ww'Kjiviy'. BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Henrg W. Sage 1891 /:2..3.r;?r.»^5> £ 2 3513-1 Cornell University Library QC 21.C96 A text-book of general physics for colle 3 1924 012 333 237 Cornell University Library The original of tiiis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012333237 A TEXT-BOOK OF General Physics FOR COLLEGES MECHANICS AND HEAT BY J. A. CULLER, Ph.D. PROFESSOR OF PHYSICS, MIAMI UNIVERSITY PHILADELPHIA J. B. LIPPINCOTT COMPANY 3) Copyright, igog By J. B. LippiNcoTT Company Printed by J. B, Lippincott Company The Washington Square Press^ Philadelphia^ U,S. A. PREFATORY NOTE In the following treatise on mechanics and heat an effort has been made to present the subject in as clear a manner as possible for use of a college student. A knowledge of plane trigonometry is necessary before undertaking this study, and the more mathematics a student knows the better he will ordi- narily succeed in physics. Some changes ' have been made in the form of the usual college text and in the method of presentation. The aim of the writer has constantly been to say the words that would help the student to understand the subject. Thus it is hoped that the book will prove to be not only a treatise but also a text -book for students. Reference matter and tables are placed in the appendix instead of being scattered through the text. This takes less room and is much more convenient for reference. A number of short lists of problems are found where they are needed to illustrate the application of principles learned. Answers to problems are given at the end of the lists, but a student should be made to understand that numerical results are not so impor- tant here as his ability to present the line of argument involved in the problem. The tables of sines, cosines, tangents, etc., are intended to make the book more desirable as a complete working text. We acknowledge our obligation to the Ball Engine Co. for cuts of the steam engine, to D. Van Nostrand Co. for cut of the Parsons steam turbine, to the T.aylor Instrument Companies for the cuts of pyrometers, and to the De Laval Steam Turbine Co, for cuts of the De Laval turbine. CONTENTS CHAPTER I. KINEMATICS. SECTION PAGE 1. Metrology i 2. Fundamental and Derived Units 5 3. Dimensions 6 4. Motion and Rest 6 5. Axes of Reference 7 6. Translation and Rotation 9 7. Measurement op Length or Distance 9 8. Measurement of Change in Direction 13 9. Velocity 15 10. Uniform and Accelerated Motion 15 11. Uniformly Accelerated Motion 16 12. Angular Velocity and Acceleration 17 13. Vectors 18 14. Composition and Resolution op Velocities 18 15. Methods op Calculating Resultants 20 16. Uniform Circular Motion 23 17. The Motion of a Projectile 26 18. Simple Harmonic Motion 29 CHAPTER II. dynamics. 19. Definition of Terms 39 20. Newton's Laws of Motion 39 21. Inertia 39 22. Force 40 23. Units of Force 40 24. Impulse and Momentum 42 25. Stress and Strain 43 26. Graphical Representation of Forces 44 27. Resultant and Equilibrant 44 28. Resolution of Forces 45 29. Moment of Force 47 30. Equilibrium op Moments 48 3 1 . The Couple 50 32. Moment of Inertia 51 V vi CONTENTS. 33. Centripetal Force 53 34. Stability op a Rotating Body 55 35. The Conical Pendulum 56 36. Effect OF Rotation of the Earth ON the Weight of Masses 57 37. The Law of Gravitation 59 38. Gravity 61 39. Equilibrium in Orbital Motion 63 40. Gravity beneath the Surface of the Earth 64 41. Weight 66 42. Centre of Gravity 67 43- Centre of Mass 6g 44. Stable Equilibrium 69 45. Determination of Mass 70 46. Work and Energy 75 47. Units of Work and Energy 77 48. Power 79 49. Potential and Kinetic Energy 79 50. Energy of a Rotating Body 80 51. Conservation of Energy 81 5s. Available Energy 82 53. The Simple Pendulum 84 54. The Physical Pendulum 85 55. Reversibility of Compound Pendulum 89 56. Use of Pendulum for Measurement of Time 90 57. Machines 92 58. Mechanical Advantage 93 5g. Kinds of Machines 93 60. Levers 93 61. Pulleys 94 62. The Wheel and Axle 97 63. The Inclined Plane 98 64. The Wedge 100 65. The Screw loi 66. Friction loi 67. Sliding Friction 102 68. Kinetic Friction 103 69. Rolling Friction 104 70. Uses op Friction , 104 71. Efficiency 105 CHAPTER III. solids. 72. Constitution of Matter 107 73. States of Matter 107 74. Elasticity op Solids 109 CONTENTS. vii 75. Volume Elasticity no 76. Shearing Elasticity in 77. Coefficient of Shearing Elasticity 112 78. Longitudinal Elasticity; Young's Modulus 114 79. Value of k in Terms of Y and n 116 80. The Torsion Pendulum 116 81. Use of a Torsion Pendulum in finding 1 117 82. Use OF Torsional Pendulum FOR finding « 117 83. The Torsion Balance 118 84. Impact of Elastic Bodies 119 85. Impact of Inelastic Bodies 122 CHAPTER IV. GASES. 86. Fluids 123 87. Character of a Gas 125 88. The Kinetic Theory of Gases 126 89. Pressure of a Gas 126 90. Avogadro's Law 128 91. Dalton's Law 129 92. Boyle's Law 130 93. Equation of Van der Waals 133 94. Elasticity of Gases 134 95. Pressure of the Atmosphere 135 96. The Barometer 136 97. Corrections of Barometric Readings 138 98. Glycerin Barometer 140 99. The Aneroid Barometer 140 100. Mechanical Air-pumps 141 loi. Mercury Air-pump 144 102. Diffusion of Gases 14S 103. Buoyancy op Air 147 CHAPTER V. liquids. 104. Liquid Pressure 150 105. Transmission of Pressure 150 106. Pressure of A Liquid ON the Walls of A Vessel 151 107. Buoyancy of Liquids 152 108. Density and Specific Gravity 1 53 109. Density of Solids 153 no. Density op Liquids 154 111. Twaddell's Hydrometer 158 112. Equilibrium in Case of Buoyancy 159 viii CONTENTS. 113. Valve Pumps for Liquids 160 114. The Siphon 161 115. Efflux op Liquids 163 116. Velocity of Efflux in Terms op p and p 163 117. Lateral Pressure of a Moving Stream 164 118. Viscosity 165 119. Surface Tension 168 120. Unit Surface Tension 168 121. Surface Tension Compared to an Elastic Membrane 168 122. Pressure dub to Surface Tension 169 123. Angle of Contact 171 124. Capillary Action dub to Surface Tension 172 125. Some Surface Tension Phenomena 173 126. Diffusion of Liquids 174 127. Osmotic Pressure 175 CHAPTER VI. HEAT. 128. Heat and Temperature 178 129. Expansion 179 130. Determination of A 181 T31. Expansion of Liquids 181 132. Maximum Density of Water 184 133. Expansion of Gas 185 134. Law op Charles 187 135. Absolute Temperature 188 136. Laws of Boyle and Charles Combined 188 137. Thermometry 191 138. Hydrogen Thermometer 191 139. Mercury-in-Glass Thermometers 193 140. Graduation of Thermometers 194 141. Calibration of Thermometers 195 142. Thermometers for Special Purposes 196 143. Pyrometry 199 144. Calorimetry 204 145. Thermal Capacity 205 146. Specific Heat 205 147. Molecular Heat 206 148. Change of Specific Heat with Change of Temperature.. 207 149. Water Equivalent 207 150. Latent Heat 208 151. Specific Heat by Method of Mixture 209 152. Specific Heat by the Method of Melting Ice 210 153. Bunsen's Ice Calorimeter 210 154. Specific Heat by the Method of Cooling 212 CONTENTS. ix 155. Specific Heat by Electric Heating 213 156. Specific Heat op Gases 214 157. Fusion and Solidification 217 158. Freezing Point of Solutions 219 159. Evaporation 221 160. Vapor Pressure 223 161. Boiling Point 224 162. Isothermals of a Vapor 225 163. Humidity 229 164. Transference of Heat 231 165. Conduction 231 166. Radiation 234 167. Source of Ether Waves 235 168. Measure of Radiant Energy 235 169. Laws op Radiation 238 170. Thermodynamics 241 171. Mechanical Equivalent op Heat 243 172. Laws of Thermodynamics 246 173. Difference of Specific Heats of Gases 247 174. Effect op Intermolecular Forces 248 175. Ratio of Specific Heats op Gases 250 176. Adiabatic Expansion 252 177. Elasticity op Gases 252 178. Carnot's Cycle 253 179. Reversible Cycle 255 180. Carnot's Theorem 256 181. Thermodynamic Scale of Temperature 256 182. Entropy 258 183. The Steam Engine 261 APPENDIX. Index of Appendix 2 69 GENERAL PHYSICS MECHANICS ^, CHAPTER I KINEMATICS 1. Metrology. — Metrology is the science of weights and measures. Convenient and well-defined units of measurement are essential in any highly organized social state. Exact units are particularly necessary to the advancement of science. The scientist is constantly trying to make exact determinations of length, area, volume, and mass. He must express these quanti- ties in fixed units. A record is thus made which can be compared with the results of other investigations and can be understood by all who are familiar with the units used. If the units em- ployed by different investigators are not exactly defined or are carelessly used, great confusion is sure to result and progress will be checked. In early times, when the people of a community were not so closely dependent on each other as now, the head of a family or clan might choose units of length and weight as he thought best. He would, however, choose some convenient and natural unit. For short distances the length of his foot, the breadth of his hand, or the length of his forearm would be chosen. Longer distances would be designated as so many paces, and still longer distances by the distance a man could travel in one day. For determination of mass he would naturally use seeds, as is evidenced by terms still in use, e.g., the grain and the carat (from carob, bean). When people began to live together in larger and more com- pact communities, it became necessary for the king or some one in high authority to fix certain standards to be used in common 1 GENERAL PHYSICS. by a great number. Thus, it is said, the English yard was first determined by the length of the king's arm. As the various nations advanced in science, arts, and indus- trial pursuits, it became necessary to fix and define certain units which all would use. The units of length having most extensive use are the yard and the metre. The yard was defined by the English Parliament in 1855. It is a solid square bar made of a special bronze, 38 inches in length and one square inch in cross section. Near each end a circular hole is sunk to half the depth of the bar. Fig. 1. At the bottom of each hole is a gold plug upon which is inscribed a transverse line. When the temperature of the bar is 16f° C, the distance between the lines is the imperial standard yard of 36 inches. This bar is carefully preserved at the standards office, Westminster. Four other bars called Parliamen- tary copies were made and deposited for safe keeping at other places. These, by law, must be compared with the original once every ten years, so that if the original is lost or destroyed, it may be exactly reproduced from its copies. A number of other standard yards were made of the same material and distributed to various in- stitutions in Great Britain and to other nations. Bronze standard No. 11 was presented to the United States. It is .000088 inch shorter than the imperial standard. This description is sufficient to show the care which has been taken to define and preserve a unit of length. The standard unit of mass in the English system is a piece of platinum marked P.-S., 1844, 1 lb. This is the avoirdupois pound, and yinnr of this mass is the grain. The unit of time is the mean solar second. A solar day is the interval between the passages of the sun across the meridian. These intervals are not equal, for the earth does not move with the same speed at all points of its orbit, but the mean of all the intervals in one year is the same as in another year. If there is any difference it has not yet been detected. A mean solar Fig. 1. KINEMATICS. 3 day is divided into 24 mean solar hours, the hour into 60 minutes, and the minute into 60 seconds, making 86,400 seconds in a mean solar day. If it is found, as some think, that the earth is rotating on its axis more slowly than formerly, a different and less variable unit of time may be selected as a standard. The English standards of length and mass are arbitrary, — i.e., they were selected by Parliamentary enactment, and their perpetuity depends on the care with which they are preserved. As standards they are probably as good as any of this character. The chief objection to the English system is the manner in which the standards are divided. The system is not a decimal one, and the various derived units are very inconvenient. In the latter part of the eighteenth century the subject of a rational system of weights and measures was strongly agitated in France. This was a time when the French people were making many changes and were in a mood to make this one, however radical the change might have seemed at other times. The plan was to agree on some natural unit of length, something that would not change while the world stands. The French Academy of Sciences recommended the length of the earth's meridian from the equator to the pole. The measure- ment of this distance was made by M^chain and Delambre (1791-1798). Of course they could not measure the entire distance, but chose the distance from Dunkirk, in the northern part of France, to Barcelona, in Spain, on the shore of the Mediterranean Sea. This distance was very carefully measured by the method of triangulation, taking into account the curva- ture of the earth. The difference of latitude between these points was found to be 9° 40' 45" Knowing the length and the number of degrees, the length of 1° is easily found, and then the length of 90°, the quadrant sought. One ten-millionth (tt^) of the length of the quadrant was called 1 metre (m.). Thus the effort was made to determine a natural standard of length, but it has since been found that the quadrant is more nearly 10,000,880 metres. The metre in use is practically an arbitrary standard, just as the yard is, and is defined as the 4 GENERAL PHYSICS. distance between two transverse lines on a certain platinum- iridium bar when the temperature is 0° C. This bar is preserved with great care by the International Metric Bureau at Sevres, near Paris. Fig. 2. The standard of mass in the metric system is the kilogram, originally intended to be the mass of one cubic decimetre of water at its greatest density, 4° C. A mass of platinum supposed to be equal to this quantity of water was selected as the stand- ard, but it has since been found that 1000 c.c. of pure water at 4° C. weighs about .04 g. less than the standard mass of platinum. Fig. 3. The subdivisions of the metric standards are made on a decimal basis. The convenience of the metric system has led to its universal adoption for scientific purposes, and in many coun- tries of Europe it is used for all purposes. KINEMATICS. 5 The unit of time in the metric system is the mean solar second, as in the English system. In the year 1866 the metric system was made lawful through- out the United States, the weights and measures in use being defined in terms of the metric units. The yard was defined as fffr nietre, and the pound avoirdupois as -^.Twrs kilogram. Thus the metric system is made the standard in the United States, but its use is not compulsory. The original advocates of the metric system failed to estab- lish a natural and invariable standard, but the arbitrary metre and kilogram as now defined are better standards than natural ones which are subject to change or for which new values are likely to be found by later and more refined processes. In recent times Prof. Michelson has determined the length of the metre in terms of wave lengths of ligiit. He found that for red light, whose wave length is .64384722/x, the length of the metre is 1,553,163.5 waves; for green light, of wave length .50858240/i, 1,966,249.7 waves; for blue light, of wave length .47999107/1, 2,083,372.1 waves. Thus the length of the metre is fixed in terms of an invari- able natural unit, — wave lengths of light, — and if for any reason the present standard metre should be destroyed, it could be exactly reproduced from the record of its length in terms of light waves. 2. Fundamental and Derived Units. — Fundamental units are those that are chosen as a basis for a system of units. The fundamental units most commonly employed are those of length, mass, and time. The unit of nearly all other physical magnitudes can be fixed in terms of these three. Derived units are those whose magnitude is expressed in terms of the fundamental units. Area, for example, is a length squared. Volume is a length cubed. Velocity is a length divided by a time. Density is a mass divided by a length cubed. A system of units fixed in this manner is called an absolute system, because physical magnitudes are thus determined not by reference to some other magnitude of the same kind which might have been adopted as a standard, but by reference to fundamental units which do not change. The most common absolute system in physical investiga- 6 GENERAL PHYSICS. tions is that in which the unit of length is the centimetre; the unit of mass, the gram ; and the unit of time, the second. This is the centimetre-gram-second or c.g.s. system. 3. Dimensions. — Dimensions of a derived unit are expressed by a power of the fundamental unit. Thus [U\ expresses the dimensions of volume in terms of length. Dimensional equations are found by calling length L, mass M, time T, and then placing these in the proper relation to express the physical quantity under consideration. Velocity, for example, is a length divided by a time; hence it is expressed by I " I or [LT"~']. Momentum is a product of mass by velocity ; hence' it is expressed by [M] [LT-^]=[MLT-^]. These expressions only show the relation between the funda- mental and derived units. When the magnitudes of the funda- mental units have been determined, the magnitude of the derived unit may be found from its dimensional equation. Thus velocity [F] = [Lr-i] and, if c.g.s. units are used, the velocity is expressed as '^'"/sec- To express the magnitude of any physical quantity we must know not only the unit used but also the number of units. In the expression X represents a pure number and is called the numeric. Dimensional formulse are valuable in many ways, as will appear in later discussions. To illustrate one of their uses, suppose it is desired to convert a velocity of 12 m. per minute to cm. per second. Let x be the numeric sought, — i.e., the number of cm. per sec. Also let L^ and T-^ be the length and time in metres per minute and L^ and T.^ be the same for centi- metres per second. Then, ■■■"-''[ErT;]-"'"'°- but by esti- mating in tenths measurement may be made to thousandths of a millimetre. To facilitate setting of this instrument and to secure the same pressure in different measurements, a ratchet, R, is provided, which will turn without turning the screw when a certain pressure has been reached. Fig. 10. Fig. 11. Another valuable instrument for measurement of small lengths is the micrometer microscope. This consists of a simple microscope with micrometer, mounted as shown in Fig. 11. The objective, o, of the microscope produces an enlarged image Fig. 12. KINEMATICS. 13 of the object within the tube in the plane MS. This image is viewed through an eye-piece, E. The micrometer, shown nearly full size in Fig. 12, is mounted in the plane of the image. The screw is attached to a light frame, across which are stretched spider lines. By turning the screw the lines are made to move across the field from one end of the image to the other. The head of the screw is a disk the circumference of which is divided into 100 equal parts. The number of turns of the screw is a measure of the length of the image. To measure an object in millimetres it is necessary first to determine the constant of the instrument, — i.e., the number of turns of the screw necessary to cause the spider lines to move over a space of one millimetre as seen through the eye-piece. Suppose this constant is found to be 9.35, then the length of an object which requires for its measurement 3.927 turns is 3.927-^9.35 or .42 mm. Numerous other devices are employed in the measurement of length, but most of them involve the principles already explained. 8. Measurement of Change in Direction. — In case the motion is a simple rotation, the displacement involves a change of direction but not of distance from the origin. Measurement is then made of the angle through which the line connecting the body with the centre of the circle moves. If a body moves from X to P along an arc whose radius is oX, it has changed in direction but not in distance from 0. The displacement , of the body may then be indicated by giving the number of degrees which the line oP makes with oX. For the construction and accurate measurement of angles, various forms of protractors are frequently used. In Fig. 14 is shown a protractor having a vernier, a transparent centre for 14 GENERAL PHYSICS. accurate placing over the origin, o, and a long arm one edge of which is in a true line with the centre. With such an instru- ment a difference of one minute in the magnitude of an angle may be measured. The degree is the unit in this method of measurement. Another unit, often much more convenient in the measure- ment of angular displacement, is the radian. A radian is the angle at the centre of a circle measured by an arc whose length is that of the radius of the circle. If the radius oa, Fig. 15, is laid off on the circumference, not as a chord but so as to coin- FiG. 14. Fig. 15. cide with the curve of the circumference, as ah, then the angle at formed by the lines aa and ho is a radian. It is evident that the angle at o is not quite 60°, as would be the case if the radius were inscribed as a chord. Since the circumference of a circle is 27tr, the radius can be laid off on the circumference 2tc or 6.2832 times, — i.e., there are 6.2832 radians in 360° The value of one radian is then 57.2958°, about 57.3°. When the angular displacement of a body is given in radians, it is easy to pass to degrees by multiplying by 57.3, or to find the linear distance passed over by multiplying radians by the length of the radius. Problems. 1. What must be the structure of a vernier that a scale graduated in J mm. may be read to -^m^ cm. ? Find value of V from (7) and substitute in (6) for n. 2. The divisions on the scale of a barometer are ^^ inch. 25 divisions on the vernier are equal in length to 24 on the scale. What is the least cotuit in fraction of an inch? KINEMATICS. 15 3. In a circle whose radius is 20 feet two bodies on the circumference are 85.96° apart. Express this angular distance in radians. What is the distance in feet as measured along the circumference 4. A wheel 10 cm. in radius rotates 360 times per minute. What is its angular velocity and what is the speed of any point on the circum- ference ? 1. 50 divisions on vernier to 49 on scale. 2. .002 inch. 3. 1.5 radians. 30 feet. 4. 37.7 radians (nearly). 377 cm. 9. Velocity. — Velocity is the rate of change of position. It is the change of position which would take place in a unit of time if the body continued its motion uniformly during that time. A body moving with a velocity of 500 cm. per minute may in fact be in motion for only one second of time or less, but while it was in motion its rate was such that it would traverse a distance of 500 cm. if its motion continued for one minute. A velocity of 10 radians per second simply indicates the amount of angular displacement which would occur in one second at that rate. Since there are two kinds of motion, there are two kinds of velocity, — linear and angular. Linear velocity is that which occurs along a line, whether that line be straight or curved. It is the distance traversed per unit of time as measured along the line of motion. Angular velocity is the rate at which the angle at the centre changes when the motion is a rotation. It is the number of degrees or radians by which a body changes its direction in a unit of time. 10. Uniform and Accelerated Motion. — A motion is uniform when the same distances or angles are traversed during each successive unit of time, — i.e., the motion is uniform when the velocity is constant. Such a moving body changes its position a certain number of centimetres or radians per second during each second of its motion. Whether the motion is actually uniform or not, it is possi- ble, when the distance and time are given, to find a uniform rate of motion by which the same space would be traversed in the given time. If the distance from A to B is 500 cm., and the 16 GENERAL PHYSICS. time required for a body to move from the one point to the other is 20 seconds, then, no matter what the nature of the motion may be between A and B, the uniform rate or average velocity is 500 -f- 20 = 25 cm. per sec. In all such cases V being the average velocity, 5 the space traversed, and t the time in seconds. When the velocity is increased or diminished a certain amount each second, the motion is said to be accelerated. The accelera- tion may be positive or negative, — i.e., the velocity may increase or decrease. If the velocity at the beginning of a certain period is known, and to this are added the successive accelerations during the time, the result is the velocity at the end of the period. If the accelerations are negative, they must be subtracted from the initial velocity. 1 1 . Uniformly Accelerated Motion. — When the change of velocity is the same for each successive unit of time, the motion is said to be uniformly accelerated. If a body starts from rest and moves with uniformly accelerated motion (U. A. M.), its velocity at the end of any period of time is found by the equation v = at (9) where v is velocity, a the acceleration, and t the time. The truth of this statement is apparent from the definition of a, for if a is the increase in velocity per second, then the increase per second times the number of seconds must be the final velocity. If under similar conditions it is desired to find the total distance through which the point moves, first find the average velocity and then multiply by t. Since the acceleration is uniform, the average is one-half the sum of the first and last velocities. This may be expressed by assuming that the motion begins from rest. This is the uniform velocity the moving point must have to traverse the space in t seconds. The total space, s, must then be s = iat.t = ^af (10) KINEMATICS. 17 These two equations, (9) and (10), are the fundamental ones for U. A. M., but a number of other important equations may be derived from them. Problems. 1. Combine (9) and (10) so as to eliminate t and show that v' = 2as (11) 2. Make use of equation (10) to find the total distance when the time is (t—1). Subtract this from the distance when the time is t. Call the distance during any one second of U. A. M, d and show that d = la(2t — l) (12) 3. If a body has a uniform motion, of velocity V, and is given, in addition, a. U. A. M., show that (i = y±ia(2f— 1) (13) 4. Show that v = V±at (14) 5. Show that when there is initial velocity V and U.A. M. the dis- tance J in time t is s = Vt±iat' (15) 6. By eliminating t from (14) and (15) show that v' = V'±:2as (16) 7. Prove that the distance passed over in the first unit of time of U. A. M. is one-half the acceleration. Use (10). 12. Angular Velocity and Acceleration. — ^Angular velocity is the rate of change in direction. This may be either uniform or accelerated, and the acceleration may be uniform or variable, just as in linear velocity. When a body rotates on an axis, its various particles have different linear velocities depending on their distance from the axis, but all have the same angular velocity, for there are 360° in any circle whatever its radius may be. Let the radian be the unit and let o) represent the number of radians per second. The angular distance in any given time is then expressed by (Ot Any particle located at a distance r from the axis has evi- dently a linear velocity v = a)r (17) for (ij is the number of radians per second and r is the length of the arc which subtends one of them. 2 18 GENERAL PHYSICS. 13. Vectors. — ^A vector is a quantity in which both magni- tude and direction are considered. Velocity is a vector quantity, because it is designated by a number and a direction. Distinction is made between velocity and speed in that speed is designated by a number without consideration of direction. A race-horse moves with a certain speed, the direction of his motion being a matter of no interest. Quantities are classified as vector and scalar. Examples of vectors are velocity, acceleration, momentum, currents of water, and force. Examples of scalars are speed, mass, and density. A Ime may be drawn to represent any given magnitude which has also a direction. In case of velocity, the length of the line drawn to any convenient scale will represent the magnitude of the velocity while its direction is that of the motion. y, i £ i t—4a Velocity in "^c Scale Toi Fig. 16. If, for example, a velocity of 500 '^^Uec due eastward is to be represented by a line, we may choose a certain scale, say 1 cm. of- length to 100 cm. of velocity, and draw the line AB, Fig. 16, assuming directions as on a map. If the direction is marked by an arrow-head, the information in Fig. 16 is complete, and any one accustomed to this form of representation will at once read 500 ^'"/sec due eastward. The lines themselves may be called vectors, but only in the sense that they represent vector quantities. 14. Composition and Resolution of Velocities. — ^Two or more velocities may combine and pro- duce a resultant velocity. The resultant is easily calculated by a consideration of the vectors which represent the component velocities. Let two component velocities be represented by the vectors oY and oX. A moving point starting at o must in one second be oy distant from oX, and oX distant from oY, — i.e., it must be at R and its velocity must be oR. KINEMATICS. 19 This is the parallelogram of velocities, in which the two componeiits are taken as the two adjacent sides of a parallelo- gram and the resultant is the diagonal drawn from the common origin. This is true no matter what the angle formed by the components may be. In case there are more than two components, as a, b, and c in Fig. 18, it is only necessary to combine a and b in the manner already shown and then combine their resultant with c. Fig. 18. Another method, known as the polygon of vectors, consists in placing the vectors end to end, — i.e., placing the end of one to the origin of the next. The line connecting the end of the last with the origin of the first is the vector of the resultant. Thus the resultant of a, b, c, and d, Fig. 19, is R. If the component vectors form a closed polygon, there is no resultant. Not only can resultants thus be found when components are given, but any given vector may be resolved into components which would produce the same effect. Let oR be a vector and oY, oX, the rectangular coordinates drawn through the origin, o (Fig. 20). It is evident that oR represents a velocity oH along the X-axis and HR along the F-axis. oH and HR may then be considered components of oR. Since the coordinates may have any position, an indefinite number of rectangular components may be drawn. The components may also be at any angle with each other, as, in Fig. 21, a and b may be taken as components of oR. 20 GENERAL PHYSICS. The kind of resolution which should be made depends on the nature of the problem which is to be solved. Numerous appli- cations of these principles will be found in later pages of this work. Fig. 20. 15. Methods of Calculating Resultants. — Two methods are commonly used in finding the magnitude and direction of the resultant when the components are known, — ^namely, the graphical and the mathematical. By the graphical method exact drawings are made in the manner just indicated, the vectors being drawn on some convenient scale and true in direction. The resultant is then measured and its value found from the scale adopted. For example, if the component vectors of velocity are drawn so that each centimetre represents one metre of velocity, and the resultant is found by measurement to be 4.5 cm. long, the resultant velocity is 4.5 m. This method is in common use in drafting rooms where problems of this character are considered. r Fig. 22. By the mathematical method the principles of geometry and trigonometry are in most cases sufficient. When two vectors are at right angles to each other, as in Fig. 22, it is sufficient to apply the well-known principle of geometry, r^^x'+y^ (18) KINEMATICS. 21 k In case the vectors are not perpendicular to each other, as a and h, Fig. 23, they may be referred to the axes in such a manner that a, say, will coincide with the X-axis. Then h may be resolved into two components, bcosd, which is along the X-axis and so can be added to a, and b sin 6, which is 2l_ parallel to the Y-axis and so is at right angles to a. Now we have two vectors at right angles and can solve as above by equation (18). A general equation for — calculating the magnitude of the resultant may be deduced as follows: Referring to Fig. 23, where a and b are the two components whose resultant is to be found, the vector value along the X-axis is seen to be a+b cos d, and along the Y-axis it is b sin 0. Hence the square of the resultant is equal to the sum of the squares of these two rectangular components, — -that is, r^ = {a + b cos dy + ib sin ey = a' + 2ab cos O + b^ cos^ d + b^ sin^ d = d' + b\sia'' ^-Fcos^ d)+2ab cos 6 = a' + ¥ + 2ab cos d (since sin^ d + cos^ ^ = 1) •■• r=-i/a.^+b^+2abcosd (19) Fig. 23. This is the expression for what is usually called the addition of vectors. Vectors may also be subtracted by reversing the direction of one of them and then finding the resultant in the usual manner. In Fig. 24 let a and b be the two vectors which are to be subtracted. The resultant is r, — a magni- tude which is less in this par- F"5- 24. ticular case than that from addition. The operation must be made algebraically, and consequently the result of the subtraction may be a larger number than that from addition. 22 GENERAL PHYSICS. The operation of finding the resultant in case of subtraction is similar to that for addition, but the component of b along the X-axis is now — b cos 6, — i.e., its direction is opposite to that of a. Fig. 25, and must be subtracted. The equation then is r^ = (a-bcosdy + (bsmey r=i/a'+b''-2ab cos ^ (20) Fig. 25. To determine the angle which the resultant makes with one of the components, let a and b, Fig. 26, make an angle 6 with each other. Let

