IStS->-1S^-"^ i> fet*ci^;, y%s¥ Si€?\?s*.Vi;..^: :^«S:.- ■^•- Cornell University Library The original of tiiis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924062544907 CORNELL UNIVERSITY LIBRARY 924 062 544 907 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1992. CORNELL UNIVERSITY LIBRARY BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND GIVEN IN 1 89 1 BY HENRY WILLIAMS SAGE A COLLEGE TEXT-BOOK OF PHYSICS BY ARTHUR L. KIMBALL, Ph. D. PROFEBBOB OF PHYSICB IK AMHERBT COLLEGE. SECOND EDITION, REVISED NEW YORK HENRY HOLT AND COMPANY 1917 COPTRIQHT, 1911, 1917, BY HENRY HOLT AND COMPANY PREFACE In offering this work to my fellow teachers, a word of explana- tion is due. The book was undertaken some years ago when the writer felt the want of a text-book adapted to the needs of students taking the general first year course in college. As the work has slowly progressed several text-books of very similar aim have appeared, and it must be admitted that the call is not so impera- tive now as formerly; and yet it is hoped that the treatment here presented may meet some still existing demand and so justify its existence. What may be called the physical rather than the mathematical method has been preferred in giving definitions and explana- tions, because it is believed that the ideas presented are more easily grasped and more tenaciously held when the mind forms for itseK a sort of picture of the conditions, instead of merely associating them with the symbols of a formula. There are many minds that do not easily grasp mathematical reasoning even of a simple sort; and it is often the case also that a student who may be able to follow an algebraic deduction step by step has very little idea of the significance of the whole when he reaches the end. Algebra is not his native tongue and it takes considerable time and experience for him to learn to think in it. And while all will agree that for the more advanced study of physics, mathematics is quite indispensable, many will grant that in a general course, which is to furnish to most of those taking it all that they will ever know of physics as a science, the ideas and reasonings should be presented as directly as possible and in the most simple and familiar terms. This then has been the central aim in the preparation of this book; to give the student clear and distinct conceptions of the various ideas and phenomena of physics, and to aid him in think- ing through the relations between them, to the end that he may see something of the underlying unity of the subject; and to carry out this aim in such a manner that students may not be repelled by any unnecessary prominence of symbolic methods, and yet that the treatment may have all the exactness and iv PEEFACE precision in statement and deduction which the subject demands. This is a large ambition and I cannot hope to have been wholly successful, but I shall be grateful if my attempt is found in any degree to have subserved its purpose. My grateful acknowledgements are due to Dr. G. S. Fulcher of the University of Wisconsin, who has read nearly all the manuscript with great care, and to whom I am indebted for important suggestions, and to my colleague Professor J. O. Thompson whose criticism at all stages of the work and pains- taking correction of the proof has been most helpful. Amherst, A. L. K. March, 1911. PREFACE TO REVISED EDITION Recent advances in physical science having made it necessary to rewrite some paragraphs of the earlier edition, especially those relating to X-rays and the electron theory of matter, advantage has been taken of the opportunity to make a few additional changes which class-room experience has shown to be desirable. Certain paragraphs relating to force and motion, which had been introduced before the section on statics, are now placed among the introductory paragraphs to kinetics, where they fall in better with the logical development of the subject. The electro- magnetic units, volt, ampere and ohm, are defined and introduced earlier than before. The sections on wireless telegraphy have been made more complete and wireless telephony is touched upon. A section also has been added treating of the flicker photometer. At the end of the volume a short discussion of Carnot's cycle and the thermodynamic basis of the absolute scale of temperature has been introduced as an appendix, also a proof is given of Newton's wave formula. Quite a number of new problems have been added, but the old problems have been found to serve their purpose well and are for the most part retained. The author gratefully acknowledges his indebtedness to Dr. G. S. Fulcher, and to Professors W. E. McElfresh and D. C. Miller, for valuable suggestions and criticisms. Amheest, Mass. A. L. K. Jvly, 1917. CONTENTS INTRODUCTION Page Aim and Method of Physical Sciences 1 Fundamental Conceptions 4 Units and Measurement 6 MECHANICS I. General Principles. Ideas and Definition of Kinematics 10 Displacement, Velocity, Acceleration 10 First Principles of Dynamics 18 Inertia Mass and Force 19 II. Statics. Equilibrium of a Particle 24 Equilibrium of a Rigid Body 29 Moment of Force, Parallel Forces 30 Torque, Center of Mass 35 Work and Energy 39 Friction 44 Machines 48 III. Kinetics op a Particle. Rectilinear Motion of a Mass 54 •Laws of Motion 58 Motion with Constant Acceleration 60 Falling Bodies 62 Units of Force 59, 64 Kinetic Energy 66 Impact 68 Motion in a Curved Path 72 Projectiles 72 Motion in Circle 73 Centripetal and Centrifugal Force 77 Vibratory Motion 81 Simple Harmonic Motion 81 Force in S. H. M 84 Pendulum 85 V vi CONTENTS Paoe IV. Rotation of Rigid Bodies. Kinematics of Rotation 88 Kinetics of Rotation about Fixed Axis 91 Angular Momentum . 93 Kinetic Energy of Rotating Body 94 Moment of Inertia 95 Compound Pendulum 96 Some Cases of Motion with Partly Free Axis 99 Foucault Pendulum, Top and Gyroscope 99 V. Univkrsali Gravitation. Kepler's laws .... 103 Gravitation Constant. . . 104 Masses of Earth and Planets 106 Variation of Gravity on Earth's Surface .... ... 107 LIQUIDS AND GASES I. Fluids at Rest. Pressure in Liquids and Gases 110 Liquid Surfaces 114 Buoyancy and Floating Bodies 117 Specific Gravity 120 Gases and Atmospheric Pressure and Buoyancy 123 Compressibility of Gases, Boyle's Law 128 Pumps and Pressure Gauges 133 II. Fluids in Motion. Momentum . 140 Water Wheels and Hydraulic Ram . 141 Torricelli's Theorem . . . . 143 Bernoulli's Theorem . . ... .144 Pressure in Fluid Stream . ... . 145 PROPERTIES OF MATTER AND ITS INTERNAL FORCES Structure 150 Elasticity and Viscosity. 151 Diffusion and Solution . 169 Capillarity and Surface Tension 163 Kinetic Theory .173 WAVE MOTION AND SOUND Surface Waves 178 Compressional Waves 182 Sound 185 CONTENTS vii Page Reflection and Refraction of Waves 190 Sound Characteristics, Intensity, Pitch, Quality . . .... 195 Doppler's principle 200 Resonators and Analysis of Sound 201 Interference and Beats 208 Standing Waves and Vibrating Bodies ... ... 211 Velocity of Wave in a Cord . . . 213 Organ Pipes and Wind Instruments 217 Vibration of Rods and Plates 225 Laws of Similar Systems 227 Musical Relations of Pitch 228 The Ear and Hearing -231 Helmholtz's Theory of Dissonance 235 HEAT Thermometry 238 Expansion of (a) Solids 246 (b) Liquids 250 (c) Gases 255 Absolute Scale of Temperature 256 Calorimetry and Specific Heat 260 Dulong and Petit's Law 267 Sources and Mechanical Equivalent of Heat . . 268 Cooling due to Work of Expansion 273 Nature of Heat ... 274 Transmission of Heat 276 Change of State 281 Fusion 281 Vaporization 288 Sublimation 297 Condensation of Gases 300 Heat Engines 307 Radiation and Absorption. 311 MAGNETISM Properties of Magnets . 3'/20 Law of Force and Magnetic Field 324 Terrestrial Magnetism 330 Unit Tubes of Force 337 Magnetic Induction 341 Permeability, Diamagnetism, and Influence of Medium 344 ELECTROSTATICS Electrification 348 viii CONTENTS Paqj" Law of Force and Distribution of Charge 352 Induction 356 Potential 368 Electrometers .... 372 Electron Theory . .' 374 Condensers and Capacity . 378 Energy of Charge . 383 Calculation of Potential 387 Capacity of Sphere and Condenser 392 Electric Discharge 394 ELECTRIC CURRENTS Electric Current . . . . 399 Voltaic Cell ... 402 Chemical Effects of Current ... . . . . 409 Faraday's Laws 410 Battery Cells 417 Primary Cells 418 Secondary Cells 420 Modes of Connecting Cells 423 Fall of Potential in Circuit . 425 Ohm's Law 427 Resistance and its Measurement ... . . . 429 Energy and Heating Effect of Currents 435 Thermo electricity . . 442 Magnetic Effects of Currents 447 Hysteresis and Electromagnets . 453 Interaction between Current and Magnetic Field . 457 Instruments Measuring Currents and Potentials . . 462 Application to Bells and Telegraph 471 Electromagnetic Induction . . 475 Electromotive Force of Induction . ... 479 Energy in Induction and Lenz Law .... 482 Self-induction 485 Induction Coil and Telephone 487 Electromagnetic Units 492 Dynamo Electric Machines and Motors 495 I. Direct Current Dynamos 495 II. Direct Current Motors 502 III. Alternating Currents 505 Alternating Current Motors 511 Electric Oscillations and Waves 516 Wireless Telegraphy and Telephony 521 Electric Discharge through Gases 525 Cathode Rays, Rontgen Rays, Canal Rays 527 CONTENTS ix RADIOACTIVITY Page Radioactivity 532 The Alpha, Beta, and Gamma Rays 534 Radioactive Transformations 535 Electron Theory of Matter 539 LIGHT Shadows and Photometry . . ■ 541 Velocity of Light ... . 548 Wave Theory 554 Reflection of Light and Mirrors 557 Refraction of Light . 568 Prisms and Lenses 575 Dispersion of Light 589 Achromatism , . 592 Rainbow ... 594 Optical Instruments ... . . 596 Photographic Camera and Projecting Lantern . 597 The Eye 600 Microscope and Telescopes 604 Analysis of Light . 612 The Spectrum . 612 Kinds of Spectra . . . . 617 Doppler's Principle in Spectroscopy 622 Colors of Bodies 625 Fluorescence 629 Color Vision, Young-Helmholtz Theory .... . 631 Interference of Light . . 632 Newton's Rings, Interferometer ..... 634 Diffraction . 639 Resolving Power of Optical Instruments . . . 645 Diffraction Grating . . 645 Measurement of Wave Lengths of Light . 650 Polarized Light 653 Polarization by (a) Absorption 653 (b) Reflection and Refraction 655 (c) Fine Particles 658 (d) Double Refraction 659 Huygen's Wave Surface 662 Rotation of Plane of Polarization . . ... . 664 Circular and Elliptical Polarized Light . . . 667 Electricity and Light 670 Magnetic Rotation , . ■ 670 Kerr Effect 671 X CONTENTS Paqb Electromagnetic Theory 671 Zeeman Effect 671 Appendix I. — Garnet's Theorem and the Absolute Scale of Tempera- ture ... 673 Appendix II. — Proof of Newton's Wave Formula 683 Tables op Constants ... 686 Index 687 INTRODUCTION Aim and Method op Physical Sciences 1. Physical Sciences. — The study of nature includes two great divisions, biological and physical sciences. The former includes those that involve the complex phenomena of life, while the latter are concerned with the investigation of the fundamental phenomena of matter. Physics and Chemistry are the funda- mental physical sciences and form the basis upon which Astron- omy, Geology and Meteorology rest in investigating their special realms in the world of nature. Formerly Physics was called Natural Philosophy, in distinction from Natural History which described the world of plants and animals. Physics deals with the properties and phenomena of inani- mate matter as affected by forces, and is expecially concerned with the properties common to all kinds of matter and those changes of form and state which matter undergoes without being changed in kind, as well as such general phenomena as sound, heat, electricity and magnetism. Chemistry is distinguished from Physics in that it is chiefly concerned with the phenomena that result when different kinds of matter are brought together and enter into combination. It deals largely with the qualities in which one kind of matter differs from another. There are, how- ever, many points where these sciences merge into each other, and the domain of physical chemistry lies largely in this borderland. 3. The Aim of Physical Science. — It is the aim of physical science so to systematize our knowledge of the material world that all its phenomena shall be seen as special instances under a few far-reaching and more inclusive generalizations called laws. And when a given phenomenon is analyzed in this way into sepa- rate parts or phases each of which is but a special case under some general law, the phenomenon is said to be explained. In seeking an explanation we determine the cottses of the X 2 INTRODUCTION phenomenon in question; that is, the essential circumstances or those circumstances without which the given event does not occur; and then we seek to determine the effect of each of these circumstances separately, and exhibit, if possible, each such effect as a special instance under some general law. For example, the complex motion of a ball struck by a bat is found to be dependent on the motion given to it by the blow of the bat, on the presence of the earth, and on air resistance. We first try and determine how a body moves when set free in the presence of the earth without any initial blow or impulse and in a vacuum. We find in this way an unvarying rule of motion that applies to all bodies of whatever size or shape, and we call it the law of falling bodies. That part of the motion of the ball which depends only on the nearness of the earth is but a special instance under this law. Now, making allowance for the motion due to the earth, we seek to determine that part of the motion due to the initial blow, and here again we find that the actual motion seems to be exactly according to a general rule which is found to hold whenever an impulsive force acts on a mass. And finally we investigate the effect of air resistance, de- termining how it affects a body at rest and how it modifies the motion of a body moving through it, and here again certain gen- eral rules are found which apply not only to the special case under consideration, but to all cases of bodies moving through air. When the effects of all three circumstances are taken into account, the motion is found to be exactly accounted for, and is then said to be explained. Leverrier and Adams, in analyzing the motion of the planet Uranus, found that after taking account of all the known circum- stances, such as the attractions of the sun and other planets upon it, there still remained a part of its motion which was not ac-- counted for, and assuming it to be due to an unknown planet they computed its position and mass, and thus the planet Neptune was discovered. But in analyzing our problem we may go deeper and show that the motion of the ball near the earth is such as would result from a force urging the two bodies together, and we may then discover that it is merely a special instance of the law that all bodies are influenced by forces urging theni together or, in other words, that INTRODUCTION § all bodies attract each other. When we can show also that the forces between the air and the moving body are due to the motion given to the air and so are simply particular exhibitions of the general rule which holds whenever matter is set in motion, we feel . that a still higher degree of understanding is reached. By such a process all the complex facts of nature are assigned their places in an orderly system. But a limit is soon reached beyond which the mind cannot go, because thinking is conditioned by experience, and even in its profoundest theories and specula- tions the mind must employ those conceptions which it has obtained from the- world about it. 3. Experiment. — Physics is an experimental science, its gen- eralizations rest solely upon experiment, and although reason- ing upon established facts has often led to the discovery of new truths of great importance, the final appeal must always be to experiment. If the deduction is thus disproved, it appears either that the reasoning was wrong or that there are certain elements entering into the problem that were neglected. In seeking for the causes of such discrepancies new truths have often been discovered. An experiment is a combination of circumstances brought about for the purpose of testing the truth of some deduction or for the discovery of new effects. The usual course of an experimental inquiry is to modify the circumstances one by one, noting the corresponding effect until the influence of each is thoroughly understood. 4. Necessary Assumptions. — In every experimental science it is assumed that the same causes always produce the same effects and that the position of the event as a whole in either time or space only affects the absolute time and position of the result, provided there is no change in the relative time or space relations of the various circumstances involved. For example, if all the other circumstances are the same, a stone will fall in exactly the same manner next week as it does to-day, or if the solar system be changing its place in space no change in the manner of the stone's falling will take place from that cause alone. Experience up to this time has justified these assumptions, and without them progress in physical science would be impossible. 4: INTRODUCTION Fundamental Conceptions 6. Force. — Our ideas of force are derived primarily from muscular effort. It requires an effort to lift a weight, to throw a ball, or to compress a spring. The upward pull upon a weight at any instant while it is being lifted is a force acting on it, and the downward tendency of the weight which the pull opposes is also a force. Anything that serves to accomplish what would require mus- cular exertion to bring about exerts a force. Thus a support exerts a force on the weight which rests upon it, and the weight exerts an equal and opposite force on the support, compressing it. A bat exerts a force against a ball in giving it motion, a clock spring exerts a force against the stop that prevents it from unwinding. 6. Matter. — Through our muscular sense and sense of touch we are made conscious of bodies around us which resist compres- sion, and may, therefore, be said to occupy space. Such bodies are said to be material substances or made of matter. Every object that we know of possesses weight; that is, it re- quires some muscular effort to support it, or if it is hung on a spring the spring is stretched. What we call its weight is a force urging it toward the ground, and as the weight of two quarts of water is twice that of one quart we are led to think of the weight of a body as a proper measure of the quantity of matter which it contains. But besides weight all bodies have inertia; that is, to produce a definite change per second in the motion of a body a certain force is required. If a given body is isolated from other portions of matter, it may be heated or cooled or bent or twisted or compressed into small volume or allowed to expand into a large one, but in all these changes its weight and its inertia remain unchanged. It will be seen later that if the body is taken from one place on the earth to another its weight also may change, so that the only general property of a given portion of matter that cannot be changed is its inertia. It is this property, therefore, by which quantities of matter are defined, and two bodies which have equal inertias are said to have equal masses or to contain equal quantities of matter. INTRODUCTION 5 The actual comparison of two masses, however, is usually made by weighing, since under the ordinary circumstances of weighing, bodies which have equal weights have equal inertias. 7. Conservation of Matter. — The mass of a given portion of matter as measured by its inertia cannot be changed by any process known to man. Not only may a piece of wood be bent and twisted or compressed without changing its mass, but it may be burned in the fire, and chemistry shows that if the ashes and vapors and gases that have come from it are collected and sepa- rated from the gases of the air with which they may have united, it will be found that the united mass of the ash and the gases and vapors is the same as the mass of the original piece of wood. This principle is known as the conservation of matter, and is established by innumerable experiments, both physical and chemical. 8. States of Matter. — Different kinds of matter differ greatly in the power of preserving their shape. Some, such as steel or copper, offer very great resistance to any attempt to change their forms. Such bodies are said to be rigid or solid bodies. Others, like water or air, have no permanent shape, but flow under the action of the weakest forces and take the shapes of the vessels containing them; they are called fluids. There are no substances that are either perfectly rigid or that are perfect fluids, for the most rigid bodies may be distorted, and those substances that flow most freely offer some resistance to change of form. ' In some cases it is difficult to say whether a substance is to be regarded as solid or fluid. Fluids are again divided into liquids and gases. Liquids are those fluids that can have a free surface and do not change much in volume under great changes in pressure. A mass of liquid has a nearl / definite bulk though no permanent shape. Water is an exampl- of a liquid. Gases, on the other hand, are fluids that do not have a free sur- face, but completely fill the containing vessel, however much it may be enlarged. A mass of gas may be regarded as having neither permanent shape nor size, since both of these are entirely determined by the vessel which contains it. Air is a familiar example of a gas. 6 INTRODUCTION Units and Measurements 9. Measurements. — The exact measurement of all the quan- tities involved in any phenomenon is a very important part of its study. It is largely owing to the recognition of this that such great advances have been made in physics during the last two hundred years. Every measurement is essentially a comparison. A quantity to be measured is compared with another quantity of the same kind called the unit. Thus to measure a length it is necessary to find how many times the unit of length is contained in the given length. The unit must be of the same nature as the quantity which is to be measured, since only like things can be compared. There must, therefore, be as many different kinds of units as there are kinds of quantities to be measured. 10. Absolute Measurements. — Each of the units employed might be arbitrarily chosen without reference to any other; the inch might be taken as the unit of length, the square foot as the unit of surface, and the quart as the unit of volume; but such a practice would lead to endless complications, especially when several different units are used in the same calculation, for it would be necessary in such a case to keep constantly in mind the number of square inches in a square foot, the number of cubic inches in a quart, etc. It is far simpler after choosing the inch as the unit of length to take the square inch as the unit of surface and the cubic inch as that of volume. The same principle applies in the case of all other units; none should be chosen arbi- trarily which can be directly derived from those which have been previously selected. A system such as this in which there are a few arbitrarily chosen fundamental units, between which no known connection exists and from which all other units are derived without intro- ducing any new arbitrary factors, is known as an absolute system of units. .11. Fundamental Units. — All the phenomena of nature are manifested to us in time and in space, through the agency of matter. It is natural, then, that the fundamental units adopted as the basis of the system of measurement used in physics, should INTRODUCTION 7 be the units of time, length, and mass. These are also convenient units, for lengths, times, and masses may be compared with great ease and precision, and all units that relate only to mass, motion and force or that depend on these by definition may be derived directly from them. Physicists usually employ what is called the centimeter- gram-second (C. G. S.) system of absolute units in which the centimeter, gram, and second are taken as the units of length, mass, and time, respectively. This system has the advantage of being in use by physicists all over the world, and therefore results expressed in its units are intelligible everywhere. But an absolute system might be based on any three units of length, mass, and time whatever. Thus a foot-pound-second system is used extensively in English-speak- ing countries. 12. Unit of licngth. — The unit of length in the C. G. S. system of absolute measurement is the centimeter, or one-hundredth part of a meter. The meter is the distance between the ends of a bar of platinum which, is kept in Paris and known as the Metre des Archives, the bar being measured when at the temperature of melting ice. This bar was constructed by Borda for the French Government, and was adopted by them in 1799 with the view to its becoming a universal standard of length; it was intended to be exactly one ten-millionth part of the distance from the equator to the pole measured along a meridian on the earth. It is now known that the earth's quadrant is about 10,000,856 meters in length, but as distances can be more easily and accur- ately compared with the length of the bar at Paris than with the length of the earth's quadrant, the former still continues to be the standard of length. The English standard of length from which the foot and inch are determined is the standard yard, which is the "distance be- tween the centers of the transverse lines in the two gold plugs in the bronze bar deposited in the office of the Exchequer" measured at the temperature of 62°F. This standard yard represents about the average length of the early yard measures that were in use, which were probably adopted as being half the distance which a man can stretch with his arms. 8 INTRODUCTION 1 yard = 91.43835 cm. 1 foot = 30.47945 cm. 13. Unit of Mass. — The unit of mass in the C. G. S. system is the gram, or one-thousandth part of the standard kilogram, which is a mass of platinum kept at Paris and known as the Kilogramme des Archives. The standard kilogram was intended to represent the exact mass of a cubic decimeter of distilled water at its greatest density or at the temperature 4°C. The gram is, therefore, equal to the mass of a cubic centimeter of pure water at 4°C. This relation between the cubic centimeter and the gram is exceedingly convenient, for it enables us to de- termine the volume of an irregular vessel from the weight of water which it can contain. But it is not a direct relation like that between the unit of length and unit of volume. Aside from con- venience, there is no reason why a cubic centimeter of copper or mercury or of anything else might not have been taken as the unit of mass. Since two masses may be compared with a far higher degree of accuracy than that with which the weight of a cubic centi- meter of water can be determined, the Kilogramme des Archives is the real standard on which all metric weights are based. 14. Unit of Time. — Intervals of time are always compared by the motions of bodies. Two intervals of time are defined as equal when a body, moving under exactly the same circumstances in both cases, moves as far in the one time as in the other. The heavenly bodies have in their motions always furnished measures of time. One of the simplest natural units of time is the period of rotation of the earth, which is the interval of time between two succesive meridian passages of the same star. This is known as the sidereal day, and time reckoned in this way is called sidereal time. By considering the possible effect of tidal friction in re- tarding the earth's motion, Adams concludes that the period of rotation of the earth has not changed by more than one-thirtieth of a second in 3000 years. The ordinary day is determined not by the rising and setting of the stars, but by the motion of the sun. When the sun is on the meridian it is said to be solar or apparent noon. The interval of time between two successive apparent noons is called the appar- INTRODUCTION 9 ent or solar day. It is this time which is indicated by the sun dial. By means of clocks, which are machines constructed to run with great uniformity, one solar day may be compared with another, and it is thus found that they are not of equal length. The average length of the solar days in a year is known as the mean solar day. The ordinary standard time used in everyday life is mean solar time. The unit of time in the C. G. S. system is the mean solar second or the 86400th part of a mean solar day. MECHANICS I. GENERAL PRINCIPLES 15. Definitions. — Mechanics treats of the motions of masses and of the effect of forces in causing or modifying those motions. It includes those cases where forces cause relative motions of the the different parts of an elastic body causing it to change its shape or size, as when a gas is compressed or a spring bent. Such changes in size or shape of different portions of a body are called strains. Bodies which do not suffer strain when acted on by forces are said to be rigid. All known bodies yield more or less to distorting or compressing forces, but when considering the motion of a body as a whole, all bodies in which the strains are small may be regarded as practically rigid. Thus we may treat the motion of a grindstone or of a shell from a rifled gun as though these bodies were rigid, though we know that they are slightly strained by the forces acting. Mechanics is usually subdivided into kinematics and dynamics. Kinematics treats of the characteristics of different kinds of motion, and of the modes of strain in elastic bodies without refer- ence to the forces involved. Dynamics treats of the effect of forces in causing or modifying the motions of masses and in producing strains in elastic bodies. It is usual to treat dynamics under the heads statics and kinetics. Statics is that part of dynamics which deals with bodies in equilibrium or when the several forces that may be involved are so related as to balance or neutralize each other, so far as giving motion to the body as a whole is concerned. Kinetics is that part of dynamics which treats of the effect of forces in changing the motions of bodies. Ideas and Definitions of Kinematics 16. Motion Relative. — When a body is changing its position it is said to be in motion. There is no way of fixing the position 10 KINEMATICS 11 of a body except by its distance from surrounding objects. When it is said, therefore, that a body has moved, it is always meant that there has been a change in its position with reference to some other objects regarded as fixed, or in other words, there has been relative motion. Thus we know only relative motion, and when we speak of an object as at rest we usually mean with reference to that part of the earth's surface in our vicinity. 17. Displacement. — The distance in a straight line from one position of the body to another is called its displacement from the first position. To completely describe any displacement, its amount and direction must both be given. If an extended rigid body is displaced, as when a book is moved on a table, it may be moved in such a way that its edges will re- main parallel to their original directions, in which case the dis- placements of all points in the body will be the same both in amount and direction. The motion is said to be one of simple translation without rotation. But in general when a rigid body is moved there is rotation as well as translation, so that to bring it into the second position from the first we may first imagine it to be translated till some point in the object is brought into its second position. Then by a rotation about a suitable axis through that point the whole body may be brought into the sec- ond position. 18. Vectors and Their Representation. — All quantities wnich involve the idea of direction as well as amount are said to be vector quantities or vectors. Such are displacements, velocities, forces, etc. While quantities having magnitude only, without any reference to direction, are known as scalar quantities. Vol- ume, density, mass, and energy are scalar magnitudes. A vector quantity is represented by a straight line which indicates by its direction the direction of the vector, and by its length the magnitude of the vector, the length being measured in any con- venient units, provided the same scale is used throughout any one diagram or construction. It must be remembered, however, that a vector represented by a line AB is not the same as that represented by BA, one is the opposite of the other, or AB = —BA. This will be evident if AB represent a displacement from A to B. A displacement BA will exactly undo what the other accomplished, and bring the 12 MECHANICS Fig. 1. body back to its starting point. The straight line representing a vector is, therefore, commonly represented with an arrow-head indicating its positive direction. 19. Composition of Displacements. — If a man in a railway car were to go directly across from one side to the other, say from AtoB (Fig. 1), then the hne AB wiU represent both in amount and direction his displacement con- sidered only with respect to the car. But if the car is in motion and in the meantime has advanced through the distance AC, the man will evidently come to D instead of to B. The displacement of the car with reference to the earth is AC and the displacement of the man relative to the earth is AD. This is called the resultant displacement of the man, of which AB and AC are the components. Another way of stating this is that the man received simulta- neously two displacements AB and AC, for if he had not been displaced in the direction AC he would have gone to B, while if he had not had the displacement AB he would have been carried to C. From the above it is evident that the resultant of any number of simul- taneous displacements may be found just as if they had been taken suc- cessively. For example, let it be required to find the resultant of four displace- ments represented in amount and direction by the vectors A, B, C, D. If A were the only displacement, the body would be brought from to a, but B is also a component displacement, therefore draw B' equal and parallel to B, and the result of the two displacements will be represented by the distance b. Then in like manner draw C and D' equal and parallel, re- spectively, to C and D, and it is clear that the result cf the four KINEMATICS 13 displacements on a body originally at would be to transfer it to 0'. Therefore the resultant of the four displacements is the single dis- placement R, and this is so whether the component displacements occur simultaneously or successively. The particular order in which the several components are taken is quite immaterial. This construction by which the resultant is found is called the diagram of displacements, it is perfectly general and applies whether the components are in the same plane or not. 30. Composition of Vectors. — The above construction is a par- ticular instance of the addition or composition of vectors. By a precisely similar process the resultant of any set of vectors may be obtained whether they represent forces, velocities, momenta, or any other quantities having direction as well as magnitude. 21. Resolution of Dis- placements. — There is only one resultant displace- ment that can be found when the components are given, in whatever order they may be taken. If it is required, however, to re- solve a given displacement into its components, there are an infinite number of ways in which it may be done. For example, the displacement AS (Fig- 3) may be regarded as having Ac and cB as its components, or Ad and dB or Ae and eB, or it may be considered the resultant of the three displacements Ag, gh, hB. Or if any broken line whatever be taken starting at A and terminating at B, AB will evidently be the resultant of the displacements which are represented in amount and direction by the several parts of the broken line. 33o Resolving of Vectors. — What has been just said of the resolving of a displacement into components is equally true of the resolving of any other vectors whatever into component vec- tors, and applies to the resolution of velocities, forces, etc. ,ff 33. Velocity. — The velocity of a body is the rate at which it Tjasses over distance in time. It is a vector quantity, its direc- tion being as important as its amount. The term speed is famil- 14 MECHANICS ilary used to express the amount of velocity without reference to its direction. Two bodies may be moving with the same speed, but if they are not going in the same direction their velocities are different. This is the strict use of the word velocity; it is often somewhat loosely used to express merely the speed of motion. 34. Constant Velocity. — When a body moves in a straight line always passing over equal distances in equal times it is said to have constant or uniform velocity. It is evident that the motion must be in a straight line, otherAsdse the direction of the velocity would not be constant. In this ease of motion if the length of any part of the path be divided by the time taken for the body to traverse that portion, the result is what is called the rate of motion, or the distance passed over per unit time, and is the same whatever part of the path may be chosen. It is this quantity which is the speed or the amount of the velocity. Thus when a train is moving with constant velocity, the number ot miles run in a given time divided by that time expressed in hours, is the speed in miles per hour. 25. Variable Velocity. — When either the rate or direction of motion of a particle is changing, it 6 10 -g gg^jjj ^Q jjg moving with variable velocity. Thus the velocity is vary- ing in case of a fallmg body which constantly gains in speed or in case of a railway train rounding a curve where the direction of motion is changing. To understand what is meant by the speed of motion at a par- ticular point when the velocity is constantly changing we may consider a short portion b c (Fig. 4) of the path of the body having at its middle the point a at which the speed is to be determined. Divide the length of 6 c by the time taken by the body in travers- ing it. The result will be what may be called the average speed over that part of the path. If, now, the part chosen is taken smaller and smaller, always having the given point at its center, the average velocities thus found will approximate more and more nearly to the true velocity at the given point, and that value, which these successive approximations continually approach as a*" limit, as the distance b c approaches zero, is the speed of motion at KINEMATICS 15 the point a. At each instant a body has a certain speed, but it may not be constant even for the shortest interval of time that can be conceived. So also with regard to the direction of motion. If the body moves in a curved path, its direction of motion at any point is the direction of the tangent to the curve at that point, and as the direction of the tangent constantly changes as we pass along the curve, so the direction of the velocity in such a case may be differ- ent at one point from what it is at a neighboring one, however near together the two points may be. 36. Composition of Velocities. — If a body has at any instant several component velocities, the resultant velocity may be found by the vector diagram as in the case of the composition of displacements. For instance, suppose a ball is thrown in a moving railway car, it is required to find the velocity of the ball with reference to the earth. Let AB represent the velocity of the railway car, say 50 ft. per second, and let AC be the velocity of the ball as thrown obliquely across the car with a velocity of, say, 40 ft. per second. Then, laying off the vectors AB and BD with the proper relative direction and length, the resultant velocity is represented by the vector AD, which is found by measurement (using the same scale as in laying off AB and BD) to be 76 ft. per second, and this is the resultant speed of the ball relative to the earth. If the angle between AB and AC is given, the side AD of the triangle ABD may be calculated by trigonometry, using the formula AZ)2 = AB-" + AC2 + 2AB. AC. cos CAB. 27. Resolution of Velocities. — Any given velocity may also be resolved into component velocities. For instance, suppose a man is rowing a boat with a velocity of 10 ft. per second in a^ direction making an angle of 30° with the straight shore of a lakeX and it is required to determine how fast he is moving along thel shore and how fast he is moving out into the lake. Let AB | (Fig. 6) represent a velocity of 10 ft. per second. Dra^*ii,4^ paral-_ ^^ i<~j h i. j^ J li 1 50 ft. sec. -** B Fig. 6. 16 MECHANICS lei with the shore and making an angle of 30° with AB. Draw CB perpendicular to AC. Then AC and CB will represent two velocities, one parallel to the shore and one at right angles to it, whose resultant is AB. Therefore the boat may be regarded as having a velocity AC parallel with the shore and a velocity CB at right angles to it, and the amounts of these may be found by measurement, using the same scale as in laying off AB. Or we may calculate them by trigo- nometry, for Fig. 6. AC = AB.cos 30" CB = AB^in 30° AC = 8.66 ft. per second CB = 5 ft. per second It is frequently necessary in practice to resolve a velocity or other vector into two components which are mutually at right angles as in the case just discussed, and so this case, while one of the simplest, is one of much importance. 38. Acceleration. — When the velocity of a body changes either in amount or direction the motion is said to be accelerated, and ', '-^^ ^1 "l^.^ji -^ ^ Vi --J !^ D I Fig. 7. the change in velocity per unit time, or the time rate of change of the velocity, is called the rate of acceleration or simply the acceleration. Change in velocity may always be thought of as due to the boci> receiving an additional component velocity which is compounded KINEMATICS 17 with the original velocity, the velocity after the change being the resultant of the two. For example, in the upper diagram of figure 7 a body moving in a straight line is represented as having a velocity Vi at A and a greater velocity Vi at B. The gain in velocity is represented by the vector C which must be added to Vi to give Vi. The average rate of acceleration between A and B is therefore found by divid- ing C, the increase in velocity, by the time taken by the body in passing from A to B. In the second diagram v^ is less than vi, and so the change in velocity is represented by the arrow D and is negative or opposite to the original motion. In the third case figured the motion is along a curve and the velocity at S is not in the same direction as the velocity at A, but a velocity represented by E if compounded with Vi will give V2 as the resu tant. The velocity E, is therefore the change in velocity between A and B, and dividing it by the time during which the change has taken place or the time of motion from A to B, the average rate of acceleration between A and B is found. 39. Acceleration in Rectilinear Motion. — If the motion is in a straight line the velocity changes only in amount and not in direction, and the acceleration is calculated by dividing the change in speed during a given interval of time by the time inter- val. Or, expressing it in a formula, V — u where u represents the velocity at the beginning of the interval of time t while v represents the velocity at its end. This formula gives in general the average rate of acceleration during the interval of time t, but if the acceleration is constant it gives the actual rate. Thus if a ball with a velocity of 50 ft. per second has, after one-half second, a velocity of 40 ft. per second in the same direc- tion, the average rate of the acceleration during the interval is '^ = -20, 72 the negative sign indicating that the acceleration is opposite in direction to the original velocity and therefore the velocity is decreasing. 18 MECHANICS 30. Composition and Resolution of Accelerations. — When a moving body has several different accelerations, as when a man in a railway car starts to walk in the car while the speed of the train is changing or while it is rounding a curve, the several accelera- tions may be compounded and their resultant found just as with other vectors. So also an acceleration may be resolved into two or more components. Problems 1. A man walks H mile in 10 minutes. What is his average velocity in feet per second? 2. A train has a velocity of 30 miles per hour; what is its velocity in feet per second? 3. A bicycle rider is traveling north at the rate of 10 miles per hour. If the wind is blowing from the east at the rate of 6 miles per hour, what is its apparent direction and velocity to the rider? Show the direction by a diagram. 4. A man rows a boat at the rate of 4 miles per hour, making an angle of 30° with the straight shore of the lake. How fast is he moving away from the shore? 5. Draw a diagram to scale showing the direction in which a man must row across a river in order to reach a point directly opposite, if he rows 3 miles per hour while the speed of the current is 2 miles per hour. 6. If the river in the last problem is J^ mile broad, how long does it take to cross it as described, and what is the velocity of the boat relative to the shore? 7. A ball rolling down an incline has a, velocity of 60 cm. per sec. at a certain instant, and 11 seconds later it has attained a velocity of 181 cm. per sec. Find its acceleration. 8. A body having an initial velocity of 60 ft. per sec. has an acceleration — 32 ft. per sec. per sec. Find its velocity at the end of 1, 2, and 3 seconds. 9. A railroad train having a velocity of 40 miles per hour is brought to rest in 1 minute. Find the acceleration in feet per second per second. FiBST Principles of Dynamics 31. First Law of Motion. — When a ball on a table starts to roll, experience convinces us either that the table is not level or that some external force has caused the motion. On the other hand, when we see a ball that has been set rolling on a level table, gradually losing its speed, we are equally satisfied that there is some force resisting its motion. For it is found in such a case DYNAMICS 19 that if the table is made smoother and if air resistance is gotten rid of, the ball loses speed much more slowly than before. We are thus satisfied that if there were no force resisting its motion the speed of the ball would remain unchanged. This convictioni, arrived at through experience, was clearly enunciated by Sir Isaac Newton in the first of his celebrated Laws of Motion, published in his Principia, in 1686. First Law of Motion. — Every body continues in its state of rest, or of moving with constant velocity in a straight line, unless acted upon by some external force. 32. Discussion of the First Law of Motion. — The first law asserts that force is not required to keep a body in motion, but simply to change its state of motion. After a railroad train has attained a constant speed the entire force of the locomotive is spent in overcoming the various resistances that oppose the motion, such as friction of wheels and bearings and air resistance. But for these the train would maintain its speed without aid from the locomotive. Therefore, when any object is observed to be at rest or moving with constant speed in a straight line, we conclude either that no external force acts upon the body, or that whatever forces act are so related as to neutralize or balance each other. Since we measure equal times by the equal angles through which the earth has moved, the law that freely moving bodies move through equal distances in equal times may seem simply a consequence of the mode of defining equal times and without any physical significance. But the statement of the law really asserts the physical fact that in case of any two bodies whatever, unacted on by external forces, while one body moves through successive equal distances, the distances traversed simultaneously by the other body are also equal among themselves. That this is true whatever the nature of the bodies concerned is a fact of nature that rests on experience, and cannot be regarded as known a priori. 33. Inertia. — The property, common to all kinds of matter, that no material body can have its state of rest or motion changed without the action of some force, is known as inertia. The amount of force required to produce a given change in the motion of a body depends both on the body and on the suddenness of the change to be produced. Any force however small can give as great a velocity as may be 20 MECHANICS desired to any mass however great, provided it acts for a long enough time. If a weight rests on a sheet of paper on a table it may be drawn along by means of the paper, for there is friction between the two and it requires a certain force to slip the one over the other. //, therefore, we do not attempt to accelerate the weight too rapidly, the friction will move the weight along with the paper. But if we attempt to start the weight suddenly, or change its velocity suddenly while moving, the paper will at once slip from under it, for the force required to produce the sudden change of motion is greater than the friction between the two. Again if a 10-lb. weight (Fig. 8) is hung by a cord from a fixed support and if it is drawn steadily downward by a piece of the same cord attached to it underneath, the cord will break above the weight, for the force exerted by the lower cord upon the weight will cause it to move downward, straining the upper cord with the com- bined force due both to the weight and the pull. But a sudden pull will break the cord below the weight. For in consequence of its inertia the weight cannot be set in motion as suddenly as the cord is pulled without the exertion of a greater force than the cord is able to bear, so that the cord breaks even before the weight has moved downward enough to strain the upper cord to the breaking point. The complete statement of the effect of a force upon the motion of a body is embodied in Newton's Second Law of Motion, and will be discussed when we take up the study of Kinetics (§93). 34. Measure of Mass. — The inertias of bodies may be com- pared quantitatively by the amounts of force required to acceler- ate them at the same rate, and when they are thus compared it is found that the inertia of a given portion of matter is always the same and cannot be increased or diminished by any known process. Consequently the inertia of a body is the sum of the inertias of its several parts, the inertia of two quarts of water is twice that of one quart whether they are combined or separate. Fig. 8. — Inertia. DYNAMICS 21 It is for this reason that the quantity of matter in a body, or its mass as it is called, is measured by its inertia as compared with that of some standard piece of matter taken as the unit of mass. Therefore, masses are said to be equal which acquire equal ve- locities when acted on by equal forces for the same length of time. , For example, suppose two masses are drawn side by side over a frictionless surface by two spring balances at such a rate that each balance is kept constantly stretched, say to the 4-oz. point, so that they exert equal forces; then, if the masses after starting together keep pace with each other, they are acquiring velocity at the same rate and consequently are equal. Of course such an experiment serves chiefly to illustrate what is meant by saying that equal masses have equal inertias, for it would be impossible to directly compare masses in this way with any degree of accuracy. The actual comparison of masses is accomplished with great accuracy by weighing; for it is found that masses which have equal inertias have also equal weights, provided they are weighed in a vacuum at the same point on the earth. (§102.) 35. Measure of Force. — A force may be measured in three ways : 1. By the weight that it can support. This is the gravitation method. 2. By its power to strain an elastic body, as in the ordinary spring balance. 3. By its power to give motion to a mass. This is the dynamical method. The first method is very convenient and forms the basis of most measurements of force in engineering and ordinary life, but it has the disadvantage that the force required to support a pound weight varies from place to place on the earth. The second method is convenient for comparing forces, but the elastic properties of one substance differ from those of another, besides being dependent on temperature and physical condition, so that a standard force could not be preserved or accurately defined by this method. The third method is difficult to apply except indirectly, but furnishes a unit of force which depends only on the inertia of matter and is, therefore, absolutely invariable and well suited to be a standard force. 22 MECHANICS 36. Equal Forces. — Two forces are said to be eqiuil when the velocity of a given mass is increased at the same rate per second by one force as by the other. When two such forces act in oppo- site directions on a given mass they neutrahze or balance each other so far as any effect on the motion of the mass is concerned. Thus when a cord is stretched horizontally between two springs, the forces exerted by the springs are equal and opposite so long as the cord remains at rest or moves with uniform velocity. 37. Stress. — When a weight is supported by a uniform cord, every part of the cord is stretched, and if the weight of the cord itself is so small that it may be neglected, the stretch of every inch of it is the same whether it is near the upper or lower end and whatever may be the total length. The section AB (Fig. 9) is pulled up by the cord above A and is pulled down by the cord below B and is, therefore, stretched until the contractile force of its own elas- ticity balances the external stretching force. In this way every portion of the cord is subject from without to an external stretch- ing force, and it exerts in opposition to this an internal contractile force and is said to be in a state of stress. In this case the stress is called tension, and every portion is subject to forces which tend to elongate it. When a weight is supported on a vertical rod or column the whole support is in another state of stress called pressure or compression, for the forces that act on any part of the rod tend to shorten it. Besides tension and pressure there is a third kind of stress, called shearing stress, which tends to distort or force out of shape the parts of a body. This is the stress in a rod that is being twisted. But the further discussion of this matter must be left until the elasticity of bodies is considered. (§234.) It is believed that all forces are transmitted by stresses. Even the attraction between a magnet and a piece of iron is ex- plained by stress in the ether around them, and the gravitation attraction which exists between all masses is also supposed to :20 lbs Fig. 9. DYNAMICS 23 be due in some unexplained way to a stress in the surrounding medium. 38. Action and Reaction. — Every stress has a double aspect. Thus when a weight rests on a table the force between the two may be regarded as a pressure down on the table or an upward push against the weight. When a cord is supporting a weight, at every cross section in the cord there is a downward pull on the cord above the section, and an. upward pull on the cord below and these two are exactly equal. When a magnet attracts a piece of iron the force may be regarded as drawing the iron toward the magnet or the magnet toward the iron. These two aspects of a stress are known as the action and reaction; they are exactly equal and opposite. This fundamental fact was stated by Newton as the Third Law of Motion. Third Law of Motion, — To every action' there is an equal and opposite reaction. 39. Discussion of Third Law of Motion. — When a weight rests upon a table it is pushed up with a force equal to that which it exerts upon the table. The table, therefore, presses a heavy weight upward with more force than it exerts on a small weight. The only limit of the power of the table to react is its strength. It is instructive to consider what happens when a weight heavy enough to crush the table is placed upon it. As it is lowered upon the table it presses more and more until the limit of the table's power of resistance is reached, when in breaking down it begins to move away from the weight at such a rate that the reaction which it exerts is at every instant exactly equal to the pressure to which it is subjected by the weight. For if a body moves away fast enough from' another which is pressing upon it the pressure may be diminished to any extent. When a ball is struck by a bat the force upon the bat at every instant while they are in contact is the same as that which the bat exerts upon the ball. 40. Composition and Resolution of Forces. — When several forces act simultaneously on a particle the single resultant force may be found by the diagram of vectors (§ 19) just as in case of accelerations; for the several component forces each cause a corresponding component of acceleration, and the resultant acceleration corresponds to the resultant force. 24 MECHANICS So also any force may be considered as the resultant of two or more component forces, and these components may be found just as the components of an acceleration are found. 41, A Special Case. — Suppose the vector AB, five units long, represents a force of five pounds, acting obliquely on a block of wood resting on a table, and it is required to find how much force is pressing the block against the table and how much is urging it along its surface. The vector .AS may be resolved as just ex- plained into the two components AC and CB, where AC repre- sents the force pressing the block against the surface and CB rep- resents the force pushing it along the surface. The amount of these components may be de- termined either by direct meas- ,„„,,. , ., , urement, using the same scale as Fig. 10. — Oblique force on block. .,.-.. -r^ , m laymg off AB, or they may be calculated as follows: If F is the amount of the force AB and if a is the angle between AB and the table top, then AC = F. sin a CB = F. cos a II. STATICS Equilibrium of a Particle 43. Equilibrium. — Before taking up the study of the motions of bodies as determined by forces, we shall consider some cases in which the various forces concerned are so related as to balance each other, so far as the motion of the body on which they act is concerned. A body is said to be in equilibrium when any forces wliich act on it are so related that the body is not accelerated. Thus a body at rest is in equilibrium, also a body moving with constant velocity in a straight line, also a body turning with constant speed of rotation about an axis through its center of mass, as in case of a well-balanced wheel, also when such a wheel is not only turning with constant speed, but moving along with constant speed. Thus a wheel rolling in a straight line along a level sur- STATICS 25 face or a wheeled vehicle like a car on a straight level track is in equilibrium if moving with constant speed.* We shall first consider the equilibrium of a particle or a body so small that the forces acting on it may all be considered as acting at one point. Afterward the conditions of equihbrium of an ex- tended rigid body will be taken up. "W hether a body is to be treated as a particle or not depends on circumstances. For instance, in astronomy the sun and planets are treated as particles when their shapes and distribution of mass do not affect the question considered. 43. Equilibrium of a Particle. — A particle is in equilibrium when the resultant of the forces acting on it is zero. Evidently in this case the diagram of the forces must be a closed triangle or polygon. For let the forces acting in a given case be a 6 c (Fig. 11), then if we draw the diagram of forces as in the lower part of the figure and if the vectors abc form a closed triangle, as shown, the resultant is zero and the particle is in equilibrium. So also in case of any number of forces in equihbrium, the diagram of forces, formed by drawing successively the several vectors repre- senting the forces, must be a closed polygon; that is, the last vector drawn must terminate at the starting point. An example of four forces in equilibrium is shown in figure 12, the several forces abed fic. ii.— Three forces in equilibrium. forming a closed polygon. It is interesting also to observe that if we resolve each of these forces into two components, one directed toward the top or bottom of the page and the other sidewise, as, for example, b is resolved into b' and b", c into c' and c", etc., we find that the components a' b' directed from left to right exactly balance the components c' d' directed from right to left, so also the upward compo- nents a" and d" are together equal to the sum of the downward components b" and c". In the above diagram the four forces have for convenience all been represented in one plane. This restriction is not necessary, * When moving as just described, a wheel considered as a whole is in equilibrium, but its particles are jtot in equiiibnuui, lur tney muve in circles and are theretore acnelerated ( §28). 26 MECHANICS the same construction is the test of equilibrium in whatever di- rections the four forces may act. 44. Illustrations. — If three cords joined at P suspend weights of 3, 4, and 5 lbs. respectively, those supporting the 3-lb. and 4-lb. weights passing Fig. 12. — Four forces in equilibrium. Fig. 13. over frictionless pulleys as shown in figure 13, then the point of junction P will assume a definite position to which it will return if pushed aside and the cords PA and PB will be at right angles to each other. For the point P is in equihbrium under the three forces 3, 4, 5, and therefore the force diagram must be the closed triangle PCD, the three FiQ. 15. sides of which are in the ratio of 3:4:5. But such a triangle is right- angled and therefore the force 4 and force 3 must be at right angles to each other. Suppose a cord is fastened at A and B and is then stretched by a weight W hung at P (Fig. 14). As before, the diagram of forces is PCD, where STATICS 27 PC represents the stress on the cord between P and B while CD represents the stress on AP, and DP represents the weight W. Evidently the more nearly AP and PB are to being in a straight Une the larger will CD and PC be in comparison with the force W which is represented by DP. So that a comparatively small pull down at P, if APB is nearly straight, may produce a force great enough to break the cord between A and B. Thus the stress brought to bear on hammock ropes may be much greater than the weight of the person supported if it is hung with insufhcient sag. The student may easily determine under what conditions the stresses on AP and PB will be each equal to the weight W. The jointed device used in hand printing presses, and shown in figure 15 as applied to the brakes on a locomotive, illustrates the same principle. Here when compressed air is admitted to the cylinder C the piston is forced upward, thus straightening the two connecting pieces A and B, thereby forcing the two brake shoes against the wheels with a force which is greater the more nearly the connecting pieces are pulled into a straight line. Fig. 16. — Bridge truss. 45. Bridge Stresses. — ^Let it be required to find the stresses, tensions or pressures, in case of the various parts of the truss which is shown in figure 16, supporting at its center the weight W. Consider what forces are acting on the end of the truss at j4 . It will be shown later (§54) that in such a case half the total weight will be borne by one abutment and half by the other. The end of the truss at A therefore W presses down on the abutment with a force equal to -2, if we neglect the weight of the truss itself. Now A is in equilibrium under the three forces represented by the arrows, P indicating the upward pressure of the abutment, S representing the oblique downward thrust of the strut AB, while T repre- sents the inward pull of the tie rod AC; therefore the diagram of these forces must be a triangle as shown above. But this triangle is similar to the triangle ACB, for the sides are respectively parallel, and so the forces P, S, and T are in the same proportion as the sides BC, AB, AC, and since the W pressure P is equal to -^, the other forces are at once known by proportion when the sides of the triangle ABC are known. In the somewhat more complicated case shown in figure 17 where the total 28 MECHANICS weight of 10 tons is supported between the two abutments the upward pressure P will equal 5 tons. The stresses on AB and AC may be found as in the preceding case, but to find the stress on the rod BC we must make a diagram of the forces under which the point B is in equilibrium as shown in the diagram. 46. Crane Problem. — A weight of 100 lb. is suspended from a crane of dimensions shown in the figure. It is required to find the tension on the tie rod AB and the compression on the strut BC. The point B is in equilibrium under three forces, the downward weight W = 100 lb., the pressure P of the strut which acts outward, and the ten- sion T of the tie rod which acts in the direction BA. The diagram of forces must therefore be a triangle with sides parallel to the directions of the three forces as shown in the figure. Fig. 17. y 1 A •^''y 1 r1 V 1 P B eft. f""'^! k^ Fia 18. — Crane. Problems A force of 300 grams and a force of 400 grams act at right angles to each other on the same point. Find the single force to which they are equiva- lent and also its direction, by a diagram. Four forces of 3, 4, 5, and 6 lbs. act on the same point in directions east, northeast, north, and northwest, respectively. Construct the force diagram and find by measurement the amount of the resultant and the angle which it makes with the north line. Three cords fastened together at a point free to move, have tensions 60, 70, and 80 grams respectively. Construct the force diagram and find by measurement the angles between the cords when at rest. Two forces of 10 lbs. each act upon a single point in such a way that they are equivalent to a single force of 10 lbs. Find the angle between their lines of action. How much force must be exerted at an angle of 45° to the top of a table to push along a weight \vhen the frictional resistance to be overcome is a force of 2 kgms.? When a force of 20 lbs. is required to draw along a sled by a rope making an angle of 30° with the ground, find the force moving the sled forward, and the force diminishing its pressure on the ground. STATICS 29 7. A weight of 2 lbs. hung from a nail by a cord 30 in. long is pushed aside by a horizontal force of 1 lb. How far will it be moved away from the vertical line through its point of support? 8. A 32-gram weight is hung by a cord 60 cm. long from a point on a ver- tical wall. How far will it be pushed out from the wall by a force of 24 grams acting perpendicular to the wall? J9. Make a diagram showing the angle between the two ropes of a hammock when the tension on each rope is twice the weight of the person in the hammock. 10. A rope supporting a weight of 180 lbs. at its middle point is hung between two hooks which are on the same level and 18 ft. apart. If the middle point sags 3 ft. below the level of the hooks, find the force on each hook. 11. In case of a crane, like figure 18, in which the horizontal strut is 5 ft. long and the vertical distance 4 C is 3 ft., find the tension on the oblique tie rod and the pressure on the strut when a weight of 270 lbs. is supported. 12. Suppose the wall in figure 18 overhangs so that .A is 6 ft. vertically above a point 1 ft. to the left of C on the bar BC, which is horizontal and 9 ft. long. Find the stresses on AB and BC when a weight of 240 lbs. is suspended at B. Equilibbium of Rigid Body 47. Equilibrium of a Rigid Body. — A rigid body is in equilib- rium when its velocity of translation is not changing in any direction and when its velocity of rotation is not changing about any axis. Or, in other words, a rigid body is in equilibrium when it has no acceleration either of translation or rotation. 48. Condition for Translational Equiiibrium. — In order that there may be no translational acceleration, the relation between the forces acting on the body must be exactly the same as that required for the equilibrium of a particle. For if there is to be no acceleration in any direction the resultant force in any one direction must be zero, and this is evidently the case when the diagram of forces is a closed polygon. 49. Case of Two Forces. — The relation which must hold be- tween the forces in order that there may be no rotational accelera- tion may be most easily reached through the study of some simple cases of equilibrium. That a body may be in equilibrium under two forces it is necessary that the two forces P and Q (Fig. 19) should be equal and opposite in order to satisfy the con- dition of no translational acceleration as just shown. In order 30 MECHANICS that there maybe no tendency of the forces to rotate the body it is clearly necessary that they shall act in the same straight line, as shown in the figure. The only effect of the forces applied a.tA and B in the figure is to compress the body between these points. 60. Resultant of Two Oblique Forces In the Same Plane. — Let P and Q represent two forces acting at A and B upon an ex- tended body, and let their lines of action when produced intersect at C. A force equal and opposite to P if applied at C will exactly balance P, as shown in the preceding paragraph, and a force equal and opposite to Q, also applied at C, will balance Q, therefore a single force R equal and opposite to the resultant of P and Q as found by the triangle of forces, will, if applied at C, exactly bal- ance both P and Q and produce equilibrium. Evidently the force R may be applied to the body at any point in its line of Fig. 19. action CD. The resultant of P and Q is, therefore, a force equal and opposite to R and acting along the line CD. 51. Moment of Force. — In the case of equilibrium just dis- cussed we may imagine the force R to be produced by pressure against a pivot fixed somewhere in the line CD, say at E (Fig. 21). The forces P and Q then balance each other, so far as causing rotation about the axis at E is concerned. Two forces so related to any axis are said to have equal and opposite moments with re- spect to that axis. Common experience shows us that the farther the line of action of a force is from the pivot or axis, the greater will be its ability to rotate the body about that axis. Thus in opening a heavy gate we take hold of it as far as possible from its hinges and pull at right angles to the gate. A pull in line with the hinges would have STATICS 31 no effect to turn it whatever. The moment of a force to turn a body about an axis depends, therefore, both on the amount of the force and the distance of its Hne of action from the axis. Let X be the perpendicular distance from E to the hne of action of the force P, and let y be the perpendicular distance from E to the line of action of Q. We shall now show that in this case, where the moments of P and Q about the axis E balance each other, Px =Qy. Construct the parallelogram CFGH having CF = P and CH = Q. Draw FE and HE; then the area of the triangle CFE is equal to 3^Pa; for P is its base and x is its altitude. So also the area of CHE is equal to }4Qy- But the two triangles CFE and CHE have equal areas, for they have a common base CE and equal altitudes HI and FK; therefore, Px = Qy. It has thus been shown that when the axis E lies on the line CG the two forces P and Q have equal and opposite moments about it and also in that case Px = Qy. These products Px and Qy may, therefore, be taken as repre- senting the abilities of the forces to pro- duce rotation about the given axis, and are, therefore, used to measure the moments of the forces. The moment of a force with reference to a given axis is its ability to produce rotation about that axis, and is measured by the product of the force by the perpendicular distance from the axis to the line of action of the force. 52. Second Condition of Equilibrium. — The second condition of equilibrium for an extended rigid body is that the various forces must be so related that there is no rotational acceleration about any axis in the body. Since the ability of a force to produce rotation is measured by its moment, this condition is satisfied when the sum of the moments of the forces tending to produce clockwise rotation about any axis whatever is equal to the sum of the counter- clockwise moments about that axis. Fig. 21. 32 MECHANICS 63. Forces in One Plane. — When the forces acting on a body all lie in one plane, such as the plane of the paper in figure 22, they can have no tendency to rotate the body except about an axis at right angles to that plane. In this case if the diagram of forces is a closed polygon showing that there is no translational acceleration, and if the clockwise and counter-clockwise moments are equal about some one axis at right angles to the plane of the forces, then the body is in equilibrium and the resultant moment of the forces is also zero about any other axis that may be chosen in the body. Fig. 22. For example, in case of a board 2 ft. square, with forces applied to it as shown in figure 22, the diagram of forces is a closed figure, as the student may easily verify. The forces, therefore, balance so far as transla- tion is concerned. Now calculate the moments of the forces about an axis perpendicular to the plane of the forces, say through the point A . Designating clockwise moments plus and counter-clockwise minus, and taking the forces in order beginning at the top, we have the moments Sum of the moments = 0. Therefore the board is in equilibrium under these forces and consequently the sum of their moments will be zero if reckoned for any axis whatever. Compute in this way the moments about an axis through the center of the board and show that their sum is zero. -2X1 = = -2l -1X2 = = -2 +2X2 = = + 4 3X0 = = 2 XO = = oj B Fig. 23. 64. Three Parallel Forces. — When a bar is in equilibrium under three parallel forces, as in figure 23, to satisfy the con- dition of no translational acceleration the up forces must be equal to the down forces, otP + Q = R. While to satisfy the second condition, that the moments of the forces shall balance, we have Px = Qy, for these are the moments about the point B, and R has zero moment about that point. Or we may take moments about A and find Rx = Q{x+y). If moments are taken about any point other than A, B, or C, there will be three . . . f-i p t* c 4 I 7 3 J/t. '/(• >/f- s STATICS 33 moments to reckon. If, for example, the point D is taken as the axis the clockwise moment of Q must be equal to the sum of the counter-clockwise moments of P and R. 55. Parallel Forces in General. — Any case of equiUbrium with parallel forces may be discussed in a similar way, two con- ditions being met, namely, the sum of the forces acting in any one direction must be equal to the sum of the forces in the opposite direction, and the sum of the clockwise moments about any axis must be equal to the. sum of the counter-clockwise moments. 56. Illustration. — A certain bar having no weight is acted on by four forces as shown in figure 24, forces of 4 lbs. and 2 lbs. acting upward and 3 lbs. and 5 lbs. acting downward, and it is required to find the single force necessary to produce Fio. 24. equilibrium and the point on the bar where it must be applied. Since the total upward force is 6 while the downward force is 8, the required force F must be an upward force 2 to satisfy the first condition of no translational acceleration. This force must be applied at such a point on the bar as to satisfy the second condition, and make the clockwise moments balance the counter- clockwise moments about any axis. Take an axis through P, for instance. The moments about P are 3X0= 1 2 X 1 = 2 I , r r, , -.nf Sum = — 4 counter-clockwise. -|-0 X .2 = +10 I -4X3 =-12j therefore, to produce equilibrium the applied force 2 must produce a clock- wise moment 4. Since it must also act upward, it must be applied at a distance 2 to the left of P, and consequently the bar must be extended 2 feet in that direction. Any point whatever on the bar might have been taken as the origin of moments, and the reader should show that the same conclusion is reached taking moments about some point such as C. 57. Couple and Torque. — If in the case just treated the upward force 4 is changed to 6, we have a case that calls for special con- sideration. The upward forces are exactly equal to the downward forces and yet the bar is not in equilibrium, for taking moments about P we find that the clockwise moment is 10 while the counter- clockwise moment is 2 -f- 18 = 20. Here, then, there is a com- 34 MECHANICS bination of forces that does not tend to produce translation, but simply rotation. Such a combination is known as a couple, and its moment is commonly known as a torque. It cannot be bal- anced by any single force, for any force applied either upward or downward would cause translation. A couple can be balanced only by another couple having an equal and opposite moment, or torque. The simplest case of a couple is when two equal parallel forces act in opposite directions not in the same straight line. For instance, the forces Fi^, figure 26, constitute a clock- 3 ift. J ft. I ft. ^i^^ couple the moment of which is * Fx where x is the distance between ^ the lines of action of the forces. The '°' moment of a couple about any axis is the same as about any parallel axis. For, take an axis perpen- dicular to the paper and through P at a distance y from the nearer force, then the moments are Fy counter-clockwise and F{x-\-y) clockwise, hence subtracting we have Fx clockwise, as the result- ant of the two. The moment of such a couple about an axis perpendicular to the plane in which the two forces lie is, therefore, measured by the product of the amount of either force by the per- pendicular distance between their lines of action. To produce equilibrium, then, in the case under consideration a couple having a clockwise moment 10 must be applied to the bar, and it may be applied at any point we choose. The following figure illustrates different modes of producing „ .,., . . ,, . *^ ^ Fig. 26— Couple. equilibrmm in this case. In every case of equilibrium the forces acting may be re- solved into a number of balancing couples. 58. Center of Gravity. — The weight of a mass is the force with which it is drawn toward the earth. All parts of a body have weight and so the weights of the several parts into which a body may be conceived to be divided constitute a system of parallel forces acting downward toward the earth. The resultant of this system of forces is a single force equal to their sum and is the total weight of the body. It may be proved that there is a STATICS 55 Fig. 27. eettaiin point in the body through which the resultant force due to weight always acta whatever may be the position of the body.- This point is called the center of gravity of the body. In all problems that involve the weight of a body we may ig- nore the fact that the weight is distributed throughout the body, and treat it as a single force applied at the center of gravity. 59. Proof of Center of Gravity. — ^Let M and m be the masses of two parts of a body and let the line joining them be inclined as shown in figure 28. Since the weights of masses are proportional to the masses themselves (§38), the single upward force necessary to balance the weights of the two masses must be applied in the vertical line AB, so situated that Mx = my. But AB intersects at P the line joining the two masses, dividing it into the two segments a and 6 which, by similar triangles, are in the same ratio as x and y, and consequently a-.h :: M: m; and since this ratio does not depend on the inclination of the line joining M and m, it follows that the balancing force must pass through the point P whatever the inclination may be. P is, therefore, the center of gravity of M and m. Now conceive the masses M and m concentrated at P and find similarly a point P' through which the resultant weight of (M-{-m) and of another mass m' must always pass. Continue in this manner until account has been taken of all the masses into which we may conceive the body to have been subdivided. The point through which the final resultant passes is the center of gravity of the body. 60. Center of Mass and of Inertia. — The center of gravity as has just been explained is determined by the distribution of mass in a body or system of bodies. It has certain remarkable / — ^ b,^ 77 ^ B y 'y Fig. 28. 36 MECHANICS properties quite independent of weight, and is therefore also called the center of mass or center of inertia of the body or system. For example, a freely rotating body like a spinning projectile will always rotate about an axis through its center of mass. Fig. 29. 61. Position of Center of Gravity. — When a body is hung by a cord or balanced on a point the center of gravity must be in the vertical line passing through the point of support. For two equilibrating forces must act in the same straight line. If, therefore, a body is hung first from one point and then from another the intersection of the two lines thus determined marks the position of the center of gravity, as shown in the figure, where it is seen to be a point outside the actual substance of the chair. The center of gravity of a uniform bar is at its center; in a uniform thin plate, square, rectangular, or in the form of a parallelogram, it is at the intersection of the diagonals. In case of any homogeneous symmet- rical body it lies in the plane of symmetry. Thus it is at the center of a sphere, circle, or circular disc and at the center of a cube. 62. Equilibrium under Grav- ity — That a body may be in equilibrium under gravity, it must be supported by a force equal to its own weight and acting upward through its center of gravity. Thus the two cones and sphere shown in the figure as resting on a Fig. 30. STATICS 37 B[± Fig. 31. level table are in equilibrium, the upward force being supplied by the reaction of the table. But the first cone is said to be in stable equilibrium, because if slightly tipped it will fall back to its original position. The second cone is said to be in unstable equilibrium because if disturbed it will fall away from its original position, while the sphere is said to be in neutral equilibrium because it remains in equilibrium when displaced. It will be observed that in the first case the center of gravity of the cone is raised when it is tipped; in the second it is lowered, and in case of the sphere it is neither raised nor lowered. The weight of a body being considered as acting at its center of gravity will always cause that point to move down or toward the earth unless opposed by some other force. In case -of a loaded wagon on a hillside the vertical line through its center of gravity may remain between the wheels if the center of gravity is low, when if it were high the line of action of the weight might fall outside the wheel base causing the load to overturn. 63. Balances. — The beam of a balance rests on a sharp steel "knife edge" A, while the pans are hung on the knife edges B and C. These three knife edges are rigidly fixed in the beam and should be parallel, and in the same plane, and the arm AC should be equal to the arm AB. (Fig. 31.) Now i{ A, B, and C are all in the same straight line, and if the weight P is greater than the weight Q, the balance will tip entirely over unless some counteracting force is available. This is found in the weight of the balance beam itself which is so adjusted that its center of gravity does not lie exactlj'' on the edge A, but slightly below it, say a distance x. Then when P is greater than Q the balance beam inclines and its center of gravity is dis- placed until the restoring moment due to the weight of the beam W, acting down through its center of gravity just balances the deflecting moment due to the difference. between P and Q. The deflection is therefore very nearly proportional to P — Q, and the greater the deflection for a given difference between P and Q, the greater the sensitiveness of the balance is said to be. Fig. 32. 38 MECHANICS 64. Double Weighing. — If the arms of a balance are not of equal length, the true weight of a body may be obtained by the method of substitu- tion, in which the body is placed in one pan of the balance and some con- venient counterpoise, such as sand or shot, which will exactly balance it, placed in the other pan. The body is then removed and weights substituted for it until equilibrium is again reached, the substituted weights are then equal to the weight of the body. Or the method of double weighing may be employed. Let r be the length of the right arm of the balance and I the length of the left arm, then if a body whose real weight is P is placed in the right pan and balanced by weights W in the left pan, we have by the equality of moments Pr = Wl. Then interchanging body and weights, it is found that a weight W is required to balance it when it is in the left pan, which gives PI = W'r. Multiplying the two equations together we have PHt — WW'lr or P' = WW; therefore P = y/WW', or if W is very nearly equal to W, P = W+W 2 Problems 1. A wooden bar 5 ft. long and weighing 2 lbs., the ends of which are supported by spring balances, has a 10-lb. weight hung on it 2 ft. from one end. Find the force exerted on each balance. 2. A beam 20 ft. long is carried by three men, one at one end and the other two supporting it between them on a cross-bar at such a point that each man carries an equal weight. Find where the cross-bar must be placed. 3. A man standing on a uniform beam, 16 ft. long and weighing 120 lbs., at a point 1 ft. from its end causes it to just balance as it lies horizon- tally across a support 4 ft. from that end. What is the man's weight? 4. At what point on a pole must a weight of 52 lbs. be hung so that a boy at one end may carry % as much as the man at the other end, and how much does each carry? Neglect the weight of the pole. 6. If the pole in problem 4 is uniform and weighs 10 lbs., where must a 50-lb. weight be hung so that the man may carry twice as much weight as the boy? 6. Find the center of gravity of a uniform bar weighing 6 lbs. and having a 2-lb. weight on one end and a 7-lb. weight on the other. 7. Forces 2 and 4 acting upward are applied to a horizontal bar at 2 ft. and 4 ft. from the left-hand end, respectively, also forces 3 and 1 acting downward are applied at 1 ft. and 5 ft. from the same end. Find amount and point of application of a single force producing equilibrium. 8. How produce equilibrium in problem 7 when an additional force 2 acts downward at a point 3 ft. from the left-hand end of the bar. WORK AND ENERGY 39 9. A board 2 ft. square is acted on by five forces applied at the same points as shown in figure 22, but the forces instead of being 2, 1, 2, 3, 2, begin- ning at the top, are 4, 2, 4, 5, 3, respectively. Find the direction and amount of the force needed to produce equilibrium and how far from the center of the board its line of action must lie. 10. A ladder standing 6 ft. from a smooth vertical wall rests against it at a point 30 ft. from the ground. If the ladder weighs 60 lbs. and its center of gravity is J^ of its length from the bottom, find the force with which it presses against the wall, also the amount and direction of its force against the ground; that is, its vertical and horizontal components. Note. — The force between ladder and wall must be perpendicular to the latter if there is no friction between them. 11. When a man weighing 150 lbs. is halfway up the ladder in problem 10, find the pressure of the ladder against the wall and also the two compo- nents of its force against the ground. 12. When is the ladder in problem 11 more liable to slip, when a man is near the top or bottom, and why? Work and Energy. 65. Work. — A man digging a ditch is said to work, so also a team of horses drawing a load, and a carpenter supporting tem- porarily the end of a beam may also, in ordinary speech, be said to be working; for in each case a useful end is secured by the exer- tion of force. But there is a difference between these cases. In the first two there is motion and a permanent change is effected ; while in the third case the beam is not moved but simply supported, and any prop would have served as well as the carpenter. In physics the term work is restricted to such cases as the first two where motion results from the action of force, and the amount of work is measured by the product of the force by the distance through which the body moves along the line of action of the force. Thus when in digging a ditch a ton of earth is thrown to an average height of 6 ft., the work done is 2000 X 6 = 12,000 foot-pounds. If the motion of the body is not in line with the resultant force, then in estimating work only that component of the motion which is in the direction of the force is to be taken into account. For instance, in raising a barrel into a wagon the work done is the same whether the barrel is lifted directly from the ground or rolled up an inclined plane. For the weight of a body is a force 40 MECHANICS that acts vertically downward, consequently in estimating work done against weight, only the vertical distance through which the body is moved is to be considered. When a body yields to a force work is said to be done by the force or upon the body; but when a moving body is retarded by some resisting force, work is then said to be done by the body or against the force. The work done in raising a weight or compressing a spring is the same whether done in a second or in an hour. The time re- quired to do the work determines the rate of working, but has nothing to do with the amount of work. It is remarkable that although force and distance are both vector quantities, work, which is their product, is not a vector quantity. It has nothing to do with direction, and consequently to get the total work done upon a body by several different forces, the work of each may be reckoned separately and then the sum taken. Motion is essential to work. A great weight may rest on a support, but no work is done in supporting the weight though a great force is exerted. 66. Bate of Working. Horse-power. — A given amount of work may be done either in a short time or a long time, and in commercial operations the rate of working, or the work done per second or per hour, is an important consideration. Thus in case of an engine we wish to know how much work it can do in a given time, and its rate of working is known as its -power. Power may be measured by the number of grams weight that can be raised one centimeter per second, or by the number of pounds that can be raised one foot per second; but the unit of power introduced by James Watt and commonly used in engi- neering practice is the horse-power (written H.P.). One horse-power = 550 foot-pounds per second, or 33,000 foot-pounds per minute. That is, a 10 H.P. engine can raise 330 lbs. through a height of 100 ft. in one-tenth of a minute, or 3300 lbs. through a height of 10 feet in the same time. 67. Energy. — The importance of the idea of work lies in the fact that a body upon which work is done acquires thereby capacity to do an equal amount of work in returning to its original WORK AND ENERGY 41 state. The capacity to do work is called energy. Thus work is done when a spring is bent, and the spring acquires energy which is measured by the work that it can do as it unbends. Also a 10-lb. weight raised 100 ft. above the earth has had 1000 ft.- Ibs. of work expended in raising it, and it has gained the power to do that same amount of work in returning to its original position. The energy of a bent spring resides in the spring itself in virtue of its internal stresses; but in case of the raised weight the energy belongs not to the weight alone, but to the system of two bodies, the earth and the weight, which are separated in opposition to the stress or attraction between them. 68. Kinds of Energy. — In both illustrations given above the energy depends on the relative positions of bodies or parts of bodies between which there exist stresses. There is another form of energy which depends not upon stress, but upon the motion of matter. Suppose the raised weight is set free and allowed to fall with nothing to resist it, the force of the earth's attraction is exerted upon the mass as it falls and consequently work is done and energy expended, but in this case the work is all spent in giving velocity to the falling mass. When the weight reaches the bottom it has lost all its advantage of position, but it still has power to do work in virtue of its motion, and experiment shows that the work it can do before coming to rest is exactly equal to the work that was done upon it in giving it motion. The mass, therefore, still retains the energy that it had in the raised position, but it is now energy of motion. The energy which a body or system of bodies has in virtue of stresses is called potential energy. The energy which a body has in consequence of the velocity of its mass is called its kinetic energy. 69. Illustration. — If a mass is hung so that it can freely swing a3 a pendulum, when it has been raised to the position A (Fig. 33) it has been raised through the vertical distance h from B to D, and, therefore, has more potential energy at A than at B by the work done in raising it from B to A. If allowed to fall freely it will reach the bottom, moving with suf- ficient velocity to carry it up to C on the same level as A. At the bottom the mass has energy of motion or kinetic energy. It has entirely lost the 42 MECHANICS advantage of position which it had at A, the work done in raising it to A being now wholly transformed into energy of motion. But as the mass rises from B toward C it loses velocity for it is doing work and using up the store of kinetic energy that it received in falling, changing it again to potential energy. The pendulum has thus a constant store of energy which changes back and forth from one form to the other, the sum of the two being always constant, except as energy is gradually lost through friction and air resistance. 70. Work against Friction. — There is one case, however, in which the work done upon a body does not seem to increase its energy or power to do work. When a weight is pushed from one point to another on a level table force has to be exerted to over- come friction. The weight, however, remains at the same level above the earth and has no more power to do work in the new position than before it was moved. The work expended seems to be quite lost. But investigation has shown (§405 et seq.) that whenever work is done against friction heat is de- veloped in amount exactly propor- tional to the work done; and also that when work is obtained from a heat engine a precisely correspond- ing amount of heat disappears. It is, therefore, concluded that the work which seems to be lost in friction is not really lost or annihilated, but is transformed into heat as into another form of energy. When, therefore, a pendulum comes to rest in consequence of friction (at its point of support, or between it and the air through which it swings) the original energy of the pendulum is not lost but transformed into heat. 71. Forms of Energy. — From the results of innumerable ex- periments physicists have concluded that not only is heat a form of energy, but sound, light, and all electrical and magnetic actions are manifestations of energy, and require energy to be expended in causing them, just in proportion as they are capable of doing mechanical work or developing heat. The different manifestations of energy may be summarized as follows : Energy of masses Energy ot molecules and atoms Energy of ether ENERGY 43 Masses in motion — akinetic. Elastic bodies in a state of stress 1 Gravitation, energy of attracting > potential. masses j Sound, both kinetic and potential. Heat. Molecular and atomic energy Chemical action. Electric and magnetic phenomena. Light and radiation. When the energies involved in all these varied phenomena are studied it is found that one form of energy may be transformed into another, and that again into a third, but in every change the amount of the energy as measured by its power of doing work or of developing heat is unchanged. 72. Conservation of Energy. — The recognition of these varied forms of energy and careful measurements of the transformations from one form to another have led to the enunciation of a great principle or law known as the Conservation of Energy, which may be thus stated : In any system of bodies which neither receives energy from without nor gives up any, the total amount of energy is unchanged whatever actions or changes may take place within that system, whether the energy manifests itself in mechanical forms, in sound, heat, light, electric, or magnetic effects, or in chemical action or molecular or atomic changes. In most cases the tracing of all the changes is a difficult matter. For example, a cannon ball receives energy from the work done by the powder gases as they expand forcing the ball from the gun. As it travels it is resisted by the air, losing kinetic energy exactly equivalent to the heat energy developed by friction in the air. On striking the target, sound waves carry off a small part of the energy, there may also be a flash of light which also takes away some energy, and the rest will be found in the form of heat devel- oped in the target and in the ball itself and also in the form of kinetic energy in the fragments which may be thrown off. The principle of the conservation of energy asserts that if we add to- gether all the energy that is derived from the motion of the ball the sum will be eactly equal to the amount of work which was required to give it its motion. 44 MECHANICS This law is the most important and extensive generalization of the science of physics, and much of the progress of modern physics is due to its recognition. Every experiment in which the quan- tities of energy can be accurately determined is a test and confirmation of its truth, and no principle of physics is better established. In consequence of this law, the determination of the energy involved in any action assumes new importance and is an essential part of the study of every physical phenomenon. 73. Availability of Energy. — The presence of friction and analogous forms of resistance everywhere in nature causes a constant transformation of various forms of energy into heat, in which stage it is conducted from one body to another and gradu- ally becomes uniformly diffused so that although the energy still exists it is no longer available for the purpose of obtaining other forms of energy that may be desired. There is thus a constant degradation of energy going on throughout the universe, more available forms being constantly frittered away into heat. Fbiction 74. Friction. — When one body slides over another the motion is resisted by a force which is called friction. It is always a resistance, acting against the motion, and depends on the char- acter of the surfaces in contact and on the force pressing them together. It is a force of the greatest importance in daily life. If it were not for friction, nails and screws and knots would not hold, ropes could not be made, nor could we even walk across a floor. On the other hand, we would gladly be rid of friction in machines, for it is the cause of a large proportion of energy being lost in heat. Friction appears to be due to the interlocking of minute rough- nesses on the surfaces, together with the clinging together or adhesion of the points of closest contact. It is, therefore, di- minished by polishing the surfaces, which diminishes the rough- ness and also makes the points of contact broader so that the film of air or oil is more effective in preventing adhesion. When two surfaces have been resting in contact the friction at starting is greater than after the motion has been established. It seems probable FRICTION 45 that this may be due to the closer contact due to the film of air or oil being squeezed out by the continued pressure. Friction also resists the rolling of one body on another, though rolling friction in case of two given surfaces is much less than sliding friction. Rolling friction when surfaces are well polished appears to be due both to cohesion and to a slight deformation both of the surface and of the roller at the point of contact; for the surface is compressed as it passes under the roller, and though it may spring back again it does not exert quite as much force in re- covering as it opposed to the deformation. 75. Laws of Friction. — ^Let the block P be drawn along by the weight F, which is not sufficient to start it in motion, but will keep it moving with constant velocity when once started. The weight F is then equal to the force of friction, for it just balances it, neutralizing the resistance to the motion. It is found in this way that the friction between two given surfaces is proportional to the force pressing them together. If the block P weighs 5 lbs. a.nd if an additional weight of 5 lbs. is placed on the block, the force of friction is doubled. It is also found that the force of friction, within wide limits, is independent of the area of the surface of contact. For instance, the friction of the block P is almost the same whether it slides on a narrow or a broad side, provided they are equally smooth. The velocity with which one surface slides over the other makes little difference, the friction being appreciably the same for all moderate speeds; but the resistance to starting, or static friction, is greater than the friction after the motion is established. It is evident, however, that these laws do not hold without limit. For if one surface is very small, as in case of a point resting on a plane surface, or if the pressure is so great that one body presses into the other, then one cannot move on the other without tearing or injuring the surface, and the law no longer holds. 76. Coefficient of Friction. — It follows from the first law of friction that the force of friction divided by the force pressing the surfaces together is a constant, this constant is called the 46 MECHANICS coefficient of friction of the surfaces concerned; it is a fraction which when multiplied by the force pressing two surfaces to- gether gives the force of friction to be overcome. Thus if the coefficient of friction in case of iron wheels on iron rails is 0.004, then, if the wheels weigh 1000 lbs., a force of 4 lbs. will be required to overcome the friction. When an engineer wishes to know how much force will be re- quired in moving a house to cause it to slide on its ways, he has only to multiply the coefficient of friction for the soaped beams on which the house rests by the weight of the house itself. Some Coefficients of Friction Sliding Friction _ , , ci- 11 1 [ without lubricant 0.42 Oak upon oak, fibers parallel, s , , i ..i , n i^ ' [ rubbed with dry soap 0. 16 Oak upon oak, fibers crossed without lubricant . 29 - , f without lubricant 0. 25 \ thoroughly lubricated, may be as small as 0.06 Rolling Friction Cast-iron wheels on rails 0. 004 77. Limiting Angle of Repose. — The angle at which a surface may be inclined before a body resting on it begins to slip down is determined by the coefficient of friction between the surfaces. Thus let a weight W rest on a sur- face inclined at an angle a. The earth attracts the weight with a force W which acts vertically downward. We may resolve this force into two components, one P which is perpendicular to the inclined surface and represents the pressure of the weight against the surface, and another F which is parallel to the surface and represents the force urging the weight down along the slope. The force of friction between the weight and the inclined plane is equal to the product kP, where k is the coefficient of friction. If the friction is less than F the weight will slide with increasing speed down the incline, while if it is greater than F the weight will remain at rest. It will be noticed that P is made smaller by increasing the slope of the incline, and since k remains constant, the force of friction is less the greater the slope, and is zero when the slope is vertical. At one particular angle a, which may be called the limiting angle of repose, the force of friction balances the force F, and we have kP = F. At that angle tiie weight does not start to slide of itself but if started, slides down with FRICTION 47 F F h constant speed. In this case fc = „> and by similar triangles p= r; there- fore k = J- or k = tan a. Hence by finding the limiting angle of repose in a given case the coeffi- cient of friction is at once determined. 78. Means of Diminishing Friction. — To make friction small the surfaces should be very hard and of fine even polish. Where there is much wear it is customary to make one of the bearing surfaces of a harder material than the other. Thus the crank pins on steam engines are made of polished steel and turn in brass boxes, the friction between the brass and steel being less than it would be between two parts of steel. Rolling friction is very much less than sliding friction, therefore, wheels are used on carriages, etc. It depends to some extent on the diameter of the wheels, being less when the diameter is greater. But even when wheels are used there is sliding friction in the hubs. The resistance to the motion of the vehicle due to this sliding friction is diminished by making the axles of small diameter, but the length of the axle in the hub of the wheel or the length of its bearing surface does not affect the frictional resistance. To avoid the friction due to the sliding '°" between wheel and axle, ball bearings are used ; but even in these bearings there is some sliding friction where adjoining balls rub against each other. In some cases the axle is made to rest on the rims of two smaller wheels which are called friction wheels (Fig. 36). This is a common practice in mounting grindstones. Friction is greatly diminished by the use of lubricants, of which those most in use are oil, grease, soap, and black lead. The substances used as lubricants cover or wet the surfaces so that the rubbing takes place between layers of these substances instead of between the original surfaces. When the bearing surfaces are subjected to great pressure, as in heavy machinery, a thick oil is used that is not driven out by the pressure; in very light machinery, as in clocks and watches, a very thin oil is used. If oil were used on wooden bearings it would only increase the friction, for it would soak into and swell the wood; dry soap or paraffin may be used as a lubricant for wood surfaces. 48 MECHANICS Machines 79. Machines. — Machines are devices by which the amount or mode of application of a force is changed for the sake of gaining some practical advantage. Simple machines, known also as the mechanical powers, are the rope and pulley, lever, wheel and axle, inclined plane, and screw. All afford interesting cases of forces in equilibrium; but they may also be discussed from the point of view of the conservation of energy, for the work done on a machine must be equal to the work done by it if there is no loss of energy in friction. The ratio of the force exerted by a machine to the force ap- plied is called its mechanical advantage. 80. Rope and Pulley. — In all tackles where ropes are used the tension or force is the same at every point in a continuous rope, whether it passes over pulleys jQQ or not, if there is no friction. Let us apply this principle to a, few cases. In case 1 (Fig. 37) there is only one rope and the 100-lb. weight is supported by it, therefore all parts of the rope are under a Fig. 37. tension of 100 lbs., and that force must be exerted at P in whatever direction the pull may be made. In case 2 the 100-lb. weight is supported by the rope A, aU parts of this rope are therefore under that tension; but B is attached to a pulley which is drawn up by two parts of A. Since the pulley is in equilibrium, it follows that the upward pull of the two parts of A must be equal to the down- ward pull of B together with the weight of the pulley. If we neglect the latter the tension on B must be 200 lbs. and similarly that on C must be equal to twice that on B. Hence, neglecting friction and the weight of the pulleys, a weight of 400 lbs. on C will balance a weight of 100 lbs. on A. In case 3 there is one continuous rope which is fastened at the top and passes over two sheaves in each pulley, the lower pulley is therefore sus- tained by four parts of one rope, hence when a weight of 400 lbs. is supported by the lower pulley the tension on the rope is 100 lbs. MACHINES 49 The mechanical advantage in the first case is 1, while in the second and third cases it is 4. 81. Principle of Worlt Applied. — From the conservation of energy it is clear that the work done by a machine must be equal to the work done upon it, provided there is no friction and the energy stored in the machine is not changed. In illustration of this principle consider the various tackles of the preceding para- graph, and let x represent the distance that W is raised in a given case while the end of the rope at P is pulled through a distance y. Then the work done by the machine when W is raised is Wx, and the work spent in raising the weight is Py, and therefore Wx = Py. In the first case x = y, therefore P = W. In the second case x = }^y, therefore }/iW = P. In the third case also x = ^j/, therefore M^ = P- It should be noted that in the last two cases if the weights of the pulleys are taken account of we cannot say that Wx = Py, for some of the work done is spent in raising the movable pulleys. Thus, in case 2, if each pulley weighs w, we have P?