GIFT OF President's Office THREE-COLOR PRINTING (See pane 236) FIRST PRINCIPLES OF PHYSICS BY HENRY S. CARHART, LL.D. \> PROFESSOR EMERITUS OF PHYSICS, UNIVERSITY OF MICHIGAN AND HORATIO N. CHUTE, M.S. INSTRUCTOR IN PHYSICS IN THE ANN ARBOR HIGH SCHOOL Boston ALLYN AND BACON 1912 COPYRIGHT, 1912 BY HENRY S. CARHART AND HORATIO N. CHUTE J. S. Cashing Co. Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE NOT many years ago the ideal textbook in elementary physics was little more than a concise, scientific statement of the fun- damental principles of the subject, illustrated by experiments and reenforced by numerous problems designed to emphasize the expression of physical relations in mathematical form. Teachers, however, came to recognize that books of this type do not make the subject sufficiently attractive to pupils who are familiar with picturesque, popular applications of physics, and who are not inclined to devote the requisite time and effort to overcome the difficulties of a less alluring side of the subject. Attempts have been made to meet this vague dissatisfaction by presenting many familiar illustrations of the practical appli- cations of physics, but with an unsatisfactory treatment of the fundamental principles on which these applications are based. As a result, pupils reached the end of their study with few definite ideas and little knowledge of the science itself. This condition was plainly so much worse than the former one that teachers, who have been drawn into the experiment, have gen- erally preferred to return to the earlier type of book. The present volume is the result of an attempt to make a book which shall have a strong element of interest and attrac- tiveness, and at the same time be so clear and definite in the treatment of principles that pupils may carry away from the course some useful acquisitions for daily life and a preparation for continuing the subject with success in the college or tech- nical school. Every effort has been made to have the language of the book clear and simple. Cordial acknowledgment is made to many teachers whose helpful criticisms have been an aid to 239420 IV PREFACE this end. The problems given require for their solution only such simple algebraic operations as offer no difficulty to pupils who have taken a course in elementary algebra. Yielding to the advice of teachers in whose judgment the au- thors have the greatest confidence, a somewhat radical change has been made in the order of subjects by placing the Mechanics of Fluids before the Mechanics of Solids. The reason assigned is that the facts and principles of the former are more familiar than those of the latter ; they thus form a natural approach to the abstract laws of motion, the composition and resolution of forces and velocities, and the formulae connected with them as the shorthand expression of physical laws. H. S. C. H. N. C. April, 1912. CONTENTS Chapter I. Introduction PAGE I. Matter and Energy 1 II. Properties of Matter . . . . . . . 3 III. Physical Measurements - . . . . . .10 Chapter II. Molecular Physics I. Molecular Motion . . . . . .16 II. Surface Phenomena .19 III. Molecular Forces in Solids 25 Chapter HI. Mechanics of Fluids I. Pressure of Fluids 31 II. Bodies Immersed in Liquids 40 III. Density and Specific Gravity 44 i IV. Pressure of the Atmosphere 49 V. Compression and Expansion of Gases ... 57 VI. Pneumatic Appliances . . . . .66 '"Chapter IV. Motion I. Motion in Straight Lines 71 II. Curvilinear Motion ....... 79 III. Simple Harmonic Motion ...... 80 Chapter V. Mechanics of Solids I. Measurement of Force ...... 83 II. Composition of Forces and of Velocities ... 86 III. Newton's Laws of Motion ...... 93 IV. Gravitation . .98 V. Falling Bodies 104 VI. Centripetal and Centrifugal Force . . . .109 VII. The Pendulum . . Ill vi CONTENTS Chapter VI. Mechanical work PAGE I. Work and Energy .... II. Machines . 117 . 126 Chapter VII. Sound I. Wave Motion . . .143 II. Sound and its Transmission . 148 III. Velocity of Sound .... IV. Reflection of Sound . . 150 . 153 V. Resonance ...... . 155 VI. Characteristics of Musical Sounds . 157 VII. Interference and Beats . 160 VIII. Musical Scales . 162 IX. Vibration of Strings .... X. Vibration of Air in Pipes . . 167 . 171 . 174 Chapter VIII. Light I. Nature and Transmission of Light . II. Photometry III. Reflection of Light .... IV. Refraction of Light .... V. Lenses ...... . 179 . 184 . 187 . 203 . 210 VI. Optical Instruments .... . 219 . 2 9 6 VIII. Color . 233 . 238 Chapter IX. Heat I. Heat and Temperature . . . . II. The Thermometer .... . 242 . 244 III. Expansion ...... IV. Measurement of Heat . 248 . 257 V. Change of State VI. Transmission of Heat . 260 . 267 VII. Heat and Work. 274 CONTENTS Vll Chapter X. Magnetism PAOK I. Magnets and Magnetic Action . . 281 II. Nature of Magnetism . 286 III. The Magnetic Field .... . 287 IV. Terrestrial Magnetism . 290 ^Chapter XI. Electrostatics I. Electrification . 293 II. Electrostatic Induction . 298 III. Electrical Distribution . 300 IV. Electric Potential and Capacity . 302 V. Electrical Machines .... . 308 VI. Atmospheric Electricity . . 313 Chapter XII. Electric Currents I. Voltaic Cells . 316 II. Electrolysis ..... . 327 III. Ohm's Law and its Applications . 332 IV. Heating Effects of a Current . . 338 V. Magnetic Properties of a Current . . .340 VI. Electromagnets . 344 VII. Measuring Instruments . 347 Chapter XIII. Electromagnetic Induction I. Faraday's Discoveries . 354 II. Self-induction ..... . 358 III. The Induction Coil . . . 359 Chapter XIV. Dynamo-Electric Machinery I. Direct Current Machines . . 371 II. Alternators and Transformers . . 378 III. Electric Lighting ... . .381 IV. The Electric Telegraph . . 384 * V. The Telephone . 388 VI. Wireless Telegraphy .... . 390 viii CONTENTS Appendix PAGE I. Geometrical Constructions 394 II. Conversion Tables . . . . . . . 398 III. Mensuration Rules 400 IV. Table of Densities .401 V. Geometrical Construction for Refraction of Light . 402 Index ... 405 PORTRAITS OF EMINENT PHYSICISTS FACING PAGE Blaise Pascal 32 Galileo Galilei . . . . 32 Sir Isaac Newton .94 Lord Kelvin (Sir William Thomson) 124 Hermann von Helmholtz ........ 158 James Clerk-Maxwell 180 James Prescott Joule 274 James Watt 274 Michael Faraday 282 Alessandro Volta 304 Georg Simon Ohm . 304 Benjamin Franklin ......... 314 Hans Christian Oersted 322 Joseph Henry .......... 358 Sir William Crookes 366 Wilhelm Konrad Rontgen 366 Madame Curie '" 370 Sir Joseph John Thomson 380 Heinrich Rudolf Hertz .390 Lord Rayleigh (John William Strutt) 400 FIRST PRINCIPLES OF PHYSICS CHAPTER I INTRODUCTION I. MATTER AND ENERGY 1. Physics Defined. Physics is often defined as the science of matter and energy. Matter is everything we can see, taste, or touch ; such as air, water, earth, gas, wood, steam in short, everything which occupies space. Energy is the capacity for doing work, that is, for producing any change in the position or condition of matter, especially against resistance opposing such a change. Water in an elevated tank, steam under pressure in a boiler, a flying cannon ball, all have the capacity for doing work, over- coming resistance, or changing the position and motion of bodies. So if everything which we recognize by the senses is matter, and every change in matter involves energy, it is plain that physics, which is the science of matter and energy, is a universal science, touching our life at every point. Whatever we see or touch is matter ; whatever we do exhibits energy. Countless physical phenomena are tak- ing place about us every day : a girl playing tennis, a boy rowing a boat, the school bell ringing, the sun giving light and heat, the wind flapping a sail, an apple falling from a tree, a train or an automobile whizzing by, all are examples of matter and associated energy. 1 2 v v ^INTRODUCTION Physics is hofrsa mubls iC&ucetii^ct 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 interest is not so much in the engine itself as in the engine with steam up ready to drive it. No one would care to purchase an automobile to stand in a garage ; its attractiveness lies in the fact that it becomes almost a living thing when its motor is vitalized by the heat of combustion of gasoline. 2. Physical Principles and their Applications. The phe- nomena of physics appear in everything we see or do. It will be impossible to learn about all of them in a single year. This is especially true because the phenomena of physics, and in particular the applications of physical principles, are constantly changing. But the principles 1 remain the same. So this book lays emphasis on the prin- ciples of physics and their common applications, leaving it to the enthusiasm and ingenuity of both teacher and pupils to supplement the applications with illustrations drawn from life and from scientific and technical journals. 3. States of Matter. Matter may exist in three states, exemplified by water, which may assume either the solid, the liquid, or the gaseous form. Ice, water, and water vapor may all exist together at the same tempera- ture. Briefly described, Solids have definite size and shape. Liquids have definite size, but the shape is that of the con- taining vessel. 1 An hypothesis is a supposition which serves to explain phenomena. The more varied the phenomena it explains, the greater the probability of its truth. When the evidence in support of it becomes large, it is raised to the rank of a theory ; and when its truth is fully established, it becomes a law or principle. PROPERTIES OF MATTER 3 G-ases have neither definite size nor shape, both depending on the containing vessel. 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 pressure it flows like a liquid, though slowly, and may be molded at will. II. PROPERTIES OF MATTER 4. Properties General and Special. The properties of matter are the qualities which serve to describe and define it. They are either general, that is, common to all kinds of matter; or special, that is, found in some kinds of matter but not in others. Thus, all matter has extension, or occupies space. On the other hand, a piece of 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 inelastic. So we see that while extension is a general property of matter, transparency and elasticity are special properties. A. General Properties 5. 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 relatively small, if its thickness should actually become zero, it would cease to be either a sheet of paper or a piece of gold leaf. Extension is the .property of occupying space or having dimensions. INTRODUCTION 6. Impenetrability. Matter occupies space, but no two portions of matter can occupy the same space at the same time. The volume 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 impene- trability. 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 before and after putting in the coal; the difference is the volume of the water displaced, or that of the piece of coal. 7. Inertia. The most characteristic gen- eral property of matter is inertia. Inertia is the property which all matter possesses of re- sisting any attempt to start it if at rest, to stop it if in motion, or to change either the direc- tion or the amount of its motion. If a moving body stops, its arrest is always owing to some- thing outside of itself ; and if a body at rest is Fig. 2 PROPERTIES OF MATTER set moving, motion must be imparted to it by some other- body. 8. 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 before 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 without disturbing it. The violent jar to a water pipe when a faucet is suddenly closed is accounted for by the inertia of the stream. The persistence with which a spinning top maintains its axis of rotation in the' same direction is due to its inertia. Bren nan's monorail car (Fig. 2) is kept in an up- right position by the inertia of a rapidly revolving wheel of great weight. Tall columns, chimneys, and buildings are sometimes twisted around by violent earthquake movements (Fig. 3). The sudden circu- lar motion of the earth under a column leaves it standing still, while the slower return motion carries it around. Suspend a heavy weight by a cotton string, as in Fig. 4, 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 Fig. 3 Fig. 4 6 INTRODUCTION 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. 9. Mass. Another general property of matter is mass. We are all familiar with the fact that the less matter there is in a Nbody, the more easily it is moved, and the more easily it is stopped when in motion. One 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. Mass is the measure of the resistance which a body offers to motion or change of motion; it is therefore the measure of the body's inertia. Mass must not be confused with weight ( 116) because mass is independent of 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 sur- face ; but its mass would be the same everywhere. For this reason, and others which will appear later, in discuss- ing the laws of physics we prefer to speak of mass when a student thinks the term weight might be used as well. 10. Cohesion and Adhesion. All bodies are made up of very minute particles, which are separately invisible, and are called molecules. Cohesion is the force of attraction be- tween molecules, and it binds together the molecules of a substance so as to form a larger mass than a molecule. Adhesion is the force uniting bodies by their adjacent sur- faces. When two clean surfaces of white-hot wrought iron are brought into close contact by hammering, they cohere and become a single body. If a clean glass rod be dipped into water arid then withdrawn, a drop will adhere PROPERTIES OF MATTER J to' it* Glue, adhesive plaster, and postage stamps stick by adhesion. Mortar adheres to bricks and gold plating to brass. Suspend from one of the scalepans of a beam balance a clean glass disk by means of threads cemented to it (Fig. 5). After counterpois- ing 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 bal- ance 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 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 cohe- sion between the molecules of the mercury is greater than the adhesion between it and the glass. Cut a fresh, smooth sur- face on two lead bullets and hold these surfaces gently to- gether. 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 worth- less stones, with an occasional diamond, 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. 11. Porosity. Sandstone, unglazed pottery, and similar bodies absorb water without change in volume. The water Fig. 5 8 INTRODUCTION fills the small spaces called pores, which are visible either to the naked eye or under a microscope. All matter is 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 exuded 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. B. Special Properties 12. 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. 6). 3 The breaking weights for wires of different mate- rials but of the same cross section differ greatly. A knowledge of tensile strength is essential in the design 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 tenacity of all solid metals, and cast steel the greatest. Even the latter is exceeded by fibers of silk and cotton. Single fibers of cotton Fig. 6 can support millions of times their own weight. PROPERTIES OF MATTER 13. Ductility. Ductility is the property of a substance which permits it to be drawn into wires or filaments. Gold, 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. 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. 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 transparent and transmits 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. Malleable iron is made from cast iron by heating it for several days in contact with a substance which removes some of the carbon from the cast iron. 15. 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 10 INTRODUCTION 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 uniformly 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 must be carefully annealed to prevent fracture and warping during the process of grind- ing and polishing. III. PHYSICAL MEASUREMENTS 16. Units. To measure any physical quantity a certain fixed amount of the same kind of quantity is used as the unit. For example, to measure the length of a body, some arbitrary length, such as a foot, is chosen as the unit of length; the length of a body is the number of times this unit is contained 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). PHYSICAL MEASUREMENTS 11 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. 17. Units of Length. The two systems of units in common use are the metric and the English. In the for- mer the unit of length is the meter. It is the distance Fig. 7 between two transverse lines on each of two bars of plati- num-iridium at the temperature of melting ice. These bars, called national prototypes, are preserved at the Bureau of Standards in Washington (Fig. 7). The meter (m.) is divided into 10 decimeters (dm.), the decimeter into 10 centimeters (cm.), and the centimeter into 10 millimeters (mm.). The only multiple of the meter in general use is the kilometer (km.), equal -to 1000 meters. It is used to measure such distances as are expressed in miles in the English system. In the metric system areas are measured in square millimeters (mm. 2 ), square centimeters (cm. 2 ), etc. In like manner volumes are measured in cubic millimeters (mm. 3 ), cubic centimeters (cm. 3 ), etc. The cubic deci- meter (dm. 3 ) is called a liter (1.) ; it is equal to 1000 cm. 3 . The weights and measures in common use in the United States were denned by Act of Congress in 1866 in terms of those of the metric system. By this act the legal value of the yard is f f $$ of a meter ; conversely, the meter is 39.37 inches. The inch is, therefore, 2.540 cm. The INTRODUCTION relation between the centimeter scale and the inch scale is shown in Fig. 8. 100 MILLIMETERS = 10 CENTIMETERS = 1 DECIMETER = 3. 937 INCHES. 1 2 3 4] 5 6] 7 8 9| 10| I I I I I I I I II I II I I II I I ! ! I I I I I ! I I I I I I I I I I I I I I i i I I II I I I I I I I I | -I I I I till 111 t 2 3 4 INCHES AND TENTHS Fig. 8 The unit of length in the English system for the United States is the yard, defined as above. One third of the yard is the foot, and one thirty-sixth is the inch. The gallon of 231 cubic inches is the unit of volume for liquid measure. Tables for the conversion of quantities from one system of units into the other may be found in Appendix II. 18. Units of Mass. The unit of mass in the metric system is the kilogram (kgm.). The United States has two prototype kilograms made of platinum-irid- ium and preserved at the Bureau of Stand- ards in Washington (Fig. 9). The gram (gm.) 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. (centi- grade scale). For prac- tical purposes this is the Fig. 9 kilogram. The gramas PHYSICAL MEASUREMENTS 13 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 volume. The unit of mass in the English system is the avoirdu? pois pound (lb.).. The ton of 2000 pounds is its chief multiple; its submultiples are the ounce (oz.) and the grain (gr.). The avoirdupois pound is equal to 16 ounces and to 7000 grains. The " troy pound of the mint " con- tains 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. 19. The Unit of Time. The unit of time in universal use in physics is the second. It is -g^TTo of a mean solar day. The number of seconds between the instant when the sun's center crosses the meridian 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. The sidereal day used in astronomy is nearly four minutes shorter than the mean solar day. 20. The Three Fundamental Units. Just as. the meas- urement of areas and of volumes reduces simply to the measurement of lengths, 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 14 INTRODUCTION length, mass, and, time. For this reason these three are considervtTfundamental 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. Exercises and Problems Problems involving the relations between the two systems of measurement should be solved, using the exact values of 17 and 18. 1. Could the volume of a lump of sugar be determined by the method of 6? f 2. The circus rider standing on the back of a moving horse jumps straight up and not forward in order to go through a paper-covered hoop. Explain. 3. An ax handle is driven into the ax by pounding the end of the handle rather than the ax. Explain. 4. To keep from falling the conductor runs by the side of the moving car before he jumps on. Why? 5. A man standing on a moving car jumps vertically upward. Does he come down on the spot from which he jumped, or back of it ? 6. Why will a bullet from a rifle shot at a window cut a smooth hole through the glass, but if thrown against it by hand shatter it? 7. Lead bullets fired from a rifle against a thick plate of lead are found welded to the plate. Why ? 8. If a horse in starting a loaded wagon is permitted to jump, he is likely to break the harness ; but if he pulls steadily, the harness is sufficient to stand the pull. Explain. 9. Why does the flywheel cause an engine to run more uniformly ? 10. Why does a pendulum keep moving when the bob reaches the lowest point of its swing? 11. How many meters in a rod ? EXEECISES AND PROBLEMS 15 12. The Washington monument is 555 ft. high. Find its height in meters. 13. How many gallons in 100 liters? 14. Calculate the number of liters in 10 gal. 15. A motor boat has a speed of 20 mi. an hour. Express it in kilometers per hour. 16. At 10? per pound, what will be the cost of 10 kgrn. of rice? 17. A cubic foot of water weighs 62.4 Ib. ; what is the weight of a cubic inch in grains? 18. A cubic foot of water weighs 62.4 Ib. ; what is the weight of a cubic inch in grams? 19. Find the difference in centimeters between one foot and 30 cm. 20. The greatest allowable weight of a package by foreign parcels post is 5 kgm. What is the nearest whole number of pounds? CHAPTER II MOLECULAR PHYSICS I. MOLECULAR MOTION 21. 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 pos- sible. 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. 10) with P. _ 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 of ammonia. Instead of these vapors air and illuminating gas may be used, and after diffusion, the presence of an explosive mixture in both jars may be shown by applying a flame to the mouth of each separately. 16 MOLECULAR MOTION 17 22. Effusion through Porous Walls. The passage of a gas through the pores of a solid is known 'ds Affusion. The rate of effusion for different gases is nearly inversely proportional to the square of their relative densities. Hydrogen, for example, which is one six- teenth as heavy as oxygen, passes through very small openings 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. 11. Invert over the porous cup a large glass beaker or bell jar, and pass into it a stream of hydrogen 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 invisible pores in the walls of the porous cup and produces gas pressure in the flask. If the beaker pj~ j j 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. 23. Molecular Motion in Gases. The simple facts of the diffusion and effusion of gases lead to the conclusion that their molecules are not at rest, but are in coiisl^tjim 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 18 MOLECULAR PHYSICS a highly important one. Witness the operation of a gaso- line engine, in which the inlet valve presents only a nar- row opening for a small fraction of a second ; and yet this brief period suffices for the explosive mixture to enter and fill the cylinder. 24. Pressure Produced by Molecular Bombardment. 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 surface. The clatter of an in- definitely large number 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 innumerable minute blows against the walls of the containing vessel and these blows compose a contin- uous pressure. This, in brief, is the kinetic theory of the pressure of a gas. 25. 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 atmosphere, or 1033 gm. 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. 26. 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. 8 UEFA CE PHENOMENA 19 Let a tall jar be nearly filled with water colored with blue litmus, and let a little strong sulphuric acid be introduced into the jar at the bottom by means of a thistle tube (Fig. 12). The density of the acid is 1.8 times that of the litmus solu- tion, and the acid therefore remains at the bottom 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. Fig- 1 27. 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. II. SURFACE PHENOMENA 28. Molecular Forces in Liquids. A primitive idea of force is derived from the sense of muscular exertion in lifting a weight, pushing a cart, stretching a spring, catch- ing a ball, throwing a stone, or making intense muscular effort in running at top speed. By an easy transition of ideas we carry this conception over to forces other than those exerted by men and animals, such as those between the molecules of a body. Molecular forces act only through insensible distances, such as the distances separating the molecules of solids and liquids. A clean glass rod does not attract water until there is actual contact between the 20 MOLECULAR PHYSICS two. If the rod touches the waiter, the latter clings to the glass, and when the rod is withdrawn, 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 glob- ule 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 as that of the oil by varying the proportions. The globule then assumes a truly spherical form and floats anywhere in the mixture (Fig. 13). Cover a smooth board with fine dust, such as lyco- podium 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. 14). In all these illustrations the spherical form is accounted 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. 29. 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 EEEE depressed and the needle rests in a little hollow (Fig. 15). Fig. 15 Let two bits of wood float on water a few SUE FACE PHENOMENA 21 millimeters apart. If a drop of alcohol is let fall on the water be- tween 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. 30. Surface Tension. The molecules composing the surface of a liquid are not under the same ^conditions of equilibrium as those within the B liquid. The latter are attracted equally in all directions by the sur- rounding molecules, -while those at the surface are attracted downward and laterally, but not upward (Fig. 16). The result is an unbalanced molecular force toward the interior lg * lc 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 as small a volume as possible. Liquids in small masses, therefore, always tend to become spherical. 31. Illustrations. Tears, dewelrpps, 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 mercury on a clean glass plate are slightly flattened by their weight, but the smaller the globules the more nearly spherical they are. 22 MOLECULAR PHYSICS Fig. 17 Make a stout ring three or four inches in diameter with a handle (Fig. 17). 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 solution 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 solution. The films in Fig. 18 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, quitr> free from an oily film, will spin around in a most erratic manner. The camphor dissolves unequally at different points, and thus pro- duces unequal weaken- ing of the surface tension in different di- rections. Make a tiny wooden boat and cut a notch in the stern ; in this notch put a piece of camphor gum (Fig. 19). The camphor will weaken the tension astern, while the tension 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. 20). The ex- Fig. 20 SURFACE PHENOMENA pelled 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 hold the water. A film fills the meshes of the gauze and makes the cylinder air-tight ; if it is broken by blowing sharply on it, the water will quickly run out. 32. 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 level outside. The top of the little column of water is concave, while that of the column of mercury is convex upward (Fig. 21). Familiar examples of capillary action are numer- pig. 21 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, like water in a siphon. Many salt solutions con- struct their own capillary highway 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 by capil- lary action still higher. This process may continue until the liquid flows over the top and down the outside of the vessel. 33. Laws of Capillary Action. Support vertically several clean glass tubes of small internal diameter in a vessel of pure water (Fig. 22). The water will rise in these tubes, highest in the one of small- Fig. 22 24 MOLECULAR PHYSICS est diameter, and least in the one of greatest. With mercury in place of water, the depression will be the greatest in the smallest tube. If two chemically clean glass plates, inclined 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 between the plates, and the water line will be curved as in Fig. 23. These experiments illus- trate the following laws : I. Liquids ascend in tubes when they wet them, that is, ^vhen the surface is concave; and they are depressed when they do not wet them, that is, when the surface is convex. II. For tubes of sTnall diameter, the elevation or de- pression is inversely as the diameter of the tube. 34. 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 may be too large for free capillary action. The moisture then remains at a lower level, where it is needed for the growth of plants. 