IC-NRLF SB 2M 733 1 I BH I ..,.;/;-,:'..';<'. : >:,;?:** ?-v ^/^^'i ' '"- RIEfl ' UNIVERSITY OF CALIFORNIA LIBRARY OF THE tTCPARTMEHI U* PHIBILi "* Received. Accessions No. Book No. ELEMENTARY TREATISE ON NATURAL PHILOSOPHY. BY A. PRIVAT DESCHANEL, FORMERLY PROFESSOR OF PHTSICS IN THE LYCEE LOUIS-LE -GRAND, INSPECTOR OF THE ACADEMY OF PARIS. TRANSLATED AND EDITED, WITH EXTENSIVE MODIFICATIONS, BY J. D. EVERETT, M. A., D. C. L., F. R. S., F. R. S. E., PROFESSOR OF NATURAL PHILOSOPHY IN THE QUEEN'S COLLEGE, BELFAST. IN FOUR PARTS. PART I. MECHANICS, HYDROSTATICS, AND PNEUMATICS, ILLUSTRATED BY 180 ENGRAVINGS ON WOOD, AND ONE COLORED PLATE, NEW YORK : D. APPLETON AND COMPANY, 1, 3, AND 5 BOND STKEET. 1883. P72, AUTHOR'S PREFACE. THE importance of the study of Physics is now generally acknowledged. Besides the interest of curiosity which attaches to the observation of nature, the experi- mental method furnishes one of the most salutary exercises for the mind constituting in this respect a fitting supplement to the study of the mathematical sciences. The method of deduction employed in these latter, while eminently adapted to form the habit of strict reasoning, scarcely affords any exercise for the critical faculty which plays so important a part in the physical sciences. In Physics we are called upon, not to deduce rigorous consequences from an absolute principle, but to ascend from the particular consequences which alone are known to the general principle from which they flow. In this operation there is no absolutely certain method of procedure, and even relative certainty can only be attained by a discussion which calls into profitable exercise all the faculties of the mind. Be this as it may, physical science has now taken an important place in educa- tion, and plays a prominent part in the examinations for the different university degrees. The present treatise is intended for the assistance of young men preparing for these degrees; but I trust that it may also be read with profit by those persons who, merely for purposes of self-instruction, wish to acquire accurate knowledge of natural phenomena. Having for nearly twenty years been charged with the duty of teaching from the chair of Physics in one of the lyceums of Paris, I have been under the necessity of making continual efforts to overcome the inherent difficulties of this branch of study. I have endea- voured to turn to account the experience thus acquired in the preparation of this volume, and I shall be happy if I can thus contribute to advance the taste for a science which is at once useful and interesting. I have made very limited use of algebra. Though calculation is a precious and often indispensable auxiliary of physical science, the extent to which it can be advantageously employed varies greatly according to circumstances. There are in fact some phenomena which cannot be really understood without having recourse to measurement: but in a multitude of cases the explanation of phenomena can be rendered evident without resorting to numerical expression. The physical sciences have of late years received very extensive developments. Facts have been multiplied indefinitely, and even theories have undergone great modifications. Hence arises considerable difficulty in selecting the most essential points and those which best represent the present state of science. I have done my best to cope with this difficulty, and I trust that the reader who attentively peruses my work, will be able to form a pretty accurate idea of the present position of physical science. 673253 TRANSLATOR'S PREFACE TO THE SIXTH EDITION. I DID not consent to undertake the labour of translating and editing the ** TRAITE ELEMENTAIRE DE PHYSIQUE " of Professor Deschanel until a careful examination had convinced me that it was better adapted to the requirements of my own class of Experimental Physics than any other work with which I was acquainted; and in executing the translation I steadily kept this use in view, believing that I was thus adopting the surest means of meeting the wants of teachers generally. In the original English edition, the earlier portions consisted of a pretty close translation from the French; but as the work progressed I found the advantage of introducing more considerable modifications; and Parts III. and IV. were to a great extent rewritten rather than translated. I have now, in like manner, rewritten Part I., and trust that in its amended form it will be found better adapted than before to the wants of English teachers. Several additional subjects have been introduced, and the order of the chapters has been rearranged. The marks of distinction which were made in the earlier editions between new and old sections have now been dropped; but Professor Deschanel's foot-notes are still distinguished by the initial "D." The numbering of the sections is entirely new. All accurate statements of quantities have been given in the C.G.S. (Centimetre-Gramme-Second) system, which, by reason of its simplicity and of the sanction which it has received from the British Association, and the Physical Society of London, is coming every day into more general use, but rough statements of quantity have generally been expressed in British units as being more familiar. A complete table for the conversion of French and English measures will be found at the end of the Table of Contents. In Part II. , the subject of Heat as a measurable Quantity is introduced at a much earlier stage than before, the chapter on Calorimetry being placed immediately after those on Thermometry and Expansion. Latent Heat and Heat of Combination are not now included in this chapter, but are treated later in connection with the subjects of Fusion, Vaporization, and Thermo-dynamics. Among the new matter may be mentioned: An investigation of the temperature of minimum apparent volume of water in a glass envelope ; An account of Guthrie's results on the freezing of brine ; A proof that the pressure of vapour in the air at any time is equal to the maximum pressure for the dew-point ; Descriptions of Dines' hygrometer, and of Symons' Snowdon rain-gauge ; A full explanation of " Diffusivity " or " Thermometric Conductivity;" Some recent results on the conductivity of rocks, and on the conductivity of water ; A note on the mathematical discussion of periodical variations of under- ground temperature ; A proof of the formula for the efficiency of a perfect thermo-dynamic engine ; Several investigations relating to the two specific heats of a gas, and to adiabatic changes in gases, liquids, and solids; A description of the modern Gas Engine. Every chapter has been carefully revised, with a view to clearness, accuracy, and consolidation; and the result has been that, with the excep- tion of Melloni's experiments, and the Steam Engine, the treatment of nearly every subject has been materially changed. Part III. also contains extensive changes. In the electro-statics, the chapter on potential has been recast and made more demonstrative. There are also additions relating to Dr. Kerr's dis- coveries, charge by cascade, and some minor points. Under the head of Magnetism, investigations have been introduced relating to bi filar suspension, and to the directive tendency of soft-iron needles. In the department of Current Electricity, there has been a complete rearrangement of subjects. The chemical relations of the current are discussed as early as possible, while thermo-electricity is reserved for a chapter on relations between electricity and heat. The chapter on induced currents, which was formerly the last of all, has been put next to that on electro-dynamics, and is followed by two chapters on telegraphs and other applications of electricity. Additional matter has been introduced under the following heads : General law for magnetic force due to current in given circuit; Helmholtz's galvanometer ; Swing produced by instantaneous current; The galvanometer a true measurer of current; Rowland's experiment on the motion of a charged body; Plante's secondary battery ; Chemical relations of electro-motive force ; Resistance coils and boxes ; "Wheatstone's bridge, and conjugate branches ; Vi TRANSLATOR S PREFACE. Clark's method for electro-motive force ; Thomson's method for resistance of galvanometer; Mance's method for resistance of battery ; Thermo-electric diagrams ; Convection of heat by electricity ; Pyro-electricity ; Effect of light on resistance of selenium ; Deduction of law of induced currents from electro-dynamic law; Superposition of tubes of force ; Stratified discharge from galvanic battery ; Siemens' and Gramme's magneto-electric machines; Cowper's writing telegraph ; Duplex telegraphy ; Edison's electric pen ; The telephone, the microphone, and the induction balance. A collection of examples on electricity has been added. Part IV. contains no radical changes. The numbering of the chapters and sections has been altered to make it consecutive with the other three Parts, but there has been no rearrangement. Additions have been made under the following heads (those marked with an asterisk were introduced in a previous edition) : Mathematical note on stationary undulation ; Edison's phonograph ; Michelson's measurement of the velocity of light; Astronomical refraction ; *Ref raction at a spherical surface ; Refraction through a sphere ; Brightness of image on screen ; Field of view in telescope ; *Curved rays of sound ; *Retardation-gratings and reflection-gratings ; Kerr's magneto-optic discoveries; besides briefer additions and emendations which it would be tedious to enumerate. The whole volume has been minutely revised; and a copious collection of examples arranged in order, with answers, has been introduced at the end of each Part, in place of the " Problems " (translated from the French) which appeared in some of the earlier editions. The dates of revision of the four Parts were, October, 1879, November, 1880, December, 1880, and May, 1881. J. D. R BELFAST, September, 1SS1. CONTENTS-PART I. (THE NUMBERS REFER TO THE SECTIONS.) CHAPTER I. INTRODUCTORY. Natural History and Natural Philosophy, 1, 2. 'Divisions of Natural Philosophy, 3. CHAPTER II. FIRST PRINCIPLES OF DYNAMICS, STATICS. Force, 4. Translation and rotation, 5, 6. Instruments for measuring force, 7. Gravita- tion units of force, 8. Equilibrium ; Statics and kinetics, 9. Action and reaction, 10. Specification of a force, point of application, line of action, 11. Rigid body, 12. Equilibrium of two forces, 13. Three forces in equilibrium at a point, 14. Resul- tant and components, 15. Parallelogram of forces, 16. Gravesande's apparatus, 17. Resultant of any number of forces at a point, 18. Eqtiilibrium of three parallel forces, 19. Resultant of two parallel forces, 20. Centre of two parallel forces, 21. Moments of resultant and components equal, 22. Resultant of any number of parallel forces in one plane, 23. Moment of a force about a point, 24. Arithmetical lever, 25. Couple, 26. Composition of couples; Axis of couple, 27. Resultant of force and couple in same plane, 28. General resultant of any number of forces; Wrench, 29. Application to action and reaction, 30. Resolution, 31. Rectangular resolution ; Component of a force along a given line, 32. CHAPTER III. GRAVITY. Direction of gravity ; Neighbouring verticals nearly parallel, 33. Centre of gravity, 34. Centres of gravity of volumes, areas, and lines, 35. Methods of finding centres of gravity, 36. Centre of gravity of triangle, 37. Of pyramids and cones, 38, 39. Condition of standing or falling, 40. Body supported at one point, 41. Stability and instability, 42. Experimental determination of centre of gravity, 43, 44. Work done against gravity, 45. Centre of gravity tends to descend, 46. Work done by gravity, 47. Work done by any force, 48. Principle of work; Perpetual motions, 49. Criterion of stability, 50. Illustration, 51. Stability where forces vary abruptly, 52. Illustrations from toys, 53. Limits of stability, 54. CHAPTER IV. THE MECHANICAL POWERS. Enumeration, 55. Lever, 56-58. Mechanical advantage, 59. Wheel and axle, 60. Pulleys, 61-63. Inclined plane, 64-66. Wedge and screw, 67-69. CHAPTER V. THE BALANCE. General description, 70. Qualities requisite, 71. Double weighing, 72. Investigation of sensibility, 73. Advantage of weighing with constant load, 74. Details of con- struction, 75. Steelyard, 76. CHAPTER VI. FIRST PRINCIPLES OF KINETICS. Principle of Inertia, 77. Second law of motion, 78. Mass and momentum, 79. Proper selection of unit of force, 80. Relation between mass and weight, 81. Third law of Vlll TABLE OF CONTENTS. motion; Action and reaction, 82. Motion of centre of gravity unaffected, 83. Velocity of centre of gravity, 84. Centre of mass, 85. Units of measurement, 86. The C.G.S. system ; the dyne, the erg, 87. CHAPTER VII. LAWS OF FALLING BODIES. Fall in air and in vacuo, 88. Mass and gravitation proportional, 89. Uniform accelera- tion, 89. Weight of a gramme in dynes; Value of g, 91. Distance fallen in a given time, 92. Work spent in producing motion, 93. Body thrown upwards, 94. Resistance of the air, 95. Projectiles, 96. Time of flight, and range, 97. Morin's apparatus, 98. Atwood's machine, 99. Theory of Atwood's machine, 100. Uniform motion in a circle, 101. Deflecting force, 102. Illustrations, stone in sling, 103. Centrifugal force at the equator, 104. Direction of apparent gravity, 105. CHAPTER VIII. THE PENDULUM. Pendulum, 106. Simple pendulum, 107. Law of acceleration for small vibrations, 108. General law for period, 109. Application to pendulum, 110. Simple harmonic motion, 111. Experimental investigation of motion of pendulum, 112. Cycloidal pendulum, 113. Moment of inertia about an axis, 114. About parallel axes, 115. Application to compound pendulum, 116. Convertibility of centres, 117. Centre of suspension for minimum period, 118. Kater's pendulum, 119. Determination of g, 120. CHAPTER IX. ENERGY. Kinetic energy, 121. Static or potential energy, 122. Conservation of mechanical energy, 123. Illustration from pile-driving, 124. Hindrances to availability of energy ; Principle of the conservation of energy, 125. CHAPTER X. ELASTICITY. Elasticity and its limits, 126. Isochronism of small vibrations, 127. Stress, strain, coefficients of elasticity; Young's modulus, 128. Volume-elasticity, 129. Oersted's piezometer, 130. CHAPTER XI. FRICTION. Friction, kinetical and statical, 131. Statical friction, limiting angle, 132. Coefficient = tan 6 ; Inclined plane, 1 33. CHAPTER XII. HYDROSTATICS. Hydrodynamics, 134. No statical friction in fluids, 135. Intensity of pressure, 136. Pressure the same in all directions, 137. The same at the same level, 138. Differ- ence of pressure at different levels, 139. Free surface, 140. Transmissibility of pressure; Pascal, 141. Hydraulic press, 142. "Principle of work" applicable, 143. Experiment on upward pressure, 144. Liquids in superposition, 145. Two liquids in bent tube, 146. Pascal's vases, 147. Resultant pressure on vessel, 148. Back pressure on discharging vessel, 149. Total and resultant pressures; Centre of pressure, 150. Construction for centre of pressure, 151. Whirling vessel; D'Alem- bert's principle, 152. CHAPTER XIII. PRINCIPLE OF ARCHIMEDES. Resultant pressure on immersed bodies, 153. Experimental demonstration, 154. Three cases distinguished, 155. Centre of buoyancy, 153, 155. Cartesian diver, 156. Stability of floating body, 157, 158. Floating of needles on water, 159. TABLE OF CONTENTS. IX CHAPTER XIV. DENSITY AND ITS DETERMINATION. Absolute and relative density, 160. Ambiguity of the word "weight," 161. Determination of density from observation of weight and volume, 162. Specific gravity flask for solids, 163. Method by weighing in water, 164. With sinker, 165. Densities of liquids measured by loss of weight in them, 166. Measurement of volumes of solids by loss of weight, 167. Hydrometers, 168. Nicholson's, 169. Fahrenheit's, 170. Hydrometers of variable immersion, 171. General theory, 172. Beaumd's hydro- meters, 173. Twaddell's, 174. Gay-Lussac's alcoholimeter, 175. Computation of densities of mixtures, 176. Graphical method of interpolation, 177. CHAPTER XV. VESSELS IN COMMUNICATION. LEVELS. Liquids tend to find their own level; Water-supply of towns, 178. Water-level; Levelling between distant stations, 179. Spirit-level and its uses, 180, 181. CHAPTER XVI. CAPILLARITY. General phenomena of capillary elevation and depression, 182. Influencing circum- stances, 183. Law of diameters, 184. Fundamental laws of capillary phenomena; Angle of contact; Surface tension, 185. Application to elevation and depression in tubes, 186. Formula for normal pressure of film, 187. Film with air on both sides, 188. Drops, 189. Pressure in a liquid whose surface is convex or concave, 190. Interior pressure due to surface action when surface is plane, 191. Phenomena illus- trative of differential surface tensions; Table of tensions, 192. Endosmose and diffusion, 193. CHAPTER XVII. THE BAROMETER. Expansibility of gases, 194. Direct weighing of air, 195. Atmospheric pressure, 196. Torricellian experiment, 197. Pressure of one atmosphere, 198. Pascal's experi- ment on Puy de Dome, 199. Barometer, 200. Cathetometer, 201. Fortin's Barometer ; Vacuum tested by metallic clink, 202. Float adjustment, 203. Baro- metric corrections; Temperature; Capillarity; Capacity; Index errors; Reduction to sea-level; Intensity of gravity; and reduction to absolute measure, 204. Siphon, wheel, and marine barometers, 205. Aneroid, 206. Counterpoised barometer; King's barograph ; Fahrenheit's multiple-tube barometer, 207. Photographic regis- tration, 208. CHAPTER XVIII. VARIATIONS OF THE BAROMETER. Measurement of heights by the barometer, 209. Imaginary homogeneous atmosphere, 210. Geometric law of decrease, 211. Computation of pressure -height, 212. For- mula for determining heights by the barometer, 213. Diurnal oscillation, 214. Irregular variations, 215. Weather charts, 216. CHAPTER XIX. BOYLE'S (OR MARIOTTE'S) LAW. Boyle's law, 217. Boyle's tube, 218. Unequal compressibility of different gases, 219, 220. Regnault's experiments, 221. Results, 222. Manometers or pressure gauges, 223. Multiple-branch manometer, 224. Compressed air manometer, 225. Metallic manometers, 226. Pressure of gaseous mixtures, 227. Absorption of gases by liquids and solids, 228. CHAPTER XX. AIR-PUMP. Air-pump, 229. Theoretical rate of exhaustion, 230. Mercurial gauges, 231. Admission cock, 232. Double-barrelled pump, 233. Single barrel with double action, 234. English X TABLE OF CONTENTS. forms, 235. Experiments ; Burst bladder ; Magdeburg hemispheres ; Fountain, 236. Limit to action of pump and its causes, 237. Kravogl's pump, 238. Geissler's, 239, Sprengel's, 240. Double exhaustion, 241. Free piston, 242. Compressing pump, 213. Calculation of its effect, 244. Various contrivances for compressing air, 245. Practical applicatioi.s of air-pump and compressing pump, 246. CHAPTER XXI. UPWARD PRESSURE OF THE AIE. Baroscope, 247. Principle of balloons, 248. Details, 249. Height attainable by a given balloon, 250. Effect of air on apparent weights, 251. CHAPTER XXII. PUMPS FOR LIQUIDS. Invention of pump, 252. Reason of the water rising, 253. Suction pump, 254. Effect of untraversed space, 255. Force necessary to raise the piston, 256. Efficiency, 257. Forcing pump, 258. Plunger, 259. Fire-engine, 260. Double-acting pumps, 261. Centrifugal pumps, 262. Jet-pump, 263. Hydraulic press, 264. CHAPTER XXIII. EFFLUX OF LIQUIDS. Torricelli's theorem, 265. Froude's calculation of area of contracted vein, 266. Con- tracted vein for orifice in thin plate, 267. Apparatus for illustrating Torricelli's theorem, 268. Efflux from air-tight space, 269. Intermittent fountain, 270. Siphon, 271. Starting the Siphon, 272. Siphon for sulphuric acid, 273. Tantalus' cup, 274. Mariotte's bottle, 275. EXAMPLES. PAGE Parallelogram of Velocities, and Parallelogram of Forces. Ex. 1-11, . . . 239 Parallel Forces and Centre of Gravity. Ex. 10*-33, 239 Work and Stability. Ex. 34-43, 241 Inclined Plane, &c. Ex. 44-48, 242 Force, Mass, and Velocity. Ex. 49-59 242 Falling Bodies and Projectiles. Ex. 60-83, 243 Atwood's Machine. Ex. 84-89, 244 Energy and Work. Ex. 90--98, 245 Centrifugal Force. Ex. 99-101, 245 Pendulum, and Moment of Inertia. Ex. 101*-107, 246 Pressure of Liquids. Ex. 108-123, 246 Density, and Principle of Archimedes. Ex. 124-159, 247 Capillarity. Ex. 160-164, 249 Barometer, and Boyle's law. Ex. 165-181, 250 Pumps, &c. Ex. 182-189, 251 ANSWERS TO EXAMPLES, 252 FRENCH AND ENGLISH MEASURES. A DECIMETRE DIVIDED INTO CENTIMETRES AND MILLIMETRES. 4-1 5 . C 7 -I ' I- 1 I I I I I i I I I I i I I I I M I I I I i I M M l| 2 3|. INCHES AND TENTIia REDUCTION OF FRENCH TO ENGLISH MEASURES. LENGTH. I millimetre == '03937 inch, or about 7 \ inch. 1 centimetre^ '3937 inch. 1 decimetre = 3 -937 inch. 1 metre = 39 -37 inch = 3 '281 ft. = l'0936 yd. 1 kilometre =1093 '6 yds., or about | mile. More accurately, 1 metre =39 '370432 in. =3-2806093 ft. = 1-09362311 yd. AREA. 1 sq. millim. '00155 sq. in. 1 sq. centim. = '155 sq. in. 1 sq. decim. 15 '5 sq. in. 1 sq. m?tre = 1550 sq. in. = 10704 sq. ft. =: 1-19G sq. yd. VOLUME. 1 cub. millim. = -000061 cub. in. 1 cub. centim. = '06 1025 cub. in. 1 cub. decim. =61'0254cub. in. cub. mctre=61025 cub. in.=35'315G cub. ft. = 1-303 cub. yd. The Litre (used for liquids) is the s.ime as the cubic decimetre, and is equal to 1701/ pint, or '22021 gallon. MASS AND "WEIGHT. 1 milligramme = '01543 grain. 1 gramme = 15'432 grain. 1 kilogramme =15432 grains =2 -205 Ibs. avoir. More accurately, the kilogramme is 2-20462125 Ibs. MISCELLANEOUS. 1 gramme per sq. centim. =2'04S1 Ibs. per sq. ft. 1 kilogramme per sq. centim. = 14 -223 Ibs. per sq. in. 1 kilogrammetre=7;2331 foot-pounds. 1 force do cheval = 75 kilogrammetres per second, or 542.^ foot-pounds per second nearly, whereas 1 horse-power (English)=550 foo't- pounds per second. REDUCTION TO C.G.S. MEASURES. (See page 48.) [cm. denotes centimetre (s); gm. denotes gramme(s).] LENGTH. 1 inch. s=2'54 centimetres, nearly. 1 foot 2= 30 '48 centimetres, nearly. 1 yard =91 "44 centimetres, nearly. 1 statute mile = 10093-3 centimetres, nearly. More accurately, 1 inch = 2"539977'2 ccuLi- metres* AREA. 1 sq. inch =6-45 sq. cm., nearly. 1 sq. foot =929 sq. cm., nearly. 1 sq. yard = 8361 sq. cm., nearly. 1 sq. mile =2'59 x 10 10 sq. cm., n^trly. VOLUME. 1 cub. inch =16-39 cub. cm., nearly. 1 cub. foot =28316 cub. cm., nearly. 1 cub. yard =764535 cub. cm., nearly. 1 gallon. = 45ff cub. cm. , nearly. MASS. 1 grain ='0048 gramme, nearly. 1 02. avoir. = 28'35 gramme, nearly. 1 Ib. avoir. =453-6 gramme, nearly. 1 ton = 1 "016 x 10 6 gramme, nearly. More accurately, 1 Ib. avoir. = 453 '59205 fe.ni. VELOCITY. 1 mile per hour =447.04 cm. per sec. 1 kilometre per hour =27 '7 cm. per sec. DENSITY. 1 Ib. per cub. foot = '016019 gm. per cub. cm. 62 '4 Ibs. per cub. ft. =1 gm, per cub. cm. Xll FRENCH AND ENGLISH MEASURES. FORCE (assuming g - 981 ). (See p. 43. ) Weight of 1 grain 63 -57 dynes, nearly. loz. avoir. =278 x 10 4 dynes, nearly. 1 Ib. avoir. = 4'45 x 10 5 dynes, nearly. 1 ton =9 "97 x 108 dynes,nearly. 1 gramme =981 dynes, nearly. 1 kilogramme = 9 '81 x 10 5 dynes, nearly. WORK (assuming ^=981). (See p. 48.) 1 foot-pound =T356x 10 7 ergs, nearly. 1 kilogram metre =9 '81 x 10 7 ergs, nearly. Work in a second \ by one theoretical > 7 '46 x 10 9 ergs, nearly, "horse." STRESS (assuming g =981). 1 Ib. per sq. ft. =479 dynes per sq. cm., nearly. 1 Ib. per sq. inch =6*9 x 1C 4 dynes per sq. cm., nearly. 1 kilog. per sq. cm. =9'81 x 10 5 dynes per sq. cm., nearly. 760 mm. of mercury at 0C. = 1 '014 x 10 6 dynes per sq. cm., nearly. 30 inches of mercury at C. = 1'0163 x 10 6 dynes per sq. cm., nearly. 1 inch of mercury at C. =3 '388 x 10 4 dynes per sq. cm., nearly. TABLE OF DENSITIES, IN GRAMMES PER CUBIC CENTIMETRE. LIQUIDS. Pure water at 4 C., - - .". . - 1-000 Sea water, ordinary, - 1'026 Alcohol, pure, '791 proof spirit, -916 Ether, -716 Mercury at C., 13 "596 Naphtha, -848 SOLIDS. Brass, cast, 7'8 to 8'4 ,, wire, 8'54 Bronze, 8'4 Copper, cast, 8 '6 sheet, 8'8 hammered, 8'9 Gold, ]9 to 19-6 Iron, cast, 6-95 to 7 '3 wrought, 7'6 to 7'8 Lead, 11-4 Platinum, 21 to 22 Silver, - 10'5 Steel, 7-8 to 7'9 Tin,- 7-3 to 7-5 Zinc, 6'8 to 7'2 Ice, -92 Basalt, 3'00 Brick, 2 to 2-17 Brickwork, 1*8 Chalk, 1-8 to 2-8 Clay, 1-92 Glass, crown, ........ 2*5 flint, 3-0 Quartz (rock-crystal), 2"65 Sand, 1-42 Fir, spruce, '48 to "7 Oak, European, -(59 to * 99 Lignum-vitse, --.... -65 to 1'33 Sulphur, octahedral, ------ -2 '05 prismatic, -1*98 GASES, at C. and a pressure of a million dynes per sq. cm. Air, dry, -0012759 Oxygen, '0014107 Nitrogen, -0012393 Hydrogen, '00008837 Carbonic acid, '0019509 ELEMENTAEY TREATISE ON NATURAL PHILOSOPHY. CHAPTER I. INTRODUCTORY. ' > ' * J o > i ' , . ^ , % 1. Natural Science, in the widest sense of the term, comprises all the phenomena of the material world. In so far as it merely describes and classifies these phenomena, it may be called Natural History; in so far as it furnishes accurate quantitative knowledge of the relations between causes and effects it is called Natural Philosophy. Many subjects of study pass through the natural history stage before they attain the natural philosophy stage; the phenomena being observed and compared for many years before the quantitative laws which govern them are disclosed. 2. There are two extensive groups of phenomena which are con- ventionally excluded from the domain of Natural Philosophy, and regarded as constituting separate branches of science in themselves; namely: First. Those phenomena which depend on vital forces; such phenomena, for example, as the growth of animals and plants. These constitute the domain of Biology. Secondly. Those which depend on elective attractions between the atoms of particular substances, attractions which are known by the name of chemical affinities. These phenomena are relegated to the special science of Chemistry. Again, Astronomy, which treats of the nature and movements of the heavenly bodies, is, like Chemistry, so vast a subject, that it forms a special science of itself; though certain general laws, which its phenomena exemplify, are still included in the study of Natural Philosophy. 2 INTRODUCTORY. 3. Those phenomena which specially belong to the domain of Natural Philosophy are called physical; and Natural Philosophy itself is called Physics. It may be divided into the following branches. I. DYNAMICS, or the general laws of force and of the relations which exist between force, mass, and velocity. These laws may be applied to solids, liquids, or gases. Thus we have the three divisions, Mechanics, Hydrostatics, and Pneumatics. IT. THERMICS; the science of Heat. III. The science of ELECTRICITY, with the closely related subject cf MA.GNETISM. IV. Aco STIES'; ^the science of Sound. |3- ;8fPTics; ;the s'cie'nce of Light. v> Th'e bi t andien l ere 'numbered I. II. III. are treated in Parts I. II. I II. respectively, of the present Work. The two branches numbered IV. V. are treated in Part IV. CHAPTER II. FIEST PRINCIPLES OF DYNAMICS. STATICS. 4. Force. Force may be defined as that which tends to produce motion in a body at rest, or to produce change of motion in a body which is moving. A particle is said to have uniform or unchanged motion when it moves in a straight line with constant velocity; and every deviation of material particles from uniform motion is due to forces acting upon them. x 5. Translation and Rotation. When a body moves so that all lines in it remain constantly parallel to their original positions (or, to use the ordinary technical phrase, move parallel to themselves), its movement is called a pure translation. Since the lines joining the extremities of equal and parallel straight lines are themselves equal and parallel, it can easily be shown that, in such motion, all points of the body have equal and parallel velocities, so that the movement of the whole body is completely represented by the move- ment of any one of its points. On the other hand, if one point of a rigid body be fixed, the only movement possible for the body is pure rotation, the axis of the rotation at any moment being some straight line passing through this point. Every movement of a rigid body can be specified by specifying the movement of one of its points (any point will do) together with the rotation of the body about this point. 6. Force which acts uniformly on all the particles of a body, as gravity does sensibly in the case of bodies of moderate size on the earth's surface (equal particles being urged with equal forces and in parallel directions), tends to give the body a movement of pure translation. In elementary statements of the laws of force, it is necessary, for 4 FIRST PRINCIPLES OF DYNAMICS. the sake of simplicity, to confine attention to forces tending to produce pure translation. 7. Instruments for Measuring Force. We obtain the idea of force through our own conscious exercise of muscular force, and we can approximately estimate the amount of a force (if not too great or too small) by the effort which we have to make to resist it; as when we try the weight of a body by lifting it. Dynamometers are instruments in which force is measured by means of its effect in bending or otherwise distorting elastic springs, and the spring-balance is a dynamometer applied to the measure- ment of weights, the spring in this case being either a flat spiral (like the mainspring of a watch), or a helix (resembling a cork- screw). A force may also be measured by causing it to act vertically downwards upon one of the scale-pans of a balance and counter- poising it by weights in the other pan. / 8. Gravitation Units of Force. In whatever way the measurement of a force is effected, the result, that is, the magnitude of the force, is usually stated in terms of weight; for example, in pounds or in kilogrammes. Such units of force (called gravitation units) are to a certain extent indefinite, inasmuch as gravity is not exactly the same over the whole surface of the earth; but they are sufficiently .definite for ordinary commercial purposes. 