view distant objects, the objective MN is of FIGURE 274. TRACING RAYS TO FORM IMAGE large aperture and long focal length. The real image given by it is the object for the eyepiece, which again forms a virtual image for the eye of the observer. The magnification is the ratio of the focal lengths of the objec- tive and the eyepiece. The objective must be large, for the purpose of collecting enough light to permit large or A ^ FIGURE 275. IMAGE IN ASTRONOMICAL TELESCOPE. magnification of the image without too great loss in brightness. Figure 275 shows that the image in the astronomical telescope is inverted. In a terrestrial telescope the image is made erect by introducing near the eyepiece two double- convex lenses, in such relation to each other and to the first image that a second real image is formed like the first, but erect. 258 LIGHT 292. Galileo's Telescope. The earliest form of telescope was invented by Galileo. It produces an erect image by the use of a diverging lens for the eyepiece (Fig. 276). This lens is placed between the objective and the real image, ab, which would be formed by the objective if the eyepiece were not interposed. Its focus is practically at the image ab, and the rays of light issue from it slightly FIGURE 276. GALILEO'S TELESCOPE. divergent for distant objects. The image is therefore at A 1 B 1 instead of at ab, and it is erect and enlarged. This telescope is much shorter than the astronomical telescope, for the distance between the lenses is the difference of their focal lengths instead of their sum. In the opera glass two of Galileo's telescopes are attached together with their axes parallel. 293. The Projection Lantern is an apparatus by which a greatly enlarged image of an object can be projected on a screen. The three essentials of a projection lantern are a strong light, a condenser, and an objective. The light may be the electric arc light, as shown in Fig. 277, the calcium light, or a large oil burner. The condenser E is composed of a pair of converging lenses; its chief pur- pose is the collection of the light on the object by refrac- tion, so as to bring as much as possible on the screen. The object AB, commonly a drawing or a photograph :m- MOVING PICTURE FILM. Most moving picture cameras take from 16 to 120 pictures per second. THE EYE 259 on glass, is placed near the condenser SS, where it is strongly illuminated. The objective, MN, is a combina- tion of lenses, acting as a single lens to project on the screen a real, inverted, and enlarged image of the object. FIGURE 277. PROJECTION LANTERN. 294. The Photographer's Camera consists of a box BQ (Fig. 278), adjustable in length, blackened inside, and provided at one end with a lens or a combination of lenses, acting as a single one, and at the other with a holder for the sensitized plate. If by means of a rack and pinion the lens U be properly focused for an ob- ject in front of it, an inverted image will be formed on the sensitized plate E. The light acts on the salts con- tained in the sensitized film, producing in them a modifi- cation which, by the processes of "developing" and "fix- ing," becomes a permanent negative picture of the object. When a " print " is made from this negative, the result is a positive picture. 295. The Eye. The eye is like a small photographic camera, with a converging lens, a dark chamber, and a FIGURE 278. CAMERA. 260 LIGHT FIGURE 279. SECTION OF EYE. sensitive screen. Figure 279 is a vertical section through the axis. The outer covering, or sclerotic coat H, is a thick opaque substance, except in front, where it is ex- tended as a transparent coat, called the cornea A. Behind the cornea is a dia- phragm D, consti- tuting the colored part of the eye, or the iris. The cir- cular opening in the iris is the pupil, the size of which changes with the intensity of light. Supported from the walls of the eye, just back of the iris, is the crystalline lens E, a transparent body dividing the eye into two chambers; the anterior chamber between the cornea and the crystalline lens is a transparent fluid called the aqueous humor, while the large chamber behind the lens is filled with a jellylike substance called the vitreous humor. The choroid coat lines the walls of this posterior chamber, and on it is spread the retina, a membrane traversed by a network of nerves, branching from the optic nerve M. The choroid coat is filled with a black pigment, which serves to darken the cavity of the eye, and to absorb the light reflected internally. 296. Sight. When rays of light diverge from the ob- ject and enter the pupil of the eye they form an inverted image on the retina (Fig. 280) precisely as in the photo- graphic camera. In place of the sensitized plate is the sensitive retina, from which the stimulus is carried to the brain along the optic nerve. THE BLIND SPOT 261 In the camera the distance between the lens and the screen or plate must be adjusted for objects at different distances. In the eye the corresponding distance is fixed, and the adjustment for distinct vision is made by uncon- sciously changing the curvature of the front surface of FIGURE 280. IMAGE ON RETINA. the crystalline lens by means of the ciliary muscle JF, Gr (Fig. 279). This capability of the lens of the eye to change its focal length for objects at different distances is called accommodation. 297. The Blind Spot. There is a small depression where the optic nerve enters the eye. The rest of the retina is covered with microscopic rods and cones, but there are none in this depression, and it is insensible to light. It is FIGURE 281. To FIND BLIND SPOT. accordingly called the blind spot. Its existence can be readily proved by the help of Fig. 281. Hold the book with the circle opposite the right eye. Now close the left eye and turn the right to look at the cross. Move the book toward the eye from a distance of about a foot, and a position will readily be found where the black circle will disappear. Its image then falls on the blind spot. It may be brought into view again by moving the book either nearer the eye or farther away. 262 LIGHT 298. The Prism Binocular. - - While the opera glass ( 292) is compact and gives an erect image, it has only a small field of view, and is usually made to magnify only three or four times. For the purpose of obtaining a larger field of view with equal compactness, the prism binocular has been devised. The desired length has been obtained by the use of two total reflecting prisms (Fig. 282), by means of which the light is reflected forward and back again in the tube. Not only is compactness secured in this manner, but FIGURE 282. PRISM BINOCULAR. . . the reflections in the prisms increase the focal length of the objective and serve to give an erect image without " perversion." 299. Defects of the Eye. A normal eye in its passive or relaxed condition focuses parallel rays on the retina. The defects of most frequent occurrence are near-sightedness, far-sightedness, and astigmatism. If the relaxed eye focuses parallel rays in front of the retina (Fig. 283), it is near-sighted. The length of the eyeball from front to back is then too great for the focal length of the crys- talline lens. The correc- tion consists in placing in /. ,-, -,. FIGURE 283. NEAR-SIGHTEDNESS. iront ot the eye a diverg- ing lens that makes with the lens of the eye a less con- vergent system than the crystalline lens itself. If the focal length of the diverging lens is equal to the greatest distance of distinct vision for the near-sighted eye, and if ANALYSIS OF WHITE LIGHT 263 this lens is held close to the eye, parallel rays from a dis- tant object will enter the eye as if they came from the principal focus of the lens, the image falls on the retina, and vision is made distinct. If the relaxed eye focuses parallel rays from distant objects behind the retina, it is far-sighted. The length of the eyeball is then too short to correspond with the focal length of the crystalline lens. The correction con- sists in placing in front of the eye a converging lens (Fig. 284), making with FIGURE 284. FAR-SIGHTEDNESS. the lens of the eye a more converging system than the eye lens alone. Light from a near object then enters the eye as if it came from a dis- tant one and vision becomes distinct. Sometimes the front of the cornea has different curva- tures in different planes through the axis ; that is, it has a somewhat cylindrical form. Persons with such an eye do not see with equal distinctness all the figures on the face of a watch. This defect is known as astigmatism. Tt is corrected by the use of a lens, one surface of which at least is not spherical but differs from it in the opposite sense to that of the defective eye. The astigmatism of the two eyes is not usually the same. VII. DISPERSION 300. Analysis of White Light. The Solar Spectrum. Darken the room, and by means of a mirror hinged outside the window, reflect a pencil of sunlight into the room. Close the opening in the window with a piece of tin, in which is cut a very narrow vertical slit. Let the ribbon of sunlight issuing from the slit be incident obliquely on a glass prism (Fig. 285). A many-colored band, gradually chang- ing from red at one end through orange, yellow, green, blue, to violet 264 LIGHT at the other, appears on the screen. If a converging lens of about 30 cm. focal length be used to focus an image of the slit on the screen, and the prism be placed near the principal focus, the colored images , of the slit will be more distinct. This experi- ment shows that white or colorless light is a mix- ture of an infinite number of differ- ently colored rays, of which the red is re- fracted least and the violet most. The brilliant band of light consists of an indefinite num- ber of colored images of the slit ; it is called the solar spectrum, and the opening out or separating of the beam of white light is known as dispersion. 301. Synthesis of Light. Project a spectrum of sunlight on the screen. Now place a second prism like the first behind it, but re- versed in position (Fig. 286). There will be formed a colorless image, slightly displaced on the screen. FIGURE. 285. ANALYSIS OF WHITE LIGHT. FIGURE 286. REFORMING WHITE LIGHT. The second prism reunites the colored rays, making the effect that of a thick plate of glass ( 275). The recomposition of the colored rays into white light may also be effected by receiving them on a concave mirror or a large convex lens. 302. Chromatic Aberration. Let a beam of sunlight into the darkened room through a round hole in a piece of cardboard. Pro- THE ACHROMATIC LENS 265 ject an image of this aperture on the screen, using a double-convex lens for the purpose. The round image will be bordered with the spectral colors. This experiment shows that the lens refracts the rays of different colors to different foci. This defect in lenses is known as chromatic aberration. The violet rays, being more re- s frangible than the red, will have > their focus nearer to the lens than 5 the red, as shown in Fig. 287, where v is the principal focus for FlGURE A 287 - ~ CHROMATIC T i f TP ABERRATION. violet light and r for red. If a screen were placed at x, the image would be bordered with red, and if at y with violet. 303. The Achromatic Lens. With a prism of crown glass pro- ject a spectrum of sunlight on the screen, and note the length of the spectrum when the prism is turned to give the least deviation ( 276). Repeat the experiment with a prism of flint glass having the same re- fracting angle. The spectrum formed by the flint glass will be about twice as long as that given by crown glass, while the position of the middle of the spectrum on the screen is about the same in the two cases. Now use a flint glass prism whose refracting angle is half that of the crown glass one. The spectrum is nearly equal in length to ^^ that given by the crown glass -prism, but the devia- tion of the middle of it is considerably less. Finally, place this flint glass prism in a reversed position against the crown glass one FIGURE 288. ACHROMATIC PRISM. "5,. nnn ^ _, (Fig. 288). The image of the aperture is no longer colored, arid the deviation is about half that produced by the crown glass alone. In 1757 Dollond, an English optician, combined a double- convex lens of crown glass with a plano-concave lens of 266 LIGHT flint glass so that the dispersion by the one neutralized that due to the other, while the refraction was reduced about half (Fig. 289). Such a lens or system of lenses is called achromatic, since images formed by it are not fringed with the spectral colors. 304. The Rainbow. Cement a crystallizing beaker FIGURE 289. 12 or 15 cm. in diameter to a slate slab. Fill the beaker With Water throu n a hole drilled in the slate - Sa P- port the slate in a vertical plane and direct a ribbon of white light upon the beaker at a point about 60 above its hori- zontal axis, as SA (Fig. 290). The light may be traced through the water, part of it issuing at the back at B as a diverging pencil, and a part reflected to C and issuing as spectrum colors along CD. If other points of incidence be tried, the colors given by the reflected portion are very indistinct except at 70 below the axis. After re- fraction at this point, the light can be traced through the water, is- suing as spectral colors after having suffered two refractions and two reflections. 7 FIGURE 290. ILLUSTRATING RAINBOW. The experiment shows that the light must be incident at definite angles to give color effects. The red constituent of white light in- cident at about 60 keeps together after reflection and subsequent refraction ; that is, the red rays are practically parallel and thus have sufficient intensity to produce a red image. The same is true of the violet light incident at about 59 from the axis. The other spectral colors ar- range themselves in order between the red and violet. For light incident at about 70 from the axis a similar DISCONTINUOUS SPECTRA 267 ct _ FlGURE >--PAWAHD SECONDARY Bows. spectrum band is formed by light which has suffered two refractions and two reflections. So when sunlight falls on raindrops the light is dispersed and a rainbow is formed. Two bows are often visible, the primary and the secondary. The primary is the inner and brighter one, formed by a single internal reflection. It is distinguished by hav- ing the red on the out- side and the violet on the inside. The sec- ondary bow, formed by two internal reflec- tions, is fainter, and has the order of colors reversed. Figure 291 shows the relative po- sitions of the sun, the observer, and the raindrops which form the bows. It should be noted that all drops in the line vE send violet light to the eye, those along rE send red light, and those between the two send the intermediate colors. 305. Continuous Spectra. Throw on a screen the spectrum of the electric arc, using preferably for the purpose a hollow prism filled with carbon bisulphide. The spectrum will be composed of colors from red at one end through orange, yellow, green, blue, and violet at the other without interruptions or gaps. The experiment illustrates continuous spectra, that is, spectra without breaks or gaps in the color band. Solids, liquids, and dense vapors and gases, when heated to incan- descence, give continuous spectra. 306. Discontinuous SpeQtra. Project on the screen the spectrum of the electric light. Place in the arc a few crystals of sodium nitrate. 268 LIGHT The intense heat will vaporize the sodium, and a spectrum will be obtained consisting of bright colored lines, one red, one yellow, three green, and one violet, the yellow being most prominent. The experiment illustrates discontinuous or bright line spectra, that is, spectra consisting of one or more bright lines of color separated by dark spaces. Rarefied gases and vapors, when heated to incandescence, give discontinuous spectra. 307. Absorption Spectra. Project on the screen the spectrum of the electric light. Between the lamp and the slit S (Fig. 292) vaporize metallic so- dium in an iron spoon so placed that the white light passes through the heated sodium vapor before dispersion by the prism. A dark line will appear on the screen in the yellow of the spectrum at the place where the bright line was ob- tained in the preced- ing experiment. The experiment illustrates an absorption, reversed or dark line spectrum. The dark line is produced by the absorption of the yellow light by sodium vapor. G-ases and vapors absorb light of the same refrangibility as they emit at a higher temperature. 308. The Fraunhofer Lines. Show on the screen a carefully focused spectrum of sunlight. Several of the colors will appear crossed with fine dark lines (Fig. 293). Fraunhofer was the first to notice that some of these lines coincide in position with the bright lines of certain FIGURE 292. ABSORPTION SPECTRUM. THE SPECTROSCOPE 269 artificial lights. He mapped no less than 576 of them, and designated the more important ones by the letters A, B, (7, D, U, F, 6r, H, the first in the extreme red and the last in the vio- let. For this reason they are referred to as the Fraunhofer ,. T FIGURE 293. FRAUNHOFER LINES. lines. In recent years the number of these lines has been found to be practically unlimited. In the last experiment it was shown that sodium vapor absorbs that part of the light of the electric arc which is of the same refrangibility as the light emitted by the vapor itself. Similar experiments with other substances show that every substance has its own absorption spec- trum. These facts suggested the following explanation of the Fraunhofer lines : The heated nucleus of the sun gives off light of all degrees of refrangibility. Its spec- trum would therefore be continuous, were it not sur- rounded by an atmosphere of metallic vapors and of gases, which absorb or weaken those rays of which the spectra of these vapors consist. Hence, the parts of the spec- trum which would have been illuminated by those par- ticular rays have their brightness diminished, since the rays from the nucleus are absorbed, and the illumination is due to the less intense light coming from the vapors. These absorption lines are not lines of no light, but are lines of diminished brightness, appearing dark by contrast with the other parts of the spectrum. 309. The Spectroscope. The commonest instrument for viewing spectra is the spectroscope (Fig. 294). In one of its simplest forms it consists of a prism A, a telescope .B, and a tube called the collimator C, carrying an adjust- 270 LIGHT able slit at the outer end J9, and a converging lens at the other J$, to render parallel the diverging rays coming from A B . .nan /^ftu^^L D FIGURE 294. SPECTROSCOPE. the slit. The slit must therefore be placed at the princi- pal focus of the converging lens. To mark the deviation of the spectral lines, there is provided on the supporting FIGURE 295. IRON VAPOR IN THE SUN. table a divided circle F, which is read by the aid of verniers and reading microscopes attached to the tele- scope arm. The applications of the spectroscope are many and various. By an examination of their absorption spectra, normal and diseased, blood VARIOUS SPECTRA COLOR OF OPAQUE BODIES 271 are easily distinguished, the adulteration of substances is detected, and the chemistry of the stars is approximately determined. Figure 295 shows the agreement of a number of the spectral lines of iron with Fraunhofer lines in the solar spectrum; they indicate the pres- ence of iron vapor in the atmosphere of the sun. VIII. COLOR 310. The Wave Length of light determines its color. Extreme red is produced by the longest waves, and ex- treme violet by the shortest. The following are the wave lengths for the principal Fraunhofer lines in air at 20 C. and 760 mm. pressure : A Dark Red . 0.0007621 mm. E l Light Green 0.0005270 mm. B Red . . . 6884 mm. E 2 5269 mm. C Orange . . 6563 mm. F Blue . . . 4861 mm. D l Yellow . . 5896 mm. G Indigo . . 4293 mm. Z> 2 5890 mm. H l Violet . . 3968 mm. In white light the number of colors is infinite, and they pass into one another by imperceptible gradations of shade and wave length. Color stands related to light in the same way that pitch does to sound. In most artificial lights certain colors are either feeble or wanting. Hence, artificial lights are not generally white, but each one is characterized by the color that predominates in its spectrum. 311. Color of Opaque Bodies. Project the solar spectrum on a white screen. Hold pieces of colored paper or cloth successively in different parts of the spectrum. A strip of red flannel appears bril- liantly red in the red part of the spectrum, and black elsewhere ; a blue ribbon is blue only in the blue part of the spectrum, and a piece of black paper is black in every part of the spectrum. The experiment shows that the color of a body is due both to the light that it receives and the light that it reflects ; 272 LIGHT that a body is red because it reflects chiefly, if not wholly, the red rays of the light incident upon it, the others being absorbed wholly or partly at its surface. It cannot be red if there is no red light incident upon it. In the same way a body is white if it reflects all the rays in about equal proportions, provided white light is incident upon it. So it appears that bodies have no color of their own, since they exhibit no color not already present in the light which illuminates them. This truth is illustrated by the difficulty experienced in matching colors by artificial lights, and by the changes in shades some fabrics undergo when taken from sunlight into gaslight. Most artificial lights are deficient in blue and violet rays; and hence all complex colors, into which blue or violet enters, as purple and pink, change their shade when viewed by artificial light. 312. Color of Transparent Bodies. Throw the spectrum of the sun or of the arc light on the screen. Hold across the slit a flat bottle or cell filled with a solution of amraoniated oxide of copper. 1 The spectrum below the green will be cut off. Substitute a solution of picric acid, and the spectrum above the green will be cut off. Place both solutions across the slit and the green alone remains. It is the only color transmitted by both solutions. In like manner, blue glass cuts off the less refrangible part of the spectrum, ruby glass cuts off the more refrangible, and the two together cut off the whole. This experiment shows that the color of a transparent body is determined by the colors that it absorbs. It is colorless like glass if it absorbs all colors in like proportion, or absorbs none; but if it absorbs some colors more than others, its color is due to the mixed impression produced by the various colors passing through it. 1 It is prepared by adding ammonia to a solution of copper sulphate, until the precipitate at first formed is dissolved. THESE PRIMARY COLORS 273 313. Mixing Colored Lights. Out of colored papers cut several disks, about 15 cm. in diameter, with a hole at the center for mounting them on the spindle of a whirling machine (Fig. 296), or for slipping them over the handle of a heavy spinning top. Slit them along a radius from the circumference to the center, so that two or more of them can be placed together, exposing any pro- portional part of each one as desired (Fig. 297). Select seven disks, whose colors most nearly represent those of the solar spec- trum ; put them together so that equal por- tions of the colors are exposed. Clamp on the spindle of the whirling machine and rotate them rapidly. When viewed in a strong light the color is an impure white or gray. FIGURE 296. MIXING COLORED LIGHTS. This method of mixing colors is based on the physiological fact that a sensation lasts longer than the stimulus producing it. Before the sensation caused by one stimulus has ceased, the disk has moved, so that a different impression is produced. The effect is equivalent to superposing the several colors on one another at the same time. 314. Three Primary Colors. If red, green, and blue, or violet disks are used, as in 313, exposing equal por- tions, gray or impure white is ob- tained when they are rapidly rotated. If any two colors standing opposite each other in Fig. 298 are used, the re- sult is white; and if any two alternate ones are used, the result is the intermediate one. By using the red, the green, and the violet disks, and exposing in different pro- portions, it has been found possible to produce any color FIGURE 297. COLORED DISKS. 274 LIGHT of the spectrum. This fact suggested to Dr. Young the theory that there are only three primary color sensations, and that our recognition of different colors is due to the excitation of these three in vary- ing degrees. The color top is a standard toy provided with colored paper disks, like those of Fig. 296. When red, green, and blue disks are combined so as to show sec- tors of equal size, the top, when spinning in a strong light, ap- pears to be gray. Gray is a 30LORD.SK. wh j te of low intengity The colors of the disks are those of pigments, and they are not pure red, green, and blue. 315. Three- and Four-color Printing. The frontispiece of this book illustrates a four-color print of much interest. Such a print is made up of very fine lines and dots of the four pigments, red, yellow, blue, and black. The various colors in the picture are mixtures of these four with the white of the paper. The picture is made by printing the four colors one on top of the other from four copper plates, each of which represents only that part of the picture where a certain color must be used to give the proper final effect. These plates are made from four negatives. The process of pre- paring these negatives is as follows : Each negative is made by taking a picture of the origi- nal colored drawing through a colored "filter," which cuts out all the colors except the one desired. A blue filter is used to prepare the plate that prints with yellow ink, a green filter to prepare the red printing plate, a red filter MIXING PIGMENTS 275 for the blue printing plate, and a chrome yellow filter fot the black printing plate. A cross-lined glass screen, dividing the image into small dots, is placed in front of the negatives in the camera. Glue enamel prints are then made on copper, and the plates are etched, leaving the desired pri ting surface in relief. In the frontispiece the yellow is printed first, the red over the }^ellow second, the blue third over the yellow and red, and the black last over the yellow, red, and blue. When no black is used the process is known as the three-color process. 316. Complei entary Colors. Any two colors whose mix- ture produces on the eye the impression of white light are called complementary. Thus, red and bluish green are complementary ; also orange and light blue. When complementary colors are viewed next to each other, the effect is a mutual heightening of color impressions. Complementary colors may be seen by what is known as retinal fatigue. Cut some design out of paper, and paste it on red glass. Project it on a screen in a dark room. Look steadily at the screep for several seconds, and then turn up the lights. The design will appear on a pale green ground. This experiment shows that the portion of the retina on which the red light falls becomes tired of red, and refuses to convey as vivid a sensation of red as of the other colors, when less intense white light is thrown on it. But it retains its sensitiveness in full for the rest of white light, and therefore conveys to the brain the im- pression of white light with the red cut out ; that is, of the complementary color, green. 317. Mixing Pigments. Draw a broad line on the blackboard with a yellow crayon. Over this draw a similar band with a blue crayon. The result will be a band distinctly green. 276 LIGHT The yellow crayon reflects green light as well as yel- low, and absorbs all the other colors. The blue crayon reflects green light along with the blue, absorbing all the others. Hence, in superposing the two chalk marks, the mixture absorbs all but the green. The mark on the board is green, because that is the only color that sur- vives the double -absorption. In mixing pigments, the resulting color is the residue of a process of successive absorptions. If the spectral colors, blue and yellow, are mixed, the product is white instead of green. So we see that a mixture of colored lights is a very different thing from a mixture of pigments. IX. INTERFERENCE AND DIFFRACTION 318. Newton's Rings. Press together at their center two small pieces of heavy plate glass, using a small iron clamp for the purpose. Then look obliquely at the glass ; curved bands of color may be seen surrounding the point of greatest pressure. This experiment is like one performed by Newton while attempting to determine the relation between the colors in the soap bubble and the thickness of the film. He used a plano-convex lens of long focus resting on a plate of plane glass. Figure 299 ' B shows a section of the apparatus. Between the lens and the plate FIGURE 299. NEWTON s there is a wedge-shaped film of air, very thin, and quite similar to that formed between the glass plates in the above experi- ment. If the glasses are viewed by reflected light, there is a dark spot at the point of contact, surrounded by sev- eral colored rings (Fig. 300) ; but if viewed by trans- mitted light, the colors are complementary to those seen by reflection ( 316). NEWTON'S KINGS 277 FIGURE 300. COLORED RINGS. The explanation is to be found in the interference of two sets of waves, one reflected internally from the curved surface ACS, and the other from the surface D CE, on which it presses. If light of one color is incident on AB, a portion will be reflected ivomACB, and another portion from DOE. Since the light reflected from DOE has traveled farther by twice the thickness of the air film than that from AGB, and the film gradually increases in thickness from C out- ward, it follows that at some places the two reflected portions will meet in like phase, and at others in opposite phase, causing a strengthening of the light at the former, and extinction of it at the latter. If red light be used, the appearance will be that of a series of concentric circular red bands separated by dark ones, each shading off into the other. If violet light be employed, the colored bands will be closer together on account of the shorter wave length. Other colors will give bands intermediate in diameter between the red and violet. From this it follows that if the glasses be illuminated by white light, at every point some one color will be destroyed. The other colors will be either weak- ened or strengthened, depending on the thickness of the air film at the point under consideration, the color at each point being the result of mixing a large number of colors in unequal proportions. Hence, the point C will be surrounded by a series of colored bands. 1 1 The light from ACB differs in phase half a wave length from that reflected from DE, because the former is reflected in an optically dense 278 LIGHT The colors of the soap bubble, of oil on water, of heated metals which easily oxidize, of a thin film of varnish, and of the surface of very old glass, are all caused by the in- terference of light reflected from the two surfaces of a very thin film. 319. Diffraction. Place two superposed pieces of perforated cardboard in front of the condenser of the projection lantern. The projected images of the very small holes, as one piece is moved across the other, are fringed with the spectral colors. With a fine diamond point rule a number of equidistant parallel lines very close together on glass. They compose a transparent dif- fraction grating. Substitute this for the prism in projecting the spec- trum of sunlight or of the arc light on the screen ( 300). There will be seen on the screen a central image of the slit, and on either side of it a series of spectra. Cover half of the length of the slit with red glass and the other half with blue. There will now be a series of red images and also a series of blue ones, the red ones being far- ther apart than the blue. Lines ruled close together on smoked glass may be used instead of a "grating." These experiments illustrate a phenomenon known as diffraction. The colored bands are caused by the inter- ference of the waves of light which are propagated in all directions from the fine openings. The effects are visible because the transparent spaces are so small that the inten- sity of the direct light from the source is largely re- duced. Diffraction gratings are also made to operate by reflecting light. Striated surfaces, like mother-of-pearl, changeable silk, and the plumage of many birds, owe their beautiful changing colors to interference of light by diffraction. medium next to a rare one, and the latter in an optically rare medium next to a dense one. This phase difference is additional to the one above described. QUESTIONS 279 Questions 1. How many degrees is it from the sun to the highest point of the primary rainbow ? 