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Les diagrammes suivents illustrent la mAthode. 1 2 3 1 2 3 4 5 6 LOVELL'S SERIES OF SCHOOL BOOKS. NATURAL PHILOSOPHY, PART I, INOLUDINQ 'TATICS, HYDROSTATICS, PNEUMATICS, DYNAMICS, HYDRODYNAMICS, THE GENERAL THEORY OF UNDULATIONS, THE SCIENCE OF SOUND, THE MECHANICAL THEORY OP MUSIC, ETC. , DESIGMBD F0& THE USE OF NORMAL AND GRAMMAR SCHOOLS, AND THE HIGHER* GLASSES IN COMMON SCHOOLS. BY JOHN HERBERT SANCISTER, H.A., NATBBUATIOAL MA8TIB AND LBOTUBBB IN 0HBMI8TBT AND KATUBAL PHILOSOPHY IN TBI NOBMAL SCHOOL FOB UPPIB CANADA, 8BOOND EDITION-REVISED AND ENLARGED. JJPlontreal: PRINTED AND PUBLISHED BY JOHN LOVELL ; AND lOLD BT B. A A. IIILLIB. R. A A. MILLER, 62 KING STREET EAST. 1861. t :: ! Entered, according to the Act of the Previncial Parliament, in the year one thousand eight hundred and siztj^ bj John LoYBLL, in the Office of the Registrar of the ProTince of Canada. I I I ! I . ■ . * ;- '■■ .| . PREFACE TO FIRST EDITION. iment, in , by John OYince of ( ; I 1 Thx following Treatise was originally designed to serve as a band-book or companion to the lectures on Natural Philosophy, delivered to the junior division in the Normal School. Although numerous text-books on the subject were already in existence, it was found that tney were either too abstruse and technical for beginners, or too general and superficial to be of much prac tical use. The aim of the present little work is to occupy a position between these extremes — to present the leading &cts of the science in a form so concise as to be readily remembered, and at the same time to give that thorough drilling upon the principles which is absolutely essential to their full comprehen- sion. As a hand-book to lectures fully illustrated by apparatus, it was not necessary to introduce many wood-cuts, and according- ly they have been given only where absolutely required. The chief peculiarity of this book consists in the intro- duction to a large extent of problems calculated to impart that intimate and practical knowledge of the facts and principles of Mechanical Science, without which the student's information on the subject is, comparatively speaking, useless. Plow frequently do we meet with a pupil who has read carefully through one of the common text-books on Natural Philosophy without acquiring any very clear or definite ideas of the science ! And what should we say of a work professing to IV PREFACE. teach the principles of arithmetic or algebra by mere rules and explanations, without an appropriate selection of exam- ples and problems ? The exercises are therefore deemed an important feature of the following pages, and it is thought that the science maj be taught by their aid more thorougbl j and in less time than otherwise. ,^ ^,^,>. ., Toronto, January, 1860. ■iLn^L t<3> "iiCk •#M( 163 CHAPTER IX. Theory of Undulations, 165 Vibrations of strings, 166 Vibrations of rods, 167 Vibrations of plates, 168 Kodal figures, 169 Undulations in liquids, 169 Undulations in elastic fluids, 161 CHAPTER X. Acoustics, 161 Velocity of Sound, 162 Echoes, 165 Whispering Galleries, 166 CHAPTER XI. Mechanical Theory of Music, 167 CHAPTER XIL The Organs of Voice, 177 The Organs of Hearing, 181 Miscellaneous Problems, • IM Bzaaination Papers, • » Hi Aniwert to Examination Papers, k....**!^ SztmiaAUon Questions, • •;lQi' Paoi ....146 ....146 ....146 ....Ut ....147 ....148 ....150 ....150 ....151 ....151 ....153 ....155 ....156 ....157 ....158 ....159 ....169 161 ...161 ...162 ...165 ...166 ...167 ...177 ...181 ... loo ...]t6 • »t ^ # ,.nW9^ NATURAL PHILOSOPHY. CHAPTER I. SUBDIVISIONS—GENERAL PROPERTIES OP MATTER- ATTRACTION. 1. Natural Science, in its widest sense, embraces the study of all created objects and beings, and the laws by which they are governed. 2. Natural objects are divided into two great classes, viz., organic and inorganiCj the former being distinguished from the latter by the exhibition of vital power or life, 3. Organic existences are separated into animals and vegetables^ the former distinguished from the latter by the possession of sensibility and volition, 4. The different subdivisions of natural science and their objects are as follows : — / Zoology describes and classifies animals. Botany teaches the classification/ use, habits, structure, il;c, oi plants. Mineralogy describes and classifies the various mineral constituents of the earth's crust. Astronomy investigates the laws, (&c., of celestial phe- nomena. Geology has for its object the description, drc, of tha crust of the earth. Chemistry teaches us how to unite two or more elemenP- a4f%i>0iu8 into one compound, or how to decompose com- pofmdMies into their simple elements. ^ . •JiilM Philosophy or Physics has for its object the ittlMlteiioii of the general properties of all bodiea and thmmAittd laws by which they are regulated. 10 PROPERTIES OP MATTER. [Aets. 5-ui. 6. Natural Philosophy is divided into — I. General Mechanics — including Statics, Hydrostatics, Dynamics, Hydrodynamics, and Pneumatics. H. Heat. ; , . j ' * in. Light — including Perspective, Catoptrics, Dioptrics, Chromatics, Physical Optics, Polarization, and Actino- Chemistry. IV. Electricity—including Statical Electricity, Galvan- ism, Magnetism, Thermo-Electricity, and Animal Elec- tricity. . . ♦ V. Acoustics, rr ♦ PROPERTIES OF MATTER. 6. Matter exists in three separate forms, — I. Solid; II. Liquid ; and III. Gaseous, Note.— The same body may exist in all three forms, as is the case with water, mercury, sulphur, Ac. The amount of heat or caloric present deter- mines-the form of the body— if heat be applied, the attraction of cohesion existing among the particles is gradually overcome, and the bodypasses from a solid to a liquid, and from a liquid to a gas. If heat be abstracted, the attraction of cohesion gradually draws the particles into closer proximity and the body passes from a gas to a liquid, and finally flrom a liquid to a solid. Hence heat and cohesion are called antagonitiie forces. 7. Matter is distinguished by the possession of certain distinctive properties. 8. The properties of matter are divided into — 1st. Essential Properties. 2nd. Accessory Properties. 9. The essential properties of matter are those without which matter could not possibly exist. 10. The essential properties of matter are Extension, Impenetrahility , Divisibility, Indestructahility, Porosity^ Compfessibility, Inertia, and Elasticity, 11. Extension implies that every body ror^t. fill a certain portion of space. XoTB.~The Dimeniloni of Exteniton are Ungth, breadth and tkiaknmi, 12. Impenetrability implies that no two bodies caa oe* cnpy the same portion of space at the same tinie. rs. 5-1^. statics, jptrics, Vctino- 3aivan- l Elec- Solid ; case with Bnt deter- ' cohesion dy passes istracted, )roxiraity aid to a certain urithout TTOsity, oerttun 00* Arts. 13-16.] PROPERTIES OE MATTER. 11 NoTB.— Eiamples of the impenetrability of matter will readily suncest themselves. Among the more common may be mentioned the impossiDility of filling a bottle with water until the air is displaced— the fact that when the hand is plunged into a vessel filled with water, a portion of the liquid overflows, &c. All instances of the apparent penetrability of matter are merely examples of displacement. Thus, when a nail is driven into a piece of wood, it displaces the particles of wood, driving them closer together. 13. Divisibility is the capability of being continually divided and subdivided, and is an essential property only of masses of matter. Note 1.— The ultimate particles of matter ; i. e., those inconceivably minute molecules which cannot be further subdivided, are termed atoms, ( Or. a ** not" and temno "to cut" ; i. e., that which cannot be cut or divided.) Note 2.— The foUoMdng may be given as examples c/ the extreme divisi- bility of matter :— I. Gk)ld leaf is hammered so thin that 300000 leaves placed one on another, and pressed so as to exclude the air, measure but one inch in thickness. II. Wollaston's micrometric wire is so fine that 30000 wires placed side by side, measure but one inch across— 160 of these wires bound together do not exceed the diameter of a filament of raw silk, 1 mile of the wire weighs but a grain, and 7 ounces would reach from Toronto to England. III. Insects' win^ are some of them so fine tliat they do not exceed the 'STTotjinr ^' ^^ ^^^^ ^^ thickness. IV. The thinnest part of a soap bubble is only the 2500000th part of an inch in tliickness. V. Blood corpuscles are so small that it requires 50000 corpuscles of human blood, or 800000 corpuscles of the blood of the musk*deer to cover the head of a common pin. Yet these corpuscles are compound bodies and may be resolved, by means of chemistry, into their simple elements. VI. There fire animalcules so minute that millions of them heaped together do not equal the bulk of a single grain of sand, and thousands might swim side by side through the eye of the finest cambric needle. Yet tliese creatures possess, in many cases, complicated organs of locomotion, nutritioii, &c. VII. At Bilin in Bohemia, a huge mountain consists entirely of shells, so minute that a cubic inch contains -ll billions— a number so vast that counting as rapidly as possible, day and night without intermission, it would require 780 years to enumerate it. VIII. The filament of the spider's web is so fine that 4 miles of it weigh oMly about a grain— yet this thread is formed of about 6000 filaments united together, Ac, Ac. 14. Indestructahility implies that it is as impossible for H finite creature to annihilate as to create matter. Note.— We can change the form of matter at pleasure, but we cannot destroy it. When fuel, for example, is burned, not a particle is lott, ai is f)rovea by the fkct that if we collect all the products of the combustion ; .e.. the smoke, soot, ashes, Ac.,and weigh them, we shall find their aggregate we»|t oiMtly eqau to that of the wood or ooal consumed. We may safely conaonle that there is not a single atom of matter, more or less, attached to our eirtik now Uian at the time of Adam. lii'lPbiN^'ty implies that the constituent atoms of mat- tev^4^itil'«ibuoh each other, but are separated by emaU i n t i i tiiikiy tpaoes called pares. 12 PROPERTIES O:^ MATTER* [Abtb. 16-28. As: ! I ■* XoTB.— The atoms even of the densest bodies are much smaller than the spaces which separate them. Newton rwards them as infinitely smaller^ as beiuff in fact mere mathematical points, and Sir J. Herschel asks why the particles of a solid may not be as thinly distributed through the space it occupies as the stars that compose a nebula, and he compares a ray of light penetrating glass to a bird ttireading the mazes of a forest. 16. Compressibility implies the capability a body pos- sesses of being forced into a smaller bulk without any di- minution in the quantity of matter it contains. NoTB.— Since all matter is porous, it follows, as a necessary consequence, that all matter must be compressible. 17. Inertia means passivencss or inactivity, or that mat- ter is incapable of changing its state, either from rest to motion or from motion \o rest. Note.— Bodies moving on or neat' the surface of the earth soon come to a state of rest, unless some constant propelling force is applied to them. This is owing to the action of certain resisting forces, as the resistance of the atmosphere, friction, and the attraction of gravity. 18. Elasticity is the capability which all bodies possess, more or less, of recovering their former dimensions after compression of after having, for a time, been compelled to assume some other form. Note.— As applied to solids, elasticity is divided into— 1. Elasticity of compression, 2. Elasticity of tension, 8. Elasticity of flexure, and ...<.;. 4. Elasticity of torsion. , Some bodies, as putty, seem to possess very little elasticitv. In glass all four kinds appear to exist almost perfect within certain limits— no force however great or long continued will cause glass to take a aet, as it is termed. 19. The accessory properties of matter are those which merely serve to distinguish one kind of matter from another. 