ornia lal MANUALS OF TECHNOLOGY. EDITED BY PROF. AYRTOI,\ F.R.S.> and R. WORMELL % D.Sc., M.A. PRACTICAL MECHANICS. EY JOHST PEEET, M.E., Professor of Mechanical Engineering and Applied Mathematics at the Ciiy and, Guilds of London Technical College, Finslunj. WITH NVMEKOlJS ILLUSTRATIONS. JOHN S. PRELL Civil & Mechanical Engineer. SAN FRANCISCO, CAL. CASSELL. FETTER, GALPIN & Co. LONDON, PAEIS $ NEW YORK. [ALL RIGHTS RESERVED.} 1883. Eagineering Library PREFACE. THIS is an attempt to put before non-mathematical readers a metlwd of studying mechanics. The student will not benefit much by merely reading the book, nor will he benefit much even if he supplements his reading by listening to lectures on mechanics ; but I believe that if by means of lectures he obtains a thorough comprehen- sion of the book, and then makes common-sense experi- ments with the simple apparatus which is to be found even in the poorest laboratories, but which has hitherto been xised merely to illustrate lectures ; if, in fact, he uses this book to study mechanics in the manner herein recommended, he will gain in a short time such a working knowledge of the subject as will well repay his labour. I am quite sure also that the mental training acquired in this way is of a kind not inferior to that the belief in which retains in our schools the study of ancient classics and Euclid. The principle of my method is one which I have tested in practice during the last twelve years, in an English Public School, at the Imperial College of Engineering in Japan, and in other places. It is simply the practical recognition of the fact that all experiment- ing must be quantitative. It may exercise the wonder 733254 JAN 2 3 1956 of a child, it may please his senses to see certain well- known lecture illustrations which are but little better than tricks, and possibly for young children there may be great instruction in such exhibitions, but they con- tain no instruction for thinking men who can obtain sufficient amusement elsewhere. Our subject must be studied through quantitative experiments, and when this method of study is adopted it is but of little consequence at what part of the subject the student begins, so long as he begins from his own natural standpoint, the stand- point given him by all his experience. The primary fact in technical education not yet sufficiently recognised is this that illiterate men often acquire and possess a useful knowledge of the principles underlying their trade. But the theory usually acted upon is that a man must be quite ignorant of the principles of his trade unless he has been led up to them through weary years' study of the elementary principles of science. On the contrary, the wise apprentice sees for himself that some illiterate journeyman gets good wages, is well thought of by his master, is able to do his work better, perhaps, than any other man in the shop, can be trusted in emergencies, and has a confidence in himself which experience justifies; and the apprentice feels that although the journeyman might be a better workman if he knew the elementary principles of science, still, some- how or other, he has gained an exact knowledge of those more complicated laws with which he has to deal in his trade. PRKFACB. vii It is good for a man to know the well-established elementary principles of science, to which all complicated laws can be reduced ; this enables him to compare his own experience with that of all other people, and enables him to make better use of his own observations in the future. In giving this knowledge, however, the usual plan of operations is to act on the assumption that the man knows nothing, because he did not begin his previous study with Euclid's axioms, and to teach him the elemen- tary principles as schoolboys of no experience are taught. Now, the standpoint of an experienced workman in the nineteenth century is very different from that of an Alexandrian philosopher or of an English schoolboy, and many men who enei'getically begin the study of Euclid give it up after a year or two in disgust, because at the end they have only arrived at results which they knew experimentally long ago. I am inclined to believe that if, instead of forcing the workman to study like a schoolboy, we were to teach the boy as if he had already acquired some of the experience of the workman, and made it our business to give him this experience, we should do better than at present. That is, let the boy work in wood and metal, let him gain experience in the use of machines, let him use drawing instruments and scales, and you put him in a condition to understand and appreciate the truth of the fundamental laws of nature, such a condition as boys usually arrive at only after years of study. It is Vlll I'UEFACK. true lie may regard the 47th proposition of the First Book of Euclid as axiomatic; he may think the important propositions in the Sixth Book as easy to believe in as those of .the First; he may have greater doubts as to the universal truth of these propositions than mathe- maticians usually have ; but it is possible that these evils are not unmixed with good. The readers of this book are supposed to have some previous knowledge of the l>ehaviour of materials and machinery. My aim is to give the student such a training as will cause him to think exactly, to give him a method of studying whatever phenomena happen to come before his eyes. Phenomena which, when carelessly considered in the light of elementary principles, appear to follow complicated laws are often found to follow approximately simple laws of their own. A man who knows these roughly correct laws is in a good position for learning the fundamental principles of mechanics ; but his teacher must try to view the subject from his student's standpoint, else he cannot take advantage of the fact that his pupil may already possess an excellent foundation on which a superstructure of knowledge may be built. I believe that the most illiterate men may be rapidly taught practical mechanics if we take the right way to teach them approach the subject from their point of view rather than compel them to approach it from ours. In a book which is to be used as a general class book by boys and men it is impossible to assume that the PREFACE. IX reader has an extensive previous practical acquaintance with natural phenomena ; but it will be seen that some such past experience is assumed, much more than is usually ascribed to the ordinary student of mechanics. Moreover, he is credited with the possession of common sense, and with the feeling that all human knowledge, instinctive and rational, is the result of experience. There is much in this book which may seem new to the reader, but inasmuch as I have been, and still am, a student, and as no man can go through life without gathering to himself and regarding as his own many notions of other men, it is probable that there is nothing here, either in the matter or method, which is wholly my own. My partnership in scientific work with Professor Ayrton during the last seven years would in itself preclude any thought of such ownership. For the treatment of some parts of the subject I know that I am indebted to my recollection of the lectures of Professor James Thomson, delivered when I was one of his students fourteen years ago. How much is owing to Sir William Thomson, to Thomson and Tait, and to Professor Ball it is quite impossible to say. For the careful correction of proofs I have to thank my assistant, Mr. William Robinson, M.E., who has taken as much pains to eliminate numerical errors, puzzling sentences, and crudities of language as if the book had been his own. I would recommend the student to omit from a first reading the part of the text contained in small type. Jc PREFACE. This comprises a more detailed and generally more difficult treatment of the matters referred to in the other parts of the text, and it will therefore be best understood by one who has already grasped the general principles of the book. JOHN PERRY. 71, Quetn Street, London, E.G. CONTENTS. CHAPTER I. INTRODUCTOR Y. PACK I. What I expect the Reader to Know 2. Equilibrium of Forces The Triangle of Forces and the Polygon of Forces 3. Our one Theory insufficient Friction, a Passive Force 1 CHAPTER IL FHICTION IN MACHINES. 4. Law of Work Velocity Ratio 5. Effect of Friction 6. Use o* Squared Paper 7. Law connecting Two Variable Things found by means of Squared Paper 8. Law of Friction 9. Force of Friction 10. Loss of Energy due to Friction 11. Friction at, Bearings of Shafts 12. Friction in Parallel Motion 13. Friction in Quick- Moving Shafts 14. Msehanical Advantage 15. Rate of doing Work I'd. Economical Efficiency Law for a Machine .... 5 CHAPTER HI. MACHINES SPECIAL CASES. 17. Blocks and Tackle Law of Work 18. Inclined Plane 19. The Screw 20. Differential Pulley-block 21. Wheel and Axle 22. Equi- librium in One Position 23. Body turning about an Axis Law of Moments 24. The Lever Weighted Safety-valve Weighbridge 25. Hydraulic Press 17 CHAPTER rV. MACHINERY IN OENEDAIi. 26. Mechanism 27. Velocity Ratio Toothed Gearing 28. Shapes of Teeth Worm and Worm-wheel 29. Skeleton Drawings Crank and Connecting Rod Pure Harmonic Motion Eccentric 30. How a Shaft transmits Power Couplings 31. Belts Friction between Cord and Pulley 32. Transmission and Absorption Dynamometers 26 CHAPTER V. FLY-WHEELS. 33. Kinetic Energy Potential Energy 34. Energy Indestructible- Simple Pendulum Law for Kinetic Energy stored up iu a Moving licdy 35. Test of the LnwAtt wood's Machine 36. Energy iu a Xll CONTENTS. PiOB Rotating Body 37. Energy in a Wheel when Rotating, experiment- ally Found 38. Friction never is Negligible The M of a Wheel 39. Counting the Number of Revolutions 10. Method to be Em- ployed 41. The M of a Wheel Ite Value in different Cases 42. Formula for Energy stored np in a Rotating Body 43. Examples of Kni'r.'y stored up in Wheels 44. Steadiness of Machines Examples of Similar Wheels 45. Total Kinetic Energy stored up in any Machine 46. Facts useful to Know 38 CHAPTER VL xxrnrsiOH AHD COMPRESSION. 47. How a Pull is Exerted 48. Strain Extension of a Wire 49. Stress Extension of a Tie-rod 50. Shortening of a Strut Struts and Ties 51. A Short Strut Young's Modulus of Elasticity 52. Per- manent Set Limits of Elasticity 53. Nature of Strain 54. Illus- tration of the Nature of Strain 55. Modulus of Elasticity of Bulk 56. Lateral Contraction 57. Tensile and Compressive Strength Examples 58. Strength of Pipes and Boilers 59. Tendency of a Boiler to Burst Laterally and Longitudinally Stress on a Spherical Boiler 51 CHAPTER VII. PECULIAR BEHAVIOUR OF MATERIALS. 60. Elastic Strength affected by State of Strain Killing of Wire An- nealing 6L Fatigue of Materials 62. Effect of Load Suddenly pplied 63. Strain Energy 64. Effect of Tensile and Compressive Stresses often Repeated 65. Tempering of Steel Strengthening of Metal Wires and Cast Iron 66. Curious Properties of Materials which Workmen know of ......... 62 CHAPTER VIII. MATERIALS. 67 Importance of the Study of Scientific Principles 68. Stones Struc- tural Characters of Rocks Preservation of Stone An Artificial Stone 69. Bricks 70. Lime Cement Mortar and Concrete 71. Pressure of Earth against a Wall 72. Pressure of Wat*r Total Energy of Water Flow of Water in Pipes and Pumps Frictional Loss 73. Discharge from Orifices and Pipes 74. Timber Struc- ture of Timber Firwoods Larch Cedar Oak Teak Mahogany Ash Elm Beech Time for Felling Timber Effect of Seasoning Preservation of Timber 75. Gloss 76. Cast Iron 77. Patterns and Moulding 78. The Coolii g of Castings 79. Wrought Iron 80. Steel -81. Copper 82. Alloys of Copper Brass Muntz Metal- Bronze and Gun-MetalPhosphor Bronze 70 CHAPTER EC. SHEAR AND TWIST. 83. Shear Stress and Shear Strain 84 Deflection due to Shearing 85 Breaking Shear Stress and Working Shear Stress 86. Punching and Shearing 87. Pure Shear Strain 88. Nature of Shear Strain CONTENTS. XU1 PAGE 89. Investigation of the Relation between Shear Stress and Shear Strain Modulus of Rigidity 90. General Results- 91. Twisting Angle of Twist Twisting Moment Rule used by Engineers 92. Investigation of the Twist in a Round Shaft 93. Strength of Shafts 94. Effects of a Twisting Couple in Shafts other than Circular 95. Greatest Distortion at that part of the Surface nearest the Axis- Re-entrant Edge or Angle, a. Weak Point Effects of Twisting on Different Sections 96. Elastic Strength varies with the State of Strain Two Limits of Elasticity for Loads Twisting in Opposite Directions 97. Stiffness necessary as well as Strength . . .86 CHAPTER X. BENDING. 98. St ite of Strain in a Loaded Beam 99. Distribution of Strain 100. Bending Moment and Shearing Force 101. Case of Pure Bending 102. Neutral Axis passes through the Centre of Gravity 103. Mo- ment of Inertia and Strength Modulus 101. Stress at any part of Section Best Section for a particular Material 105. Radius of Curvature of Bent Beam Investigation and Example 103. Elastic Curve and How to Draw it 107. Hydrostatic Arch 108. Alteration of Cross Section of Elastic Strip due to Bending 108 a. Relation between Bending and Twisting 101 CHAPTER 'XI. BEAMS. A 109. Distribution of the Loads on a Beam 110. Methods of Supporting Beams 111. Supporting Forces at Ends of Beam 112. Diagrams of Bending Moment 113. Shearing Force in Beams and Girders 114. Flanges made to Resist Bending Moment 115. Beam of Uniform Strength 116. Uniform Rectangular Beam Loaded in Various Ways Rule for Breaking Load 117. Factors of Safety 118. Strength of Beams 119. Curvature of a Loaded Beam 120. Deflection of a Beam Formulae and Examples 121. Experiments on Deflection of Beams 122. Rule and Example 123. Stiffness of Beams 124. How Strength and Stiffness vary with the Dimensions 125. Results of Experiments with Testing Machine 112 CHAPTER XII. BENDING AND CRUSHING. 126. Stress over a Section 127. Resultant Compressive and Tensile Stresses 128. Struts and Pillars 129. Strength of Struts 130. The Teeth of Wheels 131. Strength of FJat Plates 132. Similar Struc- tures Similarly Loaded 128 CHAPTER XIII. GRAPHICAL STATICS, 133, 134. Methods of Calculation 135. Forces acting at a Point The Force Polygon 136. The Link Polygon 137. Interpretation of Force and Link Polygon 138. Propositions to be Proved by Actual XIV CONTEXTS. MM Drawing 138. Forces acting on a Ladder 140. Graphical Deter- mination of the Centre of Gravity and Moment of Inertia 141. Formula for C< ntre of Gravity and Moment of luertia of any Area 1-.2. Poinsot's Theorem regarding the Mouieut of Inertia . . 135 CHAPTER XIV. EXAMPLES IN GRAPHICAL STATICS. 143. Diagrams of Bending Moment 144. Shape of -a Loaded Beam 145. Hinged Structures 145 a. Straight-line Figures 146. Recipro- cal Figures 147. Calculation of the Stresses in any Piece of a Hinged Structure 148. Method of Determining the Stresses in a Roof-principal or other Structure 149. Roofs Practical Example of a Roof Examples of Stress Diagrams 150. Stiff Joints in a Structure 151. Stresses at any Section of a Loaded Structure cal- culated by Method of Moments 144 CHAPTER XV. SUSPENSION BRIDGES, ARCHES, AND BUTTRESSES. 152. Loaded Links 153, 154. Loaded Chain Curve in which it Hangs and the Pull in Any Part 155. Arched Rib 156. Distribution of the Load on an Arch Link Polygon must lie Within Mi;r. Investigation of Stiffness of Spring 168. Energy stored up in a Spiral Spring 169. Readings of Formulas 170. Investigation of Elastic Strength 171. Spiral Spring under the Action of a Weight lu'7 CHAPTER XVII. PERIODIC MOTION. 172, Periodic Motion and Periodic Time 173. Pure Harmonic Motion 174. Aver.ige Velocity Represented 175. Acceleration at any Place Rule for Finding the Periodic Time 176. Example Hi-uvy Bail carried by Spiral Spring 177. The Simple Pendulum Time oi' an Oscillation 178. Example Strip of Steel 179. Liquid iu Beut Glass Tube 180. Continual Conversion of Energy Office of Main- springEscapements 179 CHAPTER XVIH. OTHKB EXAMPLES OF PERIODIC MOTION. 181. Combination of Pure Harmonic Motions Thomson's Tide Pre- dicter 182. Blackburn's Pendulum 183. Representation of Motiou on Paper Experimental Illustrations 184. Periodic RotutiuniU CONTENTS. XV FAQS Motion Balance of Wutch 186. Compound Pendulum Equiva- lent Simple Pendulum Kater's Pendulum Determination of Axes of Oscillation and Suspension are Interchangeable 186. Ex- amples Experiments on the 'twisting Moments of Wires and Plat Spiral Springs Determination of Viscosity Friction in Fluids Bifllar Suspension Time of Vibration of a Magnet 187. Stillii.g of Vibrations Representation of Damped Vibrations Relative Viscosities of Fluids 188 CHAPTER XIX. THE EFFECT OP A BLOW. 183. Average Force of an Impact 189. Example Pile Driver 190. Total Momentum unaltered by Impact 191. Examples Recoil of Cannon The Steamship Watcrwitch Principle of Barker's Mill- work, and Hero's Steam-Engine 192. Mean Pressure during Im- pact 193. Communication of Momentum through a Liquid 194. Impact of Free Bodies Storage of Energy during- Impact 195. Communication of Strain Energy dependent on Shape of Body 196. Example Candle Fired through Board 197. Earthquake Ef- fects 198. Examples Mentioned 199. Quasi-Rigidity produced by Rapid Motion 209. Motion produced by a Blow Centre of Percus- sion 201. Ballistic Pendulum 206 CHAPTER XX. THE BALANCING OF MACHINES. 202. Effects of Centrifugal Force on Bearings 203. Permanent Axis- All Axes of Rotation in Machines ought to be Permanent Axes 204. The Balancing of a Machine 205. Example Locomotive Eugine Considerations in Designing 206, 207. Balance- Weights on Wheels of Locomotive 208. Rules for Locomotives ..... 218 CHAPTER XXI. GLOSSARY. 209. Introductory 210. Vertical Line 211. Level Surface 212. Curva- ture 213. Mass 214. Velocity 215. Acceleration 216. Momentum 217. Impulse or Blow 218. Resultant and Equilibrant 219. Equi- librium 220. Moment of a Force Law of Moments 221. Example of Moments 222. Torque 223. Lever 224. Couple 225. Work 226. Example of Work 227. Honse-Power Indicator Diagram Exercise 228, 229. Energy Kinetic Energy and Potential Energy Examples 230. Angle 231. Angular Velocity 232. Angular Ac- celeration 233. Comparison of Linear Motion and Angular Motion 234. Centrifugal Force 235. Centrifugal Force Apparatus 236. Conical Pendulum 237. Friction Coefficient of Friction 238. Kinetic Friction Apparatus 239. Friction and Abrasion 240. Fric- tion is often very Useful 241. Fluid Friction 242, 243, 244. Appa- ratus for Measuring Viscosity of Liquids Experiments 245. Com- parison of the Laws of Fluid and Solid Friction 223 APPENDIX Kules in Mensuration 252 257 LIST OF TABLES. MM I. MOMENTS OF INEHTIA AND THE M OF ROTATING BODIES 47 II. MODULUS OF ELASTICITY OF BULK, K . . . 57 III. MELTING POINTS, SPECIFIC GRAVITY, STIIKNUTH, ETC. OF MATERIALS 68-9 IV. DIAGRAMS OF BENDING MOMENT, WITH STRENGTH AND STIFFNESS OF A UNIFORM BEAM, WHEN Sri-- PORTED OR FIXED AND LOADED IN VARIOUS WAYS 116-17 V. FACTORS OF SAFETY FOR DIFFERENT MATERIALS AND LOADING 121 VI. STRENGTH AND STIFFNESS OF RECTANGULAR BEAMS SUPPORTED AT THE ENDS AND LOADED IN THE MIDDLE 121 VII. BREAKING STRESS OF IRON AND TIMIIER STRUTS . 131 VIII. VALUES OF THE CONSTANT B FOR DIFFERENT LENGTHS OF STRUT AND DIFFERENT MATERIALS . 132 IX. VALUES OF THE CONSTANT n FOR STRUTS OF DIF- FERENT SECTIONS 132 X. NORMAL PRESSURE OF WIND ON ROOFS . . l.Vj PEACTIOAL MECHANICS. CHAPTER L INTRODUCTORY. 1. What I expect the reader to know already. In tliis book I mean to consider the principles of mechanics as they are applied in many trades. It is important that I should state in the beginning what is the amount of knowledge which I expect my reader to be possessed of beforehand. In the first place, he must know the meaning of decimals in arithmetic. It is a very strange thing that the meaning of a decimal, which might easily be taught to children before they begin addition, is usually regarded as a part of arithmetic which ought to come after vulgar fractions. Secondly, my reader must own a box of drawing instruments ; he must be able to set off at once any angle when it is stated in degrees ; he must be able to draw a triangle to any scale when one side and two angles, or two sides and one angle, or when three sides are given to him : in fact, he must know as much of the use of drawing instruments and of the use of a scale as a good teacher can give him in one lesson. Thirdly, he must be aware of the fact that a letter of the alphabet or any other symbol may be used to represent a physical magnitude. Probably there is nothing more annoying to the person who attempts to give lessons in applied science than the fact that few workmen know the meaning of the simple symbols + x -=- V. Even B 2 PRACTICAL MECHANICS. [Chap I when they are aware of the meaning of these symbols they will often get frightened at a mathematical ex- pression, although, if they spent one hour in learning the meanings of such expressions they would never feel afraid of them again. A mathematical expression is simply a very concise way of writing a rule. There are many books of rules, such as Molesworth's, and any workman who knows decimals in arithmetic might be able to use every rule in Molesworth with an hour's study ; but he never attempts to learn the key to these secrets, and when he goes to a teacher, it is usually for a long dry course in algebra which he does not really need. Please remember that very few men who use a book of logarithms know how a logarithm is calculated ; and just as a man may use a watch or a slide rule, or any other calculating machine, who does not know how to make one, so you may be able to calculate from a mathematical formula, although you do not know how it has been arrived at. I expect that the reason why men have so little practical knowledge of such matters, lies in the fact that their teachers know only one way of studying the way in which they themselves have been taught the way of the Universities, which has unfortunately become crystallised in England, and is now in use in Science Classes. It is forgotten that practical men have an experience and an amount of common sense which children have not, and there is a way of giving new ideas to men which we cannot employ with children. This brings me to my fourth requirement, namely, that my reader must know the elementary principles of mechanics. If his acquaintance with mechanics is merely derived from books or lectures, he has not the knowledge of which I speak. He cannot know the parallelogram of forces till he has proved the truth of the law half a dozen times experimentally with his own hands. I have met with men who, when given two sides of a triangle in inches, and the angle between them in degrees, could calculate readily the length of the third side and the Chap. I.] EQUILIBRIUM OF FORCES. sizes of the other angles of the triangle, and yet who had never tried with their instruments and drawing paper whether their calculations were correct. Such is not the sort of knowledge which I want my readers to have. I want them to think of things as being measurable, of laws as statements which ought to be submitted to the test of their own experiments, however rough, before they are accepted as true. 2. Equilibrium of Forces. Take as an example the law called "the triangle of forces" if three forces act on a small body, and just keep it at rest ; then if we draw on a sheet of paper three straight lines paral- lel to the directions of the three forces, and let them form a triangle, in such a way that arrow- heads representing the directions of the forces go round the triangle in Fig. 1. the same way, it will be found that the lengths of the sides of the triangle are proportional to the amounts of the forces. Now the statement of this law has very little meaning to the student until he gets three strings and three weights, as in Fig. 1, and by means of pul- leys allows the strings to pull at a small body, P, all at the same time. But when, after all sorts of trials of different weights and different positions of the small body,_ he finds that even his rough tests of the law prove to be satisfactory every time, he has obtained an exact notion of what the law means, which he can never lose again. If, instead of three forces, he lets four or more forces act upon the small body, and, when they balance one another, he draws straight PRACTICAL MECHANICS. [Chap. I. lines parallel to the directions of the forces and represent- ing their amounts, according to any scale he pleases, taking care that, in whatever order he draws them, the arrow- heads run all in the same sense, he will draw a polygon ; he will find the polygon to be closed and complete when he finishes, and he will prove roughly the law of " the polygon of forces." He will at once see that the triangle of forces is merely a simple case of the more general polygon of forces. 3. Our one Theory insufficient. Now, no reasoning man can make these trials without finding that there is a great deal to be observed beyond what his teacher or his book has taught him. A force has been represented by the pull in a string passing over a little pulley with a weight at its end. He finds that as his pulley works more easily, and as its pivots are better oiled, his proof of the law is better and better : in fact, he finds that the pull in a string is not exactly the same on the two sides of a pulley. If he takes one pulley and one string, and two weights, called A and B, Fig. 2, at its ends, he will find that there is equi- librium even when the two weights are not exactly equal. If A is slightly greater than B, and he increases the weight of A till it is just able to over- come B, then the difference between the weights represents what may be cnlli-d the friction of the pulley. If now he increases the weights which he uses, he will find that the friction is proportionately increased, and he will get to understand that this is a general law in machinery : " friction is proportional to load." Again, he sees that this friction, which is a resistance experienced in the rubbing together of any two surfaces, is a force which always opposes motion, always acts against the stronger Fig. 2. Chap. II.] FRICTION IN MACHINES. influence. Suppose, for example, that he found that a weight of 5'1 ounces was just able to overcome a weight of 5 ounces ; he will find that a weight of about 4 '9 ounces will just be overcome by a weight of 5 ounces, and that there is equilibrium with 5 ounces and any weight varying from 5 ! to 4 - 9. Friction is then a pas- sive force, which always helps the weaker to produce a balance. CHAPTER II. FRICTION IN MACHINES. 4. Law of Work. Take any machine, from a simple pulley to the most complicated mechanism. Let a weight, A, hung from a cord round a grooved pulley or axle in one part of the mechanism, balance another weight, B, hung from a cord round another axle or pulley somewhere else. In Fig. 3 we have imagined that the mechanism is enclosed in a box, and only the two axles in question make their appearance. Now move the me- chanism so that A falls and B rises, and observe their motions. Sup- pose that when A falls 1 foot B rises 20 feet, then if there were no friction in the machine a weight at A is exactly balanced by one- twentieth of this weight at B. This is the law which you will find proved in books on mechanics. The reason why it is true is this. The work or mechanical energy given out by a body in falling is measured by the weight of the body multiplied into the distance through Fig. 3. PRACTICAL MECHANICS. [Chap. II. which it falls. It is in this way that we get the energy derivable from the fall of a certain quantity of water down a waterfall, and it is in this way that we find out whether a certain waterfall gives out enough power to drive a mill (see Chapter VII.). Similarly the energy given to a body when we raise it, is measured by the weight of the body multiplied by the vertical height through which it is raised. Now every experiment we can make shows that energy is indestructible, and con- sequently, if I give energy to a machine, and find that none remains in it, it must all have been given out by the machine. Therefore the energy given out by A in falling slowly must be equal to the energy received by B in rising, and as A falls 1 foot when B rises 20 feet, the weight of A must be twenty times the weight of B. If, then, there were no friction in the machine, and if a weight of 20 Ibs. were hung at A and a weight of 1 Ib. at B, we should find that if we start A downwards or upwards there will be a steady motion produced. Any excess at A will cause it to overcome B, the weights moving more and more quickly as the motion continues. Now, in our machine, Fig. 3, we can always find by trial what is the velocity ratio; that is, the speed of B as compared with the speed of A. This cannot alter. But when we try to balance a weight at B by a weight at A, we find that the above relation is quite untrue. Hang a weight of 1 Ib. at B, hang a weight of 20 Ibs. at A, there is certainly a balance, but when we have some- what less or more than 20 Ibs. at A there still is balance. The reason for this is, that there is friction in the mechanism, and this friction always tends to resist motion, always acts against the stronger influence. 5. Effect of Friction. We shall now proceed to find out in what way friction modifies the law given in the books which I have just spoken about. You must make actual experiments with some machine, if you are to get any good from your reading. Hang on a weight, B, and find the weight, A, which will just cause a slow, steady Chap. II.] USE OP SQUARED PAPER. motion. Do this when a number of different weights are placed at B. Now, suppose you have measured the velocity ratio, that is, suppose you find that B rises four times more rapidly than A falls. Then, according to the books, there would be an exact balance if A were four times the weight of B. On actual trial, however, I find in a special case the following table of values : A overcomes B when A is 23-4 ounces and B is 5 ounces 44-7 10 65-4 86-8 107-5 128-8 149-6 171-0 15 20 25 30 35 40 But if there had been no friction in the first experi- ment, A would have been 20 ounces instead of 23-4, hence the friction is represented by this 3-4 ounces. For every experiment let this be done, subtract four times B from A and call this difference the friction. ISTow how shall we compare this friction with the corresponding load 1 6. The use of Squared Paper. And here we come to a matter of the greatest importance to the practical man, which the old-fashioned books on mechanics and old- fashioned teachers of Science Classes seem not to know anything about. How do we practically compare two things whose values depend on one another ? How do we find out the law of their dependence 1 It is a strange fact that there should be a class in the community who have a little difficulty in manipulating decimals in arith- metic, but it is almost a stranger evidence of neglected education that so many people should be ignorant of the great uses to which a sheet of squared paper may be put. A sheet of squared paper can be bought very cheaply. It has a great number of horizontal lines at equal distances apart, and these are crossed by a great number of vertical lines of the same kind, so that the sheet is PRACTICAL MECHANICS. [Chap. It covered with little squares. This sheet will enable me first of all to correct for errors of observation in the above series of experiments ; and, secondly, to discover the law which I am in search of. A miniature drawing is shown Chap. It] USE OF SQUARED PAPER. 9 in Fig. 4, many lines being left out because of the difficulties of wood-cutting. At the bottom left hand corner I place the figure 0, and I write 10, 20, &c., to indicate the number of squares along the line, A. In- stead of 10, 20, &c., I might write 1, 2, &c., or 100, 200, &c., according to the scale I am going to use. Indeed, on account of the friction being so much less than the weight A with which it is to be compared, I number the squares along the vertical line F by 1, 2, &c. instead of 10, 20, &c. We can employ any scale we please in repre- senting either of the things to be compared, and it is usual to multiply all the numbers of one kind by some number, so as to represent all our experiments on one sheet of paper, and on as much of this sheet as possible. Having sub- tracted four times B from A, I find the following numbers: A. Friction. 23-4 . . . .3-4 44-7 . . . .4-7 65-4 . . . .5-4 86-8 . . 6-8 A. Friction. 107-5 . . . .7-5 128-8 . . . .8-8 149-6 . . . .9-6 171-0 .... 11-0 I now find on my sheet of paper the point p, which is 23'4 horizontally and 3'4 vertically, and mark it with a cross in pencil. Q is 4 4 '7 horizontally and 4 '7 vertically, and so for the others. The last point, w, is 171 horizontally and ll'O vertically. We guess at the decimal part of a small square. The point p represents my first experiment, and every other point represents one experiment. Now we are certain that if there is any simple law connecting load and friction, the points P, Q, to w, lie in a simple curve or in a straight line. You see that, in this case, no simple curve will suit the points ; they would evidently lie in a straight line, only that we made some errors of observation. You must now find what straight line lies most evenly among all the points ; this you can do by means of a ruler or a fine stretched string, and the line M K seems to me to answer best. It tells me, for instance, that when A is 44 -7 the friction is really 4-5, instead of 4 '7. Take any point in the line, its 10 PRACTICAL MECHANICS. [Chap. tt. vertical measurement gives me the true friction corres- ponding to a load represented by its horizontal measure- ment. Thus, for instance, you see that friction 5*0 corresponds to load 55. This is a simple way of correcting errors made in experiments, but you cannot hope to understand much about it till you actually make experiments and use the squared paper. You will find the matter all very simple when you try for yourself ; my description of it is as com- plicated as if I were teaching you to walk. 7. If, at any time, you make a number of measure- ments of two variable things which have some relation to one another, plot them on a sheet of squared paper, and correct by using a flexible strip of wood or a ruler, to draw an easy curve or a straight line so that it passes nearly through all the points. If the line is straight, the law connecting the two things will prove to be a very simple one. In the present case it means that any increase in the load is accompanied by a proportionate increase in the amount of friction. Thus, when the load is 0, the friction is 2-3 ; when the load is 100, the friction is 7 '2. That is, when the load increases by 100 the friction increases by 4'9, so that the increased friction is always the fraction, '049, of the increased load. In fact, it is evident that we can calculate the friction at any time from the rule Friction = 2 -3 + -049 A. That is, multiply the load A in ounces by -049, and add 2-3, the answer is the friction. 8. Law of Friction. Our result is that the total fric- tion is equal to the friction, 2*3, of the machine unloaded, together with a constant fraction, -049, of the load. Now when a similar series of experiments is tried on any- machine, be it a watch or clock, or be it a great steam- engine, we always find a similar simple law. If you clean all the bearings or pivots, or if you use a different kind of lubricator, you will get other values for Chap. II.] FRICTION. 11 the two numbers in the above rule, but the law will remain of the same simple kind. 9. Force of Friction. We have in all this used the term " friction," or the term " effect of friction," to mean the difference between the weight which would balance another through the mechanism if there were no resist- ance to the rubbing of surfaces, and the weight which will just overcome the other when there is such resist- ance to rubbing of surfaces. At any rubbing surface there is a force of friction which is proportional to the pressure between the surfaces. At the rubbing surface itself we can speak of this force of friction, but when we speak generally of the whole friction of a machine, we are speaking, not of the force of friction at any one surface, for this is different probably from the force of friction at any other rubbing surface, but we speak of an effect which is produced, somehow, by the friction everywhere in the machine, wherever there is a pivot, and wherever the tooth of one wheel rubs on another. 10. Loss of Energy due to Friction. Thus, in the simple case with which we began (Art. 3), the difference of pull in a cord on the two sides of a pulley was what we called the friction of the arrangement, whereas, really, the friction takes place at every point of the pivot where rubbing occurs, and the force at one point may not be the same as the force at another point. Again, any force acting in the cord has a greater lever- age about the axis than any of the forces of friction has. The real connection between the two things is, then, this : what we have generally called " the effect of friction," or " the friction of the arrangement," multiplied by the velocity of the cord on which it is measured, is equal to the sum of all such products as the friction at any point in a rubbing surface multiplied by the velocity of rubbing. In fact, if the weight A in falling cause the weight B to rise, the work done by A is greater than the work done on B by an amount which is called the 12 PRACTICAL MECHANICS. [Chap. n. work lost in friction, and this is the work done against the forces of friction at all the rubbing surfaces. If we know the force of friction at any place, in pounds, and the distance, in feet, through which this force is overcome, that is, the distance through which nibbing has occurred, the product of force by distance measures the work or energy spent in overcoming friction, in what is called foot-pounds. This energy is all wasted, or, rather, it is all changed into heat and does not come out of the machine as mechanical work, the shape in which it was when we put it into the machine. And inasmuch as no machine can be constructed which will move without friction, we never get out of a machine as much mechanical work as we put into it. 11. Friction at Bearings of Shafts. At almost every rubbing surface which you can consider, the force of friction is different at every point of the surface, and it is generally acting in different directions at different points. Consider, for example, a horizontal shaft and its bearing (Fig. 5). The force of friction at c, per ty square inch of area of rubbing sur- face, is probably not the same as at A. A very little difference in the size of the shaft or its bearing will cause a considerable difference in the pressure per square inch at c or at A. Now the force of friction at c, multiplied by the velocity of rubbing, gives the work or energy lost per second in friction at c ; and this, added to the energy lost at every other place where rubbing occurs, gives the total loss of energy per second at all the points. It is not, then, a simple matter to investigate the force of friction at every point of such a bearing ; and the rigidity of the metal, and a number of other important matters, must be taken into account in investigating the force of friction every- where when the shaft is transmitting different amounts of power. As we have already seen, however, experiment Chap. II.] FRICTIOV AT BEARINGS. 