xv •ei'-ii'-c PHYSIC ^■^■Hl3UI3aQtaMgu»gmm!»WIllTiMIT1in i;»l>iyTimHlt lllH>a?lfla»»t WHaHlH H»WHII fe^' Physics 1-5. Course in Univ. -t ^' . L-ri-t_i _ _ Instructor. CORNELL UNIVERSITY LIBRARY FROM S.H.B-arohani Cornell University Library arV17318 Lecture outlines. 3 1924 031 303 997 olin,anx The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031303997 LECTURE OUTLINES Physics Course 1 NOTES FOR THE VSE OF STUDENTS IN COURSES 1 AND 5 WRITTEN BY J. S. SHEARER 1911 "" CORNELL UNIVERSITY CARPENTER & CO. ITHACA, N. Y. COPYRIGHT f9II CARPENTER & CO. INTRODUCTION These outlines give, approximately, the topics treated in a lecture course intended to cover the essential principles of experimental Physics. The exact nature of the experir ments chosen to illustrate the various concepts is not given and the outlines are to be regarded only as a general indi- cation of the points illustrated. It, is important to note the fact that our entire knowledge of Physics is based directly upon experimental evidence to which reference must con- tinually be made as the subject advances. A considerable part of the work required of the student consists in tracing the connection between the so called physical laws and the experiments from which they are deduced. Some portions of the subject have been briefly treated in the notes and questions in the latter part of the book. Numbers in the outlines refer to the articles in the notes and their study constitutes a part of the required work. The rough notes should he taken on the blank pages in this' book which is to be handed in when required for inspec- tion. A verbatim report of the lectures is not expected and is not even desirable. The stiident's notes taken during the lecture should show the essential features of the apparatus used in demonstration, and as soon as possible after the close of the lecture the articles should be studied and the perma- nent lecture notes written up. Should it be found desirable to give the lectures in an order different from that indi- cated, make rough notes as near the corresponding headings as possible. The final notebook should give a systematic ac- count of the subject as presented in the lectures and not be simply memoranda of separate, and apparently unrelated, experiments. 4 NOTE TAKING. The notes following the lecture outlines are to be studied as a part of course 1 and each student will be pre- sumed to be familiar with their content. The problems in the back of this book will be assigned in connection with course -5. Some good textbook may be read with advantage in connection with these courses. Some of the following may be found useful. Magies' Principles of Physics. Carhart 's College Physics. Carhart's University Physics. Watson's Physics. Ames ' Theory of Physics. Edser's Heat. Maxwell's Heat. Edser's Light. Larden's Electricity. Righis' Modern Theory of Physical Phenomena. Crew's General Physics. Nichols' Outlines of Physics. Reed & Guthe's Physics. TAKING NOTES. To save time and writing is the object of these outlines. Properly used they give a basis for both rough and perma- nent notes. Take outline sketches of the essential features of the apparatus and copy all diagrams given on the board. Keep rough notes parallel with outlines, if the lecturer does not follow the order as printed look up the proper place and put rough notes where they belong. If permanent notes are written up promptly only a small amount of writing need be done during lectures. N. B. Bead the introduction before taking and writing up your notes. 8 TU.^^^ .SURFACE TEmON.^^-t, p. Forces Due to Motion of Matter in Contact. • Static friction. -'-^^ v.,-y;^- r->-.^^-- ' Moving friction. Increase of friction with pressure. p , Movement on an inclined plane. /*^ ' 'p ~ ^^ . Resolution of forces. Determination of the coefficient of friction. Determination of critical angle. Force diagrams for uniform motion with friction. ■ Friction in flow of liquids and gases. Surface Tension and Capillary. 18. General properties of fluids. Conditions determining shape of the free surface of a liquid. Elevation or depression of liquids at rest at sides of con- taining vessel. .. Rise of water in capillary tubes, and between glass plates. Experiments showing that a liquid behaves as though its surface were an elastic membrane under tension. ' The formation of drops. No such tension on the interior of a liquid. Contractile tendency of soap films. Relation of tension to curvature and consequences of this peculiarity. Uniformity of surface tension on a flat surface. Varies with the nature of the liquid. Liquid Pressure. 16, 17. Difficulty of compressing a liquid. Transmission of pressure by liquids. ATMOSPHERIC PRESSURE. 9 Demonstration of Pascal's principle. The hydrostatic press. Archimedes principle and its use in determining densi- ties. Principle of the hydrometer. Atmospheric Pressure. 19, 20, 21, 22. Magdeburg hemispheres. Demonstration that atmospheric pressure acts in all directions. Methods of measuring gas pressure. The barometer. Conditions for accurate barometric readings. Manometers and pressure guages. The weight of air and of hydrogen. Correction for weighing bodies in air. Properties of a mass of air in rapid motion. Air pumps. Advantages of various sorts. Diifusion of fluids. Molecular motion. Flow of gas in a tube. Variation of pressure and velocity. Heat. Sensation of warmth. The heating of one body by another. Temperature. 23, 24. "When a body A is able to increase the warmth of an- other body B, A is said to be of higher temperature than B. Sensation not an accurate method for measuring tem- perature. 12 HEAT. Fusing points for crystalline substances. Agreement in temperature of solidification and fusion. Absorption of heat without change of temperature dur- ing fusion. Under-cooling of liquids. Change of volume on fusion or solidification. EfEeet of pressure upon melting point. Heat of fusion. Vaporization. Conditions which determine the boiling point of a liquid. (a) Effect of pressure change. (b) Effect of impurities. Vapor pressure. Boiling and freezing points of water may be brought to coincidence by rapid reduction of pressure. Gas Laws. 34, 35. Conditions under which a gas may be liquefied. Three quantities which determine the physical state of a body. Other transformations. Methods of Heat Transfer. 37, 38. Conduction of heat. Difference in conducting power of various bodies for heat. The spheroidal state. Convection currents in liquids and gases. Transfer of heat by wave motion. Radiation. Higher velocity of energy transmission than in conduc- tion or convection. MAGNETISM. 13 Radiation from a flame to a thermopile. Radiation from a vessel of warm water. Radiating power of different surfaces at the same tem- perature. Increase of radiation with rise of temperature. Difference in transparency of glass, hard rubber and other substances to different radiations. Reflection of heat. Production of High and of Low Temperatures. Combustion of certain substances. Electrical heating. Measurement of extreme temperatures. Freezing mixtures. lee and salt. Solid COj and ether. Cooling by expansion of gases. Refrigerating processes. Production of liquid air. Refrigerating capacity of liquid air. Construction of vessels to prevent vaporization of liquid air. Properties of matter at extremely low temperatures. Lowest temperatures yet attained. Liquefaction of hydrogen and helium. Magnetism. 39, 40, 41. Peculiar force. action in certain minerals and in pieces of iron or steel which have been subjected to special treat- ment. Selective action of this force. Behavior of a piece of iron or steel which retains its magnetism, i. e., which is constantly able to exhibit this force action. Difference between the ends. 14 MAGNETISM. Peculiar behavior of different portions of magnetic ■bodies upon each other. General idea of poles. Behavior of a magnet free to move and not near other magnets. Magnetic properties not necessarily dependent upon the amount of material involved. Magnetic moment. Method of measuring magnetic action by balancing against other forces or by means of the vibration of mass. Law of magnetic force. The Magnetic Field. Experiments showing that in the space surrounding a magnet, magnetic forces are developed in suitable materials. Rapidity with which this ability to develop magnetic force decreases with increased distance from magnet. Study of the field of a small magnet by the use of an extremely small suspended magnet. Magnetic field of the earth. Use of iron filings in giving an approximate map of a magnetic field. Production of a magnetic field through the use of an electric current. Case of a straight wire and of a coil of wire. General properties of lines of magnetic force. Should be considered as closed curves extending through the magnet itself. May be supposed to tend to push each other apart and to become as short as circumstances will permit. Experiments showing the impossibility of separating the poles of a magnet. Use of lines of force to indicate strength of a magnetic field. MAGNETISM. 15 Induced Magnets. Soft iron in the neighborhood of a powerful magnet acquires magnetic properties. These properties more or less completely lost by re- moval from the field. Magnetic action pr,oduced by the proximity of perma- nent magnets or by electric currents spoken of as induced magnetization. Conditions for high magnetization. Effect of tapping steel in the field. Reduction of magnetization by jarring steel when out of magnetic field. Change in a magnetic field by the introduction of pieces of soft iron. ^explanation of the statement that iron is a better car- rier of lines of force than air. Experiment showing combination of various magnetic fields into a resultant field. Case of the field due to a magnet in combination with a striaght wire carrying electric current. Molecular Theory. Brief statement of the molecular theory of magnetism. Magnetization and demagnetization by jarring a piece of iron in the field of the earth. High molecular activity produced by bringing iron to a red heat renders the iron non-magnetic. (Note. The magnetic behavior of iron is to be dis- cussed more in detail after the study of the electric current and in connection with principles of the dynamo and the motor. 16 ELECTRICITY. Electric Current. 42. Two things essential to the production of an electric current. Generator and circuit. Generator may be regarded as any agency capable of producing electrical separation. A conducting body capa- ble of allowing the movement of electricity between the terminals of the generators completes the circuit. Electric current always produces a magnetic field and more or less heat energy. It may produce also chemical or physiological action. Any of its effects other than the physiological may be used for the purpose of measurement. Most convenient means of measurement is by utilization of magnetic action. Study of the magnetic field due to a wire and its modi- fication when the wire is wound into a coil. Behavior of a freely suspended magnet at any point within a coil carrying a current. Magnetic effect does not depend upon the size of the wire or the heat developed, but only upon the magnitude of the current. A steady current in a wire is not modified by bending the wire into a coil of any shape. Practical method of measuring current by its magnetic effect. Galvanometers. 43. Simple tangent galvanometer. Field along the axis of a circular coil of wire. Force diagram for the magnet of a tangent gal- vanometer. Proper position for tangent galvanometer. Means of changing the sensibility. ELECTRICITY. 17 (a) Position of coils. • (b) Variable number of turns in use. (c) Change in value of controlling field. Instead of a moving magnet and a fixed coil of wire we .may use a fixed magnet and a movable coil balancing the twisting effect of the magnetic fields upon the coil against the torsion of a wire. The simplest forin of D'Arsonval galvanometer. Diagram of a field tending to twist the coil of a D 'Ar- sonval galvanometer. Portable D'Arsonval galvanometers as ammeters. Advantages and disadvantages of two forms. Heat Effect of an Electrical Current. 44. Dependent upon the nature of the conductor carrying the current. Series of wires of different materials and diameters show unequal heat development. Electric lamp used to show that a slight change in the current brings a considerable change in heat developed. Determination of the amount of heat developed by a given current passing through a given resistance. Resistance. Conditions which determine the resistance of a wire. Change of resistance with temperature. Primary standard of resistance. Secondary resistance standards and resistance boxes or rhesotats. Analogy between an electric current and the flow of water in a system of pipes. Correspondence of pump pressure and electromotive force. 5 18 ELECTRICITY. Frictional resistance and electrical resistance. Electrostatic forces shown by an ordinary lighting circuit. (a) When no current is flowing. (b) During the passage of a current. Steady change in indication of electroscope according to number of lamps between the terminals. Electrical potential. Further study of the analogy between electric and water circuits. Water pressure as related to potential. A generator, may be regarded as a device for causing potential difference. Maximum potential difference attained when no current passes through the circuit. (Note. As the study of the electric current is facili- tated by a kaowledge of electrostatics, current electricity will be further discussed after consideration of this subject.^ Electrostatics. 45, 46. Steady fall of potential with resistance in a simple circuit as shown by the electroscope. Demonstration showing that a circuit carrying an electric current is electrically charged. Transfer of charge from a high potential circuit to an electroscope. Other means of showing the development of electro- static force. Insulated metal charged by contact with fur. Identity of force action with that produced by metal brought in contact with a 500 volt circuit. A peculiar force action by all sorts of substances under proper conditions and which differs from gravitational and magnetic action defined as electrical force. ELECTRICITY. 19 Experiments showing the general attraction of electri- fied bodies for apparently tincharged bodies. Experiments indicating the development of positive and negative charges. Fundamental law of development; equal quantities of positive and negative always produced. Law of electrical force action; like charges repel and unlike charges attract. Amount of electric force depends upon the amount and distribution of the charges and on the medium in which they are placed. Use of the gold leaf electroscope to indicate the quality of charges produced by various means. Conditions which determine whether positive or nega- tive will be produced by contact and separation of two given kinds of material not well known. Conductors and Non-conductors. "Whenever two electrical charges are separated, thereby producing a difference of potential, these charges tend to move in such a way as to neutralize each other and destroy the potential difference. Difficulty of this electrical transfer differs according to the material between the two points of different potentials. All substances allow the passage of an electrical charge to a certain extent depending very largely upon the tem- perature and physical condition and upon the difference of potential tending to urge the movement of the charge. Bodies allowing the passage of the charge in consider- able amounts with small potential differences are spoken of as conductors. Bodies which, under ordinary potential con- ditions, practically prevent the transfer of charge are spoken of as non-cohductors, or insulators. 