2. Why is the red on the outside of the primary bow and on the inside of the secondary bow ? 3. If there are but three primary sensations, red, green, and violet, what effect would it have on a person's vision if the nerves for red sensations were inoperative ? 4. Why is the secondary rainbow less bright than the primary bow? 5. Are the two images of an object as formed on the retina of the two eyes identical ? Explain. 6. Account for the crossed bands of light seen by looking through the wire screening of the window at the full moon. 7. Account for the change in color of aniline purple when viewed by the light of a common kerosene lamp. 8. Under what conditions could a rainbow be seen at midday? 9. Account for the colors on water when gasoline is poured on it. 10. Why does each person in using a microscope have to focus for his own eyes ? CHAPTER IX HEAT I. HEAT AND TEMPERATURE 320. Nature of Heat. For a long time it was believed that heat was a subtle and weightless fluid that entered bodies and possibly combined with them. This fluid was called calorie. About the beginning of the last century some experiments of Count Rumford in boring brass can- non, and those of Sir Humphry Davy in melting two pieces of ice at freezing temperature by the friction of one piece on the other demonstrated that the caloric theory of heat was no longer tenable ; and finally about the middle of the century, when Joule proved that a definite amount of mechanical work is equivalent to a definite amount of heat, it became evident that heat is a form of molecular energy. The modern kinetic theory, briefly stated, is as follows : The molecules of a body have a certain amount of inde- pendent motion, generally very irregular. Any increase in the energy of this motion shows itself in additional warmth, and any decrease by the cooling of the body. The heating or the cooling of a body, by whatever process, is but the transference or the transformation of energy. 321. Temperature. If We place a mass of hot iron in contact with a mass of cold iron, the latter becomes warmer and the former cooler, the heat flowing from the hot body 280 MEASURING TEMPERATURE 281 to the cold one. The two bodies are said to differ in tem- perature or 4 ' heat level," and when they are brought in contact there is a flow of heat from the one of higher tem- perature to the one of lower until thermal (heat) equilib- rium is established. Temperature is the thermal condition of a body which determines the transfer of heat between it and any body in contact with it. This transfer is always from the body of higher temperature to the one of lower. Temperature is a measure of the degree of hotness ; it depends solely on the kinetic energy of the molecules of the body. Temperature must be distinguished from quantity of heat. The water in a pint cup may be at a much higher temperature than the water in a lake, yet the latter con- tains a vastly greater quantity of heat, owing to the greater quantity of water. 322. Measuring Temperature. Fill three basins with mod- erately hot water, cold water, and tepid water respectively. Hold one hand in the first, and the other in the second for a short time ; then transfer both quickly to the tepid water. It will feel cold to the hand that has been in hot water and warm to the other. Hold the hand successively against a number of the various objects in the room, at about the same height from the floor. Metal, slate, or stone ob- jects will feel colder than those of wood, even when side by side and of the same temperature. These experiments show that the sense of touch does not give accurate information regarding the relative tem- perature of bodies, and some other method must be re- sorted to for reliable measurement. The one most ex- tensively used is based on the regular increase in the vol- ume of a body attending a rise in its temperature. This method is illustrated by the common mercurial ther- mometer. 282 HEAT 70! II. THE THERMOMETER 323. The Thermometer. The common mercurial ther- mometer consists of a capillary glass tube of uniform bore, on one end of which is blown a bulb, either ft V spherical or cylindrical (Fig. 301). Part of the air is expelled by heating, and while in this condition the open end of the tube is dipped into a vessel of pure mercury. As the tube cools, mercury is forced into the tube by atmospheric pressure. Enough mer- cury is introduced to fill the bulb and part of the tube at the lowest temperature which the thermometer is designed to measure. Heat is now applied to the bulb until the ex- panded mercury fills the tube ; the end is then closed in the blowpipe flame. The mercury contracts as it cools, leaving the larger portion of the capillary empty. 324. The Necessity of Fixed Points. No two thermometers are likely to have bulbs and stems of the same capacity. Conse- quently, the same increase of temperature will not produce equal changes in the length of the thread of mercury. If, then, the same scale were attached to all thermometers, their indications would differ so widely that the results would be worthless. Hence, if ther- mometers are to be compared, corresponding divisions on the scale of different instruments must indicate the same temperature. This may be done by graduating every thermometer by comparison with a standard, an expensive proceeding and for many purposes unnecessary, since mer- cury has a nearly uniform rate of expansion. If two FIGURE 301. C. AND F. THERMOME- TERS. MARKING THE FIXED POINTS 283 points are marked on the stem, the others can be obtained by dividing the space between them into the proper num- ber of equal parts. Investigations have made it certain that under a constant pressure the tem- perature of melting ice and that of steam are in- variable. Hence, the temperature of melting ice and that of steam under a pressure of 76 cm. of mercury (one atmosphere) have been chosen as the fixed points on a thermometer. 325. Marking the Fixed Points. The ther- mometer is packed in finely broken ice, as far up the stem as the mercury extends. The contain- ing vessel (Fig. 302) has an opening at the bot- tom to let the water run out. After standing in the ice for several minutes the top of the thread of mercury is marked on the stem. This is called the freezing point. The boiling point is marked by observ- ing the top of the mercurial column when the bulb and stem are enveloped in steam (Fig. 303) under an atmospheric pressure of 76 cm. (29.92 in.). If the pressure at the time is not 76 cm., then a correction must be applied, the amount being de- termined by the approximate rule that the temperature of steam rises 0.1 C. for every increase of 2.71 mm. in the barometric reading, near 100 C. 326. Thermometer Scales. The dis- tance between the fixed points is divided into equal parts called degrees. The number of such parts is wholly arbitrary, and several different scales have been FIGURE 302 . FREEZING POINT. FIGURE 303. MARKING THE BOIL- ING POINT. 284 HEAT introduced. The number of thermometer scales in use in the eighteenth century was at least nineteen. Fortunately all but three of them have passed into ancient history. The Fahrenheit scale, which is in general use in English- speaking countries, appears to have made its first appear- ance about 1714, 'but the earliest published description of it was in 1724. At that time this scale began at and ended at 96. Fahrenheit describes his scale as deter- mined by three points : the lowest was the and was found by a mixture of ice, water, and sea salt ; the next was the 32 point and was found by dipping the ther- mometer into a mixture of ice and water without salt ; the third was marked 96, the point to which alcohol expanded "if the thermometer be held in the mouth or armpit of a healthy person." When this scale was ex- tended, the boiling point was found to be 212. The space between the freezing and the boiling point is there- fore 180. The Centigrade scale was introduced by Celsius, pro- fessor of astronomy in the University of Upsala, about 1742. It differs from the Fahrenheit in making the freez- ing point and the boiling point 100, the space between being divided into 100 equal parts or degrees. The sim- plicity of Celsius's division of the scale has led to its general adoption in all countries for scientific purposes, and in many for domestic and industrial use. The scale in both thermometers is extended beyond the fixed points as far as desired. The divisions below are read as minus and are marked with the negative sign. The initial letters F. and C. denote the Fahrenheit and Centigrade scales respectively. 327. The Two Scales Compared. In Fig. 304 AB is a thermometer with two scales attached, P is the head THE CLINICAL THERMOMETER 285 of the mercury column, and F and O are the readings on the scales respectively. On the Fahrenheit scale, AB = ISQ and AP ^^^ * = ^32, since the < ^^* zero is 32 spaces be- FahreDheit 3 , 2 Centigrade O 100 low A' 9 on the Centi- FIGURE 304.-- SCALES COMPARED. grade, AB = 100 and AP = O. Then the ratio of AP to AB is JZf = _i_ . By substituting the reading on either scale in this equa- tion the equivalent on the other scale is easily obtained. For example, if it is required to express 68 F. on the ^O OO i~1 Centigrade scale, then = ; whence C 20 328. Limitations of the Mercurial Thermometer. As mercury freezes at 38.8 C., it cannot be used as the therm ometric substance below this temperature. For temperatures below 38 C. alcohol is substituted for mercury. Under a pres- sure of one atmosphere mercury boils at about 350 C. For temperatures approaching this value and up to about 550 C. the thermometer stem is filled with pure nitrogen under pressure. The pressure of the gas keeps the mercury from boiling ( 356). 329. The Clinical Thermometer. -^ The clinical thermometer is a sensitive instrument of short range for indicating the temperature of the human body. It is usually graduated from 95 to 110 FIGURE ^., or from 35 to 45 C. There is a constriction 305. in the tube just above the bulb (Fig. 305), which T L ^ C ^ causes the thread of mercury to break at that point MOMETER. when the temperature begins to fall, leaving the 286 HEAT top of the separated thread to mark the highest tempera- ture registered. A sudden jerk or tapping of the ther- mometer forces the mercury down past the constriction and sets it for a new reading. The normal temperature of the human body is 98.6 F. or 37 C. Questions and Problems 1. Why do the degree spaces differ in length on different ther- mometers of the same scale ? 2. What advantages does a thermometer with a cylindrical bulb have over one with a spherical bulb ? 3. Why is mercury preferable to other liquids for use in ther- mometers ? 4. Why is it necessary to have fixed points in thermometers? 5. The bulb of a thermometer generally contracts a little after the thefmometer is completed. What is the result on the readings? 6. Why is nitrogen used in preference to oxygen in thermometers for high temperatures ? 7. Why should the thermometer tube be of uniform bore? 8. Express in Fahrenheit degrees the following 4 C., 30 C., - 38 C. 9. Express in Centigrade degrees the following 39 F., - 40 F., 68 F. 10. The fixed points on a Centigrade thermometer were found to be incorrect; the freezing point read 2 and the boiling point 99. When this thermometer was immersed in a liquid the reading was 50. What was the correct temperature of the liquid ? (NOTE. Compare this incorrect thermometer with a correct one just as a Fahrenheit thermometer is compared with a Centigrade in 327.) 11. A thermometer read 40 C. in a water bath. When tested it was found to read at the freezing point, but 95 instead of 100 at the boiling point. What was the correct temperature of the bath ? 12. If a Fahrenheit thermometer read 210 in steam and 31 in melting ice, what would it read as the equivalent of 70 F. ? EXPANSION OF SOLIDS 287 13. A correct Fahrenheit thermometer read 70 as the temperature of a room. An incorrect Centigrade thermometer read 20 in the same room. What was the error of the latter? 14. A certain Centigrade thermometer reads 2 in melting ice and 100 in steam under normal atmospheric pressure. What is the correct value for a reading of 25 on this thermometer ? III. EXPANSION 330. Expansion of Solids. Insert a long knitting needle A in a block of wood so as to stand vertically (Fig. 306). A second needle D is supported paral- lel to the first by means of a piece of cork or wood C. The lower end of D just touches the mercury in the cup H. An electric circuit is made through the mer- cury, the needle, an electric battery, and the bell B, as shown. Now apply a Bun- sen flame to A ; D will be lifted out of the mercury and the bell will stop ringing. Then heat D or cool A, and th e contact of D with the mer- cury will be renewed as shown by the ringing of the bell. FIGURE 306. SHOWING EXPANSION. This experiment shows that solids expand in length when heated and contract when cooled. To this rule of expansion there are a few exceptions, notably iodide of sil- ver and stretched india-rubber. Rivet together at short intervals a strip of sheet copper and one of sheet iron D (Fig. 307). Support this compound bar so as to play between two points A and C, which are connected through the battery P and the bell B. Apply a Bunsen flame to the bar. It will warp, 288 HEAT throwing the top over against either A or C, and will cause the bell to ring. FIGURE 307. UNEQUAL EXPANSION. The experiment shows that the two metals expand unequally and cause the bar to warp. Figure 308 illustrates a piece of ap- paratus known as Gravesande's ring. It consists of a metallic ball that at ordinary temperatures will just pass through the ring. Heat the ball in boiling water. It will now rest on the ring and will not fall through until it has cooled. FIGURE 308. EXPANSION OF BALL. We conclude that the ex- pansion of a solid takes place in every direction. 331. Expansion of Liquids. Select two four-inch test tubes, fit to each a perforated stopper, through which passes a small glass tube about six inches long. The capacity of the two test tubes after stoppers are inserted should be equal. Fill one tube with mercury and the other with glycerine colored with an aniline dye. Set the test tubes in a beaker of water over a Bunsen flame (Fig. 309) and note the change in the height of the liquids in the tubes. Two facts are illustrated: first, liquids _ Qno _. . ' FIGURE OUV. E.X- are affected by heat in the same way as PANSION OF LIQUIDS. COEFFICIENT OF LINEAR EXPANSION 289 solids ; second, the expansion of the liquids is greater than that of the glass or there would be no apparent increase in their volume. Some liquids do not expand when heated at certain points on the thermometric scale. Water, for example, on heating from C. to 4 C. contracts, but above 4 C. it expands. 332. Expansion of Gases. Fit a bent delivery tube to a small Florence flask (Fig. 310). Fill the flask with air and place the up- turned end of a delivery tube under an inverted graduated glass cylinder filled with water. Heat the flask by immersing it in a vessel of moderately hot water. The air will expand and escape through the delivery tube into the cylinder; note the amount. Now refill the flask with some other gas, as coal gas, and re- peat the experiment. The amount of gas collected will be nearly the same. FIGURE 310. EXPANSION OF GASES. Investigation has shown that all gases which are hard to liquefy expand very nearly alike at atmospheric pressure, approaching equality as the pressure is diminished. Gases that are easily liquefied, as carbon dioxide, show the largest variation in the expansion. 333. Coefficient of Linear Expansion. Nearly all solids expand with increase of temperature, but they do not expand equally. Assume three rods of the same length, zinc, brass, and steel. With the same rise of temperature, the zinc rod will increase in length 50 per cent more than the brass, and the brass nearly 50 per cent more than the steel. A brass bar will expand in length 20 times as much as a bar of " invar " (nickel steel) if the bars are 290 HEAT of the same length and undergo the same change of temperature. The coefficient of linear expansion, or expansion in length, expresses this property of expansion in a numerical way. It is the increase in a unit length of a substance per degree increase in temperature. This is equivalent to the ex- pression : Coefficient of linear expansion increase in length original length x temp, change If Zj and 2 denote the lengths of a metallic rod at tem- peratures j and 2 respectively, then 2 ~ 1 = 2 ~ 1 is the ^2 ~~ *i * whole expansion for 1 ; t is the difference of temperature. If a denotes the coefficient of expansion, then a = -2 1 ; whence 1 2 = ^(1 + at). V Since this coefficient is a ratio, it makes no difference what unit of length is used. Coefficients of expansion are usually given in terms of the Centigrade degree. For the Fahrenheit degree the coefficient is, of course, J as great as for the Centigrade. SOME COEFFICIENTS OF LINEAR EXPANSION Invar 0.0000009 Copper 0.0000172 Glass 0.0000086 Brass 0.0000188 Platinum . . . . 0.0000088 Silver 0.0000191 Cast Iron .... 0.0000113 Tin 0.0000217 Steel 0.0000132 Zinc 0.0000294 334. Illustrations of Linear Expansion. Many familiar facts are accounted for by expansion or contraction attending changes of temperature. If hot water is poured into a thick glass tumbler, the glass will probably crack by reason of the stress produced by the BRIDGE OVER THE FIRTH OF FORTH. Allowance for the expansion and contraction of the steel must be made in the construction. COMPENSATED CLOCKS AND WATCHES 291 sudden expansion of its inner surface. On the other hand, crucibles and other laboratory utensils are now made of clear fused quartz ; fused quartz has so small a coefficient of expansion that a red-hot crucible may be plunged into water without cracking. The coefficients of glass and platinum are so nearly equal that platinum wires may be sealed into glass without cracking the latter when it cools. Crystalline rocks, on account of unequal expansion in different directions, are slowly disintegrated by changes of temperature ; and for the same reason quartz crystals, when strongly heated, fly in pieces. The outcropping granite hills of the celebrated South African Matopos have been broken into huge boulders and irregular masses by the large expansion in the fervid heat of midday and the subse- quent rapid contraction during the low temperature of the succeeding night. In long steel bridges built in cold climates considerable expansion occurs in summer, and a certain freedom of motion of the parts must be provided for. Long suspension bridges are several inches higher at the middle in midwinter than in the heat of summer. Long steam pipes are fitted with expansion joints to permit one part to slide into the other ; bends or elbows in the pipe are also used, so that the pipe may ac- commodate itself to the expansion. 335. Compensated Clocks and Watches. If the length of a pendulum changes with temperature, the period of vibration will also change and the clock will not have a constant rate. The balance wheel of a watch serves a similar purpose of regulating the period of vibration and is similarly affected by changes in temperature. To com- pensate for these changes so as to keep the period of vi- bration constant, the principle of unequal expansion is em- ployed. The bob of a compensated mercurial pendulum consists of one or more glass jars, nearly filled with mercury, and attached to the lower end of a steel rod (Fig. 311). A of temperature lengthens the rod and lowers the rse FIGURE 311. M ERCU- RIAL PEN- DULUM. center of oscillation ; but the mercury expands upward and compensates by raising the center of oscillation. By a proper adjustment of the quantity of mercury in the tubes, its expansion may be made to compensate for that of the rod. 292 v HEAT The rate of a watch depends largely on the balance wheel. Unless this is compensated, it expands when the temperature rises and the watch loses time, the larger wheel oscillating more slowly under the force supplied by the elasticity of the hairspring. Compensation is secured by making the rim of the wheel in two sections, each being made of two materials and supported by one end on a separate arm (Fig. 312). The more expansible metal is on the out- side. When the temperature, rises and the radial FIGURE 312. COM- arms expand, the loaded free ends a, a' of the PENS AT ED BALANCE sec tionsmove inward, thus compensating for the increased length of the radial arms. The final adjustment is made by screwing in or out the studs on the rim. 336. Cubical Expansion. In general, solids and liquids when heated expand in all directions with increase of volume. This expansion in volume is called cubical ex- pansion. The coefficient of cubical expansion is the increase in volume of a unit volume per degree rise of temperature. Precisely as in the case of linear expansion, the coeffi- cient of cubical expansion Jc may be expressed by the equation Whence The coefficient of cubical expansion of a substance is three times its coefficient of linear expansion. Thus if the coefficient of linear expansion of cast iron is 0.0000113, its coefficient of cubical expansion is 0.0000339. 337. Expansion of Water. Water exhibits the remark- able property of contracting when heated at the freezing point. This contraction continues up to 4 C., when ex- pansion sets in. The greatest density of water is therefore at 4 C., and its density at 6 is nearly the same as at 2. THE ABSOLUTE SCALE 293 In a lake or pond water at 4 sinks to the bottom, while water below 4 is lighter and rises to the top, where the freezing begins. Ice forms at the surface of a body of cold water, which freezes from the surface downwards. Fishes are thus protected from freezing. 338. Law of Charles. It was shown by Charles, in > 1787, that the volume of a given vnass of any gas under constant pressure increases by a constant fraction of its volume at zero for each rise of temperature of 1 C. The investigations of Regnault and others show that the law is not rigorously true, and that the accuracy of Charles's law is about the same as that of Boyle's law. The coeffi- cient of expansion k of dry air is 0.003665, or about gy-j. This fraction may be considered as the coefficient of ex- pansion of any true gas. 339. The Absolute Scale. The law of Charles leads to a scale of temperature called the absolute scale. By this law the volumes of any mass of gas, under constant pres- sure, at C., and at any other temperature t C., are connected by the following relations ( 336) : At any other temperature, t r , the volume becomes , 273 Divide (a) by (6) and v Suppose now a new scale is taken, whose zero is 273 Centigrade divisions below the freezing point of water, 294 HEAT and that temperatures on this scale are denoted by T. Then 273 + 1 will be represented by T, and 273 + 1' by v 1 273 + *' T f1 or he volumes of the same mass of gas under constant pres- sure are proportional to the temperatures on this new scale. The point 273 below C. is called the absolute zero, and the temperatures on this scale, absolute temperatures. Up to the present it has not been found possible to cool a body to the absolute zero; but by evaporating liquid hydrogen under very low pressure, a temperature esti- mated to be within 9 of the absolute zero has been ob- tained by Professor Dewar; and Professor Onnes, by liquefying helium, believes that he obtained a tempera- ture within 2 or 3 of the absolute zero. At these low temperatures steel and rubber become as brittle as glass. 340. The Gas Equation. The laws of Boyle and Charles may be combined into one expression, which is known as the gas equation. It has a wider application even than its method of derivation would indicate. Let v , jt? , T Q be the volume, pressure, and absolute tem- perature of a given mass of gas. Also let v, JP, T be the corresponding quantities for the same mass of gas at pressure p and temperature T. Then applying Boyle's law ( 87) to increase the pres- sure to the value p, the temperature remaining constant, we have V* ....... oo v' p Q where v r is the new volume corresponding to the pres- sure p. QUESTIONS AND PROBLEMS 295 Next apply the law of Charles ( 339), keeping the pressure constant at the value jo, and starting with the new volume v f . Then since the volumes are directly proportional to the temperatures, we have <> where v is the new volume corresponding to temperature T. Multiply (a) and (5) together member by member, and ^o 9 or PA = 2. = a constant, since p and ^Tare any pressure and temperature and v corresponds. This con- stant is usually denoted by R. We may therefore write pv = RT. . . . (Equation 33) To illustrate the use of the above relation : If 20 cm. 8 of gas 'at 20 C. is under a pressure of 76 cm. of mercury, what will be the pressure when its volume is 30 cm. 8 and temperature 50 C.? From equation (33), ^ is a constant, or 273 from which Questions and Problems 1. Telegraph wires often " hum " in the wind. Why is the pitch higher in winter than in summer ? 2. When a piece of ice floats, about ^ of its volume projects out of the water. If a pan is level full of water and a piece of ice floats in it, both at C., why is there no change of level when the ice melts? 3. Why is a fountain pen more likely to leak when nearly empty ? 296 BEAT 4. If the bulb of a mercurial thermometer is plunged into hot- water, the top of the thread of mercury first falls and then rises. Explain. ' 5. A copper rod 125 cm. long at C. expands to 125.209 cm. at 100 C. Find the coefficient of linear expansion of copper. 6. The coefficient of linear expansion of steel is 0.0000132. What will be the variation in length of a steel bridge 250 ft. long between the temperatures - 10 C. and 40 C.? 7. The coefficient of linear expansion of steel is 0.0000132 and that of zinc is 0.0000294. What relative lengths of rods of these metals will have equal expansions in length for the same changes of temperature ? 8. The coefficient of the volume expansion of glass is 0.000258. A density bottle at 15 C. holds 25 cm. 8 . What will be its capacity at25C.? 9. Why should the reading of the mercurial barometer be cor- rected for temperature? If the relative volume coefficient of expan- sion of mercury in glass is 0.000155, and the barometer reads 755 mm. at 20 C., what would be the reduced reading at C. ? 10. The volume of a given mass of gas at 740 mm. pressure is 1200 cm. 8 ; find its volume at 760 mm. 11. The mass of a liter of air at C. and 76 cm. pressure is 1.3 g. Find the mass of 10 liters of air at 20 C. and 74 cm. pressure. 12. A liter of hydrogen at 15 C. is heated at constant pressure to 75 C. Find its volume. 13. A quantity of gas is collected in a graduated tube over mer- cury. The reading of the mercury level in the tube is 20 cm., the volume of the gas is 60 cm. 3 , the temperature is 20 C., and the ba- rometer reading is 74cm. How many cubic centimeters of gas are there at C. and 76 cm. pressure? 14. Three cubic centimeters of air are introduced into the vacuum of a mercurial barometer. The barometer read 76 cm. before intro- ducing the air and 57 cm. after. What volume does the air occupy in the barometer ? SPECIFIC HEAT 297 IV. MEASUREMENT OF HEAT 341. The Unit of Heat. The unit of heat in the c. g. s. system is the calorie. It is defined as the quantity of heat that will raise the temperature of one gram cf water one de- gree Centigrade. There is no agreement as to the position of the one degree on the thermometric scale, although it is known that the unit quantity of heat varies slightly at different points on the scale. If the degree' interval chosen is from 15 to 16 C., the calorie is then the one hundredth part of the heat required to raise the tempera- ture of one gram of water from to 100 C. In engineering practice in England and America the British thermal unit (B. T. U.) is commonly employed. It is the heat required to raise the temperature of one pound of water one degree Fahrenheit. 342. Thermal Capacity. The thermal capacity of a body is the number of calories required to raise its temperature one degree Centigrade. The thermal capacity of equal masses of different substances differs widely. For example, if 100 g. of water at C. be mixed with 100 g. at 100 C., the temperature of the whole will be very nearly 50. C. But if 100 g. of copper at 100 C. be cooled in 100 g. of water at C., the final temperature will be about 9.1 C. The heat lost by the copper in cooling through 90.9 is sufficient to heat the same mass of water only 9.1, that is, the thermal capacity of water is about ten times as great as that of an equal mass of copper. 343. Specific Heat. The specific heat of a substance is the number of calories of heat required to raise the tem- perature of one gram of it through one degree Centigrade. It may be defined independently of any temperature scale as the ratio between the number of units of heat required 298 HEAT to raise the temperature of equal masses of the substance and of water through one degree. The specific heat of mercury is 0.033, that is, the heat that will raise 1 g. of mercury through 1 C. will raise 1 g. of water through only 0.033 C. The specific heat of water is twice as great as that of ice (0.505), and more than twice as great as that of steam under constant pressure (0.477). 344. Numerical Problem in Specific Heat. The principle ap- plied in the solution of such problems is that the gain or loss of heat by the water is equal to the loss or gain of heat by the body introduced into the water. The gain or loss of heat by the body is equal to the product of its mass, its specific heat, and its change of temperature. To illustrate : 20 g. of iron at 98 C. are placed in 75 g. of water at 10 C. contained in a copper beaker weighing 15 g., specific heat 0.095. The resulting temperature of the water and the iron is 12.5 C. Find the specific heat of iron. The thermal capacity of the beaker is 15 x 0.095 = 1.425 calories. The heat lost by the iron is 20 x s x (98 12.5) calories, in which s represents the specific heat of iron, and (98 12.5) its change of tem- perature. The heat gained by the water and the copper vessel is (75 + 1.425) x (12.5 10) calories ; the second factor is the gain in temper- ature of the water and the beaker. It follows by equating these two quantities that 20 x s x (98 - 12.5) = (75 + 1.425) x (12.5 - 10). Solving for s, we have s = 0.112 calorie per gram. Questions and Problems 1. What is the specific heat of water? 2. A pound of water and a pound of lead are subjected to the same source of heat for 10 inin. Which will be at the higher tem- perature ? 3. If equal quantities of heat are applied to equal masses of iron and lead, which will show the greater change of temperature ? 4. Equal balls of iron and zinc are heated in boiling water and are placed on a cake of beeswax. Which will melt the further into the wax ? THE MELTING POINT 299 5. Why is water better than any other liquid for heating purposes ? 6. Why is a rubber bag filled with hot water better for a foot warmer than an equal mass of any solid ? 7. A copper beaker has a mass of 25 g. The specific heat of cop- per is 0.095. What is the thermal capacity of the beaker ? 8. How much heat will it take to raise a liter of water from 20 C. to 100 C.? 9. The specific heat of iron is 0.112. How much heat will be re- quired to raise 250 g. of iron from 10 to 45 C. ? 10. 120 g. of water at 5 C. are mixed with 200 g. of water at 50 C. Assuming that no heat is lost, what will be the resulting temperature? 11. 89.2 g. of iron at 90 C. are placed in 70 g. of water at 10 C. ; the resulting temperature is 20 C. Find the specific heat of iron. 12. A copper ball weighing 1 kg. has a specific heat of 0.095. It is heated in a furnace to the temperature of the furnace and dropped into a liter of water at 10 C. The temperature of the water rises to 93.1 C. Find the temperature of the furnace. 13. How many calories in the British Thermal Unit ? 14. A glass beaker weighs 100 g. If the specific heat of the glass is 0.177, how much water will have the same thermal capacity as the beaker? 