/ = w| + (w + TF)| or P = ]4w + }4,{w + W). 83. Lever. — In the lever a rigid bar resting on a point of support, OT fulcrum, is used to ^_^ exert a great force near the fwi^ \p fulcrum when a smaller force V-^ . — ^ is exerted at the end of the ^._ longer arm of the lever. A y Q/j/^ crowbar as used in moving a 'a ^^ Tl stone, a hammer in drawing a p ^^rxx ' nail, are examples of levers. ^ ^--^ Levers are sometimes divided T'' into three classes depending on j^q 38.— Classes of levers. the relation between the posi- tion of the fulcrum and the points where the weight is raised and the force applied, as shown in the figure, where P represents the force applied to support the weight W, and F is the fulcrum. The upper lever in the figure belongs to the first class; the next to the second class; and the lowest to the third class. The distance from P to the fulcrum is called the power arm and that from W to the fulcrum is called the weight arm, and the 50 MECHANICS principle of moments tells us that if these distances are measured perpendicular to the lines of action of the forces P and W, then the product ofP by the power arm is equal to the product of W by the weight arm. In other words, the moments of the two forces about the fulcrum as axis must be equal and opposite. The pressure F against the fulcrum, since the three forces P, W, and F must be in equilibrium, is represented by the vector necessary to form a triangle with P and W. Of course if P and W are parallel, F must be either their sum or difference, depending on circumstances. 83. Crank and Axle. — In case of the crank and axle, shown in figure 39, the relation between the weight and the force applied at the crank to support it, is at once obtained from the principle that the moments of F and W about the axis must be equal, since the only motion that the system can have is one of rotation. Hence if R is the length of the crank arm and r the radius of the axle or drum on which the rope support- ing W is wound, we have FR = Wr in case the force F acts at right angles to R. Here, again, we may apply the princi- ple of work, for in one revolution of the crank the weight W is raised a distance equal to the circumference of the drum or 2irr, while the balancing force F acts through a distance 2TrR. We have, therefore, in case of equilibrium W 2irr = F 2wR or Wr = FR. Fig. 39. 84. Inclined Plane. — Barrels or casks are sometimes rolled up inclined planes and thus raised where they could not be directly lifted. The advantage of the inclined plane may be understood from figure 40, where W represents a weight resting on the inclined plane having length I, height h, and base b. The attraction of the earth is a force vertically downward on W, but it may be resolved as is shown into the components N at right angles to the inclined plane and F parallel to it. The component N is balanced by the pressure of the plane. MACHINES 51 while the component F represents the force that must be balanced by the push P necessary to support the weight on the plane. From the similarity of the two triangles it is clear that W,N, axiAF are proportional to I, b, and h, respectively. That isF :W: -.h -.1, or in words, the force required to support the weight on the in- clined plane is to the whole weight as the height of the plane is to its length. The same conclusion may also be reached by the principle of work, for if the weight is pushed up the plane the supporting force P acts through the length I, while the weight W is only raised against the earth's attrac- tion through a distance A. HencePZ = Wh. If the force P, instead of acting parallel to the length of the inclined plane, were parallel to its base we should resolve the weight W into components N and F as in figure 41, where F is parallel to the base. Then 85. Screw. — The screw as used in the ordinary letter press may cause enormous pressures by the application of a very moderate force to the lever arm. In one complete revolution of the screw it advances the distance between consecutive threads measured par- allel to the axis. This distance is called the pitch of the screw. FiQ. 41. Fig. 42. The mechanical advantage of the screw may be determined by considering the thread as a sort of inclined plane wrapped around the axis, but we may deduce it more conveniently from the prin- ciple of work; for if the force P operating the screw acts at right angles to the end of a lever arm of length R, in one revolution of 52 MECHANICS the screw the force P acts through a distance 2TrR, while the screw advances through a distance h equal to the pitch of the screw. Hence if W is the force exerted by the screw we have by the principle of work 2TrRP = Wh or P 2irR h 86. Chinese Capstan and Differential Pulley. — In the Chinese capstan a drum or axle having two parts of somewhat different diame- FiG. 43. Fig. 44. — Differential pulley. ters is operated by lever arms or capstan bars, so that one end of a rope is wound up on the drum of larger diameter while the other end unwinds from the smaller drum. The rope passes around a pulley S which is at- tached to the anchor or other weight to be raised. The force W is divided between the two parts of the rope pulling on S, so that the rope is under a W tension -r-. Mr and B are the radii of the small and large drums, respeo- W tively, the moments of the forces exerted by the rope on the drum are -tj-t W and -^R and the difference between these two moments must be balanced by the moment of the force P acting on the end of the capstan bar of length I. Hence we have in case of equilibrium MACHINES 53 W PI = ~(R - r). The advantage of such an arrangement is evidently the same as if one end of the rope were fixed and the other, after passing around S, were wound up on an axle whose radius was B — r. But such an axle being of small diameter would not have the strength of the larger axle with two drums. The differential pulley is a similar device used for raising heavy weights. There is an upper pulley having a single sheave with two grooves of different diameters like the two drums of the Chinese capstan. An endless chain passes over one groove in the upper pulley then around a pulley attached to the weight to be raised, and then around the second groove of the upper or fixed pulley. The grooves of the upper pulley have notches to receive the chain so that it cannot slip, and the chain is passed over it in such a way that it is wound up on one groove at the same time that it unwinds from the other. If the difference in diameters of the two grooves in the upper sheave is small, a small pull on the chain may suffice to support a large weight. Problems 1. A 180-lb. barrel is rolled up an incUned plane 12 ft. long to a platform 4 ft. above the ground. How much force must be exerted along the plane and how much work is done? Find also the force and work when the plane is 20 ft. long, the height being the same. 2. Find the force which the barrel exerts against the plane in both the cases specified in the first problem. 3. How much force parallel to the plane is required to support a weight of 39 kgms. on a frictionless inclined plane 13 meters long and 5 meters high? Also find the force with which the weight presses against the plane. 4. If the coefficient of friction between weight and plane in the last ques- tion is 0.20, find the force of friction and how much force must be exerted parallel to the plane in drawing the weight up, also in lowering it. 6. When the coefficient of friction between a weight and the inclined plane on which it rests is 0.30, find the ratio of its height to length when the plane is so steep that when the weight is started it slides down without acceleration. 6. A certain jack-screw has a screw 2 in. in diameter with three threads to the inch, and is operated by a lever arm 2 ft. long. What weight can be raised by a force of 48 lbs. applied at right angles to the end of the lever arm, neglecting friction? 7. When the coefficient of friction of the oiled surfaces of the jack-screw described in problem 6 is 0.06 and when a weight of 5 tons is raised, find the force required at the end of the lever arm to overcome friction, and the additional force required to raise the weight. 8. In problem 7, find the ratio of the work required to raise the weight 1 ft. without friction, to the actual work with.friction, and thus determine 54 MECHANICS the efficiency of the screw. Would the efficiency be the same if one-half as large a weight were being raised? 9. Find the tension on a bicycle chain when the pedal is pressed down with a force of 120 lbs.; the crank arm being 6 in. long and the sprocket wheel 8 in. in diameter. 10. If a force of 40 lbs. must be exerted on the arm of a windlass in rais- ing a weight of 120 lbs. while a force of only 20 lbs. is required in lowering the same, find the force expended in overcoming friction, and the efficiency of the windlass, and what per cent, of the work done is lost in friction. 11. How much force must be exerted on the crank of a windlass to raise a weight of 180 lbs., if the crank arm is 20 in. long and the drum on which the rope is wound is 8 in. in diameter. 12. Find the direction and amount of the force on the bearings of the wind- lass in the previous question, first, when the crank is in a horizontal position and being pressed down; second, when the crank arm is vertical. 13. A man weighing 150 lbs. raises himself in a sling by means of a rope passing over a movable pulley attached to the sling and a fixed pulley overhead. With how much force must he pull? Show also how to obtain your result by the principle of work. 14. A man weighing 180 lbs. runs up 24 steps, each 7 in. high, in 8 seconds. How much work does he do and what horse-power does he expend? 15. A donkey-engine is required to raise by means of a tackle a 2-ton weight to a height of 100 ft. in J^ minute. What horse-power is required if the efficiency of the tackle is 70 per cent. 16. When 1 H.P. is expended by a horse in pulling a load at the rate of 6 miles per hour, find the force with which the horse pulls the load. 17. What load can two horses draw along a level road at the rate of 3 miles an hour if they spend 2 H.P. in pulUng the load, when the coefficient of friction of wagon on road is Ho- Ans. 2500 lbs. 18. A locomotive drawing a train along a level track at 30 miles per hour expends 75 H.P., find the total air and frictional resistance overcome. Ans. 937,5 lbi=. 19. A locomotive draws a 300-ton train along a level track at the rate of 20 miles per hour; while working at the same rate it draws it up a J^ per cent, grade at 15 miles per hour; what horse-power is expended, sup- posing the frictional and air resistances the same in both cases, and what is the resistance in pounds. Ans. Resistance = 4500 lbs.; H.P. = 240. ni. KINETICS OF A PARTICLE Rectilinear Motion of a Mass 87. Introductory. — Up to this point we have studied espe- cially cases of equilibrium; where the forces acting are so balanced KINETICS 55 that there is no acceleration. We must now examine in some detail the various forms of motion where forces are involved in such a way as to cause acceleration. This part of mechanics, as Mach says, "is a wholly modern science. All that the Greeks achieved in mechanics belongs to the realm of statics. Dynamics was first founded by Galileo." Before 1638, when Galileo first published the results of his experiments, so little progress had been made in this direction that it was currently held that heavy bodies fell faster than light ones. In studying the effect of force in giving motion to matter, the simplest case to examine is where a definite portion of matter is acted on by a constant force. This is the case with falling bodies; for while a body is falling freely it is being urged downward by a constant force which we call its weight. Therefore, Galileo care- fully studied the motion of falling bodies, and of bodies rolling down inclined planes, and showed that in each of these cases the motion was with constant acceleration. As pendulum clocks had not been invented at that time, he made use of a simple water clock to measure short intervals of time in his experiments. This consisted of a large vessel of water having a jet closed by the finger, from which water was allowed to escape during the time interval to be measured. Thus the weight of water escaping while a body rolled down an inclined plane served to measure the time of descent. These experiments also showed that when a plane was inclined at such an angle that the force parallel to the plane required to keep a body from sliding down was one-half the weight of the body, then its acceleration in sliding down was one-half its acceleration when falling vertically. That is, the acceleration was proportional to the force causing the motion. 88. Atwood's Machine. — A convenient device for studying the effect of forces in giving motion to masses is the apparatus known as Atwood's machine (Fig. 45). Two equal weights A and B are hung over a very light carefully balanced wheel mounted so that it shall run with as little friction as possible. An additional weight or rider w, having two pro- jecting arms, is laid on top of the weight A, which is supported so that it can be liberated at any instant. When the weight A is freed it moves down accelerated by the rider w, until it reaches the ring C which picks off the rider w and allows A to pass freely through. After passing the 56 MECHANICS ring C there is no longer any accelerating force, since the rider is removed, and the weight A continues to move with the velocity which the rider had given to it. Thus if the ring C is so adjusted that A passes through it exactly 2 seconds after being liberated, and if Dis so placed that A moves from C to D in the next second, then if C and D are found to be 30 cm. apart, we conclude that A acquired a velocity of 30 cm. per second by a force which acted steadily for 2 seconds. If the same force is now allowed to act for 1 second, a velocity of only 15 cm./sec. will be acquired. By varying the weight of the rider or using instead of A and B a pair of weights, having double the mass, the following conclusions may be estab- lished : (a) The motion is with constant acceleration. (b) The acceleration is proportional to the weight of the rider so long as the total mass A+B+iu is constant. (c) If the mass of the moving system is doubled, a given rider will cause only half as great acceleration as before. 89. General Principle. — The effect of a force in giving motion to a body, as brought out in the ex- periments just described, may be thought of as due to a general prin- ciple which may be thus stated : the effect of a force in changing the motion of a mass is not in any way affected by the state of rest or motion of the mass which is acted upon. For instance, while a force If acting on a mass and increasing its velocity, suppose a second and equal force to act in the same direction upon the same mass. The second force being equal to the first will produce just as great an increase in velocity per second as \B Pig. 45. — Atwood's machine. KINETICS 57 is being produced by the first; and since both effects take place simultaneously and without interference, the total change in velocity will be twice that which would have been produced by the original force. It follows that the change in velocity per second when a force acts on a body is proportional to the amount of the force. And the change in velocity of a body when acted on by a force is also -proportional to the length of time during which the force acts, for suppose a mass has acquired velocity by a force acting upon it for 1 second, if the force now acts for another second it will increase the velocity of the mass as much more in the same direction, since the effect of a force is in no way conditioned by the state of rest or motion of the body upon which it acts. 90. Impulse. — The change in velocity which a given mass experiences is proportioned therefore both to the amount of the force and to the time during which it acts. A large force acting for a short time may produce the same change in the velocity of a mass as a small force acting for a longer time. A billiard ball may be made to roll as fast by pushing it as by striking it with the cue; the force in the second case is very much greater than in the first, but is exerted during an exceedingly short time; the impulse in both cases must be the same. The product of the amount of a force by the time during which it acts is called the impulse. 91. Force and Motion. — Again, suppose two equal masses moving side by side are acted on by equal forces in the same direction, they will both gain in velocity equally and will accord- ingly continue to move side by side, and their motion will evi- dently not be affected in any way if the two masses are connected forming a single large mass.* From this consideration we see that if a force gives a certain acceleration to a given mass then twice the force will be required to give the same acceleration to a mass twice as great, etc. Or, in order that different masses may all have the same change in velocity per second, the forces acting on them must be propor- tional to the masses. But if the mass is doubled without any corresponding change in • This cannot be regarded as known a priori for it results from the experimental facft that the inertia of one body is not afifected by its proximity to another. 58 MECHANICS the force which acts upon it, the gain in velocity will be only half as great as before, for the motion in that case will be the same as if the original mass were acted on by half the original force. 92. Momentum. — A given impulse may produce a great change in the velocity of a small mass, or a proportionally small change in the velocity of a greater mass; therefore, to measure the effect of an impulse, a quantity is employed which is proportional both to the mass and velocity of the moving body; this is called its momentum. The momentum of a body is the product of the amount of its mass by the amount of its velocity, and is a directed or vector quantity. 93. Three Laws of Motion. — The relations between forces, masses and motion, were first clearly enunciated in the form of three laws of motion, by Sir Isaac Newton in his celebrated Principia, published in 1686. Two of these laws have been al- ready discussed (§§31,38), but are here repeated in order that all three may be presented together. First Law. — Every body continues in its state of rest or of mx)ving with constant velocity in a straight line, unless acted upon by some external force. Second Law.— Change of momentum is proportional to the force and to the time during which it acts, and is in the same direction as the force. Third Law. — To every action there is an equal and opposite reaction. 94. Discussion of Second Law.— This law may be also ex- pressed in the formula mv — mu oc Ft where F is a force acting on a mass m for a time t, and u is the velocity at the beginning of the time interval t, while v is its velocity at the end of that time. Thus mv is the momentum after the force has acted, while mu is the original momentum of the mass. The gain in momentum is, therefore, mv — mu, and accord- ing to the law this is proportional to the force F and to the time t jointly, or to their product Ft. The above formula may be written : (V — u\ — —j KINETICS 59 where fc is a constant, the value of which depends on the particular units which are employed in measuring the various quantities concerned. In the above equations F represents the average value of the force during the time t in which the velocity of the mass has changed from u to v; but when t is exceedingly short —7— approaches as its limit the actual rate of acceleration at the given instant, while F is the corresponding force at that same instant, and we may write, F = kvia (1) that is, the acceleration of a body is proportional to the force acting upon it and inversely proportional to its mass. This may be called the fundamental formula of dynamics as it is a direct expression of the second law of motion, is absolutely general, and enables us to determine the forces acting in any case where the mass and motion of a body are known, since the accel- eration is determined from the motion. Thus it follows that if the force acting on amass is constant the mass moves with constant acceleration, while if the force varies the acceleration varies in the same proportion. 95. Dyne and Poundal. — In dealing with cases of equilibrium we have used the ordinary gravitation measures of force, the weight of a pound or gram, but in studying the accelerating effect of forces it will be found more convenient to use as the unit a force which will make the constant k equal to unity in the above expression, so that we may write simply F = ma Defined in this way, unit force is one which will give unit acceleration to unit mass, or unit force acting for unit time on unit mass will change its velocity by unity. When the centimeter gram and second are the fundamental units as in the C. G. S. system, the unit force is called the dyne, from the Greek word for force. It is a force which, acting on a mass of one gram for one second, will change its velocity by one centimeter per second. Hence to find the force in dynes in a given case of motion it is 60 MECHANICS only necessary to multiply the mass in grams by its rate of accel- eration measured in centimeters per second per second. Thus a force of 100 dynes acting on a mass of 10 grams will give it an acceleration 10, or in one second will give it an increase in velocity of 10 cms. per second. A unit of force similarly based on the foot, pound, and second as units of length, mass, and time, respectively, is sometimes used and is called the poundal, it is the force which acting on a mass of one pound will increase its velocity one foot per second for every second that it acts. The dyne and poundal have the advantage of being absolutely definite units of force, and do not vary from point to point on the earth as the weight of a gram or pound varies. 96. Unit of Work or Energy. — The unit of work on the C. G. S. system of units where the force is measured in dynes and the dis- tance in centimeters is known as the erg (from the Greek word for work). It is the work done when a body moves one centimeter in the direction in which it is urged by a force of one dyne. The corresponding unit of work or energy on the foot-pound- second system is the foot-poundal, and is the work done when a body moves one foot in the direction in which it is urged by a force of one poundal. 97. Motion in a Straight Line with Constant Velocity. — When a body moves in a straight line with constant velocity the accel- eration is zero and therefore the force must be zero according to the formula F = ma. The moving mass is, therefore, in equilibrium. Tliis is the case considered in Newton's first law of motion. A railway train while running at constant speed is in a state of equilibrium. The force of the locomotive urging it on is exactly balanced by the resistance of the air and friction of the wheels. So when a bucket is drawn up out of a well with constant speed it is in equilibrium and the upward pull on the rope is exactly equal to the weight of the bucket of water. 98. Motion in a Straight Line with Constant Acceleration. — When a body moves in a straight hne with a velocity which is increasing or diminishing at a constant rate, it has a constant acceleration in the direction of the motion in one case and oppo- site to it in the other. KINETICS 61 When the acceleration a is constant, the change in velocity of the moving body in t seconds is at. And if the velocity at the beginning of the time t is u, and that at the end of the time is v, then V = u -\- at when the speed is increasing; V = u — at when the speed is decreasing (1) The space passed over in t seconds will be found by multiplying the average velocity during the interval by the time t. But since the acceleration is constant the velocity increases uniformly with the time, and therefore the average velocity is the arithmetical mean of the initial and final velocities, or — ^ — The space traversed in time t may, therefore, be expressed by the formula V -{- u s = -~-t (2) Substituting for v its value V = u + at, we find s = ut+ }i af^ (3) But equation (1) may be put in the form V — u t and this multiplied by (2) gives 2as = 1)2 — ^2 (4) By the use of these formulas (1 to 4) any two of the quantities u, V, a, t, s may be determined when the other three are given. The student should thoroughly memorize these formulas and exercise himself in applying them to simple problems, such as those on page 70. 99. Force Causing Rectilinear Motion with Constant Acceiera- tion. — The kind of motion just discussed is produced whenever a mass is acted on by a constant force in one direction; for in such a case the acceleration is constant and given by the relation F a = —■ m Thus when a car is drawn along a track by a stretched spring 62 MECHANICS which is kept constantly at the same tension, the motion is with constant acceleration. So also a falling body has this kind of motion, for it is constantly urged downward by its own weight, which is a nearly constant force. When a body slides. down an inclined plane, the force urging it down along the plane is the same at one point as at another, and, therefore, in this case also the acceleration is constant. 100. Falling Bodies. — Freely falling bodies are the most familiar examples of bodies moving with constant acceleration. For a body near the surface of the earth is attracted or urged downward with a certain constant force which we call its weight, and when it is set free so that its weight is the only force acting, it falls with constantly accelerated motion. In ordinary experi- ence, however, where bodies fall through air, the resistance of the air is another force which modifies the motion. If the resistance in a given case were constant, the body would still fall with con- stant acceleration, but the air resistance increases greatly with the velocity of the falling body, so that in case of a light body, as the speed increases the air resistance may become equal and opposite to its weight, and when that is the case it falls without acceleration. This is the case with scraps of paper and rain drops. Strictly speaking, even the weight of a body is not constant as it falls, but increases as it approaches the surface of the earth. The weight of a kilogram one mile above the earth's surface is less by 3^^ a gram than at sea level, and at the ceiling of a room 3 meters high a kilogram weighs about one milligram less than at the floor. This variation of force with height causes a corre- sponding increase in the acceleration of a falling body as it ap- proaches the earth's surface; but this is so small, however, that except in case of great heights it may be neglected. 101. Acceleration of Gravity. — The early philosophers specu- lated as to why bodies fell; Galileo was the first to carefully de- termine how bodies fell. He also showed, contrary to the univer- sally accepted opinion of his day, that except for air resistance all falling bodies are equally accelerated. A large stone or a small one, an iron cannon ball, a lump of lead, or block of wood when dropped from the top of a tower reach the ground in the same time. If a feather, scraps of paper, and some bits of metal or lead shot KINETICS 63 are placed in a long tube (Fig. 46) from which the air is exhausted, on quickly inverting the tube all reach the bottom at the same instant. Hence the rate of increase in velocity, or acceleration, is constant at any given place on the earth for all kinds and sizes of bodies. This constant acceleration is called the acceleration of gravity at the given place, it is usually represented by the symbol g and is measured most accurately by pendulum experiments. The value of g at the sea level for the latitude of New York is 980.2 cm./sec.^, or 32.16 ft./sec.2 The table on page 108 gives also the values at some other places. The formulas for falling bodies are, there- fore, obtained from those of §98 by making the acceleration equal to g. Thus V = u + gt s = ut + }4gt^ 2gs = v^ — u^. When a body is simply dropped, with no initial velo,city, u is zero, and we have y2gi' V = gt s 2gs = v'^- In approximate calculations and in work- ing problems for practice, g may be taken as 980 cm./sec.^ or 32. ft./sec.^ 102. Mass Proportional to Weight. — Galileo's discovery that all kinds and sizes of bodies when dropped to the earth at the same place are accelerated at the same rate except for air resistance, leads to an important con- clusion. For when two bodies are equally accelerated their masses must be proportional to the accelerating forces (§91), which forces, in the case under consideration, are the weights of the bodies; therefore the masses of bodies are proportional to their weights, if weighed at the same place. FiQ. 46. — Fall in vacuo. 64 MECHANICS 103. Relation between Dyne and Gram. — The force urging downward a freely falling mass m is expressed by the formula F = mg the force being in dyTies if C. G. S. units are used. Suppose m = 1 gram and g = 980, then F = 980 dynes; but the force with which a mass of one gram is attracted toward the earth is called the weight of one gram therefore the weight of one gram = 980 dynes, or the force which we have called a dyne is slightly more than the weight of a milhgram at the earth's surface. The student may show similarly that one poundal is about equal to the weight of a half-oimce; that is, one pound weight at New York = 32.16 poundals. 104. Gravitation Units of Force. — The weight of a gram or pound is often a convenient unit of force; indeed, engineers in English speaking countries almost always measure forces in pounds; for though the weight of a pound varies from place to place on the earth, its weight at some selected spot may be taken as standard. For example the standard force of a pound may be defined as the weight of a pound mass at New York where the acceleration of gravity is 32.16 ft./sec.^, and in that case it will be equal to 32.16 poundals. So also the standard force of a gram might be defined as the weight of a gram at a point where g = 980 cm./sec.^, in which case it is equal to exactly 980 dynes. If these gravitation units of force are used the constant fc in formula (1) §94 is no longer unity, but we have [F, in pounds) = (m, in pounds) X (a, in ft./sec.2) or {F, in grams) = — - (m, in grams) X {a, in cm./sec.*) But most of the formulas in this book are based on the relation F = ma; it will therefore be best for the student in working prob- lems to use consistently either the centimeter-gram-second system with the force in dynes, or the foot-pound-second system with the force in poundals, changing, when required, dynes or poundals into grams or pounds weight by dividing by 980 or 32 as the case KINETICS 65 may be; 32 being used instead of 32.16, as the value of g in ft-Zsec^", for convenience in numerical work. 105. Atwood's Machine Problem. — Suppose two masses, one 40 and the other 50 grams, are connected by a cord running over a light frictionless pulley as in Atwood's machine, and suppose for simplicity that the mass of the cord and of the pulley may be neglected. It is required to find the acceleration and the tension on the cord. In this case the whole mass 40 + 50 moves together and the resultant force which gives it motion is the weight of 50 — 40 = 10 grams, or lOg dynes. Since force = mass X acceleration we have lOg = 90 X a, therefore a = }4g, hence the acceleration is one-ninth that of a freely falling body. This result may also be reached by considering that a force of 10 grams acting on a mass of 10 grams gives it an acceleration g, and therefore if that same force act on a mass 9 times as great it will give it an accelera- tion )4 g. To find the tension on the cord consider the forces acting on the mass 40. It is urged downward by its own weight, 40 grams, and upward by the tension of the cord, which we may call T grams. It moves upward with an acceleration 3^ g, as has been shown, hence the resultant force must be upward and equal to (T — 40) grams or {T—4Q)g dynes, and we have, since F = ma, (r-40)g = 40X3^ff whence T = 44^^ grams' weight. 106. Motion on an Inclined Plane. — When a mass M rests on an inclined plane, the force due to gravity, or its weight, may be resolved into two components, as shown in figure 48, one N perpendicular'to the plane and the other F parallel to it. If M is the mass in grams, its weight in dynes is Mg. And from the similarity of the two triangles, we have Mg, N, and F respectively proportional to the sides of the large triangle formed by I, b, and h. That is, F:Mg::h: I or F = Mgj dynes. Thus the force 66 MECHANICS F causing the motion is constant, and is the same fractional part of the whole weight of the body as the height of the inclined plane is of its length. The acceleration is therefore constant. Since F = Ma, we have a = gj or a = g sin c. To find the velocity which the body acquires in sUding the length of the plane I, we have only to use the formula (4) of §98. 2as = v^ — u^. The body starts from rest, hence u = o and s = I in this case, therefore 2g -J I = v'' or v^ 2gh; but this is precisely the velocity which a freely falling body will gain in falling through a vertical distance h, and there is nothing Fig. 48. Fig. 49. in the result which depends on the slope of the plane, therefore the velocity gained by a body in sliding down a frictionless in- clined plane of any slope whatever is the same as that gained by a body in falling freely the same vertical distance. Since the velocity does not depend on the slope of the plane, it will be the same at B (Fig. 49) for any smooth, frictionless curve down which it may slide from A, and it will be the same at jB as at C or D. The time of descent, however, from Ato B depends on the curve and may be proved to be a minimum when A and B are joined by the arc of a cycloid. 107. Kinetic Energy. — We will now calculate the effect of a certain amount of work in giving motion to a mass m. Suppose a force of F dynes acts on m in the direction of its motion while it is moving through a space of s centimeters; the work done is by definition, Fs dyne-centimeters or ergs. But while the constant yimv^ KINETICS 67 force F acts there is a constant acceleration a and the equations of §98 therefore apply to the motion, and we have 2as = j;2 — w^, also F = ma. Multiplying these equations together we obtain Fs = }4mv^ - }4mu^ (1) The change from J^twm^ to }imv^ therefore expresses the amount of work required to change the velocity of the mass m from u to V. Starting from rest, the energy required to give it velocity v is }^mv^; this is also the measure of the work that the body can do before coming to rest again, therefore the quantity }/^mv^ is the measure of the kinetic energy or energy of motion possessed by a mass m moving with velocity v. , . ,. . , [m is in grams, = kinetic energy in eras when i . . [v IS in cms. per sec. 1 . j^. ■ J. J 7,7 fm is in pounds = kinetic energy in foot-poundals when { . . ,, y •> •> ^ I V is in ft. per sec. 108. Velocity at Foot of Inclined Plane. — The principle of the conservation of energy may be applied to motion on an in- clined plane and leads at once to the conclusion previously stated, §106, that the velocity of a body at the foot of an inclined plane depends only on its height and is independent of the slope. For the work done in lifting the body from the bottom of the plane to the top depends only on the height of the plane, since the work is done only against gravity and serves to increase the potential energy of the body. In sliding down the plane, if no work is done against friction, all the potential energy gained will be transformed into kinetic energy, so that when it reaches the bottom its kinetic energy will be equal to the work that was done in lifting it. The kinetic energy, of the body and consequently its velocity will therefore be independent of the slope of the plane. The work done in lifting the mass m the height of the plane h is mgh, for mg is the weight of the mass expressed in dynes. The kinetic energy of the mass at the bottom is J^mv^, hence mgh = J'^my^ and v^ = 2gh. 68 MECHANICS 109. Kinetic Energy and Momentum Compared. — Kinetic energy and momentum are both quantities that depend on the mass and velocity of the moving body, but while kinetic energy is expressed by ^^my^ and measures the work done on the body in giving it motion, momentum, expressed by mv, measures the impulse given to it, or the product of the force by the time during which it was acting on the body, for the second law of motion (§94) gives the relation Ft = mv — mu, or change in momentum is equal to the impulse when the force is measured in the appropriate unit. Hence if a force acts upon a body through a certain distance and it is required to find the change in velocity of the body, the formula for kinetic energy must be used, Fs = J^my^ — ^■2mu^; while if the time of action of the force is given, the change in velocity is found from the equation of momentum. Ft = mv — mu. 110. — ^Impact. — When one freely moving body strikes against another there is said to be impact. When the ball B is at rest and A is allowed to swing against it, if the bodies are inelastic like two balls of lead or putty, they will keep together after impact, the forward momentum of the combined mass being equal to the momentum of A before impact. If the two balls are perfectly elastic or resilient and of equal masses, like two ivory billiard balls, A will come to rest giving up its whole momentum to B, which will, therefore, swing out just as far as A has fallen. If the masses are elastic but not equal, then A may continue forward or have its motion reversed at the instant of impact depending on whether B is the less or greater mass. In all cases of impact, whether the masses are elastic or in- KINETICS 69 elastic, the total momentum of the two bodies is not changed by the impact. That is, if one body loses forward momentum the other gains an exactly equal forward momentum. Stated algebraically, Av + Bu = AV + BU where A and B are the two masses, respectively, while v and u are their velocities before impact, and V and U are their velocities after impact. This law is easily seen to be a direct consequence of the laws of motion. For at each instant during impact the forward pres- sure of A upon B is equal to the backward pressure of B against A, as expressed in the statement that action and reaction are equal and opposite. Hence the total forward impulse given to B is equal to the backward impulse sustained by A, and by Newton's second law the change in the momentum of A must be equal and opposite to the change in the momentum of B, con- sequently the sum of the momenta of the two is not changed. If the two are inelastic they move together after impact with a common velocity x, whence Av + Bu = (A + B)x. In case of elastic bodies there is a certain instant during the impact when the compression is a maximum and the two bodies are neither approaching nor receding from each other. At that instant they are moving with the same velocity x ^^ ^ which they would have acquired if '' quite inelastic. But suppose they are perfectly resilient and the pressure between them at any instant as they spring apart is exactly equal to what it was during the corresponding instant of compression. The total backward im- pulse given to A will then be twice what it would have been if the bodies had been inelastic, hence the total change in velocity of A will be twice as great sls v — x, ov 2{v — x), and its final velocity V will be V =- V -2{v - x) = V -\-2{x - v), so also C7 = M,-f- 2(x - u). 70 MECHANICS If the above expressions are written V = V -\- ij.{x — v) U = u + ii{x — u) the coefficient fi will serve to indicate the degree. of resiliency, If M = 1 the bodies are quite inelastic for F = a; and U = x, bul if /i = 2, the resiliency is perfect. Problems 1. A ball is thrown vertically upward with a velocity of 64 ft. per sec; how soon will it reach the ground again and how high will it rise, and ■what will be its velocity when half-way up? 2. A falling body has a velocity 200 cm. /sec; how far will it drop before its velocity becomes 10,000 cm. /sec? Take g = 980. 3. A weight thrown forward on ice with velocity 60 ft. per sec. is resisted by a constant force, and after 5 seconds has half its original velocity; how far has it gone in that time? 4. Knd the acceleration in the previous problem, also how far the weight will go before coming to rest. 6. A mass of 10 gms. is acted on by a constant force which changes its velocity from 100 to 500 cm. per sec. in 5 seconds. Find the accelera- tion and amount of the force. 6. What steady forward pull must be exerted by a locomotive in starting a 200-ton train to give it a velocity of 20 miles per hour in 5 minutes, neglecting friction. Find force in poundals and then in pounds. 7. A weight of 10 lbs. is thrown forward on ice with a velocity 60 ft. per sec; if the coefficient of friction between it and the ice is 0.10, how far will it go and in how many seconds will it stop. 8. A 300-lb. mass is lowered by a rope with uniform velocity. What is the tension on the rope? If it is lowered with a constant acceleration of 10 ft. per sec. per sec. what is the tension? AVhat if it is lowered with acceleration g7 9. An elevator weighing 2000 lbs. is pulled upward with a force of 3000 lbs. What is its acceleration, and how long will it take to gain an upward velocity of 2 ft. per sec? 10. A mine bucket weighing 2000 lbs. and being lowered with a velocity of 3 ft. per sec. is stopped in a distance of 1 ft. What is the average force on the supporting cable while stopping? 11. If a man weighing 75 kgms. is in an elevator which is going up with constant velocity, how much force does he exert on its floor? What if the elevator has an upward acceleration of 3 meter/sec^? 12. What is the least acceleration with which a man weighing 150 lbs. can sUdc down a fire-escape rope which can only sustain a weight of 100 lbs. 7 And what velocity will he have after sliding 50 ft.? KINETICS 71 13. A 30-gm. weight is drawn up by a 70-gm. weight by means of a cord over a frictionless pulley. Find the acceleration (taking g = 980) and also the tension on the cord. How far will the weights move in 3 seconds from the start? 14. A 38-lb. weight resting on a level, frictionless table is drawn along by a 4-lb. weight by means of a cord over a frictionless pulley. Find the acceleration and also the tension on the cord. 15. If in the previous problem the friction between the weight and table is a force of 2 lbs. find acceleration and tension as before. 16. How many foot-poundals of work are required to give a 500-lb. shell a velocity of 2000 ft. per sec? Find the work also in foot-pounds. If this work is done by the powder gas in a gun 25 ft. long, find the average force in pounds against the shell as it is discharged. 17. How much energy in foot-pounds must be expended in giving a 300- ton train a velocity of 30 miles an hour? If tl e locomotive works at the rate of 100 H.P., how long will it take to bring the train up to speed? 18. A 3-kgm. hammer with a velocity of 5 meters per sec. drives a nail 4 cm. into a plank. Find the average resistance in dynes and grams and how much weight resting on the nail would be required to force it into the wood. 19. A bullet weighing 1 oz. and having a velocity of 1000 ft./sec. is fired through a plank 3 in. thick which resists it with a force of 800 lbs. With what velocity will it come out, and how many such planks could it pierce? 20. A bullet weighing 1 oz. is shot into a suspended block of wood weighing 18 lbs. 11 oz. and gives it a velocity of 6 ft. per sec. What is the combined momentum of block and bullet after impact? What was the momentum of the bullet before impact? Thence find velocity of bullet before impact. 21. What was the kinetic energy of the bullet in problem 20 before impact? What is the kinetic energy of the block containing bullet after impact? How much energy in foot-pounds was expended by the bullet in pene- trating into the block? What proportional part of the original energy of the bullet remains as energy of motion after impact? 22. A bullet weighing 15 gms. is shot into a suspended block of wood weighing 2985 gms. and gives it a velocity of 200 cm. per sec; find the velocity of the bullet. 23. How high above its original level will the suspended block, in the last question, swing in consequence of the velocity given to it? 24. If all the energy of a 640-lb. shell having a velocity of 2000 tt./sec could be spent in raising a 10,000-ton battle ship, how high would it lift it? 25. A bullet weighing 10 gms. has a velocity of 600 meters per sec and penetrates 30 cms. into a pine log. What is the force in kilograms with which the bullet is resisted, and how far would it penetrate if it had half the original velocity? 26. A monkey clings to one end of a rope passing over a frictionless pulley, and is balanced by an exactly equal weight on the other end of the rope. 72 MECHANICS Explain what will happen to the counterpoise if the monkey climbs 10 ft. up the rope and then suddenly stops. The mass of the rope and wheel are to be neglected. 27. A cord passes over two fixed pulleys and hangs down vertically between them supporting a movable pulley which with attached weight weighs 5 lbs. A 3-lb. weight is hung on one end of the cord and a 4-lb. weight on the other end. Find the accelerations of all three weights and the tension on the cord. Note. — First find a simple relation between the accelerations of the three masses from the fact that the cord is inextensible. Motion of a Particle in Curved Path 111. Motion of a Projectile. — When a body near the surface of the earth is thrown in any direction, such as AB, it is subject to the steady force of the earth's attraction vertically downward, and, therefore, it has constantly the downward acceleration of g gravity g. The initial impulse, however, gave it a forward velocity V in the direction AB, in which direction it would have continued to move with constant velocity if no force had acted on it . The actual path in which it moves may then be regarded as the resultant of motion with constant velocity V in the direc- tion AB combined with a motion downward with constant accel- d £ H Fio. 52. — Curve of projectiles and jets. eration g. Thus after a time t the body will have traveled a distance AC = Vt in the direction AB, but it will also have fallen from C to D a distance s = j^gt^. If a is the angle of elevation of AB above the horizontal, and if d is the distance AE which the projectile has advanced in a horizontal direction and h is its height, we have d = Vt cos a h = Vtsina - }4gt'. The path traversed may be shown to be parabola with its axis vertical and passing through the highest point of the path. The highest point is half-way between the point of projection and the point G where the pro- jectile again reaches the earth. The distance AG is called the range, and is a maximum when the angle a is 45°. KINETICS 73 These results are easily deduced from the above equations, but it must be borne in mind that the influence of air resistance has been neglected. This force in rapidly moving bodies, like bullets, may be very great and changes the form of the trajectory to something like that shown by the second curve from A to H. In consequence of this the maximum range in gunnery is found at a much smaller elevation. The form of the path of a, projectile or ball is beautifully shown by a water jet, for each particle in the jet is a freely falling body. 112. Curved Pitching. — If a ball when thrown forward is rapidly rotated the resistance of the air causes it to swerve from the path that it would otherwise take. This is seen in the curv- ing of a pitched ball and in the drifting of projectiles from rifled guns. It results from the viscosity of air in consequence of which the rotating ball drags air in on one side and flings it out on the other as it advances. Suppose, for example, that a ball is spinning about an axis perpendicular to the paper as shown by the curved arrow in figure 53, while it is moving forward in the direction of the straight arrow cd; the rotation of the ball drags air in from *" ^ ^"3' the side at a and carries it jjq, 53._curving of pitched bail, around toward the front of the ball at c, giving it a greater forward momentum than is given to the air between c and h where the surface of the ball is spin- ning backward. The force against the ball is, therefore, greater between a and c than it is between h and c. Let P and p represent these pressures against the ball. Their resultant as shown in the diagram of forces is the oblique force R which in part resists the forward motion of the ball, but also has a component represented by F which is at right angles to the path of the ball and causes it to swerve to one side in the direction of the dotted line. As the force F acts constantly it causes the ball to move side- wise with constantly accelerated motion, and, therefore, the curv- ing rapidly increases as the ball advances. 113. Motion Around the Earth. — Suppose it were possible to shoot a cannon ball in a horizontal direction from the top of some high mountain on the earth with a velocity so great that 74 MECHANICS while it advanced a mile it would drop just enough to follow the curvature of the earth. Then, if there were no air resistance, the ball would continue around the earth and return to its origi- nal point of projection with undiminished velocity and would, therefore,, continue to circulate forever around the earth as a satellite. For suppose A (Fig. 54) is the point from which the projectile is shot in the direction AB. As it advances it drops away from the line AB; but the earth's surface also drops away from AB in consequence of its curvature, by about 8 in., in the first mile. If, therefore, the cannon ball has a velocity which will carry it a mile in the same time that it will drop 8 in., it will, on reach- FiG. 54. Fig. 55. ing the end of the mile, say at C, be just as high above the earth as at the start and be moving in the direction CD, tangent at C. As the ball moves forward the force due to the earth's attrac- tion is always at right angles to the direction of motion, and hence the speed of the ball is neither increased nor diminished and all the conditions of the motion remain constant. The time required for a body to drop 8 in. toward the earth is found from the formula (§101) s = V2gt\ Since s = 8 in, or 0.66 ft., and taking ? = 32 ft./sec.^ we find t = 0.20 sec; hence our cannon ball must have a velocity of a mile in 0.2 sec. or 5 miles per second. KINETICS 75 The complete calculation may be made thus. Let the ball drop a distance BC = s (Fig. 55) in going forward the distance AB = d. li R is the radius of the earth the relation between s and d may be found from the similarity of the triangles ACE and ADC from which we find AD -.AC-.-.AC -.AE or s : d :: d : 2R and ' = 2R' (1) but s is the distance fallen with constant acceleration g in t seconds, therefore s = y2gt', and d is the distance which the ball, moving with constant velocity v, advances in t seconds; that is d = vt. Substituting in (1) we have y29t^ = 2R whence '^ - R Taking g = 32 ft./sec.^ and R = 5280 X 4000 ft., v^ = 32 X 5280 X 4000 and we obtain V = 26,000; ft./sec. = 4.92 miles per sec. 114. Motion in Any Circle with Constant Speed. — The case just discussed is in no way different from any case when a mass moves in a circle with constant speed. To cause it to constantly change its direction of motion there must be a force constantly acting at right angles to the direction of motion, and if the speed does not change there can be no force at all in the direction of motion. The relation between the velocity in the circle, the 76 MECHANICS radius of curvature of the path, and the acceleration are given as above in the formula a = — ■ r It will be interesting to derive this relation in another way, from the simple conception of acceleration as the change in ve- locity per second. As a particle moves from A to B in the circle (Fig. 56) its velocity changes only in direction from v\ to v^. But this change in velocity is equivalent to compounding with the original velocity Vi, another velocity represented by the vector /. This vector therefore represents the change in velocity between A and B and when divided by t, the time taken by the body in moving from A to B, gives the average rate of acceleration between A and B or Let d represent the distance AB, and since the sector OAB is almost exactly similar to the triangle formed by vi, vi, and / we have the proportion r -.d y.v -./where v is the amount of Vi or V2. I substituting r wt :: V -.at from which v' r and this relation is exad, and not approximate, for as B ap- proaches A, the triangle and sector approach exact similarity as a limit and the average acceleration between A and B approaches the actual acceleration at A. It will be noticed also that / is parallel to a line bisecting the angle AOB, hence as B approaches A the direction of / approaches the direction AO as a limit. We conclude therefore that the acceleration at any point of the circle is directed toward the center and is equal to — r 115. Acceleration in any Curved Path. — Since the accelera- tion jiist found depends only on the instantaneous relation of / Vi or V2. But the distance d = vt and a =- or f = at hence KINETICS 77 where the various quantities involved, it applies to any curved path whatever. The acceleration at any point in a curved path may be resolved into two components, one along the curve or tangent to it and the other at right angles to it. The component along the curve is the rate of change of speed of the moving body, while the component at right angles to the path, is a V is the speed at .the given point and r is the radius of curvature of the path at that point. 116. Force In Circular Motion. — Whenever a mass is accelerated it is acted on by a force determined by the relation F — ma, hence when a mass m moves in a circle with con- FiG. 57. stant velocity it is acted on by a force F = m -directed toward the center of the circle ; or, more generally, whenever a mass is moving in a curved path it is subject at any point to a force F = m ~ directed toward the center of curvature, r being the radius of curvature of the path and v the velocity of the mass m at the given point. 117. Centripetal and Centrifugal Force. — The force by which a mass is constrained to move in a curved path is, as has just been shown, always directed toward the center of curvature of the path, and is therefore called the centripetal force. The reaction aginst this force by the moving body is called centrifugal force. Both are different aspects of the same stress, and are of course equal and opposite. For example, when a weight is whirled in a circle by a cord, it is held in the circle by the tension of the cord which supplies the centripetal force, while the reaction or outward pull of the weight against the cord is called the centrifugal force. When the string breaks both forces instantly disappear, there is no tendency for the weight to fly outward, it simply keeps moving in the tangential direction in which it was moving when freed. Fig. 58. 78 MECHANICS 118. Other Expressions for Centripetal Force. — If a mass moves in a circle and makes n complete revolutions per sec- ond, since the distance traveled in one revolution is 27rr, in n revolutions it will be 2Trrn and hence v = 2irrn, and the centri- petal force ¥ = becomes F = m4TVr (1) Or we may express the velocity in terms of the time required to make one complete revolution, which may be called the period P. In that case 2irr and F = m^ (2) Or if o) represents the angular velocity of the rotating body, or the arc in radians traversed per second, we have or = v (§130) and therefore F = mojV (3) 119. Illustrations. — When a railway train rounds a curve it is kept in the curve by the pressure of the rail against the flanges of the wheels. The weight of the train is balanced by the upward pressure of the track, represented at A, figure 59, while the centripetal force exerted by Ij?) the rails is represented by B; the re- SS|/ / sultant force R is therefore inclined — Z- iki and the track is tilted toward the center «ai___ gQ ^jjg^|. ^jjg pressure may be equal on ^"^- 59- both rails. When a fly wheel rotates, it is under great tension due to the centrifugal force of the heavy rim, and great destruction may result from the bursting of such a wheel. A grindstone also may burst if driven at too high a speed. A glass of water held in a sling may be swung in a vertical circle without spilling the water. For the acceleration down- ward due to gravity when it is at the top of the circle may not be enough to hold it in the circular path, consequently the bottom KINETICS 79 of the glass must exert an additional force upon the enclosed water, so that even at the top of its path the water presses against the bottom of the glass. In this case the tension on the cord when the mass is at the top of the circle is lessened by the weight of the body, while at the bottom the tension is increased by the same amount. The tension on the cord will therefore be F = mv'' - mg at the top. at the bottom. If a cylinder of wood having a length three or four times its diameter is suspended from the axle of a whirling machine by a short cord attached to one end, on setting it in rotation it will ""^ soon be disturbed from its initial position (Fig. 60) and will rotate in the oblique position as shown, in consequence of the centrifugal force of the two ends of the cylinder balancing the force toward the axis due to the cord ^'°- ®°- and weight of the cylinder. With rapid rotation the cylinder comes into a horizontal position rotating now about an axis at right angles to its length. So a ring suspended by a cord from a rotating point of sup- port will tip up into the horizontal position and rotate like a wheel about a vertical axis through its center. And even a chain when similarly suspended will first widen out into a loop in consequence of centrifugal force and then take the form of a horizontal ring rotating as though it were rigid; the required centripetal force in these cases being supplied by the tension on the chain or ring. In all these cases it will be observed that equilibrium is attained when the mass is on the whole as far as possible from the axis of revolution. 120. Conical Pendulum. — Suppose a mass m, as in an old- fashioned steam engine governor, is swung aroung in a circle with uniform speed, it will swing out from the axis and come into equilibrium at a certain angle depending on the speed of rotation. 80 MECHANICS Evidently in order that there may be equilibrium the mass must be acted on by a force F directed toward the axis and just sufficient to hold it in the circle. But if the mass, or conical pendulum as it may be called, makes one revolution in P seconds, then the centripetal force is _, m4xV F = > '^ pi and this force is the resultant of the tension on the cord T and the weight of the mass, which is mg dynes. Constructing the diagram of forces as in the figure, it is clear that F -.mg :: r -.h; therefore, mgr We have, therefore mgr m 4 tt V h whence = ->f- Consequently if we have two masses m and m' hung by cords of different lengths, they will have the same period of rotation if the height h is the same in both cases. The more rapid the rotation the higher the mass rises, and in the steam-engine governor the rising of the weights operates a lever through which steam is cut off and the speed of the engine decreased. Problems 1. A stream of water from a horizontal nozzle falls 3 ft. below the level of the nozzle in a distance of 20 ft., measured horizontally. Find the velocity of the escaping jet. 2. A jet of water is directed upward at an angle of 45° to the vertical, and strikes the ground at a distance of 64 ft. from the nozzle. Find the time taken by a water particle in passing from nozzle to ground, and the velocity of the jet. 3. How much weight can a cord sustain by which a mass of 100 gms. can be whirled in a circle of 1 meter radius making 2 turns per sec. neglecting the effect of gravity in the circular motion? KINETICS 81 4. A stone weighing 1 lb. is whirled by a string in a circle 6 ft. in diameter. The string breaks and the stone flies off with a velocity of 30 ft. per sec. Find the strain on the string when it broke. 5. A mass of 50 gms. has a velocity of 750 cms. per sec. in a circle of radius 60 cms. Find the acceleration in amount and direction and centripetal force in dynes. Also find angular velocity. 6. A 100-ton locomotive rounds a curve at a uniform speed of 40 miles per hour. Find the acceleration if the radius of curvature of the track is 1000 ft. Also find the horizontal force exerted against the rails. 7. In case of the last problem, how much higher must the outer rail be than the inner, in order that the resultant force due both to the weight of the locomotive and its centrifugal force, may be perpendicular to the road bed? 8. A mass of 1 lb. is whirled in a circle of 2 ft. radius on a smooth level table, being held in the circle by a cord which passes without friction through a hole in the center of the table and supports a 2-lb. weight. Find the angular velocity and revolutions per sec. of the 1-lb. mass necessary to support the weight. 9. A 200-gram mass is whirled in a vertical circle of radius 80 cms. with a uni- form angular velocity 8 radians per sec. Find the period of revolution and the acceleration. Also what is the tension on the cord in grams when the mass is at the top of the circle and what when it is at the bottom? 10. A weight of 2 lbs. is whirled in a vertical circle. If its velocity is 100 cms. per sec. at the top of the circle, what will be its velocity at the bottom, the gain being due to the acceleration of gravity as it falls, just as in an inclined plane (see §106)? Radius 80 cms. 11. A 10-lb. mass is hung as a pendulum by a cord 4 ft. long. How high must it swing in order that the tension on the cord at the lowest point of its swing may be double the tension when hanging at rest? 12. In case of "looping the loop," how high above the level of the top of the circle must the car start that it may just have speed enough to keep to the circle, neglecting friction? Circle 30 ft. in diameter. 13. Find the angular velocity and period of a conical pendulum hung by a cord 1 meter long and swinging around in a horizontal circle of 60 cms. radius. Vibratory Motion 131. Simple Harmonic Motion. — If a body moving with con- stant speed in a circular path is observed from a distant point in the plane of the circle, it appears to oscillate back and forward in a straight line. The kind of vibratory or oscillatory motion that the particle appears to have in this case is known as simple harmonic motion, 82 MECHANICS it may be defined as the projection upon a straight line of uniform motion in a circle. There are other kinds of vibratory motion that are not simple harmonic, such, for example, as the particle would appear to have in the above instance if it moved around the circle in any manner whatever except with constant speed. Simple harmonic vibra- tion is, therefore, one particular mode of oscillation; but.it is by far the most important, for it IIP \ B! Fig. 62. is the most common of all, and all other modes of vibra- ' tion may be expressed as the resultant of a sum of simple harmonic vibrations, as was shown by the French mathe- matician Fourier. A simple mechanical device illustrating this kind of motion is shown in figure 62. A pin P projects from the face of a ro- tating disc D and fits in a slot in a cross head which is attached to rods that can slide back and forth in the bearings BB'. When the disc rotates with uniform speed every point in the rods and cross head will move back and forth with simple harmonic motion. The amplitude of the vibration is the distance that the vibrating body moves on each side away from its central or mean position. 133. Velocity in Simple Harmonic Motion. — ^Let a particle A move around the circle (Fig. 63) with constant speed, and let another particle B move back and forth along a diameter DC in such a way that the line joining A and B is always perpendicular to DC. Then B oscillates with simple harmonic motion. Let Vo represent the velocity of A. It may be resolved into two components, as shown in the diagram, one at right angles to the direction in which B moves and the other parallel with B's motion. Since B always keeps abreast of A, the velocity of B at any point must be equal to that component of A's velocity which is parallel to DC, namely to the KINETICS 83 component v. Letting r represent the radius of the circle and y the distance AB, we have by similar triangles r :y = Vo '.V whence y V = -Vo or V = Vo sm e where e is the angle AOC. The velocity of B is, therefore, zero at the ends of its path at G and D, for there y = 0. While at the center y = r and the velocity of B is equal to Vo, its maximum value. The complete period of an oscillation of B is evidently the same as the time in which A goes completely around the circle. Let P represent this period, and the velocity of A is 2irr Vo = — p-> which also expresses the velocity of B at its middle point. 1133. Acceleration In Simple Harmonic Motion. — Since A and B have exactly the same motion in the direction DC, the acceleration of B must be the same as that component of the acceleration of A which is parallel to DC. The accelera- tion tto oi A, moving with uniform speed in a circle, is directed toward the center of the circle and is equal to -pj- (§118). '°" ^ " Resolving the acceleration ao into two components and letting a represent the component parallel to DC and 6 that perpendicular to it, we have by similar triangles. and, therefore, or smce a •.ao = x:r a = — X r 4 7r2 a- p, X. 84 MECHANICS The acceleration of B is, therefore, proportional to its distance from the center, it is greatest when x = r and is zero when B is at the center. It will also be noticed that the acceleration of B is always directed toward the center; that is, B is always losing velocity as it moves away from the center and gaining velocity as it moves toward the center, and consequently its velocity is greatest at the center as we have already seen. 134. Force In Simple Harmonic Motion. — The fundamental dynamical equation F = ma enables us to express at once the force in simple harmonic motion. When the mass of the oscil- lating particle is m and its complete period of oscillation is P, the acceleration at the instant when the particle is a distance x from its central position has just been shown to be 4ir^x The force at that instant is, therefore, F = m-p^ (1) and is directed always toward the center, or equilibrium position. Therefore when a mass in equilibrium is so situated that if displaced it is always urged back toward its equilibrium position by a force which is proportional to the displacement, it will, on being displaced and then set free, oscillate with simple harmonic motion about its position of equilibrium as a center. Now the force required to cause a small strain in almost any elastic body is proportional to the amount that the body is strained, whether the body is bent or stretched or twisted (Hooke's Law, §237), hence when such bodies are strained and then let go they oscillate to and fro in simple harmonic motion, as in case of the small vibrations of a tuning fork. 125. Problem. — ^Let a mass of 1 kgm. be supported by a steel spring of such stiffness that an additional weight of 100 grams will stretch it just 1 cm. It is required to find the period of oscillation of the weight if disturbed, neglecting the mass of the spring. If the kilogram weight is pushed up or pulled down as it hangs on the spring, it will move through a distance which is proportional to the force used, a force of 100 gms. being required to displace it 1 cm. To produce a KINETICS 85 displacement of x cms. the force required is 100 x gms. or 100 xg dynes. But from equation (1) above we have, since m = 1000, P = 1000 — but F = 100 gx. 4Tr2 4000 TT" Therefore, . 100 s = 1000 ^^ and P' = """ r IW.g which gives P = 0.634 sec. 126. Simple Harmonic Motion Isoclironous. — It will be 47r xwi noticed that the expression P^ = — = — does not contain r t and is, therefore, independent of the amplitude of the vibration, so that it does not naake any difference in the period of vibration whether the amplitude is large or small, provided the ratio p is constant, in which case the motion is truly simple harmonic. When vibrations have this property they are said to be isochronous. 137. Energy of a Vibrating Mass. — The energy of an oscillating mass is all potential at the ends of its vibration, but in the middle where the velocity is greatest it is all kinetic and so may readily be computed. For we have seen, §122, that the maximum velocity of the vibrating body is 2Trr Vo = -pr' and since kinetic energy = J^my we find, kinetic energy at middle or total energy = — p^ — where m is the mass of the vibrating particle, P is its period of vibration, and r is the amplitude of its motion. 128. Simple Pendulum. — A mass suspended from a fixed point so that it can swing freely in a circular arc about the fixed point as a center, is called a pendulum. As a simple or ideal case we may suppose the whole mass of the pendulum to be concen- trated at the point B, the mass of the suspending cord or wire be- ing so small as to be neglected. The forces acting on the mass m are its weight mg and the tension T of the suspending cord. The weight mg may be resolved into two components, one in line with the cord and opposing its tension and one at right angles to the suspending cord and in the direction in which the mass m moves. It is this latter component F (Fig. 65) which gives it motion alqng 86 MECHANICS the circle. Since the diagram of forces is a triangle similar to BCO we have F -.mg = BC : BO; but BO = I, and if the angle a through which the pendulum swings to and fro is small, BC is very nearly indeed equal to the arc BA, the length of which may be represented by x. Then approximately F -.mg = X -.1 and Therefore the force F urging m along the arc toward A is pro- portional to the displacement x measured along the arc. But with such a law of force there is simple harmonic vibration (§124) and the relation of force to period of vibration is expressed in the formula Att'^x F = m- Equating (1) and (2) we have (2) 47r' p2 from which Fig. 05. =-J. The period of vibration therefore depends only on the length of the pendulum and the acceleration of gravity at the place where it is swung, and is independent of its mass and of the length of the arc, provided the arc is so small that the approxi- mation made above is justified. The effect of the length of arc upon the period is shown by the following more exact formula, —aK-S)^ where a is the arc AB measured in radians. Thus if a pendulum 1 meter KINETICS 87 long swings 5 em. on each side of the lowest point, the arc o = J^o and r^ = gjjjg' so that the period is greater by one part in 6400 than if the arc had been infinitely small. 129. Pendulum Clocks. — The pendulum affords a valuable means of regulating the motion of a clock, since when it swings through a small arc its oscillations are nearly isochronous; i.e., its period of oscillation is nearly independent of the amplitude of its swing. When an ordinary clock driven by a spring is just wound up it gives a greater impulse to the pendulum through the escape- ment than when it is nearly run down, and even in clocks driven by weights the friction is not always constant and the swing of the pendulum will vary accordingly. It must be remembered also that the pendulum of a clock is not free, but the little backward and forward impulses which it receives from the escapement hasten somewhat its motion. To secure regularity of motion, therefore, the pendulum should be heavy, so that its natural period will be only slightly affected by the pushes of the escapement. In the finest astromonical clocks what is known as a gravity escapement is used, in which the pendulum does not receive any impulse directly from the spring or weight that drives the clock, but its motion is kept up by a small weighted lever which is set free just as the pendulum reaches the end of its swing, and in falling gives a slight push to the latter. Between successive impulses the lever is raised and set in position by the action of the clockwork. Full details as to some forms of gravity escapement will be found in the article Clocks in the Encyclopedia Britannica. Problems 1. Show that the motion of the piston of a steam engine when the crank is turning with uniform velocity is not simple harmonic. At which end of the piston's motion is the acceleration greatest and why? 2. Assuming that the motion of the piston is simple harmonic, find its velocity in the middle of its stroke when the crank is 8 in. long and makes 200 revolutions per minute. Also find acceleration at middle and end of its stroke. 3. If the piston and connecting rod weigh 100 lbs. in the last problem, find the maximum force against the crank pin due to their inertia alone, neglecting the effect of steam pressure. 88 MECHANICS 4. A mass of 4 lbs. is made to oscillate to and fro by a spring at the rate of 2 vibrations per sec. Find the force on the mass when it is 2 in. from its middle position. 6. A pendulum 1 meter long swings 10 cms. on each side of its lowest point; find the direction and amount of the acceleration at the ends of its swing and at middle. 6. How long must a pendulum be to beat seconds at a place where g = 980. If made 1 mm. too long will it gain or lose and how much per day? 7. A clock having a pendulum which beats seconds where g is 980, is taken to another place where g = 981 ; will it gain or lose, and how much in one day? 8. Each prong of a tuning fork making 100 complete vibrations per second vibrates to and fro through a distance of 1.5 mm. Find the velocity of the prong in the middle of its swing. 9. A 400-gm. weight when hung on a long and light helical spring stretches it 30 cms. What will be its period of oscillation if drawn down a little and then set free? Take g = 980 and neglect mass of spring. Ans. 1.099 sec. IV. ROTATION OF RIGID BODIES Motion of a Rigid Body 130. Translation and Rotation. — If a rigid body moves in such a way that any straight hne joining two points in the body remains parallel to itself as it moves along, the motion is said to be a translation without rotation. A book slid about on a table keeping one edge always parallel to one edge of the table is a case of pure translation. If the edge of the book changes its direction there is said to be rotation. Any motion of a rigid body may be considered as made up of the motion of its center of mass combined with rotation about an axis through that center. Motion of the Center of Mass. — It may be proved that when any external forces act on a rigid body the center of mass of the body moves just as though the whole mass of the body were concentrated at that point and all the forces were applied directly to it, and it makes no difference at what points on the body the forces may be apphed. When a top spins on a smooth frictionless table its center of gravity remains at rest, for the external forces acting on the top are its weight due to the attraction between it and the earth and ROTATION 89 the upward pressure of the table on its point. These two forces are equal and opposite and consequently the center of gravity has no translational acceleration even when the top is inclined ati in figure 76 (p. 102). When a stick of wood is hurled through the air its center of mass moves in a simple parabolic curve (§111) just as a particle wouW move, except as affected by air resistance. Besides this translational force which depends only upon the amounts and directions of the several forces and so is found by the simple force polygon, there is usually also a couple which causes the body to rotate about an axis through its center of mass. This couple depends not only on the amounts and directions of the forces, but also upon their points of application to the body. 131. Angular Velocity. — When any line in a body is at rest while other points in the body move in circles about that fixed line or axis, the motion is called rotation. In case of a rigid body, like a wheel, all parts whether near the axis or far from it must rotate through equal angles in the same time. The rate at which the body is turning at any instant, measured in radians per second, is known as its angular velocity and is represented by w (the Greek letter omega). Since the length of a radian of arc at a distance r from the axis is equal to r, we have V = wr where v is the linear velocity of a particle at a distance r from the axis. Example. — If a wheel of radius 15 cms. is making 3 revolutions per sec, its angular velocity is 3 X 2x radians per sec, and the linear velocity of a point on the rim is 3 X 27r X 15 cms. per sec. 133. Angular Acceleration. — When the angular velocity of a body is changing, the rate of change per second is known as its angular acceleration, and may be represented by A. If the angular velocity wi changes to 012 in t seconds, then C02 — wi or £02 = CiJl + -At where A is the average rate of acceleration during the time t. 90 MECHANICS The direction of the axis of rotation may change, and this also constitutes an angular acceleration, even though the speed of rotation about the axis may remain constant. This is illustrated by the motion of a spinning top when its axis is inclined, for the axis swings around in a circle keeping a constant inclination to the vertical. 133. Vector Representation of Angular Velocity. — The angular velocity of a body may be represented by a vector or arrow drawn along the axis of rotation and having a length propor- tional to the amount of the angular velocity, and pointing in the direction that a person must look along the axis to see the body rotating in a clockwise direction. For example, if the rotating disc shown in figure 66 has an angular velocity 10, it will be represented by a vector 10 units long drawn in the direc- tion of the arrow. 134. Change in Direction of Angular Velocity. — If the axis of rotation is changing in direction the angular velocity at one FiQ. 66. instant might be represented by the vector B and a short time later by C (Fig. 67). The change in angular velocity would then be represented by the vector D, for this combined with B gives C according to the composition of vectors. If t is the time dur- ing which the change has taken place, then D = At where A is the angular acceleration. An angular acceleration of this char- acter is found in the motion of a top. (§147.) 136. Rotation witli Constant Acceleration. — The equations for rotation with constant acceleration are exactly analogous to those for simple translation (§98), as may be seen thus: Translalion in Straight Line Rotation about a Fixed Axis s = displacement in time (. a = angle through which body turns in time t. •'1 = velocity at bejmnmffo/ interval 4. wi = angular velocity at beginning of interval t. ROTATION 91 «2 = velocity at end of interval I. a>2 = angular velocity at end of in- terval t. a = acceleration. A = angular acceleration. a=?^'- A=^^^^^. (1) ^=(^)ior s = M +f ■ ^=(^^)t or . = .,t + ^t^ (2) 2as = V2' - vi\ 2Aa = W2' - a,i2 (3) Problems 1. If a wheel revolves 1800 times per minute, what is its angular velocity; and if it is 6 in. in diameter what is the linear velocity of a point on its periphery? 2. What is the linear velocity of a point 1 ft. from the axis of a wheel making 2.5 turns per sec? Also the velocity of a point 1.4 ft. from axis? What is the angular velocity of each? 3. Find angular velocity of a wheel in which a point 6 in. from the axis has a velocity of 4 ft. per sec. 4. A locomotive rounds a curve having a radius of 800 ft. at 15 miles per hour; what is its angular velocity? 6. A wheel is given a speed of 100 revolutions per min. in 2 minutes; what is its angular acceleration in radians per sec. per sec? 6. How many revolutions will a fly wheel make in 20 seconds, while its angular velocity is changing from 3 to 10 radians per sec, if the ac- celeration is constant? 7. A body rotates about an axis with constant angular acceleration 8 radians per sec. per sec; how many turns will it have made in 10 sec- onds from the start? 8. How many revolutions will a body make starting from rest with angular acceleration 4 radians per sec. per sec. before it will be revolving at the rate of 20 turns per sec? Kinetics of Rotation about a Fixed Axis 136. Angular Acceleration Caused by Torque. — Suppose the bar shown in figure 68 is acted on by a force F at a distance d from the axis; it is required to find how rapidly the speed of rota- tion of the bar about the axis will increase in consequence of the moment of force, or torque Fd. Imagine the bar divided into little masses mi, ms, rriz, etc., and suppose the effect of the force F is to cause an angular accel- eration A in the rotation of the bar; that is, its angular velocity is increased at the rate of A radians per sec. per sec. The linear acceleration of the mass mi at distance ri from the axis will then 92 MECHANICS be TiA, and consequently the force acting on mi must be miTiA and may be represented by /i. This force /i is due to F and is transmitted to mi by the rigidity of the bar. So also nii must be acted on by a force f^ = miViA since it has the acceleration r^A. And siinilarly every one of the masses mi, rrii, mz, etc., into which the bar is divided is acted on by the force needed to give it its acceleration, as indicated by the small arrows in the figure. Now, if a force equal and opposite to /i is applied to mi, and a force equal and opposite to fi is applied to m^, and so on, apply- ing to each of the little masses a force just such as to counteract its acceleration, it is clear that there will be no acceleration and the bar will be in equilibrium. That is, a system of forces equal and opposite to /i, fi, etc., will just balance the turning moment of the force F about the axis 0. Con- sequently the sum of the moments of /i, /2, etc., about must be equal to the moment of F about that axis. Thus, Fig. 68. Fd = firi + fir 2 + fsri +, etc. But it has been shown that /i = miriA, fi = viiTiA, etc. Therefore or Fd = miTi^A + miri^A + msr3^A + , etc., Fd = A (miri^ -f- miTi' + mtr^ -)-, etc.). The quantity in the parenthesis, which depends only on the mass of the body and its distribution with reference to the given axis is called the moment of inertia of the body about that axis, and may be represented by the symbol I. The torque or sum of the moments of whatever forces may be acting to rotate the body around the given axis may be repre- sented by L, and we have then, L L = IA or A=j (1) That is, the angular acceleration caused by a given torque is ROTATION 93 equal to the torque divided by the moments of inertia of the body about the given axis. Notice the analogy to the formula F = ma, moment of force or torque corresponds to force, moment of inertia corresponds to mass, and angular acceleration corresponds to linear acceleration. The effect of torque in causing angular acceleration may be illustrated by the apparatus shown in the figure. A light bar carrying two masses M and M' is mounted on a horizontal axis perpendicular to the bar and is set in motion by a weight TThungfrom a cord wrapped around a drum on the axis. When the masses M and M' are in the position shown, the bar gains angu- lar velocity slowly, for the farther the masses are from the axis, the greater the moment of inertia of the rotating system. When the masses are close to the axis the moment of inertia is smaller and the bar gains angular velocity very much more rapidly than before. The calculation of moments of inertia will be discussed in paragraphs 139 to 141. 137. Angular Momentum. — The formula of the last paragraph see (§135) (1) (§94) The product of the moment of inertia by the angular velocity about an axis is known as the angular momentum of the rotating body about that axis, and equation (1) above, states that the change in the angular momentum of a body about any axis is equal to the moment of force or torque about that axis multiplied by the time during which it acts. When the axis of torque is perpendicular to the axis of rotation of the body its only effect is to change the direction of the axis may be put in the form L L = lA ~ '' t ■ 0)1 or Lt = Iu>2 — Zoji which is exactly analogous to t Ft ■ = mv2 — mvi 94 MECHANICS of rotation, but the amount of the angular momentum remains un- changed. This is illustrated by the top (§147). 138. Kinetic Energy of a Rotating Body. — When all parts of a body have the same velocity the kinetic energy of the body as we have already seen is H-^^^ where M is the mass of the body and V its velocity. But in case of a rotating rigid body the velocity of any part depends on its distance from the axis. In this case we may imagine the whole mass to be divided into small portions, and calculate the kinetic energy of each of these portions sepa- rately and then add them together to find the total energy of rotation. The body represented in figure 70 is supposed to rotate about an axis perpendicular to the paper. Imagine the whole body cut up into little rods parallel to the axis whose ends are seen as the re- ticulation in the diagram. Let the mass of one of these rods be m, its distance from the axis r, and its velocity due to the rotation of the body v. Then its kinetic energy is J^ww^. But if o) is the angular velocity of the body, cor will be the linear velocity of a mass at a distance r from the axis. FiQ. 70. Thus, = V and y^mv^ 3^mcoV^- Now, let mi represent the mass of another of the rods into which the body has been imagined divided and ri its distance from the axis, then its kinetic energy is J^wiicoVi^, and so the total kinetic energy of the body is K. E. = i^OTcoV -f- y^miw^ri' +, etc., there being one term for each part into which the body is con- ceived to be divided. Or we may write K. E. = Mw^imr^ + rmri' +, etc.), since the angular velocity of every part of the body is the same. But the quantity in parenthesis is the moment of inertia / of the bar about the axis, therefore K. E. = K7a,2 ROTATION 95 Notice again the analogy between this expression and the formula for kinetic energy of translation }iMv^ Moment of inertia corresponds to mass. Angular velocity corresponds to linear velocity. 139. Moment of Inertia of a Rod. — The method of computing moments of inertia may be illustrated by the case of a straight uniform rod with the axis at one end. Let I be the length and M the mass of the rod. Conceive it to be divided into n equal parts, each part having a mass m. Then the length of each part will be -, and if the distance of any part from the axis is taken as the distance' of its farther end, the dis- l 21 31 tances of the successive parts are -i — i — > etc., and the moment of inertia is, therefore, 72 2H' SV mP I = m-,+m-^ + m-^+etc.,orI =~^iV + 2^ + S'^+ . . . n^). Now, it may be shown that the larger n is taken the more closely does the sum in the parenthesis approach the value -^, and accordingly if the rod is supposed to be divided into an infinite number of parts, mP n' MP . I = ■ — 5 — X = — ;r" smce mn = M. The moment of inertia of the bar is , therefore, the same as though its mass P were concentrated at a distance k from the axis, where k' — ^• The distance k is known as the radius of gyration of the rod about the given axis. 140. Formulas for Moment of Inertia. — In case of bodies of simple figure and having the mass uniformly distributed throughout the volume the moments of inertia may be calculated by the methods of calculus. But in more complicated cases they must be determined by experiment. The following formulas are given for reference : Thin rod, of mass M and length I, having a transverse axis at one end _ MJl ^ ~ 3 ■ Thin rod, of length I, having a transverse axis through the center, _ MP ' ~ 12 ■ Rectangular block, of width a and length h and of any thickness whatever, about an axis through the center perpendicular to a and 6, 1,. /aM-6'\ 96 MECHANICS Circular disc or cylinder, of any length and of radius r, about an axis through the center and perpendicular to the circular section of the disc or cylinder, _ Mf_ ' ~ 2 ' Circular cylinder, of length I and radius r, about a transverse axis through its center perpendicular to its length, -(? + 5)' Sphere, of radius r about an axis through its center, I =M~ 141. Moment of Inertia about a Parallel Axis. — If the mo- ment of inertia of a body is known about an axis through its center of mass, it may readily be calculated about any parallel axis. For if /o is the moment of inertia about the axis through its center of mass and if M is the mass of the body, then the moment of inertia about a parallel axis at a distance h from the center of mass of the body is / = 7o + Mh'^; that is, the moment of inertia I about any axis is equal to the moment of inertia which the whole mass would have about that axis if it were concentrated at the center of mass of the body, added to the moment of inertia of the body about the parallel axis through its center of mass. 143. The Compound Pendulum. — In discuss- ing the simple pendulum it was assumed that the oscillating mass was so small that it might be consdered as concentrated at a point, and the mass of the suspending system was entirely neglected. A pendulum which has distributed mass and so does not satisfy either of the above simple conditions is said to be a compound or physical pendulum. All actual pendulums belong to this class. Let it be required to find the length of a simple pendulum having the same period of oscillation as a given physical pendulum. Suppose the pendu- lum to have mass M and let its axis of suspension be a distance h above its center of gravity C (Fig. 71). Then, when a line joining and C makes an angle a with the vertical, the pendulum may be considered as acted upon by a force Mg acting downward through its center of gravity and producing a moment of force about the axis equal to Mgd or Mgh sin a. If 7 is the moment of inertia of the pendulum about O, we have by equation (1) §136, therefore, Mgh sin a = lA _ Mgh sin a A — f ■ ROTATION 97 But in case of a simple pendulum of length I the moment of the force mg about the axis 0' is mgl sin a and the moment of inertia of m about 0' is ml'; therefore, mgl sin a = ml' A' and the angular acceleration is, _ g sin a ^ -~l If the two pendulums are to have the same period of vibration their angular accelerations A and A' must be equal when both pendulums make equal angles with the vertical; that is, Mgh sin a g r = T sm a and, therefore, '■ Mh The length of the equivalent simple pendulum calculated from the above formula will always be greater than h, since the moment of inertia / of the pendulum is always greater than if the whole mass were concentrated at its center of gravity (see §141); that is, I is greater than Mh' and, consequently, I is greater than h. The point P in line with and C and at a distance I from is called the center of oscillation. Each portion of the mass of the pendulum between P and O is con- strained to swing slower than it would if it were free to oscillate by itself about as a center, while all portions of the pendulum below P have to swing more quickly than if they were free. The mass between P. and 0, therefore, tends to quicken the motion of the pendulum while the mass below P tends to retard it, while the mass situated at P is neither hastened nor retarded, but swings exactly as it would if freely suspended from O. 143. Center of Percussion. — If a rod or pen- dulum is suspended from an axis A (Fig. 72) and if that axis is given a sudden sidewise impulse or if it is moved Fiq. 72. rapidly back and forth from side to side, the inertia of the rod will cause it to move as though a certain point B was fixed and the rod turned about that point as axis. This instantaneous center of the motion is not the center of gravity C, but is the center of oscillation corresponding to the axis of suspension at A. A marble placed on a little shelf at B is scarcely disturbed by the sudden to-and-fro movements of the axis A, while at any other point it would be instantly thrown off. On the other hand, when the pendulum suspended from the axis A is hanging at rest, if a sudden sidewise impulse is given to the bar at B, as when it is struck a blow at that point, no sidewise impulse is communicated to A in consequence, but the bar simply tends to turn about A as an axis. For this reason the point B is also called the center of percussion correspond- ing to the axis A. 98 MECHANICS In case of a baseball bat the blow is given to the ball with the least jar to the hands when the ball is struck at the center of percussion of the bat corresponding to an axis at the point where it is grasped. Problems 1. A cylinder weighing 30 kgms. and having a diameter of 1 meter is mounted on an axis and set rotating by a pull of 2 kgms. on a cord wound on an axle 10 cms. in radius. Find the acceleration produced and the speed of rotation 3 sec. from the time of starting. The moment of inertia of a cylinder about its axis is M ^ (§140) or 30 X 1000 X SD" I = 2 = 37,500,000 grm.cm.2 The force acting is 2 kgms. or 2000 gms. or 2000 X 980 dynes and the moment of the force. is 2000 X 980 X 10 dyne-cm. Substitute in the formula L = I A 2000 X 980 X 10 = 37,500,000A .". A = 0.523. Hence the system will gain in 1 second an angular velocity of a little more than half a radian per sec. In 3 seconds it will acquire an angular velocity u = 3A = 1.569; that, is, it will be turning at the rate of about 1 revolution in 4 seconds, since &> = -p- where P is the period of revolution. 2. What is the kinetic energy of a wheel which has a moment of inertia 20 lb. ft.2 and is rotating at the rate of two turns per sec? 3. If a 5-lb. weight is raised by means of a rope wound on the axle of the wheel in problem 2, how high will it be raised before the wheel comes to rest? 4. A uniform rod 40 cms. long and weighing 200 gms. can rotate about a transverse axis through its middle point. How many ergs of work will be required to make it revolve at the rate of three turns per sec? 5. Suppose the rod in question 4 is set in rotation by means of a 200-gm. weight attached to a cord wrapped around a cylindrical axle 4 cms. in diameter. How far will the weight have descended in giving a speed of rotation of 3 revolutions per sec. Note. — First solve neglecting the kinetic energy acquired by the 200-gm. weight as it sinks. Then obtain the more exact solution, taking account of this energy. 6. The fly wheel of an engine weighs 1200 lbs., the bulk of the weight being in the rim of the wheel at a distance of about 3 ft. from the axis. What is approximately its moment of inertia and how many ft.-lbs. of work must be done bj' the engine to set it rotating 3 times per sec? 7. How much energy will be given out by the fly wheel in problem 6 in slowing down from 3 to 2.5 revolutions per sec. ROTATION 99 8. A uniform bar 3 ft. long swings as a pendulum about an axis at one end. Show that the equivalent simple pendulum is 2 ft. long. 9. A uniform spherical steel ball 6 cms. in diameter is hung as a pendulum by a steel wire so that the center of the ball is just 100 cms. below the axis of suspension. Find how far the center of oscillation is below the center of the ball and what is the length of the equivalent simple pendu- lum, neglecting the mass of the suspending wire. 10. A rectangular bar of steel 1 X 1 X 12 cm. and weighing 90 gms., when suspended in a horizontal position by a wire attached to its middle point, is set oscillating about a vertical axis through its center and makes 4 complete vibrations in 10 sec. Find the moment of force or torque due to the twist in the wire when the bar is at right angles to its equilibrium position. 11. Find the period of oscillation of a solid metal sphere 6 cms. in diameter and weighing 800 gms. when hung by the same wire as the bar in problem 10 and set oscillating about a vertical axis through its center. Some Cases op Motion with Partly Free Axis 144. Foucault's Pendulum Experiment. — It occurred to the French physicist Foucault that since a pendulum undisturbed by external forces must persist in its original direction of vibration, if one were swung at the north pole by some suspension which could not transmit torsion, its direction of vibration would remain constant while the earth turned around under it, so that to an observer moving with the earth the pendulum would seem to change its direction of vibration at the rate of 15° per hour. At the equator the direction of the meridian remains parallel to itself as the earth rotates, and consequently the plane of vibration of the pendulum would remain unchanged* At any intermediate latitude the tangents to the meridians at two points differing in longitude by 15°, such as A and B (Fig. 73), will meet the axis at 0, and the angle AOB measures the change in direction of the meridian per hour. Consequently a Foucault pendulum in that latitude will shift in one hour through an angle equal to AOB. This interesting experiment was car- FiG. 73. 100 MECHANICS ried out by Foucault in 1851. He used as pendulum a massive ball of copper, hung by a wire more than 50 meters long, from the dome of the Pantheon in Paris. 145. Conservation of Angular Momentum. — In any body or system of bodies the total angular momentum of the system can- not be changed by any internal forces: for suppose A and B (Fig. 74) are two parts of the system which act on each other, since action and reaction are equal and opposite the force on A is equal and opposite to the force on B ; and since the distance from the axis to the line of action of the forces ° '/"^ is the same for both, the moments of the \o' ^'''B forces about the axis will be equal and op- \'''^ posite, so that in the same time they will (^fi give equal and opposite angular momenta Pjq 74 to the system, and consequently the total angular momentum will not be changed. For example, in the solar system the planets have not only angular momenta about their own axes, but also angular mo- menta about the common center of gravity of the system. These angular momenta may be represented as vectors and their re- sultant found from the vector diagram, and neither the direction nor amount of this resultant is changed by any internal forces, such as the attraction of one planet for another or any possible collisions between them. 146. Angular Momentum of Projectiles. — A body having angular momentum tends to keep the direction of its axis of revolution constant, and the greater the angular momentum the harder it is to disturb the direction of the rotation; that is, the slower its axis of revolution will change in direction under any given torque. So the spin of the rifle bullet or shell from a rifled gun causes it to keep pointing in a nearly constant direction as it flies through the air in spite of the tendency of a long bullet to turn sidewise in consequence of air resistance. 147. Motion of a Top. — When a rotating body is acted on by forces which tend to turn it about an axis perpendicular to its axis of rotation the effect is to change the direction of the axis of rotation without producing any change in the amount of the angu- lar momentum about that axis; precisely as when a force acts on ROTATION 101 Fig. 75. — Top with point fixed. a body at right angles to the direction of its linear motion (§114) it changes the direction, but not the speed of the motion. The motion of a top affords an excellent illustration of this principle. The top in figure 75 is represented as spinning in the direction indicated by the arrow, but in the inclined position shown it is subject to a downward force W due to its own weight acting through its center of gravity G, and the upward pressure of the floor against the point of the top at A. These two forces are equal and consti- tute a couple which tends to turn the top about an axis DA perpendicular to its axis of revolution. The effect of the couple is to cause a steady change in the direction of the axis of revolution, the upper end of the top moving around in the circle EFLH. This change in the direction of the axis of the top may be called its pre- cessional motion. The precession of the top may be explained as follows : let the vector AB represent in amount and direction the angular mo- mentum of the top about its axis, the vector being drawn so that the top is seen to revolve clockwise by an observer looking along the vector AB in the direction in which it points. Similarly the vector AD may represent the angular momentum which would be given to the top in a very small interval of time t by the couple consisting of the forces W and W. The resultant of the two vectors AD and AB is the vector AC, showing that the resultant angular momentum will have AC as its axis, and the axis of the top will accordingly move through the angle BAG in the time t. And as the vector DA is always at right angles to the plane EKA, the top will move at right angles to this plane, and there- fore its upper end E will describe a circle about the vertical axis AK. In the case just discussed the friction of the floor is supposed to be sufficient to keep the point of the top fixed at A. But when the top spins on a frictionless level surface it re- mains at a constant inclination and its precessional motion is 102 MECHANICS .'fsf^ =^ ^^ ^^ 'An / ^\ V \j p ■M Fig. 76. about a vertical axis through its center of gravity, as shown in figure 76. How it is possible for a top to rise to a vertical position as it spins was first explained by Lord Kelvin. It depends on the fact that the peg of the top is rounded and the friction between it and the floor causes it to roll around in a circle; and when this rolhng of the peg on the floor urges the top around faster than the regular processional motion, it causes the inclination of the top to gradually diminish until it stands vertical, and "goes to sleep." On a perfectly frictionless surface a top could not rise in this way. 148. Gyroscope. — In the gyroscope shown in figure 77 a wheel with heavy rim is mounted in two pivoted rings so that the axis of rotation of the wheel may be inclined at any angle and the whole may also turn freely about a vertical axis. When the wheel is in rapid rota- tion a sharp blow given with the hand to one of the rings as if to change the di- rection of the axis of rotation, will cause the wheel to vibrate as though it were held in its position by stiff springs. When a small weight is hving on near one end of the axis of rotation, the wheel, instead of tipping down, rotates slowly around the vertical axis as indicated by the arrow; if the weight is hung from the other end of the axis this precessional motion is reversed, wheel serves admirably as a gyroscope. References A. Gray: "Gyrostats and Gyrostatic Action," Smithsonian Reports, 1914, p. 193. "The Sperry Stabilizer for Aeroplanes," Scientific American, Aug. 8, 1914. "Gyro-compass," Scientific American Supplement, v. 72, p. 200, 1911. John Perbt: Spinning Tops. H. Ckabtkee : Spinning Tops and Gyroscopic Motion. WoHTHiNGTON : Dynamics of Rotation. 149. Precession of the Equinoxes. — The earth itself illustrates the precessional motion of the gyroscope. It is a rotating body with enor- FiG. 77. A bicycle GRAVITATION 103 mous angular momentum. But as it is not a sphere and its axis is not per- pendicular to the plane of its orbit, the attraction of the sun on the bulging equatorial belt tends to turn it over and make its axis perpendicular to the ecliptic. The effect of this rotational force is a slow precessional motion of the axis of the earth, just as in the gyroscope. The axis remains inclined 23K° to the pole of the echptic, but describes a circle about that pole in a period of about 25,800 years. If we take the pole of the ecliptic as a center and describe a circle of 23J^° radius it will pass through the present pole star and will mark the path which is being described by the polar axis of the earth. In about 13,000 years the bright star Vega in the constellation of the Lyre will be very nearly at the pole. V. UNIVERSAL GRAVITATION 150. Kepler's Laws. — The German astronomer Kepler in the year 1609, having made a careful study of the observations made by Tycho Brahe, came to the conclusion that the orbits of the planets were not circular as had been supposed, but elliptical, and announced his discovery in the following laws: 1. The orbits of the planets are ellipses having the sun at one focus. 2. The area swept over per hour by the radius joining sun and planet is the same in all parts of the planet's orbit. Hence the planet moves faster in its orbit when near the sun than when farther away. After nine years more of persistent search for some relation between the periodic times of the planets and their distances from the sun, he discovered and announced his third law: 3. The squares of the periodic times of the planets are propor- tional to the cubes of their mean distances from the sun. 151. Newton's Principla. — In 1686 Sir Isaac Newton pub- lished his great work, the Prmcipia, in which he clearly enunciated the fundamental principles of mechanics and applied them to a great variety of important problems. In this work he showed from the laws of mechanics that if the planets moved about the sun in ellipses in the manner described in the first two laws of 104 MECHANICS Kepler, then each planet as it moves in its orbit must be subject to a force which is directed toward the sun, and varies inversely as the square of the distance between them. 153. Universal Gravitation. — From the above result Newton concluded that probably all masses, great and small, attract each other with a force proportional to their masses and inversely proportional to the square of the distance between them. According to this law, the attractive force between any two masses m and M is expressed by the formula where r is the distance between the centers of the masses if they are spherical. The quantity C is an absolute constant for all kinds of matter and depends only on the units in which force, mass, and distance are measured. It is called the gravitation constant and is equal to the force with which two imit masses attract each other when placed unit distance apart. 163. Moon's Motions Connected with Fall of Apple. — Newton conceived that the weight of a body near the surface of the earth is due to this gravitation attraction between the earth and the body, and that an apple drops toward the earth in accordance with the same gravitation law which determines the motion of the moon in its orbit. To test this point let us, following Newton, find the accelera- tion which the apple would have if it were dropped toward the earth when as far off as the moon, and compare this acceleration with that which the moon is known to have. According to the law of gravitation (§152), the earth attracts a body at its surface with 3600 times the force that it would if the body were 60 times as far from its center, or at the distance of the moon. Consequently the acceleration toward the earth of a body at the distance of the moon should be 3'i600 of the ac- celeration of gravity at the earth's surfaces. But the acceleration of the moon toward the earth may be computed from the formula o = - or a = -pj- (§114; GRAVITATION 105 Fig. 79. where R is the radius of its orbit (240,000 miles) in feet and P is its period of orbital revolution (27.322 days) in seconds. Substituting, we have 4x2 X 240,000 X 5280 „ ^„„„„, , , , « = (2,360,620)^ = 0-008974 ft./sec.^ which is Meoo of 32.30 ft./sec.,^ while the acceleration of gravity at the pole, where it is not af- fected by the earth's rotation is 32.26 ft./sec.^ The two results therefore agree as exactly as could be expected with the data used. We conclude, then, that the mo- tion of the moon and the fall of an apple or stone are both according to the same law of gravitation. 154. Determination of the Grav- itation Constant. — To determine the constant of gravitation the force of attraction between two known masses must actually be measured. The extreme minute- ness of this attraction between small masses makes the exact determination of its value very difficult. It was first accomplished by Cavendish in 1798, using a form of apparatus indicated in figure 79. Two small spherical balls m and m' were mounted on the ends of a light crossbar which was suspended by a fine silver wire at its center. Two large spherical balls of lead M and M' weighing 158 kilograms apiece were sus- pended one near m and the other near m' but on opposite sides so that their attractions tended to turn the bar in the same direc- tion. To protect the suspended bar from being disturbed by air currents it was entirely enclosed in a narrow box, its deflec- tions being observed by a telescope through a glass window. Having observed the deflection of the bar when the large masses were in the positions shown, the masses were moved into the dotted positions where their attractions produced a deflec- tion of the bar in the opposite direction. From these observa- tions, combined with a measurement of the force required to turn the suspended bar through a given angle, the force of attraction between the masses was determined. 