T^ y=lh3,lh -li- -3, -li. li Hence the curve begins on the ordinate, 1^ cm. above o. At the end of the first unit of time y is3; so an ordinate is erected at a, 3 cm. long. This is the amplitude of the vibration. At c the ordinate is 1^ again. At / it is — li, and so is drawn below the abscissa, as are also hi and kn. At the end of -^ sec. one cycle is completed, and the point is at E', for the period is yis sec. This cycle is twice repeated, bringing the particle to E'". This 36 GENERAL PHYSICS. makes ^ sec, but the time given is -^ or f^ sec, hence the point will be at P where y is 1\ cm. This curve shows the whole movement of the vibrating particle for ^ sec. Let the student take the intervals of time as xiir sec, find corresponding values of y, and insert them in Fig. 38. Fio. 40. If two harmonic curves are referred to the same axis, as abed and ABCD, Fig. 39, their resultant may be found by add- ing the ordinates. The resultant in the figure is the dotted line. The line mr is the sum of mn and ms. The line hB is composed of the positive line ib and the equal negative line iB ; hence their sum is zero and the resultant curve here crosses the axis. In a similar manner the resultant curve may be found at any point. When two S. H. M.'s at right angles are compounded, a large variety of resultants may be obtained, depending on the period, KINEMATICS. 37 the phase, and the amplitude of the components. An experi- mental illustration may be made by suspending two pendulums so that they will swing in planes perpendicular to each other, Fig. 41. (i) Fig. 42. as shown in Fig. 40. One pendulum should, preferably, beat seconds. The other can be made any length by sliding a heavy bob up or down. Both are suspended from knife edges, that Fig. 43. (j) Fig. 44. the friction may be as little as possible. The rods extend a few centimetres above the knife edges, and to the tops of these, light rods, ac and be, are attached by universal joints. These horizontal rods are connected at c by a hinge, from which a 38 GENERAL PHYSICS. needle projects downward. The needle point must then trace the resultant motion of both pendulums. By placing a smoked glass beneath c, this motion may be recorded. In Fig. 41 the period of one pendulum was twice that of the other. A number of periods are required to make a figure such as this. The period of each pendulum is constant, but, because of friction and a very slight variation in the ratio of the periods, the trac- FiG. 45. ings are separated from one another. In Fig. 42 the periods are as 2 to 3, as indicated by the fraction beneath the figure. In Fig. 43 the pendulums are made of such length that while one is making three vibrations the other makes four. The ratios in the other figures are as indicated. The ratio in figures such as these may be determined by beginning at one of the sharp points of the tracing and counting the number of times across in a horizontal and then in a vertical direction to a corresponding symmetrical point on the other side of the figure. CHAPTER II DYNAMICS 19. Definition of Terms. — The preceding chapter has been devoted to a discussion of motion, without consideration of the cause of the motion or the mass of the moving body. The prob- lem there was to find the path of a moving point under certain given conditions. The chapter was entitled Kinematics because xivTina (kinema) means motion. In this chapter attention is directed mainly to a considera- tion of force and its effect in changing motion or causing strain in masses of matter. Force as a cause of motion is often treated tmder the title kinetics, — that which causes motion, — while the subject of equilibrium under stress is often called statics. Kinetics and statics are subdivisions of dynamics. 20. Newton's Laws of Motion. — Sir Isaac Newton, in the latter part of the seventeenth century, announced three im- portant laws which contain the fundamental principles of dynamics. These laws were written in Latin, and their inter- pretation is about as follows : 1. Every body of matter persists in its state of rest or motion. 2. The effect of an impulse in changing the momentum of a mass of matter is independent of other impulses which may be applied at the same time and of the momentum which the mass may already have. 3. The application of a force is always accompanied by an equal resistance in the opposite direction, and the energy ex- pended by any body acting as agent is equal to the energy received by another body which resists the agent. 21. Inertia. — Inertia is that property by virtue of which matter persists in whatever state of rest or motion it may have. This is a general property of all matter. Newton's first law is simply a statement of the principles of inertia. In accordance with this property, a body resting at any point in space will remain there forever, and a body in motion will continue its motion forever in a straight line, provided it receives no impulse 39 40 GENERAL PHYSICS. from an external force. In other words, a mass of matter sepa- rated from all other masses cannot change its state of rest or motion. It is impossible to supply the conditions of a moving body completely unaffected by outside influences, but we assume that the law is true in reference to moving bodies, and on this assump- tion we solve problems in mechanics the results of which con- form with experimental tests. For example, if a body is pro- jected vertically upward with a certain velocity, we assume that it will continue its velocity uniformly. We also know the effect which gravity will have on the body at the same time. By subtracting the opposing gravity effects we obtain results which are in accordance with the facts of experience. Hence the assumption is regarded as correct. 22. Force. — The word "force" is derived from the word fortis, meaning strong, and the primary idea of force was prob- ably related to muscular ability. The muscular effort required to lift a weight, for example, would be a measure of force or strength. Then a push or pull, by muscular effort against any resistance, would be called a force. Then inanimate bodies which by their motion or otherwise would do what might be done by muscular effort, would by a kind of personification be said to exert force. As applied to inanimate objects, force is a consequence of the laws and properties of material bodies. Thus, expanding gas or steam may move adjacent bodies only to secure more room for itself. A steel spring in a strained condition may push or pull other bodies to restore its own molecular equilibrium. One body may collide with another, and the motion of both will be changed as a result of the inertia of the bodies. Numerous examples of this kind may be cited where force is said to be exerted by one mass on another. The common effect of force, as the term is used in mechanics, is motion or strain in bodies of matter. Hence force may be defined as that which produces or tends to produce motion. 23. Units of Force. — A force is measured by the effect which it produces. Any force, however small, will set in motion any mass, however large, provided no resistance is offered except that of the inertia of the mass. A constant force — i.e., one which continues to be of the same magnitude — will produce in a mass DYNAMICS. 41 a uniformly accelerated motion. If the mass is small, the accel- eration may be large, and if the mass is large, the acceleration may be small. In any case, for a given constant force, the prod- uct of the mass and acceleration is a constant quantity. This is usually expressed by F=ma (41) By this equation it is seen that for any given mass, m, the force, F, varies directly as the acceleration, a. Therefore the accelera- tion which is produced may be employed as a measure of the force applied. There are two common units based on this prin- ciple. They are the dyne and the poundal. The dyne is a force which will cause a mass of one gram to have an acceleration of one centimetre per second every second. The poundal is a force which will cause a mass of one pound to have an acceleration of one foot per second every second. These two units are absolute, because they are not affected by surrounding conditions. The mass of a body is the same whether it is located in the vicinity of the earth, the moon, or is alone in space. Such units are of great value in science. There are also other units of force, such as the pound, the gram, the kilogram, and in fact any of the units by which the quantity of mass is ordinarily determined may be used as units of force. The pound or gram of force is that force which will support a mass of one pound or one gram. This gives the primary idea of what is meant by one pound or one gram of force, — i.e., the unit is thus determined. The force may then be exerted in any direction, whether gravity is considered or not, and it will be such a force that if it were exerted against gravity it would lift one pound or one gram as the case may be. Such are known as gravitational units. They are not absolute, because gravity changes with change of distance from the centre of the earth and with change of location on the surface. The relation between the dyne and the gram as units of force is found from a consideration that if 1 dyne gives Ig (mass) an acceleration of 1 ^/sec^, and Ig (force) gives to Ig (mass) an acceleration of 980 "^^/sec^, as determined by the acceleration of falling bodies, then Ig (force) must be equal to 980 dynes. In the same manner it may be shown that 1 lb. (force) is equal to 42 GENERAL PHYSICS. 32.2 poundals. These numbers are not exact except for certain locations on the earth's siirface. At points near the equator the acceleration due to gravity is 978 "^^/sec^; near the poles, about 983; at Cincinnati, 980. (See page 298.) The value of the poundal in terms of dynes, found by mtdti- plying the number of centimetres per foot (30.48) by the number of grams per pound (453.593), is 13,825.5. The same result is found by use of the dimensional equations, as follows: Since F = ma, the dimension of force is the product of the dimensions of m and a, [M][LT-^=[MLT-^ Let the dimensions when the unit is the poimdal be [M,Lir,~^], and when the unit is the dyne be [MjLjTj"^]. The numeric x will be the number of dynes in the poundal. Hence M T T 2 ■^=453.593 4^ = 30.48 4^=1 .-. a; = 453.593 ■ 30.48 • 1 = 13825.5 .•. 1 poundal =13,825.5 dynes The purpose of this kind of solution of so simple a problem is not to obtain a result, but to give exercise in the use of dimen- sional equations. 24. Impulse and Momentum. — Impulse involves two factors, — a force and the time dturing which the force acts. It is a matter of common experience that the velocity of a freely moving body will be increased by increasing either the force which causes the motion or the time during which the force is applied. The product of these two factors is called impulse. Thus, impulse = Fi The effect of the impulse is a quantity of motion in a mass of matter which is free to move. The amoxint of motion in any moving body depends on two factors, — the mass of the body and its velocity. The product of these is momentum = mv DYNAMICS. 43 The momentum is therefore proportional to the impulse, and we may write Ft=mv (42) This equation may also be derived from a consideration of the fact that F = ma and "^ ^ T „ mv or Ft = mv There are no names for the units of impulse and momentum, but the symbols of the units may be found from the dimensional formulae. Impulse is the product of force by time, hence its dimensions are [MLT-^][T] = [MLT-'] Momentum is the product of mass by velocity, hence the dimen- sions are [M][LT-'] = [MLT-'] Hence the symbol of the unit for either impulse or momentum Icf 1cm in c.sf.s. units is — f , — i.e., the unit momentum is the amount *" 1 sec of motion in 1 g. moving with a velocity of 1 cm. per sec. , and unit impulse is that which will produce this effect. To use equation (42), F must be in dynes or poundals, de- pending on the units employed. By Newton's second law, any impulse will produce a certain change of momentum independent of other impulses applied at the same time, and independent of the momentum which the body may already have. Momentum involves not only a quantity of motion (mv), but also a direction. It is therefore a vector quantity. Change of momentum is always in the direction in which the force is applied. 25. Stress and Strain. — Stress is a mutual action between two forces or between a force and that which resists it. Strain is the deformation resulting from a stress. 44 GENERAL PHYSICS. For illustration, if two forces are applied in opposite direc- tions at the ends of a wire, the wire is subjected to a stress called tension. The effect is the same when one end of the wire is fastened to a rigid support, the resistance of the support being equal to the active force at the other end of the wire. The strain in this case is the change in length of the wire, and its measure is the ratio of the increase in length to the original length. If a body of gas is enclosed in a cylinder and a piston is pressed down upon it, there is the mutual action of the force upon the piston and the reaction of the gas within the cylinder. This stress is called a pressure. The strain in this case is the decrease in volume, and it is measured by the ratio of the diminution of volume to the original volume. If an elastic rod is rigidly fastened at one end and the other end is twisted through a few degrees, there is the mutual rela- tion of force and resistance called shearing stress. The strain in this case is measured in a manner which is described later under elasticity. In all cases, whenever a force is applied, an equal and oppos- ing force or resistance is offered to it. Without resistance there could be no such thing as force. If a force is applied to a body which does not move, the effect is a deformation resulting from a stress. If the body is free to move, the resistance due to the inertia of the body is equal to the force which causes the motion. The force need not be a "little greater" than the resistance to produce motion. 26. Graphical Representation of Forces. — Since force is a vector quantity, it may be represented in magnitude and direc- tion by a line whose length, drawn to proper scale, represents the magnitude and whose direction is that in which the force is applied. The composition and resolution of forces are effected in a manner which has already been described for velocities. The parallelogram and polygon of forces, the methods of cal- culating resultants and their direction, and the resolution of forces are the same as in the case of velocities. 27. Resultant and Equilibrant. — A resultant force is one which may be substituted for two or more other forces and which will produce the same effect as the others combined. An equilibrant is equal and opposite to the resultant. DYNAMICS. 45 The resultant of two parallel forces in the same direction is their sum; when parallel but opposite in direction, the result- ant is their difference; when the two forces are perpendicular to each other, the resultant is the hypotenuse of a right-angled triangle of which the two forces are the legs. When the angle between the forces is acute or obtuse, the resultant is found by use of the well-known equation r' = d' + b^ + 2ab cos ^ (compare equation 19) Fig. 46. The graphical method of solution may also be used in any of these cases. An illustration is here given. Let oa,o6,oc, and od rep- resent forces acting as indicated in Fig. 46. Through o draw rec- tangular coordinates and resolve each force into its components along the X- and Y-axes. The components oe and of are posi- tive, while oh and og are negative. Also, ea and hd are positive and fb and gc are negative. By measurement and addition, all the forces may thus be composed into two at right angles. The re- sultant can then be easily found. When the angles made by the vectors with the axes are known, the values of the components can most easily be cal- culated from the simple trigonometrical relations. 28. Resolution of Forces. — The principles and methods of resolution of vector quantities have already been discussed. These principles have many important applications, and addi- tional illustrations will here be given. Suppose a force is applied in the direction BC, Fig. 47, on a rope attached to the top of a pillar AB, and it is desired to know the component of this force which is effective in pulling the pillar over. BC can be resolved into the two components BD and DC. An infinite number of components may be drawn, but these are the two which are desired in this case, for DC has no effect in pulling the pillar over, while BD is acting at the greatest advantage for the pur- pose in view. Hence the effective part of the force is BC sin d. The larger d becomes the greater the sine will be, and at 90° 46 GENERAL PHYSICS. the sine is tinity. Hence the longer the rope is the more effective the force will be. Another illustration of resolution may here be given, in case of wind or water directed against a turbine wheel. Let TT', Fig. 48, represent one blade of the wheel. Let the plane of rotation be as indicated by the arrow D and let the force of the wind be represented by the vector wo. Resolve wo into wp. Fig. 47. parallel to the blade, and po, perpendicular. It is evident that po is the only effective component, but it is not in the direction of the rotation. Let po, for convenience, be extended to m, making om equal to po. Resolve om into sm and os. Then sm is the component whose value is to be found. Inspection of the figure shows that, since the plane of rotation is kept per- pendicular to the direction of the wind, sm = om sin ^ = po sin

^ = 2 + 3 + 1+6 = ^^ '='"• *'"°^ " The point of reference o may be taken at any distance from the system of bodies; as here taken, the centre of gravity of the system is 5^^ cm. from o, or 4^ cm. from the mass of 2 g. When the particles are not in a straight horizontal line, but in a vertical plane, as represented in Fig. 70, the distance of the centre of gravity from both the X and Y-axis must be found. Thus — j, will locate the centre of gravity some- where in the line a parallel to the Y-axis, and — „ will locate 2m it in line b parallel to the X-axis. It is therefore at their point of intersection. DYNAMICS. 69 In case the particles are so arranged that three dimensions must be considered, the centre of gravity is located by finding its distance from three planes of reference, XZ, XY, and YZ, Fig. 71. Thus — ^^ will determine its distance from the plane YZ; its distance from XZ\ and its distance from XY. The centre of gravity is therefore at the point of intersection of the three planes. z X _I — >: Fig. 70. Fig. 71. 43, Centre of Mass. — The centre of gravity and centre of mass coincide, but their definitions are different. The centre of mass is a point whose distance from the three planes of reference is equal to the mean distance of the particles, supposed equal, from the same planes. Centre of mass may be found by the same methods as have just been described for centre of gravity. When a body is regular in shape, as a sphere or cube, and is uniform in density, the centre of mass or centre of gravity is at the centre of figure. 44. Stable Equilibrium. — In reference to the action of gravity on masses of matter, a body is said to be in stable equilibrium when it is so situated in reference to any axis that a moment of force tending to produce rotation on that axis causes a rise in the centre of gravity. Let A, Fig. 72, be a body suspended from 5, the centre of gravity being at c, in a line drawn from 5 toward the centre of the earth. This line is the direction of the resultant of all the forces of gravity acting on the body, and consequently there is here no moment tending to cause rotation on the axis s. Let the body be now changed to the position B. There will then be a moment equal to the product of the resultant 70 GENERAL PHYSICS. by the distance sa, tending to restore the body to its position at A. Any disturbance of the body that will cause rotation about 5 will cause a rise in the centre of gravity. Consequently the position at A is said to be stable. If a body is placed as in Fig. 73, where the centre of gravity is directly above the support, the condition is one of unstable equilibrium, for the least disturbance will result in a moment that will bring c below s, and the condition will then be stable again. A B Fio. 72. When the axis of rotation passes through the centre of gravity, the body is said to be in neutral equilibrium, for rota- tion can neither raise nor lower the centre of gravity. In general, whether gravity is concerned or not, a system is said to be stable when it tends to restore its configuration after certain disturbing forces have been removed. 45. Determination of Mass. — One method of determining the quantity of matter or mass of a body is to apply to it a certain force and note the acceleration produced. Then, from the relation F = ina, m can be found. Thus, if a force of 500 dynes produces an acceleration of 5 '^'"/sec^, the mass must be 100 g. This method has the advantage of being independent of the force of gravity, but it is impractical in ordinary operations. The practical method is to determine the force of gravity, or weight, and consider the mass as proportional to the weight. That this proportionality exists has been shown experimentally by dropping masses of different size and kind from a height and noting that they reach the ground in the same time. To produce the same acceleration in a large mass as in a small one. DYNAMICS. 71 the force must be proportional to the mass, for F — ma. Newton performed an experiment in which he used a pendulum with a hollow bob. By filling the bob at different times with various kinds of matter and counting the vibrations through a con- siderable length of time, he was not able to detect any difference in period. From this he concluded that the force of gravity is directly proportional to the mass and independent of the kind of matter. If the force of gravity exerted on 1 g. of matter is g, — 980 dynes, — then the force on m grams is mg, which is the weight. At any point on the earth g may be considered a constant quantity ; hence any change in the weight, mg, must be a change in mass. In instruments commonly used for determination of mass, two principles are employed. First, the principle stated in Hooke's law, where the amount of displacement caused by a mass suspended from the end of an elastic coil or rod is taken as . ^ ^ a measure of the mass. All spring i\ " scales are based on this principle. Second, that there is equilibrium when the sum of the moments _ Fig. 74. tending to produce positive and negative rotation about an axis is zero, and that, when the ratio of the distances from the axis to the line of direction of the forces is known, the ratio of the weights will be inversely as these distances. Thus, in Fig. 74, let a and b be distances from the axis o to the line of direction of the weights w and w' respectively. When the system is balanced, wa = w'b (66) w b w' a b w When the ratio — is known, that of — r is also known, and when a w' the mass in one scale pan is known, as is usually the case, the value of the unknown mass is easily determined. If a and b are now made equal in length, w will equal w' whenever there is equilibrium. Balances for accurate weighing are constructed in this manner. or 72 GENERAL PHYSICS. with very little strain. M In Fig. 75 is a diagram of one style of beam used in physical and chemical balances. Its shape permits considerable stress At the centre is a knife-edge o made of agate and resting on a plate of agate, p. The knife may be two or three centimetres long and is here shown only in cross section. Near the end of the beam are other agate knife- edges o' and o". Upon these rest agate plates from which scale pans are suspended. A long pointer is rigidly fastened to the beam and sweeps over the scale 5 when there is any rotation on o. The beam must be so constructed that it will be stable, — i.e., it will when disturbed rotate from side to side on o and finally come to rest in its original position. The distances oo' and oo" must be equal, to make the balance c Trrmfr TTTTTTJ Fig. 75. true. The same results must be obtained when equal masses are weighed at different times. A small difference of mass in the pans should cause a movement of the pointer, — i.e., the balance should be sensitive. The movement of the pointer should be sufficiently rapid, so that too much time may not be consumed in the operation of weighing. StabiUty is secured by constructing the beam in such a manner that its centre of gravity is a short distance below o. In the diagram, Fig. 76, let the centre of gravity be at c, directly below o when the beam is in a horizon- tal position. If the beam is turned through an angle 0, c will move to c', the Une oc also turning through an angle d. Let W,, be the weight of the beam, then the moment tending to restore it to a horizontal position is Fig. 76. Wi.f^' = Wt,.oc' smO Let a and h be the arms of the beam, and let weights be placed in the pans such that the one suspended from o' is greater than DYNAMICS. 73 that from o" by a difEerence d. Then the moment tending to produce positive rotation is d . a cos d while that tending to produce negative rotation is Wt . oc' smO^Wt, . a;sin(? by letting x stand for oc or oc' Hence d . a cos = Wj, . « sin 5 sin d , „ a or cos d = tan 6 — W^x (67) (68) From this equation it appears that to make the balance very sensitive, — i.e., to construct the beam so that a small difference d may produce as great a value for d as possible, — a should be made long, and W,, and x as small as possible. If a, however, is made very long, the time of the swing becomes very long. If Wft or X are decreased, the moment tending to restore the beam to a horizontal position is decreased and again the time of the swing is increased. These conflicting results are all considered, and a balance is constructed best suited to the purpose for which it is intended. Mounted at the centre of the beam is a nut which may be raised or lowered, thus changing the position of the centre of gravity. In this way different degrees of sensibility may be obtained at the will of the operator. That there may be the same sensibility when the differences in the weights on the pans are not the same, it is necessary that the three knife-edges be in the same plane. In Fig. 77 let cabd be a rigid beam with the pans hung from c and d. Let the beam be turned to the position ef. It is observed that the moment of the force F has de- creased, for oa has decreased to oi, but the moment due to F' has, in the position shown, actually increased. Hence the moment tending to restore the beam to a horizontal position will be increased, and therefore the pointer 'tF Fig. 77. 74 GENERAL PHYSICS. will not move over a number of divisions of the scale proportioned to the difference in load, as it will do when the knife-edges are in the same plane. To obtain the true weight, the length of the arms of the beam must be equal if they are assumed to be so. If they are not Fig. 78. equal, the true weight may be found by double weighing. Let w' be the apparent weight as indicated by standard weights placed in the pan A, Fig. 78, while the body whose true weight is w is in B. Then w'a= wb (69) Now let w be placed in A and standard weights w" in B ; then w"h—wa (70) The product of these two equations is w'w"ab = w^ab (71) hence w=i/i(:;'w" (72) The true weight is therefore equal to the square root of the product of the apparent weights. In very accurate weighing this method is often used even when the lengths of the arms are supposed to be equal. Another method for the elimination of error due to difference in arm length is to place the object whose mass is desired in one pan and counterbalance it with any convenient mass in the other. Then remove the object and put in its place standard weights until equilibrium is again restored. The mass of the standard weights and that of the object must be the same. Problems. 1. Calculate the moment of inertia 7 of a cylinder rotating on its own axis, the radius being 10 cm. and the weight 3 kg. 2. Find 7 of a wheel, radius 1 m., when a force of 500 dynes applied to the rim causes an acceleration of »r radians per sec'. DYNAMICS. 75 3. A mass of 500 g. is made to revolve 120 times per minute in a circle of radius 50 cm. Find the centripetal force. 4. What must be the speed of a conical swing that a car suspended from the top of the central pole by a rod 50 feet in length may incline at an angle of 45°? / v^\ {F^ — mgta.n0 F„ = nia a = ~) 5. What velocity will a body acquire in falling from a point 1000 miles above the surface of the earth to the surface, on the supposition that the body will encounter no resistance? 6. Show that, if three equal masses are placed at the comers of an equilateral triangle, their centre of gravity will be at a point one-third of the distance from the middle of the base to the opposite vertex. 7. A weight of 500 lbs. is suspended from a rope and allowed to descend from the top of a building with an acceleration of 10 ^^^^/sec''. What is the tension of the rope? 8. A uniform beam weighing 100 lbs. is at rest in a horizontal posi- tion on a fulcrum placed one-fourth of its length from one end. A weight of 500 lbs. hangs from the end nearest the fulcrum. What is the weight on the other end. 1. 150,000 g. cm'. 2. 15,915.45 g. cm^ 3. 4028.4 g. 4. 33.7 feet/jec. 6. 3.12 miles/sec 6. . 7. 344.7 pounds. 8. 133.33 pounds. 46. Work and Energy. — When force applied to a body produces motion in the direction in which the force acts, work is done. The magnitude of the force times the distance through which it is appUed is a measure of the work done. This is expressed by W = Fs where W is the work, F the force, and s the space or distance. No work is done in the application of force unless motion results. The figure of a giant cut from stone and placed as one of the supports of a building does not do any work, however great the weight upon his shoulders. It is not necessary that the body as a whole should be moved that work may be done. If the body is elastic, the force may compress, elongate, or twist it, and thus the point of application 76 GENERAL PHYSICS. of the force may move through a certain distance. When this force is removed, the body may in turn do work on other bodies. By virtue of work done upon them bodies are able to do work, and are then said to possess energy. Energy is the capacity for doing work. Work in general may be defined as the process of transferring energy from one body to another. It requires work to lift a stone to the top of a building, but the energy expended is then in the stone by virtue of its position relative to the ground. Work must be done in setting a mass of matter in motion, and the mass then contains energy by virtue of its motion. It re- quires work to pass a current of electricity through a storage cell, and the cell then contains the energy expended, by virtue of chemical changes which have been made. While energy exists as a distinct physical quantity, it is here assumed to have no inde- pendent existence. Wherever it is found, it is associated with matter, matter here being taken in a sense to include ether. Whatever difference of opinion may exist in regard to the essential character of matter Fig. 79. ^^^ energy, yet for purposes of physical investigation no false conclusions will be reached by assuming that matter has an objective existence and that energy is a rela- tive condition of matter by virtue of which work may be done. The chief function of matter, then, is to serve as the vehicle of energy. The mill owner cares nothing for the water in the mill-pond as a body of water, but he highly prizes the fact that the water is on a higher level than the mill-wheel. The engineer cares nothing for steam as a vaporized mass of matter, but he makes use of the rapidly moving molecules to drive the piston of his engine. The value of foods consists not so much in the material of which they are composed as in the work which has been done upon them by sunlight. Matter in this respect may be defined as anything capable of possessing energy as a result of work done upon it. The two factors which determine the quantity of work are the force applied and the distance through which it is applied. DYNAMICS. 77 The product of these factors is a measure of the quantity of work done. In case the direction of the force is at an angle to the direction of the motion, only that component of the force which is in line with the motion is considered. Let F, Fig. 79, be applied to raise a mass m. Let F make an angle with the direction of the motion. Resolve F into two components, F^ and Fj. The component Fj causes only pressure against the support while F, is effective in lifting the mass. Hence the work done is W = F^=F cose .s (73) where s is the vertical distance through which the mass is raised. The greater part of the energy expended in raising this mass against the force of gravity is now possessed by the mass. A certain quantity of energy may be said to have been trans- ferred to it, and it as a consequence is capable of doing work. 47. Units of Work and Energy. — ^As in the case of force, so for work and energy there are both absolute and gravitational units of measurements. The absolute units are the erg and the foot-poundal. The erg is the work done by a force of one dyne acting through a distance of one centimetre. The foot=poundaI is the work done by a force of one poundal acting through a distance of one foot. These units are absolute because the factors which enter into their definitions are invariable in time and place. They are not dependent on a variable force such as gravity. Since work is the product of a force by a distance, the di- mensions of its units are [MLT-'] [L] = [MUT-'] 1 ST 1 cm^ The symbol of the c. g. s. unit, then, is — y j— , — i.e., the erg is the product of 1 cm. by a force which will give to 1 g. an accelera- tion of 1 '^"/sec!- The symbol for the foot-poundal is — — . The relation of these two tmits may be found in the usual manner by finding the numeric for dimensions in c.g.s. units 78 GENERAL PHYSICS, when that for foot-poundals is unity. Thus, hence x= \TL . ^ ^1 -1X453.59X30.48=' = 4.214(10)= hence there are 4.214(10)* ergs in a foot-poundal. The erg is inconveniently small, being only about the amount of work required to raise one milligram to a height of one centi- metre. For this reason, ten million (10') ergs are employed as a practical unit called the joule. The gravitational units of work are the foot-pound, the gram- centimetre, the kilogram-metre, or any convenient product of a gravitational force by a distance. A foot=pound is the work done by a force of one pound exerted through a distance of one foot. The gram-centimetre is defined in a similar manner. In the gravitational units the force is that of gravity exerted on a pound or gram of matter. Since gravity is not the same at all points on the earth, these units will differ with location. Gravitational units are in common use by engineers, for in most structural work gravity is the chief agent against which force must be exerted, and for rough work the slight variations in the force of gravity in different localities may be neglected. It is possible, however, to select a certain locality, say latitude 45° at sea level, and consider the force of gravity per unit mass at that point as the standard. Then, if a mass m at that place is raised a distance h, the work done is mh in foot-pounds or gram- centimetres, depending on the units selected. The standard force of gravity per gram is 980.6 dynes. Then, in another locality where g is, say, 978 dynes, the work performed by the same operation is 978 W= mh ft.-lbs. or g.-cm. Since weight is the product of mass, m, by the force of gravity, g, per unit mass, the work W done in raising a mass to a height h is W = mh gram-centimetres or foot-pounds and W = mgh ergs or foot-poundals DYNAMICS. 79 From this relation it is easy to pass from the absolute to the gravitational system or vice versa. One foot-pound equals approximately 1.355(10)' ergs. Work may be represented by an area, as shown in Fig. 80. Let divisions on the abscissa represent distance; and those on the ordinate, intensity of force. Suppose each millimetre on the abscissa stands for one centimetre of actual distance, and those on the ordinate represent dynes. Then 30 dynes acting through 30 cm. would do work which is here represented by the rec- tangle oahc, — 900 square millimetres, each representing one erg of work. If the force is not uniform but is constantly changing, the work may be represented by the number of unit squares enclosed by the abscissa and the curve def. This curved line is the locus of all the points whose distances from the abscissa represent the intensity of the force at each successive small change in the distance. This graphical method of representing work will be used in later discussions. 48. Power. — Power is the rate of doing work. If the quan- tity of work only is specified, there is no indication of the time within which it was performed. A small force acting for a long time may accomplish as much work as a large force acting for a short time. A small boy may do as much work as a strong man. But when the element of time enters into the considera- tion, then the time-rate is a measure of the power, or, as it is sometimes called, the activity of an agent. Steam engines differ in power because they differ in the rate of doing work. A common unit of power is the horse=power. An engine or other agent is said to have one horse-power when it can do 33,000 foot-pounds of work in one minute or 550 ft. -lbs. in one second. Another unit of power is the watt, which is the rate of doing work when one joule is performed in one second. There are approximately 746 watts in one horse-power. 49. Potential and Kinetic Energy. — Energy, as has already been defined, is the capacity for doing work, and work is an operation of transferring energy from one body of matter to another. The amount of energy lost by an agent in the per- 80 GENERAL PHYSICS. formance of work is a measure of the work done. Hence the units for energy are the same as those for work. All energy is potential in the sense that it is a capacity for doing work, but a division is commonly made into potential and kinetic energy. The energy which a body possesses by virtue of its position in relation to other objects, or by virtue of a strain which has been caused by work done, is called poten= tial. Examples of this are particles of carbon in coal in relation to the oxygen in the air, the coiled spring of a watch or clock, a mass raised to a position from which it may fall. Kinetic energy is the energy which a body has by virtue of its motion. A heavy projectile in flight possesses a great deal of energy, by virtue of the fact that the exploding powder did a great deal of work upon it. It is desirable to have an equation by which the kinetic energy can be calculated when the mass and velocity of the moving body are known. To do this we assume that any mov- ing body has the energy which was given to it by a force that started it from rest and accelerated the motion until the velocity was as now observed. While the velocity is uniform, no energy is gained or lost, as may be inferred from Newton's first law. The force F which caused the acceleration a in mass m, is The distance, s, through which the force acted to give the mass the observed velocity, v, is s = ^ (from v^ = 2as) The formula for energy or work is W or E=Fs (74) v^ hence Ejdn = '"^^q- = i***^^ (75) From (75) the kinetic energy in ergs or foot-poundals may be found when velocity and mass are known. 