9. Equilibrium, Statics, Kinetics. When a body free to move is acted on by forces which do not move it, these forces are said to be in equilibrium, or to equilibrate each other. They may equally well be described as balancing each other. Dynamics is usually divided into two branches. The first branch, called Statics, treats of the conditions, of equilibrium. The second branch, called Kinetics, treats of the movements produced by forces not in equili- brium. I 10. Action and Reaction. Experiment shows that force is always a mutual action between two portions of matter. When a body is urged by a force, this force is exerted by some other body, which is itself urged in the opposite direction with an equal force. When I press the table downwards with my hand, the table presses my hand upwards; when a weight hangs by a cord attached to a beam, the cord serves to transmit force between the beam and the weight, so that, by the instrumentality of the cord, the beam pulls the weight upwards and the weight pulls the beam downwards. Electricity EQUILIBRIUM OF TWO FORCES. 5 and magnetism furnish no exception to this universal law. When a magnet attracts a piece of iron, the piece of iron attracts the magnet with a precisely equal force. 11. Specification of a Force acting at a Point. Force may be NX applied over a finite area, as when I press the table with my hand; or may be applied through the whole substance of a body, as in the case of gravity; but it is usual to begin by discussing the action of forces applied to a single particle, in which case each force is supposed to act along a mathematical straight line, and the particle or material point to which it is applied is called its point of applica- tion. A force is completely specified when its magnitude, its point of application, and its line of action are all given. 12. Rigid Body. Fundamental Problem of Statics. A force of \ finite magnitude applied to a mathematical point of any actual solid body would inevitably fracture the body. To avoid this complication and other complications which would arise from the bending and yielding of bodies under the action of forces, the fiction of a perfectly rigid body is introduced, a body which cannot bend or break under the action of any force however intense, but always retains its size and shape unchanged. The fundamental problem of Statics consists in determining the conditions which forces must fulfil in order that they may be in equilibrium when applied to a rigid body. 13. Conditions of Equilibrium for Two Forces. In order that two \\ forces applied to a rigid body should be in equilibrium, it is necessary and sufficient that they fulfil the following conditions: 1st. Their lines of action must be one and the same. 2nd. The forces must act in opposite directions along this common line. 3rd. They must be equal in magnitude. It will be observed that nothing is said here about the points of application of the forces. They may in fact be anywhere upon the" common line of action. The effect of a force upon a rigid body is not altered by changing its point of application to any other point in its line of action. This is called the principle of the transmissi- bility of force. It follows from this principle that the condition of equilibrium for any number of forces with the same line of action is simply that the sum of those which act in one direction shall be equal to the sum of those which act in the opposite direction. G FIRST PRINCIPLES OF DYNAMICS. 14. Three Forces Meeting in a Point. Triangle of Forces. If three forces, not having one and the same line of action, are in equilibrium, their lines of action must lie in one plane, and must either meet in a point or be parallel. We shall first discuss the case in which they meet in a point. From any point A (Fig. 1) draw a line AB parallel to one of the two given forces, and so that in travelling from A to B we should be travelling in the same direction in which the force acts (not in the opposite direction). Also let it be understood that the length of AB repre- sents the magnitude of the force. From the point B draw a line BC c representing the second force in direc- tion, and on the same scale of magnitude on which AB represents the first. Then the line CA will represent both in direction and magnitude the third Fig. i.-iriangie of Forces. force which would equilibrate the first two. The principle embodied in this construction is called the triangle of forces. It may be briefly stated as follows : The condition of equilibrium for three forces acting at a point is, that they be repre- sented in magnitude and direction by the three sides of a triangle, taken one way round. The meaning of the words " taken one way round " will be understood from an inspection of the arrows with which the sides of the triangle in Fig. 1 are marked. If the directions of all three arrows are reversed the forces represented will still be in equilibrium. The arrows must be so directed that it would be possible to travel completely round the triangle by moving along the sides in the directions indicated. When a line is used to represent a force, it is always necessary to employ an arrow or some other mark of direction, in order to avoid ambiguity between the direction intended and its opposite. In naming such a line by means of two letters, one at each end of it, the order of the letters should indicate the direction intended. The direction of AB is fro'm A to B; the direction of BA is from B to A. \ I 15. Resultant and Components. Since two forces acting at a point can be balanced by a single force, it is obvious that they are equiv- alent to a single force, namely, to a force equal and opposite to that which would balance them. This force to which they are equivalent EQUILIBRIUM OF THREE FORCES. 7 is called their resultant. Whenever one force acting on a rigid body is equivalent to two or more forces, it is called their resultant, and they are called its components. When any number of forces are in equilibrium, a force equal and opposite to any one of them is the resultant of all the rest. The "triangle of forces" gives us the resultant of any two forces acting at a point. For example, in Fig. 1, AC (with the arrow in the figure reversed) represents the resultant of the forces represented by AB and BC. 16. Parallelogram of Forces. The proposition called the " parallel- ogram of forces" is not essentially distinct from the "triangle of forces," but merely expresses the same fact from a slightly different point of view. It is as follows: If two forces acting upon the same rigid body in lines which meet in a point, be represented by tu'o lines drawn from the point, and a parallelo- gram be constructed on these lines, the diagonal drawn from this point to the opposite corner f's- 2.-paraiie:o S ram of of the parallelogram represents the resultant. For example, if AB, AC, Fig. 2, represent the two forces, AD will represent their resultant. To show the identity of this proposition with the triangle of forces, we have only to substitute BD for AC (which is equal and parallel to it). We have then two forces represented by AB, BD (two sides of a triangle) giving as their resultant a force represented by the third side AD. We might equally well have employed the triangle ACD, by substituting CD for AB." 17. Gravesande's Apparatus. An apparatus for verifying the par- allelogram of forces is represented in Fig. 3. ACDB is a light frame in the form of a parallelogram. A weight P" can be hung at A, and weights P, F can be attached, by means of cords passing over pulleys, to the points B, C. W r hen the weights P, P 7 , F' are proportional to AB, AC and AD respectively, the strings attached at B and C will be observed to form prolongations of the sides, and the diagonal AD will be vertical. 18. Resultant of any Number of Forces at a Point. To find the resultant of any number of forces whose lines of action meet in a point, it is only necessary to draw a crooked line composed of straight lines which represent the several forces. The resultant will be represented by a straight line drawn from the beginning to the 8 FIRST PRINCIPLES OF DYNAMICS. end of this crooked line. For by what precedes, if ABODE be a crooked line such that the straight lines AB, BO, CD, DE represent four forces acting at a point, we may substitute for AB and BC Fig. 3. Gravesaude's Apparatus. the straight line AC, since this represents their resultant. We may then substitute AD for AC and CD, and finally AE for AD and DE. One of the most important applications of this construction is to three forces not lying on one plane. In this case the crooked line will consist of three edges of a parallelepiped, and the line which repre- sents the resultant will be the diagonal. This is evident from Fig. 4, in which AB, AC, AD represent three forces acting at A. The resultant of AB and AC is AT, and the resultant of AT and AD is AT. The crooked line whose parts represent the forces, may be either ABrr', or ABGr', or ADGr', &c., the total number of alternatives being six, since three things can be taken in six different orders. We have here an excellent illustration of the fact that the same final resultant is obtained in whatever order the forces are combined Fig. 4. Parallelepiped of Forces. PAKALLEL FORCES. 9 19. Equilibrium of Three Parallel Forces. If three parallel forces, P, Q, R, applied to a rigid body, balance each other, the following conditions must be fulfilled: Q 1. The three lines of action AP, BQ, CR, Fig. 5, must be in one plane. 2. The two outside forces P, R, must act in the opposite direction to the middle force Q, and their sum must be equal to Q. 3. Each force must be proportional to Fig - 5 - the distance between the lines of action of the other two; that is, we must have 2.=J-=:JL, (1) EC AC AB The figure shows that AC is the sum of AB and BC; hence it fol- lows from these equations, that Q is equal to the sum of P and R, as above stated. 20. Resultant of Two Parallel Forces. Any two parallel forces being given, a third parallel force which will balance them can be found from the above equations; and a force equal and opposite to this will be their resultant. We may distinguish two cases. 1. Let the two given forces be in the same direction. Then their resultant is equal to their sum, and acts in the same direction, along a line which cuts the line joining their points of application into two parts which are inversely as the forces. 2. Let the two given forces be in opposite directions. Then their resultant will be equal to their difference, and will act in the direc- tion of the greater of the two forces, along a line which cuts the production of the line joining their points of application on the side of the greater force; and the distances of this point of section from the two given points of application are inversely as the forces. 21. Centre of Two Parallel Forces. In both cases, if the points application are not given, but only the magnitudes of the forces and their lines of action, the magnitude and line of action of the resul- tant are still completely determined; for all straight lines which are drawn across three parallel straight lines are cut by them in the same ratio; and we shall obtain the same result whatever points of application we assume. If the points of application are given, the resultant cut? the line 10 FIRST PRINCIPLES OF DYNAMICS. joining them, or this line produced, in a definite point, whose posi- tion depends only on the magnitudes of the given forces, and not at all on the angle which their direction makes with the joining line. This result is important in connection with centres of gravity. The point so determined is called the centre of the two parallel forces. If these two forces are the weights of two particles, the "centre" thus found is their centre of gravity, and the resultant force is the same as if the two particles were collected at this point. 22. Moments of Resultant and of Components Equal. The follow- ing proposition is often useful. Let any straight line be drawn across the lines of action of two parallel forces P 1? P 2 (Fig. 6). Let O be any point on this line, and x lf x 2 -r^ -r f z the distances measured from to the 4- >/ / points of section, distances measured in opposite directions being distin- guished by opposite signs, and forces in opposite directions being also distinguished by opposite signs. Also let R denote the resultant of P l and P 2 , and x the distance from O to its intersection with the line; then we shall have Px X L + P 2 *, = R X. For, taking the standard case, as represented in Fig. 6, in which all the quantities are positive, we have OA 1 = Xi, OA 2 = x 2) OB = x, and by 19 or 20 we have Pi. AxB-Pj.BAa, that is, P 1 (X-X 1 )='P Z (X,-X), whence (Pi-f?< that is, Rac = 23. Any Number of Parallel Forces in One Plane. Equation (2) affords the readiest means of determining the line of action of the resultant of several parallel forces lying in one plane. For let P!, P 2 , P 3 , &c., be the forces, R x the resultant of the first two forces P I} P 2 , and R 2 the resultant of the first three forces P lf P 2 , P 3 . Let a line be drawn across the lines of action, and let the distances of the points of section from an arbitrary point on this line be expressed according to the following scheme: Force P, P 2 P 3 R, R 2 Distance a?, x. x aj ~x MOMENT OF A FOKCE. 11 Then, by equation (2) we have P 2 afc Also since R 2 is the resultant of ^ and P 3 , we have and substituting for the term R x o?j, we have This reasoning can evidently be extended to any number of forces, so that we shall have finally R#j = sum of such terms as Px, where R denotes the resultant of all the forces, and is equal to their algebraic sum; while ~x denotes the value of x for the point where the line of action of R cuts the fixed line. It is usual to employ the Greek letter S to denote "the sum of such terms as." We may therefore write K=S (P) B* = S (Px) whence -_SJPaO /ox 2 (T*\ 24. Moment of a Force about a Point. When the fixed line is at V right angles to the parallel forces, the product Px is called the moment of the force P about the point 0. More generally, the moment of a force about a point is the product of the force by the length of the perpendicular dropped upon it from the point. The above equations show that for parallel forces in one plane, the moment of the resultant about any point in the plane is the sum of the moments of the forces about the same point. If the resultant passes through 0, the distance x is zero; whence it follows from the equations that the algebraical sum of the moments vanishes. The moment of a force about a point measures the tendency of the force to produce rotation about the point. If one point of a body be fixed, the body will turn in one direction or the other according as the resultant passes on one side or the other of this point (the direction of the resultant being supposed given). If the resultant passes through the fixed point, the body will be in equi- librium. The moment Px of any force about a point, changes sign with P and also with x; thereby expressing (what is obvious in itself) that 12 FIRST PRINCIPLES OF DYNAMICS. the direction in which the force tends to turn the body about the point will be reversed if the direction of P is reversed while its line of action remains unchanged, and will also be reversed if the line of action be shifted to the other side of the point while the direction of the force remains unchanged. 25. Experimental Illustration. Fig. 7 represents a simple appar- atus (called the arithmetical lever) for illustrating the laws of par- Fig. 7. Composition of Parallel Forces. allel forces. The lever AB is suspended at its middle point by a cord, so that when no weights are attached it is horizontal. Equal weights P, P are hung at points A and B equidistant from the centre, and the suspending cord after being passed over a very freely mov- ing pulley M, has a weight F hung at its other end sufficient to pro- duce equilibrium. It will be found that P' is equal to the sum of the two weights P together with the weight required to counter- poise the lever itself. In the second figure, the two weights hung from the lever are riot equal, but one of them is double of the other, P being hung at B, and 2 P at C; and it is necessary for equilibrium that the dis- tance OB be double of the distance OC. The weight P' required COUPLES. 13 to balance the system will now be 3 P together with the weight of the lever. 26. Couple. There is one case of two parallel forces in opposite * directions which requires special attention; that in which the two forces are equal. To obtain some idea of the effect of two such forces, let us first suppose them not exactly equal, but let their difference be very small compared with either of the forces. In this case, the resultant will be equal to this small difference, and its line of action will be at a great distance from those of the given forces. For in 19 if Q is very little greater than P, so that Q-P, or R is only a small fraction -r> TJ of P, the equation gQ=^g shows that AB is only a small fraction of BC, or in other words that BC is very large compared with AB. If Q gradually diminishes until it becomes equal to P, R will gradually diminish to zero; but while it diminishes, the product R . BC will remain constant, being always equal to P . AB. A very small force II at a very great distance would have sensibly the same moment round all points between A and B or anywhere in their neighbourhood, and the moment of R is always equal to the algebraic sum of the moments of P and Q. When Q is equal to P, they compose what is called a couple, and the algebraic sum of their moments about any point in their plane is constant, being always equal to P . AB, which is therefore called the moment of the couple. A couple consists of tiuo equal and parallel forces in opposite directions applied to the same body. The distance between their lines of action is called the arm of the couple, and the product of one of the tivo equal forces by this arm is called the moment of the couple. 27. Composition of Couples. Axis of Couple. A couple cannot be ^ balanced by a single force; but it can be balanced by any couple of equal moment, opposite in sign, if the plane of the second couple be either the same as that of the first or parallel to it. Any number of couples in the same or parallel planes are equiva- lent to a single couple whose moment is the algebraic sum of theirs. The laws of the composition of couples (like those of forces) can be illustrated by geometry. Let a couple be represented by a line perpendicular to its p]ane, marked with an arrow according to the convention that if an 14 FIRST PRINCIPLES OF DYNAMICS. ordinary screw were made to turn in the direction in which the couple tends to turn, it would advance in the direction in which the arrow points. Also let the length of the line represent the moment of the couple. Then the same laws of composition and resolution which hold for forces acting at a point will also hold for couples. A line thus drawn to represent a couple is called the axis of the couple. Just as any number of forces acting at a point are either in equilibrium or equivalent to a single force, so any number of couples applied to the same rigid body (no matter to what parts of it) are either in equilibrium or equivalent to a single couple. 28. Resultant of Force and Couple in Same Plane. The resultant of a force and a couple in the same plane is a single force. For the couple may be replaced by another of equal ft moment having its equal forces P, Q, each equal ~~~] to the given force F, and the latter couple may * then be turned about in its own plane and carried into such a position that one of its two forces destroys the force F, as represented in Fig. 8. There will then only remain the force P, which is equal and parallel to F. By reversing this procedure, we can show that a force P which does not pass through a given point A is equivalent to an equal and parallel force F which does pass through it, together with a couple; the moment of the couple being the same as the moment of the force P about A. 29. General Resultant of any Number of Forces applied to a Rigid Body. Forces applied to a rigid body in lines which do not meet in one point are not in general equivalent to a single force. By the process indicated in the concluding sentence of the preceding section, we can replace the forces by forces equal and parallel to them, acting at any assumed point, together with a number of couples. These couples can then be reduced (by the principles of 27) to a single couple, and the forces at the point can be replaced by a single force; so that we shall obtain, as the complete resultant, a single force applied at any point we choose to select, and a couple. We can in general make the couple smaller by resolving it into two components whose planes are respectively perpendicular and parallel to the force, and then compounding one of these components (the latter) with the force as explained in 28, thus moving the GENERAL RESULTANT. 15 force parallel to itself without altering its magnitude. This is the greatest simplification that is possible. The result is that we have a single force and a couple whose plane is perpendicular to the force. Any combination of forces that can be applied to a rigid body is reducible to a force acting along one definite line and a couple in a plane perpendicular to this line. Such a combination of a force and couple is called a wrench, and the " one definite line " is called the axis of the wrench. The point of application of the force is not definite, but is any point of the axis. 30. Application to Action and Reaction. Every action of force that one body can exert upon another is reducible to a wrench, and the law of reaction is that the second body will, in every case, exert upon the first an equal and opposite wrench. The two wrenches will have the same axis, equal and opposite forces along this axis, and equal and opposite couples in planes perpendicular to it. 31. Resolution the Inverse of Composition. The process of finding the resultant of two or more forces is called composition. The inverse process of finding two or more forces which shall together be equivalent to a given force, is called resolution, and the two or more forces thus found are called components. The problem to resolve a force into two components along two given lines which meet it in one point and are in the same plane with it, has a definite solution, which is obtained by drawing a triangle whose sides are parallel respectively to the given force and the required components. The given force and the required com- ponents will be proportional to the sides of this triangle, each being represented by the side parallel to it. The problem to resolve a force into three components along three given lines which meet it in one point and are not in one plane, also admits of a definite solution. 32. Rectangular Resolution. In the majority of cases which V occur in practice the required components are at right angles to each other, and the resolution is then said to be rectangular. When "the component of a force along a given line" is mentioned, without anything in the context to indicate the direction of the other component or components, it is always to be understood that the resolution is rectangular. The process of finding the required component in this case is illustrated by Fig. 9. Let AB represent the given force F, and let AC be the line along which the com- ponent of F is required. It is only necessary to drop from B a 16 FIRST PRINCIPLES OF DYNAMICS. perpendicular BC on this Jne; AC will represent the required component. CB represents the other component, which, along with B AC, is equivalent to the given force. If the total number of rectangular components, of which AC represents one, is to be three, c then the other two will lie in a plane per- Fig. 9.- Component along a given pendicular to AC, and they can be found by again resolving CB. The magnitude of AC will be the same whether the number of components be two or three, and the component along AC will be F -^ or in trigonometrical language, F cos . BAG. We have thus the following rule: The component of a given force along a given line is found "by multiplying the -force "by the cosine of the angle between its own direction and that of the required component. CHAPTER III. CENTRE OF GRAVITY. 33. Gravity is the force to which we owe the names "up" and " down." The direction in which gravity acts at any place is called the downward direction, and a line drawn accurately in this direc- tion is called vertical; it is the direction assumed by a plumb-line. A plane perpendicular to this direction is called horizontal, and is parallel to the surface of a liquid at rest. The verticals at different places are not parallel, but are inclined at an angle which is approximately proportional to the distance between the places. It amounts to 180 when the places are antipodal, and to about 1' when their distance is one geographical mile, or to about V for every hundred feet. Hence, when we are dealing with the action of gravity on a body a few feet or a few hundred feet in length, we may practically regard the action as consisting of parallel forces. ^ 34. Centre of Gravity. Let A and B be any two particles of a rigid body, let w^ be the weight of the particle A, and w 2 the weight of B. These weights are parallel forces, and their resultant divides the line AB in the inverse ratio of the forces. As the body is turned about into different positions, the forces w-^ and w 2 remain unchanged in magnitude, and hence the resultant cuts AB always in the same point. This point is called the centre of the parallel forces w l and w 2 , or the centre of gravity of the two particles A and B. The magnitude of the resultant will be w l -\-w 2) and we may substitute it for the two forces themselves; in other words, we may suppose the two particles A and B to be collected at their centre of gravity. We can now combine this resultant with the weight of a third particle of the body, and shall thus obtain a resultant , passing through a definite point in the line which joins 2 18 CENTRE OF GRAVITY. the third particle to the centre of gravity of the first two. The first three particles may now be supposed to be collected at this point, and the same reasoning may be extended until all the particles have been collected at one point. This point will be the centre of gravity of the whole body. From the manner in which it has been ob- tained, it possesses the property that the resultant of all the forces of gravity on the body passes through it, in every position in which the body can be placed. The resultant force of gravity upon a rigid body is therefore a single force passing through its centre of gravity. , 35. Centres of Gravity of Volumes, Areas, and Lines. If the body is homogeneous (that is composed of uniform substance throughout), the position of the centre of gravity depends only on the figure, and in this sense it is usual to speak of the centre of gravity of a figure. In like manner it is customary to speak of the centres of gravity of areas and lines, an area being identified in thought with a thin uniform plate, and a line with a thin uniform wire. It is not necessary that a body should be rigid in order that it may have a centre of gravity. We may speak of the centre of gravity of a mass of fluid, or of the centre of gravity of a system of bodies not connected in any way. The same point which would be the centre of gravity if all the parts were rigidly connected, is still called by this name whether they are connected or not. ?< 36. Methods of Finding Centres of Gravity. Whenever a homo- geneous body contains a point which bisects all lines in the body that can be drawn through it, this point must be the centre of gravity. The centres of a sphere, a circle, a cube, a square, an ellipse, an ellipsoid, a parallelogram, and a parallelepiped, are ex- amples. Again, when a body consists of a finite number of parts whose weights and centres of gravity are known, we may regard each part as collected at its own centre of gravity. When the parts are at all numerous, the final result will most readily be obtained by the use of the formula * = 's ( (py' (3) where P denotes the weight of any part, x the distance of its centre of gravity from any plane, and He the distance of the centre of gravity of the whole from that plane. We have already in 23 CENTRE OF GRAVITY OF A TRIANGLE. 19 proved this formula for the case in which the centres of gravity lie in one straight line and x denotes distance from a point in this line; and it is not difficult, by the help of the properties of similar triangles, to make the proof general. 37. Centre of Gravity of a Triangle. To find the centre of gravity of a triangle ABC (Fig. 10), we may begin by supposing it divided into narrow strips by lines (such as be) parallel to BC. It can be shown, by similar triangles, that each of these strips is bisected by the line AD drawn from A to D the middle point of BC. But each strip may be collected at its own centre of gravity, that is at its own middle point; hence the whole triangle may be collected on the line AD; its centre of gravity must therefore be situated upon this line. Similar reason- ing shows that it must lie upon the line Fig - 10 - BE drawn from B to the middle point of AC. It is therefore the intersection of these two lines. If we join DE we can show that the triangles AGB, DGE, are similar, and that AG_AB GD ~ DE ~ DG is therefore one third of DA. The centre of gravity of a triangle therefore lies upon the line joining any corner to the middle point of the opposite side, and is at one-third of the length of this line from the end where it meets that side. It is worthy of remark that if three equal particles are placed at the corners of any triangle, they have the same centre of gravity as the triangle. For the two particles at B and C may be collected at the middle point D, and this double particle at D, together with the single particle at A, will have their centre of gravity at G, since G divides DA in the ratio of 1 to 2. 