106 MECHANICS In the year 1889 C. V. Boys, who had discovered the remark- able elastic properties of fine quartz fibers, devised an apparatus similar in principle to that of Cavendish, but much more com- pact, in which the small suspended masses were hung by a quartz fiber so fine that larger deflections and greater accuracy of measurement were attained. According to Boys* determination, C = 6.6576 X 10~* in C. G. S. units. That is, the attraction between two masses of one gram each concentrated at two points a centimeter apart, or of two spherical masses of one gram each with a distance of one centimeter between centers, is 0.000,000,066,6 dyne. Two kilogram masses 10 cms. between centers attract with a force of 0.000666 dyne, or about seven ten-millionths of a gram weight. The constant of gravitation has also been reckoned by esti- mating the mass contained in an isolated mountain and then measuring its deflecting effect on a plumb-line near its base. 155. Mass of the Earth. — When the gravitation constant is known the mass of the earth itself may readily be determined. For consider the earth as attracting a gram mass at its surface. The force of attraction is g dynes or approximately 980, and from the law of gravitation Take M = mass of the earth, F = 980, m = 1 , r = radius of earth in centimeters, and C = 6.66 X 10~*- All of these quantities are known except M, which may be calculated. In this way the mean density of the earth is found to be 5.527, a result which is especially interesting as the average density of the surface materials of the earth is only about 2.5. 156. Mass of a Planet. — So also the mass may be found of any planet having a satellite whose distance and period of orbital revolution about the planet can be observed. For the attraction between the planet and satellite is expressed by — ^C, while the centripetal force in case of a 47r^77l satellite of mass m and period P and moving in a circle of radius r, is „„ r, and since it is the attraction which holds the satellite in its orbit we have mM _ 4irhn, J.2 ^ ~ pi '"• GRAVITATION 107 In the equation the mass of the satellite m cancels, and aa all the other quantities except M are known, the mass of the planet may be computed. 157. Significance of Kepler's Third Law.— Let M represent the mass of the sun, E the mass of the earth, r the mean distance between them, and P the period of the earth's revolution about the sun. Then, as in the last paragraph So also it J is the mass of some other planet, such as Jupiter, and if ri and Pi represent its distance from the sun and period of revolution in its orbit, respectively, we have MJC _ 4Tr'Jr, MC _ rj^ If the constant of gravitation C has the same value in case of the sun and earth as it has in case of the sun and Jupiter, then r' ri' pi = Y? which is precisely what Kepler's third law asserts to be true throughout the solar system. It is concluded, therefore, that the same gravitation constant holds everywhere throughout the solar system and probably throughout the material universe. 158. Variation of Gravity on Earth. — The force of gravity is not the same everywhere on the earth's surface. There are three circumstances which determine ^^^^ this variation, namely, the fact that the earth is not a sphere, its rotation, and the height above sea level of the given * ■ station. The earth is approximately an oblate spheroid having its polar radius less than its equatorial by 13.2 miles ov o^ . 21.2 kilometers and in consequence of pj^ gQ this the value of g at the poles is greater than at the equator by 1.6 cm./sec.'', due to this cause alone. But there is another circumstance which still further reduces the value of g at the equator. The rotation of the earth affects both the direction and amount of the acceleration g. For the resultant attraction F of the earth on a gram of matter situated at A (Fig. 80) is directed toward the center 0, but this resultant attraction serves both to supply the centripetal force /, which holds the mass on the earth as it rotates, and also the 108 MECHANICS force which we call its weight which gives it acceleration g when dropped. The centripetal acceleration / is directed perpendicu- lar to the polar axis and is equal to -pj-r, where P is the period of rotation of the earth and r is the distance AB. The distance AB = B cos I where R is the radius of the earth and I the latitude of A. Evidently then, / is a maximum at the equator and has zero value at the poles. Since F is the resultant of / and g, and is directed toward the center of the earth, it is clear from the diagram that g cannot be directed toward the earth's center except at the poles or equator. The direction of g is the direction in which a plumb-line will hang or a body will fall at A. Also a liquid surface, as the surface of the ocean, must be at right angles to g (see §172). At latitude 45° the plumb-hne points away from the center of the earth about 6.9 miles. At the equator the centrifugal force of a mass of one gram is 3.36 dynes. Hence the acceleration of gravity is less at the equator than at the poles by 3.36 cm./sec." on this score alone. The height of a place above sea level also affects the value of g, as it must diminish with the increase in distance from the center of the earth. If h represents the height in centimeters or in feet, the corresponding change in ^ is (0.000003) A. Though on account of the irregular shape and distribution of the earth's mass the exact value of g at any place can be deter- mined only by pendulum experiments, an approximate value may be calculated for any place on earth by the following for- mula due to Clairaut: g = 980.6056 - 2.5028 cos 2X - 0.000003ft. where X represents the latitude of the place and h its height above sea level. Some Values op g at Sea Level Place Cm./aec.^ Ft./seo.' Place Cm./see.2 Ft./Bec.s Pole London 983.1 981.2 980.9 32.25 32.19 32.18 New York .... Washington. . . Equator 980.2 980.0 978.1 32.16 32 15 Paris 32.09 MECHANICS OF LIQUIDS AND GASES PART I.— FLUIDS AT REST Pressure in Liquids and Gases 159. Fluids. — Certain substances, such as air, water, glycerin, etc., are characterized by great mobility, changing their shapes and flowing under the smallest forces. They are known as fluids. Fluids are divided into two classes, liquids and gases. Liquids change but slightly in volume when subjected to great pressure and may have a free surface. Gases are far more compressible than liquids and fill all parts of the containing vessel. Water is a type of liquid, and air of gas. 160. Density. — The mass of any substance contained in unit volume is known as its density. In the C. G. S. system of units density is expressed in grams per cubic centimeter, while in the foot-pound-second system it is expressed in pounds per cubic foot. Thus the density of water is 1.0 on the first system, while it is 62.5 on the latter system. A table showing the densities of some substances will be found on page 148. 161. Viscosity. — Fluids differ greatly in mobility. If a dish of water is tilted, the flow is so rapid that it gives rise to waves that surge to and fro, while in case of glycerin or syrup the flow is slow and the liquid only gradually settles to the new level. This difference in mobility is due to viscosity or internal friction (§245). Substances like pitch or tar are very viscous, while water, alcohol, and ether are but slightly so. A perfect fluid is one that has no viscosity and is an ideal. All known fluids, even gases, have some viscosity. 163. Force in Fluid at Rest. — The force exerted by a fluid at rest against any surface is perpendicular to that surface. Otherwise, owing to the mobility of the fluid, flow must take 109 no LIQUIDS AND GASES place along the surface, which of course cannot be in a liquid at rest. This law is true of all fluids, even those which are very viscous, after they have settled into equilibrium. 163. Pressure. — Let a very small fiat surface be imagined at some point in a fluid. The fluid on one side of that surface exerts a force perpendicular to the surface against the fluid on the opposite side. This force is proportional to the surface, and the force per unit surface is called the pressure. In C. G. S. units pressure is measured in dynes per square centimeter; it may also be measured in grams per square centi- meter, pounds per square inch, etc. 164. Hydrostatic Pressure. — At any point in a fluid at rest the pressure is the same in every direction. This is a direct consequence of the mobility of fluids, for a little sphere of liquid at the given point could not be in equihbrium if the pressure against its surface were not the same in everj^ direction. 165. Pressures on Same Level. — In a liquid at rest the pressure is the same at all points on the same level. — For a horizontal cylindrical column of liquid reaching from ^ to S is in equilib- rium under the pressure of the surrounding liquid. The pressure against its sides is perpendicular to the line AB, and therefore has no influence Yia. 81. to move the column toward A or B. And since it is level it has no tendency to slide toward A or B by reason of its weight. The force against the end at A must therefore be balanced by the force against the end at B. These forces are due to the pressures at A and B, and since the ends have equal areas the pressure at A must be equal to the pressure at B. 166. Pressures at Different Depths. — The difference in pres- sure between two points at different levels in a mass of fluid at rest under gravity, is equal to the weight of a column of the fluid of unit cross section reaching vertically from one level to the other. For a vertical cylindrical column of the fluid of unit cross section reaching from £ to C is in equilibrium under the pressure of the surrounding fluid. The pressure against the sides PRESSURE 111 of the vertical column is horizontal and has no power to support its weight, consequently the upward force at C must balance the weight of the column in addition to the downward force at B. Hence, since the force against the end of a unit column is equal to the pressure, the pressure at C is greater than the pressure at B by the weight of the column of fluid of _^ unit cross section reaching from B to C. ^ ^z zlE^ ^^^ ^ If h is the height of the column in centi- .^^ meters and d is the weight of one cubic centi- =^ meter of the fiuid in grams, then hd is the weight == - of the column and is thus the difference in ^= pressure between B and C in grams per sq.cm. =^2 The difference in pressure expressed in dynes , ^ per sq.cm. is hdg where g is the acceleration of _fI?^^^^^^^^=- gravity in cm. /sec.'' The total pressure at a ~^=.'~=£^±.^E= =^ point h centimeters below the surface, is j-j^ g2 therefore as follows: Pressure in grams per sq.cm. = Ad+pressure on surface in grams per sq.cm. Pressure in dynes per sq.cm. = Wj+pressure on surface in dynes per sq.cm. Note as to Units. — In calculating pressure by the use of the formula hd, it must be remembered that if the pressure is to be found in pounds per square inch, then h must be expressed in inches and d is the weight of one cubic inch of the liquid in pounds. The student is advised, however, to compute directly the weight of a column of the substance of unit cross sec- tion without thinking of any formula. In gases the density is so small that the pressure is practically the same everywhere throughout a small volume. Pascal's Principle. — Pressure is transmitted equally in all directions throughout a mass of fluid at rest, or if the pressure at any point is increased, it is increased everywhere throughout the fluid mass by the same amount. 167. Hydraulic or Hydrostatic Press. — An important mechan- ical device known as the hydraulic press is a good illustration of the application of the laws of fluid pressure. It was first con- structed by Bramah in 1796, and is sometimes known as Bramah's press. It consists of a strong cylinder in which works a cylindrical piston or ram of larger diameter. A collar of oiled leather or copper surrounds the piston in such a way that the greater the pressure of the liquid filling the cylinder, the more closely does the collar fit the piston. By means of a small pump, oil or water is 112 LIQUIDS AND GASES forced into the large cylinder, a check-valve preventing its return. In consequence of the lawof pressure just enunciated, whatever pressure is conununicated to the liquid by the pump will be ex- erted everwhere equally against the walls of the containing cyl- inders. So that if the large piston has 100 times the area of the other it will exert a force 100 times as great as that applied to the pump piston. HydrauUc jacks act on this principle : they contain a reservoir of oil which may be pumped into the main cyHnder, thus forcing up the ram; opening a small stop- cock permits the flow of oil back to the reservoir. Oil is used as it keeps the machine lubricated and does not freeze. It is to be observed that when the Hquid in the hydraulic press is incompressible as much work is done by the large piston as is expended upon the smaller one. 168. Pressure Independent of Shape of Vessel. — It has been shown that the pressure at any point in a liquid under gravity depends only on the depth of the point below the surface, on the density of the liquid, and on the pressure on its surface. The total force exerted against the bottom of a vessel by the pressure of the liquid which it contains is the product of the pressure at the bottom by its area, and may therefore be very different from the actual weight of liquid which the vessel con- tains; and when a vessel is filled with water to a given height the force against its bottom is the same whether the upper part of the vessel is flaring, cylindrical, or narrow. The reasonableness of this result will be evident from the following considerations. In the case of the vessel with flaring sides we may think of a cylindrical column resting on the bottom and pressed upon by the surrounding water as shown in the figure (Fig. 84). This pres- sure is necessarily perpendicular to the surface of the cylindrical column and, therefore, can have no effect in either supporting it or Fig. 83. — Hydrostatic preas. PRESSURE 113 pressing it down. The whole weight of the cyhndrical column is, therefore, supported by the bottom plate. In case of the vessel which is narrow at the top, the liquid exerts a downward force on the bottom greater than its weight because the sides of the vessel press the liquid down. Just as a man in a box may brace himself Fig. 84. Fig. 85. against the top and press against the bottom with a force far greater than his own weight. This fact that the force exerted on the bottom of a vessel may be greater than the weight of all the liquid in the vessel has been called the hydrostatic paradox. Pascal succeeded in bursting a strong cask by the pressure produced by a column of water in a narrow pipe 40 ft. high. 169. Center of Pressure. — The center of pressure of a surface is the pomt of application of the resultant force due to the pressure against the surface. The pressure is so distributed that the sur- face will just balance if supported at that point. In case of a tank having rectangular sides and filled with water, the center of pressure on a side will evi- dently be nearer the bottom than the top, because the pressure increases with the depth. Suppose the side to be divided into narrow horizontal strips of equal widths, the force exerted on each strip by the liquid pressure may be represented by an arrow as in the diagram, and it is clear that each of these forces will be proportional to the depth, since the force on any strip is the product of the area of the strip by the pressure at that depth. By the methods employed in finding the result- ant of parallel forces it may be shown that the center of pressure in this case is at P, K of the total depth from the bottom. It is not difficult to see that the center of pressure P must be on the same level as the center of gravity G of the triangle ABC formed by the lines representing the forces against the equal horizontal strips. 114 LIQUIDS AND GASES If a cylindrical water tank were to be bound by a single hoop, this should be situated }4 the height of the tank from the bottom. The hoops on water tanks are placed closer together at the bottom than at the top for the same reason. Liquid Surfaces 170. Free Surface of a Liquid. — When a liqiiid is at rest or in equilibrium the force which a surface particle exerts against the adjoining liquid must be perpendicular to the free surface at that point, otherwise the particle would move along the surface. This force depends upon gravity, on the attraction of neigh- boring particles, and on the atmospheric pressure on the surface, and also upon any acceleration which the particle may have. 171. Level Surface. — When a liquid is at rest on the earth, all parts of the surface which are not too near the walls of the containing vessel are at right angles to the direction of gravity or to the direction in which a plumb-line points. Such a surface is called level. A level surface is not a flat surface, but has the same curvature as the earth. In a pond 1 mile in diameter the center is 2 in. higher than a plane passing through the edges. The force is not necessarily the same at all points of a level surface. This is well illustrated in case of the earth, for the force of gravity at sea level near the poles is decidedly greater than at the equator. 173. Surface of a Rotating Liquid. — When a vessel containing a liquid is rotated by a whirling machine, the liquid by virtue of its viscosity soon comes into equilibrium,^ and turns at the same rate as the vessel. If the speed is slow the upper surface of the liquid is slightly concave, at greater speed it will become deeply hollowed, but it always has the form of a paraboloid of revolution. Here a little mass m exerts against the adjoining liquid a downward force mg due to gravity, and an outward cen- trifugal force^ equal to mcoV. The components q due to gravity (Fig. 87) are the same at all points of the surface, while the centrifugal components h, h, h increase in proportion to the dis- tance of the particle from the axis of rotation. The resultant * That is, it is in equilibrium considered as a whole, though the individual particles move in circles and are therefore accelerated. 2 The pressure of the adjoining parts against any little liquid mass supplies the centripetlal force urging it toward the axis as it rotates. Its outward reaction against that preasure is the centrifugal force. LIQUID SURFACES 115 forces oi, 02, as will therefore be differently inclined, and the sur- face must be of such a curve as to be at right angles to them. It will be noted that the resultant force is greater at points higher up on the surface, so that a surface particle near the top presses against the surrounding liquid with far more force than it would if at the bottom of the curve. The oblate form of the earth is similarly explained. A unit mass at the earth's surface exerts a downward force a toward the center of the earth due to attraction, and also a centrifugal force c due to rotation. The latter component is zero at the poles and reaches a maximum at the equator and is always at right angles to the polar axis. The resultant downward force g is, therefore, |C^ --5ti /a ^ "£!"-='_ 1 ! 1"=^ IS| .__. =■= = C Fig. 87. — Surface of rotating liquid. FiQ. 88. Fig. 89. directed exactly toward the center only at the poles and at the equator, and the surface of the ocean when calm must be every- where perpendicular to g. 173. Surface in Connected Vessels. — In a continuous mass of one kind of liquid all points on the same level must be at the same pressure, even though they may be in separate branches of the containing vessel. Thus the pressure at B (Fig. 89) is the same as at B', and that at C is the same as at C. It is clear that the enclosed air is under greater pressure than that of the atmosphere at A. When communicating parts of a vessel of liquid are open to the air the free surfaces must lie all on the same level because all are at the same pressure. 174. Case of Two Liquids. — If a bent tube contai aing mercury, as shown in the figure, have some other liquid, as water or oil, 116 LIQUIDS AND GASES poured into the longer arm, the mercury will be pressed down on that side and raised on the other. Since all below A is one con- tinuous liquid, the pressure at A must be the same as at A' on the same level, hence the column of mercury BA' must produce the same pressure as the column of liquid CA. Letting h and h' represent the heights of the two columns of liquid and d and df their densities, then, since the pressures of the two columns must be equal, M = h'd'. 175. Spirit-level. — The ordinary spirit- level consists of ■ a glass tube hermetically sealed, nearly filled with alcohol or ether, PiQ. 90. ^ bubble of air or vapor being left. The tube is bent slightly, forming the arc of a large circle, and the bubble always rests in equilibrium at the highest point. A level is said to be sensitive when a small inclination will cause a large motion of the bubble. In a sensitive level the curvature of the tube is very slight, and the bubble is usually large, otherwise it would be sluggish in its movements. For fine levels the tube is carefully ground on the inside so as to have a uniform curvature. FiQ. 91. — Spirit level. Problems 1. Find the pressure 3.50 meters below the surface in a pond of water; in grams per sq. cm. and in dynes per sq. cm. 2. Find the pressure in pounds per sq. in. 30 ft. below the surface of a pond, taking the weight of 1 cu. ft. of water as 62.5 lbs. 3. A piston 1 ft. in diameter carries a weight which together with that of the piston amounts to 200 lbs. How high a column of water will be required to produce enough pressure under the piston to support the weight. 4. What is the pressure 1 mile below the surface of the ocean, in pounds per sq. in., taking the relative density of sea water as 1.03. 6. Find the difference between the pressure at the bottom of a vessel 75 cms. deep filled with water, and the pressure when the vessel is full of mercury. Density of mercury = 13.6. BUOYANCY 117 6. A jar has a square cross section 5 cms. each way and is 30 cms. deep. It is half -full of mercury and half -full of water; find the pressure halfway down and also at the bottom, also the total force due to pressure against the bottom. 7. Find the total force against one side due to pressure in the preceding problem. 8. If a cubical tank 4 ft. each way is level full of water, find the pressure in pounds per sq. in. on bottom. Also the total force against one side in lbs. weight. Where is the center of pressure on the bottom 7 Where the center of pressure on one side? 9. Oil of density 0.7 is poured into one branch of a U-tube which contains enough mercury to keep the bend full. When the column of oil is 39 cm. high, how much higher will it stand than the mercury in the other branch? 10. When the atmospheric pressure is just 1,000,000 dynes per sq. cm., how far below the surface of a pond of water will the total pressure be just twice as much as at the surface? 11. In a pail of water spinning about a vertical axis through its center the surface of the water is hollowed so that at a point 10 cms. from the axis the surface is inclined 45°. Find the number of revolutions per sec. which the pail is making. Buoyancy and Floating Bodies 176. Buoyant Force of a Fluid. — Suppose that a mass of wood or iron is immersed in a liquid and it is required to find the force exerted upon it by the surrounding liquid. Imagine the given substance removed and its place filled by the liquid, and conceive of this portion as separated from the surrounding liquid by an imaginary surface ABC of the same shape as the original body. The liquid is in equilibrium, and since the mass enclosed ^^^ 92. in the surface ABC is urged down by its own weight, this weight must be exactly balanced by the force due to the pressure of the surrounding liquid on the surface ABC. Hence the resultant force due to pressure on the surface is an upward force equal and opposite to the weight of the enclosed mass of liquid, and since the whole weight of the enclosed mass acts down through its center of gravity G, the center of pressure must also be at the same point. Now, neither the amount nor direction of the pressure will 118 LIQUIDS AND GASES be changed at any point of the surface ABC if it is filled with wood or iron instead of the liquid. Therefore when any object is wholly or partially immersed in a liquid it is buoyed up by a force equal to the weight of the displaced Uquid, and the center of pressure is where the center of gravity of the submerged portion would be if it were homogeneous. There is nothing in the above reasoning which restricts this conclusion to liquids, it may therefore be stated as a general law of fluids and is known as Archimedes' principle, from its discoverer. 177. Experimental Illustration. — A brass cylinder which ex- actly fits into and fills a cup is suspended together with the cup from one pan of a balance and exactly counter- poised by weights. A vessel of water is raised under the cylinder until it is quite immersed, and the weights will now greatly overbalance the cup and cylinder; but if the cup is just filled with water the balance is restored. 178. Buoyancy at Great Depths. — Since buoyant force depends on the weight of the liquid displaced and not directly on the pres- sure, it makes no difference whether the im- mersed body is 1 in. or 100 ft. below the sur- face of the liquid except for the compression due to increased pressure. If the immersed body is more compressible than the surround- ing liquid it will displace less liquid where the pressure is great than at the surface and so will be less buoyed up at great depths. If it is less compressible than the liquid, it will be more buoyed up at great depths than when near the surface. The heavy iron shot used in deep sea soundings is buoyed up slightly more at great depths than at the surface because water is more compressible than iron. 179. Cartesian Diver. — The Cartesian diver is a small bulb of glass open at the bottom and containing just enough air to cause it to float in a jar full of water. A sheet of rubber is tied firmly over the mouth of the jar, and by pressing on the rubber the pressure in the liquid is increased and the air in the bulb compressed into smaller volume. The bulb with the contained Fia. 93. FLOATING BODIES 119 air may tKus be made to displace less than its own weight of water and will then sink to the bottom, but rises again when the pressure is relieved and the air expands. 180. Equilibrium of Floating Bodies. — A floating body may be considered as acted on by two forces : its own weight acting down through its center of gravity and a buoyant force equal to the weight of the dis- placed liquid acting up through the center of pressure. It can be in equilibrium only when these two forces are equal and opposite. The conditions for equilibrium may then be thus stated: 1. The weight of the displaced liquid must be equal to the weight of the floating body. 2. The center of gravity of the floating body must be in the same vertical line as the center of pressure. The displacement of a ship is the weight of water which it displaces, and is therefore the total weight of the ship and equipment. Fig. 94. FiQ. 95. FiQ. 96. 181. Stability of Equilibrium. — If, when a floating body is slightly inclined from its position of equilibrium, the couple resulting from its own weight and the buoyant force of the liquid tends to turn it back into its original position, the equilibrium is said to be stable. In figure 95, G is the center of gravity and P the center of pressure of the floating block. When it is tipped slightly P is displaced to one side in such a way that the combined action of the forces through G and P tends to turn the body in the direction of the arrow, bringing it back into its original state of equilibrium, which is therefore stable. In figure 96 is shown a state of equilibrium such that when the body is slightly displaced the couple acts to increase the displacement and to turn the body away from its original position. In this case the equilibium is unstable. A floating homogeneous sphere may be turned in any way and the center of 120 LIQUIDS AND GASES pressure P will always be directly under the center of gravity, and the equi- librium will remain undisturbed. Here the equilibrium is neviral. Specific Gravity and Its Measurement 183. Specific Gravity. — The relative density of a substance as compared with some standard substance is known as its specific gravity. Solids and liquids are usually compared with water as a standard, while gases are often referred to air or hydrogen. The specific gravity of a substance referred to water is found by dividing the weight of the given substance by the weight of an equal volume of pure water at the temperature of 4°C. The specific gravity of a substance is a ratio and is therefore the same whatever system of units is employed. Since 1 c.c. of pure water at 4°C. has a mass of 1 gram, the density of a substance in grams per cubic centi- meter is equal to its specific gravity referred to water. 183. Specific Gravities by Balance. — The substance, of which the specific gravity is to be determined, is sus- pended by a fine fiber from one arm of a balance and weighed, first in air and then when immersed in water. The second weighing will be less than the first by the weight of the water displaced by the substance. The difference between the two weighings will then give the weight of a mass of water of the same volume as the substance, and therefore if the weight in air is divided by the difference between the weights in air and water the specific gravity is obtained. 184. Mohr's Balance. — A convenient balance for determining the specific gravity of liquids is that shown in figure 99. A glass bulb weighted so as to sink in liquids is hung from one arm of a balance and exactly counter- poised by the weight P on the other arm. The glass bulb is hung in the liquid to be examined and the buoyant force of the liquid balanced by riders hung on the balance arm. From the weight and position of the riders the specific gravity of the liquid is obtained directly without calculation; for the several riders are so adjusted that each has one-tenth the weight of the FiQ. 98. SPECIFIC GRAVITY 121 next larger, and the position of each on the balance arm gives the figure for the corresponding decimal place in the result. 185. Hydrometers of Constant Weight. — These instruments are usually made of glass and consist of a rather long light bulb having a slender stem above and a weighted bulb below so that the instrument floats in a vertical position in the liquid whose den- sity is to be determined. By means of a scale on the stem the specific gravity of the liquid may be read directly from the point on the scale to which the instrument sinks. In such a case the weight of the whole hydrometer must be equal to the weight of the displaced liquid, so that if v is the FiQ. 99. Fio. 100. volume of the hydrometer below the mark to which it sinks in a given Uquid and if d is the weight of unit volume of the liquid, then W = vd where W is the weight of the hydrometer. The specific gravity scale of a hydrometer is not a scale of equal parts, corresponding divisions being farther apart at the upper end of the stem than at the lower. The Beaum^ scale is an arbitrary scale of equal parts in which hydrometers are often graduated. Hydrometers are made for liquids lighter than water and also for liquids heavier than water. 122 LIQUIDS AND GASES If the stem of a hydrometer is slender (compared with the volume of the immersed portion of the instrument), it will be sensitive and a small change in density will cause a large change in its immersion. 186. Specific-gravity Bottle. — When the specific gravity of a powdered substance is to be determined, the specific- gravity bottle may be used. This is a small flask having a carefully ground tubular stopper. The powder to be experimented upon, after being weighed, is put into the flask which is then filled with water up to a certain mark on the stopper. The weight of the whole is then deter- mined and also the weight of the flask when filled with pure water alone up to the same mark. From these three weighings the weight of water displaced by the powder may be determined, and so its specific gravity may be obtained. If the powder is soluble in water some liquid in which it is insoluble must be used. Problems 1. A piece of metal weighs 300 gms. in air and 260 gms. in water; what is its volume, specific gravity, and density? 2. A certain block of wood has a volume of 80 c.c. and specific gravity 0.8. Find the volume and weight of water which it will displace when floating. 3. A certain body weighs 240 gms. in air, 160 gms. in water, and 140 gms. in another liquid. Find the specific gravity of the body and also of the second liquid. 4. A block of wood floats in water with ^ of its volume above the surface. What is its density? 5. A block of wood floats in water with J^ of its volume above the surface, but when floating in oil % of its volume is submerged. Find the specific gravity of the wood and of the oil. 6. A piece of metal weighs 16 gms. in air and 14 gms. in water. Another substance B weighs 8 gms. in air, and the two when fastened together weigh 2 gms. in water. Find the specific gravity of each. 7. A sinker weighing 38 gms. is fastened to a cork weighing 10 gms. and the two together are in equilibrium when immersed in water. Find the specific gravity of the sinker if that of the cork is 0.25. 8. The stem of a hydrometer is graduated upward from to 100 in equal parts, and the volume of the instrument below the zero of the scale is three times that of the graduated stem. When placed in water it sinks to the 20-mark. Find the density of a liquid in which it sinks to the 80- mark, also of a hquid in which it sinks to the 0-point. v/ GASES 123 9. A mass of 80 grm., having a density 8, balances a mass of 140 grm. when both are suspended in water from the arms of a balance. Find the density of the larger mass. Gases and Atmospheric Pressure and Buoyancy 187. Gases. — The second great division of fluids is that of com- pressible fluids or gases. Those mechanical properties of liquids which are due simply to their fluidity are also possessed by gases. The compressibihty of gases, however, is so great that their changes of density due to variations in pressure cannot be neglected. 188. Density of Gases. — Gases possess weight as is shown by the following experiment. Pump the air out of a globe of glass or of metal until it is well exhausted, suspend it from one pan of a balance and weigh it. Now open the stopcock in the globe, ad- mitting air, and when it is full weigh it again. The difference be- j tween the two weights is the weight of the felobe full o f) air. At • temperature 0°C. and when the barometric pressure is 76 cms., a cubic foot of dry air weighs about Ka lb. or a cubic liter 1.293 gms. A room 30 ft. long, 30 ft. wide, and 10 ft. high contains under ordinary conditions about 700 lbs. of air. The following table gives the densities of some familiar gases at 0°C. and a pressure of 76.0 cms. of mercury. Gas Air Oxygen Nitrogen Hydrogen Chlorine Carbon-dioxide Ammonia Density in gms. per c.c. 0.001293 0.001430 0.001257 0.00008988 0.003133 0.001974 0.000761 Sp. gr. referred to air 1.0000 1.1057 0,9720 0.0693 2.423 1.527 0.589 189. Torricelli's Experiment. — The first measurement of the pressure of the atmosphere was made in 1643 by Torricelli, a pupil of Galileo. It was known that water could not be raised more than 34 ft. 124 LIQUIDS AND GASES by a suction pump. Torricelli believed that this was because water was raised in such a pump by the pressure of the atmos- phere. He concluded that as mercury was 13.6 times as dense as water the atmospheric pressure would be able to support a column of mercury only Hs-e times as high, or about 30 in. in length, and to test it tried the following experiment. A tube nearly 3 ft. long and closed at one end was filled with mercury and then the open end being closed with the finger to prevent the escape of mercury the tube was inverted and placed with its open end below the surface of mercury in a dish, after which the finger was withdrawn. The mercury at once sank in the tube till it stood at a height of about 30 in. or 76 cms. above the level in the dish. The space above the mercury in the tube was a vacuum except for the presence of mercury vapor. As 1 c.c. of mercury weighs 13.6 grams., the atmospheric pressure able to support a column 76 cms. high must be 76 X 13.6 = 1033.6 gms. per sq. cm., and would, there- fore, sustain a column of water 1033.6 cms. high, or 33.9 ft. Pascal, reasoning that if the pressure of the atmosphere was due to its weight the pressure should be less on top of a mountain than at its base, caused the experiment to be tried and estab- lished the fact. 190. Magdeburg Hemispheres. — Otto von Guericke, of Magde- burg, shortly after he had invented the air pump, demonstrated the pressiu-e of the atmosphere by means of two hemispherical cups of copper carefully fitted together to form a spherical vessel about 2 ft. in diameter. When the air was exhausted from the vessel two teams of horses were unable to pull the cups apart. Fig. 102. PRESSURE 125 The force with which the cups are pressed together in such a case is found by multiplying the area of the circular opening of the cups by the difference between the air pressure on the inside and outside. 191. Barometer. — Instruments for the measurement of the atmospheric' pressure are known as barometers. The best barometers usually employ a column of mercury, as in Torricelli's experiment. A form much used is the Fortin barometer, the reservoir of which is shown in the figure. The tube containing the mercury is sheathed with brass to protect it from injury, the height of the column being read through an opening by means of a vernier which slides on a scale graduated on the brass sheath. As the mercury sinks in the barometer tube it flows out into the vessel at the bottom and raises the level there, it is therefore necessary to provide some means of adjusting the height of the mercury in the lower vessel. This is accomplished by the screw C, on turning which the flexible leather bottom of the vessel is raised or lowered until the surface of the mercury exactly touches the ivory point a, which is the zero point from which the scale is graduated. As the lower vessel is not air-tight, the external air pressure is freely trans- mitted to the surface of the mercury. The greatest care is taken in filling such a barometer that no air is leftclinging to its sides, the mercury being usually heated and even boiled in the tube. FiQ. 103. — Fortin barometer. 193. Capillary Correction. — The upper surface of the mercury col- umn in a barometer tube is rounded upward in a meniscus, higher at the cen- ter than at the edges, and the height of the barometer is measured to the highest point of this curved meniscus. The effect of the curvature is to make the column stand slightly lower than if the surface vSjn flat. Hence to obtain the true height a small cor- rection, called the capillary correction, which depends on the curvature of the surface, must be added to the apparent height. In a standard barometer the tube should be so large (2 cms. in diameter) that there is no curvature at the center of the surface, in which case there is no capillary correction. 126 LIQUIDS AND GASES Capillary Correction Millimeters Capillary Depression 1.4 0.8 0.5 0.3 0.2 mm. Internal Diameter of Tube 4.0 6.0 8.0 10.0 12.0 mm. 193. Temperature Correction. — It must be remembered that the scale by which the height of a barometer is read is correct at only one tem- perature, and also that the density of the mercury itself varies with the temperature; in order, therefore, that barometer readings may be definite, what is known as the reduced reading is always given; this is the height at which it would stand if the mercury had the density which it has at 0°C. Effect of Gravity. — It might be supposed that if the reduced heights of the barometers at two places were the same that the atmospheric pressures at those places would be equal, but this is not necessarily so. The pressure in grams per square centi- meter would be the same, but the weight of a gram depends on the force of gravity. Near the equator 'a" gram weighs 978 dynes, while near the poles it weighs over 983 dynes. If the reduced height of the barometer in centimeters be multiplied by the density of mercury at 0°C. and the product by the accelera- tion of gravity at the given place, the pressure recorded by the barometer will then be deter- mined in dynes per square centi- meter, which is absolutely def- inite. 194. Aneroid Barometer. — An exceedingly convenient and por- table form of barometer is known as the aneroid (from the Greek, meaning without liquid). A disc-shaped metal box, like a small blacking box, is provided with a top made of thin metal corrugated so as to be extremely flexible. The air is exhausted from the box and it is permanently sealed, the top being sup- ported by a stout steel spring which prevents it from collapsing. As the atmospheric pressure increases the spring yields a little and its point moves downward, acting by means of levers and a delicate chain to give a greatly increased motion to the pointer which moves over a graduated dial. A hair-spring serves to take up the slack of the chain. Such an instrument may be Fig. 104. — Diagram of mechanism of aneroid barometer. BUOYANCY 127 made as compact and portable as a watch. It is subject to change, however, and needs to be compared with a mercurial barometer from time to time. Also the elasticity of the spring varies with the temperature. 195. Standard Atmospheric Pressure. — It is customary in stating the densities of gases to give them at what is called at- mospheric pressure. This standard atmospheric pressure, some- times called a pressure of one atmosphere, is the pressure of a column of mercury 76 cms. high at 0°C. When the acceleration of gravity has the value that it has at Paris (980.94) this pressure is 1,013,600 dynes per square centimeter. At London its value is 1,013,800 dynes per square centimeter. 196. Buoyancy. — The law of buoyancy, known as Archimedes' principle, that bodies immersed in a fluid are buoyed up with a force equal to the weight of the displaced fluid, holds for gases as well as for liquids. This may be easily illus- trated by the apparatus shown in the figure. A hollow globe is balanced by a solid mass of lead or brass hung from the other arm of the balance. When the .globe is closed and the whole is placed under the bell jar of an air pump, it is observed that as the air is exhausted from the receiver the globe settles down; when air is readmitted, however, the globe is again balanced by the weight. The globe with its greater volume displaces a greater volume of air than the weight, and by the law of buoyancy it must be buoyed up with a greater force. If a solid mass of brass is being weighed, using brass weights, the buoyant force of the air on both sides of the balance will be the same. But if the density of the weights is greater than that of the body weighed, the apparent weight of the body will be less than its true weight. When the apparent weight of a body is w, its true weight W may be found by the formula. Fia. 105. 128 LIQUIDS AND GASES W = w + ws(^^-}y where 3 is the density of air, d the average density of the object being weighed, and di the density of the weights used. 197. Balloons. — Balloons ascend in consequence of the buoy- ancy of the surrounding atmosphere. The gas within the en- velope simply supplies the pressure to keep. the balloon distended; in so far as it has weight it is a disadvantage. To find the sup- porting power of a balloon we must determine the weight of the balloon itself together with the enclosed gas and subtract this from the weight of an equal volume of atmospheric air. The difference is the portative force of the balloon. As the balloon rises the pressure of the atmosphere decreases and the gas in the interior expands and completely fills the bal- loon, and then as it expands still farther the excess escapes through an opening at the bottom. Expansion of Gases 198. Expansion of Gases. — When a vessel containing gas is enlarged the gas expands, keeping the vessel full however great its volume may become, and at the same time the pressure of the gas diminishes. If a small thin rubber bag containing a little air is closed and placed under the bell jar of an air pump, and the air exhausted from the space around the bag, the latter will be distended by the expansion of the enclosed air as the pressure upon it diminishes. 199. Boyle's Law. — The exact way in which the pressure of a gas changes when its volume is varied was first investigated by the English physicist Robert Boyle in 1662 and by Mariotte in France in 1679. The form of apparatus used by Boyle is illustrated in figure 106. The short arm of the tube is closed and contains a mass of air separated from the outer air by the mercury in the bend of the tube. The enclosed air is at the same pressure as the outer air since the mercury stands at the same level in each branch. Mercury is now poured into the long arm of the tube until the EXPANSION 129 enclosed air is compressed to one-half its original volume, as shown in figure 107. The height of the mercury in the long branch above that in the closed branch is then found to be just equal to the height of the barometric column. That is, the enclosed air is under a pressure of two atmospheres, one due to Fig. 106. Fro. 107. the external air pressure and the other due to the height of the mercury column. If more mercury is added the air is still further compressed, and when the total pressure is three atmospheres, the mercury column' having twice the barometric height, the air is found to be compressed to one-third of its original volume. The law of compressibility of air, which is also found to be approximately true for all the more perfect gases may then be stated thus; 130 LIQUIDS AND GASES Boyle's Law. — When the volume of a mass of gas is changed, keeping the temperature constant, the pressure varies inversely as the volume; or the product of the pressure by the volume remains constant. That is, if a mass of gas has a volume w at a pressure p and if the volume is changed to v' while the temperature is kept con- stant, the pressure will become p' such that •pv = p'«' = constant (1) This constant is evidently proportional to the mass of gas used, for if the pressure is kept constant we must take twice the original volume in order to get double the mass of gas. We may, therefore, express Boyle's law by the equation, py = mk or m (2) where A; is a constant which depends only on the kind of gas and its temperature. Thus if we have a mass of gas m having pressure p and volume v, and another mass m! of the same gas at the sa me t emperature, bub with pressure p' and volume v' , we have by (2) p. pV ^^^ m m Letting d represent the density of the gas, since d = — , we have from formula (3) V 2 d d' k; (4) FiQ. 108. that is, the density of a gas is directly proportional to its pressure when the temperature is constant. This is directly shown by Boyle's experiment, for with doubled pressure th^^rouime is diminished to one-half and the density is consequently doubled. To study the relation between pressure and volume for pres- sures less than one atmosphere, Mariotte used the apparatus shown in figure 108. EXPANSION 131 A long tube of glass closed at the upper end and plunged in a deep bath of mercury contains a small mass of air or other gas. The volume of the air or gas is given by graduations on the tube while its pressure is found by subtracting the height of the mer- cury column CD from the barometric height which measures the pressure of the external air. The volume and pressure are varied by raising or lowering the tube in the bath. 300. Variations from Boyle's Law. — Boyle's law is not exactly true in case of any actual gas. The following table will indicate the degree of departure from the law, with increasing pressures, of some common gases: Pressure in meters of mercury Air Nitrogen COa Hydrogen 1 1.0000 1.0000 1.0000 1.0000 y2 1.9978 1.9986 1.9824 2.0011 y^ 3.9874 3.9919 3.8973 4.0068 Vs 7.9456 7.9641 7.5193 8.0339 Ko 9.9161 9.9435 9.2262 10.0560 Ho 19.7198 19.7885 16.7054 20.2687 It will be noted that air and nitrogen are slightly more com- pressible than if they followed Boyle's law exactly, while hydrogen is rather less compressible; the departures from the law are, how- ever, less than 1 per cent, up to 10 atmospheres' pressure. Car- bon dioxid shows marked increase in compressibility as the pressure increases and it approaches its point of condensation. jThe French physicist Amagat has made an exhaustive study of the com- essibilities of gases at different temperatures and up to pressures as great as 3000 atmospheres. His results show that as pr'ys^iire is increased the prod- uct pv slightly diminishes at first, but when the pressure exceeds a certain amount, which depends on the gas and its terafoeiature, the product pv steadily increases up to the highest pressures usedT The Dutch physicist Van der Waals has shown that the formula Ip + ~i) (w — 6) = constant, in which a and 6 are small constants depending on the kind of gas, expresses quite exactly the relation of pressure to volume in gases at constant tem- perature for a far wider range of pressures than the simple formula of Boyle. 132 LIQUIDS AND GASES 301. Measurement of Heights by Barometer. — The difference in pressure at two different heights in the atmosphere is equal to the weight of the unit column of air reaching from one level to the other. If the average density of the air between the two levels were known then the height could easily be ascertained by\ dividing the difference in pressure by the average weight of unit volume of the air. Let H represent the height in centimeters, P and p the two pressures measured in grams per square centimeter, and d the average density in grams per cubic centimeter, then Hd^P-p and H =^-^ (1) As the average density of the air between the two levels depends on pressure, temperature, and moisture, it is clear that the chief difficulty lies in determining this quantity. An approximate rfesult may be obtained by assuming that the average pres- P+ P sure between the two levels is — ^ — Then, if do is the density of the air at standard atmospheric pressure po, and at the average temperature between the two stations, we have by Boyle's law therefore and by (1) If we take the average temperature at 15°C. and neglect moisture, we find do = 0.00122 and po = 76 X 13.6 = 1033.6, hence P + p Po 2 . do - d ■ d = do P+P Po" 2 Ff - 2po P- V Approximate height „ 2 X 1033.6 P - p P - p " = 0.00122" pTp = 1'69^'°00 ■ pT^ "'"«• H = 55,600 ^-^ ft. P+P Since the final expression involves the ratio of P — p to P + p, the pres- sures may be measured In any units whatever, centimeters of mercury or inches of mercury or whatever unit is most convenient. AIR PUMPS 133 Pumps and Pressube Gauges 203. Air Pump. — Air pumps were first 'made by Otto von Guericke, of Magdeburg, in 1650. For rapid exhaustion when a vacuum of 0.1 mm. of mercury is sufficient, a very convenient pump is Gaede's rotary air pump, shown in figure 109, in which the cyhnder A mounted close to one side of a somewhat larger cylindrical cavity, is rapidly rotated by an electric motor and sweeps out the air from the crescent shaped space by means of two sliding vanes ss, which are carried in slots in A and are pressed against the walls of the cavity by means of springs. In this way air is drawn in at C and forced out at D finally escaping at J. For higher exhaustion, pumps are used in which oil or mercury prevents leakage. In figure 110 the cyhnder of the Geryk pump is J'lG. 109. — Gaede rotary pump. Fio. 110.— Cylinder of Geryk air pump. shown in which a deep layer of oil covers the piston and valves so that no leakage of air back through the pump is possible. When the piston is raised the air above it is forced out through the valve V which is finally lifted by the shoulder S when he piston reaches the top, permitting the last bubbles of air to escape through the oil into the upper chamber, while at the same time oil flows down through the valve, filling the small space above the piston. In this way the air in the cylinder is completely expelled in each stroke. Oil pumps for high exhaustions should never be operated without 134 LIQUIDS AND GASES a drying tube to absorb all water vapor from the air before it reaches the pump, as moisture absorbed in the oil prevents the securing of a high vacuum. A most effective pump of this type is one devised by Gaede in which three cylinders, connected in series and mounted one above the other, form a single o c= Fig. 111. — Rotary mercury pump. long cylinder and are operated with one piston rod. Air is drawn in at the bottom and forced successively through the three cylinders and escapes at the top. Only a small amount of oil is used and the presence of water vapor does not interfere with the action as it does in most oil pumps. 203. Rotary Mercury Pump. — One of the most perfect pumps for high exhaustion is a rotary pump, also devised by Gaede, in which a peculiar spiral-shaped drum of porcelain T (Fig. Ill) is rotated in a cylindrical case rather more than half full of mercury. As the spiral drum rotates in the direction of the arrow, the space W, inside the spiral and above the level of the mercury, enlarges and air is drawn in through the opening L which is con- nected by the curved tube R with the vessel to be exhausted. But as the motion continues L passes below the surface of the mercury into such a position as Li and the air that has been drawn into the spiral is caught in the space W2 whence as the drum rotates it is driven out by the mercury through the narrow space between the turns of the spiral and escapes into the space surrounding the drum, from which it is removed by an auxiliary pump connected at R'. This pump will not act unless a vacuum of a few millimeters of mercury is maintained in the space around the drum, and for this purpose the rotary pump described in the last section is very well suited, both pumps being conveniently driven by the same electric motor. 204. Mercury Air Pumps. — A simple form of air pump with which high vacua may be obtained is shown in Fig. 112. The vessel fl to be Fio. 112. — Geissler- Toepler air pump. PRESSURE GAUGES 135 exhausted is connected with B. On raising A which is an open vessel containing mercury, the mercury rises into B driving out the air through the narrow tube / which dips into a cup containing a Uttle mercury. On lowering A the mercury sinks out of B, but air cannot re-enter through / because the mercury rises in that tube balancing the external pressure. The air in R expands filling B again and is removed as before. The rare- faction may thus be pushed as far as desired by alternately raising and lowering A. 305. MacLeod Gauge. — For measuring the very low residual pres- sures in the vacua produced by air pumps, a device, shown in Fig. 113, and known as the MacLeod gauge* is employed. It is connected by the tube C with the exhausted vessel, so that the pres- sure in the bulb A is the residual pressure to be determined. The bulb D is raised causing the mercury to rise into A and C and compressing the air in A into the upper part of the narrow tube B. Suppose the air is thus com- pressed into one-thousandth part of the original volume A + B, the pressure in B will then be 1000 times the original pressure, while the pressure in the tube C is unchanged. The difference between the mercury levels in B and C will then measure the difference between the pressures, which in the case supposed is 999 times the pressure in C, so that 1 mm. difference in level corresponds to an original pressure of q ^ only 0.001 mm. of mercury. B K? 206. High Vacua. — In obtaining the highest vacua chemical means also are employed. Sir Humphrey Davy was the first to use this method. Having put into the vessel to be exhausted some caustic potash and then filled it with carbonic acid gas, he pumped out the gas as far as possible, and, having sealed the vessel, left the residual gas to be absorbed by the caustic potash, and thus obtained a very good vacuum. Or the tube, in which some copper filings are introduced, may be filled with oxygen and when exhausted, sealed and heated, the oxygen combining with the copper leaves a high vacuum. By these means vacua higher than a millionth of an atmosphere may be obtained. These high exhaustions are called by courtesy vacua, as they are the nearest approaches to an absolute vacuum that physicists have been able to make by the most refined methods known to science; and yet there is reason to believe that in every cubic inch of such a vacuum there are 400 miUion million molecules of gas. To form Fig. 113. — McLeod gauge. " Pronounced MacLoud. 136 LIQUIDS AND GASES some idea of the vastness of this nmnber, we may consider that if through the side of a little glass biilb of 1 cu. in. capacity, exhausted to this extreme degree, a minute hole were to be made through which a mUUon molecules should enter in every second, it would take 10 years for the pressure in the bulb to be doubled. The highest vacua are now conveniently obtained by enclosing ia a bulb connected with the exhausted tube some fragments of cocoanut or box-wood char- coal, which when cooled to the temperature of liquid air absorbs powerfully the residual gas. 307. Pressure Gauges. — One of the simplest forms of pressure gauge is the open manometer. It consists of a bent tube containing mercury, one arm being open to the air and the other connected with the vessel in which the pressure is to be measured. The difference between the pressure in the vessel and tliat of the atmosphere is measured by the height of one end of the mercury column above the other. If the dif- erence in pressure to be measured is very small, it is often best to use water or even kerosene oil instead of mercury on accoimt of their small densities. 208. Bourdon Spring Gauge. — A device commonly used in steam gauges is the Bourdon spring, so called from its inventor. It consists of a ^i°- us tube of brass of elliptical section, bent into a nearly complete ring, the flatter sides of the tube forming the inner and outer sides of the ring. One end of the tube is closed and into the other the fluid under pressure is admitted by a pipe. This end of the tube is firmly fixed, while the closed end is free though connected with a pointer by levers and rack work or by a fine chain wrapped around a small spindle (Fig. 115) by which the motion is greatly amplified. Suppose the pressure to increase, the flattened tube will spring a little and become more nearly circular in cross section, FiQ. 114.— Open manometer. —Bourdon spring gauge. PUMPS 137 and in so doing it will slightly unbend as if to straighten out, causing the pointer to move over the scale. When this device is employed as a steam gauge the pipe leading to it is usually bent downward so that it fills with condensed water, preventing the hot steam from reaching the gauge. 209. Common Suction Pump. — In this pump there are two valves opening upward, one in the piston and one at the bottom of the cylinder. As the piston is raised, its valve being shut, the atmospheric pressure forces water from the cistern to rise through the pipe and follow the piston, the lower valve opening and per- mitting this flow. As the piston descends the lower valve closes, preventing return to the cistern, and the valve in the piston opens allowing the water to pass through. Such a pump cannot raise water from a level more than about 34 ft. below the piston. T" i h E : i, 1 : B '■ M ^^fepTlillH Fig. 116. — ^Lift pump. Fio. 117. — Force pump. FiQ. 118.— Siphon. 210. Force Pump. — Water may be raised, however, to any desired height by the use of the force pump. In this pump the water is drawn into the cylinder as in the suction pump, but the downward stroke of the solid piston forces the liquid in the cylinder out through the side tube into the rising pipe, which may be extended to any height. A valve in the side tube prevents flowing back, and an air chamber is provided which acts as a spring, the air yielding to sudden movements of the piston, which the water column on account of its great inertia could not do. 311. Siphon. — If a bent tube is filled with a Uquid and one end is introduced into a vessel of the liquid while the other end is open and held at a lower level than the surface, the liquid 138 LIQUIDS AND GASES will escape through the tube. Such an arrangement, known as a siphon, is represented in figure 118. The upper surface of the hquid in the vessel and also the open end of the syphon are subject to the atmospheric pressure, but this is partly balanced on the short side by the column of liquid of height h, while on the other it is opposed by the longer column H. The pressure which is effective in causmg the flow is, there- fore, that of a column of liquid of height H — h. From this it appears that the velocity of liquid through a siphon would be the same as from an opening directly into the vessel at the level of the outer end of the siphon, if it were not for the loss due to friction in the pipe. Clearly the liquid can only rise in the siphon to a height where it can be supported by the atmospheric pressure; water, therefore, cannot be lifted by a siphon more than 34 ft. above its level and mercury not more than 30 in. Problems 1. How high ■would the atmosphere have to be to cause the barometej to stand 76 cm. high, if its density was the same throughout as at the earth's surface, taking this density as 0.0012 gms. per c.c. 2. How much higher will a barometer stand at the base of a mountain than at a station 1000 meters higher, taking the average density of air between the stations as 0.0012. 3- The air chamber of a force pump contains at the start 600 cu. in. of air at pressure 75 cms. of mercury. What volume will the air occupy while water is being forced to a height of 150 ft. above the pump? 4. How deep must a pond be that an air bubble on reaching the surface may have twice the volume that it had at the bottom? Suppose the barometric pressure at the surface to be 75 cm. of mercury. 5. How deep must a pond be when a bubble having a volume of 12 c.c. at the bottom has a volume of 30 c.c. as it reaches the surface. Barometer reading 75 at the surface. 6. A barometer on top of a tower stands at 75.20, at the bottom it stands at 75.40. How high is the tower if the average density of' the air .be- tween the top and bottom is 0.0012 gms. per c.c? 7. A barometer having a little air in the top of the tube stands at 72; but if the level of the mercury is raised so that the air space is half as great as before, it stands at 70. What is the correct barometric height. 8. If the tube in the apparatus shown in figure 108 contains 100 c.c. of air, and the mercury stands in the tube 15 cm. above the level in the outer vessel, while the barometer stands at 75, find what would be the volume of the enclosed air if it were at atmospheric pressure, also what will the FLUIDS IN MOTION 139 volume of the enclosed air become when the tube is raised sufficiently to make the mercury stand 20 cm. high inside the tube? 9. A glass bottle containing 100 c.c. of air floats at the surface of a pond with its open mouth downward. The bottle weighs 130 grm. and the density of the glass is 2.6. If the barometric pressure is 75 cm. of mercury, how deep below the surface must the bottle be pushed that it may just float in equilibrium, neither tending to rise nor sink? Neglect the weight of the enclosed air. Will the equilibrium be stable or un- stable and why? 10. What force must be exerted on the piston of a force pump 3 in. in diame- ter to raise water 100 ft. ? PART II.— FLUIDS IN MOTION 312. Steady Flow. — When a fluid is in motion if the pressure, velocity and direction of flow remain unchanged at every point in a certain region, the motion there is said to be steady. A line drawn in the fluid so that at every point it is in the direction of the flow at that point, is called a stream line. 313. Continuity. — In case of steady flow as much fluid must flow into any region as flows out of it in the same time. Let the figure represent either an open channel or a pipe con- veying water. The total volume of water crossing the section of A per second will be vs cu. ft. per second if the velocity is v ft. per second and the cross section of the stream at that point is s sq. ft. If d represents the density at A, or the number of pounds mass per cubic foot, then vsd is the mass of water crossing A per second and similarly v's'd' is the corresponding mass crossing B in the same time, and therefore vsd = v's'd'. This equation holds for .the steady flow of any fluid whether gas or liquid. But for liquids since the density does not appreciably change during the flow, we may take d = d' and so vs = v's' or the velocity is inversely as the cross section of the stream. If at a narrow place in a stream the velocity is not correspond- ingly great, we may be sure that the stream is deep at that point. The extremely small cross section of a stream at the edge of a dam is due its great velocity at that point. 140 LIQUIDS AND GASES 314. Momentum of Liquid Stream. — When a liquid is in motion each moving particle has momentum and kinetic energy. When a jet escapes through an opening in the side of a vessel the pressure which gives the jet its forward momentum acts at the same time as a reaction press- ing the vessel in the opposite direction. If the orifice is free to move backward it will do so, as in case of the device known as Barker's mill shown in the figure. In case of the end of a hose the rush of water around a curve will by its centrifugal force tend to straighten the hose. If the end is free it will very probably swing over too far, in consequence of its inertia, when it will be flung back again, thus thrashing to and fro. 315. Turbine Water Wheels. — The centrifugal force of a stream as it moves by curved guides is made use of as a means of obtaining power in turbine water wheels. Such a wheel is shown in section in the diagram. The water flows inward toward the wheel through the fixed guides, which cause it to enter in the proper direction, and then driving the wheel forward and sweeping by the wheel guides BB, it escapes at the center of the wheel. The guides AA may be made adjust- able so as to regulate the flow of water. The entering water from the flume is conducted to the turbine by a pipe which is kept constantly full, thus giving the advantage of its pressure. The turbine may be set at the Fig. 120. — Barker's mill." Fig. 121.- -Turbine water wheel diagram. FLUIDS IN MOTION 141 lowest level so that the water escapes directly into the tail race, or it may be set higher if the water escaping from the wheel enters a closed draft pipe which leads down to the tail water. The sinking of the water in this draft pipe produces a suction which increases the efficiency of the wheel. In the great 5000 H. P. turbines in use at Niagara the water enters the wheel from below in such a way that the weight of the wheel and shaft are almost exactly balanced by the upward pressure of the water, making the friction in the bearings extremely small. 316. Efficiency of Water Wheels. — When water flows from one level down to another it loses potential energy. That pro- portion of the potential energy lost by the water which is trans- formed into useful work in a water wheel is called its efficiency. It is clear that to be efficient a wheel must as far as possible let the water down from the higher to the lower level without dash- ing, and the water escaping at the bottom should have little velocity, its energy having been expended in useful work. 317. Various Water Wheels. — The old-fashioned overshot wheel, taking water from the upper level and lowering it to the bottom of the fall, uses the whole energy of the fall, but its size and weight cause great frictional loss. Where a small supply of water at high pressure is available, some form of jet wheel is often best. Here the wheel is driven at high speed by the force of a jet escaping against cups set around the periphery of the wheel. 218. Hydraulic Bam. — The hydraulic ram is an appliance by which a small quantity of water may be raised a considerable height by using a small fall in a stream. The water is conducted to the ram through a straight, smooth, inclined pipe offering little resistance to the flow. At C is a valve opening downward through which the water at first escapes; but as its speed in- creases, it catches the valve in its rush and shuts it. This sudden stoppage of the stream causes a great pressure at this end of the pipe in consequence of the forward momentum of the stream, and the valve d which opens upward is forced open and some water driven into the pipe e. The valve d then closes and prevents any return of water from e. But with the sudden stoppage of the stream the valve C if properly weighted rebounds and opens 142 LIQUIDS AND GASES again, the stream again escapes at C with increasing velocity until the valve is again caught and closed, when water is again driven through the valve d by the hammer-hke blow of the column of water in A. The action is thus kept up indefinitely, water being gradually forced up the pipe e until it may reach many times the height through which the stream falls. The air Fig. 122. — Hydiaulic ram. chamber B is essential to the action of the ram as it presents an elastic cushion with but Uttle inertia, enabUng the valve d to yield instantly. At / there is a minute opening, the air sniff, through which, in the recoil of the water, air is drawn in, main- taining the supply in the air chamber. 7/ a hydraulic ram were ^ perfectly efficient, it would raise one-tenth of the amount of water flowing into it through ten times the height of the fall or one-half the water twice the height of the fall. But in /■/I practice the efficiency of a good ram is about 50 per cent. Rams are now jnade in which the supply pipe is as much as 4 ft. in diameter. In these rams the valve which arrests the flow is moved by a piston operated by water from a small branch of the main pipe. 219. Velocity of a Jet. — While a hquid is escaping from a vessel through an opening which is small compared with the upper surface of the liquid, no change takes place within the vessel except the gradual lowering of the surface or disappear- ance of liquid from the top, while a corresponding mass appears outside in the escaping jet. If no energy is lost in friction or SSill I -.^==.= ■= :=. -^^ ^^ = — ^^r^—-.:^ ■= -= Fig. 123. FLUIDS IN MOTION 143 viscosity, the energy of a mass escaping at B must be the same as the energy of an equal mass at A. But since the potential energy at B due to gravity is less than at A the kinetic energy at B must be correspondingly greater; that is, it must be great enough to cause the escaping mass to rise from B to A when the jet is directed upward. If h is the height of A above B we have — The difference between the potential energy of a mass m at A and B = mgh ergs. The kinetic energy of mass m escaping at B with velocity V = 3^my^ ergs. Therefore mgh = J^mw^ and v' = 2gh (1) This velocity is the same as that which a freely falling body would acquire in a distance h, a conclusion known as TorricelU's theorem. TorricelU's Theorem. — The velocity of an escaping jet is equal to the velocity which a body will acquire in falling from the level of the upper surface to that of the opening. The density of the liquid and direction of the jet do not affect its velocity. When the pressure alone is known, the height of the escaping liquid required to produce the given pressure may be calculated and then used in the above formula. Thus the pressure on the level of B is p = hdg in dynes; using this to eliminate h from equation (1) we obtain v^ = $ (2) a 330. Vena Contracta. — The hquid as it ap- proaches the opening moves in from all sides along stream lines like those shown in the dia- gram. Liquid coming from each side has a certain momentum toward the axis of the jet, ^^^ ^^4 hence the jet narrows and does not become cyl- indrical until just after it has left the orifice. A short cylin- drical neck of the size of the opening is found to increase the 144 LIQUIDS AND GASES quantity escaping per second, and if the neck is somewhat flared out the flow is still greater. 221. Efflux of Gases. — The velocity with which a gas escapes through a small opening when the difference between the pressures on the two sides of the opening is p, is also determined by Torricelli's theorem. Since for a given pressure the velocity of eflSux is inversely proportional to the square root of the density of the gas, the densities of gases may be compared by observing the times in which measured quantities escape through a small opening. 222. Energy Due to Pressure. — When a liquid is forced into a vessel against pressure, the work done is equal to the product of the pres- sure by the volume of the liquid which is introduced. This expenditure of work is not wasted in friction, but exists as energy in the mass, ready to be transformed into energy of motion if an opening allows the mass to escape. The amount of this energy, since the volume of the mass to, equals m . ^' = ~d 323. Energy Equation. — Consider a small mass of liquid at a in the vessel shown in the diagram; it is in equilibrium, and may be moved without offering any resistance from a up to the surface. Clearly there is no change in its total potential energy as it is moved from one part of the vessel to another. At the top its gravitation potential energy referred to the earth is a maximum, but then it has no energy due to pressure, while at a its gravi- tation energy is less but its pressure energy is correspondingly greater. If h represents the height of the mass m above some fixed plane, say the surface of the earth, its gravitation potential energy referred to that plane is mgh. We have seen that its pressure energy is —j-; and if the mass is in motion it will have kinetic energy }^m.v'^, and its total energy may be written -J — h mgh + }4mv' = energy of mass m. If the stream is flowing in conduits or channels without doing work, the energy of the mass will remain constant except as it is wasted in internal or external friction. The fact that in steady irrotational motion of a friction- TOT3 less fluid, the expression — j- + mgh + }imv'' remains constant for a little mass TO as it moves along, is known as Bernoulli's Principle. As an illustration of the above equation, conceive the vessel A in the figure to be kept filled to a constant level while the liquid is flowing out freely through the pipe D, and follow the changes in the mass m. As it moves FLUIDS IN MOTION 145 Fig. 125. downward h grows less and so its gravitation energy diminishes while the pressure energy increases, the kinetic energy being scarcely changed; but as it approaches the opening B its velocity increases and consequently more of its energy is kinetic and less due to pressure than at the same level farther in the vessel, the pressure at B must then be less than at 6. When it reaches Cits velocity will be less, and consequently the pressure there will be greater than at B. Throughout D the cross section, and consequently the velocity, is constant, and since it is all at the same level the pressure must be constant except as influenced by friction in the pipe. 334. Friction in Pipes. — When water escapes from a reservoir through a horizontal pipe of uniform section, as ab in figure 126, the velocity will be the same at all points in the pipe, and if there is no friction the pres- sure will be constant throughout the length of the pipe and equal to the atmospheric pressure at the end b. In that case the water will not rise in any of the gauge tubes shown. In practice, however, there is always some friction in a pipe, and, therefore, a constant expenditure of energy. But the energy equation is E ='^ + mgh + yim.v% and if the pipe is level and cylindrical h and v cannot change, consequently if there is any decrease in E there must be an equal decrease in the first . — 4: — 1 ii f — „ r_j; ,• » — , j{ tjjg friction is uniform through- out the pipe the pressure will decrease uniformly, becoming equal to the atmospheric pressure at the opening 6. The height h (see figure) which determines the pressure at 6 when the opening is stopped up so that there is no flow, is called the pressure head. When b is open the head required to produce the observed velocity of escape reckoned from the law t = ■\/2gh, is called the velocity head. In the above case it is h', and the remaining head (h — h') is spent in overcoming friction. The loss of pressure when water is flowing in pipes is a fact that has to be constantly taken into account in practice. The friction and consequent loss of pressure, increase with the velocity of flow. 235. Pressure Varies witli Velocity. — The fact just demon- strated that in a horizontal pipe of variable section the pressure term —r, and therefore, a fall in pressure. h< 146 LIQUIDS AND GASES FiQ. 127. will be greatest when the cross section is the greatest and velocity least is sq interesting, and important that it merits a brief ex- amination from another point of view. Consider a little mass of liquid at A (Fig. 127) where its velocity is clearly diminishing. It is under pressure on all sides, but since its velocity is diminishing the pressure backward on its forward side must be greater than the pressure on its left which urges it forward. When the mass reaches B, however, its velocity is increasing, hence the pressure behind it which urges it forward must be greater than the pressure in front which is opposite to its motion; the point of slowest mo- tion must therefore be a point of maximum pressiure. 336. Aspirating Pumps. — The principle just established is made use of in aspirators for exhausting air. Such an instrument is shown in figure 128. It provides a narrow channel through which water flows with great velocity, the stream widening out and moving slower be- fore it reaches the atmospheric pressure at the open end. The pressure at the narrowest point must then be very much less than that of the atmosphere, and air is accordingly drawn in through the side tube C and carried out at B by the rush of water. 227. Ball on Jet, etc. — A jet of water or even of air may support in stable equilib- rium a light ball. The explanation is that a slight shifting of the ball, say to the right, would cause the main stream to rush on the left, the velocity of flow would be greatest there, and there- fore, the pressure less than on the right, and so the ball would be pressed back again. A card with a pin through it and laid over the open end of a spool cannot be blown off by blowing through the spool because the velocity of the air stream as it spreads out under the card is least at the outer edge where it comes to the atmospheric pres- sure, the pressure nearer the center where the velocity is greater Fig. 128. — Chapman exhaust pump. FLUID MOTION 147 will, therefore, be less than that of the atmosphere and the card will accordingly be pressed against the end of the spool. Similarly a ball will be held in a cup by a jet escaping around it, as in the ball nozzle used for fire hose. If a stream of air is directed between two sheets of paper they are drawn together. So also leaves and other light objects are drawn toward a moving train as it passes. Problems 1. Find the velocity of the stream of water in a pipe having a cross sec- tion of 3 sq. in. and discharging 450 cu. ft. of water per hour. 2. How much water will escape per minute from a 2-in. hole in the side of a water tower 50 ft. high. 3. With what velocity will water spurt out of a hole in a boiler in which the pressure is 80 lb. to the sq. in., in addition to atmospheric pressure? 4. The stream of water below a certain dam has a cross section of 10 sq. ft. and a velocity of 5 ft. per sec. Find the horse-power available if the dam is 15 ft. high. 6. What horse-power would be obtained from a 20-ft. fall by a turbine wheel of 80 per cent, efficiency, when the flow is 300 cu. ft. per minute? 6. If the water escaping from a turbine water wheel which uses the water from a 10-ft. fall has a velocity of 6 ft. per sec, what is the greatest possible efficiency of the wheel? 7. If the efficiency of a hydraulic ram is 50 per cent., how much water per day will it raise to a tank at a height of 100 ft. above the ram, when the supply pipe has a fall of 8 ft. and discharges 1 gallon per minute? 8. While water is flowing with a velocity of 2.2 ft. per sec. in a pipe 1 in. in diameter the pressure drops of! from 70 to 10 lbs. per sq. in. in a length of 500 ft. Find the energy in foot-pounds spent in overcom- ing friction per cu. ft. of water. 9. Find the horse-power spent in friction in the 500 ft. length of pipe specified in problem 8. 10. Derive formulas (1) and (2) of §219 for the velocity of an escaping jet, from the energy equation of §223. 11. A horizontal water pipe of 1 sq. in. cross section widens out to 3 sq. in. in section. If the velocity is 5 ft. per sec. in the narrower pipe and the pressure 5 lbs. to the sq. in., what will be the pressure in the adjoining part of the wider pipe? Ans. 5.14 lbs. per sq. in. The pressure given is gauge preaaure, or the excess above that of the atmosphere. The total or absolute pressure is 5 + 14.7 = 19.7 lbs. per sq. in. 12. The nozzle of a fire hose has an opening 2 in. in diameter, while the pipe just back of it is 3 in. in diameter. Find the pressure just back of the nozzle when it can throw a jet 60 ft. vertically upward. Ans. 20.9 lbs. per sq. in. Note. — At the opening of the nozzle the pressure is that of the atmosphere, or 14.7 lbs. per sq. in. absolute, while the velocity is found from the height to which the water is thrown. Use energy equation of §223. PROPERTIES OF MATTER AND ITS INTERNAL FORCES Structube 338. Density. — On comparing a block of wood or aluminum with an equal weight of lead or gold, it is clear that substances differ greatly in the quantity of matter concentrated in a given volume. The mass of any substance contained in unit volume is known as its density. Densities of some Substances in Grams per Cubic Centimeter Solids Aluminum. . Iron Tin Copper Lead Gold Silver Platinum . . . Brass Glass Oak wood. . . .8 2.7 7.2-7 7.3 8.8 11.4 19.3 10.5 20.5-22.0 8.3-8.6 2.&-3.5 0.84 Liquids Mercury 13.596 Sea water 1.026 Water at 4°C. 1 . 00 Alcohol 0.8 Ether 0.72 Gases at Q°C. and 1 Atm. Air 0.001293 Oxygen. . Nitrogen . Hydrogen 0.001430 0.001256 0.00008988 239. Molecular Forces. — When a lead bullet is divided by a clean cut, if the two halves are pressed together they will cling with considerable force. This is an imperfect exhibition be- cause of poor contact of the force which originally held the two parts together. This force is known as cohesion, or when the attraction is between different substances it is known as adhesion. A drop of water is held together by cohesion, but it clings to a glass rod by adhesion. 230. Molecular Theory. — All matter is conceived as made up of separate molecules which are the smallest portions of the sub- stances that can exist in a free state, as in gas or vapor. It is be- 148 STRUCTURE 149 lieved that the molecules of any particular substance are all alike, and, in substances not at the absolute zero of temperature, are in more or less active motion or vibration, the energy of vibration depending on the temperature. In solids the vibrating molecules are held by their mutual attractions in such a way that they can- not move far away from their mean relative positions. In Hquids the phenomena of diffusion, and the Brownian move- ment (§273), show that molecules move about in the mass, and are not held in fixed positions relative to each other, though the force of cohesion may be very great. In gases or vapors there is the greatest freedom of motion of the molecules, and their average distance apart is much greater than in liquids or solids, while there is scarcely any cohesion. It is supposed that any two molecules of matter attract each other, according to the Newtonian law of gravitation, with a force varying inversely as the square of the distance between them for all considerable distances, but when very near each other the force of attraction varies with the distance according to some unknown law, giving rise to the phenomena of cohesion and ad- hesion, until the molecules come into what is called contact, when a force of repulsion opposes nearer approach. The experiments of Quincke indicate that molecules must be less than 5 X 10~* cm. apart in order that the cohesive force may be perceptible. The idea that matter is molecular in its structure is supported by a great variety of evidence found especially in the phe- nomena of heat, gases, and radiation, as well as in chemical phenomena. 231. Molecular Equilibrium. — The molecules of a substance may be regarded as in a state of equilibrium under three forces: external pressure, cohesive force, and an internal pressure due to the rebounding of adjoining molecules against each other as they vibrate to and fro. This latter force may be considered to bal- ance the other two. 150 PROPERTIES OF MATTER We may form a conception of these forces by the following model. Im- agine a row of small rubber balls drawn together by springs stretched between them (Fig. 129). Let two outer springs also press them together, and let the balls be thought of as rapidly vibrating to and fro, rebounding against each other, and so keeping a greater distance apart than if they were at rest. The force of the outer springs represents the external pressure and that of the springs joining the balls together the cohesive force,* and these are balanced by the repulsion due to the impacts of the balls against each other. In solids and liquids the pressure due to cohesion is that which chiefly balances the internal repulsion, the external pressure being usually quite insignificant in comparison. But in gases the case is different. In consequence of the great average distance between the molecules, the cohesion is so insig- nificant that the external pressure alone may be said to balance the internal pressure due to the motions of the molecules. This theory of gaseous pressure is more fully discussed in §270 et seq. 332. Structure. — When the properties of any one portion of a mass are exactly like those of any other portion, the mass is said to be homogeneous. Whether a substance is called homogene- ous or not depends on the point of view. One part of a brick wall is just like another part, and so it may be said to be homogeneous; but if we compare minute parts we find in some spots brick and others mortar and so there is a limit to its homogeneity. So water is regarded as homogeneous unless we are dealing with por- tions so small that the molecular structure is significant. If the various physical properties of a substance are the same in all directions throughout its mass, it is said to be isotropic. Water, glass, and mercury are isotropic. Most crystalline sub- stances are not isotropic, and may be called anisotropic. 233. Crystals. — In solids which pass slowly into the solid state, either directly from vapor or as the result of the slow cooling of a fused mass or of separation from a solution, there are often formed masses called crystals which have regular and distinctive forms and are bounded by plane faces. The crystallization begins at certain isolated points and the minute crystals gradually grow in size, until they may meet and form a solid agglomeration. * The springs representing the cohesive force should be conceived as exerting less force the more they are stretched, for cohesive force diminishes as particles separate. ELASTICITY 151 The study of the fundamental crystal forms has led miner- alogists to divide them into six classes or systems. Some idea of the cause of the formation of crystals may be obtained by considering the forms which may be built up of shot when placed together so that each shall touch as many others as possible. Suppose such a pyramid as that represented in figure 130, where one layer is incomplete; if we think of it as a growing crystal in which the balls represent the molecules and suppose it im- mersed in a medium in which there are free molecules surround- ing it, there will clearly be a ten- dency for these to fill out the in- complete surface, for a molecule will touch more neighbors when pj^ ^^o! placed along the incomplete edge than anywhere else, and so may be conceived to be more powerfully attracted into that position. In consequence a figure bounded by plane surfaces would result. The piling of balls would give the crystal forms characteristic of the first or regular system, but to explain the variety of crystal groups it is necessary to suppose that the molecules themselves have properties different in one direction from what they have in another, and that when built up into crystal forms they are all similarly oriented or directed. Elasticity and Viscosity 334. stress and Strain. — When a portion of matter is acted on by forces tending to change its size or shape it is said to be under stress, and the accompanying distortion or change in vol- ume is called the strain. A stress tending to stretch any portion of matter is called a tension, while a stress tending to shorten it is called a pressure. Stress is measured by force per unit surface, as in pounds per square inch, or in grams or dynes per square centimeter. 335. Strain Ellipsoid. — When a body is strained, a small spherical portion of it is in general distorted into an ellipsoid, and the axes of the ellipsoid are the three principal directions of strain at that point. 152 PROPERTIES OF MATTER When the strain is the same everywhere throughout a body, as in case of a stretched wire, it is said to be homogeneous. In such a case the strain eUipsoids are all alike and similarly situated, as shown in figure 131. When a fluid is compressed the strain is homogeneous and the ellipsoids are spheres slightly smaller than in the unstrained state. The distribution of strain in a bent beam is shown by the ellipsoids in figure 132. The strain in this case is not homoge- neous and there is a surface of no strain indicated by the dotted line. 336. Resistance to Strain. — A body is said to be elastic if after having been strained it springs back to its original form when the stress is removed. If the stress is the same for a given Fia. 131. Fig. 132. amount of strain whether the strain is increasing or diminishing, the body is said to be perfectly elastic. When strained beyond a certain point called the limit of elastic- ity, substances yield permanently and do not return to the original state when the straining forces are removed. In this case there may be a great internal stress while the body is being strained, but on a very slight diminution of strain the stress entirely disappears. Putty, wet clay, and lead all exhibit this permanent distortion under comparatively small forces and even when the strain is small; while india-rubber is remarkable for the great strain which it can experience without passing its elastic limit. It is said to have a wide limit of elasticity. Even within the limits of elasticity most substances show a time lag in returning to their original state after having been strained. Thus when a steel wire is firmly clamped at its upper end, if the lower end is twisted through an arc well within its limit of elasticity, the wire when set free returns at once nearly to its original position, but creeps very slowly back through the re- ELASTICITY 153 maining distance. This lag is found in metals and in glass., but quartz fibers are remarkably free from it. 337. Hooke's Law. — In small strains of elastic bodies the stress is proportional to. the strain. This is known as Hooke's law, having been enunciated by him in 1676. According to this law, a long spring when stretched 2 cm. will exert twice the force that it would if stretched 1 cm., and the tension required to stretch a spring' a small distance is equal to the pressure when the spring is compressed an equal amount. Careful experiment; however, shows that the law is not ex- actly true. Most substances offer slightly more resistance to a given small compression than to an equal extension. An illustration of this law is afforded by the ordinary spring balance in which equal divisions of the scale correspond to equal increments of weight. In this case the elongation or compression of the helical spring may be relatively very great, yet because of its shape the distortion or strain of any little portion is ex- tremely minute and Hooke's law holds very nearly true. 338. Elasticity. — In elastic bodies the elasticity is measured by the ratio of the stress to the corresponding strain. _,, ,. ., stress Elasticity = -r — ^• •^ strain In bodies which are homogeneous and isotropic there are two principal kinds of elasticity, that in virtue of which the body resists change of volume and that resisting change of shape. The first is called volume elasticity and the second rigidity. Volume elasticity is possessed by all bodies, fluids as well as solids, but rigidity is a characteristic of solids. In some strains both of these elasticities are involved; for instance, when a wire is stretched there is a sidewise contraction as well as an elongation, so that the resistance to stretching depends on both the rigidity and volume elasticity of the sub- stance. The elasticity of stretching or compression is so im- portant in engineering that it has received a special name and is known as Young's modulus. 339. Volume Elasticity. — When a body is so strained that every little cubical portion is compressed into a smaller cube the corresponding stress must be a pressure equal in all directions, 154 PROPERTIES OF MATTER provided the substance is isotropic or equally compressible in every direction. This kind of stress is called hydrostatic pressure because it is the only kind of stress that can exist in fluids at rest. The voluvie elasticity or bulk modulus of a substance is the ratio of the increase in pressure to the corresponding compression per unit volume. Thus this elasticity will be represented by E = Volume elasticity pressure increase change in unit volume t V V = F?. where p is the increase of pressure causing a contraction v in a total volume V. The volume elasticity a of solid maybe found by subjecting a long bar of the sub- stance to hydrostatic pressure in a strong tube having thick glass windows through which its change in length may be ob- served by fixed microscopes. 240. Compressibility of Liquids. — Liquids are, as a rule, somewhat more compressible than solids, but on the other hand so great is their resistance to com- pression that for most practical purposes they may be treated as if incompressible. The compressibility of a liquid may be measured by the apparatus shown in figure 133, known as Oersted's piezometer. In this instrument the liquid to be tested is contained in a bulb of glass terminat- ing in a long narrow tube of uniform dia- meter, open at the end and carefully graduated. This bulb A is surrounded by water in a stout cylindrical vessel of glass and subjected to pressure by means of a piston forced in by a screw. A globule of mercury in the narrow tube separates the liquid in the bulb from the surrounding water. From the number of scale Fig. 133. — Oersted's piezometer. ELASTICITY 155 divisions through which the mercury moves down toward the bulb as pressure is applied, the apparent compressibihty of the contained liquid is determined, the relation between the volume of the bulb and the volume contained in one division of the capillary tube having been previously ascertained. Although the pressure is the same on the outside of the bulb as on the inside, its volume diminishes in consequence of the compression of the glass of which it is made, so that the experiment gives the difference between the compressibihty of the liquid and that of the glass bulb. A thermometer gives the temperature of the liquid examined, and the pressure may be determined from the amount of com- pression observed in a tube M containing air and placed open end downward in the cylinder. 341. Elasticity of Gases. — In case of a gas it is necessary to distinguish between its elasticity when the temperature is kept constant during the compression, and its elasticity when com- pressed so suddenly that there is no time for the flow of heat to take place. The first is called isothermal elasticity and the second adiabatic elasticity; the latter is always greater, being, in case of air, oxygen, hydrogen, and nitrogen, about 1.40 times as great as the isothermal elasticity. The isothermal elasticity of a gas may be calculated from Boyle's law. Suppose the pressure is increased from p to p', the decrease in volume will be V — o' and we have E = vt-^ (1) But by Boyle's law pv = p'u', therefore p v' p' V and p' — p _ V — v' p' V substituting in (1) we find E = p'. But the difference between p and p' is supposed extremely small, so that for gases kept at constant temperature the volume elasticity is equal to the pressure. 156 PROPERTIES OF MATTER Volume Elasticity and Compressibility Substance Temp. Compressibility in miUionths of volume per at- mosphere Volume Elasticity Dynes per sq. cm. Lbs. per sq. inch Steel 0.65 2.44 3.00 25.00 46.00 191.00 1,000,000.00 188.00 X 10'° 41.00 X 10" 33.00 X 10'° 4.0 X 10'° 2.2 X 10" 0.52 X 10'° 27 00 X 10* Glass 6 00 X 10« Mercury 0° 20° 20° 20° ■ 4.8 X 10« Glycerin 0.58 X 10» Water 32 X 10" Ether 0.07 X 10« , . /At normal \ Air < } [ pressure J 1.00X10» 14.7 B Bi 134. 842. Rigidity. — If a cylindrical rod or wire is twisted about its axis without change of length it may be imagined divided into sections of equal thickness, in each of which there has been no change in volume but simply a distortion of the little elements of which it may be conceived as made up. Take such a little block as that represented in figure 134. If the base CD is firmly fixed, a force F applied to the upper surface will strain it into the position A'B', just as a thick book lying on a table may be pushed out of shape by force applied to the upper cover. The strain in this case is a pure distortion without any change in volume and is called a shear, and the forces bringing it about constitute a shearing stress. The strain is measured by the ratio of the displacement A A' or x to the height h, while the stress is the force applied per unit area; or if S is tne F area of the upper surface of the block the stress is The rigidity n, or elastic resistance to distortion, may therefore be expressed thus: F Rigidity = w = S = '*"«"" Fh T = strain h Sx (1) In case of a wire clamped at one end and twisted at the other, ELASTICITY 157 it may be mathematically demonstrated that the moment of the force of torsion T is expressed by the formula irr*a where n is the coefficient of rigidity of the substance of which the wire is made, r is the radius of its cross section, and I its length, while a is the angle (in radians) through which it is twisted. By measuring the moment of force required to twist a given wire through a measured angle, the coefficient of rigidity of the substance of which the wire is made may be determined by the ^ use of this formula. Rigidity Steel 82 X 10" dynes per sq. cm., 12.0 X 10« lbs. per sq. inch. Brass 38 X 10'» " " " " 6.5 X 10« " " " " Glass 24X10" " " " " 3.5X10= " " " " 343. Young's Modulus. — When a rod or wire is stretched by a weight the elongation is very nearly proportional to the stretch- ing force, and exactly proportional to the length of the wire. In this case the stress is the force per unit cross section and the corresponding strain is the elongation per unit length. The elasticity of stretch for the substance of which the wire is made, or Young's modulus, as it is commonly called, may be represented by Y and is determined by the ratio g = stress „ ~ e ^ . ~ Se -j = stram where F represents the stretching force, e the elongation of the wire, I its length, and S its cross section. Young's Modulus Copper 12 X 10" dynes per sq. cm., 17 X 10* lbs. per sq. inch. j Brass 11X10" " " " " 16 X 10« " " " " Iron 19X10" " " " " 27 X 10« " " " " Steel 22X10" " " " " 32 X 10« " " " " 344. Beams. — When a floor beam sags under a load the upper part is compressed and the lower part stretched. But the re- 158 PROPERTIES OF MATTER sistance to longitudinal stretching or cornpression is measured by Young's modulus, so that the stiffness of a beam is proportional to the modulus of elasticity of the material of which it is made. In case of a bieam supported at both ends and loaded at the middle, the sag or deflection y at the middle is expressed by the formula where I is the length of the beam, h is its breadth, and h, its height, F is the load, and Y is Young's modulus for the material of the beam. From this it appears that a beam having twice the breadth of another would sag half as much, other things being equal, while if its depth t^ere twice that of the other it would only sag one-eighth as much. For this reason floor beams are placed on edge, the breadth having but little influence on the stiffness compared with the depth. 245. Viscosity. — When a solid is strained beyond its elastic limit the strain may go on increasing indefinitely at a rate which depends on the stress to which it is subjected. Most metals show a viscosity of this kind when the distorting force is great enough, as seen in wire drawing and in the making of lead pipe. A strip of lead when stretched with a moderate weight will con- tinue slowly elongating year after year. A glass fiber fastened at one end and having a small twisting force applied at the other will twist more and more as times goes on. But when a substance yields continuously in this way to the very smallest forces, as in case of tar, pitch, or syrup, it is said to be a viscous fluid. In such a fluid one layer slides over an- other with a velocity which depends on the stress and on the vis- cosity, the slower the motion for a given stress the more viscous the substance is said to be. Viscosity may be considered a kind of internal friction be- tween contiguous layers, and the energy spent in overcoming it appears as heat. All known liquids and even gases are more or less viscous, and in consequence energy is spent in heat and there is loss of pressure (§224) whenever a fluid flows through a long pipe. The outer layers next the wall of the pipe are nearly stationary DIFFUSION 159 in such a case, while the velocity of flow increases toward the center or axis of the stream. The usefulness of a lubricating oil depends, among other things, upon its viscosity* If not viscous enough it will be squeezed out of the bearing, while if too viscous it will oflfer need- less resistance to the motion. The viscosity of a fluid may be determined from the tinie re- quired for a given quantity to escape through a long tube of small diameter, or it may be found by an apparatus called a viscosi- meter in which a long inner cylinder is supported by a torsion' wire in the axis of an outer cylindrical tube which can be rotated. The space between the two tubes is filled with the oil or other liquid to be Tested and the torsion effect on the inner cylinder is measured when the outer one is turning at a constant rate of speed. The viscosity of a substance depends on its temperature, and it is noteworthy that heating a liqwid makes it less viscous; while the opposite is true of gases. 246. Energy Absorbed by Viscosity. — The absorption of energy through viscosity is well shown by the following experiment, due to Lord Kelvin (Sir Wm. Thomson). Take two eggs, one raw and one hard-boiled, and suspend each like a tor- sion pendulum by means of a fine wire attached to a wire sling enclosing the egg, the long axes of the eggs being vertical; then give each egg a turn or two and let it go. The boiled egg will continue oscillating for a long time, while the raw egg will almost immediately come to rest. The oscillating motion of the shell is so rapid that the inner layers of the raw egg slip on the outer ones by their inertia, and the internal friction or viscosity of the egg causes the energy of vibration to be lost in heat. This principle has been applied by Lord Kelvin to prevent the violent swinging of a mariner's compass, due to the motion of the ship. The com- pass box, hung on gimbals so that it can swing freely in any direction, is made with a double bottom, and the space between the two bottoms is partly filled with a viscous liquid, such as glycerin. After any disturbance the glycerin flowing between the two surfaces of the box transforms the energy of motion into heat, and the box is promptly brought to rest. Diffusion and Solution 247. Diffusion. — If a strong solution of copper sulphate is introduced by a tube into the bottom of a tall vessel containing 160 PROPERTIES OF MATTER pure water, the denser blue solution will at first be sharply sepa- rated from the clear water above. By degrees the sulphate will be seen to steal upward into the water until in time it will be uni- formly diffused throughout the liquid, just as a gas expands and fills a vessel in which it is set free, though diffusion in liquids is extremely slow. Stirring a mixture of two Uquids increases the surface through which diffusion takes place and so greatly quickens the process of complete mixture. When no diffusion takes place between the liquids they will not mix. 348. Interdiflusion of Gases. — Gases diffuse into each other very freely, as shown by the following experiment. Two globes are connected together, the upper containing hydrogen and the lower carbonic acid gas. In spite of the density of the carbonic acid being 22 times that of hydrogen, it will diffuse upward and the hydrogen downward till finally a uniform mixture will fill both vessels. Each expands and fills the whole space as if it alone were present. 249. Solution of Solids In Liquids. — When a solid is placed in a liquid a certain amount will be dissolved, after which no I more will be taken up, and the liquid is said to be saturated. The per cent, that can be dissolved depends not only on the sub- stance, but on the temperature, solubiUty usually increasing with rise in temperature. The volume of the solution is usually less than the combined volumes of the two constituents and the process of dissolving is often accompanied by a change in temperature. Solution of Liquids in Liquids. — Two liquids that diffuse into each other may either mix in any proportion, as in case of water and alcohol, or one may only dissolve a limited amount of the other. Thus if water and ether are stirred together at a tempera- ture of 10°C., the mass will separate into a lower layer of water containing 10 per cent, of ether and an upper layer of ether con- taining 1)4 per cent, of water. At 10°C. ether will dissolve any amount of water less than 13^ per cent, and water will dissolve 10 per cent, or less of ether. As the temperature is raised water will dissolve less ether, while ether dissolves more water. ^ 350. Solution of Gases in Liquids. — Some liquids, such as water, dissolve all gases more or less freely. When there is DIFFUSION 161 simple solution without chemical union the gas is absorbed most freely when the liquid is cold and is driven off when the liquid is heated. Thus when water is heated the absorbed air escapes in bubbles before boiling takes place. The amount of gas absorbed by a given liquid is proportional to the pressure. Soda water is charged with carbon dioxid gas under pressure, and when the pressure is relieved the gas escapes in bubbles, causing effervescence. The power of water to absorb various gases is shown in the following table, the figures giving the volume of gas at one atmos- phere pressure absorbed by unit volume of water. Ahsorplion of Gases in Water Gas At 0°C. At IS'C. Oxygen Hydrogen Nitrogen Carbonic acid gas Ammonia 0.049 0.021 0.023 1.79 1140.00 0.030 0.019 0.015 1.00 756.00 361. Absorption of Gases in Solids. — Certain porous solids have a great power of absorbing gases. Boxwood charcoal will absorb 90 times its volume of ammonia and 35 volumes of car- bonic acid gas. This absorption seems to be due to the conden- sation of a layer of gas on the surface of the body. It is by the condensation of a surface film of gas over a body that the so-called Moser's breath figures are explained. If an engraved die lie for some time on a polished plate of metal or glass, on removing the die and breathing on the plate the engraved image is seen. Platinum in the porous state, known as spongy platinum, absorbs hydrogen gas so powerfully that if placed in an escaping jet of hydrogen the heat developed by the condensation is suffi- cient to ignite the jet. There is what seems to be a true solution of gases in some solids discovered by Graham and called by him occlusion. By heating iron wire and then allowing it to cool in an atmosphere of hy- drogen, it was found that it occluded 0.44 times its volume of the 162 PROPERTIES OF MATTER gas, while platinum occluded 4 times its volume of the same gas. It is probably in consequence of this that hydrogen readily dif- fuses through iron and platinum when they are red-hot. The most remarkable substance in this respect is palladium which absorbs 960 times its own volume of electrolytically developed hydrogen and at the same time expands about Ho of its volume. 