15. Why do islands in the sea have smaller extremes of temperature than inland areas ? V. CHANGE OF STATE 345. The Melting Point. A body is said to melt or fuse when it changes from the solid to the liquid state by the application of heat. The change is called melting, fusion, or liquefaction. The temperature at which fusion takes place is called the melting point. Solidification or freezing is the converse of fusion, and the temperature of solidifi- cation is usually the same as the melting point of the same substance. Water, if undisturbed, may be cooled a .number of degrees below C., but if it is disturbed it 300 HEAT usually freezes at once, and its temperature rises to the freezing point. The melting point of crystalline bodies is well marked. A mixture of ice and water in any relative proportion will remain without change if the temperature of the room is C. ; but if the temperature is above zero, some of the ice will melt; if it is below zero, some of the water will freeze. Some substances, like wax, glass, and wrought iron, have no sharply defined melting point. They first soften and then pass more or less slowly into the condition of a viscous liquid. It is this property which permits of the bending and molding of glass, and the rolling, welding, and forging of iron. 346. Change in Volnme accompanying Fusion. Fit to a small bottle a perforated stopper through which passes a fine glass tube. Fill with water recently boiled to expel the air, the water ex- tending halfway up the tube. Pack the apparatus in a mixture of salt and finely broken ice. The water column at first will fall slowly, but in a few minutes it will begin to rise, and will continue to do so until water flows out of the top of the tube. The water in the bottle freezes, expands, and causes the overflow. Most substances occupy a larger volume in the liquid state than in the solid; that is, they expand in liquefying. A few substances, like water and bismuth, expand in solidifying. ' When water freezes, its volume increases 9 per cent. If this expansion is resisted, water in freezing is capable of exerting a force of about 2000 kg. per square centimeter. This explains the bursting of water pipes when the water in them freezes, and the rending of rocks by the freezing of water in cracks and crevices. The expansion of cast iron and type metal when they solidify accounts for the exact reproduction of the mold in which they are cast. BEAT OF FUSION 301 347. Effect of Pressure on the Melting Point. Support a rectangular block or prism of ice on a stout bar of wood. Pass a thin iron wire around the ice and the bar of wood, and suspend on it a weight of 25 to 50 Ib. The pressure of the wire lowers the melting point of the ice immediately under it and the ice melts ; the water, after passing around the wire, where it is relieved of pressure, again freezes. In this way the wire passes slowly through the ice, leaving the block solidly frozen. A rough numerical statement of the effect of pressure on the freezing point of water is that a pressure of one ton per square inch lowers the freezing point to 1 C. Familiar examples of refreezing, or regelation, are the hardening of snowballs under the pressure of the hands, the formation of solid ice in a roadway where it is compressed by vehicles and the hoofs of horses, and frozen footforms in compact ice after the loose snow has melted around them. The ice of a glacier melts where it is under the enormous pressure of the descending masses F|CURE 3,3. -MERDE GLACE, CHAMOUN.X. above it. The melting permits the ice to accommodate itself to abrupt changes in the rocky channel, and a slow iceflow results. As soon as the pressure at any surface is relieved, the water again freezes (Fig. 313). 348. Heat of Fusion. When a solid melts, a quantity of heat disappears ; and, conversely, when a liquid solidi- fies, the amount of heat generated is the same as dis- 302 HEAT appears during liquefaction. The heat of fusion of a substance is the number of calories required to melt a gram of it without change of temperature. The heat of fusion of ice is 80 calories. As an illustration of the heat of fusion, place 200 g. of clean ice, broken into small pieces, into 500 g. of water at 60 C. When the ice has melted, the temperature will be about 20 C. The heat lost by the 500 g. of water equals 500 x (60 - 20) = 20,000 calories. This heat goes to melt the ice and to raise the resulting water from C. to 20 C. To raise this water from to 20 requires 200 x 20 = 4000 calories. The remainder, 20,000 - 4000 = 16,000 calories, went to melt the ice. Then the heat of fusion of ice is 16,000 -r- 200 = 80 calories per gram. 349. Heat lost in Solution. Fill a beaker partly full of water at the temperature of the room, and add some ammonium nitrate crystals. The temperature of the water will fall as the crystals dissolve. This experiment illustrates the fact that heat disap- pears when a body passes from the solid to the liquid state by solution. The use of salt in soup or of sugar in tea absorbs heat. The heat energy is used to pull down the solid structure. 350. Freezing Mixtures. Freezing mixtures are based on the absorption of heat necessary to give fluidity. Salt water freezes at a lower temperature than fresh water. When salt and snow or pounded ice are mixed together, both become fluid and absorb heat in the passage from the one state to the other. By this mixture a tempera- ture of 22 C. may be obtained. Still lower tempera- tures may be reached with other mixtures, notably with sulpho-cyanide of sodium and water. 351. Vaporization. Pour a few drops of ether into a beaker and cover closely with a plate of glass. After a few seconds bring a lighted taper to the mouth of the beaker. A sudden flash will show that the vapor of ether was mixed with the air. COLD BY EVAPORATION 303 Support on an iron stand a Florence flask two-thirds full of water and apply heat. In a short time bubbles of steam will form at the bottom of the flask, rise through the water, and burst at the top, pro- ducing violent agitation throughout the mass. Vaporization is the conversion of a substance into the gaseous form. If the change takes place slowly from the surface of a liquid, it is called evaporation ; but if the liquid is visibly agitated by rapid internal evaporation, the process is called ebullition or boiling. 352. Sublimation. When a substance passes directly from the solid to the gaseous form without passing through the intermediate state of a liquid, it is said to sublime. Arsenic, camphor, and iodine sublime at atmos- pheric pressure, but if the pressure be sufficiently in- creased, they may be fused. Ice also evaporates slowly even at a temperature below freezing. Frozen clothes dry in the air in freezing weather. At a pressure less than 4.6 mm. of mercury, ice is converted into vapor by heat without melting. 353. The Spheroidal State. When a small quantity of liquid is placed on hot metal, as water on a red-hot stove, it assumes a globular or spheroidal form, and evaporates at a rate between ordinary evaporation and boiling. It is then in the spheroidal state. The vapor acts like a cushion and prevents actual contact between the liquid and the metal. The globular form is due to surface ten- sion. Liquid oxygen at 180 C. assumes the spheroidal form on water. The temperature of the water is rela- tively high compared with that of the liquid oxygen. 354. Cold by Evaporation. Tie a piece of fine linen around the bulb of a thermometer and pour on it a few drops of sulphuric ether. The temperature will at once begin to fall, showing that the bulb has been cooled. 304 HEAT In the evaporation of ether, some of the heat of the thermometer is used to do work on the liquid. Sprinkling the floor of a room cools the air, because of the heat expended in evaporating the water. Porous water vessels keep the water cool by the evaporation of the water from the outside surface. Liquid carbon dioxide is readily frozen by its own rapid evaporation. Dewar liquefied oxygen by means of the temperature obtained through the successive evaporation of liquid nitrous oxide and ethylene. Similarly, by the evapora- tion of liquid air he has liquefied hydrogen. The evapo- ration of liquid hydrogen under reduced pressure has 'I 1 To Sewer Regulating Valve FIGURE 314. ICE PLANT. enabled him to obtain a temperature but little removed from the absolute zero, 273 C. More recently Pro- fessor Onnes of Leyden, by the evaporation of liquid helium, has reached the extremely low temperature of -271.3 C. or 1.7 absolute. 355. Ammonia Ice Plant. The low temperature produced by the rapid evaporation of liquid ammonia is utilized in the manufac- ture of ice and for general cooling in refrigerator plants. Ammonia may be liquefied by pressure alone. At a temperature of 80 F. the EFFECT OF PRESSURE ON THE BOILING POINT 305 required pressure is 155 pounds per square inch. The essential parts of an ice plant are shown in Fig. 314. Gaseous ammonia is com- pressed by a pump in condenser pipes, over which flows cold water to remove the heat. From the condenser the liquid ammonia passes very slowly through a regulating valve into the pipes of the evapo- rator. The pressure in the evaporator is kept low by the pump, which acts as an exhaust pump on one side and as a compressor on the other. The pump removes the evaporated ammonia rapidly and the evapora- tion absorbs heat. The pipes in which the evaporation takes place are either in a tank of brine, or in the refrigerating room. Smaller tanks of distilled water are placed in the brine until the water in them is frozen. The pipes in the refrigerating room are covered with hoar frost, which is frozen moisture from the air. The temperature of the brine is reduced to about 16 to 18 F. The brine does not freeze at this tempera- ture. The process is continuous because the gaseous ammonia is returned to the con- densing coils, which are cooled with water. It thus passes repeatedly through the same cycle of physical changes. 356. Effect of Pressure on the Boil- ing Point. Place a flask of warm water under the receiver of an air pump. It will boil violently when the receiver is ex- hausted* Fill a round-bottomed Fj^rence flask half full of water and heat until it boils vigorously. Cork the flask, invert, and sup- port it on a ring stand (Fig. 315). The boiling ceases, but is re- newed by applying cold water to the flask. The cold water condenses the vapor, and reduces the pressure within the flask so that the boil- ing begins again. The effect of pressure on the boiling point is seen in the low temperature of boiling water at high elevations, and in the high temperature of the water under pressure in digesters used for extracting gelatine from bones. The boiling point of water falls 1 C. for an increase in eleva- FIGURE 315. BOILING UNDER REDUCED PRESSURE. 306 HEAT tion of about 295 in. At Quito the boiling point is near 90 C. 357. Heat of Vaporization. The heat of vaporization is the number of calories required to change one gram of a liquid at its boiling point into vapor at the same tempera- ture. Water has the greatest heat of vaporization of all liquids. The most carefully conducted experiments show that the heat of vaporization of water under a pressure of one atmosphere is 536.6 calo- ries per gram. Set up apparatus like that shown in Tig. 316. The steam from the boiling water is conveyed into a beaker containing a known quantity of water at a known temperature. The increase in the mass of water gives the amount of steam con- densed. The " trap " in the delivery tube catches the water that condenses before it reaches the beaker. Sup- pose that the experiment gave the following data: Amount of water in the beaker, 400 g. at the begin- ning, 414.1 g. at the end, including the thermal capacity of the beaker in terms of water ; temperature at the beginning, 20 C., and at the end, 41 C. ; observed boiling point, 99 C. ; there were 14.1 g. of steam condensed. Now, by the principle that the heat lost or given off by the steam equals that gained by the water, we have 400 x (41 - 20) = 14.1 x / + 14.1 x (99 - 41) ; whence I = 537.7 cal. per gram. 358. Formation of Dew. The presence of clouds and the " sweating " of pitchers filled with ice water show that the atmosphere contains water vapor. The amount of water FIGURE 316. HEAT OF EVAPORATION. HUMIDITY AND HEALTH 307 vapor that the atmosphere can hold in suspension depends on its temperature. After sunset, if the sky is clear, bodies on the earth's surface, such as grass, leaves, and roots, soon cool below the temperature of the surrounding air, and water in the form of dew collects on them. Clouds act as blankets and prevent the cooling off process, so that little or no dew collects. Wind promotes evaporation and dew fails to collect. If the temperature falls sufficiently low, the dew is deposited as frost. 359. The Dew Point. The dew point is the temperature at which the aqueous vapor of the atmosphere begins to con- dense. If water at the temperature of the room be poured into a polished nickel-plated beaker and small pieces of ice be added with stirring, a mist will soon collect on the out- side of the beaker. The temperature of the water is then the dew point. 360. Humidity. The terms dryness and moisture ap- plied to the air are purely relative. Usually the air is not saturated, that is, it does not contain all the water vapor it can hold. If it is near the saturation point, it is moist; if it is very far from saturation, it is dry. The relative humidity of the air is the ratio between the amount of water vapor actually present and the amount that would be present if the air were saturated at the same temperature. The air is saturated at the dew point. A dry day is one on which the dew point is much below the temperature of the air ; a damp day is one on which the dew point is close to the temperature of the air. Humidity is expressed as a per cent of saturation. 361. Humidity and Health. The humidity of the air has an important bearing on health. The dry air of a furnace-heated house promotes excessive evaporation from the bodies of the occupants, producing sensations of chil- 308 HEAT liness and discomfort. On the other hand, excessive humidity retards healthful evaporation, gives a sensation of depression, and in hot weather checks Nature's method of keeping cool by evaporation. The humidity conducive to health is about 50 per cent. Questions and Problems 1. Why does a drop of alcohol on the hand feel cold? 2. Why does a shower in summer cool the air? 3. Why is there less dew on gravel than on the grass? 4. Why can blocks of ice be made to adhere by pressure? 5. Why do your eye glasses fog over when you go from the cold air outside into a warm room ? 6. Water in a porous vessel standing in a current of air is colder than water in a glass pitcher. Why? 7. Why does warming a room make it feel dryer ? 8. Why does water boil away faster on some days than on others? 9. Why does wind dry up the roads after a rain? 10. Why does moisture collect on the carburetor of a gasoline engine when it is in operation unless it is heated? 11. How much heat does it take to convert 50 g. of water at 100 C. into steam at 100 C.? 12. How much ice will 100 g. of water at 100 C. rnelt? 13. How much ice will 100 g. of steam at 100 C. melt? 14. 100 g. of ice and 20 g. of steam at 100 C. are put into a calorimeter. If no heat is lost, what will be the temperature of the water after all the ice is melted ? 15. How much water at 80 C. will just melt a kilogram of ice? 16. How much steam will be required to raise the temperature of a kilogram of water from 20 to 50 C. ? 17. How much ice will it take to cool a kilogram of water from 50 to 20 C. ? 18. Mt. Washington is 6288 ft. above sea level; at what tempera- ture does water boil on its top ? CONDUCTION 309 19. Water boils in the City of Mexico at 92.3 C. What is its elevation above the sea ? 20. 50 g. of ice at C. are put into 50 g. of water at 35 C. How much of the ice will melt ? VI. TRANSMISSION OF HEAT 362. Conduction. Twist together two stout wires, iron and cop- per, of the same diameter, forming a fork, with long parallel prongs FIGURE 317. DIFFERENCE IN CONDUCTIVITY. and a short stem. Support them on a wire stand (Fig. 317), and heat the twisted ends. After several minutes find the point on each wire, farthest from the flame, where a sulphur match ignites when held against the wire. This point will be found farther along on the cop- per than on the iron, showing that the former has led the heat farther from its source. Prepare a cylinder of uniform diameter, naif of which is made of brass and half of wood. Hold a piece of writing paper firmly around the junction like a loop (Fig. 318). By applying a Bunsen flame the paper in contact with the wood is soon scorched, while the part in contact with the brass is scarcely injured. The metal conducts the heat away and keeps the temperature of the paper below the point of ignition. The wood is a poor conductor. These experiments show that solids differ in their conductivity FIOURE 3 , g'J^IL HALF for heat. The metals are the best WOOD, HALF BRASS. 310 HEAT conductors ; wood, leather, flannel, and organic substances in general are poor conductors-; so also are all bodies in a powdered state, owing doubtless to a lack of continuity in the material. 363. Conductivity of Liquids. Pass a glass tube surmounted with a bulb through a cork fitted to the neck of a large funnel. Sup- port the apparatus as shown in Fig. 319. The glass stem should stand in colored water. Heat the bulb slightly to expel some air, so that the liquid will rise in the tube. Fill the funnel with water, covering the bulb to the depth of about one centi- meter. Pour a spoonful of ether on the water and set it on fire. The steadiness of the index shows that little if any of the heat due to the burning ether is conducted to the bulb. This experiment shows that water is a poor conductor of heat. This is equally true of all liquids except molten metals. 364. Conductivity of Gases. The conductivity of gases is very small, and its determination is very diffi- cult because of radiation and convection. The conduc- tivity of hydrogen is about 7.1 times that of air, while the conductivity of water is 25 times as great. 365. Applications of Conductivity. If we touch a piece of marble or iron in a room, it feels cold, while cloth and wood feel dis- tinctly warmer. The explanation is that the articles which feel cold are good conductors of heat and carry it away from the hand, while the poor conductors do not. The good heat-conducting property of copper or brass is turned to practical account in Sir Humphry Davy's miner's lamp (Fig. 320). FIGURE 319. WATER POOR CONDUCTOR. APPLICATIONS OF CONDUCTIVITY 311 The flame is completely inclosed in metal and fine wire gauze. The gauze by conducting away heat keeps any fire damp outside the lamp below the temperature of ignition and so prevents ex- plosions. The action of the gauze is readily illustrated by holding it over the flame of a Bunsen burner (Fig. 321). The flame does not pass through unless the gauze is heated to redness. If the gas is first allowed to stream through the gauze, it may be lighted on top without being ignited below. The handles on metal instruments that are to be heated are usually made of some poor conductor, as wood, bone, etc. ; or else they are insulated by the insertion of some non-conductor, as in the case of the handles to silver tea- pots, where pieces of ivory are inserted to keep them from becoming too hot. The non-conducting character of air FlGU * E 320. is utilized in houses with hollow walls, * in double doors and double windows, and in clothing of loose texture. The warmth of woolen articles and of fur is due mainly to the fact that much air is inclosed within them on account of their loose structure. FIGURE 32 1 . FLAME STOPPED BY WIRE GAUZE. Thermos Patent Reinforcements Shook Absorber The thermos bottle consists of a glass bottle with double walls (Fig. 322). The space be- tween the two walls is exhausted of air, and the inner walls of this vacuum are silvered to lessen radiation from one to the other. Either hot or cold liquids may be kept in a thermos bottle with little change of tempera- ture for several hours. The "Jireless cooker " is a box of wood or steel with a metallic vessel inside. The two are separated by heavy felt or other poorly conducting material (Fig. 323). After the material to be cooked has been raised to the proper temperature, it is placed in the cooker and the latter is tightly closed. The high tem- FIGURE 322. -THERMOS BOTTLE. 312 HEAT perature is maintained for three hours with a drop of not more than 10 or 15 C. The cooking may be completed without further appli- cation of heat. The conductivity of the lining and of the inclosed air is so low that heat escapes very slowly. Additional heat is often supplied by means of hot soapstone or cast iron disks. 366. Convection. Set up apparatus as shown in Fig. 324, and support it on a heavy iron stand. Fill the flask and connect- ing tubes with water up to a point a little above the open end of the vertical tube at C. Apply a Bunsen flame to the flask B. A circulation of water is set up in FIGURE 323. FIRELESS COOKER. the apparatus, as shown by the arrows. The circulation is made visible by coloring the water in the reservoir blue and that in the flask red. The process of conveying heat by the transference of the heated matter itself is known as convection. Currents set up in this manner are called convection currents. 367. Heating by Hot Water. The heating of buildings by hot water conveyed by pipes to the radiators and thence back again to the heater in the basement is an application of convection by liquids (Fig. 325). The hot water pipe extends to an open tank at the top of the building to allow for expansion. The circulation is maintained because the hot water in the pipes leading to the radiators is hotter and therefore Blighter than the cooler water in the return pipes beyond the ra- diators. 368. The Hot Water Heater. The simplest arrangement for heating water for general CURRENTS. CONVECTION IN GASES 313 domestic purposes is shown in Fig. 326. The cold water enters the tank at the top through a pipe which reaches nearly to the bottom. The pipe in the bottom leads to a heating coil in the gas heater. The hot water rises and enters the tank at or near the top, while heavier cold water takes its place in the heating coils. The circulation thus set up con- tinues as long as heat is applied. 369. Convection in Gases. Set a short piece of lighted candle in a shallow beaker and place over it a lamp chimney FlGURE 325. HEATING BY HOT WATER. Pour into the beaker enough water to close the lower end of the chimney. Place in the top of the chimney a T-shaped piece of tin as a short partition (Fig. 327). If a piece of smoldering paper be held over one edge of the chimney, the smoke will pass down one side of the partition and up the other. If the partition be removed, the flame will usually go out. Convection currents are more easily set up in gases than in liquids. Convec- FIGURE 326 "WATER tion current s of air on a large scale are HEATER. present near the seacoast. The wind is 314 HEAT FIGURE 327. CONVECTION IN GASES. a sea breeze during the day, because the air moves in from the cooler ocean to take the place of the air rising over the heated land. As soon as the sun sets, the ground cools rapidly by radiation, and the air over it is cooler than over the sea. Hence the reversal in the direction of the wind, which is now a land breeze. 370. Heating and Ventilating by Hot Air. The hot air furnace in the basement is a heater for burning wood, coal, distillate, or gas, and surrounded by a jacket of galvanized iron (Fig. 328). Cold air from outside is heated between the heater and the jacket and rises through the hot air flues to registers in the rooms of the building. In houses the extra air often finds an outlet through crevices and up open chimneys. ,A better way is to provide ventilation by means of separate flues. Since the heated air rises to the top of the room, it follows that if provision is made for the escape of the colder air by flues at the floor, the incom- ing air will force out the foul air, thus FIGURE 328. HEATING BY HOT AIR. THE RADIOMETER 315 changing the air of the room and warming it at the same time. Large public buildings must have positive means of supplying fresh air to the extent of about 50 cubic feet per minute for each person. For this purpose large fans driven by power draw in fresh air from the outside and force it through flues throughout the building. The air is often washed or filtered on its way in, and in cold weather is heated by steam pipes. The foul air is forced out through openings near the floor. Sometimes exhaust fans draw out the vitiated air through the ventilating ducts. 371. Radiation. When one stands near a hot stove, one is warmed neither by heat conducted nor conveyed by the air. The heat energy of a hot body is constantly passing into space as radiant energy in the ether ( 243). Radiant energy becomes heat again only when it is ab- sorbed by bodies upon which it falls. Energy transmitted in this way is, for convenience, referred to as radiant heat, although it is transmitted as radiant energy, and is trans- formed into heat only by absorption. Radiant heat and light are physically identical, but are perceived through different avenues of sensation. Radiations that produce sight when received through the eye give a sensation of warmth through the nerves of touch, or heat a ther- mometer when incident upon it. The long ether waves do not affect the eye, but they heat a body which absorbs them. 372. The Radiometer. Long heat waves may be de- tected by the radiometer, an instrument invented by Sir William Crookes in 1873 while investigating the properties of highly attenuated gases. It consists of a glass bulb from which the air has been exhausted until the pressure 316 HEAT does not exceed 7mm. of mercury (Fig. 329). Within the bulb is a light cross of aluminum wire carrying small vanes of mica, one face of each coated with lampblack. The whole is mounted to rotate on a vertical pivot. When the instrument is placed in the sun- light or in the radiation from any heated body, the cross revolves with the blackened faces of the vanes moving away from the source of heat. The infrequent collisions among the molecules in such a vacuum pre- vent the equalization of pressure throughout the bulb. The black- ened sides of the vanes absorb more heat than the bright ones, and the gas molecules rebound from the warmer surfaces with a greater ve- locity than from the others, thus giving the vanes an impulse in the opposite direction. This impulse is the equivalent of a pressure, which causes the vanes to revolve. 373. Laws of Heat Radiation Place a radiometer about 50 cm. from a small lighted lamp and note the effect on the radiometer. Support a cardboard screen between the lamp and the radiometer ; the rotation of the radiometer at once becomes slower. Hence, Radiation proceeds in straight lines. This law is illustrated in the use of fire screens and sunshades. Lay a meter stick on a table and place the radiometer at one end of it and the lamp at the other. Count the number of revolutions of the vanes in one minute. Move the radiometer to a distance of 50 cm. FIGURE 329. THE RADI- OMETER. HEAT TRANSPARENCY 317 from the lamp and count the number of revolutions again for a minute. It will be about four times as many as before. Hence, The amount of radiant energy received by a body from any small radiant area varies inversely as the square of the distance from it as a source. Note that this law is the same as that relating to the intensity of illumination in light ( 252). Support a plane mirror vertically on a table. At right angles to it and distant about 5 cm. support a vertical cardboard screen about 50cm. long and 20cm. wide. On one side of this screen place a lighted lamp and on the other the radiometer. The vanes will revolve rapidly whenever the lamp and the radiometer are in such a position that the screen bisects the angle made by lines drawn from them to the same point on the mirror. The angles between these lines and the screen are the angles of incidence and reflection of the radiant energy. Hence, Radiant energy is reflected from a polished sur- face so that the angles of incidence and reflection are equal. Select two concave wall lamp reflectors of the same size and blacken one of them in the smoke from burning camphor gum. Place a lighted lamp about one meter from the radiometer and observe the rate of rotation of the vanes. Hold the clear reflector back of the radiometer, so as to concentrate the radiation from the lamp upon it, and again note the rate of rotation. Now substitute the blackened reflector for the clear one; the rate of rotation will be greatly reduced. Hence, The capacity of a surface to reflect radiant energy depends both on the polish of the surface and on the nature of the material. Polished brass is one of the best reflectors and lampblack is the poorest. 374. Heat Transparency. Select two flat twelve ounce bottles ; fill one with water and the other with a solution of iodine in carbon disulphide. Cut an opening in a sheet of black cardboard of such a 318 HEAT size that either bottle will cover it. Place this cardboard between the lamp and the radiometer and note the effect on the radiometer as the opening is closed successively by the bottles. This experiment and others similar to it show that The transmission of radiant energy through various sub- stances depends on the temperature of the source, and the thickness and nature of the substance itself. Substances that transmit a large part of the heat energy, such as the solution of iodine and rock salt, are said to be diathermanous ; those absorbing a large part, such as water, are athermanous. Glass is diathermanous to radiations from a source of high temperature, such as the sun, but athermanous to radiations from sources of low tempera- ture, such as a stove. The radiant energy from the sun passes readily through the atmosphere to the earth, and warms its surface ; but the radiations from the earth are stopped to a large extent by the surrounding atmosphere. This selective absorption is due in large measure to tho vapor of water in the air. Questions 1. Why will newspapers spread over plants protect them from frost ? 2. Why does a tall chimney have a stronger draft than a short one? 3. Explain how it is possible to boil water in a paper pail with- out burning the pail. 4. Should the surface of a steam or hot water radiator be rough or polished ? 5. In what way does a stove heat a room ? 6. Why does a woolen garment feel warmer than a cotton or a linen one ? 7. Why is glass an effective screen ? HEAT FROM MECHANICAL ACTION 319 8. Why does steam burn more severely than hot water ? 9. Why should the registers for removing impure air be placed at the floor level ? 10. What principles of heat are applied in the radiator of an automobile ? 11. Why will the moistened finger or the tongue freeze quickly to a piece of very cold iron, but not to a piece of wood? 12. Why is the boiling point of water in the boiler of a steam engine above 100C.? VII. HEAT AND WORK 375. Heat from Mechanical Action. Strike the edge of a piece of flint a glancing blow with a piece of hardened steel. Sparks will fly at each blow. Pound a bar of lead vigorously with a hammer. The temperature of the bar will rise. In the cavity at the end of a piston of a fire syringe place a small piece of tinder, such as is employed in cigar lighters (Fig. 330). Force the piston quickly into the barrel. If the piston is immediately withdrawn the tinder will prob- ably be on fire. These experiments show that mechanical en- ergy may be transformed into heat. Some of the energy of the descending flint, the hammer, and the piston has in each case been transferred to the molecules of the bodies themselves, increasing their kinetic energy, that is, raising their tern- r IGURE perature. 330. Savages kindle fire by rapidly twirling a dry FlRE SYR - stick, one end of which rests in a notch cut in a second dry piece. The axles of carriages and the bearings in machinery are heated to a high temperature when not properly lubricated. The heating of drills and bits in boring, the heating of saws in cutting timber, the burning 320 HEAT of the hands by a rope slipping rapidly through them, the stream of sparks flying from an emery wheel, are instances of the same kind of transformation ; the work done against friction produces kinetic energy in the form of heat. 376. The Mechanical Equivalent of Heat. In 1840 Joule of Manchester in England began a series of experiments to determine the numerical relation between the unit of heat and the foot pound. His experiments extended over a period of forty years. His most successful method consisted in measuring the heat produced when a meas- ured amount of work was expended in heating water by stirring it with paddles driven by weights falling through a known height. His final result was that 772 ft.-lb, of work, when converted into heat, raise the temperature of 1 Ib. of water 1 F., or 1390 ft.-lb. for 1 C. The later and more elaborate researches of Rowland in 1879 and of Griffiths in 1893 show that the relation is 778 ft.-lb. for 1F., or 427.5 kg.-m. for 1 C. ; that is, if the work done in lifting 427.5 kg. one meter high is all converted into heat, it will raise the temperature of 1 kg. of water 1 C. This relation is known as the mechanical equivalent of heat. Its value expressed in absolute units is 4.19 x 10 7 ergs per calorie. 