38. Centre of Gravity of a Pyramid. If a pyramid or a cone be divided into thin slices by planes parallel to its base, and a straight line be drawn from the vertex to the centre of gravity of the base, this line will pass through the centres of gravity of all the slices, since all the slices are similar to the base, and are similarly cut by this line. In a tetrahedron or triangular pyramid, if D (Fig. 11) be the centre of gravity of one face, and A be the corner opposite to this 20 CENTRE OF GRAVITY. face, the centre of gravity of the pyramid must lie upon the line AD. In like manner, if E be the centre of gravity of one face, the centre of gravity of the pyramid must lie upon the line joining E with the oppo- site corner B. It must therefore be the intersection G of these two lines. That they do intersect is otherwise obvious, for the lines AE, BD meet in C, the middle point of one edge of the pyramid, E being found by taking CE '- - A one third of CA, and D by taking CD Fig. 11. Centre of Gravity of Tetral.eJron. Q^Q third of CB If D, E be joined, we can show that the joining line is parallel to BA, and that the triangles AGB, DGE are similar. Hence AG _ AB _ EC GL> ~ DE ~ DC ~ That is, the line AD joining any corner to the centre of gravity of the opposite face, is cut in the ratio of 3 to 1 by the centre of gravity G of the triangle. DG is therefore one-fourth of DA, and the dis- tance of the centre of gravity from any face is one-fourth of the distance of the opposite corner. A pyramid standing on a polygonal base can be cut up into tri- angular pyramids standing on the triangular bases into which the polygon can be divided, and having the same vertex as the whole pyramid. The centres of gravity of these trian- gular pyramids are all at the same perpendicular distance from the base, namely at one-fourth of the distance of the vertex, which is therefore the distance of the centre of gravity of the whole from the base. The centre of gravity of any pyramid is there- fore found by joining the vertex to Fig i2.-centre of Gravity of pyramid. fa Q centre of gravity of the base, and cutting off one-fourth of the joining line from the end where it meets the base. The same rule applies to a cone, since a cone may be regarded as a polygonal pyramid with a very large number of sides. CENTRE OF GRAVITY OF PYRAMID. 21 Fig. 13. Equilibrium of a Body supported on a Horizontal Plane at three or more Points. 39. If four equal particles are placed at the corners of a triangular pyramid, they will have the same centre of gravity as the pyramid. For three of them may, as we have seen ( 37) be collected at the centre of gravity of one face; and the centre of gravity of the four particles will divide the line which joins this point to the fourth, in the ratio of 1 to 3. 40. Condition of Standing or Falling. Wlien a heavy body stands on a base of finite area, and remains in equili- brium under the action of its own weight and the reaction of this base, the vertical through its centre of gravity must fall with- in the base. If the body is supported on three or more points, as in Fig. 13, we are to understand by the base the convex 1 poly- gon whose corners are the points of support; for if a body so supported turns over, it must turn about the line joining two of these points. 41. Body supported at one Point. When a heavy body supported at one point remains at rest, the reaction of the point of support equilibrates the force of gravity. But two forces cannot be in equilibrium unless they have the same line of action; hence the ver- tical through the centre of gravity of the body must pass through the point of support. If instead of being supported at a point, the heavy body is supported by an axis about which it is free to turn, the vertical through the centre of gravity must pass through this axis. 42. Stability and Instability. When the point of support, or axis of support, is vertically below the centre of gravity, it is easily seen that, if the body were displaced a little to either side, the forces act- ing upon it would turn it still further away from the position of equilibrium. On the other hand, when the point or axis of sup- port is vertically above the centre of gravity, the forces which would 1 The word conrcx is inserted to indicate that there must be no re-entrant angles. Any points of support which lie within the polygon formed by joining the rest, must be left out of account. CENTRE OF GRAVITY. act upon it if it were slightly displaced would tend to restore it. In the latter case the equilibrium is said to be stable, in the former unstable. When the centre of gravity coincides with the point of support, or lies upon the axis of support, the body will still be in equilibrium when turned about this point or axis into any other position. In this case the equilibrium is neither stable nor unstable but is called neutral. 43. Experimental determination of Cen- N tre of Gravity. In general, if we suspend a body by any point, in such a manner that it is free to turn about this point, it will come to rest in a position of stable equilibrium. The centre of gravity will then be vertically beneath the point of Fig. 14. Experimental Determination Support. If W6 nOW SUSpend the body from another point, the centre of gravity will come vertically beneath this. The intersection of these two verticals will therefore be the centre of gravity (Fig. 14). 44. To find the centre of gravity of a flat plate or board (Fig. 15), we may suspend it from a point near its circumfer- ence, in such a manner that it sets itself in a ver- tical plane. Let a plumb-line be at the same time suspended from the same point, and made to leave its trace upon the board by chalking and "snap- ping" it. Let the board now be suspended from another point, and the operation be repeated. The two chalk lines will intersect each other at that point of the face which is opposite to the centre of gravity; the centre of gravity itself being of course in the substance of the board. 45. Work done against Gravity. When a heavy body is raised, work is said to be done against gravity, and the amount of this work is reckoned by multiplying together the weight of the body and the height through which it is raised. Horizontal movement does not count, and when a body is raised obliquely, only the vertical component of the motion is to be reckoned. I Suppose, now, that we have a number of particles whose weights Fig. 15. Centre of Gravity of Board. WORK DONE AGAINST GRAVITY. 23 are w lt w 2> W B &c., and let their heights above a given horizontal plane be respectively h l} h^ h 3 &c. We know by equation (3), 23, that if h denote the height of their centre of gravity we have (ivi + w, + &c.) h = v>i hi + w z h z + &c. (4) Let the particles now be raised into new positions in which their heights above the same plane of reference are respectively H 1} H 2 , H 3 &c. The height H of their centre of gravity will now be such that (Wi + w 2 + &c.) H = wi H! + w-i H 2 + &c. (5) From these two equations, we find, by subtraction (i + >2 + &c. ) (if- h) = Wi (H! - hi) + io. 2 (H 2 - fa) + &c. (6) Now HJ h^ is the height through which the particle of weight w 1 has been raised; hence the work done against gravity in raising it is w l (H. 1 h 1 ) and the second member of equation (6) therefore expresses the whole amount of work done against gravity. But the first member expresses the work which would be done in raising all the particles through a uniform height H Ti, which is the height of the new position of the centre of gravity above the old. The work done against gravity in raising any system of bodies will therefore be correctly computed by supposing all the system to be collected at its centre of gravity. For example, the work done in raising bricks and mortar from the ground to build a chimney, is equal to the total weight of the chimney multiplied by the height - t of its centre of gravity above the ground. 46. The Centre of Gravity tends to Descend. When the forces x which tend to move a system are simply the weights of its parts, we can determine whether it is in equilibrium by observing the path in which its centre of gravity would travel if movement took place. If we suppose this path to represent a hard frictionless surface, and the centre of gravity to represent a heavy particle placed upon it, the conditions of equilibrium will be the same as in the actual case. The centre of gravity tends to run down hill, just as a heavy particle does. There will be stable equilibrium if the centre of gravity is at the bottom of a valley in its path, and unstable equilibrium if it is at the top of a hill. When a rigid body turns about a horizontal axis, the path of its centre of gravity is a circle in a vertical plane. The highest and lowest points of this circle are the positions of the centre of gravity in unstable and stable equilibrium respectively; 24 CENTRE OF GRAVITY. except when the axis traverses the centre of gravity itself, in which case the centre of gravity can neither rise nor fall, and the equili- brium is neutral. A uniform sphere or cylinder lying on a horizontal plane is in neutral equilibrium, because its centre of gravity will neither be raised nor lowered by rolling. An egg balanced on its end as in Fig. 16, is in unstable equilibrium, because its centre of gravity is at the top of a hill which it will descend when the egg rolls to one side. The position of equilibrium shown in Fig. 17 is stable as regards rolling to left or right, because the path of its centre of gravity in Fig 16. Unstable Equilibrium. Fig. 17. Stable Equilibrium. such rolling would be a curve whose lowest point is that now occu- pied by the centre of gravity. As regards rolling in the direction at right angles to this, if the egg is a true solid of resolution, the equili- brium is neutral. 47. Work done by Gravity. When a heavy body is lifted, the lifting force does work against gravity. When it descends gravity does work upon it; and if it descends to the same position from which it was lifted, the work done by gravity in the descent is equal to the work done against gravity in the lifting; each being equal to the weight of the body multiplied by the vertical displace- ment of its centre of gravity. The tendency of the centre of gravity to descend is a manifestation of the tendency of gravity to do work; and this tendency is not peculiar to gravity. 48. Work done by any Force. A force is said to do work when its point of application moves in the direction of the force, or in any direction making an acute angle with this, so as to give a component displacement in the direction of the force; and the amount of work done is the product of the force by this component. If F denote PRINCIPLE OF WORK. 25 the force, a the displacement, and the angle between the two, the work done by F is F a cos 0. which is what we obtain either by the above rule or by multiplying the whole displacement by the effective component of F, that is the component of F in the direction of the displacement. If the angle 6 is obtuse, cos 6 is negative and the force F does negative work. If is a right angle F does no work. In this case F neither assists nor resists the displacement. When is acute, F assists the dis- placement, and would produce it if the body were constrained by guides which left it free to take this displacement and the directly opposite one, while preventing all others. If 6 is obtuse, F resists the displacement, and would produce the opposite displacement if the body were constrained in the manner just supposed. 49. Principle of Work. If any number of forces act upon a body V which is only free to move in a particular direction and its opposite, we can tell in which of these two directions it will move by calcu- lating the work which each force would do. Each force would do positive work when the displacement is in one direction, and nega- tive work when it is in the opposite direction, the absolute amounts of work being the same in both cases if the displacements are equal. The body will upon the whole be urged in that direction which gives an excess of positive work over negative. If no such excess exists, but the amounts of positive and negative work are exactly equal, the body is in equilibrium. This principle (which has been called the principle of virtual velocities, but is better called the principle of work) is often of great use in enabling us to calculate the ratio which two forces applied in given ways to the same body must have in order to equilibrate each other. It applies not only to the "mechanical powers" and all combinations of solid machinery, but also to hydrostatic arrangements; for example to the hydraulic press. The condition of equilibrium between two forces applied to any frictionless machine and tending to drive it opposite ways, is that in a small movement of the machine they would do equal and opposite amounts of work. Thus in the screw-press (Fig. 30) the force applied to one of the handles, multiplied by the distance through which this handle moves, will be equal to the pressure which this force produces at the foot of the screw, multiplied by the distance that the screw travels. 26 CENTRE OF GRAVITY. This is on the supposition of no friction. A f rictionless machine gives out the same amount of work which is spent in driving it. The effect of friction is to make the work given out less than the work put in. Much fruitless ingenuity has been expended upon contrivances for circumventing this law of nature and producing a machine which shall give out more work than is put into it. Such contrivances are called " perpetual motions." 50. General Criterion of Stability. If the forces which act upon a body and produce equilibrium remain unchanged in magnitude and direction when the body moves away from its position, and if the velocities of their points of application also remain unchanged in direction and in their ratio to each other, it is obvious that the equality of positive and negative work which subsists at the beginning of the motion will continue to subsist throughout the entire motion. The body will therefore remain in equilibrium when displaced. Its equilibrium is in this case said to be neutral. If the forces which are in equilibrium in a given position of the body, gradually change in direction or magnitude as the body moves away from this position, the equality of positive and negative work will not in general continue to subsist, and the inequality will increase with the displacement. If the body be displaced with a constant velocity and in a uniform manner, the rate of doing work, which is zero at first, will not continue to be zero, but will have a value, whether positive or negative, increasing in simple proportion to the displacement. Hence it can be shown that the whole work done is proportional to the square of the displacement, for when we double the displacement we, at the same time, double the mean working force. If this work is positive, the forces assist the displacement and tend to increase it; the equilibrium must therefore have been unstable. On the other hand, if the work is negative in all possible displace- ments from the position of equilibrium, the forces oppose the displacements and the equilibrium is stable. 51. Illustration of Stability. A good example of stable equili- brium of this kind is furnished by Gravesande's apparatus (Fig. 3) simplified by removing the parallelogram and employing a string to support the three weights, one of them P" being fastened to it at a point A near its middle, and the others P, P' to its ends. The point A will take the same position as in the figure, and will return to it again when displaced. If we take hold of the point A and STABILITY. 27 move it in any direction whether in the plane of the string or out of it, we feel that at first there is hardly any resistance and the smallest force we can apply produces a sensible disturbance; but that as the displacement increases the resistance becomes greater. If we release the point A when displaced, it will execute oscillations, which will become gradually smaller, owing to friction, and it will finally come to rest in its original position of equilibrium. The centre of gravity of the three weights is in its lowest position when the system is in equilibrium, and when a small dis- placement is produced the centre of gravity rises by an amount proportional to its square, so that a double displacement produces a quadruple rise of the centre of gravity. In this illustration the three forces remain unchanged, and the directions of two of them change gradually as the point A is moved. Whenever the circumstances of stable equilibrium are such that the forces make no abrupt changes either in direction or magnitude for small displacements, the resistance will, as in this case, be propor- tional to the displacement (when small), and the work to the square of the displacement, and the system will oscillate if displaced and then left to itself. 52. Stability where Forces vary abruptly with Position. There are other cases of stable equilibrium which may be illustrated by the example of a book lying on a table. If we displace it by lifting one edge, the force which we must exert does not increase with the displacement, but is sensibly constant when the displacement is small, and as a consequence the work will be simply proportional to the displacement. The reason is, that one of the forces concerned in producing equilibrium, namely, the upward pressure of the table, changes per saltum at the moment when the displacement begins. In applying the principle of work to such a case as this, we must employ, instead of the actual work done by the force which changes abruptly, the work which it would do if its magnitude and direction remained unchanged, or only changed gradually. 53. Illustrations from Toys. The stability of the "balancer" (Fig. 18) depends on the fact that, owing to the weight of the two leaden balls, which are rigidly attached to the figure by stiff wires, the centre of gravity of the whole is below the point of support. If the figure be disturbed it oscillates, and finally comes to rest in a position in which the centre of gravity is vertically under the toe on which the figure stands. 28 CENTRE OF GRAVITY. The "tumbler" (Fig. 19) consists of a light figure attached to a hemisphere of lead, the centre of gravity of the whole being between the centre of gravity of the hemisphere and the centre of the sphere to which it belongs. When placed upon a level table, the lowest position of the centre of gravity is that in which the figure is upright, and it accord- ingly returns to this position when displaced. 54. Limits of Stability. In the foregoing discussion we have em- ployed the term "stability" in its strict mathematical sense. But there are cases in which, though small displacements would merely produce small oscillations, larger displacements would cause the body, when left to itself, to fall entirely away from the given position of equilibrium. This may be expressed by saying that the equilibrium is stable for displacements lying within certain limits, but unstable for displacements beyond these limits. The equilibrium Pig. 18. Balancer. Fig. 19. Tumblers. of a system is practically unstable when the displacements which it is likely to receive from accidental disturbances lie beyond its limits of stability. CHAPTER IV. THE MECHANICAL POWERS. 55. We now proceed to a few practical applications of the fore- gcing principles; and we shall begin with the so-called "mechanical powers," namely, the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. 56. Lever. Problems relating to the lever are usually most con- veniently solved by taking moments" round the fulcrum. The general condition of equilibrium is, that the moments of the power and the weight about the fulcrum must be in opposite directions, and must be equal. When the power and weight act in parallel directions, the conditions of equilibrium are precisely those of three parallel forces ( 19), the third force being the reaction of the fulcrum. It is usual to distinguish three " orders " of lever. In levers of the first order (Fig. 20) the fulcrum is between the power and the h Fig. 20. Fig. 21. Fig. 22. Three Orders of Lever. weight. In those of the second order (Fig. 21) the weight is between the power and the fulcrum. In those of the third order (Fig. 22) the power is between the weight and the fulcrum. In levers of the second order (supposing the forces parallel), the weight is equal to the sum of the power and the pressure on the fulcrum; and in levers of the third order, the power is equal to the sum of the weight and the pressure on the fulcrum; since the middle one of three parallel forces in equilibrium must always be equal to the sum of the other two. 30 THE MECHANICAL POWERS. 57. Anns, The arms of a lever are the two portions of it inter- mediate, respectively, between the fulcrum and the power, and between the fulcrum and the weight. If the lever is bent, or if, though straight, it is not at right angles to the lines of action of the power and weight, it is necessary to distinguish between the arms of the lever as above defined (which are parts of the lever), and the arms of the power and weight regarded as forces which have moments round the fulcrum. In this latter sense (which is always to be understood unless the contrary is evidently intended), the arms are the perpendiculars dropped from the fulcrum upon the lines of action of the power and weight. 58. Weight of Lever. In the above statements of the conditions of equilibrium, we have neglected the weight of the lever itself. To take this into account, we have only to suppose the whole weight of the lever collected at its centre of gravity, and then take its moment round the fulcrum. We shall thus have three moments to take account of, and the sum of the two that tend to turn the lever one way, must be equal to the one that tends to turn it the opposite way. 59. Mechanical Advantage. Every machine when in action serves to transmit work without altering its amount; but the force which the machine gives out (equal and opposite to what is commonly called the weight) may be much greater or much less than that by which it is driven (commonly called the power). When it is greater, the machine is said to confer mechanical advantage, and the mechanical advantage is measured by the ratio of the weight to the power for equilibrium. In the lever, when the power has a longer arm than the weight, the mechanical advantage is equal to the quotient of the longer arm by the shorter. 60. Wheel and Axle. The wheel and axle (Fig. 23) may be regarded as an endless lever. The condition of equili- brium is at once given by taking moments round the common axis of the wheel and axle ( 24). If we neglect the thickness of the ropes, the condition is that the power multiplied by the radius of the wheel must A l\ e( l ua l the weight multiplied by the radius of the axle; but it is more exact to regard the lines of action of the two forces as coinciding with the axes of the two ropes, so that each of the two radii should be increased by half the thick- ness of its own rope. If we neglect the thickness of the ropes, the PULLEYS. 31 mechanical advantage is the quotient of the radius of the wheel by the radius of the axle. 61. Pulley. A pulley, when fixed in such a way that it can only turn about a fixed axis (Fig. 24), confers no mechanical advantage. It may be regarded as an endless lever of the first order with its two arms equal. The arrangement represented in Fig. 25 gives a mechanical advantage of 2 ; for the lower or movable pulley may be regarded as an endless lever of the second order, in which the arm of the power is the diameter of the pulley, and the arm of the weight is Fig. 24. Fig. 25. Fig. 26. Fig. 27. half the diameter. The same result is obtained by employing the principle of work; for if the weight rises 1 inch, 2 inches of slack are given over, and therefore the power descends 2 inches. 62. In Fig. 26 there are six pulleys, three at the upper and three at the lower block, and one cord passes round them all. All por- tions of this cord (neglecting friction) are stretched with the same force, which is equal to the power; and six of these portions, parallel to one another, support the weight. The power is therefore one- sixth of the weight, or the mechanical advantage is 6. 63. In the arrangement represented in Fig. 27, there are three movable pulleys, each hanging by a separate cord. The cord which supports the lowest pulley is stretched with a force equal to half the weight, since its two parallel portions jointly support the weight. The cord which supports the next pulley is stretched with a force half of this, or a quarter of the weight; and the next cord with a force half of this, or an eighth of the weight; but this cord is directly attached to the power. Thus the power is an eighth of the 32 THE MECHANICAL POWERS. weight, or the mechanical advantage is 8. If the weight and the block 1 to which it is attached rise 1 inch, the next block rises 2 inches, the next 4, and the power moves through 8 inches. Thus, the work done by the power is equal to the work done upon the weight. In all this reasoning we neglect the weights of the blocks them- selves; but it is not difficult to take them into account when necessary. 64. Inclined Plane. We now come to the inclined plane. Let AB (Fig. 28) be any portion of such a plane, and let AC and BC be drawn vertically and horizontally. Then AB is called the length, AC the height, and CB the base of the inclined plane. The force of gravity upon a heavy body M resting on the plane, may be represented by a vertical line Fig. 28. MP, an( j mav be reso lved by the parallelogram of forces ( 16) into two components, MT, MN, the former parallel and the latter perpendicular to the plane. A force equal and oppo- site to the component MT will suffice to prevent the body from slip- ping down the plane. Hence, if the power act parallel to the plane, and the weight be that of a heavy body resting on the plane, the power is to the weight as MT to MP; but the two triangles MTP and ACB are similar, since the angles at M and A are equal, and the angles at T and C are right angles; hence MT is to MP as AC to AB, that is, as the height to the length of the plane. 65. The investigation is rather easier by the principle of work ( 49). The work done by the power in drawing the heavy body up the plane, is equal to the power multiplied by the length of the plane. But the work done upon the weight is equal to the weight multiplied by the height through which it is raised, that is, by the height of the plane. Hence we have Power x length of plane = weight X height of plane ; or power : weight : : height of plane : length of plane. 66. If, instead of acting parallel to the plane, the power acted parallel to the base, the work done by the power would be the product of the power by the base; and this must be equal to the product of the weight by the height; so that in this case the con- dition of equilibrium would be 1 The " pulley " is the revolving wheel. The pulley, together with the frame in which it is inclosed, constitute the "block." SCREW. 33 Power : weight : : height of plane : base of plane. 67. Wedge. In these investigations we have neglected friction. The wedge may be regarded as a case of the inclined plane; but its practical action depends to such a large extent upon friction and impact 1 that we cannot profitably discuss it here. 68. Screw. The screw (Fig. 29) is also a case of the inclined plane. The length of one convolution of the thread is the length of the corresponding inclined plane, the step of the screw, or distance between two successive convolutions (measured parallel to the axis of the screw), is the height of the plane, and the circumference of Fig. 29. Fig. 30. the screw is the base of the plane. This is easily shown by cutting out a right-angled triangle in paper, and bending it in cylindrical fashion so that its base forms a circle. 69. Screw Press. In the screw press (Fig. 30) the screw is turned by means of a lever, which gives a great increase of mechanical advantage. In one complete revolution, the pressures applied to the two handles of the lever to turn it, do work equal to their sum multiplied by the circumference of the circle described (approxi- mately) by either handle (we suppose the two handles to be equi- distant from the axis of revolution); and the work given out by the machine, supposing the resistance at its lower end to be constant, is equal to this resistance multiplied by the distance between the threads. These two products must be equal, friction being neglected. 1 An impact (for example a blow of a hammer) may be regarded as a very great (and variable) force acting for a very short time. The magnitude of an impact is measured by the momentum which it generates in the body struck. 3 CHAPTER V, THE BALANCE. 70. General Description of the Balance. In the common balance (Fig. 31) there is a stiff piece of metal, A B, called the beam, which turns about the sharp edge O of a steel wedge form- ing part of the beam and resting upon two hard and smooth supports. There are two other steel wedges at A and B, with their edges upwards, and upon these edges rest the hooks for supporting the scale pans. The three edges (called knife-edges) are parallel to one another and perpen- dicular to the length of the beam, and are very nearly in one plane. 71. Qualities Requisite. The qualities requisite in a balance are: 1. That it be consistent with itself; that is, that it shall give the same result in successive weighings of the same body. This depends chiefly on the trueness of the knife-edges. 2. That it be just. This requires that the distances A 0, OB, be equal, and also that the beam remain horizontal when the pans are empty. Any inequality in the distances A 0, OB, can be detected by putting equal (and tolerably heavy) weights into the two pans. This adds equal moments if the distances are equal, but unequal Fig. 31. Balance. SENSIBILITY OF BALANCE. 35 moments if they are unequal, and the greater moment will prepon- derate. 