20. The accessory properties of matter are hardness^ softness^ flexibility^ brittlenesSj transparency^ opacity ^ mal- leability , ductilityy tenacity^ <&c. 21. Malleability expresses the susceptibility, possessed by certain kinds of matter, of being hammered out into thin sheets. NoTi.— The most malleable metals are gold, silver, iron, copper and tin . 22. Ductility is susceptibility of being drawn out into fine wire. NoTi.--The most ductile metals are platinum, gold, inm, and ooppv. 28. Tenacity or toughness implies that a certain foiof is necessary to pull th« particles of a body asunder. ^j^^^, t. ¥ f AETS.lfi-21 I ABT8. 24.27.] ATTRACTION. IS ler than the lely smaller, lel asks why ;h the space kres a ray of tr. body pos- iit any di- M>nsequence, that mat- in rest to toon come to ied to them, resistance of es possess, sioDS after npelled to In glass all Its — no force lit is termed. )8e which another. \hardnesSy "Aty^ mal- possessed out into and tin. out into ooppw* kin fore« NoTB.— The following table shows the relative tenacity of different substances. The first column shows the number of pounds weight required .to tear asunder a prism of each substance, having a sectional area of one square incht and the second column gives the length of the rod of any given diameter which, if suspended, would be torn asunder by its own weight :— rfv TABLE OF TENACITY, i - r Weight in pounds. (Section of rod 1 sq. in.) Metals. Cast lead, Cast tin. Yellow Brass, "-<,• ^ ■. Cast Copper, Cast Iron, English Malleable Iron, Swedish do. Cast steel. Pine, Elm, . Oak. ^ Beech, Ash, 1824 4736 17968 i : 19072 19096 66872 < - . 72064 184is66 Woods. 9640 9720 11880 12226 14130 Length in feet, (any diameter.) 848 1496 6180 5003 6110 16988 19740 89466 40600 86800 32900 88940 42080 ATTRACTION. 24. Attraction is that power in virtue of which particles and masses of matter are drawn towards each other. 26. Attraction is of several kinds, viz : I. Attraction of Gravity. II. Attraction of Cohesion. III. Attraction of Adhesion^ IV. Capillary Attraction. V. Electrical Attraction. VI. Magnetic Attraction. VII. Chemical Attraction. 26. Attraction of Gravity (Lat. gravitas^ "weight") is that force hy which masses of matter tend to approach each other. It is sometimes spoken of at* gravitation^ or when Applied to the force hy which hodies are drawn towards the centre of the earth, terrestrial gravity. 97* The intensity of the force of gravity varies directly as Uiir iMM of the bodies, and inversely as the square of thair diiteace apart. 14 ATTEACTION. CASTS, 28. 29. i ' NoTB.— If we suppose two spheres of any kind of matter,lead, for example, to be placed in presence of each other, and under such conditions that being themselves free to move in any durection they are entirely uuinflu- enc^ 1^ any other bodies or circumstances they will approach each other and:— ist. If their masses are equal, their velocities will be equal. 2nd. If one contain twice as much matter as the other, its velocity will be only half as great as that of the other. 3rd. If one be infinitely great in comparison vrith the other, its motion will be infinitely small in comparison with that of the other ; and 4th. The more nearly they approach each other, the more rapid will their motion become. 28. By saying the intensity of the force of gravitation varies inversely as the square of the distance between the attracting bodies, we merely mean that if the attractive force exerted between two bodies at any given distance apart be represented by the unit 1, then, if the distance apart be doubled, the force of attraction will be reduced to J of what it was before ; if the distance between the bodies be increased to three times what it was, the force of gravity will be decreased 9 times, or will bo only ^ of what it was, gether. ARTS. 30-86,3 STATICS, ^ NoTB.— The attraction of cohesion acts only at insensible distances ; i. e., at distances so minute as to be incapable of measurement. The attraction of gravity, on the other hand, acts at sensible distances. 30. Attraction of Adhesiort is that force by which* the particles of dissimilar bodies adhere or stick together. SI, Capillary Attraction (Lat. capilla^ " a hair") is the force by which fluids rise above their level in confined situations, such as small tubes, the interstices of porous substances, &c. *■ NoTB.— It is by capillary attraction that oil and burning fluid» melted tallow, 4;o., rise up the wick of a lamp or candle. 82. Electrical Attraction is the force developed by fric- tion on certain substances, as glass, amber, sealing-wax, &c. 33. Magnetic Attraction is the force by which iron, nickel, &c., are drawn to the load-stone. 34. Chemical Attraction^ or Chemical Affinity, is the force hy which two or more dissimilar bodies unite so as to form a compound essentially different in its appearance and properties from either of its constituents. Thus Potash and Grease unite to form Soap— Sulphur and Mercury unite to form Termillion, &c. CHAPTER II. STATICS. ^ 36. The Science of general mechanics (Greek michani, ** a machine ") has for its object the investigation of the action of forces on matter whether they tend to keep it at rest or to set it in motion. 36. The Science of general mechanics is usually sub- divided as follows : — I. Statics, (Greek statosy " standing") or the science by which the conditions of the equilibrium of solids are determined. II, Hydrostatics, (Greek hMor^ " water," and statos, *' standing,") or the science by which the conditions of the equilibrium of liquids are determined. III. Dtmamios, TGreek dimamisi '*force,") or the science by which the laws that determine the motions of solids are investigated. STATICS. [Asm 87-44. I •i':ll IV. Hydrodynamics ^Greek hilidor and diinamU) or the science by which the laws that determine the motions ' of liquids are investigated. V, Pneumatics {Greek pneuma, "air," &nd staioSy "stand- ing") or Pueuma-statics, the science by which the conditions of the equilibrium of elastic fluids^ as «tD[iospheric air, are investigated. Pneumatics may be regarded as a branch of Hydrostatics. 37. A body is said to be in equilibrium when the forces which act upon it mutually counterbalance each other or are counterbalanced by some passive force or resistance. 38. Forces that are balanced so as to produce rest are called statical forces or pressures to distinguish them from moving^ deflecting, accelerating or retarding forces, 39. A force has three elements, viz., magnitude, direc- tion, and point of application, • - - — 40. A force may be represented either by saying it is equal to a certain number of lbs., oz., &c., or by a line of definite length. A line has the advantage of completely defining a force in all its three elements, while a number can merely represent its magnitude. 41. Whatever number of forces may act npon one point of a body, and whatever their direction, they can impart tb the body only one single motion in one certain direction. 42. When several forces (termed components) act on a point, tending to produce motion in different directions, they may be incorporated into one force, called the result" ant, which, acting alone, will have the same mechanical effect as the several components. 43. When any number of forc3S act on a point in the same straight line, the resultant is equal to their sum, if they act in the same direction ; but if they act in oppo- site directions, the resultant is equal to the difference be- tween the sum of those acting in one direction and the sum of those acting in the other. 44. If two forces acting upon the same point be repre- sented in magnitude and direction by two lines drawn through that point, then the resultant of such forces will J.S7-44. >r the 'stand- 5h the IdSj as » may 3 forces her or ance« est are m from e, direc- ing it is El line of npletely number ne point impart lirection. ict on a ictions, [e result" ihanical It in the sum, if in oppo- jnce be- laud the repre- drawn ircea nviU ABts.4{Hej STATICS. 17 16 be r^resented in magnitude and direction by the diagonal of the parallelogram^ of which these lines are the sides, 46. If any number of forces, A, B, C, D, &c., act upon the same point in any direction whatever, and in any plane whatever, by first finding the resultant of A and B, then of this resultant and C, then of this resultant and D, and so on, we shall finally arrive at the determination of a single force, which will be mechanically equivalent to, and will therefore be the resultant of the entire system. 46. If the components act in the same plane, the resultant is found by means of what is technically termed the 'parallelogram of forces^ if in different planes by the parallelopiped of forces, 47. The resultant of two forces, which act on different points of the same body in parallel lines and in the same direction, is a single force equal to their sum, acting paral- lel to them, and in the same direction, at an intermediate point which divides the line joining the two points of ap- plication of the components in the inverse ratio of the mag- nitudes of these components. 48. The resultant of two forces, which act on different points of the same body in parallel lines but in opposite directions, is a single force equal to their difference, acting parallel to them and in the direction of the greater force, and at a point beyond the greater of the two forces, so situated, that the point of application of the greater of the two forces divides the distance between the points of appli- cation of the smaller force and of the resultant in the inverse ratio of the magnitudes of the smaller force and of the resultant. 49. When any number of parallel forces. A, B, C, D, &c., act on a body, at any point whatever, and in any planes whatever, by first finding the resultant of A and B, next of this resultant and C, then of this last resultant and D, and so on, we shall finally arrive at the determination of a single force, which will be mechanically equivalent to, and will therefore be the resultant of the entire system of parallel forces. i 18 STATICS. tAsn-MHHS. "I 60. When a system of forces consists of two equals opposite^ and parallel forces, it is called a Couple, 4 im ^ 61. Two equal and parallel forces acting on a body in contrary directions, have a tendency to make that body revolve round an axis perpendicular to a plane passing through the direction of such two parallel and opposite forces ; and such tendency is proportional to the product obtained by multiplying the magnitude of the forces by the distance between their points of application : and, conse- qucntly, all couples, in which such products are equal, and which have their planes parallel, are mechanically equivalent, provided their tendency is to turn the body round in the same direction ; but if two such couples have a tendency to turn the body in contrary directions, then they have equal and contrary mechanical effects, and would, if simultaneously applied to the same body, keep it in equilibrium. 62. If any two forces, not parallel in direction, but which are in the same plane, be applied at any two points of a body, they admit of a single resultant, which may be determined by producing the lines, that in magnitude and direction represent the two forces, until they meet in a- point and then applying the principle of the parallelogram of forces. 63. If two forces not parallel in direction act in different planes on two points of a body, they are mechanically equal to the combined action of a couple and of a single force, and their effect will be two-fold — 1st, a tendency to produce revolution; 2nd, a tendency to produce progres- sive motion, so that, if not held in equilibrium oy some antagonistic forces, the body will at the same time move forward, and revolve round some determinate axis. 64. The process of incorporating or compounding two or more forces into one, is called the composition of forces ; that of separating or resolving a single force into two or more, is tiS W ' Ih 90 feet laired the ■Ji' [power of order to SOLUTIOir. Here ir=600.P=20uid«=l6. WX» 500XW ^^. . . Henoe S =— s— =-- ExAMPLS 7.— A power of 21 lbs. moTing through 76 feet car* ries a certain weight through 11 feet. Required the weight. SOLUTION, Here P = 21, S^ 75 and « =11. Then W=^^^^^=zl4^\h».Ans. EzAMPLB 8. — A power of 204 lbs. moving through 30 feet is made to move a weight of 1000 lbs. Through how many feet does the weight moTe ? SOLUTION. HeM P s 204, Tr=i000and S=so. _. ^ PXS 204X30 «, j^ . Then »=-^=-rj— =63iVffc.