35. Capillarity related to Surface Tension. The attrac- tion of glass for water is greater than the attraction of water for itself ( 10). When a liquid is thus attracted by a solid, the liquid wets it and rises with a concave MOLECULAR FORCES IN SOLIDS 25 surface upward (Fig. 24). The surface tension in a curved film makes the film contract and produces a pres- sure towards its center of curvature, as shown in the case of the soap bubble ( 31). When the surface of the liquid in the tube is concave, the resultant of this normal pres- sure 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 downward just equals the resultant of the normal forces of the film upward. When the liquid does not wet the tube, the normal pressure of the film 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 36. 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. 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. 26 MOLECULAR PHYSICS 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 gm. 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. 37. 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 in- sures the formation of more. The process of the sep- aration 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- 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 MOLECULAR FORCES IN SOLIDS 27 formed from the vapor of water. They are of various forms, but all hexagonal in outline (Fig. 25). 1 38. Molecular Strength. To tear apart a solid, a stretching force must be applied in excess of the forces holding together the molecules on opposite sides of the Fig. 25 fracture. Tenacity ( 12) is accounted for by this force of cohesion and is a measure of it. Steel has the greatest tensile strength of all metals ; a steel rod 1 sq. in. in cross section requires a stretching force of about 65 tons to break it. For the same cross section, hard drawn copper breaks under a tension of from 23 to 34 tons. The breaking tension varies as the cross section. 39. Molecular Equilibrium. When a semiliquid solid, such as glass at a high temperature, is suddenly chilled, the molecules do not have time to arrange themselves under the cohesive forces acting on them, and the molecular grouping is one of more or less unstable restraint. Prince Rupert drops (Fig. 26) 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 accommodate itself to the dimensions of the outer skin. Fig. 26 1 These figures were made from microphotographs taken by Mr. W. A, Bentley, Jericho, Vt. 28 MOLECULAR PHYSICS 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 bottom, and has been 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. 40. 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 of size or a change of shape. As soon as the distorting force, or stress, has been withdrawn, these bodies recover their initial volume and dimensions. This property of recovery from a strain when the stress is .removed is called elasticity. It is called elasticity of form when a body recovers 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 they recover their former volume when the original pressure is restored. They have no elasticity of form. Some solids, such as shoemaker's wax, 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. 41. Hooke's Law. Solids have a limit to their distor- tion, called the elastic limit, beyond which they yield and are incapable of recovering their form or volume. The QUESTIONS AND PROBLEMS 29 elastic limit of steel is very high ; steel breaks before there is much permanent 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. This relation is known as Hooke's law. Clamp a meter stick to a suitable support (Fig. 27), 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. P. The amount of bend- ing or distortion of the bar is proportional to the weight. 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 Problems 1. When a glass tube 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 can you not write with ink on blotting paper ? 4. Explain the action of blotting paper ; of a towel in drying the wet hands ; of a sp'onge. 5. A soap bubble is filled with air. Is the air inside denser or rarer than the air outside ? 6. If dry wooden wedges are driven into holes or a channel in a stone, and are then wet with water, what is the effect ? 7. Why does an automobile ride easier with pneumatic tires than with solid rubber ones ? 30 MOLECULAR PHYSICS 8. Are the divisions on the scale of a spring balance equal? What law is illustrated? 9. If an iron wire one tenth inch in diameter will safely support 300 pounds, how many pounds will one support one fifth inch in diameter ? 10. If a steel rod 10 ft. long is stretched within its elastic limit 0.15 in. by a certain weight, how much would a rod 20 ft. long be lengthened by the same weight ? By half the weight? CHAPTER III MECHANICS OF FLUIDS I. PRESSURE OF FLUIDS 42. 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 main- 1 tain 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 alid is traversed by both in opposite directions. It will even' flow very slowly down a tortuous channel. At the same time, sealing wax and shoemaker's wax when cold break readily under the blow of a hammer. 43. Viscosity. The resistance of a fluid to flowing under stress is called viscosity. It is due to molecular friction. The slowness with which a fine precipitate, thrown down by chemical action, settles in water is owing to the vis- 31 32 MECHANICS OF FLUIDS cosity of the liquid; and the slow descent of a cloud is 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. 44. Liquids and Gases. Fluids are divided into liquids and gases. Liquids, such as water arid 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 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. 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. 45, Pascal's Principle. A solid transmits pressure only in the direction in which the force acts ; but a fluid trans- mits pressure in every direction. Hence the law: 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 con- taining 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. It was first announced by Pascal in 1653. 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 1658 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. PRESSURE OF FLUIDS 33 Fig. 28 ILLUSTRATIONS. Fit a perforated stopper to an ounce bottle, pref- erably with flat sides, and mounted in a suitable frame (Fig. 28). 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 pressure ap- plied to the plunger will be transmitted to the water as a bursting pressure; and the whole pressure trans- mitted to the inner surface of the bottle will be as much greater than the pressure applied as the area of this surface is greater than that of the end of the plunger. Figure 29 is a form of syringe made of glass ; the hollow sphere at the end has several small openings. Fill with water and apply pres- sure 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 expand equally in all directions, forming a sphere and showing equal pressures in all directions. A large soap bubble shows the same thing. 46. The 'Hydraulic Press. An impor- tant application of Pascal's principle is the hydraulic press. Figure 30 is a section showing the prin- A heavy piston P works water-tight in the larger cyl- inder A, while in the smaller one the piston p is moved up and down A 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 Fig. 30 Fig. 29 cipal parts. 34 MECHANICS OF FLUIDS 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 #, and that of the large one by A, the ratio between the forces acting on the two pistons is JR A = 1P E a #' where D and d are the diameters of the large and small pistons respectively. Figure 31 illustrates a press driven by a belt on the pulley P. F and F are pumps forcing water into the cylinder B. 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 pis- ton is greater than the effort applied to the small one. 47. The Hydraulic Ele- vator. A modern appli- cation of Pascal's prin- ciple is the hydraulic elevator. A simple form is shown in Fig. 32. A long piston P carries the cage A, which runs up Fig. 31 and down between guides and is partly counterbalanced by a weight W. The piston runs in a tube or pit sunk to a PRESSURE OF FLUIDS 35 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 /\-?\ ,~, .-->16d Id we notice that if t is one second in equation (6), then = 1 a ; that is, the distance traversed in the first second is one half the acceleration. But the acceleration is the same as the velocity acquired the first second. Hence s = J v and d\v. Therefore the velocity >9d at the end of the first second on the inclined plane is 2d. Since 35d by equation (5) the velocities are proportional to the time, the succeeding velocities are 4d, 6d, etc. The numbers in the second 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 Fi S- 79 in successive seconds are as the odd numbers 1, 3, 5, etc. The results may be shown graphically as in Fig. 79. 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 ? Compare the result with the record for a mile. 78 MOTION 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 wheels on the track ? 8. If a locomotive driving wheel 2 m. in diameter is making 200 revolutions per minute, what is the greatest linear velocity of a point on its rim ? What is the least ? 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 ? 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 min. ? 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 accelerated motion ? 14. An electric car starting from rest has uniformly accelerated motion for 3 min. At the end of that time its velocity is 27 km. per 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 mo- tion were uniformly accelerated ? CURVILINEAR MOTION 79 II. CURVILINEAR MOTION 94. 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, the same as the path of many comets. So also is the mo- tion of a baseball when batted high in air. The thrown "curved ball," too, illustrates curvilinear motion. When the motion is along a curved line, the direction of motion at any point, as at E (Fig. 80), is that of Fig * 8 ' the line CD, tangent to the curve at the point. This is the same as the direction of the curve at the point. 95. 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 a velocity may remain unchanged in value, it may vary 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 acceleration is constant, the motion is uniform in a circle. Hence, in uniform circular motion there is a con- 80 MOTION stant acceleration directed toward the center of the circle. It is called centripetal acceleration. Uniform circular motion consists of a uniform motion around 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 centripetal acceleration is O jit a = V , . . . . (Equation 7) r . . . 7 7 _ l . square of velocity in circle or centripetal acceleration = -4 y t . radius of circle III, SIMPLE HARMONIC MOTION 96. 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. If the motion returns periodically to the same value and is as often re- versed in direction, it is said to be vibratory. The motion of the earth around the sun is periodic, but not vibratory. A hammock swinging in the wind, the pendulum of a *Let ABC (Fig 81) 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 the motion along the arc is uniform, s = vt. AB is the diago- nal 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 z by equation (6). The two triangles ABE and AB are simi- lar. Hence AB 2 = AExAC. Calling the radius of the circle r and substituting for AB, AE, and AC their values, x2r = at 2 r. Then a = . SIMPLE HARMONIC MOTION 81 clock, a bowed violin string, and the prong of a sounding tuning fork illustrate both periodic and vibratory motion. 97. Simple Harmonic Motion Described. Simple harmonic motion is a name given to all pendular motions of small amplitude. The name appears to be due to the fact that simple musical sounds are caused by bodies vibrating in this manner. Suspend a ball by a long thread and set it swing- ing in a horizontal circle (Fig. 82). Place a white screen back of the ball; standing several feet away and with the eye E on a level with the ball, watch its moving pro- jection on the screen. The eye dis- cerns the Fig. 82 motion to the right and to the left, but not the motion toward the observer and away from him. The apparent motion of the ball, as viewed against the screen, is simple harmonic. Though the ball is moving uniformly around the circle, its projected motion is vibratory. The velocity of this simple harmonic motion is greatest at the middle of its path A G, and decreases to zero at either extremity. Let the circle of Fig. 83 represent the path of the ball, and ABCD, etc., its projection on the screen. When the ball moves along the arc adg, it appears to the observer to move from A through B, C, etc., to (7, where it momentarily comes to rest. It then starts back toward A, at first very slowly, but with increasing velocity until it passes D. Its velocity then decreases, and at A it is again zero, and the motion Fig. 83 82 MOTION reverses. At k and d the ball is moving across the line of sight and the apparent motion on the screen is the fastest. 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 pro- jected motion. The frequency of the vibration is the re- ciprocal of the period. For example, if the period is J a second, the frequency is two complete vibrations a second. This relation finds frequent illustration in musical sounds, where pitch depends on the frequency ; in light, where frequency determines the color ; and in alternating cur- rents 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 - 1. At what speed must a cyclist ride around a circular track one inile in diameter in order to go around it in half an hour ? 2. The circumference of the earth at the equator is about 25,000 mi. What is the linear velocity in feet per second of a point on the equator, owing to the earth's rotation on its axis ? 3. The diameter of the 40th degree parallel of latitude is about 6000 mi. What is the speed in feet per second of a point on this parallel on account of the earth's rotation on its axis? 4. A flywheel 6 ft. in diameter runs at the rate of 60 revolutions per minute. What is the centripetal acceleration in feet per second per second of a point on the outer rim of the wheel ? 5. A locomotive driving wheel 2 m. in diameter makes 200 revo- lutions per minute on a straight track. What is the centripetal ac- celeration in meters per second per second for a point at the instant of contact with the rail ? 6. A locomotive driving wheel 2 m. in diameter makes 200 revo- lutions per minute ; what is the instantaneous centripetal acceleration for the highest point of the wheel? (Note. The radius of rotation for the highest point is at that instant 2 m.) CHAPTER V MECHANICS OF SOLIDS MEASUREMENT OF FORCE 98. Force. The effects of force in producing motion are among our commonest experiences. We drop a knife and it falls by the force of gravity ; a mountain stream rushes downward by reason of the same mysterious force ; the leaves of the trees rustle in the breeze, the branches sway violently in the wind, and their trunks are even twisted off by the force of the tornado ; powder explodes in a rifle and the bullet speeds to its mark; loud thunder shakes the ground and vivid lightning rends a tree or shatters a flagstaff. From such familiar facts is derived the conception that force is anything that produces motion or change of motion in material bodies. 99. 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 downward pull of gravity. The same is true of the met- ric 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 83 84 MECHANICS OF SOLIDS the poles of the earth is greater than at the equator ; it is less on the top of a high mountain than in the neighbor- ing valleys, and still less than at the level of the sea. Gravitational units of force are convenient for the com- mon purposes of life and for the work of the engineer, but they are not suitable for precise measurements. The so-called " absolute" unit of force in the e.g. s. system is the dyne (from the Greek word meaning force} . The dyne is the force which imparts to a gram mass an acceleration equal to one centimeter per second per second. This unit is invariable in value, for it is independent of the variable force of gravitation. It is indispensable in framing the definitions of modern electrical and magnetic units. 100. 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 (the place on the earth's surface is not specified). 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 _i_^ f 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 suffi- ciently 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. 101. How a Force is Measured. The simplest device for measuring a force is the spring balance (Fig. 84). The MEASUREMENT OF FORCE 85 common draw scale is a spring balance graduated in pounds and fractions of a pound. If a weight of 15 lb., for ex- ample, be hung on the spring and the position of the pointer be marked, then any other 15 lb. (f"\ of force will stretch the spring to the same ex- tent in any direction. If a man by pulling in any direction stretches a spring 3 in., and if a weight of 150 pounds also stretches the spring 3 in. the force 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. Flg * 84 102. 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 applica- A ~ ~ ~~ s tion of the force in the direc- tion 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 AB (Fig. 85), will represent 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 2kffm \ ^ at which the force is applied. A ~^o If it is desired to represent graphi- cally the fact that two forces act on a body at the same time, for example, 4 kgm. of force horizontally and 2 kgm. 86 MECHANICS OF SOLIDS of force vertically, two lines are drawn from the point of application A (Fig. 86), one 2 cm. long to the right, and the other 1 cm. long toward theJEbp of the page. The lines AB and A G represent the forces in point of applica- tion, direction, and magnitude, on a scale of 2 kgm. of force to the centimeter. II. COMPOSITION OF FORCES AND OF VELOCITIES 103. 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, 106.) The process of finding the resultant of two or more forces is known as the composition of forces. It will be convenient to consider first the composition of parallel forces, and then that of forces acting at an angle. 104. The Resultant of Parallel Forces. Suspend two draw scales, A and B (Fig. 87), from a suitable support by cords. Attach to them a graduated bar supporting the weight W. Adjust the draw scales and the attached cords so that they are all vertical. Read the scales and note the distances CE and ED ; also compare the value of W with the sum of the readings of A and B. Change the position of the point E and repeat the observations. It will be found in each case that = Hence the will! \1 P CE following principle : Fig. 87 The resultant of two parallel forces in the same direction 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. COMPOSITION OF FORCES AND OF VELOCITIES 87 105. Equilibrium. It will be apparent that the weight IF is equal and opposite to the resultant of the two forces measured by the draw scales A and B (Fig. 87). The three forces produce neither motion nor change of motion and are said to be in equilibrium; each of the forces is equal and opposite to the resultant of the other two and is called their equilibrant. The equilibrium of a body does not mean that its velocity is zero, but that its accel- eration is zero. Rest means zero velocity ; equilibrium, zero acceleration. 106. Parallel Forces in Opposite Directions. If two parallel forces, as A and W (Fig. 87), act in opposite directions, their resultant is their difference (equal and opposite to the force -B), and it acts in the direction of the larger force (in this case downward in the same direc- tion as W). 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. 88 MECHANICS OF SOLIDS 107. The Resultant of Two Forces Acting at an Angle. - Tie together three cords at D (Fig. 88) 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 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 Scale : SO gm. to 1 cm. Fig. 88 lines, on some convenient scale, distances to represent the readings of the draw scales, DP 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 B, and its length on the scale chosen will be found equal to that of Z)C, 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 re- sultant lies between the two ; its position and value may be found by applying the following principle, known as the parallelogram of forces : COMPOSITION OF FORCES AND OF VELOCITIES 89 If two forces are represented by two adjacent sides (DF and DE) of a parallelogram* their resultant is represented by the diagonal (^DCr) of the parallelogram drawn through their common point of application (-0). When the two forces are equal, their resultant by the principle of symmetry lies midway between them. If the two forces are at right angles (Fig. 89), the parallelogram becomes a rectangle and the two forces and their resultant are represented by the three sides of a right triangle, AB, BD, AD. The value of the resultant in this case may Fig. 89 be found by computing the hypotenuse of the triangle. For example, if the forces at right angles are 6 kgm. of force and 8 kgm. of force, their resultant is V6 2 + 8 2 = 10 kgm. of force. 108. Component of a Force in a Given Direction. frequently occurs that if a force produces any motion, must be in a direction other than that of the force itsel For example, suppose force AB (Fig. 90) appl to cause a car to move alon the rails mn. The force evidently produces tw Fig. 90 effects ; it tends to move the car along the rails, and it in- creases the pressure on them. The two effects are pro- duced by the two forces OB and DB respectively. They are therefore the equivalent of AB. The force OB is called the component of AB in the direction of the rails mn, and DB is the component perpendicular 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 90 MECHANICS OF SOLIDS 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 side parallel to the given direction represent* the component sought. EXAMPLE. Let a force of 200 Ib. be applied to a truck, as AB in Fig. 90 ; and let it act at an angle of 30 with the horizontal. Find the horizontal component pushing the truck forward. Construct a parallelogram (see Appendix) with the angle ABC equal to 30 and measure the side CB. Or, remembering that the side opposite an angle of 30 is half the hypotenuse, it follows that AC is 100 Ib. of force, and the. third side of the right triangle is then CB = - 100 2 = 173.2 Ib. of force. The kite and the sailboat are two familiar illustrations of the prin- ciple. In the case of the kite, the forces acting are the weight of the kite AB (Fig. 91), the pull of the string AC, and the force of the wind. The force of the wind may be resolved into two- components, one perpendicular to the face of the kite HK, and the other parallel to the face. If the component per- pendicular to the face equals AD, the resultant of AB and AC, then the kite will be at rest ; if greater, then the kite will move upward ; if less, the kite 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. 92, BS represents the sail, and AB the direc- tion and force of the wind. This force may be re- solved into two rectangular components, CB and DB, of which CB represents the intensity of the force that drives the boat forward. Fig. 92 COMPOSITION OF FORCES AND OF VELOCITIES 91 109. 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- 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. 93) 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 parallelogram ABDC, and draw the diago- nal AD through the point A common to the two component velocities. A D repre- sents the actual velocity of the boat; its length on the same scale as that of the other lines is 5.83: The resultant velocity Fig< 93 is therefore 5.83 miles an hour in the direction AD. When the angle between the components is aright 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 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- 92 MECHANICS OF SOLIDS 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, 108. Problems NOTE. Solve graphically the problems involving forces or velocities at an angle. Consult Appendix I for methods of drawing. 1. A body is acted on by two parallel forces of 20 and 30 Ib. of force in the same direction ; their points of application are 60 in. apart. Find the value of the resultant and the distance of its point of appli- cation from either force. SUGGESTION. Let x be the distance of the point of application of the re- sultant from that of the force 20. Then 60 x is its distance from the point of application of the force 30, and by 104, = 60 ~ x . oO X 2. A weight of 210 Ib. is carried at the middle of a bar 6 ft. long by a boy at one end and a man at the other side of the middle. Where must the man take hold so that he shall carry twice as much as the boy ? 3. A horse and a colt are hitched side by side to a loaded 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 three pounds of force pulled by the horse, the doubletree being 40 in. long? 4. Three parallel forces are acting on a bar 104 cm. long. Two of them, 500 and 800 dynes respectively, are applied at the ends. What must be the third force and where must it be applied so that the bar may remain at rest and the three forces be in equilibrium ? 5. Resolve a force of 100 dynes into two parallel forces with their points of application 20 and 30 cm., respectively, from that of the original force. 6. Two parallel forces of 100 and 150 dynes, respectively, have their points of application 50 cm. apart. What third parallel force will produce equilibrium, and where must it be applied so that the three points of application are in the same straight line ? 7. Two forces, 30 and 40 grams of force, act at an angle of 60. Find the resultant. 8. A train is running with a speed of 30 mi. an hour. A package is thrown perpendicularly from it with a velocity of 20 ft. a second. What is the velocity of the package with respect to the ground? NEWTON'S LAWS OF MOTION 93 9. 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 is blowing 12 mi. an hour, what is its component perpen- dicular to the sail ? 10. A mass of 30 gm. is suspended by a string. It is pulled aside by a horizontal force of 17.32 gin. of force, and the string then makes an angle of 30 with the vertical (Fig. 94). Find the tension in the string. (Note that in a right triangle with a 30 angle, the side opposite this angle is half the hypotenuse.) 11. Horses attached to a car pull at an angle of 30 with the track and with a force of 1200 Ib. What is the force moving the car? 12. A cord supporting a picture weighing 15 Ib. passes over a knob so that the angle between the two parts of the cord is 60. What is the tension in the cord ? III. NEWTON'S LAWS OF MOTION 110. Momentum. So far we have considered motion without reference to the mass moved, and without con- sidering the relation between force on the one hand and the moving mass and its velocity on the other. 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 1 the linear velocity of a moving body. Momentum = mass x velocity, or M = mv. (Equation 8) In the e.g. s. system, the unit of momentum is the mo- mentum of a mass of 1 gm. 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. 94 MECHANICS OF SOLIDS 111. Impulse. Suppose a ball of 10 gm. 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 weigh- ing $0 kgm. moves. .at the rate of 10 cm. per second, its momentum is also 500,000 units. But the ball has acquired its 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, because 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 con- tinues to act. In estimating the effect of a force, the time element and the magnitude of the force are equally im- portant. The term impulse takes both into account. 112. The Laws of Motion. The laws of motion, first enunciated by Sir Isaac Newton, are to be regarded as physical axioms, incapable of rigorous experimental proof. They must be considered as resting on convictions drawn from observations and experiment in the domain of physics and astronomy. The results derived from their application have so far been found to be invariably true. They may be stated as follows : I. Every body continues in Us state of rest or of uni- form motion in a straight line, except in so far as it may be compelled by applied force to change that state. II. Change of momentum is proportional to the im- pulse which produces it, and takes place in the direction in which the force acts. 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 Prmcipia, 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. NEWTON'S LAWS OF MOTION 95 III. To every action there is always an equal and con- trary reaction; or the mutual actions of two bodies are always equal and oppositely directed. 113. Discussion of the First Law. This law is known as the law of inertia ( 7), since it asserts that a body per- sists in its condition of either rest or 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 proportion to its mass. Hence the term mass is now commonly used to denote the measure of the body's inertia ( 9). From this law we also derive the Newtonian definition of force, for the law asserts that force is the sole cause of change of motion. 