3. Delicacy or sensibility (that is, the power of indicating in- equality between two weights even when their difference is very small). This requires a minimum of friction, and a very near approach to neutral equilibrium ( 40). In absolutely neutral equilibrium, the smallest conceivable force is sufficient to produce a displacement to the full limit of neutrality; and in barely stable equilibrium a small force produces a la^ge displacement. The condition of stability is that if the weights supported at A and B be supposed collected at these edges, the centre of gravity of the system composed of the beam and these two weights shall be below the middle edge 0. The equilibrium would be neutral if this centre of gravity exactly coin- cided with 0; and it is necessary as a condition of delicacy that its distance below be very small. 4. Facility for weighing quickly is desirable, but must sometimes be sacrificed when extreme accuracy is required. The delicate balances used in chemical analysis are provided with a long pointer attached to the beam. The end of this pointer moves along a graduated arc as the beam vibrates; and if the weights in the two pans are equal, the excursions of the pointer on opposite sides of the zero point of this arc will also be equal. Much time is con- sumed in watching these vibrations, as they are very slow; and the more nearly the equilibrium approaches to neutrality, the slower they are. Hence quick weighing and exact weighing are to a certain ex- tent incompatible. 72. Double Weighing. Even if a balance be not just, yet if it be consistent with itself, a correct weighing can be made with it in the following manner: Put the body to be weighed in one pan, and counterbalance it with sand or other suitable material in the other. Then remove the body and put in its place such weights as are just sufficient to counterpoise the sand. These weights are evidently equal to the weight of the body. This process is called double weighing, and is often employed (even with the best balances) when the greatest possible accuracy is desired. 73. Investigation of Sensibility. Let A and B (Fig. 32) be the points from which the scale-pans are suspended, the axis about which the beam turns, and G the centre of gravity of the beam. If when the scale-pans are loaded with equal weights, we put into one .,/] 36 THE BALANCE. of them an excess of weight p, the beam will become inclined, and will take a position such as A'B', turning through an angle which we will call a, and which is easily calculated. In fact let the two forces P and P + p act at A 7 and B' respec- tively, where P denotes the less of the two weights, including the weight of the pan. Then the two forces P destroy each other in conse- quence of the resistance of the axis O; there is left only the force p applied at B', and the weight ?r of the beam applied at G', the new position of the centre of gravity. i'-f P These two forces are parallel, and are in equilibrium about the axis O, that is, their resultant passes through the Fi s- 32 - point O. The distances of the points of application of the forces from a vertical through O are therefore inversely proportional to the forces themselves, which gives the relation IT. G'U=p. B'L. But if we call half the length of the beam I, and the distance OG r we have G-'K = r sin a, B'L = I cos a. whence wr sin a pi cos a, and consequently tan a.-^. (a) irr The formula (a) contains the entire theory of the sensibility of the balance when properly constructed. We see, in the first place, that tan a increases with the excess of weight p, which was evident be- forehand. We see also that the sensibility increases as I increases and as TT diminishes, or, in other words, as the beam becomes longer and lighter. At the same time it is obviously desirable that, under the action of the weights employed, the beam should be stiff enough to undergo no sensible change of shape. The problem of the balance then consists in constructing a beam of the greatest possible length and lightness, which shall be capable of supporting the action of given forces without bending. Fortin, whose balances are justly esteemed, employed for his beams bars of steel placed edgewise; he thus obtained great rigidity, but SENSIBILITY. 37 certainly not all the lightness possible. At present the makers of balances employ in preference beams of copper or steel made in the form of a frame, as shown in Fig 33. They generally give them the shape of a very elongated lozenge,, the sides of which are connected by bars variously arranged. The determination of the best shape is, in fact, a special problem, and is an application on a small scale of that principle of applied mechanics which teaches us that hollow pieces have greater resisting power in proportion to their weight than solid pieces, and consequently, for equal resisting power, the former are lighter than the latter. Aluminium, which with a rigidity nearly equal to that of copper, has less than one-fourth of its density, seems naturally marked out as adapted to the construction of beams. It has as yet, however, been little used. The formula (a) shows us, in the second place, that the sensibility increases as r diminishes; that is, as the centre of gravity approaches the centre of suspension. These two points, however, must not coin- cide, for in that case for any excess of weight, however small, the beam would deviate from the horizontal as far as the mechanism would permit, and would afford no indication of approach to equality in the weights. With equal weights it would remain in equilibrium in any position. In virtue of possessing this last property, such a balance is called indifferent. Practically the distance between the centre of gravity and the point of suspension must not be less than a certain amount depending on the use for which the balance is designed. The proper distance is determined by observing what difference of weights corresponds to a division of the graduated arc along which the needle moves. If, for example, there are 20 divi- sions on each side of zero, and if 2 milligrammes are necessary for the total displacement of the needle, each division will correspond to an excess of weight of -^ or -^ of a milligramme. That this may be the case we must evidently have a suitable value of r, and the maker is enabled to regulate this value with precision by means of the screw which is shown in the figure above the beam, and which enables him slightly to vary the position of the centre of gravity. 74. Weighing with Constant Load. In the above analysis we have supposed that the three points of suspension of the beam and of the two scale-pans are in one straight line; in which case the value of tan a does not include P, that is, the sensibility is independent of the weight in the pans. This follows from the fact that the resultant of the two forces P passes through O, and is thus destroyed, because 38 THE BALANCE. the axis is fixed. This would not be the case if, for example, the points of suspension of the pans were above that of the beam; in this case the point of application of the common load is above the point O, and, when the beam is inclined, acts in the same direction as the excess of weight; whence the sensibility increases with the load up to a certain limit, beyond which the equilibrium becomes unstable. 1 On the other hand, when the points of suspension of the pans are below that of the beam, the sensibility increases as the load diminishes, and, as the centre of gravity of the beam may in this case be above the axis, equilibrium may become unstable when the load is less than a certain amount. This variation of the sensibility with the load is a serious disadvantage; for, as we have just shown, the displacement of the needle is used as the means of estimating weights, and for this purpose we must have the same displacement corresponding to the same excess of weight. If we wish to employ Fig. 33. Beam of Balance. either of the two above arrangements, we should weigh with a con- stant load. The method of doing so, which constitutes a kind of double weighing, consists in retaining in one of the pans a weight equal to this constant load. In the other pan is placed the same load subdivided into a number of marked weights. When the body 1 This is an illustration of the general principle, applicable to a great variety of philo- sophical apparatus, that a maximum of sensibility involves a minimum of stability ; that is, a very near approach to instability. This near approach is usually indicated by exces- sive slowness in the oscillations which take place about the position of equilibrium. BALANCES OF PKECISION. 39 to be weighed is placed in this latter pan, we must, in order to main- tain equilibrium, remove a certain number of weights, which evi- dently represent the weight of the body. We may also remark that, strictly speaking, the sensibility always depends upon the load, which necessarily produces a variation in the friction of the axis of suspension. Besides, it follows from the nature Fig. 34. Balance for Purposes of Accuracy. of bodies that there is no system that does not yield somewhat even to the most feeble action. For these reasons, there is a decided advantage in operating with constant load. 75. Details of Construction. A fundamental condition of the cor- rectness of the balance is, that the weight of each pan and of the load which it contains should always act exactly at the same point, and therefore at the same distance from the axis of suspension. This important result is attained by different methods. The arrange- ment represented in Fig. 33 is one of the most effectual. At the 40 THE BALANCE. extremities of the beam are two knife-edges, parallel to the axis of rotation, and facing upwards. On these knife-edges rests, by a hard plane surface of agate or steel, a stirrup, the front of which has been taken away in the figure. On the lower part of the stirrup rests another knife-edge, at right angles to the former, the two being together equivalent to a universal joint supporting the scale-pan and its contents. By this arrangement, whatever may be the position of the weights, their action is always reduced to a vertical force act- ing on the upper knife-edge. Fig. 34 represents a balance of great delicacy, with the glass case that contains it. At the bottom is seen the extremity of a lever, which enables us to raise the beam, and thus avoid wearing the knife-edge when not in use. At the top may be remarked an arrangement employed by some makers, consisting of a horizontal graduated circle, on which a small metallic index can be made to travel; its different displacements, whose value can be determined once for all, are used for the final adjustment to produce exact equilibrium. 76. Steelyard. The steelyard (Fig. 35) is an instrument for weighing bodies by means of a single weight, P, which can be hung at any point of a graduated arm OB. As P is moved further from the fulcrum O, its moment round O increases, and there- fore the weight which must be hung from the fixed point A to counterbalance it in- creases. Moreover, equal movements of P along the arm pro- duce equal additions to its moment, and equal additions to the weight at A produce equal additions to the opposing moment. Hence the divisions on the arm (which indicate the weight in the pan at A) must be equidistant. Fig. 35. CHAPTER VI FIRST PRINCIPLES OF KINETICS. 77. Principle of Inertia. A body not acted on by any forces, or only acted on by forces which are in equilibrium, will not commence to move; and if it be already in motion with a movement of pure translation, it will continue its velocity of translation unchanged, so that each of its points will move in a straight line with uniform velocity. This is Newton's first law of motion, and is stated by him in the following terms: " Every body continues in its state of rest or of uniform motion in a straight line, except in so far as it is compelled by impressed forces to change that state." The tendency to continue in a state of rest is manifest to the most superficial observation. The tendency to continue in a state of uniform motion can be clearly understood from an attentive study of facts. If, for example, we make a pendulum oscillate, the amplitude of the oscillations slowly decreases and at last vanishes altogether. This is because the pendulum experiences resistance from the air which it continually displaces; and because the axis of suspension rubs on its supports. These two circumstances combine to produce a diminution in the velocity of the apparatus until it is completely annihilated. If the friction at the point of suspension is diminished by suitable means, and the apparatus is made to oscillate in vacua, the duration of the motion will be immensely increased. Analogy evidently indicates that if it were possible to suppress entirely these two causes of the destruction of the pendulum's velo- city, its motion would continue for an indefinite time unchanged. This tendency to continue in motion is the cause of the effects which are produced when a carriage or railway train is suddenly stopped. The passengers are thrown in the direction of the motion, 42 FIRST PRINCIPLES OF KINETICS. in virtue of the velocity which they possessed at the moment when the stoppage occurred. If it were possible to find a brake sufficiently powerful to stop a train suddenly at full speed, the effects of such a stoppage would be similar to the effects of a collision. Inertia is also the cause of the severe falls which are often received in alighting incautiously from a carriage in motion; all the particles of the body have a forward motion, and the feet alone being reduced to rest, the upper portion of the body continues to move, and is thus thrown forward. When we fix the head of a hammer on the handle by striking the end of the handle on the ground, we utilize the inertia of matter. The handle is suddenly stopped ' by the collision, and the head con- tinues to move for a short distance in spite of the powerful resist- ances which oppose it. 78. Second Law of Motion. Newton's second law of motion is that " Change of motion is proportional to the impressed force and is in the direction of that force." Change of motion is here spoken of as a quantity, and as a directed quantity. In order to understand how to estimate change of motion, we must in the first place understand how to compound motions. When a boat is sailing on a river, the motion of the boat relative to the shore is compounded of its motion relative to the water and the motion of the water relative to the shore. If a person is walk- ing along the deck of the boat in any direction, his motion relative to the shore is compounded of three motions, namely the two above mentioned and his motion relative to the boat. Let A, B and C be any three bodies or systems. The motion of A relative to B, compounded with the motion of B relative to C, is the motion of A relative to C. This is to be taken as the definition of what is meant by compounding two motions ; and it leads very directly Fig. 36.-Composition of Motions. fo the result thftt two rectilinear motions are compounded by the parallelogram law. For if a body moves along the deck of a ship from O to A (Fig. 36), and the ship in the meantime advances through the distance OB, it is obvious that, if we complete the parallelogram OBCA, the point A of the ship will be brought to C, and the movement of the body in space will be from O to C. If the motion along OA is uniform. SECOND LAW OF MOTION. 43 and the motion of the ship is also uniform, the motion of the body through space will be a uniform motion along the diagonal OC. Hence, if two component velocities be represented by two lines drawn from a point, and a parallelogram be constructed on these Lines, its diagonal will represent the resultant velocity. It is obvious that if OA in the figure represented the velocity of the ship and OB the velocity of the body relative to the ship, we should obtain the same resultant velocity 00. This is a general law; the interchanging of velocities which are to be compounded does not affect their resultant. Now suppose the velocity OB to be changed into the velocity OC, what are we to regard as the change of velocity? The change of velocity is that velocity which compounded with OB would give OC It is therefore OA. The same force which, in a given time, acting always parallel to itself, changes the velocity of a body from OB to OC, would give the body the velocity OA if applied to it for the same time commencing from rest. Change of motion, estimated in this way, depends only on the acting force and the body acted on by the force; it is entirely independent of any previous motion which the body may possess. No experiments on forces and motions inside a carriage or steamboat which is travelling with perfect smoothness in a straight course, will enable us to detect that it is travelling at all. We cannot even assert that there is any such thing as absolute rest, or that there is any difference between absolute rest and uniform straight movement of translation. As change of motion is independent of the initial condition of rest or motion, so also is the change of motion produced by one force act- ing on a body independent of the change produced by any other force acting on the body, provided that each force remains constant in magnitude and direction. The actual motion will be that which is compounded of the initial motion and the motions due to the two forces considered separately. If AB represent one of these motions, BC another, and CD the third, the actual or resultant motion will be AD. The change produced in the motion of a body by two forces act- ing jointly can therefore be found by compounding the changes which would be produced by each force separately. This leads at once to the " parallelogram of forces," since the changes of motion produced in one and the same body are proportional to the forces which produce them, and are in the directions of these forces. 44 FIRST PRINCIPLES OF KINETICS. In case any student should be troubled by doubt as to whether the "changes of motion" which are proportional to the forces, are to be understood as distances, or as velocities, we may remark that the law is equally true for both, and its truth for one implies its truth for the other, as will appear hereafter from comparing the formula for the distance s = %ft 2 , with the formula for the velocity v = ft, since both of these expressions are proportional to /. 79. Explanation of Second Law continued. It is convenient to distinguish between the intensity of a force and the magnitude or amount of a force. The intensity of a force is measured by the change of velocity which the force produces during the unit of time; and can be computed from knowing the motion of the body acted on, without knowing anything as to its mass. Two bodies are said to be of equal mass when the same change of motion (whether as regards velocity or distance) which is produced by applying a given force to one of them for a given time, would also be produced by applying this force to the other for an equal time. If we join two such bodies, we obtain a body of double the mass of either; and if we apply the same force as before for the same time to this double mass, we shall obtain only half the change of velocity or distance that we obtained before. Masses can therefore be compared by taking the in- verse ratio of the changes produced in their velocities by equal forces. The velocity of a body multiplied by its mass is called the momen- tum of the body, and is to be regarded as a directed magnitude hav- ing the same direction as the velocity. The change of velocity, when multiplied by the mass of the body, gives the change of momentum; and the second law of motion may be thus stated: The change of momentum produced in a given time is propor- tional to the force which produces it, and is in the direction of this force. It is independent of the mass; the change of velocity in a given time being inversely as the mass. 80. Proper Selection of Unit of Force. If we make a proper selec- . tion of units, the change of momentum produced in unit time will be not only proportional but numerically equal to the force which produces it; and the change of momentum produced in any time will be the product of the force by the time. Suppose any units of length, time, and mass respectively to have been selected. Then the unit velocity will naturally be denned as the velocity with which unit length would be passed over in unit time; the unit momentum will be the momentum of the unit mass moving with this velocity; UNIT OF FORCE. 45 and the unit force will be that force which produces this momentum in unit time. We define the unit force, then, as that force which acting for unit time upon unit mass produces unit velocity. 81. Relation between Mass and Weight. The weight of a body, strictly speaking, is the force with which the body tends towards the earth. This force depends partly on the body and partly on the earth. It is not exactly the same for one and the same body at all parts of the earth's surface, but is decidedly greater in the polar than in the equatorial regions. Bodies which, when weighed in a balance in vacuo, counterbalance each other, or counterbalance one and the same third body, have equal weights at that place, and will also be found to have equal weights at any other place. Experiments which we shall hereafter describe ( 89) show that such bodies have equal masses; and this fact having been established, the most convenient mode of comparing masses is by weighing them. A pound of iron has the same mass as a pound of brass or of any other substance. A pound of any kind of matter tends to the earth with different forces at different places. The weight of a pound of matter is therefore not a definite standard of force. But the pound of matter itself is a perfectly definite standard of mass. If we weigh one and the same portion of matter in different states; for instance water in the states of ice, snow, liquid water, or steam; or compare the weight of a chemical compound with the weights of its components; we find an exact equality; hence it has been stated that the mass of a body is a measure of the quantity of matter which it contains; but though this statement expresses a simple fact when applied to the compari- son of different quantities of one and the same substance, it expresses no known fact of nature when applied to the comparison of different substances. A pound of iron and a pound of lead tend to the earth with equal forces; and if equal forces are applied to them both their velocities are equally affected. We may if we please agree to mea- sure "quantity of matter" by these tests; but we must beware of assuming that two things which are essentially different in kind can be equal in themselves. 82. Third Law of Motion. Action and Reaction. Forces always occur in pairs, every exertion of force being a mutual action between two bodies. Whenever a body is acted on by a force, the body from which this force proceeds is acted on by an equal and opposite force. The earth attracts the moon, and the moon attracts the earth. A magnet attracts iron and is attracted by iron. When two 46 FIRST PRINCIPLES OF KINETICS. boats are floating freely, a rope attached to one and hauled in by a person in the other, makes each boat move towards the other. Every exertion of force generates equal and opposite momenta in the two bodies affected by it, since these two bodies are acted on by equal forces for equal times. If the forces exerted by one body upon the other are equivalent to a single force, the forces of reaction will also be equivalent to a single force, and these two equal and opposite resultants will have the same line of action. We have seen in 29 that the general resultant of any set of forces applied to a body is a wrench; that is to say it consists of a force with a definite line of action (called the axis), accompanied by a couple in a perpendicular plane. The reac- tion upon the body which exerts these forces will always be an equal and opposite wrench; the two wrenches having the same axis, equal and opposite forces along this axis, and equal and opposite couples in the perpendicular plane. 83. Motion of Centre of Gravity Unaffected. A consequence of the equality of the mutual forces between two bodies is, that these forces produce no movement of the common centre of gravity of the two bodies. For if A be the centre of gravity of a mass m l} and B the centre of gravity of a mass m 2 , their common centre of gravity C will divide AB inversely as the masses. Let the masses be originally at rest, and let them be acted on only by their mutual attraction or replusion. The distances through which they are moved by these equal forces will be inversely as the masses, that is, will be directly as AC and BC ; hence if A' B' are their new positions after any time, we have AC _ AA' _ AC AA' _ A/C BC ~ BB' ~ LC BB' " B'C* The line A'B' is therefore divided at C in the same ratio in which the line AB was divided; hence C is still the centre of gravity. 84. Velocity of Centre of Gravity. If any number of masses are moving with any velocities, and in any directions, but so that each of them moves uniformly in a straight line, their common centre of gravity will move uniformly in a straight line. To prove this, we shall consider their component velocities in any one direction, let these component velocities be u^ u 2 u 3 &c., the masses being ^ m 2 m 3 c., and the distances of the bodies (strictly speaking the distances of C.G.S. SYSTEM OF UNITS. 47 their respective centres of gravity) from a fixed plane to which the given direction is normal, be x l x% x 3 &c. The formula for the distance of their common centre of gravity from this plane is - mi Xj + m z y 2 + &c. TO! + ws + &c. In the time t, x l is increased by the amount Uit, x 2 by u 2 t, and so on; hence the numerator of the above expression is increased by mi u\ t + wij 'U-2 1 + &c., and the value of x is increased in each unit of time by m\ Ui + m> u 2 + &c. , v ! -f ra 2 4- &c. which is therefore the component velocity of the centre of gravity in the given direction. As this expression contains only given constant quantities, its value is constant; and as this reasoning applies to all directions, the velocity of the centre of gravity must itself be constant both in magnitude and direction. We may remark that the above formula (2) correctly expresses the component velocity of the centre of gravity at the instant con- sidered, even when u lf u. 2 , &c., are not constant. 85. Centre of Mass. The point which we have thus far been speaking of under the name of " centre of gravity," is more appro- priately called the "centre of mass," a name which is at once suggested by formula (1) 84. When gravity acts in parallel lines upon all the particles of a body, the resultant force of gravity upon the body is a single force passing through this point; but this is no longer the case when the forces of gravity upon the different parts of the body (or system of bodies) are not parallel. 86. Units of Measurement. It is a matter of importance, in scientific calculations, to express the various magnitudes with which we have to deal in terms of units which have a simple relation to each other. The British weights and measures are completely at fault in this respect, for the following reasons: 1. They are not a decimal system; and the reduction of a measurement (say) from inches and decimals of an inch to feet and decimals of a foot, cannot be effected by inspection. 2. It is still more troublesome to reduce gallons to cubic feet or inches. 3. The weight (properly the mass) of a cubic foot of a substance in Ibs., cannot be written down by inspection, when the specific gravity of the substance (as compared with water) is given. 48 FIRST PRINCIPLES OF KINETICS. 87. The C.Gr.S. System. A committee of the British Association, specially appointed to recommend a system of units for general adoption in scientific calculation, have recommended that the centimetre be adopted as the unit of length, the gramme as the unit of mass, and the second as the unit of time. We shall first give the rough and afterwards the more exact definitions of these quantities. The centimetre is approximately ^9 of the distance of either pole of the earth from the equator; that is to say 1 followed by 9 zeros expresses this distance in centimetres. The gramme is approximately the mass of a cubic centimetre of cold water. Hence the same number which expresses the speci- fic gravity of a substance referred to water, expresses also the mass of a cubic centimetre of the substance, in grammes. The second is 24 x 60 x 60 ^ a mean s l ar day. More accurately, the centimetre is defined as one hundredth part of the length, at the temperature Centigrade, of a certain stand- ard bar, preserved in Paris, carefully executed copies of which are preserved in several other places; and the gramme is defined as one thousandth part of the mass of a certain standard which is preserved at Paris, and of which also there are numerous copies preserved elsewhere. For brevity of reference, the committee have recommended that the system of units based on the Centimetre, Gramme, and Second, be called the C.G.S. system. The unit of area in this system is the square centimetre. The unit of volume is the cubic centimetre. The unit of velocity is a velocity of a centimetre per second. The unit of momentum is the momentum of a gramme moving with a velocity of a centimetre per second. The unit force is that force w^hich generates this momentum in one second. It is therefore that force which, acting on a gramme for one second, generates a velocity of a centimetre per second. This force is called the dyne, an abbreviated derivative from the Greek Svvamc (force). The unit of work is the work done by a force of a dyne working through a distance of a centimetre. It might be called the dyne- centimetre, but a shorter name has been provided and it is called the erg, from the Greek tpyov (work). CHAPTER VII. LAWS OF FALLING BODIES. 