353. Colloids and Crystalloids. — It was found by Graham that a diaphragm of parchment or bladder would allow certain sub- stances in solution, such as salts, to freely diffuse through it, while other substances, such as albumen, starch, gum, glue, or gelatin, could not pass through or only very slowly. He was led, there- fore, to divide substances into two classes, crystalloids and colloids (from the Greek word for glue), which could be separated in this way. 353. Osmotic Pressure. — If a strong solu- tion of sugar or glucose is placed in a vessel opening above in a tube and closed at the bottom with a membrane, such as a bladder, and if the whole is set in a vessel of pure water, diffusion of water takes place through the membrane, and the solution accordingly rises in the tube until the pressure Of the solution in the vessel is sufficient to prevent any more water from entering. The increase pressure in the interior when equilibrium is reached is known as the osmotic pressure of the dis- solved substance. Such a membrane, however, is not suited for measurements of osmotic pressure, for sugar diffuses through it to some extent, though not as rapidly as water, also it lacks the strength and rigidity needed to support the pres- sure developed. Pfeffer showed how to form by chemical means within the substance of a porous earthenware cup or cell, a membrane which was entirely impervious to sugar in solution while it freely transmitted water. A semipermeable membrane formed in this way has great mechanical strength, and H. N. Morse who has greatly improved the Pfeffer cell has been able by its means to measure osmotic pressures as high as 28 atmospheres, or 400 lbs. per square inch. Investigation shows that osmotic pressure is proportional to the number of molecules of the dissolved substance in 1 gram of solvent, and when aqueous solutions of different substances have equal osmotic pressures, the number of molecules of dissolved substance per gram of pure solvent is the same in Fro. 135. — OBmotio pressure. SURFACE TENSION 163 one as in the other, just as equal volumes of different gases at the same temperature and pressure contain equal numbers of molecules (Avogadro's law). Capillahity and Surface Tension 354. Capillarity. — Under this head are grouped a number of phenomena depending on the force of cohesion at Uquid sur- faces. Some of these are the upward curvature of the surface of water where it meets the side of a glass vessel, the clinging together of light bodies floating on the surface of water, the forms of drops and bubbles, and the rise of liquids in fine hair-like or capillary- tubes. (Latin Capillus, a hair.) To understand these phenomena we must first examine how the conditions at the surface of aliquid differ from those in its interior. 255. Surface Tension. — If a mass of olive oil is floated in a mixture of alcohol and water of the same density as the oil, it will gather itself up into a spherical ball. Draw it out by means of a glass rod into any long or irregular shape, and as soon as it is left to itself it returns to its spherical form, exactly as if covered with an elastic skin. When a camel's- hair brush is immersed in water the bristles stand apart as freely as in air, but when it is withdrawn they cling together by the contrac- tion of the surrounding water surface. Wet threads cling together when drawn out of water. So also a drop of mercury resting on a table is drawn up into a smooth rounded mass by the Fi°- 136.— Water contraction of the surface, in spite of the weight of the mercury which tends to flatten it out. 356. Cause of Surface Tension. — That the particles of a liquid attract each other and are held together by a strong force of cohesion may be shown by the water hammer, which is a bent tube (Fig. 136) partly filled with water from which the air has all been boiled out, the tube having been sealed up while boiUng so that it contains simply water and its vapor with scarcely any air. The absence of air causes the water to strike the end of the tube with a sharp metallic click when it is shaken, hence its name. If the water is all run into one arm of the tube, completely filling it, and is jarred into good contact with the sides of the tube, it may 164 PROPERTIES OF MATTER then be held in the position shown in the figure and the water will remain in the full arm in spite of the fact that the pressure of the vapor in the other branch is quite insufficient to sustain the column of water; for if a slight jar is given to the tube, the liquid sinks to the same level in each branch. In this case the column of water is sustained by clinging to the walls of the tube and by the cohe- sion of one particle to another, and it is under negative pres- sure or tension as much as is a rope supporting a weight. But this force of cohesion can be de- ^^ ^^^^S n ^^^^P^= tected only between particles that are "^^F^g g ""^^^ ^^^ ^^:^ exceedingly close together. -■^^^^-=: ^^Tli?^=rr==nr: The small distance within which all those particles lie that have any sensi- 137. j^jg attraction for a given particle may be called the radius of the sphere of molecular attraction. If the spheres of action are represented by the circles about A and B, it is clear that the particle at A is in equilibrium, so far as the cohesive forces are concerned, being equally drawn in all directions, while B, which is nearer to the surface than the radius of the sphere of action, has the downward attraction of liquid below the line ab balanced only by the upward attraction of the medium above the liquid surface. If the latter is a free surface with air or vapor above, the downward attraction will ^__^ be in excess and the tendency is to ' — -- - drag particles away from the sur- — ^^^- face and into the interior of the ^^^ ^^ t— -,- liquid. In consequence of this the surface tends to contract. If the upper surface of the liquid were in contact with a substance whose attraction for the particle B was greater than that of the liquid below ab, the upward attraction would be in excess and the surface would tend to enlarge. Thus a drop of oil on a clean glass plate will spread out over the whole surface of the glass. Some idea of the size of the spheres of attraction may be formed from the study of soap films. Let figure 138 represent a section of a soap film in which the circles indicate the spheres of attraction. Particles in the middle of the film, between the two dotted lines, are farther from the surface than the radius of the sphere of action, and no change in SURFACE TENSION 165 the contractile force of the film is to be expected so long as the two surfaces are thus comparatively independent of each other, but if the film is made thinner than the diameter of the sphere of action a change in the tension of the film is to be expected. Such a change — a, sudden decrease in tension — is observed when the thickness of a soap film is about 100 millionths of a millimeter, indicating that the radius of the sphere of action in the soap solution is about 50 millionths of a millimeter. 357. Measure of Surface Tension. — The surface tension of a liquid is the force with which the surface on one side of a line one centimeter long pulls against that on the other side of the line. Thus it is the contractile force which a square centimeter of surface (Fig. 139) exerts on each of its bounding sides. In the diagram the arrows marked T indicate the outward forces due to the tension of the surrounding surface required to balance the contractile force of the surface inside the square. 258. Surface Energy. — If a strip of surface one centimeter wide is made one centimeter longer than at first, the work done is measured by the contractile force or surface tension T multiplied by the distance that it is drawn out. Thus if it is drawn out one centimeter, increasing the area of the surface by one square centimeter, the work expended is T ergs. This wOrk is stored up as energy of the surface and is expended when the surface contracts. Particles of liquid near the surface have thus more energy than particles in the interior, and the increase in energy in ergs per square centimeter of sur- face is numerically the same as the surface tension in dynes per linear centimeter. The following table gives some values of surface tensions at 20°C. Surface Tensions in Dynes per Cenlimeter FiQ. 139. Substance Air Water Mercury Water 73.5 539.0 34.3 24.5 17.6 412 Mercury Olive oil Alcohol Ether 412.0 20.6 335 166 PROPERTIES OF MATTER The tensions just given are for the interface between the sub- stance at the top of the column and the one at the side. 359. Variations of Surface Tension. — The surface tension of a liquid depends on temperature, in general being less with higher temperatures; it is also in case of water greatly affected by im- purities. Pour a Uttle water into a flat-bottomed porcelain tray, but not quite enough to cover the bottom. If a few drops of alcohol are added at any point, the water will rush away from that spot in every direction leaving the porcelain surface bare. If a drop of ether or alcohol is held near the surface of clean water on which lies some lycopodium powder, this will be dragged away from near the drop. The surface tension of the liquid is weak- ened by alcohol and even by the vapor of alcohol or ether and the surrounding uncontaminated liquid with its greater surface tension contracts and draws the other after it. In the same way are to be explained the lively movements that are noticed when minute fragments of camphor are dropped on clean water. The surface tension is weakened by the impurity at the first point of contact and as the liquid is drawn away from that point the camphor also moves, leaving a trail of contaminated surface behind it. 260. Pressure Due to Surface Tension. — a^ftA^^^^ft The contractile force of a curved surface pro- duces a pressure inward on the concave side. Let the figure represent a drop of water which Pig. 140. ^^ may imagine free from gravity. It will at once assume a truly spherical form and the liquid will be under pressure in consequence of the tension of the surface. To determine the amount of this pressure consider the drop as in two halves separated by the plane ab. All around the circum- ference of this section the surface tension is pulling the two halves together. The amount of this force will be 2TrrT if r is the radius of the drop and T the tension. But this force is balanced by the pressure of one-half of the drop against the other. Calling p the pressure per square cen- timeter, the total pressure is irr^p. But since these two forces balance we have 2TrrT — irr^p; SURFACE TENSION 167 and therefore V = 2T Notice that this pressure increases as the size of the drop dimin- ishes, and in a water drop one-hundredth of a millimeter in diame- ter it amounts to 150 grams per square centimeter. This same expression gives the pressure, due to surface tension, of the air or vapor inside of a bubble in a mass of liquid. In a soap bubble there is a thin film of the soap solution having two sur- faces to be considered. Each of these has contractile force and consequently the total pressure inside of the bubble is P 2T , 2T > where ri and r^ are the radii of the outer and inner surfaces of the bubble, respectively. Since these radii are practically equal, we have II. r P = FiQ. 141. — Contact angle. 361. Contact Angle. — When a clean plate of glass is dipped into water the liquid rises in a curve against the glass. The free surface of the water is here enlarged in spite of its contractile force by the expanding force of the surface of contact between the water and glass. This expanding force or negative tension is due to the great attraction between the water and glass, as ex- plained in paragraph 256. Let E represent the amount of this expansive force or negative tension of the surface between water and glass, and let T represent the tension of the water surface. Then it is clear that the liquid will rise until the upward and downward forces are in equilibrium; that is, until E = T cos x. The particular angle x at which equilibrium takes place is known as the contact angle for the two substances involved. In case of kerosene oil and glass E seems to be greater than T, and hence there can be no equiUbrium, the angle x becomes 0°, and still the edge of the oil creeps up. It is in this way that a film of oil spreads over the whole surface of a glass lamp. 168 PROPERTIES OF MATTER The case of mercury and glass is different, the merciiry curves down instead of up at the hne of contact and the angle x is about 140°, indicating that the surface between mercury and glass has a positive tension, or contractile force. 363. Rise in Capillary Tubes. — If a large glass tube, say, 2 in. in diameter, is introduced into water, the water rises around the edge on the inside until the weight of the water raised above the original level is just equal to the total upward pull of the contracting surface. This upward pull is equal to the tension per unit length multiplied by ,the inner circumference of the tube or 2TrrT, where r is the radius of the tube and T the surface tension. If small tubes are taken the weight of liquid required to balance this force cannot be secured without raising even the middle of the Hquid above the original level, and the smaller the tube the greater the height to which the Hquid will rise. Let h represent the average height of the column, then rV/i = the volume of liquid raised; and if d is its density, its weight in grams will be r^irhd and its weight in dynes r^irhdg, and this is equal to the sustaining force 2TrrT. 2TrT = r^irhdg, 2r f'< k_ h = 2T rdg Fig. 142. — Rise in capil- lary tube. Hence the height to which liquid rises in a capillary tube varies inversely as the radius of the tube, a relation known as Jurin's law. Of course if the contact angle is not zero, we must write r cos a; in the above formula instead of T. The pressure within the hquid in a capillary tube is less than the atmospheric pressure at all points above the level surface of the liquid in the open vessel, decreasing according to the hy- drostatic law, toward the top. The curved surface at the top exerts a back pressure against the atmosphere equal to the pres- sure of a column of hquid of the height h. The height at which a liquid will stand in a capillary tube is independent of its shape, depending only on the size of the tube at the point where the curved surface or meniscus stands. SURFACE TENSION 169 A liquid cannot rise and overflow the top of a capillary tube, however short it may be, for as it reaches the top the curvature of the surface changes, becoming less until its upward pressure is just balanced by the column of liquid below. The rise of oil in lamp wicks and of sap in vegetable fibers are familiar instances of capillary action. 363. Depression of Liquids in Tubes. — In cases where the contact angle is greater than 90°, as between mercury and glass, the surface of the liquid is roundeid upward in a small tube and the level of the liquid is depressed. In this case the surface being concave downward produces a downward pressure. Thus in a barometer the mercury column stands lower than it would normally do unless the tube is so large that the center of the column is sensibly flat. (See §192.) 364. Effect of Curvature of Surface. — In a conical tube a drop of water will be concave outward at both ends, but since the Fig. 143. DROP OF WATER IN TUBE Fig. 144. B C ^ Pig. 145. Fig. 146. smaller surface has the smaller radius of curvature, it will exert the greater pressure against the air and the drop will move toward the small end of the tube, while a drop of mercury which rounds outward at both ends will be driven toward the larger end (Fig. 143). Consequently a drop of mercury will be in stable equili- brium at a widening in a narrow tube, while a drop of water will seek the narrowest point (Fig. 144). If a capillary tube is connected at the bottom with a larger vessel of water (Fig. 145), when the water in the large vessel is at A, level with the top of the capillary tube, the surface of the water at E is flat. If the level is raised to B in the large vessel, the surface at the end of the capillary tube will round up, taking exactly the curvature necessary to balance the hydrostatic pres- 170 PROPERTIES OF MATTER sure due to the height of B above E. On the other hand, if the level is lowered to C the surface at E will become concave with a curvature that will give an upward pressure equal to that of a column of water from A to C. When a narrow glass tube is dipped in water and withdrawn a short column of water will be held in the lower end, the lower con- vex surface acting together with the upper concave one to sup- port the liquid. 365. Small Floating Bodies. — When a floating body is wet by a liquid, the liquid rises around it and drags it down by the weight of this raised mass. So a hydrometer with a glass stem around which the liquid rises will sink lower than it otherwise would. Around bodies which are not wetted the liquid curves down, and consequently the body is buoyed up by the weight of the liquid displaced in consequence of the curva- ture. For instance, if a clean needle is laid ~^^^'^~"^" carefully on water it will float. The liquid is bent down where it meets the needle and therefore the volume of water displaced is much greater than the volume of the needle itself and so the weight of the displaced water may be equal to the weight of the needle. 366. Attraction and Repulsion. — When the liquid rises around floating bodies they are drawn together as soon as they are near enough for the curvature of the surface due to one to affect the other. Notice how small floating pieces of wood or cork cling together as soon as they are wet, so also bubbles in a cup of cocoa cling together, and are also drawn to the sides of the cup where the surface curves up. If the cup is fllled to the brim and enough added to make the surface curve down slightly at the edges, the bubbles will at once rush away from the edge. Bodies which are not wetted and around which the surface curves down are also drawn together, but are driven away from bodies around which the surface curves up. 267. Explanation of Attraction of Floating Bodies. — When two small floating objects are both wetted the liquid rises higher be- tween them than it does on the outside, as shown in the upper part of figure 148, and since the pressure at any point in the liquid SURFACE TENSION 171 higher than the level surface is less than the atmospheric pressure, the pressure on the outside is greater and they are forced together. In the second case the liquid stands higher around the floating bodies on the outside than it does between them, and since the pressure in the liquid at points below the level surface is greater than the atmospheric pressure, they are pushed toward each other in this case also. But when the liquid wets one and not the other, the surface is lowered on the inside of the wetted one and raised on the inside of the other so that Fig. 148. in each case the pressure on the inside is greater ^^^^^^^^ than on the outside and the two are urged apart. 368. Soap Films. — Some most interesting illustrations of surface tension are found in the phenomena of soap films. When a loop of thread is. laid on a soap film formed in a wire ring and the film is broken inside the loop, the latter will be drawn into an exact circle, for it is pulled equally in every direction by the con- tracting film. And this circular loop may be moved from one part of the film to another without changing shape, showing that the tension does not depend on the width of the film. Fio. 149. — ^Loop in soap film. Fio. 150. — Cylindrical bubble. If wire frames forming the outlines of cube, tetrahedron, or cylinder are dipped into a soap solution and then carefully with- drawn, symmetrical figures of great beauty are formed by the films. By blowing a bubble between two rings and then drawing the rings apart until it becomes cylindrical, the ends will be seen to bulge out, showing that the air within is under pressure, and 172 PROPERTIES OF MATTER the radius of curvature of the spherical ends will be found to be twice that of the cyhndrical surface. Why is this? 369. Equilibrium of Cylindrical Film and Formation of Drops. — If such a cylindrical film is short relative to its length, it is in stable equiUbrium; but if it is longer than its circumference, it is un- stable and will collapse at one end and bulge out at the other because by so doing the surface will become smaller. This will result in its breaking into two bubbles, a large and a small one (Fig. 151), with a very small one between them; a result which is of interest, because it illustrates why a thin jet of water breaks up into drops. Imagine a thin cylindrical jet escaping from a vessel of water. It is under pressure sidewise from the contrac- Fia. 151. — Breaking of unstable cylindrical bubble. o o o o O o o O Fig. 152. — Jet breaking into drops. tile force of the surface, but since it is long enough to be unstable it yields, becoming first undulatory, as shown in the upper part of the figure, then finally breaking up into alternate big and little drops which are elongated when first separated and vibrate from this to the flattened form, finally settling down to spherical shape. These details were first made out by Savart; they may best be studied from photographs made when the stream is instantane- ously illuminated by an electric spark. Problems 1. Find the diameter of a drop of water in which the pressure is twice the atmospheric pressure on its surface, taking surface tension o/ water as 74 dynes per cm. KINETIC THEORY 173 2. Two flat glass plates, 10 X 10 cm., placed face to face in a vertical posi- tion and separated only by bits of tinfoil, have the lower edges immersed in water which rises and fills the space between the plates. Find how much less the average pressure is between the plates than on the outside and thence find the force with which they are pressed together. 3. In case of a soap bubble 6 cm. in diameter, how much greater is the pres- sure within the bubble than without? Take surface tension 70 dynes per cm. Give answer in dynes and also in grams per sq. cm. 4. A glass hydrometer having a stem 8 mm. in diameter floats in water. With what force due to surface tension of water wetting its stem is it pulled downward 7 6. How much deeper will the hydrometer in the last problem sink than if it had floated in a liquid of the same density that did not rise on its stem? Ans. 3.8 mm. Kinetic Theory 270. Kinetic Theory of Gases. — It was shown by Daniel Bernoulli (1700-1782) that the pressure of a gas could be best explained as due to the impacts of its molecules against each other and the walls of the vessel. In recent years Clausius and Maxwell especially have developed this theory, showing that the characteristic properties of gases are in harmony with it, and it is now generally accepted as giving a true conception of their structure. In this theory it is assumedf|)that the molecules of gas are constantly striking against each other or the walls of the vessel and rebounding. (^^"JVhen two molecules approach each other, at a certain distance they experience a repulsive force which increases as they come nearer until further approach is stopped by the force and they are repelled apart or rebound. The distance between their centers when they are nearest together and about to re- i bound is called the diameter of the molecule. The molecule on rebounding soon gets out of the influence of the other and then flies in a straight line until it meets another from which it rebounds, either directly or glancing off sidewise, changing both its own motion and that of the molecule against which it strikes, and so it continues its path zig-zagging about. The average distance that a molecule travels between two successive impacts is called its mean free path. The velocity of a particular molecule is doubtless changed at every impact not only in direction, but in amount, sometimes increased and sometimes diminished, but A ^ /)i_ "* • / 174 PROPERTIES OF MATTER there is no loss of energy on the whole; whatever one molecule loses the one impacting against it gains. The average kinetic energy of the molecule, and consequently its average velocity, remains unchanged unless energy is in some way commimicated to the gas from outside. 371. Pressure of a Gas. — An expression for the pressure of a gas may be deduced in an elementary way bj(^eglecting the size of the molecules and their impacts against each other and con- sidering each molecule as rebounding only from the walls of the vessel. Imagine a cubical vessel one centimeter each way, and for simplicity conceive the whole number of molecules iV con- tained in it to be divided into three equal groups, one group rebounding between the sides AD and BC and producing pressure against them, the other groups being directed against the other pairs of sides. If Y is the velocity of a molecule, it will strike against the sid&J^C once every time thjat it. travels across the vessel and back again^ STffistance of 2 cms. FiQ. 153. The number of impacts per second of one molecule Y against the side BC will therefore be-p* The momentum of the molecule before •impact is mY toward the side, after impact it is mY in the oppositeyirection ; the total change in momentum of the molecule in one impact is 2mY where m is the mass of the Y . molecule. But there are ^ unpacts per second, so that each molecule in rebounding from one side experiences a change of Y momentum per second 2mY X w= mV^ and since the whole N number of molecules impacting agamst the side BC is -^r the total change in momentum produced by that side in one second is }^NmY^, and this is, therefore, the force against the side. But since the side BC has unit area^the force against it equals the pressure, hence ^ V = ViNmY-" (1) where p represents the pressure of the gas. Now, the product Nm is the total mass of gas in one cubic centi- meter, or its density d, and hence : I = VsV (2) KINETIC THEORY 175 Maxwell's Law. — It has been shown on mechanical grounds, by Maxwell and others, that when two masses of gas are at the same temperature, the average kinetic energy of a molecule of the one is equal to the average kinetic energy of a molecule of the other. That is Hm.V,^ = Hm^Fa^ (3) where mi and m2 are the masses of the molecules of the two gases and Vi and Vi are their velocities. Boyle's Law. — According to the lav/ just stated the kinetic energy of the molecules in a mass of gas is determined by its temperature, and hence V changes only when the temperature of the gas changes. Formula (2) above, then, is in agreement /swith Boyle's law and expresses the fact that the pressure of a gas4s proportional to its density when the temperature is constant. This formula may be used to calculate the average molecular velocities, giving as follows: Mean Velocity of Molecules in Gases at 0°C. Hydrogen 1843 meters per sec. Nitrogen 492 " " " Oxygen 461 Carbon dioxide 392 Avogadro's Law. — When two different gases have the same -pressure we have by equation (1) HNm.Vx'' = VsN^m^Vi'' (4) If the two masses of gas are also at the same temperature, we have by (3) i^miFi^ = M'WjFa'' (5) and combining the two equations we find iVi = iVa; that is, the number of molecules per cubic centimeter is the same in all gases at the same temperature and pressure. This is known as Avogadro's law, and was reached by him from purely chemical considerations. 273. Molecular Magnitudes. — The number of molecules in a gas at one atmosphere pressure and 0°C. is found to be 26.5 X 10'^ 176 PROPERTIES OF MATTER per cubic centimeter, or 434 million million million per cubic inch. Several different methods lead to approximately this re- sult, but the most accm-ate determination is by an electrical method to be explained later (§614). The mean free path or average distance that a molecule travels before striking against another may be deduced by the kinetic theory when the viscosity of the gas is known. Also by several different lines of reasoning that cannot here be discussed the effective diameters of gaseous molecules have been approximately determined. Molecular Magnitudes in Gases at Atmospheric Pressure and 0"C. Gas Mean free path Diameter of mole- cule in millionth3 of a millimeter Number per c.c. Nitrogen Hydrogen Carbon dioxide. . .0000098 mm. .0000185 mm. . 0000068 mm. 1 0.28 0.21 0.37 26.5 X lOi! X 10=^ X 10« (( H 11 n ti K li (. The numbers representing the diameters must be regarded as only approximations to the truth, but which doubtless ex- press the true order of magnitude, being probably neither 10 times too large nor too small. In case of nitrogen at "atmospheric pressure the mean free path is about 13 times the diameter of the molecule. More than one million such molecules in a row would be required to make a length of 1 mm. Lord Kelvin has estimated that if a drop of water were magnified to the size of the earth, "the structure of the mass would then be coarser than that of a heap of fine shot, but probably not so coarse as that of a heap of cricket balls." 273. Brownian Movement. — The English botanist Brown in 1827 on observing with the microscope very fine particles held in suspense in a mass of water, discovered they were in constant irregular motion, and the smaller the particle the more lively was the motion observed. It is a spontaneous motion that never ceases, and is believed to be caused by the incessant motion of the molecules of the liquid, which bombard the particle on all sides driving it hither and thither. KINETIC THEORY 177 The French physicist Perrin has made a careful study of this phenomenon using an emulsion in water of exceedingly fine grains of mastic. He finds by exact measurement of the dis- tribution of the grains and the amount of their motions that they distribute themselves just as should be expected from the kinetic theory, and even deduces by inference from his measurements the number of molecules in a cubic centimeter of gas under stand- ard conditions, finding 30.5 X 10'^, in good agreement with determinations by other methods. References _Soap Bubbles, C. V. Boys. Brownian Movement and Molecular Reality, J. Perrin, translated by F. SODDY. WAVE MOTION AND SOUND Surface Waves 374. Wave Motion. — The phenomena of wave motion are perhaps most easily grasped from the study of water waves. Looking upon a series of waves coming across a smooth lake from a passing vessel, we notice the steady advance of a definite form of motion, having a velocity which is quite independent of that of the vessel, and carrying energy, for the water in a wave is in motion and has kinetic energy. The water over which the waves have passed is left calm, and if a floating cork is observed it will be seen to rise and move forward with the crest of the wave, then sink and move back- ward in the trough, repeating the motion with the next wave, and coming to rest in its original position when the disturbance has passed. The motion of the cork, which is that of the water in which it lies, shows that the wave does not carry along with it a mass of water, but that the motion and energy are passed along from one mass of liquid to the next. FiQ7i54.-Origin'of a water wave. ^ wave may be defined as a form or configuration of motion advancing with a finite velocity through a medium. By means of waves energy is transmitted, being passed along from one part of a medium to the next by the interaction of adjoin- ing parts. 375. Origin of a Series of Water Waves. — When a stone is dropped into a smooth pond, water is carried down by the motion of the stone, as shown at .4, figure 154, and also thrust out in a ridge at B; beyond C the surface is undisturbed. Then it begins to rush back toward A from the surrounding parts and B sinks, at the same time the forward part of the wave between B and C is urged forward, in part by its forward momentum 178 WATER WAVES 179 and in part by the pressure. The rush toward A does not stop when A has risen to the level, but continues until the kinetic energy of the flow toward A is spent in heaping up water at A at the expense of the hollow at B, as shown in 2, figure 154. Thus there are set up oscillations at A, the energy of which is gradually spent in sending out waves. Each, wave takes with it a certain definite quantity of energy which remains with it as it advances. 376. Motion in a Water Wave. — In a series of water waves the motion of the particles is as shown in figure 155, each particle moving around in a closed curve, which in the simplest form of wave is a circle. In the diagram are shown the paths of motion of nine water particles which were originally equidistant and one- eighth of the whole wave length apart. Each moves clockwise in a circle and all with the same uniform velocity; but while particle a is at the top of its path, b is back of the top by one- FlG. 155. — Motion in a simple oscillatory water wave. eighth of a circumference. The position of each in its path is called the phase of its motion; a and i are said to be in the same phase, while the phase of e is opposite to that of a and i; also c and g are opposite in phase because they are a half-circumference apart in their motion. Each particle in the diagram differs in phase from the next one by one-eighth of a complete revolution. Of course all the par- ticles which were between a and b in the undisturbed condition of the surface will still be between a and b and will have interme- diate phases, thus forming the surface of the wave between those points. It will be seen that the wave is advancing in the direction of the long arrow at the top, for an eighth of a period later b will be at the top and a will have passed beyond, and the position of the crest of the wave will be as shown by the dotted line. In the time of a complete revolution or period of the particles the wave will ad- vance from a to i and a new crest will have come to a. 180 WAVE MOTION AND SOUND The wave length is the distance between particles in the same phase of motion, in this case from a to i. The amplitude of the wave is the radius of the circles, which is the distance that a particle is displaced from its equihbrium po- sition ; it is one-half the vertical height of the wave from trough to crest. The velocity of a wave is the velocity with which a particular phase of motion moves along; for example, it is the velocity with which the crest of the wave moves along. Since a wave travels the whole wave length X, in the period of revolution of a particle P, we have where V represents the velocity of the wave. The frequency of a series of waves is the number passing a par- ticular point per second. If n waves pass per second there must DP n complete waves in the distance V traversed by the waves in jne second, or V = n\. The velocity of the particle in its circular orbit must not be confused with the velocity of the wave. It is always less than the wave velocity and depends on the amplitude of the wave. A mechanical wave model devised by Lyman exhibits the motions in figure 165 and admirably illustrates the motion in water waves. 377. Decrease of Amplitude with Depth. — In consequence of their difference of phase, the particles a and b, near the crest of Fig. 156. — Amplitude of wave decreases with increased depth. the wave, are nearer together than their positions of equihbrium, while near the trough of the wave e and / are farther apart. But since water is practically incompressible its volume does not change and a little mass of water which was originally of cubical form must become elongated vertically near the crest and hori- zontally near the trough as shown in figure 156. The water par- ticles on the lower surface of the cube must, therefore, have less WATER WAVES 181 amplitude of motion than those at the surface, as is evident from the figure. At a depth- equal to the wave length the amplitude is only /^35 of that at the surface. 378. Velocity of Oscillatory Waves. — The type of water wave described is known as a simple oscillatory wave and the varieties of form observed in ocean waves are due to a number of such waves superposed. In a smooth pond it may easily be observed that two independent series of waves may cross each other, producing a complicated resultant motion; but after they have passed, each is found to have kept its original motion undis- turbed. It is shown in treatises on hydrodynamics that the velocity of a simple oscillatory surface wave is expressed by the formula \27r From which the following velocities are found : Wave length Velocity 75 ft. 13 . 3 miles per hour 300 ft. 26 . 6 miles per hour 1200 ft. 53 . 2 miles per hour From this it is clear that a group of waves, as we see them in the ocean, resulting from the superposition of a variety of waves of different lengths, must continually change in form, as the com- ponent simple waves travel with different velocities. 279. Ripples. — The formula just given for the velocity of a wave assumes that forces due to the weight of the liquid are the only ones involved. But the surface tension of the liquid also plays a part, though it is entirely insignificant except in very short waves or ripples. The complete formula for the velocity of a wave is '4 ^ , 27rr 27r "•" \d' where T represents the surface tension of the liquid and d its density. The effect of surface tension is to increase the velocity of shorter waves. When water waves are about 17 mm. long their velocity is a minimum, being 23.3 cms. per second. Longer waves travel faster because gravitation force predominates, while in shorter waves surface tension has the principal effect. 182 WAVE MOTION AND SOUND COMPKESSIONAL WaVES 380. cfompressional Waves. — Water waves of the type just considered are surface waves, and can only exist at the surface of a medium. But the kind of wave now to be studied can travel in every direction through an elastic medium. Consider the model shown in figure 157, which represents a series of equal masses resting in a frictionless groove and con- nected by springs. If the first mass is moved toward the second, the latter will move because the spring between the two is com- pressed. But it will not begin to move until after the first mass has approached it; for if the two moved exactly together there would be no compression of the spring between them, and consequently no force exerted on the second mass to move it. As the sec- ond mass moves forward there is compression of the second spring, followed by motion of the third mass, and so on, the masses being set in motion one after the other as the wave of compression reaches spring after spring. So also if the first mass is drawn ^^~\^2^^^■^^ away from the others, the first spring ^^ ^-^ ^-^ ^-^ ^-^ is stretched, causing motion of the Fio. 157 —Illustrating elasticity ggcond mass which stretches the and inertia of medium. . ... second sprmg. The motion is there- fore communicated through the whole series as a wave accompanied by stretching or expansion of the springs. Such a wave of expansion or compression is set up whenever a material object is set in motion or brought to rest; for all bodies may be considered as made up of massive particles in elastic equilibrium with each other, like the balls and springs in the diagram. Thus, when a chair is lifted, a wave of expansion runs down through it, and when it is set on the floor a wave of compression runs up. 381. Newton's Formula for Velocity of a Compressional Wave. — The velocity of such a wave depends on the elasticity and den- sity of the medium. Recurring to the illustration, it is evident that making the springs between the balls stiffer will increase the speed with which the motion will be communicated from one ball to the next, while if the masses of the balls are made greater the effect will be to make the speed of the wave less. It was proved by Newton from the principles of mechanics that COMPRESSIONAL WAVES 183 the velocity of a wave of compression or expansion in a medium of which the volume elasticity is E and the density d is expressed by the formula, 4- A simple proof of this formula will be found on page 683. 282. Motion in a Series of Compressional Waves. — If the first of the series of balls represented in figure 157 is made to oscillate regularly backward and forward, now moving toward the second ball and now away from it, a series of waves will be sent along the row of balls, alternately waves of compression and expansion; and each ball will oscillate just as the first one does, though the second will always be in a phase of motion a little be- hind the first, the third will lag behind the second and so on. This is precisely the kind of motion which is set up in air by a tuning-fork or other rapidly vibrating body, and which excites i- 12 3 4 5 6 7 8 9 10 n 72 fiG. 158. — Motion of particles in a sound wave. in our ears the sensation of sound. Such waves in air are accord- ingly known as sound waves. The details of the motion in a sound wave may be understood from figure 158. The diagram illustrates the relative phases of motion of a series of particles in the wave one-eighth of a wave length apart. Each particle is shown as a black dot on a short straight line which rep- resents the path in which it oscillates, the center of the line being its equilibrium position. Suppose the first particle is made to oscillate in simple harmonic motion, then that will be the mode of vibration of all the particles, and each will move back and for- ward keeping vertically under an imaginary companion particle that is supposed to move with uniform velocity around in the corresponding circle shown in the diagram. It will be seen from the positions of the companion particles in the circles that, taken in the order in which they are numbered, 184 WAVE MOTION AND SOUND each is one-eighth of a complete vibration behind the phase of the preceding particle; indeed, the associated circles are only used to show the relative phases of the numbered particles below, which represent actual particles in the medium. Particle 1 is at the end of its path, while the second particle is moving toward the end and will be there an eighth of a period later, when 3 will be in the phase now shown by 2, and so on; therefore the wave will have moved forward in the direction of the long arrow underneath. It will be seen that particles 1 and 9 are in the same phase, and accordingly the distance between them is the wave length. The particle at 7 is in the center of a condensed region, where the particles are closer together than normal, while those at 3 and 11 are the centers of rarefied or expanded regions. In the condensed region the particles are moving forward in the di- rection in which the wave is advancing; in the rarefied region they are moving opposite to the wave. There are intermediate points where H^-t+^h^t-f /? c /? Fig. 159. — Motion of air layers in* sound wave. the medium is neither condensed nor rarefied where the particles are for the instant at rest at the end of their paths of vibration, as at 1, 5, and 9. It must not be forgotten that each of the particles considered is only one of a layer all vibrating in the same way. This is rep- resented in figure 159, where the dots of the previous diagram are replaced by heavy lines which represent successive layers of par- ticles, differing in phase by one-eighth of a period. The small arrows indicate the velocities of the layers at the given instant, and the instantaneous position of the regions of condensation and rarefaction are marked by the letters C and R, respectively. To sum up, in a compressional wave the particles vibrate longitudinally, or back and forth in the direction of advance of the wave, and there is a progressive change in phase, in conse- quence of which alternate regions of compression and rarefaction are produced. SOUND 185 The amplitude in such a wave is the distance that a particle oscillates on each side of its equilibrium position, or half the whole distance through which it vibrates. 283. Illustration. — The propagation of a wave of compression or rarefaction may be very well shown in a regular spring a meter and a half long which is supported by threads in a horizontal position, as shown in figure 160. The turns of the spring should be rather large and it should be of such a stiffness that a wave will take a second or two to travel its length. Fig. 160. — Spring wave model. SOTIND 384. Sound Communicated by Waves. — There are three principal evidences that sound is communicated by compressional waves through material bodies. First, sound is not communi- cated through a vacuum; second, the motions of sounding bodies are such as might be expected to set up compressional waves; and, third, the observed velocity of sound is the same as that of compressional waves, both in air and in other media. 385. Sound Requires a Material Medium. — Place an alarm bell rung by clockwork under the receiver of an air pump, as in figure 161, so that it may rest on a mass of soft cotton, or is other- wise supported so that no vibrations can be transmitted through its supports to the plate of the air pump. When the air is ex- hausted from the receiver the bell is no longer heard, however vigorously it may be ringing. Sound waves, therefore, do not pass, through a vacuum, they require a material medium. 186 WAVE MOTION AND SOUND 386. Sound Originates in Vibrating Bodies. — All sources of sound are vibrating bodies capable of setting up air vibrations. A brass plate supported at the center and covered with sand if set in vibration by a bow may be made to give out a variety of differ- ent sounds, but in each case there is a characteristic arrangement of the sand showing that a particular mode of vibration of the plate corresponds to each sound. (See Fig. 199.) Tuning-forks are set in vibration by being struck, strings by being bowed, the vibrations of the string being evident to the eye or causing a buzz- ing sound when the string is touched with a piece of paper. In reed instruments the metal tongue of the reed vibrates strongly when sound- ing, and even in flutes and organ pipes it may easily be shown that the air is set in strong vibration. 387. Velocity of Sound. — The veloc- ity of sound in air was determined by two Dutch observers, Moll and Van Beck, in 1823, by timing the interval between seeing the flash of the dis- charge of a distant cannon and hear- ing its report. Cannons were set up on two hills nearly 11 miles apart, and by observ- ing alternately first from one hill and then from the other the observers sought to eliminate the influence of any air currents which might exist. At the same time the temperature of the air was observed at a number of points between the two stations. The velocity was thus found to be 1093 ft. or 333 meters per second at 0°C. Regnault (1810-1878) conducted an extensive series of inves- tigations on the velocity of sound in the Paris water mains, which afforded large tubes free from wind disturbance and at a uniform temperature. He made use of an automatic apparatus by which the instant of discharge of a pistol was recorded electrically on the rotating drum of a chronograph, while the arrival of the sound at the distant station, where it caused a thin stretched membrane to vibrate, was electrically recorded on the same drum. By these Fig. 161. — Bell in vacuo. SOUND 187 experiments he found that the velocity was influenced by the size of the pipes to a small extent, and also that very intense sounds traveled sUghtly faster than feebler ones. He also conducted experiments in the open air, using the same recording apparatus, and found the velocity of sound in dry air at 0°C. to be 330.6 meters per second. Bosscha determined the velocity of sound by causing two little hammers to give simultaneous taps at regular intervals, the fre- quency of the taps being determined by a pendulum which made electrical connections at every swing. If one of the sounding instruments is placed beside the observer while the other is moved away, the taps are no longer heard simultaneously, but those from the more distant one come later; if moved far enough apart so that the sound from the tap of the distant hammer reaches the ear at the same instant as the next succeeding tap of the nearer hammer, the two are again heard simultaneouly, and the distance between the two sounders divided by the time interval between the taps gives the velocity of sound. 288. Velocity of Sound in Water and in Solids and Gases. — Colladon and Sturm measured the velocity of sound in the water of Lake Geneva by causing a bell to sound under water and using as a receiving instrument a sort of ear trumpet with the outer end closed by a rubber diaphragm and placed beneath the surface of the lake. The velocity was found to be 1435 meters per second. In solids the velocity of sound is usually measured by the longitudinal vibrations of rods or wires as explained later, §342. The velocity of sound in various gases and vapors has been determined by comparison with that in air by the method of Kundt, §335. The velocities of sound in some common media are given in the following table. The velocity of sound in wood and steel is so great that a person standing near one end of a long beam or rail that'is struck at the farther end hears two sounds in quick succes- sion, first that transmitted by the solid and then that through air. 188 WAVE MOTION AND SOUND Velocities of Sound Medium Meters per sec. Ft. per sec. Air at 0°C. and 76 cms. pressure { t> u 330.6 331.6 mean Hydrogen at 0°C. and 76 cms. pressure Carbon dioxide at 0°C. and 76 cms. pressure. . . Water at 13°C 331.1 1,286.0 261.0 1,437.0 3,600.0 4,950.0 5,000.0 3,300.0 1,086.7 4,219.0 856.0 4,715.0 11,800.0 Iron rod 16,240.0 16,410.0 10,830.0 289. Velocity of Compressional Waves. — The velocity of a compressional wave in air may be readily calculated by Newton's formula d V = It was shown by Newton that the elasticity of a gas at constant temperature is equal to its pressure (see §241). But on substitu- ting pressure for elasticity in the above formula the calculated velocity was found to be too small. Laplace pointed out that though the average temperature of air is not changed by the passage of sound waves, yet in the com- pressed part of a wave the air is heated for the instant, and where it is rarefied there is cooling, and that these changes take place so rapidly that there is no time for heat to flow from one part to another, so that the air is practically in an adiabatic condition (§241). The effect of heating during compression is to resist the compression, and cooling during expansion acts to oppose the ex- pansion, the effective elasticity in this case is therefore increased and in case of air has been found to be 1.40 times as great as if the temperature had remained constant. The formula thus becomes for a gas Uke air, '4 1.4. SOUND 189 Substituting the values for air at normal temperature and pres- sure, and expressing both pressure and density in C. G. S. units, we have 4 f7Q X 13.6 X 980 X 1.40 oo ion — = 33,120 cms. per sec, 0.001293 which is in good agreement with the velocity of sound as found by experiment. In case also of solids and liquids the results obtained by the formula agree with velocities obtained by direct experiment. The elasticities of these substances are so much greater than that of air that the velocities of sound in them are large in spite of their great densities. Thus in water the elasticity or ratio of pressure increase to cor- responding decrease in volume is, in C. G. S. units, 76 X 13.6 X 980 oi«-^inioj QQQQQ^y = 2.16X101° dynes per sq. cm. or 15,230 times that of air, while it has only 773 times the density of air. 290. Influence of Temperature and Pressure on Sound Veloc- ity in Air. — From the formula in the preceding paragraph it is clear that the velacity-oisaund in air is independent of the pres- sure., for when the pressure is increased the density increases in the same proportion, by Boyle's law, and the ratio j remams constant, and consequently the velocity is constant so long as the temperature is not changed. But if the temperature is raised, pressure being constant, the density diminishes and the ratio -% increases. Hence the velocity of sound in air is increased J^46, or about 2 ft. or 0.60 meters per second per degree Centigrade rise in temperature. 391. Influence of Pitch on Velocity of Sound. — It may be easily noticed that the notes of music coming from a distant band are heard in the same relation to each other as if the band were near. There is no confusion of the melody such as would result if high-pitched sounds traveled faster or slower than low ones. Regnault made careful observations on this point and concluded that the^velqcity of sound is the same whatever the pitch may be. 190 WAVE MOTION AND SOUND It will be shown later that the pitch of a sound depends upor wave length, hence we conclude that the velocity of sound is the same for all wave lengths. Reflection and Refbaction op Waves 39a. Reflection of Water Waves. — When a water wave meets an immovable obstacle it is turned back or reflected. Since the obstacle does not move, it cannot receive energy from the incident wave, and therefore the reflected wave carries the energy away. Each point of the obstacle reacts against the waves which meet it and so produces a periodic disturbance and may be regarded as a center from which waves are sent out. The reflected wave as a whole is the resultant of these little waves coming from each point of the obstacle. Suppose a wave from a center 0, figure 162, meets a straight wall BC. When in the position AED the disturbance has just reached the wall at E and is about starting back. By the time the wave at A has advanced to B and at D has reached C, the part of the wave reflected at E will have returned an equal distance to F. If the wall had not been there the wave would have advanced to the position of the dotted line BGC, but since after reflection it has the same velocity as before, the reflected wave Fig. 162. — Reflection of waves, will at each point have gone back from the wall as far as it would have passed the line BC if the wall had not been there. The front of <' the returning wave BFC has therefore the same curvature as BGC if the wall is flat. The returning wave is therefore circular having its center at a point 0' which is as far back of the wall as the center is in front of it; and the line 00' is at right angles to the wall. Another method of looking at this subject is interesting. The effect of the vertical wall is to oppose any forward or backward motion of the water particles next to it without interfering with vertical motions. Let us now imagine the wall removed and that whenever a wave starts from O an REFLECTION 191 exactly equal wave sets out from 0'. The waves will meet along the line BC and the forward or backward movement due to the one will be exactly balanced by that of the other, while their vertical movements will be added. There results, therefore, an up-and-down oscillation along the line BC exactly as if the wall were there. On each side of the line BC there will be waves coming toward the line and others going back from it exactly as if reflected from it. And indeed they may be properly regarded as reflected, for there is no transfer of energy across the line BC because there is no forward or backward motion across that line, and if a thin wall were slipped in along BC separating the two sys- tems of waves the motion would not be changed on either side. 393. Angle of Reflection. — When a wave front meets a re- flecting surface obliquely, the direction of the wav« front and its direction of propagation are changed as shown in figure 163. At Wi is shown a portion of a wave front approaching P where it is reflected, afterward ad- vancing as shown at Wi as if it came from 0'. The angle i between the di- rection of advance of the incident wave and the normal to the surface is called the angle of incidence, while the angle r between the direction in which the j-jq, i63. reflected wave moves and the normal N is called the angle of reflection. The angle of reflection is equal to the angle of incidence. For the angles a and h are clearly equal and the angle i is equal to a, and r is equal to h, since the lines 00' and NP are parallel. It is interesting to see how the reflected wave may be re- garded as the resultant of little waves coming back from each point of the reflecting surface. Let AB (Fig. 164) be a wave front meeting the reflecting surface ^C at A. The disturbance at A causes a little circular wave to go back which will have a radius AD equal to BC by the time that the wave front at B has reached C, since the velocity of the returning wave is the same as that of the advancing one. So also the circular wavelet starting backward from F when the advancing wave has reached the position FG, will have a radius FK equal to GC when the wave front at B reaches C From each point in succession of the reflecting surface between A and C these elementary waves spread out just as from A and 192 WAVE MOTION AND SOUND mm/m///m//////m9W Fig. 164. F, in arcs of circles whose centers lie on the line AC. The envelope of these circles is the line DC which is tangent to all of them. All these elementary waves therefore act together and combine to produce a new wave front along the line DC. Since AD is equal to BC and is at right angles to the tangent DC, just as BC is at right angles to AB, the two triangles ADC and ABC are equal and the angle of incidence BAC is equal to .the angle of reflec- tion DC A. While the elementary waves are conceived as spreading out in circles in every direction, they produce an effect only along the wave front DC where all act together, for it may be shown that they interfere (§319) with each other in other directions, such as off to one side, unless the reflecting surface is very small. If the distance AC is not more than the length of a wave, regular reflection will not take place, but the reflected wave will spread out in all directions as if the reflecting surface were the center of disturbance. 294. Refraction of Waves. — When waves pass from one medium into another there is generally a change in velocity, which causes the direction of the wave to change when it meets the surface of separation obliquely. Let AB represent the ad- vancing wave front, meeting at A the second medium where the velocity is less. While the wave front advances from B to D in the first medium, the wave from A will have gone a less distance AC in the second medium in consequence of the smaller velocity. The new wave front will be CD, tangent to the elementary waves from A and from all points between A and D. The direction of advance of the wave is changed toward the perpendicular, from BD to DE. An intermediate position of the wave, where part is in one medium and part in the other shows a sharp bend at G. If the velocity in the second medium were greater than that in the first the waves would become by refraction more oblique Pig. 165. REFLECTION AND REFRACTION 193 to the surface of separation instead of less oblique as in the case illustrated. The law of refraction will be more fully discussed in connec- tion with the study of light. The refraction most commonly noted in water waves is when they run obliquely into shoal water near shore, where their velocity is retarded. The effect is to swing the wave front around more nearly parallel with the shore. 395. Reflection of Sound. — In the reflection of sound the same principles apply as in case of water waves. Sound waves reflected from a large flat surface appear to come from a point as far behind the surface as the sounding body is in front of it. Echoes from buildings, cliffs, and even from a wooded hillside are familiar examples of the reflection of sound. If there are a series of cliffs or shoulders of rock at different distances multiple echoes are heard. A pistol shot from a boat on a smooth lake comes as a single sharp sound followed by faint echoes from the distant shores, but if the water is rough the shot is followed by a reverberating roar as the sound comes back reflected from wave after wave. By means of a large parabolic mirror the tick of a watch placed at its focus is reflected so that it may be heard 50 ft. f^way by an observer having his ear at the point on which the rsflected waves are converged. The proper position to hold the ear may be found by observing where the image is formed of a light placed at the focus of the mirror, showing that the law of reflection is the same for sound as for light. A watch is used in this experiment because the waves of sound which it gives out are so short, even relative to the size of the mirror, that the law of regular reflection holds. In rooms with arched ceilings focal points may sometimes be found such that sounds going out from one point are converged toward the other. A person holding his ear at one point can hear the slightest whisper coming from the other. There are other kinds of whispering galleries in which the effect depends not on regular reflection, but on the gradual deflection of a wave as it runs along a smooth surface. 296. Reflection and Refraction of Sound Waves. — Whenever sound waves meet the surface between two media usually both 194 WAVE MOTION AND SOUND reflection and refraction take place. If there is a very great change of density and elasticity most of the energy goes into the reflected wave, and the refraction will be slight. On the other hand, if the two media, like two strata of air at different tempera- tures, differ only slightly in their properties, most of the energy will be transmitted in the refracted waves into the second medium and but little will be reflected back from the surface. If a lenticular bag of thin rubber is filled with carbonic acid gas (CO2) in which sound A^-ll \>^c travels more slowly than in air, the sound from the ticks of a watch will be concen- trated at a focus conjugate to the position of the watch, just as light is converged by a lens of glass which retards its waves. For in passing through such a lens the middle part of the wave AB is more retarded than its edges, so that it is transformed into the form CD which is concave toward the ear at E. 297. Effect on Fog Signals. — On account of reflection and refraction from strata of air of tures, layers, the sound of a fog horn may be entirely un- heard by a vessel near the shore and in danger. If the lower portion of a horizon- tally moving sound wave is in warmer air than the upper part it will travel faster and Fig. 166. different tempera- or from foggy Fig. 167. — Sound waves changed by wind. to cause the wave front change its direction and may even cause it to curve upward. So also currents of air, causing one part of a wave front to move faster than another will change its form and consequently the direction in which it advances. Thus the observer at A (Fig. 167) might not hear the church bell when the air was still because of the screening effect of the ridge, but if a breeze were blowing which being stronger above would carry the upper SOUND CHARACTERISTICS 195 part of the waves along faster than their lower part, the wave fronts would be tipped over so that they might come down to A causing the bell to be heard. Sound Characteristics 398. Sound and Noise. — Sounds that have a sustained and simple character and do not seem to be a mixture of various different sounds may be called tones or musical sounds. Abrupt and sudden sounds that do not last- long enough to convey any idea of musical pitch, or mixtures of discordant sounds are noises^ 399. Tone Characteristics. — A musical sound or tone has intensity, pitch, and quality or timbre, and each of these depends upon a physical property of the sound wave. The intensity of a sound depends upon the amplitude or the energy of the vibration, the pitch depends on the frequency of the waves, and the quality depends on the particular manner of vibration of the particles. 300. Intensity. — Intensity of a sound, as the term is ordi- narily used, refers to the strength of the sensation excited by the sound wave. It depends upon the amplitude of vibration in the wave, for increasing the amplitude of vibration of the sound- ing body increases the loudness of the sound. But one is not simply 'proportional to the other, and if two sounds of different pitches are equally intense, it by no means follows that the ampli- tudes of vibration are the same. Usually the higher pitched tone will be more intense for a given amplitude than one of lower pitch. The term intensity as applied in physics to sound waves refers to the energy of the motion and is measured by the energy transmitted per second through i square centimeter of surface. Just how the intensity of the sensation is related to the energy of the sound vibration is a question for the psychologist. The energy per cubic centimeter in a sound wave depends on the density of the medium d, the square of the frequency of vibration n^, and the square of the amplitude a^, and is ex- pressed by the formula E = 2TrHn^a\ It has been found by Lord Rayleigh that when the amplitude of vibration of the air particles is as small as one-millionth of a 196 WAVE MOTION AND SOUND Fig. 168. millimeter the sound is barely audible, while an amplitude as great as a millimeter would occur only in the very loudest sounds. 301. Decrease in Intensity with Distance. — When sound waves can spread out in every direction from a sounding body forming a series of spherical waves the intensity varies inversely as the square of the distance from the source. For the same amount of energy is transmitted across every spherical surface having its center at the source of sound, and the larger the surface of such a sphere the smaller will be the energy trans- mitted per unit surface. The spherical wave front at C, for example, will have four times the area of a sphere at one-half its distance from the source as at A, consequently the energy transmitted per square centi- meter of surface will be only one-fourth as great at C as s,i A. The energy or intensity therefore varies inversely as the square of the distance from the source. Of course if the wave is prevented from spreading out, as in a speaking tube or when the source of sound is near the surface of a smooth lake, this law does not hold. 302. Speaking Tubes and Ear Trumpets. — The ordinary speaking tubes connecting distant rooms in buildings depend not on regular reflection, but on the fact that the air particles next the inner surface of the tube vibrate most easily parallel with the surface, this causes the direction of vibration to be deflected by gradual bends in the tube, and consequently the wave runs along the tube without reflection. Sharp bends should be avoided in such tubes as they give rise to reflected waves which run back. In such a tube since the wave is pre- vented from spreading out, the vibrations do not become less energetic as the wave advances, except from back reflections and from friction against the walls of the tube, and therefpre the sound is heard with only slightly diminished intensity at the distant end. When one end of a rod or wire of metal or of a long uniform PITCH 197 beam of wood is struck the sound is carried along the rod or beam just as in a speaking tube with very little loss through waves sent out sidewise, and is therefore very distinctly heard at the farther end. In ear trumpets, by the constraint of the smooth walls of the tube, the wave entering the wide end is gradually diminished in area tiU it emerges at the small end conveying all the energy that entered at the large end. Thus if the large end is 100 times that of the small end the energy per cubic centimeter in the emergent wave is 100 times as great as in the wave which entered the trumpet, neglecting the loss by friction, etc. 303. Megaphone and Speaking Trumpet. — In the megaphone sound waves coming from the speaker instead of spreading out in all directions from the mouth are limited by the walls of the in- strument, so that the wave emerging at the wide end has the whole energy of the voice. It will be shown in connection with the diffraction of light that when light waves pass through an opening which is not more than a wave length in diameter, the waves spread out in every direction from the opening, while if the opening is much larger, the waves, on account of interference, will not spread out so much but will travel straight forward, illumi- nating a spot directly opposite the opening. For precisely the same reason sound waves coming directly from the mouth spread out in every direction while waves from the larger opening do not spread out so much, and produce a more intense effect directly ahead. Precisely how this effect is caused by the interference of waves will be better understood after studying the diffraction of Ught. 304. Pitch. — The pitch of a sound depends on the frequency of the vibrations. — This is well shown by Savart's wheel. If a card is held so that it is struck by the teeth of a rotating cogged wheel a sound is given out which rises steadily in pitch as the speed of the wheel increases. If a device indicating the number of revolutions is attached to the wheel the number of taps per second producing a sound of a given pitch is readily determined. Another instrument by which the number of vibrations may be determined is the siren devised by Cagniard de la Tour, shown in figure 170. A disc having a circular row of equidistant holes is mounted on an axis so that it can rotate almost in contact with 198 WAVE MOTION AND SOUND the upper surface of a fiat circular box in which holes are made exactly corresponding to those in the disc, so that as the disc ^^tr^ Fio. 169. — Savart's wheel. rotates the holes are alternately opened and closed as many times in each revolution as there are holes in the series. The box is connected by a tube with a bellows and the puffs of air that come through the holes of the disc as it rotates give rise to a tone which is higher in pitch the faster the disc rotates. A revolution counter is attached to the axle so that the speed may easily be determined. The holes of the disc are in- clined one way and those in the upper plate of the box are op- positely inclined, so that the blast of air through the holes causes the rotation to take place auto- matically, the speed being controlled by the strength of the blast and a brake if necessary. For finding the number of vibra- tions of a tuning-fork the graphic method may be used, illustrated in figure 171. A point or stylus is fixed to one prong of a tuning-fork which is mounted so that the stylus just touches a sheet of smoked paper stretched over Fig. 170. — Siren. PITCH 199 a cylindrical drum. The axle of the drum is a coarse screw by which the drum is moved slowly lengthwise as it rotates. If the fork is set vibrating, on rotating the drum a wavy curve will be drawn in helical form around the drum, each wave corresponding to a vibration of the fork. To find the number per second a second curve may be simultaneously drawn alongside of the first by a tuning-fork whose frequency of vibration is known; or a small electric marker connected with a clock may be mounted Fig. 171. — Vibrations recorded on blackened drum. with its point touching the drum close beside the stylus of the fork, so that its marks made every second lie close to the curve drawn by the fork. The number of vibrations of the fork per second are found by counting the undulations in the curve be- tween two consecutive time marks. These various methods of experiment show that the pitch of a sound depends only on the frequency of vibration, and that it makes no difference whether the sound comes from a tuning-fork or from the puffs of a siren or the taps of Savart's wheel, all will have the same pitch if the frequency is the same. 200 WAVE MOTION AND SOUND 305. Doppler's Principle. — The pitch of a sound as heard depends on the number of waves that reach the ear per second. Con- sequently if the ear is moving toward the sounding body the apparent pitch will be raised, since more waves per second will meet the ear; and conversely if the ear is moving away from the sounding body it will receive fewer waves per second than if it were at rest and the pitch wiU appear lower. Let E (Fig. 172) represent the position of the ear and S that of a sounding body making n vibrations per second, and let the V Fig. 172. — Case of ear moving toward a sounding body. distance from A to E be V, the distance that sound travels per second. Then between A and E there are n sound waves which will reach the ear in one second if it remains at E, but if in one second the ear advances a distance v to E' it will meet in addition the waves between E and E'. Let x be the number of waves between E and E', and n' the number reaching the ear per second, then n' = n + X and and X :n = V : V . . x = ^ n = n ± -^ =n(l± -y\ nv the signs being plus or minus according as the ear has a velocity f toward, or away from, the sounding body. A similar change in pitch is observed when the sounding body is moving toward or away from the observer. But in this case the formula is somewhat different as the wave length of the sound is changed in consequence of the motion. Let S, the source of sound, have a velocity v toward the ob- server at E. In one second as it advances from S to ^S' it gives out n waves. The first of these waves leaving it at )S has reached A, having advanced a distance V equal to the velocity of sound DOtPLEIi'S iE>KINCiPLE 20l in the medium, by the time that the sounding body giving out the nth wave has reached S'. All n waves, therefore, lie between, V-v A and *S', and the wave length, X', is • But the number of waves that will reach E per second will be the number of wave lengths that are contained in the distance V that the waves travel per second, or n' = -r-,; A hence nV Y -V V where the minus sign is to be taken when the velocity of the sounding body is toward the hearer. E A s' s V Fia. 173. — Moving source of wavea. Doppler's principle explains the sudden lowering in pitch observed in a locomotive- whistle as it passes, and why a bicycle bell on an approaching wheel is heard of a higher pitch than when it is receding. It has also a most interesting application to light waves (§909). Resonators and ANAiiTsis of Sound 306. Quality of Sound. — The ear readily observes the differ- ence in quality or timbre between the sound of a violin and that of a flute or between notes of an organ and those of a piano though of the same pitch. This individual character of tones depends in part on certain superficial characteristics. The note of the piano comes im- pulsively, suddenly strong, and then rapidly dying out, while the tones of an organ do not come instantly to full strength, but are then sustained and steady. But tones equally sus- tained and steady may yet differ greatly in quality, as for ex- ample, the tones of tuning-fork, organ-pipe, and violin. To in- vestigate the cause of this difference we shall need resonators which vibrate in sympathy with the tones studied. 202 WAVE MOTION AND SOUND 307. Sympathetic Vibration. — A bell ringer by timing his pulls on the rope to correspond to the swing of the bell is able to set a heavy bell strongly swinging, while mere random pulls would accomplish very little; so it is that sound waves or other comparatively slight impulses may set up strong vibrations in a body if they are exactly timed to correspond to its natural period of vibration. This fact of sympathetic vibration may be illus- trated by tuning two strings on a sonometer to the same pitch, and then sounding one strongly; the other will be set in vibration by the impulses communicated to it through the supporting bridges. Again, if the dampers are raised from the strings of a piano and a clear strong note is sung near the instrument, the cor- responding string will be heard sounding after the singer's voice is silent. A very interesting case of sympathetic resonance is that in which a tuning-fork is set in vibration by the sound waves from a similar-fork placed 20 or 30 ft. away. The two forks are mounted on suitable resonance boxes and must be of ex- actly the same pitch; if they are thrown out of unison even very slightly, as may be done by affixing a bit of beeswax to a prong of one of them, they will no longer respond to each other. 308. Resonators. — When water is poured into a tall cylindrical jar the noise produced has a noticeable pitch which grows higher as the water level rises in the jar. This pitch is due to the air column in the jar, which has a natural time of vibration of its own and responds to any component vibration of the same pitch which may exist in a noise produced in its vicinity. On blowing sharply across the mouth of the jar the same pitch is noticed, the confused rustling noise having some component to which the jar can respond. The roaring heard in sea shells is explained in the same way. If an ordinary tuning-fork, not mounted on a resonance box, is held by its stem and struck, it will scarcely be heard a few- feet away, but if it is held, as shown in the figure, over the mouth RESONATORS 203 of a jar tuned to respond, a strong tone will be given out.- The arrangement shown in the figure permits the tuning to be easily effected by raising or lowering the connected water reservoir, thus changing the level of the water in the resonance tube until the response is most powerful. The pitch of the air column may be lowered also by partially closing the opening of the jar. Such an air column being'easily set in vibration by the proper tone is known as a resonator and may be useful in detecting in a mass of tones the presence of the particular one to which it is tuned . 309. Helmholtz Resonators. — Helmholtz, in the analysis of composite tones, made use of spherical resonators, each having a large opening and also a small one adapted to the ear. A resonator of this form is particularly useful because it responds easily to vibrations of one pitch only and so is well suited to the analysis of sounds. 310. Complex and Simple Tones. — The following experiment will now give us a clue to the cause of the difference in quality of tones. Take a series of tuning-forks mounted on resonating boxes, the frequencies of the forks being in the order of the series of whole numbers, 1, 2, 3, 4, 5, etc., which is known as the harmonic series. If the deepest toned fork in the series makes 250 vibrations per second, the next will make 500, and the next 750, etc. Provide also a set of Helmholtz resonators, one adapted to each fork. Now, on sounding the lowest pitched fork alone, a deep tone is obtained to which the corresponding resonator alone will respond. If the next lowest is now sounded at the same time with the other, the tones blend and come to the ear as a single tone of the same pitch as before but of a different quality. And so by sounding along with the deepest or fundamental fork any or all of the others, making some of the component tones strong and some weak, great variety of tones may be obtained differing in quality but all of the same pitch. But if any of these tones is tested by the resonators it is found that all those resonators respond which correspond to the forks used in producing the tone. Such a tone is called complex, while a tone to which only one resonator will respond is called a simple tone. 204 WAVE MOTION AND SOUND 311. Analysis of Sounds. — In the case just considered it is evident from the way in which the sounds of various quahties were produced that they were complex and consisted of sounds of different pitches blended together. But if we now sound an op)en organ pipe of the same pitch as the deepest toned resonator we find that not only does that resonator respond, but so also to a greater or less degree do the whole series of resonators, showing that though the sound comes from a single pipe it is just as truly complex as though originating in a series of tuning- forks. The component simple tones which imite to form a complex tone are known as its partial tones, the lowest of these in pitch is the fundamental, and the others are the upper partial tones or upper harmonics. The latter term is especially applicable when the upper partial tones are members of the harmonic series which starts with the fundamental. From the laws of dynamics as well as from experiment there is reason to believe that a simple tone, to which a resonator of only one certain pitch will respond is one in which the vibrations of the air are simple harmonic (§121). The ear seems to hear the simple harmonic components of a complex tone as separate simple tones, for persons with ears trained to the analysis of sound can often detect the different harmonics in a tone without the aid of resonators. 313. Synthesis of Sounds. — Helmholtz devised an interesting apparatus by which complex sounds might be built up from their simple components. This consisted of a set of ten tuning- forks, corresponding to the first ten terms of a harmonic series, which were kept continuously vibrating by means of electro- magnets, each fork being mounted in front of an appropriate resonator as shown in figure 175. The resonators were cylindrical brass boxes, each mounted with its opening close to the prongs of the corresponding fork, the openings being closed by covers which could be drawn back by pressing the keys of the key-board. When the resonators were closed scarcely any sound came from the forks, but drawing back the cover from any resonator by depressing its key brought out the corresponding tone with an intensity which depended upon the amount that the key was depressed. By means of HARMONICS 205 such an apparatus the sound of an open or closed organ pipe, a violin, or reed instrument can be closely imitated. An interesting modern instance of the synthesis of sounds is found in the ingenious "telharmonium" of Mr. Cahill in which the separate harmonics are transmitted by means of alternating currents of electricity of different frequencies which combine to Fig. 175. — Helmholtz apparatus for the synthesis of sound. form a single resultant current which acts on the telephone re- ceiver at the end of the line. By combining the proper simple harmonic components all the instruments of an orchestra are imitated. 313. Quality of a Musical Tone. — The quality of a musical tone may then be said to be determined by the pitch and in- tensity of the different simple tones or harmonics into which it may be resolved. 314. Fourier's Analysis. — It was shown by the distinguished French mathematician Fourier that any regular periodic vibra- tion, such as can take place in a sound wave, may be resolved into a sum of simple harmonic components all of which belong to a harmonic series, in which the fundamental has the same 206 WAVE MOTION AND SOUND period as the vibration analyzed. Thus, according to this theorem of Fourier, it is possible to analyze any sound wave into its simple harmonic components, and it is these simple harmonic components which are the simple partial tones detected by resonators. For example, the upper curve in figure 176 represents a simple sine wave. The three lower curves represent waves having the Fig. 176. — Vibration curves. Fig Combination of vibrations. same wave length and therefore the same periodicity as the upper curve, but they represent entirely different modes of vibration. Now, according to Fourier's theorem, each of these curves can be resolved into simple harmonic components. In figure 177, for example, the wave forms expressed by the heavy lines are the resultants of the simple harmonic waves represented by the dotted sine curves. The resultant curve in each case is obtained by adding the corresponding ordinates of the two component curves. In the first two cases one component has the same wave length as the resultant, while the other has half that wave length. These two components have the same amplitude in the first case as in the second, but their relative phases are different in the two cases and hence the resultant curves are different. In the lower curve one component has one-third the wave length of the resultant, while the component having half the wave length is absent or has zero amplitude. 315. Musical Tones. — In tones suitable for music the upper partial tones almost exactly fall into the harmonic series, start- ANALYSIS 207 Fio. 178. — Resonator and maaonjetric flame. ing with the fundamental; that is, their frequencies are very nearly exact multiples of the frequency of the fundamental. But the partial tones given out by bells and plates wher struck do not correspond, even approximately, to the lower terms of the harmonic series, and are quite unsuited for music. 316. Koenlg Resonators and Manometric Flames. — The French acoustician Koenig made use of resona- tors in which the front part was cylindrical and could be pushed in or drawn out so that each could easily be adjusted in pitch. To observe the vibrations of the resonators he employed mano- metric flames. A small, flat, disc-shaped box or capsule of wood, was divided into two chambers by a thin membrane, such as gold beater's skin. The cavity on one side of the diaphragm was con- nected by a short tube with a resonator, while the cavity on the other side had two openings, through one of which illuminating gas was admitted, while the other was connected with a fine jet where the gas burned in a small flame. The vibrations of the air in the resonator were transmitted through the diaphragm in the manometric capsule to the illum- inating gas, causing the flame to dance. The image of such a flame viewed in a rotating mirror, is drawn out in a band of light which when the flame is in oscillation shows serrations like saw teeth, as shown in figure 179; the particular form of the serrations revealing the mode of vibration of the flame. Fig. 179. — Rotating mirror. 208 WAVE MOTION AND SOUND 317. Sensitive Flames. — Under some conditions a gas flame may be very sensitive to sound. A small cylindrical jet is required having an aperture about 0.5 mm. in diameter, and the pressure must be such as to produce a long flame just on the brink of roaring. A pressure of about 9 in. of water is commonly reqxiired. Such a flame is sensitive to the vibrations of exceed- ingly short waves of sound, and breaks into a shorter flaring or roaring flame when a bunch of keys is jingled in its vicinity or a sharp hiss given or a very high-pitched whistle sounded. By means of sensitive flames sound waves so short as to be quite inaudible may be detected, and their interference and reflection studied. Interference and Beats 318. Superposition of Waves. — When the same portion of fluid is traversed by two waves, the motion of the particle will be the resultant of the two and may thus become very complicated. In case of surface waves in a hquid the eye can readily observe a series of short waves running over longer ones and preserving their motion as though over an undisturbed medium. 319. Interference of Waves. — Suppose A and B are the centers of two series of waves of the same wave length and amplitude. Then there will be certain points, as at C, where waves from the two sources act together so as to produce great disturbance. Suppose the lines in the diagram represent the crests of waves, then at the points C the crest of one wave is superposed on another, while at C the troughs of two waves come together; a half period later the crests will be at C" and the troughs at C. There will result, therefore, along the line CC a series of waves of double the amplitude of the original waves. At the points D, however, the crest of a wave from one center coincides with the trough of a wave from the other, there- FiQ. 180. — Interference of waves. INTERFERENCE 209 fore there will be at least a partial neutralization at points along the line DD'. If the waves coming together at D have equal amplitudes and equal wave lengths and are also simple harmonic waves they would separately produce at D equal and opposite displacements at every instant and will therefore com- pletely neutralize each other. This interaction of two sets of waves by which at certain points one is more or less completely neutralized by the other is known as interference. 330. Energy is Not Lost In Interference. — When there is complete interference at any point there is no motion of the medium and no energy at that point, but the energy of the two interfering waves is not lost or destroyed but appears at neigh- boring points (such as C, Fig. 180) where the amplitude of the component waves are added. For at these points the energy of the resultant vibration is four times what it would be if one of the trains of waves were suppressed. There results from the interaction of the two wave systems a different distribution of energy, but the total energy remains unchanged. 331. Interference of Sound Waves. — The interference of sound waves is well shown in the following experiment. The sound waves from a tuning-fork (Fig. 181) enter a suitable receiver which is connected to an ear- piece by means of two tubes, one of which has a sliding por- fjl] tion by which its length can (f ~ j ) be varied. When the tubes are ^•^:=s=^ ''- ' " ~ ^ adjusted to be of equal length the sound of the fork is dis- tinctly heard by the observer at E. As the sliding tube is drawn Fig. i8i. out, making one tube longer than the other, the sound grows fainter and reaches a mini- mum when one tube is longer than the other by a half wave length of the sound waves sent out by the fork, for in this case the waves reach the ear through the two tubes in opposite phases and interfere. If the slider is drawn out still farther the sound increases in strength reaching a maximum when one tube is just a whole wave length longer than the other. 210 WAVE MOTION AND SOUND If two similar organ pipes of the same pitch are mounted on a rather small air chest, as shown in figure 182, and sounded simultaneously they will usually sound in opposite phases, owing to an oscillation of the air in the air chest itself. The sound waves coming from one pipe will thus interfere with those from the other and the fundamental tones will be almost completely neutralized, the higher harmonics will, however, still be heard. 322. Beats. — If two organ pipes sounding together are not exactly of the same pitch the sound comes in pulses or throbs called beats. For in this case one pipe is giving out more vibrations per second than the other and, consequently, the relative phases of the two are constantly changing, as shown in the following figure where the dotted curves represent the waves from the two pipes, one of which is supposed to give out eleven vibrations for every ten of the other. The full line represents the resultant mo- tion. It is clear that while one pipe is gaining Fig "i 82— °°® complete vibration on the other, there will be Interference an instant when the waves are in opposite phase pfp^**" ""^^^"^ ^^^ interfere and another instant when they will be in the same phase and stengthen each other. There will, therefore, be one hundred beats in the time in which one pipe has made one hundred more vibrations than the other. Or if one pipe makes m vibrations per second and the other n, the number of beats per second is m — n. Beats are easily heard when two adjoining notes on the piano or organ are simultaneously struck, and the lower the notes are on the scale the slower will be the beats. Beats do not occur however, between notes that are very different in pitch. For example, no beats would be heard in case of two simple tones, one making 200 vibrations per second and the other 300 vibra- tions, though if one made 2000 vibrations and the other 2100 there would be 100 beats per second, heard as a distinctly jarring roughness. The explanation of this was given by Helmholtz (§358). In tuning two strings or two forks to unison they are adjusted until no beats are heard. If it is required that two tuning-forks shall be very accurately of the same pitch they may be tuned to BEATS 211 make the same number of beats per second with a third fork, as it is easier to count accurately three or four beats per second than to distinguish between no beats at all and very slow beating. Fig. 183. — Formation of beats. Standing Waves and Vibrating Bodies 333. Transverse Vibration of a Cord. — Take a long flexible rubber tube or other elastic cord fixed at one end, as at P (Fig. 184), and, holding it slightly taut, let the hand give a sudden movement to one side and back again. A wave is set up as at A which runs the length of the cord, is reflected at P, and returns on the opposite side of the cord as shown at B. On reaching the hand it is reflected to A and the Fig. 184. — Waves in cord. motion is repeated. Thus the cord makes a complete vibration and returns to its original form in the time in which a wave runs the length of the cord and returns. If the wave is not sent out by so sharp a movement it may take the form shown in the lower part of the figure where the cord simply swings to and fro or vibrates sidewise. But this vibration is just as truly due to a wave running along the cord and back again as in the first case and the period of one complete vibration is the time required jor a transverse wave to run the length of the cord and return. 212 WAVE MOTION AND SOUND Now, suppose waves are sent out by the hand with tvdce the former frequency, a wave will start from the hand at the same instant that the preceding wave starts back after reflection. The two will meet at the middle and, as they are on opposite sides of the cord in consequence of reflection, they will exactly neutraUze each other at the middle point, which will, therefore, remain at rest and the cord will vibrate in two segments. The vibrating segments are called loops and the points of rest nodes. The period of a complete vibration is in this case only half that of the fundamental period when there is only one loop. If waves are sent out with three times the frequency of the first case, the cord will break up into three vibrating segments, with intermediate nodes, and with greater frequency of vibration a still greater number of segments may be produced. 324. Standing Waves. — These vibrating 'segments are a particular case of what are called standing waves, which are set up in water or air or in other elastic bodies by the interaction of similar trains of waves running in opposite directions, and are usually due to reflected waves meeting those which are advancing. Standing waves are easily observed on the surface of water in a circular vessel in the center of which a periodic disturbance is produced. If the period of the disturbance is so adjusted that the wave length produced has the proper relation to the size of the vessel, a steady state is produced in which waves going outward meet the reflected waves, causing nodal rings where the water is at rest. Between these rings the surface oscillates up and down. Standing waves are also produced in organ pipes by the re- flection of the air waves from the ends of the pipes. 335. Formation of Standing Waves in Cord. — The diagram (Fig. 185) illustrates the mode of formation of nodes and loops in a cord. The dotted line represents a wave traveling from right to left along the cord, while the broken line represents an equal train of waves moving from left to right as indicated by the arrows. The resultant wave is shown by the continuous curved line, and its ordinate at any point is the sum of the ordinates ot the two component curves at that point. It will be observed VIBRATION OF STRINGS 213 that the crests of the two component waves are approaching each other at the points marked L, and a moment later will coincide. The resultant wave will then be at a maximum. A quarter period later the two component waves will exactly neutralize each other, the crest of one coming exactly over the trough of the other, and the cord which takes the resultant form will at that instant be straight. As the waves move still further the crests A and C will come together in the middle of the diagram and the N L N' ij N L N Fig. 185. — Formation of nodes and loops. resultant wave will then show a form just opposite to that in the figure, being bent up in the middle and down on each side. A little consideration will show that there will never be any dis- placement at the points N, N'; for in any position of the two com- ponent waves one is always as much above such a point as the other is below it; these points are threfore nodes, and the dis- tance between consecutive nodes is one-half the complete wave length. 336. Velocity of Wave In a Cord. — The velocity of a transverse wave along a stretched cord nlay be deduced as follows. Suppose that an infinitely long cord having tension T and a mass per unit length m is drawn rapidly through a bent glass tube as shown in the figure. If the cord were Fig. 186. — Solitary wave in cord. at rest it would produce a pressure against the tube in consequence of its tension. At a point where the radius of curvature of the tube is r the pres- T sure against the tube, or force per unit length, is -. being greater the more sharply curved the tube is at that point. But if the cord is drawn through the tube with velocity V, its centrifugal force per unit length as it runs over the curved part of the tube is . and this acts to diminish the pressure. 214 WAVE MOTION AND SOUND If the speed is just right, one will exactly balance the other and we shall have = — or K^ = — • r r m Since the radius of curvature cancels out of the result, the speed at which there will be no pressure against the tube will be the same whatever its radius of curvature may be, and consequently whatever its shape. Suppose the cord is now drawn along at this critical speed, the tube may be made to vanish and the bend in the cord will remain unchanged. If, now, the ob- server moves along at the same rate as the cord from left to right, the cord will appear to him to be at rest while the bend will be seen to travel along the cord as a wave from right to left with a velocity V, where The velocity mlh which a transverse wave runs along a perfectly flexible cord is thus determined from the relation IT T -i r J.I. ■ I Tension in dynes Velocity of the wave m cm. /sec. = -v/i-= ^^ . \Mass per cm. m grams 337. Transverse Vibration of Cord. — It has already been shown that the time of vibration of a stretched cord when it vibrates' as a whole is the time required for the wave to run from one end of the cord to the other and back again. Or if it is vibrating in segments the period of vibration in one of the segments is the time required for the wave to run twice the length of the segment. If I be the length of the cord or the distance between consecutive nodes, the period of vibration P is ^ V and if n is the frequency of the cord, or number of vibrations per second, we have _ J. _ 7 " P~ 21 where V is the velocity of a transverse wave along the cord. Substituting the value of V from the preceding paragraph we have 1 !f a formula which expresses in compact form the following three laws: 1. The number of vibrations per second made by a string VIBRATION OF STRINGS 215 under a given tension is inversely proportional to the length of the vibrating segment. 2. In case of two strings of the same length and mass per unit length, the frequencies are proportional to the square roots of the tensions; thus if one has four times the tension of the other it will make twice as many vibrations per second. 3. If two strings have the same length and are under equal tensions, their frequencies will be inversely proportional to the square roots of their masses per unit length. Thus if one is four times as heavy as the other, it will have only half the frequency. 328. Upper Harmonics of Cord. — It has already been seen that a stretched cord may vibrate not only as a whole, but it may also vibrate in two segments or in three, and so on. These modes of vibration may be easily established by touching the cord lightly at a point where a node is desired and at the same time bowing it at a loop. For instance, if a cord is touched at one-fourth of its length from one end and then bowed half-way between the end and the point where it is touched it will vibrate in four segments and give a tone which has a frequency four times that of its fundamental mode of vibration. That the cord actually vibrates in this way was prettily shown by Tyndall as follows: Little riders of bent paper were hung on the cord at the points where nodes were to be established and others midway between them on the loops. On sounding the cord as above described all the little riders on the loops were "unhorsed," while those at the nodes remained undisturbed. These partial modes of vibration of a cord are known as its harmonics, because if the fundamental mode of vibration of the cord has a frequency of n vibrations per second the partial modes of vibration will have frequencies 2n, 3n, 4n, 5n, etc., according as the cord vibrates in 2, 3, 4, or 5 segments, respect- ively; thus the frequencies of the partial modes of vibration are related to the fundamental as the terms in a harmonic series. If cords were perfectly flexible this would be exactly the case, but as actual cords always have a certain amount of stiffness, which affects the higher harmonics where the vibrating segments are short more than the lower harmonics where the vibrating segments are longer, the above is simply a close approximation to the truth, and the cord when vibrating in four segments, for 216 WAVE MOTION AND SOUND example, will have slightly more than four times the frequency of the fundamental. 329. Superposition of Vibrations. — When a string is struck or bowed, a number of these different possible modes of vibration are in general set up simultaneously. The form of vibration assumed by a cord in which two such modes of vibration coexist is shown in figure 187. The dotted lines show three positions of 3 Fig. 187. — Two simple vibrations combined. the cord simply vibrating as a whole in its fundamental mode. The full lines, however, show its form when it is at the same time vibrating in two segments, the two halves vibrating with reference to the dotted lines just as they would have done about the middle straight line if the fundamental mode of vibration had been absent. 330. Experimental Demonstration. — That such a coexistence of different modes of vibration commonly exists in a string may be shown by the following experiment. Pluck the string of a sonometer strongly at, say, one-fourth of its length from one end; then touching the string lightly at the middle, its fundamen- tal mode of vibration will be damped, but it will still be free to vibrate in two segments and the tone characteristic of that mode of vibration, an octave higher than the fundamental, will be heard still sounding. Or if struck one-sixth of its length from one end and then touched lightly at one-third of its length, the fundamen- tal mode will be damped while it will still be heard sounding a tone having three times the frequency of the fundamental. 331. Young's Law. — -When a string is struck at any point only those modes of vibration are set up which do not have nodes at that point. When a string is touched at any point all vibrations are damped that do not have nodes at that point. It is clear from these laws that in order to stop all the vibrations of string it should be damped at the same point where it is struck. This is done in the piano. VIBRATION OF STRINGS 217 332, Quality of Tone. — Strings are suited for musical instru- ments because the partial tones that they give out have fre- quencies which form a harmonic series with the fundamental, and all the lower tones of a harmonic series as far as the seventh harmonic are pleasing when sounded together. The seventh and ninth harmonics are, however, decidedly inharmonious with the others, and therefore it is desirable that in musical instruments the strings should be struck or bowed in such a way that these harmonics may not be developed. This is accomplished in the piano by striking the strings about one-seventh or one-eighth of their length from one end, so that all the partial modes of vibration having nodes near that point are weak. The hardness of the hammer in a piano also has a decided in- fluence on the tone. The harder the hammer the more sharply the string is bent when struck and the more prominent are the higher harmonics. If the hammer is too soft the tone is soft and lacking in the richness that comes from the proper strength of the harmonics. Organ Pipes and Wind Instruments 333. Organ Pipes. — ^An organ pipe may be considered as made up of two parts — a vibrator and a resonator. There are two types in use, flvie pipes, in which the vibrations are caused by a stream of air rushing against an edge, and reed pipes, in which the vibrator is a thin strip of metal. The construction of a flute pipe is shown in figure 188. At the lower end of the pipe is the em- bouchure or mouth which is like that of an ordi- nary whistle. Air, forced into the air chamber at the bottom, escapes through a narrow slit against an edge just opposite. The upper part of the pipe is a tube which may be either open or stopped at the upper end, and constitutes a res- onator which reinforces the vibrations set up at the embouchure. If a blast of air is sent through a skeleton pipe, which has a mouth-piece but no resonating chamber, a soft whistUng noise is heard which rises in pitch with the force of the blast. The pitch of- this tone also depends on the bluntness mi the edge against Fig. 188. — Section of organ pipe. 218 WAVE MOTION AND SOUND which the blast strikes and its distance from the opening. If the pipe is now provided with a resonating chamber of a proper shape and size to reinforce the vibration, a strong, clear tone will be given out. The vibrations appear to be caused by the friction of the stream of air against the edge, together with its inertia, just as little waves are formed on the surface of a stream of water in front of a wire or rod which cuts the surface. The vibrations caused in this way are taken up by the air column in the pipe, which as it vibrates reacts on the stream of air at the mouth-piece, causing it to deflect alternately inward and outward in rhythm with the vibration of the air column ; in this way regular impulses are received by the air column, and a strong vibration is maintained. In case, then, oi flute pipes it appears that the size and shape of the resonating cavity is what chiefly determines pitch, though a certain adaptation in the form of mouth-piece and strength of blast is required in order to evoke a good tone. 334. Nodes and Loops in Organ Pipes. — The vibration of the air column in an organ pipe is a case of standing waves and is due to the mm)) f, , ffl 1 1 1 c, 1. f>3 -f 1 1 — msfii i /v. N, *, N4 Fio. 189. — Formation of nodes and loops in organ pipe. interaction between waves running up the pipe and reflected waves moving in the opposite direction, forming nodes and loops just as in case of the vibrations of a cord. The upper part of figure 189 represents sound waves advancing from left to right in the direction of the upper large arrow. In condensed portions of the waves at Ci and C2 the particles are moving forward in the direction of advancement of the wave. In the rarefied portions at Ri and Ri the particles have a backward velocity. Immediately under this is represented the state of things in waves returning from right to left. The particles in the con- densed portions of these waves have a velocity from right to left as shown by the small arrows, and from left to right in the rarefied regions. If, now, these two sets of waves pass simultaneously through the same mass of gas the par- ORGAN PIPES 219 tides take the resultant motion and nodes and loops are formed in the posi- tions indicated in the lower part of the diagram. For it is clear that the rarefactions Ri and Rz will reach Na simultaneously, tending to produce opposite displacements, and a half period later the condensations d and Ci will come together at the same point, tending also to produce opposite displacements, and a little consideration will show that at every instant the two sets of waves will balance each other at Ni so that the particles there I I IV, Ns Ns Pig. 190. — Opposite phases at nodes. will remain at rest. So also at A^i, Nz, and iVi, which are thus nodes. But at Li the rarefaction Ri of the advancing wave will arrive at the same instant as the condensation Ct of the returning wave, and since in both of these the velocity of the particles is from right to left the resultant velocity at Lj will be from right to left at that instant. A halt period later when Ci and Ri come together at Li. the particles there will have a maximum velocity from left to right. Thus the air between nodes surges back and forth, in one-half vibration swinging toward Ni. on both sides and producing a compression there as shown in the lower part of figure 190. While in the next half vibration the air layers swing away from Ni and toward Ni and Ns, producing rarefaction at Nz and condensations at A'^i and Nz, as in the upper part of figure 190. At nodes, therefore, the greatest changes in pressure take place, although the nodal layer itself remains at rest, while the motion of the particles is greatest in the loops midway between nodes. Successive nodes are a half wave length apart and are in opposite phases, one being a point of rarefaction at the instant when the other is a point of condensation. So also the phases of motion in successive loops are opposite. 