377. The Steam Engine. The most important devices for the conversion of heat into mechanical work are the steam engine and the gas engine. The former in its assential features was invented by James Watt. In the reciprocating steam engine a piston is moved alternately in opposite directions by the pressure of steam applied first to one of its faces and then to the other. This re- ciprocating or to-and-fro motion is converted into rotatory motion by the device of a connecting rod, a crank, and a flywheel. James Watt (1736-1819) was born at Greenock, Scotland, and was educated as an instrument maker. In studying the defects of the steam engines then in use, he was led to make many very important improvements, culminating in his invention of the double-acting steam engine. He invented the ball governor, the cylinder jacket, the D-valve, the jointed paral- lelogram for securing recti- linear motion to the piston, the mercury steam-gauge, and the water-gauge. He is also to be credited with the first compound engine, a type of en- gine extensively used to-day. James Prescott Joule (1818-1889), the son of a Manchester brewer, was born at Salford, England. He became known to the scientific world through his contributions in heat, elec- tricity, and magnetism. His greatest achievement was es- tablishing the modern kinetic theory of heat by determining the mechanical equivalent of heat. His experiments on this subject were continued through a period of forty years. In recognition of his great work he was presented with the Royal Medal of the Royal So- ciety of England in 1852. THE STEAM ENGINE 321 R In Fig. 331 are shown in section the cylinder, piston, and valve of a slide-valve steam engine. The piston B is moved in the cylinder A by the pressure of the steam ad- mitted through the inlet pipe a. The slide valve d works in the steam chest cc and admits steam al- ternately to the two ends of the cylinder through the steam ports at either end. When the valve is in the position shown, steam passes into the right-hand end of the cylinder and drives the piston toward the left. At the same FIGURE 331. CYLINDER OF STEAM ENGINE. time the other end is connected with the exhaust pipe ee through which the steam escapes, either into the air, as in a high-pressure non- condensing engine, or into a large condensing chamber, as in a low pressure condensing engine. The slide valve d is moved by the rod R, connected to an eccentric, which is a round disk mounted a little to one side of its center, on the engine shaft. It has the effect of a crank. The flywheel, also mounted on the shaft of the engine, has a heavy rim and serves as a store of energy to carry the shaft over the dead points when the piston is at either end of the cylinder. There is in the flywheel a give-and-take of energy twice every revolution, and a fairly steady rotation of the shaft is the result. The eccentric is set in such a way that the rod R closes the valve admitting steam to either end of the cylinder 322 HEAT before the piston has completed its stroke ; the motion of the piston is continued during the remainder of the stroke by the expansive force of the steam. Corliss valves are commonly used in large slow speed engines. As distinguished from the slide valve, the Corliss valve is cylindrical and opens and closes by turn- ing a little in its seat, first in one direction and then the other. In the Corliss engine there are four such valves, two at each end of the steam cylinder. One of each pair admits steam to the cylinder and the other is the exhaust valve. When the inlet valve is open at one end of the steam cylin- der, the exhaust valve is open at the other end. All four valves are opened and closed automatically by the motion of the engine itself. Each valve can be adjusted separately. 378. The Indicator Diagram. The steam indicator is a device for the automatic tracing of a diagram representing the relation be- tween the volume and the pressure of the steam in the cylinder during one stroke. This dia- gram is known as an "indicator card " (Fig. 332). From a to b the inlet port is open and the full pressure Volume of steam is on the piston ; at FIGURE 332. INDICATOR DIAGRAM. & the inlet port closes and the steam expands from b to c, when the exhaust port opens ; at d the pressure is reduced to the lowest value and remains sensibly constant during the return movement of the piston until e is reached, when the exhaust port closes and the remaining steam is compressed from e to/ At / the inlet port opens and the pressure rises abruptly to the initial maxi- mum, thus completing the cycle. The work done during the stroke is represented by the inclosed area abcdef. The indicator card is used also in adjusting the valves. 379. The Steam Turbine. The steam turbine has the great advantage of producing rotary motion directly with- ABOVE : SECTION THROUGH THE STEAM TURBINE, SHOWING NOZZLES AND BUCKETS. BELOW : THE ROTOR OF A TURBINE, SHOWING BUCKETS INCREASING IN SIZE FROM LEFT TO RIGHT. THE GAS ENGINE 323 out the intervention of a connecting rod and crank to convert the back and forth motion of the piston in a reciprocating engine into rotary motion. In the latter the piston stops and starts again twice during each revo- lution of the flywheel, and the stopping and starting gives rise to disagreeable vibrations. In the steam turbine the rotor revolves continuously and the impulses it receives are constant instead of intermittent. Steam enters the turbine through a set of stationary nozzles, shown in section in the half tone. Here it expands and acquires a high velocity. It then strikes the entrance edge of the first row of buckets in the rotor, gives up en- ergy to them, and drives them forward as it passes through. It then passes through the second set of sta- tionary nozzles, of greater area than the first ; here it again expands, increases its velocity, and enters the second row of buckets. The process is repeated in successive stages until it reaches the exhaust outlet. By its im- pulse on each row of buckets it gives up energy to the rotor. The half tone of a complete rotor shows the in- creasing size of the buckets from left to right. The buckets are curved openings through the rotor, as shown in section in the half tone. 380. The Gas Engine. The gas engine is a type of in- ternal combustion engine, which includes motors using gas, gasoline, kerosene, or alcohol as fuel. The fuel is intro- duced into the cylinder of the engine, either as a gas or as a vapor, mixed with the proper quantity of air to produce a good explosive mixture. The mixture is ignited at the right instant by means of an electric spark. The explo- sion and the expansive force of the hot gases drive the piston forward in the cylinder. In the four-cycle type of gas engine, the explosive mix- 324 HEAT ture is drawn in and ignited in each cylinder only every other revolution of the engine, while in the two-cycle type an explosion occurs every revolution. The former type is used in most motor car engines, and the latter in small motor boats. The operation of a four-cycle engine is illustrated in 1, 2, 3, and 4 of Fig. 333, which shows the four steps in a complete cycle. The inlet valve a and the exhaust valve b are operated by the cams c and d. Both valves are kept normally closed by springs surrounding the valve stems. The small shafts to which the two cams are fixed are driven by the spur wheel e on the shaft of the engine. This wheel engages with the two larger spur wheels on the cam shafts, each having twice as many teeth as e and forming with it a two- to-one gear, so that c and d rotate once in every two revolutions of the crank shaft. The piston m has packing rings ; h is the con- necting rod, k the crank shaft, and I the spark plug. In diagram 1 the piston is descending and draws in the charge through the open valve a ; in 2 both valves are closed and the piston compresses the explosive charge 5 about the time the piston reaches its highest point, the charge is ignited by a spark at the spark plug, and the working stroke then takes place, as in 3, both valves remaining closed ; in 4 the exhaust valve b is opened by the cam d, and the products of the combustion escape FIGURE 333. SHOWING FOUR STEPS IN CYCLE. FRONT AND REAR VIEWS OF AN AERIAL "FLIVVER. One of the smallest practical airplanes made. THE AIRPLANE 325 through the muffler, or directly into the open air. The piston has now traversed the cylinder four times, twice in in each direction, and the series of operations begins again. 381. Two-Cycle Engine. Figure 334 is a section of a two-cycle gas engine. During the up-stroke of the piston P a charge is drawn through A into the crank case 0. At the same time a charge in the cylinder is compressed and is ignited by a spark when the compression is greatest. The piston is forced down, and when it passes the port E the exhaust takes place. When the admit port / is passed, a charge enters from the crank case. To prevent this charge from passing across and escaping at E, it is made to strike against a projec- tion J2 on the piston, which deflects it upward. The momentum of the balance wheel carries the piston up- ward, compresses the charge, and , j. i i_ , j.iT i FIGURE 334. SECTION OP draws a fresh charge into the crank TWO-CYCLE ENGINE case. The piston has now traversed the cylinder twice, once in each direction, and the same series of operations is again repeated. For a more complete discussion of gas engines and auto- mobiles, see Chapter XV, page 460. 382. The Airplane. The principle of the airplane has already been described in 124. It is a " heavier than air " machine and is lifted as the kite is lifted ; but instead of the wind blowing against it, it is forced against the air by a powerful gas engine, driving a high speed propeller. Formerly the engine and the propeller were at the rear end, but recent practice is to mount them in front. 326 HEAT Questions and Problems 1. Why does the temperature of the air under the bell jar of an air pump fall when the pump is worked ? 2. Is there a difference in the temperature of the steam as it enters a steam engine and as it leaves at the exhaust ? Explain. 3. Lead bullets are sometimes melted when they strike a target. Explain. 4. Does clothing keep the cold out or keep the heat in ? 5. Is there any less moisture in the air after it has passed through a heated furnace into a room than there was before ? 6. Why does the rapid driving of an automobile heat the air in the tires? 7. A mass of 200 g. moving with a speed of 50 m. per second is suddenly stopped. If all its energy is converted into heat, how many calories would be generated ? NOTE. A calorie equals 4.19 x 10 7 ergs. 8. If all the potential energy of a 300 kg. mass of rock is con- verted into heat by falling vertically 200 m., how many calories would be generated ? 9. How high could a 200 g. weight be lifted by the heat required to melt the same mass of ice, if all the heat could be utilized for the purpose ? 10. If the average pressure of steam in the cylinder of an engine is 100 Ib. per square inch, the area of the piston is 80 sq. in., and the stroke 1 ft., how many horse powers would be developed if the engine makes two revolutions per second ? CHAPTER X MAGNETISM I. MAGNETS AND MAGNETIC ACTION 383. Natural Magnets or Lodestones. Black oxide of iron, known to mineralogists as magnetite, is found in many parts of the world, notably in Arkansas, the Isle of Elba, Spain, and Sweden. Some of these hard black stones are found to possess the property of attracting to them small pieces of iron. At a very early date such pieces of iron ore were found near Magnesia in Asia Minor, and they were therefore called magnetic stones and later magnets. They are now known as natural magnets, and the properties peculiar to them as magnetic properties. Dip a piece of natural magnet into iron filings ; they will adhere to it in tufts, not uniformly over its surface, but chiefly at the ends and on projecting edges (Fig. 335). Suspend a piece of natural magnet by a piece of untwisted thread (Fig. 336), or float it on a wooden raft on water. Note its posi- tion, then disturb it slightly, and again let it come to rest. It will be found that it invariably returns to the same position, the line connecting the two ends to which the filings chiefly adhered in the preceding experiment lying north and south. FIGURE 335. NATURAL MAGNET. FENDED. This directional property of the natu- ral magnet was early turned to account 327 328 MAGNETISM in navigation, and secured for it the name of lodestone (leading-stone). 384. Artificial Magnets. Stroke the blade of a pocket knife from end to end, and always in the same direction, with one end of a lodestone. Touch it to iron filings ; they will cling to its point as they did to the lodestone. The knife blade has become a magnet. Use the knife blade of the last experiment to stroke another blade. This second blade will also acquire magnetic properties, and the first one has suffered no loss. Artificial magnets, or simply magnets, are bars of hard- ened steel that have been made magnetic by the applica- tion of some other magnet or magnetizing force. The form of artificial magnets most commonly met with are the bar and the horseshoe. 385. Magnetic Substances. Any substance that is at- tracted by a magnet or that can be magnetized is a mag- netic substance. Faraday showed that most substances are influenced by magnetism, but not all in the same way nor to the same degree. Iron, nickel, and cobalt are strongly attracted by magnets and are said to be mag- netic; bismuth, antimony, phosphorus, and copper act as if they are repelled by magnets and they are called dia- magnetic. Most of the alloys of iron are magnetic, but .- MAGNET T OT H manganese steel is non-mag- IRON FILINGS. netic. 386. Polarity. Roll a bar magnet in iron filings. It will be come thickly covered with the filings near its end. Few, if any, will adhere at the middle (Fig. 337) . The experiment shows that the greater part of the mag- netic attraction is concentrated at or near the ends of the magnet. They are called its poles, and the magnet is said MAGNETIC TRANSPARENCY 329 to have polarity. The line joining the poles of a long slender magnet is its magnetic axis. 387. North and South Poles. Straighten a piece of watch spring 8 or 10 cm. long, stroke it from end to end with a magnet, and float it on a cork in a vessel of water (Fig. 338). It will turn from any other position to a north and south one, and invariably with the same end north. FIGURE 338. FLOATING MAGNET. The end of a magnet pointing toward the north is called the north-seeking pole, and the other, the south-seeking pule. They are commonly called simply the north pole and the south pole. 388. Magnetic Needle. A slender magnetized bar, sus- pended by an untwisted fiber or pivoted on a point so as to have freedom of motion about a vertical axis is a mag- netic needle (Fig. 339). The direction in which it comes to rest without torsion or friction is called the magnetic meridian. Fasten a fiber of unspun^ silk to a piece of magnetized watch spring about 2 cm. long so that it will hang horizontally. Suspend it inside a wide-mouthed bottle by attaching the fiber to a cork fitting the mouth FIGURE 339. -MAGNETJC NEEDLE. of the bottle ' The little magnetic needle will then be protected from currents of air. It may be made visible at a distance by sticking fast to it a piece of thin white paper. 389. Magnetic Transparency. Cover the pole of a strong bar magnet with a thin plate of glass. Bring the face of the plate oppo- site the pole in contact with a pile of iron tacks. A number will be 330 MAGNETISM found to adhere, showing that the attraction takes place through glass. In like manner, try thin plates of mica, wood, paper, zinc, copper, and iron. No perceptible difference will be seen except in the case of the iron, where the number of tacks lifted will be much Magnetic force acts freely through all substances except those classified as magnetic. Soft iron serves as a more or less perfect screen to magnetism. Watches may be pro- tected from magnetic force that is not too strong by means of an inside case of soft sheet iron. 390. First Law of Magnetic Action. Magnetize a piece of large knitting needle, about four inches long, by stroking it from the middle to one end with the north pole of a bar magnet, and then from the middle to the other end with the south pole. Repeat the operation several times. Present the north pole of the magnetized knitting needle to the north pole of the needle suspended in the bottle. The latter will be repelled. Present the same pole to the south pole of the little magnetic needle ; it will be attracted. Repeat with the south pole of the knitting needle and note the deflections. The results may be expressed by the following law of magnetic attraction and repulsion : Like magnetic poles repel and unlike magnetic poles attract each other. 391. Testing for Polarity. The magnetic needle affords a ready means of ascertaining which pole of a magnet is the north pole, for the north pole of the magnet is the one that repels the north pole of the magnetic needle. Repul- sion is the only sure test of polarity for reasons that will appear in the experiments that follow. 392. Induced Magnetism. Hold vertically a strong bar magnet and bring up against its lower end a short cylinder of soft iron. It will adhere. To the lower end of this one attach another, and so on UNLIKE POLARITY INDUCED 331 in a series of as many as will stick (Fig. 340). Carefully detach the magnet from the first piece of iron and withdraw it slowly. The pieces of iron will all fall apart. The small bars of iron hold together because they become temporary mag- nets. Magnetism produced in mag- netic substances by the influence of a magnet near by or in contact with them is said to be induced, and the action is called magnetic induction. Magnetic induction precedes attrac- . ' FIGURE 340. INDUCED MAGNETISM. 393. Unlike Polarity Induced. Place a bar magnet in line with the magnetic axis of a magnetic needle, with its north pole as near as possible to the north pole of the needle without appreciably repelling it (Fig. 341). Insert a bar of soft iron between the magnet and the needle. The north pole of the needle will be immedi- ately repelled. The repulsion of the north pole of the needle by FIGURE 341 . - POLARITY BY INDUCTION. the end of the soft ipon bar next to it shows that this end of the bar has acquired a polarity the same as that of the magnet, that is, north polarity. Then the other end adjacent to the magnet must have acquired the opposite polarity. When a magnet is brought near a piece of iron, the iron is magnetized by induction, and there is attraction because the adjacent poles are unlike. When a bunch of iron fil- ings or tacks adhere to a magnet, each filing or tack be- comes a magnet and acts inductively on the others and all become magnets. Weak magnets may have their polarity reversed by the inductive action of a strong magnet. 332 MAGNETISM 394. Permanent and Temporary Magnetism. When a piece of hardened steel is brought near a magnet, it acquires magnetism as a piece of soft iron does under the same conditions : but the steel retains its magnetism when the magnetizing force is withdrawn, while the soft iron does not. In the experiment of 392 the soft iron ceases to be a magnet when removed to a distance from the bar magnet. In addition, therefore, to the permanent magnetism exhibited by the magnetized steel, we have temporary magnetism induced in a bar of soft iron when it is brought near a magnet or in contact with it. II. NATURE OF MAGNETISM 395. Magnetism a Molecular Phenomenon. If a piece of watch spring be magnetized and then heated red hot, it will lose its magnetism completely. A magnetized knitting needle will not pick up as many tacks after being vibrated against the edge of a table as it did before. A piece of moderately heavy and very soft iron wire of the form shown in Fig. 342 can be magnetized by stroking FIGURE 342. BENT MAGNET. * J T , . it gently with a bar magnet. If given a sudden twist, it loses at once all the magnetism imparted to it. A piece of watch spring attracts iron filings only at its ends. If broken in two in the middle, each half will be a magnet and will attract filings, two new poles having been formed where the original magnet was neutral. If these pieces in turn be broken, their parts will be magnets. If this division into separate magnets be conceived to be carried as far as the molecules, they too would probably be magnets. It is worthy of notice that magnetization is facilitated by jarring the steel, or by heating it and letting it cool under the influence of a magnetizing force. If an iron bar is rapidly magnetized and demagnetized, its tempera- ture is raised. A steel rod is slightly lengthened by MAGNETIC FIELDS 333 magnetization and a faint click may be heard if the magnetization is sudden. 396. Theory of Magnetism. The facts of the preceding section indicate that the seat of mag- netism is the molecule, that the individual mole- cules are magnets, that in an unmagnetized piece of iron the poles of the molecular magnets are turned in various directions, so that they form stable combinations or closed magnetic chains, and hence exhibit no magnetism external to the bar (Fig. 343). In a magnetized bar the FIGURE larger portion of the molecules have eeee eeee eiii iiee .... ey BB iiiB BBBB .... .... BBBB BBBB BBBB their magnetic axes pointing in the same NETIZED direction (Fig. 344), the completeness BAR - of the magnetization depending on the complete- ness of this alignment. III. THE MAGNETIC FIELD 397. Lines of Magnetic Force. Place a sheet of paper over a small bar magnet and sift iron filings evenly over it from a bottle with a piece of gauze tied over the mouth, tapping the paper gently to aid the filings in ar- FIGURE ranging themselves under the influence of the magnet. 344. They will cling together in curved lines, which diverge MAGNET- f rO m one pole of the magnet and meet again at the oppo- IZEDBAR. site pole These lines are called lines of magnetic force or of mag- netic induction. Each particle of iron becomes a magnet by induction ; hence the lines of force are the lines along which magnetic induction takes place. 398. Magnetic Fields. A magnetic field is the space around a magnet in which there are lines of magnetic force. 334 MAGNETISM Figure 345 was made from a photograph of the magnetic field of a bar magnet in a plane passing through the magnetic axis. These lines branch out nearly radially from one pole and curve round through the air to the other pole. Faraday gave to them the narne FIGURE 345. MAGNETIC FIELD OF BAR MAGNEI. lines of force. The curves made by the iron filings "represent visibly the invisible lines of magnetic force." Figure 34G shows the field about two bar magnets placed with their unlike poles adjacent to each other. Many of the lines from the north pole of the one extend across to the south pole of the other, and this connection denotes at- traction. Figure 317 shows the field about two bar mag- nets with their like poles adjacent to each other. None of the lines spring- ing from either pole ex- tend across to the neigh- boring pole of the other magnet. This is a pic- FIGURE 346. MAGNETIC FIELD, Two UNLIKE POLES. ture of magnetic repul- sion. 399. Properties of Lines of Force. Lines of magnetic force have the following properties : (a) They are under tension, exerting a pull in the direction of their length ; PERMEABILITY 335 (>) they spread out as if repelled from one another at right angles to their length ; (IAO -M- A- TV FIGURE 351. MAGNETIC DIP. 403. Magnetic Dip. Thrust two unmagnetized knitting needles through a cork at right angles to each other (Fig. 351). Support the apparatus on the edges of two glasses, with the axis in an east-and-west line, and the needle adjusted so as to rest horizontally. Now magnetize the needle, being careful not to displace the cork. It will no longer assume a horizontal position, the north pole dipping down as if it had become heavier. The inclination or dip of a needle is the angle its magnetic axis makes with a horizontal plane. A needle mounted so as to turn about a horizontal axis through its center of gravity is a dip- ping needle (Fig. 352). The dip of the needle at the magnetic poles of the earth is 90, at the magnetic equator, 0. In 1907 Amundsen placed the magnetic pole of the northern hemi- sphere in latitude 75 5' N. and longi- tude 9647'W. The magnetic pole of the southern hemisphere is probably near latitude 72 S. and longitude 155 E. Isoclinic lines are lines on the earth's surface passing through points of equal dip. They are irregular in di- FIGURE 352. DIPPING NEEDLE. 338 MAGNETISM rection, though resembling somewhat parallels of lati- tude. 404. Magnetic Declination. The magnetic poles of the earth do not coincide with the geographical poles, and consequently the direction of the magnetic needle is not in general that of the geographical meridian. The angle between the direction of the needle and the meridian at any place is the magnetic decimation. To Columbus be- longs the undisputed discovery that the declination is dif- ferent at different points on the earth's surface. In 1492 he discovered a place of no declination in the Atlantic Ocean north of the Azores. The declination at any place is not constant, but changes as if the magnetic poles oscil- late, while the mean position about which they oscillate is subject to a slow change of long period. The annual change on the Pacific coast is about 4', and in New Eng- land about 3'. At London in 1657 the magnetic declina- tion was zero, and it attained its maximum westerly value of 24 in 1816 ; in 1915 it was 15 19' W. 405. Agonic Lines. Lines drawn through places where the needle points true north are called agonic lines. In 1910 the agonic line in North America ran from the mag- netic pole southward across Lake Superior, thence near Lansing, Michigan, Columbus, Ohio, through West Vir- ginia and South Carolina, and it left the mainland near Charleston on its way to the magnetic pole in the south- ern hemisphere. East of this line the needle points west of north ; west of it, it points east of north. Lines pass- ing through places of the same declination are called isogonic lines. QUESTIONS 339 Questions 1. Given a bar magnet of unmarked polarity; determine which end is its north pole. 2. Out of a group of materials, how would you select the mag- netic substances? 3. Given two bars exactly alike in appearance, one soft iron and the other hardened steel. Select the steel one by means of magnetism. 4. Magnetize a long darning needle, then break it in the middle. Will there be two magnets, each with one pole ? 5. How would you magnetize a sewing needle so that the eye is the north pole ? 6. What effect would it have on a compass to place it within an iron kettle ? 7. Will an iron fence post standing in the ground have any in- fluence on the needle of a surveyor's compass ? 8. Float a magnet on a cork. Will the earth's magnetism cause it to float toward the earth's magnetic pole ? 9. Is the polarity of the earth's magnetism in the northern hemi- sphere the same as that of the north pole of a magnet? 10. Suppose you wish to make a magnetic needle. If it is bal- anced on a point so as to rest in a horizontal position before magnet- ization, will it rest horizontally after it is magnetized ? CHAPTER XI ELECTROSTATICS I. ELECTRIFICATION 406. Electrical Attraction. Rub a dry flint glass rod with a silk pad and bring it near a pile of pith balls, bits of paper, or chaff. They will at first be at- tracted and then repelled (Fig. 353). The simple fact that a piece of amber (a fos- sil gum) rubbed with a flannel cloth acquires the property of at- tracting bits of paper, pith, or other light bodies, has been known since about 600 B.C. ; but it seems not to have been known down to the time of Queen Eliza- beth that any bodies except amber and jet were capable of this kind of excitation. About 1600 Dr. Gilbert dis- covered that a large number of substances, such as glass, sulphur, sealing wax, resin, etc., possess the same peculiar property. These he called electrics (from the Greek word for amber, electron). A body excited in this manner is said to be electrified, its condition is one of electrification, and the invisible agent to which the phenomenon is re- ferred is electricity. 340 FIGURE 353. ELECTRICAL ATTRACTION. TWO KINDS OF ELECTRIFICATION 341 407. Electrical Repulsion. Suspend several pith balls from a glass hook (Fig. 354). Touch them with an electrified glass tube. They are at first attracted but they soon fly away from the tube and from one another. When the tube is removed to a distance, the balls no longer hang side by side, but keep apart for some little time. If we bring the hand near the balls they will move toward it as if attracted, showing that the balls are electrified. From this experiment it ap- pears that bodies become elec- trified by coming in contact with electrified bodies, and that elec- trification may show itself by re- pulsion as well as by attraction. FIGURE 354. ELECTRICAL REPULSION. 408. Attraction Mutual. Electrify a flint glass tube by fric- tion with silk, and hold it near the end of a long wooden rod resting in a wire stirrup suspended by a silk thread (Fig. 355). The sus- pended rod is attracted. Now, replace the rod by the electrified tube. When the rod is held near the rubbed end of the glass tube, the latter moves as if at- tracted by the former. The experiment teaches that each body attracts the other ; that is, the action is mutual. 409 . Two Kinds of Electrification. Rub a glass tube with silk arid suspend it as in Fig. 355. Rub a second glass tube and hold it near one end of the suspended one. The suspended tube will be repelled. Bring near the sus- pended tube a rod of sealing wax rubbed with flannel. The suspended tube is now attracted. Repeat these tests with an electrified rod of sealing wax in the stirrup instead of the glass tube. The electrified FIGURE 355. ATTRAC- TION MUTUAL. 342 ELECTROSTATICS sealing wax will repel the electrified sealing wax, but there will be attraction between the sealing wax and the glass tube. The experiment illustrates the fact that there are two kinds of electrification: one developed by rubbing glass with silk, and the other by rubbing sealing wax with flannel. To the former Benjamin Franklin gave the name positive electrification ; to the latter, negative electrification. It appears further that bodies charged with the same kind of electrification repel each other, and bodies charged with unlike electrifications attract each other. Hence the law : Like electrical charges repel each other ; unlike electri- cal charges attract each otlwr- 410. The Electroscope. An instrument for detecting electri- fication and for determining its kind is called an electroscope. Of the many forms proposed the one shown in section in Fig. 356 is typical. The indicating system consists of a rigid piece of brass B, to which is attached a narrow strip of gold, leaf G. This system is supported by a block of sulphur /, which in turn is suspended by a rod fitting tightly in a block of ebonite E. A charging wire W passes through the ebonite and is bent at right angles at the bottom. By rotating the upper bent end of W, the arm at the bottom may be brought in contact with the brass strip for charging. ELECTROSCOPE Instead of a ball the supporting rod may end in a round flat plate. When the instrument has flat glass sides, the gold leaf may be projected on the screen with a lantern. FIGURE 357. PROOF PLANE. 411. Charging an Electro- scope. To charge an electroscope an instrument called a proof plane is needed. It consists of a small metal disk attached to an ebonite handle (Fig. 357). To use it, touch the disk to the electrified body and then apply it to the knob of the electroscope. The angular CONDUCTORS AND NONCONDUCTORS 343 separation of the foil from the stem will indicate the intensity of the electric charge imparted. This method is known as charging by contact in distinction from charging by induction to be described later. 412. Testing for Kind of Electrification. Charge the electroscope, by means of the proof plane, with the kind of electrification to be identified, until the leaf diverges a moderate distance. Then apply a charge from a glass rod electrified by rubbing with silk. If the divergence in- creases, the first charge was positive ; if not, recharge the electroscope from the unknown and apply a charge taken from a stick of sealing wax excited by friction with flan- nel. No certain conclusion can be drawn unless an in- creased divergence is obtained. 