20 ELECTRICITY. Hard rubber rod rubbed at one end shows charge only at that end. An insulated metal rod shows charge over its entire sur- face, but not distributed uniformly. An insulated metal body is capable of transferring charge to other conductors. Transmission of charge along a wire. Continuous production of electrical charge by friction machine. •Experiments showing the absence of charge on the in- side of hollow conductors. Distribution of charge as related to curvature of sur- face. Charging and discharging of conductors by points. Induced Charges. Insulated conductor in the neighborhood of a charged body. Effect of grounding at different points. Attraction and repulsion of light uncharged bodies explained. ' Method of charging gold leaf electroscopes by induc- tion. Electrostatic field of force. Similarities and differences between electrostatic force lines and magnetic lines. The action of the electrophorus. Explanation of the actiod of the Holtz machine as a continuous electrophorus. I Potential. 47. Term already introduced in connection with the trans- fer of electrical charges which constitutes a current. ELECTRICITY. 21 More exact meaning of the term in relation to work in moving an electrical charge. Choice of the earth as a region of zero potential. Analogy between potential and water level. Positive charge always tends to move, and if conditions permit, does move, from higher to lower potential. Difference between charge and potential. Bodies at zero potential with positive or negative charges. Bodies at positive potential having negative charges and the reverse. Equality of potential at all points of a conductor in which, electric charges are at rest. Ability of one body to give electric charge to another is determined in part by potential differences. Potential is dependent upon the position of a body with respect to surrounding bodies and their charges as well as upon the charge on the body itself. Capacity. 55. The amount of electrical charge on a conductor or a system of conductors at a given difference of potential is spoken of as the capacity of the conductor. Note that a body may have an electrical charge numer- ically larger than its capacity. Eelation, between quantity of the charge, potential and capacity. Various types of condensers and conditions which de- termine their capacity. Capacity not determined wholly by size of conductors and relative position, but also dependent upon the medium through which the electrical forces act. Specific inductive capacity. 22 ELECTRICITY. Explanation of condenser action on the basis of electri- cal attraction and repulsion. Method of charging and discharging condensers. Explanation of the process of gradually discharging a Leyden jar by alternate contact with coatings. Experiments showing the impossibility of charging a condenser by contact with only one coating. Effect of capacity upon the discharge of a Holtz ma- chine driven at a constant speed. Energy stored in a charged condenser. Electrical Discharges. Short duration of the illumination from an electric spark as shown by the appearance of the rotating wheel of a Holtz machine. Opposition of glass, rubber and other bodies to the pas- sage of spark discharges. Production of an electric current by the Holtz machine. Amount of current depends upon the condition of the machine and the rate at which the plates are driven and their number. Discharge of a condenser through a galvanometer. Current not continuous. Discharge of a condenser by a vibrating ball through a galvanometer. Transfer of charge along a wire caused by difference of potential as previously indicated by behavior of electroscope connected to a 500 volt circuit. Voltmeters. Measurements of potential necessary in order to control electrical circuits and to estimate work done. A more sensitive instrument than the gold leaf electro- ELECTRICITY. 23 scope is required for the measurement of difference of poten- tial in ordinary electrical circuits. Electrostatic voltmeters or electrometers explained. Note that no current passes through this instrument during use, action being entirely because of electrostatic forces between the fixed and moving parts. ' Electrostatic voltmeter used on the Holtz machine. Note failure of machine to maintain potential difference on closed circuit. Three fundamental quantities to be measured in elec- tricity. (a) Current or quantity of electrical charge trans- ferred per second. (b) Potential differences or the work per unit charge between two points in a circuit. (c) Resistance. Potential and E. M. F. 52. Distinction between electromotive force and potential difference. Study of the rise and fall of potential in circuit as shown by an electrostatic instrument. Demonstration that potential difference is equal to the product of resistance and current. Difference in ability of generators to produce potential differences at their terminals on open circuit and to main- tain potential differences when current is flowing. Use of high resistance galvanometer or other current measuring instruments as a voltmeter. Ohm's Law. 48. Statement of Ohm's law for a complete electrical circuit. Experiment showing that the resistance of generator must always be considered in current flow. 24 ELECTRICITY. Connection of generators to produce different electro- motive forces. Internal resistance. Extreme cases of generators. ';• Storage cell ; electromotive force about two volts, inter- ?| nal resistance a small fraction of an ohm and having current capacity of several amperes. Holtz machine; electromotive force many thousand volts, internal resistance extremely large, current not in ex- cess of .001 ampere. Failure of the Holtz machine to maintain potential dif- ference between its terminals when they are joined by a con- ductor explained. Danger to Life from Electricity. Current required for fatal results probably^ about one- tenth of an ampere. Resistance of the body depends upon the points, of con- tact, dryness of the skin and pressure. Seldom less than 5,000 ohms. In general it may be stated that power circuits con- nected to generators of 500 or more volts electromotive force are dangerous. Conditions under which it is safe to touch an electric circuit with one hand. Danger in case of grounded circuits. Distribution of Electric Current. 51, 52, 53, 54, 56, 57, 58. Series and multiple connection. (a) Generators. (b) Resistance. Examples in incandescent and arc lighting. Measurement of voltage and resistance. ELECTRICITY. 25 The Production of Electromotive Force. By changing the relative position of two charged plates. The mechanical separation of electric charges as in the Holtz machine. By means of chemical action charging the terminals of a battery. Unequal heating of junctions of two metals as in ther- mo- junction or thermo-pile. Currents and potentials for useful working purpose could hardly be obtained without the action of a generator depending upon another class of phenomena. Induced Electromotive Force. 60, 61. Straight piece of wire connected in series with a gal- vanometer and moved in a magnetic field, produces tempo- rary deflection provided wire moves across the lines of force of the field. Increase in effect by increasing the field strength. Increase in effect by increasing the length of the con- ductor used. Continuous cutting of the field of a magnet by rotation of a small copper disk. Dependence of the current on the speed of rotation. Effect of increasing the intensity of the field. .Small coil of wire suddenly removed from a magnetic field. Deflection of galvanometer in exact proportion to num- ber of turns if field is constant. Electromotive force developed by relative motion of a magnetic field and a series of conductors depends upon the rate of cutting lines of force and to produce an electromotive force of one volt the field must be cut at the rate of 10' lines per second. 26 ELECTRICITY. Note that production of this electromotive force giving the energy necessary to force current through the circuit re- quires an expenditure of work. We should therefore expect that force must be exerted in moving the wire across the magnetic field in excess of that expended to move the body against friction or to give it acceleration. The apparent inertia of a conductor is increased when- ever it is moved in a magnetic field and the currents pro- duced by induced electromotives forces' always brings into play forces resisting the motion. Force acting on a straight conductor carrying current in a magnetic field. Fataday's disk used as a motor. Further study of electromotive forces induced by the movement of a wire in a magnetic field. Force on a wire carrying current in such a field. Use of a magnetic model to show the various cases. Note that in order to produce motion it is necessary to have lines of force of the two fields at angles differing from 90 degrees. Dynamos and Motors. Simple dynamo; single rectangular coil rotating in a strong magnetic field. Dependence of electromotive force developed upon the strength of the field and the speed of rotation. Use of the same rectangular coil as a motor. Essential features of a dynamo or motor. (a) Many conductors capable of moving at high speed across a magnetic field. (b) Suitable methods for the maintenance of the requi- site magnetic field. Model and slides showing the construction of armatures. ELECTRICITY. 27 Necessity for the use of iron in the production of strong magnetic fields. Magnetization of Iron. 65. A bar of iron initially in a non-magnetic condition sub- jected to an increasing magnetic field by means of a current passing through a solenoid surrounding the iron. Meaning of the symbols E and B. Magnetization curve and hysteresis curve indicated. Ewing's model approximately explaining the behavior of iron in a varying magnetic field. Choice of iron for different purposes. Great lifting power in case of a nearly closed magnetic circuit. Great residual magnetization in a closed magnetic cir- cuit. Counter electromotive force developed in a motor. Least current in a motor when its speed is the highest. Work done on a motor and power delivered. Local or Foucault currents in a dynamo or motor. Kesistance to the movement of a bar of copper through a strong magnetic field. Effect of slitting the copper. Use of laminated iron in the construction of dynamos and motors. The induction motor as a case of pull in a magnetic field. Mutual Induction. 66, 67. Case of two coils of wire, one of which is connected permanently to a galvanometer while in the other an elec- tric current is started, increased, decreased or stopped. Effect of inserting iron in either or both of the coils. 28 ELECTRICITY. Relative position of the coils for which the induction is a maximum and a minimum. Principle and use of the transformer. The induction coil. Self Induction. Increased illumination of a lamp connected in multiple with a powerful magnet on breaking the exciting circuit. Reason for small spark when duration of contact of the wires carrying current to the electromagnet is very short. Alternating Currents. Transformation to increase voltage and decrease cur- rent or the reverse. Various experiments with transformers and induction coils. Effect of self induction in an alternating current cir- cuit. Oscillatory discharge. Experiment showing that with high frequency of al- ternation self induction is of great importance. Electrolysis. 68. Decomposition of sodium sulphate solution containing litmus. Decomposition of copper sulphate. Experiment showing chemical, magnetic, and heating effect of current simultaneously. Decomposition of lead acetate. Brief explanation of the laws of electrolysis. Counter electromotive force of cells. Variation with the nature of the cell. Brief explanation of the storage battery. Electrical Discharge. Electric are between carbons shown in projection. ELECTRICITY. 29 Peculiarities of the arc. Effect of a magnetic field on the arc. Arc between various metals. Difficulty of maintaining the arc in many cases espe- cially with an alternating current. Disruptive discharges in air shown with induction coil connected to high capacity. Effect of reduction of pressure of the air on electric discharge. Experiments with vacuum tubes. Experiment showing that the discharge in vacuum tubes when subjected to a magnetic field behaves like a cur- rent. Heat produced by discharge in vacuum tube. Brief reference to the ionic theory of electricity. Cathode rays and X-rays. Excitation of vacuum tubes in the neighborhood of a high frequency coil. Electrical resonance. Electrical oscillations. 30 MECHANICS. MECHANICS. Motion. Eelation between change of position and time. Terms used in describing motion, as, rectilinear, ro- tary, oscillatory. Composition of motions. Velocity. 78. Uniform linear velocity. Equal spaces traversed in equal time intervals, no change of direction. Uniform angular velocity. Equal angles turned through in equal time intervals, direction of motion of each particle constantly changing. Specification of velocity. Composition of velocities. Graphic representation of velocities. Resolution of velocities. Acceleration. 77. The number of velocity units gained or lost in one unit of time. Requires statement of amount and direction. Composition and resolution of accelerations. Uniform acceleration. The same amount of velocity acquired by the moving body each unit of time. Variable acceleration. More common than any other and average velocity is not equal to the mean of the initial and final velocities. May be readily treated by graphic methods. Production of Acceleration. 79. The agency causing acceleration in mass is named force. 32 MECHANICS. of matter to move in a straight line tends to maintain the plane of rotation of a body constant. Illustration of forces used to change the direction of moving masses. Illustra- tions of effort to maintain a fixed plane of rotation. Third Law. Forces acting between bodies always mutual and op- positely directed. Equality of action and reaction. Illus- trated by solids, liquids and gases set, in motion. Reason for small amount of motion observed in the large body when bodies are of very different mass. Experiments show- ing pull required for acceleration. Reaction force in a mass having acceleration is given by the product of the mass of the body and the acceleration and is directed oppositely to the resultant force producing the acceleration. Force diagram for an accelerated mass. Example of variations of acceleration with increased mass when the force is kept con- stant. Example of vftriable acceleration when the force is varied, mass remaining constant. Gravitational Force. 84, 85. Depends on the amount and distribution of matter and is entirely independent of the physical or chemical nature of the. mass on which it acts. Varies inversely as the square of the distance between the centers in case of two spherical masses. May be considered as constant for short distances when centers are far apart. Case of the earth and falling bodies. The behavior of a body falling approximately freely. Gravitation balance. Resolution of gravitational force. Demonstration of uniform acceleration. (a) The inclined plane. (b) The Atwood's machine. MECHANICS. 33 Demonstration of Newton's laws of motion. Composition of motions due to gravity and other farces. Projectiles. Experiment showing that amount of fall from line of projection is independent of the direction in which the body is thrown. Momentum. 86. Product of the mass of a body by its velocity. In all cases of interaction of two masses, momentum developed in one mass is identical, except as to direction, with that de- veloped in the other, hence the velocity of the smaller body is greater than that of the larger. The third laiw of motion and equality of momenta. Work. 87. The estimation of work. Only force components in the actual direction of motion to be considered. No work what- ever is done by any force whose point of application does not move. Case of a variable force. Work units. Energy. 88, 89, 91. The ability of things to accomplish work, i. e., to set masses in motion. For convenience frequently classified as potential and kinetic energy. Potential energy is the ability to do work because of relative position or state of strain. Elevation. Elastic strain. Energy of compressed gas stored but due to kinetics. Kinetic energy is the ability to do work because of mass in motion. 