88. Effect of the Resistance of the Air. In air, bodies fall with unequal velocities; a sovereign or a ball of lead falls rapidly, a piece of down or thin paper slowly. It was formerly thought that this difference was inherent in the nature of the materials; but it is easy to show that this is not the case, for if we compress a mass of down or a piece of paper by rolling it into a ball, and compare it with a piece of gold-leaf, we shall find that the latter body falls more slowly than the former. The inequality of the velocities which we observe is due to the resistance of the air, which increases with the extent of surface exposed by the body. It was Galileo who first discovered the cause of the unequal rapidity of fall of different bodies. To put the matter to the test, he prepared small balls of different substances, and let them fall at the same time from the top of the tower of Pisa; they struck the ground almost at the same instant. On changing their forms, so as to give them very different extents of surface, he observed that they fell with very unequal velocities. He was thus led to the conclusion that gravity acts on all substances with the same intensity, and that in a vacuum all bodies would fall with the same velocity. This last proposition could not be put to the test of experiment in the time of Galileo, the air-pump not having yet been invented. The experiment was performed by Newton, and is now well known as the " guinea and feather " experiment. For this purpose a tube from a yard and a half to two yards long is used, which can be exhausted of air, and which contains bodies of various densities, such as a coin, pieces of paper, and feathers. When the tube is full of air and is inverted, these different bodies are seen to fall with very unequal velocities; but if the experiment is repeated after the tube 4 50 LAWS OF FALLING BODIES. has been exhausted of air, no difference can be perceived between the times of their descent. 89. Mass and Gravitation Proportional. This experiment proves that bodies which have equal weights are equal in mass. For equal masses are denned to be those which, when acted on by equal forces, receive equal accelerations; and the forces, in this experiment, are the weights of the falling bodies. Newton tested this point still more severely by experiments with pendulums (Principia, book III. prop. vi.). He procured two round wooden boxes of the same size and weight, and suspended them by threads eleven feet long. One of them he filled with wood, and he placed very accurately in the centre of oscillation of the other the same weight of gold. The boxes hung side by side, and, when set swinging in equal oscillations, went and returned together for a very long time. Here the forces concerned in producing and checking the motion, namely, the force of gravity and the resistance of the air, were the same for the two pendulums, and as the move- ments produced were the same, it follows that the masses were equal. Newton remarks that a difference of mass amounting to a thousandth part of the whole could not have escaped detection. He experimented in the same way with silver, lead, glass, sand, salt, water, and wheat, and with the same result. He therefore infers that universally bodies of equal mass gravitate equally towards the earth at the same place. He further extends the same law to gravi- tation generally, and establishes the conclusion that the mutual gravitating force between any two bodies depends only on their masses and distances, and is independent of their materials. The time of revolution of the moon round the earth, considered in conjunction with her distance from the earth, shows that the relation between mass and gravitation is the same for the material of which the moon is composed as for terrestrial matter; and the same con- clusion is proved for the planets by the relation which exists between their distances from the sun and their times of revolution in their orbits. 90. Uniform Acceleration. The fall of a heavy body furnishes an illustration of the second law of motion, which asserts that the change of momentum in a body in a given time is a measure of the force which acts on the body. It follows from this law that if the same force continues to act upon a body the changes of momentum in successive equal intervals of time will be equal. When a heavy UNIFORM ACCELERATION. 51 body originally at rest is allowed to fall, it is acted on during the time of its descent by its own weight and by no other force, if we neglect the resistance of the air. As its own weight is a constant force, the body receives equal changes of momentum, and therefore of velocity, in equal intervals of time. Let g denote its velocity in centimetres per second, at the end of the first second. Then at the end of the next second its velocity will be g + g, that is 2#; at the end of the next it will be Zg+g, that is 3g, and so on, the gain of velocity in each second being equal to the velocity generated in the first second. At the end of t seconds the velocity will therefore be tg. Such motion as this is said to be uniformly accelerated, and the constant quantity g is the measure of the acceleration. Accelera- tion is defined as the gain of velocity per unit of time. 91. Weight of a Gramme in Dynes. Value of g. Let m denote the mass of the falling body in grammes. Then the change of momentum in each second is m$r, which is therefore the measure of the force acting on the body. The weight of a body of m grammes is therefore mg dynes, and the weight of 1 gramme is g dynes. The value of g varies from 978*1 at the equator to 983*1 at the poles; and 981 may be adopted as its average value in temperate latitudes. Its value at any part of the earth's surface is approximately given by the formula g = 980-6056 - 2-5028 cos 2\ - -000,003A, in which \ denotes the latitude, and h the height (in centimetres) above sea-level. 1 In 79 we distinguished between the intensity and the amount of a force. The amount of the force of gravity upon a mass of m grammes is mg dynes. The intensity of this force is g dynes per gramme. The intensity of a force, in dynes per gramme of the body acted on, is always equal to the change of velocity which the force produces per second, this change being expressed in centimetres per second. In other words the intensity of a force is equal to the acceleration which it produces. The best designation for g is the intensity of gravity. 92. Distance fallen in a Given Time. The distance described in a given time by a body moving with uniform velocity is calculated by multiplying the velocity by the time; just as the area of a rect- angle is calculated by multiplying its length by its breadth. Hence if we draw a line such that its ordinates A A', BB', &c., represent the 1 For the method of determination see 120. 52 LAWS OF FALLING BODIES. velocities with which a body is moving at the times represented by OA, OB (time being reckoned from the beginning of the motion), it , can be shown that the whole distance B described is represented by the area OB'B bounded by the curve, the last ordinate, and the base line. In fact this area can be divided into narrow strips (one of which is shown at AA', Fig. 37) A each of which may practically be re- garded as a rectangle, whose height represents the velocity with which the body is moving during the very small interval of time represented by its base, and whose area therefore represents the distance described in this time. This would be true for the distance described by a body moving from rest with any law of velocity. In the case of falling bodies the law is that the velocity is simply proportional to the time; hence the ordinates AA', BB', &c., must be directly as the abscissae OA, OB; this proves that the line OA' B' must be straight; and the figure OB' B is therefore a triangle. Its area will be half the product of OB and BB'. But OB represents the time t occupied by the motion, arid BB 7 the velocity gt at the end of this time. The area of the triangle therefore represents half the product of t and gt, that is, represents ^gt 2 , which is accordingly the distance described in the time t. Denoting this distance by s, and the velocity at the end of time t by v, we have thus the two formulae v = M (i) 8 = \gt\ (2) from which we easily deduce gs = {v\ (3) 93. Work spent in Producing Motion. We may remark, in pass- ing, that the third of these formulae enables us to calculate the work required to produce a given motion in a given mass. When a body whose mass is 1 gramme falls through a distance s, the force which acts upon it is its own weight, which is g dynes, and the work done upon it is gs ergs. Formula (3) shows that this is the same as %v z ergs. For a mass of m grammes falling through a distance s, the work is |mi> 2 ergs. The work required to produce a velocity v (cen- timetres per second) in a body of mass m (grammes) originally at rest is Jmt> 2 (ergs). 94. Body thrown Upwards. When a heavy body is projected ver- WORK IN PRODUCING MOTION. * 53 tically upwards, the formulae (1) (2) (3) of 92 will still apply to its motion, with the following interpretations: v denotes the velocity of projection. t denotes the whole time occupied in the ascent. s denotes the height to which the body will ascend. When the body has reached the highest point, it will fall back, and its velocity at any point through which it passes twice will be the same in going up as in coming down. 95. Resistance of the Air. The foregoing results are rigorously applicable to motion in vacuo, and are sensibly correct for motion in air as long as the resistance of the air is insignificant in compari- son with the force of gravity. The force of gravity upon a body is the same at all velocities ; but the resistance of the air increases with the velocity, and increases more and more rapidly as the velocity becomes greater; so that while at very slow velocities an increase of 1 per cent, in velocity would give an increase of 1 per cent, in the resistance, at a higher velocity it would give an increase of 2 per cent., and at the velocity of a cannon-ball an increase of 3 per cent. 1 The formulae are therefore sensibly in error for high velocities. They are also in error for bodies which, like feathers or gold-leaf, have a large surface in proportion to their weight. 96. Projectiles. If, instead of being simply let fall, a body is pro- jected in any direction, its motion will be compounded of the motion of a falling body and a uniform motion in the direction of projection. Thus if OP (Fig. 38) is the direction of projection, and OQ the vertical through the point of pro- jection, the body would move along OP keeping its original velocity unchanged, if J it were not disturbed by gravity. To find Q Fig. ss. where the body will be at any time t, we must lay off a length OP equal to V, V denoting the velocity of projec- tion, and must then draw from P the vertical line PR downwards equal to ^gf 2 , which is the distance that the body would have fallen in the time if simply dropped. The point R thus determined, will be the actual position of the body. The velocity of the body at any time will in like manner be found by compounding the initial : This is only another way of saying that the resistance varies approximately as the velocity when very small, and approximately as the cube of the velocity for velocities like that of a cannon-ball. 54 LAWS OF FALLING BODIES. velocity with the velocity which a falling body would have acquired in the time. The path of the body will be a curve, as represented in the figure, OP being a tangent to it at O, and its concavity being down- wards. The equations above given, namely show that PR varies as the square of OP, and hence that the path (or trajectory as it is technically called) is a parabola, whose axis is vertical. 97. Time of Flight, and Range. If the body is projected from a point at the surface of the ground (supposed level) we can calculate the time of flight and the range in the following way. Let a be the angle which the direction of projection makes with the horizontal. Then the velocity of projection can be resolved into two components, Y cos a and V sin a, the former being horizontal, and the latter vertically upward. The horizontal component of the velocity of the body is unaffected by gravity and remains constant. The vertical velocity after time t will be compounded of V sin a up- wards and gt downwards. It will therefore be an upward velocity V sin a gt, or a downward velocity (/ V sin a. At the highest point of its path, the body will be moving horizontally and the ver- tical component of its velocity will be zero; that is, we shall have \r - f\ r. V sin a V sin a g t =. ; whence t - . This is the time of attaining the highest point; and the time of flight will be double of this, that is, will be ?X|Ef. As the horizontal component of the velocity has the constant value V cos a, the horizontal displacement in any time t is V cos a multiplied by t. The range is therefore 2V- sin a cos a V 2 sin 2a - or . 9 The range (for a given velocity of projection) will therefore be greatest when sin 2a is greatest, that is when 2o = 90 and a = 45. We shall now describe two forms of apparatus for illustrating the laws of falling bodies. 98. Morin's Apparatus. rMorin's apparatus consists of a wooden cylinder covered with paper, which can be set in uniform rotation about its axis by the fall of a heavy weight. The cord which sup- PROJECTILES 55 ports the weight is wound upon a drum, furnished with a toothed wheel which works on one side with an endless screw on the axis of the cylinder, and on the other drives an axis carrying fans which serve to regulate the motion. In front of the turning cylinder is a cylindro-conical weight of cast-iron carrying a pen- cil whose point presses against the paper, and having ears which slide on vertical threads, serv- ing to guide it in its fall. By pressing a lever, the weight can be made to fall at a chosen moment. The proper time for this is when the motion of the cylinder has become sensibly uniform. It fol- lows from this arrange- ment that during its vertical motion the pencil will meet in succession the different generating lines 1 of the revolving cylinder, and will conse- quently describe on its surface a certain curve, from the study of which we shall be able to gather the law of the fall of the body which has traced it. With this view, we describe (by turning the cylinder while the pencil is stationary) a circle passing through the commencement of the curve, and also draw a vertical line through this point. We cut the paper along this latter line and develop it (that is, flatten 1 A cylindric surface could be swept out or "generated" by a straight line moving nnind the axis and remaining always parallel to it. The successive positions of this generating line are called the " generating lines of the cylinder." Fig. 39. Morin's Apparatus. 56 LAWS OF FALLING BODIES. it out into a plane). It then presents the appearance shown in Fig. 40. If we take on the horizontal line equal distances at 1, 2, 3, 4, 5 . . . , and draw perpendiculars at their extremities to meet the curve, it is evident that the points thus found are those which were traced by the pencil when the cylinder had turned through the dis- tances 1, 2, 3, 4, 5. . . . The corresponding verticals represent the spaces traversed in the times 1, 2, 3, 4, 5. . . . Now we find, as the figure shows, that these spaces are represented by the numbers 1, 4, 9, 1G, 25 . . . , thus verifying the principle that the spaces described are proportional to the squares of the times employed in their description. We may remark that the proportionality of the vertical lines to the squares of the horizontal lines shows that the curve is a parabola. The parabolic trace is thus the consequence of the law of fall, and from the fact of the trace being parabolic we can infer the proportionality of the spaces to the squares of the times. The law of velocities might also be verified separately by Morin's apparatus; we shall not describe the method which it would be necessary to employ, but shall content ourselves with remarking that the law of velocities is a logical consequence of the law of spaces. 1 99. Atwood's Machine. Atwood's machine, which affords great facilities for illustrating the effects of force in producing motion, consists essentially of a very freely moving pulley over which a fine cord passes, from the ends of which two equal weights can be sus- pended. A small additional weight of flat and elongated form is laid upon one of them, which is thus caused to descend with uni- form acceleration, and means are provided for suddenly removing 1 Consider, in fact, the space traversed in any time t ; this space is given by the formula = Kt 2 ; during the time t + the space traversed will be K(t + 6) 2 = Kt' 2 + 2Kt0 + K0 2 , whence it follows that the space traversed during the time 6 after the time t is 2K 6 + K0 2 . The average velocity during this time is obtained by dividing the space by 6, and is 2Kt + K0, which, by making 6 very small, can be made to agree as accurately as we please with the value 2Kf. This limiting value 2Kt must therefore be the velocity at the end of time t. D. Fig. 40. Parabolic Trace. ATWOODS MACHINE. this additional weight at any point of the descent, so as to allow the motion to continue from this point onward with uniform velocity. The machine is re- presented in Fig. 41. The pulley over which the string passes is the largest of the wheels shown at the top of the apparatus. In order to give it greater freedom of movement, the ends of its axis are made to rest, not on fixed supports, but on the circumferences of four wheels (two at each end of the axis) called friction- wheels, because their office is to dim- inish friction. Two small equal weights are shown, suspended from this pulley by a string passing over it. One of them P is represented as near the bottom of the supporting pillar, and the other P as near the top. The latter is resting upon a small : platform, which can be j suddenly dropped when j it is desired that the 1 motion shall commence. f A little lower down and 1 vertically beneath the platform, is seen a ring, Fl 's- 4i.-Atwood- 8 Machine, large enough to let the weight pass through it without danger of 58 LAWS OF FALLING BODIES. contact. This ring can be shifted up or down, and clamped at any height by a screw; it is represented on a larger scale in the margin. At a considerable distance beneath the ring, is seen the stop, which is also represented in the margin, and can like the ring be clamped at any height. The office of the ring is to intercept the additional weight, and the office of the stop is to arrest the descent. The up- right to which they are both clamped is marked with a scale of equal parts, to show the distances moved over. A clock with a pendulum beating seconds, is provided for measuring the time; and there is an arrangement by which the movable platform can be dropped by the action of the clock precisely at one of the ticks. To measure the distance fallen in one or more seconds, the ring is removed, and the stop is placed by trial at such heights that the descending weight strikes it precisely at another tick. To measure the velocity acquired in one or more seconds, the ring must be fixed at such a height as to intercept the additional weight at one of the ticks, and the stop must be placed so as to be struck by the descending weight at another tick. 100. Theory of Atwood's Machine. If M denote each of the two equal masses, in grammes, and m the additional mass, the whole moving mass (neglecting the mass of the pulley and string) is 2M + m, but the moving force is only the weight of m. The accel- eration produced, instead of being g, is accordingly only 2M m g. In order to allow for the inertia of the pulley and string, a con- stant quantity must be added to the denominator in the above for- mula, and the value of this constant can be determined by observ- ing the movements obtained with different values of M and m. Denoting it by C, we have m + 2M + c (A) as the expression for the acceleration. As m is usually small in comparison with M, the acceleration is very small in comparison with that of a freely falling body, and is brought within the limits of convenient observation. Denoting the acceleration by a, and using v and s, as in 92, to denote the velocity acquired and space described in time t, we shall have v = at, (I) s = at*, (2) *s = i< 2 , (3) FORCE IN CIRCULAR MOTION. 59 and each of these formulae can be directly verified by experiments with the machine. 101. Uniform Motion in a Circle. A body cannot move in a curved path unless there be a force urging it Fig - 42> towards the concave side of the curve. We shall proceed to in- vestigate the intensity of this force when the path is circular and the velocity uniform. We shall denote the velocity by v, the radius of the circle by T, and the intensity of the force by /. Let AB (Figs. 42, 43) be a small portion of the path, and BD a perpendicular upon AD the tangent at A. Then, since the arc AB is small in comparison with the whole circumference, it is sensibly equal to AD, and the body would have been found at D instead of at B if no force had acted Fig 43 - upon it since leaving A. DB is accordingly the distance due to the force; and if t denote the time from A to B, we have AD = vt (I) DB = \ft\ (2) The second of these equations gives /- 2DB J ^2 and substituting for t from the first equation, this becomes /-^?rt (3) AD 2 But if ATI (Fig. 43) be the diameter at A, and Bm the perpendicular upon it from B, we have, by Euclid, AD 2 = mB 2 =Am.m?i=2r. Am sensibly, 2r . DB. Therefore -^, - -, and hence by (3) /=; W Hence the force necessary for keeping a body in a circular path without change of velocity, is a force of intensity v - directed towards the centre of the circle. If m denote the mass of the body, the and this is called centrifugal force. It is not a force acting upon the stone, but a force exerted by the stone upon the string. Its direction is from the centre of curvature, whereas the deflecting force which acts upon the stone is towards the centre of curvature. 104. Centrifugal Force at the Equator. Bodies on the earth's surface are carried round in circles by the diurnal rotation of the earth upon its axis. The velocity of this motion at the equator is about 46,500 centimetres per second, and the earth's equatorial radius is about 6*38 x 10 8 centimetres. Hence the value of v - is found to be about 3*39. The case is analogous to that of the stone APPARENT GRAVITY. 61 at the highest point of its path in the preceding article, i instead of a string which can only exert a pull we suppose a stiff rod which can exert a push upon the stone. The rod will be called upon to exert a pull or a push at the highest point according as % - is greater or less than g. The force of the push in the latter case will be and this is accordingly the force with which the surface of the earth at the equator pushes a body lying upon it. The push, of course, is mutual, and this formula therefore gives the apparent weight or apparent gravitating force of a body at the equator, mg denoting its true gravitating force (due to attraction alone). A body falling in vacuo at the equator has an acceleration 97810 relative to the surface of the earth in its neighbourhood; but this portion of the surface has itself an acceleration of 3'39, directed towards the earth's centre, and therefore in the same direction as the acceleration of the body. The absolute acceleration of the body is therefore the sum of these two, that is 981 '49, which is accordingly the intensity of true gravity at the equator. The apparent weight of bodies at the equator would be nil if - were equal to g. Dividing 3*39 into 98.1 '49, the quotient is approxi- mately 289, which is (17) 2 . Hence this state of things would exist if the velocity of rotation were about 17 times as fast as at present. Since the movements and forces which we actually observe depend upon relative acceleration, it is usual to understand, by the value of g or the intensity of gravity at a place, the apparent values, unless the contrary be expressed. Thus the value of g at the equator is usually stated to be 97810. 105. Direction of Apparent Gravity. The total amount of centri- fugal force at different places on the earth's surface, varies directly as their distance from the earth's axis ; for this is the value of r in the formula (5) of 101, and the value of T in that formula is the same for the whole earth. The direction of this force, being per- pendicular to the earth's axis, is not vertical except at the equator; and hence, when we compound it with the force of true gravity, we obtain a resultant force of apparent gravity differing in direction as well as in magnitude from true gravity. What is always understood by a vertical, is the direction of apparent gravity; and a plane per- pendicular to it is what is meant by a horizontal plane. CHAPTER VIII. THE PENDULUM. 106. The Pendulum. When a body is suspended so that it can turn about a horizontal axis which does not pass through its centre of gravity, its only position of stable equi- librium is that in which its centre of gravity is in the same vertical plane with the axis and below it ( 42). If the body be turned into any other position, and left to itself, it will oscillate from one side to the other of the position of equilibrium, until the resistance of the air and the friction of the axis gradually bring it to rest. A body thus suspended, whatever be its form, is called a pendulum. It frequently consists of a rod which can turn about an axis O (Fig. 44) at its upper end, and which carries at its lower end a heavy lens-shaped piece of metal M called the bob; this latter can be raised or lowered by means of the screw V. The applications of the pendulum are very impor- tant: it regulates our clocks, and it has enabled us to measure the intensity of gravity in different parts of the world; it is important then to know at least the fundamental points in its theory. For explaining these, we shall begin with the consideration of an ideal body called the simple pendulum. 107. Simple Pendulum. This is the name given to a pendulum consisting of a heavy particle M (Fig. 45) attached to one end of an inextensible thread without weight, the other end of the thread being fixed at A. When the thread is vertical, the weight of the particle i. 44. -Pendulum, acts in the direction of its length, and there is equilib- SIMPLE PENDULUM. num. Fig. 45. Motion of Simple Pendulum. But suppose it is drawn aside into another position, as AM. In this case, the weight MG of the particle can be resolved into two forces MC and MH. The former, acting along the prolongation of the thread, is destroyed by the resistance of the thread; the other, acting along the tangent MH, produces the motion of the particle. This effective com- ponent is evidently so much the greater as the angle of displacement from the vertical position is greater. The particle will there- fore move along an arc of a circle described from A as centre, and the force which urges it forward will continually diminish till it arrives at the lowest point M'. At M' this force is zero, but, in virtue of the velocity acquired, the particle will ascend on the opposite side, the effective component of gravity being now opposed to the direction of its motion; and, inas- much as the magnitude of this component goes through the same series of values in this part of the motion as in the former part, but in reversed order, the velocity will, in like manner, retrace its former values, and will become zero when the particle has risen to a point M" at the same height as M. It then descends again and performs an oscillation from M" to M precisely similar to the first, but in the reverse direction. It will thus continue to vibrate between the two points M, M" (friction being supposed excluded), for an indefinite number of times, all the vibra- tions being of equal extent and performed in equal periods. The distance through which a simple pendulum travels in moving from its lowest position to its furthest position on either side, is called its amplitude. It is evidently equal to half the complete arc of vibration, and is commonly expressed, not in linear measure, but in degrees of arc. Its numerical value is of course equal to that of the angle MAM 7 , which it subtends at the centre of the circle. The complete period of the pendulum's motion is the time which it occupies in moving from M to M" and back to M, or more generally, is the time from its passing through any given position to its next passing through the same position in the same direction. What is commonly called the time of vibration, or the time of a single vibration, is the half of a complete period, being the time of G4 THE PENDULUM. passing from one of the two extreme positions to the other. Hence what we have above defined as a complete period is often called a double vibration. When the amplitude changes, the time of vibration changes also, being greater as the amplitude is greater; but the connection between the two elements is very far from being one of simple proportion. The change of time (as measured by a ratio) is much less than the change of amplitude, especially when the amplitude is small; and when the amplitude is less than about 5, any further diminution of it has little or no sensible effect in diminishing the time. For small vibrations, then, the time of vibration is independent of the amplitude. This is called the law of isochronism. 108. Law of Acceleration for Small Vibrations. Denoting the length of a simple pendulum by I, and its inclination to the vertical at any moment by 0, we see from Fig. 45 that the ratio of the effective component of gravity to the whole force of gravity is ^, that is sin 0; and when is small this is sensibly equal to 6 itself as measured by - Let s denote the lenth of the arc MM' inter- To. vening between the lower end of the pendulum and the lowest point of its swing, at any time; then is equal to -^ and the intensity of the effective force of gravity when is small is sensibly equal to gO, that is to ^. Since g and I are the same in all positions of the pendulum, this effective force varies as s. Its direction is always towards the position of equilib- rium, so that it accelerates the motion during the approach to this position, and retards it during the recess; the acceleration or retardation being always in direct pro- portion to the distance from the position of equilibrium. This species of motion is of extremely common occurrence. It is illus- trated by the vibration of either prong of a tuning-fork, and in general by the motion of any body vibrating in one plane in such a manner as to yield a simple musical tone. 109. General Law for Period. Suppose a point P to travel with uniform velocity round a circle (Fig. 46), and from its successive Fig. 46. Projection of Circular Motion. SIMPLE HAKMONIG MOTION. 