114. Discussion of the Second Law. The first law teaches that a change of motion is due to impressed force. The second law points out, in the first place, what the measure of a force is. It was restated by Maxwell 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 momentum (mass x velocity) = impulse (force x time). Expressed in symbols, mvft. . . . (Equation 9) 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 C change of momentum. Force is therefore measured by the rate of change of momentum. Since - is the rate of change t 96 MECHANICS OF SOLIDS of velocity, or the acceleration a (see equation 5), we may write f=ma. . . . (Equation 10) 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 ( 99). This law teaches, in the second place, 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 ( 107). 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. 95). When the hammer falls and strikes the spring, it projects 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. 115. Discussion of the Third Law. 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 directions. The stress in a stretched elastic cord pulls the two bodies to which it is attached equally in opposite directions ; the stress in a NEWTON'S LAWS OF MOTION 97 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 di- rections. The same is obviously true if two men pull at the ends of 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. 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. 96). When one half of this bent sheet is exposed to the air cur- rent, 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 Fig. 96 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. 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 98 MECHANICS OF SOLIDS gained by the one is equal to that lost by the other, or the momenta in opposite directions are the same. Problems 1. Why will not the same impulse impart to a 2-lb. ball the same velocity as it does to a 1-lb. ball ? 2. What is the momentum of a mass of 250 gm. moving with a velocity of 75 cm. per second ? 3. Calculate the ratio of the momentum of a ball whose mass is 5 Ib. and speed 1500 ft. per second to that of a 50-lb. ball moving with a speed of 1800 ft. per minute. 4. What velocity must be given to a mass of 25 kgm. that it may have the same momentum as a 75-kgm. ball moving with a velocity of 1500 cm. per second ? 5. A 10-lb. gun is loaded with a half-ounce ball. When fired the ball has a velocity of 1800 ft. per second. What is the velocity of recoil of the gun ? 6. Two balls have equal momenta. The first has a mass of 200 gm. and a velocity of 20 m. per second ; the second has a velocity of 600 m. per minute ; what is its mass ? 7. An unbalanced force acts on a mass of 100 gm. for 5 seconds, and imparts to it a velocity of 2.5 cm. per second. What is the value of the force in dynes ( 99) ? 8. What is the acceleration when a force of 50 dynes acts on a mass of 8 gm.? How far will the mass of 8 gm. move in 4 sec. V 9. An unbalanced force of 60 dynes acts on a body for 1 min. and gives to it a velocity of 1200 cm. per second. What is the mass of the body ? 10. Two grams of force act continuously on a mass of 49 gm. for 10 seconds. Find the velocity acquired and the space traversed. (To apply equations (9) and (10) the force must be expressed in dynes.) IV. GRAVITATION 116. Weight. The attraction of the earth for all bodies is called gravity. The weight of a body is the measure of this GRAVITATION 99 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 #nd denoted by g. If we represent the weight of a body by w and its mass by m, by equation (10), w = mg. From this it ap- pears that the weight of a body is proportional to its mass, and that the ratio of the weights of two bodies at any place 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 bal- ance shows equality of weights, it shows also equal- ity of masses. 117. 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 deter- mined by suspending a weight by a cord passing through the point. The cord is called a plumb line (Fig. 97), and its direction is a vertical line. A fjf plane or line perpendicular to a plumb line is said p . 97 to be horizontal. Vertical lines drawn through neighboring points may be considered parallel without sensible error. 118. Center of Gravity. A body is conceived to be composed of an indefinitely large number of particles, 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 100 MECHANICS OF SOLIDS center of gravity depends on its geometrical 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 of its diagonals ; (5) of a cylinder or a cylindrical pipe, the middle point of its axis. It is necessary to guard against the idea that the force of gravity on a body acts at its center of 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. 119. Law of Universal Gravitation. It had occurred to Galileo and other early philosophers that the attraction of gravity extends beyond the earth's surface, but it re- mained 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 pro- portional to the square of the distance between them. 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 dis- tance between them is the distance between their centers. Calculationsunade to find the centripetal acceleration of the moon in its orbit show that it is attracted to the earth GRAVITATION 101 with a force which follows the law of universal gravita- tion. 1 120. Variation of Weight. Since the earth is flattened at the poles, it follows from the law of gravitation that the ac- celeration of gravity, and the weight of any body, increase in going from the equator toward the poles. If the earth were a uniform sphere and stationary, the value of g would be the same 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 5- j-^ the acceleration of gravity g. Since 289 is the square of 17, and the 'centripetal acceleration varies as the square of the velocity (95), 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, or 32.16 feet, per sec- ond per second. 121. 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 center of gravity must therefore fall within its base of support. If this vertical line falls outside the base, the weight of the body and the reaction of the plane form a couple ( 106), and the body overturns. The three kinds of equilibrium are, (1) stable, for any displacement which causes the center of gravity to rise ; iCarhart's College Physics, 58. 102 MECHANICS OF SOLIDS (2) unstable, for any displacement which causes the center 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. 98). Tip the flask over ; after a few os- cillations it will return to an upright position. Repeat the experiment with a similar empty flask ; it will not stand up, but will rest in any posi- tion on its side and with the top on the table. The loaded flask cannot be tilted over Fig 98 without raising its center of gravity ; in a vertical position it is there- fore stable and when tipped over, unstable, for it returns to a ver- tical position. For the empty flask, its center of gravity is lower when it lies on its side than when it is erect. Rolling it around does not change the height of its center of gravity and its equilibrium is thus neutral. The three funnels of Fig. 99 Fig. 99 illustrate the three kinds of equilibrium 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 equilibrium. Any such body may become stable by attaching weights to it in such a manner as to lower the center of grav- Fig. 100 ity below the supporting point (Fig. 100). 122. Stability. Stability is the state of being firm or stable. The higher the center of gravity of a body must be QUESTIONS AND PROBLEMS 103 lifted to put the body in unstable equilibrium or to over- turn 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 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- B ,' 1 bility is still greater. Let / Fig. 101 represent a brick ly-; / ing on its narrow side in A and standing on end in B. In both cases to overturn it Fig. 1 01 its center of gravity c is lifted to the same height, but the vertical distance 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. 102 ? 3. Can you devise a method of find- ing where the center of gravity of a uniform triangle is ? Fig. 102 104 MECHANICS OF SOLIDS 4. Show by a figure why a ball rolls down hill the faster, the steeper the hill. 5. Why are low wagons better adapted for use on hillsides than high ones ? 6. If a body at the equator weighs 9781 gm. on a spring balance, what would it weigh at the north pole ? (Weight is proportional to <70 7. If a body weighs 16 kgm. at sea level on a spring balance, how many grams less will it weigh on the top of a mountain 2 mi. above sea level, if the radius of the earth is 4000 mi.? (The value of g is inversely as the square of the distance from the center of the earth.) 8. If the acceleration of gravity g is 32.2 ft. per second per second on the earth's surface, what is it at a distance equal to that of the moon, if the earth's radius is 4000 mi. and the distance of the moon 240,000 mi. ? 9. If the acceleration of gravity on the earth is 980 cm. per second per second, what is it on Mars, the radius of the earth being 4000 mi., that of Mars 2000 mi., and the mass of the earth nine times that of Mars ? (The accelerations are directly proportional to the masses and inversely proportional to the squares of the radii.) 10. If a body weighs 45 Ib. on a spring balance on the earth, what would it weigh on Mars, the radius of the earth being 4000 mi., that of Mars 2000 mi., and the mass of the earth nine times that of Mars ? 11. How far above the earth's surface would a pound ball have to be taken to reduce its weight to 1 oz., if the earth's radius is 4000 mi.? V. FALLING BODIES 123. Rate at which Different Bodies Fall. It is a familiar fact that heavy bodies, such as a stone or a piece of iron, 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- FALLING BODIES 105 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. 103) in the presence of professors and students of the uni- versity 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 to- gether 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. 124. 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. 104). Hold the tube in a vertical position and suddenly invert 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 Fig. 103 106 MECHANICS OF SOLIDS Fig. 104 body will now fall as fast as the heavier one. This experiment, known as the " Guinea and Feather Tube," was first performed by Newton. It demonstrates that if the resistance of the air were wholly removed, all bodies at the same place would fall with the same acceleration. An interesting modification of the Newtonian ex- periment is the following : Cut a round piece of paper slightly smaller than a cent and drop the cent and the paper side by side ; the cent will reach the floor 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 ve- locity. A cloud floats, not because it is lighter than the atmosphere, for it is actu- ally heavier, but because the surface friction 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. Such is the explanation of the Staubbach fall (dust brook) at Lauterbrunnen in Switzerland (Fig. 105). The precipice is 300 m. high, and the fall viewed from its face resembles a magni- ficent transparent veil, kept in movement by currents of air. Fig- 105 125. Laws of Falling Bodies. Galileo verified the fol- lowing laws of falling bodies : FALLING BODIES 107 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, 91 and 92. 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 marble will fall freely under gravity and the acceleration will become g ( 120). 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 s = | fft 2 . . . (Equation 12) 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. 126. 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 108 MECHANICS OF SOLIDS the quotient of v by #, or time of ascent = veloci ^ o acceleration of 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 -^g^, 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 starting 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 In the following problems, unless otherwise stated in the problem, g is to be taken as 980 cm. or 32 ft. per second per second. 1. A brick falls to the ground from the top of a chimney 64 ft. high. What will be the time of descent, and with what velocity will it strike the ground? (Use equation (12) for the time, and equation (11) for the velocity.) 2. A stone dropped from a bridge strikes the water in 3 sec. after leaving the hand. Find the height of the bridge above the water. 3. If a stone thrown vertically upward returns to the ground in 4 sec., how high does it ascend ? v 4. A ball is thrown over a tree 100 ft. high. How long before it will return to the ground ? 5. A ball fired horizontally reaches the ground in 4 sec. What was the height of the point from which it was fired ( 114) ? 6. A cannon ball is fired horizontally from a fort at an elevation of 78.4 m. above the level of the neighboring sea. How many sec- onds before it will strike the water ? 7. An iron ball was dropped from an aeroplane moving eastward at the rate of 45 mi. an hour. It reached the ground 528 ft. east of CENTRIPETAL AND CENTRIFUGAL FORCE 109 the vertical line through the point at which it was dropped. What was the elevation of the aeroplane? 8. With what velocity must a ball be fired upward to rise to the height of the Washington Monument, 555 ft., and how long will it be in the air ? 9. An inclined plane 40 ft. long is elevated at one end 2 ft. In what time will a ball roll down it, neglecting all resistance ? (The acceleration down the plane is the component of g parallel to the in- cline.) 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 127. Definition of Centripetal and Centrifugal Force. Attach a ball to a cord and whirl it around x 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. The constant pull which deflects the body from a rectilinear path and compels it to move in a curvilinear one is the centrip- etal force. The resistance which a body offers, on account of its inertia, to deflection from a straight line is the centrif- ugal 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. 128. Value of Either Force. The centripetal accelera- a tion for uniform circular motion ( 95) is a = , where v 110 MECHANICS OF SOLIDS is the uniform velocity in the circle, and r is the radius. Further, in 114 the relation between force and accelera- tion was found to be, 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) r 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 ; / is then in dynes. To obtain / in grams of force, divide by 980 ( 100). 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. For example : If a mass of 200 gm. is attached to a cord 1 m. long and is made to revolve with a velocity of 140 cm. per second, the tension in the cord is 200 >< 14 2 = 39,200 dynes = = 40 grams of 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 /= /^ * =25 pounds O X o-ij.^ of force. 129. Illustrations of Centrifugal Force. Water adhering to the surface of a grindstone leaves the stone as soon as the centrif- ugal 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 centrip- THE PENDULUM 111 etal force is insufficient 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 subject 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 insufficient, " skidding " takes place. Centrifugal machines are used in sugar refineries to separate sugar crystals from the syrup, and in dyeworks and laundries to dry yarn and wet clothes by whirling them rapidly in a large cylinder with openings in the side. Honey is extracted from the comb in a similar way. When light and heavy particles are whirled together, the heav- ier 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 milk is whirled in a dairy separator, the cream and the milk form dis- tinct layers and collect in separate chambers. When a spherical vessel containing some mercury and water is rapidly whirled on its axis (Fig. 106), both the mercury and the water rise and form separate bands as far as possible from the axis of rotation, the mercury , ., J Fig. 106 outside. The figure of the earth is an oblate spheroid, flattened at the poles. This flattening was doubtless caused by the centrifugal force of rota- tion when the earth was in a plastic state, before it reached its present more rigid condition. Centrifugal force causes the water of the ocean to flow toward the equatorial regions, exposing lands at the north which would be covered with water if the earth were stationary. VII. THE PENDULUM 130. 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 defined as a mate- rial particle without size suspended by a cord without weight. A small lead ball suspended by a long thread 112 MECHANICS OF SOLIDS without sensible 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 excursions on either side 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. 131. The Motion of a Pendulum. AN in Fig. 107 is a nearly simple pendulum with the ball at N. When the ball is drawn aside to the position 13, its weight, represented by BG, may be resolved into two components, BD in the direction of the thread, and EC 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 com- ponent BC becomes smaller and smaller and vanishes at N, where the whole weight of the ball is in the direction of the thread. In falling from B to JV, the ball moves in the arc of a circle under the influence of a force which is greatest at B and becomes zero at N. The motion is there- fore 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 con- tinues to oscillate with a periodic and pendular motion. Fig. 107 THE PENDULUM 113 132. 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 N to B (Fig. 108) and back to N again. A complete or double vibration is the motion between two suc- cessive passages 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 vibra- tion is the time consumed in making a Flg * ! complete or double vibration. The amplitude is the arc SNor the angle BAN. 133. Laws of the Pendulum. The following are the laws of a simple pendulum : 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. 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 a pendulum, made when he was about twenty years of age. This was long before their theoretical investigation. 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 re- sult will hold even when the amplitudes are so small that the vibra- tions can only be observed with a telescope. 114 MECHANICS OF SOLIDS To illustrate law II. Mount three pendulums (Fig. 109), making the lengths 1 m., \ m., and \ m. respectively. Observe the period of a single vibration for each. They will be 1 sec., 'F'p'r'T* \ 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. 134. 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 hori- zontal axis through the staple (Fig. 110). With a ball and a thread make a simple pendulum that will vibrate in the same period as the meter stick. Measure the length of this pendulum and lay it off on the meter, beginning at the staple. 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 Fig 110 axis of vibration is called the center of oscillation. The distance between the center of suspension and the center of oscillation is the length of the equivalent simple pendulum that vibrates in the same period as the com- pound pendulum. The centers of suspension and of os- cillation are interchangeable without change of period. 135. 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 Fig. 109 THE PENDULUM 115 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 swinging without jar. A base- ball 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. 136. Application of the Pendulum. Gali- leo's discovery suggested the use of the pendulum as a timekeeper. In the com- mon clock the oscillations of the pendu- lum 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. 111). The pendulum rod, passing between the prongs of a fork a, communicates its mo- tion to an axis carrying the escapement, which terminates in two pallets n and m. These pallets engage alternately with the teeth of the escapement wheel R, one tooth of the wheel escaping from a pallet every double vibration of the pendulum, ment 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. Fig. 1 1 1 The escape- 137. Seconds Pendulum. A seconds pendulum is one making a single vibration in a second. Its length in New 116 MECHANICS OF SOLIDS York is 99.31 cm. This is the length of the equivalent simple pendulum vibrating seconds. The value of gravity g increases from the equator to the poles, and the length of the seconds pendulum increases in the same proportion. Questions and Problems 1. Why can a boy throw a stone so much farther with a sling than without it ? 2. Why can an athlete throw a hammer so much farther than he can " put the shot " of the same mass V 3. Why is the outer rail on a railway curve elevated above the inner one ? 4. Why does a bicycle rider lean inward when running round a curve ? 5. A string attached to a mass of 100 gm. broke when the mass was whirled about the hand at a distance of 1 m., and at the rate of 10 revolutions in 3 seconds. Compute the breaking force in dynes. 6. A mass of 50 gm. is connected to a fixed point by a string 2 m. long, and is whirled round in a circle once in 3 seconds. Find the tension in the string in dynes ; also in grams of force. 7. A ball was mounted to swing as a conical pendulum (Fig. 112) ; its mass was 2 kgm., its distance CA from the center of its circular path was 50 cm., and it made 10 revolutions in 5 seconds. What horizontal force in grams would be necessary to hold the ball out at A if it were not revolving ? 8. Two pendulums of the same length have different bobs, one of lead, and the other of aluminum. Will their periods be the same ? Why? Fig. 112 CHAPTER VI MECHANICAL WORK I. WORK AND ENERGY 138. Work. A man does work in climbing a hill by lifting himself against the pull of gravity ; a horse does work in drawing a wagon up an inclined roadway ; a locomotive does work in hauling a train on the level against frictional resistances ; gravity does work against the inertia of the mass when it causes the weight of a pile driver (Fig. 113) to descend with increasing velocity ; steam does work on the piston of a steam engine and moves it by pressure against a resistance ; the electric current does work by means of a motor when it drives an air compressor on an electric car and forces air into a compression tank. Whenever an agent exerts force on a body and causes the point of application to move in the direction of the force, the agent is said to do mechanical work. Unless the point of ap- plication of the force has a com- ponent of motion in the direction in which the force acts, no work in a physical sense is done. The columns in a modern steel building do no work, though they sustain great weight ; the pillars supporting a pediment over a portico do no work ; a person holding a weight suffers 117 Fig. 113 118 MECHANICAL WORK fatigue, but does no work in the sense in which this word is used in physics, where it is employed to describe the result and not the effort made. Work is the act of effecting a change in the state of a sys- tem against a resistance which opposes the change. 139. Measure of Mechanical Work. Mechanical work is measured by the product of the force and the displace- ment of its point of application in the direction in which the force acts, or work = force x displacement. In symbols w =/ x * . . . . (Equation 15) Since force is equal to the product of mass and accelera- tion ( 114), TF= ma x s. . . (Equation 16) 140. Units of Work. Three units of work are in com- mon use : 1. The foot pound, or the work done by a pound of force working through a distance of one foot. 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. 2. The kilogram meter, or the work done by a kilogram of force working through a distance of one meter. It is the gravitational unit of work in the metric system, and varies in the same manner as the foot pound. 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. s. system and is invariable. Since a gram of force is equal to 980 dynes ( 100), if 1 The erg is from the Greek word meaning work. WORK AND ENERGY 119 a gram mass be lifted vertically one centimeter, the work done against gravity is 980 erg^. Hence one kilogram meter is equal to 980 x 1000 x 100 = 98,000,000 ergs. The mass of a "nickel" is 5 gm. 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. 1 Its value is 1 joule = 10 7 ergjfz 10,000,000 ei Expressed in this larger unit, the work done in lifting the " nickel " is 0.245 joule. 141. 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 coal is lifted by a steam engine out of a coal mine through a vertical shaft 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 coal will be lifted in half the time ; but the work done against gravity re- mains 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 first. Time is an important element in comparing the capacities of agents to do work. Such a comparison is made by measuring the power of an agent. Power is the time rate of doing work^ or work f x 8 ,^ power == -7 = - . . (Equation 17) The unit of power in common use among American and 1 From the noted English investigator Joule. 120 MECHANICAL WORK English engineers is the horse power (H. P.) ; it is the rate of working equal to 33,000 ft. Ib. per minute, or 550 ft. Ib. per second. Hence H. P. = {*' . . (Equation 18) oou x c, in which /is in pounds of force, s in feet, and t in seconds. In the c.g.s. system, the unit of power is the watt. 1 It is the rate of working equal to one joule per second. A kilowatt (K.W.) is 1000 watts. Hence In equation (19) /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 dynamo electric generators is now universally expressed in kilowatts ; the steam en- gines 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. 142. 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 1 From the eminent English engineer, James Watt. WORK AND ENERGY 121 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. Energy is the capacity for doing work. It is therefore measured in the same units as work. 143. 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. The storage of ^nergy is seen also in the lifted weight of the pile drives, the coiled-spring of the clock, the bent bow of the archer, the pent up waters behind a dam, the chemical changes in a charged storage battery, and the mixed charge oK gasoline vapor and air in the cylinder of a gas engine. In all such cases of the storage of energy a stress is present. The compressed air pushes outward in the air gun ; gravity tugs at the lifted weight ; the spring tends to uncoil in the clock ; the bent bow strives 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 is called energy of stress, or, more com- monly, potential energy. The energy of an elevated body, of bending, twisting, deformation, of chemical separation, and of the stress in a magnetic field are all examples of potential energy. 144. Kinetic Energy. The energy which a body has in consequence of its motion is known as kinetic energy. The descending hammer forces the nail into the wood; the rushing torrent carries away bridges and overturns buildings ; the swift cannon ball, by virtue of its high 122 MECHANICAL WORK speed, demolishes fortifications or pierces the harveyized steel armor of a battleship ; the energy stored in the massive rotating flywheel keeps the engine running and does work after the steam is shut off. Kinetic energy must not be confused with force. A mass of moving matter carries with it kinetic energy, but it exerts no force until it encounters resistance. Energy is then transferred to the opposing body, and force is ex- erted only during the transfer. 145. 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. =/ x *. . . (Equation 20) The potential energy of a body of water, for example, of weight w and at an elevation A, is wh gravitational units. For kinetic energy, suppose a force / to act on a mass m for a period of time t ; the measure of the effect is the impulse ft ( 111). If the velocity acquired from rest in the time t is v, the momentum produced is mv. By the second law of motion impulse equals the momentum im- parted. Therefore ft = mv (a) A constant force applied to a body gives rise to uni- formly accelerated motion ; and if the body starts from rest, the mean velocity is J v. It is also the space traversed divided by the time, or -. Hence, equating the two ex- t pressions for the mean velocity, WORK AND ENERGY 123 Multiply (a) and (6) together, member by member, and the result is fs = mv*. . . (Equation 21) But/s measures the work done by the force /on the mass m to give to it the velocity v, while working through the distance s; and since the kinetic energy acquired by a body is measured by the work done on it to give it motion, it, follows that the energy of the mass m moving with the velocity v is \ mv 2 , or K. E. = %mv*. . . (Equation 22) If m is expressed in grams and v in centimeters per second, the result is in ergs. To reduce to gram centi- meters, divide by the value of g in centimeters per second per second, or 980. 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 feet per second per second, or 32.2. 146. 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 when the pendulum reverses its motion. All physical processes involve energy changes, and such changes are in ceaseless progress. 124 MECHANICAL WORK 147. 