10 34 MECHANICS. Kinetic energy is measured by the work required to bring a moving mass to rest. Change in energy of a single mass impossible. Difference between energy and momentum. Absorption of momentum by a mass free to move. Removal of energy and force exerted. Experiment showing that a small force acting on a mass through a considerable distance may enable the mass to exert a much larger force through a corresponding smaller distance. Principle of the injector. Conservation of Energy. Statement as a useful working principle. Care re- quired in its application because of the great variety of forms of energy. Power. 90. Special meaning of the term power as used in mechan- ics and engineering. Does not refer to the total ability to do work, but to the rate at which work may be done. Measurement of power by brake method. Moment of a Force. 92. The measure of the tendency of a force to twist a body to which it is applied. Equilibrium of moments. Eesultant moment. Conditions for complete equilibrium of a rigid body. Effect of a system of parallel forces on an extended mass. Center of gravity. (a) Regular solid. MECHANICS. 35 (b) Irregular bodies. / (c) Non-homogeneous bodies. Stability of solids under the action of gravity. The balance. General principle. Conditions for stability and sensitiveness. Practical means of controlling sensibility. The balance is used to compare masses and not neces- sarily to determine weights. Rotating Bodies. 93-96. Angular velocity. Angular acceleration. The equality of angular velocity of aU parts of a rigid body at any instant. Change in velocity of the elements of a rigid body in rotation with increased distance from the center. Energy due to rotation. Potential energy of falling weight absorbed as kinetic energy of a wheel. Total kinetic energy of a rotating body is always to be fo\ind by multiplying the mass of each small element by the square of its distance from the axis of rotation and by % the square of the angular velocity. ^ Reason for naming the sum of such terms as mr^ the moment of inertia. The effect of increasing the moment of inertia without changing the moment of the applied forces. Great amount of energy in a body of large moment of inertia rotating at high speed. Behavior of spinning masses. The top. The gyroscope. The monorail car. 36 MECHANICS. Vibratory Motion. 97. By reason of elastic forces or constraint motion is re- peated. Path diminishes in size unless work is done on the vi- brating mass either continually or at intervals. Force conditions necessary to cause vibratory motion. Kelation of acceleration to displacement. Specification of vibratory motion. Type. The geometric description of the path followed. Amplitudes. The dimensions necessary to indicate the size of the path. Period. The time required to go completely over the path once. Frequency. The number of vibrations per second. Relation between the period and the amount and dis- tribution of material to be moved. Relation between the force producing motion and the period for a given distribution of material. Vibration of springs. Pendulums. Torsional pendulum. Effect of change of distribution of mass on period of a torsional pendulum. Force diagram for gravity pendulum. The ideal or simple pendulum. Relation between period of vibration and length of sus- pension of the simple pendulum. Simple Harmonic Motion. 98. Description of S. H. M. Conditions for the execution of simple harmonic vibra- tion. Combination of S.H.M. and a uniform motion. The sine curve. MECHANICS. 37 The actual or physical pendulum. The period of vibration is not dependent on the ma- terial of which the pendulum is made. Dependence of the time of vibration upon the position of the center of gravity. The Kater pendulum. Determination of g by use of the pendulum. Composition of several simple harmonic motions. Wave Motion. 99-101. Methods of transmitting energy through a medium. (a) By means of projectiles. (b) By setting up currents. (c) By means of wave motion. Characteristics of wave motion. Vibratory motion of all the disturbed elements of the medium. Type of vibration identical so long as the medium is homogeneous. No permanent displacement of any portion of the medium. The transfer of a form of arrangement of particles or a particular physical state from one point to another by means of wave motion. Successive taking up of vibration by portions of the medium. Meaning bi the term wave length. Distinction between wave velocity and the velocity of individual elements of the transmitting medium. Eelation between wave velocity, wave length and vibra- tion frequency. Transmission of a single wave along a spiral spring. Transmission of a torsional wave along a loaded wire. 38 SOUND. Reflection of waves. Various illustrations of wave reflection. Diagrams showing change of condition at reflection. Independent existence of different waves in a medium at the same time. Production of standing waves by use of a direct and a refleoted system. Three essential elements in study of energy transmis- sion by wave motion. (1) Vibratory source sending energy into the sur- rounding medium. (2) A suitable transmitting medium. (3) A receiving mechanism. Sound. 102; 103, 104, 105. Experiments showing that a vibratory source is neces- sary for the production of sound. Failure to transmit sound in a vacuum. Experiment showing that sound is not propogated by a current of air. Experiments showing that sound is hot propogated by means of projected masses of air like vortex rings. Nature of the gas motion in a sound wave. Velocity of sound in air and other gases. Velocity of sound in liquids and solids. Characteristics of a musical tone. • Relation between loudness and amplitude of vibration of the air wave when received. Relation between pitch and frequency. Character or "tone color" due to complexity of the vibrations. Meaning of the terms fundamental and dominant pitch. Experiments showing that all sounds have a dominant pitch. SOUND. 39 Limits of audition. Transmission of waves in air of frequency too high to be heard. Change in the quality of the sound of a tuning fork according to the relative prominence of the fundamental and overtones. The musical scale. Sources of Sound. Study of plates by means of sand figures. Sounding bells. Rods and bars. Nodal points in tuning forks. Laws governing the pitch of strings. Production of standing waves. Complexity of string vibrations. Reasons for variation in quality of the tone produced by' various stringed instruments. Longitudinal vibration of rods and strings. Vibratory columns of gas. Stationary waves in tube, closed at one end. Position of greatest and 'least freedom of air motion. Position of greatest and least change of pressure. (a) Open pipes. (b) Closed pipes. Use of manometrie flames. Experiment showing th§ series of tones m an open pipe as the driving pressure is varied. Difference in quality and pitch in open and closed pipes. Effect of change of temperature and nature of gas on the pitch of a pipe. Reed and flue pipes. LIGHT. 41 Light. 108-111. The approximate rectilinear propogation of light. Corpuscular or emission theory of light. Other assumptions as to the cause of light. Relation of rays to wave fronts. Light of itself invisible. Relation of wave convergence or divergence to vision. The pinhole camera as ah illustration of rectilinear propagation. Means of changing the curvature of a wave and the formation of images by such changes. Reflection of light. Reflection at a plane surface. Position of image by plane mirror. Reflection from curved mirrors. Production of images by spherical mirrors. Conditions necessary to secure accuracy in form and focus. Refraction. Explanation of refraction by difference of wave' velocity in two media. Change in apparent position of an object within a re- fracting medium. Conditions for total reflection. Refraction of white light passing through a prism. Change in effect when surrounding medium is changed. Explanation of the production of the series of colored images which constitute a spectrum. Production of deviation without dispersion. Optical Instruments. 112. Lenses, converging and diverging. Focal length of a lens depends on the surrounding medium. LIGHT. 43 Change in amount and nature of absorption with change in thickness. ' Color. 113. Means of color production. Dispersion. Absorption. . Selective reflection. Selective radiation. Interference. Color sensation variable with different individuals. Dependent upon the condition of the eye as a receiving mechanism. Three quantities necessary to specify color. (1) Hue or tint. (2) Ltuninosity or brightness. (3) Purity. Failure of the eye to judge of the purity of a color. Tints produced by various combination of colors. The Toung-Helmholtz color theory. Color mixture. Difference between color, mixing and mixing of pig- ments or combination of absorbing glasses. Persistence of vision. Color blindness. Agreement between the selection of colored worsteds by a red-blind person using white light and by those of normal vision using light lacking certain wave lengths. Color contrast. Production of photographs in color. Interference. Colors produced by thin films. 44 LIGHT. Approximate thickness of the soap film when certain tints are observed. Newton's rings. Impurity of the colors produced by the soap film. Effect of change of incident angle. Effect of change of pressure. Diffraction. ' Huyghen's principle of wave propagation. Failure of small objects to cast definite shadows. Directive action of a series of small linear obstacles in the path of a wave. Spectra produced by diffraction gratings. Mechanism required in the construction of good grat- ings. Rowland's photographs of the solar spectrum. Determination of the wave length of light. Reason for believing light to be transmitted by a medium of peculiar properties. The velocity of light. Polarization. Peculiar effect upon a beam of light by reflection from a glass surface at a particular angle of incidence. Detection of this effect by second reflecting mirror. Peculiar action of thin plates of tourmaline upon a beam of light as depending on the relative position of the two plates. Comparison of the effect produced by tourmaline with that produced by reflection from a glass mirror. General meaning of the term polarization. Rough illustration of plane and of circular polariza- tion by means of a vibrating string. Reasons for believing that light vibrations are trans- verse to the line of propagation. LIGHT. 45 Double refractioil. Production of two beams by refraction from a single beam in certain crystals. Polarization of each of these beams as indicated by failure of a mirror to reflect the light under certain condi- tions. Introduction of a second crystal producing either four beams of light or only two, according to relative position. - Means of producing a single beam of plane polarized light. Reason for only one transmitted beam in case of. tour- maline. The Nlcol prism. ' Effect of passing light through two Nicol prisms. Conditions for complete extinction. Effect of introducing a thin crystal between two Nicol prisms. Explanation of the production of the interference col- ors observed. Dependence of these colons upon the thiclmess and po- sition of the crystals and the distances traveled by the beams of light in the crystal. Applications of polarized light. Testing optical instruments for strain. Glass strained by sudden cooling. Strain by unequal pfessure and by uneqjial^ temperature distribution. Polarization of light by passing through a thin film qf rubber under tension. Rotation of the plane of polarization by quartz and by sugar. Use of rotary power in the' determination of the amount of sugar present in a transparent solution. 46 LIGHT. w Phosphoresence and Fluorescence. General statement of the extent of the spectrum. Tendency to name certain regions according to the mechanism used in their study. Extremely long waves studied electrically. "Wave lengths considerably shorter studied by heating effect. Wave lengths between certain limits producing visual sensation. Evidence of waves much shorter than those giving luminous effects producing actinic or chemical action and capable, in certain cases, of being transformed into lumin- ous radiation. Production of luminous effects in regions beyond the visual spectrum. Excitation of fluorescence by radiation from certain sources. Ability of a body to fluoresce dependent upon its tem- perature. Distinction between phosphorescence and fluorescence. NOTES. 47 NOTES. The following notes and questions have been prepared in order to give to students in Courses 1 and 5 some practice in connection with their lecture work. Care has been taken as far as space would permit to call attention to the methods used in applied physios. A clear grasp of the fundamental phenomena and of the relation of things is an essential part of the elementary work and details of design and manipulation are left for development in later courses. 1. Introduction. The fundamental concepts of mechanics, although deal- ing with all the materials and movements in our daily life, always present difficulty to the student. This is in part due to our easy familiarity with the words used and partly to the fact that our actual experience has not usually been such as to enable us to discriminate as to the causes of phenomena. There is a wide range of phenomena having a mechanical explanation for which at least a limited knowledge of mechanics is absolutely required. Our concept of mass is commonly that of bulk, as a "mass" of iron, meaning a piece of unusually large size or an unshaped piece, a massive structure, etc. In physical science this word does not refer to dimensions or bulk, but to the difficulty experienced in starting or stopping the body. Any material thing has this quality of resisting our efforts to change its motion. And the "mass" of the body refers to this quality from which we infer the amount of matter with which we are dealing. The notion of force is likewise one which presents difficulties because of our com- mon use of the word in ways not. permissible, in the more exact Avork of applied science. The exertion of- muscular effort as pushes and pulls gives the initial concept of force. We are inclined to estimate the magnitude of the force by reference to our sensation of effort which is not a very good basis for comparison. When we regard the effect of the 48 NOTES. force as a means for its specification, we are confronted with a great variety of possible methods. We may find what amount of certain mass standards (pounds, kilograms, etc.) the force is able to sustain against the puU of the earth, or a "weight" measure. "We might use the force to stretch a spring or otherwise distort an elastic body, a strain measure of force. There is a third way not dependent on position or condition of any ma- terial, but based on the resistance of mass to change of velocity and may be mentioned as the dynamical, measure of force. This will be discussed more in detail later. The third concept which has come to have most im- portant use in all branches of science and to which our un- trained experience gives us little useful information is that of work or of energy. "We have a more or less well defined notion that the exertion of muscular effort is work which is not the case unless some thing is moved by reason of the force. To hold the arm extended in a horizontal position for some time would result in fatigue and discomfort, but no work would be done. The time during which a force is exerted does not have any necessary connection with the work done. To measure work we must determine in some units, the force and the distance through which the force moves something which opposes it. If the pull of the earth on a pound, of mass is chosen as the unit force and the foot as the imit of length, the product, pounds, tveight X distance in feet is named the work done in foot-pounds. No matter what units we choose to measure the two quantities concerned, work is their product. When any body or system of bodies is able to exert force through distance, i. e., to do work they are said to NOTES. 