413. Conductors and Nonconductors. Fasten a smooth metal button to a rod of sealing wax and connect the button with the knob of the electroscope by a fine copper wire, 50 to 100 cm. long. Hold the sealing wax in the hand and touch the button with an electrified glass rod. The divergence of the leaf indicates that it is electrified. Repeat the experiment, using a silk thread instead of the wire ; no effect is produced on the electroscope. Now wet the thread with water and apply the electrified rod ; the effect is the same as when the wire was used. It is clear from these experiments that electric charges pass readily from one point to another along copper wire, but do not pass along dry silk thread. It is therefore cus- tomary to divide substances into two classes, conductors and nonconductors, or insulators, according to the facility with which electric charges pass in them from point to point. In the former if one point of the body is electrified by any means, the electrification spreads over the whole body, but in a nonconductor the electrification is confined to the vicinity of the point where it is excited. Sub- stances differ greatly in their conductivity, so that it is 344 ELECTROSTATICS not possible to divide them sharply into two classes. There is no substance that is a perfect conductor ; neither is there any that affords perfect insulation. Metals, car- bon, and the solution of some acids and salts are the best conductors. Among the best insulators are paraffin, tur- pentine, silk, sealing wax, India rubber, gutta-percha, dry glass, porcelain, mica, shellac, spun quartz fibers, and liquid oxygen. Some insulators, like glass, become good conductors when heated to a semi-fluid condition. II. ELECTROSTATIC INDUCTION 414. Electrification by Induction. Rub a glass tube with silk and bring it near the top of an electroscope. The leaves begin to diverge when the tube is some dis- tance from the knob (Fig. 358) and the amount of divergence increases t ;.> : as the tube approaches. When the tube is removed the leaves col- Since the leaves do not re- main apart, it is evident that there has been no transfer of electrification from the tube to the electroscope. The electrification produced in the electroscope when the electrified body is brought near it is owing to electrostatic induction. This form of elec- trification is only a temporary one and it is brought about by the presence of a charged body in its vicinity. 415. Sign of the Induced Charges. Lay a smooth metallic ball on a dry plate of glass. Connect it with the knob of the electro- FIGURE 358. ELECTRIFICATION BY INDUCTION. EQUALITY OF THE TWO CHARGES 345 scope by means of a stout wire with an insulating handle (Fig. 359). The ball and the electroscope now form one continuous conductor. Bring near the ball an electrified glass tube ; the leaves of the elec- troscope diverge. Before with- drawing the excited tube, remove the wire conductor. The electro- scope remains charged, and it will be found to be positive. A similar test made of the ball will show that it is negatively charged. o FIGURE 359. WIRE WITH INSU- LATING HANDLE. Hence, we learn that when an electrified body is brought near an object it induces the opposite kind of electrification on the side next it and the same kind on the remote side. 416. Charging an Electroscope by Induction. Hold a finger on the ball of the electroscope and bring near it an electrified glass tube (Fig. 360) . Remove the finger before taking away the tube ; the electroscope will be charged negatively. If a stick of elec- trified sealing wax be used in- stead of the glass tube, the electroscope will be charged positively. 417. Equality of the Two Charges. Using the appa- ratus of 415, charge the ball and the electroscope by induction. Then replace the wire conductor. The leaves of the electroscope will col- lapse, showing that the elec- troscope is discharged. If the P.GURE 360. -CHARGING ELECTROSCOPE bal1 be tested ' !t wil1 als BY INDUCTION. be found to be discharged. Hence, The inducing and the induced charges are equal to each other. 346 ELECTROSTATICS III. ELECTRICAL DISTRIBUTION > 418. The Charge on the Outside of a Conductor. Place a round metallic vessel of about one liter capacity on an insulated support (Fig. 361). Electrify strongly and test in succession both the inner and the outer surface, using a proof plane to convey the charge to the electro- scope. The inner surface will give no sign of electrification. Hence, it appears that the electrical charge of a conductor is confined to its outer surface. 419. Effect of Shape. Charge electrically an insulated egg-shaped conductor (Fig. 362). Touch the proof plane to the large end, and convey the charge to the electroscope. Notice the Test the side and the small end The greatest divergence of the FIGURE 361. CHARGE ON OUTSIDE. amount of divergence of the leaves. of the conductor in the same way. leaves will be produced by the charge from the small end and the least from the sides. The experiment shows that the surface density is greatest at the small end of the conductor. By surface density is meant the quantity of electrification on a unit area of the sur- face of the conductor. FIGURE 362. SURFACE DENSITY DEPENDENT ON CURVATURE. The distribution of the charge is, therefore, affected by the shape of the conductor, the surface density "being greater the greater the curvature. ACTION OF POINTS 347 420. Action of Points. Attach a sharp-pointed rod to one pole of an electrical machine ( 433), and suspend two pith balls from the same pole. When the machine is worked there will be little or no separa- tion of the pith balls. Hold a lighted candle near the pointed rod; the candle flame will be blown away as by a stiff breeze (Fig. 363). The experiment shows that an elec- tric charge is carried off by pointed conductors. This conclusion might have been drawn from the preceding experiment. When the curvature of the egg-shaped conductor becomes FIGURE 363. FLAME ,, ,, /. i BLOWN AWAY BY Dis- very great so that the surface be- CHARQE FROM POINT comes pointed, the surface density also becomes great and there is an intense field of electric force in the immediate neighborhood. The air particles touching the point become heavily charged and are then repelled ; other particles take their place and are in turn repelled and form an electrical wind. The conductor gives up its charge to the repelled particles of air. Questions 1. When a charge is conveyed by a proof plane to an electroscope, does the proof plane give up its entire charge ? 2. Why will not an electroscope remain charged indefinitely ? 3. If the ball of an electroscope were hollow with an aperture so that the charged proof plane could be introduced, would any charge remain on the proof plane after touching the inside of the ball ? 4. Will dust have any effect on the working of electrical apparatus ? 5. Why should electrical apparatus be warmer than the room if we are to get good results in electrostatic experiments ? 6. Why does electrostatic apparatus wort better in cold weather than in warm ? 7. Place an electroscope in a cage of fine wire netting. Why is it not affected by an electrified glass rod held near it? 348 ELECTROSTATICS 8. Why does not a metal rod held in the hand and rubbed with silk show electrification ? 9. With a positively charged globe, how could another insulated globe be charged without reducing the charge on the first one? 10. In charging an electroscope by induction, why must the finger be withdrawn before removing the inducing charge ? IV. ELECTRIC POTENTIAL AND CAPACITY 421. The Unit of Electrification or Charge. Imagine two minute bodies similarly charged with equal quantities of electricity. They will repel each other. If the two equal and similar charges are one centimeter apart in air, and if they repel each other with a force of one dyne, then the charges are both unity. The electrostatic unit of quantity is that quantity which will repel an equal and similar quantity at a distance of one centimeter in air with a force of one dyne. It is necessary to say " in air " because, as will be seen later, the force between two charged bodies depends on the nature of the medium be- tween them ( 427). This electrostatic unit is very small and has no name. In prac- tice, a larger unit, called the coulomb, is employed. It is equal to 3 x 10 9 electrostatic units. 422. Potential Difference. The analogy between pressure in hydro- statics and potential in electrostatics FIGURE 364.- ILLUSTRATING ig ft convenient and helpful POTENTIAL DIFFERENCE. / one. W ater will now from the tank A to the tank B (Fig. 364) when the stopcock S in the connecting pipe is open if the hydrostatic pressure at a is greater than at b ; and the flow is attributed directly to this difference of pressure. ZERO POTENTIAL 349 In the same way, if there is a flow of positive electricity from A to B when the two conductors are connected by a conducting wire r (Fig. 365), the electrical potential is said to be higher at A than at B, and the difference of electrical potential between A. and B is assigned as the cause of the flow. In both FIGURE 365. CONDUCTOR A OF cases the flow is in the di- ^"J* P TENT1AL AN C NDUC - rection of the difference of pressure or difference of potential, irrespective of the fact that B may already contain more water because of its large cross section, or a greater electric charge because of its larger capacity ( 425). If the electric charge in a system of connected conduc- tors is in a stationary or static condition, there is then no potential difference between different points of the system. The potential difference between two conductors is meas- ured by the work done in carrying a unit electric charge from the one to the other. 423. Unit Potential Difference. There is unit potential difference between two conductors when one erg of work is required to transfer the unit electric charge from one conductor to the other. This is called the absolute unit ; for practical purposes it has been found more con- venient to employ a unit of potential difference (P. D.), which is -%fa of the absolute unit, and which is called the volt, in honor of the Italian physicist, Alessandro Volta. 424. Zero Potential. In measuring the potential differ- ence between a conductor and the earth, the potential of the earth is assumed to be zero. The potential difference is then numerically the potential of the conductor. If a 350 ELECTROSTATICS conductor of positive potential be connected with the earth by an electric conductor, the positive charge will flow to the earth. If the conductor has a negative poten- tial, the flow of the positive quantity will be in the other direction. 425. Electrostatic Capacity. If water be poured into a cylindrical jar until it is 10 cm. deep, the pressure on the bottom of the jar is 10 g. of force per square centimeter. If the depth of the water be increased to 20 cm., the pres- sure will be 20 g. of force per square centimeter ( 53). It thus appears that there is a constant relation between the quantity of water Q and the pressure P ; that is, .j = (7, a constant. Again, if a gas tank be filled with gas at atmospheric pres- sure, it will exert a pressure of 1033 g. of force per square centimeter ( 81). If twice as much gas be pumped into the tank, the pressure by Boyle's law ( 87) will be doubled at the same temperature ; that is, there is a constant re- lation between the quantity of gas Q and the pressure P of the gas in the tank, or -^ = (7, a constant as before. In the same way, if an electric charge be given to an insulated conductor, its potential will be raised above that of the earth. If the charge be doubled, the potential difference between the conductor and the earth will also be doubled. Precisely as in the case of the water and of the gas, there is a constant relation between the amount of the charge Q and the potential difference V between the conductor and the earth ; that is, -j^= O. This ratio or constant O is the electrostatic capacity of the conductor. If V= 1, then = Q ; from which it follows that the INFLVENCE OF THE DIELECTRIC 351 electrostatic capacity of a conductor is equal to the charge re- quired to raise its potential from zero to unity. From-;= we have#= CV, and F= - (Equation 34) 426. Condensers. Support a metal plate in a vertical position on an insulating base (Fig. 366). Connect it to the knob of an elec- troscope by a fine copper wire. Charge the plate until the leaves of the electroscope show a wide divergence. Now bring an uninsulated conducting plate near the charged one and parallel to it. The divergence of , . _ Electroscope the leaves will decrease; remove the uninsulated plate and the divergence will increase again. The capacity of an insulated conductor is increased by the presence of another conductor FIGURE 366. CONDENSER connected with the earth. The effect of this latter conductor is to decrease the potential to which a given charge will raise the insulated one. Such an arrangement of parallel conductors separated by an insulator or dielectric is called a condenser. A. condenser is a device which greatly increases the charge on a conductor without increasing its potential. In other words, the plate connected with the earth greatly increases the capacity of the insulated conductor. 427. Influence of the Dielectric. Charge the apparatus of the last experiment, with the uninsulated plate at a distance of about 5 cm. from the charged plate and parallel to it, thrust suddenly be- tween the two a cake of clean paraffin as large as the metal plates or larger, and from 2 to 4 cm. thick. Note that the leaf of the electro- scope (Fig. 356) collapses slightly. Remove the paraffin quickly, and the divergence will increase again. A cake of sulphur will produce a more marked effect on the divergence of the leaf. 352 ELECTROSTA TICS FIGURE 367. LEY- DEN JAR. The presence of the paraffin or the sulphur increases the capacity of the condenser and, hence, decreases its potential, the charge remaining the same. Paraffin and sulphur, as examples of dielectrics, are said to have a larger dielectric capacity or dielectric constant than air. Glass has a dielectric capacity from four to ten times greater than air. 428. The Leyden Jar is a common and convenient form of condenser. It con- sists of a glass jar coated part way up, both inside and outside, with tin-foil (Fig. 367). Through the wooden or ebonite stopper passes a brass rod, ter- minating on the outside in a ball and on the inside in a metallic chain which reaches the bottom of the jar. The glass is the dielectric separat- ing the two tin-foil conduct- ing surfaces. 429. Charging and Discharg- ing a Jar. To charge a Ley- den jar connect the outer surface to one pole of an electrical machine ( 433), either by a metallic conductor or by holding the jar in the hand. Hold the ball against the other pole. To discharge a Leyden jar bend a wire into the form of the letter V. With one end of the wire touching the outer surface of the jar (Fig. 368), bring the other around near the ball, and the discharge will take place. FIGURE 368. DISCHARGING A LEYDEN JAR. THEORY OF THE LEYDEN JAR 353 FIGURE 369. SEAT OF THE CHARGE. 430. Seat of Charge. Charge a Leyden jar made with movable metallic coatings (Fig. 369) and set it on an insulating stand. Lift out the inner coating, and then, taking the top of the glass vessel in one hand, remove the outer coating with the other. The coatings now exhibit no sign of electrification. Bring the glass vessel near a pile of pith balls ; they will be attracted to it, showing that the glass is electrified. Reacji over the rim with the thumb and forefinger and touch the glass. A slight discharge may be heard. Now build up the jar by putting the parts together ; the jar will still be highly electrified and may be discharged in the usual way. This experiment was devised by Franklin ; it seems that electrification is a phenomenon of the glass, and that the metallic coatings serve merely as conductors, making it possible to discharge all parts of the glass at once. Some claim that the moisture condensed on the glass acts as a conductor when the metallic coatings are removed. 431. Theory of the Leyden Jar. A Leyden jar may be perforated by overcharging, may be discharged by heat- ing, and if heavily charged is not completely discharged by connecting the two coatings ; if left standing a few seconds, the two coatings gradually acquire a small potential difference and a second small discharge may be obtained, known as the residual charge. It appears, therefore, that the glass of a charged jar is strained or distorted ; like a twisted glass fiber, it does not return at once to its normal state when released. The two surfaces of the glass are oppositely electrified, the one charge acting inductively through the glass and producing the opposite electrification on the other surface. The two charges are held inductively and are said to be 354 ELECTROSTATICS " bound," in distinction from the charge on an insulated conductor, which is said to be "free." Questions 1. Will a charged Leyden jar be discharged by touching the knob while the jar rests on a sheet of hard rubber ? 2. Will a Leyden jar be appreciably charged by applying charges to the knob while the jat rests on a sheet of hard rubber? 3. In discharging a Leyden jar with a bent wire, why not touch the wire to the knob before touching the outside surface ? 4. Cuneus tried to charge a bowl of water by holding it in his hand, while the chain of an electrical machine dipped into the water. When he lifted the chain with the other hand he got a shock. Why? 5. Explain why a small metal ball suspended by a silk thread between two bodies, the two being near together and charged, one negatively and the other positively, flies back and forth between the two bodies. V. ELECTRICAL MACHINES 432. The Electrophorus. The simplest induction elec- trical machine is the electrophorus (Fig. 370), invented by Volta. A cake of resin or disk of vulcanite A rests in a metallic base B. Another metallic disk or cover C is provided with an insulating handle D. The resin or vul- canite is electrified by rub- bing with dry flannel or striking with a catskin, and the metal disk is then placed FIGURE 370. ELECTROPHORUS. ., . ,, on it. Since the cover touches the nonconducting resin or vulcanite A in a few points only, the negative charge due to the friction is not removed. The two disks with the film of air be- tween them form a condenser ( 426) of great capacity. INFLUENCE ELECTRICAL MACHINES 355 Touch the cover momentarily with the finger, and the repelled negative charge passes to the earth, leaving the cover at zero poten- tial. Lift it by the insulating handle, the positive charge becomes free ( 431), and a spark may be drawn by holding the finger near it. This operation may be repeated an indefinite number of times with- out sensibly reducing the charge on the vulcanite. When the cover is lifted by the insulating handle, work is done against the electrical attraction between the nega- tive charge on the vulcanite and the positive on the cover. The energy of the charged cover represents this work. The electrophorus is, therefore, a device for transforming energy in some other form into the energy of electric charges. 433. Influence Electrical Machines. There are many in- fluence or induction electrical machines, but it will suffice to describe only one, as the principle is always the same. FIGURE 371. TOEPLER-HOLTZ MACHINE. The Holtz machine, as modified by Toepler and Voss, is illustrated in Fig. 371. There are two glass plates, e 1 and e, about 5 mm. apart, the former stationary and the 356 ELECTED ST A TICS latter turning about an insulated axle by means of the crank h and a belt. The stationary plate supports at the back two paper sectors,. c and : ->T33M : M>vvA < liM^^i^SISli I. Parallel cur- rents flowing in the same direction at- tract. II. Parallel cur- rents flowing in op- posite directions re- pel. III. Currents t FIGURE 412. MAGNETIC FIELD ABOUT PARAL- matong an angle LEL CURRENTS IN THE SAME DIRECTION. with each other tend to become parallel and to flow in the same direc- tion. 483. Magnetic Fields about Parallel Currents. Figure 412 was made from a photograph of the magnetic field about two parallel currents in the same direction perpendicular to the figure. Many of these lines of force surround both wires, and it is the tension along them that draws the wires together. Figure 413 was made from a photograph of the field when the currents were in opposite direc- tions. The lines of force are crowded together between the wires, and their reaction in their effort to recover .... JSplfi :~vkv>^ their normal posi- ~H&r*">;>fS>Vif3l A . tion forces the FIGURE 413. MAGNETIC FIELD ABOUT PARALLEL CURRENTS IN OPPOSITE DIRECTIONS. wires apart. 392 ELECTRIC CURRENTS VI. ELECTROMAGNETS 484. Effect of Introducing Iron into a Solenoid. Fill the lower half of the helix of 480 with soft straight iron wires, and again pass the same current as before through the coil. The mag- netic field will be greatly strengthened by the iron. A helix of wire about an iron core is an electromagnet. It was first made by Sturgeon in 1825. The presence of the iron core greatly increases the number of lines of force threading through the helix from end to end, by reason of FIGURE 414. IRON INCREASES MAGNETIC LINES. the greater permeability of iron as compared with air (Fig. 414). If the iron is omitted, there are not only fewer lines of force, but because of their leakage at the ^ides of the helix, fewer traverse the entire length of the coil. The soft iron core of an electromagnet does not show much magnetism except while the current is flowing through the magnetizing coil. The loss of magnetism is not quite complete when the current is interrupted; the small amount remaining is called residual magnetism. 485. Relation between a Magnet and a Flexible Conductor. Iron filings arranged in circles about a conductor may be regarded as flexible magnetized iron winding- itself into a helix around the current; conversely, a flexible conductor, carrying a current, winds James Clerk-Maxwell (1831-1879) was a remarkable physi- cist and mathematician. He was born in Edinburgh and studied in the University of that city. Later he attended the University of Cambridge, graduating from there in 1854. In 1856 he be- came professor of natural philosophy at Marischal College, Aber- deen, and in 1860 professor of physics and astronomy at King's College, London. In 1871 he was appointed professor of experi- mental physics in Cambridge. His contributions to the kinetic theory of gases, the theory of heat, dynamics, and the mathemati- cal theory of electricity and magnetism are imperishable monu- ments to his great genius and wonderful insight into the mysteries of nature. THE HORSESHOE MAGNET 393 itself around a straight bar magnet. The flexible conductor of Figure 415 may be made of tinsel cord or braid. Directly the circuit is closed, the conductor winds slowly around the vertical mag- net ; if the current is then re- versed, the conductor unwinds and winds up again in the re- verse direction. 486. The Horseshoe Mag- net. The form given to an electromagnet depends on the use to which it is to be put. The horseshoe or U-shape (Fig. 416) is the most common. The ad- vantage of this form lies in the fact that all lines of magnetic force are closed curves, passing through the core from the south to the north pole, and completing the circuit through the air from the north pole back to the south pole. The U-shape lessens the distance through the air and thus increases the number of lines. Moreover, when an iron bar, called the armature, is placed across the poles, the air gap is re- duced to a thin film, the number of lines is increased to a maximum with a given current through the helix, and the magnet exercises the greatest pull on the arma- ture. When the armature is in contact with the poles, the magnetic circuit is all iron, and is said to be a closed FIGURE 415. FLEXIBLE CONDUCTOR WINDS ITSELF AROUND A MAGNET. II FIGURE 416. HORSESHOE MAGNET. 394 ELECTRIC CURRENTS magnetic circuit. The residual magnetism is then much greater than in the case of an open magnetic circuit with an air gap. Bring the armature in contact with the iron poles of the core, and close the electric circuit ; after the circuit is opened, the armature will still cling to the poles and can be removed only with some effort. Then place a piece of thin paper between the poles and the armature. After the magnet has again been excited and the circuit opened, the armature will not now " stick." The paper makes a thin air gap between the poles of the magnet and the armature, and thus reduces the residual magnetism. 487. Applications of Electromagnets. The uses to which electromagnets are put in the applications of electricity are so numerous that a mere reference to them must suffice. The electromagnet enters into the construc- tion of electric bells, telegraph and telephone instruments, dynamos, motors, signaling de- vices, etc. It is also extensively used in lift- ing large masses of iron, such as castings, rolled plates, pig iron, and steel girders (Fig. 417). The lifting power depends chiefly on the cross section of the iron core and on the ampere turns that is, on the product of the number of amperes of current and the number of turns of wire wound on the magnet. VII. MEASURING INSTRUMENTS 488. The Galvanometer. The instrument for the com- parison of currents by means of their magnetic effects is called a galvanometer. A galvanoscope ( 446) becomes a galvanometer by providing it with a scale so that the deflections may be measured. If the galvanometer is calibrated, so as to read directly in amperes, it is called an ammeter. In very sensitive instruments a small mirror is FIGURE 417. LIFTING MAGNET. THE D'ARSONVAL GALVANOMETER 395 FIGURE 418. PLAN OF D'AR- SONVAL GALVANOMETER. attached to the movable part of the instrument; it is then called a mirror galvanometer. Sometimes a beam of light from a lamp is reflected from this small mirror back to a scale, and sometimes the light from a scale is reflected back to a small tele- scope, by means of which the de- flections are read. In either case the beam of light then becomes a long pointer without weight. 489. The d'Arsonval Galvanom- eter One of the most useful forms of galvanometer is the d'Arsonval. The plan of it' is shown in Figure 418 and a com- plete working instrument in Figure 419. Between the poles of a strong permanent magnet of the horseshoe form swings a rectangular coil of fine wire in such a way that the current is led into the coil by the fine suspending wire, and out by the wire spiral running to the base. A small mirror is attached to the coil to reflect light from a lamp or an illuminated scale. Some- times the coil carries a light aluminum pointer, which traverses a scale. Inside the coil is a soft iron tube supported from the back of the case. It is designed to concentrate the lines of force in the narrow openings between it and the poles of the magnet. In the d'Arsonval galvanometer the coil is movable and the magnet is fixed. Its chief advantages are simplicity FIGURE 419. SIMPLE D'ARSON- VAL GALVANOM- ETER. 396 ELECTRIC CURRENTS of construction, comparative independence of the earth's magnetic field, and the quickness with which the coil comes to rest after deflection by a current through it. 490. The Voltmeter. The voltmeter is an instrument designed to measure the difference of potential in volts. For direct currents the most convenient port- able voltmeter is made on the principle of the d'Arsonval galvanom- eter. One of the best- FIGURE 420. VOLTMETER. known instruments of this class is shown in Figure 420. The interior is rep- resented by Figure 421, where a portion of the in- strument is cut away to show the coil and the springs. The current is led in by one spiral spring and out by the other. Attached to the coil is a very light aluminum pointer, which moves over the scale seen in Figure 420 where it stands at zero. Soft iron pole pieces are screwed fast to the poles of the per- manent magnet, and they are so shaped that the divisions of the scale in volts are equal. In circuit with the coil of _ . ...'. FIGURE 421. INSIDE OF VOLTMETER. the instrument is a coil of wire of high resistance, so that when the voltmeter is placed in circuit, only a small current will flow through it. DIVIDED CIRCUITS SHUNTS 397 491. The Ammeter, designed to measure electric currents in amperes, is very similar in construction to the voltmeter. A low resistance shunt is connected across the terminals of the coil to carry the main current, so that when the ammeter is placed in circuit, it will not change the value of the current to be measured. Questions 1. Why must the article to be electroplated be attached to the negative pole of the generator ? 2. How can you determine the positive pole of a storage battery? 3. Why will it ruin a pocket ammeter to connect its terminals to the poles of a storage battery ? 4. Why does the heating in an electric circuit manifest itself at a point where the conductor is defective? 5. If in Figure 415 the north pole of the magnet is at the top, which way will the flexible tinsel wrap around the magnet? 6. Why should the ammeter be of low resistance and the volt- meter of high resistance ? 7. Why should a Daniell cell when not in use either be taken down or placed on closed circuit ? 8. Why must the wire used in winding an electromagnet be insulated ? 9. What is the least number of gravity cells that might be used to charge a storage battery and how must they be connected? 10. What would be the harm of leaving a dry cell on a closed circuit ? 11. Why will a low resistance voltmeter give the E.M.F. of a storage battery more nearly correct than it will that of a dry cell ? 12. Why will cotton-wound wire be sufficiently insulated for a battery, but not for a Holtz machine ? 49 Divided Circuits Shunts. When the wire leading from any electric generator is divided into two branches, as at B (Fig. 422), the current also divides, part flowing 398 ELECTRIC CURRENTS FIGURE 422. DIVIDED CIRCUIT. by one path and part by the other. The sum of these two currents is always equal to the current in the undivided part of the circuit, since there is no accumulation of electric- ity at any point. Either of the branches between B and A is called a shunt to the other, and the currents through them are inversely propor- tional to their resistances. 493. Resistance of a Divided Circuit. Let the total resist- ance between the points A and B (Fig. 422) be represented by R, that of the branch BmA by r, and of BnA by r r . The conductance of BA equals the sum of the conductances of the two branches ; and, as conductance is the reciprocal of resistance, the conductances of BA, BmA, and BnA are , -, and respectively; Mr r rr' then - = + . R r r' From this we derive R = To illustrate, let a galvanometer r + r' whose resistance is 100 ohms have its binding posts connected by a shunt of 50 ohms resistance; then th$ total resistance of this divided circuit is 100 x 50 = 331 ohms. B\ rowr^ 100 + 50 The introduction of a shunt always lessens the resistance be- tween the points connected. 494. Loss of Potential along a Conductor. Stretch a fine wire of fairly high resist- ance, such as a German silver No. 30, along the edge of a meter stick (Fig. 423). Connect the ends P and Q to a storage cell with a contact key in circuit. At P connect a galvanometer in circuit with a high resistance R and a slide contact S. The galvanometer will indicate the difference of potential between P and S, the point of contact on PQ. If S be placed successively on PQ at 10 cm., 20 cm., 30 cm., etc., from P, and FIGURE 423. FALL OF POTENTIAL ALONG A CONDUCTOR. WHEATSTONE'S BRIDGE 399 the galvanometer reading be recorded each time, the ratio of the readings will be as 1:2:3, etc. Since resistance is proportional to length, these potential differences are as the resistances of the succes- sive lengths of the wire PQ, or the loss of potential is proportional to the resistance passed over. This is equivalent to another statement of Ohm's law ; IT for since 1= , and the current through the conductor is H> the same at all points, it follows that E must vary as R to make I constant. 495. Wheatstone's Bridge. The Wheatstone's Bridge is a de- vice for measuring resistances. The four conductors, R v R z , R z , R 4 are the arms and BD the bridge (Fig. 424). When the circuit is closed by closing the key K v the current divides at A, the two parts reuniting at C. The loss of potential along ABC is the same as along ADC. If no current flows through the galvanometer G when the key K } is. also closed, then there is no potential difference between B and D to produce a current. Under these conditions the loss of potential from A to B is the same as from A to D. We may then get an expression for these potential differences and place them equal to each other. Let 7j be the current through R l ; it will also be the current through R, because none flows across through the galvanometer. Also let 7 2 be the current through the branch A DC. Then the poten- tial difference between A and B by Ohm's law ( 471) is equal to RJi ; and the equal potential difference between A and D is R z l y Equating these expressions, R 1 l l - R 2 J 2 (a) In the same way the equal potential differences between B and C and D and C give FIGURE 424. WHEATSTONE'S BRIDGE. 