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 observations show that it is as impossible to create energy as it is to create matter. So the law of conservation of energy means that no energy is created and none destroyed by the action of forces we know anything about. 148. 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. 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. Why will a cord supporting a weight generally break if the weight be lifted and then let fall ? 2. A stick resting across two blocks may be strong enough to bear your weight, but will break if you jump on it. Explain. 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. QUESTIONS AND PROBLEMS 125 3. Lake Tahoe, in the Sierra Nevada, is at an elevation of 6225 ft. above the sea. Account for the energy stored there in the water. 4. In what form is the energy for driving a steamship taken on board? Is the energy driving a sailing vessel potential or kinetic ? In what form is it supplied to an automobile ; to an aeroplane ; to a man? 5. How much work is done in lifting a stone weighing 100 kgm. to the top of a building 20 m. high? What is its potential energy at the top ? 6. A rectangular marble slab 6 ft. 3 in. long, 3 in. thick, and weighing 500 Ib. lies on a level floor. How much work must be done to set it vertically on end ? 7. A baseball whose mass is 150 grn. is moving with a velocity of 5000 cm. per second. What is its kinetic energy in ergs ? How much work would be done in stopping it? 8. What is the kinetic energy of a 5-gm. bullet when fired with a velocity of 300 m. per second? 9. A force of 200 dynes moves a mass of 100 gm. through a dis- tance of 50 cm. in 10 sec. How much work is done ? 10. A pull of 50 Ib. moves a 100-lb. truck through a distance of 200 ft. How much work in foot pounds is done ? 11. A load weighing 2 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 against gravity ? 12. How much work can a 40 H. P. engine do in an hour ? How much coal can it lift out of a mine 400 ft. deep in 10 hrs. ? 13. A thousand-barrel tank at a mean elevation of 50 ft. is filled with water. How much work was done in filling it, assuming a barrel of water to weigh 260 Ib. ? How long would it take a motor, working at a 2 H. P. rate, to pump it full ? 14. An electric motor rated at 100 K. W. is used to operate a pump. The water has to be lifted to a mean height of 100 m. How many liters can the motor pump in an hour? 15. What is the power of an agent that lifts 1000 kgm. 10 m. high in 10 min. ? Express the result in K. W. 16. Express in joules the work done by 100 kgm. of force in moving a mass of 100 kgm. through a distance of 100 m. in the direction of the force. 126 MECHANICAL WORK 17. The average pressure of steam on the piston of a steam engine is 120 Ib. of force per square inch. The area of the piston is 50 sq. in. The piston travels 30 in. during one complete revolution of the fly- wheel, and the flywheel makes 220 revolutions per minute. What is the H.P. of the engine? 18. A railway train weighs 250 tons, and the resistance to its motion on a level track is 15 Ib. of force per ton. What H. P. must the locomotive develop to maintain a speed of 40 mi. an hour on the level? 19. How many H. P. are transmitted by a rope passing over a wheel 33 ft. in circumference and making one revolution per second, the tension in the rope being 100 Ib. of force ? 20. A ball whose mass is 100 gm. is struck with a club and is given a velocity of 40 m. per second. How much energy is imparted by the blow? 21. A train weighing 150 tons and running at the rate of 30 mi. an hour is brought to rest by the air brakes within a distance of 500 ft. Find the force of the brakes. 22. A constant resistance of 1000 dynes is applied to a body of 200 gm. mass, moving with a velocity of 6 m. per minute and brings it to rest. How far did the body move after the resistance was ap- plied? II. MACHINES 149. What a Machine is. A machine is a device de- signed to change the direction or the magnitude of a force required to do useful work, or one to transform and transfer energy. ILLUSTRATIONS. By the use of a single pulley, the direction of the applied force may be changed, so as to lift a weight, for example, while the force acts in any desired direction. A water wheel trans- forms the potential and kinetic energy of falling water into mechan- ical energy represented 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 mechanical work. MACHINES 127 150. 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 ap- plied energy may be dissipated as heat or may not appear in mechanical form. A machine can never produce an in- crease of energy so as to ^ive out more than it receives. Denote the applied force, or effort, by E and the resist- ance by M, 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) 151. 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 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- pendent of the area of the surfaces in contact within certain limits, and ( a' 288 320 341.3 384 426 287.3 322.5 341.7 383.6 430 hfcr iTD 9 Ifto 1 > the lips of the player act as reeds, as in the trumpet, ^ | trombone (Fig. 173), the French horn, and the cornet. Fig. 173 172 SOUND Wind instruments may be classed as open or stopped pipes, according as the end remote from the mouthpiece is open or closed. 216. Fundamental of a Closed Pipe. Let the tall jar of Fig. 174 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 Z>, 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 bottom. 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 in- stant when the prong begins to move from b back to a, and to send a rarefaction down AB. This in turn will run down the tube and back, as the prong completes its vibration; the co-vibration Fig. 174 is then repeated 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 tim,es the length of the pipe. 217. 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. 175). Pour in them melted paraffin to close the bottom. A musical note may be pro- duced 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 reinforcement the frequency Fig. 175 VIBRATION OF AIR IN PIPES 173 corresponding to its own rate. Hence the pitch may be varied by pour- ing in water. Adjust all the tubes with water until they give the eight notes of the major diatonic scale. The measured lengths of the col- umns of air will be found to be nearly as 1, f, f , , f, f, T 8 3 , |. The notes emitted have the frequencies 1, f, |, f, |, |, Y, 2 ( 204). Hence, The frequency of a vibrating column of air is inversely as its length. This is the principle employed in playing the trombone. Blow gently across the end of an open tube 30 cm. long and about 2 cm. in diameter and 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 obtained 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. 218. State of the Air in a Sounding Pipe. Em- ploying an open organ pipe, preferably with one glass side (Fig. 176), 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 at the middle. There is, therefore, a node at the middle of an open pipe. A node is a place of least motion and greatest change of density; an antinode is a place of greatest motion and least change of density. The closed end of a pipe is necessarily a node, and the Fig. 176 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 174 SOUND the stopped pipe, there is a node at the closed end and an antinode at the other end. 219. 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, &> etc. times that of the fundamental. In stopped pipes only those overtones are possible whose frequencies are 3, J, 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. XL GRAPHIC AND OPTICAL METHODS 220. Kecord of Vibrations. Graphic methods of study- ing sound are of service in determining the fre- quency of vibration. Figure 177 shows a practical device for this purpose. A sheet 'KULW^S of P a P er is wrapped s== ^^5 r \ '"-'1P^a around a metal cylin- der, and is then smoked with lampblack. A Fig 177 large fork is securely GEAPHIC AND OPTICAL METHODS 175 mounted, so that a light style attached to one prong touches the paper lightly. The cylinder is 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 between successive marks made by the spark is equal to the fre- quency of the fork. 221. Manometric Flames. A square box with mirror faces is mounted so as to turn around a vertical axis (Fig. 178). In front of the revolving mir- rors is supported a short cylinder, which is divided into two shallow chambers by a partition of gold- beater's skin or thin rubber. Illuminat- ing gas is admitted to the compartment on the right through the tube with a stop- cock, and burns at the small gas jet on Ri v \ 7R the little tube run- ning into this same compartment. The speaking tube is connected to the compartment on the other side of the flexible partition. When the mirrors are turned by twirling with the thumb and finger the milled head at the top, the image 176 SOUND Fig. 179 of the gas jet is drawn out into a smooth band of light. Any pure' tone at the mouthpiece produces alternate com- pressions and rarefactions in both chambers separated by the membrane, and these aid and retard the flow 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 ex- amined by reflection from t he rotating mirrors, its image is a serrated band (Fig. 179). Koenig fitted three of these little cap- sules with jets to the side of an open organ pipe (Fig. 180), the membrane on the inner side of the gas chamber form- ing part of the wall of the pipe. When the pipe is blown so as to sound its funda- mental tone, the middle point is a node with the greatest 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. 179. 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 Fig- QUESTIONS AND PROBLEMS 177 tongues of flame 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 over- tone are produced at the same time. This same figure may be obtained by singing into the mouthpiece or funnel of Fig. 178 the vowel sound o on the note -B, showing that this vowel sound is composed of a fundamental and its octave. 222. Kundt's Dust Tube. The division of a resonant pipe into segments may be beautifully shown by means Fig. 181 of a glass tube about 2 cm. in diameter and 40 cm. long. One end is closed, and a common whistle is attached to the other (Fig. 181). Within the tube is evenly sifted a little dry cork dust, or amorphous silica. When the whis- tle is blown, the powder is caught up by the moving air at the antinodes, and settles down in small circles at the nodes. At the same time, between the nodes it is divided into thin, airy segments, with vertical divisions, the agita- tion being sufficient to support the dust in opposition to gravity. The subdivision changes when the blast of air is increased to give overtones. 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 pitch of the sound made by pouring water into a tall cylindrical jar rise as the jar fills? 178 SOUND 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 the 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 ( 203) ? 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 1 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 is 20 C. 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 wil) be the frequency of its first overtone ? i CHAPTER VIII LIGHT I. NATURE AND TRANSMISSION OF LIGHT 223. 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. Physicists have agreed to call this medium the ether. It exists everywhere, even penetrating between the molecules of ordinary matter. Little is known about its nature and the exact way in which light travels through it, but it is generally agreed that light is a wave motion in the ether and that the vibrations are not longitudinal as in sound waves, but transverse ( 174). The theory that light is a wave motion in the ether was proposed by Huyghens, a Dutch physicist, in 1678; Fresnel, a French physicist, showed that the disturbance must be transverse ; and Maxwell modified the theory to the effect that these disturbances are probably not trans- verse physical movements of the ether, but transverse alterations in its electrical and magnetic conditions. 224. Transparent and Opaque Bodies. When light falls on a body, 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 glass, air, pure water. Translucent bodies 179 180 LIGHT transmit light, but so imperfectly that 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 classification is one of degree. Water when deep enough cuts off all light ; the bottom of the deep ocean is dark. Stars which are invisible at the foot of a mountain are often visible at the top. 225. 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 veloc- ity. But in that year Roemer, a young Danish as- tronomer, made the very impor- tant discovery that light travels with finite speed. Roemer was en- gaged at the Paris Observatory in observing the eclipses of the inner satellite or moon of the planet Jupiter. At each revolu- tion of the moon, M (Fig. 182) in its orbit round 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 much earlier recorded ones, Roemer found that the mean interval of time be- tween two successive eclipses was 42.5 hours. From this Fig. 182 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, arid 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. NATURE AND TRANSMISSION OF LIGHT 181 it was easy to calculate in advance the time at which suc- ceeding eclipses would occur. But when the earth was going directly away from Jupiter, as at JE V the eclipse interval was found to be longer than anywhere else ; and at E y 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 mi. 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. 226. 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. 227. Ray, Beam, Pencil. Light is propagated outward from the luminous source in concentric spherical waves, as sound waves in air are 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 incident on any surface are sensibly parallel. A number of parallel rays form a beam of light. For example, in the 182 LIGHT 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. Fig. 183 228. 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. Fig. 184 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. When the source of light is a point L (Fig. 188), 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. 184), NATURE AND TRANSMISSION OF LIGHT 183 the space ABDO behind the opaque body receives no light, and the parts between AC and AC 9 , and between BD and BD', receives some light, the amount increasing as AC 9 and BD 9 are approached. From these figures the cases when the luminous body is larger than the opaque body, and when it is of the same size, may be understood and illustrated by the student. 229. Images by Small Openings. Support two sheets of card- board (Fig. 185), in vertical planes. In the center of one cut a hole 2 mm. square, and place in front of it a lighted candle at a distance of 20 or 25 cm. An in- verted image of the flame of the candle will appear on the other sheet. If a second small open- ing be made near the first, a second image will be ob- tained not coincid- ing exactly with the first one. The shape of the opening, so long as it is small, has no effect on the image. With a larger opening the image gains in bright- ness but loses in distinctness. 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 opening, the overlapping of the images of the aperture Fig. 185 184 LIGHT destroys all resemblance between the image and the ob- ject, the resulting image having the shape of the aperture. 230. 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 ; the light passes through this and forms an image on the sensitized plate placed on the opposite 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 assume a cres- cent shape. II. PHOTOMETRY 231. 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. C Fig. 186 Cut three cardboard squares, 4, 8, and 12 cm. on a side respectively, and mount them on supports (Fig. 186). The centers of these screens should be at the same distance above the table as the source of 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 PHOTOMETRY 185 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 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 squares 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. 232. The Bunsen Photometer. A photometer is an in- strument for comparing the intensity of one light with that Fig. 187 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. 187), 186 LIGHT having a translucent spot made by applying a little hot paraffin, is supported on a graduated bar between a standard candle B and the light O 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 enclosed 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 position of A or of B may then be adjusted until both sides of the screen look alike. Then the intensity of Q is to the in- tensity of B as AC 2 is to Alf. Questions and Problems 1. What is the cause of an eclipse of the moon? 2. What is the cause of an eclipse of the sun ? 3. Why is the duration of a lunar eclipse greater than that of a solar eclipse ? 4. Atropine placed in the eye enlarges the pupil. Why is vision then less distinct ? 5. An illuminated vertical object 4 feet long is at a distance of 12 feet from a shutter in which there is a minute hole. Inside is a vertical screen 4 feet from the small aperture. How long is the image of the illuminated object given by the small aperture? 6. The pupil of the eye of a cat is elliptical rather than round. Is the form of the image on the retina of the eye affected by the shape of the pupil? 7. If a yardstick standing vertically casts a shadow 2 ft. long, how high is a flagpole that at the same time casts a shadow 100 ft. long? 8. The following data were obtained in measuring the candle power of a Welsbach gas flame : Distance of standard candle from photometer disk, 15 cm.; distance of lamp, 100 cm. What is the candle power of the gas flame ? REFLECTION OF LIGHT 187 9. Two lamps of equal candle power are placed 100 m. apart. Where must a screen be placed between them so that one side re- ceives four times as much light as the other ? 10. Two electric lights of 16 and 64 candle power, respectively, are placed in front of a picture so as to illuminate it equally. The 16 candle power lamp is 10 ft. from the picture ; at what distance is the other one ? III. REFLECTION OF LIGHT 233. Regular Reflection. When a beam of light falls on a polished plane surface, the greater part of it is reflected e in a definite direction. This reflection is known as regu- lar reflection. In Fig. 188 a beam of light IB is incident on the plane mirror B and is reflected as BE. IB is the incident beam, BR is the re- flected beam, the angle IBP between the incident beam and the normal (perpendicu- lar) to the reflecting surface is the angle of incidence, and the angle PBR between the .V Fig. 188 reflected beam and the normal is the angle of reflection. 234. Law of Reflection. On a semicircular board are mounted two arms, pivoted at the center of the arc (Fig. 189). 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 Fig. 189 plane mirror M 188 LIGHT is mounted at the center of the semicircle, with its reflecting surface parallel to the diameter at the ends of the arc. On the 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 thr-eads. 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 two angles lie in the same plane. 235. 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 reflec- tion. To a greater or less extent all reflecting surfaces scatter light in the same way as the smoke particles. Figure 190 illustrates in an ex- aggerated way the difference between a perfectly smooth surface and one somewhat uneven. It is by diffused reflection that objects become visible to us. Perfect re- flectors 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 re- flect the light from the sun in every direction, and thus fill the space about us with light. If the air were free Fig. 190 REFLECTION OF LIGHT 189 from all floating particles and gases, the sky would be dark in all 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 absence of floating particles. 236. Image of an Object 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 mir- ror 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 candle in front but not lighted. 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. The image of an object in a plane mirror is a virtual image because the light only apparently comes from it. This image is of the same size as the object and is as far back of the mirror as the object is in front. 237. Geometrical Position of the Image of a Point. Let A (Fig. 191) be a luminous point in front of a plane mirror MN. Any ray AB in- cident on the mirror is re- flected in the direction BD, making the angle of reflection FBD equal to the angle of incidence FBA. In like man- ner, a second ray AC is re- flected along CE. From A drop a perpendicular AK to the reflecting surface and pro- Fig. 191 190 LIGHT duce it behind the mirror. Since the incident and re- flected rays AB and BD are in the same plane with the normal AK, it follows that BD produced meets the perpendicular in some point A'. The point A' is as far behind the mirror as A is in front. 1 The ray CE produced backwards also meets the perpendicular at A'. But AB and AC are any two rays incident on the mirror. It follows that all rays from A, incident on MN^ are reflected from MN as if they came from a point as far back of the mirror as A is in front. Hence the eye placed in the region of D or E will receive the reflected rays as if they came from A 1 . The point A is the image of A in the mirror MN, and it is a virtual image because the light only apparently or virtually comes 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 of it. 238. 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 lo- cated by finding those of its points. Let AB (Fig. 192) 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. A B f is the image of N Fig. 192 AB. It is virtual, erect, and of the same size as the object. 1 In the two right triangles ABK and A'BK the angle KAB = ABF = FBD = BA'K\ that is, angle KAB = angle BA'K. Hence the two right triangles are equiangular ; and since they have one side in common, they are equal and AK = A'K. The same result is reached by the triangles A'CK; the point A' is therefore common to the two rays. REFLECTION OF LIGHT 191 239. Path of the Rays to the Eye. Let A'B f (Fig. 193) be the image of AB in the plane mirror MN, and E the position of the eye of an observer. To find the path of the rays which enter the eye at E, draw straight lines from A f and B' to E. The intersections (7 and D of these lines with MN&YG the points of in- cidence of the rays from A and B which are re- flected to the eye at E. In the same way we may trace the path of the Fig. 193 rays for any other position of the eye. Thus we see that while the image does not change, the rays which form it for one observer are not those which form it for another. X* 240. Uses of the 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 covering 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 unsil- vered plate glass, with its edges hidden by curtains, is so placed that the audience have to look obliquely through it to see the actors on the stage. Other actors, hidden from direct view, and strongly illuminated, are seen by re- flection in the glass as ghostly images on the stage. 192 LIGHT 241. Multiple Reflection. Place two mirrors so that their re- flecting surfaces form an angle (Fig. 194). 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 Fig. 1 94 perpendicular to the mirrors. The image in one mirror 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. 195 the two mirrors are at right angles. 0' is the image of in AB, and is found as in 238. 0'" is the image of 0' in A C, and is found by the line O'O'" drawn perpendicular to AC produced. 0" is the image of in AC, and since the mirrors are at right angles, O' n is also the image of 0" in AB. O nf is situated behind the plane of both mirrors, and no images of it can be formed. All the images are o'-'---' situated in the circumfer- ence of the circle whose cen- Fig. 195 ter is A and radius A 0. If E is the position of the eye, then O f and 0" are each seen by one reflection, and O llf by two reflections, and for this reason it is less bright. To trace the path of a ray for the image O" 1 , draw 0"'E, cutting AB at 6, and from the intersection b draw bO", cutting AC at a. Join a ; the path of the ray is OabE. It is interesting to find the images when the mirrors are at various angles. REFLECTION OF LIGHT 193 242. Applications. The double image of a bright star and the several images of a gas jet in a thick mirror (Fig. 196) 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 ac- count of their faintness only a limited number is visible. The kaleidoscope, a toy invented by Sir David Brewster, is an inter- esting application of the same principle. It consists of a tube containing three mirrors extend- ing its entire length, the angle between any two of 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 whole at any source of light, multiple images of these pieces of glass are seen, symmetrically arranged around the center, and forming beautiful figures, which vary in pattern with every change in the position of the pieces of glass. 243. 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 sur- face is used as a mirror. In Fig. 197 the center C of the mirror MN is the center of curvature of the sphere of which the reflecting Fig. 197 surface is a part. The middle Fig. 196 194 LIGHT point A of the reflecting surface MN is the pole or vertex of the mirror, and the straight line AB passing through the center of curvature C and the pole A of the mirror is Us principal axis. Any other straight line through the center and intersecting 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. Fig. 198 Fig. 199 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. 244. Principal Focus of Spherical Mirrors. A focus is the point common to the paths of all the reflected rays of a pencil of light. It is a real focus if the rays of light actually pass through the point, Fig. 198, and virtual if they only appear to do so (Fig. 199). 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 smallest. 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. REFLECTION OF LIGHT 195 If a pencil of parallel rays falls on a concave spherical mirror, parallel to its principal axis, the point to which the rays converge after reflection is called the principal focus of the mirror. In the case of a convex spherical mirror, the principal focus is the point on the axis behind the mirror from which the reflected rays diverge. The distance of the principal focus from the mirror is its principal focal length. 245. The Position of the Principal Focus. Let MN (Fig. 200) be a concave mirror whose center is at O 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 EDO, the angle of incidence. Since the ray BA is normal to the mirror, it will be re- flected back along AB. The reflected rays DF and AB have a common point F, which is the principal focus. The triangle CFD is isosceles with the sides OF 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 par- allel to BA will pass after reflection nearly through F. Hence, the principal focus of a concave spherical mirror is ~"c real and is halfway between the center of curvature and the vertex. Fig. 201 Let MN (Fig. 201) be a convex spherical mirror. ED and BA are rays parallel to the principal axis. When produced back of the mirror, 196 LIGHT after reflection, their common point F is back of the mirror and halfway between A and C. (Why?) Hence, the principal focus of a convex spherical mirror is virtual and halfway between the center of curvature and the mirror. 246. Conjugate Foci of Mirrors. When a diverging pencil of light ABD (Fig. 202) falls on the spherical M Fig. 202 mirror MN, it is focused after reflection at a point F on the axis AB which passes through the radiant point or source of light; after reflection the rays diverge from this Fig. 203 focus F as a new radiant point. When rays diverging from one point converge to another, the two points are called conjugate foci. In Fig. 203, 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' is a virtual focus and B and B 1 are conjugate foci. REFLECTION OF LIGHT 197 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. 1 247. Images in Spherical Mirrors. In a darkened room support on the table a concave spherical mirror, a lamp, and a small white screen. Place the lamp anywhere beyond the focus, and move the screen until a clear image of the flame .is formed on it (Fig. 204). Notice the size and posi- tion of the image, and whether it is erect or in- verted. When the lamp is Fig * 20 ' 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. The same is true for the convex mirror, whatever be the position of the lamp ; 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 depend- ing on the position of the object with respect to the mirror. If these positions are carefully noted it will be seen that there are six distinct cases as follows : i In Fig. 202, CD bisects the angle SDH. 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 = q, CA = r = 2/. Then BC = p-r, B'C = r q, and 2 = JH-T from which - + - = - = - By measuring p q r-q p q r f and g, we may compute r and /. For the convex mirror, q and r are negative. 198 LIGHT First. When the object (AB y Fig. 205) is at a finite distance beyond the center of curvature, the image is real, inverted, smaller than the object, and between the center of curva- ture and the prin- cipal focus. Fig. 20; / N Second. 