49 iave energy. Bodies above the earth's surface, water at high level, coiled springs, compressed gas, etc., are able to set up motion by force exertion and have energy. A mov- ing mass is able to force its way for a greater or lesser dis- tance against opposing forces and thus has energy. For convenience the latter is named kinetic energy, the former potential energy. Careful measurements of the energy involved in any mechanical change indicates that whatever energy is lost by one inass or system of masses, an exact equivalent appears in the remainder of the masses concerned. By no means can a mass do work without loss of its capacity for working and the mass on which- work is done increases its ability to do work. This is known as the law of the "conservation of energy," one of the most important generalizations iii scientific work. The manifestations of energy are so varied and its transformations so complicated that it is often difficult to include all changes in any given case, but such problems are of fundamental value in the study of energy. To illustrate the meaning of the terms used, consider a boy drawing a sled up hill. By means of the energy stored as the result of digestive processes, he exerts force aga,inst his own and the sled's weight. This force is in exact pro- portion to the mass raised. The force multiplied by the vertical height raised gives the work done. A small part of this is lost by friction but the greater portion is stored as potential energy. On sliding down hill, this potential is changed to kinetic energy and is ultimately lost in heat by friction. 2. Constitution of Matter. Numerous experiments have indicated that matter of every kind is made up of minute portions to which the 50 NOTES. name molecules has been given. For example, a molecule of water is an amount so small that if divided it would no longer be water, but two molecules of hydrogen and one of oxygen. Most physical properties of materials depend on the forces acting between molecules. Solids are characterized by forces returning molecules to their original position when slightly displaced, giving rigidity. "When these forces are less pronounced, the body may be plastic or viscous and with more molecular freedom we have the liquid state. 3. Molecular Motion. It seems to be clearly established that molecules of every known mass are in incessant vibration through minute distances. "When the vibrations of the molecules are sufS- ciently active, the forces holding them together are over- come and we have the gaseous state of matter characterized by a tendency to indefinite expansion. 4. Density. The 7nass of unit volume of a substance is named its density. When the mass of unit volume of water is taken as the mass unit, density and specific gravity have the same numerical value. To say the density of iron is 7.5 means that 1 cu. cm. of that iron would weigh 7.5 grams or any volume of it would weigh 7.5 times as much as the same volume of water. 5. Graphic Methods. Much of our knowledge in all branches of applied physics is deduced from a comparison of the values of re- lated quantities. Experiment enables us to tabulate these values as a series of observed quantities from which the required relations may be found. The problem is made much easier by the use of certain graphic methods which NOTES. 51 have a wide application not only in physics and engineering, but in all departments of science and in economics. "When a series of values of one quantity depend on those of another their relation may be exhibited to the eye by the use of a diagram constructed as indicated in the following example. The depth of water in a stream varies with time. Suppose we take a sheet of paper ruled into small squares and choose two lines perpendicular to each other as lines of reference. We may represent one hour by any convenient number of units measured along one of these lines and depth of water in feet by units along the other. Then erecting at each time point a line representing, to the chosen scale, the depth at that time, we secure at a glance what would require considerable study from tabu- lated data. Joining the points marked with small circles in Fig. 1, we have a fairly accurate picture of the depth variation provided large changes do not occur dm-ing the time be- tween' observations. A pencil operated by a float and Ft 8 ^«~~* 1^^ 6 i / \ / V 4 (J / % \ ,,,-— — * y ^"—^ 2 1 i I : 3 % t 5 12 M Hours Fig. 1. 52 NOTES. writing on a sheet moved uniformly in a horizontal direc- tion would give the complete record. Figures 2 and 3 show the application of this method to "temperature measurements. In 2 we have a record of the temperature effect of sunshine and in 3 that of the working AM 7 8 9 10 1 1 12 1 2 3 4 5 6 7 PI CAULENDAR RECORDER s ^^^^ , , ^ -- V ? .^ / \ t. Q J \ y r 1 / \ 5 lU / ^ A o J V ■^ Fig. 2. !300 (\ J \ 1250 / \ A / / 120 / \ / \ r\ / \ f \ f\. ilso^' \ / \ V \ / VJ \ \ I v> \^~y J no \ / \ 105 J \ ) 12 1 2 3 A 5 6 7 8 Fig. 3 NOTES. 53 temperature; of a furnace. Such records are of great use in the control of operations and often give much valuable in- formation. For example, the points of maximum and minimum temperature are clearly indicated and the heat conditions prevailing may be readily deduced. We may expect in any actual case certain peculiarities in the curve, depending on the physical relations which determine it, and, conversely, any peculiarity, as a maximum or minimum, change of curvature, asymptote, etc., will usually have a physical meaning. Every change of temperature requires a certain time interval, so that no portion of the time -temperature curve can be. vertical. Time never decreases, and tempera- ture has only one value at a given instant, so there are nO' "loops'." or multiple points in such a curve. The student should not rest content with simply draw- ing the curve but should endeavor ito associate the changes or peculiarities in form with the underlying physical condi- tions. If familiar with the methods of analytic geometry and calculus, these may be applied to their study. In particular, if the curve is a graphic representation of a general law, he should note whether all portions of the curve have an actual physical interpretation; whether the physical condition indicated by certain portions of the curve could be realized; if it cuts the axis, what the intercepts mean; whether the direction of the tangent line at any point has a physical interpretation ; does the area of a given portion represent some physical quantity, etc. "When it is drawn from observed values, the relation between the co- ordinates may often be expressed as an algebraic equation, either from its general appearance or from a knowledge of the physical law involved. 54 NOTES. 6. Elasticity. When a body is able to regain its original form after it has been distorted by a force it is said to be elastic. This has no reference to the amount of stretch or other distortion it may stand without rupture or to the amount of force that must be used in order to produce the distortion. In the case of hard steel or iron, ivory, glass, etc., the return is very complete if the change of shape is not too great while in the case of rubber any measurable stretch results in an elongation that is fairly permanent but may gradually be lost in part. In the use of many materials it is of great importance to know that the breaking point is not closely approached. As this point is beyond the elastic limit the determination of the stress at which the body iegins to indicate a permanent distortion is a matter of general interest. The elastic modulus showing the ratio of the stress to the strain is then an indicator of the elastic behavior of the material and so long as the maximum load does not cause a variation in the modulus concerned there is no danger of rupture. The actual strains in structures are very compli- cated and in the ease of machinery it is not possible to pre- dict with certainty what the actual strain may be in all eases or whether the material is entirely homogeneous so that a fair allowance must be made for safety. If the estimated actual strain is only one-third of that which would corres- pond to the elastic limit the factor of safety would be three. Permanent distortions without rupture, such as kinks in saws, etc., result from strains beyond the elastic limit, but below the breaking point. 7. Stress. The ratio of force applied to a body to the area on NOTES. 55 which it acts is defined as stress. When force per unit area is constant we have a uniform stress. A force of 50 lbs. weight applied to an area of 1 square inch would be a stress of 50 lbs. per square inch. If applied to .1 square inch the stress would be 50/.1 = 500 lbs. per square inch. 8. Strain. If the application of a force to a body produces a change in its dimensions or its volume the ratio of the change to the original value of the thing changed is called the strain. A wire 100 cm. long when increased in length .1 inoh. would have a strain of .1/100 = .001. While the same in- crease in length for a wire 10 cm. long would be .1/10 = .01. For elementary discussion it is assumed that a given stress will cause only one kind of strain. For example a stretching force causing no contraction, a compression caus- ing no expansion at right angles to the line of pressure, etc. This is not exactly true in any case and is far from correct for extreme strains. 9. Elastic Limit. When a highly elastic body is deformed it returns to its original shape or volume on the removal of the stress only when the strain has not exceeded a particular value ; which depends on the nature of the body and its physical state as regards temperature, previous treatment, etc. This limit- ing value is named the elastic limit for the substance. It does not follow that the body could not stand a greater strain without rupture but rather that it will be permanently deforiped if this limit is passed. The break- ing point, is in many eases, rather close to the elastic limit. 56 NOTES. The relation of stress to strain, so long as the elastic limit is not exceeded, is expressed by Hooke's Law, -viz. stress -f- strain = constant. The value of this constant de- pends on the material. F/A-^aL/L = K Force X length ^ LF =: constant, AL = „ . — Area X change in length ^-^ Or within the elastic limit distortion is proportional to force applied. The quantity K is called the modulus of stretching or Young's Modulus. When pounds weight, inches and square inches are used as units the following are approximate values of K for some common substances : K Steel 29,800,000 Wrought Iron 29,600,000 Cast iron 16,000,000 Glass 9,600,000 Oak ■ 1,450,000 10. Stress and Strain Diagrams. Elasticity is one of the physical problems which are simplified by the use of diagrams showing the relation of two or more quantities. Thus if we take two lines at right angles as in Fig. 4 and plot the force applied to stretch a bar as lines measured parallel to one axis and the resulting elongation parallel to the other, we have a force-stretch dia- gram. Had force/area and elongation/original length been plotted, we would have had a stress-strain diagram. PiQi = force applied for elong. OPi P2Q2 = force applied for elong. OP^ NOTES. 57 y ^ ^ . B e^ / G V I ^ So long as QP/OP is constant, the line "joining successive Q's is straight and the elastic limit has not been reached. What does the portion AB signify ? "What difference in the angle QOX if the rod were easier to stretch ? 11. Work Diagram. The work done in stretching or otherwise distorting an elastic body involves a variable force. On stretching a spring the force is in proportion to the elongation. Fig. 4 enables us to compute the work as follows : for an increase of elongation P^Pj the force varied from PjQi to P2Q2 the average force was % [PiQi + P2Q2] ^^^ this multiplied by P1P2 gives the work in changing the length by P1P2. But PiQi P^Q. X PiQi^areaPiPjQaQi Hence the area between the force-displacement curve and the axis along which displacements are plotted is the work done in displacement. This sort of diagram will be met many times during the course and should be clearly understood. 15 58 NOTES. 12. Elasticity of Fluids. Fluids have no elasticity of form as they simply con- form to the containing vessel. They have volume elasticity since they regain their original volume after compressing forces have been removed. Liquids give very great oppo- sition to compression, so that in common speech, they are designated as "incompressible," but this is not strictly true. In the case of water the ratio force change of volume . , . .; r IS very large as compared area ' original volume ' with the same quantity for iron. The name pressure is given to force per unit area. The compressibility of a fluid is given by change of volume fi= — 7—. — -, -. X pressure original volume For water at room temperature p = .000047 approximately, wh^n the pressure is measured in atmospheres (15 lbs. weight per inch) . Thus one cubic inch of water under a pressure of 30,000 lbs. per sq. inch would contract .000047 X 1 X 2000 cu. in. = .094 cu. in. 13. Elasticity of Gases. While solids and liquids differ widely as to elastic be-' havior all gases behave very nearly alike. When the pres- sure on a gas is increased, its volume decreases. If the tem- perature is kept constant the product of pressure and vol- ume remains constant. Since temperature must be consid- ered, further discussion will be deferred until the work in heat. MECHANICS. 59 14. Friction. The force required to produce uniform motion of one surface sliding upon anotlier is just equal and opposite to that produced by molecular action between the surfaces. Any increase in normal pressure increases the friction in the same proportion within wide limits. Friction is a force always directed in opposition to the existing motion. It never acts to produce motion but is called into play when outside forces attempt to move one body in contact with another. Work done against friction appears as heat and is measured by the product friction force X distance moved. This distance is always measured on the rubbing surface. Coefficient of friction is the number which multiplied by the normal pressure between surfaces gives the force required to just maintain motion at uniform velocity. The force diagram for a mass on an inclined plane when the inclination is such that when started the mass will eon- r\ fc A. ^^ fe^ Ap ^^^ A ^^\ Fig. 5. tinue in motion without change in velocity is shown in Pig. 5. In this case we have to deal with a force partly compen- 60 MECHANICS. sated by the pressure between the plane and the mass. While due to its weight, the mass would fall along OA it can only move along OD. But the force OA might be re- garded as equivalent to a force OB and a force BA = OD. OB is annulled by the plane pressure 00, and if OD is just equal to the friction force OF, the motion will be uniform. By trigonometry OB weight X sin normal pressure coef. of BA weight X cos ^ moving force friction 15. Friction Work. Friction always absorbs energy from moving bodies, this energy goes to increase irregular vibrational energy of the molecules of the bodies in contact, and is no longer available. The work is computed by taking the product, pressure be- tween surfaces X coef. of friction X distance one surface slides on the other. Example : A normal pressure of 100 lbs. is exerted on the rim of a wheel 3 ft. radius, coefficient of friction .2, making 16 revolutions per second. The working force equals .2 X 100 lbs. weight; distance movgd per second equals 27r X 16 X 3. Hence work per second, or power, equals 20 X 96 X i- ft. lbs. per second. "When a liquid or a gas flows in a pipe work is done against friction between the moving fluid and the pipe walls. This results in a loss of pressure or ' ' head. ' ' 16. Force System for Immersed and for Floating Masses. When a body is wholly or partly immersed in any fluid the loss of weight is exactly equal to the weight of the fluid actually displaced. The student is advised to draw carefully the force system in such problems, bearing in mind that this loss of weight may be considered as a force directed ver- MECHANICS. 