13 shows that the energy lost in friction for a certain amount of motion increases proportionately with the energy actually transmitted by the shaft. Keeping in mind, then, the general law " the force of friction is proportional to load" it is easy to see how to reduce the frictional loss in any machine. For instance, when a wheel is transmitting power, the load on the rubbing surfaces of its bearings or pivots depends on the power transmitted. Now, the actual force of friction at the rubbing surface is about the same, whatever be the size of the bearing ; but the distance through whicli nibbing occurs when the wheel makes one revolution is less as we have a less diameter of bearing ; in fact, the force of friction, multiplied by the circumference of a cylindric bearing, is the energy in foot-pounds lost in one revolu- tion. Our rule is, then, to make this diameter as small as possible, consistently with sufficient strength. The wheel of a carriage is made large, and the axle, where rubbing occurs, is made as small as possible, because in this way the cai'riage moves over a great distance for a small amount of rubbing. There is another reason, however, for the use of large wheels in carriages on common roads namely, their being better able to get over obstacles, such as stones. In some machines, where it is important that there should be very little friction at the bearings of axles, the axles are made to lie at each end, in the angle formed by two wheels with plain rims. The main axle rolls on these wheels, and it is only at the axles of the wheels that there is rubbing. This rubbing is a very slow motion, and as the force of friction is but little increased in consequence of the weights of the friction wheels, the energy lost in friction may be made very small in this way. 12. If you compare the parallel motion which is still used in some large steam-engines to cause the piston-rod to move in a straight line, with the slide which is now so common, you will see that there is very much less loss of energy by friction when the parallel motion is em- 14 PRACTICAL MECHANICS. [Chap. II. ployed because, whereas in the slide the rubbing motion is as much as the motion of the piston, in the parallel motion rubbing only occurs at the pivots of the arrange- ment. Unfortunately, this arrangement does not allow the piston-rod end to move exactly in a straight line, and produces some friction between the piston and its cylinder, and between the piston-rod and stuffing-box ; and it is also much more costly and less compact than slides. Hence slides are coming into general use. 13. In quick-moving shafts, it is usual to make the journals or bearings longer in proportion to their diameter than in slow-moving shafts. This is rendered necessary by the fact that, as all the energy wasted through friction is converted into heat, when there is more power wasted in friction we ought to let the heat get away more rapidly. Giving greater rubbing surface has this effect. Also, as the materials are more abraded when the velocity is too great, lengthening the journal diminishes the pres- sure at every place, tending thus to counteract the effect of increased velocity. 14. Mechanical Advantage. In books on mechanics you will usually find that, when machines are described, they are only considered in relation to their MecJianical Advantage. That is, suppose a small weight, p, usually called the power, is able by means of the mechanism to cause a larger weight, w, to rise, the ratio of w to P is called the mechanical advantage. Now, in nearly all cases you will find that, when there is a mathematical in- vestigation of a machine, the assumption is made that there is no friction. I have already shown you that the problem of taking friction into account is a very difficult one. But, as we have seen, a practical mau can experi- ment on the effect of friction, and obtain results which the mathematician never attempts to deduce ; and, happily for us, these results are generally very simple. Let the reader make a few experiments himself, or let him by means of squared paper find the relation between P and Q from the following results, taken from a crane Chap. II.] MECHANICAL ADVANTAGE. 15 whose gearing was well oiled, and whose handle was re- placed by a grooved wheel, round which was a cord sup- porting P: w. Weight just Overcome. 100 Ibs. 200 300 400 500 600 700 800 P. Power just able to Overcome Weight, 8-5 Ibs. 12-8 17-0 21-4 25-6 29-9 34-2 38-5 The weight capable of being lifted slowly by the crane we call w. We found that P fell forty times as rapidly as w rose, and you may have imagined that the me- chanical advantage was forty, or that a weight, P, could lift a weight, w, forty times as great as itself. This would be true if there were no friction ; but we see that in practice it is not the case. Plot the above values of p and w carefully on squared paper, and you will find that, if the weight w is increased 1 lb., p must be increased -0429 Ib. ; and also that when w is 0, a power, p, of 4-21 Ibs. is needed to cause a slow motion of the crane ; so that the law is p = 4-21 + -0429 w. Namely, multiply the weight w in pounds by the fraction 0429, and add 4 '21 : the answer is the power required to lift w. When you have worked out this rule, employ it in finding how much power, p, is required to lift a ton with such a crane. Answer, 100 '3 Ibs. 15. Rate of doing Work. I have been using the word power in a very wrong sense in Art. 14, because you will find it used in this way in books on mechanics. I have used it as a mere name for the weight p, which causes the weight w to rise. But the word power will be used by us in future in a very different sense. If a weight of 1,000 Ibs. fall 100 feet in two minutes, it gives out 1,000 x 100, or 100,000 foot-pounds of 16 PRACTICAL MECHANICS. [Chap. IT. work in two minutes, or 50,000 foot-pounds of work in one minute. Now, 33,000 foot-pounds of work done in one minute is called a liorse-power, and hence our falling weight gives out 50,000 -=- 33,000, or 1-5 horse- power. Ten horse-power means ten times 33,000 foot- pounds of work done in one minute. The idea, then, of power is an idea of work done in a certain time. 16. Economical Efficiency. Take any pair of numbers from the above table, say p = 8 -bibs., when w = lOOlbs. Let us suppose that p is moving at the rate of forty feet per second, then we know that w is rising at the rate of one foot per second, p is giving out the power 8-5 x 40, or 340 foot-pounds per second ; w is receiving 100 foot-pounds per second. The ratio of the power usefully employed to the power given to the machine is called the efficiency of the machine, so that our crane has an efficiency 100-f 340, or '294. Sometimes the efficiency is put in the form of a fraction ; sometimes we say that it is 29 - 4 per cent., meaning that it employs usefully 29-4 per cent, of the work given to it. Now take another pair of numbers, say p = 38*5, w = 800, and let p fall forty feet in one second, as before. We now get as our answer '519 that is, more than half, or 51 '9 per cent., of the power given to the crane is usefully employed. We see, then, that as the power given to the crane is greater for a given speed, the efficiency is also greater. This arises from the fact that the friction of the unloaded crane is always entering into the calculation ; and if we take the case where no weight, w, is being lifted, and p must be 4 '21 Ibs., we shall find an efficiency, 0, because work is being given to the crane, and none is coming out usefully. You will always find that the power usefully given out is a certain fixed fraction of the total power given to the machine, minus the power required to drive the crane at the given speed when it is unloaded. Choose some speed, say that p falls forty feet per second ; find the total power or 40 p ; find Chap. IIL] INCLINED PLANE. 17 the usefully employed power 1 x w for eve*y case of the above table. Plot your answers on squared paper, and yOU will find this law : ( Power required to drive crane at same speed when unloaded. you will find this law : c ; Total power = useful power x 1'716 + \ CHAPTER III. MACHINES SPECIAL CASES. 17. Blocks and Tackle. It is very good to have a general law telling us about machines in which there is no friction. That law you now know. The mechanical work given to a machine is equal to the work given out by it, unless it is stored up in the machine itself by the coiling of a spring or in some other way. But, besides knowing the law itself, it is well to know what it leads to in certain special cases. Take, for in- stance, a pulley-block, Fig. 6. It is evident here that if we have three pulleys in the block B, if the cord p is pulled six inches, w will only rise one inch, and therefore P will balance six times its weight at w if there is no friction. The mechanical advantage is therefore six. 18. Inclined Plane. Again, take the inclined plane, Fig. 7; w is a weight which will roll down the plane with- out friction, let us suppose ; P is the pull in a cord which just prevents w from falling. The cord is parallel to the plane. Evidently when w rises from level A to level B the cord is pulled the distance A B ; that is, w multiplied by the height of the plane is equal to P multiplied by c Fig. 6. 18 PRACTICAL MECHANICS. [Chap. in. the length of the plane. Thus, if w is 1,000 Ibs., and the length of the plane 10 feet for a rise of 2 feet, then ten times P is equal to 2,000, or P is 200 Ibs. 19. The Screw. Again, suppose there is no friction in the screw A B, Fig. 8, if it rise it lifts a weight say of 3,000 Ibs. Now, if the screw make one turn it rises by a distance equal to its pitch, that is, the distance between two threads. Say that the pitch is 02 foot, then when the screw makes one turn it does work on the weight 3,000 x -02, or 60 foot-pounds. But to do this, P must fall through a distance equal to the circum- ference of the pulley A, about which I suppose the / cord to be wound. Sup- pose the cir- cumference of the pulley to be 6 feet, then p multiplied by 6 must be GO, or p is 10 Ibs. The rule, then, rig. B. for i screw is this power multiplied by circumference of the pulley equals weight multiplied by pitch f MfMP. It is not usual to have a pulley and a cord working Chap, in.] DIFFERENTIAL PULLEY-BLOCK. 19 a screw ; it is more usual to have a handle, and to push or pull at right angles to the handle. Instead of the cir- cumference of the pulley, we should take, then, the circumference of the circle de- scribed by the point where the power is applied to the handle. Exercise. A steam-engine gives to a propeller shaft in one revolution 60,000 foot-pounds of work; the pitch of the screw is 12 feet. What is the resistance to the motion of the vessel 1 Answer : The resistance in pounds multiplied by 12 gives the work done in overcoming this resistance, and this work (leaving friction out of account)must be equal to60,000 foot- pounds, hence the resistance to the motion of the vessel is 5,000 Ibs. (We have here assumed that there is no slip in the screw.) 20. A differential pulley-block is shown in Fig. 9. When the chain A is pulled, it turns the two pulleys, or rather one pulley with two grooves, B and c. Now c is a little smaller than B, so that, although at D the chain is lifted, it is lowered at E. If the circumference of B is 2 feet, and that of c is 1-99 foot, then, when A is pulled 2 feet, D is lifted 2 feet, but E is lowered 1'99 foot, hence the pulley F, although it will turn a con- siderable distance, will only rise OO1 foot, carrying the weight w with it. If w is 2,000 Ibs., then 2,000 x -01, or 20, must Fi 3 . 9. be equal to p, the pull in A, multiplied by 2, hence p is 10 Ibs., or a power of 10 Ibs. is able to lift a weight of 2,000 Ibs. The general rule, then, for the differen- tial pulley-block is, poiver multiplied by circumference of larger groove B is equal to weight multiplied by difference between the circumference of the two grooves B and c. 20 PRACTICAL MECHANICS. [Chap. HI. You will find that this rule comes to the same thing power multiplied by diameter of 'R is equal to weight multiplied by the difference between the diameters of B and c. The grooves are furnished with ridges to catch the links of the chain, so that there shall be no slipping. 21. Wheel and Axle. If A and B, Fig. 10, are two pulleys or drums on the same axis and having cords round them, a small weight, p, hung from A, will balance a larger weight, w, hung from B. For, suppose that one complete turn is given to the axis, p falls a distance equal to the circumference of A, whilst w is rising a distance equal to the circumference of B. Hence p x circumference of A = w x cir- cumference of B, or, what really comes to the same thing, p x diameter of A = w x diameter of B, or P x radius of A = w x radius of B. 22. Equilibrium in one Position. In all the machines which we have hitherto considered, we could give motion without altering the balance of Fig. 10. p an d w, but there are many machines in which the mechanical advantage alters when motion is given. In such cases you will employ your general principle, but you must make your calcula- tion from a very small motion indeed. For instance, in the inclined plane, if the cord which prevents the weight from falling is not parallel to the plane say that it is like M, Fig. 11 you will find that the necessary pull depends on the angle the cord makes with the plane. Now, suppose that the cord pulls the carriage from b to c, evi- dently the angle of the cord alters. The question is, what is Chap. III.] EQUILIBRIUM IN ONE POSITION. 21 P, that it may support w in the position shown in the figure ? We know that it will be different after a little motion, but what is it now 1 Imagine such a veiy small motion from b to c to occur that the angle of the cord does Fig. 11. not alter perceptibly, and now make a magnified drawing, Fig. 12. P has not fallen as much as the distance b c, it has only fallen the distance b a (c a is perpen- , \ dicular to b a). In the meantime the weight \ \ has been lifted the distance k c. Hence, w x k c ought to be equal to P x b a. Thus, if you measure k c and b a on your magnified drawing to any scale you will find the relation between P and w. Another way of finding the same relationship is this. We know that ,< the weight of w' acting downwards, the pull in /' I the cord, and a force acting at right angles to '' j* the plane, are the three forces which keep w \ | where it is. Draw a triangle whose three sides \' are parallel to the directions of these three forces, Fig. is. Fig. 13, with arrow-heads going round in the same way ; then x and y are in the proportion of w and p. Here we have used the principle called " the triangle of forces" to find p. 22 PRACTICAL MECHANICS. [Chap III. Fig. 14. 23. Body turning about an Axis. In Fig. 14 we have a body which can move about an axis. It is acted on by a num- ber of cords exerting forces which just balance one another. Now, if you make this experi- ment you will find that you must keep your finger on the body, because it is in such a state that a very small mo- tion either way causes the forces to no longer balance. Suppose, however, you were to let the cord A be wound on a pulley whose radius is equal to the distance A o ; the cord B on a pulley whose radius is equal to B o, and so on, you would have the arrangement shown in Fig. 15, which dif- fers from Fig. 14 in that a small motion has no effect on the balance. Fig. 15. Now what is the condition of balance in this case ? Suppose one complete turn given to the axis, Chap. III.] PRINCIPLE OP THE LEVER. 23 every cord shortens or lengthens by a distance equal to the circumference of the pulley on which it is wound. Let A and B lengthen, and let c shorten, then we know that the work done by A and B must be equal to the work done against c. Hence, Pull in A x circumference of A'S pulley, together with pull in B x circumference of B'S pulley, must be equal to pull in c x circumference of c's pulley. We might, however, use the diameters or radii of the pulleys, and so we see that in Fig. 14 there is balance if Pull in A x AO together with pull in B x BO, equals pull in c x CO. The pull in A X AO is really the tendency of A to turn the body about the axis, and in books on mechanics it is called the moment of the force in A about the axis o. The law is then, if a number of forces try to turn a body and are just able to balance one another, the sum of the moments of the forces tending to turn the body against the hands of a watch must be equal to the sum of the moments of the forces tending to turn the body with the hands of a watch. 24. The Lever. Thus, for example, a lever is a body such as I have spoken about, capable of turning about an axis. You will find that our general rule of work, and this rule of moments, will give the same result. If two forces act on a lever, they will balance when their moments about the axis are equal ; that is, when p, multi- plied by the shortest distance from the fulcrum or axis to the line in which p acts, is equal to w multiplied by the distance of the fulcrum from the line in which w acts. If a, number of forces balance ichen acting on a lever. the sum of the moments tending to turn the lever against tJie hands of a watch must be equal to the sum of the mo- ments tending to turn the lever with the hands of a watch. It must be remembered that, if the body acted upon 24 PRACTICAL MECHANICS. [Chap. III. has its centre of gravity somewhere else than in its axis, then we must consider that the weight of the body is a force acting through its centre of gravity. Exercise. The safety valve, Fig. 16, must open when the pressure on the valve is just 10U Ibs. per square inch. The mean area of the valve A, on which we assume that the pressure acts, is 3 square inches; CD is 2 inches, E is 50 Ibs., the weight of the lever is 6 Ibs., and its centre of gravity is 6 inches from D where must E be o Fig. 18. placed? Here the upward force is 100 x 3, or 300 Ibs., and its moment about D is 300 x 2, or 600. The moment of the weight of the lever is 6 x 6, or 36. The moment of the weight E is 50 x the required distance from D Hence, 600 36, or 564 divided by 50, is the answer ; 11 '28 inches from D. Exercise. A weighbridge consists of three levers whose mechanical advantages help each other ; I mean, the short arm of each supports the long arm of the next. Suppose that the weights of all parts are arranged so as just to be balanced when no weight is on the bridge, and that the mechanical advantages of the three levers are 8, 10 and 12, what weight will be balanced by a power of 15 Ibs. ? Answer, 14,400 Ibs. Suppose that it is the first of these levers that is alterable (that is, the power is a sliding weight), what is its mechanical ad van- Chap. III.] HYDRAULIC PRESS. 25 tage altered to when the load is 16,000 Ibs. ? Answer, It was 8, it now becomes increased in the proportion of 16,000 to 14,400, so that it becomes 8-8889 feet. 25. Hydraulic Press. A hydraulic press is a machine which enables great weights to be lifted, or great pressures exerted, but in which, instead of levers and wheels, we use water to transmit the energy. In Fig. 17 the labourer exerts a force of, say 20 Ibs. at p. If PO is 15 times Q o, then Q and the plunger QS move at one-fifteenth of the speed of P. Now, let us suppose that the labourer has been working for a few minutes, the 26 PRACTICAL MECHANICS. [Chap. IV. water filling the whole tube and cylinder space from the ram M to s may be regarded as incompressible, and the cylinder B B is unyielding ; and if we force the plunger into this space, the ram must rise if there is no leakage. The cubic contents of the water displaced by the plunger must somehow or other be pi-ovided for, and it is the motion of the ram which provides for it. If the mm has an area of cross section of 200 square inches, and the plunger has an area of cross section of only one square inch, then the plunger must move 200 times as rapidly as the ram ; hence, the hand of the labourer must move 15 x 200, or 3,000 times as rapidly as the ram. Hence, if there were no friction, the ram would lift a weight of 20 x 3,000, or 60,000 Ibs. The mechanical advantage of the hydraulic press is then found by multiplying the area of cross section of the ram by the mechanical advantage of the lever, and dividing by the area of cross section of the plunger. CHAPTER IV. MACHINERY IN GENERAL. 26. Mechanism. When the power of a steam engine is distributed through a factory, the distribution is per- formed by means of shafts, spur and bevil wheels, belts and pulleys, and other kinds of gearing. As I am writing for men who have observed such transmission of energy, it is no part of my object to describe here what can be seen in any workshop. Perhaps no study is more useless from books alone than the study of mechanism ; whereas, it is very useful and easy if you examine the actual thing, or make a skeleton model or a skeleton draw- ing. What I shall say, then, is to help you in your obser- vation rather than to give you a knowledge of mechanism. Clinp. IV.J TOOTH GKARING. 27 27. Velocity Ratio. In any machinery the velo- city of any point may be calculated when the velocity of any other point is known. The number of revo- lutions per minute made by a shaft tells us the velocity of any point on any wheel or pulley fixed on the shaft ; the circumference of the circle described by such a point, multiplied by the number of re- volutions, is evidently the distance moved through by the point in one minute. Now, when one shaft drives another by means of spur or bevil wheels, or by two pulleys and a strap, it is evident that the number of revolutions per minute made by one of the shafts, multiplied by the number of teeth of the wheel, or by the circumference or diameter of the wheel or pulley, is equal to the number of revolutions made by the other shaft, multiplied by the number of teeth, or by circumference or diameter of the other wheel or pulley. This is evidently true, supposing that the strap does not slip on the pulley. Hence the rule to find the speed of a shaft, driven from another by means of any number of wheels or pulleys, multiply the speed of the driving shaft by the product of the diameters or numbers of teeth in all the driving wheels or pulleys, and divide by the product of the diameters or numbers of teeth in all the driven wheels or pulleys. By the diameter of a spur wheel we mean the diameter of its pitch circle. Two spur wheels enter some distance into one another, and the circle on one which touches a circle on the other, the diameters of these circles being proportional to the numbers of teeth on the wheels, is called the pitch circle. The circumference of the pitch circle, divided by the num- ber of teeth, gives the pitch of the teeth. 28. Shapes of Wheel Teeth. We know that if two spur wheels gear together, however badly their teeth are formed, so long as a tooth in one drives past the line of centres of a tooth in the other, their average speeds follow the above rule. But if we want the speed at any instant to be the same as at any other instant, it is necessary to form th'e teeth in a certain PRACTICAL MECHANICS. [Chap. IV. way. The curved sides of teeth ought to be cycloidal curves. The proof of this is not very difficult, but I shall not give it to you. It is not usual to employ these cycloidal curves, for it is found that certain arcs of circles approximate very closely to the proper curves. The method of drawing rapidly the curved tooth of a wheel you will find taught by every teacher of mechanical draw- ing, you will find described in a great number of books, and you will see it in use in the workshop.* You must remember that no study of books, and I may also say, no fitter's or turner's work that you may engage in, will make up for want of the experience which you would gain by actually drawing to scale a spur or bevil wheel, a bracket or pedestal with brasses, and a few other contrivances used in machinery. A worm and worm- wheel, that is, a screw, every revolution of which causes one tooth of a wheel to be driven forward, is sometimes used when we wish to drive a shaft with a very slow speed. If the worm-wheel has 30 teeth, it evidently makes one-thirtieth of the number of revolutions of the driving shaft. 29. Skeleton Drawings. When we consider the relative motions of, say, a piston and the crank which it drives, we come to something which it is not so easy to state without some little knowledge of mathematics. It is the same with all sorts of combinations of link work, and with cams. Even a good knowledge of mathematics is only sufficient to give one a rough general idea of the relative motion in such cases ; and for the study of any special case there is nothing so good as a skeleton drawing or a model. I give one example of the use of skeleton drawings a crank and connecting rod. Let A and B, Fig. 18, be the ends of a connecting rod. As A moves from a t to c and back again, B describes the complete circle, 6j d 6 t . Set off equal distances to 6,, by, 6 3 , fec., and make 6 a,, 6 4 a 4 , &c. equal to the length of the connect- ing rod. Then the points a t , a^ &c. and & 6 a , and the pulley F is keyed on the shaft. A belt from P, therefore, will drive any machine. When much torque is acting, the springs B become extended, causing a relative motion of E and H, and this motion is shown by the bright bead A, at the end of the lever i A, approaching the axis of rotation. A fixed scale attached to the frame c allows the motion of A to be measured. * The law is this. If k is the coefficient of friction between the cord or belt and the pulley ; if I is the length of the cord or belt which touches the pulley, say in inches; and r the radius of the pulley in inches ; then Log J =0-4343 l N and M being the pulls in the belt or cord on the two sides of the pulley. 36 PRACTICAL MECHANICS. [Chap. IV. I employ another kind of instrument to measure the horse power given out by any steam-engine, or other o motor. The steam-engine drives the pulley A, Fig. 25,* and the pulley B turns along with A. A cord hangs lapping round part of B, and carries at its one end a scale pan, M, * Charpeutier's or Professor James Thomson's Dynamometer. Chap. IV.] DYNAMOMETERS. 37 containing a weight. The other end, N', is pulled by means of a piece of metal fastened to the rim of a loose pulley, c, which has a weight, N, always acting upon it, tending to turn it i-ound. Evidently the cord is pulled with a weight, M, at one end, and a weight, N, at the other. If now there slipping between is the cord and B, the friction is measured by the difference of the weights x and M. If M is 1,000 Ibs, and N is 4,000 Ibs. the friction is 3,000 Ibs. If the pulley has a circumference of 2 feet, and makes 80 turns per minute, the amount of slipping is 80 x 2, or 160 feet per minute, and the work done against friction is 160 x 3000, or 480,000 foot-pounds per minute, that is, 14 '545 horse power. In this case all the power is wasted in friction, and this is called an Absorption Dynamometer because it measures the power but absorbs it in doing so ; whereas the coupling of Fig. 20 and the dynamometer of Fig. 24 are called Transmission Dyna- mometers, because they measure the power transmitted through them whilst working any machines. Any altera- tion in the torque is shown by a change in the amount of lapping of the cord N' B, and one of the weights must be altered if the speed is to be maintained constant. You will ask, perhaps, why we do not simply put a rope round B, with spring balances or weights at each end? The answer is, because of slight alterations in speed, little vibrations in the cord and changes in the coefficient of friction ; these produce large effects, and you would 38 PRACTICAL MECHANICS. [Chap. V. find that even if you used a dash pot to still the vibra- tions, the readings on the balances would continually alter; and if you use weights they will jump about in a dangerous manner.* Again, you must take two readings instead of one. In the absorption instrument which I have described to you, if the coefficient of friction diminishes there is an instantaneous alteration in the amount of lapping of the cord on B which is invisible to your eye, but which makes the weights keep quite steady, and their difference is an accurate measure of the friction. CHAPTER V. FLY-WHEELS. 33. Kinetic Energy. When a weight, A, Fig. 3, in falling lifts a weight, B, by the use of a machine inside the box c, let us consider the store of energy at any instant. The store of energy consists in First. The potential energy of A, that is, the weight A in pounds, multiplied by the distance in feet through which it is possible to let it fall. Second. The potential energy of B, which is the weight of B multiplied by the distance through which it is possible to let B fall. Third. The energy of motion, or kinetic energy, of everything which is moving, namely, A, B, and the parts of the mechanism. We are supposing that there are no other weights which can fall or rise, and that there are no coiled springs or other stores of energy in the mechanism. Now, if A is just heavy enough to maintain a steady motion, the kinetic energy remains the same ; so that, whatever energy is given out by A in fall- * This effect has been observed by several well-known experi- menters. However, in recent experiments made (August, 1882) since the above was written, we have found that a very slight guiding of one end of the rope, with the hand or otherwise, so as to keep the whole rope in a plane perpendicular to axis of rotation, is quite sufficient to prevent any jumping, without interfering with the accu- racy of the observations. We used a spring balance at one end. Chap. V.J ENERGY. ing is in part being given as potential energy to B, and is in part being wasted in friction. But suppose A to be heavier than this, then there is more potential energy being lost by A than is being stored by B or wasted in friction, and it must be stored up in some other form. The surplus stock shows itself in a quicker motion of everything ; it is being stored up as kinetic energy. 34. Energy Indestructible. We have now to con- sider an important question. When a certain amount of potential energy (measurable in footpounds) dis- appears, and becomes kinetic energy, how quickly must all the parts of the machinery move to store it all up? This problem is very troublesome, because everything in Fig. 3 is in motion in a different way; some parts of the mechanism are moving slowly, others quickly. It is, however, easy to find out how much kinetic energy a body has if we know its weight and its velocity. Let there be a small ball hung from the point o, Fig. 26, by a silk thread, so that, when it vibrates, we can call it a simple pendulum. Now, you know that when it reaches the end of its swing at A it is, for a very short interval of time, motionless, and has no kinetic energy. It falls from A to B ; and, as there is almost ... no friction, we may suppose that - : " the potential energy which it loses in falling through the vertical height from A to B, is all stored up as kinetic energy when the ball reaches B. Now, suppose the body to have a certain velocity in feet per second when it reaches B. You know how to calcu- late* the vertical height in feet through which a body must * Laws of falliiiff bodies. In. the following rules v means the velo- Fig. 26. 40 PRACTICAL MECHANICS. [Chap. V. fall to acquire this velocity; it is the square of the velocity in feet -f 6 4 '4. This is the vertical height from A to B. But the body has lost potential energy equal to its weight in pounds, multiplied by this height ; and this is now stored up as kinetic energy. Hence, to find the kinetic energy of a moving body* divide the weight of the body in pounds by 64*4, and multiply the quotient by the square of t/te velocity of t/te body in feet per second ; the result will be t/te kinetic energy in foot-pounds. In the case of the pendulum, this is the total energy of the bob. When the bob is at A all its energy is potential. When at B, all its energy is kinetic; and when it is any- where between these positions, its total amount of energy is exactly the same as before, but part is potential and part is kinetic. During the swing- ing of a pendulum there is a constant change going on, potential energy chang- ing into kinetic or kinetic into potential, and the sum of these two would always remain the same only that friction is constantly reducing this sum by convert- ing part of it into energy of another order, namely, heat. 35. Test of the Law. We now have a rule to find the energy stored up in a moving body, every part of which is moving with the same velocity. You can test this rule in the following way : Get a city of a body in feet per second, h is the height in feet from which the body has fallen, t is the time in seconds since the body began to fall ; g is 32'2, and represents the effect of gravity in England. a _ o i f The square of the velocity of the body is found by multi- /fl t plying the height by 64 4. ( Square the number of seconds during which the body has A = \ eter as axis ... . J wdf>x -001626 wd* + 112,166 o Spherical shell, whose \ outside diameter is d, f and inside is di,rotating 1 about diameter as axis / W ( its axis ) wld* X -00305 wld* - 59,814 Hollow cylinder, outside "j wKd^-dS) ) W l (tfdS) f b diameter d, inside > diameter d\, length 1 . ) x -00305 f * 59,814 = Thin rim, mean radius ) r of weight w . . . J wr 2 -f- 32'2 wr^ 5,873 : u i Thin rod, of length M rotating about axis through its middle ' point, at right angles j to its length. Weight | of rod w . . . . J wZ 2 -00258 wi! 2 -h 70,474 < $ > Thin rectangular plate,"] rotating about axis through its centre 1 *M parallel to side b, the > side d being at right w?2-00258 wP H- 70,474 angles to axis. Weight 1 of plate w . . . . J 48 PRACTICAL MECHANICS. [Chap. V multiplying the weight of the rim by the square of its average diameter, and dividing by 23,492. It will be found that if a fly-wheel has light arms and a heavy rim, as we often see on such wheels, a fairly good approximation to its M is found by multiplying the weight of the rim by the square of t/te mean diameter of the rim, and dividing by 23,000. Example. The rim of a fly-wheel weighs 15 tons; its mean diameter is 20 feet. Calculate approxi- mately what energy is stored up in it when it makes 60 revolutions per minute. Here you will find the M of the fly-wheel to be about 584, and hence the stored energy is 584 x 60 x 60, or 2,102,400 foot-pounds. 44. Steadiness of Machines. A fly-wheel is put upon a riveting or shearing machine, or other machine, because the supply of energy to the machine is not given regularly, or else because the demand for energy from the machine is irregular. The fly-wheel enables the machine to maintain a more constant speed. In cal- culating the proper size of a fly-wheel for any machine we must know two things. First, what is the greatest altera- tion of speed allowable in the case ; and secondly, the greatest fluctuation of the demand and supply of energy. Thus, suppose we wish never to have the speed of the fly-wheel more than fifty-one nor less than forty-nine revolutions per minute, and that during some interval of time the fly-wheel has to give out 20,000 foot-pounds more than it receives during that time ; then, although the fly-wheel will afterwards have this deficiency made up to it by some steady supply, it is obvious that its speed must diminish. We wish its speed to diminish only from fifty-one revolutions to forty-nine revolutions per minute in this interval of time. Now, when the fly- wheel runs at fifty-one revolutions, it has stored up an amount of energy equal toitsMx51x51; and when it runs at forty-nine revolutions, its store is Mx49x49, and the difference between these two ought to be 20,000. Hence, subtracting 49x49, or 2401, from 51x51, or Chap. V.] STEADINESS OF MACHINES. 49 2,601, we get 200; and dividing 20,000 by 200, we find 100 as the required value for M. Subtract, then, the square of the least speed from the square of the greatest, and divide the greatest excess of demand or supply by this remainder ; the quotient is the M of the fly-wheel. Having found M, the question is, how can you tell from it the size and weight of the wheel 1 ? Find the M of any wheel of the same shape and material as that which you want to use. It is obvious that the diameters of the wheels are as the fifth roots of their M's.* We want a wheel whose M is 100. Suppose I find a wheel of the shape I wish to use whose outer diameter is 8 feet, and I calculate its M, and find it to be 1 1 ; then The fifth root of 11 : fifth root of 100 :: 8 : answer. Log. 11 = 1-0413927; divided by 5 it is 0-2082785, which is the logarithm of 1-615. Log. 100 = 2-0; divided by 5 it is 04, which is the logarithm of 2-512. Hence 1-615 : 2-512 :: 8 : answer. This is an easy exercise in simple proportion. I find my answer to be 12-44 feet, or 12 feet 5^ inches, the diameter of the required fly-wheel, which is to be similar in form to the smaller specimen used by me for calculation. * If we have any two similar wheels, or other rotating bodies of the same material ; if we consider any similar small portions of them ; it is evident that their weights are proportional to their cubic con- tents, or to the cubes of any similar linear measurements. Hence, if one is, say, twice the diameter of the other, as every dimension of the one is twice that of the other, the weight of one must be 2 x 2 x 2, or eight times that of the other. Now, the M of any rotating body de- pends, not merely on the weight of each portion of the body, but on the square of its distance from the axis, so that the M of one must be 8x2x2, or thirty -two times the M of the other. Similarly, if the linear dimensions were as 3 to 1, the values of M would be as 243 to 1 for a pair of similar wheels. Example : We want a wheel which will have a store of 1,000 foot- pounds when rotating at twenty revolutions per minute, and it is to be of the same shape as that of an already existing wheel, which is four feet in diameter, and which contains a store of 1,350 foot-pounds when running at thirty revolutions. Evidently the M of this second wheel is 1,350 * 900, or 1'5, and the M of the first wheel is to be 2'5. Using logarithms, we find that the fifth root of 1*5 is to the fifth root of 2*5 as 4 feet is to 4 "4 feet, the answer. E 50 PRACTICAL MECHANICS. fChap. V. 45. The total kinetic energy stored up in any machine is found by calculating the energy in every wheel and in every moving part, and adding all together. But suppose that in the machine there is some shaft of more importance than any other, it is usual to give the speed of this shaft only, because if its speed be doubled, the speed of every other is doubled. Thus, in a steam-engine we state the number of revolutions per minute of the crank shaft, and this tells us the speed of every part of the engine. Let, then, the number of re- volutions of some such principal axle of a machine be found. If this number of revolutions is doubled, the kinetic energy stored up in the machine is quadrupled ; and, in fact, the kinetic energy stored up is equal to a certain number which can be found for the machine, and which loe shall call its M, multiplied by t/te square of tlie number of revolutions of this particular axle per minute. The M of any machine may be experimentally determined in exactly the same way as we have shown above. If we know the M of any machine, then the M of any other machine made to the same drawings, and of the same materials, but with all its dimensions twice as great, is thirty-two times as great, because the M's of the two machines are proportional to the fifth powers of their corresponding dimensions. 