50. Energy of a Rotating Body. — When the mass and angular velocity of a rotating body are given, the energy of each particle of the mass is foimd by (75), — namely, E]nn = \mv'. In Fig. DYNAMICS. 81 81 let a solid disk be represented as rotating on axis o. Con- sider the energy of a particle m^ which is moving with angular velocity w at distance r from the axis. Since w is the number of radians per second, oir is the linear velocity of m^, hence the kinetic energy of the particle Wj is Ekin = iw^miH (76) The total energy is the sum of the energies of all the particles, hence E)^-^ = ^'' {rn^r^ +m^r^ -k-m^r^ .... -|-m„0 Fig. 81. The sum of the quantities within the parenthesis is the moment of inertia of the rotating body, hence E^n = ¥^' (77) Since I = Mk^ where M is the total mass and k is the radius of gyration, E^^ = iMaj'k' (78) Equation (78) simply states that the kinetic energy of a rotating body is equal to one-half the mass times the square of the linear velocity of that point where the total mass is supposed to be concentrated. If e.g. s. units are used and the value of o) is taken in radians, the result will be in ergs. 51. Conservation of Energy. — The total quantity of the energy in the universe is constant. The operations of life and the activities in nature show a constant change of energy from place to place and from body to body, but what is lost at one point is gained at another, and the total quantity remains unchanged. This is the doctrine of the conservation of energy. The material changes which are observed in the world or universe are the transferences of energy from body to body, or the result of such transferences. The potential energy, for example, which exists in coal by virtue of the relation between 6 82 GENERAL PHYSICS. carbon and oxygen may, by burning the coal beneath a boiler, be converted to kinetic energy of countless numbers of particles of water, and these may in turn transmit their energy to an engine, causing mechanical motion. The engine then passes the energy on to various machines whch it operates. The sum of all the work done in this instance, counting the energy lost by friction or otherwise, is exactly equal to the energy which was originally in the coal. Another illustration may be given in the case of a falling body. Let a mass m be located at a height 5. The body is then said to possess potential energy, the quantity of which in gravi- tational units is ms, and in absolute units mgs. This may be expressed by Ep=-mgs (79) where E^ is potential energy. If now the body falls through the distance s, it will acquire a velocity v, and its kinetic energy will be But v^ = 2gs, hence ■Ekin = \m . 2gs = mgs = E^ Hence the quantity of energy is the same whether it is kinetic or potential. When the body has fallen part of the distance, the energy is part kinetic and part potential. At one-third the distance from the top, for example, two-thirds of the energy is poten- tial and one-third kinetic. The same is true in case the body de- scends along an inclined path AB, Fig. 82, for the length of the path is —. — ^, where sina s is the vertical height, and the value of F Fig. 82. along the incline is mg sin 6, hence E = -1 — ;; tng sin d = mes sm c* 52. Available Energy. — While it is an established fact of science that all the energy expended in the performance of work is conserved, yet it is not all available for the performance of other work. For example, when a heavy stone is raised by DYNAMICS. 83 means of a wheel and axle and pulleys, a part of the resistance is due to friction in the various parts of the machinery. Extra work must be done because of friction, and this results in heat, which consists in an increased motion of the molecules. This portion of the energy is thus dissipated throughout the mass of matter and into space. The energy is not now available by man for the performance of other work. It is not lost in the sense that it has been destroyed, but only in the sense that it was not applied in increasing the potential energy of the stone. The energy which is transferred to the stone is available, for it may, by proper connection with a machine, be made to do work while it descends. Only a small part of the total energy of the world is available for work, for, no matter how great the quantity may be, it is only while energy is being transferred that work can be done. If all water were at the same level, there would be no rivers or water-falls; but when there is a difference of level, the fall is accompanied by a change from potential to kinetic energy, result- ing finally in heat at the bottom of the precipice. This energy in a sense may be said to be lost because it is not available for the performance of work. If, however, a wheel of proper construc- tion is placed in the path of the falling water, much of the energy may be made to do useful work. It is only while the water is falling that its energy is available. In a similar manner, if all bodies were of the same tempera- ture, none of the great store of molecular energy would be avail- able. But when the temperature of one body is higher than that of another, then, in the process of transference of heat, energy becomes available and work may be done. The steam engine, as will be shown later, is a device for utilizing the mo- lecular energy of steam while heat is being transferred from a hot to a cold body. Numerous examples of this kind may be given, showing that the constant tendency is to diminish the quantity of potential energy and produce a condition of uniformity under which no energy would be available for the performance of work. The sun is the chief source of energy on the earth. To it alone we are indebted for that store of potential energy which makes life possible. 84 GENERAL PHYSICS. Problems. 1. A ladder 30 feet long stands at an inclination of 30° to a vertical wall. How much work will be done in carrying a mass of 50 pounds to the top of the ladder? 2. A force of 5000 dynes is applied at the end of a rope to drag a mass along a horizontal surface. The rope is inclined 35° to the surface. How much work is done in moving the mass a distance of two metres ? 3. If one joule of energy is expended in lifting 1 kg., to what height will the mass be raised? (g = 980). 4. What is the horse-power of an engine capable of lifting 2 tons of brick to the top of a 50-foot building in 5 minutes? 5. A mass of 200 g. is thrown vertically downward with a velocity of 50 cm/ggg at a point where the acceleration due to gravity is 980.5 <=™/sec'. If its fall is not obstructed for 10 sec. what is its energy at the end of that time ? 6. A cylinder whose mass is 2 kg. and radius 3 cm. rotates on its own axis ten times per second. What energy does it possess ? 1. 1299 ft.-lbs. 2. 819,150 ergs. 3. 10.2 cm. 4. 1.21 h.-p. 5. 9.712(10)' ergs. 6. 1.776(10)' ergs. 53. The Simple Pendulum. — A simple pendulum consists of a particle suspended from a point and capable of oscillation under the influence of gravity. The rod or chord connecting the particle to the point from which it is suspended is supposed to be without weight. Gravity exerts y^ no influence upon any part of the pendulum except the suspended particle. Such a pendulum cannot be real- ized in actual construction, but a ~^^^ near approach to it can be made by suspending a body, whose mass is supposed to be concentrated at a point, by a fine thread or wire of '"' ^^' negligible mass. Let a mass m. Fig. 83, be suspended from o, and let its position at any instant during oscillation make an angle Q with its position of rest at os. The force of gravity acting on m is mg, directed vertically downward. This may be resolved into two DYNAMICS. 85 components, F^ and F^. The latter causes only a tension of the thread and has no effect in moving m along the arc ms. The other component, F^, acts in a direction tangent to the arc at the point m, and thus is the part of the force of gravity that causes the pendulum to swing. This force is called the force of restitution, because by its action the pendulum is given an acceleration, either positive or negative, which restores it to the position of rest. When the pendulum swings to m or m^, the mass m is raised a distance sR, which we will call h in this discussion, and its potential energy is then mgh. When it falls to the lowest posi- tion, s, the mass m is moving in a horizontal direction with a velocity v. The energy is then all kinetic and is expressed by ■^w^. At the extreme limit of the oscillation the energy is all potential. At the lowest point it is all kinetic. At intermediate points it is partly potential and partly kinetic. The total quan- tity of energy is the same at all points of the swing. Hence or v^ = 2gh (80) Equation (80) shows that the velocity of the horizontal motion at 5 is the same as the vertical velocity of a mass falling from R tos. The force of restitution is expressed in terms of force of gravity and displacement by F^ = mg sin 6 (81) Now, in S. H. M. the force of restitution as well as the result- ing acceleration must be proportional to the displacement. Here the force varies as sin d and not as d. Hence the motion of the pendulum is not a S. H. M. The displacement is the distance from s to m measured along the arc, — i.e., 6 measured in radians, — and, according to equation (81), Fi varies as the sine of this angle. If, however, the amplitude is small, so that 6 is not greater than 2° or 3°, d and sin 5 will differ so little that one may be used for the other without appreciable error. 86 GENERAL PHYSICS. The period of a simple pendulum may be determined in terms of its length and the value of g in the following manner : The force of restitution, F^, Fig. 84, is ** F^ = mg sin d It has been shown in equation (40) that the acceleration in S. H. M. is 4^ where x is the displacement of the particle. Hence the force of restitution is Fig. 84. ^ In Fig. 84, X is the displacement as measured along the arc sm. Since d is measured in radians, x:=ld and we may write for the force of restitution F^= f5r— Substituting this value in (81) = mg sin d (82) Since the arc is assumed to be so small that d and sin d will not sensibly difEer, 47r^/ ■=g whence ' = 2^V (83) 54. Tiie Physical Pendulum. — A physical or compound pendulum is one in which the mass is distributed over the entire body of the pendulum, as a bar of wood or metal suspended from o, Fig. 85. It is evident that, as this bar oscillates, the particles near o would, as simple pendulums, move faster and those near s more slowly than they do when all are rigidly DYNAMICS. 87 connected. There must then be some point between o and 5 where a particle oscillates naturally, — i.e., as it would if it ^o were the particle of a simple pendulum. The length of the compound pendulum is the distance from this point to the point of suspension, or, in other words, it is the length of a simple pendulum which has the same period of oscillation. The compound pendulum is the only kind that can be realized in construction, and hence the only kind that is actually used. Any body suspended so that its centre of gravity is below the point of support is a compound pendulum. The problem, then, is to find the length of a simple pendulum that will vibrate in the same time. Let a body be suspended from o, Fig. 86, and let its centre of gravity be Fig. 85. at P. The force of gravity will be mg. The moment of force causing rotation about o is mg . Ps Let r be the distance from the point of suspension to the centre of gravity, then Ps = r sin 6 and mg Ps = mgr sin 6 this is the restoring moment when the angle is d. In the discussion of moment of inertia it was shown that Fr = IA (see equation 45) — i.e., the moment of force (Fr) tending to produce rotation is equal to the product of moment of inertia / and angular accelera- tion A. Hence mgr smd = IA (84) or mr g sin d (85) Suppose the whole mass to be concentrated at the point which vibrates naturally, and let the distance of this point from o be 88 GENERAL PHYSICS. represented by I. This is the length of a simple pendulum of the same period. The moment of inertia of this mass at the distance I from the axis is mP, and r has, under this assumption, become I. Hence l^^l^l^ (86) ml A ^ ^ hence Z = — (87) mr Substituting this value of I in equation (83), ^/_jL (88) V mer mgr or, since — is the square of the radius of gyration, we may write m P = 2n^ (89) From this it appears that, if the square of the radius of gyration is divided by the distance from the point of suspension to the centre of gravity, the quotient will be the length of a simple pendulum of the same period. The equation for the pendulum shows that the period of vibration is independent of the mass and,_within certain limits, of the amplitude, but varies directly as Vl and inversely as l/g. The pendulum furnishes a very accurate means of determin- ing the value of g at any locality. By a change in the form of (83) g = 5? (90) hence, if the values of / and P are accurately determined, g can readily be calculated. The point where the whole mass of the pendulum may be supposed to be concentrated without change in the period — i.e., the point which vibrates naturally — is called the centre of oscillation, as c in Fig. 85. This point is also called the centre of percussion, because, if an impulse is applied here in a direc- tion at right angles to the line from o to c, the axis of suspension will not be strained as it would be if struck above or below that point. DYNAMICS. 89 55. Reversibility of Compound Pendulum. — If a compound pendulum is reversed and suspended from its centre of oscilla- tion, the period of vibration will not be changed. To prove this it will be shown that the length of the equivalent simple pendulum is not changed by the reversal. Let OS be a rod suspended from o, its centre of gravity being at G and centre of oscillation at c. Let r be the distance from o to G and let li be the length of an equivalent simple pendulum. It has been shown that the length of an equivalent simple pendulum is — (see equation 89). It has also been shown that the square of the radius of ^1. ,1 gyration about any axis is equal to the square of the radius of gyration about a parallel axis through the centre of gravity increased by the square of the distance between the axes (see equa- tion 48). Hence h-^ (91) Lsr. or k^ = r{l^-r) (92) where k^ is the square of the radius of gyration about G. Now let the pendulum be reversed, and suspended from c. The distance from c to G is l^—r. Let Zj be the length of the equivalent pendulum after reversal, then i= = ^^±^ (83) just as in (91) above. Substituting the value of k^ from (92) in (93), :^^ ril.-r)^iyry ^^^l^_,^:^ (94) Hence the length, and consequently the period, is the same in either position. This principle is utilized in finding the length of a simple pendulum whose period is the same as that of the compound one. In Fig. 88 is shown a physical pendulum, usually known as Kater's pendulum, which consists of a brass rod having an 90 GENERAL PHYSICS. adjustable knife-edge near each end, and cylindrical masses which may be shifted on the bar. It is possible by proper ad- justment to find positions of the knife-edges at different distances from the centre of gravity such that the period will be the same when the pendulum is supported by either knife-edge. The Fig. 88. distance between the knife-edges can then be measured. This distance is the length of a simple pendulum of the same period. 56. Use of Pendulum for Measurement of Time. — It has been shown that the vibrations of a pendulum are practically isochronous, — i.e., the period is independent of the amplitude. By proper mechanism a pendulum may be made to record its DYNAMICS. 91 Fig. 89. own vibrations and so is an excellent means of keeping time. As the pendulum swings from side to side, it allows a tooth of the escapement wheel to pass the pallet at each swing. The escapement wheel is connected through a train of wheels to the weights or springs which are the source of energy, and also to the hands of the clock. The motion of the hands is thus controlled and regulated, but not operated, by the pendulum. Were it not for friction and resistance of the air, a pendulum once started would never come to rest. To keep the pendulum going, a wire extends from the pallet down a short distance along the pendulum rod, by which a slight impulse is given while the bob is at the lowest point of each swing. It is essential that the impulse be given while the pendulum is at the lowest point of its swing, L, Fig. 89. Let oa be a force applied while the pendulum has a position A. There will then be a component ah which will act in conjunction with the force of gravity, and consequently, during the time of the impulse, the force of restitution would be increased. The force F, however, acts in a horizontal direction and so has no component with or against the force of gravity. Problems. 1. The bob of a pendulum while passing its lowest point has a velocity of 100 c™/sec. To what height will it rise where g = 980 c^^sec^? 2. The displacement of a simple pendulum is 30° and its mass is 10 g. What is the force of restitution? 3. If in a certain locality a pendulum 99.3 cm. long beats seconds, what is the value of g? 4. If a pendulum loses 20 sec. per day at a place where g is 980.3 cm/ggg!, what is its length? 5. The length of a uniform cylindrical brass rod is 216.7 cm., its radius is 7.2 mm., and its mass is 2977 g. Its moment of inertia about an axis perpendicular to it through its centre of gravity is ^ma' + ^mb', where OT = mass, a = half the length, and i = the radius. If this rod is suspended at one end and made to oscillate as a physical pendulum, what will the period be? (g = 980). 1. 5.1 cm. 2. 4900 dynes. 3. 980.05 cm/sec!. 4. 99.37 cm. 5. 2.4 sec. 92 GENERAL PHYSICS. 57. Machines. — A machine is any contrivance through which energy may be advantageously expended in doing work. A machine is simply a medium for the transmission of energy. It receives energy by virtue of work done upon it or energy transmitted to it, and then may expend this energy in doing work. The kind of energy transmitted may be very different from that received, but the quantity, including the so-called lost energy, is exactly the same. A machine of itself can neither do work nor assist in doing work. It is only a convenient medium through which work may be done. This principle of conserva- tion of energy in machines makes the "perpetual motion" machine impossible. Work is defined as the product of a force or resistance by a distance. If the force applied to a machine is F, and the force applied by the machine is R, also if the respective distances through which they move are Sj. and S^, then FS^=RSr (95) This equation expresses the machine principle and states the equality between energy received and energy expended. Also, since each member of the equation contains two factors, either factor may be changed in value provided the other is at the same time changed in an inverse ratio. In this fact consists the chief advantage in the use of ma- chines. Thus, let FSf repre- sent a certain quantity of work. It is possible by use of a machine to make F one-fifth as great, for ex- Ci) Fig. 90. ample, and at the same time make Sf five times as great, the amount of work remaining the same. This principle may be further illustrated by reference to Fig. 90. Suppose it is desired to do the work of lifting a weight through the distance ab. This may be done by use of a simple form of machine, — the lever. If the force F is applied at c, c and a being equidistant from o, then F and R must move through equal distances and so F equals R as before. But if F is applied at d, od being, say, twice oa, then the arc df, or Sf, is DYNAMICS. 93 twice the arc ab, or 5,. Hence F=— . This illustrates how a machine may be an advantage, and also that, whatever changes may be made in F, the value of Sf changes in such a manner that the product of the two is unchanged. 58. Mechanical Advantage. — Mechanical advantage is a ratio expressing the number of times the force is multiplied by the use of a machine. It is the ratio of R to F. The ratio of Sf to 5^ gives the same result. Also, the ratio of any parts upon which the values of R and F depend, such as the arms of levers and the radii of wheel and axle, may be used to determine the mechanical advantage. If, for example, the ratio in any case is found to be 24, this means that the resistance against which the machine is capable of working is 24 times as great as the force applied to the machine. It is not to be imderstood from the term advantage that a force applied to a machine is always increased by that machine. The number expressing the advantage may be a fraction. It is often desirable that F be greater than R, but in that case Sf is proportionately less than 5,. 59. Kinds of Machines. — Machines are usually classified as simple and compound, the compound being a combination of simple machines. Simple raachines are of two kinds, — namely, the lever and the inclined plane. All simple machines may be classed with one or the other of these two. Thus, pulleys and wheel and axle are levers, while the wedge and screw are inclined planes. The ordinary hand pump is a simple machine of the lever form, by which the energy expended in the operation of the pump is transmitted to the water which is raised from the well. The dynamo is a simple machine of the lever form, which causes a flow of electricity. The various kinds of compound machines are combinations of levers and inclined planes. 60. Levers. — A lever is a mechanical device such that a force applied at one point produces or tends to produce rotation about an axis called the fulcrum, against a resistance which tends to prevent such rotation. The commonest form of lever is a straight or bent bar, the axis or fulcrum having various positions relative to the points of application of the force and the resistance. 94 GENERAL PHYSICS. ot Levers are usually divided into three classes, distinguished by the relative positions of the force F, the resistance R, and the fulcrum o. When o is between F and R, the lever is one of the first class; when R is between F and o, second class; when F is between R and o, third class. The mechanical advantage in any case is determined by the ratio of the distance fromF to o to that of R from o. If the direction of the force is not perpen- dicular to the lever, then, as already explained under the subject of moments, only that com- ponent which is perpendicular is to be considered as effective in producing rotation. Thus, in a bent lever aob. Fig. 92, let a force F be applied at a, its line of direction making an angle ^ with the arm oa. The only part of F that causes rotation about o is the component ac, equal to F sin tan (9 or F = i?-tan e (106) By use of either (105) or (106) the mechanical advantage may be found. 66. Friction. — ^When there is a relative motion between two bodies that are in contact, the resistance to this motion resulting from the contact is called friction. Fig. 103 102 GENERAL PHYSICS. Friction is observed on every hand. In all movements of liquids and gases friction enters into the calculation. This will be more definitely considered in a later discussion. The most important cases of friction are those of solids on solids. Friction is encountered in the operation of all machines. As a result of friction much energy is lost, as has already been explained. In the operation of most machines friction is, as far as possible, avoided by lubrication, by ball bearings, by pivotal bearings, and in many other ways. Friction is not, properly speaking, a force, for it does not exist until there is relative motion of bodies in contact, and when the direction of the motion changes, the direction of the resistance also changes. Since, however, the effect is the same as if an active force were exerted, the term force of friction is often employed. Friction between solids may be classified as sliding and rolling, the former being subdivided into static and kinetic. 67. Sliding Friction. — To illustrate a simple case of sliding friction, let a block, of mass w, rest on a horizontal plane AB, Fig. 104. The pressure P of the block, normal to the plane, is mg. Instead of the force of gravity it may be any other pressure normal to the plane. Let a force F act parallel to the plane. Y^9 Fig. 104. Pig. 105. then, as found by experiment, the relation between F and P for any given surfaces is such that when F is just sufficient to start the motion of the block, the ratio of F to P is a constant. This is a case of static friction, for it is the friction while standing and just on the point of starting. This constant ratio is called the coefficient of friction, and for static friction may be desig- nated by II. Then F ^=P (107) DYNAMICS. 103 — i.e., friction is independent of the area of the surfaces in con- tact, and varies only with the pressure normal to the surface in any given case under consideration. When the value of fi is once determined, the value of F for any given value of P can readily be found. One method of finding fi for any given material is to elevate one end of a plane, AB, Fig. 105, until the block m just begins to sUde. The inclination of the plane is called the angle of repose. Let d be this angle. The pressure normal to AB is mg cos d and the force which causes the block to slide is mg sin d Hence wgsm^ ^^^^^ mg cos ^ ' Thus the coefficient of static friction is equal to the tangent of the angle of repose. 68. Kinetic Friction.— The friction between two soUds when actually in motion relative to each other is called kinetic friction. This is usually less than static friction. The coefficient of kinetic friction can be found in a manner just described for static friction. The inclination of the plane. Fig. 105, is varied so that the block, once started, continues to slide with a uniform motion. Since the movement is not accelerated, the friction must just equal the component of the force of gravity which causes the sliding. Hence, if k is the kinetic friction, «=^M^ = tan^ (109) mg cos a ^ ' For illustration, suppose a mass of 1000 g. slides uniformly down a plane when d is 32° Then K = tan 32° = .6249 If now the same plane be placed horizontally, the force required to drag the mass along may be found from (109). Thus F or F = kP = .6249X1000 = 624.9 g. 104 GENERAL PHYSICS. — i.e., a force of 624.9 g. will be required to drag the mass* along with a uniform motion. If a force of 724.9 g. be applied, then 100 g. of the force will be expended in giving the mass an acceler- ated motion, and since F = ma 100X980 = 1000a or a = 98<='"/sec2. 69. Rolling Friction. — When a wheel, cylinder, or any circular body rolls on a horizontal plane, there is a resistance which causes the body to come to rest. The cause of the resist- ance is a depression in the plane at the point where the wheel is in contact with it, and also a slight flattening of the wheel at this point. The resistance is not friction, then, in the sense just explained. Let a wheel of mass m be rolled along a plane P, Fig. 106. Just in front is a slight bulge which is virtually an inclined plane up which the wheel must ascend. The resistance is mg sin 0, which is equal to a force that would cause the wheel to be in equilibrium on the incline. The smaller d becomes, the more nearly this resistance vanishes. Fig. 106. Resistance due to "rolling friction" is small compared to that of sliding friction, if the plane and wheel do not readily yield to pressure. An extreme example of resistance of rolling friction is the movement of a carriage wheel through a bed of sand or gravel. 70. Uses of Friction. — Friction is commonly considered as something to be avoided or reduced to a minimum. It, however, like all other conditions in nature, is an advantage in many ways. It makes possible the use of belts in transmitting energy from pulleys. Brakes applied to wheels would be useless with- out friction. Walking on a pavement or driving on a street or highway would be hazardous if there were no friction. Numer- ous examples of this kind may be given. DYNAMICS. 105 A special use of friction for determining the power of a machine is here described. The power of a steam engine, for example, may be found experimentally by use of a friction dynamometer. Let A, Fig. 107, be a broad-faced wheel at- tached to the shaft of an engine. A strap thrown over the wheel is fastened at one end to the spring scales s, and weights are hung on the other end. Sufficient weight is used to cause the engine to work at its full capacity against the friction of the strap on the wheel. When the wheel turns in the direction indicated by arrows, the friction tends to lift the weight w and relieve the strain of the spring in s. Hence the reading of j subtracted from the known weight w is the force of friction R. Hence R—w—s Let the number of rotations per minute be n, then the distance through which the force of friction is exerted is 27crn where r is the radius of the wheel. The work done is therefore 2KrnR or 27irniw—s) Since one horse power (h. p.) is 33,000 foot-pounds per minute, then if r, w, and 5 are measured in feet and pounds, and n is the number of revolutions per minute, 27:rn(w—s) ^■P" 33000 (110) 71. Efficiency. — Efficiency of a machine is the ratio of the quantity of energy which a machine transmits to that which it receives. Since some friction is always present, the efficiency can never be unity. If, for example, 500 foot-pounds of work are done on a machine and it in turn can do but 300 foot-pounds, its efficiency is 60 per cent. Any reduction of friction increases the efficiency of a machine. 106 GENERAL PHYSICS. Problems. 1. A weight of 50 kg. is suspended from the block of a single movable pulley. The strands of the cable are at an angle of 70° to each other. What force applied at one end of the cable will support the weight? 2. What is the mechanical advantage in a chain hoist where the radii of the wheels are 10 and 11 inches and the weight is suspended from a single movable pulley ? 3. The elevation of an inclined plane is 37° and the direction of a. force applied to hold a body on the incline makes an angle of 40° with the plane. What is the mechanical advantage ? 4. A chain is hung over an inclined plane in the manner shown in Fig. 108. Show that the force of gravity tending to cause the portion ha to slide down the incline is equal to that on he, and that there will be no movement of the chain even when friction is completely eliminated. 5. How much energy has been lost in friction if a mass of 100 g. sliding down a plane 80 cm. high acquires a velocity of 300 cn»/sec? 6. The slope of an inclined plane is 16°. What force will be required to drag a mass of 20 kg. up the incline when the coefficient of kinetic friction is .385? 7. If the radius of the body of a cylindrical screw is 1.5 inches, and the slope of the thread is 24°, what force applied to the best advantage 3 feet from the head of the screw will cause a pressure of 2 tons? 30.52. 1 .786:1. 1. 2. 3. 4; . 5. 3,340,800 ergs. 6. 12.9128 kg. 7. 74.2 lbs. CHAPTER III SOLIDS 72. Constitution of Matter. — The scientific conception of a body of matter is that it is composed of a great number of very small particles called molecules. These particles are never in permanent contact; consequently all substances are porous in the sense that there are spaces between the molecules. The particles are in a continual state of agitation or rapid motion, colliding with and rebounding from their neighbors or other bodies with which they come in contact. Many of the properties and phenomena of matter are the result of molecular arrange- ment and relations. Crystallization, tenacity, temper, rigidity, heat, expansion, elasticity, and many other subjects might be classed under the general head of molecular physics. The molecule is considered to be the smallest particle into which a substance can be divided without destroying the identity of that substance. Thus, a molecule of limestone, CaCOj, is limestone — as much so as a ton of it. If, however, this mole- cule is separated into its constituents by any chemical process, it becomes calcium, carbon, and oxygen. These parts are called atoms, and were originally supposed to be the limit of divisi- bility of matter, as the word "atom" indicates. There is now, however, very strong evidence that the atom is composed of many small parts called corpuscles or electrons. These may be the ultimate particles of which all matter is composed. This conception of matter furnishes a convenient model with which the mind can grasp and explain many of the observed phenomena of nature. The fact that by its aid satisfactory explanations can be made and new truths discovered, is suffi- cient justification for its existence. 73. States of Matter. — Matter is found in various states, depending on the relation and condition of its molecules. A solid is a body which will, under ordinary conditions, retain its shape and size by virtue of its molecular structure. The mole- cules of a solid can not, apparently, move beyond a limited space in which they vibrate. To furnish a mental picture, the 107 108 GENERAL PHYSICS. molecules of a solid may be considered as so related that they form a rigid framework. Within certain limits which are different for difEerent substances, the framework is of itself able to with- stand the stress to which it is subjected. It may yield, but will return to the original position when the stress is removed. This property is called elasticity and is discussed in succeeding paragraphs. A liquid is a substance in such a state that its molecules appear to move among their neighbors without any permanent restraint. A liquid when not confined by a solid will change its shape under the influence of a force however small. Liquids when exposed to air or other gases, or in a vacuum, have a definite free surface and are capable of being formed into drops. In these two respects liquids are clearly distinguished from gases. A gas is a substance in such a state that the molecules appear to repel each other and to move with freedom from point to point throughout the body of the substance. As a consequence, the shape of a gas is that of the total interior of a containing vessel. Many substances may easily be made to assume any one of the three states — solid, liquid, or gas — by application of the proper quantity of heat. Some bodies in their natural state partake of the nature of both solids and liquids. Such substances as ice and asphaltum, for example, have the rigidity of solids when subjected to a momentary stress, but if the stress is continued for a time they exhibit properties of liquids and will slowly flow. If asphaltum is placed in a funnel, it will, under the force of gravity, slowly yield and adapt itself to the shape of the funnel, flowing on through like a liquid. Such substances are said to be viscous. Ether is a substance which is assumed to fill all space, includ- ing even the interspaces of the molecules of a body. Ether is not matter in the ordinary sense of the term, — i.e., our senses furnish no evidence of the objective existence of ether; but there is strong evidence that ether exists and that it possesses some of the most important properties of matter. For example, light is known to be, not a substance, but a periodic disturbance of some kind in the ether. If light, then, is transmitted as a wave motion on ether, the ether must possess elasticity, for otherwise there would be no restoring force after the deforma- SOLIDS. 