61 tically upward. It is also convenient to remember that if a body has a density d, its loss of weight in water is the d ih part of its weight in vacuum. Since the loss of weight in air is extremely slight, we may say, in general, that if a body has a specific gravity of 3, it will lose one-third of its weight in air when immersed in water, and two-thirds of its weight when immersed in a liquid whose density is 2. In a liquid of density 5 the loss would be five-thirds of its weight in air or a buoyant force equal to two-thirds of its weight must be overcome to keep it submerged. Example. In Fig. 6 we have a body volume = 3, 8 ^ 5, in liquid 8 = 2. The force diagram for equilibrium is shown at the right. T LiauiD 6=2 Force. Diagram T=9-G-Wr. 6g-Wt. 15-g-Wt. Fig. 6. 17 Hydrostatic Pressure. The peculiar force system between the walls of a con- taining vessel and a fluid or between the surface of a solid immersed in a fluid and the fluid is commonly spoken of as a hydrostatic pressure. Its most important characteristic is that it acts at right angles to the surface at every point and fi2 MECHANICS. is simply a push. Its amount depends on the density, of the fluid and the distance of the point considered below the free surface of the fluid. In the case of a variable density such as the atmosphere the pressure is still equal to the weight of the column of fluid above the point in question. The pressure of the atmosphere might be produced by a sea of mercury, about 76 em. deep all over the earth, or by a sea of water about 76 X 13.6 cm. in depth. If the air were all liquefied it would form a sea a little deeper than such a sea of water. 18. Surface Tension; The term tension is often applied to a force of the na- ture of a pull. Although molecules of liquid are much more easily disturbed than are grains of fine sand, yet when sand runs through a small opening the grains scatter and faU as individuals while a liquid forms in drops. When water is placed on an oily surface, it forms flattened globules more nearly spherical as their size is reduced. The general be- havior can be compared to that of a fluid in a thin rubber bag whose surface is stretched by the liquid. Or we say that Fig. 7. along the surface of a liquid there acts in all directions a molecular force which we name surface tension. Its numeri- MECHANICS. 63 cal value is the pull perpendicular to a line one unit in. length drawn in the surface. . When a liquid wets the surface of a solid, the adhesive force, or pull between molecules of solid and liquid exceeds pulls between the molecules of liquid themselves. The liquid surface adjusts itself so that at the point of contact of the edge of the liquid film with the solid surface the pulls are in equilibrium, giving a definite angle of contact. Fig. 7. 19. Manometers. The pressure due to the weight of a fluid at any point in the fluid is found by multiplying the density of the fluid by the vertical distance between two horizontal planes, one Fig. 8. passing through the given point and the other through the highest point in the liquid which can be reached by a path entirely through the liquid. The forces in a U manometer 64 MECHANICS. closed at one end and containing liquids as shown (Fig. 8) are: (1) Gas pressure down on surface of liquid at A, balanced by an equal liquid pressure acting upwards at all points in the A level. (2) At A' we have downward gas pressure Pj due to the surrounding atmosphere, and a liquid pressure h8g. As liquid at A' does not move, P,= P.^+h (J g, if Pj and d are given and h is observed Pi may be computed. 20. Atmospheric Pressure. The pressure of the atmosphere is about 15 lbs. per sq. in. [equivalent to a column of Hg 30 inches or 76 cm. deep] at sea level and decreases with altitude but not in a very simple ratio. The total force exerted on the side of a building by the atmosphere is very large and is ordinarily balanced by the air pressure inside. If the outer pressure is suddenly reduced, the building must sustain a large out- ward pressure. Thus if one wall were 20 X 50 ft. a reduc- tion of 2 inches in the barometer would reduce the external pressure 1/15 of 15 lbs. per sq. in. and if no air escapes from the inside the force thrusting outward is 20 X 50 X 144 = 144,000 lbs. weight. 21. Weight of Gases. For purposes of comparison, gas densities must he given at the same pressure and temperature. At 0°C. and normal atmospheric pressure (760 mm. Hg.) dry air weighs .00293 grams per cc. A liter of air weighs 2.93 grams. A room 4 m. X 5 m. X 6 m. filled with air at (0°, 760 mm.) would contain a weight of 400 X 500 X 600 X .00293 = 351600 grams or 351.6 Kg., about 775 lbs. of air. MECHANICS. 65 22. Emergency and Pressure in a Flowing Fluid. Any fluid in motion may have energy of three general kinds [neglecting heat] : (1) Kinetic energy of moving mass, (2) Potential energy due to elevation, (3) Energy due to compression under pressure. Neglecting friction these three energies may shift in relative proportions as a given mass of liquid moves from one region to another. Thus in the ease of flow along a pipe whose axis is horizon- tal type (2) would be constant. In the tube shown in Fig. 9, the mean flow line is regarded as horizontal. If the flow Fig. 9. is steady as much fluid must pass one point as another. Since the section at B is less than at A or C, we must have a higher velocity at B than at either A or C. Or the kinetic energy per unit mass of fluid is greater at B, hence its pressure-volume energy is less i. e. pressure at B is less than that at A or at C. 23. Temperature. Most definitions of temperature are unsatisfactory al- though we have a more or less definite idea of the meaning of the term. The statement that when a body A is hotter than another body B heat will pass from A to B is a useful 66 HEAT. working idea. Temperature is measured by some physical property which changes when the body involved is heated or cooled. Solids and liquids increase in volume when heated, gases either expand or increase in pressure, or do both ac- cording to conditions. The electrical properties of bodies, their ability to give out or absorb light and heat and many other properties change with temperature. "We may also assume that if a body is simply heated or cooled and re- turns to its initial condition it will regain its original vol- ume. The volume of a given mass of m,ercury at the tem- perature of melting ice is constant and at the tem,perature of boiling water is increased iut on bringing it repeatedly to either of these temperatures it attains the volume charac- teristic of that temperature. Measurement of extreme temperature is complicated because of radical changes in the bodies used for measure- ment. For example, mercury freezes at a moderately low temperature ( — 39°) and boils at a temperature much be- low those used in practice. Alcohol freezes at a much lower temperature but its boiling point is below that of water. For very high or very low temperati^res no determination based on the expansion of solids or liquids is practical. By the use of suitable gases we may measure from the lowest observed temperatures up to about 1100°C. Above this electrical and optical methods, only, are applicable. 24. Temperature Scales. In the sense in which we use the term measurement in other cases, we cannot measure temperatures, i.e. we can not set aside a standard amount of temperature and use it as a measure, any more than we can measure color or loud- ness. "What we actually do is to subject some substance to the temperature to be "measured" and note the value of HEAT. ' 67 some physical quantity which changes with temperature, then by reference to some arbitrary scale, a number indi- cating relative temperature is found. Thus if we choose the temperatures of melting ice and of boiling water at 1 A. pressure, as fixed points of reference and the expansion of mercury as the means of measurement, we may divide the interval between ice temperature and steam- temperature into as many parts called degrees as we choose. If we call ice 0° and steam 100°, the change in volume is said to cor- respond to 100°, and 1° corresponds to 1/100 of this stan- dard change. Formulated this becomes, where V == volume * 100 * 1° = 100 If the physical change is not constant with temperature, the degrees will be unequal in various parts of the scale. The scale mentioned above is known as the centigrade and is more convenient than the Fahrenheit, in which the ice point is named 32°, the steam point 212° so that the change in th& thermometer from ice to steam temperature is divided into 180 parts instead of one hundred. No matter what physical change is chosen for the comparison of temperatures, the scale is found in the same manner. 25. Thermoelectric Thermometer. "When two wifes of different materials as for example iron and constantan, platinum and platinum-rhodium are connected tO' form a loop, any difference in temperature of the junctions develops an electric current which is able to exhibit a small force in a properly designed instrument. This force increases and decreases with rise and fall of the temperature of one junction if that of the other is kept constant. The effect of the force on the indicator may be shown by reflecting a beam of light from a mirror turned 68 HEAT. by the force to a scale. This device is a great convenience in many ways as the relative temperature can be shown and the sensibility can be easily controlled. Low and moder- ately high temperatures can be measured in this way, by the latter named couple up to 1600°C. 26. Heat. Heat is to be regarded as kinetic energy due to molecu- lar vibratory motion. While it is convenient to measure heat by reference to a definite change which may be pro- duced by heat action on a specific substance, it is to be borne in mind that such a unit of measurement may be expressed in terms of energy units. The calorie is defined as the heat necessary to raise the temperature of one gram of water one degree centigrade. Strictly, the amount of energy required to make the change varies slightly with the temperature of the water, but this variation is so small that, for practical purposes, we may regard the water as having an initial temperature anywhere between zero degrees and the boiling point of water at that particular pressure. Heat energy is frequently used as follows : (1) To change the temperature of material. (2) To change physical state as from solid to liquid or liquid to gaseous. (3) To cause chemical action. (4) To produce electrical energy. The law of conservation of energy should be remem- bered in the solution of all problems involving heat changes. It follows from the law that the entire amount of work ex- pended on a system of bodies must be accounted for by changes in that system whether mechanical, thermal, or of any other nature. If none but strictly thermal changes are considered, we then say that the total amount of work given HEAT. 69 to the system is equal to the heat used in making the various changes. Any loss of heat energy is always exactly equal to the energy of various kinds which appears in other forms. And all other forms of energy show a tendency to change to heat during a transformation. 27. Heat Constants. "When any rearrangement of the molecules of a mass or change in their activity occurs there is an absorption or evolution of energy in the form of heat. (1) If only a rise or fall of temperature takes place the vibratory energy of the particles only is changed. (2) If a rearrangement with no change of tempera- ture takes place only energy in displacement of molecules against molecular forces is involved. There are many changes, due to heat, the more important constants for our purposes are the following: — • The specific heat of a substance is the number of calories required to raise the temperature of unit mass of the material 1°. The heat of fusion is the number of calories required to melt unit mass of the suhstance without changing the tem- perature of either solid or liquid. The heat of vaporization is the number of calories re- quired to vaporize unit mass of liquid without changing the temperature of either liquid or vapor. The heat of combustion is the number of calories ob- tained by burning under specified conditions unit mass of a substance. In the above cases no change of temperature is consid- ered except when heat is used for temperature change ex- clusively, since these and many other physical and chemical changes of state take place at fixed temperatures, i. e. are isothermal. 70 HEAT. Hence, heat used to change temperature = mean spe- cific heat X no degrees rise in temperature X mass of ma- terial involved. Heat used to change state = appropriate constant X mass of material changed. As an example of the application of the above facts, suppose 500 g. of molten lead at 400° and 50 g. of ice at — 20° to be brought together in a vessel impervious to heat. Find the final condition if the specific heat of the molten lead is .04. Melting point of lead 326° Heat of fusion 5.4 cal. Specific heat of solid lead 03 Heat of fusion of ice 80 cal. Specific heat of ice 5 Assume that ice is melted and raised to t°G. Heat Supply. Cooling molten lead to melting point 500X.04 (400-326) cal Solidification of lead 500X5.4 cal. Cooling of solid lead 500X.03X (326 — t) cal. Heat Used. Baising temperature of ice to melting point. . 50 X -5X20 Melting ice 50X80 Heating v^ater to t°G 50X^ .-. 500(2.96 + 5.4 + 9.78 — .03* = 50(10 + 80 X 0- 29.6 + 54 + 97.8 — .Zt = 90 + i. 91.4 = 1.3*. .-. * = 70°. + . 28. Had too little lead been present to melt all the ice, how may one compute the amount melted 1 Had so little ice been present that in cooling to 100° more heat would be HEAT. 71 given up by the lead than would be required to bring result- ing water to 100° what would be the final result? 29. Specific Heats. The plausible assumption of increased kinetic energy of the molecules in the explanation of the work required in heating a mass does not quite cover the facts of the case. The expansion requires work against molecular forces within the mass and the outside pressure has to be over- come through some distance to accommodate the increased volume. In the case of a gas the first of these is very small but the latter is large on account of the high expansion coeffi- cient of gases. Worli per gram per degree rise in temperature = increase in molecular kinetic energy -\- work done in pulling molecules further apart -\- work done in ■pushing back the atmosphere or containing walls. 30. Two interesting practical cases may be mentioned. A gas may expand, do work and not fall in temperature provided heat is supplied exactly equivalent to the work done. This is called isothermal expansion. In the other extreme case no heat passes through the enclosing walls in either direction and if external work is done it is at the expense of the internal heat energy with fall in tempera- ture. This is named adiabatic expansion. (Note that com- pression is regarded as negative expansion) . 31. In computing the work done by or on a gas in ex- pansion or compression the graphic representation of the relation between pressure and volume is essential both in practical and in theoretical work. 72 HEAT. Consider a gas enclosed in a cylinder as in Fig. 10 the piston being assumed to move without friction. The gas exerts a pressure of p force units per unit area throughout the inside of the cylinder. The total push on the piston is' t ^HHi H H tlP^f^^r^ v^^^V}tt\V I « 3— t-— -t-^-4 ' I « 3— i As an example: Consider the circuit shown in Fig. 20 and the corresponding diagraia Fig. 21. Effective E.M.F. Current =- Rise in voltage Total resistance = % ampere. Portion of circuit A— B B— C 7 C— D D— B E— F Fall in voltage 4x1/4 = 1.00 3X1/4 = 0.75 6x1/4 = 1.50 2+4X1/4 = 3.00' 3X1/4 = 0.75 Total "Volts 7.00 96 ELECTRICITY. 53. Electrical Energy. The computation of electrical energy is based directly on the concept of electrical potential. The measurement of the actual force encountered during the transfer of a given electric charge from one point to another is entirely imprac- tical except in a few extremely simple cases. Since electric potential is readily measureable we reach directly the work per unit charge due to the transfer. Energy = charge X change in potential, = QXV. "When Q is transferred at a steady rate by conductors and I is the measure of the current then Q ^ I X time, since cur- rent is charge transferred in one second. Thus energy =^ IV t. Power = Energy / time =^ IV. When this energy is used to develop heat exclusively, I = V/R, or V = IE. Power expended in heating = P R. When counter or opposing E.M.F.'s are overcome in the cir- cuit Power =: IV = I X drop in potential due to resistance +1 X Counter E.M.F. or. Power = IXIR+IB = PR+IE. The power used to overcome opposing E.M.F. appears as chemical energy in some cases (viz. storage batteries) some- times as electrical potential energy (condenser charge) and as mechanical energy in others (motors). ELECTBICITT. B •^ WWWWWVVVWV : ' Fig. 22. 