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. 206). This is the converse of case one. Third. When a small object is at the center of curvature, the image is real, inverted, of the same size as the object, and at the center of curvature (Fig. 207) . Fig ' 206 Fourth. When the object is at the principal focus, the rays are reflected parallel to the principal axis, and no image is formed (Fig. 208). 4. ^ ^ Fig. 207 Fig. 208 REFLECTION OF LIGHT 199 Fifth. When the object is between the principal focus and the mirror, the image is virtual, erect, and larger than the object (Fig. . 209). . Sixth. When the mirror is convex, the image is al- ways virtual, erect, and smaller than the object (Fig. 210). N 248. 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, and the second through the 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. 205, AC is the path of both the incident and the reflected ray, while the ray AD is reflected through the principal Fig ' 210 focus F. Their intersec- tion is at a. The rays BO and BE are reflected similarly through b. Hence, ab is the image of AB. In Fig. 209, the ray AC along the secondary axis, and AD reflected back through F as DF, must be produced to meet back of 200 LIGHT the mirror at the virtual focus a. A and a are conjugate foci ; also B and 5, and ab is a virtual image. For the convex mirror (Fig. 210) the construction is the same. From the point A draw AC along the, normal or secondary axis, and AD parallel to the 'principal axis. The latter is reflected so that its direction passes through F. The intersection of these two lines is at a. The image ab is virtual and erect. 249. 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. 211. At right angles to the board at one end place a vertical sheet of cardboard containing three parallel slots. Send Fig. 211 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 mirror. The experiment shows that rays .^ incident near, the margin of a spher- ical mirror cross the axis after reflec- tion between the principal focus and the mirror. This spreading out of the focus is known as spherical aber- ration by reflection. It causes a lack of sharpness'in the outline of image's formed by spherical mirrors. It is Fig. 212 / /l 1 /v ' f Pi / \^/ \F c v> v V \ \ ; NX * REFLECTION OF LIGHT 201 Fig. 213 reduced by decreasing the aperture of the mirror by means of a diaphragm to cut off marginal rays, or by decreasing the curvature of the mirror from the vertex out- ward. The result then is a parabol- ic mirror (Fig. 212), which finds use in search- lights, light- houses, head- lights of loco- motives, and in reflecting tele- scopes. 250. Caustics by Reflection. Use the tin reflector of the last experiment as shown in Fig. 213. 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 by letting sunlight fall on a tin milk pail partly full of milk, or on a plain gold ring on a white surface. The caustic curve may be constructed by drawing a series of parallel rays incident on a concave mirror of large aperture (Fig. 214), and tracing the reflected 202 LIGHT rays by means of tta law of reflection, that is, making in each case the angle m reflection equal to the angle of in- cidence. The caustic is the curved line to which all of the reflected rays are tangent. It should be noticed that in the case of a concave spherical mirror the caustic is a surface. Problems 1. Two mirrors are placed at an angle of 45. Find graphically how many images there are of a point situated between the mirrors. 2. Show by a diagram that a person can see his whole length in a mirror of half his height. 3. Which will give the stronger illumination of your book, a 16 candle power light at a distance of 2 ft., or a 32 candle power light at a distance of 4 ft. ? 4. An object placed 8 ft. in front of a concave spherical mirror gave an image 2 ft. from the mirror. What was the principal focal length of the mirror? 5. An object is placed 20 cm. in front of a concave mirror having a principal focal length of 30 cm. How far back of the mirror is the image ? 6. 'Where must an object be placed in front of a concave spherical mirror to get an image halfway between the center of curvature and the principal focus ? 7. An object is placed 20 in. in front of a concave spherical mir- ror of 10 in. radius. Find the position of Ttte image. 8. Find the radius of curvature of a concave spherical mirror when an object 100 cm. from the mirror gives a real image 50 cm. from, the mirror. 9. By making/, r, and q negative in the formula for a concave mirror, it applies to a convex mirror. If the radius of curvature of a convex spherical mirror is 20 in., what is the position of the image when the object is 5 ft. from the mirror? 10. If a plane mirror is moved parallel to itself directly away from an object in front of it, how much faster does the image move than the mirror? REFRACTION OF LIGHT 203 IV. REFRACTION OF LIGHT 251. Refraction. Fasten a paper protractor scale centrally on one face of a retarigular battery jar (Fig. 215), and fill the jar with water to the horizontal diameter of the scale. Place a slotted card- board over the top. With 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. 216). The changes in the apparent depth of a pond or a stream, as the observer moves away from it, are caused by refraction. 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 Fig. 2 1 6 suffer as they emerge into the air. 252. Cause of Refraction. Foucault in France and Michelson in America have measured the velocity of light in water, and have found that it is only three fourths as 204 LIGHT M 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 practically the same as in a vacuum. If now a beam of light is incident obliquely on the sur- face MN of water (Fig. 217), all parts of a wave do not enter the water at the same time. Let the par- allel 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 propaga- tion BG, which is .perpendicular to the wave fronts, is change^ ; in other words, the beam is refracted. The refraction of light is therefore owing to its change in velocity in passing from one transparent medium to another. 253. The Index of Refraction. Let a beam of light pass obliquely from air to water or glass, and let AB (Fig. 218) be the incident wave front. From A as a center and with Fig. 217 Fig. 218 a radius AD equal to the distance the light travels in the second medium while it is going from B to C in air, draw REFRACTION OF LIGHT 205 the dotted arc. This limits the distance to which the dis- turbance spreads in the second medium. Then from C draw CD tangent to this arc and draw AD to the point of tangency. CD is the new wave front. The distances BO 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 index of refraction A- _ the velocity of light in air _ v l the velocity of light in the second medium v' The angle NCB 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 7 AD.> It is equal to the angle ACD 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 . 1 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- 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 SAC is , and the sine of ACD is "==- Dividing one by the other, the common AC AC TiC 11 term A C cancels out, and the index of refraction equals = -. , as before. AD v The two definitions are therefore equivalent to each other. For the con- struction to find the refracted ray, see the Appendix. 206 LIGHT 254. 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. III. The planes of the angles of incidence and refrac- tion coincide. 255. 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 bro- ken at the edge of 4ihe plate, the part under the glass ap- pearing laterally dis- placed (Fig. 219). Fig. 219 Fig. 220 To explain this, let JZV(Fig. 220) 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 emer- gent ray will be parallel to the incident ray. Hence, the apparent position of an object viewed through a plate is at one side of its true position. 256. A Prism. Let ABQ (Fig. 221) represent a section of a glass prism made by a plane perpendicular to the re- v REFRACTION OF LIGHT 207 fracting edge A. Also, let LI be a ray incident on the face BA. This ray will be refracted along IE, and enter- ing 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 Fig. 221 prism. Two lines of light may be traced through the 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 emer- gence are equal ; this occurs when the path of the ray through the prism is equally inclined to the two faces. 257. Atmospheric Refraction. Light coming to the eye from any heavenly body, as a star, unless it is directly s^ overhead, is gradually x \ bent as it passes through the air on account of the increasing density of ^Horizon J !#^w\ ^ ne atmosphere near the earth's surface. Thus, if 8 in Fig. 222 is the real position of a star, its apparent position will be AS" 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 208 LIGHT Fig. 223 desert, and the looming of distant objects are phenomena of atmospheric refraction. 258. Total Internal Reflection. Take the apparatus of 251 and place the cardboard against the end of the jar so that the slit is near the bottom (Fig. 223). Re- flect a strong beam of light up through the water and incident on its under surface just back 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. 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. 259. The Critical Angle. The critical angle is the angle of incidence corresponding to an angle of refraction of 90. This angle varies with the index of refraction of the "sub- stance. 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' in- cident at a larger angle undergo total internal reflection (Fig. 224). Fig. 224 PROBLEMS 209 Hence, an observer under water sees all objects outside as if they were crowded into this cone ; beyond this he sees by reflection objects on the bottom of the pond. Total reflection in glass is shown by means of a prism whose cross section is a right-angled isosceles triangle (Fig. 225). A ray inci- dent normally on either face about the right angle enters the prism without refraction, and is incident on the hypotenuse at an angle of 45, which is greater than the critical angle. The ray therefore suffers total in- ternal reflection and leaves the prism at right angles to the incident ray. A sim- ilar prism is some- times used in a Fig. 226 . . projecting lantern for making the image erect (Fig. 226). It would other- wise be inverted with respect to the object. Problems 1. Where would you place a lamp in front of a concave spherical mirror to get an image of the lamp on the wall larger than the lamp itself ? Illustrate by a figure. 2. Why does the full moon as it rises appear to be oblong with the major axis horizontal? 3. What deviation is produced by reflection from a plane mirror when the angle of incidence is 60? 4. Paste diamonds are made of flint glass and have about the same index of refraction as carbon bisulphide. A diamond is visible in carbon bisulphide and the paste diamond is not. Explain. 210 LIGHT 5. A bottle filled with pounded glass is opaque, and is translucent when spirits of turpentine are added. Explain. 6. Show by a diagram that objects viewed obliquely through a plate glass window are not seen exactly in their true position. 7. Show by a diagram the effect of a hollow prism filled with air and submerged in water on a beam of light passing through the water and incident on the prism. 8. The index of refraction for water is f . If the velocity of light in air is 186,000 mi. per second, what is it in water? 9. If the index of refraction for crown glass is f, and for water is f , compare the speed of light in crown glass with that in water. What simple fraction represents the relative speed ? 10. Will a pencil of light passing obliquely from water into flint glass be bent toward or away from the perpendicular in the glass? V. LENSES 260. Kinds of Lenses. A lens is a portion of a trans- parent substance bounded by two surfaces, one or both Fig. 227 being curved. The curved surfaces are usually spherical (Fig. 227). Lenses are classified as follows: LENSES 211 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. The concavo-convex and the convexo-concave lenses are frequently called meniscus lenses. The double-convex lens may be regarded as the type of the converging class of lenses, and the double-concave lens of the diverging class. 261. 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 direction. In lenses whose surfaces are of equal curvature, the optical center is their center of volume,, as 0, in Fig. 228. In piano- lenses, the optical center is the middle point of the curved face. The straight line, CO', through the centers of cur- vature, is the principal axis, and any other straight line through the optical center as EH, is a secondary axis. The normal at any point of the surface is the radius of the sphere drawn to that point; thus CD is the normal to the surface AnB at D. 22S 212 LIGHT 262. Tracing Rays through Lenses. A study of Figs. 229 and 230 shows that the action of lenses on rays of light traversing them is similar to that of prisms, and conforms to the principle illustrated in 256. A ray is always refracted towar-d 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. 230) bends it away from this axis. 263. 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. LENSES 213 Converging lenses are sometimes called burning glasses because of their power to focus the heat rays, as shown in the experiment. .Figure 231 shows that parallel rays are made to converge toward the principal focus F by a converging lens, and the focus is real; on the other hand, Fig. 232 illustrates the diverging effect of a concave lens on parallel rays ; the focus F is now virtual because the rays after passing H Fig. 232 through the lens only apparently come from F. In gen- eral, converging lenses increase the convergence of light, while diverging lenses decrease it. 264. 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. 214 LIGHT These points are called conjugate foci, for the same reason as in mirrors. In Fig. 233 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 -ffare conjugate foci. 265. Images by Lenses. Place in 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 betw.een 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 iii 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 image is real, inverted, at a distance from'the lens of more than once and less than twice the focal length, and smaller than the object (Fig. 234). 215 LENSES II. When the object is at a distance of twice the focal length from a converging lens, the image is real, inverted, Fig. 234 at the same distance from the lens as the object, and of the same size (Fig. 235). III. When the object is at a distance from a con- verging lens of less than twice and more than once its Fig. 235 focal length, the image is real, inverted, at a distance of more than twice the focal length, and larger than the object (Fig. 236). Fig. 236 216 LIGHT IV. When the object is at the principal focus of a con verging lens, no distinct image is formed (Fig. 237). Fig. 237 V. When the object is between a converging lens and its principal focus, the image is virtual, erect, and en- larged (Fig. 238). Fig. 238 VI. With a diverging lens, the image is always virtual, erect, and smaller than the object (Fig. 239). Fig. 239 LENSES 217 266. 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 necessary to find only the images of its extremities. This 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 ( 261). Second. Through the ends of the arrow draw rays parallel to the principal axis. After leaving the lens, these pass through the principal focus ( 263). The intersection of the two refracted rays from each extremity will be its image. To illustrate. Let AB be the object and MN the lens (Figs. 234-239). 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. 267. Spherical Aberration in, Lenses. If rays from any point be drawn to different parts of a lens, and their directions be determined after refraction, it will be found that those incident near the edge of the lens cross the 218 LIGHT principal axis, after emerging, nearer the lens than those incident near the middle. 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 distinct- ness 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 curva- ture of the lens is made less toward the edge, so that all parallel rays are brought to the same focus. Fig. 240 268. Formula for Lenses. The triangles A OK and aOL in 4 ~K KT) Fig. 240 are similar. Hence, ==. If the lens is thin, a straight aL LO line connecting D and H will pass very nearly through the optical center 0. Then DFO is a triangle similar to aFL, and 52 =s ~L; aL Lr Since DO is equal to AK, the first members of the two equations K ' O O F above are equal to each other, and therefore = . Put KO = p, LO Lb LO-q, and OF =/. Then LF = q -/, and P = f q ?-/' Clearing of fractions and dividing through by pqf, we have i = i + -. . . (Equation 32) />--.;? By measuring p and q we may compute /. For diverging lenses / and q are negative. OPTICAL INSTRUMENTS 219 Questions and Problems 1. Given a spectacle lens, how will you determine whether it is converging or diverging ? 2. Where must the observer place himself so as to see his own image in a concave mirror ? 3. Why is the image of yourself in the bowl of a silver spoon distorted? 4. What kind of mirrors and lenses always produce virtual images ? 5. If one half of a converging lens is covered by an opaque card, what will be the effect on the real image ? 6. When an object moves from a great distance up to twice the focal length of a converging lens, how far does the image move? ; x7. A candle is placed 10 ft. from a white wall. Find the position of a converging lens that will give an enlarged image of the candle on the wall, the focal length of the lens being 20 in. 8. Why does a mirror made of plate glass give a better image than one made of common window glass? 9. What is the smallest distance between an object and its real image in a converging lens,*expressed in terms of the focal length? 10. The focal length of a camera lens is 20 cm. How far from the lens must the sensitized plate be placed when the object is 200 cm. from the lens ? 11. An object 5 cm. long in front of a converging lens has an image 20 cm. long on a screen 100 cm. from the lens. What is the focal length of the lens? / 12. An object is placed 100 cm. from a diverging lens whose focal length is 33 cm. What is the distance of the virtual image? VI. OPTICAL INSTRUMENTS 269. 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. 220 LIGHT If AJ3 is the object in Fig. 241, the virtual image is ab ; and if the eye be placed near the lens on the side opposite M the object the virtual image will be seen in the position of the intersection of the rays produced, as at ab. 270. The Compound Microscope (Fig. 242) is an instru- ment designed to obtain a greatly enlarged image of very small objects. In its simplest form it consists of a converging lens MN (Fig. 243), called the object glass or objective, and another con- verging lens RS, called the eyepiece. The two lenses are mounted in the ends of the tube of Fig. 242. The ob- ject is placed on the stage just under the objective, and a little beyond its prin- cipal focus. A real image ab (Fig. 243) is formed slightly nearer the eyepiece than its focal length. This image formed by the objective is viewed by the eye- piece, and the latter gives an enlarged virtual image. 242 OPTICAL INSTRUMENTS 221 (Why ?) Both the objective and the eyepiece produce magnification. 271. The Astronomical Telescope. The system of lenses in the refracting astronomical telescope (Fig. 244) is simi- lar to that of the compound microscope. Since it is in- tended to view distant objects, the objective MN is of 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 magnification of the image without too great loss in brightness. Figure 244 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- 222 LIGHT 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. 272. 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. 245). Fig. 245 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 J, and the rays of light issue from it slightly divergent for distant objects. The image is therefore at A J B' 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. 273. 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. 246, the calcium light, or a large oil burner. The condenser E is OPTICAL INSTRUMENTS 223 composed of a pair of converging lenses; its chief pur- pose is the collection of the light on the object by refraction, so as to bring as much as possible on the screen. The object AB, commonly a drawing or a photo- graph on L glass, is placed near the condenser SS, where Fig. 246 it is strongly illuminated. The objective, MN, is a com- bination of lenses, acting as a single lens to project on the screen a real, inverted, and enlarged image of the object. 274. The Photographer's Camera consists of a box L8 (Fig. 247), 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 ^m'-fl^w sensitized plate. Fig. 247 If by means- of rack and pinion the lens L' be prop- erly focused for an object in front of it, an inverted image will be formed on the sensitized plate E. The light acts on the salts contained in the sensitized film, producing in them a modification which, by the processes 224 LIGHT of " developing " and " fixing," becomes a permanent negative picture of the object. When a "print" is made from this negative, the result is a positive picture. 275. The Eye. The eye is like a small photographic camera, with a converging lens, a dark chamber, and a sensitive screen. Figure 248 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 extended 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, Fig. 248 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. OPTICAL INSTRUMENTS 225 276. 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. 249) precisely as in the photo- Fig. 249 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. 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 the crystalline lens by means of the ciliary muscle F, Gr (Fig. 248). This capability of the lens of the eye to change its focal length for objects at different distances is called accommodation. 277. 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 accordingly called the blind spot. Its existence can be readily proved by the help of Fig. 250. Hold the book Fig. 250 with the circle opposite the right eye. Now close the left eye and turn the right to look at the cross. Move 226 LIGHT 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. VII. DISPERSION 278. 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. 251). A many-col- ored band, gradually changing from red at one end through orange, yellow, green, blue, to violet at the P. ~c- . other, appears on the screen. If a converg- ing 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 experiment shows that white or colorless light is a mixture of an infinite numjber of differently colored rays, of which the red is refracted least and the violet most. The brilliant band of light consists of an indefinite number of colored images of the slit ; it is called the solar spectrum, and the opening oufTor separating of the beam of white light is known as dispersion. DISPERSION 227 279. 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. 252). There will be formed a colorless image, slightly displaced on the screen. The second prism reunites the colored rays, making the effect that of a thick plate of glass ( 255). 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. 280. Chromatic Aberration. Let a beam of sunlight into the darkened room through a round hole in a piece of cardboard. Project 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. A The violet rays, being more re- frangible than the red, will have their focus nearer to the lens than the red, as shown in Fig. 253, where v is the principal focus for violet light and r for red. If a screen were placed at a?, the image would be bordered with red, and if at y with violet. 281. The Achromatic Lens. With a prism of crown glass project a spectrum of sunlight on the screen, and note the length of the spectrum when the prism is turned to give the least deviation (256). Repeat the ex- periment with a prism of flint glass having the same refracting angle. The spec- trum formed by the flint glass will be about twice as long as that given by crown Fig. 254 228 LIGHT 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 deviation of the middle of it is considerably less. Finally, place this flint glass prism in a reversed position against the crown glass one (Fig. 254). The image of the aperture is no longer colored, and 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 flint glass so that the dispersion by the one neutralized that due to the other, while the refraction was reduced about half (Fig. 255). Such a lens or system of lenses is Fig. 255 called achromatic, since images formed by it are not fringed with the spectral colors. 282. The Rainbow. Cement a crystallizing beaker 12 or 15 cm. in diameter to a slate slab. Fill the beaker with water through a hole drilled in the slate. Support the slate in a vertical plane and direct a ribbon of white light upon the beaker at a point about 60 above its horizontal axis, as SA (Fig. 256). 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 refraction at this point, the light is internally reflected twice and then emerges in front as a diverging pencil of spectrum colors. The experiment shows that the light must be incident at definite angles to give color effects. The red constitu- 7 DISPERSION 229 ent of white light incident at about 60 keeps together after reflection and subsequent refraction; that is, the red rays are practically parallel and thus have sufficient in- tensity to produce a red image. The same is true of the violet light incident at about 59 from the axis. The other spectral colors arrange themselves in order between the red and violet. When sunlight falls in this manner on raindrops, they disperse the light and the spectral colors produced form the rainbow. Two bows are often visible, the primary and the second- ary. The primary bow is the inner and brighter one formed by a single internal reflection ; it is distinguished by being red on the outside and violet on the inside. The secondary low, formed by two internal reflec- tions, is fainter, and has the order of colors re- versed. Figure 257 shows the relative position of the sun, the observer, and the raindrops which form the bows. 283. 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 baud. Solids, Fig. 257 230 LIGHT liquids, and dense vapors and gases, when heated to incan- descence, give continuous spectra. 284. Discontinuous Spectra. Project on the screen the spec- trum of the electric light. Place in the arc a few crystals of sodium nitrate. 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 coJLer separated by dark spaces. Rarefied gases and vapors, when heated to incandescence, give discontinuous spectra. 285. Absorption Spectra. Project on the screen the spectrum of the electric light. Between the lamp and the slit S (Fig. 258) vaporize metallic sodium in an iron spoon so placed that the white light passes through the heated sodium vapor before disper- sion by the prism. A dark line will ap- pear on the screen in the yellow of the spectrum at the place where the bright line was ob- tained in the pre- ceding experiment. ment illustrates an absorption, reversed, or dark line spectrum. The dark line is produced by the absorption of the yellow light by sodium vapor. Gases and vapors absorb light of the same refrangibility as they emit at a higher temperature. DISPERSION 231 286. 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. 259). iBC Eb Bed Orange Fellow Green Blue Indigo Twlet Fig. 259 Fraunhofer was the first to notice that some of these lines coincide in position with the bright lines of certain artificial lights. He mapped no less than 576 of them, and designated the more important ones by the letters A, B, 0, D, E, F, G-, H, the first in the extreme red and the last in the violet. For this reason they are referred to as the Fraunhofer 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 spectrum. 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 spectrum would therefore be continuous, were it not surrounded 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 spectrum which would have been illuminated by those particular 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 232 LIGHT not lines of no light, but are lines of diminished bright- ness, appearing dark by contrast with the other parts of the spectrum. 287. The Spectroscope. The commonest instrument for viewing spectra is the spectroscope (Fig. 260). In one of its simplest forms it consists of a prism A, a telescope B, and a tube called the collimator C, carrying an adjustable Fig. 260. slit at the outer end _D, and a converging lens at the other E, to render parallel the diverging rays coming from the slit. The slit must therefore be placed at the principal focus of the converging lens. To mark the deviation of the spec- tral lines, there is provided on the supporting table a divided circle F, which is read by the aid of verniers and reading microscopes attached to the telescope arm. The applications of the spectroscope are many and various. By an examination of their absorption spectra, normal and diseased blood COLOR 233 are easily distinguished, the adulteration of substances is detected, and the chemistry of the stars is approximately determined. Figure 261 shows the agreement of a number of the spectral lines of iron with Fraunhofer lines in the solar spectrum ; they indicate the pre- sence of iron vapor in the atmosphere of the sun. HIIIHIIII Fig. 261 VIII. COLOR 288. 