109 tion produced by the passage of a wave. Also, it is known that Hght has a finite speed of about 3(10)"' cm. per second; hence the medium on which it travels must possess inertia, for otherwise time would not be required. There are no direct raethods by which ether may be studied, and it is not known whether it is continuous or not. The substance which remains in a tube from which the air has been almost completely exhausted exhibits properties not observed in other states of matter. Sir William Crookes, who made an extensive study of this subject, referred to the condi- tion, when a current of electricity was passed through it, as a "fourth state of matter." This so-called fourth state appears to be one in which the constituent parts of atoms have been separated from their ordinary group arrangement and made to flow in a stream through the medium within the tube. 74. Elasticity of Solids. — Most solids are to some extent elastic, though some, such as lead and gold, are so slightly elastic that they are called plastic, — i.e., even a slight stress causes a permanent deformation. Others, such as ivory and steel, are very elastic, — d-.e., when the force which causes their deforma- tion is removed, they will regain their original shape and volume. A substance like rubber is fairly elastic, but is remark- able for its limit of elasticity, — i.e., it may be greatly deformed and yet will recover its shape when the deforming force is removed. Three kinds of elasticity may here be considered. 1. Elas- ticity of volume, where all parts of a body are subjected to an equal and normal stress. A body subjected to hydrostatic pressure is an example of this kind of stress. 2. Shearing elas- ticity, where stress causes a change in shape without change in volume, as, for example, the torsion of a rod. 3. Longitudinal elasticity, where the stress is in only one direction while at right angles to this the body is free to expand or contract, as, for example, a wire subjected to a longitudinal stress. The coefficient of elasticity is the ratio of a stress to the resulting strain. Thus, if a stress is denoted by F and the strain by 5, the coefficient k may be expressed by .=5 (111) V V and this is a measure ! of the strain. 1 'hen of elasticity is , _ stress _ P _ strain v V VP V or VP ^= k 110 GENERAL PHYSICS. 75. Volume Elasticity. — Let a body be subjected to a iini- form normal stress over its entire surface. Call this stress P. Let V be the original volume and v the change of volume result- ing from the stress. Then the change per unit volume is Then the volume coefficient (112) (113) When the value oi k, a constant quantity, is once determined for any given substance, the decrease in volume for any given pressure may easily be fotmd. The pressure is usually given in dynes per square centimetre and the volume in cubic centimetres. For example, a pressure of one atmosphere is 76 cm. of mercury, or 1033.6 g. per square centimetre, or 1.013(10)* dynes. A volume of water subjected to this pressure is known to decrease in volume by 5(10) ~^ of the original volume. Using 1 c.c. of water and substituting in (112) This quantity, 2.02(10)'°, is the volume elasticity of water. It is sometimes called the bulk modulus. The formula shows that it is the ratio of the pressure in dynes per square centimetre to the resulting change in volume per cubic centimetre. Volume elasticity is the only kind that liquids and gases can have, while solids have all three kinds named above. The volume elasticity of a gas is found in a manner described in §§ 94 and 177. That for a solid may be calculated by equation (124). The volume elasticity of liquids may be determined by means of a piezometer, which, in the form shown. Fig. 109, consists of a strong glass tube filled with water and provided SOLIDS. Ill with a screw and pltuiger for increasing the pressure. The liquid under exaniination is in the bulb B, which is provided with a fine capillary stem open at the top. The capacity of B and the capillary tube are known, and any change of volume of the liquid due to pressure may be indicated by the movement of a short thread of mercury in the stem. The pressure is calculated from the rise of liquid in the manometer M, which is a glass tube closed at the top but open at the bottom and filled with air. When the volume of air is reduced one-half, for example, the pressure is twice as great. The ap- parent change of volume of the liquid must be corrected for the decrease in the capacity of the bulb due to com- pression of the bulb, for although the hydrostatic pressure is equal on the inside and outside, yet each small cube of which the walls may be supposed to consist is pressed into smaller volume and hence the total volume is decreased just as if the bulb were solid glass, ^i Hence the correction is added. The piezometer may also be used to find the bulk modulus of solids. 76. Shearing Elasticity. — If several metal disks were piled one on top of another, a force applied in a horizontal direction to any one of them would cause a sliding of one relative to the others. If the disks were welded together and a similar force then applied, the effect would be to cause one portion of the solid to slide on an adjacent portion, but because of the rigidity of many solids the extent of the sliding is very limited. The change in the relative position of the molecules is called a shear. The stress is called a shearing stress, and the effect of the stress is a shearing strain. There is no change in the volume of the body as a result of the strained condition, for the molecules in each imaginary layer of the solid have not changed their distances from one another and the layers have come no closer together. If Fig. 109. 112 GENERAL PHYSICS. the strain does not exceed a certain limit, the molecules will return to their original positions when the force is removed. This ability to recover from a shearing strain is called shearing elasticity. 77. Coefficient of Shearing Elasticity. — The coefficient of shearing elasticity, also called the coefficient of rigidity, is the ratio of the force per unit area to the strain produced by that force, — i.e., it is the ratio of the stress to the strain. To find an expression for this coefficient in terms of measur- able quantities, let a thin cylindrical shell C, Fig. 110, be fast- ened at one end to a support A and let a force be applied at the other end to twist it through an angle 0. Each layer of the cylinder will thus be made to slide on an adjacent layer, the amount of sliding or shear layers being the same. The as a measure of the strain. ^G*' (^ f Fig. 110. between any two successive angle

4y6d3 If the bar is clamped at one end and loaded at the other, 79. Value of k in Terms of Y and n. — It is difficult to deter- mine by experiment the coefficient of volume elasticity, k, of solids; but if Y-, which is easily found, and the coefficient of rigidity, w, are known, the value of k may be found from «^ ^=9^^ (124) 80. The Torsion Pendulum. — A torsion pendulum is not a pendulum at all in the sense the term has already been used, for its operation is not dependent on gravity but on the elasticity of a twisted wire. If a heavy cylinder, for example, is suspended from a wire and is turned on its axis through a few degrees and then released, it will execute torsional vibrations whose period depends on the dimensions and rigidity of the wire and the moment of inertia of the cylinder. The vibrations are simple harmonic, for the force of restitution is proportional to the displacement, in accordance with Hooke's law for elastic bodies. Consequently the acceleration at any point of the vibration is, from equation (40), A=^ (125) where A is the acceleration, P is the period, and d is the dis- placement. By equation (45) A^^ (126) where h. is put in place of r to avoid confusion in later formulae. SOLIDS. 117 Combining (125) and (126), Fh 4nU e - p2 (127) ' = 2j:-Jjk_ whence P = 2n^l Fh (128) Fh The expression — r— is called the moment of torsion, and is a constant for a wire of any given length and radius. The con- stancy results from the fact that in an elastic body the strain (0) is proportional to the stress {Fh). 81. Use of a Torsion Pendulum in finding I. — If a cylinder of known moment of inertia is suspended from a wire, Fig. 113, and Fh the period of torsional vibration counted, the value of —r- may be determiaed once for all for that particular wire by use of equation (127). If any other body is then suspended from this same wire in place of the cylinder, its moment of inertia can be found by observing P and substituting in (127). Other methods of finding I by use of the torsional pendulum may be found in manuals for laboratory work. 82. Use of Torsional Pendulum for finding n. — From equation (119), 2Fhl or From (127), hence or n = nr*d Fh 6 Ttnr* 21 Fh e " 47tU pi nnr* inU 21 pz n = Snll r*P^ Fig. 113. (129) 118 GENERAL PHYSICS. 83. The Torsion Balance. — A torsion balance is a mechanism by which it is possible to measure a moment of force tending to twist a wire. A light rod, ab, Fig. 114, is suspended from a fine wire and the whole enclosed in a glass case. Any horizontal rotation of ab is resisted by the rigidity of the wire, and the angle through which the wire is twisted is proportional to the force which causes the twist, as required by Hooke's law for elastic bodies. Hence two forces may be compared by measuring the angle through which they will separately cause a6 to rotate. By means of a small mirror attached to ab at o, minute deflections may be read by use of a telescope and scale, or the upper end of the wire may be provided with a torsion head by which it is possible to read the angle through which the wire must be twisted at the upper end to balance the moment or torque at the lower end. It has been shown in equation (118) that the twisting moment is Fh^"^ (130) Fig. 114. This shows that a force applied at a or 6, tending to produce rotation in a horizontal plane, varies directly as the fourth power of the radius. If the radius is reduced one-half, for exam- ple, the force F need be only one-sixteenth as great to produce the same angular displacement. Thus by using very fine wire for suspension, very small forces may be detected. It was by use of an instrument of this kind that Cavendish first determined the value of the gravitation constant G. Refer- ring to equation (55), let the two small masses at a and b (Fig. 114) be denoted respectively as m and m'. Let two large masses, M and M', be placed at c and d. The force of gravitation between a and c at one end of ab and d and b at the other end will produce a couple which will cause a twisting of the wire. SOLIDS. 119 Since the moment of a couple is the product of either force by the distance between the two forces, the moment in this case is „wM -r Lr — ;- . ab where r is the distance between the centres of a and c or of 6 and d. Let the angle through which the wire is twisted be denoted by (p. The angle through which a unit moment of force (1 dyne at a distance of 1 cm. from the axis) will twist the wire is determined by a preliminary experiment. This may be done by causing ah, with masses m and m' attached, to vibrate as a torsion pendulum, the large masses M and M' meanwhile being removed. The moment of inertia of the system is known or can readily be found, and the period can be counted. Conse- quently the value of Q produced by unit moment can be deduced by use of equation (127), for all the terms of the second member of this equation are known, and, assuming that Fh is unity, the value of d is found. If unit moment of force will cause a deflection of d radians, then a moment which will cause a deflection

' (153) ^=V? (154) P By use of (154) it is possible to calculate the average velocity of the molecules of a gas when the pressure and density are known. Thus, for example, 1 c.c. of hydrogen, at 0° C. and under standard atmospheric pressure of 1.013(10)° dynes per square centimetre, has a mass of 8.96(10) ~^ g. Hence _ 3X1.013(10)° ^'= 8.96(10) -5 or w = 184,100'='"/seo 90. Avogadro's Law. — According to this important law, announced early in the nineteenth century by an Italian named Avogadro, equal volumes of gases under the same conditions of temperature and pressure contain the same number of molecules. The experimental facts of chemistry lead up to an establish- ment of this law, for the ratio of the densities of two gases tinder the same conditions of temperature and pressure is the same as the ratio of their combining equivalents or molecular weights. It follows that the number of molecules in two equal volumes is the same, and their difference in density results from a differ- ence in the weights of the individual molecules. Proof of Avogadro's law, based on the kinetic theory of gases, may be given as follows: Consider a cubic centimetre of each of two gases under the same conditions of temperature and pressure. Let them be GASES. 129 designated as 1 and 2. Since their pressures are equal, then, from (151), |MiWi1;,2 = |m2W2u/ (155) Since the gases are at the same temperature, the average kinetic energies of the individual molecules are equal; hence \m'^^ = \m^.^^ (156) A comparison of (155) and (156) shows that This fact, in turn, serves as a basis for the determination of molecular weights in chemistry. Molecular weight, so called, is not an absolute weight, but a ratio. Thus, if the molecular weight of hydrogen is assumed to be 2, that of oxygen is 32. If, then, the density of any gas is determined, its molecular weight may be found by comparison with the density of another gas of known molecular weight, assuming that the nuraber of molecules in the two gases is the same. 91. Dalton's Law. — Many phenomena of gases indicate that the distance between their molecules is much greater than the diameter of the molecules themselves, — i.e., the space actually occupied by the molecules is small as compared to the volume of the gas. On this assumption it would be expected that dif- ferent gases would readily mingle with each other, apparently occupying the same space at the same time. If some ether or other liquid that will readily evaporate be introduced into a closed vessel already filled with air, just as much of the liquid will evaporate as when no air is present. If a quantity of other liquid, alcohol say, be now introduced, it will evaporate in the presence of both air and ether just as it would do in a vacuum, except that the rate of evaporation is slower when other gases or vapors are present. A gas or vapor acts as a vacuum to another gas. Such facts show that there is between the mole- cules of a gas ample room for molecules of another gas. It has been shown above that the pressure of a gas is due to the activity of the molecules. It is a natural inference, then, that if two or more gases are enclosed in the same vessel, the molecules of each would continue their motion, and their im- pacts against the sides of the vessel, either direct or indirect, 9 130 GENERAL PHYSICS. would not be changed. In case of collision with one another there would be the exchange of velocity which results from the impact of perfectly elastic bodies, so that the result will be the same as if the impact had been made directly upon the sides of the vessel. Consequently the pressure would be the sum of the pressures which each would exert if it occupied the space alone. Dalton was the first to investigate this subject, and the result of his experi- ments may be stated as follows: The quantity of a liquid which will evaporate into a given space is the same, for the same tem-perature , whether the space is a vacuum or is already filled by a gas, and the press- ure exerted by a mixture of two or more gases or vapors is the sum. of the pressures which each would exert if it occupied the space alone. 92. Boyle's Law. — The relations of pressure, volume, and density of a gas were first systematically investigated by Robert Boyle about the middle of the seventeenth century A. D. By use of a glass tube of the form shown in Fig. 119, a quantity of gas enclosed in ae may be subjected to various pressures by pouring mercury into the long arm be. Let a small amount of mercury be first intro- duced and adjusted so that it stands in each arm at the level ab. A certain volume of gas is thus entrapped in ae and its pressure on the mercury at a must be equal to that in the other arm at b. If, now, mercury is poured into the long arm until it stands at c, it will also rise in the short arm to some point, d, — i.e., the gas will be compressed to the volume de. The original pressure on the gas was the pressure of the atmosphere, but now there is an additional pressure of the column of mercury fc. By varying the pressure in this manner and noting the corresponding volume in each case, Boyle was able to announce Fig. 119. GASES. 131 that the product of the pressure and volume of a gas is a constant quantity if the temperature is constant. This law was announced several years later by Mariotte, and it is known on the continent of Europe as Mariotte's law. If P is pressure and V is volume, then PV=k (157) where fe is a constant. 73S. 6S- y T : i I"!;: ".x |^ ' = '^. ■ .-■ :--i . is^>fit/fatr,ir/^»fi)^'\ ,':..■■■■ J i ■ . L ji; ■ i ..,|_^:j_.ij. .;■..! . ij.. .1.. .■[.■»....;. ..: L ■ ' '"'"p:.: — -t'i.-. I :'.'■' I , ! ■ ».. ' !.: ": ci-.;- ...,■: ;..;_: N; - -:■ — •^•Vrt= 176- «9- J77 V- - ^ ;■:::,,.: . ■>...._. Fig. 120. The same result may also be deduced from the kinetic theory of gases. Thus, it has been shown that or, since density is equal to mass divided by volume. PV = iMv' (158) The second member of this equation is a constant quantity if v' is constant. This is the case when the temperature does not change. Thus Boyle's law may be derived from a theoretical consideration alone. 132 GENERAL PHYSICS. The data obtained from an experiment such as that described above may be conveniently recorded on co-ordinate paper as shown in Fig. 120, the pressures being the ordinates and the volumes the abscissas. This may be called the PV diagram. At all points of the curve thus plotted, PV is the same or varies only slightly due to errors of experiment. This constant product is a distinguishing property of a curve called the equilateral hyperbola. The data in the figure were obtained from an experiment when the barometric pressure was 73.5 cm. of mercury and the volume of air at that pressure was 15 c.c. More elaborate experiments than those of Boyle were made later by other investigators, principally Regnault and Amagat, who subjected gases to much greater pressure. They showed that hydrogen was less compressible than Boyle's law would require, for the product PV became larger and larger as the pressure increased., Amagat experimented with other gases also, and found that, although at moderate pressures the gases were more compressible than Boyle's law would require, at high pressure they, like hydrogen, became less and less compressible as the pressure increased, — i.e., a greater pressure was neces- sary to produce the same diminution of volume. This is what would be expected from the kinetic theory of gases. When the gas is subjected to a moderate pressure, the molecules occupy but little of the space in which the gas is enclosed. Their mean free path is limited, practically, only by the sides of the vessel. If, however, the pressure is greatly increased, so that the space occupied by the molecules becomes an appreciable part of the volume of the gas, the mean free path is diminished. Since the molecules are assumed to J CiW have sensible dimensions, then when Y there is an impact of a and b, for ex- ample, against the wall of the vessel. Fig. 121, the centre of the molecules ^"'- ^^^- would not become coincident with the wall by nearly the radius of the molecules. Consequently as the pressure is increased the number of impacts on the sides increases at a greater rate than the increase in the number of molecules per unit volume, for the molecules are assumed to remain of the GASES. 133 same size and their radii bear a greater ratio to a short distance than to a longer one. Also, when the molecules are crowded by pressure they will come into more frequent collision with one an- other, and their mean free path will be less for this reason also, for the centres of c and d, for example. Fig. 121, do not become coincident on collision, but rebound from each other so that the space cd is not traversed by either one. This decrease in dis- tance causes a more frequent reversal of momentum and con- sequently a greater pressure. The fact that at moderate pressures the product of pressure by volume is less than Boyle's law requires, is probably due to the attractive force between the molecules, — a force which is not negligible in comparison with the impacts of the molecules at that stage of compression. It may be shown that this force would vary inversely as the square of the volume. Its effect is to decrease the pressure due to the impacts of the molecules. 93. Equation of Van der Waals. — An equation which expresses more accurately the relation of volume to pressure of gases is [p-\ — ^ J (u — 6) = constant (temp, const.) (159) This is known as Van der Waal's equation. The value of h, a constant, depends on the size of. the molecule, while the value of a, another constant, depends on the attractive force between the molecules. For carbon dioxide, for example. Van der Waals gives a = .00874 and & = .0023 for a specified mass and temperature. It has been shown above that the virtual decrease in the volume due to the fact that molecules have sensible dimensions increases the pressure, hence h is subtracted from v to obtain a value which will hold true for v in Boyle's equation. Likewise, since the intermolecular attraction depends on the mutual force between the attracting and attracted molecules, the force varies as the square of the number of molecules, — i.e., as the square of the density or inversely as the square of the volume. Expressed in formula, Foo—j or F = — ^ where a is a constant. Since this force decreases the pressure, — ^ niust be added to p. 134 GENERAL PHYSICS. If in equation (159) v becomes very large, — i.e., if the gas becomes very rare, ^ becomes negligible in comparison with p, and b has a similar relation to v. When a gas is in this condi- tion Boyle's law may be assumed to express the correct relation of p and V. 94. Elasticity of Gases. — As already explained, a gas can have only volume elasticity, expressed as a ratio of the stress to the strain per unit volume, — that is, t V ^ = constant, K where p is the change of pressure which caused the volume V to decrease by a volume v. Let the change of pressure and the consequent change of volume be very small. P will then become P+p and V will be V—v. According to Boyle's law, PV={P+p){V-v) or PV = PV-Pv+pV—pv Since p and v are very small, their product may be neglected, and therefore It appears therefore that the coefficient of volume elasticity of a gas is equal to the pressure to which the gas is at any time subjected. This may be roughly verified by substituting in (160) values given in Fig. 120, keeping in mind that p is change of pressure. This coefficient is constant for the same pressure, but is different at different pressures. The greater the pressure, the greater the elasticity. If the added pressure p is very small, the original pressure is practically the total pressure, — i.e., the pressure which just equals the coimter-pressure of the gas. This may be called isothermal elasticity, since the temperature is as- sumed to be constant during the change of volume. If the gas is suddenly compressed so that heat is developed, the product of pressure by volume is increased and so likewise the elasticity. An opposite effect will be obtained if the gas is GASES. 135 suddenly expanded. The discussion of elasticity under these conditions is deferred to the chapter on heat (§ 177). 95. Pressure of the Atmosphere. — After the invention of the air-pump by Von Guericke, it was demonstrated by experiment that the atmosphere exerts great pressure on bodies at the surface of the earth. Air is matter, and so it is subject to gravitational forces which bind it as a great gaseous envelope on the earth. The weight of a small mass of air causes very little pressure on the bottom of a vessel in which the air is confined, but a great number of such masses piled one on top of another to a height of several miles would press at the bottom with a force equal to the weight of all. One cubic centimetre of air at 0° C. and at sea-level atmo- spheric pressure in latitude 45° weighs .001293 g., or one litre weighs 1.293 g. If the air were of the same density - at all altitudes, the pressure at the surface of the 1 1 earth would be the weight of 1 c.c. times the height na""^ of the atmosphere in centimetres. But since air is very compressible, it is most dense in the lower layers and becomes more and more rare in the upper regions. The pressure of the atmosphere at any point may be found by balancing it against a column of mercury. If a glass tube A, Fig. 122, about 80 cm. long and closed at one end,, is filled with mercury and then inverted with its open end in a pool of the same liquid, the mercury will stand at a certain height h above the surface in the vessel. The pressure at the bottom of this column is p=pgh ^ where p is the density and g is the force of gravity in dynes per gram. This pressure is in equilibrium with atmospheric pressure, and so is a measure of that pressure. The proper unit of pressure is 1 dyne per cm.'', but it is often more convenient to employ as a unit the weight of 1 c.c. of mercury at 0° C. in latitude 45°, where the value of g is 980.6. The mass of this cube is 13.596 g., hence p = 13.596 X 980.6 X 1 = 13332.24 dynes 136 GENERAL PHYSICS. Pressure is frequently indicated by simply naming the height of a column of mercury. Thus, a pressure of 73 cm. of mercury means that the pressure per square centimetre is the weight of 73 c.c. of mercury. This is true, no matter what the actual cross section of the column may be. Another unit often employed is one atmosphere, which is the pressure at the bottom of a column of mercury 76 cm. high. Expressed in dynes, this unit is 76X13,332.24 = 1,013,250 dynes For most purposes it is a sufficient approximation to say one atmosphere = 1.013(10)° dynes A pressure of 75 cm. of mercury is almost exactly (10)* dynes, or one megadyne. This is sometimes used as a unit of pressure and is called the barie. 96. The Barometer. — A barometer is an instrument used to indicate atmospheric pressure. The principle on which it oper- ates has been explained in the preceding section. One of its standard forms is illustrated in Fig. 123. A glass tube, t, is filled with mercury and inverted with its open end in a cistern partly filled with the same liquid. The upper part of the cistern is a glass cylinder, but the bottom is a leather bag N. The graduations at the top of the tube show the correct height of the colunan of mercury only when the mercury in the cistern is at the zero level indicated by the "ivory point" h. When air pressure increases, mercury will be forced into the tube and consequently the surface in the cistern will be below the zero level, but when pressure decreases mercury will run back into the cistern and the surface will rise above the zero level. For this reason it is necessary, before reading the barometric height, to raise or lower the level in the cistern to the zero point. For this purpose a screw, o, is provided, by means of which the bottom of the leather bag may be raised or lowered until the point h just touches the surface of the mercury. The glass tube is enclosed in a tube of brass upon which are placed the graduations in millimetres or fractions of an inch, or both. For convenience and accuracy of reading, a sliding vernier is provided, as shown in Fig. 124. The lower edge of the vernier is moved down to the top of the meniscus of mercury. GASES. 137 and, to avoid error of parallax, a screen at the back of the tube is fixed to and moves with the vernier. The proper position for reading is reached when the top of the meniscus and the lower edge of both screen and vernier are in the same plane. The scale is then read up to the zero on the vernier, in a common form of which, as shown in the figure, 25 divisions are equal in length to iBI Hit ri i «« 1 i « 1 Fig. 123. Fig. 124. 24 on the scale in English units, while 20 vernier divisions cover 19 mm. in metric measure. One inch on the scale is divided into 20 equal parts, hence each division on the vernier is -^-^ of ^'0- or .002 inch shorter than one of the scale divisions, — i.e., the least count is .002 inch. The figures 1, 2, 3, 4, and 5 on the vernier, therefore, show the number of hundredths of an inch to be added to the scale reading. On the metric side the least count is -jV or .05 mm. The height of the mercurial column as shown in the figure is 74.18 cm. or 29.212 in. 138 GENERAL PHYSICS. 97. Corrections of Barometric Readings. — To reduce the reading of a barometer to standard conditions, several correc- tions must be made, as follows: 1. In most instruments the scale is made on the brass tube. If this scale is correct at 0° C, its reading at f will be too low, for each millimetre space has expanded and so the number of such spaces equal to the height of the mercury column is not so great as at 0° C. If, for sake of illustration, each millimetre should expand to twice its true length, there would need be only half the number to cover the same space, and the true height would be twice the reading. Such an extreme expansion, of course, cannot occur, biit the coefficient of linear expansion of brass is .0000178, — i.e., brass will expand .0000178 of its length when its temperature is raised 1° C. The length of each milli- metre at t° will then be 1 +.0000178/°, and the number of these false units decreases, for any given length, in the same proportion as the length of each increases. Hence the ratio of the observed to the true height is equal to the ratio of the true to the false scale division. This may be expressed by h 1 ha 1 +.0000178/° or /fo = /j,(l + .0000178/°) (161) where ht is the observed height and \ is the height at 0°C. Thus the correct reading as far as the brass scale is concerned may be obtained. 2. When the temperature rises, the density of the mercury becomes less. Consequently the column of mercury must rise to a greater height to balance the same atmospheric pressure. The height of the colunan at 0°C. is less than at any higher temperature for the same pressure. The coefficient of cubical expansion of mercury is .0001818, — i.e., the volume of 1 c.c. of mercury at t° is 1 + .0001818/° cubic centimetres. Since the height of the column varies inversely as the density, and the density varies inversely as the volume, K _ fa ^ 1 + -0001818 <° Ih" p, " 1 GASES. 139 where h^ is the height as measured by the corrected brass scale, H„ is the correct height when the temperature of the mercury is 0°C., p^ is the density at 0°C., and p,^ the density at tem- perature t° From this relation, h ^"^ 1 + .0001818 i° ^^^^^ But the value of ——-ri-5-^-5 is l-.0001818/° + (.00018180'- (.0001818^°)^, etc. Since the coefficient is very small, all powers above the first may, without sensible error, be neglected, and we may write Ho = fe„(l-. 0001818 i°) (163) By substituting the value of h„ as found above, i?o = ^.(1 + .0000178 1°) (1- .0001818 1°) (164) Thus by use of equation (164) correction is made for tempera- ture effects in both the brass scale and the mercury. It is not necessary to consider the expansion of the glass tube, for pressure depends only on the height of the mercury, being independent of the area of cross section. 3. The barometric height may also be reduced to a corre- sponding height for the same pressure at sea level in latitude 45°, where g is 980.6. At any point of observation where the value of g is greater, the height of the column will be less; if g is less, the height will be greater. Hence to reduce any observed reading to that at sea level, latitude 45°, it is only necessary to a take the fractional part expressed by -^—. the numerator being the value of g at the point of observation, and the denominator the value at sea level in latitude 45°- Equation (164) then becomes H„ = -^(1-1-. 0000178 i°)(l-. 0001818 i°) (165) The value of g for various points on the surface of the earth is given in the tables, or a fairly accurate value can be found from g = 978(1 + . 00531 sin^L) (166) where L is the latitude of the place. 140 GENERAL PHYSICS. 4. Correction should also be made for depression resulting from capillarity. When the diameter of the column is about 2.5 cm., the depression is negligible, but for smaller tubes a correction should be made. This can best be done by comparison with some standard instrument or by reference to tables of correction provided for this purpose. 98. Glycerin Barometer. — Any liquid may be used in the construction of a barometer, but, all things considered, mercury is the best. A water barometer would not be satisfactory, because it would need to be more than 34 feet high (13.6 X 30 in.) , and water vapor would fill the vacuum at the top of the tube, thus causing a depression of the column which would vary with every change of temperature. An excellent barometer, however, can be made by use of glycerin. The density of this liquid is 1.28 ^/cc ; hence, when the mercury column is 30 inches high, the glycerin would be raised by the same pressure to a height of 318.5 inches. For this instrument, then, the tube must be nearly 27 feet long. Its construction is possible where the lower part of the tube may extend into a room or basement below, only the upper end being exposed in the room where the readings are to be made. The great advantage in the use of this instrument is that, since its height is about 10.5 times as great as that of mercury, its variation in height for change of pressure is also 10.5 times as great. A change of 1 cm. in the height of mercury corresponds to a change of 10.5 cm. in the height of glycerin. Glycerin does not evaporate, and so the Torricellian vacuum at the top of the tube is maintained. 99. The Aneroid Barometer. — The aneroid barometer is so called because no liquid is used in its construction. It consists of a cylindrical metal box of German silver. Fig. 125, with a flexible, corrugated top. The air is nearly all pumped from the box, and the collapse of its sides is prevented by a stifE spring, R, attached to the central post on the top. Any increase in the pressure of the air will cause a further depression of the top, while a decrease will permit the top to spring out. In either case the movement will be such as to restore equilibrium between the air pressure on one hand and the elastic force of the top and spring on the other. This movement is multiplied by means of a system of levers, and is communicated to a hand which is GASES. 141 made to move over a graduated dial. The chief advantages of the aneroid are portability and sensitiveness. If carefully cali- brated by comparison with a mercurial barometer, it will record variations in atmospheric pressure with a fair degree of accuracy. FiQ. 125. 100. Mechanical Air=pumps. — A mechanical air-pump in its simplest form does not differ in principle from the common lifting pump used in raising water. A receiver, R, Fig. 126, fits air-tight on a plate, P. A tube leads from the interior of the receiver to the base of the pump. There are two valves, a and b, both of which open upward. By raising the piston the air pressure is removed from the top of the valve a and the impacts of the molecules beneath cause it to open. A portion of the air from the receiver thus passes into the space o. When the piston is pushed down, a is closed by the excess of molecular impacts from above, and b is opened by an excess of impacts from below. This operation is re- peated in each successive stroke of the piston, a certain constant fractional part of the air remaining in the receiver being removed by each cycle of operations. Let the volume of the receiver, including the connecting tube, be V, and the volume of the cyUnder between the limits of motion of the piston be v. When the piston is raised in the first stroke, the air in the receiver expands to a volume V+v. Now, when the piston is pushed Fig. 126. 