400 ELECTEJC CURRENTS Dividing (a) by (6) gives -2 ...... (Equation 38) In practice three of the four resistances are adjustable and of known value. They are adjusted until the galvanometer shows no deflection when the key K\ is closed after key K y The value of the fourth resistance is then derived from the relation in Equation 38. Problems 1. Calculate the resistance of 200 ft. of copper wire (k = 10.19) No. 24 (diameter = 0.0201 in.). 2. A coil of iron wire (k = 61.3) is to have a resistance of 25 ohms. The diameter of the wire used is 0.032 in. How many feet will it re- quire ? 3. What diameter must a copper trolley (k = 10.19) have so that the resistance will be half an ohm to the mile ? 4. A current of one ampere deposits by electrolysis 1.1833 g. of copper in an hour. How long will it take a current of 5 amperes to deposit a kilogram of copper ? 5. A current of two amperes passes through a solution of silver nitrate for one hour. How much silver will be deposited ? 6. A current of 10 amperes is sent through a resistance of 4 ohms for 10 minutes. How many calories of heat are generated ? 7. What current will 6 dry cells connected in series, each having an E. M. F. of 1.5 volts and an internal resistance of 0.1 ohm, give through an external resistance of 2 ohms ? 8. A certain dry cell has a voltage of 1.5 volts and when tested with an ammeter gives 20 amperes. What is its internal resistance ? 9. A certain lamp requires 0.5 ampere current and an E. M. F. of 110 volts to light it. What is its resistance ? 10. A projection lantern requires a current of 15 amperes. The voltage of the supply is 110 volts and the loss in the lamp is 40 volts. What resistance must be inserted in the line to the lantern? Joseph Henry (1797-1878) was born at Albany, New York. The reading of Gregory's Lectures on Experimental Philosophy interested him so greatly in science that he began experimenting. In 1829 he constructed his first electromagnet. In 1832 he was appointed professor of natural philosophy at Princeton College. In 1846 he became secretary of the Smithsonian Institution in Washington. It is almost certain that he anticipated Faraday's great discovery of magneto-electric induction by a whole year but failed to announce it. His principal investigations were in electricity and magnetism, and chiefly in the realm of induced currents. Michael Faraday, 1791-1867, was born near London, England. He was the son of a blacksmith and received but little schooling, being apprenticed to a bookbinder when only thirteen years of age. While employed in the bindery he became interested in reading such scientific books as he found there. Later he applied to Sir Humphry Davy for consideration and was made Davy's assistant. From this time his rise was rapid ; in 1816 he published his first scientific memoir; in 1824 he became a member of the Royal Society; in 1825 he was elected director of the Royal Institution ; in 1831 he announced the discovery of magneto- electric induction, the most important scientific discovery of any age. In 1833 he was elected professor of chemistry in the Royal Institution. He was a remarkable experimenter and a most inter- esting lecturer, and amid all his wonderful achievements, he was utterly wanting in vanity. CHAPTER XIII FIGURE 425. FARADAY'S ORIGINAL EXPERIMENT ON INDUCED E. M. F. ELECTROMAGNETIC INDUCTION I. FAEADAY'S DISCOVERIES 496. Electromotive Force Induced by a Magnet. Wind a large number of turns of fine insulated wire around the armature of a horseshoe magnet, leaving the ends of the iron free to come in contact with the poles of the permanent magnet. Connect the ends of the coil to a sensitive galvanometer, the armature being in contact with the magnetic poles, as shown in Figure 425. Keeping the mag- net fixed, suddenly pull off the ar- mature. The galvanometer will show a momentary current. Suddenly bring the armature up against the poles of the magnet; another momentary current in the reverse direction will flow through the circuit. This experiment illustrates Faraday's original method of producing an electric current through the agency of magnetism. Connect a coil of insu- lated copper wire, at least fifty turns of No. 24, in circuit with a d'Arsonval galvanometer (Fig. 426). Thrust quickly into the coil the north pole of a bar magnet. The galva- nometer will show a transient current, which will flow only during the motion of the magnet. When the magnet is suddenly withdrawn 401 FIGURE 426. CURRENT INDUCED BY THRUST- ING MAGNET INTO COIL. 402 ELECTROMAGNETIC INDUCTION a transient current is produced in the opposite direction to the first one. If the south pole be thrust into the coil, and then withdrawn, the currents in both cases are the reverse of those with the north pole. If we substitute a helix of a smaller number of turns, or a weaker bar magnet, the deflection will be less. The morrientary electromotive forces generated in the coil are known as induced electromotive forces, and the cur- rents as induced currents. They were discovered by Faraday in 1831. 497. Laws of Electromagnetic Induction. When the armature in the first experiment of the last article is in contact with the poles of the magnet, the number of lines of force passing through the coil, or linked with it, is a maximum. When the armature is pulled away, the num- ber of magnetic lines threading through the coil rapidly diminishes. When the magnet in the second experiment is thrust into the coil, it carries its lines of force with it, so that some of them at least encircle, or are linked with, the wires of the coil. In both experiments an electromotive force is generated only while the number of lines so linked with the coil is changing. The E.M.F. is generated in the coil in accordance with the following laws : I. An increase in the number of lines of force linked with a conducting circuit produces an indirect E. M. F. ; a decrease in the number of lines produces a direct E. M. F. IT. The induced E.M.F. at any instant is equal to the rate of increase or decrease in the number of lines of force linked with the circuit. A direct E.M.F. has a clockwise direction to an observer looking along the lines of force of the magnet ; an indirect E. M. F. is one in the opposite direction. Thus, in Figure INDUCTION BY CURRENTS 403 FIGURE 427. DIRECTION OF IN- DUCED E.M.F. 427 the north pole of the magnet is moving into the coil in the direction of the arrow ; there is an increase in the number of lines passing through the coil, and the E. M.F. and current are indirect or opposite watch hands, as shown by the arrows on the coil, to an observer looking at the coil in the direction of the arrow on the magnet. 498. Induction by Currents. Connect the S coil of Figure 428 to a d'Arsonval galvanometer, and a second smaller coil P to the terminals of a battery. If the current through P is kept constant, when P is made to approach S an E. M. F. is generated in S tending to send a current in a direction opposite to the current around P; removing the coil P generates an oppo- site E. M. F. These E. M. F.'s act in S only so long as P is To moving. battery Next insert the coil P in S with the battery circuit open. If then the battery circuit is closed, the needle of the gal- vanometer will be deflected, fi|!;:: but will shortly come again to rest at zero. The direction of this momentary current is opposite to that in P. Open- ing the battery circuit produces another momentary current through S but in the opposite direction. Increasing and decreasing the current through P has the same effect as closing and opening the circuit. FIGURE 428. CURRENTS INDUCED BY ANOTHER CURRENT. 404 ELECTROMAGNETIC INDUCTION If while P is inside S with the battery circuit closed, a bar of soft iron is placed within P, there is an increase of magnetic lines through both coils and the inductive effect in S is the same as that produced by closing the circuit through P. The coil P is called the primary and S the secondary coil. The results may be summarized as follows: I. A momentary current in the opposite direction is induced in the secondary conductor by the approach, the starting, or the strengthening of a current in the primary. II. A momentary current in the same direction is in- duced in the secondary by the receding, the stopping, or the weakening of the current in the primary. The primary coil becomes a magnet when carrying an electric current ( 480) and acts toward the secondary coil as if it were a magnet. The soft iron increases the magnetic flux through the coil and so increases the in- duction. 499. Lenz's Law. When the north pole of the magnet is thrust into the coil of Figure 427, the induced current flow- ing in the direction of the arrows produces lines of force running in the opposite direction to those from the magnet ( 479). These lines of force tend to oppose the change in the magnetic field within the coil, or the magnetic field set up by the coil opposes the motion of the magnet. Again, when the primary coil of Figure 428 is inserted into the secondary, the induced current in the latter is opposite in direction to the primary current, and parallel currents in opposite directions repel each other. In every case of electromagnetic induction the change in the mag- netic field which produces the induced current is always opposed by the magnetic field due to the induced current itself. JOSEPH HENRY'S DISCOVERY 406 The law of Lenz respecting the direction of the induced current is broadly as follows: The direction of an induced current is always such that it produces a magnetic field opposing the motion or change which induces the current. II. SELF-INDUCTION 500. Joseph Henry's Discovery. Joseph Henry discov- ered that a current through a helix with parallel turns acts inductively on its own circuit, producing what is often called the extra current, and a bright spark across the gap when the circuit is opened. The effects are not very marked unless the helix contains a soft iron core. Let a coil of wire be wound around a wooden cylinder' (Fig. 429). When a current is flowing through this coil, some of the lines of force around one turn, as A, thread through adjacent turns ; if the cylinder is iron, the number of lines threading through adjacent turns will be largely increased on account of the superior permeability of the iron ( 401). Hence, at the make of the circuit, the production of magnetic lines threading through the parallel turns of wire induces a counter-E.M.F. opposing the current. The result is that the current does not reach at once the value given by Ohm's law. At the break of the circuit, the induction on the other hand produces a direct E.M.F. tending to prolong the current. With many turns of wire, this direct E.M.F. is high enough to break over a short gap and produce a spark. FIGURE 429. SELF-INDUC- TION. 406 ELECTS OMA GNETIC IND UCTION 501. Illustrations of Self-induction. Connect two or three cells in series. Join electrically a flat file to one pole and a piece of iron wire to the other. Draw the end of the wire lengthwise along the file ; some sparks will be visible, but they emit little light. Now put an electromagnet in the circuit to increase the self-induction ; the sparks are now much longer and brighter. Connect as shown in Figure 430 a large electromagnet M, a storage battery B, a circuit breaker K, and an incandescent lamp L of such a size that the battery alone will light it to nearly its full candle power. The circuit divides between the lanip and the electromagnet, and since the latter is of low resistance, when the cur- rent reaches its steady state most of it will go through the coils of the magnet, leaving the lamp at only a dull red. At the in- stant when the circuit is closed, the self- induction of the magnet acts against the current and sends most of it around through the lamp. It accordingly lights up at first, but quickly grows dim as the current rises to its steady value in M. Now open the circuit breaker K, cutting off the battery. The only closed circuit is now the one through the magnet and the lamp ; but the energy stored in the magnetic field of the electromagnet is then converted into electric energy by means of self-induction, and the lamp again lights up brightly for a moment. M ) 4 1 1 B 1 1 1 K 1 FIGURE 430. LAMP LIGHTED BY SELF-INDUCTION OF MAGNET. III. THE INDUCTION COIL 502. Structure of an Induction Coil. The induction coil is commonly used to give transient flashes of high electro- motive force in rapid succession. A primary coil of com- paratively few turns of stout wire is wound around an iron core, consisting of a bundle of iron wires to avoid induced or eddy currents in the metal of the core ; outside ACTION OF THE COIL 407 of this, and carefully insulated from it, is the secondary of a very large number of turns of fine wire. The inner or primary coil is connected to a battery through a cir- cuit breaker (Fig. 431). This is an automatic device for opening and closing the primary circuit and is ac- tuated by the magnetism of the iron core. At the "make" and "break" of ., . , FIGURE 431. INDUCTION COIL. the primary circuit elec- tromotive forces are induced in the secondary in accord- ance with the laws of electromagnetic induction ( 497). Large induction coils include also a condenser. It is placed in the base and consists of two sets of interlaid layers of tin-foil, separated by sheets of paper saturated with paraffin. The two sets are connected to two points of the primary circuit on opposite sides of the circuit breaker (Fig. 432). 503. Action of the Coil. Figure 432 shows the arrange- ment of the various parts of an induction coil. The cur- rent first passes through the heavy primary wire PP, thence through the spring A, which carries the soft iron block F, then across to the screw 5, and so back to the negative pole of the battery. This current magnetizes the iron core of the coil, and the core attracts the soft iron block F, thus breaking the circuit at the point of the screw b. The core is then demagnetized, and the release of F again closes the circuit. Electromotive forces are thus induced in the secondary coil SS, both at the make and the break of the primary. The high E.M.F. of the secondary is due to the large number of turns of wire in 408 ELECTROMAGNETIC INDUCTION it and to the influence of the iron core in increasing the number of lines of force which pass through the entire coil. The self-induction of the primary has a very important bearing on the action of the coil. At the instant the cir- cuit is closed, the counter E.M.F. opposes the battery FIGURE 432. STRUCTURE OF INDUCTION COIL. current, and prolongs the time of reaching its greatest strength. Consequently the E.M.F. of the secondary coil will be diminished by self-induction in the primary. The E.M.F. of self-induction at the "break" of the pri- mary is direct, and this added to the E.M.F. of the battery produces a spark at the break points of the circuit breaker. 504. Office of the Condenser. When the primary circuit of an induction coil is broken, the self-induction tends to sustain the cur- rent as if it had inertia; hence it jumps the break as a spark and prevents the abrupt interruption of the primary current, which is essential to high induction in the secondary. The condenser connected across the break gap acts as a reservoir into which the current surges instead of jumping across the break. Thus the spark is nearly elimi- nated and the secondary E. M. F. increased. Further, after the break, the condenser, which has been charged by the E.M.F. of self-induction, discharges back through the primary EXPERIMENTS WITH THE INDUCTION COIL 409 coil. The condenser thus causes an electric recoil in the current in the reverse direction through the primary, demagnetizing the core and increasing the rate of change of the magnetic flux, and so in- creasing the E. M. F. in the secondary. Hence, when the secondary terminals are separated, the discharge is all in one direction and occurs when the primary current is broken. 505. Experiments with the Induction Coil. 1. Physiological Effects. Hold in the hands the electrodes of a very small induction coil, of the style used by physicians. When the coil is working, a peculiar muscular contraction is produced. The " shock " from large coils is dangerous on account of the high E.M.F. The danger decreases with the in- crease in the rapidity of the impulses or alternations. Experiments with induction coils, worked by alternating currents of very high frequency, have demonstrated that the discharge of the secondary up to .an ampere may be taken through the body without injury. 2. Mechanical Effects. Hold a piece of cardboard between the electrodes of an induction coil giving a spark 3 cm. long. The card will be perforated, leaving a burr on each side. Thin plates of any nonconductor can be perforated in the same manner. 3. Chemical Effects. Place on a plate of glass a strip of white blotting-paper moistened with a solution of potassium iodide (a com- pound of potassium and iodine) and starch paste. Attach one of the electrodes of a small induction coil to the margin of the paper. With an insulator, handle a wire leading to the other electrode, and when the coil is in action, trace characters with the wire on the paper. The discharge decomposes the potassium iodide, as shown by the blue mark. This blue mark is due to the action of the iodine on the starch. If the current from the secondary of an induction coil be passed through air in a sealed tube, the nitrogen and oxygen will combine to form nitrous acid. This is the basis of some of the commercial methods of manufacturing nitrogen compounds from the nitrogen of the air. 4. Heating Effects. Figure 433 shows the plan of the " electric bomb." It is usually made of wood. Fill the hole with gun powder 410 ELECTEOMAGNETIC INDUCTION FIGURE 433. ELECTRIC BOMB. as far up as the brass rods and close the mouth with a wooden ball. Connect the rods to the poles of the induction coil. The sparks will ignite the powder and the ball will be projected across the room. The heating effect of the current in the secondary of a large induction coil may be shown by stretching between its poles a very thin iron wire with a small gap in it. The discharge will melt the part con- nected to the negative pole of the coil, while the other part will re- main below the temperature of ignition. 506. Discharges in Partial Vacua. Place a vase of uranium glass on the table of the air pump, under a bell jar provided with a brass sliding rod passing air-tight through the cap at the top (Fig. 434). Connect the rod and the air pump table to the terminals of the induction coil. When the air is exhausted a beautiful play of light will fill the bell jar. The display will be more beautiful if the vase is lined part way up with tin-foil. This experiment is known as Gassiol's cascade. The experiment may be varied by admitting other gases and exhausting again. The aspect of the colored light will be entirely changed. The best effects are obtained with discharges from the secondary of an in- duction coil in glass tubes when the exhaustion is carried to a pressure of about 2 ram. of mercury, and the tubes are permanently sealed. Platinum elec- trodes are melted into the glass at the two ends. Such tubes are known as Geissler tubes. They are made in a great variety of forms (Fig. 435), and the luminous effects are more intense in the narrow connecting tubes FIGURE 434. GAS- SIOT'S CASCADE. THE DISCHARGE INTERMITTENT 411 FIGURE 435. GEISSLER TUBES. Stratifications have been than in the large bulbs at the ends. The colors are de- termined by the nature of the residual gas. Hydrogen glows with a brilliant crimson; the vapor of water gives the same color, indicating that the vapor is disso- ciated by the discharge. An examination of this glow by the spectroscope gives the characteristic lines of the gas in the tube. Geissler tubes often ex- hibit stratifications, which consist of portions of greater brightness sepa- rated by darker intervals, produced throughout a tube 50 feet long. These stratifi- cations or striae present an unstable flickering motion, re- sembling that sometimes observed during auroral displays. 507. The Discharge Intermittent. On a disk of white card- board about 20 cm. in diameter paste disks of -black paper 2 cm. in diameter (Fig. 436). Rotate the disk rapidly by means of a whirling table or an electric motor and illuminate it by a Geissler tube in a dark room. The black spots will be sharp in outline because each flash is nearly instantaneous; and the spots in the different circles will either stand still, rotate forward, or rotate backward. If in the brief interval be- tween the flashes the disk rotates through an angle equal to that between the spots in one of the circles, the spots will appear to stand still; if it rotates through a slightly greater angle, the spots will appear to move slowly forward ; if through a smaller angle, they will appear to move slowly backward. FIGURE 436. DISK FOR IN- TERMITTENT ILLUMINATION. 412 ELECTROMAGNETIC INDUCTION Mount a Geissler tube on a frame attached to the axle of a small electric motor (Fig. 437). Illuminate the tube by an induction coil while it rotates. Star-shaped figures will be seen, consisting of a number of images of the tube, the number depending on the speed of the motor as compared with the period of vibration of the circuit breaker. 508. Cathode Rays. When the gas pressure in a tube is reduced below about a millionth of an atmos- phere, the character of the discharge is much altered. The positive column of light extending out from the anode gradually disappears, and the sides of the tube glow with brilliant phosphorescence. With English glass the glow is blue ; with German glass it is a soft emerald. The luminosity of the glass is produced by a radiation in straight lines from the cathode of the tube ; this radiation is known as cathode rays. They were first studied by Sir William Crookes, and the tubes for the purpose are called Crookes tubes. FIGURE 437. ROTATION OF GEISSLER TUBE. Many other substances besides glass are caused to glow by the impact of cath- ode rays (Fig. 438), such as ruby, diamond, and va- rious sulphides. The color of the glow depends on the substance. Cathode rays have a mechanical effect. In fact they consist of electrons ( 518) moving with yery high velocity approaching that FIGURE 438. FLUORESCENCE BY CATHODE RAYS. Sir William Crookes, a dis- tinguished English chemist, was born in 1832. In 1873 he began a series of investi- gations on the properties of high vacua. While engaged in this work he invented the radiometer, developed the Crookes tubes, and dis- covered what he called "ra- diant matter." His investi- gations led him very close to the discoveries of Rbntgen. He edited the Quarterly Jour- nal of Science from 1864 until his death 'in 1919. Wilhelm Konrad Rontgen was born in 1845. It was at Wiirzburg, Germany, in 1895, that he discovered while passing electric charges through a Crookes tube, that a certain kind of radiation was emitted capable of pass- ing through many substances known to be opaque to light. The nature of these rays being unknown, he called them ' X-rays." They differ from the cathode rays dis- covered by Crookes, in that they affect a sensitized photo- graphic plate. CATHODE BATS 413 of light. When they strike a target, their motion is arrested and their energy of motion is largely trans- , .' FIGURE 439. RAILWAY TUBE. ferred to the target. The light paddle wheel in Figure 439 runs smoothly on glass rails. It may FIGURE 440. MAGNETIC DEFLECTION OF CATHODE RAYS. be made to traverse the tube in either direction by projecting elec- trons from the cathode against the paddles on top. When the cathode is changed from one end to the other by reversing the current in the induction coil, the little wheel stops promptly and reverses its di- rection. Its paddles are driven as if by a blast from the cathode disk. Cathode rays, unlike rays of light, are deflected by a magnet, and when once deflected they do not regain their former direction (Fig. 440). Cathode rays proceed in straight lines, except as they are deflected by a magnet or by mutual repul- sion. A screen placed across their path interrupts them and casts a shadow on the walls of the tube. When the cathode is made in the form of a concave cup, the rays are FIGURE 441. Focus TUBE. brought to a focus near its center of 414 ELECTROMAGNETIC INDUCTION curvature; platinum foil placed at this focus is raised to bright incandescence and may be fused (Fig. 441). Glass on which an energetic cathode stream falls may be heated to the point of fusion. It has been conclusively shown that cathode rays carry negative charges of electricity. Hence the mutual repul- sion exerted on each other by two parallel cathode streams. 509. Roentgen Rays. The rays of radiant matter, as Crookes called it, emanating from the cathode, give rise to another kind of rays when they strike the walls of the tube, or a piece of platinum placed in their path. These last rays, to which Roentgen, their discoverer, gave the name of " X-rays" can pass through glass, and so get out of the tube. They also pass through wood, paper, flesh, and FIGURE 442. -ROENTGEN TUBE. man 7 other sub ' stances opaque to light. They are stopped by bones, metals (except in very thin sheets), and by some other substances. Roentgen discovered that they affect a photographic plate like light. Hence, photographs can be taken of objects which are entirely invisible to the eye, such as the bones in a living body, or bullets embedded in the flesh. A Crookes tube adapted to the production of Roentgen rays (Fig. 442) has a concave cathode K, and at its focus an inclined piece of platinum A, which serves as the anode. The X-rays originate at A and issue from the side of the tube. 510. X-Ray Pictures. The penetrating power of Roentgen rays depends largely on the pressure within the tube. With high exhaus- THE FLUOROSCOPE 415 With somewhat lower tion the rays have high penetrating power and are then known as "hard rays." Hard rays can readily penetrate several centimeters of wood, and even a few millimeters of lead. exhaustion, the rays are . less penetrating and are then known as "soft rays." The possibility of X-ray photographs depends on the variation in the pene- trability of different sub- stances for X-rays. Thus, the bones of the body ab- sorb Roentgen rays more than the flesh, or are less penetrable by them. Hence fewer rays traverse them. Since Roentgen rays cannot be focused, all photographs takenby them are only shadow pictures. A Roentgen photograph of a gloved hand is shown in Figure 443. The ring on the little finger, and the cuff studs are conspicu- ous. The flesh is scarcely visible because of the high penetrating power of the rays used. The photographic plate for the purpose is inclosed in an ordinary plate holder and the hand is laid on the holder next to the sensitized side. * 511. The Fluoroscope. Soon after the discovery of X-rays it was found that certain fluorescent substances, like platino-barium-cyanide, and calcium tungstate, be- come luminous under the action of X-ra}^s. This fact has been turned to account in the construction of &fluoro- scope (Fig. 444), by means of which shadow pictures of concealed objects become visible. An opaque screen is FIGURE 443. X-RAY PICTURE. 416 ELECTROMAGNETIC INDUCTION covered on one side with the fluorescent substance ; this screen fits into the larger end of a box blackened inside, and having at the other end an opening adapted to fit closely around the eyes, so as to exclude all outside light. When an object, such as the hand, is held against the fluorescent screen and the fluoroscope is turned FIGURE 444. FLUOROSCOPE. toward the Roentgen tube, the bones are plainly visible as darker objects than the flesh because they are more opaque to X-rays. The beating heart may be made visible in a similar manner. IV. RADIOACTIVITY AND ELECTRONS 512. Radioactivity. Wrap a photographic plate in black paper. Flatten a Welsbach mantle and lay it on the paper next to the film side of the plate. Place the whole in a light-tight box for about a week. If the plate be now developed, a photographic picture of the itiantle will appear on*it. The mantle contains the rare metal thorium. This ,met&l 'po^jsesses tKte property- of emittiri all the time radiations that act like . A-rays oh a ^hqto^aphic .plate. Substances having'this property are known as radioactive. The principal ones are uranium, polonium, actinium, tho- rium, radium, and their compounds. 513. Discovery of Radioactivity. The activity of X-rays in producing photographic changes led directly to the dis- covery of the radioactivity of uranium by Becquerel in 1896. He found that uranium salts give off spontaneously radiations capable of passing through black paper and thin THREE KINDS OF RADIUM RATS 417 sheets of aluminum foil, and that they affect photographic plates as X-rays do. These radiations are not modified in any way by the most drastic treatment of the uranium, whether by heat or cold or other physical changes. 514. Radium. Two years after Becquerel's discovery Madame Curie found in pitchblende (an impure oxide of uranium) a constituent much more highly radioactive than uranium itself. She succeeded by chemical means in ex- tracting this remarkable substance from pitchblende and named it radium. Radium is a million times more radioactive than ura- nium. Although widely distributed, the total quantity of radium in the earth is undoubtedly small. It takes 150 tons of pitchblende to furnish one ounce of radium. It is a hard white metal, resembling barium. It is very un- stable and is usually prepared and used as a chloride or a bromide. Its radiations excite strong fluorescence in several substances, notably zinc sulphate, diamond, and ruby; and they produce on the human body sores difficult to heal. 515. Three Kinds of Radium Rays. Rutherford has shown that radium emits three kinds of "rays," which can be separated by means of a strong magnetic field. Their difference in behavior in a magnetic field is illus- trated in Figure 445. The radium is placed at the bottom of a small hole in a block of lead, so that only a thin pencil of rays escapes in a vertical direction. A strong magnetic field is applied so that the lines of force run away from the observer. The radiations are then separated into three kinds, known as alpha, beta, FIGURE 445. ALPHA, BETA, AND GAMMA RAYS. 418 ELECTROMAGNETIC INDUCTION and gamma rays. The alpha rays are slightly deflected to the left, the beta rays strongly to the right, while the gamma rays are not affected in the least. The fact that the alpha and beta rays suffer deviations in opposite direc- tions shows not only that they are charged particles, but that they are oppositely charged, the former positively and the latter negatively. The alpha rays are positively charged particles emitted with an average velocity about one-fifteenth the speed of light. They have little penetrating power and are ab- sorbed by a sheet of ordinary writing paper. The beta rays are negatively electrified, highly penetra- tive, and identical in nature with cathode rays ( 508). They travel with an average speed from about one-half down to about one-tenth that of light. The gamma rays are of very high penetrating power, they travel with the velocity of light, and appear to be identical with X-rays. They show no trace of electrification. 