54. If in Fig. 22, G is a direct current generator, B a storage battery, M a motor and the resistance in the circuit is distributed as shown, how is energy used ? Power developed by generator = IXE.M.F. of gener- ator. Power transformed to heat ^ P(ri+r2+r5+I^) ; Power transformed to chemical form = IBj. Power transformed to mechanical form = IE3. All measured in watts. 55. Condensers. When electric energy is stored as cnarge in a condenser we measure the final potential between the coatings and the final charge. The final potential stands for the work done per tinit charge when the last element of charge was trans- ferred. Since charge and potential are proportional, twice the average work per element of charge was done on the last •element moved, and the work is the same as would be re- •quired to carry the total charge through one-half of the final potential. Energy in condenser = % Q^ = % I t V where I = average current while charging. Charging a condenser may be considered as analogous to pumping water into an open vertical stand pipe. The pump used can force water in until the "head" H due to water pressure is equal to P. Or the potential difference 98 ELECTRICITY. between condenser plates is finally equal to the applied E.M.F. A series of stand pipes of different cross-sections would require different amounts of water to fill to the same level or head. We might, if we were unable to measure otherwise, rate the capacity of the reservoirs as the quantity of water contained when iinit pressure was exerted by the pump. Electrically, capacity is amount of charge when potential. difference between plates is unity. (1 volt) . B_^ ■t- + + 'HI Fig. 23. If (Fig. 23) the pressure exerted by the pump is twice- as great twice as much water would be needed to maintain equilibrium [or no flow either way] or quantity = capacity X potential, Q = CV. The work required to fill such a vessel with water to a; given height would increase with the capacity and with the- head, or work = total weight of water X average height lifted, and h/2 = average height. 56. Electrical Measurements. In electrical measurements it is desirable as far as pos- sible to use means which do not depend on the calibration of" instruments, whose constants are apt to change. No volt- meter or ammeter can be relied upon to retain its calibration: permanently when subjected to jars, changes in tempera- ture or to variable and powerful magnetic fields. ELECTRICITY. 99 In the "null" or "zero" methods dependence is placed upon the constancy of resistance ratios which are very re- liable if properly constructed. The galvanometer or volt- meter is then only used as a sensitive means to determine when no current flows in a portion of the circuit. Two important illustrations of zero methods may be mentioned. The Wheatstone bridge for measurement of resistance and the potentiometer for comparison of E.M.F.'s. Both of these depend on the inability of charge to move when there is no dijference in potential between points joined by a conductor. Such lack of potential difference might be due to opposing electromotive forces or the divi- sion of current between conducting paths. 57. Wheatstones Bridge. The Wheatstone bridge for the comparison of resist- ances is illustrated in Fig. 24 and consists of four resistances Fig. 24. and two lines connecting to a battery and a galvanometer. No source of B.M.F. is included in any of the four resist- ances AB, BC, AD, DC. 100 ELECTRICITY. The current divides at A part passing along ABC part .along ADC. The total drop in potential along one path is exactly equal to that along the other. There may always be found a point in each path having any given potential be- tween that at A and that at C, and, for any potential in one branch a point having the same potential may be found on the other branch. If B and D are such points then no cur- rent wiU flow through G. The potential drop along AB == 1^& == IjC = drop along AD The potential drop along BC = IiX = Ijb :=: drop along DC Ija IjC ch — = — or a; ^ — . lib IjX a Exactly the same principle would apply to the water circuit of Fig. 25. If a gauge at A should indicate 40 lbs. Fig. 25. per sq. in., at B, 12 lbs. per sq. in. the pressure drop along each pipe is 28 lbs. per sq: in. Nowhere between A and B is the pressure above 40 or below 12 if there is no pump in the pipes. As all values between 40 and 12 occur on each path only once, a pipe joined to one at C could be joined to some point D, in the other of the same pressure and hence no flow from C to D. ELECTRICITY. 101 58. Comparison of E.M.F.s by the Potentiometer. If Pj, Fig. 26, is a pump forcing water along BA and Fig. 26. thus creating a difference of pressure between A and B of hi — hj, this pressure difference from B to A, may be con- trolled by adjustment of the frictioh f and F. If the branch circuit APjB is connected at B and A, and contains a pump Pj applying pressure from right to left, there will be no cur- rent in AP2 if P2 = hi — ■ h;,. In Fig. 27, we have the same principle in the electrical -mmmmmMmHnmmmmmmmmmr-^ 10000 Ol-IMS R ' I — Nmmmm- HH + Fig. 27. case. The main circuit battery driving current from A to B causes a fall of potential from B to A. The shunt circuit with a cell B shows no current when Vd — Va^E. 102 ELECTRICITY. But Vd — VA^main current X resistance between D and A. ==I X V. For another cell E^ the corresponding resistance might be Vj, E V Then -r" = —r. if the current between B and A has not E, V; been changed. 59. Magnetic Properties of Currents. The general properties of the lines of force of a mag- netic field may serve as a basis for the discussion of the phenomena of indu^ced currents and the force action be- tween currents and magnetic fields. Two statements are especially useful: (1) That lines of force parallel and in the same direc- tion are always coincident with a mechanical force of repul- sion between the masses with which they are associated. (2) The general tendency of these lines is to become as short as possible. In addition to the facts regarding magnetic fields de- veloped in the treatment of magnetism, it is essential to recognize that when a long, straight conductor carries an electric current there are invariably lines of magnetic force in the form of closed circles which lie in planes perpendicu- lar to the wire. On the other hand, if lines of force are made to encircle a conductor, a tendency to displace electricity will he caused in the conductor; i. e. an E.M.F. will ie pro- duced. The direction of current and the positive direction of the lines of force due to it are related to each other in the same way as ai-e the direction of motion parallel to its length of a right-handed screw and the direction in which it is turned. Or, if we imagine current to fiow from the eye to a clock-face, lines of force around the current would ELECTRICITY. ■ 103 be such that a + pole would go around it like the hands of the clock or ' ' clockwise. ' ' 60. Induced E.M.F. Relative motion of a conductor and a magnetic field which causes a transfer of magnetic lines of force from one side of the conductor to the other (or so that the conductor cuts the lines) always produces an E.M.F. in the conductor. This lasts only so long as the cutting continues. The magnitude of the E.M.F. in C.O.S. units is nu- merically equal to the number of lines cut per second. "When this rate is variable (as is usually the case), we may find the average E.M.F. during a time t by dividing the entire number of magnetic lines cut by t. Since 1 volt = 10^ C.G.S. units, the average E.M.F. in volts, is, number of lines cut per second The student should note carefully the various ways in which this motion niay be produced, among which may be mentioned : (1) Movement of a conductor across a stationary field. (2) Movement of permanent magnets near conductors. (3) Increase or decrease of current through conductors especially when these are wound on iron cores. (4) Movement of iron in a magnetic field near con- ductors, or any combination of these. In (8) when a current increases we are to regard mag- netic lines as thrown out in closed curves surrounding the conductor. On decreasing current these lines move in to- ward the conduct6r with decreasing perimeter. The rate at which a change in magnetic field moves in space is equal to the velocity of light. 104 ELECTRICITY. 61. Illustrations. (a) A straight wire 10 cm. long is moved perpendicu- lar to a magnetic field having 1200. lines per sq. cm. at a velocity of 300 cm. per second. "What E.M.F. would be produced ? The area swept over by the wire in 1 second is 10 X 300 = 3000 sq. em. Through this area there pass 3000 X 1200 lines of force or 36 X 10° all of which are cut. E = 36 X lOyiO' ^ .036 volts. Fig. 28. (b) A rectangular coil 15 X 30 cm. of 40 turns makes 1800 R.P.M. about an axis perpendicular to a magnetic field of 9000 lines per sq. cm. During each quarter of a ro- tation 1 turn cuts 450 X 9000 lines. 40 turns cut 450 X 9000 X 40 X 30 X 4, per sec. Fig. 29. E = 194.4 volts. ELECTRICITY. 105 Fig. 29. It is very important to note clearly the dual nature of electro-magnetic phenomena. All relative motion of a conductor resulting in cutting a magnetic field causes an E.M.P. and requires mechanical force in order to maintain the cutting. If the E.M.P. acts on a closed circuit the force demanded increases with the current. The mechanical work needed because of the magnetic field = electrical energy developed = IV. "When an E.M.P. forces current through a movable con- ductor, that, moving through a magnetic field cuts lines of force, it requires electric energy to overcome the oppos- ing E.M.P. due to the motion. Or a motor is at the same time an opposing dynamo. 62. Self-induction. When the applied E.M.P. or the current in an electric circuit changes in amount, there are brought in play electro- motive forces which do not exist during the passage of a steady current. Such an E.M.P. is said to be due to self- induction. These E.M.P. 's become of great importance in the utilization of alternating currents. No matter how the/ electric circuit is designed it invariably has a certain amount of resistance, self-induction, and capacity. Self-induction is due to the fact that a certain magnetic field must be established no matter how simple the circuit may be. The 18 106 ELECTRICITY. magnitude of the field so developed depends upon the shape of the circuit and the distribution of the masses of iron in its neighborhood. While the capacity of most ordinary circuits may be considered as small, nevertheless it becomes of importance, inasmuch as a definite quantity of charge is necessary to bring the various parts to the potential differ- ences required for operation. "We may look upon the flow of currents through the circuit as being caused by the im- pressed E.M.F. and; as being opposed by resistance and counter-electromotive forces. The E.M.F. , brought into play by reason of self-induction, is always proportional to the rate at which current is changing at that instant, since the rate at which the lines of force are being cut is directly proportional to this change, and its direction is such as to always oppose the change. If L volts are developed when the current changes at the rate of one ampere per second, L is named the coefficient of self-induction. If the rate of WS^ H) Fig. 30. change of current was one hundred amperes per second, the counter E.M.F. would be 100 L. The E.M.F. due to induc- tance is a maximiim at the instant the current is changing the fastest. "Why? The E.M.F. due to capacity is greatest at the instant that the difference of potential between the parts concerned is a maximum. Why? The simplest and most important application of the above facts is in connection with E.M.F. 's which vary ac- cording to the harmonic law and are consequently repre- sented by sine or cosine functions. • ELECTRICITY. 107 Illustration: In Fig. 30, G is an alternating current generator,, which is in series with an electromagnet and a condenser. State the various E.M.F.'s involved. Where are they located? To what are they due? Are they con- stant or variable? Do they reach their maximum at the same or at different instants? 63. Counter E.M.F. In a circuit containing an electric motor, we must re- gard the rotating armature as cutting magnetic field lines and thereby tending to produce a current in opposition to that driving the motor. This tendency to drive current against the generator is spoken of as the counter electromo- tive force of the motor, and depends upon the rate at which magnetic lines of force are being cut by the conductors in its armature. The power delivered to the motor is the product of the current passing through it and the difference in potential at its terminals. Part of this Vork is wasted in heating the armature, and the remainder is utilized in forc- ing the armature to revolve against resisting forces. It follows at once from these considerations that in case the motor was doing no work its rate of rotation would be very high, the drop in potential at its terminals would be large and the current very small. (^- -wwvwwvi (wwywvwv Fig. 31. 64. All except very small motors must be started with a resistance in series with. the armature. This starting resistance prevents burning out the arma- 108 ELECTRICITY. ture by excessive current while the counter E.M.F. is low. The resistance is gradually, cut out as the motor increases its speed. Pig. 31 shows connection for starting a shimt motor. 65. Magnetization in Iron. The practical transformation of mechanical energy to electrical and the reverse is dependent on the peculiar mag- netic behavior of iron. The introduction of iron into a magnetic field increases the number of lines of force. This increase depends on the field strength and on the quality and previous treatment of the iron. The relation between magnetizing field and the total number of lines per square centimeter when iron is present is shown by the magnetization curve in Fig. 32 for a par- ticular sample of iron. The statement B ^ ju H means that if there were H 'a 00 Ft P n L^ I6t 50 ^4 ' m M A t,ti on 1 Hfi to J m ^ v-l f- t-\ 1 ^ > f; oil -} PC -^1 m ) 'V 1 ■ -lii Y)l / 1 / - fr( m 1 / / / -f on to '/■ u ^ -/i Ji » Fig. 32. ELECTRICITY. 109 magnetic lines per sq. cm. of surface through a plane per- pendicular to the field when iron were absent, there will be fi times as many when iron is introduced into this field. In the case of iron /* is a variable quantity always greater than unity for values of H used in practice. The production of a magnetizing field is accomplished by passing an electric current around the space where the field is desired. The field increases with the increase of cur- rent and with the number of times the current goes around the enclosed region but decreases, with an increase in the length of this space. This is indicated by the reliation, field strength inside a coil = .4 tt X current in amperes X num- ber of turns -4- length of solenoid. Illustration : A solenoid is 100 cm. long, has 600 turns of wire 'and carries a current of 5 amperes. The field inside this coil if no iron were present would be .4X3.14X600X5 100 = ''■'' lines per sq. cm. If iron having a permeability of 300 for this magnet- izing field were inserted, B = 37.68X300 = 11304. If the iron has a cross section of 5 sq. em. the total magnetic flux (no. of lines) = 11204X5. 