142 GENERAL PHYSICS. down, a certain volume v of the air which had expanded to F +u escapes. Consequently the fractional part of the total mass, m, removed by the first stroke is V+v The mass remaining after the first complete stroke is, therefore, V V V +v V+v The sum of these is seen to be the original mass. By a second cycle of operations the same fractional part of the remaining air will be removed, — i.e., V V Vv V + v ■ V + v ^~ {V + vy^ and the mass remaining after the second cycle is V Vv V V + v"^ (V+vy"^^ {V + vr"^ In a similar manner it may be shown that the mass remaining after the third cycle is V (v+vy V and so on for any number of strokes, the exponent of ^ — always being that number. When the number of strokes is n, the mass of air remaining in the cylinder is Since this coefficient of m expresses the fractional part of the original mass occupying the volume V, it also expresses the fractional part of the original density and pressure. If the operation of this pump were exactly as here assumed, any degree of vacuum could be obtained by increasing the number of strokes. But, for mechanical reasons, a limit is reached at the end of a few strokes. It is difficult to prevent leaks between the piston and the walls of the cylinder, and after a certain degree of exhaustion has been reached, the pressure of air in the receiver is not sufficient to raise the valve a. GASES. 143 An improved mechanical pump, known as the "geryck" or Fleuss pump, is free frdm many of the defects of the ordinary pump. Its construction is illustrated in Fig. 127. A pipe. A, leads to the vessel which is to be exhausted. The air passes into the annular space B and thence, without any obstruction, through the port p into the space above the piston. A leather bucket, CC, forms part of the piston and is held against the sides of the cylin- der by the pressure of the air and oil above. The piston valve E operates only during the first stages of exhaustion, and when the vacuum is less than about 1.3 cm. it becomes inactive. A pipe, F , leads from the annular space B to the bottom of the cylinder, the purpose of which is to prevent a great difference of pressure be- tween the upper and lower sides of the piston at the beginning of the ascent during the first few strokes. Otherwise there would be a vacuum below the piston and full atmospheric pressure above. The air can pass freely from B into the cylinder, there being no valves to be pushed open. When the piston is raised, the port p is closed, and all air thus entrapped in the cylinder is carried up through the valve G into the upper chamber of the cylinder. The valve G is held on its seat by a spring K. When the piston reaches the upper end of its stroke, it raises the valve G and holds it open while all the air below passes through, the two bodies of oil meanwhile becoming one. Thus, it is not possible for any ^^^ay ^s^ 144 GENERAL PHYSICS. air to pass the piston. If oil leaks through at the valve or between the piston and cylinder, it is picked up during the next stroke of the piston. With this style of pump a very good vacuum may be obtained. When two are joined in series, as shown in Fig. 128, and the air is made to pass through a drying tube filled with phosphoric pentoxide before it enters the pump, a vacuum of about -gTnnr ^™- Fig. 128. may be obtained. The rapidity with which a vacuum may be produced gives this pump a great advantage over mercury pumps for many purposes. 101. Mercury Air=pump. — If the space at the top of a barom- eter tube is made to include the vessel which is to be exhausted of air, a very good vacuum can be obtained. A common form of pump constructed on this principle, and known as the Sprengel pump, is illustrated in Fig. 129. Mercury from, the reservoir F is made to fall drop by drop into the top of the tube T. The vessel to be exhausted is attached at A . Each drop of mercury GASES. 145 carries before it a quantity of air which escapes at o. As the exhaustion progresses the quantity of air between the drops becomes less and less until there will be a continuous column of mercury in T, equal to the height of the barometric column. The vacuum in C and also in the vessel attached to A will then be a Torricellian vacuum, so called, at the top of a barometer. The loop B is longer than the height of a barometric column, so that no air can enter C through the tube leading from the reservoir even when all the mercury has run out of F. 102. Diffusion of Gases. — If two or more gases are enclosed in the same vessel, each gas will in a short tirae be distributed to all parts of the vessel, just as if the other gases were not present, — i.e., there will be a uniform mixture of all the gases. This process is called diffusion. The rate of diffusion may be deduced from a consideration of the velocity with which a gas will issue from a small orifice in the side of a vessel. Let a body of gas, G, Fig. 130, do work by expanding and thus exerting pressure on the piston o. Let the pressure per unit area be p and the area of the piston A. Let the piston be moved by the expanding gas through the distance x. The work done is expressed by W = pAx W being the work and pA the total pressure or force. The product Axis the change of volume, hence the work is equal to the product of pressure by change Fig. 129. of volume. If, now, instead of ^^^^ moving the piston, the same increase of volume is produced by allowing the gas to escape from a small orifice, work will be done in giving kinetic energy to a stream of gas. The mass of the issuing stream is Vp, where V is the volume, equal to Ax above, and p is the density. Hence the kinetic energy is Fio. 130. E, = ^Vfw' 10 146 GENERAL PHYSICS, where v is the velocity of the stream. Hence . = ^ (167) From this it is seen that the velocity of the gas as it issues from the orifice varies inversely as the square root of the density of the gas. This deduction may be shown to be consistent with the v kinetic theory of gases, for, as already shown, there will be -^ impacts of each molecule upon one of the walls of a unit cube in which the gas is confined. If n is the total number of mole- cules and one-third of them move in each of the three possible directions, n V nv "3 '2^'Q is the total number of impacts upon one side of the vessel, v being the average velocity of the molecules. Now, if a small hole of area 5 is made in this side of the vessel, the number of molecules that formerly formed impacts against that portion of the side will now issue from the hole. The rate at which the gas will escape is, then, nvs ~6" The only variable quantity in this expression is v, n being, according to Avogadro's law, the same for all gases under the same conditions. Consequently the rate of escape of the gas is proportional to the average velocity of the molecules. But it has been shown in equation (154) that Vp — i.e., the rate of escape of the gas is inversely proportional to the square root of the density. To illustrate this principle, let a vessel. Fig. 131, be divided into two compartments by a porous partition, C, which may be made of tmglazed porcelain or plaster of Paris. A manometer, GASES. 147 T, partly filled with liquid passes through a cork into the cham- ber V. If a stream of hydrogen is made to flow through the bottom of the bottle into the space 0, it will diffuse through the partition into V more rapidly than the gas there will pass in the opposite direction, and the increased press- ure in V will be shown by the manometer. If V is filled with oxygen and with hydrogen, the rate of diffusion toward V will be four times as great as from V to 0. If V is filled with hydrogen, the manom- eter will show decrease of pressure. In a short time, however, the liquid will stand at the same height in each arm of the manometer, showing that diffusion is complete. The atmosphere is a mechanical mix- ture of several gases of different density, but each gas is uniformly distributed by diffusion. 103. Buoyancy of Air. — According to the well-known principle of Archimedes, a body is buoyed up by a force equal to the weight of the fluid which a body dis- places. When the body is dense, as iron or copper, the effect of buoyancy of air is not always apparent ; but a light body of large volume, as a balloon inflated with hydrogen, will be buoyed up by a force greater than the weight of the body. The force of buoyancy is just as great in case of a solid mass of iron as large as the balloon, but it is not so apparent. For accurate weighing allowance must be made for buoyancy of air. If a body is placed in one of the pans of a balance and is counterpoised by a known weight on the other pan, both are buoyed up by a force equal to the weight of the air which they displace. The true weight (weight in vacuum) , less the weight of air displaced, is the apparent weight. Let x be the true weight of the body and p its density. Also let w represent the standard weight in the other pan, and p^ its density. Let the density of air be a. If the body had the same density as air, its apparent Fig. 131. 148 GENERAL PHYSICS. weight in air would be zero. If twice as dense as air, its apparent weight would be -^x. Whatever the density p may be, the loss of weight is — x. Likewise the standard weight will lose w. Since the apparent weights in the pans are balanced. Pi a a X- — x=w w p Pi K'-f)=K'~) a If both sides of this equation be divided by 1 i2 or /a a a^ \ x = w[l-\ h^retc. ) ^ P Pi P I x='W+'wa( jnearly (168) \p p^l for the square or higher powers of — and — may be considered negligible quantities. " To illustrate the use of this equation, suppose a mass x of aluminum is balanced in air by a brass weight marked 500 g. Then *: = 500 + 500 X. 001293 (-^—^) or a; = 500. 174 g. — i.e., the weight in vacuum is .174 g. more than the weight in air. If the density of the object is the same as that of the standard weight, wax (=0 \p pj hence x=w and no correction is necessary. If the density of the object is greater than that of the stand- ard weight, the correction must be subtracted from w, as equa- tion (168) shows. GASES. 149 Problems. 1. If the average velocity of the hydrogen molecules at a pressure of 1.013(10)" dynes and at 0° C. is 184,100 cm/sec, what is the average velocity of oxygen molecules under the same conditions? 2. If the atmosphere were all as dense as it is at sea-level, latitude 45°, what would be its height where the pressure is 76 cm. of mercury? 3. Calculate the value of g in the latitude where you live. (Use (166).) 4. A mercurial barometer is inclined 10° to the vertical. The read- ing is 73.2 cm. What would be the reading in a vertical position? 5. What part of the mass of air remains in a receiver after five cycles of the piston of an air-pump, the ratio of the capacities of the pump and receiver being ^? 6. A sphere 10 cm. in diameter weighs 523.6 g. in air of density .0012 8/cc. What would the sphere weigh in vacuum, the standard weights being brass? 7. A glass tube used in sounding is 24 inches long and is covered on the inner walls with a brown pigment which becomes white when in con- tact with sea water. The tube, open end down, is lowered in water in which the pressure at a depth of 33 feet is equal to one atmosphere. On raising the sounder it is found that the pigment is white for a distance of 18 inches from the open end. What is the depth of the sea water? 1. 46025 cm/gec. 2. 5 miles, approx. 3. . 4. 72.08. 5. .328, approx. 6. 524.155. 7. 99 ft. of water. CHAPTER V LIQUIDS 104. Liquid Pressure. — ^One cubic centimetre of pure water at its greatest density, 4°C., weighs almost exactly one gram. Water, like other liquids, is almost incompressible, hence the pressure at any given depth, h, measured in centimetres, is hg dynes per square centimetre. For any other liquid of density p the pressure at a depth h is pgh. It has been shown that at any point in a fluid at rest the pressure is equal in all directions, and in liquids the pressure at any point may be taken as pro- portional to the depth. The pressure here considered is only that due to the weight of the liquid. 105. Transmission of Pressure. — The pressure per unit area exerted on a fluid enclosed in a vessel is transmitted to every equal unit area on the interior of the vessel. This principle is known as Pascal's law. It is a direct consequence of the fact that fluids do not resist a shearing stress. Let C be a cylinder filled . with fluid and let a and A I m Fig. 132. "■ be the respective areas of /p the pistons as shown in * Fig. 132. If the small pis- - ton is thrust against the fluid with a pressure p per unit area, every equal unit area within the cylinder will be sub- jected to the same increase of pressure. Any area 5 within the body of the fluid is subjected to an increased pressure ps. The total pressure of the small piston is pa, and that on the large pis- ton pA. Thus, whatever pressure is exerted at o is multiplied — times at R, for a pa . — =pA The mechanical advantage in this case is the ratio of the areas of the pistons. This is the principle of the hydrostatic press. By making A large, the total pressure may be enormously 150 LIQUIDS. 151 increased, but the distance moved is less in proportion as the area of A is greater. Hence the hydrostatic press conforms to the general law for machines. 106. Pressure of a Liquid on the Walls of a Vessel. — Since the pressure resulting from force of gravity is proportional to the depth of a liquid, and also since the pressure per unit area at any depth is transmitted to every equal unit area below that depth, it follows that the pressure on the horizontal bottom of a vessel is P = hpgA„ (169) where h is the vertical height and A„ is the area of the bottom. It also follows that this pressure is independent of the shape of the vessel, for, according to Pascal's law, if the box B, Fig. 133, has a tube T inserted at one side, or at any point, and the whole be filled with liquid to a height /, the pressure on the bottom of the vessel is the same as if, instead of the tube, the sides of B had been extended up to the level / and the whole filled with the same liquid. Whatever the shape of the vessel may be, the pressure on the bottom, or any part of it, may be found by equation (169). c 1 ! ! < r ,- — -f-- 1 1 1 '' ^'^ B ( °'&^ Fig. 133. Fig. 134. Pressure on the side of a vessel is zero at the surface of a liquid and maximum at the bottom. Reference is here made only to pressure due to weight. Since this pressure increases uniformly with the depth, one-half the sum of the zero and maximum pressures is the average pressure per unit area, — i.e., the average is at a point of which the depth is one-half the depth of the liquid. If then the side of the vessel is of such a shape that a horizontal line through its centre divides the area into equal upper and lower parts (e.g., square, rectangle, circle, 152 GENERAL PHYSICS. Fig. 135. etc.), the total pressure is the average pressure times the area. In Fig. 134 a rectangular box is supposed to be filled with a liquid of density p. The total pressure on AB, a portion of one side, is the area oiAB times the pressure at o. The total pressure on SD is its area times the pressure at o' So also the total pressure on the circle C is its area times the pressure at o". When the area of the side considered is not symmetri- cally divided by a horizontal line through its centre, the total pressure cannot be found in this manner. If, for example, the side is in form of a triangle, as ABC, Fig. 135, the total pressure on the triangle when the vessel is full of liquid is the area of ABC times the pressure at o, the distance from A to xpg LIQUIDS. 163 and the flow will be toward A . To transfer hydrogen from A to B the whole apparatus shown in the figure should be inverted. 115. Efflux of Liquids. — If an opening be made in the side or bottom of a vessel filled with liquid, the velocity of efflux will depend, for the most part, on the depth of the liquid. Let a vessel. A, Fig. 147, be filled to a height, h, above e, with a liquid of density p. If, now, 1 c.c. of the same kind of liquid is forced into A through e, the amount of work done is equal to that required to lift a column of h cubic centi- metres of the liquid 1 cm. high. This is the same as lifting 1 c.c. to a height of h centi- metres. If m is the mass of 1 c.c, the work done or the increase of potential energy of „ the whole mass is m-gh ergs. The actual rise of liquid in the vessel is inversely proportional to the area of the surface, but in any case the increase of energy is equal to the work done in introducing liquid at e. If now the liquid is allowed to flow from e, the potential energy of a mass m may be assumed to be all converted to kinetic energy, ^mv^. Hence or v^ = 2gh (176) Thus it appears that the velocity of efflux is the same as that of a body falling freely from the surface of the liquid to the orifice. It would seem that if the area of the orifice is a and the velocity of efflux is v, the volume of liquid which would flow out in time t would be avt. But because portions of the liquid in the vessel move from all directions toward the orifice, the issuing stream is contracted at a point, c, called the vena con- tracta. Let the area of cross section at this point be A„, then the volume of efflux in any given time is A^vt, the value of which is about .62 of avt. If a short tube (about two or three times as long as the diameter) is fitted to the orifice, the rate of flow is increased to about .82 avt. In this discussion the viscosity (internal friction) of the liquid has not been considered. 116. Velocity of Efflux in Terms of p and p. — The pressure p which causes the flow of liquid from an orifice depends on the 164 GENERAL PHYSICS. depth h and the density p of the liquid, and may be expressed by P=Pgh (177) The pressure of the atmosphere downward on the surface of the Uquid and inward at the orifice may for most cases be considered as balanced. From equation (177) h- Substituting (178) in (176), or -1^ Pg pg P (178) (179) which is the same expression as that for the velocity of a gas escaping from a small orifice. 117. Lateral Pressure of a Moving Stream. — The pressure per imit area at a depth h is mgh, and this, as shown above, is the potential energy of a unit mass (w) in vessel A, Fig. 148. Hence the potential energy per unit mass is equal to the pressure. If, then, the potential energy becomes wholly kinetic, the pressure p becomes zero. Let B be a tube, small as compared with A, and at the end of B let the liqtiid issue from a small orifice, e. If the velocity of efHux at e is the =■? same as that which would be ac- quired by a body falling freely through a distance h, — i.e., from the surface of the liquid to a point on a level with the orifice, — all the potential energy of that portion of the liquid becomes kinetic, and so there is no lateral pressure at that point. For, let P be the pressure at any point in a flowing stream, then Fig. 148. or P=p — ^mv' P=p-^fyV (180) since m is the mass of unit volume. The pressure while the LIQUIDS. 165 liquid is not flowing is p. If the value of v in (179) be substituted in (180) p=p-p = (181) The velocity of the flow of liquid through B is less than at e, for the same volume passes a larger cross sectional area. The energy of the liquid in B is therefore partly potential and partly kinetic, the pressure being found by equation (180). 118. Viscosity. — Viscosity is that property of fluids by virtue of which they partake in some measure of the properties of solids. A perfect liquid would offer no resistance to a shear- ing stress, but all liquids and gases do to some extent resist such a stress. Work must be done in sliding one layer of liquid on an adjacent layer. The degree of resistance offered to a shearing stress is a measure of the viscosity of a substance. There is a coefficient of rigidity for fluids as well as for solids. No distinct line of division can be made between liquids and solids. Some liquids, such as alcohol and ether, are very mobile, — i.e., comparatively free from viscosity. Others, such as molasses, heavy oils, pitch, and molten glass, are distinctly viscous. Other substances, such as ice and asphaltum, are classed as solids, but will permanently change their shape when subjected to a continued stress. A great mass of ice will flow down a channel just as a river of water does, though much more slowly. A lead bullet placed on a block of asphaltum will in time sink to the bottom, while a cork placed beneath the block will rise to the top. Thus a substance which is brittle when subjected to a sudden stress, behaves like a liquid under a continued stress. When a liquid, such as water, flows through a tube or pipe, it adheres to the walls of the pipe, forming a layer over which the liquid flows. When a pipe is full of flowing water, the part having the greatest velocity is at the centre of the pipe, while that adhering to the walls does not flow at all. Consequently the rate of flow is independent of the nature of the tube. It is plain from this that viscosity will retard the rate of flow, and the greater the viscosity the greater the retardation. The coefficient of viscosity may be defined as the tangential force per tmit area required to move one plane with unit velocity parallel to another plane which is fixed, the space between 166 GENERAL PHYSICS. them being filled with a viscous fluid. As shown in Fig. 149, let a and h be planes 1 cm. apart, the space between them being filled with a Uquid which adheres to the planes. Then the force in dynes required to move the plane a with a velocity of 1 '=™/sec, divided by the area of a, is the coefficient of viscosity of the liquid. It may be shown that the volume of liquid under pressure that will flow through a tube in a given time is V- TcprH 8l

^ -'■M i0a=^ ^ ir 1 Fig. 182. Then, by passing through t a stream of alcohol which is cooler than 0° C, a part of the water in B will be frozen. The alcohol may be made cooler than 0° C. by use of a mixture of ice and crystallized calcium chloride or ice and sodium chloride. The bulb B then contains ice and water both at 0° C. and kept so by the surrotmding snow. If now any warm body is dropped into t, some of the ice will be melted and so its volume becomes less. Consequently the thread of mercury in M will be with- drawn through a certain distance. To determine the value of the divisions on M, let a mass of water w at temperature t° be dropped into the tube t, and suppose that as a result the mercury withdraws through 200 mm. The amount of heat given out by the hot water is wt cal., for its temperature fell from t° to 0° C. Hence the value of each division on the scale would be wt 200 212 GENERAL PHYSICS. or, to make the conditions general, suppose the end of the thread of mercury moved from a to b. Then the number of calories indicated by each scale division would be wt Now, if a quantity, Wi, of some other substance at a tempera- ture t° and of specific heat 5 is dropped into the tube, additional ice will be melted and the mercury will further withdraw to some point c, — i.e., through b — -c divisions. The number of calories of heat given to the ice by the substance is w^t^s cal. ; hence again the value of each scale division is Hence or b-c wt w^t^s a — b b — c b-c ^ = r ■ wt W,ti (223) 154. Specific Heat by the Method of Cooling. — The specific heat of a liquid may be fotmd by noting its time of cooling as compared with the time for water when the rate is the same in both cases. A vessel in form of a bottle having thin metallic walls is suspended in air within another vessel which is surrounded with water or crushed ice. Fig. 183, the temperature of which may be assumed to remain constant. The bottle is first filled with a mass of water, m, at a temperature which may be observed by the thermometer which passes through the stopper and which also serves as a handle. The time, T, required for the contents of the bottle to cool through t° is noted. The bottle is then filled with a liquid of mass Wj, the specific heat of which is sought, and the time T^ is noted during which the temperature falls through the same range, t°. Since the conditions and range of temperature are the same in both cases, the amount of heat that escapes from Pig. 183. HEAT. 213 the bottle in each case is proportional to the time. Hence mt _ T m^ts Ti where 5 is the specific heat sought, and, since t is the same in both cases, niiT (224) The mass of the liquids must include the water equivalent of bottle and thermometer. 155. Specific Heat by Electric Heating. — An excellent method of finding the specific heat of some liquids is by use of two electric calorimeters similar to that shown in Fig. 184. Two heavy copper wires, covered with a coat of shellac to protect them from the action of the liquids, extend from binding posts down to a coil of resistance wire near the bottom of the calorimeter. If the resistance of the coil is about 2 or 3 ohms, a current of 5 amperes will cause it to be- come hot and thus heat the ^^^3^^ ^-j-- liquid in which it is placed. '^''^ImH/nmW _ i ? The two calorimeters are ex- actly alike and the current is passed through them in series. The liquid must be thoroughly stirred and the temperature is read on a delicate thermometer which passes through the cover. One calorimeter is partly filled with water of mass m, and the other with liquid of unknown specific heat the mass of which is m-^. The two masses are made such that the rise in temperature is very nearly the same in each calorimeter. This may be done by a preliminary trial or may be calculated when the specific heats are approximately known. By doing this the water equivalent of the calorimeters and the correction for radiation may be neglected. Let t° be the rise of temperature of the water and t^ that of the other liquid. Since each calorimeter will receive the same quantity of heat in the same time. 214 GENERAL PHYSICS, where 5 is the specific heat sought. Hence s = ^ (225) 156. Specific Heat of Gases. — Gases have two specific heats, (1) specific heat at constant pressure, which may be denoted by Cp, and (2) that at constant volume, C„. If a mass of gas is enclosed in a horizontal cylinder one end of which is closed by a movable piston, the pressure on the enclosed gas is that of the atmosphere, which during the time of the experiment may be considered constant. If now the gas is heated, it will expand and move the piston against this pres- sure. Hence an amount of work will be done equal to the product of the pressure by the change of volume of the gas (§ 102) . Thus, not only is the gas heated but a certain amount of work is done beside, both being at the expense of the heat applied. Conse- quently, when a gas under pressure is made to expand, more heat must be applied to change its temperature through any given number of degrees than when it is confined to a constant volume. When the process is reversed, — i.e., when heat is taken from the gas, — the quantity will be not only that which caused a rise in temperature but also that which was expended in work. If the gas is confined in a vessel which does not permit a change of volume, less heat is required to cause a given rise of temperature, for no external work is done. Hence the value of Cp is always greater than C„. Specific heat of gases cannot be accurately determined by simple devices such as are used for solids and liquids, for the thermal capacity of the containing vessel is large compared with that of the gas itself, and so it is difficult to obtain reliable data for the gas which contains only a small part of the total heat. The operation for finding Cp usually consists in passing a large quantity of gas continuously through a long spiral coil immersed in a bath of hot water or other liquid. The gas, heated as it passes through this coil, is then passed on through a second coil immersed in the water of a calorimeter. The product of the mass of water by its rise of temperature gives the number of calories of heat received from the gas. The mass of HEAT. 215 gas is calculated from its volume and pressure. The product of its mass by its fall in temperature while passing through the calorimeter and by its specific heat is the number of calories of heat given to the water. By equating the calories given up by the gas and those received by the water, the specific heat of the gas is readily found. By this method a large quantity of gas may be used, and by passing it slowly through the coils the pressure may be kept practically constant. Specific heat of a gas at constant volume is difficult to deter- mine by experiment, for reasons mentioned above; but by use of the steam calorimeter, invented by Jolly, of Dublin, fairly reliable results have been obtained. This consists of a vessel, B, Fig. 185, into which steam is admitted through a tube, 5. The mass of steam condensed on any object within S is a measure of the quantity of heat needed to raise it to the temperature of the steam. For gases two hollow globes of the same mass and material are suspended within the calorimeter, one from each end of the beam of a delicate balance. Both globes are exhausted. If steam is now admitted and the balance remains undisturbed, the heat capacity of the two globes is the same, for the same amount of steam is condensed by each. If the balance is disturbed, weights are added to com- pensate for the difference. One of the globes is now filled with a gas under a pressure of 30 or 40 atmospheres, that the mass may be as great as possible. The other globe remains exhausted. Steam is again admitted, and the condensation on the globe filled with gas will be increased, for the gas must receive sufficient heat to raise its temperature to that of the steam. The weight needed to restore the balance is the mass of water condensed by the gas. Knowing the mass of gas nig, its rise in temperature t°, and the mass of steam condensed m.. Fig. 185. nigts = Lm„ mJ .'. 5 = - (226) 216 GENERAL PHYSICS. where 5 is the specific heat of the gas and L is the latent heat of steam. No allowance for the thermal capacity of the globe need be made, for the globes are alike in this respect, nor does the change of buoyancy due to immersion in steam affect the balance when two globes are used. The steam calorimeter may be used in finding the specific heat of solids and liquids as well as of gases. The solid may be suspended by a fine wire from one end of the beam and counter- balanced by weights at the other end. Specific heat of a gas at constant volume may be deduced from the fact that the ratio of Cp to C„ is a constant quantity for any given gas. Thus where j- is a constant quantity, being 1.41 for air. The value of y may be fotmd from the velocity of soimd in any gas; the value of Cp is found in the manner described above, and C„ may then be calculated from C. = ^ (227) This ratio is more fully discussed in § 175. Problems. 1. If 50 g. of copper at 100° C. immersed in 50 g. of water at 20° C. raise the temperature of the water to 26.94° C, what is the specific heat of the copper? 2. If a copper calorimeter weighs 50 g. and contains 80 g. of water at 18° C, what will be the temperature of the water after 10 g. of melt- ing ice have been strired into it ? 3. How much steam at 100° C. must be condensed in 1 kg. of water at 20° C. to raise the temperature of the water to 75° C. ? 4. How much heat would be required to melt a block of ice 20 X 30 X 50 cm. ? 5. A room measures 3X4X6 m. The air within the room is at a temperature of 15° C. and under a pressure of 72 cm. How many degrees will the temperature of the air be raised by condensing 1 kg. of steam at 100° C. to water at 100° C. in the steam radiators? 1. .095. 2. 7.65° C. 3. 98.04 g. 4. 2367.18 large calories. 5. 26.6° C. HEAT. 217 157. Fusion and Solidification. — The state of a substance, solid, liquid, or gas, is mainly dependent on temperature. Iron, for example, is known as a solid because its temperature is com- monly such that it is found in that state. Mercury for the same reason is usually known as a liquid, and hydrogen as a gas. But any of these substances may be changed to any one of the three states by proper changes in temperature and pressure. In case of most crystalline substances there is a definite temperature known as the melting point. When this point is reached, a solid will begin to change to the liquid state. While the solid is melting, the temperature will be constant, for the heat energy applied is expended in causing a change of state. During the process of melting, the solid and liquid exist side by side in equilibrium. More heat simply changes some of the solid to liquid, and if some heat is abstracted, a portion of the liquid will change back to the solid state. The quantity of heat per unit mass required to produce this change of state is called the latent heat of fusion. (§ 150.) (Appendix 24.) Substances which are amorphous, — ^not crystalline, — such as glass, rosin, solder, paraffin, etc., gradually soften and finally become liquid as the temperature is raised, but there is no exact point at which they may be said to fuse. The melting point of alloys is lower than that of the metals which are fused together to form the alloy. Thus, by melting together tin and lead in different proportions, solder of different degrees of hardness may be made. Rose's fusible metal, com- posed of 4 parts bismuth, 1 part tin, and 1 part lead, melts at 94° C. Wood's fusible metal, 4 parts bismuth, 2 parts lead, 1 part tin, and 1 part cadmium, by weight, melts at 60.5° C. There appears to be a change in the grouping of molecules of an alloy so that not so much heat is needed to change the state. Substances which have a definite melting point will in some instances also change abruptly in volume at the moment of change of state. Cast iron has practically the same volume in the solid or liquid state, hence it will take the exact form of the pattern in moulding. Bismuth and antimony increase slightly in volume when they solidify. Most substances decrease in volume when they change from liquid to solid. Water more than any other substance increases in volume when it solidifies. 218 GENERAL PHYSICS. One cubic centimetre of water at 0° C. will in form of ice have a volume 1.0907 c.c, an increase of more than 9 per cent. This is a fact of great economic value in nature. It may readily be inferred from what has just been said that pressure affects the melting point of those substances that change in volume on change of state, for pressure would either assist or hinder that change of volume which accompanies the change of state. Phosphorus, for example, increases in volume when it is melted. It will when all pressure is removed melt at about 44° C, but under a pressure of 2000 kg. per sq. cm. it melts at about 97° C. Ice, on the other hand, melts at a lower temperature vinder pressure. Professor James Thomson showed, from theoretical considerations (see equation 259), that a pressure of one atmosphere lowers the melting point of ice .0075° C. Lord Kelvin later verified this by experiment. Under Fig. 186. a pressure of 1000 atmospheres water will not freeze above — 7.5° C. Hence, if a strong vessel is filled with water and closed, the water will either remain a liquid or the vessel will be hvLTSt when the temperature is reduced below 0° C. Many illus- trations of the effect of pressure on the melting point of ice might be given. If a strong iron cylinder, Fig. 186, be filled with fragments of ice, both cylinder and ice being below 0° C, and pressure be applied by screwing in the plug which exactly fits the bore of the cylinder, the ice will be melted, but when the pressure is then removed the water will by regelation become one solid block of ice. In accord with this same principle, snowballs are formed by the pressure of the hands, but if the snow is very cold the pressure may not be sufficient to cause any melting. An experiment due to Bottomley consists in sus- pending a weight from each end of a wire thrown over a block of ice. The ice beneath the wire is melted by pressure and flows to the upper side where it is frozen. Thus the wire will in time pass through the ice and leave the block as solid as before. The formation of "ground ice" at the bottom of streams or at HEAT. 219 points where there are eddies in the current results from the fact that water at such points may be near the freezing tem- perature, and pieces of ice carried by the water are driven against the bottom. The ice is first melted at the point of concussion and then at once is frozen and so adheres to the bottom. Other pieces in a similar manner are frozen to this and so the mass accumulates. 