^»*. f.t ■XIB0I8E. 9. A power of 7 lbs. is made to move a weight of 1000 lbs. through 11 feet ; through how many feet must&e power move ? Jlns. 157lf feet. 10. A power of 97 lbs, moving through 86 feet raises a certain weight through 10 feet. Required the weight. Ant. 834^- lbs. 11. A weight of 888 lbs. is raised by a power of 60 lbs.; through how many feet must the power move in order to raise the weight through I foot? Jins. 14^ feet. 12. A certain power moving through 27 feet is so applied that it carries a weight of 2500 lbs. through 4 feet. Required the power. Jins. 370|f lbs. 67. Any contrivance by which, in accordance with the principle of Virtual Velocities, a small force acting through a large space is converted into a great force acting through a small space, or vice versd, is a Machine. Machines are either simple or complex. 68. In the composition of machinery it is usual to speak of six mechanical powers — ^more properly termed Mechan- ical Elements or Simple Machines, viz :-^ The Lever, ) The Inclined Plane, > Primary Meclianical Elements, The Pulley and Cord, ) The Wheel and Axle, \ The Wedge, > Secondary Afechanical Mements^ The Screw, ) 22 THB LEVER. [ABT8. 0»-75, iii 69. In reality however, there are but two simple me- chanical elements/ viz. : the Lever and the Inclined Plane. The Wheel and Axle and the Pulley are merely modifica- tions of the lever J while the Wedge and the Screw are both formed from the inclined plane, 70. In theoretical mechanics levers are assumed to be perfectly riff id and imponderable — cords, ropes and chains are regarded as having neither thickness, stiffness nor weight, they are assumed to be mere mathematical lines^ infinitely flexible and injinitely strong. At first no allow<> ance is made for friction, atmospheric resistance, &c. After the problem, divested of all these complicating cir- cumstances has been solved, the result is modified by taking into consideration the effects of weight, friction, atmos- pheric resistance, rigidity of cords, flexibility of bars, r Com- inds, — itions of between ■between r-*- 76. In levera of the third class the power is between the /uleruniltLnd the weight, pjg, 3, p A pair of ooQinion tonfi^, gheep- sheftrs, the treadle of a foot lathe, a door when opened or closed by placing the hand near the hinRo, afford examples of levera of the third daas. KoTB.— In levers of the first class the power mvy be either greater or less than the weight; in levers of the second class, the power is al- watf9 leu than the weight; and in levers of the third class, the power is dhoaM greater than the weight. Hence levera of the third class are called losing levers, and are used merely to secure extent of motion. Most of the levers in the animal economy are levers of the third kind. 77. That portion of the lever included between the ful- crum and the weight is termed the arm of the weight ; that portion between the fulcrum and the power is termed the arm of the power. The power and the weight in the lever are in equilibrium when the power it to the weight ae the arm of the weight it to the arm 0/ the power. Or let P = power f W =: the weighty A =: the arm of the power, and a = the arm of the weight, ThexkP:W::a:ji. ^ ^ Wxa^ PXJ Pxj9 ^^ Wxa Hence P =: — -^ — ; W=: — — - ; a= — ^^ ; and . tf j= ■■ p ' ■ ExAHPLi 13.— The power-arm of a lever is 11 feet long, the arm of the weight 3 ft. long, the weight is 93 lbs. Beqoired the power. ^. SOLUTIOV. Here fF = 9S, ^ = 11 and a =s 3. ThenPs TFXa _ 93XS A " n = 26^ lbs. Ane. ExAMPU 14. — The power-arm of a lever is 17 feet long, the arm of the weight is 20 feet long, the power is 110 lbs. What is the weight ? SOLUTIOS. Hen P s no lbs., ^ = 17 and a = 20. rbmW^^^=^^^=mm. An,. a IV 24 THB LEVEE. [AXTS.79»79. Example 15.— By means of a lever a power of 4 oz. is made to balance a weight of 1 lbs. Avoir. ; the arm of j|he weight is 2^ inches long. Bequired the arm of the power. BOIfUTION, Here P = 4 oz., 1^= 7 lbs. = 112 ojs., and a = 24 E|^ = liip = 70inche8. Am, EXBBOISB. Then 4 16. The power-arm of a lever is 16 feet long, tfa« arm of the weight 2 feet long, and the weight is 250 lbs. Required the power. *dM. 31 i lbs. 17. The power-arm of a lever is 20 fset long, the arm of the weight 70 fbet; what power will balance a weight of 5 cwt. ? ^ns. 17J cwt. 18. The power-arm of a lever is 60 inches long, the arm of the weight 90 inohea long, the power is 76 Ifa^. Required the weight. e .. jins. 50§ lbs. 19. The power-arm of a lever is 17 feet long, the arm of the weight 19 ft. ; what power will balance a weight of 950 lbs.? jins^. 1061if lbs. 20. The power-arm of a lever is 12 ft. long, the power is 10 lbs., and the weight 75 lbs. Required the length of the arm of the weight. .Ant. If feet. 21. By means of a lever a power of 12§ lbs. is made to balance a weight of 93 lbs. ; the arm of the weight being 6} feet, what is the length of the arm of the power ? Ant, 4%^ ft. 78. When the power and the weight merely balance each other, i, e,y when no motion is produced, there is no difference between the second and third classes of levers since neither force can be regarded as the mover or the moved. In order to produce motion, one of these forces must prevail, and the lever then belongs to the second or third class, according as the force nearer to ov farther frmn the fulcrum prevails. 79; If the arms of the lever are curved or bent, their effective lengths must be ascertained by perpendionlars drawn from the fulcrum upon the lines of direction of the power and the weight ; the same rule must be adopted when the lever is straight, if the power and weight do not act parallel with one another. ABTS.78b79. ASX8.80,81.] THE COMPOUND LEVER. 25 z. is made ) weight is Eirm of the squired the M. 31i lbs. irm of the of 6 cwt. ? t. ITJ cwt. irm of the tquired the M. 60§ lbs. Einn of the >f 960 lbs.? I061i^lbs. ' is 10 lbs., be arm of u. If feet. to balance ig 6i feet, H. 41FH ft. balance Bre is no of levers >r or the sse forces econd or her from Dt, their idionlars m of the adopted t do not f-l THE COMPOUND LEVER. 80, Two or more simple levers acting upon one another constitute what is called a Compound Lever or Com- Pig. 4. IT ^r''ts I- J ' >. ^- ■■'■] position of Levers. In such a combination the ratio of the power to the weight, is compounded of the ratios existing between the several arms of the compound lever. 81. In the compound lever if W= weight j P :=^power^ a a' a" the arm* of the weighty and A A' A" the arms of the power, % Then P : IT: : a X a' X a" : ^ x ^' X A" ■(' > V „ „ Wxaxa'xa" ^ „^ P y. A x A' x A" - Hence P = — ^ .. >,/ .. a„ and W^= AxA' X A" ax a' X a" ExAMPLB 22..^In a combination of levers the arms of the power are 6, 7, and 11 feet, the arms of the weight 2, 3, and 3| feet, the weight is 803 lbs. ; what is the power ? SOLUTION. Here ir= 803 lbs., a = 2, a' = 3. a" = 3i, il = 6, A' — 7, A' = 11. mu D Wxaxa'xa" 803 X 2 X 3 X 3i ... .. . ExUiPLB 23.— In a compound lever the power is 1*7 lbs., the arms of the power 9^ 7, 6, 5, and 4 ft., and the arms of the weight 2, 3, 1, 1, and } ft. Required the weight. SOLUTIOIf. Here P=171b8.. 4=9. -i'rsT, il" = 6. wl"' = 6, 4""a«4,a=2,a' = 3, a" = l.a"' = l.anda"" = t. Then W^ PXAXA'XA"XA"'XA^'" 17X9X7X6X6X4 128820 aXa'Xa"X a'" x a"" = 64160 Ibi. Atu, 2X3X1X1X1 IXIR0I81. 24. In » compound lever the arms of the power are 9 and 17 ft. . the arms of the weight 3 aad 4 ft.) the power is 19 lbs. What is the weight 7 t^nt. 242i Ibi. / THE WHEEL AND AXLE. CABX8. 82-8(L 25. In a compound lever the arms of the power are 6, 8, 10, and 12 ft., the arms of the weight, 7, 5, 3, and 1 ft., the weight is 700 lbs. Required the power. Ans. 12^. 26. In a compound lever the arms of the weight are 11, 13, and 9 ft., the arms of the power are 4, 7, and 2 ft., the weight is 660 lbs. What is the power ? , w^ns. 12870 lbs. y;i^ii aI^: THE WHEEL AND AXLE. 82. The wheel and axle consists of a wheel with a Fig. 6. cylindrical axle passing through its centre, perpendicular to the plane of the wheel. The power is applied to the circumference of the wheel, and the weight to the circumfer- ence of the axle. 83. The wheel and axle is merely a modification of the lever with un- equal arms; the radius of the wheel corresponding to the arm of the power and the radius of the axle to the arm of the weight. 84. The wheel and axle is sometimes called the con- tinual or perpetual lever^ because the power acts continu- ally on the weight. ^^4* 01 if. 86^ The power and weight in the wheel and axle are in equilibrium when the power is to the weight as the radius of the axle is to the radius of the wheeL 86. For the wheel and axle-^let P =: the power, W = the weighty r = radius of Me^ax/e, R = radiut of the wheel, .$.,» Then P :W::r :R, Hence P == WXr R W=: PXR PXR ^„ Wxr •;andJg= p -•. W EzAMPLi 27. — ^In a wheel and axle the radius of the axle if 7 inches, the radius of the wheel is 36 inches, what powtr wi^| balance a weight of 643 (bi, 7 J" ABT. 86.] 4.f. THE WHEEL AND AXLE. t'& :'4 .? h' ly-l^ft SOLUTION. V ..TC Hero W=s 643 lbs., J2 = 35 inches, and r = 7 inches. ,, , ^' -^ „ TTXr 643X7 ,„„, , '. . Then P=; — =— = — — — =1281. Ans, .> .< - , : fr: : i.* , - . ExAUPLB 2S. — In a wheel and axle the radius of the axle is 6 inches, the radius of the wheel is 27 inches. What weight will be balanced by a power of 123 lbs. ? SOLUTION. Here P = 123 lbs., 22 = 27 in., and r = 6 in. Then W= ^^^ = lH2i!? = 553i ibg. Ans. Example 29. — 67 means of a wheel and axle a power of 11 lbs. ' is made to balance a weight of 719 lbs., the radius of the axle ia ' 3 inches. Required the radius of the wheel ? SOLUTION. Here W== 719 lbs., P = 11 lbs., and r = 3 in. Then R = ^^^ == ^4^ = 196^ inches. Ant. 11 IXBBOISB. Ti.' :-U' ■f.' *, ■ ••t ' •♦• '♦,, 30. In a wheel and axle the radius of the axle is 7 inches, the > radius of the wheel is 70 inches. What power will balance a weight of 917 lbs. ? Jint. 91-^ lbs. 31. In a wheel and axle the radius of the axle is 6 inches, and the radius of the wheel 1 Y inches. What power will balance '^^ a weight of 6950 lbs. ? Jins, 1750 lbs. 32. In a wheel and axle the radius of the axle is 9 inches and the radius of the wheel is 37 inches. What power will balance a weight of 925 lbs. ? Jlns. 225 lbs. 33. In a wheel and axle the radius of the axle is 11 inches and the radius of the wheel is 45 inches. What weight will a power of 17 lbs. balance ? »dni. 69-]|V lbs. 34. By means of a wheel and axle a power of 37 lbs. balances a weight of 700 Ibg., the radius of the axle being 8 inches, what is the radius of the wheel? Jitu. 151^^ inches. 35. By means of a wheel and axle a power of 22 lbs. balances a weight of 870 lbs. If the radius of the wheel be 67 inches what will be the radius of the u^le 7 Jm, H^ inohei • 28 THE WHEEL AND AXLE. [A£TS.87,88. 'H •; If Fig. 6. « i*«-v;f r ■.S---J :'-J THE DIPPERENTLAL WHEEL AND AXLE. 87. In the differential wheel and axle, the axle consists of two parts, one thicker than the other. By each revolution of the wheel the rope rolls once off the thinner portion and once on the thicker portion, and is consequently shortened only by the differences between the circumferences of the axles; and the distance through which the weight is raised is equal to half the shortening of the rope. The effect is therefore the same as if an axle had been used with a radius equal to half the difference between the radii of the thicker and thinner parts of the differential axle.* 88. For the differential wheel and axle let d =: the difference between the radii of the axles^ R = radius of the wheel^ P = the power f and W 5= the weight. ,. ThenP: W:: Id : R, Whence P = - j^ " , W= -r^, i2 = — p— , and d = -r^. ExAMPLi 36.— In a difforential wheel and axle the rhima of the larger axle is 4^ inches, the radius of the smaller axle is 4^ inches, ihe radius of the wheel is 70 inches. What power will balance a weight of 1000 lbs. SOLUTIOK. Here d = difference of radii = ^ — J. = ^. jr= lOOO lbs., 22 = 70 in, * Then P = — jg-i- = — ^3— = -|^ = ^g = 3^ lbs. .liM. Example 37.