46. Facts Useful to Know. In a condensing steam- engine, when the steam is cut off at from one-third to one-eighth of the stroke, there is a certain portion of the stroke during which *16 to '19 of the total work done during the stroke is given out by the steam to the engine, in excess of that given out by the engine itself as useful work. In a non-condensing engine, steam cut off at from one-half to one-fifth of the stroke, the excess work is '16 to '23 of the total work of one stroke. These facts will enable you to calculate the proper size of fly-wheel for a given steam-engine when you know the work done in one stroke, and also the greatest and least speeds chap. Vi.] EXTENSION. 51 allowable. For a punching, shearing, or riveting machine no figures of this kind are available. Observations are much needed. In the case of a pump with a fly-wheel it is easy to calculate the excess work done during any period. But in many kinds of pump great variations of speed are generally allowed. CHAPTER VI. EXTENSION AND COMPRESSION. 47. How a pull is exerted. How is it that a cord transmits force from my hand to an object when I pull the object by means of a string 1 If you study this matter you will see that every particle o[f the string coheres to the next, and although the refusal of one particle to come away from its neighbour might easily be overcome, there are so many of them to be separated at any particular section of the string that it requires a con- siderable pull to perform this operation. When a string is pulled it really lengthens a little, and it lengthens more the more force is applied, although it may not break. A stiing is not so easy to experiment with as a wire of metal, because we find that it differs more in its quality at different sections, and it is affected by dampness and many other circumstances. No doubt it is also difficult to obtain a metal wire which shall just be as willing to break at one place as another, that is, which shall be exactly of the same material everywhere; but metal wire is certainly more to be depended upon than string. 48. Strain. Take, then, a steel wire, A B (Fig. 29), fastened near the ceiling at A, between two pieces of wood, screwed together firmly so that there may be 110 tendency for the wire to break just at the fastening. Similarly fasten at B a scale-pan arrangement, and, first, place just so much weight in the pan as keeps the wire 62 PRACTICAL MECHANICS. [Chap. VT. taut. Let there be two light little pointers stuck or tied on at a and b, and let there be a vertical scale on the wall. Now read off the distance be- tween a and b on the scale and note the weight. Add more weight, and again read the distance, and continue doing this until the wire breaks. You will prove by means of squared paper that The amount of the exten- sion of a wire is proportional to the weight which produces the extension. When we speak of the strain in the wire, and want to use the term strain in an exact sense, we mean tJie fraction of itself by which a b lengthens. Thus, suppose that a b was 50 feet, and that it lengthens 1 foot, we say that the strain is , or '02, or 2 per cent. I need hardly tell you how important it is to learn the exact meaning of a word like this ; it will give clearness to your ideas. 49. Stress. If you take another wire of the same material, but of twice the sectional area of this one, you will find that it needs twice as much load to produce the same strain. The reason of this is that you have at any section twice as many particles of steel resists ing the pull. The pull produced by the load acts at every cross section in the same way, no matter how long the wire n ay be; but if the wire is thicker at one place than another, then at such a cross section the pull is distributed over a gi ^ater number of pairs of particles. We see, then, that if a wire or rod is transmitting a pull, it is well not to con- sider the total load, but rather the load per square inch of section. The load per square inch is called t/te stress. Fig. 29. Chap. VI.] STRESS AND STRAIN. 53 This is the exact meaning which we give to the word stress. Much of the difficulty you may have met with in your reading is due to the fact that you have not made a proper distinction between the meanings of thefce cwo words. Stress is the load per square inch which produces a fractional alteration of the length of a wire or rod, and this fractional alteration is called the strain. Suppose your load to be 6 Ibs., and your wire circular in section, with a diameter of 0-05 inch. Then the area of the section is 0-025 x 0-025 x 3-1416, or -00196 square inch. The stress is 6 -f '001 96, or ^3, 061 Ibs. per square inch. You will find that this thin wire gets the same strain with a total load of 6 Ibs. as a rod one square inch in section would get with a load of 3,061 Ibs. If ever you get a problem to work out, relating to the lengthen- ing of a wire or rod produced by a load, you must con- sider, not the total lengthening of the wire or rod, but its fractional amount of lengthening, and call this the strain; also consider, not the total load, but the load per square inch of section, and call this the stress, and you will find that for some kinds of wrought iron The stress = the strain x 29,000,000. Example. How much extension is produced in a wrought-iron tie-rod 80 feet long, whose cross section is 3 square inches, by a pull of 9,000 Ibs. ] Here the stress is 3,000 Ibs. per square inch, and 3,000 is equal to 29,000,000 times the strain, or 3,000 -f 29,000,000, or 0001034 is the strain. The extension is the fraction 0001034 of the total length, and 80 feet x '0001034, or 00828 foot is the answer nearly one one-hundredth of a foot, or, more nearly, the tenth of an inch. 50. It is somewhat more difficult to experiment on the shortening of a strut or column when it trans- mits a push, because you cannot use very long struts. A strut, as you know, tends to bend if it is very long; and when it breaks, unless great care is taken to keep it straight, it breaks more easily the longer it is. The 54 PRACTICAL MECHANICS. [Chap. VI. bending action causes the load to act more on one part of the cross section than another, and the stress or the pushing force per square inch is greater at one part of the section than at another. If you expeiiment, there- fore, you must take care to use struts which are in no danger of bending. In Chap. X. I shall consider the bending of beams, after which you will better under- stand the present difficulty. It is sufficient for you at present to know that, whereas the pull in a tie-bar tends to make it straighter if possible, the push in a strut tends to make it bend. Hence, in an iron railway-bridge or roof you will see that the tie-bars are thin solid rods usually, and they might be chains or ropes if these were cheap enough ; but the struts must not merely liave a proper area of cross section, this cross section must also be wide in every direction. Thus, instead of a solid cast-iron column you always see a hollow one, unless the column is very short. Also, a thin plate of iron suffices for the lower boom or flange of a railway -girder (because it resists a pull), whereas the top boom is a hollow tube, or is U-, or fl-, or I i-shaped, because it must resist a push. Long struts, therefore, must be considered in Chap. XII., after we have investigated the bending of beams. 51. A Short Strut will be found to obey exactly the same laws as a tie bar. The load per square inch is called the stress. The shortening is a fraction of tJie whole length of the strict, and this fraction is called the strain. You will find from your experiments that the strain is proportional to the stress. Thus for wrought iron struts or columns The stress = the strain x 29,000,000. The multiplying number is found to be the same for the same material, whether it resists a push or a pull. This number is called "Young's Modulus of Elasticity;" it has been measured for various materials, and is given in Table III. In using it you must remember that the stress is in pounds per square inch. Chap. VI.] LIMITS OF ELASTICITY. Exercise 1. By how much would a round bar of steel, 1 20 feet long, whose diameter is 2 inches, lengthen with a pull of 30 tons ? Answer : 0-0855 foot. Exercise 2. By how much would a column of oak, 7 feet long and 4 inches square, be compressed in supporting a weight of 2 tons 1 Answer : Q'0013 foot. 52. I have said that if you use squared paper after making your experiments, you will find that the strain is proportional to the stress, and the lengthening of a tie bar is proportional to the total pulling force. But you will find that this law is not true when the loads become too great. If your loads are less than a quarter of the breaking load, you will find on removing them that the wire on which yon are experimenting goes back to its original length.* But if your loads much exceed this amount, it will be found that the wire has taken a permanent set ; that is, if you remove the load the wire will not go back to its original length. It remains permanently longer than it originally was, and we say that we have exceeded the limits of elasticity. The load which produces this permanent set is said to be the measure of the elastic strength Q of the wire, for although it does not break the wire it 7 alters it permanently. Now, o it is only for loads less than this that the law " strain is proportional to stress," is true. Your squared paper for experiments on a steel wire would give a straight line becoming a curve, like Fig. 30. * It may not go back at once to its old length, but in a few minutes it will be found exactly where it was before you loaded it. Similarly, when the load is put on, there is first a sudden lengthening and after this there is a slight extension going on so long as the load remains, but it practically comes to an end in a few minutes. This after-action is so slight that I have not till now spoken about it, although we have reason to believe that its investigation would be of great importance, Fig. 30. 56 PRACTICAL MECHANICS. fCba;. VI. When you plot your results, making the distance TO n represent the extension of the wire for a load repre- sented by the distance m g, to any scale you please, you will find that the line passing through your points is straight only from o to M say, and then it curves upwards. The distance, Q M, represents the load which produces permanent set. For greater loads than this, the extension is more than proportional to the load, and increases more rapidly until we get at K a very rapid extension indeed, for the wire broke with the load, K x, and just before it broke its extension was K z. * 53. The nature of the strain in a wire which is being extended, or in a column which is being compressed, cannot be said to be simple. If all lines in one direction, and in one direction only, became shorter or longer the strain would be called simple, but it needs rather a com- plicated system of external pressure to produce this effect. No matter how a body is strained, if we consider a small portion of it we shall find that any strain simply consists of extensions and compressions in different directions. In fact, imagine a very small spherical portion of the body before it is strained ; the effect of strain is to convert the little sphere into a figure called an ellipsoid, that is, a figure every section of which is an ellipse, or a circle ; remember that every section of a sphere is a circle. It may be proved that there were three diameters of the sphere at right angles to ono another, which remain at right angles to one another in the ellipsoid, and are known as the principal axes of the ellipsoid. These directions are now called the principal axe* of the strain existing at that part of the strained body. Along one of these directions the contraction (or extension) is less, and in another greater, than in any other direction whatever. 54. Example. Thus if M' N' (Fig. 31) is part of a long wire subjected to a pull, the portion of matter which was en- closed in the very small imaginary spherical surface, A B c i>, * Instruments have been designed which register on a sheet of paper (as the pencil of a steam-engine indicator does) the load pulling a rod, and the extension which it produces. A little brass cylinder covered with paper is touched by a pencil on the end of the rod. The amount of rotation of the barrel is regulated so that it is proportional to the load. By this means, curves, h'ke that of Fig. 30, may rapidly be drawn as the load on the rod ia gradually made to increase till the rod breaks. (See Art. 125.) Chap. VI.] NATURE OF STRAIN. before the pull was applied is now enclosed in the ellipsoidal spherical surface, A' c' B' D'. The sphere has become an ellipsoid of revolution ; A B becomes A' u', c D becomes c' D'. The strain in the direction ABis A/B ' ~ AB and this is equal to the pull in the wire per square inch divided by Young's Modulus of Elasticity, E. As, how- ever, it is often more convenient to use a multiplier than a di- visor we are in the habit of using the reciprocal Fig. 31. of E, and denoting it by the letter a. Thus, if the pull per square inch is one pound it produces a strain of the amount, , in the direction A B ; the lateral contraction of the material is ---' and in this case is usually denoted by the letter b. 55. The diminution in bulk of a substance when it is subjected to pressure uniform all round, as, for instance, when it is surrounded by water in a hydraulic press, or sunk in the sea, has been experimented upon. The lessening in the bulk per cubic inch is called the cubical strain of the substance. The pressure in pounds per square inch all over its surface represents the stress, and it is found that the strain is proportional to the stress. In fact, in any substance the stress is equal to the strain multiplied by a certain number, for which the letter K is usually employed, called the Modulus of Elasticity of bulk. TABLE II. Substance. Ether . Cold Water Water at 130 Fahr. Mercury Flint glass . Cast iron . Wrought iron Copper Steel . Modulus of Elasticity of Bulk iu pounds per square inch. 120 thousand. 300 330 5 mill ons. 6 14 20 24 30 Imagine a cube one inch in each edge (Fig. 32), subjected 58 PRACTICAL MECHANICS. ' L Chap. VI. to a unif orm compressive force of 1 Ib. per square inch on the opposite faces, A D E F and B c L o. Evidently the edges A B, c D, L E, and o F, become 1 a inch in length, a being the reciprocal of Young's Modulus used above. Also the edges A D, B c, o L, and F E, get the length 1 + b inch. If now we give to the faces, A B c D, and E F o L, of this cube, compres- sive forces 1 Ib. per square inch, it is the edges A F, &c. , which shorten, and the edges A B, &c., which lengthen. Again, give the corn- pressive forces to the third pair of opposite faces, A B o F and c u B L, and we have the edges A D, &c., shortening and B o, &c. , lengthening. If, now, all three sets of compressive forces act at the same time, that is, the cube gets on every face a pressure of 1 Ib. per square inch, as the compressions and extensions are exceedingly small, each edge shortens by the amount a and lengthens by the amount 2 b. Hence the edge which used to be 1 inch is now 1 a -f 2b inch. The cubic contents used to be 1 cubic inch, it is now 1 3 (a 2 l>) with great exactitude. Hence 3 (a 2 It) is the amount of cubical strain produced by 1 Ib. per square inch. That is, the Modulus of Elasticity of bulk, Fig. 32. _ _ ~3(a-2b) and if we know a and b it may be calculated. 56. It is found then that when a rod is pulled, not only does it get longer, but its diameter gets less. When, for example, a rod of glass is pulled so that its length in- creases by the one-thousandth of itself ; it is found that its diameter gets less by the one three-thousandth of itself. 57. Strength. Table III., p. 68, shows among other things the pulling (or tensile) and pushing (or compres- sive) stress which a material will bear before breaking. Probably if these stresses were allowed to act on the ma- terial for some time it would break even if they were not added to. They are obtained from experiments in which the load was increased pretty quickly, and yet quietly, that is, without any jerking or sudden action. The numbers in the table are taken from many sources, and must in Chap. VI.] STRENGTH OF PIPES AND BOILERS. 59 general only be regarded as giving rough average values. The strengths of the metals I have taken from Mr. Unwin's book on machine design. The stress per square inch which will produce a permanent set in the material is sometimes called tJte elastic strength. The working stress is usually a fraction of this ; it is the stress which experience tells us to calculate upon for loads acting for a long time on materials, and which we shall be sure are perfectly safe in the case of such materials as are supplied from foundries and forges. Exercise 1. How great a pull will a round rod of brass stand before it breaks, if its diameter is 0-3 inch? What pull would produce in it a permanent set, and what is the safe working pull 1 ? Answers: 1,237, 484, and 254 Ibs. Exercise 2. A short hollow cylindric column of cast iron is 8 inches in outer diameter, 5 inches inner diameter. What is the safe load and what load will produce permanent set ] Answer : the area of cross section is 4 x 4 x 3-1416 minus 2-5 x 2-5 x 3-1416, or 30-63 square inches; 30-63x21,000 is 643,230 Ibs., or 287 tons ; 30-63 x 10,400 is 318,522 Ibs., or 142 tons. 58. Pipes and Boilers. We may consider that a pipe or other hollow cylinder, when it tends to burst with internal pressure, has twice as much tendency to burst laterally as to burst longitudinally. If, however, the cylinder is short, the ends may modify this effect, strengthening the cylinder laterally without alter- ing the endlong strength, but it is usual to have such cylinders long in proportion to their diameter, hence it is their lateral strength pig. 33. which has to be considered. Imagine a hoop of breadth one inch, the pressure per square inch inside multiplied by the diameter in inches is the total force which tends to make this 60 PRACTICAL MECHANICS. [Chap. VI. hoop break at A and B (Fig. 33), or at the ends of any other diameter. The tendency to burst is resisted by the tension at A and B, so that the area in square inches at A, together with that at B, is like the area of cross section of a tie-rod subjected to a total pull of the above amount. Hence we have the rule, the greatest safe pressure per square inch inside a boiler or pipe, multiplied by the diameter (in inches), is equal to twice the thickness of the metal multiplied by the safe work- ing tensile stress of the material per square inch. It is in this way that we calculate the strength of a boiler or large water-pipe. When the boiler has riveted joints we must, of course, regard the material as weaker than if it could resist tensile stress everywhere like a continuous boiler plate. In cast iron pipes and in steam-engine cylinders it has to be remem- bered that the difficulty in getting castings which are of the same thickness everywhere, and the allowance that must be made for tendency to cross-breaking when the pipes are handled, as well as the great allowance that must be made in steam-engine cylinders for stiffness, the difficulty of casting, and boring out, cause such cal- culations as the above to be somewhat useless. Thus it will usually be found that, whereas a large cast iron water-pipe is not much thicker than the above calculation would lead us to expect, yet a thin cast iron pipe is often of more than twice such a thickness. 89. It is easy to prove the truth of the statement made at the beginning of the last paragraph a boiler has twice as much tendency to burst laterally as longitudinally. When a boiler bursts endwise, the area of section at which fracture occurs is the circumference multiplied by the thick- ness, or 3'1416 d t if d is the diameter and t the thickness. But the total endlong pressure is the pressure on the sectional area of the boiler at the place, or '7854 d- p, if p is the fluid pressure in pounds per square inch, hence 7854 (P p -T- 3'1416 d t, or dp -i- 4 t is the stress on the material. Chap. Vt] STRESS ON A SPHERICAL BOILER. 61 In the lateral bursting tendency considered above, the total force acting at A and B (Fig. 33) is dp pounds, and the area of the metal at A and B being 2 t square inches, the stress on the material is dp -4- 2 t, or twice as much as in the other case. In this investigation I have considered that the stress at A and B is one of mere tension, and this is the case when the metal is thin in comparison with its diameter. In a thick pipe or in a gun it is found that, although the average stress may be arrived at in the above way, some portions of the metal are more severely strained. It is not my purpose to consider such cases in this book. In a spherical boiler the tensile stress anywhere is evidently dp -4- 4 t, being the same as the endlong stress in a cylindrical boiler of the same diameter and thickness. Some students may have difficulty in understanding how it is that the pressure tending to burst a spherical boiler at A B is found by multiplying the cross-sectional area at AB by the pressure of the fluid in pounds per square inch. The pressure is really exerted on every portion of the surface A c B (Fig. 33A), and it is everywhere at right angles to the surface, but if the re- sultant force is calculated it will be found to be what has been stated. This will become evident on consider- ing that if one half of such a boiler is closed by a flat plate at A B (Fig. 3 SB), as the fluid pressure does not cause motion of any kind, its resul- |C tant action on any portion of the surface must be equal and opposite to its resultant action on all the rest of the surface, there- fore the resultant pressure on A c B is equal to the pressure on A B. It is for this reason that we always cal- culate the pressure on a pump plunger as being the sectional area in inches of the plunger multiplied by the pressure per square inch of the fluid, taking no account of the fact that the end of the plunger may either be rounded or flat. Fig 33A. Fig. S3B. 62 PRACTICAL MECHANICS. [Chap. VH. CHAPTER VII. PECULIAR BEHAVIOUR OF MATERIALS. 60. In this chapter I wish to draw your attention to a subject in which the workman is more likely to obtain valuable information than any other person. I have told you that when a load continues to pull a wire the wire continues to lengthen, although for small loads the extension is practically ended after a few minutes. A load which is so great that it strains the wire per- manently will very often be quite unable to break the wire, however long it is applied, but it is never thought advisable to allow such a load to act for a long time. It is found that after getting such a permanent set a wire is more elastic, that is, its elastic strength is greater than it was previously. A man who puts up bells in a house " kills " his copper wire, that is, gives it a permanent set, as he finds that, after this operation, it obeys better the laws of elasticity referred to above. Similarly a telegraph-line man kills his iron wire before fixing it to the telegraph posts. It is probable that the eflect of this " killing " is like the straining of a piece of riveted work beyond its limits of elasticity, which makes all the rivets fit better into their beds. It is, however, very curious to see how much set can be given to some materials. For instance, a thin brass wire gently pulled may be twisted to an enormous extent, and still retain elastic properties; indeed, its elastic strength may be higher than it was in the be- ginning. Again, when a wire by being drawn through a die is reduced to a smaller size, there is a complete alteration in the arrangement of its particles, and yet we know that the drawn wire has usually greater strength than it had originally, that is, it will bear a greater load per square inch of its section ; even hardened steel wire can be drawn in this way. In the same way, a Chap. VII.] LOAD SUDDENLY APPLIED. 63 block of copper, by a series of beatings and temperings, may be shaped like a pot or boiler, and a coin take the impression of a die, without losing their strength. In fact, metals seem to be able to flow if sufficient stress is applied to them, and at the end of the operation they are as strong as ever; indeed, they are very often stronger than before. When they are a little harder than they^were, this quality, if not wanted, can be removed by heating and slow cooling, a process which goes by the name of " annealing." 61. Another noticeable fact. It is found that a watch goes faster and faster for some time after it is made, but at the end of some months the balance spring settles down into a state which does not much change afterwards. In this state then its elasticity is greater than it was in the beginning. The springs of chrono- meters are, however, often laid aside as useless after a few years' service, their elastic condition having altered so much since the beginning that they have to be replaced. It has been found that when a long wire is kept slightly twisting and untwisting except on Sundays, there is a gradual softening or an increase of internal fric- tion going on all the week, which greatly disappears during the Sunday rest. This and other facts concerning the behaviour of materials which have been overstrained are vaguely comprehended under the expression, "fatigue of materials. "* 62. When a load is suddenly applied to stretch a wire, it produces greater effects than when slowly and quietly applied. We know the reason of this. A weight which slowly applied would produce an extension of one inch, would, when suddenly applied, produce an extension of two inches. The wire now shortens to its original length ; then extends two inches and continues to get shorter and longer alternately. As * Consult Sir Wm. Thomson's article on Elasticity, in the "En- cyclopaedia Britannica." 64 PRACTICAL MECHANICS. [Chap. VII. there is friction of some kind among the particles of the wire, and there is also external friction, the lengthenings and shortenings gradually lessen till, in a short time, the wire settles down into the same state as it would have been in if the load had been slowly applied. Now, if we suppose this wire, when stretched two inches, to be strained just beyond its elastic strength, it is evident that the suddenly applied load does harm, whereas, the same load slowly applied would do no harm. The harm is greater if the weight, besides being applied suddenly, is moving before it begins to act on the wire. Take the case of a stone which is being removed by means of a crane. If the stone, happening to fall a little, be brought up by the chain, the increase in the stress on the chain is simply proportional to the height from which the stone has fallen, and is greater the less the chain is extended (see Chap. XIX.). When a wire is lengthened -1 foot by a weight of 1,000 Ibs., which has been increased gradually, we know that the pull on the wire began with 0, and, as the wire gradually extended, the pull became greater, till it is now 1,000 Ibs. The average pull was 500 Ibs. and 500 x '1, or 50 foot-pounds is the total energy stored up in the wire in the shape of a strain. If we wish to give more energy to the wire, we must strain it more ; and this is just what we do when we let the weight fall suddenly. 63. The energy stored up in any strained body may be calculated if we know the stress and the strain. The main-spring of a watch contains a store of energy which is gradually given out by the spring in returning to an unstrained condition. Each strained portion of the spring contains a portion of the store, and if at any place in the body there is too great a store, the body will break there. Let us consider why a chisel cuts into an iron plate. When I strike the head of a chisel with a hammer I give to the chisel in a very short period of time a certain amount of energy. This energy is transmitted very quickly to the plate through the edge of the chisel. The shorter and more rigid the chisel, the more quickly is the energy sent through the cutting edge into a portion of the Chap-VIL] STRAIN ENERGY. 65 plate. If it is not conveyed away rapidly from the edge, the amount contained in a small portion of material just under the edge is very great, and the material is fractured there. As the energy of strain is proportional to the product of stress and strain or to strain squared, the pos- sibility of fracture for a material is represented hy the square root of the strain energy it contains per cubic inch. If a material is brittle there is a sort of instability which causes fracture at one place to extend to all neighbouring places. And hence, if we deliver with great rapidity to a small portion of such a material a moderate supply of energy, it is sufficient to produce a large fracture. As our material becomes less and less brittle, we must have, over a larger and larger part of the volume in which we want fracture to occur, a sufficient supply of strain energy delivered. Hence, in cutting wood we use a wooden mallet and a more or less lengthened wooden-headed chisel. The mallet and chisel act as a reservoir for the energy of the blow which is delivered to the wood from the edge of the chisel with comparative slowness and just in sufficient quantity to cause rupture in front of the edge. If the wood without gaining in strength became more rigid so as to be able to carry off more rapidly the energy given to it by the chisel's edge, it would be necessary to make the supply more rapid by using a more rigid chisel and mallet, and as we do this we must take care that the chisel itself near the edge is strong enough to resist fracture (see Chap. XIX.). 64. These are facts which we can understand ; the following, however, are not so self-evident. A piston rod is subjected to tensile and compressive stresses, often repeated. It is found that its breaking strength is not 45,000 Ibs. per square inch, which, let us say, it would be for a steady pull or push, but 15,000 Ibs. per square inch. If, instead of such an action, we have a tensile stress which varies frequently, although not sud- denly, from 30,000 Ibs. per square inch to zero, the rod will break after a time. In the same way, steel which will bear a steady stress of 84,600 Ibs. per square inch, will only bear 46,500 Ibs. per square inch if the stress varies between this and zero, but is always of the same kind ; whereas, it will only bear 25,000 Ibs. per square inch if the stress is sometimes a pull of p 66 PRACTICAL MECHANlCa [Chap. VIL this amount, and is sometimes a push of the same amount. 65. We are also quite ignorant of the reason why steel hardens when suddenly cooled, and why this hardness is different according to the temperature from which this cooling starts. In every workshop the common method adopted for tempering- a fitter's chisel is as follows : Heat the chisel to a dull red colour, put the edge in water to a distance of say half an inch, quickly rub with pumice or a file, watch the edge till, as it heats by conduction from the thicker portion you know that a certain temperature has been reached by seeing a certain colour (lightish yellow for a chisel) of oxide of iron making its appearance. When this colour appears plunge the whole chisel into water. The steel is first made extremely hard at its edge, and is then brought back to the required degree of hardness by re-heating up to a certain temperature and then suddenly cooling. This simple process is in common use. In tempering other objects sometimes much greater care must be taken, since it is often necessary that every portion of the object shall be of the same hardness, and in such cases the whole may be cooled at first and then re- heated in a bath of oil, mercury, or other melted metal whose temperature is definitely known. The effect is of the same kind, however, whether the process is the rough one which I have described or a more careful one. It is usual to explain it by saying that in sudden cooling the particles of steel have not had time to get into their natural positions when cold, and that they jam each other somehow, getting into positions of instability ; but if it be remembered that we often find steel when hard to be stronger than when it was soft, you will see that there is a great deal wanting in this explanation of what occurs. As regards the influence of impurities, of gases from the atmosphere which are suddenly imprisoned among the particles of steel, very little is yet known. Again, cast iron is stronger Chap.Vn.] CURIOUS PROPERTIES OF MATERIALS. 67 if compressed in the melted condition until it solidifies, and we explain this vaguely by saying that the pressure closes up little cavities. Metal wires are strengthened in being drawn smaller through dies, but they lose this increase of strength, and gain in toughness, when afterwards heated and cooled slowly. 66. I need not give you any more items of a long catalogue of curious properties of materials which we do not yet understand. Workmen know of and depend upon many of these actions, but nobody seems to have any clear idea as to how they take place. It is not merely that workmen temper steel and find that curious changes occur in the properties of their steel when it is altered a little in its chemical state ; the philosopher and the workman are equally aware of these facts, and equally ignorant of their real nature ; but some workmen who deal with little mechanical contrivances make use in their trades of certain properties of brass and iron and steel which the philosopher is quite ignorant of, and it is possible that an observing workman who knows a little of chemistry and physics may discover the key to all the mass of hitherto unexplained facts which I have indicated. As an illustration of an explainable effect which for a long time troubled the minds of students, the reader may refer to Art. 61 in which I speak of the elastic strength of materials, which to some extent depends upon the loads to which the materials have previously been subjected. TABLE Expan- Material. Point. (Fahr.) Specific' Gravity Weight of One Cubic Foot in sion by seating from freezing ixrint to Breaking Stress, in per sq. pounds. boiling point. Tensile. Com- pressive. Cast Iron . . . . X 2,786*) to \ 2,600* j 711 441 0011 ( 30,500 \ 17,500 ( 10,800 130,000 96,000 50,000 Wrought Iron Bars 3,280*) ( 67,000 \ 57,600 I 33,500 i 50,000 Wrought Iron Plates, > with fibre to \ 3,500* j 77 460 0012 } 50,700 Ditto, across fibre 46,100 Ditto, mean . . J 48,400 Soft Steel, unhardened ( 100,000 . 90,000 I _ Ditto, hardened Cast Steel, untempered 3,300) to \ 2,850 j 7-8 489 0011 190JOOO C l-TO.OOO \ 1-30,000 , : ( 84,000 _) Ditto, tempered J 0014 Copper 2,000* 8-8 556 0018 33,000 58,000 Brass, Yellow 1,847* 7-8 to 8-4 437to524 0019 17,500 10,500 Gun Metal 1,900* 8'6 536 ( 52,000 \ 36,000 \- ( 23,000 ) Foundry Metal 8-1 505 49,000 Phosphor Bronze {,000 Cast Zino 758 7 436 0029 7,500 Lead 600 11-4 712 0028 1,900 7,300 Tin tm 7-4 462 0028 4,700 Wood, Pine 5 to 7 31 to 44 as glass. 12,000 6,000 Ditto, Oak 7 to 1-0 43 to 62 15,000 10,000 Leather 4,200 _ Red Brick Fire Brick ~ { 2to ) 2-167 f 125tol35 {0005 280 to 300 I,100to550 1,700 Granite - 27 168 0009 - f U.OOO 1 to 5,500 Cement - -I 56 (In state of dry ! -A ! r. }- 200 to 600 10,000 Marhle 2-8 175 0014 5,5fX) Limestone 4,250 Sandstone 23 144 0018 i B,MOto ( 2,200 Plate Glass 27 169 00089 9,400 Hemp Rope, in ordi- ) nary state . . / - 1-3 - - 5,600 - Slate ^ 2-8 175 0014 1 9.fiOO to ) 12,800 Ordinary Mortar 50 _ Brickwork ~ 1-8 112 ~ TEL pounds inch. Stress which produces Permanent Set. Safe Limit of Stress, in pounds per sq. inch. Modu- lus of Elas- ticity, millions of pounds per sq. inch. Modu- lus of Rigidity millions of pounds per sq. inch. Shear- ing. Pensile. Com- jressive Shear- ing. Tensile. Com- >ressive Shear- ing. i 28, 500 10,500 21,000 7,900 S.fOO 10,400 2,700 (23 ) I 17 r (14 j 6'3 50,000 24,030 24,000 20,000 10,400 10,400 7,800 29 10-5 _ _ _ 25 20,000 20.0CO 15,000 10,000 10,000 7,800 27 26 95 - 35,000 - 26,500 17,700 17,700 13,000 30 11 - 70,500 53,000 - - 30 11 - 80,000 - 64,000 52,000 52,000 38,500 30 11 190,000 4,300 6,950 3,900 145,000 2,900 5,200 3,600 3,600 3,120 2,300 2,700 36 15 92 13 5-6 3'4 - 6,200" - 4,150 3,120 - 2,400 9'9 37 - 19,700 3,200 1,500 - 14,. p ;00 9,^70 - 7,380 14 5-25 27 650 2,300 ! - - \ 1 I 1-4 1-5 025 09 08 } ~ - - - - - - 8 - - - - 16 70 PRACTICAL MECHANICS. [Chap. VIII. CHAPTER VIII. MATERIALS. 67. A little knowledge is not a dangerous thing if the owner is modest enough to feel that it is only a little. It is often very useful, for instance, to know the most elementary facts of chemistry, for these will give you clear ideas as to the changes which occur during the manufacture of metals, the cause of the rusting of metal, the buming of fuel, and many other matters which you would otherwise be unable to comprehend. Again, a little knowledge of electricity would enable you to get clear ideas as to the action by which, when two metals touch in a liquid, one of them rapidly corrodes and the other does not, and how it is that oil preserves a polished metal surface. A little knowledge of heat will give you clear ideas as to how friction wastes mechanical energy by converting it into heat. It will tell you that when a body is heated it expands uniformly in all its dimensions ; wrought iron, -0001235 of every dimension for one degree Fahrenheit; cast iron, -00001127; steel, 00001145; brass, -00001894; copper, -00001717 ; lead, 00002818; glass, -00000861 ; and Platinum, -00000884. It will tell you that when a gas is heated 490 degrees from the temperature of freezing water at constant pressure, it expands to twice its volume or cubic con- tent, and that liquids expand very much less than gases and more than solids. It will also give you clear ideas about melting and boiling, about the way in which heat is measured as a form of energy, and the properties of steam which enable it to be used in the steam-engine. It will also tell you about the giving up of heat from one body to another by conduction and radiation, things Chap. VIII.] STONE. 71 which enter into every process going on in the workshop, and of which you can only have vague and incorrect ideas unless you spend a month or two in experi- menting. I am sorry I cannot give you this clearness of ideas by anything which I can write ; I know no other way of obtaining it than through your own handling of some simple apparatus such as is usually kept unfortunately for the mere illustration of lectures. I mean in this chapter to give a rough account of the various materials used in construction. 