65 positions P p P 2 , &c., let perpendiculars P^, P 2 > 2 , &c., be drawn to a fixed straight line in the plane of the circle. Then while P travels once round the circle, its projection p executes a complete vibration. The acceleration of P is always directed towards the centre of the circle, and is equal to f ^ ) r ( 101). The component of this acceler- ation parallel to the line of motion of p, is the fraction ^ of the whole acceleration (x denoting the distance of p from the middle point of its path), and is therefore \-f) x - This is accordingly the accelera- tion of p } and as it is simply proportional to x we shall denote it for brevity by px. To compute the periodic time T of a complete vibration, we have the equation ^- (-fj , which gives T=-^. (1) V^ 110. Application to the Pendulum. For the motion of a pendulum in a small arc, we have acceleration = | *, where s denotes the displacement in linear measure. We must therefore put /* = j, and we then have (2) which is the expression for the time of a complete (or double) vibra- tion. It is more usual to understand by the " time of vibration " of a pendulum the half of this, that is the time from one extreme position to the other, and to denote this time by T. In this sense we have To find the length of the seconds' pendulum we must put T 1. * This gives If g were 987 we should have Z=100 centimetres or 1 metre. The actual value of g is everywhere a little less than this. The length of the seconds' pendulum is therefore everywhere rather less than a metre. 111. Simple Harmonic Motion. Rectilinear motion consisting of vibration about a point with acceleration px, where x denotes 5 66 THE PENDULUM. distance from this point, is called Simple Harmonic Motion, or Simple Harmonic Vibration. The above investigation shows that 2?r such vibration is isochronous, its period being -^=- whatever the amplitude may be. To understand the reason of this isochronism we have only to remark that, if the amplitude be changed, the velocity at correspond- ing points (that is, points whose distances from the middle point are the same fractions of the amplitudes) will be changed in the same ratio. For example, compare two simple vibrations in which the values of p are the same, but let the amplitude of one be double that of the other. Then if we divide the paths of both into the same number of small equal parts, these parts will be twice as great for the one as for the other; but if we suppose the two points to start simultaneously from their extreme positions, the one will constantly be moving twice as fast as the other. The number of parts described in any given time will therefore be the same for both. In the case of vibrations which are not simple, it is easy to see (from comparison with simple vibration) that if the acceleration in- creases in a greater ratio than the distance from the mean position, the period of vibration will be shortened by increasing the amplitude; but if the acceleration increases in a less ratio than the distance, as in the case of the common pendulum vibrating in an arc of moderate extent, the period is increased by increasing the amplitude. 112. Experimental Investigation of the Motion of Pendulums. The preceding investigation applies to the simple pendulum; that is to say to a purely imaginary existence; but it can be theoretically demonstrated that every rigid body vibrating about a horizontal axis under the action of gravity (friction and the resistance of the air being neglected), moves in the same manner as a simple pendu- lum of determinate length called the equivalent simple pendulum. Hence the above results can be verified by experiments on actual pendulums. The discovery of the experimental laws of the motion of pendu- lums was in fact long .anterior to the theoretical investigation. It was the earliest and one of the most important discoveries of Galileo, and dates from the year 1582, when he was about twenty years of age. It is related that on one occasion, when in the cathedral of Pisa, he was struck with the regularity of the oscilla- tions of a lamp suspended from the roof, and it appeared to him CYCLOIDAL PENDULUM. 67 that these oscillations, though diminishing in extent, preserved the same duration. He tested the fact by repeated trials, which con- firmed him in the belief of its perfect exactness. This law of isochronism can be easily verified. It is only necessary to count the vibrations which take place in a given time with different amplitudes. The numbers will be found to be exactly the same. This will be found to hold good even when some of the vibrations compared are so small that they can only be observed with a telescope. By employing balls suspended by threads of different lengths, Galileo discovered the influence of length on the time of vibration. He ascertained that when the length of the thread increases, the time of vibration increases also; not, however, in proportion to the length simply, but to its square root. 113. Cycloidal Pendulum. It is obvious from 64 that the effective component of gravity upon a particle resting on a smooth inclined plane is proportional to the sine of the inclination. The accelera- tion of a particle so situated is in fact g sin a, if a denote the inclina- tion of the plane. When a particle is guided along a smooth curve its acceleration is expressed by the same formula, a now denoting the inclination of the curve at any point to the horizon. This inclina- tion varies from point to point of the curve, so that the acceleration g sin a is no longer a constant quantity. The motion of a common pendulum corresponds to the motion of a particle which is guided to move in a circular arc; and if x denote distance from the lowest point, measured along the arc, and r the radius of the circle (or the length of the pendulum), the acceleration at any point is g sin * This is sensibly proportional to x so long as a? is a small fraction of r; but in general it is not proportional to x, and hence the vibra- tions are not in general isochronous. To obtain strictly isochronous vibrations we must substitute for the circular arc a curve which possesses the property of having an inclination whose sine is simply proportional to distance measured along the curve from the lowest point. The curve which possesses this property is the cycloid. It is the curve which is traced by a point in the circumference of a circle which rolls along a straight line. The cycloidal pendulum is constructed by suspending an ivory ball or some other small heavy body by a thread between two cheeks (Fig. 47), on which the thread winds as the ball swings to 68 THE PENDULUM. Fig 47. Cycloidal Pendulum. either side. The cheeks must themselves be the two halves of a cycloid whose length is double that of the thread, so that each cheek has the same length as the thread. It can be demonstrated 1 that under these circumstances the path of the ball will be a cycloid identical with that to which the cheeks belong. Ne- glecting friction and the rigidity of the thread, the acceleration in this case is proportional to dis- tance measured along the cycloid from its lowest point, and hence the time of vibration will be strictly the same for large as for small amplitudes. It will, in fact, be the same as that of a simple pendulum having the same length as the cycloidal pendulum and vibrating in a small arc. Attempts have been made to adapt the cycloidal pendulum to clocks, but it has been found that, owing to the greater amount of friction, its rate was less regular than that of the common pendu- lum. It may be remarked, that the spring by which pendulums are often suspended has the effect of guiding the pendulum bob in a curve which is approximately cycloidal, and thus of diminishing the irregularity of rate resulting from differences of amplitude. 114. Moment of Inertia. Just as the mass of a body is the measure of the force requisite for producing unit acceleration when the movement is one of pure translation; so the moment of inertia of a rigid body turning about a fixed axis is the measure of the couple requisite for producing unit acceleration of angular velocity. We suppose angle to be measured by ,. , so that the angle turned by the body is equal to the arc described by any point of it divided by the distance of this point from the axis; and the angular velocity of the body will be the velocity of any point divided by its distance from the axis. The moment of inertia of the body round the axis is numerically equal to the couple which would produce unit change of angular velocity in the body in unit time. We shall now show how to express the moment of inertia in terms of the masses of the particles of the body and their distances from the axis. 1 Since the evolute of the cycloid is an equal cycloid. MOMENT OF INERTIA. 69 Let m denote the mass of any particle, r its distance from the axis, and the angular acceleration. Then r is the acceleration of the particle m, and the force which would produce this acceleration by acting directly on the particle along the line of its motion is mr0. The moment of this force round the axis would be mr ? since its arm is r. The aggregate of all such moments as this for all the particles of the body is evidently equal to the couple which actually produces the acceleration of the body. Using the sign S to denote " the sum of such terms as/' and observing that is the same for the whole body, we have Applied couple = 2 (mrV) = 02 (mr 2 ). (1) When is unity, the applied couple will be equal to S (mr 2 ), which is therefore, by the foregoing definition, the moment of inertia of the body round the axis. 115. Moments of Inertia Round Parallel Axes. The moment of inertia round an axis through the centre of mass is always less than that round any parallel axis. For if r denote the distance of the particle m from an axis not passing through the centre of mass, and x and y its distances from two mutually rectangular planes through this axis, we have r z =x 2 +y 2 . Now let two planes parallel to these be drawn through the centre of mass; let I and i\ be the distances of m from them, and p its distance from their line of intersection, which will clearly be parallel to the given axis. Also let a and b be the distances respectively between the two pairs of parallel planes, so that a z + b 2 will be the square of the distance between the two parallel axes, which distance we will denote by h. Then we have 2 (wr 2 ) =- 2 -[ m (a 2 + & 2 )} + 2 {m (|* + r,')} 2o 2' (m) 26 2 (mrj) = A 2 2m -f 2 (Mi/> 2 ) 2a 2m 2Z>"^ 2m. where and 77 are the values of 5 and rj for the centre of mass. But these values are both zero, since the centre of mass lies on both the planes from which and 17 are measured. We have therefore 2 (mr 2 ) = h 2 2m + 2 (m/> 2 ), (2) that is to say, the moment of inertia round the given axis exceeds the moment of inertia round the parallel axis through the centre of 70 THE PENDULUM. mass by the product of the whole mass into the square of the dis- tance between the axes. 116. Application to Compound Pendulum. The application of this principle to the compound pendulum leads to some results of great interest and importance. Let M be the mass of a compound pendulum, that is, a rigid body free to oscillate about a fixed horizontal axis. Let h, as in the preceding section, denote the distance of the centre of mass from this axis; let denote the inclination of h to the vertical, and the angular acceleration. Then, since the forces of gravity on the body are equivalent to a single force M#, acting vertically downwards at the centre of mass, and therefore having an arm h sin with respect to the axis, the moment of the applied forces round the axis is Mgh sin 0; and this must, by 114, be equal to 0S (mr 2 ). We have therefore 2 (mr 3 ) _ //_sin B (e}] MA 0~ If the whole mass were collected at one point at distance I from the axis, this equation would become MZ 2 _ . _ {/_sin_0 m Ml ~ and the angular motion would be the same as in the actual case if I had the value 2 mr* 1 = MA I is evidently the length of the equivalent simple pendulum. 117. Convertibility of Centres. Again, if we introduce a length k such that MA; 2 is equal to S (wp 2 ), that is, to the moment of inertia round a parallel axis through the centre of mass, we have Fig. 48. s ( m ,.) - s ( mf fl) + tf z m - MP + MA 2 , and equation (5) becomes fc 9 + A 2 * = L = I" + '*! or P = (l-k) A. (7) In the annexed figure (Fig. 48) which represents a vertical section through the centre of mass, let G be the centre of mass, A the "centre COMPOUND PENDULUM. 71 of suspension," that is, the point in which the axis cuts the plane of the figure, and the " centre of oscillation," that is, the point at which the mass might be collected without altering the movement. Then, by definition, we have I = AO, h = AGr, therefore I- h = GO, so that equation (7) signifies P = AG . GO. (8) Since k 2 is the same 1'or all parallel axes, this equation shows that when the body is made to vibrate about a parallel axis through O, the centre of oscillation will be the point A. That is to say; the centres of suspension and oscillation are interchangeable, and the product of their distances from the centre of mass is k 2 . 118. If we take a new centre of suspension A' in the plane of the figure, the new centre of oscillation O' will lie in the production of A'G, and we must have A'G . GO' = F = AG- . GO. If A'G be equal to AG, GO' will be equal to GO, and A'O' to AO, so that the length of the equivalent simple pendulum will be un- changed. A compound pendulum will therefore vibrate in the same time about all parallel axes which are equidistant from the centre of mass. When the product of two quantities is given, their sum is least when they are equal, and becomes continually greater as they depart further from equality. Hence the length of the equivalent simple pendulum AO or AG + GO is least when AG = GO = k, and increases continually as the distance of the centre of suspen- sion from G is either increased from k to infinity or diminished from k to zero. Hence, when a body vibrates about an axis which passes very nearly through its centre of gravity, its oscillations are exceed- ingly slow. 119. Kater's Pendulum. The principle of the convertibility of centres, established in 117, was discovered by Huygens, and affords the most convenient practical method of constructing a pendulum of known length. In Kater's pendulum there are two parallel knife-edges about either of which the pendulum can be made to vibrate, and one of them can be adjusted to any distance 72 THE PENDULUM. from the other. The pendulum is swung first upon one of these edges and then upon the other, and, if any difference is detected in the times of vibration, it is corrected by moving the adjustable edge. When the difference has been completely destroyed, the distance between the two edges is the length of the equivalent simple pendu- lum. It is necessary, in any arrangement of this kind, that the two knife-edges should be in a plane passing through the centre of gravity; also that they should be on opposite sides of the centre of gravity, and at unequal distances from it. 120. Determination of the Value of g. Returning to the formula for the simple pendulum T = K */* we easily deduce from it g = ^, whence it follows that the value of g can be determined by making a pendulum vibrate and measuring T and I. T is determined by counting the number of vibrations that take place in a given time; I can be calculated, when the pendulum is of regular form, by the aid of formulae which are given in treatises on rigid dynamics, but its value is more easily obtained by Rater's method, described above, founded on the principle of the convertibility of the centres of suspension and oscillation. It is from pendulum observations, taken in great numbers at different parts of the earth, that the approximate formula for the intensity of gravity which we have given at 91 has been deduced. Local peculiarities prevent the possibility of laying down any general formula with precision; and the exact value of g for any place can only be ascertained by observations on the spot. CHAPTER IX. CONSERVATION OF ENERGY. 121. Definition of Kinetic Energy. We have seen in 93 that the work which must be done upon a mass of m grammes to give it a velocity of v centimetres per second is Jm^ 2 ergs. Though we have proved this only for the case of falling bodies, with gravity as the working force, the result is true- universally, as is shown in advanced treatises on mathematical physics. It is true whether the motion be rectilinear or curvilinear, and whether the working force act in the line of motion or at an angle with it. If the velocity of a mass increases from v 1 to v 2 , the work done upon it in the interval is Jm (vfv-?)', in other words, is the increase of Jmv 2 . On the other hand, if a force acts in such a manner as to oppose the motion of a moving mass, the force will do negative work, the amount of which will be equal to the decrease in the value of Jmi> 2 . For example, during any portion of the ascent of a projectile, the diminution in the value of %mv 2 is equal to gm multiplied by the increase of height ; and during any portion of its descent the increase in Jmi> 2 is equal to gm multiplied by the decrease of height. The work which must have been done upon a body to give it its actual motion, supposing it to have been initially at rest, is called the energy of motion or the kinetic energy of the body. It can be computed by multiplying half the mass by the square of the velocity. 122. Definition of Static or Potential Energy. When a body of mass m is at a height s above the ground, which we will suppose level, gravity is ready to do the amount of work gms upon it by making it fall to the ground. A body in an elevated position may therefore be regarded as a reservoir of work. In like manner a wound-up clock, whether driven by weights or by a spring, has 74 CONSERVATION OF ENERGY. work stored up in it. In all these cases there is force between parts of a system tending to produce relative motion, and there is room for such relative motion to take place. There is force ready to act, and space for it to act through. Also the force is always the same in the same relative position of the parts. Such a system possesses energy, which is usually called potential. We prefer to call it statical) inasmuch as its amount is computed on statical principles alone. 1 Statical energy depends jointly on mutual force and relative position. Its amount in any given position is the amount of work which would be done by the forces of the system in passing from this position to the standard position. When we are speaking of the energy of a heavy body in an elevated position above level ground, we naturally adopt as the standard position that in which the body is lying on the ground. When we speak of^ the energy of a wound-up clock, we adopt as the standard position that in which the clock has completely run down. Even when the standard position is not indicated, we can still speak definitely of the differ- ence between the energies of two given positions of a system; just as we can speak definitely of the difference of level of two given points without any* agreement as to the datum from which levels are to be reckoned. 123. Conservation of Mechanical Energy. When a frictionless system is so constituted that its forces are always the same in the same positions of the system, the amount of work done by these forces during the passage from one position A to another position B will be independent of the path pursued, and will be equal to minus the work done by them in the passage from B to A. The earth and any heavy body at its surface constitute such a system; the force of the system is the mutual gravitation of these two bodies; and the work done by this mutual gravitation, when the body is moved by any path from a point A to a point B, is equal to the weight of the body multiplied by the height of A above B. When the system passes through any series of movements beginning with a given position and ending with the same position again, the algebraic total of work done by the forces of the system in this series of movements is zero. For instance, if a heavy body be carried by a roundabout path back to the point from whence it started, no work is done upon it by gravity upon the whole. Every position of such a system has therefore a definite amount 1 That is to say, the computation involves no reference to the laws of motion. TRANSFORMATIONS OF ENERGY. 75 of statical energy, reckoned with respect to an arbitrary standard position. The work done by the forces of the system in passing from one position to another is (by definition) equal to the loss of static energy; but this loss is made up by an equal gain of kinetic energy. Conversely if kinetic energy is lost in passing from one position to another, the forces do negative work equal to this loss, and an equal amount of static energy is gained. The total energy of the system (including both static and kinetic) therefore remains unaltered. An approximation to such a state of things is exhibited by a pendulum. In the two extreme positions it is at rest, and has there- fore no kinetic energy; but its statical energy is then a maximum. In the lowest position its motion is most rapid; its kinetic energy is therefore a maximum, but its statical energy is zero. The difference of the statical energies of any two positions, will be the weight of the pendulum multiplied by the difference of levels of its centre of gravity, and this will also be the difference (in inverse order) between the kinetic energies of the pendulum in these two positions. As the pendulum is continually setting the air in motion and thus doing external work, it gradually loses energy and at last comes to rest, unless it be supplied with energy from a clock or some other source. If a pendulum could be swung in a perfect vacuum, with an entire absence of friction, it would lose no energy, and would vibrate for an indefinite time without decrease of amplitude. 124. Illustration from Pile-driving. An excellent illustration of transformations of energy is furnished by pile-driving. A large mass of iron called a ram is slowly hauled up to a height of several yards above the pile, and is then allowed to fall upon it. During the ascent, work must be supplied to overcome the force of gravity; and this work is represented by the statical energy of the ram in its highest position. While falling, it continually loses statical and gains kinetic energy; the amount of the latter which it possesses immediately before the blow being equal to the work which has been done in raising it. The effect of the blow is to drive the pile through a small distance against a resistance very much greater than the weight of the rani; the work thus done being nearly equal to the total energy which the ram possessed at any point of its descent. We say nearly equal, because a portion of the energy of the blow is spent in producing vibrations. 125. Hindrances to Availability of Energy. There is almost 70 CONSERVATION OF ENERGY. always some waste in utilizing energy. When water turns a mill- wheel, it runs away from the wheel with a velocity, the square of which multiplied by half the mass of the water represents energy which has run to waste. Friction again often consumes a large amount of energy; arid in this case we cannot (as in the preceding one) point to any palpable motion of a mass as representing the loss. Heat, however, is pro- duced, and the energy which has disappeared as regarded from a gross mechanical point of view, has taken a molecular form. Heat is a form of molecular energy; and we know, from modern re- searches, what quantity of heat is equivalent to a given amount of mechanical work. In the steam-engine we have the converse process; mechanical work is done by means of heat, and heat is destroyed in the doing of it, so that the amount of heat given out by the engine is less than the amount supplied to it. The sciences of electricity and magnetism reveal the existence of other forms of molecular energy; and it is possible in many ways to produce one form of energy at the expense of another; but in every case there is an exact equivalence between the quantity of one kind which comes into existence and the quantity of another kind which simultaneously disappears. Hence the problem of constructing a self-driven engine, which we have seen to be impossible in mechanics, is equally impossible when molecular forms of energy are called to the inventor's aid. Energy may be transformed, and may be communicated from one system to another; but it cannot be increased or diminished in total amount. This great natural law is called the principle of the con- servation of eneryy. CHAPTEE X. ELASTICITY. 126. Elasticity and its Limits. There is no such thing in nature as an absolutely rigid body. All bodies yield more or less to the action of force; and the property in virtue of which they tend to recover their original form and dimensions when these are forcibly changed, is called elasticity. Most solid bodies possess almost per- fect elasticity for small deformations ; that is to say, when distorted, extended, or compressed, within certain small limits, they will, on the removal of the constraint to which they have been subjected, instantly regain almost completely their original form and dimen- sions. These limits (which are called the limits of elasticity) are different for different substances; and when a body is distoited beyond these limits, it takes a set, the form to which it returns being intermediate between its original form and that into which it was distorted. When a body is distorted within the limits of its elasticity, the force with which it reacts is directly proportional to the amount of distortion. For example, the force required to make the prongs of a tuning-fork approach each other by a tenth of an inch, is double of that required to produce an approach of a twentieth of an inch; and if a chain is lengthened a twentieth of an inch by a weight of 1 cwt., it will be lengthened a tenth of an inch by a weight of 2 cwt., the chain being supposed to be strong enough to experience no permanent set from this greater weight. Also, within the limits of elasticity, equal and opposite distortions, if small, are resisted by equal reactions. For example, the same force which suffices to make the prongs of a tuning-fork approach by a twentieth of an inch, will, if applied in the opposite direction, make them separate by the same amount. 78 ELASTICITY. 127. Isochronism of Small Vibrations. An important consequence of these laws is, that when a body receives a slight distortion within the limits of its elasticity, the vibrations which ensue when the constraint is removed are isochronous. This follows from 111, inasmuch as the accelerations are proportional to the forces, and are therefore proportional at each instant to the deformation at that instant. 128. Stress, Strain, and Coefficients of Elasticity. A body which, like indian-rubber, can be subjected to large deformations without receiving a permanent set, is said to have wide limits of elasticity. A body which, like steel, opposes great resistance to deformation, is said to have large coefficients of elasticity. Any change in the shape or size of a body produced by the appli- cation of force to the body is called a strain; and an action of force tending to produce a strain is called a stress. When a wire of cross-section A is stretched with a force F, the TT longitudinal stress is ^; this being the intensity of force per unit area with which the two portions of the wire separated by any cross-section are pulling each other. If the length of the wire when unstressed is L and when stressed L+, the longitudinal strain is A stress is always expressed in units of force per unit of area. A strain is always expressed as the ratio of two magnitudes of the same kind (in the above example, two lengths), and is therefore independent of the units employed. The quotient of a stress by the strain (of a given kind) which it produces, is called a coefficient or modulus of elasticity. In the above TTT example, the quotient -^ is called Young's modulus of elasticity. As the wire, while it extends lengthwise, contracts laterally, there will be another coefficient of elasticity obtained by dividing the longitudinal stress by the lateral strain. It is shown, in special treatises, that a solid substance may have 21 independent coefficients of elasticity; but that when the substance is isotropic, that is, has the same properties in all directions, the number reduces to 2. 129. Volume-elasticity. The only coefficient of elasticity possessed by liquids and gases is elasticity of volume. When a body of volume V is reduced by the application of uniform normal pressure over its whole surface to volume V v, the volume-strain is =L and if this COEFFICIENTS OF ELASTICITY. 