— In a differential wheel and axle the radii of the axles are 2|and 2^ inches, the radius of the wheel is 100 inches. What power will balance a weight of 7234 lbs. 7 The r»dti being proportfona) to the oircunferenoM. ^TS.87,88. le consists 5. r/^W. mfr ■ tJ)>:-^ <•. .ii ts of the difference ':, P = the PXft tAlVLS of zle is 4^ wer will 70 in. 1 of the inches. ABt.89.] THE WHEEL AND AXLB. 29 SOIUTION. Here d =. | — ^3^= ^^ in. 12 = 100, and TF= 723^ ThenP— TT X id 7234 X sh = lH^flbs. ^n*; IB "' 100 Example 38. — In a dififerential wheel and axle the radii of the axles lure Sj- and 3-^ inches, the radius of the wheel is 86 inches. What weight will a power of 17 lbs. balance ? , ' sonjTioir. Here d = i—-3^ = yi^ of an inch, 22 = 86 inches, and P = 17 lbs. Then W= PXB 17X86 1462 id = -; — =897664 lbs. Ans. ITI 'STI Example 39. — In a differential wheel and axle the radius of the wheel is 32 inches, and a power of 5 lbs. balances a weight of 729. What is the difference between the radii of the axles ? SOLUTIOir. Here W= 729 lbs., P =5 lbs., and J2=32 inches. P X JB 6 X 32 160 Thend = --|yp- = ^-^^^ = y-p = f|f Of aninch. Ans, ■>"j,-%"^'^ Si As EXBBCIBB. 40. In a differential wheel and axle the radii of the axles are 7^ and 7^ inches, and the radius of the wheel is 85 inches, what power will balanee a weight of 6900 lbs. ? '^^ Jins. §i^lbs. 41. In a differential wheel anc axle the radii of the axles are 17 and 16 inches, and the radius of the wheel is 130 inches^ y-i what weight will a power of 17 lbs. balance? Jins, 4420 lbs. 42. In a differential wheel and axle^ the radii of the axles are 2^ and 2^ inches, and a power of 23} oz. balances a weight of 6400 oz. Required the radius of the wheel. jlns. 6|ff inches. 43. In a differential wheel and axle^ the radii of the axles are 4^ and 5 inches, the radius of the wheel being 120 inches, what power will balance a weight of 2430 oz. 7 ,Ans. 8i^oz. 44. In a differential wheel and axle, the radii of the axles are 1| and If feet, the radius of the wheel is 12| feet, what weight will a power of 880 lbs. balance? ,Ans. 146880 lbs. 80i Since the wheel and axle is merely a modification of the lever j a eystem of wheels and axles is simply a modification of the compound lever, and the conditions of 30 WHEEL WOBKi [ABt8.(K>-M- equilibrium are the same, i. ^., the ratio of the power to the weight is compounded of the ratios of the radii of the axles to the radii of the wheels. In toothed gear, however, owing to the diflSculty in determining the effective radii of wheel and axle, the ratio of the power to the weight is determined by the number of teeth and leaves upon the wheel and pinion. 90. Axles are made to act on wheels by various methods — as by the mere friction of their surfaces, by straps or endless bands, &c. ; but the most common method of transmitting motion through a train of wheel work is by means of teeth or cogs raised upon the circumferences of the wheels and axles. .- ^ 91. When cogged wheels and axles are employed^ that part of the axle bearing the cogs is called a /)m- ^^ff- '^• ion. The cogs raised upon the pinion are called leaves^iho^Q upon the wheel are termed teeth. 92. Wheel work may be used either to con- cenlrateor diffuse power. The power is concen- trated when the pinions turn the wheels, as is the case in the craney which is used to gain power. The power is diffused when the wheels turn the pinions, as is the case in the fanning mill, threshing machine, &c., where extent of motion \-^- sought. 93. In a system of toothed wheels and pinions^ the conditions of equilibrium are that^—the power is to the weight as the continued product of all the leaves is to the continued product of all the teeth. 94. For a train of wheel work let P = the power, W = the weight, t f i"zzthe teeth of the wheel, and 1 1' 1" = the leavtt of the pinion. -.f^TT. ABt8. 90-94. AST.9«.] WHEEL WORK. 31 wer to the ' the axles however, '^e radii of weight is upon the ^ i !> . J methods straps or lethod of ork is by Fences of )yedj that -•ET ben the nfir mill, sought. itiont of oniinued he teeth, szthe feavee of Then P: W.ilx I' Xl" :t X f X t". Hence P= wxixvxi f// -, and W=z PxtxVxtf' txt'xt" »«"--- ixvy.1" ExAHPLB 45. — The number of teeth in each of three successive wheels is 80 and the number of leaVes in each of the pinions is 5. With this machine what weight will be supported by a power of 17 lbs.? SOLUTION. HereP=rl7, f =80. <* =80,f ' = 80,2= 6. Z' = 5 and ^"=5. -z^ ../• Then]r= PXtXt'Xt" 17X80X80X80 8704000 IXVXl" "" 6X6X5 ~ 126 « 69632 lbs. Ani, " EzAMPLiB 46. — ^In a train of wheel work there are four wheels and four axles, the first wheel and the fourth axle plain, (i. e.j without cogs), and having radii respectively of 10 and 2 feet. The second wheel has 60, the third 90 and the fourth 70 teeth, the first axle has 7, the second 5 and the third 9 leaves. What power will hold in equilibrium a weight of 20000 lbs. ? :;.>r *^ " SOLVTIOir. ^ Here we have a combination of the simple wheel and axle and a systentf*^ of cogged wheels and axles« jr=a0000lbs.28=10,r=2, e=60, <'=90, <"=:70, l=7» l'=6 and J"=9. 20000X7 X 6 X 9 - Then cogged wheels and axles acting alone, P = =161 lbs. 60 X 90 X 70 and so flur as the action of the plain wheel and axles is concerned this 16^ lbs. becomes the weight. mv « Wxr 161X2 33i „, „ Then Pr= — - — = — ^ — = -j^==3j lbs. Am. fi ~ 10 ^ Example 47.-— In a train of wheel work there are three wheels and axles, the first wheel and the last axle plain, and having a radius of 9 and 3 feet respectively^the cogged wheels bate respectively 80 and 110 teeth, and the pinions 11 and 8 leaves. What weight will a power of 100 lbs. sustain ? 1-...^ ^ SOLUTION. Here P =e 100 lbs., 22 = 9, r = 3, < = 80, e'= 110, fc=ll and l'= 8. «, , ^ ^ , ^ X, , ,.r ^X^X^' 100X80X110 ^ J, Then for cogged wheel work acting aIone» ir= ^^p — \\xfi 880000 88 = 10000 lbs. For plain wheel and axle alone, W^ SOOOOlbs. An$, PXB 10000 X 9 90000 32 WBEEL WORK* [ABTSi>95^» if- ;■> BXSBCISE. 48. In a system of wheel work there are fire wheels and pinions ; the wheels have respectively 100, 90, 80, 70 and 60 teelii, and the pinions respectively 9, 7, 11, 9 and 7 leaves— with such an appliance, what weight would be sustained by a power of 77 lbs. ? Jins. 5333333^ lbs. 49. In a train of four wheels and axles the wheels have respec- tively 70, 65, 60 and 50 teeth) and the axles respectively 9, 8, 7 and 6 leaves ; with such an instrument, what power could support a weight of 13000 lbs. ? Jins. 2|§^ lbs. In a train of wheel work there are three wheels and three axles, the first wheel and last axle plain, and having radii respectively 6 and 2 feet. The second and third wheels have respectively 80 and 50 teeth, and the first and sec^ A pinions, respectively 5 and 8 leaves. With such a machine what weight will be balanced by a power of 11 lbs. ? . . ^n«. 3300 lbs. 50. 95. In ordinary wheel work it is usual, in any wheel and pinion that act on each other, to use numbers of teeth that are prime to each other so that each tooth of the pinion may encounter every tooth of the wheel in succes- sion^ that thus, if any irregularities exist, they may tend to diminish one another by constant weart This odd tooth in the wheel is termed the hunting cog. Thus if a pinion contain 10 leaves and the wheel 101 teeth, it is evident that the wheel must turn round 101 times and the pinion 10 X 101 or 1010 times before the same leaves and the teeth will be again engaged. 96. Wheels are divided into crown^ spur and bevelled gear, 97. The crown wheel has its teeth perpendicular to its plane ; the spur wheel has its teeth, which are continua- tions of its r^Mii placed on its rim ; the bevelled wheel has its teeth obliquely placed, t. e»y raised on a surface inclined at any angle to the plane of the wheel. 98i To communicate motion round parallel axes spur* gear is employed ; bevelled gear is used when the axes of motion are inclined to one another at any proposed angle. Where the axes are at right angles to one another a crown wheel working in a spur pinion or a crown pinion working in a spur wheel is usually employed. TBi>95^* pinions ; teeth, iS — ^with led by a ^33^ lbs. 5 respec- tively 9, t power 2i^ lbs. nd three ing radii 1 wheels 1 sec^ ^d machine 3300 lbs. Aets. 99-103.] THE PULLEY. 33 Fig. 8. 99. Bevelled wheels are always frusta of cones chan- neled from their apices to their bases. Note.— When bevelled wheels of different diameters are to work toother the sections of the cones of which they are to be frusta are found in the following manner :— Jjet A B be the dia- meter of the lai^e wheel and B C that of the smaller. Place A B and B C so as to include the pro- posed angle. Bisect A B in D and B G in B. Draw perpen- diculars D F, E F meeting in F and joinFA,FBandFC. Then FAB and FBC are sections of the required cones. Also drawing H G paral- a^ lei to A B and G P ^ parallel to B C, we obtain H A B G, and G B G P any required frusta. ly wheel of teeth of the succes- tendto d tooth is evident 01 or 1010 bevelled to its btinua- 3el has inclined |s spur' iaxes of angle. crown [orking i rt ■{■,•: 't THE PULLEY. 100. The Pulley is a circular disc of wood or iron, grooved on the edge and made to turn on its axis by means of a cord or rope passing over it. 101. The pulley is merely a modification of the lever with equal arms, and hence no mechanical advantage is gained by using it — the theory of its use being just as perfect if the cord be passed through rings or over perfectly smooth surfaces. The real advantage of the pulley and cord as a mechanical power is due to the equal tension of every part of the cord, i, e,, is founded upon the fact that the same flexible cord, free to run over pulleys or through smooth rings in every direction must always un- dergo the same amount of tension in every part of its length. 102. The pulley is called either fixed or movable ac- cording as its axis is fixed or movable. 103. Movable pulleys are used either singly, in which case they are called runners, or in combination. Systems of pulleys are worked either by one cord or by several cords. Pulleys worked by more than one cord are called Spanish Bar]x .,- ^ns. 100. Fig. 10. 108. In a system of pulleys such as represented in Fig. 10, where each movable pulley hangs by a separate cord, one extremity of each cord being attached to a movable pulley and the other to a hook in a beam or other fixed support, each pulley doubles the effect, and the conditions of equi- librium are that the power is to the weight asl is to 2 raised to the power indicated hy the num- ber of movable pulleys. Note. — This will become evident by attentively examining the diagram ana following up the several (^ords. The figures at the top show the portion of weight borne by the several parts of the beam, those attached to the cords show the portion of the weight sustained by each part of the cord. 16 lbs. 36 THE PULLEY. AsTS. 109, 110. mm CiTi.', ■ f, ; • .r "■ Ji. ■■' V ■i 109. I'br a system of pulleys such as exemplified in Fig. 10 let P = the power f W= ^Ae weight j and n = Mc nt«m6«r of movable pulleys. W Then P : IT : : 1 : 2". HenceP= — and W= P X 2'», Example 59. — In a system of pulleys of the form indicated in Fig. 10, there are 5 movable pulleys and a weight of 128 lbs. What is the power ? SOLUTION. Here W = 128 lbs. and w = 5. Then P = W 128 128 , ,^ ^ ■ = ^o~= 4 lbs. Ans. 2" 2« 32 Example 60. — In such a system of pulleys as is shewn Fig. 10 there are 1 movable pulleys. What weight will a power of 11 lbs. balance? SOLUTION. Here P:= 11 and « = 7. Hence TT = P X 2» = 11 X 2' = 11 X 128 = 1408 lbs. Ans, EXERCISE. 61. In the system of pulleys represented in Pig. 10, where there are 6 movable pulleys, what power will sustain a weight of 8000 lbs. ? 62. In such a system when there are 10 movable pulleys, what power will sustain a weight of 48000 lbs. ? Jns 46|- lbs. 63. In such a system when there are "7 movable pulleys, what power will support a weight of 4564 lbs. ? Jlns. 35^^ lbs. 64. In such a system when there are 3 movable pulleys, what weight will be sustained by a power of 17 lbs. ? jlns. 136 lbs. 65. In such a system what weight will a power of 70 lbs. support when there are 5 movable pulleys ? Jlns. 2240 lbs. 66. In such a system what weight will a power of 100 lbs. support when there are 11 movable pulleys? Jlns. 204800 lbs. jlns. 125 lbs. 110. In a system of pulleys such as represented in Fig. 11 where the cord passes over a fixed pulley at- tached to the beam instead of being 27 lbs rs. 109, 110. Fig. 10 let \f movable X 2", iicated in • 128 lbs. m Fig. 10 ►wer of 11 iiere there a weight ;. 