68. Stone. The rocks which have once been melted, and have cooled slowly, are usually hard, compact, strong, and durable. They are most easily worked when regard is paid to the t fact that they naturally divide up into certain regular shapes. They are all more or less crys- talline in texture. Stratified rocks are those which have been deposited at the bottom of a sea or river ; they are often easily divided in a direction parallel to the layers of which they are built up, but sometimes there are lines of easy cleavage in other directions. These rocks vary very much in appearance, according to the method of their formation, and to the heat and pressure to which they have been subjected, sometimes being very crystalline, strong and durable, like marble ; slaty rocks may be hard and durable, or soft and perishable ; sandstones are hardened sand of different degrees of com- pactness, porosity, strength, and durability ; there are limestones whose particles seem to form one continuous mass, and which, when they have been subjected to great heat and pressure, become marbles ; there are also lime- stones, which are composed of distinct grains cemented together, and which may vary very much in compactness, strength, and durability ; besides these there are con- glomerates, in which fragments of older rocks are im- bedded. A little knowledge of geology is necessary in order to understand the properties of rocks. Stones are preserved by coating them with some material such as coal-tar, various kinds of oil and paint, and soluble glass, 72 PRACTICAL MECHANICS. [Chap. VIII. which fills their pores and prevents the entrance of moisture. An artificial stone, which can be made in blocks of any required size and shape, is obtained by turning out of moulds and afterwards saturating with a solution of chloride of calcium, a mixture of clean sharp sand and silicate of soda. The chloride of calcium and silicate of soda produce silicate of lime which cements the sand together and thus gradually consolidates the whole mass. 69. Bricks. Bricks are made of tempered clay, moulded, dried gently, then raised to and kept at a white heat in a kiln for some days, and cooled gradually. Bricks should have plane parallel surfaces and sharp right-angled edges, should give a clear ringing sound when struck, should be compact, uniform, and somewhat glassy when broken, free from cracks, and able to absorb not more than one-fifteenth of their weight of water. They ought to require at least half a ton per square inch to crush them. 70. Limestone, when burnt in kilns, gives off carbonic acid. If pure it forms quick-lime, which combines readily with water, becoming larger in volume. Mixed with clean sand this forms mortar, which, in the course of time, hardens by losing its water and combining with carbonic acid from the air. If the burnt limestone were not pure, but contained certain kinds of clayey materials, iron, &c., it would not combine with much water, but when ground up fine, water enables its particles to combine chemically with one another with greater or less rapidity, depending on its composition. Such cement first sets, acquiring a large degree of firmness, and then more slowly becomes as hard as many limestones. When these natural hydraulic limestones are not available, nearly pure limestone may be mixed with a proper pro- portion of blue clay to produce, when ground and mixed in plenty of water, then drained and dried, then burnt and ground up again, an artificial cement, which is equal, if not superior, to the natural cement. Sand in mortar Chap. PRESSURE OF EARTH. 73 saves expense, and prevents the cracking of the mortar in drying, but in too great a proportion it weakens the mortar. Two measures of sand to one of slaked lime in paste is the average allowance, but every person who uses mortar ought to test a particular lime to see how much sand it will bear to have mixed with it. Concrete is a mixture of gravel or broken stones and hydraulic lime, the stones and gravel having about six times the volume of the lime. 71. Earth. It is usual to consider that the pressure of earth against a wall, A B (Fig. 34), is due to the tendency of a wedge-shaped mass of earth to slide downwards. We may sup- pose that A B c, or A B D, or A B E, is the sliding wedge, and we choose for our calculation that one which presses most against . the wall. It is the weight of the wedge of earth which urges it downwards ; friction at its face B c, B D or B E tends to support it, as well as friction against the surface A B, where it presses on the wall. This friction is usually calculated from knowing B F, the natural slope taken by the earth when not prevented from sliding. It is obvious that if the earth is very soft, or if much water gets between the earth and the wall, the pressure becomes like that of water. It cannot be said that experience has proved the untruth of this old theory; experience has shown that it is somewhat diffi- cult to find what is the natural slope of the earth immediately behind a wall, and what is the friction between the wall and the earth. Rankine, K 35 neglecting the friction against the wall, obtains from such a common- sense view as I have given, the following rule, which has been found to work fairly well in practice. Draw an 74 PRACTICAL MECHANICS. [Chap. VIII. angle x o R (Fig. 35) to represent the natural slope of the earth. Describe Y K x a semicircle touching o R. Now if A B (Fig. 36) is the vertical face of a wall sustaining a bank of this earth whose slope is A c, make the angle x o P equal to the inclination of A c to the horizon. Find B D so that P o : Q : : A B : BD. Then A B D is a wedge of earth whose weight represents the total presswre acting on A B. The pres- sures act in directions parallel to A c, and the resultant force, repre- senting the total pressure, acts a third of the way up from B to A. You must remember that this is a mere rule giving the result of a calculation, and that the wedge A B D is an imaginary thing used to help the memory. 72. Water. The pressure of still water is at right angles to any surface, and does not depend on the slope of the surface. It is greater at greater depths. If the pressure per square foot at any place is known, we can calculate the additional pressure at any lower level, for it is the weight of a vertical column of water one square foot in cross section reaching from the one level to the other. The pressure at all places on the same level is the sama Suppose that when water fills a vessel from which it cannot escape, we push in a piston or plunger until the pressure on the plunger is increased by say ten pounds per square inch, then at every place in the vessel there will be the same increase of pressure. Water is compressed about one forty -sixth- millionth of its cubic content for one atmosphere of pressure. (The pressure of one atmosphere is 147 Ibs. per square inch, or 2,117 Ibs. per square foot. ) The total pressure of water on any surface is obtained by regarding the pressure on each little portion of its area as a force, and finding the resultant of all the forces. On any Chap. VIII.] WATER IN MOTION. 75 plane surface submerged in a pond, the total pressure is found to be the weight of a column of water whose cross section is the area, and whose length is equal to the vertical depth of the centre of gravity of the area below still water level. If water is not still, but has a steady motion of any kind, let us consider the path taken by any particle. Suppose that it goes more quickly at one place than another, then we shall find that its gain of kinetic energy is accompanied by a lowering of level or else by a lessening of pressure. If it is not getting lower in level then it must be exerting less pressure. In a horizontal pipe where the section is smaller the velocity must be greater, and here the pressure must be less.* No matter how quickly water may move in a pipe, the pressure can never become equal to that of a vacuum, because the water will give off vapour and completely alter the conditions of the case. Remember that the law given in the note supposes that there is no friction. The frictional loss of energy experienced by a particle of water moving in pumps and pipes is found by experiment to be nearly proportional to its kinetic energy, f Hence in hydraulic presses, and * If a little volume of water (one cubic foot we take for simplicity), whose weight is w pounds, is h feet above some datum level, if the pressure upon it is p pounds per square foot, and its velocity is v feet per second, then h w is the potential energy, due to its merely being above the datum level. It has also, in virtue of the steadiness of the motion, pressure or potential energy, which is represented by p foot-pounds, and its kinetic energy is J 1> 2 . Its total energy is then h w + p + % - v 2 , and however its position, pressure, or velocity may change during its motion, the sum of these three terms remains the same, so that if two are given the third may be calculated. The student may object to this by saying that pressure cannot be regarded as a form of energy ; however, it is certain that in steady motion pressure enters into the expression for the total energy, and this is due to the fact that in nearly still water the pressure represents the work which all the rest of the water will do upon a particle should it rise slowly to a higher level. t The force of friction in fluids is proportional to the velocity, when the velocity is small ; it is proportional to the square of the 76 PRACTICAL MECHANICS. [Chap. VIII. in other machines where there is only a slow motion of the water, the loss through friction is much more negligible than it is in turbines and pumps. Thus, in a reciprocating pump, as the flow of the water is stopped in the barrel and valve-chest every stroke, its kinetic energy is all wasted, and hence it is advisable to make this flow as slow as possible By the use of air vessels we can prevent the flow of water being suddenly stopped, and thus prevent the total loss of the kinetic energy. At any particular kind of bend in a pipe the energy lost is a certain fraction of the kinetic energy, and this fraction is found by experiment 73. Example. Water flows from an orifice in a vessel into the atmosphere. The free water surface is twelve feet above the orifice. What is the velocity of a particle of the issuing water which is in contact with the atmosphere ? (The particles in the interior of the jet may not be at the pressure of the atmosphere.) Now, when this particle was motionless at the surface of the water in the vessel, its pressure was that of the atmosphere ; call it zero. Pressure energy, then, is zero at beginning and end. Loss of potential energy is the weight of the particle multiplied by the difference of level, and this has all been converted into kinetic energy. If the weight of a particle is 1 lb., it has 12 x 1 or 12 foot-pounds of potential energy changed into kinetic energy, but its kinetic energy is g^ x square of its velocity in feet per second ; hence the square of its velocity is 64-4 x 12, or 772-8, or the velocity is 27'8 feet per second. You will, in fact, find that the velo- city of the particle is the same as if it had fallen velocity in the case of ordinary steamers, and becomes proportional to a higher power of the velocity in very quick moving vessels. Now the energy wasted per second in overcoming friction is equal to the force of friction multiplied' by the velocity per second. Hence in water pipes, when the velocity is not great, the energy lost is proportional to the square of the velocity ; in ordinary ships it is proportional to the cube of the velocity. Chap. VIIL] TIMBER. 77 freely from the height of twelve feet. If, instead of flowing into the atmosphere, the water flowed into a place where the pressure is greater than that of the atmosphere, the velocity would have been less. If you can find a place in the issuing jet at every point of which the water flows at right angles to the cross section of the jet, and this seems to be the case at the most contracted part of the jet just outside a circular orifice, then the area of this cross section in square feet multiplied by the velocity we have calculated in feet per second gives the quantity of water in cubic feet per second. In the same way the quantity of water flowing through a pipe is the cross sectional area in square feet multiplied by the velocity. 74. Timber. A tree is made up of a great number of little tubes and cells arranged roughly in concentric circles. The process of seasoning consists in uniformly drying the timber. As each little portion dries, it con- tracts, and becomes more rigid, and it contracts much more readily in the direction of the circular arrange- ment of the tubes than it does towards the centre of the tree, and least easily in a direction along the tree. It is obvious, then, that if the tree is dried whole, there will be a tendency to splitting radially. If the tree is cut up before drying we can tell the way in which the planks will warp if we remember the above facts. Firwoods are easily wrought, and possess straightness in fibre and great resistance to direct pull and transverse load, and are largely used because of their cheapness. They differ greatly in strength, but their weak point is their inability to resist shearing. The best of these is the red pine or Memel timber from Russia, which can be had in large scantlings, and thus used without trussing. The ivhite fir or Norway spruce is suitable for planking and light framing, and is imported from Christiania in "deals," " battens," and "planks." Larch is a very strong timber, hard to work, and has a tendency to warp in drying, and is therefore not suitable for framing, but 78 PRACTICAL MECHANICS. [Chap. VHI. is largely used for railway-sleepers and fences, because of its durability when exposed to the weather. Cedar lasts long in roofs, but is deficient in strength. The English Oak is the strongest and most durable of all woods grown in temperate climates, but is very slow-growing and expensive. Its great durability when exposed to the weather seems to be due to the presence of gallic acid, which, however, in any wood corrodes iron fastenings ; trenails or wooden spikes should be used instead. Teak, which is grown in the East, is the finest of all woods for the engineer. It is very uniform and compact in texture, and contains an oily matter which contributes greatly to its durability. It is used specially in ship-building and railway carriages. Mahogany is unsuitable for exposure to the weather, but it has a fine appearance and is not likely to warp much in drying. It is chiefly used for furniture and ornamental purposes, and to some extent in pattern-making. Ash is noted for its toughness and flexibility, and a capability of resisting sudden stresses of all kinds, which make it specially adapted for handles of tools and shafts of carriages. It is very durable when kept dry. It is not obtainable in large scantlings, and is sometimes very difficult to work. Elm is valuable for its durability when constantly wet, which makes it useful for piles or foundations under water. It is noted for its toughness, though inferior to oak in this respect, as also in its strength and stiffness. It is very liable to warp. Beech is smooth and close in its grain. It is nearly as strong as oak, but is durable only when kept either very dry or constantly wet. It is very tough, but not so stiff as oak. (See also Table VI.) The best time for felling timber is when the tree has reached its maturity, and in autumn when the sap is not circulating. We want to have as little sap in the timber as possible, and in order to harden the sapwood, some foresters are of opinion that the bark should be taken off in the spring before felling. After timber is felled, it is well to square it by taking off the outer slabs. Chap. Vin.l GLASS. 79 Timber is, for the most part, dried by putting it into hot-air chambers, from one to ten weeks according to the thickness. Even when kept quite dry, ventilation is necessary to prevent dry rot. The circumstances least favourable to the durability of timber are alternate wetting and drying, as in the case of timber between high and low water mark, whereas good seasoning and ventilation are most favourable conditions. The most effective means adopted for preserving timber is by saturating it with a black oily liquid called creosote, The timber is placed in an air-tight vessel, and the air and moisture extracted from its pores as far as possible. The warm creosote is then forced into these pores at a pressure of 170 Ibs. per square inch. In this way timber may be made to absorb from Yjjth to Yjth of its weight of creosote. 75. Glass. Glass is a combined silicate of potassium or sodium, or both, with silicates of calcium, aluminium, iron, lead, and other chemical substances. Certain mixtures of flint and chemicals are melted in crucibles, formed when hot into the required shapes, and cooled as slowly as possible. The more slowly and more uniformly the cooling is effected, the more likely is it that the glass shall be without internal strains. When glass is suddenly cooled, as when a melted drop falls into water, the outside is suddenly contracted, becomes hard and brittle, and there are such internal strains that if the tapering part be broken or scratched at the point, the whole drop crumbles into a state of dust. A blow or scratch on the thick part produces no such effect. Heating and gradual cooling destroys this property. Many peculiarities in the behaviour of metals when heated and cooled seem to be caricatured in glass, possibly because they are due to the fact that all the portions of matter which are about to form one crystal must be at the same temperature, and when the substance is a bad conductor of heat there is great variation in temperature. Pure metals are good conductors, but the admixture of small quantities of 80 PRACTICAL MECHANICS. [Chap. VIIl carbon and of gases hurts their conductivity. Toughened glass is the name wrongly given to the hardened glass produced by plunging glass, in a nearly melting state, into a rather hot oily bath. This glass is somewhat in the condition of the glass in a Rupert's drop. It is so hard that it is difficult to cut it with a diamond, but if the diamond cuts too deep the whole mass breaks up into little pieces. Objects made of it may be thrown violently on the floor without breaking. 76. Cast Iron. Certain chemical changes occur when the ores of iron are smelted ; the iron ceases almost en- tirely to be in chemical combination with other substances, and impurities almost disappear, excepting carbon, which is mainly derived from the fuel. In the cupola of the foundry a greater purification is effected, and it is found that the composition of a casting is from 97 to 95 per cent, of iron, with 3 to 5 per cent, of carbon, although traces of other substances are to be found. About 2 cwts. of good coke are usually required to melt each ton of iron in a cupola. When the carbon is all chemically combined with t/te iron, the cast iron is white (specular iron) and is very hard and brittle. When only a little of the carbon is chemically combined, and most of its particles crystallize separately, the cast iron is grey in colour. Using the common names for the different varieties, No. 1 is darkest in colour, and from No. 4 to No. 1 there is a gradual darkening in colour. Nos. 1, 2, 3, and 4 are commonly used in the foundry, mixtures being made of them in various proportions according to circumstances. A greater proportion of No. 3 or 4 gives greater strength, whereas a greater proportion of No. 1 gives greater fluidity, and a better power of expanding at the moment when the metal solidifies, so that the sharp corners of the mould are better filled. Higher numbers than 4, as 8, 7, 6, and 5, the white varieties, are seldom used in the foundry, but they may be converted into grey varieties by slow cooling. To soften a hard casting it is heated in a mixture of Chap. VIII.] CASTINGS. 81 bone ash and coal dust or sand, and allowed to cool there slowly. 77. Patterns of objects are xisually made in yellow pine, about one-eighth of an inch per foot in every direc- tion larger than the object is to be, because the iron object contracts to this extent in cooling. Prints are excrescences made on the patterns to show in the mould where certain cores are to be placed. These cores are made of loam or core sand in core-boxes, which the pattern-maker supplies ; they represent the spaces in the object where the melted metal is not to flow. You must see for yourself in a foundry what are the usual methods of preparing a mould ; how the pattern is made so as to draw out easily ; how the moulder arranges his vents to let gases escape ; how he places his gates to let the metal run into the mould with just enough rapidity, and yet without hurt to the mould. You must also se6 for yourself, taking sketches in your notebook and making a drawing of the cupola, how the pig iron is melted and poured into the moulds ; how the moulder stands moving an iron rod up and down in one of the gates, producing just so much circulation and eddying motion in the melted iron, as is likely to remove bubbles of gas which may otherwise be unable to escape from the sides and corners of the mould ; how in some castings he exposes to the air certain parts which would otherwise cool too slowly for the rest of the object ; how next morning he screens his sand and wets it. You ought to observe the appearance of the castings before and after they are cleaned up next morning. 78. The Cooling of Castings. The most important matter in connection with moulding is that there shall be the same amount of contraction at the same time in every portion of the mass of metal as it cools; otherwise, when finished, there may be internal strains which very much weaken the object, and often produce fracture. In designing the shape of an object which is to be cast, care is taken that when a thin Q 82 PRACTICAL MECHANICS. [Chap. VIII. portion joins a thick one, it shall do so by getting gradually thicker, and not by an abrupt change of size. The thin piece exposes more surface, and cooling is effected through the surface. The thin rim of a pulley cools sooner than the arms, and becomes rigid sooner : when the arms cool they contract so much as some- times to produce fracture near the junction. In a thick cylindric object the outer portion becomes rigid first ; now, when the inner portion contracts, it tends to make the outer portion contract too much, and the outer portion prevents the inner from contracting as much as it ought to, so that the outer portion retains a compressive strain, and the inner a tensile strain. When a hollow cylinder is cast, and is required to withstand a great bursting pressure, that is, all the metal is required to withstand tensile stresses, it is usual to cool it from the inside by means of a metal core, in which cold water circulates. The inside now becomes rigid sooner, the outer portions as they solidify contract, and tend to make the inner portion contract more than it naturally would, and there is a permanent state of compres- sive strain in the object which materially helps it to resist a bursting pressure. This inequality of contrac- tion and production of internal strains in objects cause them to vary in their total bulk as compared with that of their patterns, but it is probable that some of this varia- tion is due to the fact that the contraction of grey cast iron is only one per cent, of its linear dimensions, whereas white cast iron contracts two to two and a half per cent. The fractional difference between size of pattern and the finished object varies from one-twenty-fifth of an inch per foot in small thin objects, to one-eighth of an inch per foot in heavy pipe castings and girders. As there is always great inequality in the rate of cooling of a casting near a sharp corner, internal strains may be expected here, and also an inequality in the nature of the cast iron, since the grey variety gets whiter the more rapidly it is cooled ; now, in nearly all bodies a re-entrant Chap. VIII.] WROUGHT IRON. 83 corner is a place of weakness (see Art. 95), and is specially to be guarded against in castings. Crystals of cast iron and other metals group themselves along lines of flow of heat. When a plate or wire of iron or steel is rolled or pulled, the crystals become more longitudinal, and the wire or plate becomes stronger, whereas annealing allows the crystals to arrange them- selves laterally, and the material is weakened. Castings which have been rapidly cooled by being cast in an iron mould (painted on its inside with loam) are white, and very hard in those parts which lie nearest the mould, whereas they are grey and strong inside. These are called chilled castings. When a casting is put in a box, sur- rounded with oxide of iron, and kept at a high temperature for a length of time, its surface, to a depth dependent on the time, loses its carbon and becomes pure or wrought iron, which is much tougher than cast iron. The teeth of wheels are sometimes heated in this way. Such are malleable castings. Melted cast iron possesses the property of dissolving pieces of wrought iron, and is then said to be toughened cast iron. 79. Wrought Iron. Cast iron is exposed to the air in a melted state for a long time, and the carbon is burnt out of it. The pig-iron really undergoes two processes, one called refining, the other puddling. It is then hammered and rolled when hot into bars of various shapes. The quality of wrought iron bars as bought in the market varies greatly. We have common iron, used for rails, ships, and bridges; best, double best, and treble best Staffordshire iron, used for boilers and forgings generally ; Lowmoor, Bowling, and other good irons for the most difficult forgings ; and lastly, charcoal iron, which is nearly pure. Up to the temperatures of ordinary boilers, the tensile strength of iron is not much diminished by heating, but at a red heat it is veiy much less than in the cold state. By rolling and hammering when hot, iron gets a fibrous texture, and becomes more tenacious. By hammering when cold, or by long continued strains 84 PRACTICAL MECHANICS. [Chap. VIII. of a vibratory kind, it is thought that wrought iron changes its fibrous and tough for a crystallised and more brittle condition. This brittle condition may be removed by heating and slowly cooling (annealing). Iron wire is stronger the thinner it is. Bar iron is generally stronger than angle or T-iron, and this again than plate iron. The toughness of an iron bar is best shown by the contraction it undergoes before it breaks. The section of a very tough bar may contract as much as forty-five per cent, in area. Case hardening of a wrought iron object is effected by heating it in a box with bone-dust and horn shavings. The iron absorbs carbon, and is partially converted into steel 80. Steel. Steel contains less carbon and impurities than cast iron, and thus lies intermediate between cast iron and wrought iron. It is produced by giving carbon to wroxight iron, keeping the iron heated for some days in contact with powdered charcoal, and then hammering it whilst hot till it is homogeneous, or else casting it when melted into ingots. Steel is also produced by taking only a portion of the carbon from very pure varieties of cast iron by a puddling process such as is employed in the production of wrought iron, or by the Bessemer process. In the Bessemer process, air is forced into the melted cast iron for a time, and very pure white cast iron is then added to help in removing bubbles of gas. I have already told you about the tempering of steel. (Art. 65.) It is more fusible than wrought iron, and some success has been met with in the production of steel castings in spite of the fact that they are apt to contain cavities. The strength of steel is greater than that of any other material, and is greater as it contains more carbon. The properties of steel depend so much on so many seemingly small things, small impurities, a little too much heating or variation in the rate of cool- ing at different places, that great care must be taken in working it. By the Bessemer and Siemens processes great quantities of steel are produced cheaply, contain- Chap. VIII.;) ALLOYS OP <30PPER. 85 ing small percentages of carbon. This steel is largely coming into use for locomotive rails, bridges, and ships, instead of wrought iron. 81. Copper is noted for its malleability and ductility when both hot and cold, so that it is readily hammered into any shape, rolled into plates, and drawn into wires. When cast it usually contains many cavities, but when pure it may be worked up by hammering into a state of great strength and toughness, whereas slight traces of carbon, sulphur, and other impurities necessitate its being refined before it loses its brittleness. The brittleness pro- duced by hammering when cold is very different, as it is re- movable by annealing. Copper is an expensive metal, and is only used now for pipes which require to be bent cold, for bolts and plates in places where iron would be more readily corroded, and for electrical purposes. Its tensile strength is more reduced by heating than that of iron. 82. Brass consists of about two parts by weight of copper to one of zinc. It is used chiefly on account of its fine appearance and the ease with which it can be worked. A little lead added in melting makes it much softer. Muntz metal contains more zinc than ordinary brass. Bronze and Gun-metal are alloys of copper and tin in varying proportions, more tin giving greater hardness. Five of copper to one of tin is the hardest alloy used by the engineer. A slight addition of zinc increases the malleability. A great many experi- ments have been made on bronze. Its strength depends very much upon the care taken in mixing the metals. It makes good castings, which are usually made in cast- iron moulds. Hard bronze is much used for the bear- ings of shafts. There are also various soft alloys of copper with lead, zinc, tin, and antimony, which are used for this purpose. Phosphor bronze is an alloy of copper and tin, to which some phosphorus has been added. It bears re-melting (unlike gun-metal), and its properties may be varied at will. It may be either strong and hard, or weaker but very tough. PRACTICAL MECHANICS. [Chap IX. CHAPTER IX. SHEAR AND TWIST. 83. Letc D (Fig. 37) be the top of a firm table, F H a long prism of india-rubber glued to the table, A B a flat piece of wood glued along the upper side of the india-rubber. We try in this way to apply a horizontal force to the whole upper surface of the india-rubber, so that if, for instance, the pull in the cord is 20 Ibs., and the upper surface of the india-rubber is 10 square inches in area, there will be a force of 2 Ibs. per square inch acting at every part of the surface, and this force will be transmitted through the india-rubber to the table. When the length of the prism is great compared with z F, we may suppose that the bending in it is very small, and in this case we say that the india- rubber is being subjected to a pure shear strain, and the force per square inch acting on its surface is also acting from each horizontal layer to the next and is called the Chap. E.] SHEAR STRESS. 8? shear stress. If you had drawn vertical lines like Y' x before the cord was pulled, you would now find them sloping like Y x. Thus, making a magnified drawing of Y x in Fig. 38, the point Y' has gone to Y, and any point like M has gone to N. Points touching the table cannot move, but the farther a point is away from this fixed part the further it can move. Now suppose that Y' Y is 0-01 inch, and we know that x Y' is 2 inches, what is the amount of motion of M if M x is 1 '7 inch ? Evidently Y' Y is greater than M N just in the proportion of Y' x to MX, or 2 to 1-7, hence M N is 0'0085 inch. Thus the motion of any point is simply proportional to its dis- tance above the fixed plane, and if we know the amount of motion at, say a distance of one inch, we can calculate what it must be anywhere else. The amount of motion at one inch above the fixed plane is called the shear strain. In this case we have supposed the force on F G to be 2 Ibs. per square inch. This is said to be the amount of the shear stress, and it produces or is produced by a shear strain whose amount is '005 inch per inch. If the shear stress were 4 Ibs. per square inch, you would find the strain to be '01, if the stress were 8 Ibs. per square inch the strain would be '02. In fact, we find experimentally that the stress and strain are proportional to one another. Thus if, instead of india-rubber, we had a block of tempered steel, we should find that the force in pounds per square inch is equal to 13,000,000 times the strain. This number is called the modulus of rigidity for steel ; it is given in Table III. 84. Example. A beam of steel has one end fixed, and at the other is a weight of 20 tons. The cross section of the beam is 2 square inches in area, and the length of the beam is 5 inches. Besides the deflection of this beam due to bending, there is a certain deflection due to shearing ; how much is it ] Answer : the shear stress is 10 tons, or 22,400 Ibs. per square inch. This produces a shear strain of 22,400 -r 13,000,000, or -00172. This is the amount of yielding at 1 inch from the fixed end, 88 PRACTICAL MECHANICS. [Chap. IX. and at 5 inches the yielding must be 5 x -00172, or 0086 inch. 85. The shear stress which will produce rupture is not well known for any substance except cast and wrought iron, but the shear stress which will produce permanent set is fairly well known, and we are also agreed as to the ordinary working shear stress of materials. For wrought iron it is usually regarded as less than the working tensile stress; but in a single- riveted lap joint in boiler-plates, as the holes are usually punched (and this weakens the metal), and as rivet iron is usually of a better quality than plate, the cross section of the iron which is left, which is resisting pull, is made to have the same area as the cross sections of all the rivets, which, of course, resist shearing. Besides breaking by either a tensile or a shear stress, a riveted joint may give way by the rivet crushing or being crushed by the side of its hole. Again, in many riveted joints, when the rivets are long, as they tend to contract in cooling and are prevented by the plates, so much tension may remain permanently in them that they are greatly weakened. In bolts there is usually a want of perfectly uniform distribution of the shear stress, and they are made larger than rivets in the same positions. 86. In the punching of rivet-holes it is a shearing force which acts on the material ; the area of the curved side of tJie hole, multiplied by the breaking shear stress of the material per square inch, represents the force with which tJie punch must be pressed down on the plate. The punch must be able to resist this force as a compressive stress on its own material. Experiments made on punching machines show that about 24 tons per square inch is the average shearing force required. This pressure has to be exerted through a very short distance indeed, for as soon as fracture occurs the punch has to overcome no more resistance to shear. In shearing machines, if the entire edges of the shears coincided Ohap.IX,] frtfRE SHEAR STRAIN. & with the plate, as soon as they touched anywhere there would be the same sort of effect produced; but by inclining the edges the shearing action does not occur instantaneously at every place, and the rupture being more gradual than in punching, the shearing resistance is usually from 10 to 30 per cent. less. It is very probable that the power lost in punching and shearing machines is wasted rather in the friction of the heavy parts of the mechanism than in the almost instantaneous effort of cutting the material. The effort required seems rather that of an impact (see Chap. XIX.) than of the more gradual action to be found in most existing machines. The only excuse for using such uneconomical machines as hydraulic bears and shears is that, although they are uneconomical, they may be worked by hand. In the fly presses used for hand-punching, and used largely in coining, the idea of an impact is already in use ; it will come much more into use in large machines when engineers become better acquainted with the distinction between force and energy. 87. However long we may make our block of india- rubber in Fig. 37, we shall still have some bending in it that is, the stress will not be uniformly distributed over each horizontal layer. To prevent this bending effect, and Pig. 39. to produce a really pure shear strain, we ought to have force distributed over the ends F z and G H of the same amount per square inch as we have now acting over F G and z H. These are shown in Fig. 39, where P is the pull in the cord of Fig. 37, P' is the equal and opposite force exerted by the table on the glued under- side of the india-rubber, and F and F' are equal and opposite forces distributed over the ends, such that the t I 90 PRACTICAL MECHANICS. [Chap. IX. couple r r 7 is able to balance the couple p p'. There can now be no bending moment at any place. As F multiplied by the length of the prism is the moment of the couple F F', and is equal to p multiplied by the vertical dimen- sion, we see that P distributed over the horizontal surface is the same stress per square inch as F distributed over the ends. From such a material then, if we cut a cubical block, A, Fig. 40, its horizontal faces Y y and x x are acted upon by equal and opposite tan- gential forces, and its faces Y x, y a; are acted upon by forces of exactly the same * * x amount. The faces parallel to the paper Fig. 40. have no forces acting on them. This will give you the best idea of pure shear stress. The material in Fig. 37 near the ends of the block does not get a pure shear, but if the block is very long, then at the middle there is a nearly pure shear acting. In Fig. 40 the cube x Y' y' a: has become x Y y .r. Suppose the side of this cube to be 1 inch, then Y 7 Y is the shear strain, which I shall call . The tangential force distributed over Y y is p lb., let us say. Then if we denote by the letter N the modulus of the rigidity of the material, p = N S. 88. Nature of Shear Strain. Now when y 7 moves to y, the diagonal x y' becomes extended to x y. Its original length was Aa* u> caJ&rf N, the modulus of rigidity of the material. 90. General Results. Referring back to Arts. 54 and 55, you will see that we have Modulus of rigidity . . . N = Modulus of elasticity of bulk . K = Young's modulus of elasticity . E = and you will also see that if we know two of these for any material we can find the third. Some French mathematicians have thought that the ratio of a to b, and therefore the ratios of H, x and E to one another, are constant for isotropic substances; a being always four times b. Experiment has shown that this is not the case, the ratio of a to b being 3 to 2-5 in glass or brass, 3'3 in iron, 4'4 to 2'2 in copper, and in other sub- stances varying from these values very much indeed. Just as Young's modulus is seldom found from experi- ments on the extension of wires, but rather from the Chap. IX.] ANGLE OF TWIST. 93 bending of beams ; so the modulus of rigidity is seldom found from experiments like that of Fig. 37, but rather from experiments on the torsion of rods. 91. Twisting. In Fig. 43, A B represents a wire held firmly at A. At B there is a pulley fixed firmly to the wire, and this pulley is acted upon by two cords, which tend to turn it without moving its cen- tre sideways. In fact, they act on the pulley with a turning moment merely. But the pulley can only turn by giving a twist to the wire, and the amount of motion it gets tells us how much the twist is. A little pointer fastened at c moves over a cardboard dial, and tells us accu- rately how much twist is given to the wire. The angle turned through by the pointer is called the angle of twist at C. If we had a pointer at each of the places G, H, and c, and if A, G, H, and c were one foot apart from one another, we should find Fig. 43. that the angles of twist at G, H, and c are as 1 : 2 : 3 ; in fact, the angle of twist is proportional to the length of wire twisted. You will find that if a twisting moment of 10 pound- feet produces a twist of 4, then a twisting moment of 20 pound-feet produces a twist of 8, and, in fact, the twist is proportional to the twisting moment which 94 PRACTICAL MECHANICS. [Chap. IX. is applied. You will also find that if you try different sizes of wire of the same material, say wires whose diameters are in the proportion of 1, 2, 3, &c., and to each of them you apply the same twisting moment, the amount of twist produced in them will be in the propor- tion 1, jg, g7, -*. couples on these ends. Consider *"'"! ~ J now the equilibrium of any por- ~~1 tion, say c D B (Fig. 60). At ^-EL. the section c D we know that pulling and pushing forces must be exerted by the material which exists at the left of c D on the material which exists at the right of c D, and the moments of these just balance the moment of the forces F and F, and this is evidently the same at any section of the strip. The bending moment at any section is then the moment of the couple acting at either end. Let Chap. X.] NEUTRAL AXIS OP SECTION. 