66. Mutual Induction. A variable current through one coil of wire varies the magnetic field through a neighboring coil. If these coils are properly placed, an induced E. M. F. will be observed in the second coil which may have no metallic connection with the first coil or any generator. This action of two coils upon each other is named mutual induction. Iron in the first coil increases the magnetic field surging out or in. 110 ELECTRICITY. iron in the second localizes or conceiitrates the field in that coil. 67. Transformers. A transformer usually consists of two separate coils of wire having a common iron core. This core may either be a straight piece (open magnetic circuit) or a closed ring or yoke (closed magnetic circuit). By mutual induction a variable current in one will cause an induced variable cur- rent in the other. As every turn of the one operated from a generator (primary) sends out magnetic lines cut by all the turns of the other (neglecting "leakage") the B.M.F.'s will be in the ratio of the number of turns. Since the en- ergy in the secondary comes from the primary, its power can not exceed that of the primary, at best IN V N L X V = I^ X V^ or — = — since — = — I. N. V N., where the subscripts p and s refer to the primary and sec- ondary currents and voltages. By this means it is possible to change the relative values of the two factors by which a ^given amount of power is indicated, increasing voltage with a reduction of current or the reverse. This enables power to be transmitted by alternating current at high voltage and sJnall current with consequently reduced heat loss and to change to moderate voltage and large current at or near the place of utilization. 68. Electrolysis. When current passes through a metal there is no effect except development of heat and a magnetic field. When solutions of certain salts are used as conductors, there are changes of a chemical nature in addition to the ordinary thermal and magnetic effects. This action is named electro- ELECTRICITY. Ill lysis and such conductors electrolytes. These processes have become of great commercial importance in manufacture as well as in plating and in storage batteries. The decomposition of chemical compounds by the elec- tric current has also acquired great significance by reason of its relation to the recently developed notion of the atomic nature of electricity. If current is passed through solutions of Hj SO4, Cu SO4 and Nag SO4 it will be found that hydrogen, copper and sodium all tend to separate from the SO4 and to go in what is called the + direction of the current. The laws of electrolysis would indicate that the SO4 is associated with — electricity and that H and the metals are associated with '-j- electricity. Since equal amounts of + and — are always produced by any electric disturbance, if we assume that SO4 always carries the same — charge when dissociated, then, Hg, Cu and Naj must carry the same + charge. Taking one atom of H as the standard electrical carrier, the amount of H needed for the transfer of a given charge [current X time] would always be the same. It would require only half as many atoms of copper to accomplish the same transfer since 1 Cu. atom carries as much as 2 H atoms. But Cu atoms are heavier. The mass of copper needed for any given transfer of charge would then be made up of % as many atoms as would have been required if hydrogen had been used ; but each atom is 63 times as heavy as an atom of H. or mass of Cu = mass of H X 63/2. The electrochemical equivalent of any metal is the m,ass transferred by 1 ampere flowing for 1 second. Or the mass per coulomb. For H this number is .00001036. For copper in a solution where the valence of copper is 2. the electro- chemical equivalent is .000329. Why? 112 ELECTRICITY. 69. Electrical Oscillations. If water is held at different levels on two sides of a U tube by some kind of a removable plug, when set free the water does not simply come to equality of level on the two sides, but the fluid overruns the equilibrium position and the final state is reached through a series of oscillations of rapidly decreasing amplitude. If the tube be very small so that the friction is considerable this does not occur, but the fluid settles to the final position with gradually decreasing velocity, the original energy of position being rapidly changed by friction into heat. "When the two plates of a charged condenser are joined by a conductor of low resistance, the charges on the plates alternate in sign, the potential reducing rapidly with each alternation. The original energy of stored charge in part sets up a magnetic field through the current passiag be- tween the plates. When the plates reach a common poten- tial or are discharged, this magnetic field has not disap- peared and the return rush of these magnetic lines con- tinues current in the original direction. Thus the plate which had an excess of -|- charge has more than that excess removed. If the resistance between the plates is high, the current is small and no considerable magnetic field exists when the plates reach the same potential. Since these vari- able currents have magnetic fields oscillating with the same frequency, oscillating electric charges are induced in outside circuits. These effects are inappreciable except where the natural frequency of the receiving circuit is the same as that of the sending arrangement. 70. Directed Quantities (Vectors). Physical phenomena are largely dependent on a class of quantities which are associated with the idea of direction. MECHANICS. 113 ¥ov the complete statement of any of these we must first ex- press their size as compared with some standard, like quan- tity taken as a unit, and second, specify in which direction they act. The familiar idea of motion from one point to .another may be regarded as typical of this class of quan- tities. Suppose, for example, one asks the way from one place in a city to another. The answer might be given in terms of a series of successive distances in specified direc- tions as 1 mile north,. 2 miles northeast, etc., or one might be told to go a certain distance in a specified direction without turns. All of the things associated with motion as velocity, acceleration, force, etc., require for their complete speeifica- "tion the same two quantities of amount and direction. A given change of position may result from an indefi- nite number of individual changes subject only to the con- dition that one of these must start from the first point and one must end at the last point in question. The final result of the separate individual motions can be expressed by stat- ing its direction with reference to some fixed line. The single directed quantity which would produce the same result as the succession of individual ones is spoken of as the resultant of the series. TMs resultant is to he found hy lay- ing off the individual quantities, as straight lines end to end and joining the initial point with the end of the last step used. The name vector (i. e., carrier) is usually applied to this class of quantities and the resultant of a system of vec- tors is spoken of as the vector sum of the components. When only two vectors are considered, it is quite common to speak of vector parallelogram, as, for example, the parallelogram of forces, etc., but it is much more convenient to simply ■draw the vectors end to end forming a polygon, the closing ■side of which is the resultant required. It is entirely im- material whether these vectors are all in one plane or 114 MECHANICS. whether they are in various planes. The term "vector sum" is only used to express the common sense fact that a given point may be reached by an infinite variety of paths. Example : A boat is moved 9 miles due S. by the wind, 4 miles N. W. by a current and 12 miles N. E. by its engines. Taking any convenient scale the diagram is shown in Fig. 33. 71. Not only is it convenient to use diagrams to show the resultant of a system of velocities, forces, etc., but the xm- derstanding of phenomena is often facilitated by regarding these quantities as actually consisting of components in MECHANICS. 115 particular directions. A case of great general usefulness is the resolution of a force or motion into two components at right angles to each other. Thus a motion, force, velocity or acceleration indicated by a line 50 units long, N. 30 de- grees W., may^ regarded as equivalent to 25 units due West and 25 V 3 units due North. A simple triangle drawn to a scale gives a sufficient approximation to these value^ and is to be used in the solution of problems given in this connection. A great variety of applications of the idea of directed quantities will be found in the notes which follow. 72. Computation of Vector Sum. The computation of vector sums is often required and may be carried out either by graphic methods or by the use of very simple trigonometry. w D. ^ y ^ ^^ /" N \ / ^ \C. ^^ y' / 1 / ! / z CO GO / / 1 / / f / \(P / A. ^._ — — B.BC COS

> irvnssma^ T[ wm//////m/f///im////m//ff/m///ff/m////i/m/mwmm//mmmm/mMimmiii Fig. 41. For uniform friction the stretch of the spring is constant in case (a) and is a force overcoming one represented to scale by OA. When accelerated, an increased stretch, always in proportion to the product mass X acceleration is observed. This increase of tension could be caused by a static force equal to mass X acceleration directed from right to left as AB. This force required for the acceleration is entirely in- dependent of the direction of the acceleration and must be applied in addition to the force needed to overcome forces of friction, weight or those due to constraints. 82. Mass and Weight. The common use of the word mass is not always con- sistent with its scientific meaning. A mass of iron is used to refer to an unorganized or unshaped aggregation of that material. A "massive" building refers to a large or })ulky structure which may or may not contain much material. The volume of a wooden block may be the same as that of a solid block of lead but their difference in mass is apparent at once when one attempts to increase or decrease their velocity. MECHANICS. 125 lyiass in physics refers to the amount of reaction shown to accelerating force. Gravitational force is proportional to mass in any given locality. When we purchase things by weight, we do not think of getting weight which is force, but rather mass which is determined by comparing gravitational forces on the stuff purchased with that on standard masses agreed upon for the purpose. Units of mass are arbitrary and have been developed according to custom and convenience (or inconvenience). The pound, the ton, ounce, dram, the gram, etc., (have be- come customary in certain cases. They all serve their pur- pose as far as interchange of commodities are concerned well enough, but are not at all equally convenient in the study of related properties of bodies as they are not cor- related with the volume of any standard substance. The gram, as a mass unit, is the mass of 1 ee. of water. And as densities are referred to water as a standard substance, mass per unit volume and density become numerically equal in the C.G-.S. system. 83. Units. Considerable difficulty is always experienced by en- gineering students in physics owing to an unfortunate dif- ference in the units commonly employed in scientific work and in engineering or applied science. The force unit in physics is based directly on the fundamental property of mass, i. e., its inertia or resistance to any change in motion. On this basis, if the foot, the pound and the second were chosen as the units in mechanics, unit force would change the velocity of one pound of matter one foot per second each second. "When the gram and centimeter are units of mass and distance the unit force is named a dyne, and a dyne will accelerate 1 gram 1 cm. per see. each sec. The engineering 126 MECHANICS. force unit is the pull exerted by the earth on a mass of one pound. This pull wiU give tp a mass of one pound a velocity 32.2 times as great as the unit force defined above. Since, no matter how definitions are constructed, the facts of na- ture remain the same, the increase in velocity in a mass in- creases with the force applied and decreases with the amount of stuff accelerated. The engineering force unit having been chosen as a force giving unit mass an acceleration of g [32.2 ft. per sec. per sec] the number representing mass is divided by g and is called mass. Where G is a number expressing weight 9 in lbs. or other units. This has the effect of making g appear in all equations as a divisor in engineering while in physics it appears as a multiplier on the other side of the equation. For example in Pig. 42 in physics the equation would be written (neg- lecting friction) I 80 L bs. -■0- 4- m 5 0/.BS Fig. 42. Total mass accelerated X acceleration = active mass X g. Or [50 + 80] a = 50 g. In engineering, 50 + 80 a = 50. MECHANICS. 127 Note that the computed acceleration is the same in each method. To change from physics equations to engineering divide by g. There are two objections to the engineering method. It seems to suggest a dependence of quantity of matter on gravity, and, it connects gravitational- force with the masses not directly active because of gravity. The difficulty of accelerating [50 + 80] lbs. above does not depend on g, but the moving force is due to g acting on 50 lbs., i.e., g enters in engineering equations just where it does not, logically, be- long. See also Church, ' ' Notes and examples in mechanics, ' ' p. 76. Art. 72a. Also Art. 43. 84. Motion Under Uniform Acceleration. . If the acceleration of a mass on the incline. Fig. 43, is A and it starts from rest at time t ^ 0. At the end of 1 second its velocity will be A. It will only have moved a distance A/2 while it ac- Fig. 43. quires this velocity as its average velocity was only one-half its final. Newton's first law would indicate that had the plsuie suddenly become level or g ceased to act it would have continued in motion with a velocity A. This velocity ac- quired during the first second will then account for a motion over two of the spaces A/2 during the next second and the continued action of g produces its effect exactly as during the first time interval. The total distance during this see- 128. MECHANICS. ond interval will then be 3A/2. The velocity at 2 = 2A which would cause a displacement of 4A/2 during 3d in- terval and the continued action of g will add A/2. The student should tabulate these distances and deduce the re- lation between time and total distance and between the num- ber of the interval and the distances during that interval. The most common motions having nearly constant ac- celeration are due to the force of gravitation. The rate at which velocity is acquired in free fall is such that observa- tion is difficult. "We may readily prevent the full action of gravity by the introduction of some constant opposing force of smaller magnitude. Suppose a mass M on a smooth, inclined plane BA, Fig. 44. The real applied force is the weight of the mass Fig. 44. M = Mg. But Mg might be replaced by any two forces which represented by lines to scale would form a triangle with Mg as a side. Since, the plane is smooth the only force it can exert on the mass is one perpendicular to their con- tact point and this neutralizes force normal to AB. The mass in Pig. 44 will have an acceleration in the di- rection BA numerically equal to g sin $ while the accelera- MECHANICS. 129 tion in free fall is g. So we may reduce the acceleration by making AB more nearly parallel to OA. 85. Acceleration with Friction. ' Force diagrams with friction should be drawn with the friction force always directed against the motion. Fig. 45 shows a typical force diagram with friction. Fig. 45. Mg. = weight of M. Mg is equivalent to a force (1) parallel to the plane and a force as shown by the perpendicular to the plane. (1) = Mg sin *. Normal pressure == M.g cos $. Friction force, \2) = coef. of friction X Mgr cos $. (3) = reaction against acceleration = Ma. T = pull required to lift against gravity, to overcome friction and to produce acceleration. ' Note that as 4> is increased Mg sin $ increases and the friction force decreases. If the gravity component parallel to the plane becomes equal to the friction force, the body would slide down the plane with uniform velocity, if once started. 