335. Kundt's Experiment. — The nodes and loops in a vibrating column of gas are beautifully shown in the following experiment due to Kundt. FiQ. 191. — Kundt's tube experiment. A glass tube about 5 cm. in diameter and a meter long is tightly stopped at one end, while in the other is fitted a light piston of cardboard attached to the end of a glass rod which is clamped firmly at the middle. The rod is set in longitudinal vibration by 220 WAVE MOTION AND SOUND drawing along it a wet cloth held firmly clasped around the rod. By adjusting the position of the piston A in the large tube a point is found where the air column between A and 5 is in resonance with the vibrations of the rod. The air is then set in such power- ful vibration that any light dust in the tube, such as lycopodium powder, is driven out of the loops and gathers in little heaps in the nodes. This will occur when the sound waves run the length of the air column and back in a certain whole number of vibrations of the rod. In the diagram there are six loops indicating that the rod makes six vibrations while the wave runs the length of the tube and returns. The stopped end is exactly a node, while the piston end, where the motion is communicated, is very nearly a node. The distance between nodes is a half wave length and may easily be measured with considerable accuracy. The tube may now be filled with some other gas and the distance AB again adjusted and the distance between nodes found for this gas also, and since the frequency of vibration of the glass rod is the same in both cases the velocity of sound in the gas is to that in air in the same ratio as the distances between nodes in the two cases. In this way the velocity of sound has been measured in a large number of gases and vapors. 336. Reflection in Organ Pipes. — Reflection of waves takes place in general when a wave meets a boundary where there is a change of medium. Fig. 192. — Reflection model. In a stopped organ pipe, a wave running up the pipe meets the unyielding end, and must therefore be reflected in such a way that the reflecting surface is a node, or point of no motion. In an open pipe it is quite the reverse, the wave advancing in the pipe on coming to the open end finds a freer medium, unconstrained by the walls of the pipe, and therefore reflection takes place, but in such a way that the end is a looj) or point of great motion. The mechanical model represented in figure 192 will serve to make clear the nature of these two kinds of reflection. The figure represents a series of balls resting in a frictionless groove. ORGAN PIPES 221 Half are larger and of greater mass than the other half, and all are connected together by springs of equal stiffness. Suppose the left-hand ball is given an impulse forward, the spring between it and the next will be compressed and the motion transmitted as a wave of compression from one to the other and so on along the line, each coming to rest after giving up its motion to those ahead. But the last of the row of large balls will move forward more freely than it would have done if there had been no change in the size of the balls, and, therefore, it stretches the spring behind it, which gives a forward pull to the ball next behind it and so sends back a wave of rarefaction. This is the kind of reflection which takes place at the open end of an organ pipe. When, on the other hand, a compressional wave is sent from right to left along the row of small balls, the last one does not move forward as far as it would have done if there had been no change of medium, and the spring behind it is, therefore, more compressed and gives a backward impulse to the preceding ball, sending a compressional wave back through the series. This is the kind of reflection which takes place at the stopped end of an organ pipe. 337. Open Pipes. — In. open organ pipes there must, therefore, be a loop at the top and also a loop at the mouth for there is great motion of the air at these points. Between them near the middle of the pipe is a node. The position of the node may be demonstrated by lowering into the pipe a horizontal tray of thin membrane covered with sand. If the tray is exactly in the node the tone is not affected, but if it is either raised or lowered a loud buzzing is heard in con- sequence of the vibration of the membrane. An open pipe sounding in this way is giving out its deepest or fundamental tone. Since consecutive loops are a half wave length apart, it follows that the length of an open pipe is half the wave length of its fundamental tone. Thus an open pipe 1 meter long gives out waves 2 meters long and 331 V accordingly makes -~- = 165.5 vibrations per second, since n = — • But an open pipe may vibrate in other ways consistent with the condition that the two ends must be loops. Thus it may vibrate having loops and nodes, as shown in figure 193, so that the 222 WAVE MOTION AND SOUND distance from loop to loop or node to node may be only one-half the length of the pipe, or it may be one-third the length, etc. The frequency of vibration in the first of these partial modes is, therefore, twice- that of the fundamental, that of the next three times, the next four times, etc. Thus, any of the harmonics of the fundamental may be pro- duced by an open organ pipe. The partial modes of vibration generally coexist to a greater or less degree with the. fundamental, giving character and richness to the tone. The relative strength of the harmonics depends on the shape of the mouth-piece and the force with which it is blown and the shape of the pipe itself. The higher harmonics are more emphasized when the pipe is strongly blown. Fio. 193. — Nodes and loops in open pipes. Fio. 194. — Nodes and loops in stopped organ pipes. The position of the nodes and loops just discussed is only approximately correct, as the open end is not exactly a loop, still less is the mouth exactly at a loop and the node is nearer the mouth of the pipe than it is to the top, the variation being most marked in wide pipes. 338. Stopped Pipes. — In stopped pipes the stopped end is exactly a node, while the mouth is nearly a loop. Thus the length of the pipe is one-fourth the wave length of the fundamen- tal tone. Suppose a compressional wave starts at the mouth of the pipe, it runs up to the top, is there reflected back as a compressional wave, but on reaching the mouth again is reflected in the opposite ORGAN PIPES 223 way as a rarefaction, then traveling up and being reflected as a rarefaction it returns to the mouth where it is again reflected as a condensation; the wave has thus traversed the length of the pipe four times before returning to its original phase. A stopped pipe one-half meter long would therefore have a wave length two meters long and would vibrate with the same frequency as an open pipe one meter long. Stopped pipes also may vibrate in other modes than the fun- damental, but there must in every case be a node at the top of the pipe and a loop at the mouth. In figure 194 is shown the distri- bution of nodes and loops in the fundamental and next two upper partial modes of vibration. It will be observed that the distance from node to loop in the first partial mode is one-third that in the fundamental and consequently the frequency of this mode of vi- bration is three times the fundamental. The next higher mode of vibration has five times the frequency of the fundamental, etc. Hence a stopped organ pipe may sound its fundamental and odd harmonics, the frequencies of its proper tones being related to each other as the series 1, 3, 5, 7, etc. The absence of the even harmonics causes the tones of stopped pipes to differ decidedly in quality from those of open ones. Fig. 195. — Free reed. 339. Reed Pipes. — In reed pipes the vibrations are caused by a metal tongue or reed. Two forms are used, the free reed and the striking reed. A free reed, represented in figure 195, con- sists of a tongue of thin metal riveted firmly at one end to a plate in which there is an aperture just under the tongue large enough to admit of its vibrating freely through it without touching. The tongue when at rest is slightly above the aperture and when the reed is blown the stream of air catches it and carries it down, nearly closing the opening, this stops the rush of air and the tongue springs back, when the action is repeated. In this way a vibration is maintained. Reeds of this type are used in cabinet organs, harmonicas, and accordions. In striking reeds the tongue is a little larger than the opening 224 WAVE MOTION AND SOUND and when at rest stands slightly above it. When blown it is carried down and clapping over the opening stops the rush of air, then rebounding is again carried down, thus being maintained in vibration. The tones from striking reeds are stronger and more penetrating than from free reeds. They are used in or- dinary tin horns and in the clarinet and in some stops of pipe organs. When reeds are used in pipe-organs they are provided with resonators which strengthen and improve the tones. The tones of reed pipes are rich in the higher harmonics, and the shape of the resonator used greatly influences the relative strength of these harmonics and hence determines the quality of the tone produced. 340. Effect of Changes of Temperature. — When the tempera- ture rises the velocity of sound in air increases and consequently the pitch of the flute pipes in an organ is raised. On the other hand, the effect of higher temperature is to diminish the elasticity of the metal tongues of reeds so that they vibrate more slowly, lowering the pitch of the reed pipes. An organ is thus thrown out of tune by great change of temperature. 341. Other Musical Instruments. — In the flute and piccolo the vibrations are produced by blowing across an opening or embouchure near one end, the pitch produced being determined by the strength of blast and by the effective length of the resonat- ing cavity which is regulated by opening or closing holes in its side. The deepest tone of a flute, as of an open organ pipe, is one whose wave length is double the length of the instrument. When blown hard the higher harmonics are sounded. The mouth-piece of a fife is like an ordinary whistle or flute organ pipe, while the clarinet and oboe have mouth-pieces in which a thin slip of wood mounted over an opening forms a striking reed. In the bugle the vibrations are due to the air being blown between the tightly drawn lips of the player as they are placed upon a suitable cup-shaped mouth-piece, the pitch being determined by the tension of the Ups and by the resonance of the tube. The long coiled tube in such instruments has a very deep fundamental tone the numerous upper harmonics of which can be easily evoked. The cornet is also provided with little valves by VIBRATION OF RODS 225 which the effective length of the tube is varied and an additional number of tones made possible. Vibration of Rods and Plates 342. Longitudinal Vibration of Rods. — If a rod of steel, say a meter long and a centimeter in diameter, is held firmly at the middle point and if a cloth dusted with powdered rosin and folded over the rod is grasped firmly with the hand and drawn off the end with a quick strong pull, a clear, high-pitched sound may be produced due to the longitudinal vibrations of the rod. That the vibrations are of this nature may be demonstrated by means of a small ivory ball hung by a cord and resting against the end of the rod. The ball will be violently driven off, swing- ing out as shown in the figure. Glass tubes held at the middle may be similarly set in vibration, using a wet cloth instead of one dusted with rosin. Tyndall was able to set a large glass tube so power- fully in vibration by this means that the tube was shivered to pieces. The middle point where the rod is held or clamped is a node and the ends vibrate lengthwise to and fro simultaneously toward the middle or away from it so that the bar is alternately lengthened and shortened. The vibrations are thus precisely like those in an open organ pipe where there is a node in the middle and loop at both ends, and, as in the organ pipe, the period of a complete vi- bration is the time required for a compressional wave to travel the length of the bar and back again. Thus if the velocity of sound or of compressional waves in steel is 5000 meters per second a bar 1 meter long will make 2500 vibrations per second, since the wave length is 2 meters. The area of cross section of the bar does not affect the result, the same pitch is obtained from bars of different diameters and shapes of cross section if they are of the same material and length. Fig. 196. — Ball driven from end of rod. 226 WAVE MOTION AND SOUND Fio -Tranverse vibration of rod. By a little dexterity such a rod may be made to give a higher harmonic, vibrating with a node in the middle, and two others, each one-sixth of the length of the bar from the end. The wave length in this case is evidently one-third of that in the former, and the frequency of vibration three times as great. Rods of other metals or of wood or glass may be caused to vibrate in this way and the velocities of sound in them may be compared by their frequencies of vibration as shown by the tones which they give out. 343. Longitudinal Vibration of Wires. — ^Longitudinal vibrations may also be set up in wires firmly clamped at both ends by rubbing them lengthwise with a bit of rosined cloth. The clamped ends of the wires are nodes in this case and the middle is a loop. The pitch depends only on the velocity of sound along the wire and on its length and is quite independent of its tension except in so far as the tension affects the elasticity of the wire. 344. Transverse Vibrations of Bars. — The trans- \|ii| verse vibrations of bars are determined by their mass and stiffness, and hence depend on Young's modulus H||' [||i of elasticity, since it is this coefficient which deter- mines a bar's resistance to bending. If a uniform free bar is struck at the middle point it tends to vibrate as shown in the figure, with a node near /"Hii'i j|i|l|-^yv each end, and it may be supported at the nodes on wooden bridges without materially affecting its vibration. In the xylophone or kaleidophone the bars of wood or metal vibrate transversely and are sup- ported at their nodes. 345. Tuning-forks. — A tuning-fork may be considered a bent bar vibrating in the mode shown in figure 197. For there are two nodal points, one on each leg of the fork near the bottom. The prongs swing alternately toward and away from each other, while the stem of the fork, being attached to the vibrating segment be- tween the nodes, vibrates up and down. This is made ap- larent by the loud tone given out when the stem of a vibrating fork is touched to a wooden table top or sounding board. Fig. 198. VIBRATION OF RODS 227 Tuning-forks are often mounted on wooden resonators, boxes enclosing an air chamber capable of responding to the vibrations of the fork. 346. Law of Similar Systems. — When two vibrating systems are made of the same material and are exactly similar in dimen- sions, though not of the same size, their periods of vibration are proportional to their linear dimensions. This law is shown by mathematical reasoning to be a consequence of mechanical prin- ciples, and is illustrated in many familiar instances. For example, if two stopped organ pipes are constructed with cubical resonating chambers, but one having half the dimensions of the other, the smaller will vibrate with twice the frequency of the larger. Two tuning- forks of equally stiff steel and exactly similar in shape will be an octave apart if one is twice as large as the other. And so, also, if we take two straight steel bars, one of which has half the dimensions of the other in each direction, the smaller will make twice as many vibrations per second as the larger when vibrating in the same manner. 345. ■ Vibration of Plates. — The vibrations of flat plates of various shapes were studied by Chladni who scattered sand on FiQ. 199. — Chladni's figures. the plates and observed the figures formed by the nodal lines in which the sand gathered when the plates were bowed. Some of these forms, known as Chladni's figures, are shown in figure 199. The upper row shows different modes of vibration that may be set up in a square plate supported at its center and bowed at some point on the edge. The slowest mode of vibration is the first, in which the vibrating segments are the four corners. Segments separated by a nodal line must always be opposite in phase, one vibrating up, while the other swings down. This opposition of phase is indicated by marking them alternately plus and minus. If a resonator or wide-mouthed bottle which can respond to the vibra- tions of the plate is held with its mouth over any vibrating segment it will respond strongly, but if moved over a nodal line so that it is simultaneously 228 WAVE MOTION AND SOUND acted on by two adjoining segments it is silent because the segments are in opposite phases. So also when the plate is vibrating as shown in the first or second diagram in figure 199, if the hands are held just above two similarly vibrating seg- ments so as to quench the sound waves coming off from them, the sound from the plate will be heard louder than before. 348. Bells. — The blow of its tongue on a bell causes the circular rim to spring out into slightly elliptical shape, from which it springs back passing through the circular form into an ellipse with its greater axis at right angles to the first; thus it oscillates in four segments with four intermediate nodes as shown in the figure. Making the rim of the bell thicker causes it to oscillate more quickly by reason of its increased stiffness and thus raises its pitch. The above is its fundamental or slowest mode of vibration, but simultaneously with this the blow of the hammer sets up higher modes of vibration in which the rim may vibrate in 6, 8, or 10 segments with inter- mediate nodes. These higher tones are not Fig. 200.— Vibration of j^ ^^^^ harmonic Series of the fundamental bell. and hence the tones of bells are unsuitable for music. When the bell is first struck the higher tones are more prominent than the fundamental, but as the sound dies away the fundamental tone persists the longest. The beating or throbbing heard as the tone of a bell dies away is due to want of uniformity in the rim, in consequence of which there are two funda- mental tones of slightly different pitch. One or the other of these is excited according to the point struck by the hammer, though in general both are simultaneously set up. Musical Relations of Pitch 349. Musical Intervals Depend on Ratios. — The musical effect of two tones when sounded tgether depends upon the ratio of their frequencies. This is well shown by means of the siren ( §304) . If four rows of holes in the siren are simultaneously used, in which the numbers of holes are proportional to 4, 5, 6, and 8, respectively, a combination of tones will be produced which will be recognized as the major chord — do mi sol do. And this tnusieal MUSICAL SCALES 229 relationship holds whatever may be the speed of the siren, show- ing that whatever the pitch may be it is the ratio of the frequen- cies of two tones which determines their musical relationship. 360. Harmonious Ratios.— Tones are harmonious whose frequencies are proportional to any two of the simple numbers i, 2, 3, 4, 5, 6. The most important harmonious ratios and their musical names are here given : 1 : 1 unison 1 : 2 octave 1 : 3 twelfth 2 : 3 fifth 3 : 4 fourth 4 : 5 major third 5 : 6 minor third The names are derived from the ordinary musical scale; thus the octave is the relation of the first and eighth tones of the scale ; the fifth, that of the first and fifth; \h& fourth, that of the first and fourth, etc. 351. Major Scale. — Three tones whose frequencies are in the ratio 4:5:6 form what is known as a major triad. The major scale is a sequence of tones so related that the first, third, and fifth tones form a major triad; also the fourth, sixth, and eighth, and the fifth, seventh, and ninth. The first note of the sequence is called the key-note and the triad starting with the key-note is the triad of the tonic. The fifth tone is known as the dominant and the fourth as the subdominant, and their triads are, respectively, known as the triads of the dominant and of the subdominant. If the tones of the scale are represented by letters as in or- dinary musical notation, their ratios for the key of C will be as follows : Designation c D E F G A B c d Triad of tonic 4 5 4 6 . . . . 4 5 5 6 2 Triad of dominant e, 1 % y^ « 94 230 WAVE MOTION AND SOUND 353. Tones and Hall Tones. — If the ratio of the vibration frequency of each tone to that of the one immediately preceding it is taken, we find C D E F G A B c d ^~jr^~^%^^~^Hr'jr~^H^~ji^~^Hr~jr etc. Tone Tone Half-tone Tone Tone Tone Half-tone Tone These ratios determine the musical character of the intervals. When the ratio of the frequencies of two tones is % or '^%, they are said to differ a whole tone, while those whose ratio is ^^{5 are said to be a half tone apart. 353. Minor Triad. — In the major triad, three tones whose ratios are 4:5:6, the interval between the first and second tone is a major third, while that between the second and third is a minor third. If we had three tones in the ratio 10 : 12 : 15, the interval between the first and second would be a minor third (5 : 6) while the interval between the second and third would be a major third (4 : 5). Such a combination of tones is known as a minor triad. 354. Minor Scale. — A scale based on minor triads in the same way that the major scale is based on major triads is known as the minor scale. In the key of C the tones C, D, F, G are the same on both scales, while E, A, B each differs from the corre- sponding note of the major scale by the interval ^^4, the minor tone being lower in each case. These tones of the minor scale may be designated E flat, A flat, B flat. 355. Temperament. — Since there are two kinds of whole- tone intervals {^i and '^%) and also two kinds of half-tone in- tervals (^^-15 and ^^^4), and since a note a half tone higher than D, for example, which is called D sharp, would not be the same as E flat, it is clear that many notes are required to admit of playing music accurately even in a single key; and this number must be greatly increased if we are also to be able to play correctly in other keys. In such an instrument as the violin the artist may indeed use true intervals, but in keyed instruments, like the piano or organ, the number of keys that would be required in such a case would make the key-board unmanageably complicated. Hence what is known as the equally tempered scale is used. In this scale the HEARING 231 whole tones are all equal and each equal to two half-tones. And as there are five whole tones and two half-tones in an octave, the octave must be equivalent to six whole-tone intervals or twelve half-tone intervals; hence since two notes an octave apart are in the ratio 1 : 2, notes a whole tone apart must be in the ratio 1 : V2, and those a half tone apart in ratio 1 : ^^^2. The following table gives in the upper row the vibration fre- quencies of notes in the true or diatonic scale, beginning with middle C of the piano, while in the lower row are shown the cor- responding frequencies in the equally tempered scale. D E 261 261 293.6 326.2 292.9 328.8 348.0 348.3 391.5 391.0 435.0 438.9 489.4 492.6 522 522 Diatonic scale. Equally tempered. The Ear and Hearing 356. The Ear. — To make clear the physical basis of hearing a short account of the structure of the ear will be required. Referring to figure 201, three principal parts will be noticed: the external ear channel closed at the end by the tympanum or drum skin, the middle ear in which is the chain of little bones or ossicles which connect the tympanum with the inner ear, and the inner ear itself in which the auditory nerve terminates and which is contained in a cavity in the massive part of the temporal bone. The small bones of the middle ear are situated in the upper part of a tube containing air, which is known as the Eustachian tube and which opens into the back of the mouth through a small valve which opens in the act of swallowing. The air pressure in the Eustachian tube is thus kept the same as that on the outside of the drumskin. Persons going down in diving bells often experi- ence a pain in the ears owing to the difference of pressure, which is relieved at once by swallowing. The inner ear consists of a long chamber coiled up like a snail shell, and hence known as the cochlea, the three semicircular canals, and the vestibule. There are two openings from the Eustachian tube into the inner ear, one of which is closed by a membrane and the other by the stirrup bone or stapes, one of the 232 WAVE MOTION AND SOUND four ossicles of the middle ear. The interior of the inner ear is filled with liquid, the endolymph. The cochlea is probably that part of the ear by which musical sounds are distinguished in pitch and quality. It is divided into two parts by the basilar membrane which runs lengthwise through all its convolutions dividing it into two chambers. This mem- brane is strongly fibrous in structure, the fibers running across from one side of the tube of the cochlea to the other. Along its inner edge, where it is attached to the walls of the cochlea, the fibers of the auditory nerve terminate, so that a disturbance of Auditory nerve Basilar membrane Eustachian tube Stirrup' Round window' Fig. 201. — Diagram showing tympanum; ossicles, and internal ear. any part of the basilar membrane causes a stimulus to the cor- responding filament of the auditory nerve. As this membrane also gradually varies in width from one end to the other, its fibers vary in length like the strings of a piano and have their own periods of vibration, which are slower for the long fibers and quicker for the shorter ones. When sound waves falling on the tympanum cause it to vibrate, these vibrations are transmitted through the ossicles to the liquid on one side of the basilar membrane, then through the membrane itself to the hquid on the other side of it which is in contact with the flexible membrane closing the second opening HEARING 233 into the Eustachian tube. The vibrations are transmitted most easily by that part of the basilar membrane which can vibrate in sympathy with the impressed vibration, and therefore the cor- responding nerve filaments are stimulated. The arrangement is such that sounds of different pitch, awaking sympathetic vibra- tions in different portions of the basilar membrane, stimulate different nerve filaments and so give rise to different sensations. It is now easy to understand why the ear should analyze com- plex sounds, hearing each simple harmonic component as a sepa- rate simple tone. For it is in accordance with the mechanical laws of sympathetic resonance that when in a complex vibration there is a simple harmonic component which has the same period as the resonator, then the latter will respond. If there are, therefore, three different harmonic components in the vibrations communi- cated to the basilar membrane, the three corresponding portions of the membrane will be set in vibration, and consequently three different nerve filaments will be stimulated, exciting three dis- tinct sensations of pitch. 357. Influence of Phase.— From the above theory of audition developed by Helmholtz, it is to be expected that the relative phases of the components in a complex tone will have no influence on the resulting sensation, for the same parts of the basilar mem- brane are set in vibration whatever the phases of the component tones. 358. Beats. — There is one important exception to this. In case of beats the pulsations of tone are certainly due to the changing relative phases of the two components which alternately act together and against each other. But this case is exceptional because the two tones are so near in pitch that they affect closely adjacent portions of the basilar membrane. It is not to be sup- posed that only a single fiber of the basilar membrane vibrates in response to a particular tone, but the adjoining portions are also set in vibration to some extent. Suppose that two tones very near together in pitch are sounded and one excites the strongest response in the basilar membrane at a, figure 202, and the other at b. The membrane on each side of a will also respond to the first tone and on each side of b to the second. If a and b are sufficiently near together, the fibers midway between them will vibrate 234 WAVE MOTION AND SOUND simultaneously in sympathy with both tones; they will therefore take the resultant motion and will vibrate alternately strongly and feebly according as the two component vibrations are in the same or opposite phases. Hence the nerve filaments connected with these fibers receive an intermittent stimulus which produces the disagreeable jarring sensation of beats, just as the intermittent stimulus of a flickering light is painful to the eye (Helmholtz). It is quite in accordance with this theory of audition that rapid beats are heard as a distinct roughness and do not merge into a tone. Thus if two Koenig forks, one making 2816 vibra- tions per second and the other 2560, are strongly sounded the beats are heard as an extremely disagreeable buzzing, though the number is 256 per second, while a tone of that frequency is wholly agreeable when sounded with either of the forks. The beating is due to the disturbance of the basilar membrane between the points where it responds to 2560 and 2816 vibrations per second while to excite the sensation of a tone having a frequency of FiQ 20^ ^^^ ^^ entirely different portion of the mem- brane must be set in vibration, viz., that part which has a natural frequency of 256 per second. 359. Combinational Tones. — Under some circumstances when two tones are strongly sounded, a tone is also heard whose fre- quency is equal to the difference between the frequencies of the two generating tones; its frequency is thus the same as that of the beats between the tones, though it is an entirely separate phe- nomenon. These differential tones, as they are called, may be very distinctly heard when two high-pitched forks are strongly sounded together, such as the two Koenig forks referred to in the last paragraph. Helmholtz showed on mechanical principles that when two simple harmonic vibrations act on a membrane to set it in vibra- tion, if the displacement of the membrane is so great that it is not simply proportional to the displacing force, then the resulting motion of the membrane will not be simply the stim of the two impressed harmonic vibrations, but will also include other com- ponents, one of which has a frequency equal to the difference of the frequencies of the two original vibrations, and the other a frequency equal to the sum of their frequencies. HEARING 235 Now, the tympanum of the ear is attached at the center to a small bone which is drawn inward by a muscle, thus keeping the drum skin tense, hence it is stretched in slightly conical form, and is therefore unsymmetrical and resists inward displacement more than it does outward. Hence, according to Helmholtz, when- two strong vibrations are simultaneously impressed upon the tym- panum, the motion which it communicates to the inner ear con- sists not simply of these, but includes also a vibration whose fre- quency is their difference and another whose frequency is their sum. These are called the differential and summational tones. The latter were first observed by Helmholtz after he had shown theoretical reasons why they should exist. They are not so easily observed as the differential tones, and some observers have disputed their existence. 360. Helmholtz Theory of Dissonance and Consonance. — Helmholtz showed that dissonance was explained by beats taking place between either the tones themselves or their upper har- monics or the differential tones that they gave rise to, and that a clearly marked consonance occurs when the ratio of two tones is such that there are no beats, but when a slight change in the ratio gives rise to disagreeable beating. For example, the most perfect consonance is unison, for then the fundamental tones and upper harmonics all agree and there is no beating, but a slight mistuning causes beating not only between the fundamental tones, but between each pair of har- monics. Suppose, for example, one tone and its harmonics have the vibration frequencies shown in the series 100 200 300 400 500 if the other tone, instead of making exactly 100 vibrations, makes 106, then it with its harmonics will form the series 106 212 318 424 530 and there will be 6 beats per second between the fundamental tones, 12 between the first harmonics, 18 between the second harmonics, etc., The ear, therefore, selects a unison as a well- marked consonance. So also with the octave: suppose two tones which with their harmonics are given by the two series 236 WAVE MOTION AND SOUND / 100 ' 200 300 400 500 600 Octoyej goo 400 600 here the fundamental of the tone making 200 vibrations per second is of exactly the same pitch as the first harmonic of the other tone, and there are no beats between any of the harmonics. But suppose the octave is mistuned, as, for example, below Tir-. J . /lOO 200 300 400 500 600 Mzstuimdoclave^ 210 420 630 here there are 10 beats per second between the fundamental of one and the first harmonic of the other, 20 per second between the harmonics 400 and 420, etc., and the result is great dissonance. It is clear from the above that the richer tones are in har- monics, the more dissonant they will be when mistuned. It is thus much easier to judge whether an octave is tuned correctly in case of two reed pipes than with two wide stopped pipes al- most free from harmonics. Problems 1. How long must a water wave be to travel with a velocity of 20 miles per hour? 2. What relation is there between the lengths of two water waves one of which has twice the velocity of the other? 3. Find the velocity of sound in dry air at 20°C. and pressure 73 cm. of mer- cury, when its velocity at 0°C. and 76 cm. pressure is 332 meters per sec. 4. What must be the amplitude of motion of the particles in a water wave, if the velocity of the particles at the wave crest is equal to the velocity of the wave? 6. If the height of a water wave from crest to trough is 3 ft. and its length is 50 f t. , find its velocity, its frequency or the number of waves that pass per sec, and the direction and amount of the velocity of the water parti- cles on the crest of the wave. 6. How many vibrations per second will be received from a bicycle whistle giving out 500 vibrations per sec. and approaching at the rate of 10 miles per hour? 7. If an observer were to move with the velocity of sound toward a sounding body at rest, what pitch would be heard? What if the observer were at rest while the sounding body approached him with the velocity of sound? 8. A tuning-fork having a frequency of vibration of 1000 per sec. is moved away from an observer and toward a flat wall with a velocity of 5 meters per sec. Find how many beats per second will be heard by the observer. 9. A cord 30 ft. long is stretched between two fixed supports with a force of SOUND 237 40 pounds' weight. How many transverse vibrations per sec. will the cord make if it weighs }4 lb.? 10. A very long cord weighing 5 gms. per meter and stretched with a weight of 5 kgs. has one end made to oscillate sidewise 4 times per sec. Find the length of the waves set up in the cord. 11. A brass wire and a steel wire of the same diameter are stretched by equal weights and their lengths adjusted to give the same pitch when vibrating transversely. When the steel wire is 1 meter long between supports, how long will the brass wire be? 12. How many vibrations per sec. will be given out by an open organ pipe 76 cm. long. Give also the frequencies of its first three upper har- monics. Take temperature of air as 20°C. 13. How long must a stopped organ pipe be in order to have the same fre- quency of vibration as the open pipe in problem 12; also what are the frequencies of its first three upper harmonics? 14. What rise in temperature would raise the pitch of a flute pipe in an organ one semitone? Take original temperature as 0°C. and semitone ratio as 15^5. 15. In a Kundt's tube filled with air the distance between the dust heaps is 17 cm., but when the tube is filled wih carbon dioxid gas, the distance between nodes is 13.4 cm. Find the velocity of sound in the carbon dioxid if that in air is 340 meters per sec, both gases being at 15°C., and vibrations being produced by the same rod in both cases. References Tyndall: Lectures on Sound (Appleton). A delightfully written exposition of the subject by a brilliant experimen- ter and lecturer. Helmholtz: Sensations of Tone, translated by ElUs (Longmans). A thorough treatise on the scientific basis of music. HEAT Thebmometkt 361. Temperature Sense. — The idea of temperature is ob- tained directly from our sense of touch. We speak of bodies as hot or cold according to the way in which they affect our tem- perature sense; and though temperatures cannot be accuratelj"^ compared in this way, we may yet roughly estimate whether one body is hotter than another or whether a body is growing warmer or colder. 362. Transfer of Heat. — When a hot body is brought into contact with a cold body the former is cooled while the latter is warmed. When a layer of copper is interposed between the hot and cold bodies the change goes on rapidly, but when a layer of felt is interposed the change is much slower. Hot water in a thermos bottle changes its temperature very slowly indeed, so that it is easy to imagine an ideal receptacle in which no change whatever in temperature could occur. These facts indicate that the temperature of a body changes only when something passes into it from without or escapes from it to other bodies. This something is called heat. Heat is said to pass from the hot to the colder body rather than that cold passes from the cold to the hot body, because experi- ment shows that when a body cools it loses something, namely, energy or power to do work, and hence heat rather than cold is considered the entity. 363. other Effects of Heat. — As bodies change in temperature other accompanying changes take place. As they grow hotter they increase in size, an enclosed mass of gas or vapor exerts a greater pressure, if heated enough a solid melts to a liquid, or a liquid is changed, to vapor; also the elastic, electric, and magnetic properties of substances are seriously modified. 364. Equal Temperatures. — When two bodies are placed in contact and no change takes place in either one such as would indicate a transfer of heat, they are said to be at the same tem- 238 THERMOMETRY 239 perature. When one grows hotter and the other colder, the latter is said to be at a higher temperature than the other. Temperature may be defined as that property of a body which determines the flow of heat. If there is no transfer of heat between two bodies when placed together, they are at the same temperature. Thus temperature plays the same part in the flow of heat that pressure does in the flow of fluids. 365. Thermometers. — To accurately compare temperatures instruments are employed called ther- mometers. Thermometers may be based on the expansive effect of heat, on the changes in pressure in a gas or vapor that are produced by change of temperature, on changes in the elec- trical properties of bodies, or, in short, on any easily measurable property of a substance, which depends on tem- perature. Ordinary thermometers depend on the expansion of a liquid, such as mercury, alcohol, or ether, contained in a bulb of glass having a long tube or stem in which the liquid rises or sinks as it expands or contracts. 366. Fixed Points. — In order that temperature observations by different observers may be comparable, all ther- mometric scales are based on two fixed temperatures. These are the tem- peratures at which ice melts and that at which water boils under standard atmospheric pressure. 1. The chief precaution to be taken in determining the freez- ing point is to see that the ice is free from salt. Ice from ponds frequently has traces of salts from the soil. Changes in baro- metric pressure do not affect the freezing poir.' of water by as much as 0.001°C., and may, therefore, be disregardc i. 2. The boiling point is determined by the use of some appa- ratus, such as that shown in the figure, so that the whole ther- 203. — Boiling-point apparatus. 240 HEAT mometer up to the point at which the mercury stands in the stem is bathed in steam as. it escapes from the boiling water. The escaping steam is made to pass down around the outside of the vessel so as to prevent the steam in contact with the ther- mometer from being cooled. Impurities in the water may cause it to boil at a temperature slightly above the point at which pure water boils, but the escaping steam will have the same temperature as that from pure water if the pressure is the same. It is for this reason that the thermometer bulb is kept in the steam and is not allowed to dip into the water itself. The boiling point is decidedly influenced by changes in atmos- pheric pressure. An increase of 27.0 mm. in the barometric height raises the boiling point by one whole degree Centigrade. C 100 =3) Centigrade 32 if 212 _, ^ ^ .^ II I M 1 1 I I M I [ I I I I 1 1 1 1 I I 1 1 1 1 1 1 1 M I J 1 1 II g) Fahrenheit jff 80 . „ -I I I I i I I I I I I I I 1 1=3 Reaumur A P B FiQ. 204. — Thcrmometic scales. 367. Scales of Temperature. — In order that a thermometer may be useful in determining intermediate temperatures it must be graduated or divided into intervals or degrees. Three scales are in general use: the Centigrade or Celsius scale, used extensively on the continent and in most scientific investigations; the Fahrenheit scale, used chiefly in English- speaking countries; and that of Reaumur, used to some extent on the continent of Europe. In the Centigrade scale the freezing point is marked zero and the boiling point 100, the interval being divided into 100 degrees. In Fahrenheit's scale the freezing point is 32° and the boiling point is 212°, so that there are 180 degrees between the two. In Reaumur's scale the freezing point is 0° and the boiling point 80°. The relation of the three scales is shown in figure 204. Since 180 Fahrenheit degrees correspond to 100 Centigrade degrees, a Fahrenheit degree is ^i of a Centigrade degree. To change Fahrenheit temperatures to Centigrade we, therefore, subtract 32° and take ^i of the remainder. On the other hand, to change Centigrade temperatures to the Fahrenheit scale add 32° to % of the Centigrade temperature. Or since the ratio of the number of degrees between A and P (Fig. 204) to THERMOMETRY 241 the whole number of degrees betwen A and B is the same in each case, we have the equations C_ ^ f - 32 ^ R 100 180 80 by which relations temperatures on any one scale may be changed to either of the others. 368. Graduation of the Scale. — A thermometer is ordinarily graduated so that each degree corresponds to an equal apparent increase in the volume of the mercury. Thus if the thermometer tube is perfectly cylindrical the degree marks are equidistant, but if the tube is not uniform in diameter the degree marks should be so spaced that the volume between consecutive marks is the same everywhere throughout its length. The graduations will, therefore, be closer together where the tube is wider and farther apart in the narrower portions of the tube. The 50° mark should be so placed that it divides in half the volume between the 0° and 100° points. 369. Arbitrary Feature of Thermometric Scales. — But it is evident that even such a scale depends upon the properties of the expanding substance and if we were to take two thermome- ters, one containing alcohol and the other mercury, and were to graduate them in this manner, while they would agree at the fixed points they might not agree anywhere else. This arbitrary element enters into every scale of temperature. Suppose, for example, we were to define 50° as that temperature which results from mixing equal weights of water at 0° and 100°, respectively. It would be found that if mercury had been taken instead of water the resulting temperature would have been different. And even if we were to define 1° as the rise in tem- perature of a given mass of water due to the addition of Koo of the whole amount of heat required to raise it from 0° to 100°, the scale of temperature obtained would be different from that obtained by using in a similar way some other substance than water. 370. Peculiarities and Defects of Thermometers. — The rise of the mercury in a thermometer when it is heated is due to the difference between the expansion of the mercury and that of the glass bulb; for if the mercury and glass expanded equally the mercury would not rise at all in the tube. Therefore, twomer- 242 HEAT curial thermometers may not agree except at the fixed points, unless they are made of the same kind of glass. The most serious defect in mercurial thermometers is the change in the zero point. When the bulb is cooled after having been heated it takes a long time to return to its original dimen- sions. Thus if a thermometer is heated to 100° and then quickly cooled, the zero point will be found lower than before it was heated; this is known as the depression of the zero point. The bulb continues slowly to contract, but it may be weeks before the original zero point is reached. This depression of the zero point may amount to two- or three-tenths of a degree and is a source of error that affects more or less all temperature observations with such instruments, for the read- ing at a particular temperature depends on whether or not the thermometer has recently been heated to a higher temperature. Researches carried on at Jena have resulted in the production of a special glass for thermometers, known as the Jena normal glass, which is almost free from this defect and is, therefore, employed in making thermometers for exact work. 371. Spirit Thermometers. — Alcohol and ether thermometers can be used at temperatures so low that mercury would freeze, their expansions also are so much greater than mercury that so fine a tube is not required. But these liquids wet the tube and if the upper part of the stem is cooler than the surface of the liquid column the liquid will distil and condense in the upper end of the tube. Alcohol expands 6 times as much as mercury and ether 8J^ times as much, they are therefore suitable for sensitive thermometers, though on account of the pressure of the vapors of these liquids an alcohol thermometer should not be used above 100°C. and an ether thermometer not above 60°C. 372. Maximum and Minimum Thermometer. — For registering maximum and minimum temperatures the form of instrument devised by Six (Fig. 205) is found convenient. In this instrument the bulb B and the tube as far as the mercury column at C is filled with phenol or some liquid having a large expansion coefficient. The mercury column fills the lower part of the tube between C and D while the tube above D is also filled with phenol reaching up to the bulb A which is only partly filled. When the tem- FiG. 205. — Six's maximum and min- imum thermometer. THERMOMETRY 243 perature rises the expansion of the liquid in the bulb B causes the mercury column to sink at C and rise at D, pushing upward a little index of iron in the tube above D which in consequence of friction remains where it is pushed and marks the maximum temperature. On cooling the contraction of the liquid in B causes the mercury to rise at C pushing upward a little index at that point which marks the minimum temperature. To set the instrument the indices are drawn down against the mercury column by means of a small magnet. 373. The clinical tliermometer used by physicians is a maximum thermometer having a short scale ranging from about 95° to 108°F.; the tube is made very flat and narrow just above the bulb. The mercury will readily pass through the constriction in rising, but as it contracts capillary force causes the column to separate at that point, leaving the upper part of the mercury column to mark the maximum point. To set the instrument the mercury is brought back to the bulb by a vigorous shake FiQ. 206. Galileo's air thermometer. 374. Air Thermometer. — Since mercury-in- glass thermometers can be used only between — 40°C. and 450°C., and their readings are so much influenced by the peculiarities of the kinds of glass of which they are made, they are not suited to be used as independent standards. For stan- dard purposes air or hydrogen or nitrogen may be used as the thermometric substance, because with a porcelain bulb such a thermometer can be used from -200°C. up to 1500°C., and the ex- pansion of these gases is so great (more than twenty times that of mercury) that the expan- sion of the glass or porcelain bulb containing the gas is quite insignificant in comparison, and can be allowed for without sensible error. A crude form of air thermometer which is interesting because it was used in 1597 by Galileo, the inventor of the thermometer, is shown in figure 206. The bulb containing air terminates in a tube dipping into a vessel of colored liquid which rises or sinks in the tube, according as the enclosed air contracts or expands. The most evident defect of this arrangement is that the liquid column will rise and fall as the atmospheric pressure changes, even though the temperature of the gas may remain constant. 375. Standard Air Thermometer of Constant Volume. — For the exact measurement of temperature by the air thermometer 244 HEAT it is found most convenient to keep the volume of the air con- stant and use its pressure to measure temperature. A form of instrument devised by Jolly is much used. The bulb A contains the gas to be used, which may be hydrogen or nitrogen or air that has been dried and freed from carbon dioxide. The bulb is connected by a capillary tube with the wider tube at B. The vertical tubes B and DE are connected by a flexible rub- ber tube, which is full of mercury, the mercury column extending up into the glass tubes at B and E. The tube DE is attached to a slide and can be raised or lowered along a fixed scale and clamped at any point. In using the instrument the air in the bulb is first cooled to 0° in melting ice and the tube DE adjusted in height until the mercury at B comes exactly to a fixed mark at the end of the capillary tube. The pressure of the en- closed air is then obtained by subtracting the height of the mercury column BE' from the barometric height which gives the pressure of the external air on E. In a similar way the pressure of the enclosed air may be measured when the bulb is heated to 100° in steam. In this case also the mercury level at B must be adjusted to the same point as before, keeping the volume of the air constant except for small changes in the size of the bulb itself. The pres- sures of the enclosed gas in these two cases may be represented by po and pi, respectively. If it is now desired to determine the temperature of a bath in which the bulb is immersed it is only necessary to measure the pressure p exerted by the gas just as in the other cases. If this pressure is found to be half-way between po and pi the tem- perature of the bath is 50°. Or, in general, if t is the temperature to be determined corresponding to the pressure p i : 100 :: p — po : pi — po In the most refined work we must make corrections for the expansion of the glass bulb itself due both to changes in temperature and pressure, and also take account of the fact that the gas just above B is not at the same tempera- ture as the bulb A. I'lG. 207. — Jolly's air thermometer. THERMOMETRY 245 Such a process would evidently be too cumbrous to employ except for the purpose of standardizing some more convenient working form of instrument, such as the mercurial thermometer or the electrical resistance thermometer. 376. Electrical Methods. — Some very important methods of meas- uring temperatures are based on electrical phenomena and will be more par- ticularly described in that connection. The thermo-electric method determines temperature by measuring the electromotive force set up when the junction of two wires made of different metals is heated. For low temperatures a copper-iron junction may be used, while for high temperatures the junction of a pure platinum wire with one of platinum-rhodium alloy is used. The resistance method depends on the increase in the electrical resistance of a coil of pure platinum wire with rise in temperature (§651). A resistance thermometer consists of a coil of platinum wire mounted in a glass or porcelain tube to protect it from injury and contamination, and provided with connections by which its electrical resistance may be tested. By means of suitable accessory apparatus the temperature of the coil may be read directly without calculations. On account of the range of temperatures that can be measured in this way (from —270° to 1500°C.), and the accuracy and ease with which the determinations may be made, this is one of the most valuable of all methods of temperature measurement. 377. High Temperatures.— For measuring high temperatures the gas thermometer, or electrical methods, or radiation pyrometers may be used. The hydrogen gas thermometer having a porcelain bulb may be used up to 1500°C. (§375). The platinum-rhodium thermo-couple and the electrical resistance ther- mometer may be used up to 1500°C. if protected by porcelain tubes. For the highest temperatures radiation pyrometers are used. These are of two types. One depends on the heating power of the radiation from a mass of molten metal or from the interior of a furnace, and is so devised that it may be used at quite a distance from the hot body if the radiating surface is large. The other type depends on a measurement of the intensity of the light from the glowing hot body or interior of a furnace. Problems 1. Find the Fahrenheit temperatures corresponding to 80°, 20°, —10°, and -50°C. 2. Find the Centigrade temperatures corresponding to 1000°, 98.6°, 0°, and -50°F. 3. What temperature reads the same on both Fahrenheit and Centigrade scales, and at what temperature is the Fahrenheit scale-reading twice that on the Centigrade scale? 4. A temperature interval of 35° on the Centigrade scale is an interval of how many degrees Fahrenheit? 246 HEAT 5. The absolute zero of temperature is —273° on the Centigrade scale; what is it on the Fahrenheit scale? 6. Calculate the Fahrenheit temperatures of the melting points of iron, copper, lead, and mercury. (See p. 288.) Expansion of Solids 378. Expansion of Solids. — Almost all solids expand when heated. Isotropic bodies, such as glass and all liquids, expand equally in every direction. Crystals in general expand differ- ently in different directions, and may even contract along one direction and expand in another, but in most cases the expansion more than makes up for the contraction so that there is on the whole an increase in volume with rising temperature. 379. Coefficient of Linear Expansion. — The fractional part of its length that a rod elongates when raised one degree in temperature is called its coefficient of linear expansion. Let the length of a bar at 0° be U, and let a be its coefficient of linear expansion, then its increase in length for a rise in temperature of 1° will be Ua, and for t degrees its increase in length is Uat, so that its total length I at the higher temperature is: I = lo -\- loal or I = U{1 + at). In this formula I may be taken as the length of the bar at a temperature t degrees higher than that at which its length is U, even though the latter may not be its length at 0°C. It must not be supposed that the coefficient of expansion of a substance is the same at all temperatures, for in general it in- creases as the temperature rises. In the above formula a represents the average value of the coefficient throughout the rise in temperature represented by t. 380. Coefficient of Volume Expansion. — If a cube of substance is taken measuring 1 cm. each way at 0°, and having a coefficient of linear expansion a, then its linear dimensions at t° will be 1 -^ at and its volume will be (1 -I- aO' = 1 + Zat + ZaH^ + aH"^ but the coefficient a is so small that the terms involving o' and o^ may be neglected and the volume may be expressed as 1 + Zat. EXPANSION 247 ■ia, therefore, respresents the increase in volume of a unit cube for one degree rise in temperature and may be called the coefficient of cubical or volume expansion; hence the coefficient of cubical expansion is three times the coefficient of linear expansion in an isotropic body. 381. Measurement of Coefficients of Expansion. — When the substance whose coefficient of expansion is to be obtained has the form of a long rod, its expansion may be measured by a comparator such as that shown in the figure. Two microscopes are set on two marks on the bar, one near each end. The microscopes are firmly clamped to a solid base which is kept free from temperature change. The bar to be examined is enclosed in a box provided with glass windows Fig. 208. — Comparator. through which the microscopes are set on the marks. The bar is first packed in melting ice and the micrometers attached to the microscopes are set on the two marks. Then water at a higher temperature is caused to circulate through the box, maintaining a constant higher temperature, and the microme- ters are again set on the two marks. The difference between the micrometer readings gives the elongation of the bar and accurate thermometers give the change in temperature. The whole length of the bar between the marks is then carefully determined. If this length is I and the elongation is e when the temperature is raised from t to t', the coefficient of expansion a is found from the relation e = la(t' — t), or lif~t) 248 HEAT This is the average value of the coefficient between the tempera- tures t and t'. 382. Expansion of Crystals. — Crystals that do not belong to the regular system expand differently in different directions. A sphere cut out of such a crystal will become an ellipsoid when its temperature is raised. In some cases two of the axes of the ellipsoid would be found of the same length and in some cases all three would be different. The directions in the crystal corre- sponding to the axes of the ellipsoid are called the axes of thermal expansion. In quartz the expansion at right angles to the axis of the crystal is nearly twice the expansion in the direction of the axis. Table of Coefficients of Linear Expansion, per Degree Centigrade Invar 00000096 Brass 0000189 Glass 089 Silver 194 Platinum 089 Aluminum ... 222 Steel 110 Lead 280 Iron 117 Zinc 298 Copper 167 Ebonite 770 These values are approximate. The exact value for any substance de- pends on the state of hardness, purity, and temperature of the specimen. 383. Some Illustrations. — An iron tire when heated expands so that it can easily be slipped over the wooden rim of the wheel, which it binds firmly on cooling. So the breeches of cannon are strengthened by having a series of tubes shrunk over the inner core, in this way producing an outside compression of the core which enables it to withstand the enormous pressure of the powder gas. Allowance has to be made for expansion in case of bridges. In a steel bridge 1000 ft. long the change in length between extremes of summer and winter may amount to 8 in. The aggregate length of the rails in a mile of track may be 4 ft. longer when hottest than when coldest, so that an allowance of about 0.3 of an inch is needed for each 30-ft. rail. The grate bars of furnaces rest loosely in their supports in order to allow ex- pansion, and long steam pipes are provided with sliding or "ex^ pansion" joints unless the bends in the pipe are such as to yield elastically to elongation and contraction. EXPANSION 249 Quartz crystals have very large expansion, and when unequally heated fly to pieces because of the great strains which result in that case. When quartz is fused, however, into a glass, its coefficient of expansion is extremely small, and vessels made of fused quartz may, when red hot, be suddenly quenched in water without breaking. A specially prepared nickel-steel, having 36.1 per cent, of nickel and known as invar, has a temperature coefficient of only 0.0000009 or J^o as large as platinum. It is of great value for measuring bars and tapes, and for pendulums. Since the expansions of glass and plati- num are nearly equal, platinum wires are used wherever wires are to be hermetically sealed into glass, as in case of the connec- tions of an incandescent-lamp filament. Wires of another metal having a greater coefficient of expansion would shrink away from the glass on cooling, leaving a crack J ^ I through which air could pass. F f F F C C C C Fio. 209. Fia.210. Pig. 211. 384. Compensated Clock Pendulums. — The elongation of a clock pendulum with rising temperature causes it to swing more slowly and the clock loses time. Dry wood pendulum rods have very small expansion and so are- sometimes used, but they are affected by moisture. For the most accurate clocks compensated pendulums are used. One of the best forms is Graham's mercurial pendulum (Fig. 210), where a reservoir of glass or steel containing mercury is hung by a steel rod. If properly designed, the rais- ing of the center of oscillation (§142) due to expansion of the mercury is balanced by the lowering due to the elongation of the steel suspending rod, so that the effective length remains constant. 250 HEAT In Harrison's gridiron pendulum (Fig. 209) the expansion of the steel bars FF will lower the bob, while the expansion of the brass rods CC will tend to raise it. If the upward elongations of C and C for a given change in temperature are together equal to the combined downward elongations of FFF the bob will neither be raised nor lowered. 385. Watch Compensation. — The balance-wheel of a watch if un- compensated will run slower as the temperature rises, because the elas- ticity of the hair spring is less at higher temperatures, and also the expansion of the wheel makes its moment of inertia greater. Compensation is secured by making the balance-wheel as shown in figure 211. The rim is made of brass on the outside and steel on the inside, and instead of being continuous it is cut in two segments which are connected rigidly by a cross-bar. When the temperature rises the brass outer side of the rim expands more than the steel inner side so that the free ends of the segments bend inward, thus carrying part of the mass in toward the axis and so tending to compensate the outward expansion of the cross-bar, and the diminished elasticity of the hair-spring. The adjustment is completed by means of little screws set in the rim of the wheel. Those near the free points tend to increase the compensation, while those near the fixed ends of the segments have the opposite effect. 386. Force of Contraction. — The force produced by the shrinking of a bar on cooling is the same as would be required to stretch it by the same amount at the same temperature. Problems 1. What is the change in length of the steel cables of a suspension bridge 2000 ft. long between the extremes -20°F. and 97°F.? 2. A brass meter bar is correct at 15°C.; what will be its length at 20°C.? 3. What is the coefficient of expansion of a 30-ft. steel rail on the Centi- grade scale and also on the Fahrenheit scale if it changes in length 0.234 in. when the temperature ranges from — 17°F. to 100°F.? 4. At 20°C. a brass plug 6 cm. in diameter is J^oo of a millimeter too large to fit a hole in a steel plate. At what temperature will it just fit? 6. A glass specific-gravity bottle has a capacity of exactly 300 c.c. at 15°C. ; what will be its capacity at 0°C.? 6. A cylindrical zinc pendulum bob has a hole running lengthwise through it in the direction of its axis through which the steel pendulum rod passes, and rests on a cross-piece at the lower end of the rod. How long must the rod and the bob be that the center of gravity of the bob may remain constant at 95 cm. below the point of support while the temperature changes? Take expansion coefficient of steel as 0.000010 and for zinc 0.000029. Expansion of Liquids 387. Expansion of Liquids. — When a liquid contained in l bulb provided with a long neck is heated, it rises in the stem by EXPANSION 251 an amount which depends on the difference between the expan- sion of the liquid and that of the bulb. The rise indicates what is known as the apparent expansion. If a bulb containing liquid is suddenly plunged into a vessel of hot water the liquid in the stem may be observed to sink at first because the bulb expands before the liquid within is fully heated. To determine the expansion of a liquid take a bulb with a gradu- ated stem like the tube of a thermometer and calibrate the stem, or determine the relation between the volume of the whole bulb and the volume of the divisions of the stem. This may be done by filling the bulb with mercury and weighing it, and then separately weighing the amount of mercury required to fill a certain number of divisions of the stem; the relative weights give the relation between the volumes. The bulb is now filled with some liquid up to a certain mark on the stem and then packed in ice or cooled to some steady low temperature and the point to which the liquid contracts is observed. It is then warmed to some higher temperature and the point at which the liquid stands is again observed. From the divisions of the stem between these two points the apparent increase in volume is deter- mined, and if this is divided by the original volume and then by the rise in temperature, the apparent coefficient of expansion is obtained. The expansion of a glass bulb of volume V, is Vat where a is the coefficient of volume expansion of glass and t is its rise in temperature, while the ex- pansion of the contained liquid is Vbt where b is its coefficient of expansion. Since the rise of the liquid in the stem is due to the excess of its expansion over that of the bulb the apparent expansion is Vbt - Vat = Vt(jb - a). The apparent coeflBcient of expansion is therefore b — a, or the difference between the coefficients of expansion of the liquid and the bulb. Hence the coefficient of expansion of the bulb must be deter- mined before that of the contained liquid becomes known. This Fig. 212.— Bulb with graduated stem. 252 HEAT may be accomplished either by studying the expansion of a bar made of identically the same glass or by observing the apparent expansion in the bulb of some liquid whose coefficient of expansion is already known. 388. Weight Thermometer Method. — ^In case of a liquid such as mercury which has great density and does not wet the glass the apparent coefficient of expansion in a bulb may be determined as follows. The bulb and stem are both completely filled to the very top at the lower tem- perature, and when the temperature is raised the expanding liquid escapes in drops at the end of the stem, where it is caught and weighed and the amount of the expansion thus determined. 389. Absolute Expansion of Mercury. — The expansion of mercury has been studied with great care because it is the liquid best adapted for use by the weight thermometer method (§388) in determin- ing the coefficients of expansion of bulbs to be used in the study of other liquids. Its coefficient of expansion was deter- mined by Dulongand Petit by the follow- ing method which is indepe'ndent of the expansion of the tube containing the mercury. Two vertical tubes (Fig. 213) connected at the bottom by a very thin horizontal cross tube, contain mercury. One is packed in ice and the other is heated to some known tempera- ture t. Then by the laws of hydrostatics the less dense liquid will stand higher, and the height of the cold column of mercury multiplied by its density is equal to the product of the height of the hot column by its density, or hd = hodo (1) But as a given mass of mercury expands in volume it diminishes in density, so that V:Vo = do:d (2) and since V:Vo = l + at:l do:d = l+at:l or Ice Cold Warm Fig. 213. — Expansion of mercury. EXPANSION 253 do = d{l + at) and by equation (1) A ho l+at (3) So that by measuring the heights h and ho and determining the temperature t of the hot column the coefficient of volume ex- pansion a of the mercury becomes known. 390. Expansion of Water. — The expansion of water has been determined with great accuracy at the German National Labora- tory or Reichsanstalt by the method just described. The curve of expansion (Fig. 214) shows that when water is heated from 0°, it first contracts and then expands, reaching its maximum density at almost exactly 4°C. VOLUME IMi 1.003 1.002- 1.001- 1.000 THg CORVE GIVES THE CHANGE IN LENGTH WITH TEMPERATURE OP A COLUMN OF WATER 30 FT. LONG / / / V -- -^^ 10 15 20 '25 TEMPERATURE 30 Fio. 214. — Expansion 0°-30° of water from FiQ. 215. — Hope's apparatus for determining the temperature of maximum density of water. This fact is of great importance in nature, for the cooling of a lake goes on rapidly at first, the cooled surface water settling to the bottom, thus aiding the cooling of the whole by convection currents. But when the water has reached 4°C. any farther cooling must be accomplished by the slow process of conduction, for the colder water being less dense will remain at the top. So ice forms at the top and only gradually thickens downward, and if the lake or pond is not too shallow the bottom does not fall below 4°C. for there is a small supply of heat flowing out from 254 HEAT the earth which makes up for that lost by conduction toward the surface. Hope made use of the apparatus shown in figure 215 to deter- mine the temperature of maximum density of water. A vessel of water provided with thermometers at the top and bottom is cooled above by being surrounded by ice. The lower part of the vessel is carefully jacketed with cotton or felt to prevent the inflow of heat through the sides. The upper thermometer will at first stand higher than the other, but finally the lower will stand steadily at 4°C. while the upper will cool below that point. The temperature of maximum density of water is lowered when salt is dissolved in it. Sea-water attains its maximum density only at —3.67° which is below its normal freezing point. Density and Volume per Gram of Water Temperature Density Volume per gram 0=0. 0.999867 1.000132 c.c. As found at the 3.98 1.000000 1.000000 Reichsanstalt by 10.00 0.999727 1.000272 the method of 15.00 0.999126 1.000874 bal a n c i n g col- 20.00 0.998229 1.001773 umns. 25.00 0.997071 1.002937 30.00 0.995673 1.004345 35.00 0.994057 1.005977 40.00 0.992241 1.007819 60.00 80.00 0.9834 0.9719 1.0169 1 . 0289 • Approximate values. 100.00 0.9586 1.0431 Coefficients of Expansion of Some Liquids atO" at 20° at 40° Average be- tween 0-40 Water Mercury Alcohol -0.000067 0.000179 0.000206 0.000180 0.000388 0.000181 0.000192 0.000180 00112 Ether 0.00151 0.00165 0.00189 0.00167 It will be observed that these coefficients are larger than those of solids, and that in general they increase with the temperature. EXPANSION 255 Expansion of Gases 391. Expansion of Gases. — The expansion of ga ses, withheat is much greater than that of soUds or Uquids and is remarkable fdrT)eing nearly the same for all gases. On account of the great compressibility of gases there are two distinct conditions under which their expansion by heat may be determined. First, the pressure may be kept constant and the volume expansion of the gas measured as the temperature rises, or, second, the volume of the gas may be kept constant and the increase in pressure with rising temperature may be measured. If the gas perfectly obeyed Boyle's law its coefficient of expansion at constant pressure would be equal to that with constant volume. 392. Expansion at Constant Pres- sure. — Gay-Lussac was the first to carefully study the expansion of gases at constant pressure, but Regnault by the apparatus indicated in the diagram obtained far more accurate results. The bulb A is filled with the gas to be studied and cooled to zero by means of melting ice. By the stopcock E' it is then shut off from the gas supply and connected with B which is com- pletely filled with merciu-y up to the opening of the small tube at the top, and if the gas in the bulb is at the same pressure as the outer air the mercury will stand in the open tube C at the same level as in B. The bulb A is then heated to any desired temperature, say to 100°, and as the gas expands mercury is allowed to flow out of the stopcock at the bottom so that it is kept at the same level in B and C, thus maintaining the pressure constant. Part of the expanded air is in A at 100° and part in B at the temperature of the water bath which surrounds the tubes. The tube B is graduated, so that the exact volume of the ex- panded gas may be determined. Fig. 216. 256 HEAT 393. Increase in Pressure at Constant Volume. — Regnault also was the first to make accurate measurements of the increase in pressure of a gas when the volume is kept constant. The apparatus used is the same as that described above (Fig. 216) but when the bulb A is heated and the expanding gas begins to force down the mercury in B, more mercury is poured into C until the additonal pressure again causes the mercury to exactly fill B. In this way the heated gas is kept confined in the bulb A, and its pressure is measured by the height of the mercury in C above that in B together with the height of the barometer. In this experiment the bulb is expanded slightly both by the rise in temperature and by the increased pressure in the interior, and on account of this change in volume a small correction must be applied. Coefficients of Expansion of Gases Gas iDcrease in volume at constant pressure per degree C. Increase in pressure at constant volume per degree C. Air 0.003671 . 003668 Oxygen 3674 Nitrogen 3671 3661 3669 3710 3903 3668 Hydrogen 3660 Carbon monoxide 3667 Carbon dioxide. . .... 3687 3845 One cubic foot of air at 0° would expand to 1.367 cu. ft. at 100°, an increase of more than ^■i of its volume at 0°C. From the above table it is clear that different gases have nearly equal coefficients of expansion. This is known as the law of Charles or Gay-Lussac. The increase in volume of a gas per degree rise in temperature is about 3^73 of its volume at 0°C. 394. Absolute Scale of the Air Thermometer. — According to Charles' law, gases, at constant pressure, expand nearly 0.00366, or M73 of their volume at zero for a rise in temperature of one degree Centigrade. Consider a cylinder filled with air or hydrogen and closed by a piston which always exerts the same pressure on the enclosed gas. When the gas is at 0° suppose EXPANSION 257 373 273' jNssmw-^ .700' the piston stands at A, then when the gas is warmed to 100° it expands and the piston rises to B. If we divide the space from A to B into 100 equal parts and continue the graduation down below A, marking off equal spaces for every degree, we shall find that there will be 273 degrees below the zero. If we now call the bottom of the cylinder the zero point we shall have a scale of temperature in which 273° will be the freezing point of water and 373° „ B will be the boiling point. This scale is called the absolute scale of the air thermometer, and its zero is called the absolute zero. It is only necessary to add 273° to any Centigrade tem- perature to obtain the corresponding tempera- ture on the absolute scale. It will be seen from the way in which the scale is obtained above, that the volume of the gas in the cylin- der is proportional to its temperature on the absolute scale, and since all gases have nearly the same coefficient of expansion, it may be stated as true in general, for all gases that are not too near their points of condensation, that the volume of a gas is very nearly proportional to its absolute temperature when the pressure j-j^^ 217. is kept constant. So also when the volume is kept constant the pressure of a gas is nearly proportional to the abso- lute temperature. At the absolute zero the pressure would be zero. There are good reasons for believing that the pressure of a gas is proportional to the energy of vibration of the molecules and therefore that at the absolute zero the molecules of gas have no energy of motion. Consequently this is the lowest possible temperature, for if a substance has no energy of motion to give up, it cannot give out any heat and be cooled farther. Of course no gas would actually be reduced to zero volume, however much it might be cooled, though its pressure might be reduced to zero. It would condense into a liquid and cease to behave as a gas before reaching zero volume. An entirely independent and more conclusive line of reasoning has led to the establishment of the absolute thermodynamic scale of temperature. (See Appendix 1.) This is independent of the -273° 258 HEAT properties of any particular substance, and its zero is the lowest possible temperature. Experiment shows that the absolute scale based on the expansion of gases agrees almost exactly with the thermodynamic scale except at the very lowest temperatures. The zero of the gas scale is therefore properly called the absolute zero. By means of liquid air, temperatures as low as — 200°C. may be obtained, and by the evaporation of liquid hydrogen — 258°C. has been reached, only 15° above the absolute zero, while the boiling point of liquid helium is found by Onnes to be — 268.5°C., only 4.5° above the absolute zero. At these low temperatures rubber and steel become as brittle as glass, lead becomes stiff and elastic, while the electrical resistances of metals are greatly reduced. 395. General Gas Formula. — As was shown in the last para- graph, the volume of a given mass of gas kept at constant pressure is proportional to its temperature on the absolute scale; that is T~ To where T = 27S+t and To = 27Z+tQ, T and To being the absolute temperatures corresponding to t and s, '^t ip A?' -^^ ,<\0 P^ .^^ „- -^v V ?f } P\<* =-» '-v. \ h $^ /f N ./if A-^ \ A A o<> V /'^' • - - +.0001 09 ■ o c -.O001 FiQ. 268. — Diurnal changes in dip, declination and intensity at Kew. The curves of figure 268 show the average variations at Kew, near London. It will be observed that the maximum changes take place in the daytime and may be due to variations in temperature of the earth's surface. 496. Irregular Disturbances. — The magnetic needle is also often disturbed by what are called magnetic storms; these dis- turbances usually accompany any marked display of the aurora 336 MAGNETISM borealis, and they also seem to be more prevalent at times of sunspot maxima. 497. The Earth a Magnet. — It was suggested by Dr. William Gilbert (1600), physician in the court of Queen Elizabeth and the first to take up the scientific study of magnetism, that the earth itself was probably a great magnet, and later observations have borne out this idea. Two well-marked magnetic poles being found, one northwest of Hudson Bay in North America and the other south of Australia. But while there is this general resemblance to a simple magnet, the direction of the magnetic force varies from place to place in a way that cannot be wholly aQcounted for by the supposition of simply two poles. The magnetism of the earth seems to be due to a variety of causes, the presence in the earth of magnetic masses is a cause of local variations and may have great influence in the surface layer of the earth, but it seems probable that the temperature in the interior of the earth is too high for it to possess any very strong magnetism. Electric currents flowing in the surface of the earth and due to its varying temperature as first one side and then another is exposed to the sun, as well as currents of elec- tricity in the upper air, probably play an important part in determining its magnetic state. But the complete explanation has not yet been given, and any theory to be satisfactory must account for the remarkable secular changes in its magnetism which go on slowly and progressively year after year. 498. Gauss' Method of Measuring the Horizontal Intensity. — The horizontal component of the earth's magnetic force may be measured by the following method due to Gauss. A small steel bar magnet is sus- pended horizontally by a fine fiber in a closed box by which it is protected from air currents. It is then set oscillating through a small arc and the period of oscillation carefully determined. This period depends on M the magnetic moment of the magnet and on H the horizontal component of the earth's magnetic force. By §488 HM = — y^ where K is the moment of inertia of the magnet, a quantity that is deter- mined by its mass, size, and shape, and T is the period of a complete oscilla- tion. The product HM is thus found. To determine the relation of ff to M a second experiment is necessary. Suppose P is the point where the magnetic force H is to be determined and UNIT TUBES OF FORCE 337 where the period of oscillation of the magnet NS was observed in the first experiment. Place at P a very short magnetic needle, while the magnet NS is placed exactly east or west of P and with its axis on the east and west line, as shown in Fig. 269. If r is the distance from the center of NS to P, then the force at P due to the magnet is, as shown in F = 2M Then at P the force H due to the earth and the force F due to the magnet are at right angles to each other, as shown by the arrows in the figure. The needle at P will take the direction of the resultant force R and will therefore be deflected through the angel a, but whence F 2M tan o = -fj or tan a = —rn H M r^ tan a Fig. 209. This expression shows that to determine the ratio of H to Af it is only necessary to measure the distance r and the angle of deflection a. Having by the first experiment determined the product HM and by the second the ratio -jr>> it only remains to multiply the two together tofind W^ and so determine H. Unit Tubes of Fokcb 499. Number of Lines of Force. — Up to this point lines of force have been regarded as simply expressing the direction of the force in the magnetic field. We must now follow Faraday in a very remarkable development of the idea. In a stream of water flowing steadily lines may be imagined drawn which at every point are in the direction of flow, and which may be called stream lines. An infinite number of such lines may be drawn. The whole stream may then be conceived to be divided up into tvbes o} flow by means of surfaces which everywhere coincide with stream lines. These tubes of flow may be taken of such a size that each will transmit the same 338 MAGNETISM quantity of water per second, say one cubic foot. Then, where the stream is most rapid, the cross sections of the tubes of flow will be smallest and they will widen out as the velocity dimin- ishes. The whole number of such tubes in the stream will be equal to the number of cubic feet of water transmitted per sec- ond. These tubes of flow may be called unit tubes, and the num- ber of them crossing perpendicularly a surface 1 sq. ft. in area is equal to the number of cubic feet of water crossing that area per second. Thus the number of unit tubes passing perpen- dicularly through a unit surface at any point in the stream is equal to the velocity at that point. Now in the same way the magnetic field may be conceived as divided up into unit tubes by means of surfaces parallel to the lines of force. And it may be proved that where such a unit tube is smaller the field is more intense, and where it widens out the strength of field is less, just as the velocity varies in case of the stream of water. So that it is possible to take these tubes of such a size that the number passing perpendicularly through a square centimeter of sur- face at any point may be equal to the strength of the magnetic field at that point. We may imagine that each unit tube is represented by a line of force drawn through its center or axis, and when the phrase number of lines of force is used it refers to such lines. Using the term in this way, it is clear that in the case shown in figure 270 more lines of force pass through a card in position C than in position A, as the force is greater at C than at A, and con- sequently there are more lines of force to the square centimeter. Clearly, also, fewer lines of force pass through the card in position B than in A and most of all in position D. If the card were placed parallel to the lines of force none at all would pass through it. The number of lines of force through A is found by taking the average strength of the field at the surface A and multiplying this by the area of A in square centimeters, since the number of lines per square centimeter is equal to the strength of the field at that Fig. 270. UNIT TUBES OF FORCE 339 point. If the surface is oblique to the Hnes of force as at B, the number of Unes of force passing through it will be found by mul- tiplying the number in the perpendicular position A by the cosine of the angle a, or, what comes to the same thing, multiply the average strength of the field at the surface B by the projection of that surface on a plane at right angles to the lines of force. Problems 1. How many lines of force pass through a square meter of floor area where the total strength of the earth's magnetic field is 0.6 and the lines of force are inclined 60° from the horizontal? 2. How many lines of force in this case would pass through an area of 1 square meter on an east and west wall, and how many in case the wall ran north and south? 3. How many lines of force pass through an area of 4 sq. cm. placed as at A in figure 270, with its center 12 cm. from each pole of a magnet 20 cm. long between poles, strength of poles being 288? 4. How many lines of force pass through a circle 1 cm. in diameter placed 8 cm. from the north pole of a bar magnet 16 cm. long, which points directly at it and has poles of strength 200? The plane of the circle is perpendicular to the axis of the magnet. 600. Lines of Force Inside a Magnet. — The lines of force of a magnet are not to be supposed as only outside of it. If we Fig. 271. imagine a minute magnetic needle placed in a crack extending across the magnet it will be acted on most powerfully by the poles N'S' on each side of the crack, but it will also be affected by the attraction of the end poles N and S of the magnet. In conse- quence of the superior influence of the poles N'S' , it will set its north pole toward the left or ;S'. If we now imagine the cleft 340 MAGNETISM shifted along toward the north end of the magnet, the force inside the cleft will become less because it will be nearer to N, which pole tends to make the needle point in the opposite direction. But still the needle will point from right to left. When the cleft comes infinitely near to the end N, the magnetism of N and of S', which form two opposite and equally magnetized layers, will neutralize each other so that the effect is the same as though the needle were just outside the magnet at N. We see in this way that the force in the cleft is absolutely continuous with that out- side of the magnet : there is no abrupt change in passing through the surface. The force in such a cleft is called the magnetic induction and the lines of force outside of a magnet form con- tinuous closed curves with the lines of induction inside of the magnet. The lines of force outside are also called lines of induc- tion, as there is no distinction between the two except inside of a magnetic medium. What is called the positive direction of these lines is from the north to the south pole outside of the magnet. Of course as many lines of force as emerge from the north pole enter at the south pole, and all the lines of force or induction in the magnet pass through its middle section. Looked at in this way, the poles are seen to be simply those regions where the lines leave or enter the magnet, and the most intensely mag- netized portion of the magnet is the center where the lines of in- duction are closest together. If a little block could be cut from the center of the magnet without disturbing its magnetism, it would be found a more powerful magnet than a similar block cut from any other part where the lines of induction are not so close. Warning. — In using the words entering and emerging with reference to lines of force nothing like flow or motion must be supposed; when what we arbitrarily call the positive direction of the line of force is toward a surface, it is spoken of as entering it; and when that direction is away from a surface the line of force may be said to leave the surface or emerge from it. 501. Influence of the Shape of a Magnet on Its Power and Betentiveness. — A short thick bar of steel is more difficult to magnetize strongly than a long thin one and loses its magnet- ism more easily. A thick magnet may be thought of as made up of a bundle of thin ones of the same length. But it is clear that in such a bundle each little magnet would tend to set up lines of MAGNETIC INDUCTION 341 force down through its neighbor in such direction as to oppose or weaken the other's magnetism. Thus there is a demagnetizing tendency which is greatest in a short thick magnet. Horseshoe magnets are long and have their poles close together and consequently there is very little demag- netizing tendency. There is, however, a tendency for the lines of force in this case to pass across on the inside of the poles instead of out at the ends. A soft-iron block placed across the poles, and called an armature or keeper, provides an easy path for the lines of force from one end around to the other and thus tends to keep the poles near the ends. 503. Ring Magnet. — A uniform ring of iron or steel may be magnetized by means of an electric current so that the lines of force are circles entirely within the substance of the ring. In such a case the magnet has no poles as there are no places „ where the lines of force enter or leave the ring. Such a magnet has no external field of force and would not act on a magnetic needle placed near it, and yet it is magnetized, as will be evident if it is broken, for in that case each half will show two poles. Magnetic Induction 603. Induction Studied by Iron Filings. — If the lines of force of a horseshoe magnet are examined by means of iron filings on a plate of glass, as described in §483, and if a bar of soft iron is then placed a short distance in front of the poles of the magnet and the field again examined in the same way, a notable change will be observed. The lines of force are bent toward the two ends of the soft-iron bar as though they could be established in the iron more easily than in the surrounding medium. And the softer the iron and the more easily it is magnetized, the greater the number of lines of force that will pass through it rather than the more re- sisting medium around it. Thus the presence of the iron makes the field of force weaker beyond it, and the nearer the iron bar is to the poles of the magnet the more lines of force will be drawn into it and the fewer there will be in other parts of the field. 604. Permeability. — The ease with which lines of force may be established in any medium as compared with a vacuum has 342 MAGNETISM been called by Lord Kelvin the permeability of the medium. Thus iron has a permeabihty several hundred times greater than air. Most other substances have a permeability which is sensi- bly the same as air or vacuum, and, therefore, the magnetic field is practically the same in wood, glass, or water as in air. A hydraulic analogy may aid in forming a clear conception of this subject. Imagine a stream of water continually flowing out of the north of the horseshoe magnet (Fig. 273) and entering its south pole. Suppose the medium surrounding the magnet was of a uniform porous nature that opposed considerable re- sistance to the flow from N to *Si. The lines along which the flow would take place would be like the lines of force in the field before the soft iron was introduced. Now imagine a cavity to be made in the porous medium having just the size and position of the soft-iron bar. The lines of flow would now tend toward this cavity through which the liquid would flow freely and a correspondingly smaller flow would take place in other regions. Fig. 273. Fig. 274. — Bar of soft iron parallel with lines of force of field. Fig. 275. — Bar of soft iroc across the lines of force. The lines of flow in this case correspond to the lines of force when the soft-iron bar with its great permeability is in the field. 505. Magnets Formed by Induction. — When a soft-iron bar is placed in front of a magnet as shown in figure 273, at the end nearest the north pole of the magnet the lines of force are directed toward the end of the bar as toward the south pole of a magnet and at the other end they are directed away from the bar as from a north pole. The bar of iron thus becomes a magnet by induction. MAGNETIC INDUCTION 343 If it were of steel it would retain some of this magnetism when taken out of the field. Suppose a long bar of soft iron to be placed in a magnetic field parallel to the direction of the lines of force. The result will be as shown in figure 274, lines of force will be drawn into the bar in consequence of its great permeability entering it at one end and leaving it at the other, so that one end becomes a south pole and one a north pole. On each side of the bar the field is weakened. When, however, the bar is placed across the field of force as in figure 275 it will have only a very slight effect on the field of force since the lines of force can pass through only a small thickness of iron. So also a thin flat sheet of iron placed perpendicular to the lines of force of the field would have practically no effect on the field. 506. Effect of Heat and Jarring in Case of Magnetizing by Induction. — The magnetism induced in an iron or steel bar placed in a magnetic field parallel to the lines of force may be increased by striking the bar with a hammer or jarring it while under the influence of the field, also by heating the bar red-hot and allowing it to cool in the magnetic field. These disturbances seem to facilitate the arrangement of the molecules under the in- fluence of the magnetic force and help to overcome the resistance to magnetization which especially characterizes hard steel. 507. Magnetic Induction in the Earth's Field. — If a bar of soft iron having no permanent magnetism is placed in the earth's field parallel to the lines of force, that is, in the direction of the dipping needle, its lower end in north latitudes will become a north pole and its upper end a south pole, as may be shown by a magnetic needle. If jarred by the blow of a hammer while in this position it will be found permanently magnetized. If, however, it is placed at right angles to the lines of force of the earth it is scarcely magnetized at all (§505). In consequence of induction iron ships are magnetized by the earth differently when pointing in different directions. In such vessels the standard compass is usually compensated by having soft-iron bars so placed near it that the magnetism in- duced in them will in every position just balance that induced in the ship, while permanent steel magnets may be used to compen- sate the permanent magnetism of the ship. 344 MAGNETISM 508. Hysteresis. — When the magnetic field in which a mass of iron is placed is varied in strength, the changes in the magnet- ism of the iron lag behind the changes in the field. This is known as hysteresis and is discussed in connection with the magnetiza- tion of iron by electric currents, §680. FiQ 276. Permeability, Diamagnetism, and Influence of Medium 509. Magnetic Substances Attracted. — When a fragment of iron is placed in a magnetic field it experiences a force in that direction in which the strength of the field increases most rapidly. If at A (Fig. 276) it is drawn directly toward the magnet in the direction of the lines of force. If at B it is drawn toward the magnet at right angles to the lines of force. If at C it will be drawn in a direction oblique to the line of force somewhat as shown. If it is in a uniform field, as in the earth's magnetic field, or is at a point in a magnetic field where the force is a maximum or a minimum, it will be in equilib- rium and have no tendency to move in any direction. Such a point of equilibrium would be found mid- way between two equal poles either like or unlike. If the fragment is long in shape it will turn and yoint in the direction of the line of force, but it will not always tend to 'mov& along that line. Any substance whose permeability is greater than vacuum will act in this way in a vacuum and such are known as para- magnetic or simply magnetic substances. 610. Diamagnetic Substances. — Faraday (1845) experimented on the behavior of a great variety of substances in the intense field between the poles of a powerful electromagnet. A little oblong of pure copper when suspended by a fine fiber in this field was found to set itself at right angles to the lines of force, as shown in figure 277. So also fragments of wood, paper, aluminum, bismuth, glass, and many other substances. These Fig. 277. — Bismuth in magnetic field. INFLUENCE OF MEDIUM 345 substances Faraday called diamagnetic. Substances like nickel, cobalt, and manganese which behave like iron, setting them- selves in the direction of the lines of force, he called paramagnetic or magnetic. Diamagnetic substances when placed in a magnetic field are driven from a stronger field toward a weaker, the force acting on a fragment of such a substance being in the direction in which the strength of the field diminishes most rapidly. This may be well shown in the following way. A ball of bismuth, which is the most strongly diamagnetic substance known, is suspended between the poles of a powerful electromagnet, being hung from one end of a light arm of wood which is itself supported in horizontal position by a delicate bifilar suspension, so that the slightest force will cause the arm to swing around carrying the ball out of the magnetic field. If while the ball hangs between the two poles the current is applied to the electromagnet, the bismuth ball will at once be driven aside out of the intense field. The setting of the diamagnetic bars across the lines of force described at the beginning of this section finds its explanation in the preceding experiment; for the field of force between the magnet poles is most intense next the poles as is shown by the crowding together of the lines of force, and so the ends of the bar are in a much less intense field when the bar stands across the lines of force than if it were to be directed along them; it therefore assumes the former position. 511. Influence of the Medium. — By the following interesting experiment Faraday showed that the medium surrounding a body in a magnetic field plays an important part in determining the magnetic force upon it. When a thin-walled glass capsule, long in shape, is filled with a weak solution of ferric chloride and suspended between the poles of a magnet, it sets itself along the lines of force showing that the ferric chloride is magnetic. This happens whether the cap- sule is hung in air or water. If, however, it is surrounded by a solution of ferric chloride stronger than that within the capsule it will act as if diamagnetic, placing its length across the lines of force. 512. Permeability of Magnetic and Diamagnetic Substances. — When the permeability of a substance is greater than that of 346 MAGNETISM the surrounding medium, the lines of force are drawn in toward the substance, as already discussed in §505 and as shown in figure 278 which represents the disturbing effect of a ball of substance whose permeability is greater than that of the medium around it. If, however, the permeability of the ball is less than that of the medium, the lines of force will be spread, as shown in figure 279. A magnetic needle placed near the ball will point aside instead of toward it. In the first case if the ball is in a field that is not uniform, as near the pole of a magnet, it will be attracted or drawn toward the stronger field. If, however, the ball has a permeability less than the surrounding medium, it will be driven away from the pole toward a weaker field. Magnetic or paramagnetic substances may then be defined as those whose permeability is greater than that of vacuum, while those whose permeability is less than vacuum are diamagnetic. Fig. 278. — Permeability of ball greater Fig. 279. — Permeability of ball less than that of medium. than that of medium. 513. Magnetism of Gases. — Faraday also studied the magnetic qualities of different gases. Oxygen gas was found to be at- tracted toward the poles, while hydrogen was repelled. Oxygen was thus shown to be more permeable than air. Later experi- ments have shown liquid oxygen to be decidedly magnetic. 514. Magnetic Alloy. — In 1903 Heusler made the very inter- esting discovery that an alloy of 25 parts manganese, 14 alumi- num, and 61 copper, had decided magnetic properties, although none of the substances of which it is made is magnetic except in the very slightest degree. It seems to indicate that magnetism depends upon molecular rather than atomic structure. The permeability of this alloy has been found to be nearly 33. 515. Effects of Heat on the Magnetism of Metals and Mag- nets. — The permeability of iron and nickel diminishes as the temperature rises. At 737°C. iron ceases to be magnetic. A small piece of iron heated to a bright red heat is not attracted INFLUENCE OF MEDIUM 347 even by a powerful magnet, but as it cools to 700° it again becomes strongly magnetic. A steel magnet when heated to bright red heat loses all trace of magnetism, and if cooled while away from magnetic influence will be found completely demagnetized. Even when a magnet is slightly heated, say to 100°C., it is not as strong as at lower temperatures. 516. Force with which a Magnet Attracts its Armature. — The force with which a magnet attracts its armature evidently depends on the fact that the permeability of the armature is greater than that of the surrounding medium. // there were no difference between them there would be no change in the lines of force on withdrawing the armature and consequently no attractive force. When the armature is of such a size that most of the lines of force from one pole to the other pass through it, the force of attraction is given very nearly by the formula Stt where A is the combined area of the two poles and B is the induction or the number of lines of force that pass from a pole into the armature across a square centimeter of surface. If these quantities are taken in C. G. S. units, the attractive force P will be found in dynes. Problems 1. Find the force 5 cm. away from a pole of strength m, the other pole being so far away in comparison that it may be disregarded. How many lines of force go through the sphere of 5 cm. radius surrounding the pole m? 2. If a magnet having poles of strength 300, and 30 cm. apart is mounted on a pivot in a uniform magnetic field of strength 0.2, how much force, applied 10 cm. from the pivot, will be required to hold it at right angles to the lines of force of the field. 3. What is the magnetic moment of the magnet in problem 2. What torque is required to hold it at an angle of 45° to the lines of force of the earth field of strength 0.2? 4. Where the total intensity of the earth's magnetic field is 0.6 and the dip 70°, how many lines of force pass through a circular hoop 50 cm. in diame- ter lying horizontally on the floor? How many if the plane of the hoop is vertical facing north and south? 6. If a compass needle oscillates 2 times per sec. when 15 cm. distant from the pole of a long magnet, how fast will it vibrate when 8 cm. from the pole, neglecting the influence of the other pole? ELECTROSTATICS Electbification 517. Electrification. — If a hard-rubber rod is rubbed with fur or flannel it will attract light fragments of pitch, paper, or gold leaf. A light ball of pith suspended by a thread, as shown in figure 280, is strongly attracted. A rod of sealing wax, or sulphur, or indeed of dry wood, will show the same power. In cold dry weather if a piece of paper is laid on a table and rubbed with flannel, it will be found to cling to the table and a slight crackling may perhaps be heard as it is pulled away; and if in this condition it is held near the wall it will be drawn toward it and cling to it. The shavings which come from a carpenter's plane in winter when the air is very dry will often behave in the same way. In all of these cases the substances are said to be elec- trified, a term which comes from electron, the Greek word for amber, a substance which was known to the ancients to possess this power. About the year 1600, Dr. Gilbert, who was also a pioneer in the study of mag- netism, found that a very large number of substances could be electrified by rubbing, though with metals he could get no results. He accordingly classified substances as electrics and non-electrics, according as they could, or could not, be electrified by rubbing. 518. Conductors and Non-conductors. — In 1729 Stephen Gray discovered that the electrification of a glass rod would leak off from it and could be communicated to a ball through a damp cord. His experiments showed that electrification could be communicated through certain bodies which he called con- ductors, while it could not be communicated through others which were named non-conductors. 348 FiQ. 280. ELECTRIFICATION 349 Metals were found to be the best conductors, wood and damp cord were fairly good, while glass, sulphur, and resin were non-conductors. Gray then showed that the substances which Gilbert had classed as non-electrics were conductors, and if they were insulated or mounted on non-conducting supports they could be electrified as other substances can. The old distinction of electrics and non-electrics was therefore abandoned, and substances were classified as conductors and non-conductors or insulators. The insulating power of bodies may be compared by the times required for a given amount of electrification to leak through similar rods of the different substances. In the following table bodies are classified according to their resistances or insulating powers. Insulators Poor Conductors Good Conductors Amber Dry wood Metals Sulphur Paper Gas carbon and graphite Fused quartz Alcohol Aqueous solutions of salts Glass Turpentine and acids. Hard rubber Distilled water Air and gases 519. Electricity. — ^Take two metal pails, each mounted on an insulating support (Fig. 281), electrify one of them and then connect the two by a con- ductor, such as a cord or a wire. Both will show electri- fication when tested by the suspended pith ball, though the electrification of the one first charged will be less than before the two were connected. (The connecting conductor must be supported on glass rods or from loops of sUk thread or otherwise insulated when placed on the pails.) If the pails are connected by a metallic wire the redistribution of the electrification is instantaneous; if by a rod of wood or by a cord a perceptible time is required. The communication of electrification from one body to another. Fig. 281. 350 ELECTROSTATICS one always losing as the other gains, suggests a transfer of something . of which electrification is the external evidence. This something is called electricity, and when electrification is communicated from one body to another, there is said to be a flow of electricity. 530. Two Kinds of Electriflcation. — Rub a rod of hard rubber or sealing wax with fur, and when strongly electrified present it to a suspended pith ball. The ball will be attracted at first, but if allowed to touch the electrified rod it may cling for a moment and then spring away, strongly repelled. If a rod of glass, electrified by rubbing with silk, is now brought near the pith ball, it will fly to the glass, but after contact it will be repelled as it had been from the rubber rod. While repelled by the glass it will be attracted by the elec- trified rubber, and vice versa. It is clear that the electrical states of the glass and rubber are different. This discovery was made by Du Fay, a French investigator, in 1733. He found that all electrified substances behave either like glass or rubber, and the two kinds of electrification were accordingly called vitreous and resin- ous. Franklin named the electrical state of the glass positive and that of the rubber negative, and these names have been universally adopted. 521. Similarly Electrified Bodies Repel. — Two strips of hard rubber electrified by fur and suspended near together repel each other. Two strips of paper if drawn through a fold of flannel in dry cold weather, or even when drawn between the fingers, will repel each other and stand apart. On the other hand, a strip of rubber negatively electrified by fur is attracted by a rod of positively electrified glass. Similarly electrified bodies repel, while oppositely electrified bodies attract each other. The pith ball attracted by the rod of glass is repelled after contact because it receives from the glass a charge of positive electricity by conduction. Rubber Fig. 282. ELECTRIFICATION 351 523. Gold-leaf Electroscope. — An instrument, such as the suspended pith ball, used to detect electrification is called an electroscope. A much more sensitive electroscope is that shown in figure 283. Two strips of thin gold leaf about }4 in. wide and 3 in. long are attached to the end of an insulated brass rod so that they hang side by side in a glass jar which screens them from air currents. The brass rod passes through the insulating stopper of the jar, and terminates above in a plate or knob. Two strips of tinfoil a a on the inside of the jar are in metallic connection with a metal base or tinfoil coating over the outside of the bottom. If a charge is given to the upper plate of the electroscope it at once distributes itself by conduction over the rod and gold leaves, causing the latter to repel each other and diverge as shown in the figure. If the charge should be too great the leaves will diverge enough to touch the side strips a through which the whole charge will escape. 