105 us suppose this to be 20 pound-feet. Magnifying the section c D, as in Fig. 61, and representing the amounts of the pulling and pushing stresses by arrows, we see that as the sum of all the forces one way must be equal to the sum of all the forces acting the " other way, and as the stress at each place is proportional to distance from o, the part where there is no stress is a line through o at right angles to the paper (called the neutral axis), and this must pass through the centre of gravity of the section. 102. One has an instinctive feeling that this must he true, but it is difficult to prove it _ without algebra. If p Ibs. is the tensile stress per square inch at the distance one inch .c above o (or rather the line through o at right k angles to the paper), then at the distance x inches above o the tensile stress is p x Ibs., Fi S- 61> and at the same distance beneath o there is a compressive stress of p x Ibs. If each little strip of area is multiplied by the pressure upon it per square inch, the sum of all the tensile forces ought to be equal to the sum of all the compressive forces. Thus, if a square inches is a very small area at the distance x inches from o, then the sum of all such terms as p x a for places above o ought to be equal to the same sum for places beneath o. Hence, as p is a constant multiplier, the sum of all such terms as x a for places above o ought to be equal to what it is for places beneath o. But this is neither more nor less than saying that the centre of gravity is neither above nor beneath o. In fact, the line through o at right angles to the paper, that is, the neutral axis of the section, passes through the centre of gravity of the section. 103. Now the force on any little portion of area is proportional to its distance above or beneath o, and hence the turning moment of this force about o is proportional to the square of this distance, but if every little area of a section is multiplied by the square of its distance from the neutral axis and the results added together, we get what is called the moment of inertia of the section, hence the bending moment at a section 106 PRACTICAL MECHANICS. [Chap. X. is equal to the stress at one inch from o multiplied by the moment of inertia of tlie section. The moment of the force p x a which acts on the area a about the neutral axis of the section is p x a x x, or p x* a, and the sum of all such terms for the section is p multiplied by the moment of inertia of the section. Hence, suppose we know the bending moment M at a section, and the moment of inertia i of the section, about the neutral lino through its centre of gravity, then M -7- i is the stress at one inch from the neutral line. If the extreme edge of the section is y inches from ' the neutral line, then y * is the greatest | stress there, and must not exceed the break- ing stress of the material, i -f- y has been calculated for a great number of sections of beams, and it is called the strength modulus of the section. The bending moment at a section divided by the strength modulus must not ex-ceed the breaking stress of the material. The strength modulus for a rectangular section is | b d 3 where b is breadth and d depth of the section. For a circular section the strength modulus is -0982 d 3 if d is its diameter. For a hollow circular section it is '0982 D - if D is outside and d inside diameter. The strength modulus is exactly the same for a hollow rectangle (Fig. 62) as for the section (Fig. 63), being Fig. 62. , where B and b are the outside and BP 3 bd' fir inside breadths, and D d the outside and inside depths, of the rectangles forming the section.* For a given amount of material we have the best arrangement for strength in either the I form, or hollow rectangle, since here the ma- Fig. 63. terial is found where most required, namely, in the top and bottom members, thus giving a greater moment of inertia than if collected near the neutral line. 104. If, then, we would know what is the amount of the stress everywhere in a section, it is necessary to find the stress at one inch from o by dividing the bending moment at the section by the moment of * A very complete list of these strength moduli will be found in Wertheim. Chap. X.] RADIUS OF CURVATURE. 107 inertia of the section. Fortunately this moment of inertia has been calculated for us in several cases. To find it, if the section is rectangular, multiply one-twelfth of the breadth by the cube of the depth ; if the section is circular, multiply the fourth power of the diameter by 0-0491. To take a numerical case, suppose the bending moment to be 8,000 pound-feet at the rectangular section of a beam whose breadth is 2 inches and depth 3 inches, the moment of inertia is 2x3x3x3-j-12, or 4 '5, hence the stress at one inch from the middle of the depth would be 8,000 -r 4-5, or 1,778 Ibs. per square inch ; and the stress at the top or bottom surfaces, which are 1*8 inch from the middle of the section, is 1,778 x 1'5, or 2,667 Ibs. per square inch; in one case being a tensile, and in the other a compressive stress. If the section of the beam is such that the centre of gravity and neutral line are nearer the bottom than the top, then at the bottom there never can be as great a stress per square inch as at the top. Sometimes a material is such that it can bear much more compressive than tensile stress cast iron, for instance. In the case of cast iron the section is made so that the centre of gravity is nearer the edge which is to be subjected to tension, in order that the tensile stress may never be so great as the greatest compressive stress. (See B, Fig. 69.) 105. In Fig. 59 the neutral line which passes through the middle of every section, being neither extended nor compressed, is of the original length of the strip. Suppose it to be 1 foot long. When the beam is bent as in the figure, A B becomes longer than this, and a b shorter, yet their ends are in the same planes A a and B b. Thus the strip may be considered as a bundle of fibres lying in arcs of circles which have the same centre and subtend the same angle at that centre. If we know their relative lengths we can tell where the centre of the circle is. Now we know the stress per square inch on a certain fibre, and we know its original length, hence we can calculate its present length (see Arts. 49 and 51), 108 PRACTICAL MECHANICS. [Chap. I. aud its length is to the length of the neutral fibre as its radius is to that of the neutral fibre. In this way we find that the radius of the neutral fibre is numeri- cally equal to the modulus of elasticity of the material in ultiplied by the moment of inertia of the section, and i/irided by the bending moment at the section. M is the stress at one inch from the axis (see Art. 103), and a fibre all along the rod at one inch from the axis is extended proportionally to this stress. Its old length was one foot, therefore its extension of length is * -=- B, a fraction of a foot (see Art. 49), if E is the modulus of elasticity of the material. Every fibre forms the arc of a circle. Now let r inches be the radius of the circle formed by the middle fibre, which is not strained, then r + 1 inch is the radius of the fibre we have been considering, and as their ends are in the same radii, we know that the lengths of the fibres are proportional to their radii. The length of the unstrained fibre is 1 foot, and that of the extended one is 1 + jr feet, hence r:r + l::l:H., from which we find that r = , the rule given above. But it is sometimes more convenient to put it in the form M = ", or M = B i x curvature of the strip, rod, or beam. (See GLOSSARY for definition of Curvature.) Now if the strip in its natural unstrained condition had been curved, instead of being straight, you would have found in exactly the same way that M = E i x change of curvature, or M = E i ( ), if r. was the radius of curvature of the \r TO' strip at any place when unstrained, and r is its present radius of curvature. Example. A straight strip of tempered steel, - 7 inch broad, O'l inch thick (this represents the depth of a beam), is subjected to a bending moment of 100 pound- inches : find its radius of curvature. Answer : the moment of inertia of the section isO-7x-lx-lx-l-rl2, or '0000583. The modulus of elasticity of steel is, say 36,000,000, and 36,000,000 x -0000583 - 100 gives 21 inches for the radius of curvature of the bent strip. Chap. X.] ELASTIC CURVE. 109 106. Elastic Curve. If you take a straight uniform strip of steel and subject it to two equal and opposite forces in the same straight line, the strip will assume one of the forms shown in Fig. 64, which all go under a common name the elastic curve. Now consider the part 110 PRACTICAL MECHANICS. [Chap. X. p B, Fig. 64 a. Neglecting its own weight, it is acted on by a force F a ts, and at P there must be a force or forces produce balance. There is a force at p tending to com- press the steel, but what is of more importance is the fact that F produces a bending moment at p, and the amount of it is the force x the distance P K. Now our strip is everywhere of the same material and section, and the only thing that can alter is its radius of curvature. This radius of curvature at any place we know to be greater when the bending moment is less, and less when the bending moment is greater ; in fact, the radius of curvature is inversely proportional to the bending moment, and this really comes to the fact that the radius of curvature at any place p is inversely proportional to the distance p K. r = ^?, or -^-, if F is the force acting at B. Now E i and F do not alter, and hence r x F K = some constant number. You can obtain the shapes shown in Fig. 64 in two ways : first, by taking a straight strip of steel and per- forming the operation ; secondly, by drawing the curves in a series of arcs of circles. Suppose we have calculated, as in the above example, that the modulus of elasticity of the material multiplied by the moment of inertia of its cross-section is, say, 200, and suppose we know that the force acting at B is 10 Ibs., then we know that the radius of curvature at P is equal to 200 divided by the bending moment at P, which is 10 x p K. In fact, tho radius of curvature at P is equal to 20 divided by P K. Choose now in Fig. 65 the point c as the middle point in the strip. Suppose c D to be 4 inches, then the radius here is 5 inches. Take c o = 5 inches, and with o as centre describe a small arc, c E. Join E o and produce it. Now at E measure E F, and suppose you find it 3 '4 Chap. X.] HYDROSTATIC ARCH. Ill inches; divide 20 by 34, and we get 5-88 inches, and set this new radius off from E to o'. Take o' as a new centre, and describe the short arc E G of any convenient small length, and in this way proceed until the curve is finished. This is not a very accurate method of drawing the curve unless the arcs are very short, and small errors are apt to have magnified evil effects, but I know of no better exercise to im- press upon you the connection between radius of curvature of a strip and the bending moment which produces it. You are therefore supposed to have actually drawn one such curve at least before pro- ceeding with your study of this subject. 107. Parts of these curves happen to be the shape taken by liquids, because of their capillary action, between two solid plates. They are also the shapes of the arches which are best fitted to withstand fluid pressure. Thus, for instance, in Fig. 66 the curved water from M to N is of the shape of the curve Fig. 64 e, from M to N, the free water level being the line AB; and in Fig. 67 the middle line of the joints of the arch M to N is the same curve inverted. The water, whose pressure ^=0=-^-^-.^=-=- = ..^- --,- ^^^4 it resists, has as free ?"2?i water level the line AB. 108. When a strip of elastic material is bent, it not only alters its shape in the well- known way, but it alters the form of its cross- section. On the convex side of the strip the breadth becomes concave, and on the concave side of the strip the breadth becomes convex. It is very easy to try this for yourself on a broad strip of steel or a bar of india-rubber. These saddle shapes of the sm-faces Fig. 67. 112 PRACTICAL MECHANICS. [Chup. XI. are due to the fact that when each fibre is pulled it gets thinner as well as longer (see Art. 56), and when it is pushed it gets broader as well as shorter, and it is very curious that this action should not interfere perceptibly with the laws of bending as I have given them to you. 108a. It may now be interesting to consider the relation between bending and twisting. We have Been (Art. 105) that a couple, M, applied to produce bending gives a curvature . Now i, the moment of inertia of a circular section about a diameter, is *^-, hence the curvature is ^ -i- ^-. We have also proved (Art. 93) that a couple, M, applied to produce twisting in a cylindric shaft, gives an angle of twist ^ -j- *~. a ' Now, if, in the above cases, the bending and twisting couples are equal, we have Angle of Twist ?L _ A _: L_ o+b _ 1 j_ JL Curvature ~2w~2a ' 2(a+b)~ a a' Experiment shows that for isotropic substances the ratio lies between o and -5- . Hence, jp- is always greater than unity. Therefore, twist must always exceed bending when the couples producing them are equal. CHAPTER XI. BEAMS. 109. To be able to calculate the state of strain of a beam it is necessary to know all the forces acting on it from the outside ; these are the loads, which include the weight of the beam, and the supporting forces at its ends. If we know the loads, it is easy to calculate the supporting forces when a beam is sup- ported at the ends. The load may be concentrated at one or more points, or it may be distributed uniformly over the whole or part of the beam. Chap. XI.] METHODS OF SUPPORTING BEAMS. 113 It may be a dead load or a live load. A dead load is one which has been applied very gradually, and remains pretty much the same for a long time ; a live load is one which has been more or less suddenly applied. Given the loads, we are always able to determine the supporting forces if there are only two, either by the ordinary rules of mechanics or by the graphic method described in Art. 143. When we know the necessary supporting forces at the ends of a beam, we can take care that there are suitable means of support for the beam. 110. Methods of Supporting Beams. In practice, whenever it is possible, the ends of a beam are not merely supported, but they are fixed by being built into brick- work or masonry, for it is known that fixing the ends strengthens and stiffens a beam very considerably (see Art. 123). Thus also the cross-beams of a railway bridge are well fixed at the ends by means of bolts or rivets. Timber structures are always attached as rigidly as possible to their supports, and this is the case with all structures in which there is no fear of unequal expansion by heat. A long iron beam merely rests upon masonry or timber supports, without being rigidly attached, because the iron expands during the summer and contracts during the winter, more than the timber or masonry, and every facility must be given for re- lative motion due to these causes. Thus one end of a long iron roof-principal or iron girder generally rests upon a carriage or frame supported on rollers. 111. The supporting force at the fixed end of a beam is often rather indeterminate, but if one end rests upon a carriage we may regard the supporting force there aa being nearly perpendicular to the plate on which the rollers rest, and this supposition enables us to find both supporting forces (see Art. 148). In what follows my attention is mostly devoted to beams which are horizontal and are supported by vertical forces at one or both ends. 114 PRACTICAL MECHANICS. [Chap. II. 112. When beams carry loads they are not usually subjected to the same bending moment everywhere, and the shearing force is also different at different places. Take any simple case for instance, a beam A B, Fig. 68, supported at the ends, and loaded uniformly all along its length. If the total load is 2,000 Ibs., then each upward supporting force at A and B is 1,000 Ibs. Now at any point, c, the bending moment is 1,000 x CB acting against the hands of a watch round c, minus the load on the part c B multiplied by half the distance c B. Erect a perpendicular, c E, and make its length represent, on some scale, the bending moment at c. Do the same for a number of points, and by joining the ends of all the perpendiculars you will get a curve which shows t at a glance the bending moment everywhere. In Table IV. the -x 1 j- figures M M are diagrams of bend- * B ing moment which have been Flg ' ^ calculated by the graphic method described in Art. 143. When the upper parts of sections of a beam are in compression, the bending moment is usually measured from A B upwards. When the upper parts of sections of a beam are in extension, the bending moment is usually measured from A B downwards. It would have been difficult to give the bending moment in every case tq ; the; same scale, as the greatest bending moment in case I. of the Table (p. 1 16) is twelve times the greatest bending moment in case VL (p. 118). Hence, if we regard the scale for case I. as 120 pound-feet per inch, the scales for the other cases are 60, 30, 15, 15, and 10 pound-feet to the inch. 113. Again, the shearing force at c (Fig. 68) is simply the upward force at B minus the whole load on the part CB. Set off on the perpendicular at c a distance eqxial to the shearing force there ; do the same for other points, and draw a curve showing the shearing force everywhere. To know the shearing force at every section of a beam is very important in railway girders, because the Chap. XI.] FLANGES IN BEAMS. 115 lattice-work that is, the struts and ties which connect the upper and lower horizontal booms is proportioned to resist the shearing force. It is the same with the thin central web of a wrought-iron girder, if the girder is formed of plates of iron riveted together. But in small beams of cast iron or timber, and even in wrought-iron girders that have been rolled in one piece, the web is usually made so thick that it is unnecessary to try if it is strong enough to resist the shearing force. 114. To resist bending moment, only those parts of a beam which are far from the neutral line are of much importance hence, in iron beams we have two flanges far apart con- nected by means of the web or by lattice- work. Thus, in Fig. 69, A is the usual section of a wrought-iron beam, and B of a cast-iron beam. The neutral line o o in i_ J.-L i_ each case passes through the centre of gravity of the section. All parts of the section below o o are subjected to tensile stress, all above o o are subjected to compressive stress. Because wrought iron will resist nearly as much tensile stress as compressive before it breaks, the two flanges of A are made equal in area. But inasmuch as cast iron will stand about 4 times as great compressive stress as tensile, the flange c c, subjected to tension, has about 4J times the area of d, which is the com- pressed flange. Thus, the total breaking stress on one of the flanges is equal to the total breaking stress on the other. Suppose the area of a flange is 10 square inches, and its breaking stress is 50,000 Ibs. per square inch, then the total breaking stress on this area is 10 x 50,000, or 500,000 Ibs. If, now, the distance between the centres of gravity of the two flanges is two feet, we can say that the bending moment which will 69. 116 PRACTICAL MECHANICS. [Chap. XI. Chap. XI.] STRENGTH AND STIFFNESS OF BEAMS. 117 fc s ii O - - po II 8 pq 1 10 fen ^ *S o H5 O 118 PRACTICAL MECHANICS. [Chap. XI. i MS 1 . Strength. - , | *_ r - co ' '._. (jj ^ 1 0} -i co "s 7 o 1 / /-(f 1 o g D /* O ' ? V P3 /' 1 / i "o / j / .K ,'f. i 1 ^- > j / Si | <- W Y ' H I 2 > S I \ i ; t ' 1 \^ \ o s a i \ 3 e= 1 'o ^ < PH \5 U H "S \ \^ I \ VN >$ \ \ 03 1 Z : Q wlii' ws/ez 1 z 1 g*3gii|ju JjvitifiiMi Jii&litJij iPPMSA ~ * to .2 g c c ^^Ill^ll E||3| J = 1 g 1 'i it's J I S s MO H lN^|!!-s J|till|lM l-l fet^' ^^^ ill i^:i^ii 1 ^ ^'U.- "S o" ft d OOjj ^CS^rrtTJO ^c* 5 ? Sc^'-S iliili^llilJ JO H Chap. XI.] STRENGTH OP BEAMS. 119 destroy the beam if it acts at this section is 500,000 x 2, or 1,000,000 pound-feet. 115. When much of the material has been left near the middle part of the section, as it is in ordinary timber beams, it is not so easy to make the calculation, for although much of the timber is in a position where it is but little capable of resisting bending moment, yet it does resist to some extent. Again, in iron beams it is usual to shape them everywhere so that those sections where there is but little bending moment to be resisted are made with smaller flanges, or else flanges which are nearer together. If we have a diagram, such as we see in the various cases of Table IV., showing the bending moment at every part of the beam, we simply vary the section, so that it is just capable of resisting the bending moment which acts there. Now, timber beams, as a general rule, are everywhere of the same rectangular section. There is one place, the place of greatest bending moment, where such a beam is likely to break; we therefore calculate the size of the section to with- stand this greatest bending moment. 116. Suppose we take a certain beam which has every- where the same section, and we load it in various ways. Thus, the load may be hung from one end of the beam, the other end being rigidly fixed, say by being built into a wall. When we say that the end of a beam is fixed, we mean that it is rigidly held in position, whereas when we say that a beam is supported at its ends, we mean that it is merely held up there. In Table IY. six ways are shown in which the same length of beam is supposed to be loaded. The total load is supposed to be the same in every case, and the length from A to B is supposed to be the same. Then, we see that when the beam is fixed at both ends, and the load spread over it, it is 12 times as strong as when one end is fixed, and the whole load hung from the other end. This means that if, with the beam fixed at one end, a load of one ton, hung at the other 120 PRACTICAL MECHANICS. [Chap. XI. end, breaks the beam, then, when fixed at both ends, and the load spread uniformly over it, the same-sized beam will carry 12 tons. Hence, if experiments are made on the strength of the beam when loaded in any of these ways, we know what its strength ought to be when loaded in any of the other ways. Now, a great many experiments have been made upon beams of rectangular section, supported at both ends and loaded in the middle, the third case given in the Table ; and from these experi- ments we know how to find the load which such a beam will carry. Having found this, we know that when loaded and supported in a different way, the beam will carry more or less according to the numbers in the column headed Strength. The rule which has been deduced from experi- ments on beams whose sections are rectangular is this : A beam supported and loaded in any of the ways shown in Table IV. will break with a total load which is found by multiplying together the breadth of the section in inches, the square of the depth in inches, the number called strength in Table IV., tlw number called strength in Table VI., and dividing the product by the length of the beam, A B, in feet. Example. A beam of English oak, 20 feet long, 9 inches broad, 12 inches deep, is fixed at the ends. What load placed in the middle will break it ? This is case V. of Table IV., and the relative strength is given as 2 in the same Table. Opposite English oak, in Table VI., we find the number 557 ; and hence, 9 x 12 x 12 x 2 x 557 -f 20, or 72,188 Ibs., or more than 32 tons, is the answer. 117. Suppose the breaking load on a beam of timber is found to be 32 tons, you would follow the usual practice if you really never placed on it a load of more than 8 tons. Thus, you divide the breaking load by 4 to get the safe load, or the working load. This number 4 is called a factor of safety. The usual factors of safety employed in structures generally are given in the follow- ing Table: Chap. XI.] STRENGTH OF BEAMS. 121 TABLE V. FACTORS OF SAFETT. A Live Load, or one that alters. A Dead Load, Material. or one that does not alter. In Tem- porary Structures. In Per- manent Structures. In Struc- tures sub- jected to Shocks. Wrought Iron ) and Steel ) 3 4 4 to 5 10 Cast Iron . . . 3 4 5 10 Timber .... 4 10 Brickwork . . . 6 Masonry . . . 20 20 to 30 118. You must specially remember that it has been found by experience that if we have beams of the same material of rectangular section loaded in the same way, the strength is doubled if ive double the breadth of the beam or halve its length ; but if we double the depth, we increase the strength four times. TABLE VI. BEAMS SUPPORTED AT THE ENDS AND LOADED IN THE MIDDLE. Nature of Material. Strength. Deflection. Teak .... 820 00018 Oak .... 450 to 600 00044 to '00020 English Oak . 557 0003 Ash .... 675 00026 Beech .... 518 00031 Pitch Pine 544 00035 Red Pine .... 450 00023 Fir 370 0005 to -0002 Larch .... 284 00041 Deal .... 600 00023 Elm 337 00061 Cast Iron 2540 000024 Wrought Iron 3470 000016 Hammered Steel 6400 000013 122 PRACTICAL MECHANICS. [Chap. XI. The numbers given in this Table are merely the. average values found by various experimenters. You may wish, however, to find for yourself whether they are correct or not. You are designing a beam of pitch- pine, say; then take a rod of pitch-pine, 1 foot long, 1 inch broad, 1 inch deep ; support it at the ends, and load in the middle till it breaks ; the Table says that the load will be 544 Ibs., but you may find it to be more or less than this. Remember also that it is near the middle that your beam is likely to break; this, then, ought to be the soundest and most evenly grained part of the timber if possible, and the specimens which you try ought to be as nearly as possible the same kind of timber. 119. When a beam is loaded in any way, you know how to find the bending moment at any place, and if you know the modulus of elasticity of the material, and the moment of inertia of the section, you can find the curvature of the beam. You may draw a bent beam, then, in the same way as you drew the springs of Fig. 64, but the beam is so little curved usually that you will have difficulty in getting compasses long enough. In this case it is usual to diminish all the radii in some large proportion, remembering that the deflection of your beam as you draw it is increased in this proportion. For a beam fixed at one end and loaded at the other you would get a curve just like the portion s T in Fig. 64 c, s being the fixed end and T the loaded end. 120. The important thing to know is the deflection of a beam that is, the greatest yielding of any part of it. It can be proved mathematically, from what has been given in Art. 105, that if D is the deflection of a beam whose cross section is the same everywhere, w the load, L the length, i the moment of inertia of the section, and E the modulus of elasticity, and if all these are in inches and pounds, or in any other units so long as they are all in the same units, then w L 8 for a beam fixed at one end and loaded at Sxi the other. Chap. XI.] STIFFNESS OP BEAMS. 123 3 wi.s for a beam fixed at one end, and loaded 8 SBI uniformly. 1 wi? for a beam supported at the ends and ~ 16 3 E i loaded in the middle. 5 l^ WL 3 for a beam supported at the ends and 8 16 SBI loaded uniformly. The thii-d of these formulae is the one most needed. It is by means of this formula that the modulus of elasticity is generally determined. Thus in careful experiments with an iron beam, 1 inch broad, 1 inch deep, carried on supports 24 inches asunder, suppose we find that a load of 2,000 Ibs. produces a deflection of one-quarter of an inch. Now 1 v 1 y 1 V 1 1 i for the beam is ^ , or i. The third formula given above becomes -25 = ~ 2000x24x24x24 and from this we find that E is 27,648,000 Ibs. per square inch. Again, taking the fifth of the cases shown in Table IV., I find that 560 Ibs. produced a deflection of 0'22 inch in a beam of wood 24 inches long, 1J inch square. Here i = 1'75 x 1-75 x 1-75 x 1-75 -f- 12, or -781, and -22 = i 560x24x . 2 l x24 , from which we find that B is 10 o E X 7ol 938,656 Ibs. per square inch. Again, from Table VI. we see that a beam of teak 12 inches long, 1 inch broad, 1 inch deep, gets a deflection of -00018 inch for a load of 1 Ib. Here the moment of inertia of the cross eection is and -00018 = Ix ^ 2xl2 1 xl2 > from 12! J.D O E X i*y which we find that E for teak is 2,400,000 Ibs. per square inch. 121. Take a small beam, A B, Fig. 70, supported at the ends, and load it in the middle. Measure carefully the deflection or lowering of the middle point. This is called the deflection of such a beam. Now this distance will usually be small, and so you had better magnify it by letting the string c w pass over the little axle E, which carries a long pointer. This pointer will show on the scale P K a magnification of the deflection. You will find that the more load you place at c, the greater is the de- flection; and in fact that the deflection is proportional 12T4 PRACTICAL MECHANICS. [Chap. XI. to the load until your loads become great enough to produce permanent set, when (Art. 52) the deflections increase more rapidly than the load. If now you use a beam of the same material but of double the breadth, then for the same load you will get one-half the old deflection. If you use a beam of double the depth, then Fig. 70. for the same load you will get only one-eighth of the old deflection. Also, if you double the length of your beam, using the same load, you will get eight times the old deflection. A very instructive series of experiments may be made very easily in this subject, and you will not thoroughly understand the matter unless you make a few such experiments. It is found that a beam of pitch pine, 1 foot long, 1 inch broad, and 1 inch deep, sup- ported at its two ends and loaded in the middle, is deflected '00035 inch by a load of 1 Ib. This explains Chap. XL] STIFFNESS OF BEAMS. 125 the numbers given in Table VI. It is found that if the same beam is fixed at one end and loaded at the other (first case of Table IV.), the deflection is 16 times as great, whereas if the beam is fixed at both ends and the load is spread uniformly (last case of Table IV.), the deflection is only -125, or one-eighth as great. This explains the " deflection " column of Table IV. 122. The rule, then, to find the deflection in inches of any beam loaded in any of the ways shown in Table IV. is this : Multiply together the cube of the length in feet, the total load in pounds, the number called deflection in Table IV., and the number called deflection in Table VI., and divide the product by the breadth of the beam in inches, and by the cube of the depth in inches. Example. A beam 20 feet long, 10 inches broad, 15 inches deep, of pitch pine, fixed at one end and having spread all over it a total load of 4,000 Ibs. what is its deflection ? Here the number in Table IV. is 6, and in Table VI. it is -00035; hence we have 20 x 20 x 20 x 4,000 x 6 x -00035 divided by 10, and again divided by 15 times 15 times 15, which gives as answer 1-99 inch. The end of the beam would be deflected this distance. 123. A beam is said to be stiff if its deflection is small, and we say that the stiffness of a beam supported and loaded in the various ways shown in Table IV. is for the various cases , -, 1, 1-6, 4, 8. In fact, a beam of 16 6 a certain length carrying a certain load is 128 times stiffer when it is fixed at the ends and loaded uniformly than when it is fixed at one end and loaded at the other end. It is well to remember that when we double the breadth of a beam we double its strength and also its stiffness, but if we double its depth we get four times the strength and eight times the stiff- ness. Beams required to be very stiff ought to be 126 PRACTICAL MECHANICS. [Chap. XI. very deep. Care must be taken, however, that they are laterally supported, else they will buckle. If you double the length of a beam you get half the strength, but you only get one-eighth of the stiffness. 124. What about beams that are not rectangular in section 1 ? Suppose we have a beam of the same section everywhere, whose strength and stiffness we know, and suppose we want to know the strength and stiffness of another beam which has the same form of section that is, suppose the new section is such that all the old lateral dimensions are increased in a certain ratio then the strength and stiffness increase in this ratio ; if all the old vertical dimensions are increased in a certain ratio, then the strength increases as the square of this ratio, and the stiffness increases as the cube of this ratio. The effect of change of length is just the same as it was with rectangular beams, and we know the effect produced by different methods of supporting and loading the beam from Table IV. From Arts. 103 and 112 it is evident that the load which a beam will carry without breaking is proportional to the strength modulus of its section divided by the length of the beam. The deflection of the beam is proportional to the load multiplied by the cube of the length, divided by the moment of inertia of the cross section. 125. At the Imperial College of Engineering, in Japan, we had a testing machine with which I have made a great many experiments with my students. It in- creased the load on a beam at a uniform rate, and registered the load and deflection of the beam at every instant that is, it drew a curve, each point of which showed the deflection and the load which produced it. Mr. George Cawley, instructor in mechanical engineer- ing at the college, lithographed a number of these curves, taken by himself ; and although the experiments were made on Japanese wood, so that the actual amounts of load and deflection are not of general interest, yet the shapes of the curves are so interesting as to be Chap. XI.] STRENGTH AND STIFFNESS OP TIMBER. 127 12S PRACTICAL MECHANICS. [Chap. XII. worthy of publication. With only one exception, two beams were broken and two curves taken for each kind of wood. The mean of these two curves has been given in Fig. 71 that is, a curve lying between the two. The specimens were all free from knots. They were all 28 inches long and 1 inch square. The distance o w represents one ton, and the distance o D represents a deflection of 2 inches, so that the scale of the diagram is known. The load was in each case added to at a uniform rate, beginning with o, and the rate at which it increased was one ton in two minutes, and we see from the figure that practically only in three cases did the breaking of the beam take more than two minutes. The end of each curve shows where the specimen broke ; it is easy to see where the curve ceases to be a straight line that is, where the law, " Deflection is proportional to Load," ceases to be true ; and this point is therefore the elastic limit. In some cases the load corresponding to the elastic limit is less than half the breaking load, and in some cases greater than this, but usually it may be seen that it is about one-half. CHAPTER XII. BENDING AND CRUSHING. 126. Stress over a Section. When any portion of a column or beam or arch on one side of a section, B c, is acted upon by loads and supporting forces, we can generally find one force, representing the resultant of the stresses at the section, which will balance them all If, instead of a force, we merely get a couple, then the section is exposed solely to bending moment, and we Fig. 72. Chap. XII. J BRNDING AND CRUSHING. 129 know now how to find the effect of this. If the force is parallel to the section, then we know that the section is either exposed to mere shearing strain or shearing and bending, as in a horizontal beam with vertical loads ; but if the force is inclined to the section, there will usually be shearing and bending, and besides this a uniform distribu- tion of compression or extension all over the section. In practice we generally find that c o na- ppes s i o n and bending alone have to be considered. Thus, if B c (Fig. 72) is the edge view of the F section of an arch ring, and if p F is the o resultant force in magnitude and direc- tion, and if o represents a line through the centre of gravity of the section at right angles to the paper ; then P K the resolved part of P F parallel to B C is the shearing force which must be resisted by the section. F K x o K is the bending moment at the section, causing the parts between o and B to be compressed, and the parts between o and c to be extended. But besides this we must suppose the compressing force F K to be distributed over the whole section. This will increase the com- pression over the part o B, and will diminish the tension over the part o C, which mere bending would have produced. 127. Thus, in Art. 103 we saw that o B . K F [ -, where i is the moment of inertia of the section about the axis through o, is the compressive stress at B, due to mere bend- ing moment, and to this we must now add " if A is the area of the section. Hence, the resistance to crashing of the material per square inch must be greater than Of course the tensile stress at c is KP "A ' 130 PRACTICAL MECHANKs. [Chap. XII. If the section is of such a nature that we never wish any portion of it to resist tensile stress, this second expres- sion must be 0, or less than 0. This is usually assumed to be the case in stone or brick bridges, and it is easy to show that if the section is rectangular it leads to the general condition that o F ought never to be more than one-sixth of OB or o c ; in fact, that the resultant force P F must fall within the middle third of every joint of the stone work. If it falls outside the middle third of the joint, you will have to depend on the resistance of the cement of the joint to tensile stresses, and this is not usually regarded as a safe thing to do. 128. Struts and Pillars. I disposed much too easily of the compression of a strut in Chap. VI. At short distances from the ends of a bar subjected to pull, the tensile stress is pretty uniformly distributed over the cross section, and whether the bar is iong or short the material has nearly as much freedom to get uniform tensile strain in one case as in another. But this is different in struts. If a strut is long it breaks by bending ; if it is just so short that we know there is no bending, the load per square inch that will break it may be taken as representing its resistance to crushing; but even this is not such a resistance as a cube of the material would offer. If we take a much shorter column, say a thin disc, the way in which the load is applied may be such as to prevent the lateral spreading which always accompanies compression, and a much greater load is required to crush the material than might have been expected. If a number of specimens of cast iron are taken one-quarter inch square, the first being a cube and the last being 1 inch in length, it will be found that the load which they will support diminishes gradually from 72 tons per square inch to 45 tons. After a certain height is passed the rupture seems to be produced rather by sliding along an oblique section than by mere crushing at a cross section. 129. When a strut or column is of considerable length Chap. XII.] STRENGTH OF STUUTS. 