516. Radium a Product of Disintegration. Uranium has the highest atomic weight of any known substance, and it is always associated in nature with other radioactive sub- stances. This association suggested that the other ra'dio- active substances are derived from uranium by its dis- integration, or loss of particles with reduction of atomic weight. Such has been found to be the case. Uranium is the parent of ionium, and ionium is the parent of radium. Radium is thus a product of disintegration. Further, the radium atom disintegrates with the expul- sion of an alpha particle ; and the alpha particle, after losing its positive charge, becomes an atom of helium. Thus a known element is produced during the transformation of radioactive matter. All alpha particles from whatever source consist of helium atoms carrying positive charges. Madame Marie Sklodowska Curie was born in Warsaw in 1867. She imbibed the spirit of scientific research from her father, a distinguished physicist and chemist. In 1895 she mar- ried Professor Curie of the University of Paris. Three times she has been awarded the Gegner prize by the French Academy for her valuable contributions to the world's knowledge of the mag- netic properties of iron and steel and for her discoveries in radio- activity. In 1903 and again in 1911 the Nobel prize was awarded her. In January of 1911 she failed only by two votes of election to membership in the French Academy of Sciences, being de- feated by Branley, the inventor of the coherer used in the Mar- coni system of wireless telegraphy. Sir Joseph John Thomson was born near Manchester, Eng- land, in 1856. He received his early training at Owens College, and acquired there some knowledge of experimental work in the laboratory of Balfour Stewart. At the age of twenty-seven he was appointed to the Cavendish professorship at the University of Cambridge, a position made famous by Maxwell and Rayleigh. The wisdom of the appointment was soon proved; for shortly after, Thomson began a series of experiments on the conduction of electricity through gases, culminating in the discovery of the "electron," out of which has developed the electron theory of matter. ELECTRONS 419 I Evidence derived from the study of uranium minerals makes it almost certain that the final product of the dis- integration of uranium is lead. 517. Heat Generated by Radium. The salts of radium exhibit an altogether new and remarkable property ; they are always maintained at a temperature several degrees higher than that of the surrounding air. They are thus always radiating heat and giving out energy. A gram of pure radium would emit heat at the rate of from 100 to 130 calories per hour. It has been estimated that before a gram of radium is exhausted it would emit enough heat to melt a gram of ice every hour for 1000 years. Also, that the energy of radium is a million and a half times greater than that of an equal mass of coal. 518. Electrons. Sir William Crookes, at the time of his discovery of the cathode discharge, regarded it as matter in a radiant state. Later it was demonstrated that the cathode discharge carries negative electricity. Still later, by a series of brilliant experiments, Sir J. J. Thomson proved that cathode "rays" consist of streams of negatively electrified particles, now called electrons. The mass of an electron is only about y^VlF ^ the mass f the hydrogen atom. Moreover, he measured their speed in a vacuum and found it to have the enormous value of about 50,000 miles per second. The electron is invariable in magnitude, and is said to be " the atom of electricity," that is, the smallest quantity of electricity that can be transferred from one atom of matter to another. It is the smallest quantity that exists in a separate state. The beta rays spontaneously emitted by radium and other radioactive matter have now been identified with the elec- trons of a Crookes tube. There is good evidence also that 420 ELECTROMAGNETIC INDUCTION the electron is identical with the single atomic charge of a negative ion in electrolysis. If positive electricity is atomic, its atom is several thousands of times greater than the atomic quantity of negative electricity. Electrons enter into the composition of all matter. An electric current is supposed to be a stream of electrons flowing under electric pressure through a conductor from negative to positive. CHAPTER XIV DYNAMO-ELECTRIC MACHINERY I. DIRECT CURRENT MACHINES 519. A Dynamo-Electric Generator is a machine to convert mechanical energy into the energy of currents of electric- ity. It is a direct outgrowth of the brilliant discoveries of Faraday about induced electromotive forces and currents in 1831. It is an essential part of every system, steam or hydro-electric, for electric lighting, the transmission of electric power, electric railways, electric locomotives, elec- tric train lighting, the charging of storage batteries, electric smelting, electrolytic refinement of metals, and for every other purpose to which large electric currents are applied. 520. Essential Parts of a Dynamo-Electric Machine. Every dynamo-electric machine has three essential parts: 1. The field magnet to produce a powerful magnetic field. 2. The armature, a system of conductors wound on an iron core, and revolving in the magnetic field in such a manner that the magnetic flux through these conductors varies con- tinuously. 3. The commutator, or the collecting rings and the brushes, by means of which the machine is connected to the external circuit. If the magnetic field is produced by a permanent magnet, the machine is called a magneto, such as is used in an automobile for ignition ; if by an electromagnet, the machine is a dynamo, which is used in electric lighting stations, and for all other purposes requir- ing large currents generated by high power. Both are often called generators. 421 422 DYNAMO-ELECTRIC MACHINERY 521. Ideal Simple Dynamo. For the purpose of simpli- fying what goes on in the revolving coils of a generator, let us consider a single loop of wire revolving between the poles of a magnet (Fig. 446) in the direction of the arrow and around a horizontal axis. The light lines indicate the magnet fluy running across from N to S. In the position of the loop drawn in full lines it incloses the FIGURE 446. IDEAL SIMPLE DYNAMO. largest possible magnetic flux or lines of force, but as the flux inclosed by the coil is not changing, the induced E.M.F. is zero. When it has rotated forward a quarter of a turn, its plane will be parallel to the magnetic flux, and no lines of force will then pass through it. During this quarter turn the decrease in the magnetic flux, threading through the loop, generates a direct E.M.F. ; and if the rotation is uniform, the rate of decrease of flux through the loop increases all the way from the first position to the one shown by the dotted lines, where it is a maximum. The arrows on the loop show the direction of the E.M.F. During the next quarter turn there is an increase of flux through the loop, but it runs through the loop in the opposite direction because the loop has turned over ; this is equivalent to a continuous decrease in the original direction, and therefore the direction of the induced E.M.F. around the loop remains the same for the entire half turn; the E.M.F. again becomes zero when the half turn is completed. After the half turn, the conditions are all reversed and the E. M. F. is directed the other way around the loop. THE COMMUTATOR 423 / s x \ / <* / f \^ . 7 / \ / y 0' u , 2 0" z 0" uo \ w / s ^ ^ / If there are several turns in the coil, the E. M. F. reverses in all of them twice every revolution. The curve of Figure 447 shows by its ordinates the suc- cessive relative values of tho induced electromotive forces when the coil rotates with uniform speed. If the coil is part of a closed circuit, the cur- rent through it reverses twice every revolution, that is, it is an alternat- FIGURE 447. CURVE OF E. M. F.'s. ing current. 522. The Commutator. When it is desired to convert the alternating currents flowing "in the armature into a current in one direction through the external circuit, a special device called a commutator is employed. For a single coil in the armature, the comm utator consists of two parts only. It is a split tube with the two halves, a and 5, insulated from each other and from the shaft 8 on which they are mounted (Fig. 448). The two ends of the coil (not shown) are connected with the two halves of the tube. Two brushes, with which the external circuit L L is connected, bear on the com- mutator, and they are so placed that they exchange contact with the two commu- tator segments at the same time that the current reverses in the coil. In .this way one of the brushes is always positive and the other negative, and the current flows in the external circuit from the positive brush back to the FIGURE 448. TWO-PART COMMU- TATOR. 424 DYNAMO-ELECTRIC MACHINERY / /" v s \ / S X \ / ^ \ s A \ / / \ / \ / 1 (}' 'i W 1 0" 90 FIGURE 449. RECTIFIED E. M. F.'s. negative, and thence through the armature to the positive again ; but with a single coil the current is pulsating, or falls to zero twice every revolution (Fig. 449). 523. The Gramme Ring. The use of a commu- tator with more than two parts is conven- iently illustrated in connection with the Grramme ring. This ar- mature has gone out of practical use, but it is useful here because it can be understood from a simple diagram ; and fundamentally its action is the same as that of the com- mon drum type. The Gramme ring has a core made either of iron wire, or of thin disks at right angles to the axis of rotation. The iron is divided for the purpose of preventing induc- tion or eddy currents in it, which waste energy. The re- lation of the several parts of the machine is illustrated by Figure 450. A number of coils are wound in one direction and are all joined in series. The coils must be grouped symmetrically so that some of them are always active, thus generating a continuous current. Each junction be- tween coils is connected with a commutator bar. Most of the magnetic flux passes through the iron ring from FlGURE 45 THE GRAMME Rma the north pole side to the south pole ; bence, when a coil is in the highest position in the figure, the maximum flux passes through it i as the ring rotates, the flux through the THE FIELD MAGNET 425 coil decreases, and after a quarter of a revolution there is no flux through it. The current through each coil reverses twice during each revolution, exactly as in the case of the single loop. No current flows entirely around the arma- ture, because the JH.M.F. generated in one coil at any in- stant is exactly counterbalanced by the E.M.F. generated in the coil opposite. But when the external circuit connect- ing the brushes is closed, a current flows up on both sides of the armature. The current has then two paths through the armature, and one brush is constantly positive and the other negative. The current is therefore direct and fairly steady. 524. The Drum Armature. This very useful form of armature is in universal use for direct current (D. C.) genera- tors. The core is made up of thin iron disks stamped out with teeth around the periphery (Fig. 451). When these are assembled on the shaft, the slots FIGURE 451. - TOOTHED DISK. form grooves in which are placed the armature windings. All the coils in the armature may be joined in series, and the junctions between them are con- nected to the commutator bars, as in the Gramme ring. 525. The Field Magnet. The magnetic field in dynamos is produced by a large electromagnet excited by the cur- rent flowing from the armature , this current is led, either wholly or in part, around the field-magnet cores. When the entire current is carried around the coils of the field magnet, the dynamo is said to be series wound (Fig. 452 a). When the field magnet is excited by coils of many turns of fine wire connected as a shunt to the exter- nal circuit, the dynamo is said to be shunt wound (Fig. 452 6). A combination of these two methods of exciting 426 DYNAMO-ELECTRIC MACHINERY the field magnet is called compound winding (Fig. 452 vT :j - V t-^ j) are produced in the neighbor- , , ,, , i ,, . . FIGURE 487. THE COHERER. hood of the tube, the resistance of the filings falls to so small a value that a single voltaic cell sends through them a current strong enough to work a relay ( 553). If the tube is slightly jarred, the filings resume their state of high resistance. A minute discharge from the cover of an electrophorus ( 432) through the filings lowers the resistance just as electric oscillations do. It is thought that minute sparks between the filings partially weld them together and make them conducting. 564. Crystal Detectors. The coherer is now obsolete and more sensitive detectors have been discovered. The object aimed at in most of them is the rectification of the rapid oscillations from the receiving antenna or aerial wire, so as to secure a unidirectional discharge which will affect a telephone. On account of its high self-inductance, a telephone acts as a choke coil to high frequency electric oscillations and will not respond to them. It has been found that certain crystals, such as polished silicon, galena, and carborundum, possess a unilateral conductivity for electricity. A crystal of carborundum may have three or four thousand times as great conductivity in one direction as in the opposite for certain voltages. Hence, if a crystal detector is inserted in the oscillation circuit of a receiver, it rectifies the oscillations in a train of electric impulses, to which a telephone will respond with a sound correspond- ing in pitch to the number of impulses per second. 456 DYNAMO-ELECTRIC MACHINERY The crystal is held in a conducting holder and is touched lightly by a metal point (Fig. 488). The brass cup shown in the figure holds the crystal securely by means of three set screws. Another method of mounting is to embed the crystal in a soft alloy which melts at a low temperature. The contact wire can be moved about so FIGURE 488. - HOLDER FOR CRYS- aS tO find the Sensitive spots TAL DETECTOR. in the crystal. 565. The Audion is a very sensitive detector, depending for its action on the fact that electrons are thrown off from the negative end of an incandescent filament in an exhausted (or partly exhausted) bulb. If the bulb has supported in it a plate surrounding the fila- ment (Fig. 489), a single voltaic cell will send a (negative) current from its negative electrode to the negative end of the hot filament, thence through the space in the bulb to the metal plate, and out to the other pole of the voltaic cell. No current will flow unless the negative pole of the cell is connected to the negative of the filament. This arrangement is therefore an electric valve or rec- tifier, which lets electric impulses through in one direction and not in the other. Fleming calls it an "oscillation valve." In the figure, oscillations in one direc- tion from the oscillation trans- former T will pass through the circuit, including the valve V and the telephone P, but not those FlGURE 489 ~ ' OSCILLATION ,. ' VALVE." in the other direction. The auction is a modification of the " oscillation valve " of Fleming, which becomes a relay for the aerial oscillations to operate receiving TRANSMITTING AND RECEIVING CIRCUITS 457 HI telephones in a circuit with a battery (Fig. 490). In addition to the hot filament and the metal plate the audion has a " grid " consisting of a coil of copper wire, which is one terminal* of the circuit from the receiving helix. The other terminal of this cir- cuit is joined to the fila- ment. The negative of the adjustable battery .B is joined to the negative end of the filament. The recti- fied train of impulses passes through from the hot fila- ment to the copper coil. The passage of these impulses causes similar impulses from the battery B to pass between the Jilament and the metal plate, and hence through the receiving telephone T. FIGURE 490. THE AUDION. 566. Transmitting and Receiving Circuits. A simple tuned transmitting circuit for wireless telegraphy is illus- trated in Figure 491, where I is an induction coil, O a con- denser, S a spark gap, H a variable helix, A the aerial or antenna, and E the earth con- nection. Figure 492 is a correspond- ing simple receiving circuit. The receiving telephones are shown at T, the detector at D, and a variable condenser at 0. These arrangements are capable of many variations. The magnetic effect of a rectified train of electric im- pulses is never reversed. Hence they pass through the FIGURE 491. TRANSMITTING CIRCUIT. 458 DYNAMO-ELECTRIC MACHINERY FIGURE 492. RECEIVING CIRCUIT. high resistance telephones and produce a distinct musical tone. Continued tones are interpreted as dashes and I - short ones as dots ; together they make up either the Morse or the Continental alphabet. The circuits in commercial wireless telegraphy are much more elaborate than those shown (Fig. 493). To avoid interfer- ence between signals from differ- ent stations, it is necessary to tune the sending and receiving circuits to the same frequency. They are then sensitive to one frequency and not to others. For detailed information the reader is advised to consult technical books on wireless telegraphy. 567. Uses of Wireless Telegraphy. In less than thirty years after Hertz's fundamental discovery, wireless teleg- raphy has grown to large proportions, especially for sig- nals between ships at sea and for international intercourse. Wireless telegraphy is in use between all steamships. They are thus in communication with one another and with stations on the land. Various government stations have been erected for the purpose of keeping each govern- ment in communication with the ships in its navy, and with other governments. Notable among these are the station in Paris, for which the Eiffel Tower is utilized to support the antenna, and the station in Arlington near Washington. Communication between these two stations is not difficult, and signals between them have been used to determine the difference of longitude between Paris and Washington. During the progress of this work, the Heinrich Rudolf Hertz (1857-1894) was born in Hamburg, and was educated for a civil engineer. Having decided to aban- don his profession, he went to Berlin and studied under Helm- holtz, and later became his assistant. In 1885 he was appointed professor of physics at the Technical High School at Karlsruhe, and while there he discovered the electromagnetic waves pre- dicted by Maxwell, who in the middle of the century had ad- vanced the idea that waves of light are electromagnetic in char- acter, In 1889 he was elected professor of physics at Bonn, where he died at the age of thirty-seven. Electromagnetic waves are called Hertzian waves in his honor. Thomas Alva Edison was born at Milan, Ohio, in 1847. Beginning life as a newsboy, he has become the greatest American inventor. He per- fected duplex telegraphy, and invented among other things the carbon telephone transmitter, the microtasim- eter, the aerophone, the megaphone, the phonograph, the kinetoscope, and the in- candescent electric lamp. Guglielmo Marconi was born at Bologna, Italy, in 1874. He studied in his native city, at Leghorn, and also, for a short time, in England. At the age of twenty-one he began his experiments in wireless teleg- raphy, and by 1895 was able to send messages across the English Channel. Since then his system has been so developed that marconigrams are sent across the Atlantic, and practically all important ships are equipped with wire- less apparatus. WIRELESS TELEPHONY 459 time of transmission of the signals between Paris and Washington was found to be 0.021 second. Signals are occasionally received at the Marconi Station, County Gal- way, Ireland, from stations many thousand miles away ; for example, from Darien, San Francisco, and Honolulu. A.- Aerial A.G.- Anchor Gap H.M. - Mill-Ammeter O.H.- Oscillation Helix R. Rotary Spark Gap T.- Transformer C. - Transmitting Condenser K. Transmitting Key G. Ground - Aerial Switch .C.- Variable Condenser* .C. - Loose-Coupled Turner Detector .C.- Fixed Condenser Receivers FIGURE 493. COMMERCIAL TRANSMITTING AND RECEIVING APPARATUS. 568. Wireless Telephony. -^- For the purpose of trans- mitting speech by wireless, it is necessary to have a source of energy that will transmit a persistent train of undamped waves. This may be accomplished either by means of an oscillating arc or by a high frequency al- ternator. These must emit continuous trains of waves with a frequency of 4000 or more per second. A special microphone carves the transmitted current and the train of waves emitted into groups of amplitudes corresponding with the sounds spoken into the microphone. The words are received with the usual telephonic receivers. CHAPTER XV THE MOTOR CAR 569. The Modern Motor Gar or Automobile has come into such extensive use in the last few years that the principles of its construction and operation should be generally understood. Motor cars are usually classified according to the power which propels them, as electric, steam, and gasoline. Since the gasoline car is so much more widely used than either of the other two, it is the only one con- sidered in this chapter. 570. The Gasoline Automobile uses the internal combus- tion engine ( 380), the four-cycle type, for its motor. The number of cylinders varies from four to twelve, and the pistons, whatever their number, act on a common crank shaft. Figure 494 shows a four-cylinder engine and Figure 495 a six. Whatever the number of cylinders, the energy from the explosion is applied intermittently. The greater the number, the more nearly continuous is the stream of energy. In the four-cylinder engine there are two explosions for each revolution of the crank shaft, 180 apart. For a six, there are three, 120 apart ; for an eight, there are four, 90 apart ; and for a twelve, there are six, 60 apart. Figure 496 illustrates a twelve -cylinder engine or " twin- six." 571. The Engine. The vital part of the motorcar is the engine, and constant and intelligent attention on the 460 THE ENGINE 461 Courtesy of Dodge Brothers FIGURE 494. CROSS SECTION OF A FOUR-CYLINDER ENGINE. Courtesy of the Buick Company FIGURE 495. CROSS SECTION OF A SIX-CYLINDER VALVE-IN-HEAD ENGINE. 462 THE MOTOR CAR part of the operator is necessary to secure smooth and uniform action. In 380 the mode of action of a four- cycle engine is described, but if continuous and uninter- rupted action is to be secured a number of points must be observed : (1) Since the explosion of the gaseous mixture heats the cylinder to a very high temperature, the operation of Courtesy of the Packard Company FIGURE 496. ONE SIDE OF A TWELVE-CYLINDER ENGINE. the engine soon becomes impossible unless some cooling device is used. This is secured, except in the "air- cooled" type of car, by the circulation of water about the cylinders and through a device called the radiator, where it is cooled by the action of a fan and the rapid radiation due to the large surface exposed. In some cases the water is circulated by a pump; in others, the so-called " thermo-siphon " system is used. By this system, the cold water entering the water-jacket from the bottom of THE ENGINE 463 the radiator forces up the hot water that is around the cylinders into the reservoir above the radiator, from which it flows through the radiator where it is cooled. Courtesy of the Cadillac Company FIGURE 497. FRONT VIEW OF V-TYPE ENGINE. Most " eights " and " twelves " are of this type with four or six cylinders placed in opposite rows. The reservoir above the radiator should always be kept as nearly full as convenient. (2) Every motor must be kept thoroughly lubricated. Practically all cars now use both the " pump " and 464 THE MOTOR CAR " splash " systems, whereby the oil is not only spattered about the inside of the engine by the rapidly revolving crank shaft, but it is also pumped to parts less likely to be reached by the splash system. Every engine has an oil gauge which should be constantly watched, as the explo- sions of the gas consume some of the oil, and the absence of lubricant causes " knocking" and laboring on the part of the motor. (3) Because of the incomplete burning of the carbon in the explosion, a gradual deposit forms on the pistons and points of the spark plugs. This accumu- lation of carbon may cause a fouling of the spark plugs to such an extent that they do not function, and the engine stops ; or it may form sufficiently to hold fire between ex- plosions, and produce pre-ignition. A pound- ing or knocking in the engine is one of the indications of the presence of carbon. It may be removed by opening the engine and scraping the carbon off. (4) The proper mixture of the air and gasoline is neces- sary to the best action of the engine. This mixing is done by the carburetor (Fig. 498), a device through which the suction of the pistons draws air and gasoline in proper proportions for the several cylinders. The manifold is the tube that conveys the mixture from the carburetor to the different cylinders. Most cars have a carburetor FIGURE 498. THE ENGINE 465 adjustment on the dash, so that any desired mixture can be obtained. A richer mixture is desirable when starting the engine than after it has become warm, and it is also possible to operate on a thinner mixture at high speeds than at low. A car which has been running satisfactorily at twenty-five miles an hour on as thin a mixture as pos- sible will often stall when " throttled down " to ten or twelve miles an hour. The carburetor is often warmed by a pipe from the exhaust of the engine and sometimes it is partly surrounded by a water-jacket. (5) The gas is ignited by an electric spark which jumps between two metal points in the spark plug, which is placed at or near the head of the cylinder. This spark SAFETY SPARK GAP BALL BEARING ARMATURE BALL BEARING FIGURE 499. DIAGRAM OF THE BOSCH MAGNETO. is furnished either by an induction coil connected with a storage battery or by a high-tension magneto (Fig. 499) which is turned by a connection with the crank shaft. The crank shaft by suitable gearing also operates a timer, 466 THE MOTOR CAR which connects the spark plugs successively and explodes the gas at the proper time. Ignition troubles are usually caused by foul spark plugs. When the action of the engine is jerky it shows that some cylinder is "missing." The one at fault can be ascertained by touching some metal part of the engine with the end of a screw-driver and holding another part of the metal of the screw-driver close to the spark plug connections. The plug where t no spark jumps across is probably the one causing the trouble. 572. The Storage Battery. So little attention is re- quired by the storage' battery that it is often too much neglected. It should be examined at least every two weeks and the plates covered with distilled water. It should be tested from time to time with the hydrometer (Fig. 60) and recharged at once if the density has fallen below 1.200. The electrolyte of a battery in good condi- tion has a density from 1.250 to 1.300. An idle battery will not remain charged but must have attention as often as once every two weeks. On long summer trips of continuous driving and also by rapid driving for a few hours a battery some- times becomes overcharged. This may be remedied by switching on the lights of the car for a while. In the winter season, on account of the difficulty of starting the car when cold, the battery is likely to be run down. This condition will be accentuated by the in- creased use of the lights. It will relieve the heavy drain on the battery to start the car by the use of the crank ; at least it is advisable to turn the engine over a few times to get it well oiled before resorting to the starter. THE RUNNING GEAR 467 573. The Chassis (pronounced "shassy") is the name applied to the skeleton body of the car (Fig. 500), as distinguished from the hood, which incloses the engine, the tonneau, which is the rear seat division of a touring car, and the running gear, as the wheels are generally FIGURE 500. A TYPICAL CHASSIS. called. The chief care required by the chassis is its lubrication. It is fully supplied with grease cups which require constant "turning up" and filling. Grease cups .which lubricate revolving parts require more fre- quent attention than those on joints and spring bolts. 574. The Running Gear consists of the wheels and tires. The rear wheels are usually lubricated from the differen- tial ( 578) and require practically no attention. The bearings of the front wheels are usually packed in grease, and at least once a year the wheels should be removed and the bearings cleaned and repacked in grease. The tires consist of a flexible inner tube, containing air under pressure, and a thick outer casing, sometimes called the shoe. Tire makers recommend a pressure of about twenty pounds for each inch of cross sectional diameter; that is, a four-inch tire should carry eighty 468 THE MOTOR CAB pounds' pressure to the square inch ; a four and a half, ninety pounds ; and a five-inch tire, one hundred pounds. Less pressure may give more comfort in riding, but there is the danger that an excessive flattening of the tire may separate the layers of fabric and rubber. This applies to the heavy outer casing wherein lies the main expense in the maintenance of a car. Oil and tar are enemies of rubber and should be removed from the tire as soon as possible by means of a cloth dampened with gasoline. Rough roads should either be avoided or traversed at as low a speed as possible. Fast driving, especially in hot weather, is particularly hard on tires in that the tires become heated and disintegration sets in. A blow-out is caused by a weakening of the tire casing or shoe, through which the inner tube is forced out by the air pressure, exploding with a loud report. A punc- ture is caused by a nail, or some sharp instrument like a piece of glass cutting through the casing and making a small hole in the inner tube, through which the air es- capes gradually without " blowing out " the casing. Sometimes a leak occurs in the valve, which is delicately constructed, and the rubber washer of which may be- come defective through heat or age. The valve may then be unscrewed by reversing the little pointed cap which protects it and a new valve may be screwed in at slight expense. 575. The Brakes. All automobiles are provided with double brakes the service brake, operated by the foot, and the emergency brake, usually operated by a hand lever. These brakes consist of steel bands lined with asbestos acting by friction on drums attached to the driving shaft or to the rear wheels. They should always be kept in THE CLUTCH 469 good condition and should always be applied gradually except in a case of great emergency. The clutch should always be thrown out when the brakes are applied. In some types of cars the clutch and service brake are on the same foot lever and in applying the brake the construction is such that the clutch is thrown out before the brake comes into action. 576. The Clutch is a friction coupling connecting the crank shaft with the transmission shaft. There are many different forms, as the multiple disk, the cone, etc., but FIGURE 501. MULTIPLE DISK CLUTCH AND TRANSMISSION. those that have proved the most satisfactory depend on friction. The clutch must always be thrown out in shift- ing the gears from " neutral," in changing the gears in any way, and in stopping the car, and it should be let in 470 THE MOTOR CAR gently to prevent jerking. Figure 501 shows a section of one type of clutch and the transmission gears. 577. The Transmission comprises all those parts which transmit power from the engine to the rear wheels, but the group of gear wheels just back of the clutch is usually re- ferred to as the transmission. Its function is to make changes in speed possible by various combinations of gears. The gasoline motor develops power in proportion to its speed, so that if great pulling power is required, a high speed of the motor must be combined with a low speed of the car, and this is obtainable only through a system of gears. In starting a car always begin in low gear, shifting to second when a moderate degree of speed has been attained, and not going into " high " until the car is well under way. 578. The Differential. The rear wheels of a car are the driving wheels and motion is communicated to them from the engine through the clutch, the transmission, the FIGURE 502. THE DIFFERENTIAL. driving shaft, and finally through a device called the differential (Fig. 502). This is an ingenious assem- blage of gear wheels so connected as to permit the drive THE 8TAETEE 471 wheels to rotate independently, as is necessary in turning a corner. The differential requires little attention, but must be examined occasionally to make sure that it is thoroughly oiled. The plan of the differential is such that one wheel may turn while the other is stationary, and for this reason on a slippery road it is necessary to place a chain on each drive wheel. If only one chain is used, the chain wheel may be standing still while the other one spins rapidly without securing any u traction." \ 579. The Steering Device is a broad wheel and shaft carrying the throttle lever and the spark lever and connected with the front wheels by an endless screw working in a worm wheel (Fig. 503). It should turn readily, but should not be allowed to have too much play, as on it depends the control of the car. 580. The Starter. Most cars are now equipped with an electric starter, a small direct current motor oper- ated by the storage bat- tery. Pressing a spiral switch by means of a plug, usually in the floor board, closes the circuit, causing the armature to revolve. A suitable reduction gear con- nects the armature shaft with the crank shaft of the engine, thus turning it over and setting it in motion. FIGURE 503. 