158. Freezing Point of Solutions. — It is a matter of common observation that a liquid when pure will freeze at a higher temperature than when it contains foreign substances in solu- tion. The principles underlying this subject were investigated in the last quarter of the nineteenth century by the French chemist Francois Marie Raoult. He found experimentally the lowering of the freezing point which resulted from a solu- tion of acids, bases, and salts in water, acetic acid, benzene, and other solvents. He used very dilute solutions, less than one gram-molecule in 2 kg. of the solvent. The advantage in this is that a considerable quantity of ice can be formed without greatly changing the concentration, the range of lowering can be made 1° C. or less and so a delicate thermometer can be used, and dissociation if such occurs is most nearly complete in very dilute solutions. From this it is possible to calculate the lower- ing which would occur in a 1 per cent, solution, assuming that the rate of lowering would continue. Thus different solutions would all be reduced to a standard for comparison. The lower- ing for 1 per cent, solution — i.e., 1 g. of a substance in 100 g. of the solvent — is called the coefficient of lowering. Raoult's expression for finding this is where A is the coefficient of lowering, K is the lowering observed in the experiment, P is the mass of the solvent,- and P' is the mass of the substance dissolved. The product of this value of A by the molecular weight M of the dissolved substance is the molecular lowering T, — i.e., T is the lowering which would be obtained if one gram-molecule of the substance had been dis- solved instead of 1 g. This is expressed by Raoult in the equation MA = T (229) 220 GENERAL PHYSICS. As a concrete illustration of this procedure, suppose 40 g. of H2SO4 is dissolved in 1000 g. of water and that the observed lowering is 1.56° C. Then, by equation (228), .4-156 ^""° -30 ^-^•^^40X100 ~-^^ The molecular weight of HjSO^ is 2 + 32+64 = 98. Hence, from equation (229), r = . 39X98 = 38.22 The values of T throughout a great variety of solutions cluster about two numbers, one of which is twice as great as the other. For solutions in water the numbers are 18.5 and 37. For the same number of physical molecules in a given solvent the lower- ing is the same whatever the character of the substance may be. Variation in experimental results can usually be explained as resulting from special causes. The purpose of multiplying A by M in equation (229) is that the lowering can thus be obtained for the same number of molecules in all cases, for in equal masses of two substances the numbers of molecules vary inversely as the molecular weights. Distinction is made between a chemical molecule and what is often called a physical molecule, the latter of which may consist of a grouping of, or may be a part of, a chemical molecule. The greater the number of physical mole- cules the greater is the lowering of freezing point. When water is used as the solvent the values of T are either 37 or 18.5. For all strong acids and bases and all salts of alkalies the number is 37. These are also the substances which cause abnormally great osmotic pressure (§ 127) and an abnormal elevation of the boiling point (§ 161). These are also the solutions known as electrolytes, — i.e., conductors of electricity. In explanation of this difference in the behavior of solutions, Svante Arrhenius in 1887 announced the dissociation theory now generally accepted. In accordance with this theory, the molecules of the dissolved substance in an electrolyte separate iato ions, which are atoms or groups of atoms charged with positive or negative electricity. + - Thus H2SO4 in water will separate into H and 80^. Each ion then acts as a physical molecule, and the molecular lowering is therefore twice as great as when such dissociation does not occur. In the experimental work of finding the freezing point the solu- HEAT. 221 tion is placed in a test tube around the bulb of a delicate ther- mometer, the whole being surrounded by a freezing mixture. When the temperature is sufficiently reduced small flakes or granules of ice will appear. The difference between this tem- perature and that at which the pure solvent freezes is the lower- ing due to the substance in solution. Since it is the solvent that freezes, the remaining liquid is a more concentrated solution than before. If the temperature is further reduced, more ice will be formed, and so on till the solution is saturated. Any further withdrawal of heat does not reduce the temperature, but causes more of the solvent to freeze and a precipitation of some of the substance in solution. This may be continued till the whole becomes a solid, — a mechanical mixture called cryohydrate. It was pointed out by Raoult that the laws governing mo- lecular lowering could be used to determine molecular weight. For illustration, the value of T for all salts of alkalies is 37. Then, if the coefficient of lowering is found for any salt of this kind, by equation (228), the molecular weight is M = ^ (230) 159. Evaporation. — The process by which many substances slowly and quietly change to a vapor is known as evaporation. The process is chiefly observed in the change of volatile liquids to aeriform fluids, as in case of water, alcohol, ether, etc. Some solids, such as snow, ice, iodine, etc., may change to an aeriform state by sublimation, — i.e., they appear to evaporate directly without change to a liquid. Evaporation is a result of molecular motion within a sub- stance, and hence may be considered as a heat phenomenon. It has already been explained that when a substance is in a gaseous state the molecules are widely separated from one another as compared with the diameter of the molecules them- selves, the interspace being something like 100 times greater than the diameter. Hence there is freedom of motion and but little constraint from neighboring molecules. In solids and liquids, however, the molecules are within the range of attrac- tion of one another. In liquids a molecule has freedom of 222 GENERAL PHYSICS. motion through the mass of a substance, but whatever position it may have, it is subjected to the attractive influence of its neighbors (§ 119). The molecules of a liquid are in motion in all directions as long as the substance contains any heat energy. Only at the theoretical, absolute zero are all supposed to be at rest. If, then, molecules move up to the surface of a liquid, as cotmt- less numbers are doing, their escape into the space above will in most cases be prevented by the attraction of their neigh- bors below. Consequently a liquid has a definite surface and a surface tension. It may readily happen, however, that some molecules moving with greater speed than others will leap from the surface and will free themselves from the attractive force of their neighbors. In this way a mass of liquid may under proper conditions of temperature and pressure be completely changed to a vapor. Not only do molecules of the liquid leap into the space above, but a number of those of the vapor re- enter the liquid. As long as the former is in excess, evaporation will continue. When the number leaving the liquid is equal to the number that re-enter it, the vapor is said to be saturated. Then, although evaporation still continues, the quantity of liquid is not diminished. A liquid is cooled by evaporation. This is as would be ex- pected, for those molecules that are moving with greatest velocity are the ones most likely to leap from the surface, hence there is a decrease in the average kinetic energy of the molecules that remain in the liquid. Hence when a liquid evaporates rapidly there will be a rapid fall in temperature. This fact is utilized in the manufacture of ice. Ammonia gas is liquefied by cooling and compression and is then allowed to evaporate rapidly into a coil which is submerged in strong brine. The temperature of the brine is thus reduced below 0° C. and is then made to flow about the metal moulds which contain the water to be frozen. The same gas is returned to the pump, where it is again lique- fied and the operation is repeated. In accordance with the theory of the cause of evaporation, it is plain that the rate at which a liquid will evaporate depends on (1) the area of the surface exposed, (2) the temperature of the liquid, (3) the removal of the vapor as soon as it appears HEAT. 223 at the free surface, as by fanning or any movement of air, and (4) decrease of pressure on the surface of the liquid. 160. Vapor Pressure. — -The molecules of a vapor are in rapid motion, and so will, like a gas, exert a pressure on the walls of a containing vessel. The vapor of each liquid will when saturated exert a pressure known as its vapor pressure at that temperature. While a saturated vapor is in presence of its liquid, any change in the volume of the vapor will not change the pressure, for any decrease of volume only changes some of the vapor to liquid and any increase of volume only allows some liquid to vaporize. This fact may be shown experimentally by use of the apparatus shown in Fig. 187. A U-shaped glass tube, each arm of which is about 76 cm. long, is closed at one end by a stopcock. By inclining the tube to one side with the end a beneath the surface of mercury, an aspirator or pump may be used to fill one arm of the tube. If the stopcock is now closed and the tube placed upright, the difference in height of the mercury columns in the arms will show the atmospheric pressure in centimetres of mercury. If now the tube a above the stop- cock is filled with ether or some other volatile liquid, and the cock is turned so that only a drop or two of the liquid is admitted to the vacuum above c, the heights of the mercury in the arms will change, as shown in B. Sufficient ether is admitted so that a small quantity in the liquid form may be seen at c'. saturated, and the difference in the levels c from the difference in level of c and b is the vapor pressure ex- pressed in centimetres of mercury. If this pressure were equal to the pressure of the atmosphere, c' and 6' would be at the same level. Now let some mercury be poured in at d'. Both c' and b' will rise, but their difference of level will be unchanged. As c' rises, more vapor will change to liquid, but the pressure will not change. (Appendices 28 and 29.) Fig. 187. ^ The vapor is then and b' subtracted 224 GENERAL PHYSICS. 161. Boiling Point. — When a mass of liquid is heated, not only is evaporation increased but at a certain temperature bub- bles of vapor formed within the liquid rise to the surface. The liquid is then said to boil, or to be in a state of ebullition. When boiling begins, temperature becomes constant, and all heat applied to maintain the process of boiling becomes latent heat of vaporization (§ 150). The temperature at which boiling begins is called the boiling point under the conditions present. Boiling point is greatly modified by pressure. This is as would be expected, for bubbles of vapor cannot form until there is equilibrium between vapor pressure and external pressure. In fact boiling point may be defined as such an equilibrium. It will be noted in appendix 28 that water boils at 100° C. when the pressure of the atmosphere is 76 cm., because the vapor pressure at that temperature is also 76 cm. The same table shows that when the external pressure is 45 cm. water will boil at 83° C. When a liquid is confined in a vessel, — e.g., water in a boiler, — ■ it may be heated far above the boiling point, for the pressure of steam above the water prevents vaporization. Any decrease of steam pressure, as when a valve to the engine is opened, permits some water to vaporize. When a confined mass of water contains all the heat necessary for its vaporization, it will, unless the containing vessel is strong enough to prevent it, explode with terrific violence, for 1 c.c. of water will when vapor- ized in air occupy nearly 1500 c.c. of space. It is safe to heat water to the boiling point, 100° C, only because an additional 536 calories must be added to convert each gram of it to a vapor, and this under ordinary conditions is a slow process. Many other conditions also modify the boiling point. The nature of the material of a vessel and the roughness or smooth- ness of the interior walls may cause a difference of several degrees. In a glass vessel the temperature of boiling water may be as much as 3° C. higher than in a metal one. A few tacks or a small quantity of sand thrown into a vessel of hot water will lower the boiling point 1° C. or more. Water must contain nuclei of some kind, such as air, which is nearly always found in water, before bubbles of vapor can be formed. Pure water from which all air has been removed may be heated far above the boiling point. It is then in an unstable condition and liable to vaporize all at once, — i.e., it will explode. HEAT. 225 Bubbles of vapor formed at the bottom of a vessel are under the pressure of water above them and also the pressure due to surface tension of the bubble. For these reasons, the tempera- ture is higher than need be for equilibrium with atmospheric pressure, but when the bubble breaks at the surface, the vapor at once expands and is thus cooled to the true boiling point. For these reasons, the bulb of a thermometer is suspended in the vapor above a liquid, and not in the liquid, when the true boiling point is sought. The boiling point of solutions was investigated by Raoult, and his results were published in 1887-8 A.D. His researches showed that when one gram-molecule of a non-volatile substance is dissolved in 100 g. of a solvent, the molecular lowering of vapor pressure is independent of the nature of the substance, — i.e., is dependent only on the number of physical molecules present; that in dilute solutions the lowering is proportional to the concentration; and that for a given solvent there is nearly a constant ratio between the molecular lowering of the freezing point and that of vapor pressure. Since vapor pressure is lowered by the presence of substances in solution, the boiling point is raised, for a higher temperature is then necessary to cause equilibrium between the vapor and the external pressure. Some substances in solution cause an abnormal lowering of vapor pressure. These are the same substances as those that cause an abnormal osmotic pressure or an abnormal depression of freezing point. These phenomena are due to dissociation (§ 158). From a knowledge of the lowering of vapor pressure or, what is more easily determined, the elevation of the boiling point, it is possible to calculate molecular weights of soluble substances that are non -volatile. The vapor above a solution is that of the pure solvent, and hence the bulb of a thermometer must be placed in the solution to obtain its boiling temperature. 162. Isothermals of a Vapor. — According to Boyle's law, the product of pressure by volume of a gas is constant as long as the temperature does not change. A curve plotted on the pres- sure-volume diagram for a gas which is very nearly true to Boyle's law is shown in Fig. 120. This is the curve for a given mass of air at 22° C. If air had been at a higher or lower tem- perature, other similar curves would have been formed either 15 226 GENERAL PHYSICS. farther from or closer to the axes of reference. Such curves are called isothermals, for the temperature remains constant dur- ing the changes in volume and pressure. In case of a vapor, however, the curve is different, because at the point of satura- tion a decrease of volume is not accompanied by an increase of pressure (§ 160). Thus, in Fig. 188 let the point a represent the pressure and volume of an tinsaturated vapor. As the pressure increases the volume will decrease, forming the curve ab, which is thus far much like the curve of a gas. Further decrease of volume does not increase the pressure (§ 160), but will convert the saturated vapor at 6 to a liquid at c. This change is therefore represented by the horizontal line he, in which the vapor and liquid exist together with a distinct plane of separation. If the state is /, the ratio of the quantity of vapor to the quantity of liquid will be as cf to fb. When all is in the state c a further increase of pressure results in only a slight diminution of volume, as indicated by cd, for liquids are only slightly compressible. The curve abed is an isothermal of a vapor. If successively higher temperatures are taken, a similar series of changes in volume and pressure will produce similar isothermals, but the horizontal parts of the curves will become shorter and shorter, for there must be a greater decrease of volume before condensation begins and the liquid has greater volume at higher temperature. The point at which the horizontal line becomes infinitely short is called the critical point. If a line is passed through the points of satura- tion and the points of complete conversion to liquid, as shown by the dotted line in Fig. 188, the highest point of the curve thus formed is the critcial point P The line Pb is sometimes called the steam line and Pc the water line. The temperature of that isothermal which passes through the critical point is called the critical temperature. The pressure corresponding to the point P is called the critical pressure, and the volume of Fig. 188. HEAT. 227 So ■ iiiii^ ii|iiiiiii lljllj|ili|i|i||i|i|| lllllljl '^ so 1 III K IH iHiHIHI!III.HIBMI e JO 5 IB 11 i Ifti 1 i B 1 m so 40 so 60 Tem^ensttjre unit mass of a substance in this state is the critical volume. As the temperature is increased more and more above the critical one, the isothermals show that the substance complies more and more closely with Boyle's law. The distinction between a gas and a vapor is as follows: A sub- stance in a gaseous form and at a temperature above the critical one is called a gas, but below the critical temperature the same sub- stance is called a vapor. That a gas may be converted to a liquid it is necessary to cool it below the critical temperature and then subject it to a pressure sufficient to saturate the vapor at that temperature. No amount of pressure will liquefy a gas above the critical temperature. Oxygen, for example, must be cooled to — 118° C. and put under a pressure of about 50 atmospheres before it will become liquid. All well-known gases have been liquefied. This whole subject of the behavior of vapors as compared wth gases was investigated by Andrews, of the University of Glasgow, previous to the year 1869. His diagram for carbon dioxide is similar to that shown in Fig. 188. At a temperature of 13.1° C. and a pressure of 49 atmospheres the COj was changed to liquid. At 21.5° C. a pressure of 60 atmospheres was required to effect the same result. The critical temperature was found to be 30.9° C. At 31.1° C. the isothermal still showed consider- able deflection, but the substance remained a homogeneous mass, — i.e., did not become part vapor and part liquid, as it would have done during part of the process if below the critical temperature. At higher temperatures the deflections became less and less until at 48.1° C. they entirely disappeared. The state of a saturated vapor is often conveniently repre- sented on a pressure=temperature diagram. The curve in Fig. 1S9 is plotted from the table in appendix 29. The point a Fig. 189. Solid 228 GENERAL PHYSICS. represents a state of tinsaturated vapor at 65° C. and a pressure of 10 cm. It is plain from this diagram that a vapor in this state may be saturated either by uicreasiag the pressure to 6, — i.e., to 20 cm., — or by reducing the temperature to c, — i.e., to 49° C. Whenever the state may be represented by a point on the saturation curve, the vapor is saturated. In Fig. 190 any point on the curve Pc represents a state in which a vapor and its liquid are in equilibrium, — i.e., they can exist together without any of either passing over into the state of the other. This is the saturation curve of Fig. 189. Any point in Ph represents the pressure and temperature at which a state of equilib- rium exists between the solid and the liquid. In this diagram Ph is drawn for ice and water, and, since ice melts at a slightly higher temperature when pressure is decreased, the line will slope a very little ^'"'° downward toward the right. Some solids, such as ice, will evaporate and pass into vapor by a process called sublimation. This process will continue until there is equi- librium between the pressure resulting from the tendency of the solid to pass into vapor and the counter pressure of the vapor against the solid. Any point on the line Pa represents a state of equilibrium between a solid and its vapor at that pressure and temperature. A point common to these three lines represents a state called the triple point. Here the vapor is in equilibrium with its liquid, the liquid with its solid, and the solid with its vapor. This condition may be realized by a simple experiment first performed by Leslie. A shallow metal dish containing 3 or 4 c.c. of water is supported over another dish containing strong sulphuric acid. These are placed on the plate of an air-pump and covered with a shallow receiver. By exhausting the air the water will rapidly evaporate and its vapor is in large measure absorbed by the acid. Satura- HEAT. 229 tion of vapor in the receiver is thus prevented and the water will be rapidly cooled to the freezing point where its vapor pressure is only .46 cm. The triple point has then been reached, and the water is observed to boil and freeze at the same time. 163. Humidity. — Humidity is the state of the atmosphere in reference to the amotmt of water vapor it contains. Rela- tive humidity is the ratio between the mass of vapor actually contained in a given quantity of air and the mass which it would contain if it were saturated. This is evidently the same as the ratio of the vapor pressures or vapor densities on the assumption that Boyle's law holds true for vapors. Absolute humidity is the mass of vapor in the unit volume of air. Relative humidity is of greater importance and is usually designated simply as humidity. Air in its natural state always contains more or less water vapor, which may be brought to a state of saturation by a reduction of temperature, thus causing clouds, fog, and dew. The dryness of air, however, depends not so much on the quantity of vapor present as on the nearness to saturation. An increase of tem- perature will cause air to appear dry though the quantity of vapor remains the same. That which relates to the determination of humidity is called iiygrometry, and an instrument used for this purpose is a hygrometer. The three classes of instruments of most impor- tance are (1) chemical hygrometers, (2) dew-point hygrometers, and (3) wet and dry bulb hygrometers. In use of a chemical hygrometer a quantity of air of known volume, temperature, and pressure is passed through a drying tube which contains calcium chloride, pumice stone soaked in sulphuric acid, or phosphorus pentoxide. The increased weight of the tube is the actual amount of vapor in the air used. The ratio of this quantity to that which the air would hold if satu- rated is the humidity. (Table 30.) This method is accurate, but is somewhat difficult and tedious and is seldom used except for special scientific purposes. The dew=point hygrometer is in common use. There are several different styles of instruments of this kind. A good form is one shown in Fig. 191, known as the AUuard hygrometer. The face of this instrument is polished nickel. The surface D is the front part of a metal tube which is filled with ether. One 230 GENERAL PHYSICS. n thermometer shows the temperature of the ether and the other that of the air. By attaching a long rubber tube to A, it is pos- sible by use of a bulb to force a stream of air bubbles up through the ether. Thus the temperature is rapidly lowered and a mist appears on the surface D. The temperature of the ether when the mist first appears is the dew-point. The thermometers are read through a telescope at a distance, so that the humidity may hot be affected by the breath or heat of the body. Knowing the dew-point and the temperature of the air, it is seen from the diagram, Fig. 189, that the pressure of a satu- rated vapor at these temperatures would be represented by ordinates erected at the proper points of tem- perature on the abscissa and limited above by the saturation curve. The ratio of the ordinate at dew-point to the one at the temperature of the air is the humidity, for the longer ordi- nate is the pressure which the vapor would exert at that temperature if saturated. Instead of using the curve the pressures may be taken from table 29 in the appendix. The wet and dry bulb hygrometer is also in common use. One form of it is shown in Fig. 192. Two ther- mometers are mounted as shown, and the bulb of one of them is covered by a hollow wick which extends into a vessel of water. If the air were saturated with moisture no water would evaporate and the two thermometers would show the same temperature. But in proportion as the air is dryer the rate of evaporation will be greater and consequently the reading of the wet bulb thermometer will be lower. Evaporation at the maximum rate will occur when the air is moving about 10 feet per Fig. 191. HEAT. 231 second. After the temperatures have become stationary the ther- mometers are read. Then, by reference to psychrometrical tables prepared by long observation and by comparison of this instrument with dew-point hygrometers, the humidity may be directly found. 164, Transference of Heat. — There are three ways by which heat is dis- tributed or moved from one point to another, — viz., conduction, convection, and radiation. There is but one way, in fact, by which heat as such distrib- utes itself through a body, and that is by conduction, where heat energy is passed from molecule to molecule. Convection is an efficient method in the distribution of heat, as when por- tions of heated gases or liquids are displaced by buoyancy, thus causing a circulation which brings the colder portions of the fluid in contact with the source of heat. Convection involves the use of some agent outside of the, heated body to effect the transference. The heated body is carried from place to place. A hot mass of iron carried into a cold room would be a method of distribution according to the principles of convection. The uses of convection are numerous, as in the heating of water by the application of heat to the bottom of a vessel; the heating of buildings by the circulation of hot air or hot water; the circulation of the atmosphere and the movemients of ocean currents. In case of radiation a heated body sets up ethereal vibrations which are not heat, though they possess energy at the expense of the body whence they came, and when they are arrested by another body heat energy will appear again. Thus heat energy may be transferred through the agency of ether waves. Conduction and radiation are more fully discussed in the sections which follow. 165. Conduction. — While we say that heat is conducted from molecule to molecule throughout a mass of matter, yet it is Fig. 192. 232 GENERAL PHYSICS. not known by what mechanism this is accomplished. There appears to be an intimate relation between heat and electricity, as is evidenced by the fact that a good conductor of heat is also a good conductor of electricity. It has been suggested that the "roaming electrons," which when set in motion along a con- ductor cause what is known as a current of electricity, are also the agents by which heat is conducted. Silver and copper are the best conductors of electricity and also of heat. A list of conductors and non-conductors of electricity stand in nearly the same order as for heat. Heat conductivity is usually measured by the number of calories of heat that will pass through a cubic centimetre of a substance in one second when the difference of temperature on two opposite faces is 1° C. It is plain that more heat, Q, will flow in proportion as the area of the faces a, the time T, and the difference of temperature, t^ — t^, are greater. Also Q will be less in proportion as the distance, /, between the two faces is greater. Hence Qcc- I If k is the constant for any given substance, — i.e., the conduc- tivity as just defined, — then Q^^ait-QT ^231) where Q=k when all the other terms in the equation become unity. From (231) a{t,-t,)T (232) By use of an apparatus like that shown in Fig. 193 the value of k may be found experimentally. A rod of copper, c, is enclosed at one end in a jacket through which steam from the tube a is made to flow. The other end is surrounded by a coil through which a steady stream of water flows. The thermometers t^ and ^2 are fitted into holes in the rod at a distance, /, from each other. The whole is packed in abestos or other non-conducting material, that radiation may be prevented. Practically all the heat communicated to the rod by the steam will be conducted HEAT. 233 to the water at the other end. After the steam and water have been flowing for some time the readings of the thermometers will become stationary. Then, by collecting a mass, M„, of Fig. 193. water which flows in time T around the end of the rod, the quantity of heat which flows through the rod in that time can be measured. The temperature of the water has been raised ti — ig degrees; hence Q = M„{t-t,) (233) Equation (232) may then be written a{h-h)T (234) in which all the terms in the right-hand member may be deter- mined by the experiment. The rate at which the temperature of a bar of metal or other substance will rise when heat is applied at one end of it depends not only on thermal conductivity but also on another property called diffusivity, — i.e., the rate at which heat spreads, causing a rise of temperature in the cooler parts of the body. This depends on the specific heat of the body as well as on the con- ductivity. A body of large coefficient of thermal conductivity 234 GENERAL PHYSICS. and also large specific heat may diffuse heat more slowly than one of low conductivity and very small specific heat. Let 5 be the specific heat and p the density of the substance, then sp is the quantity of heat required to raise the temperature of unit volume 1° C. But if the quantity of heat measured by k is applied to this unit volume, the temperature will be raised /°. Then k=spt k or t = — (235) Thus it is seen that the rise of temperature is directly propor- tional to conductivity and inversely proportional to specific heat. The ratio expressed by the right-hand member of equa- tion (235) is called the diffusivity. 166. Radiation. — Radiation is a process by which a heated body sets up waves in the surrounding ether. The body is thus cooled by a loss of heat energy which then appears as energy of wave motion. Ether serves as a medium for the transference of heat, but the medium itself is not heated. If several bodies at different temperatures are placed apart from each other in an enclosure from which the air is exhausted, all will in time have the same temperature. The transference could not have been by convection for there was no air in the vessel, nor could it have been by conduction for the bodies were not in contact. One may feel the warmth of a fire at a distance from it, though the intervening air may be at freezing temperature. Heat energy of the sun is transformed into that of ether waves which move with the velocity of light, 3(10)"* cm, through about 1.5(10)' kilometres of space to the earth, where the energy is converted back to heat. According to Prevost's theory of exchanges, all bodies are constantly giving out ether waves whether there are other bodies to receive the waves or not. When several bodies of different temperatures are considered in their relation to one another, the hotter bodies radiate more heat than they receive from the cold ones, so in time there will be thermal equilibrium. Radiation still continues, but each body receives as much heat as it loses and hence there is no change of temperature. HEAT. 235 There is abundant evidence that heat and light waves are identical in character, the only difference being the length of the -waves. Both are transmitted on the ether and travel with the same speed. Both may be reflected, refracted, or dispersed. They are distinguished only by the difference in effects produced by waves of different length. When the wave length is .0004 mm. a sensation of violet is produced in the eye. As the waves grow longer and longer all different shades of color will result until red is reached, where the wave length is .00076 mm. These are only the limiting values of wave lengths that affect the eye. Much longer waves, formerly called heat waves, may be detected below the red of the spectrum. 167. Source of Ether Waves. — Waves set up in any medium have their origin in a vibrating body. A vibrating bell or tun- ing-fork sets up waves in the air, but waves of heat and light are transmitted on ether. These travel with a speed of 3(10)"' cm. per second; hence, if the wave length is .00004 cm., as in case of violet light, the number of waves per second must be 3(10)""-^ 4(10)'^ = 7.5(10)'^ This then must be the number of vibrations of the particles which cause waves of violet. It is thought that the vibrating particles which cause heat and light radia- tions are not the molecules but the minute corpuscles of which the molecules are composed. Waves of ether may vary in length all the way from .00001 cm. to several kilometres in length. 168. Measure of Radiant Energy. — Instruments used in the measurement of etherial radiations are called radiometers. Various forms of radiometers have been devised, the most common of which are here described. The thermopile, shown in Fig. * 195, consists of a number of thermo- ' ^^ couples made preferably of bars of "" * antimony and bismuth joined so that -— the electromotive force at each joint will be in the same direction. In ^'°' ^^*' Fig. 194 three couples are thus connected, the circuit from the first to the last bar being closed by a conducting wire and gal- vanometer. If the joints at the ends h axe heated, a current of electricity will flow from bismuth to antimony, while at the ends c the current is from antimony to bismuth. When the ends c are 236 GENERAL PHYSICS. 0° C. and h is 100° C, the electromotive force for these metals is only about .01 volt for each couple, but by joining a large num- ber of these couples a very small difference of temperature may be detected by use of a sensitive galvanometer. The thermopile shown in Fig. 195 consists of 49 couples. Their ends are black- ened, that they may more completely absorb the radiant energy which falls upon them. In the instrument shown, where the galvanometer is not par- ticularly sensitive, the radia- tions from a candle flame at a distance of 10 feet will cause a deflection of several divisions on the galvanom- eter scale. The bolometer is an in- strument invented by Lang- ley for the detection and measurement of small changes in temperature. It consists essentially of a Wheatstone bridge, Fig. 196, in one arm of which is inserted a strip of platinum foil covered with lampblack. The bridge is first balanced and the galvanom- eter shows no deflection, then when radiations from a heated body fall upon the blackened strip P, the resistance in that arm is increased, thus throwing the bridge out of balance. The result- ant deflection in the galvanometer is a measure of the change of temperature of the platinum strip. The intensity of radiation from various sources may thus be compared. A change of .0001° C. may be detected. The radiomicrometer is an instrument devised by D' Arson val and later improved by Boys. It is a delicate D'Arsonval gal- vanometer in which the movable coil consists of a single turn of pure copper wire suspended between strong magnets. Fig. 197. One end of the coil is soldered to a light block of antimony, a, and the other to a similar block of bismuth, b. These are joined to a thin strip of copper, d, which extends a short distance below and is covered with lampblack. Radiations falling on d will ' J I ¥1 Fig. 195. HEAT. 237 heat the junction of the thermo-couple, thus causing a current of electricity to flow through the coil C. This causes the coil to turn in the magnetic field, and the distance through which 771 A Fig. 196. a Fig. 197. m it turns is a measure of the intensity of radiation. This instru- ment may be made exceedingly sensitive. Crookes's radiometer, shown in Fig. 198, consists of a light frame, at the ends of the arms of which are mounted light disks of mica blackened on one side. This is supported so that it will freely rotate in a glass bulb from which the air is exhausted to a vacuum of about one-thou- sandth of a centimetre of mercury. When radiant energy from some heated body passes through the glass into the bulb, the blackened faces are heated more than the polished ones, and the whole frame will rotate with the blackened faces moving away from the source of heat. The cause of the rotation is that when the molecules of air next to the black surfaces are heated they leap away with in- creased energy and by their reaction cause the disk to move in an opposite direction. Since the air is rare, the mean free path of the molecule is compara- tively long. Hence the pressure due to this increased molecular motion is not communicated through the mass of air to the opposite side of the disk, as would be the case at ordinary density. Fig. 198. 