125 lbs. ABTS. Ill, 112.] THE PULLEY. 37 fastened to a hook in the beam, each movable pulley triples the effect, and the conditions of equilibrium are that the power is to the weight as 1 to S raised to the pmoer indicated hy the number of movable pulleys. This will appear plain by a reference to the accompanying diagram where the numbers represent the same as in Art. 108. 111. In a system such as is represented in Fig. 11, let P = power. W= the weight j and n = the number of movable pulleys. Then P :W::1 :3\ W : Hence P = — and W=zP x 3". Example 67. — In the system of pulleys represented in Fig. 11, what power will balance a weight of 4500 lbs. when there are 4 movable pulleys ? SOLUTION". Here fr= 4500 and n — 4. 4500 81 3«» 3* = 55^ lbs. Ans. ExAHPLB 68. — In such a system when there are 6 movable pulleys, what weight will a power of 10 lbs. support ? SOLUTION. Here P = 10, and « =6. Then W= P X 3n = 10 X 36 =10 X 729 — 7290 lbs. EXERGISB. Ans. 69. In the system of pulleys represented in figure 11 there are 5 movable pulleys ; what weight may be supported by a power of 10 lbs. ? Jlns. 2430 lbs. 70. In such a system there are 7 movable pulleys and the weight is 24057 lbs. Required the power? Jins. 11 lbs. 71. In such a system there are 9 movable pulleys — through Pig. 12. how many feet must the power descend in order to raise the weight 10 feet ? Jlns. 19683..^ feet. 112. If the lines of direc- tion of the power and weight make with one another an angle greater than 120'=^, the power will require to be great- 3S THE INCLINED PLANE. [Aets. 113-117 er than the weight; arid as this angle approaches 180 ^ , the diflference between the power and weight will ap- proach QC . Hence it is impossible for any power P, however great, applied at P, to pull the cord ABC mathematically straight, and that however small the weight TFmay be. • . Si 4 . H' •J^ THE INCLINED PLANE. 113. The Inclined Plane is regarded in mechanical science as a perfectly/ hard, smooth, inflexible plane, in- clined obliquely, to the weight or resistance. 114. There are two ways of indicating the degree of inclination of the inclined plane : 1st. By saying it rises so many feet, inches, f Then W—' U 9 34r00 lbs. Ans, Example 74. — On an inclined plane a power of 11 lbs. acting parallel to the plane supports a weight of 150 lbs.— how much does the plane rise in 200 feet ? SOLUTION. Here P = 11 lbs., W=z 150 lbs., L i= 200 feet. * 14 feet 8 inches. Ans, mu ir— PX^ 11X200 Then 5=-^=-^^ Example 75. — The base of an inclined plane is 40 feet and the height 3 feet, — what power acting parallel to the base will sup- port a weight of 250 lbs. ? SOLUTION. Here W= 260 lbs., fl = 3, and B = 40. ThenP = W[xJ5f__ 250X3 P ~ 40 -■= 18i lbs. Ans, Example 76.— On an inclined plane a power of 9 lbs. acting parallel to the base supports a weight of 700 lbs. — the height of the plane being 18 feet, what is the length of the base ? THE WEDGE. [Arts. 118-11». i. .-'^.i- ^ s * > I" , i ' ■ i . : f ! i 'i: SOLFTIOH". Here P= 9 lbs., W= 700 lbs., and B:= 18 feet. mi- « WxH 700X18 ,^^-4, . . Then JS = — ;; — == — - — =1400 feet. Ans. 9 EXERCISE. 77. 78. On an inclined plane rising 1 foot in 35 feet what power acting parallel to the plane will support a weight of 17500 lbs. ? Ans. 600 lbs. On an inclined plane rising 9 feet in 100 feet what power acting parallel to the plane will sustain a weight of 42 3 7 lbs . ? Ans. 381iVjlbs. 79. On an inclined plane whose height is 11 feet and base 900 feet what power acting parallel to the base will sustain a weightof 27900 lbs.? ./?ns. 341 lbs. 80. On an inclined plane rising 7 feet in 91 feet what weight will be supported by a power of 1300 lbs. acting parallel with the plane ? Ans. 16900 lbs. 81. On an inclined plane a power of 2 lbs. acting parallel to the plane, sustains a weight of 10 lbs. — what is the inclination of the plane ? Ans. Plane rises 1 foot in 5 feet. 82. On an inclined plane a power of 7 lbs. acting parallel to the base sustains a weight of 147 lbs. — if the base of the plane be 17 feet what will its height be ? Ans. ^ feet. On an inclined plane rising 2 feet in 109 feet what weight will be sustained by a power of 17 lbs. acting parallel to the plane? Ans. 926^ lbs. On an inclined plane a power of 4f lbs. sustains a weight of 223^^ lbs. ; the power acting parallel to the plane what is the degree of inclination ? Ans. Plane rises 341 feet in 17199 feet. 85. What weight will be supported by a power of 60 lbs. acting parallel to the base of an inclined plane whose height is 7 feet and base 15 feet. Ans. 128f lbs. 83. 84. THE WEDGE. 118. The wedge is merely a movable inclined plane or a double inclined plane, i. e., two inclined planes joined to- gether by their bases, 119. The wedge is worked either hy presture or by 'percussion, NoTB.—When the wedge is worked bv percussion, the relation between the power and weiffht cannot be ascertained since the force of percussion differs so completefy from continued forces as to admit of no comparison with them. 88. 89. TS. 118-119. AetS. 120-122. j THE SCREW. 41 120. In the wedgt! the conditions of equilibrium are that the power is to the weight as half the width of the back of the wedge is to its lengths Note 1.— Unlike all the other mechanical powers, the practical use of the wedge depends on friction, as, were it not prevented by Motion, the wedge would recoil at every stroke. Note 2. — Razors, knives, scissors, chisels, awls, pins, needles, &c., are examples of the application of the wedge to practical purposes. 121. For the wedge, let P =: power or pressure, W=i the weight, L = the length of the wedge, and B = the width of the hack. PXL Then P :W::l B:L. Hence P = WxhB and W=: IB Example 86. — The length of a wedge is 24 inches, and its thickness at the back 3 inches, what weight would be raised by a pressure ofTSO lbs. ? : , " SOLUTION. -^;, Here P = 750 lbs., i = 24 inches, and iB = l^ inches. Then W~ PXL 750X24 = 750 X 16 = 12000 lbs. Ans. hB U Example 87. — In a wedge, the length is 17 inches, thickness of back 2 inches, and the weight to be raised is 11000 lbs. Re- quired the pressure to be applied? ^ Here W Then P SOLUTION. : V : 11000, X == 17 inches, and \B — \ inch. frxiB__ 11000X1 -,^, ,, . ; — j~~ ^ — -- — =647^ lbs. Ana. EXERCISE. 88. The length of a wedge is 30 inches and the thickness of its back 1 inch, what weight will be raised by a pressure of 97 lbs.? Ans. 5820 lbs. 89. The length of a wedge is 19 inches and the thickness of its back 4 inches, what pressure will be required to raise a weight of 864 lbs. ? Ans. 90i^| lbs. 90. The length of a wedge is 23 inches and the thickness of its back 3 inches — with this instruLoent what pressure would be required to raise a weight of 1771 lbs. ? Ans. 115^ lbs. THE SCREW. 122. The screw is a moditication of the inclined plane and may be regarded as being formed of an inclined plane wound round a cylinder. NoTB.—The lorew bears the same relation to an ordinary inclined piftne that a circular staircase does to a straight one. 42 THE SCREW. [Abts. 123-128. ■m Fig. 13. 123. The threads of the screw are either triangular or square. The distance of a thread and a space when the thread is square, or the distance between two contiguous triangular threads, is called the pitch, 124. The screw is commonly worked by pressure against the threads of an external screw, called the box or nut. The power is applied either to turn the screw while the nut IS fixed, or to turn the nut while the screw is kept immovable. 125. In practice, the screw is seldom used as a simple mechanical power, being nearly always com- bined with some one of the others — usually the lever. 126. The conditions of equili- brium between the power and the weight in the screw are the same as for the inclined plane, where the power acts parallel to the base, i.e. The power is to the weight as the pitch (i, e. height) is to the circumference of the base (i. e. length of the plane,) When the screw is worked by means of a lever, the con- ditions of equilibrium are : — The power is to the weight a>s the pitch is to the circum- ference of the circle described by the power, 127. The eflSciency of the screw as a mechanical power may be increased by two methods : Ist. By diminishing the pitch. 2nd. By increasing the length of the lever. 128. For the screw ^ let P r= the power ^ W= the weighty p = the pitch, and I = length of the lever. Then since the lever forms the radius of the circle described by the power, and the circumference of a circle is 3*1416 times the diameter, and the dicCmeier is twice the radius, P : W: :p:l X 2 X 31416. Hence P Wxp w~ PX^X2X3-1416 andp = i»X?X2X8U10 iX2X3*1416 p " W NoTK.— The pitch uid the length of the lever nrast be both expressed iin units of the same denominations, i.e. both feet, or both inches. . 123-128. ular or en the iguous against )r nuL ile the is kept ^ n^ht) is plane.) the con- ! circum- il power er. ght, p = icribed by times the :lX2X 2X81410 [pressed iin Art. 128.3 THE SCREW. 43 Example 91. — What power will sustain a weight ofTOOOO lbs. by means of a screw having a pitch of i^th of an inch, and the lever to which the power is attached 8 ft. 4 in. in length ? SOLUTION. H«^re W = 70000 lbs., p = J^- in., and ?= 8 ft. 4 in. = 100 in. Hpr»P=. WXp __ 70000 X -h =5000^=500000^.95711,8. ^W5. ^X2X31416 100X2X3-1416 628'32 62832 Example 92. — ^What weight will be sustained by a power of 5 lbs. by means of a screw having a pitch of -fVth of an inch, the power lever being 50 inches in length ? SOLUTION. Here P = 5 lbs.2> = i^ inch, and ? = 50 inches. /. ™ __ PX?X2X3-1416 5X50X2X3-1416 1570*8 , ^^„ ,^ , Then W= r= = — — =15708 lbs. Arts. Example 93. — By means of a screw having a power lever 5 ft. 10 inches in length, a power of 6 lbs. sustains a weight of 80000 lbs. ; what is the pitch of the screw ? Here P =6 lbs., Wz solution. , : 80000 lbs., and ? = 70 inches. Then p = or about ^ PX?X?X3'1416 6X70X2X8-1416 2638*944 1000 W of an inch. Ans. 80000 80000 = 0329868 inches, Example 94. — What power will sustain a weight of 96493 lbs. by means of a screw having a pitch of 1^7-th of an inch, the power lever being 25 inches in length ? SOLUTION. Here Wz=: 96493 lbs., p = i^h inch, and I ■ ,__ TTXp ___ 96493Xt^f __ ?X2X3-1416 """25X2X3-1416 "" 108*408 lbs. Ans. EXERCISE. ThenP- = 25. 167*08 17028*1764 167*08 95. What power will support a weight of 87000 lbs. by means of a screw having a pitch of J?^th of an inch, the power lever being 6 ft. 3 inches long? Ans. 31*83 lbs. 9G. What weight will be sustained by a power of 200 lbs. acting on a screw having a pitch of {^(^ih of an inch — the power lever being 15 inches long ? Ans. 314160 lbs. 9*7. By means of a screw having a power lever 50 inches in length, a weight gf 9000 lbs. is supported by a power of 12 lbs. Required the pitch of the screw. Am. -41888, or rather over f of an inch. 44 THE DIFFERENTIAL SCREW. [Aets. 129, 130. 98. What power will support a weight of 11900 lbs. by means , of a screw having a pitch of -^fth of an inch, the power lever being 10 ft. in length ? ^ns. 3-713 lbs. By means of a screw having a power lever T ft. 6 inches in length, a power of 10 lbs. supports a weight of 65400; what is the pitch of the screw ? Jns. -0864 of an inch. What weight will be supported by a power of 50 lbs. act- ing on a screw with a pitch of 4%th of an inch — the power lever being 8 ft. 4 inches in length ? Jns. 418880 lbs. 99 100 ■I THE DIFFERENTIAL SCREW. 