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 : 5269 mm. 4861 mm. 4293 mm. 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 cer- tain colors are either feeble or wanting. Hence, artificial lights are not generally white, but each one is character- ized by the color that predominates in its spectrum. A B C Do Dark Red . 0.0007621 mm. Red . . . 6884 mm. Orange . . 6563 mm. Yellow . . 5896 mm. 5890 mm. Ei Light Green F Blue . . . G Indigo . . H, Violet 234 LIGHT 289. 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 ; 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 shade some fabrics undergo when taken from sunlight into gaslight. Most artificial lights are deficient in blue and violet rays; and hence all com- plex colors, into which blue or violet enters, as purple and pink, change their shade when viewed by artificial light. 290. 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 ammoniated 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. 1 It is prepared by adding ammonia to a solution of copper sulphate, until the precipitate at first formed is dissolved. COLOR 235 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. 291. 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. 262), 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 Fig. 262 together, exposing any proportional part of each one as desired (Fig. 263). Select seven disks, whose colors most nearly represent those of the solar spectrum ; put them to- gether so that equal portions 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. This method of mixing colors is based on the physio- logical fact that a sensation lasts longer than the stim- ulus producing it. Before the sensation caused by one stimulus has ceased, the disk has moved, so that a different impression is pro- Fig. 263 duced. The effect is equivalent to superposing the several colors on one another at the same time. 236 LIGHT 292. Three Primary Colors. If red, green, and blue, or violet disks are used, as in 291, exposing equal portions, gray or impure white is obtained when they are rapidly rotated. If any two colors standing op- posite each other in Fig. 264 are used, the result is white ; and if any two alternate ones are used, the result is the interme- diate one. By using the red, the green, and the violet disks, and exposing in different pro- F . 264 portions, it has been found possible to produce any color 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 varying degrees. The color top is a standard toy provided with colored paper disks, like those of Fig. 262. When red, green, and blue disks are combined so as to show sectors of equal size, the top, when spinning in a strong light, appears to be gray. Gray is a white of low intensity. The colors of the disks are those of pigments, and they are not pure red, green, and blue. 293. Three-color Printing. The frontispiece in this book illustrates a three-color print of much interest. Such a print is made up of very fine lines and dots of the three pigments, red, yellow, and blue ; the various colors in the picture are mixtures of these three with the white of the paper. The greens come chiefly from the overlapping and mixture of the yellow and blue pigments. COLOR 237 The process is briefly as follows : Three negatives of the same original are taken through transparent screens of red, green, and blue, and each is crossed by fine lines or dots. Copper plates are made from the negatives, and each plate is inked for printing with an ink of a color which gives white, when mixed with the color of the screen through which the negative was taken. Thus, the plate made with the red screen is printed with greenish blue ink; those taken with the green and blue or violet screens are printed with crimson, red, or yellow ink, respectively. In the frontispiece the first plate was printed with yellow, the second with yellow and then with red, and the third with all three. 294. Complementary 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 com- plementary 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 screen 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 impres- sion of white light with the red cut out ; that is, of the complementary color, green. 238 LIGHT 295. 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. The yellow crayon reflects green light as well as yellow, 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 survives 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 mix- ture of colored lights is a very different thing from a mixture of pigments. IX. INTERFERENCE AND DIFFRACTION 296. 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 thick- ness of the film. He used a plano- convex lens of long focus resting G f F G on a plate of plane glass. Figure 265 shows a section of the appa- ratus. Between the lens and the plate there is a wedge- shaped film of air, very thin, and quite similar to that formed between the glass plates in the above experiment. INTERFERENCE AND DIFFRACTION 239 Fig. 266 If the glasses are viewed by reflected light, there is a dark spot at the point of contact, surrounded by several colored rings (Fig. 266) ; but if viewed by transmitted light, the colors are complementary to those seen by re- flection ( 294). The explanation is to be found in the interference of two sets of waves, one reflected internally from the curved surface AGB, and the other from the surface DOE, on which it presses. If light of one color is incident on AB, a portion will be reflected from ACB, and another portion from D OE. Since the light reflected from D OE has traveled farther by twice the thickness of the air film than that from ACB, and the film gradually increases in thickness from O outward, 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 fol- lows that if the glasses be illuminated by white light, at every point some one color will be destroyed. The other colors will be either weakened or strengthened, depending on the thickness of the air film at the point under consid- eration, the color at each point being the result of mixing a large number of colors in unequal proportions. Hence, 240 LIGHT the point C will be surrounded by a series of colored bands. 1 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. 297. 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 ( 278). 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 farther 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 reduced. Diffraction gratings are also made to operate by reflecting light. Striated surfaces, like mother-of-pearl, changeable 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 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 241 silk, and the plumage of many birds, owe their beautiful changing colors to interference of light by diffraction. Questions 1. Why is the flint glass of an achromatic lens the diverging part of the combination instead of the converging? 2. Why is the rainbow circular? 3. Do different persons see the same rainbow ? 4. Why is the rainbow not seen at midday? 5. In projecting pictures on a screen, why should the screen be white ? 6. Of which case of images by lenses is the projecting lantern an application ? 7. In enlarging a negative by photography where must the nega- tive be placed with respect to the lens? 8. Why do flowers that are purple by sunlight look red by lamp- light? 9. Account for the color on a plate of glass when it is brushed over with alcohol. 10. Account for the crossed bands of colors seen by looking through a silk umbrella at an arc electric light. CHAPTER IX HEAT I. HEAT AND TEMPERATURE 298. 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 caloric. About the beginning of the last century certain experiments of Count Rumford and Sir Humphry Davy 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. Temperature. If we place a mass of hot iron in contact with a cold one, the latter becomes warmer and the former cooler, the heat flowing from the hot body to the 242 HEAT AND TEMPERATURE 243 cold one. The two bodies are said to differ in temperature or " heat level," and when they are brought in contact there is a flow of heat from the one of higher temperature to the one of lower till thermal (heat) equilibrium 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 quan- tity of heat. The water in a pint cup may be at a much higher temperature than the water in a lake, yet the latter contains a vastly greater quantity of heat, owing to the greater quantity of water. 300. Measuring Temperature. Fill three basins with moder- ately 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 objects 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 temperature of bodies, and some other method must be resorted to for reliable measurement. The one most ex- tensively used is based on the regular increase in the volume of a body attending a rise in its temperature. This method is illustrated by the common mercurial ther- mometer. 244 HEAT II. THE THERMOMETER 301. 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 spherical or cylindrical (Fig. 267). 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 pres- sure. Enough mercury is introduced to fill the bulb and part of the tube at the lowest temperature which the thermometer is de- signed to measure. Heat is now applied to the bulb till the expanded mercury fills the tube ; the end is then closed in the blowpipe flame. The mercury contracts as it cools, leaving a vacuum at the top of the tube. 302. Necessity of Fixed Points. No two thermometers are likely to have bulbs and stems of the same capacity. Consequently, the same increase of temperature will not produce equal changes in the height of the 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 thermometers 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 mercury has a nearly uniform rate of expansion. If two points are marked on the stem, the others can be obtained by dividing the space between Fig. 267 THE THERMOMETER 245 them into the proper number of equal parts. Investigations have made it certain that under a constant pressure the temperature of melting ice and that of steam are invariable. 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. 303. Marking the Fixed Points. The thermometer is packed in finely broken ice, as far up the stem as the mer- cury extends. The containing vessel (Fig. 268) has an opening at the bottom to let the water run out. After standing in the ice for several minutes the top of the thread of mer- cury is marked on the stem. This is called the freezing point. The boiling point is marked by observing the top of the mer- curial column when the bulb and stem are enveloped in steam (Fig. 269) under an atmospheric pressure of 76 cm. (29.92 in.)- If the pressure at the time is not 76 cm., then a cor- rection must be applied, the Fi - 268 Fi s- 269 amount being determined 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. 304. Thermometer Scales. The distance 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 introduced. Three of these are in use at the present time : the Fahrenheit, the Genii- 246 HEAT grade, and the Reaumur. The Fahrenheit scale was introduced by Fahrenheit about 1714, and is the one in common use in all English-speaking countries. For some unknown reason he marked the freezing point at 32 above the zero of the scale, and the boiling point at 212, dividing the space between into 180 equal parts. The Centigrade scale was designed by Celsius about 1742. It differs from the Fahrenheit in making the freezing point and the boiling point 100, the space between being divided into 100 equal parts. This is the one in general use among scientific men. The Reaumur scale marks the freezing point and the boiling point 80. This is the household scale on the continent of Europe ; in this country its use is restricted to breweries. Each of these scales is extended beyond the fixed points as far as desired. The divisions below are read as negative ; for example, 10 signifies 10 degrees below zero. The reading according to any par- ticular scale is indicated by affixing the initial letter of the name; for example, 5 F., 5 C., and 5 R. signify 5 degrees above zero on the Fahrenheit, Centigrade, and Reaumur scales respectively. 305. The Three Scales Compared. In Fig. 270 AB is a thermometer with three scales attached, P is the head of the mercury col- A p \ n umn, and F, (7, and R are the readings Fahrenheit Centigrade on the scales respec- R eaun ^ 112 100 tively. On the Fah- I/M. i A -D Fig. 270. renheit scale AB 180 and AP = F ' 32, since the zero is 32 spaces below A; on the Centigrade JJ?=100 and AP= 0; on the THE THERMOMETER 247 Reaumur AB to AB is 80 and AP = R. Then the ratio of AP _ 39 n '72 - - = = . By substituting the read- 180 1UU oU ing on any one scale in this equation the equivalent on either of the other scales is easily obtained. For ex- ample, if it is required to express 68 F. on the Centi- grade scale, then 68 ~ 2 180 100 whence (7=20. >/ )c *' \ 6-oTC- 306. Limitations of the Mercurial Thermometer. As mer- cury freezes at 38.8 C., it cannot be used as the thermometric substance below this temperature. For temperatures below 38 C. alcohol is sub- stituted for mercury. Under a pressure 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 ( 331). 307. 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 F., or from 35 to 45 C. There is a constriction in the tube just above the bulb (Fig. 271), which causes the thread of mercury to break at that point when the temperature begins to fall, leaving the top of the separated thread to mark the highest temperature registered. A sudden lg * jerk or tapping of the thermometer forces the mercury down past the constriction and sets it for a new reading. 248 HEAT Questions and Problems 1. Why should the tube of a thermometer be of uniform bore ? 2. Is it correct to speak of one temperature as twice that of an- other? 3. Why may thermometers differ in length and still measure be- tween the same extremes of temperature? 4. Why do thermometers have a bulb on one end? Would a closed uniform tube answer just as well? 5. What would be the effect on the indications of a thermometer if the glass expanded the same as the mercury? 6. Convert into equivalent readings on the Centigrade scale: 98 F., - 40 F., 68 F. 7. Convert into equivalent readings on the Fahrenheit scale: 36 C., - 40 C., 29 C. 8. The lowest temperature yet obtained is claimed to be 271.3 C. What would this be on the Fahrenheit scale ? 9. The melting points of iron and copper are 2737 F. and 1943 F. respectively. Express these temperatures in Centigrade degrees. 10. A correct Fahrenheit thermometer registers the temperature of a room as 70; a faulty Centigrade thermometer reads 20. Find the error of the latter. 11. When the barometric pressure is 74 cm., what is the boiling point? 12. A certain Centigrade thermometer registers 2 in melting ice and 100 in steam under normal atmospheric pressure. What is the correct value of a temperature of 25 as given by this instrument? III. EXPANSION 308. Expansion of Solids. Insert a long knitting needle A in a block of wood so as to stand vertically (Fig. 272). A second needle D is supported parallel 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 mercury, the needle, EXPANSION 249 an electric battery, and the bell B, as shown. Now apply a Bunsen flame to A ; D will be lifted out of the mercury and the bell will stop ringing. Then heat D or cool A, and the contact of D with the mercury will be renewed as shown by the ringing of the bell. This experiment shows that solids ex- pand in length when heated and contract when cooled. To this rule of expansion there are a few exceptions, notably iodide of silver and stretched india- rubber. Fi g. 272 Rivet together at short intervals a strip of sheet copper and one of sheet iron D (Fig. 273). 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, throwing the top over against either A or C, and will cause the bell to ring. Fig. 273 The experiment shows that the two metals expand unequally and cause the bar to warp. 250 HEAT Figure 274 illustrates a piece of apparatus 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. We conclude that the ex- pansion of a solid takes place in every direction. 309. Expansion of Liquids. Partly fill several small air ther- mometers of the same capacity with different liquids, and support them vertically in a metallic vessel (Fig. 275). Note the height of the several liquids and then fill the vessel with hot water. The liquids will rise in the tubes but not equally. Two facts are illustrated : first, liquids are affected by heat in the same way as 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. 310. Expansion of Gases. Fit a bent delivery tube to a small Florence flask (Fig. 276). Fill the flask with air and place the upturned end of a delivery tube under an inverted graduated glass cylinder filled with water. Heat the flask by immersing it in a vessel of Fi g- moderately hot water. The air will expand and escape through the delivery tube into the cylinder; note the amount. Now refill EXPANSION 251 the flask with some other gas, as coal gas, and repeat the ex- periment. The amount of gas collected will be nearly the same. 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 car- bon dioxide, show the largest variation in their expansion. ' 311. Coefficients of Expan- sion. It appears from the preceding experiments that substances when heated ex- Fig* 276 pand in every direction. This expansion in volume is called cubical expansion, in distinction from linear expansion, or expansion in length, and superficial expansion, or expan- sion in area. The coefficient of linear expansion is the fraction of its length which a body expands when heated from C. to 1 C. ; the coefficient of superficial expansion is the fraction of its area which a body expands when heated from C. to 1 C. ; and the coefficient of cubical expansion is the fraction of its volume which a body ex- pands when heated from C. to 1 C. Since the linear expansion of most substances is found to be nearly con- stant for each degree of temperature, it is customary to determine the average coefficient for a change of several degrees. If Z x and Z 2 represent the lengths of a metallic rod at the temperatures ^ and t 2 respectively, then ~ ^ = 2 . 1 is the expansion for 1, in which t is the T 2~h difference of temperatures. If a represents the aver- age coefficient of expansion, then a = ~ - ; whence lit 252 HEAT 1 2 ^(1 -|- at). In like manner for volumes, if k is the coefficient of cubical expansion, v l and v 2 the volumes at the temperatures ^ and 2 respectively, then whence v 2 = Vj(l 4- kt). In the case of solids, superficial and cubical expansion are obtained by computation from the linear expansion, the coefficient of the former being twice the linear, and that of the latter three times. 312. Law of Charles. It was shown by Charles, in 1787, that the volume of a given mass 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 2T3^ This fraction may be considered as the coefficient of ex- pansion of any true gas. 313. The Absolute Scale. The law of Charles leads to a fourth scale of temperature called the absolute scale. By this law the volumes of any mass of gas, under constant pressure, at C., and at any other temperature t C., are connected by the following relations ( 311) : At any other temperature, ', the volume becomes EXPANSION 253 Divide (a) by (5) and Suppose now a new scale is taken, whose zero is 273 Centigrade divisions below the freezing point of water, and that temperatures on this scale are denoted by T. Then 273 + 1 will be represented by T, and 273 + 1' by T 1 , and v = 273 + = T_ ' "~ or the 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 3 of the absolute zero. 314. The Laws of Boyle and Charles Combined. If v, p, and T denote the volume, pressure, and absolute tempera- ture of a given mass of gas, then by Boyle's law ( 74) v oc -, when T is constant ; and by the law of Charles, P v oc T, when p is constant. Therefore when T and p both vary, v varies directly as T and inversely as jp, or T v oc . Whence pv oc T, or pv = constant x T. This P 254 HEAT relation is known as the " gas equation " and is written pv = RT. . . . (Equation 33) R is the constant which converts a proportionality into an equality. It follows that not only is the volume of a given mass of gas under constant pressure proportional to its absolute temperature ( 313), but the product of the pressure and volume of a given mass of gas is proportional to its absolute temperature. 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. 3 and temperature 50 C. ? From equation (33), ^ is a constant, pv p f v r ~T =2 ~T r ' Hence 76 x 20 _ p x 30 273+20~273 + 50' from which p = 55.85 cm. 315. Force of Contraction and Expansion. Fill a small test tube about one quarter full of water, and close the end by fusion. Lay it in an empty sand bath on the ring of an iron stand. Apply heat, and stand at a safe distance. In a few minutes there will be a loud report, caused by the bursting of the tube. The force of expansion or of contraction of a subtance is evidently equal to the force necessary to compress or expand it to the same extent by mechanical means, and hence can be computed by proceeding in the manner illus- trated in the following example : A bar of malleable iron, one square inch in cross-sectional area, if placed under the tension of a ton, increases in length 0.0001 of itself. EXPANSION 255 The coefficient of linear expansion of iron is 0.0000122. Since 0.0001 -r- 0.000122 = 8 + , a change of temperature of about 8 C. will produce the same change in the length of the bar as a force of one ton. It takes a pressure of 600 atmospheres to keep mercury from expanding when heated from C. to 10 C. 316. Applications of Expansion and Contraction. Many familar phenomena are accounted for by expansion or con- traction attending changes of temperature. If hot water is poured into a thick glass tumbler, the glass will prob- ably break because of the stress produced by the sudden expansion of its inner surface. The principle of unequal expansion is employed in thermometers, in the compen- sated clock pendulum and in the bal- ance wheel of a watch (Fig. 277), in which the rim is made in two sec- tions, each composed of two metals soldered together side by side, with the more expansible metal on the out- side. When the temperature rises, the ends #, a 1 move inward. Glass and Fig 2 77 platinum have nearly the same coef- ficient of expansion. For that reason platinum is in great demand in the manufacture of incandescent electric lamps, since it does not crack the glass when it cools. Iron tires are fitted to wheels and then expanded by heating so that they slip on easily ; on cooling, they contract and com- press the wheel. The rivets which hold together the plates of steam boilers are inserted red-hot, and hammered down. The contracting rivets press the plates together with great force. In all heavy iron structures, such as railroad bridges, a certain freedom of motion of the parts 256 HEAT must be provided for ; otherwise, the changes in length attending variations in temperature would have a disas- trous effect. Sidewalks of artificial stone should have spaces left for expansion to prevent " buckling. " Crys- talline rocks, on account of unequal expansion in different directions, are slowly disintegrated by changes of tem- perature ; and for the same reason quartz crystals, when strongly heated, fly in pieces. Questions and Problems 1. What is the objection in stringing telegraph wires in the sum- mer time to stretching them tight ? 2. Why will warming the neck of a bottle often loosen a glass stopper that has stuck ? 3. A quantity of alcohol that measures 20 gallons on thfc first of January might measure as much as 21 gallons on the first of July. Explain. 4. Set a pan even full of cold water on a hot stove. In a short time it will begin to overflow. Why ? 5. Fill a vessel containing a piece of ice level full of water. When the ice melts the water level neither rises nor falls. Why ? 6. An iron rod 60 cm. long at 20 C. was 60.055 cm. long at 95 C. Calculate the coefficient of expansion. 7. A brass rod was 100 cm. long at 20 C. What will be its length at C., if the coefficient of linear expansion is 0.0000186 ? 8. A copper bar 62.5 cm. long at 5C. expands by 1.1 mm. when heated to 98 C. Find its coefficient of expansion. 9. An iron bridge is 300 ft. long. Calculate the variation in length it will undergo between the temperatures 10 C. and 40 C., the coefficient of expansion of iron being 0.0000122. 10. A glass graduate holds one liter at 15 C. How much will it hold at 25 C., if the coefficient of cubical expansion is 0.000025 ? 11. Two metal bars are each one meter long at 10 C. How much will they differ in length at 40 C. if one is steel (coefficient of MEASUREMENT OF HEAT 257 linear expansion, 0.0000132) and the other aluminum (coefficient, 0.0000222) ? 12. If 1500 cm. 8 of air at 15 C. be changed in temperature to 50 C., what will the volume be if the pressure is constant? 13. In an experiment to determine the coefficient of expansion of air the following data were obtained : Volume of air at C., 145 cm. 8 ; at 100 C., 198 cm. 3 , the pressure remaining the same. Calcu- late the coefficient of expansion. 14. A flask filled with air at 20 C. and under 75 cm. pressure is stoppered and heated to 70 C. Assuming that the flask does not ex- pand appreciably, under what pressure will the air be? 15. If a liter of dry air at C. weighs 1.3 gm., how many liters will weigh 10 gm. if the pressure be reduced to one half and the tem- perature be raised to 100 C. ? IV. MEASUREMENT OF HEAT 317. The Unit of Heat. The unit of heat in the e.g. s. system is the calorie. It is defined as the quantity of heat that will raise the temperature of one gram of 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 16C., 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. 318. 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 258 HEAT masses of different substances differs widely. For example, if 100 gm. of water at C. be mixed' with 100 gm. at 100 C., the temperature of the whole will be very nearly 50 C. But if 100 gm. of copper at 100 C. be cooled in 100 gm. of water at 0C., the final temperature will be about 9.1C. 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. 319. 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 to raise the temperature of equal masses of the substance and of water through one degree. The specific heat of mercury is .033, that is, the heat that will raise 1 gm. of mercury through 1 C. will raise 1 gm. of water through only 0.033 C. 320. 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 gm. of iron at 98 C. are placed in 75 gm. of water at 10 C. contained in a copper beaker weighing 15 gm., 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 rep- resents the specific heat of iron, and (98 12.5) its change of temper- ature. 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- QUESTIONS AND PROBLEMS 259 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. If water were used as the substance in a thermometer, what would be the lowest temperature it would register ? 2. Give three reasons why mercury is a more suitable substance for thermometers than water. 3. If a pound of water and a pound of iron expose the same sur- face area to the direct rays of the sun, which will show the greater change of temperature in an hour? 4. If equal quantities of heat are applied to equal masses of iron and copper, which will show the greater change of temperature ? 5. Which would be the more efficient foot warmer, a rubber bag containing 5 Ib. of water at 80 C., or a 5-lb. block of iron also at 80 C. ? Give reasons. 6. If a number of balls of the same mass but of different materials are heated in boiling water and are then placed on a cake of wax, will they all melt the same quantity of wax ? Why ? 7. How many calories of heat will it take to raise the temperature of 50 gm. of water from 10 C. to 70 C.? If this heat were all applied to 1 liter of water at 2C C., to what temperature would it raise the water ? 8. If 90 gm. of mercury at 100 C. are stirred with 100 gm. of water at 20 C. and the resulting temperature is 22.3 C., what is the specific heat of mercury ? 9. If the specific heat of iron is 0.112, how much heat will be re- quired to raise the temperature of 2.5 kgm. of iron from 15 C. to 100 C.? 10. A copper ball weighing 5 kgm. is heated to a temperature of 100 C., and when placed in water raises its temperature from 20 C. to 25 C. How many grams of water are there, the specific heat of copper being 0.095? 260 HEAT V. CHANGE OF STATE 321. 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 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 will remain without change if the temperature of the room is C. ; but if the tempera- ture 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 welding and forging of iron. 322. Change in Volume 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. CHANGE OF STATE 261 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 kgm. per square centimeter. 323. 