238 GENERAL PHYSICS. Professor E. F. Nichols has modified this instrument by suspending a horizontal arm from a fine quartz fibre, a mica disk blackened on one side beiag supported at each end of the arm. In the wall of the glass vessel is a fluorite window which freely admits radiations of all wave lengths. One of the black- ened disks is opposite the window, and when it is repelled as explained above the quartz fibre is twisted through an angle which may be measured by aid of a light mirror attached to the arm. This is the most sensitive of all radiometers. 169. Laws of Radiation. — When a certain quantity, Q, of radiant energy falls upon a body, a portion of it, q, will be ab- sorbed and converted to heat. The ratio ~ is called the coeffi= cient of absorption, or simply the absorption. If in any case this ratio is unity, the body will absorb all the incident waves and convert them into heat. Such bodies are said to be perfectly black, or simply "black bodies," A body covered with lamp- black is nearly a "black body," though in no case is the ratio exactly unity. The ratio of the quantity of heat emitted by a body to the quantity which it would emit if it were a "black body" is called the emission. For any given body the absorption and emission are numeri- cally the same. This law was first deduced by Balfour Stewart and Kirchhoff, and is sometimes known as the Stewart=Kirch= hoff law. It may be directly deduced from Prevost's law of exchanges, for when a body is in thermal equilibrium with its surroundings its absorption and emission must be the same or its temperature would change. The relation between emission and absorption may be illustrated by a simple apparatus like that shown in Fig. 199. A glass tube, bent in the form shown, communicates with the interior of the metal cylinders p and b and is partly filled with a colored liquid. The cylinders are full of air. When the cylinders are at the same temperature, the liquid in the tube will be stationary. The face of b is covered with lampblack while that of p is polished. The face B of the central drum C is black and P is polished. Now if C is filled with boiling water or other hot Uquid, no change will be noted in the tube; consequently the air in the two cylinders is equally heated. HEAT. 239 The blackened face b absorbs all the energy radiated by the polished face P, while p absorbs only a portion of that from B. But the faces ^ and P are alike, and P emits as much heat energy as p absorbs. Hence the ratio of heat emitted by P to that which it would emit if it were black is the same as the ratio of the amount absorbed by p to that which fell upon it, — i.e., p absorbs as much energy as it would emit if its temperature were the same as P. The power of absorption possessed by matter may be ex- plained as a kind of resonance. When the motion of the minute particles of which matter is composed is synchronous with the ether waves which fall upon a body, the effect is increased motion or heat. Consequently we would expect that the waves which are thus absorbed would be the ones which would be emitted when the body is heated. Thus a body which transmits red light must absorb the waves which would give the higher colors of the spectrum. If then this same body is sufficiently heated, it will give out a bluish-green light. Heated carbon will emit waves of all lengths; hence when carbon is cold it will absorb waves of all lengths and will as a consequence be black. Lampblack and platinum black absorb nearly all the radiations that fall upon them. Polished silver reflects nearly all radiations incident on it, absorbing only about .02 of them. Rock salt transmits about .92 of the incident radiations, absorbing practically none. The Stefan=Boltzman law, already referred to under pyrom- etry (§ 143), states that the total energy emitted by a black body is proportional to the fourth power of the absolute tem- perature. By total energy is meant that due to waves of all lengths. Radiation from such a body is independent of the nature of the substance and depends on temperature only. If E represents the total energy, t the absolute temperature, and c a constant for any conditions that are selected, then E = CT* (236) Fig. 199. 240 GENERAL PHYSICS. By use of a radiation pyrometer operated according to these principles, it is possible to determine the temperature of distant bodies. Thus, the temperature of the s\m is found to be about 6000° C. This, however, is the "black body" temperature. Estimates from other data give 7000° C. or more. The tempera- ture of the electric arc is 3500° C, and, since the body is black, this is about the true temperature. The Nemst lamp is about 1950° C. and the incandescent lamp about 1500° C. The displacement law, first formulated by Wien and often known by his name, states that if the radiations from a "black body" which is heated to a high temperature are dispersed so as to form a spectrum, there will be one color, — i.e., a train of waves of definite wave length, — where the energy of radiation is maximum, — i.e., greater than for any other train of waves. As the temperature of the body rises, this region of maximum radiation shifts to waves of shorter wave length. Wien has shown that if X is the wave length of that train of waves where radiation is maximum and t is the absolute temperature, then Xt: = c, a constant (237) After c is once determined and the value of X found by observa- tion, T is readily calculated. This principle is applied in one form of optical pyrometer. A body at a high temperature will lose heat by radiation more rapidly than when it is cooler. Newton's law of cooling is that the rate of cooling is proportional to the difference in temperature between the body and its surroundings. This law, however, is not exact, as may be seen from the following data of an experiment. Difference of temperature. Rate of cooling in degrees per mm. Ratio. 68.3° C. 42.6° C. 28.5° C. 19.8° C. 1.95 1.10 .65 .40 35 39 44 50 If Newton's law were rigidly correct, the ratios in the third column would be constant, but the rate of cooling decreases more rapidly than does the difEerence in temperature. If, how- HEAT. 241 ever, the difEerence in temperature is small, only a few degrees, as is the case in many laboratory experiments, the law raay be applied to determine the loss of heat due to radiation. This may be done, with a fair degree of approximation, as follows: Suppose correction is to be made for the loss of heat of a calo- rimeter where the temperature of a mass of water or other liquid has been raised by the introduction of a hot body. The tem- perature is noted at short intervals during the rise, and the average of these gives the mean temperature i„ of the calo- rimeter. The temperature of the surrounding air t^ may be considered constant or its mean may be determined. Then the mean difference of temperature during the time of the experi- ment is t^ — ta- If the time required for the experiment is n minutes, the number of calories, q, lost during that time is q = c{t^-Qn (238) The constant of radiation, c, may be found after the tempera- ture of the calorimeter begins to fall by noting the number of calories lost per minute when the difference of temperature between the calorimeter and its surroundings is 1° C. For exam- ple, suppose that in 3 minutes the temperature falls .6° C. and that the mean difEerence of temperature between the calorimeter and air during that time is 5° C, then if the mass of water includ- ing the water eq\uvalent of the vessel is 300 g., 300X.6 ,„ , ^ = ^X^ = 12cal. This is the loss in one minute when the difEerence in tempera- ture is 1° C. Hence equation (238) gives the loss in n minutes when the difEerence in temperature is i„ — /„. 170. Thermodynamics.— Thermodynamics, as the name im- plies, treats of the relation between heat energy and mechanical or other forms of energy. The motion of the small particles of which a mass is composed represents a certain quantity of energy, just as truly as does the motion of the mass as a whole. Any kind of energy may be converted into molecular motion (heat), and this heat energy is exactly equal to the energy expended in producing it. Heat may in turn be often converted into mechanical motion. In all such changes there is an exact 16 242 GENERAL PHYSICS. equivalence between energy expended and energy received for the same amovint of heat. The fact, now well established, that heat phenomena are included in the general law of conservation of energy is one of the greatest and most important generalizations of modem science. The principle of conservation of energy is, that, while energy may appear in a great variety of forms, — ^heat, electrical, mechanical, wave motion, etc., — yet the sum total of all the energy in the universe is a constant quantity, and the various forms are so correlated that whenever energy of one form dis- appears an exact equivalent in some other form appears. This is a ftindamental principle in physical science, and includes heat energy as well as other kinds. Up to the beginning of the nineteenth century a generally accepted theory stated that heat was an imponderable, self- repellent fluid called caloric. A body was hot when it contained a large quantity of caloric. Water had a great capacity for heat because it could hold a large quantity of caloric. Friction caused a rise of temperature because the abraded particles had less capacity for heat than when in the solid body. A piece of metal was heated by concussion because the impact increased the density of the metal and so the caloric was squeezed out. Thus, heat was assumed to be a material substance. An anal- ogous theory, also generally accepted at that time, explained combustion as the escape of a material substance called phIogis= ton. Such theories prevailed up to 1800 A.D. and were not abandoned till about the middle of the nineteenth century. Even in an edition of the "Encyclopaedia Britannica" published in 1856 heat is defined as a "material agent of peculiar nature. " The first to combat publicly the caloric theory was Benjamin Thompson (Count Rumford), a native of Massachusetts, U.S.A., but at that time superintending the construction of cannon at Munich, Germany. He in 1798 performed a simple experiment which led to very important results. He noticed that the cannon were heated by boring, and was able, by using a blunt drill, to produce a large quantity of heat while only a small quantity of the metal in form of powder or shavings was bored away. As long as his engine continued to turn the drill, heat was produced. He rightly concluded that the heat was com- HEAT. 243 mtinicated to the cannon by the motion of the drill and at the expense of work done by his engine. According to the calorists the amount of heat should be in proportion to the quantity of shavings or abraded particles, but Rumford showed that the capacity of the shavings was the same as that of the solid metal. The calorists in reply insisted that while the temperature might rise as before, yet the quantity of heat was less. The claims of Count Rumford and others who believed as he did were ridiculed by the calorists for about forty years. But a line of experimental evidence has firmly established the modem theory of heat and thermodynamics. In 1799 Sir Humphry Davy performed a simple but convinc- ing experiment which the calorists could not satisfactorily explain. By rubbing together two blocks of ice he showed that the ice will be melted. The calorists admitted that water has a greater capacity for heat than ice has. Whence, then, does the heat come if not from the motion of rubbing? — for the experi- ment can be successfully performed in a vacuum or in air below the temperature of melting ice. In this case the friction did not diminish the capacity for heat. Later a series of most careful and painstaking experiments were performed by James Prescott Joule, of Manchester, to test the claim that heat is a form of energy due to motion of particles within a body and to determine the number of units of energy in a thermal tmit. Later still, in 1878-9, Rowland, of Johns Hopkins University, made a similar but more exact determina- tion as described in the next section. Then Griffiths (1893), Schuster and Ganon (1894), and Callendar and Barnes (1899) made similar tests, using the heat effects of electric currents. 171. Mechanical Equivalent of Heat. — The mechanical equiv- alent of heat is the number of units of energy which when con- verted into heat will produce one thermal \init. Thus, it may be the number of foot-pounds of energy required to raise the temperature of 1 lb. of water 1° F. at any chosen temperature of the water, or it may be the number of ergs required to raise the temperature of 1 g. of water 1° C. at any selected temperature. Joule employed a variety of experiments in his endeavor to find this mechanical equivalent. His methods were direct, — i.e., he converted a measurable quantity of mechanical energy into 244 GENERAL PHYSICS. .« (y © la W © heat and then measured the quantity of heat produced. Thus the heat effects due to work done in compressing air, in expansion of air, in friction of mercury, friction of iron plates, friction of water, etc., were measured by the rise of temperature in a given mass of water or other Uquid. The chief value of Joule's work was not so much his numerical results as the fact that the con- stancy of his results showed that heat is only another form of energy, and the quantity of heat measured in energy units is equal to the quantity of work expended in produc- ing it. In Joule's later and more accurate work he used an apparatus illustrated by dia- gram m Fig. 200. Paddles are attached to the vertical axis of the calorimeter C and are made to rotate in a meas- ured quantity of water. Pro- jecting strips fastened to the interior walls of C prevent the movement of the water as a body along with the paddles. The calorimeter is supported mainly from beneath by the vessel v, which floats on water. Thus fric- tion of the bearings is greatly reduced. The paddles were made to rotate by turning the wheel a. Cords, b, b, passed tangentially from opposite sides of a circular rim at the top of C, over pulleys, and to their ends were attached weights, w, w. When the paddles are turned with sufficient speed, the weights are just sufficient to prevent any rotation of C. By means of a recorder attached to the vertical axis, the rotations in a given time are automatically counted. To calculate the work done, let r be the radius of the rim D, then the circumference is 2nr. Although the weights do not rise or fall, yet the work done in tiuming the paddles through one revolution is just the same as if the weights had been raised through a distance 27:?', — i.e., the work is the same as if the paddles had been stationary and the calorimeter had been turned by a force 2w exerted through a distance 2Kr. Hence work = 27n- . 2w = ^-jzrw (239) Fig. 200. HEAT. 245 and in n revolutions work = 4LKrniv (240) If the quantity of water including the water equivalent of the calorimeter is M and the rise of temperature is t°, the mechanical equivalent, /, is /=^=^ (24.) This is the energy per unit mass per degree, — i.e., the mechanical energy per caloric of heat. Joule gave as his result 7 = 772.65 .j^*"",^!*!; for water at 61.69° F. ' lb. 1° F. This expressed in c. g. s. units and centigrade degrees is 7 = 4.167(10)' ergs g. 1°C. at 16.5° C. Dr. Joule used a mercury-in-glass thermometer which by com- parison with recent standards is found to be inaccurate. A recalculation with correction for errors in temperature gives as the results of Joule's experiment 7 = 4.173(10)' Henry Augustus Rowland (1848-1901) devised an apparatus similar in principle to that of Joule but much better in construc- tion. The paddles were rapidly turned by means of an engine and thus the time of each experiment was shortened. He repeated his experiment thirty times, varying the temperature of the water and using different thermometers. His original values were Temperature. J (in ergs). 5°C. 10° C. 15° C. 20° C. 25° C. 30° C. 35° C. 4.212(10)' 4.200(10)' 4.189(10)' 4.179(10)' 4.173(10)' 4.171(10 ' 4.173(10)' 246 GENERAL PHYSICS. It was by this series of experiments that Rowland estab- lished the fact that the capacity of water for heat is different for different temperatures. As seen in the table above, the value of J diminishes with rise of temperature to 30° C. and then begins to increase. The change in capacity is shown by the change in the number of ergs required to raise the temperature 1° C. Rowland's results are probably the most reliable of all the various determinations of /, and we may accordingly give as a very close approximation 7 = 4.187(10)' y-5^||^ between 15° and 16° C. Thus, our thermal iinit may be defined in terms of dynamic tmits as 4.187 joules of energy, or, if a temperature of 10° C. is chosen, 4.2 joules. Other experimenters, principally those mentioned at the end of the preceding section, sought by indirect methods to determine the value of /, — i.e., they measured the heat effects of a current of electricity when passed through a wire submerged in water. If the strength of current, i, is measured in amperes, and the resistance, R, in ohms, then the energy expended in heating the wire in T seconds is energy ='j^i?r joules (242) The number of calories of heat, Q, received by the water is determined by the mass of water and its rise in temperature. Hence JQ = i'RT(10y (243) from which the value of / can be calculated. Values obtained by this and similar methods are Griffiths, 4.187(10)' at 25° C. Schuster and Gannon, 4.190(10)' at 19.1° C. Callendar and Barnes, 4.190 (10)' at 15° C. 172. Laws of Thermodynamics. — There are two general principles underlying the phenomena of heat energy, known as the laws of thermodynamics. HEAT. 247 1. Whenever mechanical energy is transformed into heat, the heat energy thus produced is exactly equal to the mechanical energy which disappeared. This may be expressed by E=JQ where E is the mechanical energy expended or work done and Q is the number of calories of heat which appears as a consequence. This law is only a statement that heat is included in the general law of conservation of energy. 2. Heat cannot of itself pass from a cold to a hot body. This law states the direction of the flow of heat, — i.e., always from a body of higher to one of lower temperature. Since it is only during the transference of heat that its energy becomes available, it is impossible to devise any machine which will derive any mechanical effect from a body which is colder than all surrovmding bodies. The cold body may contain a consider- able quantity of energy, but no way is known by which that energy may be made available except by the transference of heat from it to a body which is still colder. 173. Difference of Specific Heats of Gases. — It has been shown (§ 136) that PV^Rmr If now the pressure is constant while the temperature is raised one degree, the volume will become V^. Hence PFi=i?w(T+l) (244) The difference between these equations is P{V^-V)=Rm (245) But m = \ g. , since specific heat is the quantity of heat needed to raise 1 g. 1° C. Hence in this consideration P{V,-V)=R (246) This is the work done, for P{V^—V) is the product of pressure by change of volume. The cause of the difference in specific heat at constant pres- sure, Cp, and that at constant volume, C„, is the external work done when the pressure is constant (§ 156). Hence, expressing 248 GENERAL PHYSICS. this difference in mechanical units, JiC^-C^)=P{V,-V)=R (247) The value of R may be foimd when the pressure, volume, and temperature of a gas are known (§ 136) ; Cp can be reUably determined by experiment; hence C„ can be calculated. 174. Effect of Intermolecular Forces. — When a gas expands without doing any external work, its temperature as a whole does not change, for it has lost none of its energy. This is on the assumption that the energy of the gas consists only in the motion of its molecules. But if there is cohesion between the molecules, then when the gas expands some of the kinetic energy of the molecules would become potential, for work would have to be done in effecting a separation against cohesion. The effect would be a fall of temperature, — i.e., a lowering of kinetic energy. If, on the other hand, there is repulsion between mole- cules, this would add to kinetic energy in case of expansion and the temperature would rise. Dr. Joule attempted to test this matter experimentally by compressing gas in one vessel and allowing it to escape through a small tube into another vessel from which the air was exhausted. Both vessels were immersed in a water bath. By this arrangement no external work would be done during expansion, but Joule was not able to detect any change in the temperature of the water and concluded that if there was any change it was very slight. In a later experiment, with an improved apparatus and more delicate thermometers. Dr. Joule and Lord Kelvin performed what is known as the porous plug experiment. They passed air and other gases through a copper coil immersed in a water bath at constant temperature and allowed the gas to escape through a plug of cotton wool, the difference of pressure on the two sides of the plug being one atmosphere. The temperature of the gas on each side of the plug was measured by delicate thermometers. An air-pump operated by an engine was made to do the external work which the gas would have had to do if it had expanded without assist- ance. Hence any change of temperature would be due to work done because of intermolecular forces. By this arrangement it was possible to maintain continuous expansion of a stream of gas. The porous plug prevents a rapid motion of the gas, which HEAT. 249 if permitted would represent a certain amount of kinetic energy obtained from the heat energy on the other side of the plug. The results for a few gases are Gas. Temperature before passing through the plug. Change of temperature. Air 17.1 91.6 8.7 93.0 12.8 19.1 91.5 6.8 90.2 -0.255° C. — 0.203° C. — 0.317° C. — 0.165° C. — 1.207° C. — 1.144° C. — 0.69° C. + 0.89° C. + 0.46° C. Air Oxygen CO2 CO2 COj Hydrogen Hydrogen All gases except hydrogen showed a fall of temperature, indicat- ing a cohesion of the molecules. Hydrogen increases slightly, indicating a repellent force between its molecules. An ideal gas is one in which there are no intermolecular forces. This is a condition which is not found to exist, but with a knowledge of the deviation in any case, an actual thermometer may be compared with ideal conditions. The greater the difference of pressure on the two sides of the plug, or the lower the temperature of the gas, the greater is the change of temperature on passing through the plug. Thus, as seen in the table above, oxygen at 93° C. fell .165° C, while at 8.7° C. it fell .317° C. This principle is utilized in the Hque- faction of air, oxygen, hydrogen, and other gases. Although hydrogen increases in temperature when it expands without doing external work, yet at about — 80° C. the effect is reversed and it is cooled. The gas to be liquefied is compressed by power- ful pumps and allowed to escape by a small opening correspond- ing to the porous plug. It is then made to return over the tube through which it advanced, thus cooling the gas, which by escape from the orifice is still further cooled. This constitutes what is called a regenerative process, which is continuous as long as the pumps are operated. When the critical temperature (table 26) is reached, the gas begins to change to the liquid state. By this process liquid air is produced in large quantities, and other gases long considered permanent have been liquefied. In this manner 250 GENERAL PHYSICS. Professor Dewar in 1898 liquefied hydrogen and by allowing it to evaporate under reduced pressure changed it to the solid state. The hydrogen must first be cooled by liquid air or other cold liquid, and then it can be further cooled by the regenerative process. The boiling point of liquid air under one atmosphere is at first — 192° C, but later, after the nitrogen has escaped, — 182° C. Hydrogen boils at — 253° under a pressure of one atmosphere. In the year 1908 Professor Onnes, of the University of Leyden, succeeded in changing helium to the liquid state. He was able to obtain a considerable quantity of this gas from monazite sand. By cooling with liquid hydrogen and by use of the regenerative process he obtained about 60 c.c. of liquid helium, the density of which he found to be .15 and the temperature of the boiling point 4.5° C. on the absolute scale. This is the lowest tempera- ture known. Temperature as low as — 200° C. may be measured with a platinum thermometer. For still lower temperature the hydro- gen thermometer is reliable provided the pressure is well above the critical pressure. 175. Ratio of Specific Heats of Gases. — ^When a gas is heated, the energy thus expended (1) will increase the translatory motion of the molecules, — i.e., give them greater kinetic energy due to their increased velocity; (2) may do external work, as when a gas expands against pressure; (3) may increase the rotary or vibratory motion within the molecule, a change which is prob- ably proportional to the change of translatory motion; or (4) may do work in separating molecules against intermolecular forces. The fourth will ordinarily consume but a small quantity of heat, as is shown by the porous plug experiment; hence we may consider theoretically what the ratio -^ should be if all heat were expended as indicated in (1) and (2), and compare this result with experimental results. The comparative influ- ence of (3) may thus be shown. _ The kinetic energy of any mass m moving with velocity V is e = \mV^ (248) If m is the mass of one molecule of a gas and V is the mean velocity, the energy e oi n molecules is i-mnV^ (249) ■V HEAT. 251 Let n be the number in 1 g. of the gas, then mn is 1 g. ; hence e = ^y2ory2 = 2e (250) One gram is taken because we are here discussing specific heat. By §89, PV = \V^ : PV==ie (251) Now if this gas is heated 1° C. under constant pressure, its volume will increase by a certain amovmt, say v, and its energy will increase to e'. Hence P{V+v)=y (252) Subtracting the preceding equation, Pv^^e'-e) (253) But Pv is the product of pressure by change of volume, and therefore it is the external work done by expansion under constant pressure. The total change of energy is e' — e, hence two-thirds of this is expended in doing external work. Now, C„ includes the external work, hence ^p ^= ' ^ + ^^^' ^) = 1+1= 1.666 = r (254) The value of y for various gases and vapors may be found from the velocity of compressional waves passing through them, as when sound passes through air. The velocity of such a wave varies directly as the square root of the elasticity E of the gas and inversely as the square root of the density p. Hence velocity of wave = , — V p But since the compressions and rarefactions are so rapid that heat has not time to enter or leave the gas during the passage of the wave, the elasticity is adiabatic, and so is not equal to the pressure (§§ 94, 176), but to yP Hence velocity of wave = — ^ Vp From this the value of y can be determined. For such gases as argon, helium, and vapor of mercury, experiment shows a value very nearly equal to 1.66. In these gases the molecules consist 252 GENERAL PHYSICS. of a single particle of matter, — i.e., they are monatomic, — hence the heat would all be expended in increasing translatory motion and doing external work. In a gas such as oxygen, O2, the value of y is 1.41. In air, 1.41. In ether, with its complex molecule, C4H10O, the value of Y is only 1.03. Thus, as might be expected, the greater the complexity of the molecule the greater the amount of heat ab- sorbed in producing motion within the molecule. 176. Adiabatic Expansion. — When a gas expands without receiving or losing any heat, the expansion is said to be adiabatic — {a-dia-^atveiv, not to pass through) . It is evident that if a gas is compressed and the heat resulting from the external work done on the gas is not allowed to escape, the pressure will increase more rapidly with decrease of volume than when the process is isothermal. Hence if curves are drawn on the pressure-volume diagram. Fig. 201, for PV = constant, isothermal, and PV' = constant, adiabatic, the adiabatic curves a, a, a will show a more rapid rise in pres- sure than the isothermals i, i, i. If the gas expands in doing external work, the pressure rapidly falls, for no heat is received from the outside. 177. Elasticity of Gases. — Since a gas may expand either isothermally or adiabatically, there will as a result be two coefficients of elasticity. It has already been shown that the isothermal elasticity (§ 94) is equal numerically to the original pressure P, for if V then, as the increase of pressure p and the resulting change of volume V approach the limit zero, the actual pressure on the HEAT. 253 gas becomes P; hence P=E, The adiabatic elasticity is greater than P because all the heat energy is retained in the gas. This coefficient, as shown in appendix 13, is equal to the product of pressure by the ratio , for since PV' — a. constant, E^ = rP (255) 178. Carnot's Cycle. — The modem theory of thermody- namics may be said to have been established by such scientists as Lord Kelvin, Helmholtz, Clausius, and Rankine about the year 1850, yet the foundations upon which these men built ^ i ' &im 1'" ■ 1 ' t H I ?^V:^^:^-^?^ v/" l'.-/.;;'^;;'^ I c i ^ i Hot i i e Fig. 202. were laid by Sadi Camot (1796-1832), a French physicist who sought to increase the efficiency of the steam engine. The so-called Carnot's engine is an imaginative one, devised for the study of ideal conditions which can never be fully realized but to which the practical engine is an approach. The true nature of heat was not known to Camot, but a study of the principles which he announced has led to valuable results in the way of a knowledge of the limits of a heat engine and of thermodynamic principles in general. To tmderstand Carnot's cycle suppose a quantity of air, steam, or other gas or vapor, called the working substance, is enclosed in a cylinder, A, Fig. 202, the walls and piston of which Fig. 203. 254 GENERAL PHYSICS. are adiabatic but the bottom a perfect conductor. Assume also that the tops of the hot body H and the cold body C are perfect conductors, the top of B being adiabatic. Let the gas in A have a pressure and volume represented by the point a on the pressure-volume diagram Fig. 203. First, having the gas at the temperature of the cold body, place the cylinder on B and press the piston down till the temperature rises to that of the hot body. Here work is done on the gas and, since no heat escapes, the change is repre- sented by the adiabatic line ab. The gas is now in the state b. Second, move the cylinder over on H and allow the gas to expand. In this change exter- nal work will be done by the gas and heat will flow from H — i-- into A and keep the tempera- ture constant. The volume will increase and pressure will fall. This change is represented by the isothermal be. Third, move the cylinder back to B and allow the expansion to continue till the temperature falls to that of the cold body. There is an increase of volume and a rapid fall of pressure. This is represented by the adiabatic line cd. Fourth, place the cylinder on the cold body C and push the piston down to the original starting point. Here work is done on the gas, but without increase of temperature, for heat flows freely into C. One cycle is now complete and the gas is in the same state as at the start. The change was from a to b,b to c, c to d, and thence back to a. The capacity of the bodies H and C are assumed to be so great that their temperature is practically tmchanged by the loss or gain of heat. Some heat was transferred from H to C during the cycle of operations, but not all of that taken from H was delivered to C. An inspection of Fig. 203 shows that the lines da and ab represent changes produced by work done on the gas, while be and cd show changes while work was done by the gas. If the mean pressure along each of these four lines is multiplied by the change of volume, the products will be the work done on or by the gas. HEAT. 255 The difference is the area abed, which is the excess of work done by the gas. Thus it is seen that of the heat taken from H part was ex- pended in doing external work and the balance was delivered to the cold body. It is also apparent from the diagram that the greater the difference between the hot and cold bodies, the greater will be the area of abed, — i.e., the greater the external work. This is true no matter what the working substance may be, for the work is accomplished only by the expenditure of the heat energy. The efficiency of an engine is defined as the ratio of the work done to the energy received, or, in other words, the ratio of the work done by the engine to what it would do if all the energy which it received were expended in work. Let Qj be the number of calories of heat received from the hot body, and Q^ that delivered to the cold body, then the work W, expressed in ergs, is W = J{Q,-Q,) (256) and the efficiency E is ^ To a ^ ^ If all the heat were expended in work, Q2 would be zero and the efficiency would be -^^ = 1. Efficiency could not be greater than unity. ' 179. Reversible Cycle. — By use of an external agent it is possible to reverse Camot's cycle and restore to H, Fig. 202, the heat which was transferred to C. Starting at a, Fig. 203, let the gas expand isothermally while the cylinder is on C. Heat will flow from C into the gas and the change will be ad. Then place the cylinder on B and push the piston down till the temperature rises to i^. This change is represented by dc. Then place the cylinder on H and push the piston still lower. This gives the isothermal cb. Now place the cylinder on B and let the gas expand till the temperature is t„. The cycle is thus completed in a reverse direction, a quantity of heat, Q.^, has been transferred from C to H, and the excess of work done by the external agent over that done by the gas is represented by the area abed. 256 GENERAL PHYSICS. Camot's cycle is exactly reversible because of the ideal conditions which are assumed for his engine. In practical operations, however, it is not possible to avoid conduction and radiation of heat and friction of the moving parts of an engine. These cannot be reversed by reversing the engine, and so the working substance cannot be brought to its original state. 