533. Electric Series. — In every case of electrifica- tion by friction the substances rubbed together become oppositely electrified, as though something was taken from the one and added to the other. When glass is rubbed with silk the glass becomes "*' positively charged and the silk negatively, but when hard rubber is rubbed with silk the rubber becomes negative and the silk positive. The silk thus becomes negative in one case and positive in the other. And in general, any substance may become either positive or negative, depending on what it is rubbed with. It is possible to arrange substances in a series such as the following in which any substance is more positive than those below it in the list, but is negative to those that precede it. Glass (surface rubbed clean and polished). Fur. Flannel. Glass (passed through a Bunsen flame). Silk. Wood. Sealing wax. Hard rubber. Sulphur. 352 ELECTROSTATICS Law of Force and Distrie'jtion of Charge 634. Coulomb's Law. — The French physicist Coulomb (1784) investigated the law of force between two electrified bodies using a torsion wire balance, illustrated in figure 284. By means of a very fine wire a light horizontal bar of shellac, glass, or other insulating material is suspended inside of a glass jar by which it is screened from air currents. On the end of the sus- pended bar is a light pith ball n which is covered with gold foil. A metal ball m mounted on the end of an insulating glass rod can be introduced into the glass jar through a hole in the cover. To use the instrument remove the ball m and by means of the graduated head from which the wire is suspended turn the wire until the ball n hangs exactly in the place of m. Now give a . charge to to and introduce it into the jar. At first n is attracted and touches to, the charge then divides between the two since both balls are conductors, and immediately n is repelled to such a distance that the twist in the wire balances the force of repulsion. The distance between the balls is observed and also the number of degrees through which the wire is twisted. Now increase the twist in the wire by means of the graduated head, thus forcing n toward to. It will be found that when the two are at one- half the first distance the force of repulsion as measured by the twist in the wire is four times as great. To study the effect of changing the quantity of charge, a second insulated brass ball is taken of the same size as m and mounted on a glass rod in the same way. The ball to with its charge is now withdrawn from the jar and touched to the other similar ball which has no charge. Imme- diately the charge divides equally between the two (since they are alike) and m now has only half the charge which it had at first. If it is carefully intro- duced without permitting it to touch n, the charge on the latter will not be changed, and if the force is observed when the balls are the same distance apart as in the first experiment it will be found that the force is only one- half as great. From many such experiments Coulomb concluded that the force between two given charged bodies, provided they are small FlQ. 284. — Coulomb balance. COULOMB'S LAW 353 compared with the distance between them, is inversely propor- tional to the square of the distance and directly proportional to the amounts of their charges. This law may be expressed algebraically thus where F represents the force, Z is a constant, and r is the distance between the centers of the two bodies whose charges are repre- sented by q and q'. The constant K depends on the units that may be used, and also, as was shown by Faraday, on the medium between the two charged bodies. 535. Unit Charge. — Unit charge or unit quantity of electricity, in the electrostatic system of units, is defined as that quantity which when placed one centimeter from an equal charge in vacuum repels it with a force of one dyne. The force in dynes between two electric charges in vacuum may therefore be expressed by the formula where the quantities q and q' are measured in electrostatic units and where r is the distance between the charges, measured in centimeters. When the charges are in any other medium the force is usually less and the formula is ^ Kr^ where iiL is a constant usually greater than one, known as the specific inductive capacity or dielectric constant of the medium. The force between two charges in air is appreciably the same as in vacuum, for the specific inductive capacity of air is greater than that of vacuum by only about one part in 2000. 526. Distribution. — The distribution of an electric charge may be examined by means of a little metal disc mounted on an insulating handle and known as a proof plane. If the disc is placed flat against the surface of the charged and insulated pail shown in figure 285 and then removed, it will carry away 354 ELECTROSTATICS a charge which may be tested by the gold-leaf electroscope. When examined in this way it is found that the greatest charge is obtained from the outer surface of the pail near its upper edge and on the outer corner at the bottom, less is found on the middle of the side and none at all in the interior except near the upper edge. When there is a metal cover on the pail, absolutely no charge can be found on any part of the interior. Other irregular bodies may be examined in the same way and it will be found that the greatest density of charge is at corners and knobs projecting outward. For example, in a con- ductor shaped as in figure 286 the greatest density will be found Fid. 285. — Distribution Fig. 286. — Density of distribution indicated on pail. roughly by dotted line. on the projecting point on the left, no charge will be obtained in the cavity even though there is a sharp point there, and very Uttle will be found toward the bottom of the dimple on the right end. 527. Charge Entirely on the Surface of a Conductor. — The following experiment was carried out independently by Cavendish and Biot. A metal ball having two closely fitting hemispherical metal cups which were provided with insulating handles, was insulated and then charged strongly with electricity. When the cups were simultaneously removed they were found to have the entire charge, the ball being left without any trace of electrification, showing that the whole charge was on the surface. 628. Discharge from Points. — The density of charge on sharp projecting points of conductors may be so great that the charge DISTRIBUTION 355 will escape to the air. This discharge is accompanied by a stream of air which, if the point is connected with an electrical machine, may be strong enough to blow out a candle or turn a FiQ. 287. little wheel with light vanes, or if the point from which the dis- charge takes place is movable it will be driven backward as illustrated in figure 288. Conductors which are designed to hold electrical charges should therefore have all pro- jecting parts or corners carefully rounded, other- wise they will be rapidly discharged. 529. Frictional Electric Machine. — The early forms of electrical machines were frictional; the one illustrated in figure 289 is a good type of this class. A circular glass plate is mounted firmly on an axle so that it can be turned between leather covered rubbers, which are pressed against the glass by springs. The charge from the glass is received by a metal conductor which is on an insulating support of glass or of hard rubber. From this conductor there are two branches which reach out, one on each side of the glass plate, and on the inside of each is a row of sharp metal points project- ing toward the glass plate, like the teeth of a comb. The elec- tric charge excited on the glass by the rubbers is carried under the combs by the turning of the plate, and through them it readily passes from the glass into the conductor. At the same time that the plate is positively electrified the rubbers become negatively charged and should be connected FiQ. 288— Elec- tric wind. 356 ELECTROSTATICS by a chain or wire with the ground to permit this negative charge to escape. It is usual also to have the lower half of the Fig. 289. — Electrical machine. glass plate covered with a silk bag which prevents the escape of electricity from the glass as it turns. Problems 1. If the charges on two small conducting pith balls are +8 and —8, will they attract or repel and with what force when 4 cm. apart? What it they are allowed to touch? 2. Two small conducting balls of the same size and 6 cm. apart have charges +36 and —4, respectively. What is the force between them? Also what will the force become if they are touched together and then placed as before? 3. Two pith balls 3 cm. apart and equally charged repel each other with a force of 16 dynes; find the charge on each. 4. Two pith balls hung from the same point and weighing J{o grm. each, are equally charged and repel so that they diverge until the threads are at right angles to each other. What is the force of repulsion in grams and in dynes? If they are 8 cm. apart, what is the charge on each? 6. Two pith balls, each weighing J-f o grm. and suspended from the same point by threads 30 cm. long, are equally charged and repelling each other, hang 8 cm. apart. What is the charge on each ball? Induction 630. Induction. — When a conductor having no charge is insulated and then brought near a positively charged body, such as A, figure 290, it is found to be negatively electrified on the side next to A and positively electrified on the farther side. This can readily be tested by means of the proof plane and electroscope. INDUCTION 357 In this case no charge has passed from A to B. The con- ductor B is insulated and no flow of electricity either into it or out of it is possible. When B is taken away from the charged body A it is found to have no charge, just as before it was brought up. Also no charge is lost or gained by the body A in the opera- tion. The charges produced in the conductor B by the prox- imity of A are said to be induced. 631. The Induced Charges are Equal. — If, instead of the one conductor B of the previous experiment, we take two insulated conductors B and C, having no charge, and bring them near the charged body A while in contact with each other, as shown in figure 291, a negative charge will be induced in the nearer one and a positive charge in the farther one. If they are now separated and then removed the positive charge is trapped in Fig. 290. Fig. 291. one and the negative in the other. On bringing them near an electroscope they are found oppositely electrified; and if while away from the vicinity of A they are touched together, the charges totally disappear, showing that the positive charge of the one was just sufficient to neutralize the negative charge of the other. The charges are therefore equal and opposite. The same result is reached whatever the shape or size of the con- ductors B and C. Thus B may be large and C small, or vice versa, and still the charges on each when removed from the in- fluence of A will be found to be equal and opposite. For if we consider the single cnductor B used in the first experiment (Fig. 290) it is clear that if it is imagined cut in two at any point, near the positively charged end for example, the positive charge on the smaller part will be equal to the excess of negative over positive on the greater part. 358 ELECTROSTATICS c J^ Fig. 292. Thus the positive and negative charges produced by induction are always equal. 533. Induction when the Conductor is Already Charged. — When the conductor has a charge to begin with the inductive action takes place in the same way, but the original charge is combined with the induced charge. Thus if B (Fig. 292) is given a positive charge and brought toward the. positively charged body A, it will be found that the positive charge is denser on the side away from A. At a certain distance there will be no charge at all on the end toward A. If brought still nearer a negative charge will be found on that end, while the positive charge on the rest of the conductor will be correspondingly increased. If A and B are both conductors and similarly charged, say positively, they will react on each other, and the greatest density will be found on the outer side of each. If, however, they are oppositely charged the greatest den- sities are on the adjacent sides. 533. Effect of Connecting with the Earth. — If, while in the presence of the positively charged body A, the conductor B is connected with the earth by a wire, or by touching it with the finger, the positive charge will escape to the earth, and at the same time the negative charge will increase some- what, so that the conductor will be left with a negative charge greater than when it had both positive and negative charges together. It makes no difference what part of B may be connected to the earth, whether the nearer end, as shown, or the farther end, the result is exactly the same. The induced charge that remains is sometimes called a bound charge because it does not flow off or disappear when B is touched, but is held by the presence of the charged body A. 534. Charging Electroscope by Induction. — When a rod of INDUCTION 359 hard rubber, negatively electrified by rubbing with fur, is brought toward a gold-leaf electroscope, the leaves will be observed to diverge strongly while the rod may be several inches from the instrument. Positive electrification is induced in the top of the electro- scope and an equal negative charge is given to the leaves pre- cisely as in the case discussed in §530. If the top of the instrument is touched for an instant by the finger while the electrified rod is still held near, the negative electrification of the leaves will escape and they will hang straight down as in b (Fig. 294). If the finger is now removed, and then the rod, the positive charge will distribute itself over the top and leaves of the electroscope and they will diverge as shown in c. Fig. 295. When the electroscope is positively charged the approach of a positively charged rod increases the divergence of the leaves, while a negatively charged body will draw them together unless it is brought too near, in which case they will again diverge with a negative induced charge. It will be noticed that there are shown in figure 294 induced charges on the metal side strips inside the electroscope. These charges are opposite to the charge on the leaves and increase their divergence. 535. Attraction of Pith Balls Explained by Induction. — When an electrified rod is brought near a pith ball (Fig. 295) an inductive action takes place as shown, and the attraction between the positively electrified rod and the negatively elec- trified side of the ball is greater than the repulsion of the posi- tively electrified side, since the negative side of the ball is nearej to the rod. 360 ELECTROSTATICS If the ball had an initial positive charge it would be repelled, though even in this case if the rod is much more strongly elec- trified than the ball and is brought very near to it there may be attraction. 536. Induction Takes Place through Non-conductors. — The interposition of a sheet of glass or hard rubber or a cake of bees- wax or any other insulator between an electrified body and an electroscope does not interfere with the inductive action. Indeed induction takes place more readily through these substances than through air, though but slight evidence of change would be observed in such a rough experi- ment. 537. Induction through Con- ductors. — If a charged rod (Fig. 296) is held over an electroscope and a small insulated sheet of metal is interposed between the two, the gold leaves will diverge as ^^°- ^^^- though the plate were not there, for if the rod is negative a positive charge will be induced on the upper side of the plate and an equal negative charge on the lower side which in turn wiU act on the electroscope. If, however, the plate is connected with the earth the charge on its lower surface will escape and the electroscope will be screened almost entirely from the effect of the rod. 638. Conductor Surrounding an Electroscope. — When an electroscope is entirely surrounded by a conducting sheet it is absolutely protected from all outside inductive action. It has already been shown (§526) that there is no electrification on the interior of a closed conductor, so also there is no induction from the outside. If a delicate electroscope is enclosed in a cage of wire gauze which is underneath as well as around and above it, the cage may be strongly electrified by a machine and there will be no disturbance of the electroscope, except such as may be due to electrified air passing into the interior. Faraday constructed a. small room about 6 ft. each way and covered with tinfoil and found that within it he was unable to detect any disturbance of. his most delicate electroscope though an assistant INDUCTION 361 was electrifying the outside by a machine so that long sparks escaped from it. 539. Electrophorus. — A simple form of induction apparatus devised by Volta is known as the electrophorus. It con- sists of a cake or plate of non-conducting material, such as resin, sulphur, or hard-rubber, supported on a metal base and having a metal cover provided with an insulating handle of glass or hard rubber. The upper sur- face of the resinous plate is negatively electrified by rub- bing it with fur and the cover is then placed upon it. A positive charge is induced on the lower side of the cover and a negative charge on its upper side, and on touching it with the finger or connecting it by a metal chain or wire with the base plate the negative charge escapes, leaving the positive charge held by the negatively charged resin. If the cover is now lifted from the resin, on presenting Fig. 297. — No electric force within enclosing conductor. ^3 Fig. 298. — Electrophorus. the knuckle the positive charge escapes in a bright spark. The cover may then be again placed on the resin, touched, and with- drawn and a second positive charge obtained, and so on indefinitely. 362 ELECTROSTATICS In this way a great number of charges may be obtained without renewing the electrification of the resinous sole plate. The resin is a good insulator and the cover touches it at so few points that there is very little direct loss by conduction. 540. Source of the Energy of the Charges. — Each of the charges obtained in this way has energy, as shown by the noise and light given out by the spark. This energy does not come from the energy spent in electrifying the plate of rubber or resin, for a spark is obtained every time the insulated cover is touched and withdrawn without any appreciable loss of electrification by the resin. The energy must be supplied in the operation of withdrawing the plate. This will be made evident by the following experiment. Suspend the cover by silk cords from a spring, and after having discharged it let it be lowered upon the resin and then with- drawn without being touched. The spring is scarcely stretched more when the plate is withdrawn than when it was lowered. But if when the cover is on the resin it is touched, the negative charge escapes and the attrac- tion between the positive charge in the cover and the negatively charged resin causes the spring to be greatly stretched when the plate is raised, showing that more work has to be ^g 299 done to raise the plate after it has been touched than before. It is this work done by the person lifting the plate that is the source of the energy of the charge that is obtained. If the cover is raised only an inch from the resin the spark will be much less energetic than if it had been raised 10 in., for less work has been done. 541. Faraday's Ice-pail Experiment. — A very important case of induction is where the charged body is surrounded by a con- ductor. This was first investigated by Faraday as follows : Taking an insulated metal pail having a metal cover and con- nected with an electroscope, it was observed that whsn a charged metal ball was brought up toward the pail the divergence of the leaves of the electroscope increased until the ball was entirely INDUCTION 363 within the pail, after which no change was observed whatever the position of the ball might be, whether it was close to the bottom or to one side or in the middle. The ball was now permitted to touch the inside of the pail, but not the slightest change in the gold leaves was observed. When the ball was withdrawn it was found completely discharged while the leaves remained diverging. The same observations may be made using a deep open pail, as in figure 300, provided the ball is not too near the open top. When the positively charged ball is introduced into the pail there is induced a negative charge on the inside of the pail and a positive charge on the outside, as may be shown by a proof plane. When the ball touches the interior of the pail the charges on the ball and on the interior of the pail disap- pear, for the ball and pail then become one conductor and there is no charge on a cavity in a conductor. If these charges were not exactly equal there would be some excess of either positive or negative charge which would pass to the outside and cause a change in the electroscope. The experiment then leads to the following conclusion: When a charged body is surrounded by a conductor a charge is in- duced on the inside of the conductor equal and opposite to that on the body. The walls, ceiling, and floors of ordinary rooms are fairly good conductors so that when we have a positively charged body in a room we may be sure that an exactly equal negative charge is distributed over the walls and neighboring objects. 642. Positive and Negative Electrlflcatlons Always Equal.— Hold a rod of sealing wax in an insulated pail connected with an electroscope and rub it with a pad of flannel which is insulated on another rod of sealing wax. They may be rubbed quite vigorously but no sign of elec- trification is shown by the electroscope, but if either one by Fig. 300. 364 ELECTROSTATICS itself is drawn out of the pail there is decided divergence of the gold leaves. It follows that the electrifications developed on the sealing wax and flannel, respectively, are equal and opposite. In every case of electrification, whether by induction or friction, equal positive and negative charges are produced. 643. Electric Charges Multiples of a Certain Unit. — Very important results have recently been obtained by the exact measurement of extremely small electric charges. A method which has proved most fruitful was developed in 1910 by R. A. Millikan, by which the minute electric charges on microscopic drops of oil spray from an atomizer were accurately measured. A drop was isolated and observed through a low power micro- scope as it slowly settled down through air in the space between two horizontal metal plates which were connected together so that there was no electric force in the region between them. When the drops had nearly reached the lower plate the two plates were electrified, one positive and the other negative, in such a way that the electric force on the charged drop carried it upward. As it neared the top the plates were once more con- nected and discharged permitting the drop to settle again — and so on indefinitely. It was possible in this way to observe a single drop for hours at a time, and to measure accurately the velocity with which it settled downward and also the velocity with which it was urged upward. From the former of these two measurements the size of the drop could be determined, and then from its upward velocity in the electric field its charge could be calculated. In several thousand such experiments the charges upon the drops, whether positive or negative, were always found to be exact multiples of a small charge e, which had the value 4.77 X lo"'" in electrostatic units. It is believed that all electric charges, whether large or small are a whole number of times this elementary charge, so that it is impossible to increase or diminish an electric charge by a fractional part of e. 644. Induction Machines. — Various forms of electrical ma- chines have been devised in which charges developed by induction from small initial charges are continually added to the original charges until powerful effects are obtained. The first powerful INDUCTION 365 and successful machine of this kind was made by Holtz about 1864. A modification of this machine, due to the labors of Voss and Toepler and known as the Voss-Holtz or Toepler-Holtz machine, is shown in figure 301 with a diagram illustrating its action. A circular plate of glass carrying on its front surface six small discs of tin foil, marked ai, 02, . . . at, is rotated rapidly in front of a fixed plate of glass, on the back of which are attached Fig. 301. — Toepler-Holtz machine and diagram. two conductors of paper A and B, called armatures (outlined by the dotted lines). In front of the rotating plate are mounted on insulating supports the two conducting combs DD' with sharp points close to the plate and directed toward it; these conductors are connected to the knobs K and K'. A con- ducting bar, called the equalizing bar, EE', crosses diagonally in front of the rotating plate and is also provided with combs directed toward the plate. At C and C there are metallic arms which are connected with the armatures on the back of the fixed plate and carry little tinsel brushes that touch the tinfoil discs on the revolving plate as they pass. Suppose, now, that A is slightly more positive than B, owing to the remains of a previous charge or to the influence of some neighboring charged body or to the brushes at C and C rubbing a little differently on the discs as the plate is turned. And suppose that the discs aa and 05 are under the combs of the equalizing bar EE' and are connected with it by the tinsel 366 ELECTROSTATICS brushes carried by that bar. The bar with the two discs thus forms one continuous conductor with the two inductors A and B opposite its ends. A negative charge will therefore be induced in the end toward A and an equal positive charge in the end toward B. If the plate is now turned, as carries its negative charge past the position a^ until it is between the brush C and the armature B with which that brush is connected, and is therefore situated almost as if inside of a conductor; it accordingly gives up almost its entire charge through the brush C to the armature B which thus becomes more negative. At the same time the disc a2 has moved past the position a-i and given up its positive charge to the armature A through the brush C The armatures with their increased charges act more power fully on the next pair of discs that pass under the bar EE', and so these discs carry still larger charges to the armatures and thus the effect rapidly increases till the armatures are so highly charged that they lose by leakage as rapidly as they gain. When the armature A is positively charged it acts inductively on the comb D through the two layers of glass, attracting a nega- tive charge on the points of the comb and repelling positive to the knob K, but the negative charge induced on D is discharged upon the surface of the revolving glass plate from the sharp points of the comb, and is carried away by the motion of the plate, leaving the conductor and knob K strongly positively charged. At the same time the positive charge induced on D' is discharged on the revolving plate, leaving K' negatively charged, and if the gap between K and K' is not too great, spark discharges will take place between them. Two small Ley den jars (§567) are connected with the conductors D and D' and act as reservoirs in which the charge accumulates between discharges. 545. Wimshurst Machine. — In the Wimshurst machine two circular plates of glass are revolved in opposite directions, one in front of the other. On the outer surface of each are a number of radial strips or sectors of tinfoil, on each of which is a Httle metal knob or button. As the plates rotate, the buttons strike the tinsel brushes of a pair of equalizing bars, one of which is fixed in front INDUCTION 367 of each plate, one inclined to the right and one to the left, so that the two are nearly at right angles to each other. In the diagram the inner and outer circles represent the two plates while the heavy lines indicate the positions of the con- ducting sectors. Suppose that the plates turn in the directions of the arrows and that the sector a is shghtly positive and 6 negative. Then c and d which are connected by the equalizing bar will become oppositely charged by induction, c negatively and d positively. The rotation of the plates carries c with its negative Pig. 302. — Wlmshurat machine. Fig. 303. — Diagram. charge to e, and d with its positive charge to /, where they are opposite the ends of the second equalizing bar and by induction attract positive charge into g and negative into h. At A and B are combs between which the plates turn, the rotation of the plates carries the positively charged sectors toward B and the negatively charged ones toward A, making the knobs N and M positive and negative respectively. 546. Summary. — The following is a summary of the main preceding facts relating to electric charges. 1. There are two kinds of charges. Bodies with like charges repel and with unlike charges attract each other. 2. The force between two small charged bodies is directly proportional to their charges and inversely proportional to the square of the distance between them. 368 ELECTROSTATICS 3. The force between charged bodies also depends on the medium between them. 4. It is impossible to produce one kind of charge without at the same time producing an equal charge of the other kind. 5. Whenever a positive charge disappears an equal negative charge also disappears. 6. The total charge in a body or the sum of the positive and negative charges which it may exhibit does not change so long as the body is truly insulated. 7. The distribution of charge in a conductor is influenced by neighboring charged bodies. (Induction.) 8. Charges are always multiples of an elementary unit e taken a whole number of times. Potential and Electbometebs 547. Potential or Electrical Pressure. — Suppose an electro- scope connected with a charged pail by a wire. It makes no difference in the indication of the electro- scope whether it is connected to the in- side of the pail where no charge can be obtained by the proof plane or to an edge where the density is greatest. There is perfect equilibrium between the electro- scope and the charged pail and no tend- ency of the charge to flow from one to the other. When two conductors are in this relation they are said to be at the same potential or to have the same electrical pressure. The potential of a conductor is that electrical condition which determines the flow of electricity. Potential determines the flow of electricity just as pressure determines the flow of fluids and temperature the flow of heat. When any conductor is connected with the earth flow takes place until the conductor comes to the potential of the earth. When two charged conductors are connected by a wire, the one that loses positive charge is said to have been at a higher potential than the other. 548. Effect of Increased Capacity. — If an insulated -conductor having no electric charge is brought up and touched to the POTENTIAL 369 charged pail shown in figure 304, part of the charge will flow into the conductor and the whole system comes to a new state of equilibrium in which the potential is less than before. This is shown by the fact that the gold leaves of the electroscope do not diverge so strongly. In this case there has been a decrease in potential though there has been no change in the total amount of the charge. The enlarged system of conductors is said to have a greater electrical capacity than the original pail and electroscope. Change in potential due to a change in capacity of the charged conductor is well shown by Faraday's ap- paratus, figure 305, in which a roller suspended by insulating silk cords- carries a conducting ribbon of tin- foil. When rolled up and charged the pith balls diverge widely, but as the ribbon is unrolled, thus in- creasing the surface and capacity of the conductor, the pith balls ap- proach each other. That this is not due to any loss of charge is shown by the fact that when the ribbon is again rolled up the pith balls diverge as at first. 549. Potential of a Conductor. — In non-conductors the poten- tial may be different at different points, but in a conductor or in connected conductors, when the electrical charge is at rest, all parts are at the same potential, since otherwise flow would take place from one part to another. Even when a conductor is hollow the potential in its interior, due to charges at rest, is everywhere the same as at its surface, provided it does not contain any insulated charged bodies ; for Faraday showed (§538) that there was no electric force inside of a hollow conductor in such a case and consequently it must be a region of uniform potential. Thus the surface of a conductor is an equipotential surface, since all points of it are at the same potential, and the interior Fig. 305. 370 ELECTROSTATICS of a conductor, provided it does not contain insulated charged bodies, is an equipotential region. 550. Zero Potential. — Bodies are usually discharged by con- necting them with the earth, and its potential is accordingly taken as the zero. Bodies which give up a positive charge when connected to the earth are said to have a positive potential, while bodies which receive a positive charge from the earth have a negative potential. The walls and floors of wood and plaster which enclose ordi- nary rooms are conductors, though they conduct rather slowly, the interiors of rooms are therefore to be regarded as cavities in conductors which are at the earth potential. When there are no insulated charged bodies inside of such a room it is an equi- potential region, all at zero potential, even though there may be electrified clouds floating overhead. 551. Potential Without Charge. — It was shown by Faraday that when an insulated conductor is touched to the inside of a hollow conductor which completely surrounds it, it receives absolutely no charge. It follows that in such a case there is no flow of electricity from one to the other and therefore both must have been at the same potential before they touched. We see, then, that merely putting a conductor without charge inside of a hollow conductor brings it to the potential of that conductor; that is, a conductor which has no charge takes the potential of the region where it is placed. 553. Potential Affected by Neighboring Charges. — The case just discussed is a special instance of the general principle that the potential of a conductor depends not only on its own charge, but on that of all neighboring objects. This is well shown in case of the electrophorus (§539), for when the metal cover of that instrument is removed and dis- charged by touching it, it comes to the earth potential. But when it is placed on the negatively charged base its potential is lowered, as is shown by the fact that if it is now connected with the earth positive electricity flows from the earth to the cover. It thus comes to the earth potential and has a positive charge. If it is now removed from the negatively charged base plate its potential is raised, for on touching it positive electricity flows from it to the earth. POTENTIAL 371 It thus appears that the potential of a conductor depends not only upon its own charge, but also upon all other charges near enough to affect it. If a conductor were removed from all other charged bodies its potential would depend only on its own charge and would be proportional to that charge. But if a positively charged body is brought near the conductor its potential is raised though its charge is not changed. A 'positive charge not only raises the 'potential of the body to •which it is given, hut it raises the potential of the whole neighboring region and of any bodies that may be near. So also a negative charge lowers the potential of all points in its vicinity. 553. What Determines the Potential of a Conductor. — From what precedes it will be seen that the potential of a conductor depends upon the following three conditions: 1. Its capacity, — determined by its size and shape. 2. Its charge. 3. The charges on surrounding bodies. 554. Measure of Difference of Potential. — When a small electric charge is moved along an equipotential surface or from one part of an equipotential region to another, no work is done, for there can be no electric force acting on the charge since there is no tendency to flow. If two conductors are at the same potential no work is done in transferring a small charge from one to the other. But if they are not at the same potential work must be done to carry a small positive charge from the one at the lower potential to the one at the higher, just as in case of two vessels, each containing a fluid under pressure, work must be done by a pump to force any fluid from the vessel in which the pressure is less into the one in which it is greater. It may be proved that if a little charge is transferred from one conductor to another the work done will be the same by whatever path the transfer may be made, and accordingly the work done may be used as a measure of the difference of potential of the two conductors. Thus the difference of potential of two conductors is measured by the number of ergs of work required to transfer unit charge from one to the other. Difference of j otential determined in this 372 ELECTROSTATICS way is in electrostatic units and one electrostatic unit of potential is very nearly equal to 300 volts, the volt being the unit of potential in what is called the practical system of units. 555. Instruments to Measure Potential. — Electroscopes and electrometers are potential measuring instruments. For instance, the deviation of the gold leaves of a gold-leaf electroscope de- pends on the difference of potential between the leaves and the side conducting strips that are connected with the earth. The leaves have the same potential as any conductor that may be connected with them. But this instrument, in the ordinary Fig. 306. — Attracted-diso electrometer. form, is not well adapted for exact measurements, though a modified form in which the deflection of a single narrow strip of gold leaf is measured by a low power microscope, is valuable for some investigations. 556. Attracted-dise Electrometer. — The instrument shown in figure 306 is known as the attracted-disc electrometer, or the Kelvin absolute electrometer, and may be used to measure large differences of potential. Two circular flat plates of metal, A and B, are mounted par- allel to each other. By means of the screw s the plate A may be raised or lowered and the distance between the two plates may be determined by the scale and vernier v. The upper plate is made in two parts, a central disc and a surrounding ring. The disc is suspended from one arm of a balance and hangs so that its lower surface is exactly flush with that of the ring, which is separately supported, though the two are in conducting communication . POTENTIAL 373 The plate B is connected with the earth, while A, which is insulated by hard rubber or glass at r, is connected with the charged conductor whose potential is to be determined. There is a charge on A and an opposite induced charge on B and conse- quently an attraction between them. By means of the balance the force with which the disc is attracted is exactly measured. The difference of potential V between the plates A and B may then be found in electrostatic units by the formula \ -s where d is the distance between the plates in centimeters, F is the force of attraction in dynes, and S is the area of the disc in square centimeters. The instrument is called an absolute electrometer, because its determinations depend directly on measurements of length and force and it may be used to standardize other instruments. The guard ring, as it is called, which surrounds the attracted disc was introduced by Lord Kelvin to cause a uniform distribu- tion over the central disc, without which the difference of poten- tial could not be calculated by the above simple formula. For in case of two parallel plates the distribution is denser toward the edges, but is extremely uniform near the center if the plates are not too far apart. The balance must be enclosed in a metal case, as shown by the dotted lines, to screen it from all outside disturbing electrical attractions. 557. Quadrant Electrometer. — The quadrant electrometer, also designed by Lord Kelvin, is shown in figure 307. A small round brass box is cut into four quadrants which are slightly separated from each other, and mounted on insulating supports as shown in the figure. The needle consists of a thin flat plate of aluminum, broad at the two ends as shown in the plan, and mounted on a light vertical wire of aluminum which passes through its center and carries on its upper end a small mirror by which the motions of the needle are observed. The flat needle is suspended by a fine quartz fiber or by two parallel fine fibers of silk constituting a bifilar suspension, so that it hangs horizontally in the middle of the box formed by the four quadrants and in the position shown. 374 ELECTROSTATICS The diagonally opposite quadrants A A are connected by wire conductors to the pole P', while the quadrants BB are connected to the pole P. The needle is given a positive charge so that if the A quadrants are connected with a positively charged body while the B quadrants are joined to earth, it will turn toward the B quadrants; while if the A quadrants are negative it will turn toward them. The deflection of the needle is read by the motion of a narrow beam of light, reflected from the attached mirror upon a graduated scale, and is nearly propor- tional to the difference of potential between the A and B quadrants. The sensitiveness of this instrument may be many times greater than that of the gold-leaf electroscope. To secure the greatest sensi- tiveness a very light paper needle is used, hung by an exceedingly fine quartz fiber. Another method of using the instrument is to connect the B quadrants to the body to be tested, while the needle and A quadrants are connected together and to the earth. In this case the needle will turn toward the B quadrants whether the charged body is positive or negative, and the deflection is nearly proportional to the square of the difference of potential measured. ELECTRON THEORY 658. Theories of Electricity. — The early investigators thought of electrified bodies as containing something which they called an imponderable fluid, because it could flow from one body to another and yet did not seem to possess weight or inertia. Fig. 307. — Quadrant electrometer. ELECTRON THEORY 375 Symmer conceived two such fluids, positive and negative electricities, which neutraUzed each other when mingled. Franklin, however, advocated the view that there was but a single elec- tricity and that for every body there was a normal amount when it showed no electrification; if there was an excess it showed one kind of electrification, while if there was a deficiency of the electric fluid the body was electrified in the opposite way. The strong points of this theory are that it explains how opposite charges neutralize each other and how it is impossible to develop a positive charge anywhere without at the same time causing an equal negative charge to appear somewhere else. But Franklin's theory assumed that each portion of the electric fluid repelled every other portion directly. 559. Faraday's Theory. — Faraday, however, conceived that electrical forces were communicated by the insulating medium between electrified bodies and showed that, while the force between two charged conductors does not depend on the kind of metal used for the conductors or whether they are solid or hollow, it does depend on the kind of insulating medium that sepa- rates them. 560. Electron Theory. — Maxwell, following out Faraday's idea of the importance of the dielectric, showed how electric phenomena might be explained by what may be called the displacement theory. He conceived all substances, conductors and insulators alike, as full of electricity which could flow freely through conductors, but in insulators experienced an elastic resistance which prevented it from being more than slightly displaced. A form of this theory based on the modern conception of electrons, is known as the electron theory. The experiments of Millikan, (§543) together with recent researches relating to electric discharge in gases, and in the field of radioactivity, have led to the belief that all electric charges whether positive or negative are exact multiples of a unit charge e having the value 4.70 X lO""* in electro- static units, which is so small that there are more than 2000 million of them in the electrostatic unit as defined in §525. The positive elementary unit charge is never found separate from atoms of matter, but the negative unit, as was shown by Sir J. J. Thomson, is carried by the small particles or corpuscles that make up the cathode-rays in a vac- uum tube (§774) and have only Ksoo the mass of the hydrogen atom. These negative particles or electrons, as they are called, exist in all kinds of matter, can pass through conductors, and may be transferred from one body to another. The electron theory supposes that every atom of matter in the neutral state is made up of a certain number of the elementary positive units and an equal number of electrons held in equilibrium by electric forces. When an atom loses an electron it becomes positive, when it gains an extra one it becomes negative. 561. Conductors and Insulators. — In conductors the slightest external electric force causes electrons to pass continuously from atom to 376 ELECTROSTATICS FiQ. 308. atom through the body, thus constituting a flow or current of electricity. In this motion the electrons are constantly checked by their impacts against atoms of matter or other electrons and in this way they are retarded by a sort of jrictional resistance, just as shot are retarded in moving through a mass of molasses ; but there is nothing like an elastic resistance to make them spring back when the displacing force is removed. In insulators, on the other hand, if electric force is applied, there is a certain yielding or displacement of the electrons, if the force is increased the electrons are displaced more, but there is no continuous flow as in a conductor, and as soon as the external force is removed they spring back to their original positions, behaving as shot would if imbedded in a mass of rubber. Now suppose that the process of charging two conductors A and B by an electric machine consists in forcing some electrons out of A into B, thereby making A positive and B negative. This will cause a crowding outward of the electrons in the dielectric immediately surrounding B, while those around A will be displaced inward to 'make up for its deficiency, and so in all the dielectric surrounding A and B the electrons will everywhere be displaced in the opposite direction to the arrows which indicate the positive direction of the lines of force. But since in a dielectric electrons are not free to move, their displacement at any point is only through a very small distance and is opposed by the internal electric forces between the positive and negative elementary charges in the dielectric which urge all the electrons back toward their original un- strained positions. There is therefore produced a back pressure on the elec- trons in B and a negative pressure on that in A, so that if A and B are now connected by a conducting wire there will be a flow of electrons from B to A, until the displaced electrons in the dielectric have sprung back into their original positions. The discharge is thus conceived as forced from £ to A by the springing back of the electrons in the dielectric in consequence of the internal electric forces in the dielectric. This difference in pressure between A and B due to the reaction of the dis- placed dielectric is the difference between their potentials. Suppose that after A and B are charged they are moved nearer together. The strain will now take place through a less thickness of dielectric and the difference in pressure between A and B will accordingly be less. The work required to produce a given charge will therefore be less when they are nearer together; that is, the energy of the charge will be less. They will therefore tend to move together; that is, there is an attraction between A and B. For if they are moved apart they will have more energy, but they can only get this addi- tional energy from the work done in separating them ; therefore there must be a force opposing the separation or a force of attraction. ELECTRON THEORY 377 563. Tubes of Force. — The electric field may be conceived as divided up into tubes by means of surfaces in the direction of the lines of force. (Compare §499). These tubes of force will always have at one end a positive charge and at the other an equal negative charge. On the electron theory there will be as many electrons displaced inward across one end of a tube of force, as will be displaced outward across the other end. 563. Induction as Explained on Electron Theory.— Suppose A and B are conductors near each other (Fig. 309) and having no charge at first. Let a negative charge be given to A. In doing this we may suppose that electrons are transferred from the ground so that the walls of the surrounding room become positive. Tubes of Fig. 309. — Induction. force in which electrons are displaced outward will extend outward from A in every direction toward the walls of the room. But since B is a conductor there is no force resisting the displacement of the electrons through it, whereas in every other direction there is the active elastic resistance of the dielectric to be overcome. Displacement can therefore take place more readily on the side of A toward B than in other directions and a number of tubes of displacement will terminate on B, and the electrons where the tubes terminate will be displaced toward B's interior, while on the farther side of B there will be an equal outward displacement of electrons. These con- stitute equal positive and negative charges respectively. 564. Why an Electrostatic Charge Appears Only on the Sur- face of a Conductor. — If a charge is given to a hollow conductor (Fig. 310) all parts of the metal shell come to the same potential and there is a displacement of electrons in the dielectric surrounding it, and this displace- ment is either away from the conductor or toward it depending on the kind of charge given to the conductor. But the dielectric A in the interior is entirely surrounded by the conductor aAd is therefore pressed upon equally in every direction consequently there can be no displacement of its electrons. The pressure or potential in the interior is, however, everywhere the same as that in the conductor which surrounds it. If a small conductor B is touched to the interior of the shell it comes to 378 ELECTROSTATICS the same potential as the shell, but the electrons in the dielectric around it, being equally urged in every direction are not displaced and accordingly B neither gains nor loses electrons; that is, it does not receive a charge. It is much as though a bottle with flexible rubber walls was filled with water and then put inside of a vessel full of water under considerable pres- sure. If the stopcock is then opened no water will flow either into the bottle or out of it. For the pressure is the same inside of the bottle as it is outside. If the stopcock is closed and the bottle is removed from the region of pres- sure it will be found neither to have gained nor lost charge. So it is also with the conductor B; when inside of A it is at the same poten- tial as A, but it does not receive any charge because no displacement in the dielectric at its surface is possible, and so when removed from A its potential changes to that of the region where it is placed, but it shows no trace of charge. Fig. 310. — Hollow conductor. Fig. 311. 665. A Case of Induction. — Suppose, however, that the conductor B while inside the charged body A and insulated from it, is connected with the earth by a wire as shown in figure 311. The electrons in the dielectric will then be displaced not only in the dielectric between A and the walls of the room, but also between A and B, and if A is positive they are displaced toward A and away from B causing a corresponding flow of electrons into B from the earth. If B is now insulated and removed from the interior of A it will be found negatively charged, for the displacement in the dielectric around it cannot disappear until the electrons that flowed into B are permitted to escape. Since the outward displacement on B must be equal to the inward displace- ment over the inner surface of A, the charge on B must be equal and opposite to thai on the interior of A ; this has previously been shown to be the case in Faraday's ice-pail experiment (§541). Condensers and Capacity 666. Condenser Experiment. — Take a tin plate, mount it bottom upward on an insulating stand and connect it with a CONDENSERS 379 gold-leaf electroscope. Cover the plate with a sheet of glass and then give it a sufficient charge to cause the gold leaves to diverge strongly. Now take another tin plate, connected to earth by the hand or by a wire, and lower it upon the glass. The gold leaves will be observed to come together as the plates approach each other, showing that the potential of the charged conductor is diminished by the approach of the grounded conducting plate. The closer the two plates are brought together, the greater will be the decrease in potential. On removing the upper plate the leaves diverge as at first, showing that there has been no loss of charge. In this experiment evidently the capacity of the first plate has been increased by the proximity of the second uninsulated plate. Such a combination is known as a condenser, because it can take a large charge at a small potential. The decrease in potential is due to the presence of an induced charge on the second plate opposite in kind to that on the first. 567. Leyden Jar. — The earliest form of condenser was de- vised in 1746 by Musschenbroek, of Leyden. That experi- menter, in attempting to charge a glass of water with electricity, held the glass in his hand while one pole of the elec- trical machine was connected with the water through a nail resting in the glass. After charging it well, the knuckle of the other hand was touched to the nail and a smart electric shock was obtained. Further experimentation showed that all that was necessary was that there should be two conducting coat- ings separated by the glass. Accordingly, a Leyden jar, as it is called, is made by coating a glass jar or bottle inside and outside with tinfoil for about two- thirds the height of the jar. Connection is made with the inner coating through a metal ro.d terminating in a knob. To charge such a jar one coating must be connected to one pole of the electrical machine while the other coating is connected to the other pole of the machine either directly or through the earth, so that the two coatings simultaneously receive opposite charges. Fig. 312, — Discharge of Leyden jar. 380 ELECTROSTATICS The jar is discharged by connecting the knob and outer coating by a conductor. If the outer coating is touched with one hand while the knuckle of the other hand is brought to the knob, the jar is discharged through the body and the sensation of shock is experienced. Slight shocks are felt in the hand and arms while stronger shocks are felt in the body. A Leyden jar is said to have greater capacity for charge than an ordinary insulated conductor, because it will receive a much greater charge from a given electrical machine than will be taken by the simple conductor. 668. Condensers. — Any contrivance in which two conductors are separated by a thin dielectric which has sufficient dielectric strength to prevent discharge between them, is a condenser and has the same properties as a Leyden jar. A convenient form of condenser, due to Franklin, may be made by coating the opposite sides of a plate of glass with tinfoil, which, however, must not reach too near the edges of the plate. 669. Insulated Leyden Jar. — If a Leyden jar has either of its coatings insulated, no more charge can be given to the other coating than to a simple metal conductor of the same shape and size. If a Leyden jar is charged and then placed on an insulating stand, it cannot be discharged by simply touching the knob. But if the finger is touched first to the knob and then withdrawn and touched to the outer coating and so on alternately a small spark will be obtained every time and the jar may thus be very slowly discharged. 570. Explanation of Action of a Condenser. — The older way of explaining the action of a condenser is as follows: The plate A receives, say, a positive charge from the electrical machine. This charge acts inductively through the glass dielectric and attracts a negative charge from the earth, which in turn reacts on the positive charge in A attracting it, and so enabling a much larger charge to be given to A by the machine. If the plate B were not connected to the earth .the positive charge which would be induced on its outside surface could not escape and would by its repulsive action on the charge in A neutralize the attractive action of the negative induced charge, so that no more charge could be given to A than if the plate B were not there. CONDENSERS 381 But it is better to look at the action in the following way, from the standpoint of the displacement theory. Let A (Fig. 314) represent the inner coating of a Leyden jar which is stripped of its outer coating. Connect A to the positive pole of an electrical machine and connect the negative pole to the floor or walls of the room. The conductor A will become positively charged and an equal and opposite negative charge will be found on the walls of the room. The tubes of force or displacement extend from A to the walls, but the difference of potential produced by the machine can cause only a small strain or displacement of the electrons in so thick a dielectric, and, therefore, only a small charge will be given to A. But if the outer coating is ^o'""' now put upon the jar and connected with the negative pole of the machine, as in figure 315, the whole strain will take place in the thin layer of glass and so a great displacement of electrons in the glass will take place involving a large flow of electrons into one coating and out of the other, leaving the jar strongly charged. If the outer coating were insulated and the negative pole of Fig. 313.— Charging a. condenser. Fig. 314. Fig. 315. the charging machine were connected to the walls of the room, no displacement could take place through the glass toward the outer coating without an equal displacement taking place outward from the outer coating toward the walls, so that the jar would take no greater charge than if there were no outer coating. 382 ELECTROSTATICS 571. Capacity of a Conductor. — From the above discussion it will be clear that the charging of every conductor is analogous to the charging of a Leyden jar; the conductor itself corresponds to the inner coating of the jar; while the surrounding walls or conductors on which the tubes of force terminate correspond to the outer coating; it differs from a Leyden jar only in the greater thickness of the dielectric. 573. Capacity. — The larger the charge given to a Leyden jar or condenser, the greater is the difference of potential between its coatings, so that we have Q^VC where Q is the charge, V is the difference of potential between the coatings of the jar, and C is a constant called the capacity of the jar. When F = 1, C = Q, and, therefore, the capacity of a condenser is the quantity of charge required to make unit difierence of potential between its coatings. Capacity depends on that area S of one surface which is op- posed by the other and varies inversely as the thickness of the dielectric d which separates them. It may be computed from the formula ^ ~4,rd where K is a constant which depends on the nature of the dielectric and is called its specific inductive capacity or dielectric constant. The derivation of this formula is given in §583. Caution. — The student is warned against thinking that the capacity of a condenser is the greatest charge which it can hold. The maximum possible charge of a condenser depends upon its insulation and the strength of the dielectric between its coatings to resist disruptive discharge. One condenser may be charged to many times its capacity before it discharges, while an- other may break down or discharge before it is charged to one-tenth of its capacity. Some Specific Inductive Capacities, or Dielectric Constants Hard rubber ....2.5 Paraffin 2.0 Air (normal pressure) . . . 1 . 00059 Glass 6 to 8 Turpentine . . 2.2 Carbon dioxide 1 . 00090 Mica 8.0 Petroleum. .. S.l Hydrogen 1.00028 CONDENSERS 383 573. Hydraulic Analogy. — It is instructive to consider the following hydraulic model of a condenser. A metal box is divided into two parts A and B by a partition of thick sheet rubber. Each side is provided with a tubular opening controlled by a stopcock, the whole is then filled with water and immersed in a pond. While the stopcocks are open the pressure is the same in A and B and the.rubber is not strained. It is like a Leyden jar unin- sulated and discharged. Now attach a pump to B and force water in while the stopcock of A is left open or, what amounts to the same thing, connect the pump both to A and B so that it pumps water out of A and into B. The rubber will be strained as shown in the figure, the side A will be at the pressure of the pond which may be called zero, while the other side is at a higher pressure p. This difference in pressure p between the two sides is due to the strain of the t^„ „,„ „ , ,. .,,.,, , , . FiG- 316. — Hydraulic rubber. If A and B are now connected by a pipe model of Leyden jar. and the stopcocks are opened there will be a flow from one side to the other as the rubber springs back into the unstrained condition. So when a Leyden jar is discharged electricity may be thought of as forced from one coating to the other by the springing back of the displaced elec- trons in the dielectric. If the rubber diaphragm were thicker more difference in pressure would be required to force in a given charge. So in a Leyden jar, the thicker the dielectric the greater the difference of potential between its coatings when it has a given charge. If the diaphragm were made of a substance that was more yielding than rubber, it would correspond to a dielectric of greater specific inductive capa- city; for a given pressure would then force in a greater charge. Also suppose the stopcock A is closed and the pump connected to B, pres- sure will be produced in B and perhaps a slight amount of water forced in due to the elastic yielding of the box itself, but the rubber diaphragm will not be appreciably strained and the pressure will be the same on both sides. This is the case of trying to charge an insulated jar. The stiff and but slightly yielding walls of the box represent the insulating dielectric that surrounds the Leyden jar and extends to the walls of the room, while the rubber diaphragm represents the thin glass dielectric between the coatings of the jar. Remembering that the dielectric surrounding the jar is slightly yielding will enable the student to explain the succession of small sparks obtained from the insulated jar as described in §569. 574. Energy of Charge. — When we begin to charge a Leyden jar or condenser the two coatings are at the same potential and, therefore no work is required to transfer the first little portion of charge from one coating to the other. But as the charging goes on the difference of potential between the two coatings increases and more work is required to produce a given increase in charge. 384 ELECTROSTATICS Suppose the final potential to which the jar is charged is F, and suppose that in charging it Q units of electricity are trans- ferred from one coating to the other, giving one a charge +Q and the other a charge — Q. If during this transfer the differ- ence of potential between the coatings werp to remain constant and equal to V the work done in charging would be QF ergs. But since the difference of potential is zero at the start and in- creases in proportion to the charge, the average potential during charging is 3^F and the work actually done in charging is MQ^i which is therefore the energy of the charge. The case is analogous to the filling of a cylindrical water- tower, the pressure is zero when the tower is empty, and increases as the water rises until the final pressure p is reached. The work done is, therefore, 3^py where v is the total volume of water pumped in. The energy of the condenser exists as electrical strain in the dielectric. 675. Dissected Leyden Jar. — That the energy of the charge is in the dielectric and not in the conducting surfaces is shown by the following experiment. Pig. 317. — Leyden jar with removable coatings. Take a Leyden jar, such as is shown in figure 317, in which the metal coatings can be removed from the glass. Charge it strongly and first remove one coating while the other is insulated, and then remove the other also. They are found to have only shght charges, but when they are again fitted upon the glass a vigorous discharge may be obtained. 676. Leyden Battery. — The Leyden jars in the conbination shown in figure 318 have their inner coatings connected together and are mounted in a box lined with tinfoil by which their outer CONDENSERS 385 coatings are also joined. Such an arrangement is known as a Leyden battery, the jars are also said to be connected in parallel or multiple, and the combination is equivalent to a single large jar having a capacity equal to the sum of the capacities of the sepa- rate jars. Fia. 318. Fig. 319. 577. Leyden Jars Connected In Cascade or Series. — In each of the two arrangements shown in figure 319 four jars on insulating stands are connected in such a way that if the discharge were to burst through the glass of the jars it would have to pierce all four jars to pass from one end to the other, as four layers of glass rl f g Fig. 320. — Condensers in series. intervene between the terminal conductors. In such a case the jars are said to be joined in cascade or in series. The diagram (Fig. 320) shows the state of electrification, the regions between the charged plates representing the layers of glass. Four similar jars joined in this way are like a single jar having a dielectric four times as thick, and the capacity of the combination is one-fourth that of a single jar. 386 ELECTROSTATICS This case is well illustrated by the hydrostatic analogue (Fig. 321) in which four models such as are described in §573 are connected in series. Clearly when water is pumped in at A and out at B the rubber diaphragms are all strained and an equal quantity of water is displaced from each into the next succeeding, thus representing the equality of the charges in each. The pres- sures represent the potentials. Evidently the pressure pi is greater than Po, and Pi is the greatest of all, and to force in a given quantity of water four times as much pressure must be used as to force it into a single one of the cells. The chief practical advantage of the cascade arrangement is that it has great dielectric strength and sparks do not easily burst through the glass; for this reason the small jars used on induction electrical machines are usually connected two in series, one being connected to one pole of the machine and one to the other, while their outer coatings are joined by a wire. When Leyden jars of different capacities Ci d Cs are joined in series, the capacity C of the combination is found from the relation C Ci d Cs To prove this formula let Q represent the charge, which will be the same for each jar when they are charged in series. The potentials of the jars will be ^, Q -,, Q ^ Q Cl t^2 *^3 The total difference of potential between the end coatings of the series win therefore be and if C is the capacity of the combination we have F = ^; therefore, Q = (f + Q' + c' Problems 1. A Leyden jar 14 cm. in diameter and made of glass 3 mm. thick is coated on the bottom and sides up to a height of 20 cm. What is its capacity and what charge is required to bring it to potential 30? Take dielectric constant of glass = 6. 2. Two Leyden jars, one of capacity 300 and charged to potential 20, the other brought to potential 30 by a charge of 7200 units, are connected POTENTIAL 387 together in parallel, positive coating being connected to positive and negative to negative. Find the resulting potential and the charge in each jar after being connected. 3. If in the preceding problem the positive coating of each jar is connected to the negative of the other, what will be the resulting difference in potential and charge in each jar? 4. A jar of capacity 1000 is charged to potential 50; find the heat developed in gram-calories when it is discharged through a long fine wire. 5. Three jars each of capacity 500 are charged each to potential 12 and then joined in series and finally discharged by connecting the end coatings of the combination. Find the difference of potential between the end coatings and the quantity of charge that passes through the discharge, and thence calculate the energy expended in the discharge. 6. Let the three Leyden jars of the previous problem be joined in parallel and then discharged. Find the difference of potential between coatings, and the quantity of discharge, and thence determine the energy of the discharge. 7. .A Leyden jar of capacity 500 is joined in series with another of capacity 200, and the combination is given a charge 3000. Find the difference of potential between the coatings in each jar. Thence find the difference of potential between the end coatings of the combination. What is the capacity of a single jar which when given the charge 3000 would have the same difference of potential as the combination? 8. A Leyden jar of capacity 600 is joined in series with one of capacity 400. Find the capacity of the combination. 9. Two Leyden jars of capacities Ci and Ct are joined in series and given a charge Q. Find the capacity of a single jar equivalent to the combina- tion, by following the method of problem 7. 10. A Leyden jar of capacity 800 is joined in series with another of capacity 200, and the combination charged to potential 20. Find the charge in each jar and the difference of potential between the coatings of each jar. Calculation of Potential and Capacity 578. Potential at a point. — Up to this point we have thought of electrical potential simply as a certain condition which de- termines the flow of electricity; and we have shown that the differ- ence of potential between two conductors may be measured by the work done in transferring unit charge from one to the other (§554). But potential is not a property of conductors only. When a little charge is brought up to any point whatever in space, work must in general be expended in bringing it to that point, on ac- count of the attractions or repulsions of neighboring charges; and this work, per unit charge, is used as the measure of the potential at that point. 388 ELECTROSTATICS Definition. — The potential at any point is measured by the work done against electrostatic forces in bringing a unit positive charge up to that point from an infinite distance. This work may be calculated as follows: Suppose there is a charge of q units of electricity at A (Fig. 322), and it is required to find the work done in carrying unit charge from C to 5 in the same straight line with A when air is the medium between the charges. Conceive the distance BC divided into n small parts at the points ffli a2, as, etc., and let r be the distance from A to B and ri, the dis- tance from A to ai, etc. Then the force with which q repels unit charge at B in air is -j (§525), while the force at ai is — ^- To get the work wi done when unit charge is moved from ai to B, the y? r^ ^2 ^ ^0 1 1 '' ^ *" "^ '>! B a, as a^ Fig. 322. average force must be multiplied by the distance from B to oi which is ri — r. The geometrical mean of —^ and ^ is — ; and this may be taken as the average force between a,y and B if these points are close together. The work done in this element of the distance will therefore be Wl JTi ' r ri and similarly the work done when unit charge is moved from 02 to tti is '^^ ri 7-2 so also 9 9 W3 = — - — ri rs and finally 9 9 ^"-r_ R Adding, we find w 1 + 9 9 W2+, etc., + w„ = ~— a POTENTIAL 389 where W1 + W2+, etc., +Wnis the whole work done in moving the unit charge from C to B. In the final result all intermediate terms have disappeared, the result is therefore the same however great the number of parts into which CB may be divided; it is therefore clear that no error was introduced by taking the geometrical mean of the forces at B and ai as the average be- tween those points. It may be shown that the work will be the same along any path whatever between B and C, even though these points may not lie in the same direction from A. Thus the work done in carrying unit positive charge from C to B (Fig. 323) against the repulsive force of a charge 9 at A is r R Now if the point C is at an infinite distance from A then -^ = 0, and the work done against the re- pulsion of q, in bringing a unit charge up to B from an infinite distance in air or vacuum, is simply -' and this, by definition, is Fig- 323. the potential at B due to the charge q. Representing this po- tential by V we have, F = — If there are a number of charges gigjga, etc., at distances 7-1 7-3 7-3, respectively, from the point B and in any directions what- ever, the potential at that point becomes, when air is the medium, y = 11 + 1^ + 2^+, etc. since the potential at a point depends only on its distance from charges and not on their directions. The signs of the terms depend on whether the charges are positive or negative. In any other medium than air the potential V may be com- puted from the formula ''4(^^^.*■) where K is the specific inductive capacity of the medium (§525). 390 ELECTROSTATICS 579. Zero Potential. — According to the definition just given, those points are at zero potential which are at an infinite distance from all electrified bodies. But the earth's potential has also been defined (§550) as zero potential. These two definitions are inconsistent if the earth has a charge, and there are reasons for thinking that it has. But any charge which the earth may have will change the po- tential of the earth and of all bodies in our laboratory rooms by the same amount, so that differences of potential will he unchanged, and it is only differences of potential that are measured by our instruments. In discussing problems that involve the electrical state of the heavenly bodies or of regions remote from the earth, of course it will not do to assume that the earth potential is zero. The zero must then be taken as defined in the preceding paragraph. 580. Equlpotential Surfaces, — Suppose there is a charge of 12 units at A (Fig. 324) which is not near any other charged body. Then the potential due to A may be calculated from the formula F = ^ r where q = 12. At 1 cm. from A in any direction the potential will be 12. The sphere of radius one having A as center is there- fore an equlpotential surface of potential 12. The sphere whose radius is 2 cms. is the surface of potential 6, the surface of po- tential one would have a radius of 12 cms., while zero potential would be at an infinitely great distance. If the charged point A is inside of a room the surface of the room will be at zero potential, for there will be an induced negative charge at each point of the surface sufficient to counteract the action of the charge A. The figure shows the position of the successive equlpotential surfaces, differing by unity, from 2 to 12. It will be noticed that they are closer together the nearer they are to A. The same amount of work must be done to move a unit charge from the sur- face 2 to 3, as from 3 to 4 or 11 to 12; in each case one erg of work is done. But the shorter the distance in which a given amount of work is done the greater the force that must be exerted, hence the surfaces are closer together near A where the electric force is greater. POTENTIAL 391 No work at all is done when an electric charge is moved along an equipotential surface, hence at every point the direction of the resultant force must be at right angles to the equipotential surfaces. Lines of force, or lines which at each point have the direction of the resultant force, must therefore cut equipotential surfaces at right angles, and in the above case are a set of radial straight lines. Fia. 324. — Equipotential surface due to a cnarge 12 at A. 581. Induction from the Point of View of Equipotential Sur- faces. — In figure 324 notice that the region B surrounded by the elliptical line reaches from a point where the potential is 3 to where the potential is 5. If 5 is a non-conductor this distribution Fig. 325. — Equipotential surfaces where B is a conductor. of potential is possible, but if 5 is a conductor flow must take place until it is all at the same potential. The left-hand end will receive a negative charge which will lower its potential, while the right- hand end will have its potential raised by a positive charge till all parts come to some potential intermediate between 3 and 5. The . lines of force and equipotential surfaces in this case are shown in figure 325. In this diagram the conductor is supposed to come 392 ELECTROSTATICS to potential 4, all other equipotential surfaces are bent outward or inward away from B. Some lines of force from A terminate on the negatively charged left end of B, while lines of force go out from the positively charged end of B to the right. This case is analogous to the formation of a level spot or pond on the side of a mountain. The ground must be cut away on the side toward the mountain, and built up on the outside. 583. Capacity of an Isolated Sphere. — Suppose an insulated sphere in the center of a large room. If it has a posi- tive charge Q, its lines of force term- inate on an equal quantity of negative electricity induced on the walls of the room. To find the poten- tial at the center of the sphere we have the formula, Fig. 326. — Isolated sphere. 7 = 21 + 2!+, etc. ri r2 ' (§578) In the present case the only charge which is near enough to pro- duce any appreciable effect at is the charge + Q. Although this charge is distributed over the sphere, it is all at the same dis- tance r from 0. Therefore the potential at the center of the sphere is 7 = ^ r But in case of a charged conductor all parts of it, inside and outside, are at the same potential, the sphere is, therefore, all at the potential V of its center. But by §572 Q = VC therefore C = r or the capacity in electrostatic units of an isolated sphere sur- rounded by air is numerically equal to its radius. If the medium surrounding the sphere has specific inductive capacity K, its capacity becomes (§572) C = Kr. CAPACITY 393 Fia. 327.— Spherical condenser. 583. Capacity of a Condenser Made of Two Concentric Spheres. — Suppose we have a condenser such as shown in figure 327, consisting of two concentric metal spheres with air between them. Let ri be the outer radius of the inner sphere and r-2 be the inner radius of the outer sphere. If a charge +Q is given to the inner sphere, an induced charge —Q will be found on the outer sphere. If the outer sphere is connected to earth it comes to zero potential and all charge disappears from its outer surface. The potential at the center of the small sphere is therefore since the charge +Q is at a distance ri from the center, and the charge —Q is at a distance ri from the center. But the potential everywhere inside of a closed conductor is the same as at its surface. Hence the potential of the inner sphere is Ti Ti \ri 7-2/ and since the outer sphere is at zero potential, V is the difference of potential between the two. But by the definition of capacity Q = CV therefore 1_ l^ Ti — Ti ri r-2 If the medium between the spheres has a specific inductive capacity K, the capacity of the condenser will be If the spheres are close together we may write ri7-2 = r^ and Ti — ri = d where r is the mean radius of the spheres and d is the thickness of the space between them. Then d 4ird Vr, 7-2/ C = 394 ELECTROSTATICS but 4irr'^ is the area of surface of a sphere of radius r; therefore ^ ~ 4,rd' In this form the formula can be used for any condensers where the two surfaces are close together, as in a Leyden jar or in a condenser made of two flat parallel plates. Problems 1. How much work must be done to carry a unit positive charge from a point 1 meter distant from a charge +100 to a point 2 cm. from it? 2. What is the potential at a point half-way between two equal spherical conductors having charges +100 and —100, respectively? 3. What is the potential at one corner of a rectangle which measures 40 X 30 cm. when there is a charge —300 at the diagonally opposite corner and + 120 at each of the adjacent ones? 4. A spherical conductor 10 cm. in diameter has a charge of +200 units and a small body having an equal plus charge is situated 1 meter from the center of the sphere. What is the potential at the center of the sphere? What is the potential at its surface? Is the charge distributed uni- formly over the sphere? 6. How much work would be done in moving the small charged body of the preceding question up to 50 cm. from the center of the sphere? Electric Discharge 684. Electric Discharge through Air at Ordinary Pressure. — Three forms of discharge are recognized through air at ordinary Fig. 328. pressures, the electric spark or disruptive discharge, brush dis- charge, and glow discharge. In the ordinary spark discharge there is a flash of light ac- companied by heat and sound and the medium is mechanically rent. The energy that was in the strained dielectric is dissi- pated in these various ways. The discharge must not be thought of as "jumping across" from one body to the other, it cannot be said to leap from posi- tive pole to negative or from negative to positive, but takes place simultaneously at every point along the path of discharge. Imag- DISCHARGE 395 ine a piece of rope AB held in position by a set of elastic bands which are attached to nails on each side of it, as shown in figure 328. If we pull the rope toward B the elastic bands are stretched and resist; but if enough force is exerted they will break, and the breaking will begin not necessarily at one end or the other, but wherever the weakest one is found. But when breaking occurs all parts of the rope move forward at once. This illus- trates very crudely what probably takes place in disruptive dis- charge; the strained medium begins to break down at the weakest point, wherever that may be, but the electric discharge takes place simultaneously at all points along the line of discharge. The hrush discharge is seen when in a darkened room the hand is brought near the positive conductor of a highly active electrical machine. If it is not held near enough for the spark discharge a luminous brush, like a little tree with branches of light ramify- ing from a short stem, extends out toward the hand from some point on the positively charged conductor. It seems to be caused by an almost continuous succession of extremely small discharges. Sometimes in the dark when an electric machine is highly ex- cited, but when the conductors are separated too far for sparks to pass, a faint velvety glow of violet light known as the glow discharge is seen on the knob of the negative conductor. 585. Oscillatory Discharge. — When a spring is bent and let fly it oscillates back and forth, coming to rest when its energy is Fig. 329. — Oscillatory discharge. finally spent in heat, sound, and air waves. So when a charged Leyden jar is discharged through a circuit of small resistance the energy of the charge cannot be dissipated, in the first rush and consequently there is a back-and-forth rush of current from one coating to the other until the energy is finally spent in sound, heat, light, and electric waves. This is known as the oscillatory discharge. If there is sufiicient resistance in the discharge cir- cuit there is no oscillation, just as a pendulum hung in molasses will sink to its lowest position without oscillation. 396 ELECTROSTATICS The oscillatory discharge was examined by Feddersen in 1863 by means of a rapidly rotating mirror. Seen in this way, each discharge showed as a group of sparks at regular intervals and rapidly dying out, as shown in figure 329; see also §758. 586. Mechanical and Heating Effects of Disruptive Dis- cliarge. — When the discharge takes place through a sheet of glass it is pulverized at the point of discharge. Pasteboard is perforated by the discharge, the edges of the hole being raised in a burr on each side as if by the sudden expansion and bursting out of the contained air or moisture. When trees are struck by lightning they are apt to be splintered, large slivers being flung violently out sidewise, perhaps due to sudden vaporization of moisture. When a living tree is struck the discharge usually takes the sap layer and frequently follows the grain. A glass tube having a fine bore filled with water and with a wire thrust a short distance in each end may be burst by the discharge of a Leyden jar. The electric discharge is accompanied by heat. Ether and bisulphide of carbon are readily ignited by it. Buildings con- taining inflammable material are occasionally set on fire when struck by hghtning. A mixture of one volume of oxygen with two volumes of hydrogen explodes with violence if even a minute electric spark passes in it. A little gunpowder placed between the ends of two wires through which a discharge is sent will usually be scattered unless the discharge is retarded by causing it to pass through a wet string or other poor conductor, in which case the powder may be ignited. Narrow strips of gold foil, 1 or 2 mm. in width, gummed to a sheet of paper so that they form a conducting strip, may be deflagrated or volatilized by the discharge of a Leyden battery. The purple stain which is left is wider than the gold-foil strips and is streaked at right angles to its length as though the vol- atilized metal had been driven violently out from the path of discharge. 687. Lightning. — The resemblance between lightning flashes and electric sparks was early noticed. Franklin, in 1752, per- formed the celebrated experiment of obtaining electric charges by means of a kite as a thunder storm was approaching. The LIGHTNING 397 kite was provided with metal points and the linen kite cord was a fairly good conductor when wet. To the lower end of the kite cord was fastened a metal key to which a silk cord was attached which was held in the hand and acted as an insulator Sparks were obtained from the key and Leyden jars weie charged, and the familiar phenomena of electric charges were observed. The so-called globe lightning, described by different observers as a ball of fire slowly moving along and then suddenly exploding with terrific violence, has never been imitated by any electrical discharges obtained in the laboratory and is so different from the ordinary phenomena of discharge that many physicists con- sider such observations illusory and due to a subjective effect of the discharge on the eye of the observer. 588. Atmospheric Electricity. — The electrical separation in thunder storms according to the theory of Simpson, is due to the disruption of rain drops in the uprushing current of air; for labora- tory experiments show that when a drop is broken up by falling on a vertical jet of air the resulting drops are positively charged while the current of air carries off negative charge. Rain from the lower part of the cloud will carry down positive charge while rain from higher regions of condensation will be negatively charged. Another circumstance that very possibly plays a part in the development of thunder storms is that condensation of moisture in the atmosphere takes place more easily around negative nuclei or electrons than it does around positive nuclei, and the fall of such drops to the earth will give it a negative charge. In fair weather the earth is usually negative, the potential being higher at points above the earth's surface, increasing at the rate of from 75 to 150 volts per meter above level ground, while in thunderstorms the atmospheric potential fluctuates greatly and may even be negative to the earth. 589. Lightning Bods. — It was shown (§538) that when an electroscope was surrounded by a conducting surface or even enclosed in a wire cage it was screened from outside electrical disturbances, and this suggests how buildings should be protected. Buildings with metal outer sheathing need no other protection, though care should be taken that the metal walls are at least as well connected to damp earth as the gas and water pipes within. 398 ELECTROSTATICS Wooden structures should have low metal points on the chim- neys and gables and other projecting portions, these points should be connected together by heavy wires or other conductors which run down the main corners of the building to the ground. At or near the ground it is well to have them connected together by a wire passing entirely around the building, and at two points on opposite sides of the building good ground connections should be made by connecting to pipes driven down to water or to a metal plate bedded in coke in damp earth. Insulation from the building is not needed, metal roofs and gutters and rain-water pipes should be connected together and may serve for lightning conductors if given good ground con- nections. Ordinary heavy galvanized iron telegraph wire will serve well for the conductor or, still better, a flat ribbon of sheet copper. ELECTRIC CURRENTS OR ELECTRODYNAMICS The Electric Current and Voltaic Cell 590. The Electric Current. — When a Leyden jar is discharged or when a series of sparks from an electrical machine pass through a conductor, in fact whenever a charge is communicated from one point to another, there is what is called a flow of electricity, or an electric current. The current is said to flow from the positive to the negative conductor. This is a convention; for vitreous electrification was called positive, and resinous was called negative, long before there was any idea of the direction of flow. Recent investiga- tions have led to the belief that in an electric current there is a flow or transfer of elementary negative charges or electrons from the negative to the positive conductor, thus what is believed to be the actual direction of flow is exactly opposite to the ordinary convention. In all the cases hitherto considered the flow has been so transitory as to be almost instantaneous. We now come to a series of discoveries which made possible the production of currents of electricity lasting for a considerable time. 691. Galvani's Discovery. — In 1786, Galvani, professor of anatomy at Bologna, in experimenting on the muscular contrac- tions produced by discharges from an electric machine, noticed that frogs' legs, hung on metal hooks in such a way that they rested in contact with a strip of another metal through which the hooks had been driven, were thrown into convulsive move- ments such as were produced by electric discharges. Following up the observation, he found that if strips of two unlike metals, such as zinc and copper were taken and one put in contact with the main nerve of the frog's leg while the other was touched to 399 400 ELECTRODYNAMICS the thigh muscles, spasmodic muscular contractions took place, provided the other ends of the metal strips were in contact with each other. 692. Volta's Discovery. — Volta, who was professor of physics in the University of Pavia, believed that the source of the elec- trical effects observed by Galvani was to be found in the contact of the dissimilar metals. But if there was any difference of potential produced in such a case it was far too small to be detected by the gold-leaf electroscope as ordinarily used. This difficulty was most ingeniously over- come by Volta's device of the con- densing electroscope. A gold-leaf electroscope was con- structed having a flat brass plate instead of a knob, as shown in figure 330, on which rested a second brass plate of the same size having an in- sulating handle of glass by which it could be raised. Both plates were given a thin coating of shellac var- nish by which they were insulated from each other and thus formed a condenser of large capacity, since the separating dielectric was thin. The lower plate was then touched by a strip of copper soldered to the end of a zinc strip held in one hand while at the same time the upper plate was touched with the other hand. When the upper plate was raised after breaking these contacts, the gold leaves diverged with negative electricity, showing that the upper plate of the condenser had been charged positively and the lower negatively by the operation. The advantage of the condenser was that although the difference of potential between the plates was exceedingly small, a considerable charge was accumulated which was set free when the plates were separated. When the two condenser plates were of brass and directly connected by the copper-zinc circuit, as in figure 331, no charge was obtained since the end metals were alike, being the two brass condenser plates; but if at any point in the circuit two dissimilar metals were connected by a dilute acid or salt solution, as shown in figure 332, the condenser plates were charged. In this case Fiu. 330. — Volta's discovery. VOLTA'S DISCOVERY 401 the solution takes the place of the body of the experimenter in the original experiment. It is now believed that the differences of potential obtained by Volta were mainly due not, as he supposed, to the contact of Copper Diluted aold or aalt tolutlon no charge Fig. 331. condenser charged Fig. 332. dissimilar metals, but to the contacts between these metals and the hands of the experimenter or the acid or salt solution. 593. Voltaic Pile. — In seeking to obtain a larger effect Volta found that when he took two cells in which strips of zinc and copper dipped into dilute acid, and joined them in series, as shown in figure 333, he obtained in the electroscope double the charge given by one cell. The effect was found to depend only on the kind of metals and acid used and not at all on the size of the plates. The Voltaic pile, based on this discovery, consists of discs of copper, zinc, and cloth or paper saturated with acid or salt solution, piled one upon another, first a disc of copper, then acidulated cloth, then zinc, then again copper, cloth, and zinc, and so on. A pile having 50 such combinations will produce 50 times the difference of potential that can be ob- tained from a single element consisting of zinc-acid-copper. What are known as dry piles are made by taking discs of gilt paper and so-called silver paper, placing them in pairs, the gilt face of one against the silver face of the other, and then Fig. 333.— Charge by two cells. 402 ELECTRODYNAMICS making a pile of such pairs, the same kind of paper being upper- most in each pair. In a moist climate the natural dampness of the paper enables it to play the part of the acidulated cloth layers in Volta's pile. 594. Voltaic Cell. — A cell having a plate of zinc and a plate of copper dipping in dilute sulphuric acid is known as a simple Voltaic cell, and several cells combined constitute a Voltaic or Galvanic battery. Since Volta's day many improved kinds of battery cells have been devised, some of which will be considered later (§§626-635). The two plates of a Voltaic cell are called the electrodes, and the terminals of the plates where the external wires are con- nected are called the poles of the cell. The copper terminal is at a higher potential than the zinc terminal and gives a positive charge, it is therefore called the positive pole, while the zinc terminal is the negative pole. On the other hand, the copper plate is often spoken of as the negative electrode or electro- negative element in the cell and the zinc as the positive electrode or electropositive element, because positive charge is transmitted through the acid of the cell from the zinc to the copper Fig. 334.— Electric plate as though repelled by the zinc and at- tracted by the copper. Electric Current in a Cell. — Since the two poles of a Voltaic cell are at different potentials, an electric current is established when they are connected by a metallic wire just as when the two coats of a charged Leyden jar are connected. This current, however, flows steadily instead of lasting only for an instant. The metallic wire together with the plates and liquid between them form a conducting circuit in which the positive direction of the current or that in which positive electricity flows is from copper to zinc through the outside wire and from zinc back to copper inside the liquid of the cell. There are three principal evidences of the existence of the current : 1. Heat is developed in all parts of the circuit. 2. Every part of the circuit affects a magnetic needle brought near it. VOLTAIC CELL 403 3. Chemical action takes place at the surfaces of contact between the metal electrodes and the liquid. If the copper and zinc plates are in dilute sulphuric acid, bubbles of hydrogen gas appear at the surface of the copper plate, while the zinc plate is eaten away by the acid, and zinc sulphate is formed. All these phenomena cease at once when the current is inter- rupted, either by breaking the metallic connection between the plates or by separating the acid around one plate from that around the other by a non-conducbing partition. 595. Contact Potentials in a Voltaic Cell. — When zinc is immersed in the acid there is what may be called a solution pressure, or tendency for the zinc to be dissolved and form zinc sulphate in solution, each atom of zinc carrying into the solution a positive charge. As the positively charged atoms of zinc pass into solution, the plate, losing positive charge with each one, becomes negative, while the solution becomes positive, in consequence of which there is an electrostatic force tending to prevent the positively charged zinc atoms from going into solution. Therefore when a certain difference of potential between the zinc and acid solution is reached there will be equilibrium between the electrostatic force and the solution tendency, and the zinc will cease to be dissolved. There is thus definite difference of potential due to the contact of zinc and acid when there is equilibrium between them, and another due to the contact of copper and acid which is less than the former since the solution pressure of copper in the acid is less than that of zinc. 596. Electromotive Force. — The diagram, figure 335, repre- sents the relative potentials of the elements in a Voltaic cell. The acid has the highest potential and is positive both to zinc and copper. The difference between acid and copper is, however, less than between it and zinc, and the copper is therefore at a higher potential than the zinc as shown. If the copper pole of the cell is connected to the earth, it comes to the earth potential or zero, and the zinc pole as tested by a quadrant electrometer is found to have a negative potential. On the other hand if the zinc pole is connected to the earth it will be at zero potential while the copper pole will be found to 404 ELECTRODYNAMICS be positive; but the difference of potential between them .will be the same in each case. Every cell can produce a certain maximum difference in potential between its two electrodes, and when this is reached there is equilibrium and the chemical action stops. The maximum difference of potential which a cell can produce is called its electromotive force; it is measured by the difference of potential between the electrodes when there is no current and the chemical action has ceased. The electromotive force of a cell depends only on the chemical relations of the constituents of the cell and is therefore the same whether the plates are large or small. A small cell formed by dipping the tips of a zinc and of a copper wire into a single drop of acid will cause as great a deflection of a quadrant electrometer as a cell of the same kind with plates a foot square. A convenient abbreviation for electromotive force is E.M.F., or in equations the symbol E is commonly used. 597. Hydraulic Analogy to Voltaic Cell. — The following anal- ogy given by Lodge is instructive. Two tall open vessels con- Uquld Potentral Copper PoUnilat Zinc Potential FiQ. 335. ^^ B F^= P ^^^ \ V Fig. 336. taining water are connected by a pipe in which is a pump P driven by a weight W (Fig. 336). The water will flow from one vessel to the other until the back pressure on the pump due to the higher level of B just balances the force of the weight. The difference in level will be the same whether the vessels are large or small. The difference of level represents the difference of ELECTRIC CURRENT 405 potential between the zinc and copper which is independent of the size of the cell, the pump with its driving weight is the electromotive force of the cell, which through chemical action can produce a certain definite difference of potential and no more. Figure 337 represents the state of things when the zinc and copper plates are connected by a wire, represented by the tube shown. The difference of pressure causes a flow through the tube from B to A, at the same time the level sinks in B and rises in A so that the difference in pressure on the two sides diminishes and is no longer able to balance the pressure of the pump, which therefore begins to act, forcing water from A to B; at the same time the weight W descends, supplying energy for the circulation, which will be maintained so long as the weight can move down- ward. Here it is seen that the electromotive force, represented by the power of the pump to produce pressure, is the same as before, but the difference of potential between the plates, shown by the difference between the levels of A and B is less than before. The work done by the pump in circulating the water is obtained from the weight, which loses potential energy as it descends. So in the Voltaic cell, the energy expended by the electric current is supplied by the chemical changes which take place at the electrodes. 698. Magnetic Effect of Current. — In 1819 Oersted discovered that when a wire connecting the poles of a Voltaic cell was held over a balanced magnetic needle and parallel to it, the needle was deflected, the north pole of the needle moving toward the west when the current was from south to north, as in the diagram, while if the current was reversed the north pole of the needle moved toward the east. The effect was reversed when the wire was placed under the needle. This discovery aroused the greatest interest, as it was the first evidence of a connection between magnetism and electricity. 599. Electric Circuit. — It was also found that the action was the same whatever part of the wire connecting the plates was brought near the needle, the deflection produced by the current in the middle of the wire being just as great as that near its ends. By this experiment also the direction of the current in the electrolyte may be shown to be opposite to that in the wire; for if 406 ELECTRODYNAMICS two vessels are used connected by a short tube containing the acid, and if a zinc plate is placed in one vessel and copper in the other, as shown in figure 339, a magnetic needle will be deflected toward the west when placed under the wire connecting the plates, but toward the east when under the tube. The experi- ment shows that the current in the electrolyte is just as strong as that in the wire, but in the opposite direction. From experiments such as the above it is inferred that steady electric currents always flow in closed circuits and are equally strong at every point, and if the circuit is interrupted at any point, whether in the electrolyte or the wire, the magnetic action and all other current effects cease everywhere at almost the same instant. It is very much as when an incompressible liquid cir- culates in a closed tube, just as much liquid must pass any one section of the tube as any other during the same time. Fig. 338. — Current and magnetic needle. Fig. 339. — Current in electrolyte. 600. Galvanometers. — When a wire is bent into a vertical circle having its plane parallel to the direction of a magnetic needle balanced at its center, if a current is established in the wire all parts of it act together to deflect the needle, turning the north pole to one side or the other, depending on the direc- tion of the current. An instrument which measures electric currents by the deflection of a magnetic needle is known as a galvanometer. 601. Galvanometers Measure Current. — Faraday showed that a galvanometer measures the quantity of charge transmitted per second, or what is called the current strength. For he found that when a Ley den jar was discharged through a sensitive galvanometer there was an instantaneous swing of the needle to one side, the amount of which depended only on the quantity of the charge; that is, the swing produced by forty turns of his ELECTRIC CURRENT 407 electrical machine was the same whether the charge was held in a small jar at high potential or in a large Ley den battery at low potential, and whether the wet string through which the discharge was sent was long or short. It was also established by Faraday that when a constant current flowed through a galvanometer producing a steady deflection of the needle, the magnetic force on the needle due to the current was proportional to the quantity of charge transmitted per second. 602. Unit Current. — Instead of measuring electric currents by the quantity of charge in electrostatic units transmitted per second, it is found better to adopt a new system of units based on the magnetic effect of a cur- rent and using magnetic units as already defined. This system is known as the C. G. S. ^^°- 340.— Galva- nometer. or absolute electromagnetic system, since it also is based on the centimeter, gram and second. In this system a unit current is one which, fl,owing in a circular coil of one centimeter radius, will act on a unit magnetic pole at its center with a force of one dyne for every centimeter of wire in the coil. The Ampere or Practical Unit of Current. — The unit of current in the practical system is called the Ampere in honor of the French physicist who first investigated the laws of the magnetic effects of currents. It is defined as one-tenth of the absolute or C. G. S. unit current, being chosen smaller than the absolute unit for reasons of convenience. The quantity of charge transmitted by one ampere in one second is called a coulomb. One coulomb is equal to 3,000,000,000 electrostatic unit charges as defined in §525. 603. Unit of ElectromotiTe Force. — In our studies of electro- statics it was shown (§554) that the difference between the po- tentials of two conductors might be measured by the work re- quired to transfer unit charge from one conductor to the other. Just so in the absolute electromagnetic system of units two points in a conductor are said to have unit difference of potential when one erg of work is required to transfer the C. G. S. unit quantity of electricity from one point to the other. Unit quantity of electricity in the C. G. S. system is of course the charge transmitted per second by unit current in that system. 408 ELECTRODYNAMICS The unit of potential in the absolute system is found to be so small compared with the electromotive forces of ordinary battery cells, that it was decided to adopt for ordinary use a unit one- hundred million times as great, called the volt in honor of Volta. The Volt is the unit of electromotive force in the practical system and is lO^ times as great as the C. G. S. electromagnetic unit of potential. It is much smaller that the electrostatic unit of potential defined in §554, the latter being almost exactly equal to 300 volts. The electromotive force of the Voltaic cell is nearly 1 volt. 604. Resistance. — ^Let a circuit be made up of two battery cells A and B joined in series with some other conductors and a galvanometer, the two cells being so connected that their electro- motive forces act in the same direction. After observing the current strength as shown by the galvanometer, let the circuit be rearranged, taking the same components in any other order whatever. If the two electromotive forces still act together the current will be found the same as before, showing that the current strength is not affected by the particular order of the parts in an electric circuit. But if one cell is turned around so that its electromotive force opposes that of the other cell, then the effective electromotive force in the circuit will be the difference between the Fig. 341. electromotive forces of the two cells in- stead of their sum as in the former case, and the current in this case will be smaller than before, just in proportion as the electromotive force is smaller. That is, the current strength is proportional to the effective electromotive force; or, in other words, the ratio of the electro- motive force to the current strength in a given circuit is a con- stant, which depends only on the make-up and physical condition (temperature, stress, etc.) of the circuit. This constant is called the resistance of the circuit and is not affected by the order in which the various conductors, cells, etc., are connected, nor by the direction in which the current flows through them. This relation, established by the German physicist G. S. Ohm, is known by his name and may be stated as follows: RESISTANCE 409 Ohm's Law: The ratio of electromotive force to current in a given circuit is a constant which may be called the resistance of the circuit. Or, in symbols, -J = K = a constant where E represents the electromotive force, 7 the current, and R the resistance of the circuit. It is also established by experiment that each battery cell and piece of wire or other conductor has a definite resistance which belongs to it individually and depends only on its temperature and state of stress (provided that the same two points on the conductor^ are always used in maldng connection with the rest of the circuit) ; and when the several parts of a circuit are joined together one after another, in series as it is called, the resistance of the whole is the sum of the resistances of the several parts. 