129. The differential screw, (invented by Dr. John Hunter,) like the differential wheel and axle, acts by dimi- nishing the distance through which the weight is moved in comparison with that traversed by the power. Fig. 14. It consists of two screws of dif- ferent pitch, working one within the other (Fig. 14), so that at each revolution of the power lever the weight is raised through a space only equal to the difference be- tween the pitch of the exterior screw and the pitch of the inner screw. It follows that the mechan- ical effect of the differential screw is equal to that of a single screw having a pitch equal to the differ- ence of pitch of the two screws. For instance, in Fig. 14, the part B works within the part A. Now, if B have a pitch of T^th of an inch and A a pitch of ^g, then at each revolu- tion of the handle the weight will be raised through -^ — ^^^^i^ of an inch, and the whole instrument has the same mechanical eil'ect as a single screw having a pitch of usirth of an inch. 130. For the differential screw, let Pzn power, Wz^. weight, I = length of lever, and d = difference of pitch of the two screws, ThenP: W::d : /X 2 y 3-1416. Hence P = WX d 1X2 X31416 and W: PX/X2X31416 129, 130. means jr lever ri3 lbs. ches in 65400 ; n inch. Ds. act- power h80 lbs. John Y dimi- moved ^ow, if B h revolu* i a single weightf screws. Abt. 131.] THE ENDLESS SCREW. 45 Example 101. — ^What power will exert a pressure of 20000 lbs. by means of a differential screw having a power lever 50 inches in length, the exterior screw a pitch of -^f of an inch, and the inner screw a pitch of ^th of an inch ? SOLUTION. an^ind,.^"^^^^''^^^'""*"*^ ^=-A— A = #o— ^2l=l^^J of ThenP: Wxd sjijisLa lX2X'6-U16 = 7.81 lbs. Ans. 2Km(iX^i^ 50X2X3-14il6~ 314-16~~ 3U-I6 2454-545 245454'54 314-16 Example 102. — ^What pressure will be exerted by a power of 1000 lbs. acting on a differential screw in which the power lever is 75 inches long, the pitch of the "exterior screw -^^^th of an inch, and that of the interior screw 3'^th of an inch ? solution. Here P = 1000 lbs., ? = 75 inches, and d = 1^7 _ an inch. _, ^ PXZX2X3*1416 1000X75X2X3*1416 Then Wz=: , = 7-— L5JJ >0 F5(J 119 — -3Lof 471240 400554000 = 1292109611 lbs. Am. Isso* EXERCISE. »50 31 103. What power will exert a pressure of 100000 lbs. by means of a differential screw in which the power lever is 100 inches long, the pitch of the outer screw ^g of an inch, and that of the inner screw 4'o of an inch ? Jlns. '102 or about t^ of a lb. 104. What pressure will be exerted by a power of 20\lbtf. acting on a differential screw in which the power lever is 50 inches long, the pitch of the exterior screw 1^ of an inch, and that of the inner screw ^^ of an inch ? ,^ns. 345576 lbs. 105: What power will give a pressure of 60000 lbs. by means of a differential screw in which the power lever is 60 inches, the pitch of the outer screw ^j, and that of the inner screw ^2^ of an inch? ^ns. 2-652 lbs. THE ENDLESS SCREW 131. The Endless Screw, Fig. 15, is an instrument formed by combining the screw with the wheel and axle. The teeth of the wheel are set obliquely so as to act as much as possible on the threads of the screw. Fig. 15. 46 THE ENDLESS SCREW. [Arts. 132, 138. 1 ' '■ hi' «« 132. In Fig. 15 each revolution of the handle makes the wheel revolve only through the space of one cog ; hence if the whole has 24 cogs, the winch must revolve 24 times in order to make the wheel revolve once. It follows that in the endless or perpetual screw the con- ditions of equilibrium are that the poioer is to the loeight as the radius of the axle is to the product of the number of teeth in the wheel multiplied hy the length of the winch ; i. e,, the radius of the circle described by the power, 133. For the endless screw let PzzzpoweVj Wzziweightj Izzzlength of winch or handle j t:=:number of teeth in the wheely and r-z^radius of axle. ThenP: Wv.r'.lv.t, Whence P= ^^f w^^^^^ IXt r . Example 106. — In an endless screw the length of the winch or handle is 25 inches, the wheel has 60 cogs, and the axle to which the weight is attached has a radius of 2 inches. What weight will be sustained by a power of 100 lbs. ? • SOLUTION, Here P = 100 lbs., r =2 inches,? =25 inches, and <= 60. ' ^ __ PxlXt 100 X25X60 150000 h^aaaiu a Then W=^ = :; =^ — :; — =75000 lbs. Ans. 2 2 Example lOT. — In an endless screw the length of the winch is 20 inches, the wheel has 56 teeth and the radius of the axle is 3 inches. What power will support a weight of 14000 lbs ? SOLUTION. Here W=z 14000 lbs., r =3 inches, 2= 20 inches, and t WXr 14000X3 42000 56. '^«^^=-TxT 20X66 1120 :=: 37i lbs. Ans. EXERCISE. 108. In an endless screw the length of the winch is 18 inches, the radius of the axle is 2 inches, the wheel has 48 teeth, and the power is 120 lbs. Required the weight. Jins. 51840 lbs. 109. What power will support a weight of a million of lbs. by means of an endless screw having a winch 25 inches long, an axle with a radius of 1 inch, and a wheel with 100 teeth ? Jlns. 400 lbs. 110. What weight will be raised by a power of 40 lbs. by means of an endless screw in which the winch is 20 inches long, the radius of the axle 2 inches, and the number of teeth in the wheel 80 7 Ms. 32000 lbs. . 132, 133. ABTB. 134rl39.] FRICTION. 47 makes e cog; revolve the con- loeight imber of winch ; • l=:length '•=radius he winch 3 axle to 3. What B winch is 16 axle is )lbs? L8 inches, 48 teeth, 51840 lbs. )f lbs. by ihes long, with 100 s. 400 lbs. by means 3hes long, f teeth in 32000 lbs. 134. The theoretical results obtained by the foregoing rules are in practice very greatly modified by several retarding forces. Thus friction has to be taken into ac- count in each of the mechanical powers — the weight of the instrument itself in the lever and in the movable pulley — the rigidity of cordage in the pulley and in the wheel and axle, &C. ^ ;:;,.,^^ .,,-; - ; - j :> ;,; FRICTION. 135. Friction aids the power in supporting the weight, but opposes the power in moving the weight, and hence materially affects the conditions of equilibrium in the mechanical powers. If P be the power neoessanr in the absence of all friction and/ the fac- tion, thenthe weight will be held in equilibrium by any power which is less than P+/, or greater than P—f. 136. Friction is of two kinds: 1st. Sliding Friction. 2nd. EoUing Friction. 137. The fraction which expresses the ratio between the whole weight and the power necessary to overcome the friction, is called the coefficient of friction. The coeffi- cient of sliding friction, in the case of hard bodies, varies from 4 to |. 138. On a perfectly level road, power is expended only for the purpose of overcoming friction, and on the same road the ratio between the power and the load is constant, — varying on common roads, according to their goodness, from tV *^ iV ^^ *^® ^^*^<1* ^^ *^ ®^®^ railway, however, it is not more than y}^ to ^}^ of the load, according to the dampness or dryness of the rail. On a good macadamized road the coefficient of friction is about J^, so that a horse drawing a load of one ton or 2000 lbs. must draw with a force of jV ^^ 2000 lbs. or 66| lbs. ; this is called the force of trcLCtion, 139. Various expedients are in common use for dimin- ishing the amount of friction, such as crossing the grain, when wooden surfaces rub on one another, using surfaces of different materials^ as wood on n^etal, or one kind of 48 FRICTION. [Aet. 139. ] I 1 ;, '% ,'!*■" w l'^' [ M '.a ':, 1^ W ■ > • r 1 V r ■'.r; ■«,!■].; 1 T ;;^4'ri 1 : (■ ' ' 1\ = 2 7 = JL 6 metal on another kind, and anointing the surface with oil, tar, or plumbago. Tallow diminishes the friction by one- half. The following are the conclusions of Coulomb on the important subject of sliding friction : — I. Friction is directly proportional to the pressure. II. Friction between the same two bodies is constant, being uninfluenced by either the extent of surface in contact or the velocity of the motion. III. Friction is greatest between surfaces of the same material. lY. Friction varies with the nature of the surfaces in contact. The friction between surfaces of wood, newly planed= ^ The friction between similar metallic surfaces =^ ' The friction of a wooden surface on a metallic surface=: ^ The fHction of iron sliding on iron The friction of iron sliding on brass V. Friction decreases as the surfaces in contact wear. In wood the Ariction is thus reduced from ^ to ^. VI. Friction is diminished between wooden surfaces by crossing the fibres. If when the fibres are in the same direction the coefficient of fric- tion is i, it is diminished to k ^y crossing them. VII. Friction is greater between rough than between polished surfaces. Hence arises the use of lubricants in machinery. Wlien the pressure is small, the most limpid oils are used. At greater pressures, the more viscid oils are preferred, then tallow, then a mixture or tallow and tar, or tallow and plumbago, then plumbago alone, and in the heaviest machinery soap- stone has been found to be the most efficacious substance. Note.— At very great velocities the fHction i' perceptibly lessened; when the pressure is very greatly increased, the fricuon is not increased in pro- portion. BOLLINQ FBIGTIO^. VIII. Friction caused by one body rolling on another is directly propor- tional to the pressure, and inversely to the diameter of the rolling body. That is, if a cylinder rolling along a plane have its pressure doubled, its friction will also be doi^bled; but if its diameter be doubled, the Mction will be only half of what it was. The friction of a wooden cylinder of 32 inches in diameter rolling upon rollers of wood is rls" of the pressure. The firiction of an iron axle turning in a box of brass and well co2|ite(jl with oil is i^G of the pressure. J.' V' AbtS. 140, 141.] UNIT OF WORK. 49 CHAPTER IV. UNIT OF WORK, WORK OF DIFFERENT AGENTS, HORSE POWER OF LOCOMOTIVES, STEAM ENGINES, AND WORK OF STEAM. , UNIT OP WORK. ■_,.:, 140. In comparing the work performed by diflferent agents, or by the same agent under diflferent circumstances, it becomes necessary to make use of some definite and dis- tinct unit of work. The unit commonly adopted for this purpose in England and America is the labor requisite to raise the weight of one pound through the space of one foot. Thus in raising 1 lb. through 1 foot, 1 unit of work is performed. If 2 lb. be raised 1 ft., or if 1 lb. be raised 2 ft., 2 units of work are performed. If 7 lbs. be raised through 9 ft., or if 9 lbs. be raised through 7 ft., 63 units of work are performed, &c. 141. The units of work expended in raising a body of a given weight are found by multiplying the weight of the body in lbs, by the vertical space in feet through which it is raised. > ExAMPLB 111.— How many units of work are expended in rais- ing a weight of 642 lbs. to a height of 70 ft. ? , « SOLFTION. ^ns. Units of work=642X 70=44940. ExAHPLB 112. How many units of work are expended in raising a weight of 423 lbs. to a height of 267 ft. 7 SOLUTION. ./3ns. iTnits of work=423X 267=112941. ExiMPLB 113. — How many units of work are expended in raising 11 tons of coal from a pit whose depth is 140 ft.? SOLUTlOir. Here, 11 tons=llx 2000=22000 lbs. Then 22000X140 =3080000 -in*. ExAH^LB 114.— iflow many units of work are expended in raising 7983 gallons of water to the height of 79 ft. ? SOLUTION. Here, since a gallon of water weighs 10 lbs., 7988 gals.=79880 lbs. Then units of work =79830X79 =6806670. ^n». ExAMPLB 115.— 'How many units of work are expended in railing 60 cubic foet of water from a well whose depth is 90 feet ? 60 UNIT OP WORK. [Abt. 142. I: '■ i' SOLUTIOH". Since a cubic foot of water weighs 62^ lbs., 60 cubic feet weigh 62iX60= 3750 lbs. Then units of work = 3750X99=337600. Ans. BXEBOISE. 116. How much work would be required to pump 60000 gallons of water from a mine whose depth is 860 ft. ? Jlns, 516000000 units. 117. How many units of work would be expended in pumping 8000 cubic feet of water from a mine whose depth is 679 feet ? Ms. 339500000 units. 118. How much work would be expended in raising the ram of a pile driving engine — ^the ram weighing 2 tons, and the height to which it is raised being 29 ft. ? Jins. 116000 units. 119. How much work would be required to raise 17 tons of coals from a mine whose depth is 300 feet 7 Ans. 10200000 units. • 120. How much work would be expended in raising 600 cubic feet of water to the height of 293 feet? .v^> Jlns. 10987500 units. 142. The most important sources of laboring force are animals^ water^ wind, and steam. The laboring force of animals is modified by various circumstances, the most important of which are the duration of the labor, and the mode by which it is applied. The following table shows the amount of effective work that can be performed under different circumstances by the more common living agents : TABLE. SHEWING THB WORE DONB PER MINUTE BT VARIOUS AGENTS. [Duration of labor eight hours per day. Horse 33000 units Mule 22000 «* Ass 8250 " Man, with wheel and axle 8600 " " drawing horizontally 3200 - ** ** raising materials with a pulley 1600 " ♦* throwingearthto theheightof5ft... 660 " Akr, 142.3 WORK OF LIVING AGENTS. 