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 about 25 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 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, "fr 6cope insulated plate, and the divergence will increase again. The capacity of an insulated conductor is not dependent on its dimensions alone, but it is in- Fig. 320 creased by the presence of another conductor 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 ar- rangement 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. 306 ELECTROS TA TICS 393. 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. 310) 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. 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 dielec- tric capacity from four to ten times greater than air. Fig. 321 394. 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. 321). 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 dielec- tric separating the two tin-foil conducting surfaces. 395. Charging and Discharg- ing a Jar. To charge a Ley- den jar connect the outer surface to one pole of an electri- cal machine ( 399), either by Fig. 322 ELECTRIC POTENTIAL AND CAPACITY 307 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. 322), bring the other around near the ball, and the discharge will take place. 396. Seat of Charge. Charge a Leyden jar made with movable metallic coatings (Fig. 323) 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. Reach 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 dis- charged in the usual way. Fig. 323 This experiment was devised by Franklin ; it shows that electrification is a phenomenon of the glass, and that the metallic coat- ings serve merely as^ conductors, making it possible to discharge all parts of the glass at once. 397. Theory of the Leyden Jar. A Leyden jar may be broken by over charging, may be discharged by heating, and if heavily charged is not completely discharged by connecting the two coatings ; if left standing a few sec- onds, 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 308 ELECTROSTATICS 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 "bound," in distinction from the charge on an insulated conductor, which is said to be "free." Questions 1. Why is not a Leyden jar standing on a cake of paraffin dis- charged by touching the ball with the finger? 2. Why is the capacity of a Leyden jar increased by connecting its outer surface with the earth? Is the result the same if one coating is connected to one pole of an electrical machine and the other coat- ing to the other pole? 3. Why will a dust-covered electrified body soon lose its charge? 4. Cuneus tried to charge a bowl of water by holding it in his hand while a chain from an electrical machine dipped into it. When he lifted out the chain he got a severe shock. Why? 5. Why is it unsafe'to touch the ball first when discharging a Ley- den jar by means of a bent wire? Would it make any difference if the wire were held by an insulating handle? 6. When a charged proof plane touches the ball of an electroscope does it give up all its charge ? 7. When a charged proof plane touches the inside of a hollow insulated conductor, does it give up all its charge? Why? 8. If a condenser whose capacity is 200 c. g. s. units is charged with 2000 c.g.s. units of electricity, what is the potential difference in volts between its two coatings ( 391)? V. ELECTRICAL MACHINES 398. The Electrophorus. The simplest induction elec- trical machine is the electrophorus (Fig. 324), invented by Volta. A cake of resin or disk of vulcanite (yi) rests ELECTRICAL MACHINES 309 in a metallic base (1?). Another metallic disk or cover ((7) is provided with an insulating handle (D). The resin or vulcanite is electrified by rubbing with dry flannel or striking with a catskin, and the metal disk is then placed 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 between them form a condenser ( 392) of great capacity. Touch the cover momentarily with the finger, and the repelled negative charge passes to the earth, P. 324 leaving the cover at zero poten- tial. Lift it by the insulating handle, the positive charge becomes free ( 397), and a spark may be drawn Jby 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 the continuous transformation of mechanical work into the energy of electric charges. 399. Influence Electrical Machines. There are many influence or induction electrical machines, but it will suffice to describe only one as the principle is always the same. The Holtz machine, as modified by Toepler and Voss, is illustrated in Fig. 325. There are two glass plates, e' 310 ELECTROSTATICS and e, about 5 mm. apart, the former stationary and the latter turning about an insulated axle by means of the crank h and a belt. The stationary plate supports at Fig. 325 the back two paper sectors, c and 354 ) ; they wil1 drop off as soon as the Fig. 354 circuit is opened. The experiment shows that a conductgrjh"gh whin.h an electric current is passing lias magnetic properties. The iron filings are magnetized by the current and set themselves at right angles to the wire. When the circuit is broken, they lose their mag- netism and drop off. 443. Mapping the Magnetic Field. Support horizontally a sheet of cardboard or of glass LB with a hole through it. Pass vertically through the hole a wire, W, con- necting with a suitable electric generator, so that a strong current can be sent through the circuit (Fig. 355). Close the circuit and sift iron filings on the paper or glass about the wire, jarring the sheet by tapping it. The filings will arrange them- selves in circular lines about the wire. Place a small mounted mag- netic needle on the sheet near the wire; it will set itself tangent to the circular lines, and if the current is flowing downward, the north pole will point in the direction in which the hands of a watch move. The lines of magnetic force aboutTaT wire through "which an"elecTric current is flowing, Fig. 356 are concentric circles. Fig. Fig. 355 MAGNETIC PROPERTIES OF A CURRENT 341 356 was made from a photograph of these circular lines of force as shown by iron filings on a plate of glass. Their direction relative to the cur- rent is given by the following rule : G-rasp the wire by the right hand so that the extended thumb points in the direction of the current; then the fin- gers wrapped around the wire indicate the direction of the lines of force (Fig. 357). Fig. 357 Fig. 358 is a sketch intended to show the direction of these circular lines of magnetic force (^o^jnagnetic whirl) which everywhere surround a wire conveying a current. Fig. 358 444. Properties of a Circular Conductor. Bend a copper wire into the form shown in Fig. 359, the diameter of the circle being about 20 cm. Suspend it by a long untwisted thread, so that the ends dip into the mercury cups shown in cross section in the lower part of the figure. Send a current through the suspended wire by connecting a battery to the binding posts. A bar magnet brought near the face of the circular conductor will cause the latter to turn about a vertical axis and take up a position with its plane at right angles to the axis of the magnet. With a strong current the circle will turn under the influence of the earth's magnetism. This experjiiKmt_sJ:iow^ acts like a disk magnet, whose poles are its faces. The lines 342 ELECTRIC CURRENTS of force surrounding the conductor in this form pass through the circle and around from one face to the other through the air outside the loop. The north-seeking side is the one from which the lines issue; and to an observer looking toward this side, the current flows around the loop counter-clockwise (Fig. 360). If instead of a single turn we take a long in- sulated wire and coil it into a number of par- allel circles close together, the magnetic effect will be increased. Such a coil is called a helix or solenoid ; and the passage of an electric current through it gives to it all the properties of a cylin- drical bar magnet. 445. Polarity of a Helix. The polarity of a helix may be determined by the following rule: Crrasp the coil with the right hand so that the fingers point in the direction of the current ; the north pole will then be in the direction of the extended thumb. 446. Mutual Action of Two Currents. Make a rectangular coil of insulated copper wire by winding four or five layers around the edge of a board about 25 cm. square. Slip the wire off the board and tie the parts together in a number of places with thread. Bend the ends at right angles to the frame, remove the insulation, Fig. 359 MAGNETIC PROPERTIES OF A CURRENT 343 and give them the shape shown in Fig. 361. Suspend the wire frame by a long thread so that the ends dip into the mercury cups. Make a second similar but smaller coil and connect it in the same circuit with the rectangular coil and a battery. First. Hold the coil HK with its plane perpendicular to the plane of the coil EF, with its edge H parallel to F, and with the currents in these two adjacent portions flowing in the same direction. The suspended coil will turn upon its axis, the edge F approaching H, as if it were attracted. Second. Reverse HK so that the cur- rents in the adjacent portions K and F flow in opposite directions. The edge F of the suspended coil will be repelled by K. Third. Hold the coil HK within EF, so that their lower sides form an angle. EF will turn until the currents in its lower side are par- allel with those in H, and flowing in the same direction. These facts may be summarized in the following laws of action between cur- Fig. 361 - 362 I. Parallel cur- rents flowing in the satne direction at- tract. II. Parallel cur- rents flowing in oppo- site directions repel. III. Currents making an a ng I e with each other tend to become parallel and to flow in the same direction. 447. Magnetic Fields about Parallel Currents. Fig. 362 was made from a photograph of the magnetic field about 344 ELECTRIC CURRENTS 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. Fig. 363 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 their normal position forces the wires apart. VI. ELECTROMAGNETS 448. Effect of Introducing Iron into a Solenoid. Wind evenly on a paper tube, about 2 cm. in diameter and 15 cm. long, three layers of No. 18 insulated copper wire. Support the tube and wire in a slot cut in a sheet of cardboard. Pass an electric current through the solenoid and note the magnetic field as mapped out by iron filings (Fig. 364). Repeat after filling the tube with straight soft iron wires. The magnetic 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 Fig. 364 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 ELECTRON A GNET8 345 the greater permeability of iron as compared with air (Fig. 365). If the iron is omitted, there are not only fewer lines of force, but because of their leakage at the sides of the helix, fewer traverse the entire length of the coil. Fig. 365 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 remain- ing is called residual mag- netism. 449. Relation between a Magnet and a Flexible Con- ductor. Iron filings arranged in circles about a conductor may be regarded as flexible mag- netized iron winding itself into a helix around the current ; con- versely, a flexible conductor, carrying a current, winds itself around a straight bar magnet. The flexible conductor of Fig. 366 may be made of tinsel cord or braid. Directly the circuit Fig. 366 346 ELECTRIC CURRENTS is closed, the conductor winds slowly around the vertical magnet; if the current is then reversed, the conductor unwinds and winds up again in the reverse direction. 450. The Horseshoe Magnet. The form given to an electromagnet depends on the use to which it is to be put. The horseshoe or U-shape (Fig. 367) is the most common. The advantage of this form lies in the fact that all lines of mag- netic force are closed curves, pass- ing through the core from the south to the north pole, and com- pleting the circuit through the air from the north pole back to the .south pole. The U-shape lessens Fig. 367 the distance through the air and thus increases the number of lines. Moreover, when an iron bar, called the arma- ture, is placed across the poles, the air gap is reduced 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 armature. When the armature is in contact with the poles, the magnetic circuit is % all iron, and is said to be a closed 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. MEASURING INSTRUMENTS 347 451. Applications of Electromagnets. The uses to which electromagnets are put in the applications of electricity are so nu- merous that a mere reference to them must suffice. The electromagnet enters into the construction of electric bells, telegraph and telephone instruments, dynamos, motors, signaling devices, etc. It is also extensively used in lifting large masses of iron, such as castings, rolled plates, pig iron, and steel girders (Fig. 368). The lifting power de- pends 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 cur- rent and the number of turns of wire wound on the magnet. Fig. 368 VII. MEASURING INSTRUMENTS 452. The Galvanometer. The instrument for the com- parison of currents by means of their magnetic effects is called a galvanometer. A galvanoscope ( 413) 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 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 telescope, by means of which the deflections are read. In either case the beam of light then becomes a long pointer without weight. 453. The d'Arsonval Galvanometer. One of the most useful forms of galvanometer is the d'Arsonval. The 348 ELECTRIC CURRENTS Fig. 369 Fig. 370 plan of it is shown in Fig. 369 and a complete working instrument in Fig. 370. Between the poles of a strong permanent magnet of the horseshoe form swings a rec- tangular 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 run- ning to the base. A small mirror is attached to the coil to reflect light from a lamp or an illuminated scale. Sometimes the coil carries a light alum- inum 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 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. 454. The Voltmeter. The voltmeter is an instrument designed to measure the difference of potential in volts. For direct currents the most convenient portable volt- meter is made on the principle of the d'Arsonval galva- nometer. The appearance of one of the best-known MEASURING INSTRUMENTS 349 instruments of this class is shown in Fig. 371. The interior is represented by Fig. 372, where a portion of the instrument is cut away to show the coil and the springs. The cur- rent is led in by one spiral spring and out by the other. At- tached to the coil is a very light aluminum pointer, which moves over the scale seen in Fig. 371, where it ^ 3j stands at zero. Soft iron polepieces are screwed fast to the poles of the permanent magnet, and they are so shaped that the divisions of the scale in volts are equal. In circuit with the coil of 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. 455. The Ammeter, de- signed to measure electric currents in amperes, is very similar in construction to the voltmeter. Its coil has only a few turns of wire and its resistance is low, so that when the ammeter is placed in circuit, it will not change the value of the cur- rent to be measured. 350 ELECTRIC CURRENTS 456. Divided Circuits Shunts. When the wire leading from any electric generator is divided into two branches, as at B (Fig. 373), the current also divides, part flowing by one path and part by the other. The sum of these two currents is always equal to the current in the undi- vided part of the circuit, since there is no accumula- tion of electricity at any point. Either of the branches be- tween B and A is called a shunt to the other, and the currents through them are inversely proportional to their resistances. 457. Resistance of a Divided Circuit. Let the total resist- ance between the points A and B (Fig. 373) be represented by R, that of the branch BmA by r, a-nd of En A 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 J5n^4 are , -, and respectively; then = + . R r r K, r r From this we derive R = rr . 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 the total resistance of this divided I r\r\ y Kf\ circuit is ^2L^ - 33 1 o hms. The introduction of a shunt always 100 + 50 lessens the resistance between the points connected. 458. Loss of Potential along a Conductor. When a cur- rent flows through a conductor a difference of poten- tial exists, in general, between different points on it. Let A, B, Q be three points on a conductor conveying a current, and let there be no source of E. M. F. be- tween these points. Then if the current flows from A to B, the potential at A is higher than at B, and the poten- tial at B is higher than at O. If the potential difference MEASURING INSTRUMENTS 351 between A and B and that between B and be measured, the ratio of the two will be the same as the ratio of the resistances between the same points. This is only TjJ another statement of Ohm's law. For since /= , and the H current is the same through the two adjacent sections of the conductor, the ratio of the potential differences to the resistances of the two sections is the same. This impor- tant principle, of which great use is made in electrical measurements, may be expressed by saying that, when the current is constant, the loss of potential along a conductor is proportional to the resistance passed over. 459. Wheatstone's Bridge. The Wheatstone's Bridge is a de- vice for measuring resistances. The four conductors /2i, R 2 , R 3 , R 4 are the arms and BD the bridge (Fig. 374). When the circuit is closed by closing the key K^ 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 KI is also closed, then there is no potential difference be- tween B and D to produce a current. Under these conditions the loss of po- tential from A to B is the same as from A to D. We may then get an Fig. 374 expression for these potential differences and place them equal to each other. Let /i be the current through 7?i ; 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 ADC. Then the poten- tial difference between A and B by Ohm's law ( 435) is equal to RI\ ; and the equal potential difference between A and D is R^I 2 . Equating these expressions, jRi/i = R 2 fz () In the same way the equal potential differences between B and C 852 ELECTRIC CURRENTS and D and C give RJi = R s I 2 ........... (b) Dividing (a) by (6) gives (Equation 37) 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 KI is closed after key K*. The value of the fourth resistance is then derived from the relation in equation (37). Questions and Problems 1. An electric bell wire passes through a room and the battery is inaccessible. How may one determine the direction of the current through the wire? 2. How can it be proved that the strength of current is the same in all portions of an undivided circuit? 3. The poles of a battery are joined by a thin platinum wire, which is heated to a dull red. If a piece of ice is applied to the wire at one end, the remainder of the wire will glow more brightly. Explain. 4. Given a charged storage battery. Determine which is the positive pole. 5. While a current is passing through a helix, a small iron rod is brought near one end of the helix and in line with its axis. The iron rod will be drawn into the helix. Explain. 6. A current passing through a long elastic spiral of wire causes it to shorten. Explain. 7. Calculate the resistance of 100 ft. of copper wire (k = 10.19) No. 24 (diam. = 0.0201 in.). 8. No. 20 wire has a diameter of 0.032 in. How many feet of German silver wire (k = 181.3) will it take to make a 20-ohm coil? 9. How many feet of iron wire (k = 61.3), No. 10 (diam. = 0.1014 in), will it take to make a coil of 50 ohms resistance? 10. A current of one ampere deposits by electrolysis 1.1833 gm. of copper in an hour. How many amperes in 10 hours will deposit l.kgm. of copper? QUESTIONS AND PROBLEMS 353 11. A current of 0.5 ampere is passed through a solution of silver nitrate for 30 min. How much silver is deposited? 12. What strength of current in amperes will deposit 10 gm. of silver by electrolysis in an hour? 13. What current will a battery having an E. M. F. of 2.2 volts and an internal resistance of 0.2 ohm supply through an external resist- ance of 5 ohms? 14. How large a current will a battery of 6 cells (E. M. F., 1.5 volts each) connected in series send through an external resistance of 6 ohms, the internal resistance of each cell being 0.5? 15. If a dry cell has an E. M. F. of 1.5 volts and sends a current of 20 amperes through an ammeter (resistance negligible), what is the internal resistance of the cell? 16. What current will be derived from a Daniell cell, E. M. F. 1.1 volts, internal resistance 1 ohm, in series with a dry cell, E. M. F. 1.5 volts, internal resistance 0.2 ohm, when the external resistance is 4 ohms ? 17. A current of 10 amperes passes through a resistance of 1 ohm for half an hour. How many calories of heat are generated? 18. A current of 2.1 amperes is sent through a divided circuit of two branches, with resistances of 5 and 10 ohms respectively. Cal- culate the current in each branch. 19. If the current through an incandescent lamp is 0.55 ampere and the potential difference between its terminals 110 volts, what is the resistance of the lamp? 20. What resistance would be necessary in circuit with an electric lamp when the potential difference between its terminals is 50 volts, the pressure in the main line 200 volts, and the current through the lamp 12 amperes? CHAPTER XIII ELECTROMAGNETIC INDUCTION I. FARADAY'S DISCOVERIES 460. Electromotive Force Induced by a Magnet. Wind a number of turns of fine insulated wire around the armature of a horse- shoe magnet, leaving the ends of the iron free to come in contact with the poles of the permanent mag- net. Connect the ends of the coil to a sensitive galvanometer, the armature being in contact with the magnetic poles, as shown in Fig. 375. Keeping the mag- net fixed, suddenly pull off the armature, show a momentarv current. Fig. 375 The galvanometer will Suddenly bring the armature up to the poles of the magnet; an- other momentary current in the reverse direction will flow through the cir- cuit. Connect a coil of insu- lated wire, consisting of a large number of turns, in circuit with a d'Arsonval galvanometer (Fig. 376). Thrust quickly into the coil the north- pole of a bar magnet. The galvanome- ter will show a transient current, which will flow only during the motion of the magnet. When the magnet is suddenly withdrawn a transient current is pro'duced in the opposite direction to the first one. If the south pole be thrust into 354 FARADAY'S DISCOVERIES 355 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 momentary 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. 461. Laws of Electromagnetic Induction. When the arma- ture 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 number of mag- netic 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 threading through a coil produces an indirect electromotive force; a decrease in the number of lines produces a direct electro- motive force. II. The electromotive force induced is proportional to the rate of change in the number of lines of force threading through the coil. These two laws give the direction and value of induced electromotive forces. 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 356 ELECTR OMA GNETIC IND UCTION \ Fig. 377 direction. Thus, in Fig. 377 the north pole of the mag- net is moving into the corl~fn 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 in- direct or opposite watch hands, as shown by the arrows on the coil, to an ob- server looking at the coil in the direction of the arrow on the magnet. 462. Induction by Currents. Connect the coil of Fig. 376 to a d'Arsonval galvanometer, and a second smaller coil to the terminals of a battery (Fig. 378). If the current through P is kept con- stant, when P is made to ap- proach S an E. M. F. is gener- ated in S tending to sejid a current in a direction opposite to the current around P; re- moving the coil P generates an opposite E. M. F. These E. M. F.'s act in S only so long as P is moving. Next insert the coil P in S with the battery circuit open. If then the battery circuit is closed, the needle of the galva- nometer will be deflected, but will shortly come again to rest at zero. The direction of this momentary current is opposite to that in P. Opening the bat- tery 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. Fig. 378 FAEADAY'S DISCOVERIES 357 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 O ( H i q oJ Pri. Fig. 411 Fig. 412 or to the low voltage side of a transformer in the case of alternating currents (Fig. 412). Single lamps are turned off usually by the key in the socket (Fig. 409), and groups of lamps by a switch 8. IV. THE ELECTRIC TELEGRAPH 493. The Electric Telegraph is a system of transmitting messages by means of simple signals through the agency of an electric current. Its essential parts are the line, the transmitter or key, the receiver or sounder, and the lattery. 494. The Line is an iron, copper or phosphor-bronze wire, insulated from the earth except at its ends, and serving to connect the signaling apparatus. The ends of this conductor are connected with large metallic plates, or with gas or water pipes, buried in the earth. By this means the earth becomes a part of the electric circuit containing the signaling apparatus. 495. The Transmitter or Key (Fig. 413) is merely a current Fig. 413 THE ELECTRIC TELEGRAPH 385 Fig. 414 interrupter, and usually consists of a brass lever A, turn- ing about pivots at B. It is connected with the line by the screws O and D. When the lever is pressed down, a platinum point projecting under the lever is brought in contact with another platinum point J3, thus closing the circuit. When not in use, the circuit is left closed, the switch F being used for that purpose. 496. The Receiver or Sounder (Fig. 414) con- sists of an electromag- net A with a pivoted armature B. When the circuit is closed through the terminals D and E, the armature is attracted to the magnet, producing a sharp click. When the circuit is broken, a spring causes the lever to rise and strike the back stop with a lighter click. 497. The Relay. When the resistance of the line is large, the current is not likely to be strong enough to oper- ate the sounder with sufficient energy to ren- der the signals t^^^_ '^^**Ht distinctly audi- ^ - "r^K ble. To remedy this defect, an electromagnet, called a relay (Fig. 415), whose helix A is composed of many turns of fine wire, is placed in the circuit by means of its terminals Q and D. As its armature moves to and Fig. 415 886 DYNAMO-ELECTRIC MACHINERY fro between the points at K^ it opens and closes a second and shorter circuit through j^and F, in which the sounder is placed. Thus the weak current, through the agency of the relay, brings into action a current strong enough to do the necessary work. 498. The Battery consists of a large number of cells, usually of the gravity type, connected in series. It is generally divided into two sections, one placed at each terminal station, these sections being connected in series through the line. The principal circuits of the great telegraph companies are now worked by means of currents from dynamo machines. 499. The Signals are a series of sharp and light clicks separated by intervals of silence of greater or less dura- tion, a short interval between the clicks being known as a "dot," and a long one as a "dash." By a combination of "dots" and " dashes," letters are represented and words are spelled out. 500. The Telegraph System described in the preceding sections is known as Morse's, from its inventor. Fig- ure 416 illustrates diagrammatically the instruments necessary for one terminal station, together with the mode of connection. The arrange- ment at the other end of the line is an exact duplicate of this one, the two sections of the battery being Sounder |i|ilh Line Battery ~~^parth Fig. 416 placed in the line, so that the negative pole of one and the positive of the other are connected with the earth. THE ELECTRIC TELEGRAPH 387 At intermediate stations the relay and the local circuit are connected with the line in the same manner as at a terminal station. 501. The Electric Bell (Fig. 417) is used for sending sig- nals as distinguished from messages. Besides the gong, it contains an electromagnet, having one terminal connected directly with a binding-post, and the other through a light spring attached to the arma- ture (shown on the left of the figure) and a contact screw, with another binding-post. One end of the arma- ture is supported by a stout spring, or on pivots, and the other carries the bent arm and hammer to strike the bell. Included in the circuit are a battery and a push-button B, shown with the top unscrewed in Fig. 418. Fig. 417 When the spring E is brought into contact with D by pushing (7, the circuit is closed, the elec- tromagnet attracts the armature, and the hammer strikes the gong. The move- ment of the armature opens the circuit by breaking contact between the spring and the point of the screw ; the arma- ture is then released, the retractile spring at the bottom carries it back, and con- tact is again established between the spring and the screw. The whole opera- tion is repeated automatically as long as the circuit is kept closed at the push-button. A " buzzer " is an electric bell without the hammer and gong. Fig. 418 388 DYNAMO-ELECTRIC MACHINERY V. THE TELEPHONE 502. The Telephone (Fig. 419) consists of a permanent magnet 0, one end of which is surrounded by a coil of many turns of fine copper wire 5, whose ends are connected with the binding-posts and t. At right angles to the magnet, and nob quite touching the pole -within the coil is an elastic diaphragm or disk a of soft sheet- iron, kept in place by the conical mouthpiece d. If the instrument is placed in an elec- tric circuit when the current is unsteady, or alternating in direction, the magnetic field due to the helix, when combined with that due to the magnet, alters intermittently the number of lines of force which branch Fig. 419 out from the pole, thus varying the attraction of the mag- net for the disk. The result is that the disk vibrates in exact keeping with the changes in the current. 