131 it usually bends before it breaks. Professor Gordon designed a formula based on this assumption which fairly well represents the results of experiments, and although it is known not to satisfy the facts of the case so well when elastic strength has to be considered, yet it is so easy of application, and is, on the whole, so correct, that I give it in preference to the more correct rule, based on the theory of Euler, which will be found in Professor Unwin's " Machine Design." Usually the total load divided by the area of cross section is regarded as the stress on the material, but by Art. 126 we see that to this must be added the stresses produced by such bending as the strut undergoes. The result is that the stress on the strut as usually calculated must be increased by a fraction of itself which depends on the square of the length of the strut divided by the moment of inertia of the cross section regarded as the cross section of a beam. The practical rule becomes then For a strut whose ends are hinged, or a column whose ends are not fixed, as A, Fig. 73, the breaking load in pounds is equal to the breaking stress per square inch given in Table VII. multiplied by the area of cross section in square inches, and divided by 1 + n B where n is given in Table IX. and B is given in Table VIII. Fig. 73. TABLE VII. Breaking Stress, in pounds per square inch. Cast Iron "Wrought Iron .... Timber 80,000 36,000 7,200 132 PRACTICAL MECHANICS. TABLE VIII. [Chap. XII. Value of B for struts of the sections shown in Table IX. The first column gives the length of the strut divided by its least lateral dimension. Length divided by Lateral Dimension d. B for Cast Iron. Bfor Wrought Iron. B for Strong Dry Timber. 10 0748 0-132 1- 15 1-68 0-300 3-6 20 3-00 0-632 6-4 25 4-64 0-832 10-0 30 6-76 1-200 14-4 35 9-20 1-632 19-6 40 12-00 2-132 25-6 45 18-72 3-332 40-0 TABLE IX. Values of n for struts and pillars of the following sections : Square of side rf, or rectangle with smallest side d Hollow rectangle, or square with thin sides Circle, diameter d "f l Thin ring, external diameter d . Angle iron, smallest side d Cruciform, smallest breadth d . S3 2-00 2-00 * Modified from Professor Fleeming Jenkin's article on Bridijes in the " Encyclopaedia Britannica." Chap. XH.] TEETH OP WHEELS. 133 If we want the breaking load for a strut whose ends are not hinged, it is necessary to find in what way it tends to bend, and to use the above rule regarding the strut as hinged at two points of contrary flexure. Thus in Fig. 73 the strut or column B is as strong as a strut hinged or rounded at both ends, whose length is only a b. The rule becomes For a strut fixed at both ends, calculate by the above rule, but take n one fourth of what I have given in Table IX. For a strut, one end of which is fixed and the other is only hinged, calculate the breaking load as if both its ends were hinged, then calculate it as if both its ends were fixed, and take the mean value of the two answers. 130. The Teeth of Wheels. When toothed wheels drive each other, their teeth tend to break like little beams fixed at one end. It is usual in considering their strength to regard the pressure between two teeth as acting at a corner, because this may accidentally occur, and it is the most trying condition. There are usually two pairs of teeth in contact at once, so we consider that only half the total horse-power has ever to be transmitted by one pair of teeth. This transmitted horse-power, multiplied by 33,000, divided by the circumferential velocity of the wheel per minute, is of course the pressure in pounds which each tooth has to withstand. Imagine the tooth to tend to break at a section making 45 with the depth, just as we know it would break if the comer were struck smartly with a hammer. This consideration leads to the rule, that the pitch is proportional to the square root of the pressure, divided by the greatest safe stress per square inch to which the material may be subjected. 131. Flat Plates. A plate, round or square, either merely supported or firmly fixed all round its edge, will carry a total load uniformly spread over it which is simply proportional to the square of the 134 PRACTICAL MECHANICS. [Chap. HI. thickness, and is not dependent on the area of the plate. Fixing the edges adds a quarter to the strength. 132. Similar Structures Similarly Loaded. If a girder is loaded mainly by its own weight, then any other girder made to the same drawing but on a different scale would be a similar structure similarly loaded ; and this is the name given to all structures made from the same drawings but to different scales, if their loads are in the same proportions to the weights of the structures themselves. It will be found that in all cases the stress at similar places is proportional to the size of the struc- ture that is, the weakness of the structure is in direct proportion to its size. This is easily seen if we imagine the structure to be such a simple one as a rod, A, Fig. 74, carrying a weighty ball, w. If there is another such arrangement, of twice the size in every direction, the area of cross section of the rod would be four times as great, but the load to be carried would be eight times as great, and therefore the stress per square inch at a section would be twice as great that is, the larger rod and ball would be twice as weak. As the stress would be twice as great and the length of the rod twice as great, the extension would be four times as great The extension of the rod per foot in length would only be twice as Fi 74. S rea ^- l n the same wa y a beam of cast iron, 1 inch square and 1 foot long, is 1,700 times too light to break with its own weight, whereas a beam of cast iron whose length, breadth, and depth are in the same proportion, if 1,700 feet long and 1,700 inches square in section, would break with its own weight. The deflection of similar beams similarly loaded is proportional to the square of their dimensions; but the deflection per foot of length is only proportional to their dimensions. Cliap. XlII.] GRAPHICAL STATICS. 135 CHAPTER XIII. GRAPHICAL STATICS. 133. The basis of all applications of mathematics in Physics and Engineering is the fact that any physical phenomenon which is directional (such as a force, a velocity, an acceleration, a stress, the flow of a fluid, \ j2 +AB.BC.MO-, OrAB.BCI + M O J 1 . The student will find it good exercise to take a few sec- tions of angle-iron, T-iron, rails, and other specimens of rolled iron, and find the position of centre of gravity of each section, and the moment of inertia of each area about any line through the centre of gravity. The exact forms ought to be taken from real specimens. If the area is symmetrical, A one line through the centre of gravity can always be found by mere inspection. 142. When the moments of inertia of an area about any three axes through a point are known, the moment of inertia about any other axis through the same point may be found; because if a distance be measured from the point along an axis which is equal to the reciprocal of the 144 PRACTICAL MECHANICS. [Chap. XIV. radius of gyration of the area about the axis, the extremities of all such measured distances lie in an ellipse. The principal axes of the area are in the directions of the major and minor axes of this ellipse. Thus, if for any areff, M N r a (Fig. 85), the least moment of inertia is about an axis, OA, and is v and if the greatest moment of inertia is about o B, and is -^3, then A B A' B' being an ellipse whose major and minor axes are A A' and B B', the moment of inertia about an axis, o c, is j. This theorem of Poinsot's is proved in all elementary treatises on dynamics. The student will find it useful to prove it by actually finding the moments of inertia of any area about a number of axes. CHAPTER XIV. EXAMPLES IN GRAPHICAL STATICS. 143. Diagrams of Bending Moment. Let A B (Fig. 86) represent the length of a beam which has three verti- cal loads 1, 2, 3. To find B the vertical supporting ML- Fig. 86. ,,''' K forces at A and B, draw the unclosed force polygon, K L (Fig. 87) before the student arrives at this part of the book he will probably have drawn other force poly- gons where all the sides were really in the same straight line 1, 2, and 3 (Fig. 87), representing in direction and magnitude the three loads of Fig. 86. Choose any point, o. Join OK, o 1 2, o 2 3, and o L. Now draw the link polygon (Fig. 86), beginning at any Chap. XIV.] SCALE OP DIAGRAM. 145 point, a, in the vertical from A, and ending in the point b. Now a b is the side wanting in the force polygon. Draw o N (Fig. 87) parallel to a b (Fig. 86). Then L N is the amount of the supporting force at B, and N K is the amount of the supporting force at A. Also, draw any vertical line, S T (Fig. 86). Then the length s T, intercepted by the sides of the force polygon, represents the bending moment of the beam at any point, P, on some scale which it is easy to find. To prove this. Draw o H horizontally. The moment at any point, P, due to the supporting force, NK at A, is N K x A p ; and this is equal to o H x FT, for, by similar triangles, as one of the polygons, we only take into account the eight triangles. It will be found that in all figures satisfying the above conditions the number of closed polygons plus the number of points equals the number of sides plus two. This is proved by taking any suitable figure and adding a new side ; it is found that the sum of the number of closed polygons and the number of points is also increased by unity. This is, indeed, the relation between the number of faces, summits, and edges of a polyhedron, and all the figures of which we speak may be regarded as projections of polyhedra. 145a. Straight-line figures generally may be divided into : 1. Deformable figures, or those which may alter in shape, the lines retaining their original lengths. 2. Figures perfectly stiff. 3. Figures which would be perfectly stiff, even if we removed one or more lines. It may be shown that a figure belongs to class 1, 2, or 3, according as the number of its sides is less than, equal to, or greater than, double the number of points minus three. It is evident that in a figure of the third class the lengths of the extra lines may be expressed mathemati- cally in terms of the other sides. Thus, if there are a points, b sides, and e polygons in any figure, we generally find that there exist b - 2 a + 3 necessary conditions regarding the lengths Fig. 93. of the sides, and if these conditions are not satisfied the figure has no existence. If in a given figure b - 2 a + 3 is negative, then 2 a b 3 further conditions must be given, to add lines to the figure before it can become stiff. Thus, Fig. 90 is deformable (class 1) ; Fig. 91 is stiff (class 2) ; Fig. 92 contains one extra side (class 3) ; Fig. 93 contains four extra sides (class 3). 148 PRACTICAL MECHANICS. [Clutp. XIV. 146. In every problem in Graphical Statics it will be found that we are concerned with two figures which have a certain reciprocal relation to one another. We shall now define this relation. Two figures are said to be reciprocal to one another when to each line in one figure there corresponds one parallel line in the other figure, and the lines which meet in a point in one figure correspond to the lines forming a closed polygon in the other figure. Thus Fig. 74 and Fig. 75 are reciprocal, and Fig. 78 and Fig. 79 are reciprocal. It may be shown that unless a figure satisfies the con- ditions given in Section 145 it cannot have a reciprocal figure, but that all figures which satisfy those conditions do admit of reciprocal figures. Thus Figs. 88 and 89 admit of reciprocal figures. So also that a figure composed of any number of closed polygons, A B c D E, A' B' c' D' E', &c., with the same number of sides, A A', B B', &c., being joined, shall admit of a reciprocal figure, it is necessary and sufficient that the points in which all agreeing sides meet, if produced, shall lie in a straight line. Thus, the points where A B and c D meet, A' B' and c 7 D', A" B" and c" D", etc., all lie in one straight line, and so for the other sides. We observe, therefore, that any number of link polygons obtained from the same forces may form a figure which admits of a reciprocal figure. Generally, we may say ; that where any figure com- posed of points joined by straight lines shall admit of a reciprocal figure, it is necessary and sufficient that it is the projection of a polyhedron with plane faces. That a given figure shall admit of one, and only one, reciprocal figure, it is in general necessary and it is sufficient that it contains one extra line. If it does not contain an extra line, then it is necessary for it to satisfy one condition, if it is to have one and only one reciprocal figure. If it is deformable, it only admits of a reciprocal figure when it satisfies as many conditions plus one as there are new lines to be traced to render it stiff. If it contains two or more extra lines, it admits of any number of reciprocal figures. In fact, a figure admits in general of none, or of one, or of any number of reciprocal figures according as the number of its points is greater than, equal to, or less than, the number of its closed polygons. There are exceptions to these rules, which I shall not enter into. 147. In any structure, such as the principal of a roof and many girders of bridges, formed of many different Chap. XIV.] FORCES AT A HINGE JOINT. 149 bars, if we assume that each bar is subjected to direct pull or direct push that is, if we assume that the forces with which a bar may act at the two joints at its ends are in the line joining those two joints then it is easy to calculate the direct push or pull which each bar has to resist when the structure is loaded. This is assuming that the external loads are all applied at the joints of the structure, and that these joints are really the centres of frictionless hinges. In actual practice, however, the joints are usually stiff that is, the bars are really subjected to bending and shearing stresses as well as to direct compressive or tensile stress. But it is found that the strength of many structures, when tested, is approximately the same as if they were hinged structures. The conditions which enable us to calculate the stresses in a hinged structure are : 1. All the external forces are in equilibrium with one another. 2. The pulls and thrusts and loads acting at any one joint form a system of forces which are in equilibrium with one another. 3. A piece connecting two joints pulls or pushes one of the joints with the same force with which it pulls or pushes the other. Having determined the amount of pull or push in each bar, we can find the most suitable cross section to give the necessary strength, if we know the material, by using the re- B suits of Chap. XL To illustrate what we mean by a joint, find the stresses in the two pieces, o A and o B (Fig. 94), which have a hinged joint at o, when there is a vertical load of 2,000 Ibs. at o. The piece o A acts upon o with a certain force, and so does o B, and the three forces balance. Draw a triangle (Fig. 95), 150 PRACTICAL MECHANICS. [Chap. XIV. with its sides parallel to o c, o B, and o A, in such a way that the side c represents on some scale the force of 2,000 Ibs. The direction of o c, being a pull downwards, shows the direction in which we must draw the arrows concurrent round the triangle c b a ; then b represents to the same scale the force in OB, and the arrow shows its action on o ; thus we see o B pushes o, so we call o B a strut. We also know that at the other joint of o B, say B, the piece o B exerts a pushing force of amount b. Similarly, A o is a tie that is, it exerts a pull upon o of the amount a. 148. Let us now consider the roof-principal shown in Fig. 96. Certain loads are given acting at the joints, and we know that the structure is supported by two forces or reactions at its two ends. Our first step is to find these two supporting forces. They must be in equili- brium with all the external loads. Now, it is well known that we must be given either the direction or the amount of one of these supporting forces, else the problem becomes indeterminate. It is usual to be told that one or other supporting force is vertical This condition is arrived at in practice by having at one end of a structure a little carnage with wheels resting on a horizontal plate of iron. The notation which we use ig due to Mr. Bow. It Chap. XIV.] STRESSES IX ROOF PRINCIPALS. 151 very materially simplifies the process of calculation. You observe that every space between two forces in Fig. 96 is indicated by a letter. The line which separates the space A from the space B is called A B, and corresponds with the line A B in Fig. 97. A point is indicated by the letters of the spaces which meet at that point. Thus, A G H is the end of the roof-principal. Suppose the supporting force at the point F G p is known to be vertical. We must first find the amount of the force r G, and the direction and amount of the force A G. Draw the force polygon, A B c D E F, Fig. 97. We see that to close it we need two lines to join F and A. Now, one of these, F G, is vertical. Take o as pole. Join o A, OB, o C, o D, o E, OF in the usual way. Draw the link polygon, shown dotted in Fig. 96, commencing at AGII. Now, o G (Fig. 97) is parallel to the last side of it, and thus we find F G and G A, the supporting forces at the end. of the principal. Having found the two supporting forces, F G and G A, we proceed as follows : We have the closed force polygon, ABCDEFGA. The arrow-heads shown on this force polygon are not to be rubbed out during 152 PRACTICAL MECHANICS. [Ch*p. XIV. the calculation, and in practice we mark them in ink. All other arrow-heads which we draw on Fig. 97 may require to be rubbed out, and ought only to be marked in pencil. We must begin our calculation at a joint where only two pieces meet, and where one force which acts there is given. Now at the joint A o H we know the force A o. In Fig. 97 draw A H and G H parallel to the pieces A H and G H of Fig. 96. Put arrows on the sides of the triangle a A H concurrent with the arrow on G A. Now we see by the arrows that the piece AH pushes the joint with a force represented to scale by the length of the line AH (Fig. 97). We know, then, that A H is a strut, since it pushes, and we know the total pushing force in it. Similarly, H G is a tie, and the total pulling force in it is repre- sented by the length of the line HG in Fig. 97. We now rub out the arrows which we are supposed to have drawn in pencil on the lines A H and H G (Fig. 97), and proceed to the joint A B i H. It must be remembered that although the pieces A H and B I are in the same straight line, we regard them as two separate pieces. Now we know the force A B, we also know that the force with which the piece AH pushes the joint is represented by the length of the line AH (Fig. 97). Draw, then, H I and B I (Fig. 97) parallel to the pieces H I and B i (Fig. 96). We have thus a polygon, A B I H. The force A B (Fig. 97) tells us how to pencil arrow-heads concurrently round this polygon. When we do this we find that the piece B I pushes the joint with a force represented by the line B I (Fig. 97), so that B i is a strut Also i H is a strut, in which the stress is represented by the line I H (Fig. 97). We proceed in this way from joint to joint, always taking care to rub out our pencilled arrow-heads when we proceed from one joint to the next. The lenytlts of the lines in Fig. 97 give the magnitudes of the forces in the pieces of the structure. It is easy to prove that, if no mistake is made, no discrepancy will Chap. XIV.] PRACTICAL EXAMPLE OF ROOF. 153 appear when the drawing is being finished that is, when we are returning to the joint with which we began. If in Fig. 97 the points K and I are found to coincide, this evidently means that the piece K I is un- necessary in the structure. If, again, we find that we cannot close one of our little polygons in Fig. 97, we ought to proceed to new joints, and, possibly, when we again consider the joint with which we had difficulty, we shall be able to close its polygon. If we still find difficulty, it must be caused by two or more joints, and the pieces connecting these are evidently unnecessary to the structure. If we find in Fig. 97 two points with the same letter, we evidently require to add a new piece to the structure, which will exert a force equal to the distance between these two points. No explanation in writing will enable the student to master this beautiful method of determining the stresses in structures. He must select structures, apply loads to the joints, and calculate the various stresses for himself. When he has made four such calculations, he will know nearly all that can be said on the subject. 149. Hoofs. It is not my object here to describe the construction of a roof or a bridge. For such information the student must examine real structures for himself; he must read Tredgold's treatise on roofs, and examine many good drawings of roofs and bridges. Suppose, for instance, that he finds a roof, somewhat like his own, to weigh including possible snow, &c. 20 Ibs. per square foot of horizontal area covered. Suppose his principals are to be placed 8 feet apart, the span being 50 feet, then each principal has to support about 8x50x20, or 8,000 Ibs. Now, if Fig. 98 is the shape of his principal, as A B, B c, CD, and D E are all equal, we may suppose that, however the roof covering may be supported by the prin- cipal, the piece of rafter, A B, or any other of the divisions, supports 2,000 Ibs. The joint B gets half the load on A B 154 PRACTICAL MECHANICS. [Chap. XIV. and half the load on BC; consequently, the load at the joint B is taken to be 2,000 Ibs., and similarly for c and D. The joints A and E do not need to get loads, because we c have afterwards to calculate the total forces at A and E by the link-polygon method. When the above verti- Fig. 98. cal loads have been given to the joints, we have to consider wind pressure on one side of the roof. If we suppose, as we reasonably may, that 40 Ibs. per square foot is the greatest pressure of wind ever likely to occur on a surface at right angles to the direction of the wind, then the normal pi-essure per square foot on roofs of the following inclinations may be taken from the following table, which is obtained from Button's experiments. TABLE X. Normal Pressure of Wind against Hoofs. Angle of Roof. 5 10 20 30 Q 40 Normal Pressura 5-0 9-7 18-1 26-4 33-3 Angle of Roof. 60 60 70 80 90 Normal I'r.-ssmv. 38-1 40-0 40-0 400 40-0 Thus, if the portion of one slant side of the roof between two principals has an area of 240 square feet, and if the inclination of the roof is 30, ^ say, then 240 x 26-4, or 6,336 Ibs., has to be !\\ supported by each bay. Transferring this to the joints, we see that at B (Fig. 99) we have the vertical load x B, or 2,000 Ibs., due to weight of roof, snow, &c., and also Y B, or one half of 6,336 Ibs., normal to the roof, and due to wind. Complete the parallelogram, and evidently z B is the load at the joint B which we must use in our calculations. Fig. 99. Chap. XIV.] EXAMPLES OP STRESS DIAGRAMS. 155 The student will find that if a roof-principal can only be supported by a vertical force at a certain end, the stresses in the structure are greatest when the other side of the roof is acted on by the wind. I would advise every student to design at least one timber and one iron roof, making detail drawings of all the joints, etc., referring much to books and drawings, and writing out complete specifications. The following examples of stress diagrams (Figs. 100, 101, 102) may be useful for reference. ir>6 PRACTICAL MECHANICS. [Chap. XIV. 150. Every joint in a real structure is usually a stiff joint ; so that every piece may really be subjected to bending, as well as to direct compressive and tensile stresses. A general method of taking stiff- ness of joints into account is quite unknown ; but as we have discussed bending we see pretty clearly what is the effect of a stiff joint, and in some cases we are able to make calculations 011 the subject. It may generally be assumed that the strength of a structure is greater if the joints are stiff than if they are merely hinges. This is not always the case, and, from the indeter- minateness of the problem of finding the stresses in a structure whose joints are stiff, many large bridge trusses are at present made with Fig. 101. Fig. 102. nearly all their joints hinged. In roof-principals the joints are made stiff, rather for the purpose of stiffening Chap. XIV.] SECTIONS OP STRUCTURES. 157 the whole structure that is, increasing its resistance to change of shape than for the sake of strength. In a roof all joints of struts are usually made stiff. What we shall now say is of more importance in bridges than in roofs. If two or more pieces of a structure are in a straight line with one another at joints where they meet, it is usual for strength to make the joints between them quite rigid. Thus the pieces A H and B I of Fig. 96, or A B and B c of Fig. 98, ought to form one bar. But this is only useful when the pieces in question are struts, and our reason for the continuity of the pieces is that a strut is stronger when its ends are fixed than when its ends are not fixed. Thus the piece B I (Fig. 96), will resist a greater thrust if it is continuous with A H and C K than if it were hinged with these pieces. (See Art. 129.) It is not good in all cases to fix the end of a strut by rivets, &c., instead of a hinge ; because the benefit due to fixing an end may be more than counter- balanced by the evil effects of bending introduced to the strut through the joints by a tendency to change the angle which the strut makes with the piece to which it is fixed. The common sense of the engineer will always enable him to decide as to the judiciousness of fixing the end of a strut. 151. Sections of Structures. It is often of consider- able importance to find immediately the stresses in pieces of a , structure which are not near the ends. If we can draw any surface which will cut through the pieces in question, we can calculate the stresses in these pieces directly, supposing the pieces are only three in number. Thus, the section AGE (Fig. 103) cuts the pieces B A, B c, 158 PRACTICAL MECHANICS. fChap. XIV. and D E. Now not only the whole structure, but every part of it is kept in equilibrium. What forces keep the part A H E in equilibrium ? They are the known forces at B, F, and H, together with three unknown forces whose direc- tions are BAA', B c c', and D E E'. Given the directions of three forces which equilibrate a number of known forces, we know that they may be determined in magnitude by the link-polygon method. Sometimes the link-polygon method is more troublesome than the following : To find the push or pull in D E E'. We know (Art. 23) that the moment of the force in E E' about the point B is equal to the sum of the moments about B of all the external forces (for the forces in the directions A A' and cc' have no moment about B, since they pass through it). Let the algebraic sum of the moments of the external forces be actually cal- culated, multiplying numerically each force by its per- pendicular distance from B. This sum, divided by the perpendicular distance from B to E E', will give the force in E E'. If the algebraic sum gives a moment tending to turn the structure about B against the direction of the hands of a watch, the force in E E' is a pulling force acting from E towards E', and therefore the piece D E is a tie. It will be observed that if we wish to know the stress at any section of any loaded structure, we must consider that the parts of the structure on any one side of this section are in equilibrium. Thus, if A and B are the two parts of the structure, consider the equilibrium, say, of B. Now, B is kept in equilibrium by the external forces or loads which act on B, and by the forces which act on B at the section. Of course it is A which causes these forces to act on B through the section ; but in calculations concerning them we do not need to consider A or the loads on A. Chap. XV.] SUSPENSION BRIDGES, ETC. 159 CHAPTER XV. SUSPENSION BRIDGES, ARCHES, AND BUTTRESSES. 152. Loaded Links. Let A c, c D, D E, and K B be four links hinged together at c, D, and E, and supported some- how by hinges at A and B, and, neglecting the weights of the links themselves, let x, y, and z be forces acting at the three joints, so as ^A to make the links take the positions shown in Fig. 104. Take any point , o (Fig. 105), and draw lines om, on, op, and o q parallel to the links, and from any point, m, in om, draw m n parallel to the force z, n p parallel to the force y, and p q parallel to the force x; then it is easy to prove that the lengths of the lines m n, n p, and p q are proportional to the forces z, y, and x, and the tensile forces in the link,s are pro- portional to the lengths of the lines o m, o n, op, and o q. For it is evident that the three forces at E, keeping the joint in equilibrium as they do, must be proportional to the sides of the triangle omn. If you put arrow- heads on o m and o n concurrent with the one already on m n, you will see that the bars B E and D E do not push the joint E, they pull it and are tie-bars. Thus, then, the lengths of the lines in Fig. 105 repre- sent to some scale all the forces acting at the joints c, D, and E. Fig. 104. 160 PRACTICAL MECHANICS. [Chap. XV. 153. Loaded Chain. If, now, we want to find the pull in every part of one chain of a suspension bridge, and to draw the shape of the chain, it is first necessary to know the weight of the bridge at every place. This weight is probably supported by two chains, so, as we have only one chain to deal with, we only take half the weight of the bridge. We will suppose that there is no long girder or other support for the bridge but the chain. It is usual to suspend the supporting beams of the roadway from the chain by vertical iron rods, placed at equal horizontal distances from one another. We may imagine the roadway to be as heavy at one place as another, so that the pull in all the rods will be the same. Suppose there are ten rods, and in each a pull of 20 tons. Draw ten equidistant vertical lines (Fig. 106) to represent the rods. We must get another condition before we can draw the chain. Let it be this, that the chain in the middle where it horizontal shall be capable of with- standing a pull of 200 tons. Now draw O H horizontally (Fig. 106A), and make its length on any scale represent 200 tons. Make H A and H B on the same scale represent each 100 tons (if your chain is to be symmetrical), and divide them up so that each portion represents 20 tons that is, the vertical load communicated to the chain by each tie-rod. Now join o with each point of division in AB. Suppose now that p (Fig. 106) is one point of support of the chain, draw pa (Fig. 106) parallel to O A Fig. 106A. Chap. XV.] ARCHED RIB. 161 (Fig. 106A), a c parallel to oc, cd parallel to OD, and so on till you reach the point Q, which I suppose to be on the same level as P. Of course, the points of support, P and Q, may be anywhere on the lines a P and in Q. It is quite evident from what you have already learnt that the pull in any part of the chain is represented by the length of the line from o, which is parallel to it in Fig. 106A, and it is also evident that the chain will take this shape without any tendency to alter. 154. We began by assuming a pull of 200 tons in the part y/i, where the chain is horizontal. We might have assumed a pull of 300 tons in/A; this would have caused the chain to hang in a natter curve. Assuming a pull of 100 tons in f h, we should have obtained a greater difference of level between P and h. It will be found that in the present case, where the load is supposed to be uniformly distributed along the horizontal, the links would just circumscribe the curve called a parabola. With any other distribution of load they will fit some other curve than a parabola, but in any case you know now how to draw the shape of such a chain, and to determine the pull in any part of it. 155. Arched Bib. If instead of a hanging chain you wanted to use a thin arched rib to support your roadway, then if you have numerous vertical rods by which to hang your load to the rib, and if the distribution of the load is known, you can draw the curve of the rib in exactly the same way, but it will now be convex upwards of course. With uniform horizontal distribution of your load you will get a parabolic rib. The difference between the two cases is this : a slight inequality in your loads or a temporary alteration will only cause the chain to take a slightly different position for the time, and it will get back to its old shape when the old loading is returned to ; whereas the arch is in a state of unstable equilibrium, and as it is very thin, so that it cannot resist any bending, a slight change of loading will very 162 PRACTICAL MECHANICS. [Chai>. XV mult*? r if H !.]/L.l...j H l i r materially alt-i it^ shape and it will get destroyed. Such a rib or series of struts is either stayed with numerous diagonal pieces or else it is made very massive, so that should the line like P h Q (inverted), which is supposed to pass everywhere along its axis, deviate a little from this position, the rib may resist altera- tion of shape by refusing to bend. 156. The load carried by an arch may either be hung from it by means of tie-rods, or else it may rest on the top of the arch, the weight being carried from the different parts by means of struts or pillars iron, stone, or brick, the arch may be levelled up to the road- way by means of a solid mass of masonry, or 4 merely by one or two pillars of masonry and a filling-in of earth. It is rather difficult in a stone or brick bridge to say exactly what is the load on every portion of the arch, but it is guessed at, and a curve or link polygon, such as P h Q, Fig. 106 (inverted), drawn. It is shown in Art. 127 that in a stone or brick arch it is dangerous to have the arch so thin that the line P h Q (inverted) passes anywhere out- side the middle third of the arch ring. Thus, in Fig. 107, we have a section of a stone arch, the various stones or voussoirs, as they are called, being separated by joints of mortar or cement. Now divide each joint into three equal parts and draw two polygons, m m m and n n n, marking out the middle third of Fig. 107A Chap. XV.] MASONRY ARCH. 163 every joint. Let us suppose we know the weight which each voussoir supports, including its own weight (it is usual to consider the arch as one foot deep at right angles to the paper), and let these weights be the weights M>J, Wy &c., shown in Fig. 107. Now draw the force polygon, Fig. 107 A; it happens to be all in one vertical line, the forces being all vertical. And now we come to the drawing of our link polygon, but we are stopped at the outset by not knowing what is the thrust at the crown of the arch. The pull at the middle of our suspension bridge chain was quite definite, but the thrust at the crown of the arch may be what we please, and the arch will remain stable if the link polygon which we draw never passes outside the middle third of any of the joints.* Suppose we draw any symmetrical link polygon to begin with, by bisecting a k in H (Fig. 107A), draw H o horizontal, and take o anywhere we please. Now o H will be the thrust in the crown of our arch, if this link polygon is the correct one. Join oa, o 1 2, o 2 3, &c. Now start from any convenient point in w- a , Fig. 107, say E, within the space which contains the middle thirds of all the joints. Draw E r>, Fig. 107, parallel to o 4 5, Fig. 107A; draw D c, c B, B A, A p, in succession parallel to the corresponding lines in Fig. 107A, and so also for E F, &c., to K Q. If any of the lines so. drawn passes outside the space m m, nn, you must choose some other point E to begin at, and if you find that no choice of E will allow the link polygon to lie altogether within the space mm, nn, then you must choose another pole, o, in Fig. 107A, until at length you find, as in the figure, a link polygon, P E Q, which cuts within the middle third of every joint. The lengths of the lines in Fig. 107 A tell us the forces acting at the joints of Fig. 107. Thus oa, Fig. 107 A, is the force P A, Fig. 107, the resistance of the abutment of the bridge. * It is obvious also that the link polygon wherever it crosses a joint must make an angle so near a right angle with the joint that there can be no slipping or rupture by shearing there. 164 PRACTICAL MECHANICS. [Chap. XV. Again, the length of o 4 5 is the force acting in the direction E D between the stones E and D. 157. Professor Fuller has made the work of drawing such a link polygon very easy. It can be shown that if a number of link polygons are drawn in Fig. 107 for different lengths, OH, Fig. 107A, then the vertical dis- tances between the points A, B, c, D, &c., are in the same proportion in all the link polygons.* Beginning with the first load w u draw (Fig. 108) A 1' 2' 3' 4' 5' E, the half of any link polygon corresponding to p A B c D E in Fig. 107. Divide A s into any number of " T Pig. ioa equal parts ; I choose six. Erect perpendiculars at 1, 2, 3, 4, 5, s. Draw horizontal lines from 1', 2', 3', vinding up, it is evident that the angle B ought to increase as fast as A ; that is, there ought to be a very considerable amount of yielding in the fastening of the spring to its case. The same effect will be produced by exerting considerable pressure on the arbor at its pivots, or in some way causing the arbor and its case to be not quite concentric with one another. The watchmaker's usual plan to get moderately good isochronism is to make one of the above errors tend to correct another ; that is, by allowing a greater yielding or greater stiffness of the outer attachment counteract the results due to centre of gravity of the spring not remaining exactly in the axis of the balance. Chap. XVI.] EXPERIMENTS ON MAINSPRINGS. 171 163. Thus we see that by applying the law given in Art. 161 to the case of a flat spiral spring fastened to a case at its outer end, N, and to an arbor or axle at its inner end, M, we find that if the spring is riveted firmly both at N and M, and if it is so long and its coils so nearly circular that its centre of gravity is always nearly in the centre of the axle, then, when partly wound up, the spring tends to unwind itself with a turning moment which is pro- portional to the amount of winding up. This is the case in the balance spring, and it is this condition that gives to the balance its character of taking almost exactly the same time to make a small swing as to make a great one. (See Art. 180.) When the end N is not riveted, but merely hinged or fastened in any way that will allow it to turn about N, the unwinding tendency is not proportional to the amount of winding up ; it is proportional to the difference between the angle of winding and this angular yielding at N. If the strip is everywhere of the same breadth and thickness, the unwinding tendency is proportional to the moment of inertia of its own section that is, to its breadth and to the cube of its thickness ; it is also proportional to the modulus of elasticity of the material used, and is inversely propor- tional to the total length of the strip. Suppose you wind a cord round the barrel A B or case containing a mainspring of a watch whose arbor is fixed firmly, and, using a scale- pan with weights, you find the turning moment of the spring Fig. 112. for various amounts of winding up. If you plot your results on squared paper, you will find that the points lie in a curve like A 0, BO, CO, or DO of Fig. 112, whereas for a balance spring we should get nearly a straight line through o. In Fig. 113 is represented an instrument which I have 172 PRACTICAL MECHANICS. [Chap. XVI. been in the habit of using in my laboratory, to show the connection between the turning moment and the angular winding in a flat spiral spring. Different weights used at the end of the string give different readings of the Fig. 113. pointer. By means of such an apparatus, we are enabled to verify the laws described above. When we have performed one set of experiments with a spring, another set may be made on the same spring with its length diminished or increased by means of the arrangement for clamping, shown in Fig. 130. In this way we can experiment with springs of different breadths and thick- nesses, as well as of different materials. Chap. XVI ] CYLINDRIC SPIRAL SPRING. 