79 effect is produced by a pressure of p units of force per unit of area, the elasticity of volume is the quotient of the stress p by the strain This is also called the resistance to compression; , or is p . and its reciprocal ~ is called the compressibility of the substance. In dealing with gases, p must be understood as a pressure super- added to the original pressure of the gas. Since a strain is a mere numerical quantity, independent of units, a coefficient of elasticity must be expressed, like a stress, in units of force per unit of area. In the C.G.S. system, stresses and coefficients of elasticity are expressed in dynes per square centimetre. The following are approximate values (thus expressed) of the two co- efficients of elasticity above denned: Young's Elasticity of Modulus. Volume. Glass (flint), 60 x 10 10 40 x 10 10 Steel, 210 x 10 10 180 x 10 10 Iron (wrought), 190 x 10 10 140 x 10 10 Iron (cast), 130 x 10 10 96 x 10 10 Copper, 120 x 10 10 160 x 10 10 Mercury, 54 x 10 10 Water, 2 x 10 10 Alcohol, 1-2 xlO 10 130. (Ersted's Piezometer. The compression of liquids has been observed by means of (Ersted's piezometer, which is represented in Fig. 49. The liquid whose compression is to be observed is contained in a glass vessel b, resembling a thermometer with a very large bulb and short tube. The tube is open above, and a globule of mercury at the top of the liquid column serves as an index. This apparatus is placed in a very strong glass vessel a full of water. When pressure is exerted by means of the piston klh, the index of mercury is seen to descend, showing a diminution of volume of the liquid, and showing moreover that this diminution of volume exceeds that of the containing vessel b. It mi^ht at first Fig. 49. OSrsted's Piezometer. 80 ELASTICITY. sight appear that since this vessel is subjected to equal pressure within and without, its volume is unchanged; but in fact, its volume is altered to the same extent as that of a solid vessel of the same material; for the interior shells would react with a force precisely equivalent to that which is exerted by the contained liquid. CHAPTER XL FRICTION. 131. Friction, Kinetical and Statical. When two bodies are pressed together in such a manner that the direction of their mutual pressure is not normal to the surface of contact, the pressure can be resolved into two parts, one normal and the other tangential. The tangential component is called the force of friction between the two bodies. The friction is called kinetical or statical according as the bodies are or are not sliding one upon the other. As regards kinetical friction, experiment shows that if the normal pressure between two given surfaces be changed, the tangential force changes almost exactly in the same proportion; in other words, the ratio of the force of friction to the normal pressure is nearly constant for two given surfaces. This ratio is called the coefficient of kinetical friction between the two surfaces, and is nearly independent of the velocity. 132. Statical Friction. Limiting Angle. It is obvious that the statical friction between two given surfaces is zero when their mutual pressure is normal, and increases with the obliquity of the pressure if the normal component be preserved constant. The obliquity, however, cannot increase beyond a certain limit, depending on the nature of the bodies, and seldom amounting to so much as 45. Be- yond this limit sliding takes place. The limiting obliquity, that is, the greatest angle that the mutual force can make with the normal, is called the limiting angle of friction for the two surfaces; and the ratio of the tangential to the normal component when the mutual force acts at the limiting angle, is called the coefficient of statical friction for the two surfaces. The coefficient and limiting angle remain nearly constant when the normal force is varied. The coefficient of statical friction is in almost every case greater 6 82 FRICTION. than the coefficient of kinetical friction; in other words, friction offers more resistance to the commencement of sliding than to the continuance of 'it. A body which has small coefficients of friction with other bodies is called slippery. 133. Coefficient = tan 0. Inclined Plane. If be the inclination of the mutual force P to the common normal, the tangential com- ponent will be P sin 0, the normal component P cos 0, and the ratio of the former to the latter will be tan 0. Hence the coefficient of statical friction is equal to the tangent of the limiting angle of friction. When a heavy body rests on an inclined plane, the mutual pressure is vertical, and the angle is the same as the inclination of the plane. Hence if an inclined plane is gradually tilted till a body lying on it slides under the action of gravity, the inclination of the plane at which sliding begins is the limiting angle of friction between the body and the plane, and the tangent of this angle is the coefficient of statical friction. Again, if the inclination of a plane be such that the motion of a body sliding down it under the action of gravity is neither accelerated nor retarded, the tangent of this inclination will be the coefficient of kinetical friction. CHAPTER XII HYDROSTATICS. 134. Hydrodynamics. We shall now treat of the laws 01 force as applied to fluids. This branch of the general science of dynamics is called hydrodynamics (vSwp, water), and is divided into hydrostatics and hydrokinetics. Our discussions will be almost entirely confined to hydrostatics. FLUIDS. TRANSMISSION OF PRESSURE. The name fluid comprehends both liquids and gases. 135. No Statical Friction in Fluids. A fluid at rest cannot exert any tangential force against a surface in contact with it ; its pressure at every point of such a surface is entirely normal. A slight tangen- tial force is exerted by fluids in motion; and this fact is expressed by saying that all fluids are more or less viscous. An imaginary perfect fluid would be perfectly free from viscosity; its pressure against any surface would be entirely normal, whether the fluid were in motion or at rest. 136. Intensity of Pressure. When pressure is uniform over an area, the total amount of the pressure, divided by the area, is called the intensity of the pressure. The C.G.S. unit of intensity of pressure is a pressure of a dyne on each square centimetre of sur- face. A rough unit of intensity frequently used is the pressure of a pound per square inch. This unit varies with the intensity of gravity, and has an average value of about 69,000 C.G.S. units. Another rough unit of intensity of pressure frequently employed is " an atmosphere " that is to say, the average intensity of pressure of the atmosphere at the surface of the earth. This is about 1,000,000 C.G.S. units. 84 HYDROSTATICS. The single word " pressure " is used sometimes to denote " amount of pressure" (which can be expressed in dynes) and sometimes " intensity of pressure" (which can be expressed in dynes per square centimetre). The context usually serves to show which of these two meanings is intended. 137. Pressure the Same in all Directions. The intensity of pressure at any point of a fluid is the same in all directions; it is the same whether the surface which receives the pressure faces upwards, downwards, horizontally, or obliquely. This equality is a direct consequence of the absence of tangential force between two contiguous portions of a fluid. For in order that a small triangular prism of the fluid (its ends being right sections) may be in equilibrium, the pressures on its three faces must balance each other. But when three forces balance each other, they are proportional to the sides of a triangle to which they are perpendicular; 1 hence the amounts of pressure on the three faces are proportional to the faces, in other words the inten- sities of these three pressures are equal. As we can take two of the faces perpendicular to any two given directions, this proves that the pressures in all directions at a point are of equal intensity. 138. Pressure the Same at the Same Level. In a fluid at rest, the pressure is the same at all points in the same horizontal plane. This appears from considering the equilibrium of a horizontal cylinder AB (Fig. 50), of small sectional area, its ends being right sections. The pressures on the sides are normal, and therefore give no component in the direction Flg<5 - of the length; hence the pressures on the ends must be equal in amount ; but they act on equal areas; there- fore their intensities are equal. A horizontal surface in a liquid at rest may therefore be called a " surface of equal pressure." 139. Difference of Pressure at Different Levels. The increase of pressure with depth, in a fluid of uniform density, can be investi- gated as follows: Consider the equilibrium of a vertical cylinder mm' (Fig. 51), its ends being right sections. The pressures on its 1 This is an obvious consequence of the triangle of forces (art. 14) ; for if the sides of a triangle are parallel to three forces, we have only to turn the triangle through a right angle, and its sides will then be perpendicular to the forces. INCREASE OF PRESSURE WITH DEPTH. 85 sides are normal, and therefore horizontal. The only vertical forces acting upon it are its own weight and the pressures on its ends, of which it is to be observed that the pressure on the upper end acts downwards and that on the lower end upwards. The pressure on the lower end therefore exceeds that on the upper end by an amount equal to the weight of the cylinder. If a be the sectional area, w the weight of unit volume of the liquid, and h the length of the cylinder, the volume of the cylinder is ha, and its weight wha, which Fig - 6L must be equal to (pp) a if p,p r are the intensities of pressure on the lower and upper ends respectively. We have therefore that is, the increase of pressure in descending through a depth h is wh. The principles of this and the preceding section remain appli- cable whatever be the shape of the containing vessel, even if it be such as to render a circuitous route necessary in passing from one of two points compared to the other; for this route can always be made to consist of a succession of vertical and horizontal lines, and the preceding principles when applied to each of these lines separ- ately, will give as the final result a difference of pressure wh for a difference of heights h. If d denote the density of the liquid, in grammes per sq. cm., the weight of a cubic cm. will be gd dynes. The increase of pressure for an increase of depth h cm. is therefore ghd dynes per sq. cm. If there be no pressure at the surface of the liquid, this will be the actual pressure at the depth h. 140. Free Surface. It follows from these principles that the free surface of a liquid at rest that is, the surface in contact with the atmosphere must be horizontal; since all points in this surface are at the same pressure. If the surface were not horizontal, but were higher at n than at n (Fig. 52), the pres- sures at the two points m, m' vertically Fi g . 52 . beneath them in any horizontal plane AB would be unequal, for they would be due to the weights 86 HYDROSTATICS. 53. of unequal columns nm, n'm', and motion would ensue from m towards m'. The same conclusion can be deduced from considering the equili- brium of a particle at the surface, as M (Fig. 53). If the tangent plane at M were not horizontal there would be a component of gravity tending to make the particle slide down; and this tendency would produce motion, since there is no fric- tion to oppose it. 141. Transmissibility of Pressure in Fluids. Since the difference of the pressures at two points in a fluid can be determined by the foregoing prin- ciples, independently of any knowledge of the absolute intensity of either, it follows that when increase or diminution of pres- sure occurs at one point, an equal increase or diminution must occur throughout the whole fluid. A fluid in a closed vessel perfectly transmits through its ivhole substance whatever pressure ive apply to any part. The changes in amount of pressure will be equal for all equal areas. For unequal areas they will be propor- tional to the areas. Thus if the two vertical tubes in Fig. 54 have sectional areas which are as 1 to 16, a weight of 1 kilo- gram acting on the surface of the liquid in the smaller tube will be balanced by 16 kilograms acting on the surface of the liquid in the larger. This principle of the perfect transmis- sion of pressure by fluids appears to have been first discovered and published by Stevinus; but it was rediscovered by Pascal a few years later, and having been made generally known by his writings is Fig. 54. Principle of the Hydraulic often called ." Pascal's principle." In his celebrated treatise on the Equilibrium, of Liquids, he says, " If a vessel full of water, closed on all sides, has two openings, the one a hundred times as large as the other, and if each be supplied with a piston which fits exactly, a man pushing the small piston will exert a force which will equilibrate that of a hundred men pushing the piston which is a hundred times as large, TRANSMISSIBILITY OF PRESSURE. 87 and will overcome that of ninety-nine. And whatever may be the proportion of these openings, if the forces applied to the pistons are to each other as the openings, they will be in equilibrium." 142. Hydraulic Press. This mode of multiplying force remained for a long time practically unavailable on account of the difficulty of making the pistons water-tight. The hydraulic press was first successfully made by Bramah, who invented the cupped leather collar illustrated in Fig. 166, 264. Fig. 165 shows the arrangements of the press as a whole. Instead of pistons, plungers are employed; that is to say, solid cylinders of metal which can be pushed down into the liquid, or can be pushed up by the pressure of the liquid against their bases. The volume of liquid displaced by the advance of a plunger is evidently equal to that displaced by a piston of the same sectional area, and the above calculations for pistons apply to plungers as well. The plungers work through openings which are kept practically water-tight by means of the cup-leather arrange- ment. The cup-leather, which is shown both in plan and section in Fig. 166, consists of a leather ring bent so as to have a semi- circular section. It is fitted into a hollow in the interior of the sides of the opening, so that water leaking up along the circumfer- ence of the plunger will fill the concavity of the leather, and, by pressing on it, will produce a packing which fits more tightly as the pressure on the plunger increases. 143. Principle of Work Applicable. In Fig. 54, when the smaller piston advances and forces the other back, the volume of liquid driven out of the smaller tube is equal to the sectional area multi- plied by the distance through which the piston advances. In like manner, the volume of liquid driven into the larger tube is equal to its sectional area multiplied by the distance that its piston is forced back. But these two volumes are equal, since the same volume of liquid that leaves one tube enters the other. The distances through which the two pistons move are therefore inversely as their sectional areas, and hence are inversely as the amounts of pressure applied to them. The work done in pushing forward the smaller piston is therefore equal to the work done by the liquid in pushing back the larger. This was remarked by Pascal, who says "It is, besides, worthy of admiration that in this new machine we find that constant rule which is met with in all the old ones such as the lever, wheel and axle, screw, &c., which is that the distance is increased in proportion to the force; for it is evident that 88 HYDROSTATICS as one of these openings is a hundred times as large as the other, if the man who pushes the small piston drives it forward one inch, he will drive the large piston backward only one-hundredth part of that length." 144. Experiment on Upward Pressure. The upward pressure exerted by a liquid against a horizontal surface facing down- wards can be exhibited by the following experiment. Take a tube open at both ends (Fig. 55), and keeping the lower end covered with a piece of card, plunge it into water. The liquid will press the card against the bottom of the tube with a force which increases as it is plunged deeper. If water be now poured into the tube, the card will remain in its place as long as the level of the liquid is lower within the tube than with- out; but at the moment when equality of levels is attained it will become detached, 145. Liquids in Superposition. When one liquid rests on the top of another of different density, the foregoing principles lead to the result that the surface of demarcation must be horizontal. For the free surface of the upper liquid must, as we have seen, be horizontal. If now we take two small equal areas n and n' (Fig. 5C) in a horizontal layer of the lower liquid, they must be subjected to equal pressures. But these pressures are measured by the weights of the liquid cylinders nrs, ritl; and these latter cannot be equal unless the points r and t at the junction of the two liquids are at the same level. All points in the surface of demarca- tion are therefore in the same horizontal plane. The same reasoning can be extended downwards to any number of liquids of unequal densities, which rest one upon another, and shows that all the surfaces of demarcation between them must be horizontal. Fig. 55. Upward Pressure. Fig. 56. LIQUIDS IN SUPERPOSITION. An experiment in illustration of this result is represented in Fig. 57. Mercury, water, and oil are poured into a glass jar. The mercury, being the heaviest, goes to the bottom; the oil, being the lightest, floats at the top; and the surfaces of contact of the liquids are seen to be horizontal. Even when liquids are employed which gradually mix with one another, as water and alcohol, or fresh water and salt water, so that there is no definite surface of demarcation, but a gradual increase of density with depth, it still remains true that the density at all points in a horizontal plane is the same. 146. Two Liquids in Bent Tube. If we pour mercury into a bent tube open at both ends (Fig. 58), and then pour water into one of the arms, the heights of the two liquids above the surface of junction will be very unequal, as shown in the figure. The general rule for the equilibrium of any two liquids in these circum- stances is that their heights above the surface of junction must "be in- versely as their densities, Fig. 57. Phial of the Four Elements, since they correspond to equal pressures. 147. Experiment of Pascal's Vases. Since the amount of pressure I on a horizontal area A f at the depth h in a liquid is whA., where w denotes the weight of unit volume of the liquid, it follows that the pressure on the bottom of a vessel containing liquid is not affected by the breadth or narrowness of the upper part of the Fig. 53. Equilibrium of Two Fluids in Communicating Vessels. 90 HYDROSTATICS. vessel, provided the height of the free surface of the liquid be given. Pascal verified this fact by an experiment which is frequently ex- hibited in courses of physics. The apparatus employed (Fig. 59) is a tripod supporting a ring, into which can be screwed three vessels of different shapes, one widened upwards, another cylindrical, and- the third tapering upwards. Beneath the ring is a movable disc Fig. 59. Experiment of Pascal's Vases. supported by a string attached to one of the scales of a balance. Weights are placed in the other scale in order to keep the disc pressed against the ring. Let the cylindrical vase be mounted on the tripod, and filled up with water to such a level that the pressure is just sufficient to detach the disc from the ring. An indicator, shown in the figure, is used to mark the level at which this takes place. Let the experiment be now repeated with the two other vases, and the disc will be detached when the water has reached the same level as before. In the case of the cylindrical vessel, the pressure on the bottom is evidently equal to the weight of the liquid. Hence in all three PRESSURE ON VESSEL. 91 Fig. 60. Total Treasure. cases the pressure on the bottom of the vessel is equal to the weight of a cylindrical column of the liquid, having the bottom as its base, and having the same height as the liquid in the vessel. 148. Resultant Pressure on Vessel. The pressure exerted by the bottom of the vessel upon the stand on which it rests, consists of the weight of the vessel itself, together with the resultant pressure of the contained liquid against it. The actual pressure of the liquid against any portion of the vessel is normal to this portion, and if we resolve it into two components, one vertical and the other hori- zontal, only the vertical component need be attended to, in computing the resultant; for the horizontal components will always destroy one another. At such points as n, ri (Fig. 60) the vertical component is downwards; at s and s it is upwards; at r and r there is no vertical component; and at AB the whole pressure is vertical. It can be demonstrated mathematically that the resultant pressure is always equal to the total weight of the contained liquid; a conclusion which can also be deduced from the consideration that the pressure exerted by the vessel upon the stand on which it rests must be equal to its own weight together with that of its contents. Some cases in which the proof above indicated becomes especially obvious, are represented in Fig. 61. In the cylindrical vessel ABDC, it is evident that the only pressure trans- mitted to the stand is that exerted upon the bottom, which is equal to the weight of the liquid. In the case of the vessel which is wider at the top, the stand is subjected to the weight of the liquid column ABSK, which presses on the bottom AB, together with the columns GHKC, RLDS, pressing on GH and RL; the sum of which weights composes the total weight of liquid contained in the vessel. Finally, in the third case, the pressure on the bottom AB, which is equal to the weight of a liquid column ABSK, must be diminished by the ._ -Rli !L ~H.ru < 1 <> R t B A B A Fig. 61. Hydrostatic Paradox. 92 HYDROSTATICS. upward pressures on HG and RL. These last being represented by liquid columns HGCK, KLSD, there is only left to be transmitted to the stand a pressure equal to the weight of the water in the vessel. 149. Back Pressure in Discharging Vessel. The same analysis which shows that the resultant vertical pressure of a liquid against the containing vessel is equal to the weight of the liquid, shows also that the horizontal components of the pressures destroy one another. This conclusion is in accordance with everyday experience. How- ever susceptible a vessel may be of horizontal displacement, it is not found to acquire any tendency to horizontal motion by being filled with a liquid. When a system of forces are in equilibrium, the removal of one of them destroys the equilibrium, and causes the resultant of the system to be a force equal and opposite to the force removed. Accordingly if we remove an element of one side of the containing vessel, leaving a hole through which the liquid can flow out, the remaining pressure against this side will be insufficient to preserve equilibrium, and there will be an excess of pressure in the opposite direction. This conclusion can be directly verified by the experiment repre- sented in Fig. 62. A tall floating vessel of water is fitted with a hori- zontal discharge-pipe on one side near its base. The vessel is to be filled with water, and the discharge-pipe opened while the vessel is at rest. As the water flows out, the vessel will be observed to acquire a velocity, at first very slow, but continually increasing, in the opposite direction to that of the issuing stream. This experiment may also be re- garded as an illustration of the law of action and reaction, which asserts that momentum cannot be imparted to any body without equal and opposite momentum being imparted to some other body. The water in escaping from the vessel acquires horizontal momentum in one direction, and the vessel with its remaining contents acquires horizontal momentum in the opposite direction. Fig. C2. Backward Movement; of Discharging Vessel. BACKWAKD MOVEMENT. 93 The movements of the vessel in this experiment are slow. More marked effects of the same kind can be obtained by means of the hydraulic tourn- iquet (Fig. 63), which when made on a larger scale is called Barker's mill. It consists of a vessel of water free to rotate about a vertical axis, and having at its lower end bent arms through which the water is discharged hori- zontally, the direction of dis- being charge nearly at angles to joining the dis- charging orifice to the axis. The unbalanced pressures at the bends of the tube, opposite to the openings, cause the apparatus to revolve in the opposite direction to the issuing liquid. 150. Total and Resultant Pressures. Centre of Pressure. The intensity of pressure on an area which is not horizontal is greatest on those parts which are deepest, and the average intensity can be shown to be equal to the actual intensity at the centre of gravity of the area. Hence if A denote the area, h the depth of its centre of gravity, and w the weight of unit volume of the liquid, the total pressure will be w Ah. Strictly speaking, this is the pressure due to the weight of the liquid, the transmitted atmospheric pressure being left out of account. In attaching numerical values to w, A, and h, the same unit of length must be used throughout. For example, if h be expressed in feet, A must be expressed in square feet, and w must stand for the weight of a cubic foot of the liquid. When we employ the centimetre as the unit of length, the value Fig. 63. Hydraulic Tourniquet. HYDROSTATICS. of w will be sensibly 1 gramme if the liquid be water, so that the amount of pressure in grammes will be simply the product of the depth of the centre of gravity in centimetres by the area in square centimetres. For any other liquid, the pressure will be found by multiplying this product by the specific gravity of the liquid. These rules for computing total pressure hold for areas of all forms, whether plane or curved; but the investigation of the total pressure on an area which is not plane is a mere mathematical exercise of no practical importance; for as the elementary pressures in this case are not parallel, their sum (which is the total pressure) is not the same thing as their resultant. For a plane area, in whatever position, the elementary pressures, being everywhere normal to its plane, are parallel and give a resul- tant equal to their sum; and it is often a matter of interest to determine that point in the area through which the resultant passes. This point is called the Centre of Pressure. It is not coincident with the centre of gravity of the area unless the pressure be of equal intensity over the whole area. When the area is not hori- zontal, the pressure is most intense at those parts of it which are deepest, and the centre of pressure is accordingly lower down than the centre of gravity. For a horizontal area the two centres are coincident, and they are also sensibly coincident for any plane area whose dimensions are very small in comparison with its depth in the liquid, for the pressure over such an area is sensibly uniform. 151. Construction for Centre of Pressure. If at every point of a plane area immersed in a liquid, a normal be drawn, equal to the depth of the point, the normals will represent the intensity of pressure at the respective points, and the volume of the solid con- stituted by all the normals will represent the total pressure. That normal which passes through the centre of gravity of this solid will be the line of action of the resultant, and will there- fore pass through the centre of pressure. Thus, if RB (Fig. 64) be a rectangular surface (which we may suppose to be the surface of a flood-gate or of the side . ,. r . of a dam), its lower side B being at the Fjg. 64. Centre of Pressure. ' bottom of the water and its upper side R at the top, the pressure is zero at R and goes on increasing uni- formly to B. The normals B6, Dd, HA, LI, equal to the depths of a CENTRE OF PRESSUEE. 95 series of points in the line BR will have their extremities b, d, h, I, in one straight line. To find the centre of pressure, we must find the centre of gravity of the triangle RB6 and draw a normal through it. As the centre of gravity of a triangle is at one-third of its height, the centre of pressure will be at one-third of the height of BR. It will lie on the line joining the middle points of the upper and lower sides of the rectangle, and will be at one-third of the length of this line from its lower end. The total pressure will be equal to the weight of a quantity of the liquid whose volume is equal to that of the triangular prism constituted by the aggregate of the normals, of which prism the triaAgle RB6 is a right section. It is not difficult to show that the volume of this prism is equal to the product of the area of the rectangle by the depth of the centre of gravity of the rectangle, in accordance with the rule above given. 152. Whirling Vessel. D'Alembert's Principle. If an open vessel of liquid is rapidly rotated round a vertical axis, the surface of the liquid assumes a concave form, as represented in . Fig. 65, where the dotted line is the axis of rota- tion. When the rotation has been going on at a uniform rate for a sufficient time, the liquid mass rotates bodily as if its particles were rigidly connected together, and when this state of things has been attained the form of the surface is that of a paraboloid of revolution, so that the section represented in the figure is a parabola. We have seen in 101 that a particle moving uniformly in a circle is acted on by a force directed towards the centre. In the present case, therefore, there must be a force acting upon each particle of the liquid urging it towards the axis. This force is supplied by the pressure of the liquid, which follows the usual law of increase with depth for all points in the same vertical. If we draw a horizon- Fig 65 _ Rotatinrr Ve8sel tal plane in the liquid, the pressure at each point of of Liquid ' it is that due to the height of the point of the surface vertically over it. The pressure is therefore least at the point where the plane is cut by the axis, and increases as we recede from this centre. Consequently each particle of liquid receives unequal pressures on two opposite sides, being more strongly pressed towards the axis than from it. 96 HYDROSTATICS. Another mode of discussing the case, is to treat it as one of statical equilibrium under the joint action of gravity and a fictitious force called centrifugal force, the latter force being, for each par- ticle, equal and opposite to that which would produce the actual acceleration of the particle. This so-called centrifugal force is therefore to be regarded as a force directed radially outwards from the axis; and by compounding the centrifugal force of each particle with its weight we shall obtain what we are to treat as the resul- tant force on that particle. The form of the surface will then be determined by the condition that at every point of the surface the normal must coincide with this resultant force; just' as in a liquid at rest, the normals must coincide with the direction of gravity. The plan here adopted of introducing fictitious forces equal and opposite to those which if directly applied to each particle of a system would produce the actual accelerations, and then applying the conditions of statical equilibrium, is one of very frequent appli- cation, and will always lead to correct results. This principle was first introduced, or at least systematically expounded, by D'Alem- bert, and is known as D'Alembert's Principle. CHAPTER XIII. PRINCIPLE OF ARCHIMEDES. 