605. Unit of Resistance. — In honor of the discoverer of this law the unit of resistance in the practical system is called the ohm; it is the resistance of a circuit in which an electromotive force of one volt will produce a current of one ampere. Ohm's law may then be expressed in units of the practical system, thus : Electromotive force in voUs Current in amperes Resistance in ohms The electrical resistance of a conductor is analogous to the frictional resist- ance which a pipe offers to the flow of liquid through it. In both cases work done against the resistance appears as heat, and in neither case does the resistance have any tendency to produce a back current. 606. Exception to Ohm's Law. — In gaseous conductors the ratio of the electromotive force to the current is not constant as in other conductors, but depends on the strength of the current. Chemical Effects of Current 607. Decomposition of Water. — When a current of electricity is passed through dilute sulphuric acid (1 part acid to 10 of water), xising platinum electrodes immersed in the acid, gas is given off at each electrode. The gases may be separately collected in tubes filled with the dilute acid and inverted over the electrodes as shown 410 ELECTRODYNAMICS ^ A A H T' 1 ^ i i Fig. 342.— Electrolysis of water. in figure 342. The gas liberated at the positive electrode is found to be oxygen while that at the negative electrode is hydro- gen, and the volume of hydrogen is just twice the volume of the oxygen. These volumes are exactly in the ratio in which the gases combine to form water, and on this account it was at first supposed that the current directly decomposed water. The decomposition of water in this way by the electric current was first accomplished in 1800 by Carlisle and Nicholson. 608. Discovery of Potassium and Sodium. — Sir Humphrey Davy, in 1807 by the use of a powerful battery of 250 cells, decomposed caustic potash, obtaining metallic potassium at the negative electrode. A frag- ment of caustic potash slightly moistened was laid on a platinum plate which was connected to tlie positive pole of the battery; on touching the potash with a platinum wire connected with the negative pole, minute globules appeared at the negative electrode which rapidly oxidized in air or took fire; these he recognized as a new metal which he named potassium. In a similar manner metallic sodium was ob- tained from caustic soda. 609. Faraday's Researches. — About the year 1833 Faraday began the systematic investiga^tion of the chemical effects of the electric current. Substances which are decomposed by the passage of the electric current he called electrolytes; the electrode connected with the positive pole of the battery or that through which (according to ordinary convention) current enters the electrolyte was named the anode (Greek, inward path), while the electrode through which the current leaves the electrolyte was named the kathode (Greek, out- ward path). The two constituents into which a molecule of the electrolyte is broken up were called ions (Greek, wanderers), that which is set free at the kathode being the hation, while that which appears at the anode was named the anion. 610. Faraday's Laws. — The following are some of the most important results of Faraday's investigations: 1. By introducing a number of electrolytic cells in different ELECTROLYSIS 411 parts of a circuit it was shown that the amount of substance decomposed is the same in each ceff through which the whole current passes, and in case of a divided cifcuit the sum of the amounts decomposed in the branches is equal to the amount in the undivided parts of the circuit. 2. The quantity of a given substance electrolyzed in a cell is proportional to the amount of charge or quantity of electricity which passes. 3. If several electrolytic cells containing different substances are connected in series in the same circuit, the quantities of the ions set free at the electrodes are proportional to their chemical combining equivalents. ' 611. Electrochemical £qulvalej|its. — The electrochemical equiv- alent of a substance is the qu^tity that is set free per second by a current of one ampere 09 by the passage of one coulomb of electricity. The following table gives the electrochemical equivalents of some well-known substances. It will be noticed that they are proportional to the combining equivalents. Elecirochemical Equivalents Substance Atomic weight Valence Combining equivalent Electrochemical equivalent. Gma. per coulomb Kations Hvdroeen 1 63.18 107.7 16 35.37 1 2 1 2 1 1 31.59 107.7 8 35.37 000010357 00032840 Silver 00111800 Anions Oxygen Chlorine 0.00008283 0.0003671 96,550 coulombs are transmitted when the number of grams liberated equals the combining equivalent of the substance. 613. Primary and Secondary Actions. — It is important to distinguish between the direct or primary effect of the current in electrolysis and the secondary chemical reactions that take place when the ions are set free. In illustration of this difference take the electrolytic apparatus containing dilute sulphuric acid, as described in paragraph 607, and connect it in series with a pre- cisely similar apparatus containing a solution of sodium sulphate 412 ELECTRODYNAMICS — O li ? ^ M H,SO, Ntt.SO, FiQ. 343. — Electrolytic cells in series. in water, colored by an infusion of purple cabbage. On sending a current through, both hydrogen and oxygen gases are set free in one cell exactly as in the other, and at the same time the coloring matter in the sodium sulphate solution turns red around the positive electrode, or anode, and green around the negative electrode, or kathode, showing that the originally neutral salt has become acid at the anode and alkaline at the kathode. Analysis shows that sodium hydroxide (NaOH) has appeared at the one electrode and sulphuric acid (H2SO4) at the other. It might seem at first that more decomposition was effected by the current in one cell than in the other, in violation of Faraday's law, for equal amounts of gas are set free in both cells, and in addition the sodium sulphate in the second cell is decomposed, while the sulphuric acid in the first cell remains unchanged. But it is believed that theprimary effect of the current is to separate precisely equivalent quantities of H2SO4 and N2SO4 in accordance with Faraday's law, the other changes being secondary chemical actions. Thus in the sulphuric acid cell the primary action of the current is to separate the H2 and SO4 ions; oxygen (O) is set free at the anode as the result of a secondary reaction in which the SO4 ion displaces the oxygen from a water molecule (H2O) ana forms sulphuric acid (H2SO4), which remains in solution. In sodium sulphate the primary action of the current is to separate the Na2 and SO4 ions. Then secondary reactions take place at both electrodes, the SO4 ions effect the liberation of oxygen at the anode exactly as in the other cell, while the posi- tively charged sodium (Na2) ions pass to the kathode, where each combines with two molecules of water 2(H20), forming sodium hydroxide 2(NaOH), which remains in solution, and setting free hydrogen (H2), which gives up its positive charge and escapes at the kathode. ELECTROLYSIS 413 These secondary reactions may be expressed symbolically thus : S04+H20 = H2S04 + Naa + 2(H20) = 2 (NaOH) + H^ 613. Theory of Electrolysis. — The earlier explanations of electrolysis supposed the decomposition of the electrolyte to be effected by the electric current, but it is now believed that a large per cent, of the electrolyte is ionized, or broken into positively and negatively charged ions, as a result of going into solution, and that the electric force in the electrolyte is simply directive, causing the positively charged ions to move with the current and the negatively charged ions to move in the opposite direction through the solution until they reach the electrodes where their charges are given up and the molecules are set free in the neutral state. The current is supposed to be made up of the charges which are thus carried convectively by the moving ions. Hittorf showed that different kinds of ions moved through the electrolyte with widely different velocities, and measured the relative velocities of anions and kations in aqueous solutions of many different salts and acids. While Kohlrausch, by measure- ment of the electric charges transmitted per second through these solutions, and assuming that the electric current is transmitted through the electrolyte wholly by the charges carried by the moving ions, has determined the actual velocities with which different kinds of ions move in aqueous solutions, and finds them proportional to the electric force, that is, to the fall of potential per centimeter in the solution. Some values found by Kohlrausch are given in the following table. Ionic Velocities for a potential gradient of one volt per centimeter Kations Anions Na 45. X 10-s cm./sec. CI 69. X 10~^ cm. /sec. H 320. " '• N03 64. Ag 57. " " OH 182. 614. Ionic Charges. — Faraday's laws show that every unival- ent ion carries a certain charge +e which is either positive or negative, depending on whether the ion is an anion or a kation ; while bivalent and trivalent ions carry charges +2e and ±3e, respectively. 414 ELECTRODYNAMICS It was suggested by Helmholtz that the charge e may be the atom of electricity from which all other charges are made up and of which they are therefore multiples. This is borne out by the experiments of MilUkan as we have already seen (§543). In the electrolysis of 1 gram of hydrogen 96550 coulombs of electricity are transmitted; and assuming that each atom of hydrogen carries the charge e as found by MiUikan, we find that in 1 gram of hydrogen there are 5.91 X 10'^ atoms. 615. Polarization. — At the electrodes where the ions are set free or enter into new combinations there are generally electro- motive forces, because at those points electric energy has to be spent to effect chemical changes. The resultant of these electro- motive forces is called the polarization of the cell. In case of the electrolysis of copper sulphate between copper electrodes the chemical change which takes place at the kathode is opposite to that at the anode: Cu and SO4, are separated at the one and united at the other. Therefore, in such a cell there is on the whole no electromotive force of polarization. But if dilute sulphuric acid is electrolyzed between platinum electrodes there is an electromotive force developed against the current, or a back electromotive force of about 1.7 volts, and unless the battery employed has an electromotive force greater than this the current cannot be maintained. While the electrodes are thus polarized the cell is in reality a battery cell, and if it is disconnected from the main circuit and its electrodes joined by a conducting wire, a current is obtained opposite to that which caused the polarization. This current flows until the gaseous layers on the electrodes disappear. The cell is thus really a storage battery cell of very small capacity. All storage battery cells or accumulators depend on the elec- tromotive force of polarization. When dilute sulphuric acid is electrolyzed with a zinc anode and copper kathode, as in the simple Voltaic cell, more chemical energy is given out at the anode where zinc sulphate is formed than is absorbed at the kathode where hydrogen is hberated from the solution, and consequently, on the whole, energy is given out by the chemical changes instead of being required to bring them about, hence the electromotive force of polarization is with the current instead of against it, and the combination is called a battery cell. ELECTROLYSIS 415 616. Measurement of Current.— Currents of electricity are con- veniently measured by their electrolytic effect. In the instrument shown in figure 344, known as a voltameter, dilute sulphuric acid is electrolyzed between platinum electrodes, and the es- caping gases are caught mingled together in the graduated tube above the electrodes. From the temperature, volume, and pressure of the collected gas its weight can be determined, and if the time during which the current was flowing is known the current can be calcu- lated, since 1 ampSre will set free in 1 minute 0.00559 gms. or about 12.2 c.c. of the mixed gases. 617. Copper Voltameter. — A more accurate in- strument for the measurement of current is the copper voltameter, in. which copper sulphate is electrolyzed between copper electrodes. From the gain in weight of the kathode while the current is flowing the amount of copper deposited per second is determined, and so the current is found from the electrochemical equivalent of copper. The form shown in figure 345 is con- venient. The alternate plates are connected into one set and form the anode, while the intermediate plates form the kathode, so that each kathode plate is between two anode plates. The number of plates used depends on the strength of the current to be measured. To secure the best results something like 40 sq. cms. surface per ampere is required in the kathode. 618. Silver Voltameter. — For standard determinations it is found that the most reliable results are obtained from a form of silver voltameter in which the liquid is a standard solution of nitrate of silver contained in a platinum cup which also serves as the kathode, while the anode is a rod of pure silver which dips into the liquid. The anode must be sur- rounded by a covering of filter paper to prevent any particles of silver that may become loosened from the anode from falling into the platinum cup. 619. Electroplating. — By means of the electric current metallic objects may be plated with gold or silver or other metals. Figure 346 shows a form of elec- troplating bath. The objects to be plated are hung on metal rods which are all connected with the negative pole of the battery or dynamo which supplies the current. If silver is to be de- posited, plates of silver connected with the positive pole are hung in the bath between the objects which are being plated, so that while silver is being deposited the anode plates lose an equal amount and the strength of Fig. 344.— Volt- ameter or Cou- lomb-meter. FiQ. 345. — Copper voltameter. 416 EI.ECTRODYNAMICS the solution is maintained constant. The thickness of the deposit will generally be greater on projecting parts of the object plated and on parts that are nearer to the anode plate. 620. Capillary Electrometer. — Let a drop of mercury rest in a. level tube turned up at the ends and full of dilute sulphuric acid (Fig. 347). If a current of electricity is passed through the acid the mercury will move Fig. 346. — Electroplating bath. along in the direction of the current; i.e., from .the positive toward the negative terminal. The surface tension where the current passes from acid to mercury is increased and that where it passes from mercury to acid is decreased, hence the motion takes place (§259). Fig. 347. — Drop of mercury in acid. Lippmann has devised an electrometer based on this principle, known as the capillary electrometer, which is useful in measuring electromotive forces of less than a volt. Problems 1. If in a copper voltameter the kathode plates gain 1.50 grm. of copper in ten minutes, find the average current strength in amperes. 2. How many grams of hydrogen and of oxygen are set free when 1 grm. of water is decomposed by electrolysis ; and how many cubic centimeters of each of these gases will there be at 0°C. and 76 cm. pressure? 3. How many coulombs of electricity will be required to effect the decom- position in the previous problem, and how long a time must a current of 0.5 amperes flow to accomplish it? ELECTROLYSIS 417 Battery Cells 631. Battery Cells. — A battery cell is a combination in which electromotive force is produced by chemical action. The simple cell of Volta is the earliest type, but it has important practical defects. An ideal cell will have: 1. Small resistance. 2. Large electromotive force. 3. A constant electromotive force whatever the current. 4. No local action or wasteful chemical action. 633. Resistance of Battery Cells. — When the electrode plates are large and close together the resistance of the cell is small. While if the plates are very small the resistance of the cell may be so great that even when the poles are short-circuited or con- nected by a short copper wire offering very little resistance, the current will be extremely small. Cells from which large currents are to be obtained must, therefore, have large plates separated by a comparatively thin layer of electrolyte. 633. Local Action. — If commercial zinc is used in a Voltaic cell hydrogen gas will be given off at the surface of the plate as soon as it is placed in the acid and before it is connected with the copper plate. This is accompanied by a corresponding wearing away of the zinc and formation of zinc sulphate, which goes into solution. This wasting of the zinc is called local action and is due to impurities. Suppose that a particle of iron or carbon imbedded in the surface of the zinc is in contact both with the zinc and acid ; it forms a minute Voltaic cell, in which the cur- rent flows from the iron or carbon to the zinc and through the acid from zinc to iron again, as indicated in the figure, and zinc is eaten away near the impurity and hydrogen set free at ^ ^^°- 34?' ... ^ ■^ Local action, its surface. To prevent local action the zinc surface is freshly amalgamated with mer- cury, which dissolves the zinc, covers up the impurities, and presents a homogeneous surface to the acid. 634. Polarization. — When the poles of a simple Voltaic cell are connected by a wire, the current does not remain constant but rapidly decreases in strength. This weakening of the current is due to polarization. The hydrogen set free at the copper electrode forms a sort of gaseous % i^ m 418 ELECTRODYNAMICS layer over the plate which interferes with the action of the cell in two ways. In the first place, the resistance of the cell is increased, for the flow of electricity is interfered with by the bubbles of gas. In the second place, the electromotive force of the cell is diminished, for the hydrogen layer is much more like zinc in its relation to the acid than is the copper which it covers. This difficulty is most effectively met by the use of two electrolytes. 635. Primary and Secondary Battery Cells. — Cells such as the Voltaic cell in which the current is obtained from the chem- icals of which the cell was originally constructed are known as primary cells, while cells in which the chemical state neces- sary for the production of a current is produced by sending through the cell a current from some outside source for a cer- tain length of time, are known as secondary batteries, storage cells, or accumulators. A few of the cells most commonly used in practice will now be considered. Primary Battery Cells 636. The Daniell Cell.— One of the first and most useful two fluid cells was devised by Daniell in 1836. It consists of a copper electrode immersed in a solution of copper sulphate and an electrode of amalgamated zinc immersed in dilute sulphuric acid, the two being separated by a partition of porous earthenware. In figure 349 the copper elec- trode with its solution is represented as con- tained in a cup of porous earthenware sur- rounded by the zinc and dilute acid. When the circuit is closed, the positively charged zinc atoms pass into solution form- ing zinc sulphate with the negative SO4 ions, while the positively charged hydrogen ions (H2) in the acid move toward the copper plate, passing through the porous cup by diffusion and forming sulphuric acid (H2SO4) with the negative SO4 ions from the copper sulphate, and displacing the positive copper ions (Cu) Fig. 349.— Daniell's cell. BATTERY CELLS 419 which give up their charges and are deposited on the copper plate. In the dilute acid In the copper sulphate Zn + H2SO4 = ZnS04 + H2. H2 + CUSO4 = H2SO4 + Cu. Thus zinc is dissolved and zinc sulphate formed, copper sul- phate is used up and copper deposited on the copper electrode. There is no hydrogen layer formed on the copper and conse- quently no polarization. The electromotive force of this cell is about 1.08 volts. 637. Gravity Cell. — A form of Daniell cell which has been extensively used in telegraphy and is still much used where a small constant current of electricity is required is the gravity cell, so called because the liquids are kept separate by gravity alone, the denser copper sul- phate solution resting at the bottom of the cell, while the lighter acid or zinc sulphate solution floats above it. If the gravity cell stands without being used the copper sulphate diffuses gradually up into the acid above and copper is de- posited on the zinc, causing extensive local action. A small current, sufficient to bal- ance the diffusion, should always be kept flowing while the cell is set up. 638. The Bichromate Cell.— In this cell the positive pole is a plate of gas car- bon in a solution of bichromate of potas- sium. The negative pole is of zinc in dilute sulphuric acid and the two solutions are separated by a cup of porous earthenware which holds one of them. The electromotive force of the cell is about 2.0 volts. In the ordinary plunge battery the carbon and zinc plates both dip into the same bichromate solution to which a little sulphuric acid has been added. The zinc plate is lifted out of the solution when not in use. 639. Leclanche Cell. — This very useful form of cell has a zinc and a carbon electrode. The carbon is packed in a porous cup with a mixture of fragments of carbon and black oxide of manganese; the zinc electrode is in a strong solution of ammonium chloride (sal ammoniac) which surrounds the porous cup. The hydrogen which would polarize the carbon electrode combines with oxygen from the manganese dioxide and forms water. But as the depolariz- ing agent is in the solid form its action is slow, and the cell polarizes tempo- rarily. It is extensively used, however, for open-circuit work, such as for Fig. 350. — Gravity or crow- foot cell. 420 ELECTRODYNAMICS bells, annunciators, and clocks, where a steady current is not required. It is entirely free from injurious or disagreeable fumes, there is but little local action, and no trouble from diffusion, so that the cells may stand set up for a year or two without attention, and ready for use m\ at any instant. Its electromotive force is about ^g^ I I 1.40 volts. JjSSSL Q 11 The chemical changes in this cell are: "^mjii^S i 2NH4+Mn02 = 2NH3+H2O+MnO ^HlH ^^^ + Zn = ZnClz I ■11 ^^' ^^^ Cells. — The so-called dry cells H H are ordinarily a form of Leclanch^ cell. The lUi III outer cylindrical cup forms the zinc electrode I 111 which is lined with thick absorbent paper and iBl I ll lllllKlly -'^- packed with the pulverized manganese dioxide and •I ^^^^^ ^g ;^-- ;_: carbon mixture surrounding the central carbon rod. — The whole is saturated with ammonium chloride FiQ. 351. —Leclanche cell, solution and sealed with pitch to keep it from drying out. 631. Edison-Lalande Cell. — In this form of cell, devised by Lalande and improved by Edison, the positive plate is a tablet of compressed black oxide of copper, while the negative plate is of zinc. These are immersed in a strong solution of caustic potash, which is covered with a thick layer of heavy oil to prevent evaporation and the creeping up of the solution on the sides and plates. The plate of copper oxide acts both as electrode and depolar- izer, the hydrogen which is set free at that pole reducing the copper oxide to metallic copper. When the cell is exhausted both plates as well as the liquid must be renewed. This cell does not polarize and may be used where a steady current is required and where a Daniell or gravity cell would have too much resistance. It has the further advantage that it is quite free from local action and may be left standing without deterioration when no current flows. Its electromotive force is low, being about 0.75 volt. Its chemical changes may be expressed as follows: 2K + CuO + HoO = Cu + 2KOH 20H + Zn + 2K0H = KjZnOa + 2H2O. Secondary Cells 632. Grove's Gas Battery. — The English physicist Grove showed that when in the decomposition of water long electrodes were used, extending to the tops of the tubes in which the gases were collected, as in figure 352, on changing the switches ss' to the dotted positions, thus disconnecting the battery B and simply BATTERY CELLS 421 joining the two electrodes together through a galvanometer G, a current was obtained which was in the opposite direction to the decomposing current. At the same time a gradual recombina- tion of the hydrogen and oxygen took place until these gases had entirely disappeared. Fig. 352. Long electrodes were necessary since each electrode must pass through the surface where the gas and electrolyte meet. 633. Plante Cell. — If a current of electricity is sent through a cell consisting of two plates of sheet lead in dilute sulphuric acid it becomes polarized, one plate becoming oxidized while hydrogen is set free at the surface of the other, reducing any oxide that may be there. In the year 1860 Plants, a French, physicist, found that secondary cells of large capacity could be made in this way. His method was to send a current through the cell in one direction until one plate was well oxidized, after which the cell was discharged and then charged by a current in the opposite direction, thus oxidizing the other plate and reducing the oxide on the first to metallic lead in a spongy form. The cell was then again discharged and charged with a current in the same direction as at first, and so by alternately charging and discharging, first making one plate positive and then the other, a deep layer of active material was formed on each plate. The plates were then said to be formed. 634. Storage Cells — Accumulators or Secondary Batteries. — Secondary battery cells, or storage cells as they are frequently called, have become extensively used in electric motor vehicles, in electric power plants, and in telegraphy. The plates are usually heavy lead grids full of holes or grooves containing the active material, which is either packed in them me- chanically or formed in them by some such process as that used by Plants. The positive plates contain a high oxide of lead, P6O2, while the active parts of the negative plate are of spongy lead. A cell is formed of a number of such 422 ELECTRODYNAMICS plates, alternately negative and positive, as shown in the figure, set in a suitable vessel containing dilute sulphuric acid. The negative plates are connected together and form one set and the positive plates form another, there being one more negative than positive plate, so that each positive plate is between two negative ones. This is to prevent buckling or bending of the positive plate, for the formation of oxide in charging is accom- panied by an increase in volume or swelling of the plate, which would warp it badly if it took place only on one side. The nearness of the plates to each other and the large surface obtained by using a number of plates cause the resistance of the cell to be very small. The greater the number and size of the plates in a cell the larger the current that can be sent through it without in- jury to the cell. About 1 ampere per 12 sq. in. of opposed surface is usually a safe rate of discharge. The commercial importance of such storage cells is due in part to their extremely small resistance and to the fact that they are renewed not by means of costly chemicals, but by a current obtained from a dsTiamo machine driven by an engine or by water power. They can be used, therefore, to store the superfluous energy of a power plant at times when but little power is used and give it back again in times of need. The electromotive force of this type of storage cell is about 2.10 volts. The chemical changes in such a cell are as follows : Fig. 353. — Storage cells. Discharging Positive plate PbOj + Hz + H2SO4 = PbSO* + 2HsO Negative plate Pb + SOi = PbSOi Lead sulphate is thus formed at each plate. Charging Positive plate PbSO, + SO, + 2H2O = PbOz + 2H2SO« Negative plate PbSOi + H2 = Pb + H2SO4 635. Edison Storage Cell. — A new form of storage cell has been devised by Edison in which the active materials of the electrodes are the oxides of nickel and iron, respectively, the electrolyte being a solution of caustic potash in water. A battery of these cells is said to weigh one-half as much as the equivalent lead cells. The cells are very durable and are not so easily injured as lead cells by overcharging or leaving uncharged. Fig. 354. BATTERY CELLS 423 Modes of Connecting Cells 636. Battery Cells in Series. — When battery cells are con- nected as shown in the figure, the positive pole of one being joined to the negative pole of the next, they are said to be joined in series, and the electromotive force of the combination is the sum of the electromotive forces of the several cells. As the whole current must pass through each cell the resistance of cells joined in series is the sum of the resistances of the separate cells. Suppose that three cells, each with electromotive force e and resistance r, are connected in this way with an external resistance R. The total electromotive force is 3e and the resistance is 3r -{- R so that by Ohm's law, I = „ I D (for 3 cells in series) where I is the current. This arrangement is advantageous when the external resist- ance R is large and a large electromotive force is required. Battery cells in series may be likened to a series of pumps, the first of which lifts water to a certain level where the second takes it and lifts it to the next higher level and then the third raises it again to a still higher level, etc. 637. Battery Cells in Parallel. — If cells are joined together as shown in figure 355, all the copper poles being connected together for the positive pole and all the zincs for the negative, they are said to be joined in parallel. p>jg 355 ' Such a combination has precisely the advantage that a large cell has over a small one. Its electromotive force is the same as that of one cell, while its resistance is less — a combination of four similar cells joined in this way having only one-fourth the resist- ance of a single cell. Only similar cells should be joined in parallel, otherwise a cell of smaller electromotive, force may have a reverse current sent through it by a stronger cell. This mode of arrangement is useful when the external resist- 424 ELECTRODYNAMICS ance in the circuit is much smaller than that of a single battery cell or where the current to be obtained is more than can ad- vantageously be transmitted through a single cell. Cells in parallel may be likened to a set of pumps which are all lifting water from the same lower canal to another at a higher level. 638. Combined Series and Parallel Arrangement of Cells. — Several similar series of cells may be combined in parallel as shown in figure 356, where two series of three cells each are con- nected in parallel. It will be 9^ 9^ £^ observed that the resistance of each of the rows is Sr and the electromotive force of each row is 3e. The resistance of the two 3r If ^^ rows in parallel is then -~- or ■mrrrrmmnnrnm — Pjq 356. one-half that of the row, while the electromotive force of the combination is the same as that of a single row. The ex- pression for current is then, for this particular arrangement, 3e l+« A square arrangement, where the number of cells in each row is the same as the number of rows, will have the same resistance as a single cell. 7/ o given number of cells are to be combined so as to give the maximum current through a given outside resistance, use that arrangement which will make the battery resistance most nearly equal to the external resistance. Commercial types of storage battery cells have such small resistances that except where very large currents are required they are connected in series. If large currents are to be obtained, several series of cells are arranged in parallel so that the current through each series will only be such as the cells are adapted to transmit without injurj^, and in each series as many cells are used as are necessary to give the required electromotive force. BATTERY CELLS 425 Problems 1. If two gravity cells and one Leclanch^ cell are joined in series with a coil of wire having a resistance of 5 ohms, what current is obtained, when the gravity cells have a resistance of 2 ohms each, while the Leclanch6 cell has resistance 0.4 ohms? 2. Find the current when the zinc pole of a gravity cell of 2 ohms resistance is connected to the zinc pole of a Leclanch6 cell of 0.5 ohms resistance while their other poles are connected by a wire of 1 ohm resistance. 3. Reverse the gravity cell in problem 2 so that its copper pole is connected to the zinc pole of the Leclanch^ cell, the other poles being connected by the 1-ohm wire, and find the current as before. 4. Make a diagram of two gravity cells of 2 ohms resistance each, con- nected in parallel to a coil of wire having 1 ohm resistance, and show what is the current in the wire and what current through each cell. 5. What is the electromotive force and internal resistance of a combination of 12 gravity cells, consisting of three series of four cells each, the three being connected in parallel? Take resistance of each cell 2 ohms and its E.M.F. 1 volt. 6. When the terminals of the battery described in question 5 are connected by a wire having a resistance of 3 ohms, find the current in the wire and also in each cell. 7. If a single storage cell has an electromotive force of 2 volts and a maxi- mum permissible discharge rate of 10 amperes, how many such cells will be required and how arranged to give a current of 30 ampferes and have an electromotive force of 50 volts? 8. If the cells in the last problem each have a resistance of 0.01 ohm, find what is the smallest resistance that can be permitted in the outside circuit. 9. How many gravity cells having a resistance of 2 ohms and E.M.F. 1 volt each, will be required to light a 50-volt incandescent 16-candle-power lamp which has a resistance of 50 ohms and requires 1 ampere of current, and what arrangement will require the smallest number of cells? Note. — The smallest number will be required when the battery resistance is equal to the external resistance. Fall of Potential and Resistance 639. Fall of Potential along a Circuit. — In every electric circuit there is a gradual decrease of potential along the external circuit from the positive to the negative pole of the battery. Faraday showed that in all parts of a simple undivided circuit the current is of the same strength. Therefore, except at points where electromotive forces are introduced, the potential must everywhere gradually change from point to point, for there is 426 ELECTRODYNAMICS no current between points at the same potential. In figure 357 is shown a circuit with a corresponding diagram of potentials, the potential at each point of the circuit being represented on the diagram by the vertical distance from the base line to the upper curve. Starting at C, it will be observed that the potentials steadily diminish till the zinc pole is reached at Z, where the lowest value is found. But passing from the zinc into the acid the potential is raised to its maximum at A by the chemical action; here there is an electromotive force and con- sequently a sudden change in potential. In the acid the poten- tial again steadily falls as we pass from zinc to copper, until at Fia. 357. — Circuit and diagram of potentials. the surface of the copper there is another electromotive force and a consequent sudden drop in potential bringing us again to the starting point at C. If in the diagram AE is laid off equal to BC, then EZ is the difference of potential which represents the total electromotive force of the cell, for it is the difference between the two oppositely directed electromotive forces AZ and BC, and the fall of potential from AtoBisi equal to that from E to C. It is therefore clear that the total fall of potential around the circuit, including that which takes place within the acid as well as that in the outside circuit, is equal to the total electromotive force of the cell. It will be noticed that CD, the difference of potential between the two poles, is equal to the fall of potential in the external circuit, and is therefore less than the total electromotive force of the cell when- ever there is any current flowing, for there is then a fall of potential within the cell itself from A to B. FALL OF POTENTIAL 427 640. Ohm's Law Applied to Part of a Circuit. — It has been seen (§604) that in any whole circuit where E is the electromotive force in the circuit and R is a. constant known as the resistance of the circuit. Similarly in any part of a circuit such as that between A and B (Fig. 358) the current / is pro- portional to the difference of potential be- tween those points and may be written / = - or /r = P (2) where P is the drop in potential between A and B and r is the resistance of that Fio. 358. part of the circuit. When there is no source of electromotive force, such as a battery cell, in a given portion of a circuit, the difference of potential between its ends in volts is equal to the product of the current in amperes by the resistance of that portion in ohms. When an electromotive force E is included in any part of the circuit considered, the difference of potential between the ends of that part may be written P = Ir — E {E.M.F. in same direction as current) (3) or P = 7r + E {E.M.F. acting against the current) (4) For Ir measures the drop in potential in the direction of the current, but when E is with the current it lifts the potential, as seen in the preceding paragraph. The sign of E is therefore opposite to that of Ir in that case. 641. Resistances in Series. — In a complete circuit made up of several conductors, having resistances rrir^, and including an electromotive force E, as in figure 358, the successive steps in potential may be written thus from A to B resistance = r P = Ir (5) from B to C resistance = ri Pi = Iri (6) from C to A resistance = fi P2 = Ir^ - -E (7) 428 ELECTRODYNAMICS The sum P + Pi + Pa is evidently the total change of potential around the circuit from A around to A again, but this must be zero for it ends at the same potential as it began. Therefore, adding 5, 6, and 7 we have Ir + Iri + Ir2-E = but this may be written 7= ^ r + n + Ti and by Ohm's Law 7 = ^ R hence R = r + ri-^ri; i.e., the resistance of several conductors connected in series is the sum of their separate resistances. 642. Combining Resistances in Ij^ — ^is^^ Parallel. — ^Let three conductors /'^j^^v;_iA^- having resistances ri, rj, rs be aL--^^ / joined in parallel in a battery /^jf^N. J _^S/ circuit as shown in figure 359. It ^T — ^g was shown by Faraday that the sum of the currents in the branches is equal to the total current 7 before it divides, 7 = 7i + 72 + 73 (1) But the drop in potential from A to B must be the same along either branch. Letting P represent this drop we have 7i _P n h < '' _p Therefore by (1) 7 = _P + ^ + ^ -(^- 7-2 r^/ But if 72 is the effective resistance of the three branches combined I- P ~ R Therefore 1 R~ 1 ^vA The reciprocal of the resistance of a conductor is called its RESISTANCE 429 conductance, hence the sum of the conductances of several con- ductors joined in parallel is the conductance of the combination. If the three resistances above considered are equal the com- bination will have one-third the resistance of one alone. 643. Galvanometer Shunt. — It frequently happens that a current is to be measured by a galvanometer adapted to smaller currents. In such a case a wire S of suitable resistance, called a shunt (Fig. 360), may be connected across from one galvanometer terminal to the other. The current then divides between the shunt and the galvanometer. If the resistance of the gal- vanometer is just 9 times that of the shunt used, the current will divide in the ratio 1 : 9, so that one-tenth of it flows through the galvanometer and nine- tenths through the shunt. 644. Resistance of Wires and Specific Resistance. — The resistance of a wire or of any conductor of uniform cross-section increases with its length and is inversely proportional to its cross section. The results of the last two articles show that this is so. For if the cross-section of a conductor is doubled it is equivalent to two of the original conductors side by side in parallel, and hence by §642 the resistance is one-half as much as before. The resistance of a cylindrical conductor of a given substance one centimeter long and one square centimeter in cross section is called the specific resistance of that substance, or its resistivity. When a wire of length I and cross section s is made of a sub- sT.ance having resistivity p, its resistance R is given by the formula R = ^^. s 645. Resistivities. — The curves given in figure 361 show the specific resistances of certain pure metals and alloys and also the variation of the resistances with temperature. If the curves for the pure metals are produced it will be found that they intersect the base line in the region of the absolute zero (—273°). The experiments of Onnes on the resistance of gold, silver, mercury, lead and tin at very low temperatures show that as the temperature is lowered they approach zero resistance at a point a few degrees above the absolute zero, the change being sud- 430 ELECTRODYNAMICS den at the last. The resistance of mercury, for example, de- creases slowly from 4.41° to 4.21° above the absolute zero, it then rapidly diminishes and practically disappears at 4.19°. The following table shows some specific resistances at 0°C. in mil- 5 o 30 r- ( aTs Tver c / / // v,^ / / / "^ V \? / A\ m / / / y 4s<> / / y / /. ^ / / ,/ / ^ / /. y x^ ■^ / ' 656. Incandescent Electric Lamps. — In the ordinary incandescent lamp used in electric lighting the current passes through a fine filament of carbon or tungsten enclosed in a glass bulb from which the air is thoroughly exhausted. The ends of the filament are joined to wires which must be of platinum where they are sealed into the glass, because that metal has very nearly the same coefficient of expansion as glass and therefore the joint does Fig. 369. — Electrical calorimeter. HEATING EFFECT 439 not crack with changes in temperature. The carbon filament possesses the advantage of being extremely infusible and an excellent radiator besides being cheap. Ordinary carbon lamps have an efficiency of about 3 watts per candle-power, and though by raising the temperature of the filament the candle-power per watt is greatly increased, the filament rapidly volatilizes, blackening the bulb, and finally breaks. The tungsten filament in vacuum is ordinarily operated at a temperature of 2400°C. and gives about 1 candle- power per watt, but if its temperature is raised to 2800°C. it gives more than 2 candle-power per watt in vacuum, though it volatilizes so rapidly that the life of the lamp is very short. Where the bulb is filled with nitrogen, how- ever, the evaporation of the filament is only about 0.01 of what it is in vac- uum, and so the lamp may be run profitably at the higher temperature. Such a nitrogen filled lamp may have a temperature of 2800°C. and an efficiency of 1.5 candle-power per watt. It will be observed that filling with nitrogen causes some loss of luminous efficiency due to the conduction of heat away from the filament. The filament is therefore made in the form of a rather closely wound helix, for the cooling effect of the gas is then much less than when the turns of the filament are widely separated. Incandescent lamps are usually connected in parallel between two conduct- ors as A and B (Fig. 370) which are maintained at a constant difference of potential by a dynamo or battery of low resistance. They .are called 50-volt lamps when they are brought to their proper luminosity by a difference of potential of 50 volts between their terminals. The candle-power depends on the power supplied to the lamp and there- fore both on the current and voltage. Good carbon lamps as ordinarily used hav6 an efficiency of about 3 watts per candle-power, so that a 16-candle- power lamp will consume 48 watts. Comparison of a 50-volt and 100-volt lamp of 16 candle-power, and equal efficiencies. Voltage Current Resistance Power 50 volts 1 amp6re 50 ohms 50 watts 100 volts J^ ampere 200 ohms 50 watts As the current in the 100-volt lamp is only one-half as great as in the 50- volt lamp, there is only one-fourth as much loss in heat in wires of a given resistance leading to the lamps. I® }@ }€) }03 ( a) (fp (H) Fig. 370. — Lamps in parallel. Fig. 371. — Lamps in series. 657. Incandescent Lamps in Series. — The mode of connecting incandescent lamps in parallel shown in figure 370 has the advantage that any one lamp can be turned on or off without particularly disturbing the others. But the current in the main wires is the sum of the currents in the lamps and, as the energy spent in the circuit is proportional to the square of 440 ELECTRODYNAMICS the current, the loss of energy in the main wires will be serious unless they are large and of low resistance. For street lighting, lamps are commonly connected in series so that however many lights there may be, the current in the conducting wires is no greater than for one lamp. There are two con- siderations which prevent this system from being used in house lighting. First, the potentials required are dangerous. If 30 lamps are connected in series and each requires 20 volts, the dynamo must have an electromotive force of 600 volts, and if a break or interruption of the circuit occurs at any point, the fuU difference of 600 volts will be experienced there. Second, if tlie filament of one lamp breaks it stops the current and all the lamps go out. This latter difficulty is overcome most ingeniously in street lighting by ar- ranging a little side circuit or by pass in each lamp which is complete except at one point where a slip of paper is interposed. When the lamp is acting the current passes wholly through the filament ; but if this breaks, the current is interrupted and immediately the whole electromotive force of the dynamo is brought to bear on the paper which is thereby punctured, permitting the current to pass through the side circuit. 658. Nernst Lamp. — Another form of incandescent lamp, devised by the German chemist Nernst, is also in use, in which the glower is a little rod made of a mixture of infusible earths, which though almost a non-con- ductor when cold becomes a suitable conductor when hot. A heating coil of platinum which is automatically cut out when the current begins to pass through the glower, is therefore used to start the lamp. 659. Electric Arc. — In the year 1801, Sir Humphrey Davy, who had constructed an immense battery of 1000 cells, observed that when the terminal wires were touched together for an instant and then drawn apart the discharge took place through the air like a stream of fire from one pole to the other, and at the same time the tips of the wires were intensely heated. The effect was most marked when carbon rods were used for the terminals. When the discharge took place horizontally it was bent upward like a bow (on account of the heated air rising) and so Davy called it the arc discharge. This tendency to curve out to one side is noticed in every long arc whatever its position. In arc lamps the carbons are only slightly separated, both tips are intensely heated, particles of carbon are carried across from the positive to the negative carbon causing a crater-like cup on the end of the positive carbon while the negative carbon is pointed; the positive carbon also is used up about twice as fast as the negative. The point of most intense luminosity is in the crater of the positive carbon where the temperature is found to be about 3500°C. The difference in luminous power of HEATING EFFECT 441 large and small carbon arcs seems to be due to the greater extent of luminous surface in one than in the other, the actual brightness of the glowing surface being the same in all. An electromotive force of 40 volts is required to maintain an arc between carbons. The temperature of the arc is the highest that has been produced by artificial means. Copper, iron, gold, silver, and platinum, if placed in it, are melted and volatilized. 660. street arc lamps have two regulating coils. The whole current flows through one, which by its magnetic action tends to draw the carbons apart if the current is too strong. A second coil is arranged through which a branch or shunt current flows, which acts to push the carbons together, and the further apart the carbons are the stronger this current will be. A sort of balance is thus maintained which keeps the lamp in adjustment. There is also for series lamps a device by which if the carbons become caught and do not make the arc at all the current can still flow through the lamp. Arc lamps are usually connected in series for street lighting because a cur- rent of only about 7 amperes is then needed ; but as 50 volts must be allowed for each lamp, to operate 50 lamps on the one circuit an electromotive force of 2500 volts is required; hence arc light circuits are dangerous. What are known as enclosed arcs, have surrounding the arc a small closely fitted glass globe, which soon becomes filled with carbon dioxide gas so that there is no ordinary combustion of the carbons. The carbons last much longer in these lamps, they require a higher electromotive force, about 70 volts, and give a softer light. 661. Electric Furnace. — In the production of aluminum and carborun- dum electric furnaces are employed. In these furnaces great carbons 2 in. in diameter are mounted horizontally and embedded in the materials that are to be heated, the whole is surrounded by walls of brick or fire clay, and the electric arc is established between the carbons. Under the com- bined influence of the enormous heat and the electrolytic action of the cur- rent the desired transformations are wrought. Problems 1. A current of 14 amplres divides between two branches, one of 2 ohms and one of 5 ohms resistance. Find the current in each branch, and the watts spent in each. In which resistance is the greater amount of heat developed per second? 2. The terminals of a gravity cell of 2 ohms resistance and 1 volt E.M.F. are connected with a coil of resistance 3 ohms. Find the watts spent in heat in the coil and also in the cell, also the total watts supplied by the cell. 3. What must be the resistance of a coU of wire in order that a current of 2 amperes flowing through the coil may give out 1200 gram-calories of heat per minute? 442 ELECTRODYNAMICS 4. If the difference in potential of the ends of a coil is 60 volts, what must be its resistance that 500 gram-calories of heat may be developed in it per second? 5. Find the gram-calories per second developed in each of two coils; one having resistance 3 ohms and current 6 amp6res, the other a resistance of 4 ohms and a difference of potential of 20 volts between its ends. 6. How many horse-power must be expended to maintain 200 100-volt lamps in operation, each lamp taking J^ ampfere of current and having a potential difference of 100 volts between its terminals? 7. How many horse-power are required to operate a series of 60 incandes- cent street lamps in series, the current in each lamp being 3 amperes and the resistance per lamp being 7 ohms? 8. In an electric railway having a total line resistance of 0.4 ohm per mile, what is the loss in horse-power in two miles of line when a current of 50 ampferes is being supplied to a distant car? 9. At 10 cents a kilowatt-hour what is the cost of heating 1000 liters or a cubic meter of water from 20°C. up to 90°C. by electricity? Therm oelectkicity 663. Seebeck's Discovery. — In 1821, Seebeck, of Berlin, dis- covered that in a circuit made of two different metals if one junction is hotter than the other there is an electromotive force which causes an electric current. This electromotive force is gen- erally very small compared with ordinary battery cells, and conse- 700° so-'a Hot Fig. 373. quently to obtain much current the circuit must have very low resistance. For example, in the copper-iron circuit shown in figure 372 when one junction is at 100° and the other at 0°, the electromotive force is about 0.001 of a volt and causes an electric current from copper to iron at the hot junction and from iron to copper at the cold one. The introduction of another metal does not make any difference provided the two junctions of the new metal are at the same temperature. THERMOELECTRICITY 443 Fig. 374. — Thermopile diagram. For example, the electromotive force is the same in the three circuits shown in figures 372 and 373. 663. Thermopile. — In order to obtain larger electromotive forces pairs of metals are combined in series to form thermopiles. The form devised by Nobili and used by Melloni in his researches on heat radiation consists of alternate strips of antimony and bismuth connected as shown in the figure, and carefully insulated from each other except at the junc- tions, where they are soldered to- gether. These metals were chosen because they give a large electro- motive force which acts from bis- muth to antimony at the hot junc- tions and from antimony to bis- muth at the cold. Rubens has improved the ther- mopile by using fine wires of iron and constantan (a nickel alloy) in place of antimony and bismuth. The mass to be heated in this case is very small so that it warms quickly when exposed to radiation. The thermopile is usually mounted in a metal case so that only one set of ends is exposed to the source of heat to be in- vestigated. If its terminals are connected to a sensitive galva- nometer of low resistance, it be- comes an exceedingly delicate means of measuring heat radiation. 664. Change of Thermoelectric Force with Temperature. — If one junction of a copper-iron circuit is kept at 0°C. while the other is steadily raised in temperature, the electromotive force is found to increase rapidly at first, then more gradually, reaching a maximum when the hot junction is at 260°C., after which the electromotive force falls off, becoming zero at 520°C. If the junction is heated still hotter the electromotive force reverses and the current flows from iron to copper at the hot / \/? ^ Ui \ 0' 100° 260° Fi 3 375 e. m Temperatu — Thermo . f . of copi res electric curve of Der and iron. 444 ELECTRODYNAMICS junction. If the observations are plotted with the temperatures of the hot junction as abscissas and electromotive forces as ordi- nates a curve such as shown in figure 375 is obtained. It is a parabola and is perfectly symmetrical about the vertical line through its vertex, which corresponds to the temperature of maximum electromotive force. This reversal of the thermoelectric current was discovered by Gumming in 1823. If the cold junction is kept at 100° instead of zero the curve will be exactly the same except that the origin of coordinates will be moved from to ^, and electromotive forces will now be measured from the base line AB. 665. Thermoelectric Powers. — It is clear from the foregoing that the inclination of the curve at any point, or the fate 6f change of electromotive force per degree change in temperature, de- pends only on the temperature of the junction which is being warmed or cooled and not at all on the , temperature of the other junction, Fig. 376 —Thermoelectric powers provided it is Constant. of iron aud copper. ^±\jv^\*\^\m. ai, ao ^v^ai-muv. This change of electromotive force -per degree change of temperature of a junction is known as the relative thermoelectric power of the substances involved. If the thermoelectric powers of iron and copper are plotted as ordinates along a scale of temperatures we shall obtain the diagram shown in figure 376. The curve is a straight line inter- secting the axes at 260°C.; for at that temperature the relative thermoelectric power is zero, as is seen also from the curve in figure 375, where the maximum point is at 260°, showing that a small change in temperature produces no change in electromotive force. This is called the neutral temperature for these two metals. 666. Thermoelectric Diagram. — In the thermoelectric dia- gram devised by Tait, the thermoelectric powers of the metals referred to lead are plotted as ordinates along a scale of tempera- tures; lead being taken as standard because in it the Thomson effect (§669) is zero. Such a diagram is shown in figure 377. It will be observed that within the limits of the diagram the THERMOELECTRICITY 445 variations with temperature of the thermoelectric powers of the metals are represented by straight lines. The electromotive force of a couple made of any two metals is expressed by the area included between the lines of the two metals and the ordinates of the temperatures of the junctions. The diagram is so constructed that the direction of the resul- tant electromotive force is clockwise; that is, in case of iron and 60° lUO" 150" 200" 2iO° 300" 351)= 100° 150° 500° 6S0° 600 Fig. 377. — Thermoelectric diagram. Thermoelectric powers are given in micro-volts per degree. copper between 0° and 100°, the current will be from copper to iron at the hot junction. 667. Peltier Effect. — It was discovered by Peltier in 1834, that if a current of electricity flows around a circuit made up of two metals heat will be given out at one junction and absorbed at the other. A beautiful demonstration of the Peltier effect was given by Tyndall by means of an ordinary thermopile. A thermopile is taken in which all parts are at the same temperature, so that it gives no current. On connecting it for a few seconds to a battery and then disconnecting it and joining it to a galvanometer a decided current is observed, showing that one set of junctions must have been more heated by the current than the other set. The current obtained is opposite to the first and tends to restore the equality of temperature disturbed by the first, for one stored 446 ELECTRODYNAMICS up heat energy in the thermopile and the other transforms that energy back again into energy of current. By a thermopile there is a direct transformation of heat energy into electrical energy, but it is not efficient because there is a serious loss of heat by conduction from the hot junctions to the cold. 668. The Conservation of Energy in Thermoelectricity. — The Peltier effect affords a beautiful illustration of the principle that energy is absorbed at those points in a circuit where there is an electromotive force acting with the current and is given out at those points where there is an electromotive force acting against the current (§653). In a circuit of two metals all at one temperature there may be electromotive forces at the two junctions, but since the temperature is the same at both, these electromotive forces are equal and opposite and consequently there is no current. If a current is now caused to flow by means of a battery, energy is given out at the junction where the electromotive force is Against the current and that junction is heated, while the other is cooled. The two junctions no longer balance each other, and it is clear that the resultant electromotive force which arises from the change in temperature must be against the current which brought it about. Otherwise in a simple closed circuit of two metals if one junction were heated a little to begin with, a current would be set up which would still further increase the difference in temperature of the junctions and would so become continually stronger and might be used to run a motor and do mechanical work until all the heat energy in the thermopile was used up and it was reduced ;■ to the absolute zero of tempera- ture. 669. Thomson Efifect. — In 1854, Lord Kelvin (Sir William Thomson) showed that in a thermoelectric circuit there must in general be electromotive forces not only at the junctions, but also in the homogenous conductors between the junctions, as they are not at the same temperature throughout. This effect was predicted by Lord Kelvin as a consequence of the law of energy and was then verified by the following experiment. MAGNETIC FIELD 447 A bar of iron was set up as shown in figure 378 so that the center was heated by boiling water while the ends were cooled with ice. When a current was established all parts of the bar were warmed, but a thermometer at A was observed to stand higher when the current was from left to right than when it was reversed, while the opposite was true at B. 670. Applications. — The thermopile as a delicate means of observing the intensity of heat radiation has already been described (§663). A particularly sensitive instrument for the same purpose was devised by Boys and is known as the radio-micrometer. In this instrument a simple circuit of bismuth and antimony is suspended between the poles of a powerful magnet by a fine quartz fiber. One of the two junctions is protected from outside radiation by the surrounding instrument, while the other hangs in an openmg so that radiation may be directed upon it. The slightest difference of temperature causes an electromotive force and since the resistance of so short a circuit is very small a comparatively large current is produced, which, reacting on the magnetic field, causes the suspended circuit to turn. A light mirror mounted on the suspended system turns with it so that the angular deflection may be read by a telescope and scale. For the measurement of high temperatures a thermal couple consisting of a wire of pure platinum joined to another of an alloy of platinum and rhodium may be used. In the Le Chatelier pyrometer such a couple, mounted in a protecting sheath of porcelain, is thrust into the furnace or oven of which the temperature is to be determined; wires from the couple lead to a suitable galvanometer graduated to read temperatures directly up to 1500°C. For the measurement of ordinary temperatures a thermal couple of iron and German silver is often convenient. Problems 1. Find the thermal electromotive force of an iron-copper circuit in which one junction is at 0° and the other at 200°C. 2. Find the increase in electromotive force in a lead-iron circuit when the temperature of the hot junction is changed from 150° to 151°. 3. What relation must the lines of two metals on the thermo-electric dia- gram bear to each other in order that the increase in electromotive force per degree rise in temperature of the hot junction may be a constant? 4. When one junction of zinc-iron circuit is at 50°C., at what different tem- perature may the other junction be without causing any current in the circuit? Magnetic Effects of Cukkents 671. Oersted's Experiment. — The first evidence of the mag- netic action of an electric current was obtained in 1819 by the Danish physicist Oersted, who discovered that when a wire 448 ELECTRODYNAMICS carrying a current is held in a north and south direction over or under a balanced magnetic needle the needle is deflected as shown in figure 379; and if the directive force of the earth's mag- netism is neutralized by means of a magnet, the needle sets itself at right angles to the current. 672. Magnetic Field Around a Straight Conductor. — The experiment of Oersted indicates that the magnetic force due to a current is in a plane at right angles to the current. To investigate, its direction more fully cause a strong current to flow Ln a wire which passes vertically through a card on which some fine iron fihngs are scattered ; on tapping the card the filings arrange themselves in circles about the wire as shown in figure 380. If the current is dovm as shown by the arrows, a small compass needle near the wire, at any point such as P, will point with its north pole in the direction of the arrow at Wire above needle Wire under needle FiQ. 379. — Oersted's experi- ment. Fig. 380. — Field around current. Fig. 381. — Right-handed screw. that point, tangent to the circle. If the current is reversed the compass needle will point in the opposite direction. The lines of magnetic force about a stiaight conductor carrying a current are circles of which the conductor is the axis. By a comparison of figures 380 and 381 it will be seen that the positive direction of the lines of force bears the same relation to the direction of the current as the direction of rotation of a right- handed corkscrew bears to the direction in which it advances. Another rule that may be given -is that if an observer looks MAGNETIC FIELD 449 along a conductor in the positive direction of the current, the positive direction of the lines of force as he sees them is clockwise. 673. strength of the Field. — The strength of the magnetic field near a straight conductor is greatest next to the conductor and diminishes as the distance increases. The strength of field H, at a distance r from the axis of a long straight wire carrying a current of strength 7, is given by the expression 27 H = — (all quantities in C. G. S. units) ^ = \0r (7 in amperes, H in C. G. S. units) since the ampere is one-tenth the C. G. S. unit of current (§602). This for- mula assumes that the return circuit is so far off that its magnetic effect at the point considered may be neglected. As the card is tapped on which the iron fihngs rest, in the experiment described in the last article, the filings work toward the center, the circles gradu- ally getting smaller, for the filings are drawn toward the stronger part of the field. If a fine copper wire carry- ing a rather strong current is dipped into some fine iron filings they will cling together in little circular filaments, forming a mass around the wire. 674. Field of a Circular Current. — When a conductor carrying a current is bent into a circle the lines of force are crowded together within the circle and spread out outside. In this case, shown in figure 382, all parts of the circuit con- spire to cause magnetic lines of force of which the direction is through the circuit perpendicular to its plane on the inside, and back again on the outside, as shown in the diagram. The lines of force very near the wire are nearly circles about the wire, while at the center they are nearly straight and perpendicular to the plane of the coil. Fig. 382. — Field of circular current. 450 ELECTRODYNAMICS If the wire carrying the current makes two turns around the circle instead of one, the magnetic force will everywhere be doubled, and so on for any number of turns. The strength of the magnetic field at the center of a circular current is proportional to the total length of the conductor wound in the circle and to the strength of the current and inversely proportional to the square of the distance of the conductor from the center. Thus if r is the radius of the coil and if n is the total number of turns, the length of the wire in the coil is 2 irrn, and when the current I is measured in C. G. S. units as defined in §602, the strength of the magnetic field H at the center in C. G. S. units is given by the formula: 2TrrnI 2iml rr ■ n t^ a ■+ ^ H = — ^ — = (i m C G. b. units) or, since the ampere is one-tenth the C. G. S. unit current, H = -jrT- {I measured in amperes). The formula assumes that the cross section of the coil is negli- gibly small compared with r. By measuring the magnetic force at the center of a coil of known dimensions, the number of amperes of current may be determined. 675. Rowland's Discovery. — Rowland discovered that a disc of ebonite, charged with electricity and rotating at high speed, acted upon a magnetic needle placed near it, just as a circular current would. The magnetic effect was found to be proportional to the speed of the disc. This remarkable experiment was car- ried out by him in 1875. 676. Solenoid. — A long helix, such as shown in figure 383, is known as a solenoid, and may be wound with one or several layers. When a current passes through such a coil all the turns act together to cause a field of magnetic force in which lines of force pass lengthwise through the interior looping back around the outside. Looking through the solenoid in the positive direction of the lines of force, the direction of the current is clockwise. Inside the solenoid a strong magnetic field of great uniformity is produced. MAGNETIC FIELD 451 The system of lines of force of a solenoid is thus like that of a bar magnet, the south pole corresponding to that end of the solenoid about which the current flows clockwise, as seen by an observer facing that end. If such a solenoid is mounted so that it can turn, it behaves like a suspended bar magnet when another solenoid or a bar magnet is presented to it. \ -|iir Fig. 383. — Solenoid. Fig. 384. — Ring solenoid. 677. Iron in a Solenoid. — If a soft-iron core is introduced into a solenoid the number of lines of force is greatly increased, it may be several hundred times, so that it acts as a strong mag- net. A hard-steel core does not have so great an effect in increas- ing the number oi lines of force, though it largely retains its magnetism after the current stops. The precise effect in such a case depends on the relative pro- portions of the solenoid and its core. In a short broad solenoid an iron core which fills it will not increase the number of lines of force so much as if the core and solenoid were longer in propor- tion, for the strong poles exert a magnetic force which in the interior of the solenoid is opposite to that of the coil, as explained in §501. 678. Ring Solenoid. — If a long solenoid is bent into a ring so that its two ends come together as shown in figure 384, a ring solenoid is obtained, and when a current flows in such a solenoid it produces a very nearly uniform field in the interior, though since the lines of force are not straight but circles, the force must really be slightly stronger toward the inside where the lines of force are smaller circles. There is no magnetic force outside of such a solenoid. 452 ELECTRODYNAMICS If the interior of the solenoid is filled with a ring of iron, all parts of the iron experience the same magnetizing force and there are no poles to complicate matters, so that the permeability (§504) of the iron can be immediately determined from the increase in the number of lines of force due to its presence. If, for example, the total number of lines of force in the iron is 1000 times what it would have been in the same space if the iron had not been there, the permeability of the iron is said to be 1000. It is easy to measure by electromagnetic induction (§710) the changes that take place in the number of lines of force through the ring and in this way the permeability of iron was studied by Rowland. 679. Magnetic Induction, Permeability and Magnetizing Force. — The strength of the field inside a ring solenoid when no iron is present may be called the magnetizing force. Let it be represented by H which will thus express the number of lines of force per square centimeter of cross section before the iron is introduced. The number of lines of force in the iron core per square centi- meter of section is called its magnetic induction or flux density and may be represented by B (see §500). We then have f^ — jr or B = jxH where ti represents the permeability of the iron. The relation between the induction B in iron and the magnet- izing force H as the latter is increased, starting at zero, is shown in the curve ab of figure 385, in which abscissas represent values of H and ordinates the corresponding values of B. From this curve it appears that at first B increases slowly, then rapidly, and finally at 6 as the iron approaches what is called saturation a considerable increase in H causes only a small increase in B. The curve shows that the permeability of iron is not a constant but increases with increase in magnetizing force up to a maxi- mum, after which it rapidly diminishes. Some values are given below. HYSTERESIS Permeability of a Sample of Soft Iron 453 H B M H B M 1 1,000 1,000 4 9,700 2,425 2 6,000 3,000 5 11,800 1,966 3 8,200 2,733 17 13,000 765 680. Hysteresis. — The changes which take place in iron when its magnetism is reversed were first thoroughly studied by Pro- fessor Ewing of Cambridge University. The curve of figure, 385 shows the changes in induction in soft iron when the magnetizing H3000. Fig. 385. — Hysteresis curve. force is changed from +13 to —13 and back again. The rise of induction when the iron is first magnetized is shown by the curve from a to b. The induction is here 13,000 while H is 13. On reducing the magnetizing force to zero the induction only falls to 10,800 following the curve be. By means of a gradually increasing reversed current the magnetizing force is made nega- tive until when H = —2.0 the induction is zero and there are no lines of force in the iron. As the force is made still more 454 ELECTRODYNAMICS negative the iron becomes oppositely magnetized, reaching the value B= -13,000 when H = -13. Then as the force is again reduced to zero the induction drops to — 10,800 following the lower curve, and does not become zero till H is +2.0. If the magnetizing force is carried up to the former maximum and then again diminished the curve rises to b and then falls back to c exactly as before. This lag of the induction in iron and steel behind the mag- netizing force was named by Ewing hysteresis. In consequence of it, when a mass of iron is put through such a complete cycle of changes, more energy is spent in magnetizing than is given back when it is demagnetized, the difference being a certain amount lost in heat. The amount lost in this way per cubic centimeter of iron is proportional to the area of the loop of the hysteresis curve, and with a maximum induction of 5000 it may amount to as much as 2500 ergs per cubic centimeter in each cycle of magnetic change. Every time the magnetism in a mass of iron is reversed it is put through such changes and since in the iron cores of trans- formers and dynamo armatures the reversals take place many times in a second it is important in such cases that soft iron should be used in which the hysteresis loss is small. The loss of energy due to this cause may amount to 2 per cent, in a trans- former made of fairly good iron. 681. Electromagnets. — Powerful magnets are made by sur- rounding soft iron cores with magnetizing coils, as was first shown by the French physicist Arago in 1820. A typical form of elec- tromagnet is shown in figure 386. On each of the two arms of a U-shaped piece of iron is fitted a cylindrical coil made of wire wound with silk or cotton insula- tion to separate one turn from another. The coils are so con- nected that the current flows in opposite directions around the two legs of the magnet, making one end a north pole and the other a south pole. When the soft-iron armature is placed across the two poles a closed circuit of iron is formed so that the magnet with its armature resembles somewhat the ring solenoid with iron core described in §678. If the armature is sufloiciently large most of the magnetic induction will be in the iron, the lines of force being closed curves. The whole number of lines of force ELECTROMAGNETS 455 established in the core of an electromagnet may be considered as due to the relation of two factors, the magnetizing power of the current in the magnet coils, called the magnetomotive force, and the resistance to magnetization offered by the iron core, called its reluctance. Number of lines of force established = Magnetomotive force ^ Reluctance of core In case of a uniformly wound ring solenoid the magnetomo- tive force may be shown to be Aiml where n is the number of turns of wire around the core and I is the magnetizing cur- rent, and we have N = Awnl ~R~ where N is the number of lines of force through the core and R is its reluctance, all the quanti- ties being in C. G. S. units. The above formula applies also approximately to ordinary electromagnets having nearly continuous iron cores. If the current I' is given in amperes while R and N are in C. G. S. units the formula becomes Fig. 386. — Electromagnet. N Airnl' lOR since the C. G. S. unit of current is equal to 10 amperes. The product nl' is called the ampere-turns and the strength of a magnet excited by a small current making many turns is the same as with a large current making few turns, provided the ampere-turns are the same in both cases. The shorter the iron circuit and the greater its cross-section the less mil be the reluctance and the more lines of force will he estab- lished by a given number of ampere turns. 456 ELECTRODYNAMICS The reluctance of an iron ring may be calculated from the formula ~ liA where I is the mean length of the ring, A is its cross section, and /i is the permeability of the iron. If a circuit is made up of parts that have different permeabilities their reluctances must be cal- culated separately and added together when the parts are in series. When the armature is not across the poles the reluctance is greatly increased because of the small permeability of the air through which the lines of force must pass. Therefore the num- ber of lines of force established in that case is very much less than with the armature across the poles. The force with which such a magnet holds its armature is proportional to the area of its poles, and is expressed by the ap- proximate formula Attraction in dynes = -q— where B represents the induction or number of lines of force per square centimeter of the surface between pole and armature, and A is the combined areas of the two poles. Problems 1. What is the strength of the magnetic field 15 cm. from a straight wire carrying a current of 6 amperes? 2. A wire 3 meters long is made into a circular coil with a mean radius of 6 cm. Find the strength of field produced at the center of the coil by a current of 0.1 ampfere in the wire. 3. How much current is flowing in one rail of an electric railway which runs in a north and south direction and causes a deflection of 45° in a compass needle held 30 cm. above the center of the rail; taking the strength of the horizontal component of the earth's magnetic field as 0.20? 4. Find the strength of the magnetic field at the center of a circular coil of 7 turns of wire 18 cm. in diameter when carrying a current of 3 amperes. 6. What will be the deflection of a magnetic needle at the center of the coil in the last problem if the coil is placed with its plane vertical and in the magnetic meridian at a point where the earth's horizontal force is 0.16? 6. A ring solenoid has a cross section of 9 sq. cm. There are 8 turns of wire per cm. of length of the solenoid and its whole length is 30 cm. measured along its axis. What magnetomotive force is produced by a PARALLEL CURRENTS 457 current of 1 ampere in the coil? How many lines of force will there be in the solenoid if it does not contain iron? And how many if filled with an iron core of permeability 100? 7. How many lines of force will be set up in a horseshoe magnet with iron armature, the iron circuit having an average cross section of 36 sq. cm., each leg being 15 cm. long and the two legs 12 cm. apart between centers? On each leg is a coil of 400 turns of wire carrying a current of 5 amperes. The permeability of the iron may be taken as 100. 8. Find the force in kilograms which an electromagnet can sustain when it is magnetized so that there are 600 lines of force per sq. cm. in the core, each pole piece having an area of 36 sq. cm. Interaction of Cuhkents and Magnets 683. Mutual Action of Parallel Currents. — Two parallel con- ductors carrying currents in the same direction attract each other, while if the currents are in opposite directions they repel. This may be shown by means of Ampere's frame, a light rectangu- lar frame of wire connected to a battery through two mercury cups so that it can freely revolve, as shown in figure 387. If a Fig. 387. second frame having a number of turns of wire through which a current passes is brought up so that one of its edges is parallel and near to one of the vertical wires of the pivoted frame, the attrac- tion or repulsion of parallel currents is easily demonstrated. Also if the frame B is held under the pivoted frame so that its 458 ELECTRODYNAMICS upper edge is at right angles to the lower wire of the movable frame the latter will then turn until the two are parallel and with the adjacent currents in the same direction. 683. Magnetic Field Around Parallel Currents. — If the lines of force of two parallel currents are studied by means of iron filings or a compass needle, they will be found as in figure 388 when the ciurrents are both in the same direction. While if the currents are in opposite directions the resultant lines of magnetic force are as shown in figure 389. According to Faraday's conception, the attraction in the first case may be explained by a tension in the magnetized medium or a tendency for it to shorten up in the direction of hues of force; on the other hand, the repulsion in the second case is also in Fig. 388. Fig. 389. Lines of magnetic force, currents perpen- Magnetic field of two currents perpen- dicular to paper and both down dicular to the paper; one down and one up. accordance with Faraday's idea that there is a pressure or tend- ency for a magnetized medium to expand at right angles to the lines of force. It is also to be noticed that in the first case the fsld of force is stronger just outside of the conductors than it is between them, for be- tween the two the magnetic effect of the one is opposed by the other; while in the second case the two act together to "produce a strong magnetic field between them and a weaker field outside. 684. Action Between Current and Magnetic Field. — In the case just considered each conductor may be thought of as acted on b}'^ the magnetic field due to the other. That there is such a reaction between a magnetic field and a conductor carrying a current may be demonstrated by presenting one pole of a bar mag- net to one of the vertical branches of Ampere's frame (§682) when the wire will move across the lines of force of the magnet. Or if a current of electricity is established in a light flexible con- ductor of tinsel cord hanging between the poles of a horseshoe CURRENT IN MAGNETIC FIELD 459 magnet the cord is repelled outward from between the poles when the current is downward and the poles are situated as shown in figure 390. If the current is reversed or if the magnet is turned over so that the poles are interchanged the cord is drawn inward. The field of force due to a current flowing across a uniform mag- netic field is shown in figure 391, where the current is supposed to flow downward in a wire which intersects the paper perpendicu- larly at O. The broken lines are the lines of force of the uniform field, the circles are those of the current, and the full lines are the resultant lines of magnetic force. Clearly a tension in the medium along lines of force and pressure at right angles will urge Fig. 390. — Current in magnetic field. Fig. 391. the conductor in the direction shown by the arrow. At points nearer the top of the diagram than 0, the force due to the currents acts with the original field, while below 0, the two are in oppo- sition, hence the field above is strengthened by the current while it is weakened below the conductor, there being a neutral point P where there is no magnetic force at all. These experiments lead to the following general rule: When the magnetic field immediately adjoining a conductor carrying a current is strengthened on one side and weakened on the other by the effect of the current, the conductor is urged toward that side where the field is weakened. If the current is not at right angles to the lines of force of the field, only that component of the magnetic force which is perpen- dicular to the conductor is effective, so that the effective magnetic force, the current, and the force acting on the conductor to move 460 ELECTRODYNAMICS it are in three directions mutually at right angles to each other, and their relation can always be determined by the rule just given. The amount of the force F experienced by the conductor is F = IHI where I is the length of the conductor in the field, H is the strength of the component of the magnetic field at right angles to the con- ductor, and I is the current strength, all being measured in C. G. S. units. 685. Magnet and Current In a Coil. — The mutual action of a magnet and a current in a coil may be studied by a little light circular coil of wire connected to zinc and copper terminals which dip into a test-tube containing dilute sulphuric acid, the whole system be- ing fioated in a tank by means of a cork.* On presenting the pole of a bar magnet the coD. will set itself so that the lines of force of the magnet are in the same direction through the coil as the lines of force of the coil itself, thus strengthening the field within the coil. It will then approach the pole and slip over it to the middle of the magnet. If the magnet is now pulled out and quickly thrust through the coil in the opposite direction, the coil will slip off from the magnet, revolve so as to present the opposite face, and then again approach it. It may be seen that these actions result from the general rule of §684. For in each portion of the circular circuit the magnetic field is strengthened on one side of the conductor and weakened on the other and each part strives to move from the stronger toward the weaker field. On the whole, therefore, the coil always turns and moves so as to increase the resultant number of Unes of force through it. It slips to the middle of the magnet and then sets itself obliquely as shown in figure 393, for in that position it • In this experiment the test-tube should be weighted with shot until the cork is entirely lubmeroed, only the upper part of the teat-tube projecting above the surface, otherwise the motions will be greatly impeded by the surface viscosity of the water. Fig. 392. — Magnet and floating current. CURRENTS AND MAGNETS 461 embraces the whole number of lines of force through the magnet and avoids also including those that turn back at the sides, which are in the opposite direction. 686. Barlow's Wheel. — In the apparatus shown in figure 394 a copper disc is balanced on an axle so that it can turn freely between the poles of a horseshoe magnet which produces a strong field perpendicular to the disc. The lower edge of the disc dips in a trough of mercury. If one pole of a battery is connected to the axle of the disc and one to the mercury trough, a current will flow through the disc between its center and the trough. This current, being perpendicular to the lines of force Fig. 393. Fig. 394. — Barlow's wheel. of the magnet, is urged to the right or left, depending on its direction, and accordingly the disc itself is set in continuous rotation. This experiment appears to show that the displacing force acts on the conductor which transmits the current and not simply on the current itself. 687. Rotation of a Magnet. — The following interesting experi- ment is due to Faraday. A strong straight steel magnet is mounted vertically between pivots. To one side of it is attached a short arm of copper which reaches out and dips into a circular trough of mercury surrounding the magnet, as shown in figure 395. If one pole of a battery is connected to the magnet and the other pole to the mercury trough, the magnet will rotate about its axis in a direction which depends on the direction of the current and on whether the north or south pole of the magnet is uppermost. 462 ELECTRODYNAMICS In this case the current in the projecting arm crosses the lines of force of the magnet at right angles and therefore tends to move across them, thus causing the rotation. The experiment is remarkable because the motion is caused by a reaction between the cross arm attached to the magnet and the magnetic field of the moving magnet itself. It shows that in a certain sense the magnetic field of the magnet does not rotate with it when it turns. Fig. 395. Fig. 396. — Tangent galvanometer. Instruments foe Measuring Current and Potential 688. Measurement of Current. — The strength of an electric current may be measured by its magnetic effect or by its heating or chemical action. Instruments which measure a current by its action on a magnetic needle are known as galvanometers. 689. Tangent Galvanometer. — In the tangent galvanometer there is a circular coil having one or more turns of wire, at the center of which a magnetic needle is either balanced on a point or suspended by a fine fiber of silk or quartz. The instrument is placed so that the plane of the coil is vertical and in the magnetic north and south plane. When a current is sent through the coil the needle turns to one side or the other, and the strength of the current is proportional to the tangent of the angle of deflection, as may be shown as follows: GALVANOMETERS 463 The force due to the current in the coil is at right angles to the plane of the coil at its center (§674) and the strength of the field at that point in a given coil is proportional to the strength of the current. Let G represent the strength of field at the center due to the coil when unit current is flowing, then IG will be the strength of field when the current strength is I. Let OA in figure 397 represent the plane of the coil and the point where the needle is placed, then when no current is flowing the needle points in the direction OA, being acted on only by the horizontal component H of the earth's magnetic force. The magnetic force Fig 397 F due to the current in the coil is IG and at right angles to H, therefore, the resultant force R is the diagonal of the rectangle whose sides are IG and H, and tan X = -jr where x is the angle which the resultant force makes with H. But the needle must point in the direction of the resultant force, and so x is the angle through which the needle turns. Therefore I =7r tan x and if H and G are known the current may be determined by measuring the angle x. 690. Coil Constant of a Tangent Galvanometer. — In case of a tangent galvanometer the magnetic force F due to the coil is expressed hy IG. But if the current is measured in electromagnetic units, ^ = ^! •■■ G = ^, (§674) And since the length of n turns of wire of radius r is 2irrn, 2irnr 2irn The galvanometer coil constant G can be calculated from this formula when the coil of the galvanometer has so large a radius compared with the length of the needle that the poles of the needle may be regarded as at the center, and when the cross section of the coil is so small that all the turns bear nearly the same relation to the needle. 464 ELECTRODYNAMICS If G is determined in this way, r being measured in centimeters, and if H is found by the method described in §498, the current will be found in TT C. G. S. electromagnetic units by the use of the formula I = -p, tan x. (j To obtain the current strength in amphres, we must take as the value of the coil constant 27rn ^ = ioTr By this method the strength of a current is determined in amperes directly from the fimdamental units of length, mass, and time, for we have already seen how the measurement of H is based on these units. A tangent galva- nometer in which the constant is determined in this way directly from meas- urements of the coil is known as a standard galvanometer. 691. Sensitive Astatic Galvanometer. — For the measurement or detection of extremely small currents of electricity the coil of wire must contain a great number of turns as close as possible to the needle, and because the turns nearest to the needle are most effective it is cus- tomary to use finer wire for these turns so that a greater number can be placed in a given space. The sensitiveness of the instrument is fur- ther increased by using an astatic needle. This is a system of two magnetic needles, as nearly as possible of the same strength, con- nected together bj'^ a light aluminum wire so that the poles of the two needles are oppositely directed, as shown in the figure. The combination is then suspended by a fine silk or quartz fiber so that the galvanom- eter coil surrounds only one needle, or the second needle may be surrounded by another coil around which the current flows in the opposite direction to the first so that any current in the coils will tend to turn both needles in the same direction. If the two needles have equal magnetic moments the earth will have no directive action on the combination. But as no system of needles will remain perfectly astatic, a directive magnet above or below the instrument serves to balance the effect of the earth on the combined system. The influence of this magnet Fig. 398. — Astatic gal vanometer diagram. GALVANOMETERS 465 and the torsion of the suspension fiber serves to give the needle a definite position of rest. A small mirror attached to the needle enables its angular deflection to be measured by the usual telescope and scale method or by the reflection of light upon a scale. 692. Moving-coil Galvanometer. — In this type of instrument, known also as the D'Arsonval form of galvanometer, the sus- pended system is a coil of fine wire which hangs in a strong magnetic field due to a permanent steel horse- shoe magnet. In figure 399 is shown a vertically placed horseshoe magnet, between the poles of which is hung a light rectangular coil of many turns of fine wire, the plane of the coil be- ing parallel to the direction of the lines of force. The coil is suspended by a fine ribbon of phosphor-bronze which also serves to connect one end of the suspended coil to the outer circuit while the other connection is made through a spiral wound strip of the phosphor-bronze ribbon attached to the lower end of the coil. A cylindrical mass of soft iron is fixed midway between the poles of the magnet so that as the suspended coil turns its vertical branches move in the gaps between the core and pole pieces. This arrangement secures a strong uni- form field across which the wires of the coil pass, and when a current is sent through it, it is deflected. A small mirror mounted just above the coil and moving with it, enables the deflection to be determined by the telescope and scale or reflected spot of light method. The moving-coil galvanometer has the advantage that it is not affected by changes in the earth 's magnetic field, and can be used near dynamo machines and where there is considerable magnetic disturbance. Also the coil damps strongly or comes almost immediately to rest when the wires leading to it are touched Fig. 399. — D'Arsonval gal- vanometer. 466 ELECTRODYNAMICS together, forming a short circuit, as it is called. This damping is due to electromagnetic induction (§720). 693. Electrodynamometers. — Instruments in which no iron or magnetic substance is used, but where the measurement depends on the mutual action of two coils carrying currents, are known as electrodynamometers. The Siemens electrodynamometer, shown in figure 400, is a Fig. 400. — Siemens electrodynamometer. good example. An oblong coil is fixed in a vertical position and surrounding it closely at right angles, but not touching it, is a rectangular suspended coil. The current passes through the fixed coil and is led into the suspended coil through two mercury cups into which its ends dip. The magnetic action of the coils upon each other causes the suspended coil to turn, but its top is attached to a helical spring the upper end of which is fastened to a knob which is turned till the torsion of the spring forces the suspended coil back again into its zero position. The strength of the current is determined from the amount of torsion required, as shown by a circular scale. GALVANOMETERS 467 In such an instrument the force of torsion T depends on the current strength in each coil or T is proportional to II', but when the current is the same in each coil T is proportional to P, whence I = k\/f where /c is a constant for the instrument which depends on the size, shape, and number of turns in the coils and the scale by which the torsion is measured. It is determined by experiment, by measuring the torsion produced by a known current. When the current is reversed in both coils the deflection is in the same direction as before; for this reason an electrodynamome- ter can be used to measure a rapidly alternating current which would give no deflection in a galvanometer. Fia. 401. — Ammeter. 694. Ammeters. — An ammeter is some form of galvanometer or electrodynamometer graduated so that the current strength in amperes may be directly read from the scale. A form of ammeter much used for direct currents is shown in figure 401. It consists of a sensitive moving-coil galvanometer in which the coil instead of being suspended is mounted in jeweled bearings and is held in equilibrium by two non-magnetic spiral springs which also serve as conductors for the current. The main current passes through a strip of metal (called a shunt) having very small resistance, only a minute portion of the current passing through the delicate movable coil. But the current in the mov- able coil is always the same proportional part of the whole 468 ELECTRODYNAMICS current, and therefore the scale over which the pointer moves may be so graduated as to show directly the number of amperes in the total current. Fig. 402. — Electrostatic voltmeter. An instrument of this type has the advantage of having a very small resistance. Fig. 403. — Voltmeter. 695. Voltmeters. — A voltmeter is an instrument designed to lueasure differences in potential, and gives the readings directly VOLTMETERS 469 in volts. There are two principal types, electrostatic voltmeters and those that depend on the flow of current. 696. Electrostatic Voltmeters. — These instruments are elec- trometers adapted to meet the requirements of ordinary engineer- ing practice. Of this type is the instrument shown in figure 402. 697. Current Voltmeters. — A voltmeter using current is a high-resistance galvanometer with a scale graduated to give directly the number of volts difference in potential between its terminals. The voltmeter shown in figure 403 is a moving-coil galva- nometer such as is used in the ammeter shown in figure 401, but there is no shunt across between the terminals as in the ammeter, and a considerable resistance is inserted in the circuit so that only a small current passes through the instrument. Voltmeters using current give correct values only in circum- stances where the current through the instrument is so small that it does not appreciably change the potentials to be measured. For instance, the difference of potential of two statically charged bodies could not be determined by such an instrument, for they would be instantly discharged through it. And if we attempt to measure the difference of potential of the terminals of a battery cell whose internal resistance is as great as that of the voltmeter itself, the deflection will indicate only one-half the total electromotive force of the cell, for the current is such that half the fall in potential takes place in the cell itself (§639). In ordinary commercial work the other resistances in the circuit are so small compared with that of a well-constructed voltmeter that there is no difficulty on this score. Such a voltmeter cannot be used for alternating currents. 698. Ammeter with Iron Core. — A simple form of ammeter is that shown in figure 404 in which a soft-iron core is drawn into a helical coil through which the current flows. Both ammeters and voltmeters are constructed on this principle, and as the soft- FlQ. 404. — Ammeter iron core. ■with 470 ELECTRODYNAMICS iron core is drawn inward when the current is in either direction they may be used either for direct or alternating currents, though the graduation must be different in the two cases. 699. Hot-wire Instruments. — In some instruments the cur- rent passes through a iine wire and the elongation resulting from its heating causes a pointer to move over a scale. The scale may be graduated to show either the current in amperes or the difference in potential between the terminals in volts. The wire is mounted in a metal case to screen it from air currents and keep it under as uniform conditions as possible. The heating effect of a current is irrespective of its direction, and therefore such an instrument may be used either for direct or alternating currents. 700. Wattmeter. — If it is desired to know the energy per second or watts spent in any part of a circuit, as in the lamps between A and B in the left diagram of figure 405, the current Fig. 405. may be measured by the ammeter and the difference of potential between A and B by the voltmeter. The watts expended are given by the product of the current in amperes by the volts. The result may, however, be obtained directly by using a wattmeter. This may be an instrument like the Siemens electro- dynamometer connected so that the main current flows through the fixed coil E (right diagram figure 405) while the suspended coil has a great many turns of fine wire and is connected at A' and B' to the main circuit, so that the current in the suspended coil will be proportional to the difference of potential in volts between A' and B'. The torsion produced by the mutual action WATTMETER 471 of the two coils is proportional to the product of the currents in each, and is, therefore, proportional to IP where I is the main current and P is the potential difference between A' and B'. The instrument may therefore be graduated to give directly the watts expended between A' and B'. The suspended coil in this case is known as the potential or pressure coil, while the fixed one is the current coil. Problems 1. What is the force of attraction between two straight parallel wires 30 cm. long and 1 cm. apart each carrying 3 amperes of current? 2. What must be the diameter of a coil of 3 turns of wire in order that a current of 5 ampferes may produce a strength of field at its center of 0.20 dynes per unit pole? 3. A sensitive galvanometer having a resistance of 25 ohms is deflected one scale division by a current of ?1 5 o o of an ampere. What resistance is required and how connected to change it into a voltmeter reading 1 volt per scale division, and what resistance and how connected to change it into an ammeter reading 1 ampere per scale division? 4. Given a voltmeter having a resistance of 800 ohms and reading 1 volt per scale division. How can it be made to read 10 volts per division? 6. How can the voltmeter described in problem 4 be used to find the current flowing through a conductor having a resistance 0.01 ohm per foot in length? Bells and Telegraph 701. Electric Bells. — Bells are rung by electricity by the method shown in the figure. When the key at fc is pressed, making a connected circuit, the current flows around the electro-magnet M, causing it to attract the soft-iron armature a to which is attached the hammer which strikes the bell. But as the armature a is drawn toward the magnet a metallic contact at h is separated, thus interrupting the cir- cuit and causing the magnet to lose its magnetism. The armature being mounted on a spring flies back, makes contact again at h, and is then again attracted by the magnet as at first. Fig. 406. — Electric bell. 472 ELECTRODYNAMICS 702. Electric Telegraph. — In the Morse telegraph, as origi- nally used, a recording instrument made a dot or dash when the key was pressed in the distant station. In this instrument a strip of paper was drawn steadily over a roller by clock work, and when the key was pressed an electromagnet drew up a lever provided with a sharp steel point which pressed against the paper making a dot or dash, depending on whether the key made an instantaneous or more prolonged contact. It was soon discovered that operators read the messages by sound, and therefore the elaborate recording instrument was replaced for the most part by the sounder, a simple electro- magnet and armature arranged so that a vigorous click is heard when the circuit is closed or broken. In consequence of the resistance of long lines the current is very small and is therefore used to operate a relay, which merely closes the connection in a local battery circuit in which the sounder is included. The relay has a magnet wound with a great many turns of wire and in front of its poles is a nicely balanced arma- ture controlled by a delicate spring so that a very small force will at- tract it. The armature is connected to the binding post a and the stop against which it is drawn is connected to b, so that when it is attracted by the influence of the main-line current, connection is made between a and b, thus closing the local circuit which includes the battery B and sounder S. The feeble motions of the relay armature are thus reproduced by the vigorous clicks of the sounder. Since the magnet of the relay must have a great many turns of wire, it must be wound with fine wire and will therefore have a large resistance; but since the resistance in the main line is al- ready large, the additional resistance of the relay will have a comparatively slight effect on the current. In the local circuit, the sounder and battery are all included in the same station and the resistance of the circuit may therefore be very small, hence the resistance of the sounder should be small, and accordingly it is wound with fewer turns of coarser wire. — Relay and sounder. TELEGRAPH 473 In the main-line circuit a single wire of galvanized iron or hard drawn cop- Iier is used, the return circuit being through the earth. The following dia- gram shows the arrangement of a main line including three stations. main line S I R>\ \j I rA [_ J- I Rs R' Fig. 408. — Diagram of telegraph line. Each station has a key, relay, sounder, and local battery indicated respec- tively, by k, R, S, L. The main-line battery Pi operates all the relays for a certain length of the line. At the last station shown in the diagram there is a relay R' which transmits the signals to a second section of the main line which is operated by the battery P2. The keys are all provided with switches by which the circuit is kept closed everywhere except in the station where the operator is sending a message. 703. Duplex Telegraphy. — By the duplex system of telegraphy the efficiency of a telegraph line is doubled as it enables messages to be trans- mitted simultaneously in both directions One arrangement is shown in the diagram, figure 409. When the operator at A presses the key, contact is Fig. 409. made with the battery and the current flows, but it divides between p and q in such a way that there is no flow across through R. This is accomplished, as in Wheatstone's bridge, by a suitable adjustment of the resistances p, g, and s. At the second station, however, part of the current will flow through the relay R', causing it to give the signal. The relay R acts only in response to the key B, just as that at R' is affected only when A is pressed, and the two may therefore be operated quite inde- pendently of each other. 704. Quadruples Telegraphy. — By the use of polarized relays or relays which act only when the current is in one direction, Edison was able to modify the old duplex method so that two messages could be simultaneously transmitted in each direction, or four altogether. 