173 164. The flat spiral spring just considered is a case of the bending of a strip of steel along its entire length. I will now take up a case in which there is no bending. Fig. 114 shows a cylindric spiral spring* whose coils are very flat. Besides its own weight, it is acted upon by two equal and opposite forces in the direction of its axis, the supporting force at N and a weight at M. Now let us consider the equilibrium of the portion of the spring from any point p to M. Suppose the wire cut at r by a plane passing through the axis ; this section will be more and more nearly a cross section normal to the axis of the wire, as the spirals are more and more nearly horizontal. Let us regard it as a normal cross section of the wire. Now, whatever may be the stresses at this cross section, they must balance all the other forces acting on P M namely, the force F at M, which is axial, and the weight of P M, which is very nearly axial. If we neglect the weight of p M, we have only to balance the force F acting at M. To do so we evidently need a shearing force, F, at p, distributed over the section, and a twisting torque which is equal to F . p H. It is easy to show that the shear is of much less im- portance than the torsion. Again, since p H is the same for every part of the spring, every section of the wire is acted on by the same twisting couple, just as the shaft of Fig. 46, or the wire of Fig. 43, and its strength is calculated in the same way. Now, what is the amount of motion at M in consequence of this twist ? As the wire is everywhere twisted, just as if it were a straight wire fastened at one end whilst at the other end there were a force, F, acting at the * Our authority on this subject is a paper by Professor James Thomson in the Cam- bridge and Dublin Mathematical Journal, Nov., 1848. F Fig. 114. 174 PRACTICAL MECHANICS. [Chap. XVI. end of an arm whose length is equal to P H the radius of the coils of the spring, the amount of the motion of M is just the same as the motion of the end of such an arm attached to the straight wire. 165. We have, then, the following pretty illustration (Fig. 115), which serves to keep the rule for spiral springs in our memory. Let two pieces of the same wire of the same length be taken ; one of them kept straight, fixed Fig. 115. firmly at A, and fastened at B to the axis of a pulley which can move in bearings. A cord, c, fastened to the rim of this pulley, carries the upper end of a spiral spring, D E, formed of the other piece of wire, the dia- meter of its coils being equal to the diameter of the pulley. Evidently, if a weight, w, is placed in the scale-pan, a point E gets just double the motion of a point c, for E gets c's motion as well as the lengthening of the spring. The scales F and G and the little pointers are for the purpose of making exact measurements. It is interesting to note how accurately the law is fulfilled, even in a roughly- constructed piece of apparatus such as any one may easily put up for himself. 166. Exercise. A spiral spring of charcoal iron spring wire, O'l inch diameter, 21-6 inches long, its coils having a radius of l - 3 inch, is extended by a weight of 10 Ibs. Supposing that a piece of wire of the same material 1 inch long and 0*05 inch diameter, gets a twist of 24-2 Chap. XVI.] STIFFNESS OF SPRING. 175 degrees wich a cwisting moment of 2 inch-pounds, what is the extension of the spring 1 We see that if the trial wire were of twice the diameter the twist would be 24-2 -f- 1C, or 1 -51 degree, and with a twisting moment of 13 inch-pounds, which is 6 -5 times as great, the angle would be 9 '82 degrees, and on a wire 21-6 times as long would be 212' degrees, or 3 '7 radians, and the arc of a circle whose radius is 1 '3 inch, subtending this angle is 3 '7x1 -3 or 4 - 8 inches, the answer. 167. In designing a cylindric spiral spring it is very im- portant to know the greatest elongation it will bear without taking a permanent set. If the material has internal strains given to it during its manufacture and this it is very diffi- cult to prevent in steel springs, unless great care is taken in tempering, and it is almost impossible to prevent in brass springs, because the elasticity added in manufacture is often regarded as a necessary quality which ought not to be destroyed by any annealing process in this case the reader must keep in mind the considerations of Art. 96. Otherwise, let s be the greatest shearing stress per square inch which the material can resist without getting a per- manent set. Let m be the greatest twisting moment which a round wire of radius r can bear without getting a permanent set, we see from Art. 93 that m = ^ war 3 (1) Now s will be approximately known from Table III, or m may be found by experiment for a given wire by any person who wishes to make a spring ; and whether m or s is used in a formula, you now know how to calculate one when given the other and the size of the wire. If, then, we have a spring made of wire whose radius is r, and if the radius of the coils as measured to the centre of the wire from the axis of the spring is , we see that when w is the greatest weight with which the spring may be elongated without producing a permanent set, fci (2) being independent of the length of wire employed. From Art. 92 we see that if N is the modulus of rigidity of the material, A the greatest angular twist in radians which we can give to a wire of radius r inches and length 176 PRACTICAL MECHANICS. [Chap. XVI. one inch, and m the twisting moment which produces this twist, then m being what we have previously measured or calculated. N is approximately known for a material from Table III., or A may be found by experiment for a given wire; and whether A or N is used in a formula, you now know how to calculate one when given the other. Putting the result of our reasoning in Art. 164 into an algebraic form, we see that a load, w, will elongate the spring by the amount 2 w ? a* JT X I' 1 ' and hence the greatest elongation which can be given to the spring without its getting a permanent set is Combining (2) and (4), we see that when a spring is stretched to its elastic limit, the mechanical energy stored up in it, which is called its "resilience," being half the product of \v into the elongation, is J*A,OT^y or^... ...(5) 4n ' irsr* \ / 168. Many interesting methods may be taken to express in words the meanings of these results. Thus, the second expression in (5) shows that the work which we can store up in a spiral spring is simply proportional to the weight or quantity of material in it. It would be easy to show that we can store more energy in a spring formed of wire of circular section than in one of equal weight of the same material whose wire has any other than a circular section. 169. The following readings of our formula may prove to be useful : 1st. If r, the radius of the wire, and a, that of the coils, be fixed, the elongation produced by any weight, w, will be proportional to I, the length coiled up to form the spring. 2nd. If a wire of a certain length and radius be given to form a spring, the elongation produced by a certain weight, w, will bo proportional to the square of the radius which we may adopt for the coil. 3rd. If the radius of the wire be fixed, and the length of the spring when closed, so that the coils may touch one another, or, what is the same, the number of coils be also fixed, / must be proportional to a, and therefore the elongation due to a weight, w, will be proportional to the third power of the radius which we Chap. XVI.] ELASTIC STRENGTH OP SPRING. 177 may adopt for the coil. 4th. If the length of the wire and the radius of the coil be fixed, the elongation due to a weight, w, -will be inversely proportional to the fourth power of the radius of the wire which we may adopt, oth. With a given weight of metal and a given radius of the coil, the elongation due to a weight, w, will be propor- tional to I 3 , or inversely to r 6 , since I must be proportional t0 ^ We see that the ultimate elongation is 1st, proportional to the length of the wire, if the radius of the wire and that of the coil be fixed. 2nd, proportional to the radius of the coil, if the length and the radius of the wire be fixed. 3rd, Inversely proportional to the radius of the wire, if the length of the wire and the radius of the coil be fixed. It will be found that a weight hung at M (Fig. 114) will tend to turn as the spring lengthens, unless the coils of the spring are very flat. This is due to the fact that the cross sections of the wire are really subjected to a little bending as well as torsion. 170. We can cause the strain in such a spring to consist altogether of bending, if, without exerting any axial force such as I have shown in Fig. 114, we exert a couple about the axis such as we exerted on the wire in Fig. 43. The wire in Fig. 43 would be twisted, but the wire in Fig. 114 is subjected everywhere to bending without any twisting or with only a very little twisting, due to the fact that the coils arc not perfectly flat. If a is the radius of the coils to the centre line of the wire when unstrained, and the length of the coiled wire is /, then the number of coils multiplied by the circumference of each is the total length, so that the number of coils is I -r- 2 IT a . If now the moment of inertia of the cross section of the wire about the axis through its centre about which it bends is i, and if M is the moment which acts at the unfixed end of the spring to twist it, then the new radius, a, of every coil is obtained from our knowledge of the fact given in Art. 160. M = E i x change of curvature, or M = E i ( ), \a oo/' E being the modulus of elasticity of the material.* * i is b dP 4- 12 for a wire of rectangular section, d being the dimension in inches of the section, measured radially out from the axis of such a spring; I is fti 4 -s- 64 for a wire of circular section of diameter d. M 178 PRACTICAL MECHANICS. [Chap. XVI. From this we have -I !,;_! _!,-_ F. i a a ' B I a ' K i where w was the original number of windings, and n is the new number of windings. But one additional winding means that the unfixed part of the spring has moved through 360 degrees, or 2 T radians ; and hence if A is the angular motion of the unfixed end of the spring in radians, _ 2lTMl " F.I * We see that it does not depend on the radius of the coils. For a spring of round wire of diameter d, the angular motion due to a turning moment, M, is A= 128 Mf-r-Ed 4 . 171. From these considerations it is evident that a spiral spring like Fig. 115 when it lengthens under the action of a weight, has all its wire subjected to torsion. The spring itself is extended, but the wire of the spring is twisted. Again, if we subject the spring to torsion as a whole, the strain really going on in the wire is a bending strain. Usually, a spiral spring, as its coils are not perfectly flat, has its wire subjected to torsion principally, and a little bending as well, when the spring is extended ; and when the spring is twisted as a whole its wire is mainly subjected to bending, but there is also a little twist in it. The extension of a spiral spring is proportional to the pulling force, and also to the length of the wire and to the diameter of the coils ; it is inversely proportional to the fourth power of the diameter of the wire if the wire is round. The twist given to a spiral spring is proportional to the moment of the twisting forces, it does not depend on the size of the coils ; it is pro- portional to the length of wire, and inversely proportional to the fourth power of the diameter of the wire if the wire is round. Chap. XVII.] PERIODIC MOTION. 179 CHAPTER XVII. PERIODIC MOTION. 172. When, after a certain interval of time, a body is found to have returned to an old position, and to be there moving in exactly the same way as it did before, the motion is said to be periodic, and the interval of time that has elapsed is said to be the periodic time of the motion. Thus, if a body moves uniformly round in a circle, the time which it takes to make one complete revolution is called its periodic time. 173. When a body moves uniformly in a circle, as, for instance, the bob of a conical pendulum (see GLOSSARY, Art. 236), if we look at it from a point in the plane of its circle, it seems merely to swing backwards and forwards in a straight line. Thus, it is known that Jupiter's satellites go round the planet in paths which are nearly circular, but a person on our earth sees them move backwards and forwards almost in straight lines. Now if we were a very great distance away from the bob of a conical pendulum in the plane of its motion, we would imagine it to b moving in a straight line, and the motion which it would appear to have slow at the ends of its path, quick in the middle would be a pure harmonic motion. To get an exact idea of the nature of this motion in fact, to define what I mean by pure harmonic motion draw a circle AO'LO" (Fig. 116), and divide its circum- ference into any even number of equal parts. Draw the perpendiculars B'B, c'c, &c., to any diameter. Now, if we suppose a body to go backwards and forwards along AOL, and if it takes just the same time to go from A to B as from B to c, or from any point to the next, then 180 PRACTICAL MECHANICS, [Cliap. XVII. its motion is said to be a pure harmonic motion. This sort of motion is nearly what we observe in Jupiter's satellites ; it is almost exactly the motion of the bob of any long pendulum or the cross head of a steam-engine ; it is the motion of a point in a tuning-fork, or a stretched fiddle-string when it is plucked aside and set free; of the weight hung from a spring balance when it is vibrating ; of a cork floating on the waves in water; and of the free end of a rod of metal when the other end is fixed in a vice and the rod is set in vibration; it tells us in all these cases the nature of the mo- tion, when such motion is of its simplest kind. Thus, for exam- ple, a cork float- ing on water may really have a very complicated motion, but if the wave in the water is of its simplest kind, the cork goes up and down with a pure harmonic motion. If you study the figure which you have drawn, and then watch the vibration of a very long pendulum, you will learn about this kind of motion what cannot be learnt by reading. 174. Now let me suppose that the body takes one second to go from A to B, or from B to c, or from any point to the next in Fig. 116. Then the length of A B in inches represents the average velocity between the points A and B, and in the same way we get the average Fig. lie. Chap. XVII.] PURE HARMONIC MOTION. 181 velocity anywhere else. Thus, in the figure from which the woodcut is drawn I find Velocity from A to B B to c 1-00 c to 1-59 D to E I to F F to o o to a a to H H to I i to j j to K 1-00 K to L Is in inches ^ per second j" 0-34 2-07 2-41 2-59 2-59 2-41 2-07 1-59 0-34 175. You will observe that the velocity increases as the body approaches the middle of the path, and diminishes again as it goes away from the middle. Now the increase in the velocity of a body every second is called its acceleration, and I want you to observe what is the acceleration at every place. You see that the velocity changes from '34 to I'OO near B in one second that is, the acceleration near B is '66 inch per second per second. Similarly subtracting 1 -00 from 1 -59 we find the acceleration at c to be 0*59, and so on. Now make a table of these values, and place opposite them the distances of the points B, c, &c., from the centre. In this way I find from my figure the following Table of Values : Distance from o to Acceleration at Displacement divided by Acceleration. B is 9-66 B is 0-66 14-6 c is 8-66 c is 0-59 14-7 D is 7-07 D is 0-48 14-7 E is 5 00 B is 0-34 14-4 F is 2-59 F is 0-18 14-4 o is o is o o is 2-59 G is 0-18 14-4 H is 5-00 H is 0-34 14-4 i is 7-07 i is 0-48 14-7 j is 8-66 j is 0-59 14-7 K is 9-66 K is 0-66 14-6 From this it is evident that when the distance of a point from the centre is divided by the acceleration at the point, you get about 14*6 in every case 182 PRACTICAL MECHANICS. LChap. XVH. that is, the acceleration at a place is pro- portional to the distance from the centre. This curious property is characteristic of the kind of motion which I am describing. If, again, you draw a number of figures, such as Fig. 116, and divide the circles into very different numbers of equal parts, you will find that in eveiy case the following law is true : The periodic time of a pure harmonic motion that is, the time which elapses from the moment when the body is in a certain condition until it gets into exactly the same condition again is equal to 6-2832 multiplied by the square root of the ratio of displacement to acceleration given in the third column of the above Table. Thus, in the Table we find the mean value of the ratio (adding all the quotients and dividing by their number we get 14*56) to be, let us say, 14-6. Now the square root of 14-6 is 3-82, and this multiplied by 6-2*32 is 24 seconds, which we see by inspection is the periodic time in Fig. 116. The acceleration is always towards the middle point that is, whilst a body is leaving the middle, its velocity is 1 'ing lessened, when it is approaching the middle its velocity is being increased. The velocity at the middle is equal to the uniform velocity in the circle from which we imagine the harmonic motion to be derived that is, the velocity in the middle is equal to 8*1416 times the distance A L divided by the periodic time. Suppose the body to be at Q, Fig. p. .._ 117, moving with a pure harmonic motion in the path AOL. Describe the circle, draw QP perpendicular to A L, then P is the position of a body which has cor- responding uniform circular motion. The acceleration at Q is equal to the resolved part along OL of the centripetal acceleration at P in the direction P o, which is known to be v 2 -7- P o where v is the uniform velocity of p. Chap. XV1T.] PERIODIC TIME. 183 The resolved part of this in the direction Q o is evidently obtained by multiplying by a o and dividing by r o, and we get for the acceleration of a, v 2 x Qo-f-p o 2 , so we see that the acceleration of Q, is proportional to Q o. The accelera- tion at L or A, the ends of the path, is of course greater than anywhere else, being v 2 p o. If at any place, Q, we divide the displacement by the acceleration, we get Q o -~ ^-^, or p o 2 -7- v 2 , and as v is the circumference of the circle 2ir-po, divided by the periodic time T, wo have Periodic time of a pm-e\_ 3.1416 A/ harmonic motion ) V r>is P lacement i Acceleration ' We see then that if the force acting on a body and causing it to move is always proportional to the distance' of the body from a certain point, and acts towards that point, the body gets a pure harmonic motion, and we have a rule for finding the periodic time. 176. Example. In Fig. 118, A is a ball of lead weighing 20 Ibs. carried by means of a spiral spring whose own weight may be neglected, let us suppose. Find by experiment how much the spring lengthens when we adtl 1 Ib. to the weight of A 01 shortens when we subtract 1 Ib. from the weight of A. Let it lengthen or shorten 0-01 foot. Evidently, if ever A is 0-01 foot upwards or downwards from its position of rest, it is being acted upon by a force of 1 Ib. tending to bring it to its position of rest. We know also (see Art. 171) that if A is 0'02 foot or 0*03 foot above or below its place of rest, there is a force of 2 or 3 Ibs. trying to bring it back. We see then that the up and down motion of A must be pure harmonic. When the displacement is, say 0'02 foot, the force acting on A is 2 Ibs., and the acceleration of A is force 2 -j- mass of A, Fig. 118. is I PRACTICAL MECHANICS. [Chap. XVIF. and as the mass of A is 20 4- 32-2, or 0-621, the acceleration of A is 3-22 feet per second per second when it is displaced 0-02 foot from its middle position. Now employing the rule given above, divide 0*02 by 3-22 and extract the square root, then multiply by 6-2832, and we get 0-495 second, or about half a second as the periodic time of the swinging ball. If you make experiments you will find that, unless the coils of the spring are very flat indeed, and the rigid support of A exactly in the axis, the ball has a tendency to turn and to vibrate laterally, which will somewhat disturb your observations if you are making careful measurements of the length of swing. 177. Example. The Simple Pendulum. A simple pendulum consists in an exceedingly small but heavy body suspended by means of a long thread whose weight may be neglected, capable of swinging backwards and forwards in short arcs. Thus, in Fig. 119, s is the point of suspension, s P a silk thread, P a small ball of lead, p will move backward and forward along the path AOL with a motion which is pure harmonic, provided the thread is so long and A L so short that A L may be regarded as nearly straight, because the force acting on the ball at any time in the direction of its motion is pro- portional to the distance of the ball from o. To show that this is so, resolve the vertically acting weight of the ball in the direc- tion of its motion along A o. You find that it is not quite proportional to A o unless A o is very nearly straight, but if this slight discrepancy is neglected the force urging the ball towards o is the weight of the ball multiplied by o A and divided by s A, the distance from Pig. 119. Cbap. XVII.] TITE SIMPLE PENDULUM. 185 the point of support to the centre of gravity of the ball. As a matter of fact, the nature of the vibration does not depend on the weight of the ball; but, to fix our ideas, let us suppose that the weight is 2 Ibs., then the mass of the ball is 2 -j- 32-2, and acceleration along A o is the force -f- mass, or * A the force -f ^-^ the mass, so that the acceleration is 32-2 x A o-r-s A. Now our rule is to divide A o by the acceleration at A, and this gives ^^ ; extract the square root, and multiply by 6-2832 for the periodic time of oscillation of the pendulum. The general rule for a simple pendu- lum swinging in short arcs is then : fir /. 7 , . 77 .. /> noon k. I length of pendulum Time of a complete osctWatasition of equili- brium, that is, the position in which the liquid is at the same level in both limbs of the tube. Thus, if c is -01 foot below the proper level, and B is -01 foot above this level, the force which tends to cause the liquid to return to its proper level is twice the weight of the liquid o B. Sup- pose the weight of the liquid is 10 Ibs. per foot in length of the tube, then the force acting on the liquid is "02 x 10, or 2 Ib. If the whole length of tube filled 6 feet, then the weight of liquid which in motion is 60 Ibs., and its mass is Fig. 121. with liquid is has to be set 60 -5- 32-2, or 1 -863 ; hence the acceleration is -2 -5- 1 -863. Chap. XVII.] THE NATURE OP A VIBRATION. 187 or 0-107 foot per second per second. The displacement is '01, and, working by our old rule, displacement divided by acceleration is -0935. The square root of this is 3058, and multiplying by 6'2832 we get 1-92 second as the periodic time of the oscillation. You will find it easy to prove that the liquid swings in the same time as a simple pen- dulum whose length is half the total length of the liquid in the tube, and that it is the same whatever be the density of the liqui d that is, whether it is mercury or water. If w Its. is the \veight of liquid per foot in length of the tube, if d is the displacement o B or o c, the force causing motion is 2 d w. If a is the total length of liquid in the tube, the weight of liquid moved is a iv, and its mass -r-ff, if g is 32 -2, which represents the effect of gravity. Hence the acceleration is 2 d w -j- *^p or ~ 9 , and the displacement divided by acceleration is d -. 9 , or - so that the periodic time is 2 it A/ y ' y' 180. It will be observed that in all these cases of vibration of bodies there is a continual conversion going on of one kind of energy into another. At each end of a swing the body has no motion ; all the energy is there- fore potential, whether it is the potential energy of a lifted weight or the potential energy of strained material. In the middle of the swing the body is going at its greatest speed, and its energy is kinetic. At any intermediate place the energy is partly potential and partly kinetic, but the sum of the two remains always the same, excepting in so far as friction is wasting the total store. Now in time-keepers the office of the mainspring is to give just such supplies of energy to the balance as are necessary to replace the loss by friction ; and we have to ask the question At what part of the swing of a pendulum or balance can we give to it an impulse which shall increase its 1 ^ PRACTICAL MECHANICS. [Chap. store of energy without disturbing its time of oscillation? The answer is this. If a blow is given to the bob of a pendulum when it is just at its lowest point, energy is given to the pendulum ; we give it power to make a greater swing, but the time which it will take to make this greater swing is just the same as the time it would have taken for a smaller swing. This middle point is the only point at which we can give an impulse to the bob without altering the time of its swing. In the lever escapement, and in other detached escapements of watches, the impulse is always given just at the middle of the swing. CHAPTER XVIIL OTHER EXAMPLES OF PERIODIC MOTION. 181. When the periodic motion of a body is not pure harmonic, we find that by imagining the body to have two or more kinds of pure harmonic motion at the same time we can get the same result. Thus, it is known that a float, employed to measure the rise and fall of the tide by marking on a moving sheet of paper with a pencil, has a motion which is periodic and not pure harmonic. Thus, if horizontal distances represent the motion of the paper (unwound from a barrel by means of clockwork), and therefore represent time, and if vertical distances ,_. ^-^ mean the rise or fall of v s \ / \ water-level in feet, we \^/ \^_y ^ get such a curve as is Fig. 122. shown in Fig. 122. Now this is not a pure harmonic motion, for if you plot on squared paper the distances o A, o B, o c, o D, o E, OF, &c. (Fig. 1 1 6), for equal intervals of time, you will get a curve like Fig. 1 23, Chap. XVHI.] THOMSON'S TIDE PREDICTEB. 189 which is easily recognised, and is called a curve of sines. But it has been found that if you take certain curves of sines whose periodic times are 1, the semi-lunar day ; 2, the semi-solar day, and some others, and draw them on squared paper, and add their ordinates together, you will get the curve shown in Fig. 122. In the very same way you can com- bine pure harmonic motions to arrive at any periodic motion. A good way of combining pure harmonic motions experimentally a body hang string which Fig. 123. let is to from passes over two or more movable, and the same number of fixed, pulleys. These pulleys are pivoted on crank pins, and their pivots are made to re- volve at any desired rela- tive speeds, and each gives to the body a, pure harmonic motion by its action on the string. The body gets a motion compounded of the mo- tions of the pulleys, and if it is an ink-bottle or pencil pressing on the paper on a revolving paper roller, we get a time curve of the periodic motion. This is the prin- ciple of the construction of Sir William Thomson's Tide Predicting Machine. Fig. 124. 190 PRACTICAL MECHANICS. 182. When a body can swing east and west under the influence of forces which have no tendency to move it except in a direction due east and west, and if forces act- ing due north and south can make it swing in their direction, then both sets of forces acting together on the body will give it a motion compounded of the two simpler motions. Thus, a ball A .(Fig. 124) is suspended by a string, PA, which is knotted at p to two other strings, PS and PS', equal in length, and fastened at s and s'. The ball may swing in the direction E o w as if it were the bob of a pendulum hung directly from the ceiling at p', but it may also swing in the direction N o s at right angles to E o w, and if it does so it swings as if the point P were the fixed end of the pendulum A p. When it swings under the influence of the two sets of forces tending to make it move both ways at once, the motion of A is compounded of the other two simpler motions. If p A is one-quarter of the length o P', then the east and west swing takes J * 7 S e twice as long as the ~ north and south swing. If p A is one-ninth of OP', then the east and west swing takes three times as long as the north and south swing. The motion of A is sometimes very beauti- ful, and the experiment is easily arranged. 183. The motion is quite easily represented on paper. Thus, in Fig. 125, A' M is the north and south direction, and A M, at right angles to it, is the east and west direction. Let the points 0, 1, 2, &c., in each of these lines be found as in Fig. 116. Let the bob be supposed to go from to 1 in A'M in the same time as Tig. 125. Chap. XVIII.] COMBINATION OF VIBRATIONS. 191 it oes from to 1 in A M. You will observe that I have twice as many points in A M as in A' M, showing a slower oscillation in the direction A M. You can begin to number ct-oooooo ccooooocc~ -1 !c 2 00 00 O O (5 O Fig. 126. your points anywhere, remembering that when the bob completes its range it comes back again in the opposite direction. Now put marks where the east and west lines meet the north and south ones, drawn through corresponding points. It is evident that the curve drawn through these successive marks is the real path traced out by the ball when 192 PRACTICAL MECHANICS. [Clmp. XVIII. acted upon simultaneously by the two sets of forces urging it in a north and south, and an east and west direction. If you have the same number of points in A' M as in AM, you will get a circle, ellipse, or straight line, as in A, B, c, Fig. 126. This represents the motion of a conical pendulum free to swing in every direction. Again, D, E, F, and many other curves that might be drawn, represent the case which I took up in Fig. 125, where one vibration is twice as quick as the other. If the time of vibration in AM is to the time of vibration in A' M as 2 to 3, we get curved paths like G, I, J, and so on. In experimenting with the pendulum, Fig. 124, it will usually be found that slight inaccuracies in the lengths Pig. 127. of the cords will cause a continual change to go on in the shape of the path traced out by the ball We can produce these motions by spiral springs, and in other ways. Thus, for example, if we use instead of the strip of steel, in Fig. 1 20, a combination of two strips, B and B', as in Fig. 127, so that the heavy bright bead A is capable of vibrating in two directions at the same time, you will get the same combinations of pure harmonic motions, depending on the point at which B is held in the vice c. 184. When a body has a periodic rotational motion about an axis like the balance of a watch or a rigid pendulum, we must no longer speak of the force causing motion, and the mass of the body, and the distance of displacement ; but if we substitute for these terms, moments of forces, moment of inertia of the body and angle of displacement, we have exactly the same rule for finding the periodic time of oscillation. The periodic Chap. XVIII.] ROTATIONAL PERIODIC MOTION. 193 time is 6 '2832 times the square root of the angular displacement of the body at any instant, divided by the angular acceleration at that instant. And \ve know that angular acceleration may be calculated by dividing the turning moment acting on a body by the moment of inertia of the body. A point in the balance of a watch swings in circular arcs, but if you only take account of the distances which it passes through, and suppose it moved in a straight line instead of in the arc of a circle, the motion is very nearly pure harmonic. If there were no friction or other forces acting on the balance except the turning moment of the balance spring (see Arts. 162-3), and if the moment of the spring were always exactly proportional to the angular dis- placement of the balance, the motion would be pure harmonic. We saw in Art. 161 that the turning moment of the spring is ^-07 A, if B is modulus of elasticity of the spring, b its breadth, t its thickness, and I its length, and if A is the angalar displacement in radians. Angular acceleration is this moment divided by moment E ?> t^ A of inertia i of the balance, or -J-ST ~- Hence, angular displacement A, divided by angular ;eleratioi balance is 1 O 7 T acceleration, is r-p> so that the periodic time of the T = 6-2832 Increasing the moment of inertia of the balance or the length of tJie spring makes the vibration slow. Increasing the breadth and, what is still more important, increasing the thickness of the spring makes the vibration quick. As we saw in Arts. 162-3 that our calculation of the turning moment of the spring is not quite right, that the dimen- sions of the balance and spring alter with temperature, and that above all the elasticity of the steel alters with temperature and with its own state of fatigue, the 194 PRACTICAL MECHANICS. [Chap. XVIII. rule given in the note is not perfectly true, nor can any balance be regarded as taking exactly the same time for its oscillation in different lengths of arc. At the same time it is of great help to the watchmaker to know that with considerable, although not with perfect ac- curacy, the time of vibration of a balance is proportional to the square root of the length of the spring, and so on. For example, suppose the spring is 3 inches long, and the balance makes one swing in 0-251 second, now if he- wishes it to make a swing in 0'25 second, he must shorten it in the ratio of -251 x '251 to '25 x '25, or in the ratio -063001 to -0625, so that the length of his spring ought to be 3 x -0625 -f -063001, or 2-976 inches that is, it ought to be shortened -024 inch. In the same way he can calculate the effect of adding little masses at any distances from the centre of the balance, so that its moment of inertia may be increased, and the balance made slower in its swing. The same law tells him how he can compensate the balance, so that when in summer the steel of the spring loses its elasticity, some of the mass of the balance will come nearer the centre, in order that the moment of inertia may diminish the same proportion. 185. Compound Pendulum. The simple pendulum described in Art. 177 is not like the pendulums used in practice. In these the bob is not so small that we can consider it as a point; the long part is not a thread but a stiff rod of metal or wood, and there is usually a knife-edge for support, about which it can turn with little friction. In common clocks, however, the top end of the pendulum is a thin strip of steel held firmly in the chops, but the easy bending of this strip is such that we may imagine the pendulum to move freely about an axis. Employing our general rule of Art. 184 we find how to calculate the time of vibration. This compound pendulum vibrates in the same time as a certain simple pendulum, called the equivalent simple pendulum, whose length you ought to find by experiment. Chap. XVIII.] COMPOUND PENDULUM. 195 In Fig. 128 let s be the axis of suspension, G the centre of gravity, and p a point in the continuation of the line s G such that s P is the length of the equivalent simple pendulum. Then p is called the centre of oscillation, and it is also known to s ( be the centre of percussion of the pendulum (see Art. 200). It can be proved that if the pendulum be inverted and made to vibrate about a parallel axis through p, it will vibrate in exactly the same time as it does about s; and it was in this way, by inverting a pendulum which had two knife-edges, and adjusting these until the pendulum took the same time to vibrate about one as about the other, and then measuring the distance be- tween them, that Captain Kater found the length of the simple pendulum which vibrates in a given time. This method is still employed in gravitation experiments everywhere to find the value of g which is 32'2 feet per second per second at London. If s is the axis of suspension, o the centre of gravity, w the weight of the pendulum, then the moment with which gravity urges the pendulum to return to its position of rest is \v x G N, but if the angle G s o be measured in radians, and if it is very small, this moment is almost exactly equal to w x s G x angle G s N. The angular acceleration is obtained by dividing this by i, the moment of inertia of the pendulum about s, and our rule becomes Fig. 128. = 6-2832 or T= 6-2832 / angli /V/ w S G an /T /y/ w-s gle G s N-J-I (1) calculations being made in pounds and in feet. When we examine this formula we see that it may be put in another form. Find a point K, such that if all the mass 1D6 PRACTICAL MECHANICS. [Chap. XVIIL of the pendulum were gathered there, its moment of inertia about s would bo the same as at present ; in fact, such that ^ the mass of the pendulum x 8 x s would be equal to i. The distance 8 K is called the radius of gyration of the pendulum (see Art. 142), and our rule now becomes, T= 6-2832 Y/J.S where g is 32-2. In the simple pendulum, 8 K and s o are equal, and if you make them equal you will find this to be the same rule which is given in Art. 177. However, in an ordinary pendulum, s K and s o are not equal, but s K S -T-S o is equal to some length such as s r, and our rule becomes = 6-2832.4 I (3) Evidently s P is the length of the imaginary simple pendulum which would vibrate in the same time as our real pendulum. The imaginary point P has been called the centre of oscillation, because when the pendulum is inverted and made to vibrate about an axis through P it vibrates in the same time as before.* * To prove this it is necessary to return to equation (1). We know that I is equal to the moment of inertia of the body calculated as if all its mass existed at o, together with the moment of inertia of the body as it is at present, but calculated about an axis through o parallel to the present axis that is 9 where k ia some length unknown to us just now, being the radius of gyration about the axis through the centre of gravity. Rule (1) becomes T = 6-2832 V^ HJ w-s c V80+ -T- ._ . SO or T = 6-2832 That is, the length of the simple pendulum which will vibrate in the same time is 8 o+ 8 - and we. have already found it to be s P in equation (3), so that GP= k * or G P x S O_= i j . But in the very same Chap. XVIII.] CONSTRAINT OP SPIRAL SPRING. 