153. Pressure of Liquids on Bodies Immersed. When a body is immersed in a liquid, the different points of its surface are sub- jected to pressures which obey the rules laid down in the preceding chapter. As these pressures increase with the depth, those which tend to raise the body exceed those which tend to sink it, so that the resultant effect is a force in the direction opposite to that of gravity. By resolving the pressure on each element into horizontal and vertical components, it can be shown that this resultant upward force is exactly equal to the weight of the liquid displaced by the body. The reasoning is particularly simple in the case of a right cylinder (Fig. 66) plunged vertically in a liquid. It is evident, in the first place, that if we consider any point on the sides of the cylinder, the normal pressure on that point is horizontal and is destroyed by the equal and contrary pressure at the point dia- metrically opposite; hence, the horizontal pres- sures destroy each other. As regards the vertical pressures on the ends, one of them, that on the upper end AB, is in a downward direction, and equal to the weight of the liquid column ABNN; the other, that on the lower end CD, is in an upward direction, and equal to the weight of the liquid column CNND ; this latter pressure exceeds the former by the weight of the liquid cylinder ABDC, so that the resultant effect of the pressure is to raise the body with a force equal to the weight of the liquid displaced. 7 Fig. 66. Principle of Archimedes. 98 PRINCIPLE OF ARCHIMEDES. Fig. 67. Principle of Archimedes. By a synthetic process of reasoning, we may, without having recourse to the analysis of the different pressures, show that this conclusion is perfectly general. Suppose we have a liquid mass in equilibrium, and that we consider specially the portion M (Fig. 67); this portion is likewise in equilibrium. If we suppose it to become solid, without any change in its weight or volume, equilibrium will still subsist. Now this is a heavy mass, and as it does not fall, we must conclude that the effect of the pressures on its surface is' to produce a resultant upward pressure exactly equal to its weight, and acting in a line which passes through its centre of gravity. If we now suppose M replaced by a body exactly occupying its place, the exterior pressures will remain the same, and their resultant effect will therefore be the same. The name centre of buoyancy is given to the centre of gravity of the liquid displaced, that is, if the liquid be uniform, to the centre of gravity of the space occupied by the immersed body; and the above reasoning shows that the resultant pressure acts vertically upwards in a line which passes through this point. The results of the above explanations may thus be included in the following pro- position: Every body immersed in a liquid is subjected to a resul- tant pressure equal to the weight of the liquid displaced, and acting vertically upivards through the centre of buoyancy. This proposition constitutes the celebrated principle of Archimedes. The first part of it is often enunciated in the following form: Every body immersed in a liquid loses a portion of its iveight equal to the weight of the liquid displaced; for when a body is immersed in a liquid, the force required to sustain it will evidently be diminished by a quantit}^ equal to the upward pressure. 154. Experimental Demonstration of the Principle of Archimedes. The following experimental demonstration of the principle of Archi- medes is commonly exhibited in courses of physics : From one of the scales of a hydrostatic balance (Fig. 68) is sus- pended a hollow cylinder of brass, and below this a solid cylinder, whose volume is equal to the interior volume of the hollow cylinder ; these are balanced by weights in the other scale. A vessel of water is then placed below the cylinders, in such a position that the lower cylinder shall be immersed in it. The equilibrium is immediately EXPERIMENTAL PKOOF. 99 destroyed, and the upward pressure of the water causes the scale with the weights to descend. If we now pour water into the hollow cylinder, equilibrium will gradually be re-established; and the beam Fig. OS. Experimental Verification of Principle of Archimedes. will be observed to resume its horizontal position when the hollow cylinder is full of water, the other cylinder being at the same time completely immersed. The upward pressure upon this latter is thus equal to the weight of the water added, that is, to the weight of the liquid displaced. 155. Body Immersed in a Liquid. It follows from the principle of Archimedes that when a body is immersed in a liquid, it is subjected to two forces: one equal to its weight and applied at its centre of gravity, tending to make the body descend; the other equal to the weight of the displaced liquid, applied at the centre of buoyancy, and tending to make it rise. There are thus three different cases to be considered: (1.) The weight of the body may exceed the weight of the liquid displaced, or, in other words, the mean density of the body may be 100 PRINCIPLE OF ARCHIMEDES. greater than that of the liquid; in this case, the body sinks in the liquid, as, for instance, a piece of lead dropped into water. (2.) The weight of the body may be less than that of the liquid displaced; in this case the body will not remain submerged unless forcibly held down, but will rise partly out of the liquid, until the weight of the liquid displaced is equal to its own weight. This is what happens, for instance, if we immerse a piece of cork in water and leave it to itself. (3.) The weight of the body may be equal to the weight of the liquid displaced; in this case, the two opposite forces being equal, the body takes a suitable position and remains in equilibrium. These three cases are exemplified in the three following experi- ments (Fig. 69): (1.) An egg is placed in a vessel of water; it sinks to the bottom iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii Fig. 09. Egg Plunged in Fresh and Salt Water. of the vessel, its mean density being a little greater than that of the liquid. (2.) Instead of fresh water, salt water is employed; the egg floats at the surface of the liquid, which is a little denser than it. (3.) Fresh water is carefully poured on the salt water; a mixture of the two liquids takes place where they are in contact; and if the egg is put in the upper part, it will be seen to descend, and, after a few oscillations, remain at rest at such a depth that it displaces its own weight of the liquid. In speaking of the liquid displaced in this case, we must imagine each horizontal layer of liquid surrounding the egg to be produced through the space which the egg occupies; and by the centre of buoyancy we must understand the centre of "LIQUID DISPLACED" DEFINED. 101 gravity of the portion of liquid which would thus take the place of the egg. We may remark that, in this position the egg is in stable equilibrium; for, if it rises, the upward pressure diminish- ing, its weight tends to make it descend again; if, on the contrary, it sinks, the pressure increases and tends to make it reascend. 158. Cartesian Diver. The experiment of the Cartesian diver, which is described in old treatises on physics, shows each of the different cases that can present themselves when a body is immersed. The diver (Fig. 70) consists of a hollow ball, at the bottom of which is a small opening 0; a little porcelain figure is attached to the ball, and the whole floats upon water contained in a glass vessel, the mouth of which is closed by a strip of caoutchouc or a blad- der. If we press with the hand on the bladder, the air is compressed, and the pressure, trans- mitted through the different horizontal layers, condenses the air in the ball, and causes the entrance of a portion of the liquid by the open- ing 0; the floating body becomes heavier, and in consequence of this increase of weight the diver descends. When we cease to press upon the bladder, the pressure becomes what it was before, some water flows out and the diver ascends. It must be observed, however, that as the diver continues to descend, more and more water enters the ball, in conse- quence of the increase of pressure, so that if the depth of the water exceeded a certain limit, the diver would not be able to rise again from the bottom. Fig. 70.- Caitesiaii Diver. 102 PRINCIPLE OF ARCHIMEDES. If we suppose that at a certain moment the weight of the diver becomes exactly equal to the weight of an equal volume of the liquid, there will be equilibrium ; but, unlike the equilibrium in the experi- ment (3) of last section, this will evidently be unstable, for a slight movement either upwards or downwards will alter the resultant force so as to produce further movement in the same direction. As a consequence of this instability, if the diver is sent down below a certain depth he will not be able to rise again. 157. Relative Positions of the Centre of Gravity and Centre of Buoyancy. In order that a floating body either wholly or partially immersed in a liquid, may be in equilibrium, it is necessary that its weight be equal to the weight of the liquid displaced. This condition is however not sufficient; we require, in addition, that the action of the upward pressure should be exactly opposite to that of the weight; that is, that the centre of gravity and the centre of buoyancy be in the same vertical line; for if this were not the case, the two contrary forces would compose a couple, the effect of which would evidently be to cause the body to turn. In the case of a body completely immersed, it is further necessary for stable equilibrium that the centre of gravity should be below the centre of buoyancy; in fact we see, by Fig. 71, that in any other Fig- 72. ij Re'ative ^Positions of Centre of Gravity and Centre of Treasure. position than that of equilibrium, the effect of the two forces applied at the two points G and O would be to turn the body, so as to bring the centre of gravity lower, relatively to the centre of buoyancy. But this is not the case when the body is only partially immersed, as most frequently happens. In this case it may indeed happen that, with stable equilibrium, the centre of gravity is below the centre of pressure; but this is not necessary, and in the majority of instances is not the case. Let Fig. 72 represent the lower part of a floating body a boat, for instance. The centre of pressure is at O, the centre of gravity at G, considerably above; if the body STABILITY OF FLOATATION. 103 is displaced, and takes the position shown in the figure, it will be seen that the effect of the two forces acting at O and at G is to restore the body to its former position. This difference from what takes place when the body is completely immersed, depends upon the fact that, in the case of the floating body, the figure of the liquid displaced changes with the position of the body, and the centre of buoyancy moves towards the side on which the body is more deeply immersed. It will depend upon the form of the body whether this lateral movement of the centre of buoyancy is sufficient to carry it beyond the vertical through the centre of gravity. The two equal forces which act on the body will evidently turn it to or from the original position of equilibrium, according as the new centre of buoyancy lies beyond or falls short of this vertical. 1 158. Advantage of Lowering the Centre of Gravity. Although stable equilibrium may subsist with the centre of gravity above the centre of buoyancy, yet for a body of given external form the stability is always increased by lowering the centre of gravity; as we thus lengthen the arm of the couple which tends to right the body when displaced. It is on this principle that the use of ballast depends. 159. Phenomena in Apparent Contradiction to the Principle of Archimedes. The principle of Archimedes seems at first sight to be contradicted by some well- known facts. Thus, for instance, if small needles are placed carefully on the surface of water, they will remain there in equilibrium (Fig. 73). Fig. 73. Steel Needles Floating on Water. It is on a similar principle 1 If a vertical through the new centre of buoyancy be drawn upwards to meet that line in the body which in the position of equilibrium was a vertical through the centre of gravity, the point of intersection is called the metacentre. Evidently when the forces tend to restore the body to the position of equilibrium, the metacentre is above the centre of gravity ; when they tend to increase the displacement, it is below. In ships the dis- tance between these two points is usually nearly the same for all amounts of heeling, and this distance is a measure of the stability of the ship. We have defined the metacentre as the intersection of two lines. When these lines lie in different planes, and do not intersect each other, there is no metacentre. This indeed is the case for most of the displacements to which a floating body of irregular shape can be subjected. There are in general only two directions of heeling to which metacentres correspond, and these two directions are at right angles to each other. 104 PRINCIPLE OF ARCHIMEDES. that several insects walk on water (Fig. 74), and that a great number of bodies of various natures, provided they be very minute, can, if we may so say, be placed on the surface of a liquid with- out penetrating into its interior. These curious facts depend on the circumstance that the small bodies Fig. 74. -insect walking on water. in question are not wetted by the liquid, and hence, in virtue of principles which will be explained in connection with capillarity (Chap, xvi.), depressions are formed around them on the liquid surface, as represented in Fig. 75. The curvature of the liquid surface in the neighbourhood of the body is very distinctly shown by observing the shadow cast by the floating body, when it is illumined by the sun; it is seen to be bordered by luminous bands, which are owing to the refraction of the rays of light in the portion of the liquid bounded by a curved surface. The existence of the depression about the floating body enables us to bring the condition of equilibrium in this special case under the general enunciation of the principle of Archimedes. Let M (Fig. 75) be the body, CD the region of the depression, and AB the corresponding portion of any horizontal Fig. 75. layer; since the pressure at each point of AB must be the same as in other parts of the same horizontal layer, the total weight above AB is the same as if M did not exist and the cavity were filled with the liquid itself. We may thus say in this case also that the weight of the floating body is equal to the weight of the liquid displaced, understanding by these words the liquid which would occupy the whole of the depression due to the presence of the body. CHAPTER XIV. DENSITY AND ITS DETERMINATION. 160. Definitions. By the absolute density of a substance is meant the mass of unit volume of it. By the relative density is meant the ratio of its absolute density to that of some standard substance, or, what amounts to the same thing, the ratio of the mass of any volume of the substance in question to the mass of an equal volume of the standard substance. Since equal masses gravitate equally, the com- parison of masses can be effected by weighing, and the relative den- sity of a substance is the ratio of its weight to that of an equal volume of the standard substance. Water at a specified tempera- ture and under atmospheric pressure is usually taken as the standard substance, and the density of a substance relative to water is usually called the specific gravity of the substance. Let V denote the volume of a substance, M its mass, and D its absolute density; then by definition, we have M=VD. If s denote the specific gravity of a substance, and d the absolute density of water in the standard condition, then D=sd and M= Vsd When masses are expressed in Ibs. and volumes in cubic feet, the value of d is about 62'4, since a cubic foot of cold water weighs about 62-4 Ibs. 1 In the C.G.S. system, the value of d is sensibly unity, since a cubic centimetre of water, at a temperature which is nearly that of the maximum density of water, weighs exactly a gramme. 2 The gramme is defined, not by reference to water, but by a standard kilogramme of platinum, which is preserved in Paris, and 1 In round numbers, a cubic foot of water weighs 1000 oz., which is 62'5 Ibs. 3 According to the best determination yet published, the mass of a cubic centimetre of pure water at 4 is 1-000013, at 3 is 1-000004, and at 2 is '999982. 106 DENSITY AND ITS DETEKMINATION. of which several very carefully made copies are preserved in other places. In the above statements (as in all very accurate statements of weights), the weighings are supposed to be made in vacuo; for the masses of two bodies are not accurately proportional to their apparent gravitations in air, unless the two bodies happen to have the same density. 161. Ambiguity of the word " Weight." Properly speaking, " the weight of a body " means the force with which the body gravitates towards the earth. This force, as we have seen, differs slightly according to the place of observation. If m denote the mass of the body, and g the intensity of gravity at the place, the weight of the body is mg. When the body is carried from one place to another without gain or loss of material, m will remain constant and g will vary; hence the weight mg will vary, and in the same ratio as g. But the employment of gravitation units of force instead of absolute units, obscures this fact. The unit of measurement varies in the same ratio as the thing to be measured, and hence the numerical value remains unaltered. A body weighs the same number of pounds or grammes at one place as at another, because the weights of the pound and gramme are themselves proportional to g. Expressed in absolute units, the weight of unit mass is g, and the weight of a mass m is 'mg. The latter is -m times the former; hence when the weight of unit mass is employed as the unit of weight, the same number m which denotes the mass of a body also denotes its weight. What are usually called standard weights that is, standard pieces of metal used for weighing are really standards of mass; and when the result of a weighing is stated in terms of these standards, (as it usually is,) the " weight," as thus stated, is really the mass of the body weighed. The standard " weights " which we use in our balances are really standard masses. In discussions relating to density, weights are most conveniently expressed in gravitation measure, and hence the words mass and weight can be used almost indiscriminately. 162. Determination of Density from Weight and Volume. The absolute density of a substance can be directly determined by weighing a measured volume of it. Thus if v cubic centimetres of it weigh m grammes, its density (in grammes per cubic centimetre) is . This method can be easily applied to solids of regular geometrical forms; since their volumes can be computed from their SPECIFIC GRAVITY BOTTLE. 107 linear measurements. It can also be applied to liquids, by employ- ing a vessel of known content. The bottle usually employed for this purpose is a bottle of thin glass fitted with a perforated stopper, so that it can be filled and stoppered without leaving a space for air. The difference between its weights when full and empty is the weight of the liquid which fills it; and the quotient of this by the volume occupied (which can be determined once for all by weighing the bottle when filled with water) is the density of the liquid. The advantage of employing a perforated stopper is that it enables us to ensure constancy of volume. If a w^ide-mouthed flask were employed, without a stopper, it would be difficult to pronounce when the flask was exactly full. This source of inaccuracy would be diminished by making the mouth narrower: but when it is very narrow, the filling and emptying of the flask are difficult, and there is danger of forcing in bubbles of air with the liquid. When a per- forated stopper is employed, the flask is first filled, then the stopper is inserted and some of the liquid is thus forced up through the perforation, overflowing at the top. When the stopper has been pushed home, all the liquid outside is carefully wiped off, and the liquid which remains is as much as just fills the stoppered flask including the perforation in the stopper. In accurate work, the temperature must be observed, and due allowance made for its effect upon volume. 163. Specific Gravity Flask for Solids. The volume and density of a solid body of irregular shape, or consisting of a quantity of small pieces, can be de- termined by put- ting it into such a bottle (Fig. 76), and weighing the water which it displaces. The most convenient way of doing j this is to observe \ (1) the weight of the solid; (2) the weight of the bottle full of water; (3) the weight of the bottle when it contains the solid, together with as much water as will fill it up. If the Fig. 76. Specific Gravity Flask for Solids. 108 DENSITY AND ITS DETERMINATION. third of these results be subtracted from the sum of the first two, the remainder will be the weight of the water displaced; which, when expressed in grammes, is the same as the volume of the body expressed in cubic centimetres. The weight of the body, divided by this remainder, is the density of the body. 164. Method by Weighing in Water. The methods of determining density which we are now about to describe depend upon the prin- ciple of Archimedes. One of the commonest ways of determining the density of a solid body is to weigh it first in air and then in water (Fig. 77) the Fig. 77. Specific Gravity of Solids. Fig. 78. Specific Gravity of Liquids. counterpoising weights being in air. Since the loss of weight due to its immersion in water is equal to the weight of the same volume of water, we have only to divide the weight in air by this loss of weight. We shall thus obtain the relative density of the body as compared with water in other words, the specific gravity of the body. WEIGHING IN WATER. 109 Thus, from the observations Weight in air, 125 ym. Weight in water, 100 Loss of weight, 25 we deduce !?? = 5 = density. A very fine and strong thread or fibre should be employed for sus- pending the body, so that the volume of liquid displaced by this thread may be as small as possible. 165. Weighing in Water, with a Sinker. If the body is lighter than water, we may employ a sinker that is, a piece of some heavy material attached to it, and heavy enough to make it sink. It is not necessary to know the weight of the sinker in air, but we must observe its weight in water. Call this s. Let w be the weight of the body in air, and w the weight of the tody and sinker together in water. Then w will be less than s. The bod^ has an apparent upward gravitation in water equal to s w', showing that the resultant pressure upon it exceeds its weight by this amount. Hence the weight of the liquid displaced is w + s w, and the specific gravity of the body is w + _ w , If any other liquid than water be employed in the methods described in this and the preceding section, the result obtained will be the relative density as compared with that liquid. The result must therefore be multiplied by the density of the liquid, in order to obtain the absolute density. 166. Density of Liquid Inferred from Loss of Weight. The densities of liquids are often determined by observing the loss of weight of a solid immersed in them, and dividing by the known volume of the solid or by its loss of weight in water. Thus, from the observations Weight in air, 200 gm Weight in liquid, 120 Weight in water, 110 we deduce Loss in liquid, 80. Loss in water, 90. Density of liquid, |j = * A glass ball (sometimes weighted with mercury, as in Fig. 78) is the solid most frequently employed for such observations. 110 DENSITY AND ITS DETERMINATION. 167. Measurement of Volumes of Solids by Loss of Weight. The volume of a solid body, especially if of irregular shape, can usually be determined with more accuracy by weighing it in a liquid than by any other method. If it weigh w grammes in air, and w grammes in water, its volume is w w cubic centimetres, since it displaces ww grammes of water. The mean diameter of a wire can be very accurately determined by an observation of this kind for volume, combined with a direct measurement of length. The volume divided by the length will be the mean sectional area, which is equal to trr 2 , where r is the radius. 168. Hydrometers. The name hydrometer is given to a class of instruments used for determining the densities of liquids by observ- ing either the depths to which they sink in the liquids or the . Fig. 79. Nicholson's Hydrometer. weights required to be attached to them to make them sink to a given depth. According as they are to be used in the latter or the former of these two ways, they are called hydrometers of constant or of variable immersion. The name areometer (from apmoQ, rare) is used as synonymous with hydrometer, being probably borrowed from the French name of these instruments, artfometre. The hydro- NICHOLSON'S HYDROMETER. Ill meters of constant immersion most generally known are those of Nicholson and Fahrenheit. 169. Nicholson's Hydrometer. This instrument, which is repre- sented in Fig. 79, consists of a hollow cylinder of metal with conical ends, terminated above by a very thin rod bearing a small dish, and carrying at its lower end a kind of basket. This latter is of such weight that when the instrument is immersed in water a weight of 100 grammes must be placed in the dish above in order to sink the apparatus as far as a certain mark on the rod. By the principle of Archimedes, the weight of the instrument, together with the 100 grammes which it carries, is equal to the weight of the water dis- placed. Now, let the instrument be placed in another liquid, and the weights in the dish above be altered until they are just sufficient to make the instrument sink to the mark on the rod. If the weights in the dish be called w, and the weight of the instrument itself W, the weight of liquid displaced is now W + w, whereas the weight of the same volume of water was W + 100; hence the specific gravity of the liquid is w +^ Q . This instrument can also be used either for weighing small solid bodies or for finding their specific gravities. To find the weight of a body (which we shall suppose to weigh less than 100 grammes), it must be placed in the dish at the top, together with weights just sufficient to make the instrument sink in water as far as the mark. Obviously these weights are the difference between the weight of the body and 100 grammes. To find the specific gravity of a solid, we first ascertain its weight by the method just described; we then transfer it from the dish above to the basket below, so that it shall be under water during the observation, and observe what additional weights must now be placed in the dish. These additional weights represent the weight of the water displaced by the solid; and the weight of the solid itself divided by this weight is the specific gravity required. 170. Fahrenheit's Hydrometer. This instrument, which is repre- sented in Fig. 80, is generally constructed of glass, and differs from Nicholson's in having at its lower extremity a ball weighted with mercury instead of the basket. It resembles it in having a dish at the top, in which weights are to be placed sufficient to sink the instrument to a definite mark on the stem. 112 DENSITY AND ITS DETERMINATION. Hydrometers of constant immersion, though still described in text-books, have quite gone out of use for practical work. 171. Hydrometers of Variable Immersion. These instruments are usually of the forms represented at A, B, C, Fig. 81. The lower end is weighted with mercury in order to make the instrument sink to a convenient depth and preserve an upright position. The stem is cylindrical, and is graduated, the divisions being frequently marked Fig. 80. Fahrenheit's Hydrometer. Fig. SI. Forms of Hydrometers. upon a piece of paper inclosed within the stem, which must in this case be of glass. It is evident that the instrument will sink the deeper the less is the specific gravity of the liquid, since the weight of the liquid displaced must be equal to that of the instrument. Hence if any uniform system of graduation be adopted, so that all the instruments give the same readings in liquids of the same densi- ties, the density of a liquid can be obtained by a mere immersion of the hydrometer an operation not indeed very precise, but very easy of execution. These instruments have thus come into general use for commercial purposes and in the excise. 172. General Theory of Hydrometers of Variable Immersion. Let V be the volume of a hydrometer which is immersed when the in- strument floats freely in a liquid whose density is d, then yd repre- HYDROMETERS. 113 sents the weight of liquid displaced, which by the principle of Archi- medes is the same as the weight of the hydrometer itself. If V, d' be the corresponding values for another liquid, we have therefore Vd = V'd', OTd:d'::V:V, that is, the density varies inversely as the volume immersed. Let d l} d 2 ,