51 Man, working with his arms and legs as in rowing 4000 units ** raising water from a well with a pail ' and rope 1054 '* " raising water from a well with an upright chain pump 1730 " Note.— The work assigned by Watt to the horse per minute was 33000 units, but this is known to be about | too great. A horse of average strength performs about 22000 units of work per minute. The number given in the table is, however, still used in all calculations in civil engi- neering. Example 121. — How many cubic feet of earth, each weighing 100 lbs., will a man throw to the height of 5 feet in a day of 8 hours? , SOLTITIOlf. Since (by the table) a man throwing earth to the height of 5 ft., does 560 units of work per minute— and from the example he works 8X60=480 minutes. Units of work done in the day=560X480. Units of work required to throw 1 cubic foot to height of 5 feet = 100X5. Then ^^=537? cubic feet. ^»w. Example 122. — How many gallons of water will a man raise in a day of 8 hours from a well whose depth is 70 feet — using a pail and rope? SOLUTION. Units of work=rl054X60X8 ; work required to raise 1 gal.=rlOx70. mu 1. * 11 1054X60X8 ^„„„„ . Then number of gallons = — — =722|f. Ans. Example 123. — How many gallons of water can a man raise hy means of a chain pump in a day of 8 hours from the depth of 80 feet? solution. Units of work performed by the man = 1730X60X8. Units of work required to raise 1 gaL of water = 10X80. * The number of gaUons=H^^i^= 1038. Ans, Example 124. — How many tons of earth will a man working with a wheel and axle raise in a day of 8 hours from a depth of 87 feet? SOLUTION. Units of work performed by the man = 2600 X 60 X 8. Units of work required to raise 1 ton to height of 87 ft. = 2000X87. rr ^- A 2600X60X8 ^ . . Tons raised = ^^^^^ = 7^^. Ans. Example 125. — How many gallons of water per hour will an engine of 7 horse powrs raise from a mine whose depth is 110 feet? 52 WORK OF LIVING AGEKTS. CAst. 14^i i. •'■s B^c/'' ; ( n SOLUTION. Units of work in one horse power =i 83000 per minute. Units of work in 7 horse powers = 33000X7. Units of work performecf by the engine per hour =: 38000X7X60. Units of work required to raise 1 gallon of water to the height of 110 ft.=: 10X110. Hence number of gallons = — ■ — = 12600. Ans, Example 126. — How many horse powers will it require to raise 22 tons of coals per hour from a mine whose depth is 360 feet ? B0LTJTI03S". Weight of coals to be raised = 22 tons r= 44000 lbs. i Units of work required per hour =44000X360. , , Units of work in one horse power per hour=:33000X60. „ „ ,, 44000X360 „ . '■ Hence. H.P.= 3^j^^j^^^=8.^n.. Example 127. — How many cubic feet of wdter will an engine of 15 horse powers pump each hour from a mine whose depth is 900 feet ? SOLUTION. Units of work performed by engine per hour =33000X60X15. Units of work required to raise 1 cubic foot = 62*5X900. TT V, * u- * 4. 83000X60X15 .__ .^ Hence, number of cubic feet =--r— — r— — = 528. ^»«. 62-5X900 Example 128. — What must be the horse powers of an engine in order that working 12 hours per day it may supply 2300 fami- lies with 50 gallons of water each per day — taking the mean height to which the water is raised as 80 feet, and assuming that \ of the work of the engine is lost in transmission? SOLUTION. . Weight of water pumped per day =: 2300X50X10. Units of work required daily = 2300 X 50 X 10 X 80. Units of work in one horse power per day =33000X12X60. But since ^ of the work of the engine is lost in transmission. Useful work of one H. P. per day = f X 33000 X 12 X 60 TT TT T. 2300X50X10X80 Hence, H. P.= ^X33i)OOXl2X60 = 4'64. Ans, EXERCISE. 129. How many cubic feet of earth, each weighing 100 lbs., will a man raise by means of a pulley from a depth of 30 feet in a day of 8 hours ? Am. 256 cubic feet. 130. How many cubic feet of water per hour will an engine of 20 H. P. raise from a mine whose depth is 450 feet, assuming that | of the work of the engine is lost in transmigsioii 7 An%, 1126J^ e«iblc feet. abt«.i«,m*.] work on a level plane. 53 131. What must be the H. P. of an engine in order that it may- raise 11 tons of material per hour ftom a depth of YOO ft.? Jns. 1-11 H. P. 132. A forge hammer weighing 890 lbs. makes 50 lifts of 4 feet each per j;ninute — what must be the horse powers of the engine that works the hammer ? ^ns. H. P. = 5*39. 133. An engine of 8 horse powers works a forge hammer, caus- ing it to make 50 lifts per minute, each to the height of 6 feet. What is the weight of the hammer ? Jtns. 880 lbs. 134. An engine of 8 horse powers gives motion to a forge hd,m- mer, which weighs 300 lbs., and makes 30 lifts per minute of 2 feet each ; and at the same time raises 2 tons of coal per hour from the bottom of a mine. Required the depth of the mine. ^ns. 3690 feet. Note.— The work of the engine =33000 X 8 units per minute. From this subtract the units of work required by the hammer ; the remainder will be the ^ork expended per minute in raising the coal. Multiplying this by 60 gives us the work required per hour for the coal ; and this last is the product of the weight in lbs. by the depth in feet, of which the former is given. WORK EXPENDED IN MOVING A OARRIAQB OR RAILWAY TRAIN ALONG A HORIZONTAL PLANE. 143. In moving a carriage, &c., along a level plane, a certain amount of power is expended in overcoming the friction of the road. This is rolling friction, and amounts, as before stated (Art. 138), to from -g^ to yV of the entire load on common roads, and from -j}^ to y}^ of the load on railway tracks. In the case of railway trains, friction is usually taken as 7 lbs. per ton of 2000 lbs. 144. In running carriages of any description, work is employed to overcome the resistances. These resistances are : — 1st. Friction — which on the same road and with the same load is the same for all velocities. 2nd. Ascent of inclined planes — in which, since the load has to be lifted vertically through the height of the plane, the work is the same, whatever may be the velocity of the motion. 3rd, The Resistance of the Atmosphere — which depends upon the extent of surface, and increases as the square of the velocity. 54 WORK ON A LEVEL PLANE. [Aits. 1«. 146. la i- •'.! Il •'if .li 145. When a railway train is set in motion, the work of the locomotive engine at first far exceeds the work of re- sistances, and the motion is consequently rapidly accelerated. But as the velocity of the train increases, the atmospheric resistance also increases, and with such rapidity as very soon to equalize the work of resistances to the work of the loco- motive. When this occurs, L e.^ when the work applied by the locomotive is exactly equal to the continued work of resistances (atmospheric resistance and friction), the velocity of the train will be uniform. In this case the train is said to have attained its greatest, or maximum speed. 146. The traction or force with which an animal pulls depends upon the rate of his motion. A horse, for example, moving only 2 miles an hour, can draw with a far greater force than when running at the rate of 6 miles an hour. The following table shows the relation between the speed and the traction of a horse : TABLE OF TRACTION OF A HORSE. f, {.. it 3 81 4 m * fi Speed. ' ' Traction. A horse moving 2 miles per hour, can draw with a force of 166 lbs. 125 " 104 " 83 « 621 *< • 4U « Example 135. — What gross load will a horse draw travelling at the rate of four miles per hour on a road whose friction is ^ of the whole load ? • SOLUTION. Here from the table the traction is 83 lbs., which by the conditions of the question is ih of the gross load. Hence load = 83 X 20 = 1660 lbs. Ans. ExAHPLB 136. — At what rate will a horse draw a gross load of 1800 lbs. on a road whose coefficient of friction is ^^g^ ? SOLUTION. Here, traction = ^^^ = 100 lbs., whence by the table the rate must bo rather over 3i miles per hour. ExAMPLB 137. — If a horse draw a load of 2500 lbs, upon a road whose coefficient of friction is ^{y, what traction will he exert and how many units of work will he perform per minute ? AbX.146.] WORK ON A LEVEL PLANE. m SOLUTION. Here, traction =^^^=8Si lbs., and hence he moves at a rate of four miles per hour. 4 X 5280* Then distance moved per minute = — — — = 352 feet. -' , 60 Hence units of work=83|X352=29333|. Ans. , v Example 138. — ^What must be the eflfective horse powers of a locomotiye engine to carry a train weighing 70 tons upon a level rail at the steady rate of 40 miles per hour, neglecting at- mospheric resistance and taking -^ as the coefficient of friction? BOIUTION. Here, weight of train =70 tons =140000 lbs. Space passed over per miuute= ^ miles = 40 X 6280 60 140000 = 3520 feet. Work of friction to 1 foot = thot 140000 = =^^ = 700 units. Work of flriction per minute = 700 X 3520 = 2464000 units. Units of work in one H. P. = 33000. Therefore H. P. of locomotive = 700 X 3620 2464000 = 74'66. Ans, 33000 33000 Ex.*> ^ ""SD.— A train weighing 120 tons is carried with a uniforE. « i Atj of 30 miles per hour along a level rail ; assum- ing the tiiuiion to be 11 lbs. per ton, and neglecting the resistance of the atmosphere^ what are the horse powers of the locomotive ? BOLUTIOK. Space passed over per minute = ^ miles = 30 X 5 280 60 = 2640 feet. Work of flfiction to each foot = 120 X 11 =1320 units. Work of friction per minute = 1320 X 2640 = 3484800 units. Hence H. P. = 2?????^ = 105"6. Ans, 33000 \'m Example 140. — At what rate per hour will a train weighing. 90 tons be drawn by an engine of 80 horse powers, neglecting the resistance of the atmosphere and taking -^hs ^«s the coeffi- cient of friction ? SOLUTION. Work done by the engine per hour = 33000 X 60 X 80. Weight of train in lbs. = BO X 2000=180000. Units of work required to move the train through 1 fool = 7^ of 180000 = 720. Work expended in moving the train through 1 mile =■ 720 X 5280. ^. , - ,, , 38000 X 60 X 80 ,, „„ . . • . Number of miles per hour = — — - — — — — = 41 66. Ans. 7aU X 6280 Example 141. — A train moves on a level rail with the uniform speed of 35 miles per hour ; assuming the H. P. of the locomotive to be 60, the friction equal to 9 lbs. per ton, and neglecting atmospheric resistance, what is the gross weight of the train 7 * 6280 is the number of feet in one mile. 56 WORX ON A LEVEL PLANE. [AsT. 146. II If ) SOXUTION. Work of engine per hour = 88000 X 60 X 50. Feet moved over per hour = 35 X 5280. Work expended per hour in moving 1 ton = 35 X 6280 X 9. . • . Weight of train in tons = ~ 33000 X 60 X 50 35 X 6280 X 9 = 59*523. Arts. ExAMPLB 142. — In what time will an engine of 100 H. P. move a train of 90 tons weight through a journey of 80 miles along a level rail, assuming friction to be equal to 10 lbs. per ton and neglecting atmospheric resistance ? SOLUTION. Work expended in moving the train through 1 foot = 90X10 = 900 units. Work expended on whole journey in moving the train =900 X 5280 X 80. Work of engine per minute = 83000 X 100. 900X5280X80 ,,i . , ', _i = 115i mmutes = 1 hour 55t . * . Number of minutes = minutes. Ans. 83000X100 SZIBOISIE. 143. What gross load will a horse draw travelling at the rate of 2 miles per hour on a road whose coefQcient of friction is t^? Ans, 2988 lbs. 144. What must be the H. P. of a locomotive in order that it may draw a train whose gross weight is 130 tons, at the uniform speed of 25 miles per hour, allowing the friction to be 7 lbs. per ton and neglecting atmospheric resistance ? Ans. H. P. 60-66. 145. A train weighs T5 tons and moves with the uniform speed of 30 miles per hour on a level rail ; taking ^hs ^s the coeffi- cient of friction and neglecting the resistance of the atmos- phere, what are the horse powers of the engine 7 Ans. H. P. = 48. 146. In what time will an engine of 160 H. P. moving a train whose gross weight is 110 tons complete a journey of 150 miles, taking friction to be equal to 7 lbs. per ton, neglect- ing atmospheric resistance and assuming the rail to be on a level plane throughout? Ans. 1 hour 65^ minutes. 