180. Carnot's Theorem. — Camot showed that all reversible engines working between the same temperatures have the same efficiency, and no other engine can have a greater efficiency under the same conditions of temperature. He proved this by showing that the operation of such an engine would be contrary to the second law of thermodynamics. Suppose an engine, M, assumed to be more efficient, is coupled to a Camot's engine, C, so that C is made to run backward as explained in the preced- ing article. Let C take Qj units of heat from the cold body and deliver Qj units to the hot body. Let M take Q^ units from the hot body and, since M is more efficient, deliver less than Q^, say Q^, units to the cold body. By this operation the quantity of heat in the hot body will remain the same, but that in the cold one will grow less and less. The engine M would thus be able to operate C and do external work besides. Work would be done by using up the heat of the cold body. But this is contrary to experience, — i.e., to the second law. Hence no engine can be more efficient than a reversible one. For the same reason no form of reversible engine can be more efficient than another. This theo- rem, in other words, states that an engine is most efficient when all the heat received from the hot body and from external work done on the working substance is used in changing the state of the work- ing substance, — i.e., there is no loss by conduction, radiation, etc. 181. Thermodynamic Scale of Temperature. — In all ther- mometers which have thus far been described temperature is measured by changes depending on some property of a substance, as expansion of a gas or mercury, change in electric conductivity, etc. Exact determinations of temperature by these methods is very difficult, because of the many different conditions which modify the results. If, however, there were a perfectly reliable standard with which ordinary thermometers could be compared, the latter could be used as effectively as the standard. A per- fect gas — i.e., one in which there are no intermolecular forces. HEAT. 257 and consequently one which is true to Boyle's law — would make such a standard, but such a gas does not exist. It has been shown above that the efficiency of a Camot's engine is independent of the character of the working substance, and depends only on the difference of temperature of the hot and cold bodies between which the engine works. The amount of work done by a working substance in one cycle depends only on the difference of tem- perature between the isother- mals of that cycle. Based on this fact. Lord Kelvin proposed a thermody- namic scale of temperature which is independent of the substance used. Thus, in Fig. 204, A, let a Camot's engine be of such dimensions that, when working between tem- peratures ij and t^, 1 joule of work will be done in complet- ing the cycle abcda. Then ti—t^ might be chosen as a degree on some new scale. If the same engine does 2 joules of work in the cycle fbcef, h~h would be two degrees on this scale, and 3 joules would be three degrees, and so on. To make this scale correspond with the centigrade scale, let the two isothermals of a cycle be tjj^ and t„, — i.e., the temperature of boiling water and that of melt- ing ice. Then the area abed, Fig. 204, B, will be the work done in one cycle between these limits of temperature. Now, if isothermals are drawn dividing this area into 100 equal areas, each isothermal will differ from the next one by 1° C. If the temperature of the hot body remains constant and that of the cold body is lowered until all the heat, JQ^ ergs, taken from the hot body is expended in work during the cycle, then the Fig. 204. efficiency = - 01 ' = 1 17 258 GENERAL PHYSICS. for 02 has become zero. This could occur only in case the cold body contains no heat. It could not then be any colder, for the efficiency cannot be greater than unity. This low temperature is the absolute zero on the thermodynamic scale. Lord Kelvin showed by experiment that this scale of tem- perature coincides very closely with the centigrade scale of a gas thermometer and would agree exactly if the gas were perfect. Hence the temperature of melting ice on this scale is almost exactly 273 and that of boiling water 373. The hydrogen thermometer most nearly coincides with the thermodynamic scale, the difference being due to Latermolecular forces in the gas. This very slight difference was determined by the porous plug experiment (§ 174), and corrections may be made accordingly in exact measurements. By inspection of the diagram, Fig. 204, B, it is seen that if each small area included by two adiabatics and two adjacent isothermals be considered as a imit of heat measured in ergs, the number of such units at any temperature, counting froin absolute zero, is to the number at any other temperature as the corresponding absolute temperatures are to each other, for each change of one degree corresponds to a change of one unit of heat. Hence, if Tj and ^2 ^^^ the absolute temperatures, 1 = 1 (258) Q!^.h=l3 (259) Thus efficiency is expressed in terms of temperature. 182. Entropy. — In considering a reversible cycle such as is represented in Fig. 203, where ab and cd are adiabatics inter- sected by isothermals, let the point a represent the state of a body in reference to pressure, volume, and temperature. If the pressure is increased, no heat being allowed to escape, the state may be changed to b. Let Q be the total quantity of heat in a substance in state a, — i.e., the total number of thermo- dynamic units as measured from absolute zero; also let r be the absolute temperature at a. Then, as Q is increased, z will increase in the same ratio ; therefore — will be constant between = constant (260) HEAT. 259 a and b. For illustration, let the total quantity of heat at a be 22,800 ergs and the absolute temperature 285° C. Then for each increase of 1° C. there is an increase of 80 ergs of heat energy. Hence 22800 _ 22880 _ 22960 285 ~ 286 287 ^^^'-^^ Whatever the value of Q may be, the ratio T Entropy may be defined as that quantity which remains con- stant during an adiabatic change, just as temperature is that which remains constant during an isothermal change. For this reason an adiabatic line is often called isentropic. Although entropy cannot be measured by any instrument, as temperature can be measured by a thermometer, yet it is a distinct physical quantity which can be calculated when the pressure, volume, and temperature are known. In passing from b to c, Fig. 203, t remains constant but Q increases, hence the ratio — increases. From c to d the ratio is constant, for Q and t decrease at the same rate. From d to a the temperature is again constant while Q decreases, so the ratio of Q to T decreases. If Q^ is the increase of heat in passing from b to c, and Qj is the loss of heat in passing from d to a, then, according to equation (258), -^+-^ = (261) — i.e., in a complete cycle of this character the sum of the ratios — is zero. T A series of changes which form a complete reversible cycle may be represented by a closed curve, as in Fig. 205. The area enclosed represents work done, and may be considered as composed of an infinite number of Camot's cycles such as abcda. Since equation (261) is true for each cycle, it is true for all the cycles of which this area is composed. Hence 2-1 = (262) T Fig. 205. 260 GENERAL PHYSICS. where q is an infinitely small change in the quantity of heat gained or lost in the small cycle and t is the absolute tempera- ture at which the change was made. By making the Camot's cycles infinitely small the change of state becomes continuous, as shown by the smooth curve. Equation (262) shows that the increase of entropy during one part of the cycle is equal to the decrease during another part, so that on the completion of the cycle the entropy is the same as at the beginning. The natural zero of entropy is — the state of a substance devoid of all heat, but it is more conven- ient to select a certain state arbitrarily as a standard and calcu- late the change of entropy from that standard, just as the tem- perature of melting ice is assumed to be 0° C. while in fact the temperature is 273° C. Thus in Fig. 203 let a be the point of reference for entropy, then the entropy at c will be the sum of an infinite number of very small additions of heat each divided by the temperature at which the addition was made. If entropy is represented by rj, rio = I^ (263) This is the case no matter whether we pass from a to c by the path ab, be or by ad, dc, and in the continuous change shown in Fig. 205 the entropy at B in reference to A as a standard is also expressed by equation (263), — -i.e., entropy depends only on pressure, volume, and temperature, and is independent of how or by what means the change of state was effected, just as a weight raised from the groimd to a certain height will possess a definite amoimt of potential energy in reference to the grotmd no matter how or by what path it was raised. In the particular case shown in Fig. 203 it is seen that in changing from o to 6 there is no change of entropy, but from b to c r remains constant and change of entropy is proportional to the increase in the quantity of heat. Here Iq==Q, where HEAT. 261 Q is the total heat added between b and c; hence -I but in changes along a path where both q and z are constantly changing, change of entropy will be found by adding an infinite number of these ratios as expressed by equation (263) . Mechanical energy is constantly being changed into heat, and heat is constantly passing by conduction, convection, and radiation from bodies at higher to those at lower temperature. If 9i is heat transferred from a body at temperature r^ to an- other body at temperature T2, the entropy of the former will be diminished — and the latter will be increased — . But since z, is less than z^, the entropy of the second body will be increased more than the first is decreased. Clausius expressed this by saying that the entropy of the universe tends to a maximum. If the maximum is ever reached, there will no longer be any available energy, for there will be no high or low and hence no transference from one to the other. The Sim is the great disturber of equilibrium, and as a result we have at present, available for work, fuel, food, running water, wind, water waves, and direct solar radiation. The amount of radiant energy intercepted by the earth is only about ^200000 P^rt of the total energy sent out continually from the sun, yet, according to experiments made at points where the sun's rays are vertical, the energy value of solar radiations on each square meter of earth's surface is about one horse-power. The radiations falling on one-half the whole surface of the earth are the same as would fall vertically on the area of a great circle of the earth. This area is about 128(10)" square metres. Hence the total energy received from the sun in any given time is equivalent to the work which could be done by 128(10)" horse- power in the same time. 183. The Steam Engine. — In a steam engine the working substance is vapor of water at a high temperature and pressure. A cycle of operations is performed similar to that of a Camot's cycle, but the walls of the steam engine are not adiabatic, and so there is considerable conduction and radiation of heat. Such 262 GENERAL PHYSICS. heat is lost as far as ef&ciency is concerned. The boiler may be considered the hot body and the condenser the cold body. The amount of work done will be, allowing for loss of heat, the difference between the thermal energy received from the boiler and that given to the condenser. The maximum efficiency is expressed by equation (259). In the reciprocating engine steam is admitted first to one side and then to the other of a piston, thus causing a forward and backward motion. As shown in Fig. 206, steam is admitted Fig. 206. to the steam chest sc above the slide valve sv. When sv is in the position shown, steam at boiler pressure is admitted through port p', thus driving the piston toward B. Before this stroke is completed, the slide valve moves back and covers the port p', and the steam in A contiaues to expand to the end of the stroke. During this time the port p has been closed, but the steam in B, used in the previous stroke, may escape beneath the slide valve and out through the exhaust pipe e. The exhaust port is closed shortly before the completion of the stroke, and the vapor thus entrapped in B serves as a cushion for the piston. Then the port p is opened to boiler pressure, and the same operation is repeated, but in opposite ends of the cylinder. When the engine is working under full load, more heat energy is needed than when the load is light. Various devices are employed to regulate the admission of steam at a rate adapted to the work being done. One method of accomplish- HEAT. 263 ing this end is to .regulate the flow of steam into the steam chest by use of a governor like that described in § 35. Another method is to regulate the motion of the slide valve so that for light work less steam will be admitted to the cylinder, and the work will be done by expansion rather than by direct boiler pressure. This may be accomplished by use of a governor like that shown in Fig. 207. A stiff spring fastened to the fly-wheel carries a heavy mass, m. When the wheel rotates, m is thrown out toward the rim. This moves the arm a and brings the eccentric Fig. 207. Fig. 208. ring e to a position more nearly concentric with the axis c. Around the eccentric and sliding upon it is a metal strap to which the eccentric rod er is attached. If e were exactly con- centric with c, the eccentric rod would have no motion and the slide valve would rest over both steam ports. If e is at maxi- mum eccentricity, the slide valve would admit steam at boiler pressure through the whole stroke of the piston. Thus, a con- stant speed with a changing load is effected by regulating the distance through which the slide valve moves. The work which is actually being done by the steam in the cylinder may be determined by use of a steam-engine indicator, shown in Fig. 208. Steam is admitted from either end of the engine cylinder to a small cylinder, C. The steam pressure raises a small piston against a stout spring, and thus the pencil P is made to move up or down according to pressure. At the HEAT. 265 spring gives the average pressure of steam — usually in pounds per square inch. This times the area of the piston in square inches gives the total pressure. The total pressure times the distance in feet travelled by the piston per minute is the number of foot-pounds of work per minute. The indicated horse=power is then found by dividing by 33,000. The exhaust steam still contains a large quantity of heat and so may do more work between its temperature and a lower one. If the steam is exhausted from the first or high pressure cylinder into another cylinder of the same construction but larger, additional work will be done. Such an arrangement is shown in Fig. 210. Both pistons are attached to the same axis and move in the same direction. The steam exhausted from one end of the high pressure cylinder passes into the opposite end of the lower pressure cylinder. The steam then passes into the condenser, or into a third, fourth, or even sixth cylinder, each larger than the preceding one, and then into the condenser. Turbine Buckets Fig. 212. Instead of having the steam pressure produce a reciprocating motion of a piston, it may be directed with great velocity against a row of buckets or blades on the periphery of a wheel. Such constitutes a rotary engine or steam turbine. In one type of such engines, steam under high pressure is projected from a nozzle against a series of buckets attached to the circumference of a wheel, the wheel being keyed to the drive-shaft. These may be called impulse turbines. The De Laval turbine, illustrated in Fig. 211, is one of this type. The turbine proper is shown to the right of the figure. Steam may be admitted to the buckets at several points around the wheel as shown. A section of one 266 GENERAL PHYSICS. of the nozzles is shown in Fig. 212. To the left of the turbine, Fig. 211, is shown the gear case. The turbine is nin at high speed to secure proper efficiency. Here the angular velocity is reduced, and the motion is communicated to two shafts which operate the dynamos shown on the left of the figure. Another type may be called impulse-reaction turbines, for they are driven not only by the impulse of the steam but also by the reaction resulting from a change in the direction of flow caused by numerous fixed and movable blades set in the path of the steam. This is illustrated in diagram, Fig. 213. 111111 Ml Fig. 213. Steam under high pressure is admitted from the side A through a row of fixed blades, 5, and directed against a row of movable ones, R, which are thus forced in the direction of the long arrows passing through them. The same operation is repeated, but with less intensity, through a number of successive rows, all fixed to the same spindle. In Fig. 214 is shown a spindle covered with rows of movable blades, and also one half of the casing. The fixed or guide blades are attached to the casing and fit in between the rows of movable ones. Steam is admitted at the smaller end of the casing and forced along the space between the casing and spindle, — i.e., the space filled with alternate rows of fixed and movable blades. The Parsons turbines are of this type and are in common use, particularly for the propulsion of large steamships. The increase in size of the spindle as the velocity of steam decreases gives an advantage similar to that in case of compotind cylinders. The turbine, when properly constructed Fig. 214. cFrom Sothcrn's " Tlit- Marinu Steam Turbine. "J HEAT. 267 and operated, is quite as efficient as other forms of engine and is without many of the objections to the reciprocating engines. Instead of converting water to steam and then using the heat energy of steam to do work, the energy of fuel may be more directly used in internal combustion engines, usually called gas engines. Here the fuel in form of a gas is mixed with oxygen of the air and exploded in a cylinder. Thus the piston which closes one end of the cylinder is driven forward. Two impor- tant types of this engine are the four-cycle and the two-cycle. In the former there are four strokes of the piston for each explosion. (1) The outward movement of the piston draws into the cylin- der a mixture of gas and air. (2) The back- ward motion compresses the mixture. (3) The explosion drives the piston out again. (4) The return motion exhausts the cylinder. There cannot in this type be more than one explosion for every two revolutions of the fly-wheel. In the two-cycle engine there may be an . . T 1 ■ Fig. 215 explosion m each revolution. In diagram. Fig. 215, (J is a space enclosed by the crank case. When the piston C moves toward A , it will close the ports p and e, compress the gas in A , and draw gas through o into the crank case. Then the gas in A is exploded by an electric spark, and as C descends, the port e is first opened for the exhaust and then p is opened for the admission of fresh gas and air. This operation is then repeated. This type has the advantages of being without movable valves and of being lighter for the same power, while in the four-cycle type the com- bustion is more complete and the cylinder is not so highly heated. The power of a gas engine may be computed in a manner very much like that for the steam engine. The average pres- sure is found from the indicator diagram and horse-power = - 33000 where ^ = average pressure per sq. m. / = length of stroke in feet a = area of piston in sq. in. e = number of explosions per min. 268 GENERAL PHYSICS. Problems. 1. If when 1 c.c. of water at ioo° C. is converted to steam under pressure of one atmosphere its volume increases to 1649 c.c, how much external work is done? How much internal work? 2. How much is the melting point of ice lowered by a pressure of one atmosphere? Use ^'~^' = '"'~'^' (see equation 259) Q, — (32=work done=P(t/j— iij) P = 1.013(10)' dynes Change of voliune on melting = .091 c.c. per c.c. (3i = 80X4.187(10)'ergs=ergs of heat energy expended in melting 1 g. i",—r2= result sought n = 273°C. 3. How much ice at 0° C. would be melted by the energy in a mass of 20 kg. moving with a velocity of 3000 cm. per second? 4. If when the temperature of air is 23° C. the dew point is 8° C, what is the relative humidity? 5. Calculate the maximum efficiency of a steam engine working between temperatures 120° C. and 10° C. (Equation 259.) 6. Calculate the indicated horse power (I.H.P.) of a steam engine from following data: Area of indicator diagram 2.7 sq. in. Length of diagram 3.0 in. Diameter of piston 20.0 in. No. of spring 50 Revolutions per min. (R. P.M.) of fly-wheel. . . .300 Length of stroke 2 ft. 7. How much work can be done by the energy obtained by the combustion of 50 litres of hydrogen at 20° C. and under a pressure of 10 atmospheres? 8. Calculate the value of / from J(Cp—Cv) =K and PV^mRr Use air at 0° C, assuming it to be a perfect gas. — = — where p is m p density of air under standard conditions. P = l. 013(10)°. r = 273. 1. 1.67(10)» ergs. 2.077(10)" ergs. 2. .00751° C. 3. 26.866 g. 4. 38.3 per cent. 5. 28 per cent. 6. 514.08 h. p. 7. 5.96(10)" ergs. 8. 4.19(10)' ergs/ig. PC. INDEX OF APPENDIX PAGE 1-14. Proofs not found in body of text 271 15. Dimensions of mechanical units 286 16. Dimensions of thermal units 287 17. Moment of inertia, 1 288 18. Density of solids, liquids, and gases 289 19. Density of air 290 20. Density of water and mercury 291 21. Reduction of barometric height to 0° C 292 22. Coefficients of elasticity 292 23. Surface tension 292 24. Heat constants of solids and liquids 293 25. Specific heat of gases and vapors 294 26. Critical temperatures 294 27. Thermal conductivity 295 28. Vapor pressures 295 29. Pressure of saturated water vapor 296 30. Weight of aqueous vapor in i cubic metre 297 31. Heat of combustion 297 32. Freezing mixtures 297 33. Value of gravity 298 34. Miscellaneous data 298 35. Sines, cosines 300 36. Tangents, cotangents 302 37. Logarithms 304 APPENDIX PROOFS NOT FOUND IN THE BODY OF THE TEXT. 1. Pressure on the vertical side of a rectangular vessel filled with liquid. It is required to find the total pressure on the side yl, Fig. 216, of a liquid whose density is p and depth h. Since pressure is proportional to depth, if the liquid is considered as composed of elementary layers of width a and depth dx, then at any depth *; the pressure is pgx, and the pressure against the element- ary strip is pgaxdx. The total pressure on all the strips is -•A / pgaxdx = ipgah^ =pgah~ for ah is the total area A. Hence the total pressure on the side is the weight of a column of liquid whose dimensions are the area of the side considered and whose depth is half the depth of the liquid. If c. g. s. units are used, the result is found in dynes. Fig. 216. This is true of any area where a horizontal line through its centre divides the area into equal upper and lower parts. In case the vertical side is a triangle, as shown in Fig. 217, the total pressure on that side is found by multiplying the area of the triangle by the pressure at two-thirds of the depth. Let ABC be the end of a prism filled with water. Let m be the 271 272 APPENDIX. length of the horizontal base and h the altitude. Also let a and b be the segments of m made by the vertical h. Then tan d = -r h hence an elementary strip at depth x and of width dx has an area -rxdx h The pressure in water at depth x is x, hence the total pressure on this elementary area is -z-x'^dx h and the total pressure on this part of the triangle is -•ft P„= / ^x'dx •=/^-' ah' ah' 3h 3 ^ah 2h ■ 3 In the same manner it may be shown that the pressure on the portion of the triangle having the base b is ^■f Pb = ^bh .^h Hence the pressure on the whole triangle is P=l(a + b)h.^h or P=—mh . -^h The pressure is thus shown to be equal to the product of the area of the triangle {^mh) by the pressure at two-thirds of the depth. This result times the density would give the pressure for any other liquid. APPENDIX. 273 2. To deduce a formula for the velocity of falling bodies when the acceleration varies inversely as the square of the distance. Fig. 218. Let r be the radius of the earth and 5 any distance greater than r. Let g be the acceleration at the surface of the earth and a the acceleration at a distance 5 from the earth's centre. Then a _r''' since the force of gravitation, and consequently the accelera- tion, varies inversely as the square of the distance. Hence of- Hence, since F = ma, the force at a distance S is The work done by this varying force is W= / Fds -I- Integrating between S and S„, ■"So ISo W = mgr^ I ~=- = mgr^\ — = ■^ \s ^ /So 1 1 .•. w =mgr'i 18 274 APPENDIX. The energy of the falling body may also be expressed by Thus the velocity acquired by a body in falling from a point S„ to S may be found. 3. A mass swinging as a simple pendulum will at its lowest point have the same velocity as when it falls vertically from the same elevation. In Fig. 219 let om be a pendulum whose length is I and point of suspension o. Let om be inclined 6° to the vertical. When m swings to P it will be moving with the same velocity as if it had fallen from Q to P. The general equation for velocity in terms of space is The value of 5 when the body falls verti- cally is QP. But QP=l-oQ=l-l cos 6 = 1(1-005 d). Hence v^ = 2gl{l- cos 6) for velocity acquired in the vertical fall. When m moves along the arc mP, the same general equation {v^ = 2gs) is appli- cable, but g becomes g sin d as shown in the figure, and this quantity varies with each small change in position, ds, along the arc mP. But ds=ldd, and the square of the velocity will then be the sum 2gl sin ddd from 6 to zero angle. That is Fig. 219. r v' = 2gl I si sin ddd = 2gl cos = 2gl-2gl cos = 2gl{l-cose) This is the same result as that found above for the vertical fall from Q to P. 4. When force is applied to a turbine wheel in a direction per- pendicular to the plane of rotation, the component causing rota- tion will be maximum when the blades are set at an angle of 45°. APPENDIX. 275 Let F be the total force and x the efEective component, then, as already shown in the text, x = F sin 6 cos d By differentiation, do- ^ = F(cos='e-sin2(?) The turning effect of the force will be greatest when —j^ is zero, consequently the angle sought is one where F(cos2|9-sin2(9)=0 or cos ^ = sin ^ This is true only when the angle is 45°, for only then does cos (? = sin(9. 5. To find an expression for coefficient of rigidity in case of a solid cylindrical rod. For the cylindrical shell it has been shown that Fhl 2:!drH The t is here the thickness of the shell.^^.e., the differential of the radius, dr, of the solid rod, and Fh is the dFh when the solid rod is considered as composed of an infinitely large number of concentric shells. Hence ,7^ 2nndrHr dFh = :.Fh = I 2i:wd I rHr .'. n = - U 21 2Fhl ndr* 6. Show that the moment of inertia, 7, about any axis is equal to / about a parallel axis through the centre of gravity plus the mass of the body multiplied by the square of the dis- tance between the axes. 276 APPENDIX. Let o, Fig. 220, be the centre of gravity of the mass. Draw rectangular coordinates through o. Let one axis, perpendicular to the paper, pass through o, and another, parallel to the former, Fig. 220. through A. Let /„, be the moment of inertia ia reference to the axis through o, and /„ in reference to that through A. Then it is to be shown that Let an elementary mass dm be chosen- at a distance r from o, and p from A. Then r.=/. /„= / p'^dm but p^ = y''-\-{x+sy = y^+x'^ + 2xs+s^ '■-\-2xs+s^)dm = I {y'' + x^)dm + 2s I xdm+s" I ■ I iy^ + x^)dm= I y'^-\-x^ = r^ r^dm=I„ 2s I xdm = for a body is in equilibrium about the centre of gravity, and hence the sum of the moments, xdm, is zero. APPENDIX. 277 / dm = s'm .■.I^=I„+ms^ 7. Find moment of inertia, I, of a circular disk with axis through its centre and perpendicular to its plane. Fig. 221. Let the mass of the disk be m and the radius r. Choose an elementary ring of radius x and width dx; then the total area of the surface of the disk is to the area of the elementary ring as the total mass is to the differential mass dm. That is, m 2nocdx dm , 2mxdx .". dm,= r — 7 'f' 'dm Substituting the value of dm and integrating between x = o and x=r, 2mx^ -P dx im^ = ^mr'' »' .". I = ^mr^ 278 APPENDIX. The moment of inertia about any other axis parallel to this one and distant s from it is I'-m{^+s^) The radius of gyration, k, is such a distance from the axis that In case of this disk .■.k'=- m 2w 2 1/2 The square of the radius of gyration in any case is the moment of inertia with the factor m omitted. The expression I = ^r^ is independent of the thickness of the disk, and hence is the expression for the moment of inertia of any cylinder ro- tating on its own axis. 8. Another method of calculating the moment of inertia of a cylinder rotating on its axis is by use of polar coordinates. Thus, in Fig. 222 let st be a section perpendicular to the axis, and let dA be a differential area in this plane. This area is located by (r, d) and its value is dA^rdrdd Fig. 222. for d is expressed in radians, and there- fore rdd is the length of the arc in this elementary area, dA, and dr is the width. If the length of the cylinder is /, an elementary prism extending the whole length of the cylinder and having a cross section dA is IdA^'lrdrdd and if density is tmiform, the elementary mass is dm = plrdrdO (p = density) APPENDIX. 279 But Jo Jo /2k rHrddpl plR 4 since i:RHp is the mass m, R being the radius of the cylinder. 9. To find moment of inertia of a sphere rotating on an axis through its centre. Let R be the radius of the sphere, and r the radius of an elementary section, st, perpendicular to the axis and at a distance x from the centre of the sphere. The thickness of this element- ary section is then dx, and its mass is pTtr'^dx where p is the density. The mass of this elementary section is the differential mass, dM, of the sphere! The moment of inertia of this elementary mass is therefore Ir'^dM for it is a cylinder of length dx. ^r'dM = ^p7tr*dx Hence the moment of inertia of the sphere is the sum of all the elementary sections between +R and —R. That is I^ipn I r*dx Fig. 223. J-B (R^—x^ydx 280 APPENDIX. Since r^ = R^—x^, as seen in the figure, r*={R^ — x^y r+R \pK I iR*-2R'x' + x*)dx=ip7r . nR' = ^7:pR' = i7:R'p . W But ^nR^p is the mass, M, of the sphere. .-. I = ^MR^ 10. Find the moment of inertia of a thin rectangular plate whose length is b and whose width is c, when the axis bisects b and is parallel to c. b 2 b 2 I — 11 Ac Fig. 224. Let e be an elementary area at a distance x from the axis. Its area is cdx. The total area is be. Let the total mass be m. The mass of the elementary part is dm. Hence be _ m cdx dm .'. dm = -j-dx "J" /m x^dx b , b Integratmg between — — and+— APPENDIX. 281 T'dx m T ■g-X .-./ = m I IT 24 "*" 24 /^2 1 1 . Find 7 of a rectangular par allelopiped with axis through its centre and perpendicular to the side ab. .•«_..:''A dx Fig. 225. Let a be the length, b the breadth, and c the depth. Let m be the total mass. Let an elementary section at a distance x from the axis have a mass dm. Then a m dx dm or dm = — dx a It has been shown above (10) that I for this elementary mass yyib' is — r^ when the axis is through its centre and parallel to c. It has also been shown (6) that its moment of inertia in reference 282 APPENDIX. to another axis at a distance x and parallel to its own axis is ( V \ »i(— jj- + af^). But the wi of this elementary mass is the dm of the parallelepiped. Hence, substituting for m in >**(— ^+a;M the value of dm = — d%, and integrating between — it and -^-7:, a ^ A 2 "^ m f b^x x^\ "2 \a ■ 12 • 2j'^ \a " 2iJ a' + b' .I = m 12 12. To prove that pv^ is a constant quantity in case of adiabatic expansion of gases, f being the ratio of specific heat at constant pressure to that at constant volume, — i.e., -—-. Let Q be the quantity of heat required to raise unit mass of a gas through dt degrees, the pressure p being constant. This heat will be expended both in raising the temperature and in increasing the volume of the gas. The quantity of heat required will be the same as if the gas had been kept at constant volume and the temperature raised dt degrees, and, in addition, the heat energy required to do the external work in increasing the volume by dv. This external work is pdv, expressed in ergs, or in calories (§ 102), / being the number of ergs per calorie. The increase of temperature per gram requires C„dt calories where C„ is the specific heat at constant volume. Hence APPENDIX. 283 But in adiabatic expansion no heat is gained or lost by a gas; hence Q = 0. Then ^ + CA = Q (1) By equation (210), pv=Rx By differentiation, pdv+vdp=Rdx , pdv + vdp or or=- R But if p is kept constant, then pdv=RdT pdv „ or ^-;—=R dr If dr is 1° C, pvd=R = J{Cr-C„) since ^dw is the excess of work done at constant pressure over that at constant volume when the mass and change of tempera- ture are unity, — i.e., it is the difference in specific heats times the number of ergs per calorie. Hence , _ pdv+vdp Substituting this value of dz in equation (1) above, pdv C^pdv + C^vdp ^ or Cppdv—C„pdv + Cypdv + C„vdp = r or -p^pdv+vdp^O Let -^ be represented by y; then, dividing through by pv, ' V p By integration, ;■ log V + log p = = constant .'. ^z;v = constant 284 APPENDIX. 13. Adiabatic coefficient of elasticity. It has been shown that when no heat enters or leaves a body of gas, pyT = constant, c (1) (2> (3) (4) (5> (6) Which shows that £,, the elasticity at constant entropy, is equal to the pressure times gamma, y, where ;- = the ratio of specific heat at constant pressure to that at constant volume. It has been shown that elasticity at constant temperature is E, = P (§94) (7) Dividing (6) by (7) -E-r-r-u: (8) 14. To calculate the change of entropy in case of a perfect gas. Let Q be a ftmction of pressure p and volume v, then by partial differentiation V = c P Differentiating, yV-^dV- -cdP = p2 or, substituting the value of c from (1) in (3) ^V^-idF = . -PVdP p2 dP p2^y7-i Pr •• dV PV V dP .-. dV =. Er, = Pr V dr Multiplying each quantity in the parentheses by -r— , APPENDIX. 285 But — = Cp = specific heat at constant pressure in the first parenthesis, but C„ in the second. Since the gas is assumed to obey Boyle's law, pv = Rz (4) /arX _v_ ' ■ \dp)^~'R and m =1- \dv)p R Substituting these values in (3), dQ^C^^dv + C^^dp (5) pv Dividing by T=-n-, T '^ V p = C,^ + C^^ (6) Integrating and remembering that / ~^:r = 'il, the change of entropy. dp •. , = C,log,^ + C„log^^ (7) .■"i , r i„„ ^2 286 APPENDIX. 15. Dimensions of mechanical units. Physical quantity. Definition. Dimension. Derivation. I L Mass m M Time t T • V U LxL Volume V U LxLxL Velocity — =D LT-^ ^=LT-^ T Linear acceleration — ^t LT-' LT-'^T = LT-' Angtilar velocity I 7--1 LT-^-i-L-T-'^ Angular acceleration T r-' LT-^-i-T=LT-' Force F=ma LMT-' MXLT-^ F P ML-'T-' MLT-^ -^U Ft LMT-^ LMT-' X T Momentum mv LMT-^ MxLT-'^ Work Fs=w UMT-^ LMT-'' XL Potential energy Fs UMT-' Kinetic energy ^mv^ UMT-' MxL'T-' Power W t UMT-' L'MT-^—T Moment of force Fl UMT-^ LMT-'- XL Moment of inertia y:mr' UM Densitv m ML-' Pressure F P L-'MT-' LMT-'^—U APPENDIX. 16. Dimensions of ttiermal units. 6 = unit of temperature Thermal unit = e.g. calorie Dynamical unit = unit of energy e.g. erg. 287 Physical quantity. Definition, Dimensions. Derivation. Quantity of heat mt° (thermal) Me Quantity of heat mfj (dynamic) MVT-' MexL'T~'e-^ Coefficient of thermal expansion . . I Lt° e-' LxL-'xe-' Thermal conductivity ATt" (thermal) ML-'T-' Thermal conductivity QL ATt" ■' (dynamic) MLT-'e-^ ML-^ r-' X UT-' e-' = MLT-' e-' Emissivity. . . Q ATt" (thermal) ML-'T-' Me X r-' L-' e-' =ml-' r- Emissivity. .. ATt" ■' (dynamic) MT-'e-^ ML-'T-"^ xL?T-'e-^ =Mr-'e-i Capacity wXsp.ht. M Latent heat.. Q m (thermal) e M-^xMe=e Latent heat.. m ' (dynamic) UT-' exUT-'e-'^ Joule's equivalent . . - energy ■'- Q ]JX-iQ-i MUT-' xM-'e-i ^jjx-^e-^ Entropy Q T (thermal) M Mexe-'=M Entropy (dynamic) MUT-'e-' Mx^T-'e-^ 288 APPENDIX. S^ .3 ^ 00 3 •3 '•B + + + 5, + °0 e c at ta 'd p ■d CI ^^ 'd M_i J o 'o l«l O ca o fi '» be f-; .t3 01 CIS w biD ^ nj.r^ +3 t^Ti $ ■g u S tu fl 18 'd CJ axi {H 2^ o o ny .d 0) ■+-> ° s — -fj -d « ca 13 •d ta .El U) 3 o u 3 o u § 8 "^ bpo 3 +J O _1 Hi ca X e -d CI o CO ■d o en ■d ci "o X ■d cl o "o K X a ta s 42 .3 ^ •3 M hfl C G ca ca Pi Hi p< a. ca U3 3 a ca rt APPENDIX. 289 18. Density of solids in e/c.c. Alum 1.7 Ivory 1.83—1.92 Aluminum < I wrought . . . . 2.56—2.58 Lead 11.3 . . 2.65—2.8 Limestone 2.46—2.86 Asphaltum . .. 1.1—1.2 Marble 2.5—2.8 Brass ... 8.1—8.7 Mica 2.6—3.2 Brick .... 2—2.2 Nickel 8.9 Copper . . 8.5—8.95 Oak 8 Cork 24 Osmium 21.4—22.4 Clay . .. 1.8-2.6 Paraffin 87— .91 Coke ... 1.1—1.7 Pine, white .... 35— .5 Diamond ... 3.5—3.6 Platinum 21.5 Ebonite 1.15 Porcelain 2.,3-2.5 ^^-{mr.;::;;:::;; . .. 2.4—2.8 Quartz 2.65 ... 2.9^.5 Silver 10.53 Gold 19.3 Sodium 97— .99 Graphite ... 1.9—2.3 Tin 7.29 Ice at o°C 91 Zinc 7.15 Iridium . 21.8—22.4 Mean density of the earth . . . 5.576 t cast 7.4 Surface density of the earth . . 2.56 Iron, ■{ wrought 7.8 V steel 7.8 Density of liquids at 0° C. in «lc Alcohol 81 Bromine 3.187 Carbon disulphide 1.293 Ether 736 Glycerin 1.26 Hydrochloric acid 1.27 Linseed oil, boiled 942 Mercury 13.596 Nitric acid 1.56 OUve oil 918 Sea water 1.025 Sulphuric acid 1.85 Turpentine 873 Density of gases at 0° C. and 76 cm. pressure in k/c.c Air 001293 Ammonia 00079 Carbon dioxide 001974 Chlorine 003133 Hydrogen 0000895 Marsh gas 00072 Nitrogen 001257 Oxygen 00143 Steam 100° C 000561 19 290 APPENDIX. 19. Density of air. Density of dry air at 0° C. and under pressure of 76 cm. of mercury is .001293. Coefficient of expansion = .00367. Hence h 76 where t is the temperature and h is barometric pressure. , .^ .001293 density = ^^_QQ3g^^ t°C. 72 cm. 73 cm. 74 cm. 75 cm. 76 cm. 77 cm. 10 .001181 .001198 .001215 .001231 .001247 .001263 baro- by a eair. point )or. 11 1177 1194 1210 1226 1243 1259 12 1173 1189 1206 1222 1238 1255 «-S5ts 13 1169 1185 1202 1218 1134 1250 5-s.a^.S 14 1165 1181 1197 1214 1230 1246 idity Iimin: apor the orate 15 .001161 .001177 .001193 .001209 .001225 .001242 5