503. The Microphone is a device for varying an electric current by means of a variable resistance in the circuit. One of its simplest forms is shown in Fig. 420. It consists of a rod of gas-carbon A, whose tapering ends rest loosely in coni- cal depressions made in blocks of .the same material attached to a sounding board. These blocks are placed in circuit with a bat- tery and a telephone. While the current is passing, the least motion of the sounding board, caused either by sound waves or by any other means, such as the ticking of a Fig. 420 THE TELEPHONE 389 watch, moves the loose carbon pencil and varies the pres- sure between its ends and the supporting bars. A slight increase of pressure between two conductors resting loosely one on the other lessens the resistance of the contact, and conversely. Hence, the vibrations of the sounding board cause variations in the pressure at the points of contact of the carbons, and consequently make corresponding fluctuations in the current and vibrations of the telephone disk. 504. The Solid Back Transmitter. The varying resist- ance of carbon under varying pressure makes it a valuable material for use in telephone transmitters. Instead of the loose contact of the microphone, carbon in granules between carbon plates is now commonly employed. The form of transmitter exten- sively used for long distance work is the "solid back" transmitter (Fig. 421). The figure shows only the essential parts in section, minor details being omitted. M is the mouthpiece, and F and C the front and back parts of the metal case. The aluminum diaphragm D is held around its edge by a soft rubber ring. The metal block W has a recess in front to receive the carbon electrodes A and B. Between them are the carbon granules. The block JS is attached to the diaphragm and is insulated from W except through the carbon granules. The transmitter is placed in circuit by the wires connected to IF and E. Provision is made for an elastic motion of the diaphragm and the block E. Sound waves striking the diaphragm Fig. 421 390 DYNAMO-ELECTRIC MACHINERY cause a varying pressure between the plates and the car- bon granules. This varying pressure varies the resist- ance offered by the granules and so varies the current. The transmitter is in circuit in the line with the primary of a small induction coil, the secondary being in a local circuit with the telephone receiver. The induced currents in the secondary have all the peculiarities of the primary current; and when they pass through a receiver, it re- sponds and reproduces sound waves similar to those which disturb the disk of the transmitter. VI. WIRELESS TELEGRAPHY 505. Oscillatory Discharges. The discharge of any con- denser through a circuit of low resistance is oscillatory. The first rush of the discharge surges beyond the condi- tion of equilibrium, and the condenser is charged in the opposite sense. A re- verse discharge follows, and so on, each successive pulse being weaker than the pre- ceding, until after a few surges the oscillations P. 422 cease. Figure 422 was made from a photograph of the oscillatory discharge of a condenser by means of a very small mirror, which reflected a beam of light on a falling sensitized plate. Such alternating surges of high frequency are called electric oscillations. Joseph Henry discovered long ago that the discharge of a Leyden jar is oscillatory. 506. Electric Waves. In 1887-1888 Hertz made the discovery that electric oscillations give rise to electric 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. WIRELESS TELEGRAPHY 391 waves in the ether, which appear to be the same as waves of light, except that they are very much longer, or of lower frequency. They are capable of reflection, refrac- tion, and polarization the same as light. Evidence of these waves may be readily obtained by setting up an induction coil, with two sheets of tin-foil on glass, Q and $', connected with the terminals of the sec- ondary coil, and with two discharge balls, F and F', as shown in Fig. 423. So simple a device as a large picture Fig. 423 frame with a conducting gilt border may be used to detect waves from the tin-foil sheets. If the frame has shrunken so as to leave narrow gaps in the miter at the corners, minute sparks may be seen in a dark room breaking across these gaps when the induction coil produces vivid sparks between the polished balls, .Fand F' . The plane of the frame should be held parallel with the sheets of tin-foil. The passage of electric waves through a conducting circuit produces electric oscillations in it, and these oscillations cause electric surges across a minute air gap. 507. The Coherer. A very sensitive device for the detection of electric waves is the coherer (Fig. 424). When metallic filings are placed loosely between solid 392 DYNAMO-ELECTRIC MACHINERY Fig. 424 electrodes in a glass tube they offer a high resistance to the passage of an electric current ; but when electric oscil- lations are produced in the neighborhood 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 ( 497). If the tube is slightly jarred, the filings resume their state of high resistance. A slight discharge from the cover of an electrophorus ( 398) through the filings lowers the re- sistance just as electric oscillations do. It is thought that minute sparks between the filings partially weld them to- gether and make them conducting. 508. Wireless Telegraph Set. The connections and parts of a simple wireless telegraph set are shown in Fig. 425. One ball of the secondary of the induction coil is con- * O Spark Gap Earth f pi Coherer gJ Filings M Local Battery Fig. 425 nected to the earth and the other to a vertical wire mast for transmitting electric waves. At the receiving station a similar wire mast is connected to one end of a coherer which is joined in series with a battery and a relay. The WIRELESS TELEGRAPHY 393 make and break circuit of the relay includes a sounder operated by another battery. When the transmitter and the receiver are both properly adjusted, the closing of the key at the sending station produces sparks at the spark gap of the secondary coil ; the electric waves sent out by the vertical mast are received by the corresponding one at the receiving station, the coherer has its resistance lowered so that the battery works the relay, arid the sounder responds. The sounder is usually made to tap the coherer for the purpose of restoring its sensitiveness for the recep- tion of the next waves. Wireless messages are transmitted further over the ocean than over the land. In fact the most conspicuous application of wireless telegraphy is between ships at sea, or between them and shore stations. It has proved of inestimable value in the timely assistance it' has brought to disabled or sinking vessels. Ocean liners are in daily communication with other vessels or with maritime sta- tions. Wireless systems of communication are now a part of the equipment of naval vessels. They may thus be kept in touch with one another and with the navy depart- ment of the governments they represent. APPENDIX I. 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. 426) 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. 427) 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 J5, and draw an arc cutting the adjacent side at C. 394 Fig. 427 GEOMETRICAL CONSTRUCTIONS 395 Cut the card into two parts along the straight line BC. 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. . To con- struct an angle of 90. Let A be the ver- D A Fig. 428 E tex of the required angle (Fig. 428). 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 "/ m PROS. 2. To construct an angle of 60. Let A be the vertex of the re- quired angle (Fig. 429), 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 line AE; this line will make an angle of 60 with AB. C Fig. 429 396 APPENDIX PROB. 3. To bisect an angle. Let BAG be an angle that it is required to bisect (Fig. 430). 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; it is required to make a second angle equal to it (Fig. 431). 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 C D Fig. 431 E mn, draw the arc op, cutting rs at G. Through D and G draw the line DF. 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. 432). Through A draw ED, GEOMETRIC A L CONS TR UCTIONS 397 cutting BC at D. At A make the angle EAG equal to EDO. Then AG or FG is parallel to BC. G Fig. 432 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. 433). With C as a center and a radius equal to AB, W r-. Fig. 433 draw the arc mri. With B as a center and a radius equal to AC, draw the arc rs, cutting mn at .D. Draw CD and BD. Then ABDC is the required parallelogram. 398 APPENDIX II. CONVERSION TABLES 1. LENGTH To reduce Multiply by To reduce Multiply by Miles to km. . 1 60935 Kilometers to mi. . . 0.62137 Miles to m. 1609.347 Meters to mi. . . . 0.0006214 Yards to m 91440 Meters to yd 1 09361 Feet to m 0.30480 Meters to ft 3.28083 Inches to cm. 2 64000 Centimeters to in. . . 0.39370 Inches to mm. 25.40005 Millimeters to in . 0.03937 2. SURFACE To reduce Multiply by To reduce Multiply by Sq. yards to m. 2 . . . 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 . 0.00155 3. VOLUME To reduce Multiply by To reduce Multiply by Cu. yards to m. 3 . . . 0.76456 Cu. meters to cu. yd. . . 1.30802 Cu. feet to m 3 . . . 0.02832 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 kgm 907.18486 Kilograms to tons . . 0.001102 Pounds to kgm. . . . 0.45359 Kilograms to Ib. . . 2.20462 Ounces to gin. . . . 28.34953 Grams to oz. ... 0.03527 ' Grains to gm 0.064799 Grams to grains . . 15.43236 CONVERSION TABLES 399 5. FORCE, WORK, ACTIVITY, PRESSURE To reduce Multiply by Lb.-weight to dynes, . 444520.58 Ft.-lb. to kgm.-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 watte .... 745.196 Lb. per sq. ft. to kgin. per m. 2 4.8824 Lb. per sq. in. to gin. per cm. 2 70.3068 Calculated for g = 980 cm., To reduce Multiply by Dynes to Ib.-weight, 22496 x 10' 10 Kgm.-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 Kgm. per m. 2 to Ib. per sq- ft 0.2048 Gm. per cm. 2 to Ib. per sq. in 0.01422 or 32.15 ft. per sec. per sec. 6. MISCELLANEOUS 0.01602 0.004329 14.69640 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 4 C. . Cu. in. to U.S. gal. . . Atmospheres to Ib. per sq. in Atmospheres to gm. 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 Gm. 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 400 APPENDIX III. 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 / Volume of cone = (base x altitude). a)(sb) (s c) where s= base x altitude. Altitude x \ sum of parallel sides. diameter x 3.1416. f circumference * 3.1416. 1 circumference x 0.3183. t diameter squared x 0.7854. I radius squared x 3.1416. product of diameters x 0.7854. | (sum of sides x apothem). circumference of base x altitude. = area of base x altitude. t diameter x circumference. 14 X 3.1416 x square of radius. ( diameter cubed x 0.5236. "if Of radius cubed x 3.1416. = | (circumference of base x slant height). = \ (area of base x altitude). Lord Rayleigh (John William Strutt) was born at Essex in 1842, and graduated from Cambridge University in 1865. In 1 884 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 contributions of the first im- portance to modern electrical measurements. TABLE OF DENSITIES 401 IV. TABLE OF DENSITIES The following tables gives the mass in grams of 1 cm. 3 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 Alurn, common .... 1.724 Aluminum, wrought . . 2.670 Antimony, cast .... 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 . .1.26 to 1.800 Coal, bituminous . 1.27 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 . . 0.0000896 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 ..... 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 0.835 Silver, wrought . . . 10.56 Sodium 0.970 Steel 7.816 Sulphuric Acid . . . 1.84 Sulphur 2.033 Sugar, cane .... 1.593 Tin, cast 7.290 Water, at C. . . . 0.999 Water, at 20 C. . . . 0.998 Water, sea ..... 1.027 Zinc, cast 7.000 402 APPENDIX M 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 MN (Fig. 434) be the surface separating air from water, AB the incident ray at JB, and BE the normal. With B as a center and a radius BA draw the arcs mn and Cs. With the same center and a radius f of AB, (-f 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 C. Draw BC. This will be the refracted ray, because = -, the index of refraction. \j Jj O 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. Fig. 434 GEOMETRICAL CONSTRUCTION 403 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. 435, 436). Draw the normal, 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, C'B, 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 N before. Through Fi S- 436 the intersection of BD produced with xy, .draw It parallel to the normal CD, cutting uv at /. Through D and I draw DH] this will be the path of the ray after emergence. INDEX [References are to pages.} Aberration, chromatic, 227 ; spherical, 200, 217. Absolute, scale of temperature, 252 ; unit of force, 84 ; zero, 253. Absorption spectra, 230. Accelerated motion, 74. Acceleration, 73 ; centripetal, 79 ; of gravity, 99. Achromatic lens, 227. Action of points, 301. Adhesion, 6 ; selective, 7. Aeroplane, 278. Agonic line, 292. Air, compressibility of, 57 ; com- pressor, 60 ; pressure of, 50 ; weight of, 49. Air brake, 69. Air columns, laws of, 172. Air pump, 61 ; experiments with, 62. Airships, 64. Alternator, 378. Altitude by barometer, 55. Ammeter, 349. Ampere, 334. Amplitude, 113. Analysis of light, 226. Aneroid barometer, 54. Annealing, 10. Anode, 318. Antinode, 170. Arc, carbon, 381 ; inclosed, 382 ; open, 382. Archimedes, principle, 41. Armature, 346, 371 ; drum, 375. Artesian well, 38. Athermanous substances, 272. Atmosphere, a unit of pressure, 53. Atmospheric electricity, 313. Attraction, electrical, 293 j molecular, 19, 25. Aurora, 315. Balance, 132. Balloons, 64. Barometer, aneroid, 54; mercurial, 54 ; utility of, 55. Baroscope, 64. Battery, storage, 331. Beam of light, 181. Beats, 161. Bell, electric, 387. Blind spot, 225. Body, 1. Boiling, 263. Boiling point, effect of pressure, 264 ; on thermometer, 245. Bottle imp, 43. Boyle's law, 58; inexactness of, 59. Bright line spectra, 230. Brittleness, 9. Buoyancy, 40; of air, 63; center of, 42. Caloric, 242. Calorie, 257. Camera, photographer's, 223. 405 406 INDEX Capacity, dielectric, 306 ; electro- static, 304 ; thermal, 257. Capillarity, 22 ; laws of, 22 ; re- lated to surface tension, 24. Capstan, 134. Cartesian diver, 43. Cathode, 318 ; rays, 365. Caustic, 201. Cell, voltaic, 317 ; chemical action, in, 318. Center, of buoyancy, 42 ; of gravity, 99 ; of oscillation, 114 ; of percussion, 114 ; of suspension, 113. Centrifugal force, 109; illustra- tions of, 110 ; its measure, 110. Centripetal force, 109. Charge, residual, 307 ; seat of, 307. Charles, law of, 252. Choke coil, 379. Chord, major, 162 ; minor, 162. Chromatic aberration, 227. Circuit, closing and opening, 318 ; divided, 350 ; electric, 318. Circular motion, 71. Clarinet, 171. Clinical thermometer, 247. Coherer, 391. Cohesion, 6. Coil, choke, 379 ; induction, 359 ; primary,. 357 ; secondary, 357. Cold by evaporation, 263. Color, 233 ; complementary, 237 ; mixing, 235 ; of opaque bodies, 234 ; of transparent bodies, 234 ; primary, 236. Commutator, 372. Composition of forces, 88 ; of velocities, 91. Compressibility of gases, 57. Concave, lens, 210 ; mirror, 193 ; focus of, 194, 213. Condenser, 305 ; action of, 361. Conduction, of electricity, 332 ; of heat, 267. Conductor, electrical, 296 ; charge on outside, 300 ; magnetic field about, 340. Conservation of energy, 124. Convection, 270. Convex, lens, 210 ; mirror, 193 ; focus of, 194, 212. Cooling, law of, 272. Corpuscles, 370. Coulomb, 302. Couple, 87. Critical angle, 208. Crookes tubes, 365. Crystallization, 26. Current, electric, 316 ; convection, 270 ; detection of, 321 ; heating effects of, 338 ; induced by cur- rents, 356 ; induced by magnets, 354 ; magnetic properties of, 340 ; mutual action of, 342 ; strength of, 334. Curvilinear motion, 79. Cyclonic storms, 55. Daniell cell, 324. Day, sidereal, 13 ; solar, 13. Declination, 291. Density, 44 ; of a liquid, 46 ; of a solid, 44 ; bulb, 47 ; tables of, 401. Derrick, 134. Deviation, angle of, 205. Dew point, 266. Diamagnetic body, 282. Diathermanous body, 272. Diatonic scale, 163. Dielectric, 305 ; capacity, 306 ; in- fluence of, 306. Diffraction, 240. INDEX 407 Diffusion, 16, 18, 19. Dimensions, 3 ; measurement of, 11. Dipping needle, 291. Discharge, oscillatory, 314, 390. Dispersion, 226. Drum armature, 375. Dry cell, 326. Dryness, 266. Ductility, 9. Dynamo, 371 ; compound, 375 ; series, 375 ; shunt, 375. Dyne, 84. Earth, a magnet, 290. Ebullition, 263. Echo, 153. Efficiency, 128. Effusion, 17. Elasticity, 28 ; limit of, 28 ; of form, 28 ; of volume, 28. Electric, bell, 387 ; circuit, 318 ; current, 316 ; current detection, 321 ; motor, 376 ; telegraph, 384 ; waves, 390. Electrical, attraction, 293 ; distri- bution, 300 ; field, 298 ; ma- chines, 309 ; potential, 302 ; repulsion, 293 ; resistance, 332 ; wind, 301. Electrification, 293 ; atmospheric, 313; by induction, 298; kinds of, 294 ; nature of, 297 ; simulta- neous development of both kinds of, 296 ; unit of, 302. Electrode, 318. Electrolysis, 327 ; laws of, 329 ; of copper sulphate, 328; of sodium sulphate, 327 ; of water, 329. Electrolyte, 317, 318. Electromagnet, 344. Electromotive force, 318, 335 ; in- duced by magnets, 354 ; induced by currents, 356. Electron theory of matter, 369. Electrophorus, 308. Electroplating, 330. Electroscope, 295. Electrostatic, capacity, 304 ; in- duction, 298. Electrostatics, 293. Electrotyping, 330. Energy, 1, 120 ; conservation of, 124 ; dissipation of, 124 ; kinetic, 120 ; measure of, 122 ; potential, 120 ; transformation of, 123. English system of measurement, 11. Equilibrant, 87. Equilibrium, 87 ; kinds of, 101 ; molecular, 27 ; of floating bodies, 42. Erg, 118. Ether, 177. Evaporation, cold by, 263. Expansion, coefficient of, 251 ; force of. 254 ; of gases, 57 r 250 ; of liquids, 250 ; of solids, 248. Extension, 3-. Eye, 224. Falling bodies, 104 ; laws of, 106. Field, electrical, 298 ; magnetic, 287. Field magnet, 371, 374. Floating bodies, 42. Fluids, 31 ; characteristics of, 31 ; pressure in, 31. Fluorescope, 368. Flute, 171. Focus, 194 ; conjugate, 196, 213 ; of lens, 212 ; of mirrors, 194. Foot, 5. Foot pound, 118. 408 INDEX Force, 83 ; composition of, 86 ; graphic representation of, 85 ; how measured, 84 ; lines of, 292 ; molecular, 19 ; moment of, 130 ; of expansion and contraction, 254 ; parallelogram of, 88 ; reso- lution of, 89 ; units of 83. Force pump, 68. Forced vibrations, 155. Fountain, 39; vacuum, 63. Fraunhofer lines, 231. Freezing point, 245; mixtures, 262. Friction, 127; uses of, 128. Fundamental, tone, 168; units, 13. Fusion, 260; heat of, 261. Gallon, 12. Galvanometer, d 1 Arson val, 347. Galvanoscope, 321. Gas engine, 276. Gases, 32; compressibility of, 32; expansion of, 250 ; media for sound, 149 ; thermal conductivity of, 268. Gassiot's cascade, 363. Gauge, water, 38. Geissler tube, 364. Geometrical constructions, 394, 402. Grain, 13. Gram, 12. Gramme ring, 373. Graphic methods in sound, 174. Gravitation, 98 ; law of, 100. Gravitational unit of force, 83. Gravity, 99; acceleration of, 99; cell, 325; center of, 99; direction of, 99; specific, 44, 48. Hardness, 9. Harmonic, curve, 145; motion, 71, 80. Harmonics, 170. Heat, 242; conduction of, 267; con- vection of, 270; due to electric current, 338; from mechanical action, 274; kinetic theory of, 242; lost in solution, 262; me- chanical equivalent of, 274; measurement of, 257 ; nature of, 242; of fusion, 261; of vaporiza- tion, 265, radiant, 271; related to work, 274; specific, 258; transmission of, 267. Helix, 342; polarity of, 342. Holtz machine, 309. Hooke's law, 29. Horizontal line or plane, 99. Horse power, 120. Hydraulic, elevator, 34; press, 33. Hydrometer, 47. Hydrostatic paradox, 37. Hypothesis, 2. Images, by lenses, 214; by mirrors, 189, 199; by small openings, 183. Impenetrability, 4. Impulse, 94. Incandescent lamp, 383. Inclination. 290. Inclined plane, 137; mechanical advantage of, 137. Index of refraction, 204. Induced magnetism, 284. Induction, charging by, 300; coil, 359; electrostatic, 298; magnetic, 285 ; self-, 358. Inertia, 4. Influence machine, 309. Insulator, 297. Intensity, of illumination, 184; of sound, 159. Interference, of light, 238; of sound, 160. INDEX 409 Intervals, 162; of diatonic scale, 163; of tempered scale, 165. Ions, 318. Isobars, 56. Isoclinic lines, 291. Isogonic lines, 292. Joule, 119. Joule's equivalent, 274. Kaleidoscope, 193. Keynote, 163. Kilogram, 12. Kilogram meter, 118. Kinetic energy, 120; measure- ment of, 122. Kinetic theory, 18; of heat, 242. Kundt's dust figures, 177. Lamp, arc, 382; incandescent, 383. Lantern, projection, 222. Law, Boyle's, 58; Lenz's 357; Ohm's 336; of Charles, 252; of cooling, 272; of electromagnetic induction, 355; of electrostatic action, 295; of falling bodies, 106; of gravitation, 100; of heat radiation, 271; of magnetic ac- tion, 284; of machines, 127; Pascal's, 32; physical, 2. Laws, of motion, 93; of air col- umns, 172; of strings, 167. Leclanch.6 cell, 326. Length, 11. Lens, 210; achromatic, 228; focus of, 212; images by, 214. Lenz's law, 357. Lever, 130 ; mechanical advantage of, 132. Leyden jar, 306 ; theory of, 307 ; charging and discharging, 306. Lift pump, 67. Light, 179; analysis of, 226; propagation of, 181 ; reflection of, 187; refraction of, 203; speed of, 180 ; synthesis of, 227. Lightning, 313 ; rod, 313. Lines, agonic, 292 ; isoclinic, 291 ; of magnetic force, 292. Liquefaction, 260. Liquid, 2, 32 ; cohesion in, 7 ; compressibility of, 32 ; density pf, 46 ; downward pressure, 35 ; expansion of, 250 ; in connected vessels, 38 ; medium for sound, 149 ; surface level in, 38 ; surface tension in, 21 ; thermal con- ductivity of, 268 ; velocity of sound in, 161. Liter, 11. Local action, 322. Lodestone, 281. Longitudinal vibrations, 144. Loudness of sound, 159. Machine, 126 ; efficiency of, 128 ; electrical, 311; law of, 127; mechanical advantage of, 129 ; simple, 129. Magdeburg hemispheres, 63. Magnet, artificial, 282; bar, 282; electro-, 344; horseshoe, 282; natural, 281. Magnetic, action, 284 ; axis, 283 ; field, 287 ; lines of force, 287 ; meridian, 283; needle, 283; polarity, 282 ; substance, 282 ; transparency, 283. Magnetism, induced, 284 ; nature of, 286 ; permanent and tem- porary, 285 ; terrestrial, 290 ; theory of, 286. Magnets, 281. 410 INDEX Major chord, 162. Malleability, 9. Manometric flame, 175. Mass, 6 ; units of, 12. Matter, 1 j properties of, 3 ; states of, 2. Mechanical advantage, 129. Mechanical equivalent of heat, 274. Mechanics, of fluids, 31 ; of solids, 83. Melting point, 260 ; effect of pressure, 261. Mensuration rules, 400. Meter, 11. Metric system, 11. Micrometer caliper, 140. Microphone, 388. Microscope, compound, 228 ; simple, 219. Minor chord, 162. Mirror, 189 ; focus of, 194 ; images by, 189, 197 ; plane, 189 ; spherical, 193. Mobility, 31. Moistness, 266. Molecular, forces, 19, 25 ; motion, 17 ; physics, 16. Moment, of a force, 130. Momentum, 93. Monorail car, 5. Motion, 71; accelerated, 74; curvilinear, 71, 79 ; harmonic, 71, 80 ; laws of, 94 ; molecular, 17 ; pendular, 64 ; periodic, 89 ; rectilinear, 71 ; rotary, 71 ; uniform, 72 ; vibratory, 80. Motor, electric, 376. Musical, scales, 162 sounds, 157. Octave, 162. Ohm, 333. Ohm's law, 336. Opaque bodies, 179. Opera glass, 222. Optical, center, 211 ; instruments, 219 ; constructions, 402. Organ pipe, 171. Oscillation, center of, 114 ; electric, 390. Ounce, 13. Overtones, 170, 174. Partial tones, 170. Pascal, experiments, 52 ; law, 32. Pendular motion, 112. Pendulum, applications of, 115 ; laws of, 113; seconds, 115; simple, 111. Percussion, center of, 114. Period of vibration, 113. Periodic motion, 8. Permeability, 289. Phenomenon, 1. Photographer's camera, 223. Photometer, 185. Photometry, 184. Physical, law, 2 ; measurements, 10. Physics, 1. Pigments, 238. Pitch, 158; limits of, 165; meas- urement of, 158 ; relation to wave length, 158 ; of screw, 139. Plumb line, 99. Pneumatic appliances, 66. Points, action of, 301. Polarization, 323. Polarity of helix, 342. Poles of a magnet, 283. Porosity, 7. Potential, difference of, 302 ; energy, 121 ; loss of, 350 ; unit difference of, 303 ; zero, 304. INDEX 411 Pound, 12. Power, 119. Pressure, of fluids, 31 ; at a point in fluids, 36 ; air, 58 ; down- ward, 35 ; effect on boiling point, 264 ; effect on melting point,'264 ; independent of shape of vessel, 36 ; rules for comput- ing, 37. Principle of Archimedes, 41. Prism, 206 ; angle of deviation, 207. Proof plane, 296. Properties of mattery 3. Pulley, 135 ; mechanical advantage of, 136 ; systems of, 135. Pump, air, 61 ; compression, 60 ; force, 68 ; lift, 67. Quality of sounds, 159 ; due to overtones, 159. Radiation, 271 ; laws of, 271. Radioactivity, 369. Rainbow, 228. Rays of light, 181. Reflection, diffused, 188 ; law of, 187 ; multiple, 192 ; of light, 187 ; of sound, 153 ; regular, 187 ; total, 208. Refraction, cause of, 21 ; atmos- pheric, 207 ; laws of, 206. Regelation, 261. Relay, 385. Resistance, 129 ; electrical, 332 ; laws of, 333 ; unit of, 333. Resolution, of a force, 89 ; of a velocity, 91. Resonance, 155, 156. Resonator, HelrnLoltz's, 157. Resultant, 86. Riveting hammer, 70. Roentgen rays, 366. Scale, absolute, 252; diatonic, 163; tempered, 164. Screw, 138 ; mechanical advantage of, 139. Second, 13. Secondary or storage battery, 331. Seconds pendulum, 115. Self-induction, 358. Shadows, 182. Sidereal day, 13. Sight, 225. Singing flame, 161. Siphon, 66 ; intermittent, 67. Solar day, 13. Solenoid, 342 ; polarity of, 342. Solids, 2 ; .density of, 44 ; ex- pansion of, 248 ; thermal con- ductivity of, 267 ; velocity of sound in, 152. Solution, 25 ; saturated, 26 ; heat lost in, 262. Sonometer, 167. Sound, 143, 148 ; air as a medium, 149 ; liquids as media, 149 ; loudness of, 157 ; musical, 157 ; quality, of, 159 ; reflection of, 153 ; sources of, 148 ; trans- mission of, 149 ; velocity of, 150 ; waves, 150. Sounder, telegraph, 385. Specific gravity, 44, 48 ; bottle, 46. Specific heat, 258. Spectroscope, 232. Spectrum, solar, 226 ; kinds of, 229. Spherical aberration, 200, 217. Spheroidal state, 263. Spherometer, 140. 412 INDEX Stability, 102 ; angular, 43. Stable equilibrium, 101. Steam engine, 275. Steelyard, 132. Storage cell, 331. Strain, 96. Strength of an electric current, 334 ; methods of varying, 336. Stress, 96, Strings, laws of, 167. Sublimation, 263. Submarine boat, 44. Substance, 1. Surface tension, 21 ; illustrations of, 21. Suspension, center of, 113. Sympathetic vibrations, 155. Synthesis of light, 227. Tables, conversion, 398 ; density, 401. Telegraph, electric, 384 ; key, 384 ; signals, 386 ; system, 386 ; wireless, 390. Telephone, 388. Telescope, astronomical, 221 ; Galileo's, 222. Temperature, 242 ; measuring, 243. Tempered scale, 164. Tempering, 10, 165. Tenacity, 8. Theory, 2. Thermal capacity, 257 . Thermometer, 244 ; clinical, 247 ; limitations of, 247 ; scales, 245. Thunder, 313. Time, 13. Tone, fundamental, 168; partial, 170. Torricellian experiment, 52. Transformer, 378. Translucent bodies, 179. Transmission of heat, 267. Transmitter, 389. Transparent bodies, 179. Transverse vibrations, 143. Trombone, 171. Tuning fork, 148. Units, 10; of heat, 257; of length, 11; of mass, 12; of time 13. Vacuum, Torricellian, 52. Vaporization, 262; heat of, 265. Velocity, 71; composition of, 91; constant, 72; of light, 180; of molecules, 18; of sound, 150 ; resolution of, 91; uniform, 72; variable, 72. Ventral segments, 170. Vertical line, 99. Vibration, amplitude of, 113 ; clas- sified, 143; complete, 113, 143; forced, 155; longitudinal, 144; of strings, 167; period of, 113; single, 113; sympathetic, 155; transverse, 143; rate measured, 195. Viscosity, 31. Volt, 335. Voltaic cell, 316; chemical action in, 318. . Voltameter, 335. Voltmeter, 348. Voss electrical machine, 309. Water, gauge, 38; waves, 147. Watt, 120. Wave motion, 144. Waves, 144; longitudinal, 146; electric, 390; length, 147; sound, 150 ; transverse, 144 ; water, 147, Wedge, 138. INDEX 413 Weight, 6; law of, 101; of air, 49. Weston normal cell, 335. Wheatstone's bridge, 351. Wheel and axle, 133; mechanical advantage of, 133. Whispering gallery, 154. Wireless telegraph, 390. Work, 47; units of, 118; useful, 128; wasteful, 128. X-rays, 367. Yard, 12. UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the last date stamped below. Fine saddle: 2 5' cents fourth day overdue** 14 1947 APR 27 1948 140ct'498C REC'D LD APR 14 1957. BP 16 1957 S REC'3 LD I3EC3-1959 25JUL'64Gf 7Jan'53SS P *5 1953 LU LD 7J957 100ct'58BB ; REC'D LD 1 7 1959 LD21-100m-12,'46(A2012s 6)4120 17Dec'59WW 4J6 rf?c C Z ' THE UNIVERSITY OF CALIFORNIA LIBRARY