130 MECHANICS. This occurs when M.g cos * X coef . of friction = Mg sin 4>. or coef. of friction ^= tan $c where *c is called the critical angle. The critical angle is larger the greater the friction. For engineering imits divide through by g. 86. Momentum. Force action being always mutual between at least two bodies, the effect of the force must show on each, either as strain or as acceleration of each mass as a whole. For some purposes, it is convenient to define quantity of motion in such a way as to include amount of mass and velocity. Ex- periment shows that the proper measure of the mutual effect of the force on this basis is found by takiag the product, mass X velocity, to which is given the name momentum. The force acting between two masses produces the same momentum in each. Thus if the subscript ^ labels mass and velocity acquired by reason of the mutual force on one body, ^ the corresponding quantities for the other. Then M^Vi = M^V^ Note that equality of momentum does not mean equality of energy. 87. Work. "Work is expressed as the product of average force in the direction of motion and the distance actually moved. The name given to the unit of work depends on the units chosen for force and for length. In the solution of problems involving work the student should analyze the question carefully, bearing in mind that : (a) Forces applied to points at rest do no work. (b) Where force is not constant throughout the mo- tion, either an average value of the force must be used or MECHANICS. 131 the force at each short path element must be multiplied by that element and the sum of such results computed. (e) Only the force components in the 'direction of motion contribute to the work. Note the agreement of the definition of mechanical work with our ordinary conceptions. Transportation costs in proportion to the product of mass, hurry, and distance j i. e., work equals the product of mass, acceleration and dis- tance or force times distance. The C.G-.S. imit of work is the erg. An amount of work done by 1 dyne moving its point of application 1 cm. Lifting 1 gram 1 cm. requires 980 ergs. In gravitational units work is expressed in foot-lbs. or in Kg. meters. 88. Energy. The amount of work required to change the condition of a body or a system of bodies from one state to another is called energy. Thus the work required to give a mass a cer- tain velocity is the energy which the mass has by virtue of its increase of velocity. The work used in stretching a spring is the energy of the spring due to elastic stretching. AH energy due to motion is spoken of as kinetic, while that due to relative position is called potential. The kinetic en- ergy of a body is the work done in giving velocity to mass. Since work is measured by force times distance. at' m mv' E^ = mad = ma — = — a^f = 2 3 2 The potential energy of a system depends on the forces called into play in changing from one state to another ex- clusive of those used to accelerate mass. Note, for example, the work done in bending springs, lifting weights, etc. 132 MECHANICS. A machine is a device for receiving energy in a con- venient form and applying it in some desired manner. The principle of the conservation of energy states that the en- ergy supplied must be exactly equal to the total energy used. The energy given to a machine may be used in various ways; as, (1) Lifti^ng weights, etc. (visible and useful work). (2) Overcoming friction (waste, transformed to heat). (3) Strain of parts of machine (potential energy). (4) Momentum of parts of machine (kinetic energy) .- (5) Transformed to other forms, as electric, chemical, thermal, etc. The complete analytical expression in case aU of these are considered is likely to be quite complicated. We some- times simplify matters by neglecting some items of rela- tively small importance, yet it should be remembered that in certain cases this may cause serious errors. The student should note carefully that all forces which do not cause motion are excluded, as they do no work. 89. Efficiency. No machine ever gives out in useful work all the energy supplied. The efficiency of a machine is the ratio of useful work to total energy supplied. 90. Power. In all questions relating to doing work the question of how much can be done per hour or day is important. The rate at which an agency can do work is named its power. To clearly express power we must name both the unit of work and that of time. In engineering the foot pound is, commonly used and power may be expressed in ft. Ihs. per MECHANICS: 133 sec. A horse power is defined as 33,000 ft. lbs. per minute or 550 ft. lbs. per second. Both work and power may be, and often are, expressed in terms which indicate the nature of the working agency as thermal, electrical. Illustra- tions of these are given under these heads. "When a force acts on a moving. body, the rate of motion at any instant multiplied by the force is the power, i. e. work per second = force ' X distance moved per sec. = force X velocity. 91. Work Diagrams. In the computation of work done by a force a graphic method in which force and distance moved are used as co- ordinates it is often very convenient. The case of a constant force is shown in Fig. 46. Here work done is the area OABC where OA = force and OC = distance moved. Y B Area = Work Distance Moved Fig. 46. Ai £ y /- \ E ^y y X v>^ \ \ \ ^ c ) A LEh ^GTf 1 D_ Fig. 47. Fig. 47 shows a case where the force is variable. At any given point where the force has a value / may be 134 MECHANICS. regarded as constant over a very short interval 81. The work done by / while it has this value is / X 81 ^ area of shaded strip. Total work is then the area ABCD. Such a system might occur in practice in maintaining uniform velocity in a mass along a surface of variable roughness. 92. Moment of Force. The ability of a force to cause rotation of a body de- pends on its leverage as well as on its amount and is ex- pressed by the product of force -i- to lever and length of lever arm. This product is called the moment of the force. Moments may be regarded as + or — about a given axis. If we assume all moments tending to set up rotation like the hands of a clock as +, counter clockwise moments must be called — . A rigid body is in complete equilibrium only when, (a) all applied forces tending to move the body with- out rotation add up to zero. (b) and also the sum of all + and — moments is zero. 93. Mechanics of Rotation. All possible motions of a rigid body may be regarded as made up of two general types. In one of these every line in the body is constantly parallel to its former position (translatory motion) while in the other type we have to deal with turning or rotation (rotary motion). When dealing with rotation use is made of the terms of angular measurement in the determination of displacements, veloci- ties, and accelerations. 94. Angular Velocity and Acceleration. When a solid body rotates about an axis, the distance MECHANICS. 135 moved by minute elements of mass vary greatly according to the radius of the circular path travelled, but the angles turned through by all lines in the solid about a given axis are alike. "When a body makes one complete revolution, the ■ angle turned is measured by the circumference of a circle of unit radius. Or one turn is an angle of 2ir where ir = 3.14 or ^;^ Any turn is expressed as a multiple of this number, i. e. a quarter turn or — , a half turn — or tt, 4 2 2 5 turns 5 X 37r or 1 Ott. The angle turned in 1 second in this notation is named the angular velocity. Thus if 4.5 turns were made in 5 seconds, 4.5x2;r the angular velocity would be =: .9 X 2ir and 1.8 X 5 3.14 would be the linear velocity of all points moving on a circle of radius = 1. The linear velocity for points moving on any other circle is found by multiplying the angular velocity by the radius of the circular path. Thus for the given case a particle moving in a circle of radius 6 would have a Telocity 6 X 1-8 X 3.14. For n turns per second and radius r, the linear velocity is, 2n7rr. If a body should make 3 revolutions during one second and 5 during the next second it would have a change of angular velocity in one second of 4ir per second. If this angular acceleration is constant it will make 7, 9, 11, etc. revolutions in successive seconds. As in the case of linear from angular velocity, the linear acceleration = radius X angular acceleration. 95. Reaction against Rotational Effort. A force F applied to a mass. Fig. 48, wliich is moving 136 MECHANICS. Fig. 48. in a circular path tends to increase the angular velocity. At every instant M is moving in the direction of F and we have Force = mass X linear acceleration = mass X radius X angular acceleration Moment of ¥ = Fr ^ mass X radius' X angular accelera- tion. This shows that the moment of the forces due to the inertia which delays the development of angular velocity grows rapidly with increase of the radius of the rotating body. Example. Suppose a hoop of radius 30 cm. has a mass of 500 grams. What force moment must be used to give it an angular acceleration of-—. Here the linear acceleration of each element is 30 X 7C The force for this acceleration is 500 X 30 X The moment of this force is 30X500 X30X -r- (?r=3.14) o MECHANICS. 137 96. Energy of Rotating Masses. The distribution of kinetic energy among the mass ele- ments of a rotating body is readily derived from the rela- tion of linear to angular velocity. Since linear velocity = radius X angular velocity we have V^ = r^ X <^^ (« = uStt, n = revol. per sec.) ffiY' =-— r^ | tension mass per unit length where S is any integer. 105. Vibrating Columns of Gas. A column of gas in a tube closed at one end must have a place of least freedom of motion at the closed end, and of maximum freedom at the open end. If no other places' of Fig. 54. maximum or minimum exist between the ends the portion of the enclosed standing wave is % of the wave length, as PC, Fig. 54. If another place of maximum and minimum are added as at A and B, the new quarter wave length is ^/j of PC. If two more are added it becomes ^/g, etc. Since frequencies are inversely as wave lengths, the series of possible pitches for this pipe are i'n the frequency ratios 1:3:5:7, etc. The same argument would give the ratios 1 :2 :3 :4, etc., for a pipe open at each end. 146 SOUND. In fact, however, the place of greatest freedom of mo- tion, or least pressure change is not just at the open end but a little beyond it maMng the effective pipe length slightly greater than the measured. 106. Forced Vibrations. In the strict sense all vibrations are forced at the start at least. "When left to themselves, vibrating masses rather rapidly lose their energy. The term resonant vibra- tion is reserved for the excitation of a vibration of a mass by a series of minute but properly timed impulses. A tun- ing fork sends out a series of air pressures alternately greater and less than normal. When an excess pressure reaches a similar fork, the slight force gives an infinitesimal movement to the prong. Immediately afterward the pressure on the same side of the fork will be below normal and a force wiU be exerted by the atmosphere to bring the fork back. If the prong is ready to move back by reason of its own elastic force, the rarefac- tion increases the motion. Otherwise, it annuls part of the effect due to the initial push. Thus if the normal frequency of the forks are alike one will start the other by the waves between them. t4 i © 1^ <^ Jl y H' Fig. 55. LIGHT. 147 107. Resonance of Air Oolumns. A long pipe with a movable piston is placed as shown in Pig. 55, and the successive quarter period points of the vibrator are marked 1, 2, 3, 4, 5, 6, 7, 8, etc. If the piston is placed at the point reached by the compression during the movement 01, the reflected compression will reach the open end just in time to add to the compression caused by the fork's motion from 2 to 3. Were the piston placed at 3', the corresponding positions of the condensation are shown below the tube. Or resonance occurs for a length of tube equal to 1/4, 3/4, 3/4, 5/4, etc., of the wave length. # 108. Light. In the phenomena of light and in electrical waves we have again to deal with the transfer of energy through space (radiation). If at a point in space some 'form of energy is changed into a form capable of being radiated this point is spoken of as a source of radiation and in case en- ergy capable of affecting the eye is sent out it is called a luminous source. Such a source placed in a homogeneous medium would send out energy in all directions thus becoming the center of a system of spherical waves. The radii of these spheres would be the directions of energy transfer constituting the rays of the older optical notions. 109. Wave Propagation. The properties of wave motion on which the explana- tion of the phenomena of light is based should be studied with some care. The first to point out these properties and indicate their application was Huyghens. Consider a radiating source at an enormous distance from the region where its waves are observed. The regions 148 LIGHT. of like disturbance (wave fronts), at a single instant would be a series of parallel planes equidistant from each other. Could we actually see these planes continuously they would all appear to travel at a uniform velocity so long as the medium was homogeneous. Each of these planes retaining its identity and its relative distance from its neighbors. If 1, 2, 3, 4, Pig. 56 are traces of such planes on the plane of the paper, and V = wave velocity. They would occupy the position V 2' 3' 4' at a time r/V later. Huy- ghens explained the persistence of form and relative posi- tion by considering every point on a wave front as an in- dependent center from which spherical waves start out and whose radii grow at the rate of V units per second. All of the waves having centers along 1 as a, b, c, d, e, will acquire ' radii equal to r in the same time so all these spheres will be tangeilt to 1 at the same instant. In this way the move- ment of the wave may be regarded as the resultant of an infinite number of smaller waves having centers on any wave front we choose. ■ Figure 57 shows the corresponding case for a spherical or circular wave. The wave shown is called a diverging wave and its curvature is spoken of as the reciprocal of its 2 ,V3 2' Fig. 86. Fig. 57. LIGHT. 149 radius. If R becomes infinity, 1/R = or we have the plane wave. If the direction of propagation in Fig. 57 were re- versed so as to move toward C instead of away the wave would be called a converging wave. The radii of these waves or in general, the normals to t|ie wave surface are the rays of geometrical optics. 110. Change of Curvature. In Fig. 56 so long as each elementary wave has the same velocity the wave fronts remain plane. If a group of them, for example, those between c and d should have a velocity less than their neighbor's that portion of the wave Fig. 58'. would become converging, if greater, divei-ging. In Fig. 57 if £t group of adjacent elementary waves entered a region where the velocity was' different that portion of the resultant wave would acquire a new curvature and the wave normals or rays would no longer intersect at C. Figures 58 and 59 show two eases where curvature is changed. Fig. 58 by reflection of a spherical wave at a plan^ surface. Fig. 59 by refraction of a plane wave at a curved surface. 150 LIGHT. Fig. 59. 111. Wave Centers. It is a great help in the study of visual phenomena to bear in mind that we see things only by means of diverging rays, i. e. by wave fronts which are convex toward the eye as they enter the pupil. The apparent location of what we see, or think we see, is always at the poini from which, the rays entering the eye seem to diverge, i. e. at the center of curvature of the portion of the diverging wave which actually enters the pupil. We have no means whatever by which to tell whether light really came from the point or not. Fig. 60 may aid in making this point clear. Consider a single luminous point S, light leaves S in all directions with spherical diverging wave fronts as W^. A small portion of this light strikes the lens L and is converted into an approxi- mately spherical converging wa-