197 186. Examples. The bar of Fig. 129 with two adjust able masses may be hung at one end of a wire, the other Fig. 129. end of which is fixed to the ceiling. By twisting and untwisting the wire the bar will oscillate with a motion way, if we considered the pendulum as vibrating about P, we should find the length of the equivalent simple pendulum to be greater than M G P by an amount equal to , and we know that s G is equal to this amount, so that s P would as before be the length of the equivalent simple pendulum. The axes of oscillation awl suspension are therefore interchangeable, 198 PRACTICAL MECHANICS. [Chap. XVIII. which is much more nearly pure harmonic than that of the balance of a watch. My students experiment with such a bar ; they can adjust the weights A and B at any Fig. 130. distance from the axis (there is an engraved scale on the bar), so that the moment of inertia can be varied. They can fasten the bar at the end of a wire, or they can use it as in Fig. 130, with a flat spiral spring, or as in Fig. 129, with a cylindric spiral spring; and the rate of its vibration gives one of the best ways of investigating the Chap. XVIII.] CONSTRAINT OF TWISTED WIRE. 199 twisting moments of wires and such springs when strained through given angles. In the case of a wire the twist always tends to bring the bar to its position of rest with a moment which is proportional to the angle of displacement from this position it is this property which causes the motion to be pure harmonic. This moment is also proportional to the fourth power of the diameter of the wire, and it becomes less as the length of the wire is increased. By means of a circular scale and a pointer we can measure the extent of each swing, and this is found to decrease gradually, due to friction with the air and the internal friction or viscosity of the metal. The amount of diminution of swing gives us a means of determining the viscosity, and the apparatus can so easily be fitted up, that no person who wishes to understand the pro- perties of materials can be excused from making these experiments. If the length of the wire is I inches, its diameter d, and if N is its modulus of rigidity (see Table III.), then from Art. 92 we see that the moment with which the wire acts on the bar, when its angle is A from the position of rest, is If the moment of inertia of the bar is i (we are neglecting the fact that the wire itself has some mass which has to be set in motion), then the moment, divided by i, is the angular acceleration, and using this quotient as denominator and A as numerator, extracting the square root, and multiplying by 6-2832 or 2w, by the general rule of Art. 184, we find the square of the period of a complete oscillation to be When motion is slow, the friction in fluids is propor- tional to the velocity, and any friction which follows this law is called fluid friction. A great many vibrating bodies tend to come to rest by the action of such friction as this ; and it is found that if the friction is numerically /times the angular velocity, then the logarithm of the ratio of the length of one swing from the middle position, to the next swing in 200 PRACTICAL MECHANICS. [Chap. XVIII the same direction, is nearly equal to k times the periodic time. Hence, this logarithmic decrement, as it is called, is proportional to the friction co-efficient. If we observe twenty-one elongations on one side of the middle position, then one-twentieth of the logarithm of the first elongation divided by the last, is k times the periodic time of oscilla- tion. Exercise. Bifilar Suspension. In many measuring instruments a body is suspended by two thin wires nearly vertical. If the vertical length of each of these is I, the distance between their ends at the top, a, and at the bottom, b, and the weight of the body, to, it is easy to show that for a small angular displacement, A, the moment tending to bring the body to its position of rest is very nearly (neglecting tension of the wires themselves) Find the time of vibration of such a body when its moment of inertia is known. Exercise. A magnet, turning on a frictionless pivot at its centre of gravity, is subjected to a turning moment, HA, due to the earth's magnetic action, if it makes only a small angle, A, with its position of rest. Find the time of a vibration if the moment of inertia is known, and show that the square of the time of vibration of the magnet in different places is in- versely proportional to H. 187. Stilling of Vibrations. When a pure harmonic motion is represented on paper in the manner described in Art. 181, we have a curve of sines. The curve may be obtained by producing the lines BB', cc', &c., of Fig. 116, cutting them at right angles by equi-distant hori- zontal lines and joining the successive points of inter- section so found. It may also be drawn by finding from a book of tables the sines of 0, 10, 20, f!<' energy must be 25,763, and evidently this, multiplied by 64 '4 and divided by 60, gives the square of the new velocity, which I find to be 168'1 feet per second. Evidently in such a question we are not concerned with the direction in which the body is moving. It may be a cannon-ball, or a falling or rising stone, or the bob of a pendulum. Given its velocity and height at any instant, we can find for any other height what its velocity must be, or for any other velocity what its height must be. Chap. XXI.") ANGULAR VELOCITY. 235 230. Angle. An angle is the amount of opening between two straight lines. An angle can be drawn : First, if we know its magnitude in degrees. A right angle has 90 degrees. Second, if we know its magnitude in radiam. A right angle contains 1-5708 radian. Two right angles contain 3-1416 radians. One radian is equal to 5 7 '29 5 8 degrees. One radian has an arc, BC, equal in length to the radius A B or A C. It sometimes gets the clumsy name " a unit of .circular measure." Third, we can draw an angle if we know either its sine, cosine, or tangent, &c. Draw any angle, B A c (Fig. 137). Take any point, p in AB, and draw PQ at right angles to A A c. Then measure P Q and A p in inches and decimals of an inch. Fig. 137. p Q -r A P is called the sine of the angle. A Q -f A P is called the cosine of the angle. p Q -r A Q is called the tangent of the angle. Calculate each of these for any angle you may draw, and measure, with your protractor, the number of degrees in the angle. You will find from a book of mathematical tables whether your three answers are exactly the sine, cosine, and tangent to the angle. This exercise will impress on your memory the meaning of these three terms. Divide the number of degrees in an angle by 57 '2958, and you find the number of radians. Suppose we know the number of radians in the angle BAG, and we know the radius A B or A c, then the arc B c is A B x number of radians in the angle. Given, then, a radius to find the arc, or given an arc to find the radius, are very easy problems. 231. Angular Velocity. If a wheel makes 90 turns per minute, this means that it makes 1-5 turn per second. But in making one round any radial line moves through the angle of 360 degrees, which is 6-2832 radians; so that 1-5 round per second means 6-2832 x 1-5, or 236 PRACTICAL MECHANICS. fCliap. XYT. 94248 radians per second. This is the common scien- tific way in which the angular velocity of a wheel is measured ; so many radians per second. If a wheel makes 30 rounds per minute, its angular velocity is 3'141G radians per second. One round is the angular space traversed in one revolution. The real velocity in feet per second of a point in a wheel is equal to the angular velocity of the wheel multiplied by the distance in feet of the point from the axis. 232. Angular Acceleration. The acceleration to angular velocity per second. If a wheel starts from rest, and has an angular velocity of 1 radian per second at the end of the first second, its average angular ac- celeration during this time is 1 radian per second per second. 233. Comparison of Linear Motion and Angular Motion. The mass of a body is its The moment of inertia of a weight divided by 32 '2. wheel] or other rotating body is found by taking the mass of each portion of it and multi- plying by the square of its distance from the axis. A linear motion is given to To produce angular mo- a body when an unbalanced tion that is, rotation it is force acts upon it. necessary to have an un- balanced force acting at a dis- tance from the axis of rotation. Force multiplied by perpen- dicular distance from axis is called the turning moment of the force. Acceleration of a body is Angular acceleration of a equal to force -=- mass. body moving about an axis is moment of force -f- moment of inertia. If a f orco continually acts on If a turning moment con- a body, the velocity is equal tinuallyacts on a body (as by Chap. XXI.] LINEAR AND ANGULAR MOTION. 237 to acceleration multiplied by time from rest. Also Space passed over is equal to half acceleration multiplied by square of time. Energy stored up in a body is half its mass multiplied by square of its velocity. If a body moves backwards and forwards under the action of' a variable force which is always proportional to the dis- tance of the body from its middle position, and which always acts towards this posi- tion, and if the force at a dis- tance of one foot is 5 Ibs., then the time of vibration is equal to 3-1416 times the square root of the quotient of the mass of the body divided by 5. If a force of 20 Ibs. acts on a body through the distance of 3 feet in the direction of the force, the work in foot-pounds done by the force on the body is equal to the force 20 multi- plied by 3, or 60 foot-pounds. If, then, a body receives power, say like a carriage, by a force acting on it in the direction of motion, the horse- power received is equal to the force in pounds multiplied by the distance in feet passed a cord wound very many times round the axle of a wheel with a weight at its end), the angular velocity is equal to the angular acceleration multiplied by the time from rest. Also The angle described by the wheel in any time is equal to half the angular acceleration multiplied by square of time. Energy stored up in a wheel is half its moment of inertia multiplied by square of angular velocity. If a wheel vibrates about its axis under the action of a spiral spring or twisted wire, so that the torque is always proportional to the angular displacement of the wheel from its mean position, and if the torque is 5 pound-feet when the wheel is 1 radian from the mean position, then the time of a vibration is equal to 3 - 1416 times the square root of the quotient of the moment of inertia of the body divided by 5. If a torque is 30 pound-feet, and it turns a wheel through the angular distance of three radians, the work in foot- pounds done upon the wheel is equal to the torque 30 multi- plied by 3, or 90 foot-pounds. If, then, a body receives power, say through a shaft, the horse-power received is equal to the turning moment in pound- feet acting on the shaft mul- tiplied by the angle in radians described in one minute (or the 238 PRACTICAL MECHANICS. [Chap. XXI. through in one minute divided by 33,000. The mass of a body multi- plied by its velocity is the momentum possessed by the body. A force multiplied by the time during which it acts in hasten- ing or stopping the motion of a body is equal to the momentum produced or destroyed, If a body has momentum represented in direction and amount by the line OP (Fig. 138), and if a force acting in the direction OQ produces a change of momentum repre- sented by the length of OQ, then OB. is the resultant mo- mentum in magnitude and direction possessed by the body after the operation of the force. Fig. 13a If forces represented in direction and magnitude by the lines o r and o Q act on a body, their action is the same as that of a force represented in direction and magnitude by the line o R. number of rounds per minute x 6-2832), divided by 33,000. The moment of inertia of a body multiplied by its angular velocity is its moment of mo- mentum. A torque multiplied by the time during which it acts in hastening or stopping the rota- tion of a body is equal to the moment of momentum pro- duced or destroyed. If a rotating body's axis is in the direction o p, and if its moment of momentum about this axis is represented by the length of o p, and if forces act upon it so as to turn it about an axis whose direction is o Q, and if the amount of moment of momentum produced by the torque is represented by the length of o Q, then the resultant motion of the body is a rotation about the axis OR, its moment of momentum being represented by the length of R. (The arrow-heads in this case mean that an eye at o sees that the rotations are in the same sense that is, all against the direction of the hands of a watch, let us suppose.) If torques act on a body, if one of these is about an axis in the direction o p, and if the amount of the torque in pound-feet is represented by the length of the line o p, and if o Q similarly represents the torque about an axis in the direction o Q, their combined action is the same as that of a torque about an axis in the Chap. XXI.] CENTRIFUGAL FORCE. 239 direction OR, the amount of the torque being represented by the length of the line OR. (The arrow-heads have the same meaning as in last case.) The change of momentum The change of moment of produced in a body which re- momentum produced by an ceives an impulse is equal to impulse is equal to the sum the sum of the products of the of the products of the moments pressures during the impulse, of the pressures during the im- each multiplied by the time pulse, each multiplied into the during which it acts. time during which it acts. 234. Centrifugal Force. If a body is compelled to move in a curved path, it exerts a force directed out- wards from the centre, and its amount in pounds is found by multiplying the mass of the body by the square of the velocity in feet per second, and dividing by the radius of the curved path. Thus a weight placed at the end of an arm like the arm of a wheel exerts a pull in the arm. If a body moves round an axis 20 times per minute in a circle whose radius is 3 feet, you can determine the centrifugal force by first finding the velocity of the body and using the above rule, or you may proceed as follows : The weight of the body multi- plied by 3 multiplied by the square of 20 divided by 2,937 is the centrifugal force.* Suppose a wheel, whose total weight is 20 tons or 44,800 Ibs, has its centre of gravity 0'4 foot away from the axis that is, suppose the wheel to be somewhat eccentric then if the wheel makes 50 revolutions per minute, the centrifugal force is 44,800 x 0'4 x 2,500 -f 2,937, or 15,253 Ibs. that is 6-81 tons. This force acts on the bearings of the shaft, always in the direction of the centre of gravity of the wheel (see Chap. XX.). 235. Any one who wants to get clear ideas about vni& "Centrifugal force = , or ma?r, or wrn 2 + 2,937; where m is mass of body, or w its weight in pounds ; r, radius of curved path ; v, velocity in feet per second ; a, angular velocity in radians per second ; and n, number of revolutions per minute. Chap. XXI.] EXPERIMENTS OX CENTRIFUGAL FORCE. 241 centrifugal force ought to make experiments of his own. Unfortunately, although there are many toys made to illustrate the effects of centrifugal force, I know of only one p4 ALANCE of watch, how compen- sated, 194 Motion of, 193 Rule for periodic time of, 193 weights on wheels of locomotive, 221 Balancing of machines, 218, 219 258 Balancing of locomotive, Eulee for, 222 Ballistic pendulum, 217 Beam, Behaviour of when loaded, 101 Deflection of, 116, 128, 145 Table of values, 116 Formulae for, 122, 123 fixed at ends, 113, 119 flanges to resist bending moment, 54, 106, 115 loaded in various ways, 116 119 Methods of supporting, 113 of uniform strength, 119 Rule for breaking load on, 120 Strength of similar, 134 Bearings of shafts, Friction at, 12 Pressure on, ilue to centri- fugal force, 218 Beech timber, 78 Behaviour of materials when over- strained, 63, 100, 193 Belt, Difference of pulls in, 32 Horse-power transmitted by, 32 Bending, 109 and twisting, Eolation between, for cylindric shafts, 112 moment, 103 at section of beam, 104, 106 Diagrams of, 144, 146 How to draw, for loaded beam, 114. Table with, 116, 118 in a spiral spring, 168 Iron flanges of beams to resist, 115 of india-rubber beam, 101 Bifilar suspension, 200 Blow, 228 Effect of, 206 Motion produced by, 211, 216 Bodies, falling, Laws of, 39, 40 Boilers, Rule for bursting pressure Boilers, Strength of spherical, 60 Boiler-plates, Eiveted, 60, 88 Brags, Composition and use of, 85 Breaking load for beams, how calcu- lated, 120 struts or columns, Rules for, 131, 133 Bricks, Manufacture of, 72 Characteristics of good, 72 Bronze, Comi>osition and use of, 85 Bulk, Modulus of elasticity of, K, 57. .> Bullet, Velocity of, how measured, 218 Buttresses, 165 /CANDLE and board experiment, J 213 Carbon in cast iron, 80 steel, 84 Cose hardeniug of iron, 84 _s, 80 Chilled, 83 Malleable, 83 The cooling of, 81 Cast Iron, 80 S3 beam, Flanges in, 54, 115 Closeness of structure of, 66, 80 Effects of carbon in, 80 Factors of safety for, 121 Grey, 80 Modulus of elasticity of bulk, K, for, 57 Strains in, due to inequality in rate of cooling, 82 Strength of, 68, 69, 131 Struts of, 54, 130 Toughened, 83 White, 80 C'cdiir timber, 78 Cement, 72 Centre of gravity, 252 graphically determined, 141 INDEX. 259 Centre of gravity of section of bent beam is in the neutral axis, 105 of wheels should be in axis of rotation, 219 Ordinary formula for, 142 Centre of percussion, 195 how found, 217 Centres of oscillation and suspen- sion, interchangeable, 197 Centrifugal force, Apparatus for ex- perimenting on, 240, 242 Definition of, 239 Effects of, on bearings of shafts, 218, 239 Rule for, 220, 239 Chain, Loaded, How to draw the curve in which it hangs, 160, 161 Chemistry, Useful, 70 Circle, Pitch, of spur wheel, 27 Coefficient of friction, 35, 244 Columns, 53, 54 breaking stress for, Table of, 131 Gordon's rules for strength of, 131, 133 Hollow cylindric, of cast iron, 54,59 Long, break by bending, .54, 130 Mode of breaking varies with length, 130 Combination of pure harmonic mo- tions, 189, 192 Comparison of laws of fluid and solid friction, 251 linear motion and angular motion, 236, 239 Compensation balance of watch, 194 pendulum, 185 Compound harmonic motiois, curves of, and how plotted, 190 191 , how produced, 189, 190, 192 pendulum, 194 Kule for time of vibration. 195 Compression of struts or columns, 53, 58, 130 loaded beam, 102, 129 Concrete, Composition of, 73 Conglomerates, 71 Conical pendulum, Method of ex- perimenting with, 242 Motion of, 179, 192 Connecting-rod and crank, Motions of, 28 30 Centrifugal force of, 220 Constraint of spiral spring, 197 twisted wire, 199 Conversion of energy, continual, 40, 187 Cooling of castings, 81 83 Copper, Alloys of, 85 Properties and use of, 85 Cord, friction between post aud, 3J 35 Core iised in castings, 81 Correction of errors in experiments, 10 Couple, Definition of, 232 equivalent to a system of forces, 140 Couplings, 31 Dynamometer, 31, 36 Crane, Efficiency of, 16 Friction in, 15 Law for, 17 Crank and connecting-rod, 28, 220 travel of piston, 233 Curvature, 223. Change of, in a spiral spring, 167 of bent beam, or strip, 108 Curve, Elastic, 109 of loaded chain, 160 sines, how drawn, 189, 200 interpreted, 200 traced by pendulum bob, 201 Curves showing elastic limit o/ loaded beams, 55, 128 260 IKDEX, Cutting and chipping, 65, 207 Cycloidal teeth give uniform motion, 27,28 T)AMPED vibrations, Law for, 201 Damping of vibrations, Investiga- tion of, 201 Representation of, 204 Decimals, Necessity for student to know, 1 Deflection of beams, 122128 Curves showing, 127 Examples of, 123, 125 Formula for, 122, 123 loaded and supported in various ways ; table of relative values, 116118 method of measuring, 124 of different materials, table, 121 whose sections are not rect- angular, 126 similar beams similarly loaded, 184 Diagram of bending moment, 144 Table of, 116-118 Link polygon, 164 Examples of stress, 155 Differential pulley-block, Mechani- cal advantage of, 19 Diminution of swing, Bate of, a measure of viscosity, 199, 205, 249 Discharge of water from orifices and pipes, 76, 77 Discs, Experiments with, 205, 249 Distribution of load on an arch, 162 Drawing, Experience to be gained by, 28 instruments, necessary for stu- dent, 1 Drawings, Skeleton, 28-30 Dynamometer coupling, 31 Dynamometers, Transmission and absorption, 35 38 T^ YOUNG'S modulus of elasti- ' city, 54, 69, 92 Earth, Pressure of, against a wall, 73,74 Eccentric-rod, 29 Edge, Re-entrant, ought to be rounded, 82, 99 Effect of friction, 6 a blow, 206 Efficiency of machines, 16 Elastic curve, 109 how drawn, 110 Examples of, 111 Elastic strength, 55, 59, 62 Elasticity affected by state of strain, 62, 63, 99, 100 Law of, 55, 128 Limits of, 55, 100, 128 Modulus of, of bulk, K, 57, 58, 92 Table of values, 57 Electricity, Some knowledge of, useful, 70 Elm timber, 78 Ends fixed, Strength of beams with, 113, 119, 120 Columns with, 133 Energy communicated by a blow, how calculated, 216 Communication of, dependent on shape of body, 212 Continual conversion of, 40, 187 Indestructible, 6, 39, 234 in pile-driver, 208 rotating body, 4250, 237 how determined, 45 waterfall, 6 Kinetic, 38-50, 75, 234, 237 INDEX. 261 Energy, Loss of, due to friction, H 14 Potential, 38-40, 75, 234 Storage of, during impact, 212 Store of, in water, 75 wasted in fluid friction, Rule for finding, 76 Engine, Locomotive, balancing of, 220 Steam, fly-wheel for, 50 giving out energy, 26 Indicated horse-power of, how found, 233 Equilibrant of two or more forces, 137,229 Equilibrium of forces, 3, 139, 159, 229 Equivalent simple pendulum, 194 Escapements, 188 Experience gained by making skele- ton drawings, 28 Experiments, Necessity for, 3 Extension, 51 58 of part of loaded beam, 102, 108 of similar and similarly loaded rods, 134 of spiral spring, experiments, 174 wrought iron tie-rod, example, 53 produced by load suddenly ap- plied, 63 proportional to load which pro- duces it, 52 TRACTOR of safety, 120 Table giving usual/values of, 121 Falling bodies, Laws of, 39 40 Fatigue of materials, 63, 100, 193 Figures, Properties of straight- line, 146, 147 Reciprocal, 148 Firwoods, 77 Fixed ends, Beams with, 113, 119 Struts with, 133 Flanges in beams and girders, 51, 106, 115 Flat plates, Strength of, 133 Flow of water in pipes and pumps, 75,77 from an orifice, 76 Fluid friction, 75, 199, 247251 Apparatus for investigating, 248 Laws of, 251 pressure, 59, 61, 74, 75, 111 Arches to withstand, 111 Fly-wheels, Calculation of the size of, 48 50 Foot-pound, 12, 232 Force, Centrifugal, 218, 239 how represented, 136 Moment of a, 23, 229 polygon, 137, 139 Shearing, in beams and girders, 104, 114, 129 Forces acting at a point, 136 140, 159,228 on arbor of spiral spring, 167 Equilibrium of, 3, 137, 139, 229 Polygon of, 4, 137, 229 Triangle of, 3, 229 Friction and abrasion, 247 a passive force, 4 5, 6, 243 apparatus, 245 at bearings of shafts, 12 14 between cord and post, 3335 Coeflicient of, 35, 244 Effect of, as distinguished from force of, 6, 11 Experiments on, 710 force of, 11, 13 in fluids, 75, 199, 247, 251 measured by heavy discs, 205,249 machines, 5 17 parallel motion, 13. 262 INDEX. Friction in quick-moving shafts, 14 Kinetic less than statical, 246 Law of, 4, 10, 15 Laws of fluid and solid, com- pared, 251 Loss of energy due to, 1117, 89, 247 never negligible, 44 of metals on metals, 246 wood, 247 often useful, 247 proportional to pressure, 11, 243 wheels, 13 QATES used in casting, 81 Geology, Principles of useful, 71 Girders (see Beams) Gloss, Composition and properties of, 79 Toughened, 80 Graphical statics, 135166 Mr. Bow's notation in, 150 Gravity, Intensity of, how found, 195 Gun metal, 85 Gun, Recoil of, 209 Gyration, Radius of, 144 Gyrostat, 215 TTARMONIC motion, Pure, 29, 179 acceleration at any point, 181 Examples of, 180 of liquid in U tube, 186 spiral-spring weighted, 183 periodic time, Rule for find- ing, 182 Representation of, 180 Harmonic motion, Pure, velocity at any point, 180 motions, How to combine, 189 190 plotted, 190 Heat, Elementary principles of use- ful, 70 Heavy disc vibrating in fluids, 205, 249 Hinged structures, 146, 149 Calculation of stresses in, 149 Horse-power, 16, 233 of steam-engine, how indi- cated, 233 transmitted by belt, 32 shafts, 2630, 237 toothed wheels, 133 through coupling. Low measured, 31 Hydraulic press, 25, 26 Hydrostatic arch, 111 TMPACT, average force of, 20j mean pressure during, 211 mutual pressure during, 208 of two free ivory balls, 211 Total momentum unaltered by, 208 Impulse and blow, 206, 20'i, 239 Inclined plane, 17, 18, 20 ludia-rubber beam, Bending of, 101 Indicator diagram, 233 Inerti i, Moment of, 46, 103, 236 for rectangular or circular section, 107 greatest and least for any area, 14 1 Internal strains due to contraction in cooling, 7983 at sharp corners, 82 Iron, Annealing of, 83, 84 Cast, 8088 INDEX. 263 Iron, Wrought, Varieties and pro- perties of, 83, 84 Isochrouism of spiral springs, 170 JOINTS, Effect of stiffness of, 156 masonry, Middle third of, 130, 163, 166 of arch, Link polygon nearly nor- mal to, 163 structures, Stresses at, 149 riveted, Strength of, 88 Treenails in wooden, 78 TRILLING wire, Meaning of, 62 ' Kinetic energy, 3850, 234, 237 converted into potential, 40, 187 stored up in any machine, 50 in water, 75 friction apparatus, 245 T ADDER, Forces acting on a, graphically determined, 141 Larch timber, 77 Law for a machine, 17 of moments, 23, 230 applied to stresses at section of loaded structure, 158 work, 17, 23, 231 Laws connecting variable things, how found, 7, 10 of falling bodies, 39, 40 of friction between solids, 251 fluid friction, 251 Level surface, Definition of, 223 Lever, 23-20, 232 Limestones, Compact and granular, 71 Limestones, Pure and hydraulic, 72 Limits of elasticity, 55, 100, 128 Linear motion and angular, com- pared, 2362:59 Lines, and what they may represent, 136 Link motion, 30 polygon, 137 a diagram of bending moment 164, for arch, Fuller's method of drawing, 164 for minimum thrust at crown of arch, 164 must cut joint of arch nearly normal, 163 pole of, 138 Liquid in a U tube, Harmonic motion of, 186 Time of vibration of, 187 Load, Breaking, Eule for finding, 120 carried by an arch, how distri- buted, 162 Live or dead, 113, 121 proportional to strength modu- lus, 106, 126 Similar, on similar structures, 134 Loaded beam, Bending moment in, 114 chain of suspension bridge, 160, 161 links, Stresses in, 159 Locomotive engine, Balancing of, 220222 Considerations in designing, 220 TIT OF A 1'LY-WHEEL, 44-49 of any machine, 50 ratio of values for similar wheel?, 264 INDEX. M, Use of, in designing fly-wheels, 48 Machines, Balancing of, 218 Efficiency of, 16 in box, 5 Law for, 17 Mechanical advantage of, 1728 Steadiness of, 48 Magnet, Time of vibration of, 200 Mahogany timber, 78 Mainspring in time-keepers, Office of, 187 Marble, Formation and character of, 71 Masonry arch, 162 Mass, Definition of, 223, 236 Reciprocating, in locomotive, 220 Materials, Behaviour of, when over- strained, 62, 63, 99, 193 Disposition of, in beams or girders, for strength, 106, 115, 119 Mean pressure during impact, 211 Mechanical advantage, 14 of blocks and tackle, 17 differential pulley-block, 19 hydraulic press, 26 inclined plane, 18, 20 lever, 23 screw, 18 wheel and axle, 20 Mechanics, method of study, 2, 28, 71 Mechanism, 26 Memel timber, 77 Mensuration, Kules in, 252 Metals, 8085 Tables giving strength of, 68, 69, 121, 131 Middle third of joint in arch ring, 130,163 Modulus of elasticity of bulk, K, 57> 58,92 Young's, E, 54, 69, 92 Modulus of rigidity, N, 69, 87, 90,92 Moment, Bending, 103, 104 Diagrams of, 116118 of a force, 23, 229 inertia, 46, 105, 107, 236 Formula for, 142 Poinsot's theorem regarding, 144 momentum, defined, 238, 239 Moments, Law of, 23, 230 Method of, applied to stresses in structures, 158 Momentum, Definition of, 227 Total, unaltered by impact, 203 Mortar, how prepared and why it hardens, 72, 73 Motion, Communication of, in li- quids, 211 Motion is either translation, rota- tion, or both, 216 Linear and angular, compared, 236 239 Periodic, 179, 188 Eotational, 192 Processional, Examples of, 215 produced by a blow, 216 Moulds for castings, 81 Muutz metal, 85 Mutual pressure during impact, JT A TUBE of pure shear strain, 90, strain, 56, 57 Neutral axis, defined, 105 '. passes through centre of gra- vity of section, 105 Radius of curvature of, 108 line, 102, 105, 107 surface, 102 Notation in graphical statics, Mr. Bow's, 15Q INDEX. 265 QAK timber, 78 Oriflce, Flow through, 7G Oscillation, Centre of, 195 and suspension, Centres of, 107 , Squared, 7, 10, 15 Parallel motion, Friction of, 13 Patterns for moulds, 81 Pendulum arranged to trace curve of sines, 201 Ballistic, 217 Blackburn's, for combining har- monic motions, 189, 190 Compensation, 185 Compound, 194 Conical, 179, 242 Equivalent simple, 194 Eadius of gyration of, lOo Simple, 37, 184, 187, 194 -- Time of vibration of, 185 Percussion, Centre of, 195 --- how found, 217, 218 Periodic motion, 179 -- Examples of, 188 -- Rotational, 192 time, 179 -- Rule for finding, 183 -- of balance of watch, 193 Permanent axes in machines, Necessity for, 219 axis, Meaning of, 219 set, 55, 100 Phosphor bronze, Composition of, 85 Pile-driver, Energy in, 208 Pillars, Gordon's fonnulse for strength of, 131, 133 Ways of breaking, 130 with ends fixed, 131 --- rounded, 133 Pipes, Flow of water in, 75 Pipes, Eule for bursting pressure of, 60 Strength of water, 59 Piston, 28, 233 Piston-rod slide, Friction of, 13, 14 Strength of, 65 Pitch circle of spur wheel, 27 of screw, 18 teeth, Eule for, 133 wheel teeth, 27 Plane, Inclined, 17, 20 Plates, Boiler, 60, 88 Strength of flat, 133 Pole of link polygon, 138 Polygon, Closed, 137 Force, 137 Link, 137 Unclosed, 137 of forces, 4, 229 Potential energy, 38, 234 stored up in water, 75 converted into kinetic, 40, 187 Power, meaning of the term, 15, 233 (see Horse-power) Precessional motion, 215 Pressure during impact, 208 Pressure-energy of water, 75 Fluid, 59, 61, 75, 111 on bearings of shafts, 218 on teeth of wheels, 133 of steam on piston, how found, oqo 6OO wind on roofs, 154 Principal axes of an ellipse, 144 Prints on patterns, 81 Propeller screw, Exercise on, 19 Puddling, 83 Pull in belt, 32 how transmitted by wire, 51 in cord, Difference of, on two sides of. pulley, 4, 11, 32 Pulley and cord, Friction between, 33 266 INDEX. Pulley block, T>iffpFfiT|^ft] pn.-/*hftni- cal advantage of, 19 Pumps, Flow of water through, 75, 78 Punching and shearing, 88 Pore harmonic motion of piston- rod, denned, 179 (see Harmonic motion) shear strain, 89 how produced, 89 Nature of, 90 QUASI-RIGIDITY produced by ^ rapid motion, 215, 216 Quaternions, 135 Quicklime, 72 Quick - moving shafts, How to diminish friction in, 14 J ADIAN, Definition of, 235 Radius of Curvature, 223 of any fibre in bent beam, how found, 106 elastic curve, 110 gyration, 141 of pendulum, 196 Batio, Velocity, 6, 27 Reaction of* fluids. Application of the principle of, 209 Reciprocal figures, 148 Reciprocating mass in locomotive engine, 220 Recoil of gun, 209 Red pine, or Kernel timber, 77 Re-entrant edges or corners, weak points, 82, 98 Relation between bending and twist- ing of cylindric abaft, 112 Resilience of cylindric spiral springs, 176 Resultant, Definition of, 136, 228 force on joint of arch ting, 129, 163 of a number of forces, how found, 137,138 Rib, Arched, is in unstable equ : l> brium, 161 Rigid bodies, Meaning of the term, 103 Rigidity, Modulus of, N, 87, 90. 92 produced by rapid motion, 215 Riveted joint may break in several ways, 88 Rivet holes usually weaken the metal, 88 Rocks, granitic, History and character of, 71 stratified, History and character of, 71 Roof, Necessity for detail drawing* of, 155 Weight of snow on, 153 Wind pressure on, 154 Roof - principal, Investigation of stresses in, 150153 Rotating body, Energy stored up in, 42-51 Rotational periodic motion, 192 Round, Meaning of the term, 236 Rupert's drop, Condition of the gloss in, 80 Rupture produced by shear stress, 88,89,98 gAFETY, Factor of, 120 Table giving usual values of, 121 ralve, Weighted, 24 Sandstones, Character of, 71 INDEX. 267 Screw, Mechanical advantage of, 18 Pitch of, 18 propeller of vessel, Exercise on, 19 Section of beam varied for uniform strength, 119 loaded beam or arch, resultant of stresses at, 128130 structures, stress at, how cal- culated, 157, 158 Set, Permanent, 55, 100 Shafts, cylindric, Relation between bending arid twisting in, 112 Strength of, 96 effects of twisting couple on different sections, 9799 Effects produced by wheels fixed eccentrically on, 218, 219 Friction at bearings of, 12 in quick-moving, 14 Power transmitted by, how measured, 31 35 Practical rule for strength of, 95 Stiffness necessary as well as strength, 100 Torsional vibration of, when transmitting power, 95, 101 Shape of a loaded beam, 145, 146 chain, 160 wheel teeth, 27, 28 Shearing and punching, 88, 89 force in beams and girders, 10i, 114, 115 Shear strain, 86101 Nature of, 90 pure, How to produce, 86, 89 and shear stress, Eelatiou between, 92 - stress, 86101 Simple pendulum, 39, 184, 187, 194 Sine of an angle, 235 Sines, Curve of, how drawn, 189, 200, 204 Sines, Curve of, interpreted, 200 traced by a pendulum bob, 201 Skeleton drawings instructive and necessary, 2830 Slide and piston-rod, 13, 14, 220 Slide valve, Motion of, 30 rule, TJse of, 136 Solids, Laws of friction between, 251 Specific gravity, Definition and examples of, 256 table of values, 68, 69 Spherical boiler, Strength of, 61 Spinning bodies, precessioual mo- tion of, 215 Spiral spring, cylindric, Angular motion of, due to turning mo- ment, 178 Behaviour of, when weighted, 178, 183 Elongation in different cases, 176 in dynamometer coupling, 31,35 Investigation of the forces in a weighted, 173 Experiment showing relation between extension of spring and twisting of wire, 174 Resilience of, 176 Strength of, 175 Twisting moments of, ez- perimentally determined, 197 Ultimate elongation, 177 Work stored up in, 176 Plat, Angle of winding propor- tional to couple in, 170 Bending moment in, 168 Change of curvature of, 167 Curve for turning moment of, 171 Isochronism, how usually obtained, 170 268 FNDKX. Spiral spring, Plat, Turning mo- ment proportional to winding, 171 Squared paper, Use of, 710, 15, . 55 State of strain affects elastic strength, 62, 63, 99, 193 Statical friction, 246 Statics, Graphical, 135-166 Steadiness of Machines, 48, 219 Steam-engine, Condensing and non- condensing, 50 Fly-wheel for, 50 giving out energy, 26 indicated horse-power, how found, 233 Steel, Carbon and impurities in, 84 Sffength of, 84 Tempering of, 66 Stiff joints in structures, 156 Stiffness important in shafting, 100, 101 of beams, how it varies with linear dimensions, 125, 126 Stilling of vibrations, 200 Stone, 71 An artificial, how mode, 72 Preservation of, 71, 72 Storage of energy during impact, 212 Store of energy in a moving body, 39,75 Straight line figures,' Properties of, 146-148 on squared paper, Meaning of ,. 9,10 what it may represent, 135 Strain, 5158 energy, Storage and transmission of, 64, 65 in bent beam, proportional to distance from neutral surface, 102 Nature of, in a wire, 56, 57 proportional to stress, 53, 55 Strains due to contraction in cool- ing, 79-83. Stratified rocks, 71 Strength and stiffness of beams, Tables giving, 116118, 121 best section of beam for, 106 Elastic, 59, 99-100 modulus for beams of different sections, 106 of rectangular beams, how it varies with breadth and depth, 121,125 supported at ends and loaded in'middle, Table, 121 of columns, long and short, 130 Gordon's rule for, 131 133 cylindric spiral spring, 175 flat plates, 133, 134 pipes and boilers, 59 61 riveted joints in boiler plates, 88 shafts, 95101 similar structures similarly loaded, 134 structures with stiff. joints, 156 teeth of wheels, 133 timber, 77, 78, 121 Table of, for different materials, 68,69 Stress, 5261 Amount of, anywhere in a section, 106,107 Breaking, for beams, 120 columns or struts, 131 table of values, 68, 69 diagrams, Examples of, 155 how represented, 135 tensile and compressive often re- peated, Effect of, 65 working shear, 88 Stresses at joints of structures, Methods of fiudiug, 157, 158. INDEX. 269 Stresses in a hinged structure, Con- ditions for calculating, 149 loaded chain of suspension bridge, 160 links, 159 roof - principal, how deter- mined, 150153 Strip of steel, Forms assumed by, 109 Time of vibration of, 186 Vibration of, in two direc- tions, 192 Structures, Hinged, 146149 similar and similarly loaded, Strength of, 134 Struts, 53, 54, 150 Gordon's rules for strength of, 131133 long, Modes of breaking, 130 with ends fixed, and hinged, 131 Suddenly applied load, Effect of, 63, 64 Supporting and loading, Effects of different methods of, 113, 116 122 Suspension, Axis of, 195, 217 Bifilar, 200 bridge, Stresses in chain of, 160 Symbols, Algebraic, 1 rpACKLE, Blocks and, mechanical advantage, 17 Teak timber, 78 Teeth of wheels, Cycloidal, 28 like beams fixed at one end. 133 Pitch and number of, 27 Bole for, 133 Pressure on, 133 Shapes of, 27, 28 Teeth of wheels, Strength of, 133 Temperature, Effects of, on beams and supports, 113 Tempering steel, Method of, 66 Tensile strength of materials, 58, 68, 69, 134 Thrust at crown of arch, 163, 165 Tie-rod, 53, 150 Timber, Ash, 78 beams, Strength of, 120, 121 Beech, 78 Cedar, 78 Durability of, 78 Elm, 78 Felling of, 78 Larch, 77 Mahogany, 78 Meinel, 77 Oak, 78 Preservation of, 79 Seasoning of, 79 Teak, 78 Warping of, 77 White fir or Norway spruce, 77 Time of vibration of balance of a watch, 193 bar hung at end of a wire, 19D compound pendulum, 195 liquid iu bent tube, 187 magnet, 200 simple pendulum, 185 spiral spring, 183 Periodic, of a pure harmonic motion, Eule for, 183 Torque, Definition of, 231 Torsional vibrations of shaft and want of stiffness, 100, 101 Torsion of shafts and beams, 93 101 Transmission dynamometer, 35, 36 of power by belts and pulleys, 270 Transmission of power by shafts, 30, 95-101 wheels, 27, 28, 133 strain energy, 64, 65, 307, 21 1 214 Triangle of forces, 3, 21, 229 Twist, Angle of, 31, 93 bow measured, 30, 31, 9396 Twisting and bending combined in shafting, 100 Relation between, 112 moment, 93101 Experimental investigation of, 197-199 TT TUBE, Vibration of liqnid in, U 186, 187 Unclosed polygon, 137 Uniform velocity ratio given by cycloidal teeth, 28 strength in beam, 119 Use of elementary principles, 70 \7ALVE, Safety, Example of, 24 slide, Motion of, 30 Variable things compared, 7, 10 Velocity, Definition of, 224 Angular, 235 at any point in pure harmonic motion, how found, 180 of bullet, how measured, 218 ratio, 6, 27 Vents, Use of, in moulding, 81 Vertical line, Definition of, 223 Vibration, Amplitude of a, 201 Nature of a, 40, 187 Time of, for simple pendulum, 185 Torbiumil, of a shaft, 101 Vibrations, Damped, Law for, 201 Investigation of damping of, 201 Eepresentation of, 204 Stilling of, 200 Viscosity measured by twist in wire, 199, 248 Eelative, Example worked out, 205 Voussoirs, Pressure on, 162 TXT" ALL, Pressure of earth against, 73,74 Watch balance, Motion of, 193 Waterfall, Energy in, 6 Water flowing through an orifice, Velocity of, 76, 77 Friction of, in pipes and ptimps, 75,76 Pressure of still, 74 pressure on any surface sub- merged, 74, 75 Quantity of, flowing through a pipe, how calculated, 77 Total energy of, 75 Waterplpes, Strength of, 59, 60 Weighbridge, Mechanical advantage of, 24 Weight of a body, 223, 224 Wheel and axle, Mechanical advan- tage of , 20 Necessity for drawings of, 28 of locomotive, Balance weights on, 221 teeth, Pressure on, 133 Shapes of, 27 White fir or Norway spruce, 77 Wind pressure against roofs, 154 Work, Definition of, 232 how measured, 5, 6, 15, 16, 31, 32, 232, 233 IttDEX. 271 Work, Law of, 5, 17, 231 lost in friction, 1117 Kate of doing, 15, 233 Working stress, 59 Worm and worm-wheel, 2 Wrought iron, 83, 84 VOTING'S modulus of elasticity, " E, 54, 69, 92 2TINC, Alloys of, with copper, 85 THE END. LoxDOiC: CASSELL, FETTER, GALPIN tt Co., BELLE SAUYAGE WORKS. LODOATK HILL, E.G. from which It was borrowed