472 THE MOTOR CAB Before pressing the starter plug, the gear lever should always be put in neutral, the spark retarded, the throttle advanced, and the " mixture " in the carburetor enriched. The starting plug should be released immediately on the engine's beginning to run. The spark should then be advanced, the motor throttled down to a moderate speed, and the carburetor adjusted. 581. On the Road. The two most important rules of the road are " Safety First " and Courtesy to All." Different cities and towns have local regulations, but the driver who is always careful and courteous will save him- self the trouble of memorizing countless specific rules. Always be prepared for every one else doing the wrong thing. In turning corners drive slowly and thus avoid becoming an example under 145. Do not try to climb steep hills on " high " just because you may be able to do so, and do not descend hills at high speed. If the motor stops or " stalls," as it is usually termed, the first thing to investigate is the gasoline supply. In nine cases out of ten lack of gasoline causes the stalling. Otherwise, some flaw in the ignition system, such as a broken wire or a short circuit, may cause the trouble. Sometimes the fan belt has worked loose so that the fan has ceased to function and the engine is overheated. This is usually detected by the boiling of the water in the water jacket. After stopping, it is best to set the emergency brake, even on level ground ; but do not forget to release it before starting. Always put the gear lever in neutral on stopping, except that after stopping on a steep down grade, it is usually wise to throw the gear lever into reverse for safety's sake, and on a steep upgrade, to throw it into low THE PEDESTRIAN 473 gear. But you should be particularly careful to put it back into neutral before attempting to start the car. The engine is a natural brake. So in descending a hill, throttle the engine down and leave the clutch in. The speed can then be governed easily by the service brake and the car be more completely under control. When de- scending very steep hills, it is well to go into second or even into first speed to brake the car. In night driving do not use bright head lights on ap- proaching another car ; always turn them down to " dim." A bright light is blinding to the driver of the oncoming car and may cause a serious accident. In general, study the car, the plan of all the parts, their office, and their adjustment. Rattles usually come from loose nuts, squeaks from empty grease cups. Look the car over often to see if everything is secure and in place. Inspect the gasoline, water, and oil supply before starting out from the garage. In this way you will have fewer accidents and less annoyance and expense. 582. The Pedestrian. In communities where motor cars are numerous, pedestrians should take the utmost care to avoid accidents. In crowded cities which have traffic officers at crossings, we should watch the signals of the officer, keeping on the sidewalk until he signals the traffic to stop. Where there is no officer the traffic usually keeps to the right. Hence we should look first to the left until halfway across, then to the right for the rest of the way. As we start across there is no danger of being hit from the right; but when halfway across, that is the side from which the danger comes. Asking for rides should be discouraged as a dangerous 474 THE MOTtiR CAR proceeding. If the driver is unfriendly, we are completely at his mercy and he can take us where he will. If he is friendly, we are subjecting him to risk, because he is liable for any injury to us, whether in the car or in getting on or off. Courtesy, common sense, and obedience to traffic regu- lations are as important for pedestrians as for motorists. APPENDIX L GEOMETRICAL CONSTRUCTIONS The principal instruments required for the accurate con- struction of diagrams on paper are the compasses and the ruler. For the construction of angles of any definite size the protractor (Fig. 504) can be used. There are, how- ever, a number of angles, as 90, 60, and those which can be obtained from these by bi- secting them and combining their parts, that can be constructed by the compasses and ruler alone. A convenient instrument for the rapid construction of the angles 90, 60, and 30, is a triangle made of wood, horn, hard rubber, or card- board, whose angles are these respectively. Such a triangle may be easily made from a postal card as follows : Lay off on the short side of the card (Fig. 505) a distance a little less than the width, as AB. Separate the points of the compasses a distance equal to twice this distance. Place one point of the compasses at B, and draw an arc cutting the adjacent side at C. 475 90 FIGURE 505. 476 APPENDIX Cut the card into two parts along the straight line BO. The part ABC will be a right-angled triangle, having the longest side twice as long as the shortest side, with the larger acute angle 60 and the smaller 30. With this triangle and a straight edge the majority of the con- structions required in elementary phys- ics can be made. PKOB. 1. To con- struct an angle of 90. Let A be the ver- tex of the required B angle (Fig. 506). Through A draw the straight line BC. Measure off AD, any convenient distance ; also make AE = AD. With a pair of compasses, using D as a center, and a radius longer than AD, draw the arc mn ; with E as a cen- ter and the same radius, draw the arc rs, intersecting mn at F. Join A and F. The angles at A are right angles. D A FIGURE 506. E PJBOB. 2. To construct an angle of 60. Let A be the vertex of the re- - quired angle (Fig. 507), and AB one of the sides. On AB take some convenient distance as AC. With a pair of compasses, using A as a center and AC as a radius, draw the arc CD. With C as a center and the same radius, draw the arc mn, intersecting CD at E. Through A and E draw the straight lineAE; this line will make an angle of 60 with AB. C FIGURE 507. GEOMETRICAL CONSTRUCTIONS 477 PROB. 3. To bisect an angle. Let BAG be an angle that it is required to bisect (Fig. 508). Measure off on the sides of the angle equal distances, AD and AE. With D and E as centers and with the same radius, draw the arcs mn and rs, intersecting at F. Draw AF. This line will bisect the angle BAG. PROB. 4. To make an angle equal to given angle. Let BAG be a given angle; FIGURE 508 ^ * s re( l u i re( l to make a second angle equal to it (Fig. 509). Draw DE, one side of the required angle. With A as a cen- ter and any convenient radius, draw the arc mn across the given angle. With D as a center and the same radius, draw the arc rs. With s as a center and a radius equal to the chord of A n CD E FIGURE 509. mn, draw the arc op, cutting rs at G. Through D and 0- draw the line DP. This line will form with DE the required angle, as FDE. PROB. 5. To draw a line through a point parallel to a given line. Let A be the point through which it is required to draw a line parallel to BG (Fig. 510). Through A draw ED, 478 APPENDIX cutting BC at D. At A make the angle EAG equal to EDO. Then AG or F# is parallel to EG. FIGURE 510. PROB. 6. Given two adjacent sides of a parallelogram to com- plete the figure. Let AB and AC be two adjacent sides of the parallelogram (Fig. 511). With C as a center and a radius equal to AB, FIGURE 511. draw the arc mn. With B as a center and a radius equal to AC, draw the arc rs, cutting mn at D. Draw CD and .BZ>. Then ABDC is the required parallelogram. CONVERSION TABLES 479 II CONVERSION TABLES 1. LENGTH To reduce Multiply by To reduce Multiply by 1 60935 Kilometers to mi. . . 0.62137 Miles to m 1609.347 Meters to mi. . . . 0.0006214 Yards to m . 0.91440 Meters to yd. . . . 1.09361 Feet to m 30480 Meters to ft. . . . . 3.28083 Inches to cm. . , . 2.54000 Centimeters to in. . . 0.39370 Inches to mm. . . . 25.40005 Millimeters to iu. . . 0.03937 2. SURFACE To reduce Multiply by To reduce Multiply by Sq. yards to m. a . . . 0.83613 Sq. meters to sq. yd. . . 1.19599 Sq. feet to m 2 . . 09290 Sq meters to sq. ft . . 10.76387 Sq. inches to cm. 2 . . . 6.45163 Sq. centimeters to sq . in. 0.15500 Sq. inches to mm. 2 645.163 Sq. millimeters to sq. in. 0.00166 - 3. VOLUME To reduce Multiply by To reduce Multiply by Cu. yards to m. 8 . . 0.76456 Cu. meters to cu. yd. . . 1.30802 Cu. feet to m. 3 . 02832 Cu meters to cu ft . . 35.31661 Cu. inches to cm. 3 . . 16.38716 Cu. centimeters to cu . in. 0.06102 Cu. feet to liters . . . 28.31701 Liters to cu. ft. . . . . 0.03532 Cu. inches to liters . 0.01639 Liters to cu. in. . . 61.02337 Gallons to liters . . . . 3.78543 Liters to gallons . . . 0.26417 Pounds of water to liters . 0.45359 Liters of water to Ib. . 2.20462 4. WEIGHT To reduce Multiply by To reduce Multiply by Tons to kg 907.18486 Kilograms to tons 0.001102 Pounds to kg. . . . 0.45359 Kilograms to Ib. . 2.20462 Ounces to g 28.34953 Grams to oz. . . 0.03527 Grains to g. 0.064799 Grams to grains 15.4323B 480 APPENDIX 5. FORCE, WORK, ACTIVITY, PRESSURE To reduce Multiply by Lb.-weight to dynes, . 444520.58 Ft.-lb. to kg.-m. . . . 0.138255 Ft.-lb. to ergs . . 13549 x 10 8 Ft.-lb. to joules . . 1.3549 Ft.-lb. per sec. to H.P. 18182 x 10~ 7 H.P. to watts .... 745.196 Lb. per sq. ft. to kg. perm. 2 4.8824 Lb. per sq. in. to g. per cm. 2 70.3068 Calculated for g = 980 cm. , To reduce Multiply bj Dynes to Ib. -weight, 22496 x 10' 10 Kg.-m. to ft.-lb. . . . 7.233 Ergs to ft.-lb. . . 0-7381 x 10~ 7 Joules to ft.-lb. ... 0.7381 H.P. to ft.-lb. per sec. . 550 Watts to H.P 0.001342 Kg. per m. 2 to Ib. per sq. ft 0.2048 G. per cm. 2 to Ib. per sq. in 0.01422 or 32.15 ft.-per-sec. per sec. 6. MISCELLANEOUS To reduce Multiply by Lb. of water to U.S. gal. 0. 11983 Cu. ft. to U.S. gal. . . 7.48052 Lb. of water to cu. ft. at 4C 0.01602 Cu. in. to U.S. gal. . . 0.004329 Atmospheres to Ib. per sq. in 14.69640 Atmospheres to g. per cm. 2 1033.296 Lb.-degrees F. to calories. 252 Calories to joules . . . 4.18936 Miles per hour to ft. per sec 1.46667 Miles per hour to cm. per sec. , . 44.704 To reduce Multiply by U.S. gal. to Ib. of water. 8.345 U.S. gal. to cu. ft. . - 0.13368 Cu. ft. of water at 4 C. tolb 62.425 U.S. gal. to cu. in. . . 231 Lb. per sq. in. to atmos- pheres 0.06737 G. per cm. 2 to atmos- pheres 0.000968 Calories to Ib. -degrees F. 0.003968 Joules to calories . . . 0.2387 Ft. per sec. to miles per hour 0.68182 Cm. per sec. to miles per hour 0.02237 MENSURATION TABLES 481 MENSURATION RULES Area of triangle Area of triangle Area of parallelogram Area of trapezoid Circumference of circle : Diameter of circle Area of circle . Area of ellipse Area of regular polygon Lateral surface of cylinder Volume of cylinder Surface of sphere : Volume of sphere : Surface of pyramid > Surface of cone I Volume of cone = (base x altitude). V( <*)(* &)(* c) where s=| (a+6+e). base x altitude. Altitude x ^ sum of parallel sides. dLJj diameter x 3.1416. t circumference -f- 3.1416. I circumference x 0.3183. f diameter squared x 0.7854. \ radius squared x 3.1416. ^ <3. product of diameters x 0.7854. \ (sum of sides x apothem). circumference of base x altitude. = area of base x altitude. ( diameter x circumference. \4 x 3.1416 x square of radius. f diameter cubed x 0.5236. " I f of radius cubed x 3.1416. = $ (circumference of base x slant height). = J (area of base x altitude). 482 APPENDIX IV. TABLE OF DENSITIES The following table gives the mass in grams of 1 cm. 8 of the sub- stance : Agate ....... 2.615 Air, at C. and 76 cm. pressure 0.00129 Alcohol, ethyl, 90%, 20 C. 0.818 Alcohol, methyl . .. ,-, . 0.814 Alum, common ... . 1.724 Aluminum, wrought . . 2.670 Antimony, cast . V. . 6.720 Beeswax 0.964 Bismuth, cast ...--. . .9.822 Brass, cast . . . 8.400 Brass, hard drawn . . . 8.700 Carbon, gas ..... 1.89 Carbon disulphide . . . 1.293 Charcoal 1.6 Coal, anthracite . .1.23 to 1.800 Coal, bituminous . 1.-J7 to 1.423 Copper, cast 8.830 Copper, sheet 8.878 Cork 0.14 to 0.24 Diamond 3.530 Ebony 1.187 Emery 3.900 Ether ....... 0.736 Galena 7.580 German silver .... 8.432 Glass, crown 2.520 Glass, flint ... 3.0 to 3.600 Glass, plate 2.760 Glycerin 1.260 Gold . 19.360 Granite. ...... 2.650 Graphite 2.500 Gypsum, crys. , , , , 2.310 Human body .... 0.890 Hydrogen, at. C. and 76 cm. pressure . . 0.000080C Ice 0.917 Iceland spar . . . . 2.723 India rubber .... 0.930 Iron, white cast . . . 7.655 Iron, wrought .... 7.698 Ivory . . ...... . 1.820 Lead, cast . Y . . . 11.360 Magnesium .... 1.750 Marble 2.720 Mercury, at C. . . 13.596 Mercury, at 20 C. . .13.558 Milk 1.032 Nitrogen, at C. and 76 cm. pressure . . 0.001255 Oil, olive 0.915 Oxygen, at C. and 76 cm. pressure . . . 0.00143 Paraffin . . . 0.824 to 0.940 Platinum 21.531 Potassium 0.865 Silver, wrought . . . 10.56 Sodium 0.970 Steel 7.816 Sulphuric Acid . . . 1.84 Sulphur 2 33 Sugar, cane .... 1.5. Tin, cast 7.290 Water, at C. . . . 0.999 Water, at 20 C. . . . 0.998 Water, sea 1.027 Zinc, cast 7.000 GEOMETBICA L CONS TB UCTION 483 V. GEOMETRICAL CONSTRUCTION FOR REFRACTION OF LIGHT The path of a ray of light in passing from one medium into another of different optical density is easily constructed geomet- rically. The following problems will make the process clear : First. A ray from air into water. Let JOT" (Fig. 502) be the surface separating air from water, AB the incident ray at B, and BE the normal. With B as a eenter and a radius BA draw the arcs mn and Cs. With the same center and a radius f of AB, (^ being the index for air to water), draw the arc Dr. Produce AB till it cuts the inner arc at D. Through D draw DC parallel to the normal EF, cutting the outer arc at 0. .Draw BC. This will be the refracted ray, because = -, the index of refraction. When the ray passes from a medium into one of less optical density, then the ray is produced until it cuts the outer or arc of larger radius, and a line is drawn through this point parallel to the normal. The intersection of this line with the inner arc gives a point in the refracted ray which together with the point of incidence locates the ray. If the incident angle is such that this line drawn parallel to the normal does not cut the inner arc, then the ray does not pass into the medium at that point but is totally reflected as from a mirror. It is immaterial whether the arcs Dr and Cs are drawn in the quadrant from which the light proceeds, or, as in the figure, in the quadrant toward which it is going. FIGURE 512. 484 APPENDIX Second. Tracing a ray through a lens. Let MN represent a lens whose centers of curvature are C and C", and AB the ray to be traced through it (Figs. 503, 504). Draw the normal, FIGURE 513. OB, to the point of incidence. With B as a center, draw the arcs mn and rs, making the ratio of their radii equal the index of refraction, f . Through p, the intersection of AB with rs, draw op parallel to the normal, OB, and cutting mn at o. Through o and B draw oBD; this will be the path of the ray through the lens. At D it will again be refracted ; to determine the amount, draw the normal CD and the auxiliary cir- cles, xy and uv, as before. Through the intersection of BD produced with xy, draw It parallel to the normal CD, cutting uv at I. Through D and I draw DH-, this will be the path of the ray after emergence. When the index of refraction is f , the principal focus of both the double convex and the double concave lens is at the center of curvature ; for piano-lenses, it is at twice the radius of vature from the lens. N FIGURE 514. INDEX [References are to pages.] Aberration, chromatic, 264 ; spherical, 235, 253. Absolute, scale of temperature, 293 ; unit of force, 105 ; zero, 294. Absorption spectra, 268. Accelerated motion, 95. Acceleration, 93; centripetal, 100; of gravity, 124. Achromatic lens, 265. Action of points, 347. Adhesion, 10; selective, 11. Agonic line, 338. Air, brake, 87 ; compressibility of, 73 ; compressor, 76 ; pressure of, 68 ; weight of, 65. Air brake, 87. Air columns, laws of, 206. Airplane, 3, 113, 120, 325. Air pump, 76; experiments with, 78. Airships, 81. Alternator, 431. Altitude by barometer, 71. Ammeter, 397. Ampere, 381. Amplitude, 138. Analysis of light, 263. Aneroid barometer, 70. Annealing, 15. Anode, 373. Antinode, 204. Arc, carbon, 442; inclosed, 443; open, 443. Archimedes, principle, 53. Armature, 421, 425; drum, 425. Artesian well, 50. Athermanous substances, 318. Atmosphere, unit of pressure, 69. Atmospheric electricity, 358. Attraction, electrical, 340; molecu- lar, 30, 35. Audion, 456. Aurora, 360. Automobiles, 1, 325, 460-474. Balance, 163. Balloons, 80. Barometer, aneroid, 70; mercurial, 69 ; utility of, 70. Baroscope, 80. Battery, storage, 376; in a motor car, 466. Beam of light, 216. Beats, 195 ; number of, 196. Bell, electric, 449. Binocular, prism, 262. Blind spot, 261. Blow-out, of a tire, 468. Boiling, 303 ; in a motor car, 472. Boiling point, effect of pressure, 305 ; on thermometer, 283. Boyle's law, 74 ; inexactness of, 75. Brake, of a motor car, 468 ; use of engine as, 473. Bright line spectra, 268. British "tank," 2, 104. Brittleness, 14. Buoyancy, 53; of air, 80; meas- ure of, 53. Caloric, 280. Calorie, 297. INDEX [References are to pages.] Camera, photographer's, 259. Capacity, dielectric, 352; electro- static, 350 ; thermal, 297. Capillarity, 32; laws of, 33; re- lated to surface tension, 34. Capstan, 165. Carbon, in a motor car, 464. Carburetor, 464. Cartesian diver, 56. Cathode, 373 ; rays, 412. Caustic, 236. Cell, voltaic, 361 ; chemical action in, 363. Center, of gravity, 123 ; of oscilla- tion, 139 ; of percussion, 140 ; of suspension, 138. Centrifugal force, 133; illustra- tions of, 135 ; its measure, 134. Centripetal force, 133. Charge, residual, 353 ; seat of, 353. Charles, law of, 293. Chassis, of a motor car, 467. Choke coil, 436. Chord, major, 197 ; minor, 197. Chromatic aberration, 264. Circuit, closing and opening, 363 ; divided, 397 ; electric, 363 ; trans- mitting and receiving, 457. Circular motion, 100. Clarinet, 205. Clinical thermometer, 285. Clutch, of a motor car, 469. Coherer, 455. Cohesion, 10. Coil, choke, 436; induction, 406; primary, 406.; secondary, 407. Cold by evaporation, 303. Color, 271 ; complementary, 275 ; mixing, 273; of opaque bodies, 271 ; of transparent bodies, 272 ; primary, 273. Commutator, 423. Composition of forces, 107; of velocities, 113. Compressibility of air, 73. Concave, lens, 249; mirror, 229; focus of, 230, 249. Condenser, 351 ; office of, 408. Conductance, of electricity, 378 ; of heat, 309. Conductor, electrical, 343 ; charge on outside, 346 ; magnetic field about, 387. Cone clutch, 469. Conservation of energy, 153. Convection, 312 ; in gases, 313. Convex, lens, 246 ; mirror, 231 ; focus of, 230. Cooling system, in a motor car, 462, 463. Coulomb, 348. Couple, 109. Crank shaft, in a motor car, 460. Critical angle, 244. Crookes tubes, 412. Crystal detectors, 455. Crystallization, 35. Current, electric, 361 ; convection, 313; detection of, 366; heating effects of, 385; induced by cur- rents, 403 ; induced by magnets, 402 ; magnetic properties of, 387 ; mutual action of, 390 ; strength of, 381. Curvilinear motion, 99. Cyclonic storms, 71. Cylinders, in a motor car, 460-463. Daniell cell, 370. Day, sidereal, 23 ; solar, 22. Declination, magnetic, 338. Density, 58 ; of a liquid, 62 ; of a solid, 60 ; bulb, 62. Derrick, 165. Deviation, angle of, 241. Dew point, 307. Diamagnetic body, 328. Diathermanous body, 318. Diatonic scale, 197. INDEX [References are to pages,} Dielectric, 351 ; capacity, 352 ; in- fluence of, 351. Differential, in a motor car, 467, 470, 471. Diffraction, 278. Diffusion, 25, 28. Dipping needle, 337. Discharge, intermittent, 411 ; oscil- latory, 453. Dispersion, 263. Drum armature, 425. Dry cell, 371. Dry dock, 57. Dryness, 307. Ductility, 12. Dynamo, 421 ; compound, 426 ; series, 425 ; shunt, 425. Dyne, 105. Earth, a magnet, 336. Ebullition, 303. Echo, 186. Efficiency, 159. Effusion, 26. Elasticity, 36; limit of, 36; of form, 36 ; of volume, 36. Electric, bell, 449; circuit, 363; current, 361 ; current detection, 366; motor, 426;. railways, 430; telegraph, 447 ; waves, 454. Electrical, attraction, 340; distri- bution, 346 ; machines, 355 ; potential, 348 ; repulsion, 341 ; resistance, 378 ; wind, 347. Electrification, 340 ; atmospheric, 358; by induction, 344; kinds of, 341 ; simultaneous, 342 ; unit of, 348. Electrode, 363, 377. Electrolysis, 373 ; laws of, 375 ; of copper sulphate, 373 ; of water, 374. Electrolyte, 362, 373. Electromagnet, 392 ; applications of, 394. Electromotive force, 365, 381 ; in- duced by magnets, 401 ; induced by currents, 403. Electrons, 419. Electrophorus, 354. Electroplating, 376. Electroscope, 342. Electrostatic, capacity, 350; in- duction, 344. Electrostatics, 340. Electrotyping, 376. Energy, 1, 148; conservation of, 153 ; dissipation of, 153 ; kinetic, 150 ; measure of, 151 ; potential, 149 ; transformation of, 152. Engine, gas, 323; steam, 320; two-cycle, 325; four-cycle, 324, 460-466. 'English system of measurement, 22. Equilibrant, 109. Equilibrium, 108; kinds of, 125; of floating bodies, 55 ; under gravity, 125. Erg, 145. Ether, 214. Evaporation, cold by, 303. Expansion, coefficient of, 289; of gases, 289 ; of liquids, 288 ; of solids, 287. Extension, 6. Eye, 259 ; defects of, 262. Falling bodies, 128, 130. Field, electrical, 387, 391 ; mag- netic, 333. Field magnet, 425. Floating bodies, 55. Fluids, 39; characteristics of, 39; pressure in, 41. Fluoroscope, 415. Flute, 205. Focus, 230; conjugate, 232; of lens, 249 ; of mirrors, 232. Foot, 18. Foot pound, 144. INDEX [References are to pages.} Force, 5, 104 ; composition of, 107 ; graphic representation of, 107 ; how measured, 106 ; molecular, 29 ; moment of, 160 ; parallelo- gram of, 110; resolution of, 111; units of, 105. Force pump, 86. Forced vibrations, 188. Fountain, siphon, 85 ; vacuum, 79. Four-cylinder engine, 460, 461. Fraunhofer lines, 268. Freezing, mixtures, 302 ; point, 283. Friction, 156 ; uses of, 158, 469. Fundamental, tone, 202 ; units, 23. Fusion, 299; heat of, 301. Gallon, 20. Galvanometer, d'Arsonval, 395. Galvanoscope, 366. Gas engine, 322-325, 460-466. Gas equation, 294. Gases, 40 ; compressibility of, 41 ; expansion of, 289; media for sound, 182 ; thermal conductivity of, 310. Gassiot's cascade, 410. Gauge, water, 49. Geissler tube, 410. Grades, 170. Grain, 21. Gram, 21. Gramme ring, 424. Gravitation, 122 ; law of, 123. Gravitational unit of force, 105. Gravity, 122 ; acceleration of, 122 ; cell, 371 ; center of, 123 ; direc- tion of, 122 ; specific, 59. Hammer, riveting, 88. Hardness, 14. Harmonic, curve, 178 ; motion, 101. Harmonics, 204. Heat, 280 ; conduction of, 309 ; con- vection of, 312 ; due to electric current, 385; from mechanical action, 319; kinetic theory of, 280; lost in solution, 302; me- chanical equivalent of, 320; measurement of, 297 ; nature of, 280 ; of fusion, 301 ; of vaporiza- tion, 306; radiant, 315; related to work, 319; specific, 297; transmission of, 309. Heating by hot water, 312. Helix, 389 ; polarity of, 389. Holtz machine, 355. Hooke's law, 37. Horizontal line or plane, 123. Horse power, 147. Humidity, 307. Hydraulic, elevator, 44; press, 42; ram, 51. Hydro-airplanes, cuts facing page 66. Hydrometer, 63, 466. Hydrostatic paradox, 47. Ice plant, ammonia, 304. Images, by lenses, 250 ; by mirrors, 225, 233 ; by small openings, 218. Impenetrability, 6. Impulse, 116. Incandescent lamp, 444. Inclination, 387. Inclined plane, 169; mechanical advantage of, 170. Index of refraction, 240. Indicator diagram, 322. Induced magnetism, 331. Induction, charging by, 345 ; coil, 406 ; electromagnetic, 401 ; elec- trostatic, 344 ; motors, 440 ; self- induction, 405. Inertia, 7. Influence machine, 355. Insulator, 343. Intensity of illumination, 219. Interference, of light, 276; of sound, 194. Intervals, 196 ; of diatonic scale, 198'; of tempered scale, 199. INDEX [References are to pages.] Ions, 364. Isobars, 71. Isoclinic lines, 337. Isogonic lines, 338. Joseph Henry's discovery, 405. Joule, 145. Joule's equivalent, 320 ; law, 385. Kaleidoscope, 229. Keynote, 197. Kilogram, 21. Kilogram meter, 144. Kinetic energy, 150; measure of, 151. Kinetic theory, 27 ; of heat, 280. Lag of current, 433. Lalande cell, 372. Lamp, arc, 442; gas-filled, 446; incandescent, 444 ; metal fila- ment, 445. Lantern, projection, 259. Law, Boyle's, 74; Lenz's, 404; Ohm's, 378; of Charles, 293; of electromagnetic induction, 401 ; of electrostatic action, 342 ; of falling bodies, 130 ; of gravita- tion, 123 ; of heat radiation, 316 ; of magnetic action, 330 ; of ma- chines, 156 ; Pascal's, 41. Laws, of motion, 117; of strings, 201. Leclanche cell, 371. Length, 17. Lens, 246 ; achromatic, 265 ; focus of, 248 ; images by, 250. Lenz's law, 404. Lever, 161 ; mechanical advantage of, 162. Leyden jar, 352; theory of, 353; charging and discharging, 352. Lift pump, 85. Light, 214; analysis of, 263; propagation of, 216 ; reflection of, 223 ; refraction of, 238 ; speed of, 215 ; synthesis of, 264. Lightning, 358 ; rod, 359. Lines, agonic, 338 ; isoclinic, 337 ; of magnetic force, 333. Liquefaction, 299. Liquid, 4, 40; cohesion in, 11; compressibility of, 40; density of, 62 ; downward pressure, 45 ; expansion of, 288; in connected vessels, 49 ; medium for sound, 182 ; surface level in, 49 ; surface tension in, 30 ; thermal con- ductivity of, 310; velocity of sound in, 185. Liter, 20. Local action, 367. Lodestone, 327. Longitudinal vibrations, 177. Loudness of sound, 192. Lubrication, of a motor car, 463, 464. Machine, 155; efficiency of, 159; electrical, 355; law of, 156; mechanical advantage of, 160; simple, 159. Magdeburg hemispheres, 79. Magnet, artificial, 328; bar, 328; electro-, 392; horseshoe, 328; natural, 327. Magnetic, action, 330; axis, 329; field, 333; lines of force, 333; meridian, 329; needle, 329; polarity, 329; substance, 328; transparency, 329. Magnetism, induced, 330 ; nature of, 332 ; permanent and tem- porary, 332; terrestrial, 336; theory of, 333. Magneto, in a motor car, 465. Magnets, 327. Major chord, 197. Malleability, 14. Manifold, in a motor car, 464. Manometric flame, 209. 6 INDEX [References are to pages.] Mass, 9; units of, 2l. Matter, 1 ; properties of, 6 ; states of, 4. Mechanical advantage, 160. Mechanical equivalent of heat, 320. Mechanics, of fluids, 39 ; of solids, 104. Melting point, 299; effect of pressure, 301. Meter, 17. Metric system, 17. Micrometer, 174. Microphone, 451. Microscope, compound, 256; sim- ple, 255. Minor chord, 197. Mirror, 225 ; focus of, 230 ; images by, 226, 233; plane, 225; spherical, 229. Mobility, 39. Molecular, forces, 29 ; motion, 26 ; physics, 25. Moment, of a force, 160. Momentum, 116. Motion, 91 ; accelerated, 95 ; curvilinear, 99 ; harmonic, 101 ; molecular, 26 ; periodic, 101 ; rectilinear, 91 ; rotary, 91 ; uniform, 92; vibratory, 101. Motor, electric, 426 ; induction, 440. Motor car, 1, 460-474. Multiple-disk clutch, 469. Musical, scales, 196 ; sounds, 191. Needle, dipping, 337 ; magnetic, 329. Newton's, laws of motion, 116; rings, 276. Nodes, 203. Noise, 191. Octave, 197. Ohm's law, 378, 382. Opaque bodies, 214. Opera glasses, 258. Optical, center, 247; instruments, 255. Organ pipe, 206. Oscillation, center of, 139 ; electric, 360. Ounce, 22. Overtones, 204, 208. Partial tones, 204. Pascal, experiments, 67; principle, 41. Pendulum, applications of, 140; laws of, 138; seconds, 141; simple, 136. Percussion, center of, 140. Period of vibration, 138. Periodic motion, 101. Permeability, 335. Phonodeik, 211. Photographer's camera, 259. Photometer, 221. Photometry, 219. Physical measurements, 16. Physics, 1. Pigments, 275. Pistons, in a motor car, 460. Pitch, 191 ; limits of, 199 ; relation to wave length, 192 ; of screw, 173. Plumb line, 123. Pneumatic appliances, 83. Points, action of, 347. Polarity of helix, 389. Polarization, 368. Polyphase alternators, 434. Porosity, 11. Potential, difference, 348; zero, 349. Pound, 21. Power, 145. Pre-ignition, 464. Pressure, 41 ; of fluids, 39 ; at a point in fluids, 46 ; air, 65 ; down- INDEX [References are to pages.] ward, 45 ; effect on boiling point, 305 ; effect on melting point, 301 ; independent of shape of vessel, 47 ; in tires, 467. Principle of Archimedes, 53. Prism, 242 ; angle of deviation, 243. Proof plane, 342. Properties of matter, 6. Pulley, 165; differential, 168; me- chanical advantage of, 167 ; sys- tems of, 166. Pump, air, 76 ; compression, 76 ; force, 86 ; lift, 85. Puncture, of a tire, 468. Quality of sounds, 193; due to overtones, 193. Radiation, 315 ; laws of, 316. Radiator, in a motor car, 462. Radioactivity, 416. Radiometer, 315. Radium, 417. Rainbow, 266. Rays of light, 216. Reflection, diffused, 224; law of, 223; multiple, 228; of light, 223 ; of sound, 186 ; regular, 223 ; total, 243. Refraction, cause of, 239 ; atmos- pheric, 243 ; laws of, 241. Regelation, 301. Relay, 448. Resistance, of air, 129 ; electrical, 378; formula for, 380; laws of, 379 ; unit of, 379. Resolution, of a force, 111; of a velocity, 113. Resonance, 188, 190. Resonator, Helmholtz's, 191. Resultant, 107. Riveting hammer, 88. Roentgen rays, 414. Rotating field, 438. Running gear, of a motor car, 467, 468. Scale, absolute, 293; diatonic, 197 ; tempered, 198. Screw, 172; applications of, 173; mechanical advantage of, 173. Second, 22. Secondary or storage cell, 376. Seconds pendulum, 141. Self-induction, 405. Shadows, 217. Shoe, in a motor car, 467 Sidereal day, 23. Sight, 260. Singing flame, 195. Siphon, 83 ; intermittent, 85. Six-cylinder engine, 460, 461. Solar day, 22. Solenoid, 389 ; polarity of, 389. 'Solids, 4; density of, 60; ex- pansion of, 287 ; thermal con- ductivity of, 309 ; velocity of sound in, 185. Solution, 34 ; saturated, 35 ; heat lost in, 302. Sonometer, 201. Sound, 176, 181 ; air as a medium, 182; liquids as media, 182; loudness of, 192 ; musical, 191 ; quality of, 193 ; reflection of, 186 ; sources of, 181 ; trans- mission of, 182 ; velocity of, 184 ; waves, 183. Sounder, telegraph, 447. Spark lever, 471. Spark plug, 465. Specific gravity, 59 ; bottle, 62. Specific heat, 297. Spectroscope, 269. Spectrum, solar, 263; kinds of, 268. Speed, 92 ; of light, 215. Speeds, of a motor car, 470. Spherical aberration, in mirrors, 235 ; in lenses, 253. Spheroidal state, 303. Spherometer, 174. INDEX [References are to pages.] Splash system, of lubrication, 464. Stability, 126. Stable equilibrium, 125. Stalling, of a gas engine, 472. Starter, in a motor car, 471. Starting resistance, 429. States of matter, 4. Steam, engine, 320 ; turbine, 322. Steelyard, 162. Storage cell, 376, 466; Edison, 378. Strain, 36. Strength, of an electric current, 381 ; methods of varying, 383. Stress, 36. Strings, laws of, 201. Sublimation, 303. Submarine boat, 57. Surface tension, 30; illustrations of, 31. Suspension, center of, 138. Sympathetic vibrations, 189. Synthesis of light, 264. Telegraph, electric, 447; key, 447; signals, 449; system, 449; wireless, 453. Telephone, 451. Telescope, astronomical, 257 ; Galileo's, 258. Temperature, 280; measuring, 281. Tempered scale, 198. Tempering, 15, 199. Tenacity, 12. Thermal capacity, 297. Thermometer, 282; clinical, 285; limitations of, 285 ; scales, 283. Thermo-siphon, system of cooling, 462. Throttle lever, in a motor car, 471. Thunder, 358. Time, 22. Timer, in a motor car, 465. Tires, 467, 468. Tone, fundamental, 202; partial, 204. Torricellian experiment, 67. Transformers, 435 ; cut facing page 439. Translucent bodies, 214. Transmission, of heat, 267 ; of power, 437 ; in a motor car, 469, 470. Transmitter, 447, 452. Transparent bodies, 214. Transverse vibrations, 176. Trombone, 205. Tuning fork, 190. Turnbuckle, 174. Twelve-cylinder engine, 460, 462. " Twin-six," 460. Units, 16 ; of heat, 297 ; of length, 17; of mass, 21 ; of time, 22. V-type engine, 463. Vacuum, Torricellian, 67. Vaporization, 302 ; heat of, 265. Velocity, 92; composition of, 113; of light, 215 ; of molecules, 27 ; of sound, 184; resolution of, 113. Ventral segments, 204. Vertical line, 123. Vibration, amplitude of, 138; complete, 138; forced, 188; longitudinal, 177 ; of strings, 201 ; period of, 138 ; single, 138 ; sympathetic, 189 ; transverse, 176. Viscosity, 39. Volt, 381. Voltaic cell, 361 ; electrochemical action in, 363. Voltameter, 381. Voltmeter, 396. Water, gauge, 49 ; supply, 50 ; waves, 180. Watt, 148. Wave motion, 177. Waves, 177; longitudinal, 179; electric, 454 ; length, 180 ; sound, 183 ; transverse, 177 ; water, 180. INDEX 9 [References are to pages,] Wedge, 172. Whispering gallery, 188. Weight, 9, 122; of air, 65; varia- Wireless telegraphy, 453; teleph- tion of, 124. ony, 459. Welding, cut facing page 16. Work, 143 ; units of, 144 ; useful, Weston normal cell, 382. 159 ; wasteful, 159. Wheatstone's bridge, 391. X-rays, 414. Wheel and axle, 164; mechanical advantage of, 164. Lrd ' 18 ' Wheels, of a motor car, 467. Zeppelin, 81, 82. YB 360C6 459995 UNIVERSITY OF CALIFORNIA LIBRARY