474 ELECTRODYNAMICS 705. Cable Telegraphy. — Ocean telegraphy presents some serious difficulties from which land lines are comparatively free. The cable acts like an enormous Leyden jar, for it consists of a central conducting core made of a bundle of copper wires twisted together surrounded by a thick coating of rubber insulating material, outside of which is a protecting sheath of hemp and steel wires. The copper core is the inner coating of the jar and the steel sheathing is the outer coating. The capacity of an Atlantic cable is about equal to 600,000 gallon Leyden jars. When one end of the cable is connected to the battery the current at the other end rises to its full strength only very slowly, as the cable is being charged at the same time. And when the current is broken the whole charge has to escape before the current dies out. In a typical Atlantic cable the current rises to Jf o of its maximum value in 0.2 second, and would require 2 seconds to come to %o of its maximum; therefore, in order to save time, exceedingly sensitive instruments must be used which will give an indi- cation as soon as the current begins to rise at the farther end. In giving a signal, connection is made to the battery for an instant and then the end is grounded, thus sending a sort of wave into the cable which is sufficient to affect the instrument at the other end without fully charging the cable. A double transmitting key is used by which the cable may be connected either to the positive or negative pole of a battery, and thus a series of waves may be transmitted, positive corresponding to dots, and negative to dashes of the telegraphic code. The receiving instrument is a sensitive galvanometer which swings to the right or left as the waves of current pass through it. On a line connecting points so far apart on the earth there is a tendency for earth currents to flow which would powerfully affect the delicate galva- nometers used and completely overpower the desired signals. To obviate this difficulty Varley devised the plan of connecting the cable at each end ]Earth Earth] Fig. 410. — Diagram of cable connections. to a condenser of large capacity which entirely prevents any steady flow through it due to earth potentials, but does not interfere with sending the signal waves. A simple arrangement of a cable is shown in the above diagram. The switches SS' are shown in position for sending by the key K and re- ceiving by the galvanometer G'. _ Pressing the upper key at K gives a positive charge to the condenser C, while the other key gives it a negative charge. One terminal of the galvanometer G' is connected to the condenser C while the other terminal is connected to earth. INDUCTION 475 706. Siphon Recorder. — The instrument now commonly used for receiving cable messages is the siphon recorder devised for the purpose by Lord Kelvin. It is a galvanometer of the type which later became known as the D'Arsonval form. A coil of wire hangs between the poles of a power- ful magnet, and through this coil the cable currents pass, causing it to turn. Attached to the suspended coil is a fine capillary tube of glass shaped like a siphon, one end of which dips into a little cup of ink. The other end of the siphon tube just touches a strip of paper which is carried along by clockwork. Aa the coil turns the siphon moves to and fro across the paper, tracing a wavy line as the paper moves along. An automatic jarring apparatus pre- vents the friction between the paper and point of the siphon from interfering with the free motion of the coil. Electromagnetic Induction 707. Faraday's Discovery. — The year 1831 was made mem- orable by the discovery of electromagnetic induction by Michael Faraday, then professor in the Royal Institution in London. In seeking to find some action of an electric current on a neighbor- ing conductor Faraday, having placed a coil of wire carrying an electric current upon another coil which was connected to a gal- vanometer, found that if the electric current was interrupted or broken there was a sudden deflection of the galvanometer lasting only for an instant, and when the battery connection was made again there was an equal deflection but in the opposite direction. But the steady flow of current in one coil had no effect whatever upon the other. These momentary currents are called induced or secondary currents, while the battery current by which they are produced is called the -primary current. The corresponding coils of wire are known as the primary and secondary coils. 708. Induction by a Moving or Varying Current. — Faraday also showed that when a coil carrying a current is moved either toward or away from another coil connected to a galvanometer, an induced current is set up. Such an arrangement as shown in figure 411 may be used, where the primary coil A has a current flowing through it from the battery and the secondary coil B is joined to the galvan- ometer. If the coil A is either pushed down inside of the coil B or withdrawn from it, an induced current is obtained which flows around B in the opposite direction to the current in A 476 ELECTRODYNAMICS when the two are pushed together, but in the same direction as in A when the coils are drawn apart. If while the coil A is inside coil B the current in A is made weaker, an induced current is set up the same as though A were Fig. 411. — Induction by a moving current. being withdrawn. But when the current in A is strengthened the effect is as though the coils were moved closer together. 709. Induction by Magnets. — Since a coil of wire carrying a current is surrounded by a magnetic field, it may be supposed that a magnet will produce a similar effect, and experiment shows this to be the case. When a bar magnet is thrust into a coil of wire connected in circuit with a galvanometer there is an instantaneous swing of the needle of the galvanometer, but the needle at once returns to its zero position and remains there so long as the magnet is held at rest; when it is withdrawn from the coil there is another instantaneous deflection opposite to the first. If the experiment is repeated with the magnet reversed, the deflections are opposite to those previously obtained. 710. General Condition of Induction. — In general an induced current is set up in a coil whenever there is a change in the number of lines of magnetic force passing through the coil. INDUCTION 477 This condition is illustrated in each of the three modes of pro- ducing induced currents just described. When the two coils of Faraday's first experiment are placed in the relation shown in figure 412 so that the lines of force due to the primary coil P instead of passing through the secondary coil pass on each side of it, there is no induced current in the secondary coil. So also there is no induction when a magnet is brought up to the coil in Fig. 412. Coils with no mutual induction. Fig. 413. the position shown in the upper diagram of figure 413 or when the plane of the coil is parallel to the magnet as shown in the coil C on the right of the magnet in the lower diagram, but when the coil is at right angles to the magnet as in the left-hand coil D there will be an induced current when the magnet is brought up or taken away, because more lines of force of the magnet pass downward through the coil when it is near the magnet than when it is at a distance. (See §499 on number of lines of force.) 711. Induction by Earth's Field. — The inductive effect of the earth's magnetism may be easily observed by means of a coil of large area and many turns of wire connected with a suitable galvanometer. If such a coil is held with its plane perpendicular to the lines of the earth's magnetic force as at .4, figure 414, the maximum number of lines of force will pass through it. If it is now turned 478 ELECTRODYNAMICS Fig. 414.— Coil in earth field. quickly into the position B parallel to the lines of force, where none pass through it, there is an induced current because of the change in the number of lines of force through the coil. If the coil, instead of being turned half-way, is turned completely over, its position relative to the lines of force is exactly reversed and the inductive effect is twice as great as when it was turned half- way over. When the coil in any position . \ \ \ '\ W^^^3^^ is rotated about an axis OX parallel to the lines of force of the field, there is no induction since no change takes place in the number of lines of force passing through it. When the coil is laid flat on a table and slipped about from one place to another there is no induction, even if the table is tipped so that its top is at right angles to the lines of force, because the same number of lines of force pass through the coil wherever it is, since the field is uniform. 712. Faraday's Disc. — The following experiment due to Fara- day shows that when a conductor moves across the lines of force of a magnetic field an induced electromotive force is developed. A copper disc is mounted on an axis so that it can rotate between the poles of a horseshoe magnet, the axis of the disc being parallel to the lines of force. The edge of the disc dips into a mercury trough con- nected to one end of a low-resistance galvanometer circuit, the other end of which is put in contact with the axle of the disc. On rotating the disc in the direction of the arrow a current is set up in the direction shown in the figure, the strength of which is proportional to the speed of revolution of the disc. If the disc is rotated in the opposite direction the current is reversed. Fig. 415. — Faraday's disc. INDUCTION 479 This experiment shows that each radial strip of the disc, as it cuts across tb.e lines of force of the magnetic fi^ld, is the seat of an electromotive force which is found to be proportional to the number of lines of force cut across per second, for it is pro- portional both to the speed of rotation of the disc and to the strength of the magnetic field. 713. Electromotive Force of Induction. — When the C. G. S. electromagnetic system of units is used (§602-603) the electro- motive force of induction is numerically equal to the number of lines of force, or unit tubes, cut across per second by the con- ductor; that is, N E = -y {E in C. G. S. units) t where E is the electromotive force induced in a conductor which is cutting across lines of force at the rate of N lines in t seconds. Or, since one volt (§603) is equal to 10^ C. G. S. units of poten- tial, £ = j^ {E in volts) To prove this relation suppose a circuit, such as is shown in figure 416, consisting of two straight parallel con- ducting rails connected together at one end and also connected by a cross con- ductor AB which can slide in the di- rection of the arrow; and let this circuit be in a magnetic field of strength ff in which the lines of force are perpendicular „ . ^ „ to the plane of the circuit. Then if AB is slid along by hand at the rate of x cm. per second, an induced electro- motive force will be produced which will cause a current / in the circuit. The energy expended per second by this current will be /£ (§652), but this energy is supplied by the work expended in moving the conductor along and must be equal to it. But a conductor of length I which carries a current J across a magnetic field of strength H, is acted on by a force F =HIl (§684) and if in one second the conductor is moved against that force through a dis- tance X the work done in one second is Fx = HIlx. We have then, IE = mix or, E=mx but Ix is the area moved over by the conductor AB in one second, and so Hlx equals the number of lines of force cut across per second. ThiLS the electromotive force o/ induction in C. G. S, units is shown to he numerically equal to the number of lines of force cut across per second by the moving conductor. c r- Magnetic Field ] of Strength H \ 1. 480 ELECTRODYNAMICS ""38!^', Fio. 417.— Wire moving across lines of force. 714. Illustration. — For example, suppose a straight conductor AB, one meter long (Fig. 417), is moved in the direction of the large arrow at the rate of 3 meters per sec, and sappose it is in a magnetic field of strength 0.5 (about as strong as the earth's field) in which the lines of force are dovm perpendicular to the paper. Then the number of lines of force cut per second will be the area in centimeters swept across per second by the con- ductor, multiplied by the number of lines of force per square centimeter which in this case is 0.5, or E = 100 X 300 X 0.5 = 15000, which is the electromotive force in C. G. S. units; to change it to volts it must be divided by 10', hence E = 0.00015 volt which is the difference of potential between the ends of the wire, since it is disconnected and no current can flow. 715. Why Induction Depends on Cliange in Number of Lines of Force tlu-ough a Circuit. — We are now prepared to under- stand why it is that the resultant electro- motive force induced in a circuit depends on the change in the number of magnetic lines of force passing through the circuit. It has already been seen that when a coil of wire lying on a table is slid along, no induced current is produced although the wires of the coil cut across the lines of force of the earth's mag- netic field (§711). The explana- tion of this is that electromotive forces are induced, but in such a way that they balance each other. For suppose the coil is moved from A to B as in figure 418 and that the lines of mag- netic force are straight down per- pendicular to the diagram, then the sides of the coil cut across lines of force in such a way as to cause electromotive forces in the direction of the arrows. The electromotive forces induced in the two sides therefore act against each other in the ring, but they are equal because each side of the ring cuts across the same number of lines of force in the same time, therefore the electromotive forces balance and there is no current. FiQ. 418.- -Coil moved sidewise in magnetic field. INDUCTION 481 7/ the field is not uniform so that more lines of force pass through the coil in the second position B than in the first position, then more lines of force must have cut into the coil across its left-hand side than have cut out of it across its right-hand side. The electro- motive force developed in the left-hand side of the coil will then be greater than the other and will cause a current to flow around the coil counter-clockwise. Therefore there must be a resultant electromotive force whenever the number of lines of force through a coil is increased or diminished. 716. Induced Electromotive Force. — Since the electromotive force developed in any part of a conductor by induction is equal to the number of lines of force which cut across it per second (§713), it follows that in any circuit or coil the electromotive force of induction is equal to the change per second in the number of lines of force included by the circuit. This is expressed by the formula TV, -Ni E = - t which gives the average electromotive force during the time interval t when Ni is the number of lines of force through the circuit at the beginning of the interval and Nz the number at the end. By taking the time interval very short we approach the in- stantaneous value of the electromotive force as a limit. If there are several turns of wire in the coil, to get the total electromotive force the above expression must be multiplied by the number of turns. It is clear from the above that the more quickly the change in the number of Hnes of force takes place the greater the electro- motive force. 717. Induced Current and Total Flow.— The induced cur- rent at any instant is by Ohm's Law / = "5 and smce h = we have ^ Rt The instantaneous value of the induced current is therefore 482 ELECTRODYNAMICS greatest when the induced electromotive force is greatest; that is, when the change in the number of lines of force through the circuit is taking place most rapidly. But It, the product of current by the time that it flows, is the whole quantity of charge or electricity that passes in time t; thus It or Q = D A simple integration shows that this expression holds true in every case, at whatever rate the lines of force through the circuit may be changing. The total quantity of electricity passing a given point in the circuit in consequence of induction is equal to the change in the number of lines of force through the circuit divided by its resistance. If C. G. S. units are used for N and R the quantity Q will also be in that system. To find it in coulombs it must then be multiplied by 10. It is to be remarked that the total quantity of the induced flow is independent of the time during which the induction takes place. It is the same when a magnet is put into a coil as when it is pulled out and whether it is moved slowly or rapidly. 718. Energy in Induction. — Every current of electricity pos- sesses energy, and therefore energy is required to produce in- duced currents. During the changes which produce an induced current energy is supplied to it, and it dies out immediately when the inductive action stops because its energy is expended in heat in the conductor if in no other way. When induced currents are set up by making or breaking the current in an adjoining primary circuit the energy comes from the primary battery. When the induced current is caused by the motion of a conductor in a magnetic field the energy is supplied by the agency which causes the motion. For instance, more energy must be expended when a magnet is thrust into a coil in which the ends of the wire are connected forming a closed circuit than if the ends had not been joined, for there is an induced current in the first case and not in the other. But in order to expend energy resistance must be over- come, and so the induced current must cause a force which resists the magnet as it is pushed into the coil. For the same reason the current which is induced when the magnet is with- INDUCTION 483 drawn must exert a force to resist the withdrawal of the magnet so that more work is done than if the current could not flow. 719. Lenz' Law. — The general law suggested in the last para- graph was first stated by Lenz and is known by his name. It may be stated thus: An induced current is always in such a direction as to resist by its electromagnetic action the motion by which it is produced. This law is a direct consequence of the conservation of energy, as has been already indicated. 730. Illustrations of Lenz' Law. — Thus in case of Faraday's disc experiment (§712) the induced current tends to rotate the disc in the opposite direction (see Barlow's wheel, §686) so that it is harder to turn the disc while the induced current is flowing than if the circuit were disconnected. If a thick strip of sheet copper is hung like a pendulum so that it can swing edgewise between the poles of a powerful electro- magnet, it may swing down with a rush, but is instantly checked as it comes between the mag- net poles, since there are induced in the copper currents of elec- tricity which resist the motion, transforming the energy of mo- tion into current energy which finally results in heat in the cop- per. In some forms of galva- nometer a bell magnet is em- ployed, so called because it is shaped like a cylindrical bell of steel slit part way up, the poles being on the two sides. If such a magnet is suspended in a slightly larger cylindrical cavity in a copper block, it generates by its motion induced currents which quickly bring it to rest. This mode of stopping the vibrations of a magnetic needle is called electrical damping. The damping of the coil of a D'Ar- sonval galvanometer (§692) is also explained in the same way. Fig. 419.- -Copper pendulum and magnet. 484 ELECTRODYNAMICS 731. Arago's Disc. — A celebrated experiment of Arago's, which was first explained by Faraday, is illustrated in figure 420. A copper disc is rotated rapidly under a magnetic needle from which it is separated by a sheet of glass or parchment which prevents air currents from having any influence on the needle, and the needle is carried around with the disc. Induced currents are set up in the disc which resist the /-'''^^-— ^^\ relative motion of the two, conse- "^(^ j' \, \ quently the needle is dragged along after the disc. The lines of force due to the needle go down through the disc under the north pole and the induced currents are as indicated by the dotted curves. It is easily seen that the current flowing under the needle will tend to cause it to turn Fig. 420.-Cu|Tents in Arago's j^ ^he direction of the disc, as in Oersted's experiment (§671). 732. Rules for the Direction of Induced E.M.F. and Current. — Lenz' Law leads to the following rules for the direction of the induced electromotive force and resulting current: Case of a Wire Moving Across Lines of Force In this case the electromotive force induced in the wire is in such a direction as to cause a current which will strengthen the field immediately in front of the moving wire and weaken the field immediately behind it. For it has been seen in §684 that such a current would urge the wire across the field in the opposite direction, thus resisting the motion. Case of a Closed Circuit When the number of lines of force through a circuit is increasing, the induced current is in such a direction as to set up lines of force through the circuit opposite to those already there, thus opposing the increase. If the number of lines of force is decreasing, the induced current is in such a direction as to set up lines of force inside the coil in the same direction as those already there, thus opposing the decrease. INDUCTION 485 783. Self-induction. — When a current of electricity is set up in a coil of wire each turn in the coil experiences the inductive effect of the current starting in all the other turns. All act to- gether to cause an induced current in the coil opposite to the cur- rent which is starting. The resultant current is therefore weaker than the steady current which will flow when the inductive action is over. When the circuit is broken thu self-induced current is in the same direction as the current which has been flowing; it acts therefore with that current and prolongs its flow, causing a bright spark across the gap where the circuit is broken. The current induced on break- ing connection is known as the extra current. In this case, as in all other cases of induction, the action is due to that relative motion of conductors and magnetic field expressed by the phrase "cutting lines of force." The coil after the current is established has a magnetic field, and includes a large number of lines of force. These lines of force form closed curves surrounding the coil and may be considered as starting in the coil and spreading out in expanding curves as the current becomes stronger. Each turn of wire in the coil is cut by all the lines of force and hence the electromotive force of self- induction depends on the number of turns of wire in the coil and the total number of lines of force that are set up by its current. What is called the coefficient of self-induction of a coil is the prod- uct of the number of its turns of wire by the number of lines of force through the coil when unit current is flowing in it. Thus even a circuit consisting of a single turn of wire has some self- induction, but it is greatest in coils which have many turns of wire and include a great number of lines of force, as in electro- magnets, where the iron core immensely increases the self- induction. 724. Experimental Illustration. — Take a large electromag- Fio. 421. 486 ELECTRODYNAMICS Fig. 422. — Self-induction. net of low resistance having an armature across its poles and connect a small incandescent lamp across its terminals, as shown in figure 422. Then join the magnet to a storage "battery which is strong enough to light up the lamp when connected to it alone, interposing a contact key. On pressing the key the lamp lights for an instant as the electromotive force of self-induction opposes the flow of current through the magnet and sends it through the lamp instead. The lamp dies out, however, as the current comes to its steady state and divides between lamp and magnet. On breaking the circuit the lamp again glows as the self-induced current rushes around through the lamp instead of leaping the gap at the key. The •phenomena of self-induction are observed only while the current is changing, hence in case of steady currents self-induction need not be considered, but in dynamo machines and all alternating current apparatus it plays a most important part. In breaking connection in a circuit containing much self-induction, such as one including' electromagnets or a dynamo machine, great care must be taken not to be touching the conductors on both sides of the gap when the contact is broken; otherwise a severe shock may be obtained from the extra current even when the ordinary voltage in the circuit is small. 725. Energy of a Magnet. — Every portion of the magnetic field whether within the iron core of the magnet or outside of it has a certain energy in consequence of its magnetization. It was shown by Maxwell that the energy per cubic centimeter in any part of a magnetic field is 3—) where B is the induction at that point or number of lines of force per square centimeter. Therefore, when a current is starting, in a coil or electro- magnet it has to supply the energy of the magnetic field besides spending energy in heat owing to the resistance of the conductor. After the magnetic field is fully established, which may take several seconds in a large magnet, the current is steady and INDUCTION 487 spends energy only in heat in the conductor. No energy is re- quired to keep up a magnetic field when it is once established. The spending of energy by a current in making a magnetic field causes the current to delay in coming to its full strength and is the cause of the self-induced current on making connection. When the circuit is broken the field loses its magnetization and therefore gives up its energy again to the current. This causes the extra current or induced current on breaking the con- nection, and the energy of this extra current is equal to the energy that was stored up in the magnet and surrounding magnetic field. ^ 726. Induction Coll. Ruhmkorff Coil. — The induction coil is a device for obtaining induced currents of very great electro- motive force from an ordinary battery current. The construc- tion is illustrated in figure 423. The primary coil, of a few layers of large copper wire so as to have small resistance, is wound about a central core which consists of a bundle of soft-iron wires. Outside of the primary coil and thoroughly insulated from it by a thick tube of hard-rubber is the secondary coil, made of an im- mense number of turns of fine wire the ends of which are brought to two insulated posts supporting the discharging rods a b. The diagram, for distinctness, shows only a few turns of wire in the secondary; but in the actual instrument there are thou- sands of turns, a coil to give a one inch spark must have some- thing like a mile of wire in its secondary coil. The primary must be thoroughly insulated from the secondary by a thick tube of hard-rubber with hard-rubber flanges at the ends. The primary coil is connected to a battery of a few storage cells and when the current is interrupted the induced electromotive force in the secondary coil may be great enough to cause a dis- charge across between the discharging rods. The primary current is automatically connected and broken. A device com- monly used is shown at d. A little block of iron on the end of a spring is mounted opposite the end of the iron core of the appa- ratus. The spring rests against the end of an adjusting screw, Fig. 423. — Induction coil. 488 ELECTRODYNAMICS the points of contact on each of them being made of platinum. The connections are made so that the primary current flows across the contact between spring and screw, and consequently as the core becomes magnetized and attracts the block of iron mounted on the spring, the connection is broken. But as the core then loses its magnetism the spring comes back and again makes the connection; and so the action is repeated, automatic- ally making and breaking the current many times in a second. The self-induction of the primary coil causes both the starting and stopping of the current to be prolonged, and consequently the E.M.F. of induction . would be comparatively small if this were not obviated. It is found that if a condenser of suitable capacity is connected to the primary circuit, its two surfaces being connected one on each side of the point where the current is broken, the electromotive force produced on breaking is greatly increased. Such a condenser is represented at C; it is usually made of alternate sheets of tinfoil and paraffined paper, the odd sheets of tinfoil being connected together for one coating and the even sheets forming the other. By this construction a large capacity is obtained in very compact form. The condenser is often contained in the base of the instrument. When the current is broken at d the extra current of self- induction rushes into the condenser and charges it instead of discharging in a spark across the gap at d. The flow of the extra current is thus very quickly stopped; but after the con- denser is charged it immediately discharges itself back through the coil in a direction opposite to the original current, and so more perfectly demagnetizes the core or even magnetizes it oppositely. By the use of the condenser, then, there is a greater change in the number of lines of force on breaking the current, and the change is more instantaneous, both effects serving to increase the electromotive force of induction; and at the same time the sparking at the gap d, which is very destructive to the platinum contacts, is greatly reduced. 737. Wehnelt Interrupter. — Instead of a mechanical inter- rupter for the circuit an electrolytic cell may be used, known as the Wehnelt interrupter from its discoverer. This cell consists of a vessel containing dilute sulphuric acid, having for the nega- TELEPHONE 489 tive electrode a plate of sheet lead and for the positive electrode a platinum wire or rod covered with a glass or porcelain sheathing so that only the tip end projects into the acid. An adjusting screw is provided by which the amount projecting may be regulated. If the voltage in the circuit is sufficient and the exposed tip of platinum wire is properly proportioned to the cur- rent, the circuit will be rapidly interrupted, bubbles of gas being given off at the plati- num wire accompanied by flashes of light. The frequency of interruption depends on the self-induction of the circuit and the electromotive force of the battery as well as upon the adjustment of the platinum point, and may be varied through wide limits. With this form of interrupter a condenser is of no advantage. 738. Telephone. — In the early telephone as devised by Bell the receiver and trans- Fig. 424.— Electrolytic . , . interrupter. mitter were alike, the construction being shown in figure 425. A hard-rubber handle contains a hard- steel cylindrical magnet, around one end of which is fixed a coil of many turns of fine wire the ends of which are brought to binding screws on the handle. A disc of thin sheet iron, sup- ported at the edges so that it is free to vibrate in the middle, is mounted so that its center comes close to the end of the magnet and surrounding coil but does not touch them. A hard-rubber cap or ear-piece having a hole in the center fits over the disc and serves to clamp it firmly at the edges as well as to improve the quality of the tone by favoring the sound waves from the center of the disc. Suppose two such instruments with the coil in one connected in closed circuit with the coil in the other. If a person speaks Fig. 425. — Telephone receiver. 490 ELECTRODYNAMICS into one the sound waves impinging on the center of the disc cause it to vibrate; but as it vibrates induced currents are set up, for when the disc approaches the magnet more hnes of force pass through the coil into the disc, and as it springs away the lines of force spread out again cutting across the coil. These induced currents flow through the coil around the magnet of the receiving telephone and by alternately opposing and strength- ening its magnetism cause the iron diaphragm of the receiver to vibrate in exact correspondence with that of the transmitter, so that the same motion is given to the air at one end as that which caused the disc to vibrate at the other, thus reproducing the sound. Fig. 426. — Transmitter. There is a serious defect in this mode of transmission. All the energy of the induced currents must come from the sound waves which cause the disc of the transmitter to vibrate, and as a part of this energy is spent in heat in consequence of the re- sistance of the circuit, the sound heard at the receiver must be faint. To meet this difficulty another form of transmitter shown in figure 426 is ordinarily used. The cell C containing carbon granules between two plates of polished carbon is mounted between the thin metal diaphragm and the solid back of the instrument, and on each side of the cell is a metal plate con- nected in circuit with a battery B and the primary winding P of a small induction coil, of which the secondary iS is connected to the line leading to the receiving station. When sound waves fall upon the diaphragm of the trans- mitter the vibrations cause a variation in its pressure on the carbon cell and a consequent change in its resistance. The other resistances in the battery circuit are small compared with TELEPHONE 491 that of the granular carbon, hence variations in its resistance cause decided changes in the strength of the current. These set up induced currents in the secondary and produce corresponding vibrations in the diaphragm of the receiver, thus reproducing the sound. By this arrangement the energy for transmission is supplied by the battery, and by taking a proper number of turns in the secondary coil the induced current can be adapted to the resistance of the line. wy7 A telephone line is usually a complete circuit of two wires in- stead of using the earth, as in telegraphy, and the two wires are carried near together so that the -p^^ 427 inductive action of neighboring telegraph lines and lighting wires on one wire may be neutralized by their action on the other. The arrangement adopted in the local battery system is shown in figure 427. When the receiver R is hung on the hook H the battery circuit is broken at D and also the secondary circuit so that no current flows from the Subscriber Line Plug O- Connection Fig. 428. — Telephone with central battery. battery except when the line is in use. The call bell is of very high resistance so that only a very small part of the current is diverted through it, and the magneto M by which the bell is rung is so devised that it is connected to the line only while being used. The local battery is often done away with and the current through the subscriber's transmitter supplied by a single battery at the central station. One method of connection is shown in figure 428. A battery B of about 24 volts is connected to the line at the central station; but when the receiver R hangs on the hook h there is no current in the line, for the circuit is broken at q and no current can flow across through the call bell M because the condenser e is interposed. The subscriber may be called. 492 ELECTRODYNAMICS however, by connecting to the terminals ab any source of altemaiing current, which causes a surging of current back and forth in the line to the condenser, charging it so that first one of its coatings is positive and then the other, alter- nately. This current as it flows alternately into the condenser and out again rings the call bell M. On the other hand, if the subscriber wishes to call "central " he has only to lift his receiver from the hook. The current is then established through the contact points at q and flowing around the relay / closes at g the circuit through the signal lamp I which flashes out and shows that a connection is desired. The correspondent's line is then connected by means of a flexible cord having two conductors and terminating in a double contact plug which connects one conductor to a and the other to &. The receiver R is so connected that only an extremely minute direct current from the battery can flow through it on account of the large resist- ance of the bell M (1000 ohms), but the alternating "talking" current induced in the secondary s is readily established through the condenser e. The " talk- ing" current and the direct current from the battery through the transmitter are thus both transmitted over the same line without interfering with each other. Electromagnetic Units 729. C. G. S. Electromagnetic Units. — The C. G. S. electro- magnetic system is based on the unit magnet pole as defined in §485, unit current as in §602, and unit electromotive force as in §603. These units are determined from the above definitions by certain measurements of length in centimeters, of mass in grams, and of time in seconds. They have the advantage of being directly connected with the fundamental mechanical units of the C. G. S. system. Thus the product of current by electro- motive force measured in these units gives the rate of spending energy in ergs per second. The C. G. S. unit of resistance is the resistance of a circuit in which the above unit electromotive force will produce unit current of the same system. 730. Practical System of Units. — The C. G. S. units are not of a convenient size for use in commercial measurements, but it is desirable that the practical units should be related to the C. G. S. units by ratios which can be expressed by simple powers of 10. Thus the volt, the ^radical unit of electromotive force, is chosen equal to 10* C. G. S. units of electromotive force, because that particular power of 10 gives a value nearer to the electromotive forces of ordinary battery cells than any other would have done. The ohm is defined as 10' times the C. G. S. unit of resistance. ELECTROMAGNETIC UNITS 493 This power of 10 was adopted because it corresponds very nearly to the Siemens unit* of resistance which was already in use and had been found convenient. The ampere is IQ-^ times the C. G. S. unit of current. It is determined by Ohm's law as the current which results from an electromotive force of 1 volt in a circuit having a resistance of 1 ohm. The coulomb is the unit of charge. It is the charge transmitted in 1 second by a current of 1 ampere. It is almost exactly equal to three thousand million electrostatic units of charge as defined in §525. The farad is the unit of capacity. It is the capacity of a con- denser which will hold a charge of one coulomb when the differ-' ence of potential between its coatings is 1 volt. This unit is so large that ordinary condensers are rated in microfarads, or millionths of a farad. Th£ henry is the unit of inductance.^ It is the inductance of a circuit in which an increase in current strength at the rate of 1 ampere per second produces a back electromotive force of 1 volt. Elaborate experiments have been made to determine how the units as above defined may be realized in practice, and the fol- lowing experimental values have been obtained : The ohm is the resistance of a column of pure mercury 106.3 cm. long and 1 sq. mm. in cross section at the temperature of melting ice. The ampere is a current which will deposit 0.001118 grm. of silver per second in a silver voltameter. The volt may be determined from a standard Clark cell, the electromotive force of which at 15°C. is found to be 1.4322 volts. 731. To Change from the Electrostatic to the Electromagnetic System. — The ratio of any electrostatic unit to the corresponding electromagnetic unit is in every case some power of the velocity of light (3 X 10") cm. per second. Electrostatic quantity of charge H- (3 X 10") = charge in C. G. S. electromagnetic units. * The Siemens unit is the resistance of a column of pure mercury 1 meter long and 1 sq. mm. in cross section, at the temperature of melting ice. Named from Sir William Siemens, the distinguished German physicist and engineer who advocated it. t Named in honor of Joseph Henry, a distinguished American physicist and 6rst Secre- tary to the Smithsonian Institution, who discovered self-induced currents. 494 ELECTRODYNAMICS Electrostatic quantity of charge h- (3 X 10') = coulombs. Electrostatic potential X (3 X 10'°) = potential in C. G. S. electromagnetic units. Electrostatic potential X 300 = volts. Electrostatic capacity -;- (3 X 10")^ = capacity in C. G. S. electromagnetic units. Electrostatic capacity ^ (9 X W) = farads. Electrostatic capacity -h (9 X 10^) = microfarads. Problems 1. A coil of wire of 10 turns, each turn enclosing an area of 900 sq. cm., is turned from position A to B (see Fig. 414) in }i second. Find the induced E.M.F. in volts when the strength of the magnetic field is 0.5. 2. A metal spoke in a wheel is 80 cm. long. If the wheel makes 300 revolu- tions per minute in a plane perpendicular to the lines of force of the earth where the field strength is 0.5, find the difference of potential be- tween the center and rim of the wheel. Which part is at the higher potential when the wheel rotates clockwise as seen by one looking in the positive direction of the lines of force? 3. Show that the work expended in producing an induced current by turn- a coil over in a magnetic field becomes 4 times as great when the time of the operation is reduced J^. 4. A railway train runs south on a straight track with a velocity of 25 meters per sec. If the vertical component of the earth's magnetic force is 0.50, find the electromotive force induced in a car axle 120 cm. long; also which end, east or west, is at the higher potential. 6. When the vertical component of the earth's magnetic force is 0.50, find the electromotive force induced in a coil of 10 turns of wire 3 meters in diameter which while lying on the ground is in ^^i second pulled out into a loop so long that the sides touch. 6. When a circular coil of 100 turns of wire 1 meter in diameter lying on the floor is turned over in 0.3 seconds, find the average electromotive force, earth field being as above. 7. If the resistance of the coil in the last problem is 2 ohms, find the total flow of electricity in coulombs, also the average current in amperes, also the energy spent in producing the current. 8. A magnet which includes 6000 lines of force is pulled out of a coil of 160 turns of wire which closely surrounds it, in J^o second. Find the induced electromotive force in volts. 9. A disc of iron 60 cm. in diameter mounted in a uniform magnetic field so that 4000 lines of force per sq. cm. pass perpendicularly through it, rotates like Faraday's disc (§712), making 30 revolutions per sec. Find the difference in potential between the edge of the disc and its center. DYNAMO MACHINES 495 10. A rectangular loop of wire 20 X 30 cm. is rotated about an axis parallel to the long sides and half-way between them, in a magnetic field of strength 2000. If the axis is perpendicular to the lines of force of the field, what will be the average electromotive force in a half rotation between reversals (§733) when the loop makes 20 revolutions per sec? 11. What is the maximum electromotive force in the case specified in the preceding problem? Dynamo Elbctbic Machines and Motors Part I. — Direct-current Dynamos 733. Introductory. — The first machine by which a continu- ous current of electricity was developed by electromagnetic induction was Faraday's rotating copper disc (§712). A machine developing current by electromagnetic induction consists of a strong magnet between the poles of which an armature rotates which contains the conductors in which the currents are induced. Such generators, as they are called, are known as magneto machines when permanent steel magnets are used, and dynamo machines when electro- magnets are employed. 733. Rectangular Armature. — Sup- pose that a simple rectangular frame of wire is rotated between the poles of a powerful magnet as shown in figure 429, and that its ends are con- nected to two rings a and 6 which Fig. 429.— Induction in simple are mounted on the axle, and against °°^ ^^"^^ ^^^' which press two springs connected to the ends of the outer cir- cuit. In the position shown the upper bar C is rapidly cutting across lines of force. By the rule of induction (§722) the induced electromotive force is in the direction of the arrow. So also electromotive force is developed in D. These two electro- motive forces act together to cause a current in the outside circuit from B to A. This will be the direction of the current so long as C is moving down across the field of force and D is moving upward. When the coil is in the vertical position both C 496 ELECTRODYNAMICS and D will be moving parallel to the lines of force so that there is no electromotive force in the circuit at that instant. Then as C comes up and D descends the electromotive force is reversed, causing a current from A toward B in the outer circuit, which reaches a maximum when the coil is horizontal, for then both C and D are cutting perpendicularly across the lines of force. The electromotive force again becomes zero when C reaches the top and D is at the bottom and then reverses again into the original direction. In the vertical position of the coil the electromotive force is zero, although it includes the maximum number of lines of force, because in that ■position a small motion of the coil does not appre- ciably change the number of lines of force which it embraces. While in the horizontal position the electromotive force is a maximum, although no lines of force pass through the coil, because the change is most rapid in that position. , /dl I I I I I I I I I IKtSO" QJO" 360 ' 0° ^^ Xlllllllll ly^ Top BottomXI^ yyTop Fig. 430. — Diagram of alternating electromotive force. The diagram (Fig. 430) exhibits what may be called the curve of electromotive force in such a case. The curve starts with C at the top, the abscissa at any point is the angle through which C has moved and the corresponding electromotive force is the ordinate, drawn above the horizontal when it is directed from B io A, and below when it is reversed. The current produced is what is known as an alternating current and goes through a complete cycle in the time of one revolution of the armature. An alternating current mxiy he compared to the surging hack and forth of water in a pipe in which a tightly fitting piston is moved to and fro. 734. Commutator. — The terminals of the coil just discussed, instead of being joined to two rings, may be connected to the two halves of a divided ring or commutator, as shown in figure 431, on which rest springs or brushes which connect to the ex- ternal circuit and are so placed that they slip from one segment DYNAMO MACHINES 497 to the other at the instant when the electromotive force in the coil is reversing. In the above diagram, whichever side of the coil is descending, is connected with A, while the ascending side is connected with B, so that the current is always from B to A in the external circuit. The current curve in the external circuit will in such a case be as in figure 432, where ordinates represent the current and abscissas the cor- responding instants of time. Each section of the curve represents half the period of a complete revolution of the armature. Such a current, though always in the same direction, is fluctuating. 735. The Ring Armature. — A valu- able armature, devised by Pacinotti, is known as the Gramme ring from the French inventor who was the first to Fig- 431.— Loop armature with , , . , , . . commutator. construct commercial machmes usmg that type of armature. It consists of a soft-iron ring made of a coil of iron wire or a pile of ring-shaped plates of thin sheet iron, wrapped around with a coil of insulated copper wire, the ends of which are joined together forming an endless-ring solenoid with an iron core. For distinctness in the diagram (Fig. 433), the turns of copper wire are shown widely separated. Suppose the ring to be mounted on an axle and rotated between the poles of a powerful magnet as shown in the figure. The lines of force of the magnetic field pass from one pole to the other chiefly through the iron ring as shown by the dotted lines. This, of course, is in consequence of its great permeability. As the armature rotates, those parts of the copper -winding which cross the outside of the ring cut across lines of force in the space between poles and arma- ture. On the right-hand side the wires cut down across the field, and the electromotive forces in these turns will be from the front toward the back of the armature. This tends to cause a current in the windings in the direction shown by the small arrows. All aeo' 498 ELECTRODYNAMICS of the turns on one side act together like so many little battery cells in series, though those in the middle are most effective. The outer sides of the turns on the left-hand side of the ring, next the south pole of the field magnet, cut up across the field of force, and hence the electromotive force in them is from the back toward the front of the armature, and so they conspire to produce a current on that side in the direction of the small arrows. But it will be observed that in consequence of the winding of the wire. FiQ. 4.33. — Gramme ring armature. the induced electromotive force on each side acts to cause a flow around the coil working from the bottom toward the top of the ring, and hence the top of the coil will be a point of high potential and the bottom a point of low potential, when the poles and winding of the armature are as shown in the diagram, but there will be no flow around the coil for the electromotive force on one side balances that on the other. To obtain a current, the top and bottom of the coil vmst be con- nected with an outside circuit. This is accomplished by the com- mutator which consists of a number of segments of copper insu- lated from each other and mounted in cylindrical form around the axis, each segment being connected with a corresponding point in the copper coil. The sections of the armature coil included between the points where connection is made to the commutator, all have the same number of turns. In the diagram only one turn is shown for each section, but any number may be used. If the ends of the external circuit are connected to the two brushes A and B there will be a current from .4 to B as indicated. DYNAMO MACHINES 499 For the brush A rests upon the upper segment in the commutator which is connected with the top of the wire coil, and is in this case a point of high potential, while similarly the brush B is in connection with the bottom of the coil where the potential is low. The flow of current within the armature coil is around on each side as shown by the arrows, the two currents coming together at the top and flowing out through the commutator f \ at A, around through the external circuit, and in at the bottom of the armature coil where the current divides, half flowing around on one side and half on the other. The case resembles an external circuit B connected to two batteries joined in par- Fig. 434.— Two batteries allel (Fig. 434), the electromotive force of each battery corresponding to that of one side of the arma- ture. 736. Drum Armature. — Of every turn of wire on a ring arma- ture part lies on the inside of the ring, and this does not contrib- ute to the electromotive force. Whatever slight effect it may have, due to the weak magnetic field inside of the ring, is in op- position to the outside part. It is desirable to have as little inactive wire as possible in an armature since it adds to its resistance. The drum armature is like a ring armature where the opening in the ring is filled up with iron and the turns of copper wire pass clear across the ring from one side to the other, so that the only inactive wire is that across the ends. The core is a cylinder of iron made of a pile of thin sheet-iron plates bolted together, around which the coils of wire pass longi- tudinally lying in grooves made for them. In winding, the wire starts at one of the commutator segments, is passed around the core lengthwise in one of the grooves the desired number of times, suppose twice, and then is connected to the next commutator segment. It is then carried right on around the core in the next groove in the same direction as before, making two more turns, and then connected to the third segment of the commutator. This process is continued until the segment is reached where the winding began and there the end is made fast. In this way an 500 ELECTRODYNAMICS endless coil is constructed just as in the Gramme ring, and be- tween each commutator segment and the one opposite there are two paths by which the current may flow within the armature, so that the current divides in the armature just as in the ring armature. 737. Foucault Currents. — In each of these armatures the in- ductive action which causes electromotive force in the copper coils also causes a similar electromotive force in the iron core tending to set up currents within the core itself. Such currents would spend energy in heat, and the double disadvantage would result that more work would have to be spent in turning the armature, and this useless expenditure of energy would go to unduly heat the machine. In order to prevent these Foucault currents, or eddy currents as they are often called, the iron core is laminated or made up of thin plates insulated from each other by varnish, or paper, and lying across the direction in which the currents would flow. The thinner the sheets of iron the more perfectly is this waste of energy prevented. 738. Electromotive Force of Armature. — The electromotive force of a ring armature is easily reckoned. The electromotive force of the ring is the same as that of one side, since the two sides of the ring act in parallel. Let N be the number of lines of force passing through the armature, n the number of revolutions per second, and C the number of turns of wire on the ring, then since each turn cuts down on one side across all N lines of force once in every half revolution, that is in ^ second, the average electro- motive force induced in each coil as it moves across the field must be N-^^ = 2Nn. 2n But all the coils on one side of the ring act together or in series, hence if there are C coils of wire on the ring the total electro- motive force must be thus E = NnC in C. G. S. units, or E = ^„g volts. DYNAMO MACHINES 501 The electromotive force depends on three factors : the num- ber of lines of force through the armature, the number of revolu- tions which it makes per second, and the number of coils of wire upon it. The electromotive force of a drum armature is calculated from the same formula, C representing the whole number of wires on the armature which cut across lines of force. 739. Field Magnets. — In most dynamo machines and motors the armature rotates between the poles of an electromagnet which receives its exciting current from the armature. Three Fig. 435. — Series and shunt field magnets. modes of winding are in use, series, shunt, and compound. In the first diagram in the figure is shown a sen'es-wound dynamo. The whole armature current passes around the field magnets and through the external circuit. Any resistance introduced into the external circuit, causing the current to diminish, weakens the magnetic field and therefore makes the electromotive force of the machine less. When there is no current flowing its elec- tromotive force is zero except for the residual magnetism. In the shunt arrangement the current in the armature divides, part flowing around the magnet and part to the external circuit. In order that but a small current may be taken for the magnet, it is wound with many turns of rather fine wire. The current through the shunt coil depends only on its resist- ance and on the difference of potential of the brushes; hence it is constant and the strength of the magnet is constant so long as the difference in potential of the brushes is unchanged. The electromotive force of such a dynamo is very nearly constant, 502 ELECTRODYNAMICS but is slightly greater when no external current is flowing, for with increasing current in the external circuit there is more cur- rent and a greater fall of potential in the armature itself. Compound winding is a combination of the shunt and series arrangements, in which there is a shunt coil and also a few turns carrying the whole current around the magnets. In this way a dynamo may be made to maintain a nearly constant potential at the terminals, though the external current may vary greatly, or it may be oe^er-compounded so that its terminal electromotive force may be greater with large currents than with small. Part II. — Direct-current Motors 740. Motors. — An electric motor is an appliance in which an electric current gives motion to an armature, thus producing Fig. 436. — Motor with ring armature. mechanical work. Small direct-current motors usually have ring or drum armatures and are in most respects like dynamos. The action of the ring armature in a motor may be under- stood from the diagram (Fig. 436). The current from a battery or other source is shown as flowing in at the upper brush and out at the lower one. Within the armature the current divides, half flowing around and down through the coils on one side and half through those on the other side as shown by the arrows, and the effect of these currents in the armature is to make each half of the ring a magnet with its north pole at the top and south pole at the bottom. The attractions and repulsions between these poles and those of the field magnet cause the armature to rotate in the direction of the large arrows. ELECTRIC MOTORS 503 Another aspect of the action is worth considering. The gaps between the pole pieces and the armature are regions of intense magnetic force, and the wires on the outside of the armature carry currents directly across these lines of force, wp (perpendicu- lar to the paper) on the left and down on the right; there is, there- fore, a force (§684) urging these wires to move across the lines of force toward the top of the diagram on the left and toward the bottom on the right. 741. Energy Spent in Motor. — While the motor is running mechan- ical work is being done in addition to the energy which is spent as heat in the armature in consequence of its resistance. But the total energy spent per second in the motor is equal to the product of the current strength by the difference of potential between the brushes. Therefore if the current is kept constant the difference of potential between the brushes must be greater when the motor is running and doing work than when the armature is at rest. This increase in the difference of potential between the brushes due to the motion of the armature is the hack electromotive force of the motor. There must be such a back electromotive force in every kind of device in which motion results from the flow of an electric current. Let Vi — Vi = difference of potential between brushes of motor. IR = drop in potential due to the resistance of the armature. Vi — Vz = E + IR where E is the back electromotive force. watts spent in turning armature + Total watts spent in motor = ^ . , i ■ v, * [ watts spent m heat or in symbols liVi-Vi) =IE + PR. 743. Back Electromotive Force. — Connect an electric motor to a battery by which it may be driven and introduce into the cir- FiG. 437. cuit an incandescent lamp which will glow with full briUiancy when the armature of the motor is held stationary. On letting the armature run the lamp grows dim, and an ammeter in circuit shows that the current has diminished, but a voltmeter connected to the brushes of the motor will show a much greater difference of potential between them than when the armature was at rest. 504 ELECTRODYNAMICS Since the electromotive force of the battery and the resist- ance of the whole circuit is unchanged by the running of the motor, it is clear that the current can have been diminished only by the development of an electromotive force in the circuit back against the driving current. The motor, in fact, while running acts like a dynamo and develops an electromotive force, called its back electromotive force, because it acts in opposition to the electromotive force of the driving battery. 743. Starting a Motor. — In starting a motor there is at first no back electromotive force to oppose the current, and in order to prevent the current being excessive and "burning out" the armature before the motor is well started some such device as shown in figure 438 is commonly used. The current is led to the motor through the wires AB, one of which is connected directly to the motor while the other is joined to the switch S. When the switch is turned from 1 to 2 the cur- rent flows through coils of wire having considerable resistance and starts the armature. As the speed increases, developing more back electromotive Fig. 43R.-starting connections j^^^^^ ^^le switch is moved on to 3 and 4, reducing with each step the extra resistance until, as the armature comes to full speed, the switch on 5 makes direct connection, and the back electromotive force keeps the current moderate even though the armature resist- ance may be extremely small. 744. Dynamo and Motor. — Suppose a transmission system consisting of dynamo and motor and connecting circuit. Let the electromotive force of the dynamo be 200 volts, and suppose the resistance of the whole circuit including the armatures of both dynamo and motor to be 1 ohm, and let the back electro- motive force of the motor be 180 volts at the working speed. Then the resultant or effective electromotive force in the circuit is 200 — 180 = 20 volts, and the current is 20 amperes. Power spent in the dynamo 200 X 20 = 4000 watts. Power used in motor 180 X 20 = 3600 watts. Loss in heat (PR) is the difference 400 watts. ELECTRIC MOTORS 505 If a motor is used which in running develops twice the back electromotive force of that just discussed, then with a current of 10 ampferes as much power will be obtained as with the 20 amperes in the former case. In this case the electromotive force of the dynamo must be 370 volts, and that of the motor being 360 volts the effective R-1 Ohm £=200 1=20 Amperes R-1 Ohm E-370 1^10 Amperes e=3eO Fig. 439. electromotive force is 370 — 360 = 10 volts. The current will therefore be 10 amperes, and we have Power spent by dynamo = 370 X 10 = 3700 watts. Power used in motor = 360 X 10 = 3600 watts. Power wasted in heat PR = 100 X 1 = 100 watts. This is evidently a much more economical arrangement than the first and illustrates the general principle that electrical energy can be transmitted with least loss by means of small currents at high voltage. Part III. — Alternating Currents 746. Alternating Currents. — Alternating currents have come extensively into use because of the ease with which a large alternating current at a low voltage can be changed to a small one at a high voltage. The small high-voltage current can be carried by comparatively small conductors to a distant point and then be transformed down again to a large current at a low enough voltage to be safely used for light or power. , 746. Alternating-current Dynamos. — Almost any direct-cur- rent dynamo will give alternating currents if it is provided with two rings mounted on the axis and connected respectively to two diametrically opposite segments of the commutator. A circuit whose ends are connected to these rings by brushes will 506 ELECTRODYNAMICS have an alternating current. Such a case was illustrated in §733. To secure good insulation, high electromotive force, and sufficient frequency of alternation, alternating-current dynamos are usually multipolar, as illustrated in figure 440. In the type shown the field magnet poles, alternately north and south, pro- ject outward from the rim of a rotating wheel and are magnetized by the current supplied by a small separate direct-current dynamo called an exciter. This rotary field, or rotor, rotates within the fixed armature or stator, in which the poles project inward from the outside circular frame. These poles are of the same number as those on the rotor, and are laminated or built up of thin plates of sheet iron to prevent eddy currents. Around the poles WIret to exciter gotatlng Field Rotor Fig. 440. of the stator the armature coils are fitted, passing through slots between them and wound alternately clockwise and counter- clockwise around successive poles. As the rotor turns and a north pole facing one pole of the armature moves over to the next, the lines of force from the pole of the rotor cut across the conducting wires lying between the poles of the stator and induce electro- motive force in them which reverses as the succeeding south pole moves across, thus causing an alternation. But as the wires in one slot return in the opposite direction through the next slot, and as a north pole is moving across the one while a south pole is moving across the other, the electromotive forces induced in all act together at every instant, so that in the case figured where there are 16 poles, an alternating electromotive ALTERNATING CURRENTS 507 force is produced 16 times as great as would be developed in a single coil. If a low electromotive force is desired the several coils of the armature may be connected in parallel instead of in series as above described. 747. Virtual Amperes and Volts. — An alternating current is constantly varying in strength, as illustrated in the curve of figure 441, its average value is zero and it will not give a steady deflection of the needle in an ordinary galvanometer. A defini- tion must therefore be given of what is meant by an alternating current of one ampere. Since the energy relations of a current are commercially the most important, an alternating current Fig. 441. — Alternating-current curve. Current = 10 Amperes. i is said to have the strength of i ampere, when it will develop the same amount of heat in a given resistance as would be pro- duced by a direct current of i ampere. The heating effect of a current at any instant is proportional to the square of its strength at that instant, so also the deflection produced by a current in an electrodynamometer is proportional to the square of the current strength (§693); therefore an electrodynamometer measures directly the virtual amperes of an alternating current just as it does a direct current. If the alternating-current curve is a sine curve, the virtual strength as defined above is to its maximum value in the ratio of 1 to 1.41; thus an alternating current of 10 amperes ranges from -f- 14.1 to — 14.1 amperes in its instantaneous values. So also the virtual value of an alternating electromotive force is said to be i volt when it will develop an alternating 508 ELECTRODYNAMICS current of i ampere in a resistance of i ohm having no self-inductance. 748. Effect of Self-induction. — It has already been shown (§723) that the effect of self-induction in a circuit is to cause an electromotive force contrary to an increasing current and with a decreasing current. In case of alternating currents, the effect is twofold. First, it causes an apparent increase in resist- ance. It may be proved that the current produced by an alternating electromotive force E in a. coil whose coefficient of self-induction is L and whose resistance is R, is E 1 = y/ R^ + {2irnLy where n is the number of complete cycles per second, the cur- rent and electromotive force being measured in virtual am- peres and volts, the resistance in ohms and the inductance in henry s. The denominator is known as the impedance of the conductor. If the self-induction of the coil is large and if there are a large number of alternations per second, the impe- dance may be large although the resist- ance is small. Second, self-induction causes the phase of the current to lag behind that of the electromotive force, so that the current does not reach its maximum value at the same instant that the elec- tromotive force is a maximum, but a certain fractional part of a period later, which is called the lag. The relations of these quantities are shown by the triangle in figure 442. If the base of the right-angled triangle represents the resistance of a coil, and the altitude, the quantity 2ir7iL, then the hypothenuse represents the impedance, and the angle a at the base, the angle of lag. That is, the current maximum lags behind the maximum of electromotive force the same fractional part of a complete period that a is of the whole angle about a pomt. 749. Theater Dimmers. — If an electric glow lamp, connected ALTERNATING CURRENTS 509 in series with a coil of wire having very low resistance, is lighted by means of an alternating current, the light may be dimmed by inserting a laminated core of soft iron inside the coil. The self-induction of the coil is greatly increased in this way and the current is decreased, but there is no waste of energy as there would have been if the current had been reduced by introducing resistance. This method is used for dimming theater lights. 750. Transformers. — Alternating currents are easily changed from low voltage to high, or vice versa, by means of transformers. Fig. 443. — Transformer and section. Fig. 444. — Transformer connections. A transformer consists of two coils side by side, having a common core of soft iron. In the form shown in figure 443, the iron core is made up of a pile of thin sheet-iron plates of the shape shown in the section. The core thus formed is a block of soft iron having two rectangular holes through it in which the two coils lie side by side, one coil having many turns of fine wire and the other a few turns of coarse wil-e as shown in the diagram. When an alternating current is set up in one coil it magnetizes the iron core, setting up lines of force which at one instant are in the direction shown by the arrows in the diagram and a half period later are exactly opposite. But the lines of force pass through the second coil as well as the first and therefore an alternating in- duced current is set up in the secondary coil. The same lines of force cut across one coil as the other and 510 ELECTRODYNAMICS consequently the electromotive force induced in one coil is to that in the other as the number of turns of wire in the coils. Suppose it is required to transform from 1000 volts down to 50. The fine wire coil which is connected to the 1000-volt circuit must have 20 times as many turns of wire as the coarse wire coil which is connected with the lamps. If no lamps are turned on, there is no current in the secondary coil and the magnetic field through the primary coil causes such a strong back electromotive force that only a very small current flows through it and there is but a small loss of energy. When lamps are turned on in the secondary, a current flows which, by the laws of induction, opposes the changes in the magnetic field by which it is produced, and therefore more current must flow in the primary coil to keep up the magnetism necessary to produce the back electromotive force in the primary which balances the electromotive force of the main line. In this way the transformer is self-regulating, the primary current being very nearly in the same ratio to the secondary as the number of turns of wire in the secondary coil is to that in the pri- mary. In the example considered, the current in the primary would be one-twentieth that in the secondary The energy spent in the secondary circuit is equal to that which the transformer takes from the main line except for a small amount, perhaps 5 per cent., which is lost as heat in the transformer. 751. Advantage of Transformers. — ^Large currents cannot be transmitted long distances without great loss in heat unless large conductors of low resistance are used, in which case the cost and interest charges are high. By means of a transformer a large electrical power may be transmitted by a small current at high voltage. Thus in districts where scattered houses are to be lighted a small current at high voltage is used on the street line and transformed down, giving large currents at low potentials at the points where lights are used. In many lines where power is to be transmitted a long distance transformers are used at both ends of the line. Thus at Niagara dynamos develop currents at 2200 volts, which are then trans- formed up to 22,000 volts, and so transmitted 20 miles to Buffalo, where they are transformed down again for power and lighting purposes- ALTERNATING CURRENTS 511 753. Electric Welding. — An important application of large electric currents is in fusing bars of metal together. Two bars of iron as large even as a man's wrist may be placed end to end and fused together in a few seconds. For such a purpose a very- large current is required just at the spot to be heated. Accord- ingly a transformer is used in which the secondary may consist of only a single turn or two built of heavy copper bars, the ter- minals of which are clamped to the bars to be welded, one on each side of the junction. The primary coil is made of many turns of wire and takes a comparatively small current at high voltage. 753. Alternating-current Motor. — There are two principal types of alternating-current motors, the synchronous motor in which the armature will not start of itself, but must be brought by some accessory motor to such a speed that its armature coils move from one field pole to the next in exact synchronism with the alternations of the driving current. When brought to speed, it will continue to work when driven by a single alternating current. A second type is the induction motor in which a rotary magnetic field is produced by polyphase currents. 754. Rotary Magnetic Field. — Suppose that a laminated ring of soft iron, having four poles projecting inward as shown in figure 445, is wound with two independent circuits, one of which magnetizes the A and C poles and the other the B and D poles. And let an alternating current be established in each circuit, the phase of the current in the E circuit being a quarter of a period ahead of that in the F circuit, as shown in the curves ^'°- 445.-Rotary field magnet. E and F in figure 446, so that one reaches its maximum value, either positive or negative, at the instant that the other is passing through its zero value. The corresponding changes in the direction of the lines of force in the field between the poles are shown in the lower dia- grams of figure 446. Thus in the first diagram the current E isa maximum, and is supposed flowing from E to E' (Fig. 445) making 512 ELECTRODYNAMICS A a north pole and C a south pole, while at that instant the F current is zero. But as the F current increases that in E dimin- ishes until F becomes a maximum and E zero. The north pole has now passed to B, while A and C have lost their polarity as shown in diagram 3. The sign of the E current is now re- versed and it begins to flow from E' toward E, maktog C a north t|t) ^g^ fe^ Fig. 446. — Diagram of two phase currents and rotary field. pole as shown in 4; at the same time the F current is decreasing and becomes zero in 5, where E reaches its maximum negative value. In this way what is known as a rotary magnetic field is produced in which the north and south poles move around the ring making one revolution for every complete period of the current. 755. Induction Motors. — Between the poles of the rotary field just described there is mounted a cylindrical -shaped arma- ture having a set of parallel rods of copper at equal intervals around the circumference, like the bars in the wheel of a squirrel cage, connected across the ends by copper plates. And to strengthen the lines of force through the armature, it is filled with a soft-iron core made of a pile of circular plates of thin sheet iron. As the lines of force of the field rotate they cut across the bars of the armature, inducing currents which by Lenz' law are in such a direction as to resist the relative motion of armature and field, and the armature is therefore carried around in the direction in which the field rotates. But clearly in such an induc- tion motor, the armature cannot rotate as fast as the magnetic field, because it is the difference between the motions of the two that causes the induction on which the rotation of the armatiire depends. ALTERNATING CURRENTS 513 756. How Currents in Different Phases are Obtained. — Imagine a Gramme ring armature as shown in figure 447 .pro- vided with four insulated brass rings mounted on its axis, each of which is connected permanently to one of the points EFE'F', which are just one-quarter circumference apart on the ring. If one circuit is now connected to the brushes e and e' and another FiQ. 447. — Connections for currents in quadrature. to the brushes / and /' which rest on the rings, the currents in the two circuits will be one-quarter period different in phase as represented in figure 446. 757. Ttiree-pliase Motors. — The usual form of induction motor uses three-phase currents, or three currents which differ in phase by one-third of a period, and requires only three line wires instead of four. The generator has three rings connected, re- FiG. 448. — Three-phase currents. spectively, to three equidistant points in the armature, so that the currents developed in the three line wires are related as shown in the curves of figure 448. It will be noted that the sum of the ordinates of any two of the three curves taken at any point along the base is equal and opposite to the ordinate of the third curve at that point; that is the sum of the currents in any two of the three line wires at any instant is equal and opposite to the cur- rent in the remaining line wire, the three are, therefore, connected 514 ELECTRODYNAMICS together at the farther end and each serves as the return wire for the other two, as shown in figure 449. Fig. 449. — Connections of three-phase generator to the poles P of the field magnet of the motor. Three-phase motors are usually multipolar, each principal pole being subdivided into three parts. The figure shows a field having twelve small poles which are so wound as to form a rotary field with two north poles and two south poles. How this is done may be under- stood from the diagram in which the field ring is supposed to be cut at one point and bent out flat so that we look di- rectly at the faces of the twelve poles. For simplicity the wire is represented as carried only once through each groove. It will be seen that when' the current in 1 is a maximum in the direction of the arrow the poles will be situ- ated as shown in the upper row of letters. A third of a period later the current in 2 will be a maximum in the same direction, Fig. 450. Fig. 451. — Windings of a three-phase field magnet. and the poles will then be as indicated in the second row. Then after another one-third of a period current 3 will have reached ALTERNATING CURRENTS 515 its maximum and the poles will have shifted to the positions indicated in the third row of letters. There is thus produced a steady movement of the poles around the ring, moving over the distance between two similar poles in the time of one complete period of alternation of the current. In the above case, if the current has a frequency of sixty periods per second, the field will make thirty revolutions per second. Problems 1. The core of a Gramme ring armature has a cross section of 6 X 10 cm. How many turns of wire must it have that it may give an electro- motive force of 20 volts when making 800 revolutions per minute in a magnetic field so strong that where the lines of force in the ring are most concentrated there are 6000 per.square centimeter? 2. A certain dynamo armature when making 1000 revolutions per minute is supplying a current of 50 amperes at 100 volts. Find the horse-power required to drive it and thence the moment of force or torque in pound- feet required to turn the armature at the given speed. 3. When the armature of a certain motor is held fixed a current of 10 am- peres through it causes a difference in potential between its brushes of 5 volts. When the armature is permitted to run at 600 revolutions per minute the current is 4 amperes and difference of potentials at the brushes is 30 volts. Determine the back electromotive force of the motor. 4. The core of a drum armature is a cylinder of iron 30 cm. long and 15 cm. in diameter, the induction through its middle longitudinal section is 6000 lines of force per square centimeter. If there are 50 complete turns of wire on the armature, or 100 longitudinal wires in grooves on its surface, what is its electromotive force when making 1200 revolutions per minute? 6. A transformer has a coil of 250 turns; what must be the size of the i;on core in order that an average electromotive force of 100 volts may be developed in this coil while the number of lines of force in the core changes from + 6000 to — 6000 per sq. cm., the current alternating at the rate of 60 complete periods or cycles per second? 6. A certain transmission line has a resistance of 20 ohms. How much power will be lost in the line when 100 kilowatts are transmitted at 2000 volts? How much when the same power is transmitted at 20,000 volts? 7. A multipolar generator having 16 poles (Fig. 440) makes an alternating current of 60 cycles per sec. How fast does it rotate? If there are 30 turns in each armature coil, what E.M.F. is developed when each pole of the rotor gives rise to 100,000 lines of force? 516 ELECTRODYNAMICS Electric Oscillations and Waves 758. Oscillatory Discharge of a Leyden Jar. — It has been already stated (§585) that when the resistance of the discharge circuit is sufficiently small the discharge of a Leyden jar is oscillatory. This was discovered by the American physicist Joseph Henry, who, as early as 1842, found that when a Leyden jar was discharged through a wire wound around a needle the latter was magnetized, but sometimes one end was made the north pole and sometimes the other, although the jar was always charged the same way. He believed that this was caused by the oscillation of the discharge current which kept reversing the magnetism of the needle back and forth until the current became too small to have a further effect. This opinion was confirmed by eating off the surface layer of the needle with acid, when the interior was found magnetized opposite to the outer layer. Lord Kelvin, in 1855, quite unaware of Henry's discovery, showed by the principle of energy that the discharge must oscillate back and forth until all the original energy of charge is expended in sound, heat, light, and radiation, and that when the resistance of the circuit is very small the period of oscillation is given by the formula P = 2iry/LC where L is the coefficient of self-induction of the circuit and C is the capacity of the jar. In case of an ordinary gallon jar dis- charged by a short discharging rod, the period of oscillation may be as small as two ten-millionths of a second, while Lodge, by using a battery of large capacity and discharging it through a very long circuit having large self-induction, was able to make the alternations so slow as to give out a distinct musical note. Feddersen, in 1859, first analyzed the spark by a rotating mirror, as already related (§585). 759. Electric Resonance. — When a Leyden jar is discharged not only may there be oscillations in the discharge circuit itself, but in consequence of induction there are set up electric oscilla- tions or surgings in neighboring conductors. In general these are but feeble, but if the free period of the surging happens to be the same as that of the oscillations in the discharge circuit, quite energetic surgings may result, just as a tuning-fork will excite ELECTRIC WAVES 517 strong vibrations in a resonator which is in tune with it. The circuits are then said to be in resonance. 760. A Case of Electrical Resonance. — The influence of elec- trical resonance is well shown in the following experiment due to Lodge. Two Leyden jars of nearly equal capacities are chosen. One which can be charged by an electrical machine or induc- tion coil is provided with a short circuit of thick wire which is attached to the outer coating and terminates in a knob separated by a short spark gap from the knob of the jar. The second jar has a strip of tinfoil reaching from the inner coating over the edge and terminating in a point at e near the upper edge of the outer coating; its inner and outer coatings are connected by a wire circuit, part of which, marked AB in the figure, can be slid along changing the length of the path. When the two jars are placed, say, a foot apart with the two circuits parallel, a position for the slider AB may be found by trial, such that whenever the first jar discharges across between the knobs, a spark leaps the gap between the tinfoil strip and the outer coat- ing of the second jar. If the slider is moved a short distance away from this position in either direction, the sparks at e cease. Lodge calls the sparks at e the "slopping over" of the powerful surgings due to the two circuits being in resonance. 761. Electric Waves in Wires. — When one end of a long straight wire is given a charge or touched to a battery pole, a wave of elec- tric pressure or potential runs along the wire with a velocity which depends on the insulating medium immediately surrounding the wire. In case of a straight bare wire in air the wave has the velocity of light; but when the wire is coiled, forming a closely wound helix, the wave travels much slower on account of the greater self-induction of the coil. On reaching the end of the wire the wave is reflected back, just as a sound wave is reflected at the end of a stopped organ pipe. FiQ. 452. — Sir Oliver Lodge's reso- nance experiment. 518 ELECTRODYNAMICS If, instead of a single impulse, a series of alternate positive and negative charges are given to the end of the wire in exactly the right frequency, it may be set in strong electrical resonance just as a stopped pipe vibrates powerfully when a tuning-fork of the proper frequency is sounded at its mouth. Resonance will occur when the period of the electrical impulses is four times as long as it takes a wave to run the length of the wire, exactly as in case of a stopped organ pipe. The resonance of waves in wires may be beautifully shown by the following experiment due to the German electrician Seibt: A large Leyden jar has its coatings connected by a circuit having a spark gap at S with zinc knobs. By moving the slider Fig. 453. — Resonance experiment. L nearer to the jar or farther away, the length and self-induction of the discharge circuit may be varied and consequently the period of the oscillatory discharge can be adjusted. Two long helical coils of wire A and B are mounted on in- sulating stands. They are both connected at the bottom to one of the coatings of the Leyden jar while each terminates above in a point. One helix is wound with a much greater length of wire than the other. If by means of a powerful induction coil the Leyden jar is caused to discharge across the gap S, each discharge will be oscillatory and consequently a series of impulses is communi- cated to the lower ends of the helices A and B, and when the slider is in such a position that the period of oscillation of the discharge is the same as the period of oscillation in the wire on A, a strong brush discharge will -be observed from the upper ELECTRIC WAVES 519 point of that helix; while by moving the shder until the jar circuit is in resonance with B, the discharge will take place from the top of B instead of from A. 763. Electromagnetic Waves in Air. — As early as 1862 Clerk Maxwell, who followed Faraday in recognizing the important function of the dielectric in all electric phenomena, showed that it was probable that when a current is stopped or started in a conductor, the inductive action on other conductors is not commu- nicated instantly, but is propagated through the intervening dielectric with a velocity equal to 1 where ix is the magnetic permeability of the medium and K is its specific inductive capacity. The quantity l/y/lKix can be determined by electrical experi- ments in a variety of ways and is found to have a value in air of very nearly 300,000,000 meters per second, which agrees with the velocity of light. Of course, if induction is propagated with a definite velocity, an alternating current sending out first one kind of inductive disturbance and then the reverse must produce a series of electrical waves, just as a tuning-fork giving a series of impulses which travel successively forward through the air produces a train of sound waves. 763. Hertz' Experiments. — Maxwell's conclusions as to elec- tric waves were not directly demonstrated until 1884, when the German physicist Hertz obtained such waves and measured their velocity. The difficulty was twofold: to set up waves short enough to be studied- — for if their velocity was 186,000 miles per second, an alternating current with a frequency of even 186,000 per second would produce waves a mile long — and, second, to devise some method of detecting and measuring them. Hertz succeeded in obtaining waves sufficiently short to measure by using those sent out in the oscillatory discharge of the apparatus shown in figure 454. Two rectangular metal plates were mounted as shown, with polished knobs close together. The plates were connected to the secondary of a powerful induction coil so that when charged by 520 ELECTRODYNAMICS the coil they discharged with oscillations across the spark gap between the knobs. Thus a group of short waves was sent out by the oscillatory discharge every time the induction coil acted, and this may have been 200 times a second, but each group died out absolutely before the next was formed. To detect the waves. Hertz used an electrical resonator, a hoop Fig. 454. — Hertz oscillator. Fig. 455. — Hertz resonator. of metal having at one point a minute spark gap between two knobs. The resonator was adjusted to be in resonance with the vibrator so that in a darkened room a small spark could be seen at the spark gap of the resonator at every discharge of the vibrator, even when it was 10 or 12 meters distant. In order to test whether the disturbance was propagated as Fig. 456. — Hertz nodes and loops. a wave motion, Hertz set up the vibrator in front of a great reflector of sheet metal, as shown in figure 456, so that the re- flected waves meeting the advancing ones might cause nodes and loops just as in any other case of wave motion. He then found by means of the resonator that there actually were points of maximum disturbance and points half-way between ihzm where the effect was a minimum. In this way the existence of electrical waves was proved and the wave length measured. ELECTRIC WAVES 521 From the wave length and period of oscillation the velocity of the waves was calculated and found to he, as nearly as could be determined, the same as the velocity of light, thus confirming the anticipations of Maxwell. 764. Other Experiments. — ^Later experimenters have devised oscillators of other forms more suitable for obtaining short waves. One of the best arrangements is that of Righi shown in figure 457. Two brass balls are mounted near each other, the space between being filled with oil contained in a surrounding glass cylinder. Just outside of these are other balls connected with the poles of the induction coil. A large difference of potential between the two inner balls is required before a spark can burst through the oil, and consequently the vibrations are so much the more energetic. Using oscillators of this form electric waves only a few milli- meters long have been obtained and measured, and have been reflected, refracted, and polarized, like waves of light. Fig. 457. — Righi oscillator. |l|l|l|l[ Battery Fig. 458a. — Simple wireless sender. Fig. 4586. — Wireless sending circuit. 765. Wireless Telegraphy. — An important application of electromagnetic waves has been made by Marconi in wireless telegraphy. The waves are sent out from a tall antenna or vertical wire which is connected to one pole of an induction coil, the other pole of which is connected to earth. Between the 522 ELECTRODYNAMICS two there is a spark gap across which the oscillation takes place. Such an arrangement sends out waves on all sides, the most energetic waves going out at right angles to the wire. No energy- is sent directly upward. The aerial wire or antenna is given a variety of forms; a common type con- sists of several copper wires stretched at a height of from 50 to 100 ft. above the ground and all connected to the earth through a single wire at one end. In order to set up more powerful oscillations the arrangement shown in figure 458b is commonly used. The antenna A is shown directly connected to the grounds through a coil of perhaps a dozen turns of heavy copper wire. The condenser C has its coatings connected by a circuit which takes a few turns close around the coil B in the antenna circuit, and includes a spark gap S. When the condenser is charged by an induction coil or high-tension transformer, discharges take place across the spark gap, accompanied by ■ Sec. -^y Sec. 25,000 500 Fig. 459. — Current in Antenna. Fifty such groups may be in one "dot." oscillations or surgings in the condenser circuit, and these, by induction between the coils of wire in B and D, set up corresponding oscillations in the antenna circuit, and if one circuit is in resonance with the other the oscilla- tions will be strong. When the key K is pressed for a tenth of a second to send a "dot" of the telegraphic code, perhaps 50 sparks will pass at S each causing a group of surgings in the antenna which rapidly die out. It is these oscillations that determine the length of the waves that are sent out. If the wave length is 600 meters each oscillation in the antenna will have a period of one five-hundred-thousandth of a second, and each group will die out after perhaps 20 such oscillations or in HsjOOO second, so that the cur- rent in the antenna may be represented by the above diagram, the current having zero value between the groups of oscillations. 766. Receiving Apparatus. — An early form of receiving apparatus using a coherer as a detector is indicated in figure 460a. The coherer consists of two small silver rods fitted closely into a short glass tube and having a narrow gap between their ends partly filled with sharply cut nickel and silver filings. When electric waves meet the antenna of the receiving station, they excite oscillations which surge alternately down through the wire to the ground and back again. These surgings pass through the filings in the coherer and have the effect of causing the particles to cling together and so their electrical resistance is greatly diminished permitting current from the battery C to pass. This operates the relay and sounder giving the signal. There is also a tapper which slightly jars the coherer and restores it to its original high ELECTRIC WAVES 523 resistance, thus interrupting the current from C so that all is ready for the next signal. Electric waves may also be detected by a simple microphone consisting of a needle laid across two sharp edges of carbon connected in series with a telephone and battery cell. Coh. Kelay Loadtng Coil FiQ. 460a. — Wireless receiver. Secondary Coil Fig. 4606. Telephopes A simple form of receiving circuit widely used in wireless telegraphy is shown in figure 4606. The antenna is connected directly to earth through a cylindrical coil known as the primary coil. Another coil, the secondary, of smaller diameter, is mounted so that it can be slid inside of the primary coil or moved away from it when it is desired to weaken the inductive action of one coil upon the ether. The terminals of the secondary coil are joined to the coatings of a variable condenser of small capacity. When electric waves of a certain period fall on the antenna, oscillations are set up which will be strongest when the natural period of electrical surg- ings in the antenna and primary coil are the same as the period of the on- coming waves. This adjustment may be effected by regulating the number of turns used of the primary coil by means of a sliding contact, or by connect- ing additional turns from a so-called "loading coil." The secondary also is brought into resonance with the primary, by the adjustment of the variable condenser and distance between the two coils. 767. Detectors. — The detector system is attached to the terminals of the variable condenser and may consist of a pair of receiving telephones in series with a crystal detector. The receiving telephones are specially wound with a great number of turns of very fine wire so that they may respond to the slightest current. The crystal detector may consist of a crystal of galena which is lightly touched by the sharp tip of a fine brass or steel wire ; or it may be a fragment of silicon against which a point of steel piano-wire is gently pressed; or an electrolytic detector may be used in which the tip of a 524 ELECTRODYNAMICS very fine WoUaston wire of platinum just touches the surface of a little dilute sulphuric acid, a battery cell being in series with the arrangement in this case. But the most sensitive type of detector in use is the so-called audion of DeForest, which makes use of the fact that a glowing incandescent fila- ment in a vacuum gives off negative electricity more readily than positive. These various detectors act as valves or rectifiers which permit current to flow easily in one direction but oppose it when it is reversed. The electrical oscillations which are received are too rapid to affect the telephone diaphragm, but the detector rectifies the oscillations in each group (Fig. 459) so that for each group of oscillations there is a little pulse of current in one direction through the telephone; and this being repeated with the frequency with which the groups come along, causes a distinct tone to be heard. 768. Wireless Telephony. — For wireless telephony it is necessary to send out a sustained series of oscillations which although too rapid to affect a telephone directly are, by means of a suitable transmitter, made to fluctuate VWWIAAAAAAA/VWWWWWWVWWVW^ '^^aN\/\I\I\I\JWvs^-^ Fig. 461. — Current curves in wireless telephony. in intensity just as an ordinary telephone current fluctuates in response to the voice, and it is these fluctuations in intensity that act upon the receiving telephone and reproduce the sounds to be transmitted. For instance, sup- pose the transmitting station sends out a continuous series of what are called undamped waves 10,000 meters long. These will come along at the rate of 30,000 to the second, and if the receiving station is in resonance with these waves a strong electrical oscillation is set up represented by the upper curve in figure 461, but these oscillations are far too rapid to set in vibration the telephone diaphragm. Now if sound waves acting through the transmitter cause fluctuations or variations in intensity in the original waves the effect will be such as is indi- cated in the middle curve. This also would not directly affect a telephone, for the positive and negative currents exactly balance each other. But if a suitable rectifying device, such as an audion, is used, which only permits currents to flow in one direction in the detecting circuit, then the current in the telephone though intermittent will on the whole vary in intensity as indi- cated in the lower curve and the diaphragm will vibrate in response, thus reproducing the sound. References Poincar:^ and Vreeland: Maxwell's Theory and Wireless Telegraphy. Rupert Stanley: Wireless felegraphy ELECTRIC DISCHARGE Electric Dicharge through Gases 525 769. Discharge at Atmospheric Pressure. — The difference in potential between two knobs required to cause a spark to pass between them at atmospheric pressure appears to be nearly the same whatever metal is used for the knobs, but it depends on their curvature. For knobs over 2 cm. in diameter and more than 2 mm. apart, the number of volts required for spark discharge is approxi- mately given by the formula V = 30,000d + 1500 where d is the distance between the knobs in centimeters. Table of Spark Polenlials between Slightly Curved Surfaces in Air Spark length Voltage Voltage per centimeter 0.0015 cm. 426 284,000 0.01 948 94,800 0.1 4,419 44,190 0.5 16,326 32,652 0.8 25,458 31,822 1.0 31,650 31,650 Heydweiler 770. Efifect of Diminished Pressure. — On diminishing the pressure, the potential necessary for discharge is lowered until a certain critical degree of exhaustion is reached. Beyond this point the higher the exhaustion, the greater the potential dif- ference required to produce discharge, until at the highest ex- haustions a spark can hardly be made to pass, but discharge will take place through several inches of air at atmospheric pressure, in preference to 1 mm. in the vacuum. The critical pressure is less than a millimeter of mercury when the electrodes are more than a few millimeters apart, but if they are very close it is somewhat greater. The appearance of the discharge may be conveniently studied in a wide glass tube 3 or 4 ft. in length, closed at the ends with caps in which the electrodes are mounted, and connected with an air-pump. If the electrodes are connected with an in- 526 ELECTRODYNAMICS duction coil or electrical machine, the discharge will take place through the tube after a few strokes of the air-pump. At first there is a crackling, flashing discharge along narrow flickering lines, but as the exhaustion proceeds the lines of discharge widen out and fill the whole tube, which glows with a steady light. The discharge at first is between certain points on the elec- trodes, but with higher exhaustion the luminous glow entirely covers the surface of the negative electrode. In this stage the characteristic features of the discharge are as follows: A faint velvety glow covers the surface of the negative electrode or cathode; just outside of this is the Crookes dark Fig. 462. — Discharge in gas at low pressure. space which surrounds the cathode and has nearly a constant width everywhere. Then comes a luminous region, called the negative glow, and then a dark space, the so-called Faraday dark space, after which a luminous column, known as the posi- tive column, reaches all the way to the anode. The positive column is commonly not continuous, but shows alternate bright and dark layers across the path of discharge; and these bright layers or strice become broader and farther apart as the exhaustion is increased. If the distance between electrodes is increased the appearance at the negative electrode is not particularly changed, the posi- tive column, however, is increased in length and reaches as before nearly to the negative glow. Professor J. J. Thomson examined the discharge in an exhausted tube 50 ft. long, and found that the positive column reached the entire length of the tube to within a short distance of the negative electrode and was stratified throughout. 771. Geissler Tubes. — Geissler tubes, so called from the name of a well-known maker who showed great skill and ingenuity in their construction, are tubes of glass especially designed for the purpose of exhibiting the phenomena of discharge. They are exhausted to a pressure of about 1 mm. of mercury, and are provided with aluminum electrodes attached to platinum CATHODE RAYS 527 wires sealed into the glass. They are usually wide near the electrodes, but often a part of the tube is quite narrow, and here the concentration of the discharge makes the illumination par- ticularly brilliant and the stratification very noticeable. In these tubes a marked fluorescence of the glass is produced by the discharge, some kinds of glass glowing with a yellowish- green light, while other kinds appear bluish. When such a tube is surrounded by a solution of sulphate of quinia or fluorescein or other fluorescent liquid, the characteristic fluorescence is strikingly brought out. The character of the light from such a tube depends on the gas which it contains, a tube containing nitrogen or atmospheric air appears of a reddish-violet color, a hydrogen tube is much bluer, while carbon dioxide gas shows a pale whitish illumination. The light of each when analyzed by a spectroscope is found to be made up of certain particular wave lengths characteristic of the gas. 772. Cathode Rays.-;— The Crookes dark space closely sur- rounding the negative electrode or cathode is the seat of a re- FiG. 463. — Crooke's tube with screen intercepting cathode rays. Fio. 464.- -Wheel driven by cathode rays. markable kind of discharge called the cathode rays, the properties of which are best studied in tubes that are exhausted far higher than ordinary Geissler tubes, or to a pressure of about one- thousandth of a millimeter of mercury. In such a tube the positive column is faint and inconspicuous, while the dark space around the cathode occupies a considerable space in the tube and is pervaded by a discharge which streams out nearly at right angles to the surface of the cathode without reference to the position of the positive electrode. In the tube shown in figure 463 a sharply defined shadow of the metal cross is cast on the 528 ELECTRODYNAMICS end of the tube opposite the cathode, for the rays excite brilliant yellowish fluorescence in the glass wherever they strike it directly. The position of the positive electrode is immaterial. A crystal of Iceland spar or calcite on which the cathode rays fall glows with orange-red light which persists some seconds after the discharge has ceased. If the cathode is concave, the discharge may be concentrated in a focus and produce intense heat. That there is also a mechanical effect produced by the cathode rays was shown by the English chemist Crookes by means of the tube shown in figure 464, in which a carefully balanced little wheel with light vanes of mica or aluminum rests with its axle on horizontal rails of glass. The electrodes are mounted at each end of the track facing the upper part of the wheel, and when the discharge passes the wheel is driven away from the cathode. J. J. Thomson has shown that the rotation of the wheel in this case is probably caused by the unequal heating of the two sides of the vanes as in the radiometer (§465) rather than by the direct mechanical effect of the cathode rays. 773. Deflection of Cathode Rays. — In the tube shown in figure 465 the rays from the cathode after passing through a narrow Fig. 465. — Cathode rays deflected by magnet. opening in the' screen 5 fall upon a sheet of mica running length- wise the tube and covered with fluorescent material so that the path of the rays is distinctly seen. The boundaries of the lumi- nous path are seen to curve outward slightly as though the discharge consisted of a stream of particles charged with elec- tricity whose mutual repulsion causes them to separate while they are streaming forward. If a horseshoe magnet is now held with its poles on opposite sides of the tube so that its lines of force are at right angles to the path of discharge, the rays are deflected to one side. If CATHODE RAYS 529 the lines of force are down, perpendicular to the paper, the rays will be bent into the position shown by the shaded curve, just as would be expected of particles of matter negatively charged and projected forward. It has also been shown by J. J. Thomson that the stream of cathode rays is deflected when a positively or negatively charged body is brought near it, being attracted by the former and repelled by the latter. 774. Nature of Cathode Rays. — The experiments described in the last paragraph all point to the conclusion that the cathode rays consist of a stream of negatively charged particles projected from the cathode. From the amount of deflection of the rays in a magnetic field of known strength, taken in connection with the deflection caused by a known electrostatic field, the English physicist J. J. Thomson has estimated that the particles forming the cathode rays have each a mass about Hsoo th^'t of a hydrogen atom and move with a velocity which depends on the fall of potential at the cathode, but may be about 3d^o of the velocity of light, or at the rate of about 18,000 miles per second, and each carries a negative charge of electricity equal to one elementary unit charge (4.77 X 10~ '° electrostatic units) as found by Millikan. The mass and charge of the cathode-ray particles have since been studied by other physicists and measured in different ways and are found to be the same whatever may be the nature of the metal electrodes or of the residual gas in the vacuum tube. These particles have received the name electrons (a contrac- tion for electrical ions), and are the smallest known particles of matter. 775. Rontgen Rays. — An entirely new kind of radiation emanating from those parts of a Crookes tube which are acted on by the cathode rays was discovered in 1896 by the German physicist Rontgen. This radiation, called Rontgen rays or X- rays, is strongly sent out from an oblique plate of platinum on which the cathode rays are converged, as in the tube shown in figure 466. The Rontgen rays are detected by photography, as they act powerfully on an ordinary dry plate, and also by their power to excite fluorescence. They are not deflected by a magnet nor refracted by prisms or lenses, and are but feebly reflected. They 530 ELECTRODYNAMICS have a remarkable power of penetrating substances of small density, such as wood, pasteboard, or flesh, which are opaque to light. They also penetrate thin sheets of metal of small densitj^, such as aluminum, while they are screened off by lead or platinum. A fluorescent screen made of pasteboard covered with fine crystals of barium platino-cyanide or calcium tungstate will Fig. 406. — Runtgen-ray tube. glow brightlj' if brought in front of a Rontgen-ray tube in a darkened room. If the hand is interposed between the tube and the screen, the flesh, being most easily penetrated by the rays, will show but faintl^y while the shadow of the bones is strongly marked. If a photographic plate enclosed in the usual plate holder with slides of hard rubber, wood, or pasteboard is sub- stituted for the fluorescent screen, a photograph is obtained on development such as is shown in figure 467. Fig. 467. — Radiograph of hand. 776. Tbe Nature of Kontgen Rays. — There has been consider- able uncertaintj- as t,o the nature of the Rontgen radiation, though the fact that these rays are not deviated by a magnet shows conclusively that they cannot consist of a stream of charged particles like the cathode rays. But the recent discovery by Laue that diffraction effects (§935) are CATHQDE RAYS 531 obtained when a pencil of Rontgen rays is passed through a thin plate of crystal, and the studies of the reflection of these rays from crystal surfaces by W. H. and W. L. Bragg, which grew out of Laue's discovery, have made it seem quite certain that this radiation is like light but having wave lengths which in some cases are less than one-thousandth the shortest visible wave lengths of ordinary Ught. It is found that the radiation from an X-ray tube is generally quite complex, being made up of radiations of very different wave lengths, the shortest being the most penetrating. 777. Characteristic X-rays. — In an ordinary X-ray tube where the rays are produced by the convergence of cathode rays upon an oblique plati- num target there are certain wave lengths in the radiation which are especially energetic. If the target is made of some other metal the wave lengths given off are different from those given by platinum, and in general it has been discovered that each element, so far as it has been possible to test the matter by experiment, when subjected to bombardment by cathode rays gives off X-rays of certain special wave lengths, designated by Barkia, their discoverer, the characlerislic X-rays of that substance. The investigations