14*7. At what rate per hour will a horse draw a load whose gross weight is 2200 lbs. on a road whose coefficient of fric- tion is ^ ? Ans. Rather over 3i miles per hour. 148. From the table given (Art. 145) ascertain at what rate p:?r hour a horse must travel, when drawing a load, in order to do the greatest amount of work ? Ans. 3 miles per hour. 149. At what rate per hour will a locomotive of 50 H. P. draw a train whose gross weight is 70 tons, neglecting atmos- pheric resistance, taking ^^^ as the coefficient of friction and assuming the rail to be level ? Ans. 26*78 miles. Aet8.147,148.] work on a level PLANE. 67 147. When a body moves through the atmosphere or any other fluid, it encounters a resistance which increases : Ist. In proportion to the surface of the moving body ; 2nd. In proportion to the square of the velocity. Thus 1st. If a board presenting a surface of 1 sq. foot in moving through the air meet with a certain resistance, a board having a sur- face of 2 sq. feet will meet with double that resistance ; a board having a surface of 3 square feet will meet with three times that resistance, &c. 2nd. If a body moving 2 miles per hour meet with a certain resis- tance, a body of the same size moving 4 miles per hour will meet with (|^) ^ , or 2 ^ , or 4 times that resistance. If the velocity be increased 3 times : i. 0., to 6 miles per hour, the resistance will be increased 9 times (t. e„ 3^ times.) If the velocity be increased 7 times, i. e., to 14 miles per hour, the resistance will be increased 7' times, i. e., 49 times, &c. 148. In the case of railway trains, the atmospheric re- sistance is about 33 lbs. when the train is moving at the rate of 10 miles per hour. It has been found, however, by recent experiment, that the atmospheric resistance encoun- tered by a train in motion depends very much upon the length of the train. ExAHPLB 150. — ^When a train is moving at the rate of 10 miles per hour, it encounters an atmospheric resistance of 33 lbs. ; what will be the resistance of the atmosphere when the train moves at the rate of 50 miles per hour ? SOLUTION. Here the velocity increases ^ times, i. «., 5 times. Hence the resistance increases 6' times = 25 times. . '. Resistance = 33 X 26 = 825 lbs., i. e., 825 units of work are expended every foot in overcoming the atmospheric resistance. ExAHPLB 151. — If a train moving 7 miles per hour meet with an atmospheric resistance equal to 5 lbs. what resistance will it encounter if its speed be increased to 49 miles per hour ? SOLUTION. Here the velocity increases 7 times, {i.e,, 4j?). Hence the resistance increases 7^ = 49 times. . *. Besisi^^nce =- 6 X 49 = 245 lbs. ; i. e., 245 units of work are expended every foot in overcoming the atmospheric resistance. Example 152. — If a railway train moving at the rate of 10 miles per hour encounters an atmospheric resistance of 33 lbs. ; what must be the horse powers of the locomotive in order that the train may move 60 miles per hour, neglecting friction an(| assuming the rail to be level ? 58 WORK ON A LEVEL PLANE. [ABT. 148. T- SOLUTION. Here the velocity is increased 6 times, since ^ g = 6. Then the resistance is increased 36 times (Art. 147.) Hence atmospheric resistance =88 X 36 = 1188 lbs. work are expended in moving the train through 1 ft. 1. e., 11S8 units of 60X6280 60 = 5280. Number of feet train moves through in a minute = Units of work required per minute") _, n oo v Koan to overcome atmospheric resistances —^^°° ^ o^o". 118R X {)280 .'. H. P. Of locomotive =-i^^^~^ = 19008. -4wff. ExAMPLB 153.— What must be the H. P. of a locomotive to move a train at the rate of 40 miles per hour on a level rail, taking atmospheric pressure as usual, (i. e., 33 lbs. when a train moves 10 miles per hour,) and neglecting friction ? SOLUTION. Here velocity increases 4 times, and hence resistance increases 16 times. Then resistance encountered = 33 X 16 = 628 = units of work required per foot. Feet moved over per hour = 5280 X 40 ; hence units of work per hour == 5280 X 40 X 528. mu * TT T» 628 X 40 X 5280 ...„,, . ..... Therefore H. P.= ^^^^ = 56-32. Ans. Example 154. — What must be the H. P. of a locomotive to draw a train whose gross weight is 80 tons, along a level rail, with the uniform velocity of 40 miles per hour, taking atmos- pheric resistance and friction as usual ? SOLUTION. X, * J '4. 40X6280 ,^„„ Feet passed over per mmute = — — — = 3620. 60 Work of ftriction per minute = 80 X 7 X 3620 = 1971200 units. Work of atmospheric resistance = 33 X 16 X 3620 = 1868509 units. mu * «« TT -D — Work of f riction + w ork o f atmospheric resistance xnereiore u. ir.— ■ _ ,. — ^ — r ._, _ . , . . Work of one H. P. 1971200 + 1868660 3829760 „„,,„, ^ 33000 = -33000- = ^^^^^' ^***- ExAMPLB 155. — What must be the H. P. of a locomotive to draw a train, whose gross weight is 125 tons, along a lerel rail with the uniform velocity of 42 miles per hour, taking friction as usual, and assuming that the atmospheric resistance encountered by the train is equal to 10 lbs. when moving at the rate of 7 miles per hour ? SOLUTION. Feet moved over per minute = 42 X 6280 60 = 3696. Work of ftiction per minute = 126 X 7 X 3696= 3234000 units. Work of atmospheric resistance per minute=10X36X369ft=1830560 units. Work of ftriction + work of atmospheric resistance Then H.P.= 3284000 4- 1330660 __ 4664660 33000 "^ 38000 Work of one H. P. = 138'32, Arts, Aets.149,160.] work on AN INCLINED PLANE. 69 157. BXBBOISH. . 156. If a train encounters an atmospheric resistance of 8 lbs. when moving at the rate of 5 miles per hour, what re- sistance will it encounter when its speed is increased to 45 miles per hour ? ^ns. 648 lbs. What must be the H. P. of a locomotive to draw a train at the rate of 30 miles per hour on a level rail, assuming that the atmospheric resistance is equal to 9 lbs. when the train moves 6 miles per hour, and neglecting friction ? Jns. H. P. = 18. What must be the H. P. of a locomotive to draw a train weighing 140 tons along a level rail with the uniform velocity of 36 miles per hour, taking fri ^ _»ix as T lbs. per ton, and the resistance of the atmosphere 12 lbs. when the train moves 9 miles per hour? ^ns. H. P. = 112'512. 159. A train weighing 200 tons moves along a level rail with a uniform speed of 30 miles per hour ; what are the H.P. of the engine — ^friction and atmospheric resistance being as usual? ^ns. H. P. = 135-76. 158. 149. If a body be moved along a surface without fric- tion or atmospheric resistance, the units of work performed are found by multiplying the weight of the body in lbs. by the vertical distance in feet through which it is raised. Thus, if a body weighing 12 lbs. be moved 200 feet along an inclined plane having a rise of 19 feet in 100, the units of work performed will be 12 X 19 X 2 = 466, because in moving up the plane 200 feet, the body is raised through 19 X 2=38 feet. 150. When a train is moving along an inclined plane, and the inclination is not very great, the pressure on the plane is very nearly equal to the weight of the body. Hence we find the work due to friction by Arts. 143-146, the work due to atmospheric resistance by Art. 148, and the work due to gravity by Art. 149. ExAMPLB 160. — A train weighing 90 tons is drawn up a gradient having a rise of 3 feet in every 1000 feet, with the uniform speed of 40 miles per hour — neglecting friction and atmospheric re- sistance, what are the H. P. of the engine ? - SOLUTION. Weight of train in lbs. = 90 X 2000 = 180000. ■r,^^ 11 J '4. 40X 5280 „.„^ Peet travelled per minute = — — — =3620. Vertical distance moved through per minute =y-Jiy^ of 3520: Units of work due to gravity per minute = 10'56 X 180000. „ -, 10-66 X 180000 _,^ ^ . •. H. P. = -^-zTTT = 67'6. Ans, 33000 *'-^r?* :10-66 ft. 60 WORK ON AN INCLINED PLANE. [Aet.IBO. •■ri ■I .!■ ly :\ ExAMPLB 161. — A train weighing 140 tons moves up a gradient haying a rise of 3 feet in 1100 feet, with the uniform velocity of 36 miles per hour— neglecting atmospheric resistance and taking friction as usual, what are the H. P. of the locomotive ? SOLUTION. Here weight of train in lbs. =140 X 2000 = 280000 ; and speed per mi- , 36X5280 „,^„, , nute = — — = 3168 feet. The units of work due per minute to friction = 140 X 7 X 3168 = 3104640. Height to which train is raised per minute = y^%^ of 3168 = 8*64 ft. Then units of work due per minute to gravity = 8*64 X 280000 = 2419200. . „ p _ work due gravity + work due friction _ 3104640 + 2 419200 _ " Work Of one H. P. "" 33000 5623840 ,^ ,„„ . . -^5^ = 167-^89.- ^«*. ExAUPLB 162.— A train weighing 100 tons moves up a gradient with a uniform velocity of 30 miles per hour, the rise of the plane being 3 feet in 1000 feet, and taking friction and atmospheric resistance as usual, what are the H. P. of the locomotive ? SOLUTIOSr. Here weight of train in lbs. = 100 X 2000 = 200000; space passed per QA sf 6280 minute = = 2640 ft., and elevation of train per minute = ^-^^ Of 2640=7'92 ft. ' Work of friction per minute = 100 X 7 X 2640 = 1848000 units. Work of atmospheric resistance per minute = S3X9X 2640 = 784080 units. Work of gravity per minute = 7*92 X 200000 = 1584000 units. my. __ TT p __ Work dne to fric, per iiun.~T'work dne to atmoi. resirt. per min.-T^work due to uraT. per min Cnita of work in on* H. P. . IT V ~ ^^^^^ + ^8^8Q + ^^844)00 _ 4216080 _ ..H.P.- ^^^ -^^^^-.127 76.^lW. Example 163. — A train weighing 130 tons descends a gradient having a rise of 7 ft. in 2000 ft. with the uniform velocity of 60 miles per hour — taking atmospheric resistance as usual, and the coefficient of friction tJ^, what are the horse powers of the loco- motive ? SOLUTION. Here weight of train in lbs. = 130 X 2000 = 260000 ; space passed over per 60 X 6280 minute = — — r — = 6280 ft. ; increase in the velocity = J|o = q . and verti- cal fall of train per minute = ^i^ <*' ^280 ft. = 18*48 ft. Then work of fiiotion per minute = ^ X 260000 X 5280 =1800 X 5280 = 6864000 units. Work of atmospheric resistance per n)inute=33X 36X5280=6272640 units. Work of gravity per minute = 18-48 X 260000 = 4804800 units. Then, since the train descends the gradient, gravity acts with the engine. „ „ ^ Wo rk of fr ictiou-f-work of atmos. resist.— work of gravity. uenoe a, r. =————— -— — — — -. Work of one H. P. .„^ 6864000 + 6272640 — 4804800 8331840 „,„ ,„ , • • ^' ^- = ^000 ^ ISOOO" = ^^'^' ^^' AST. 160.] WORK ON AN INCLINED PLANE. 61 Example 164. — A train weighing 80 tons moves along a gradient with the uniform speed of 40 miles per hour — assuming the inclination of the gradient to be 3 ft. in 1000 ft., and taking friction and atmospheric resistance as usual, what will be the H. P. of the locomotive : 1st. If the train move up the gradient, and 2nd. If the train move down the gradient ? SOLUTION. Here weight of train in lbs. =80 X 2000=160000 ; space passed over per minute = — — — =3520 ft.; velocity is increased ^ =4 times, and ver- tical ascent or descent of train j^^ of 3520 =10*66 ft. - ' * Work of friction = 80 x 7 X 3520 = 1971200 units per minute. Work of atmospheric resistance = 33 X 16 X 3520 = 1858660 units per min. Work of gravity =10-56 X 160000 = 1689600 units per minute. Then H P —^^^^ ^^ friction + work of atmos. resist.dbwork of gravity . ' Work of one H. P. Train ascending, H.P.= 1971200 + 1868560 + 1689600 5519360 = 167-268. 33000 — 33000 m • J ,. „T, _ 1971200+1858560—1689600 2140160 _«..q_„ Tram descendmg. H.P.= g^^^^^ =___, = 64 853. Example 165. — A train weighing 110 tons ascends a gradient having a rise of \ in 100 — taking friction as usual, and neglecting atmospheric resistance, what is the maximum speed the train will attain if the H. P. of the locomotive be 120 ? SOLUTION. Here weight of train in lbs. = 110 X 2000 = 22f )00. Work of fk-iction in one mile = 110 X 7 X 5280 — 4066600 units. Work of gravity in one mile = -g^ of 6280 = 6*6 X 220000 = 1452000 units. Total work of resistance in 1 mile = 4065600 + 1462000 = 6617600 units. Total work of engine per hour= 33000 X 60 X 120=237600000 units. *T t- , .1 V 237600000 ^„.^ . . •. Number of miles per hour = ^^ = 43*06 Ans. Example 166. If a horse exert a traction of 120 lbs., what gross load will he pull up a hill whose rise is 17 ft. in 1000 ft., assuming the coefOicient of friction to be t^ ? SOLUTION. Work of horse in moving the load over 1000 ft. = 120 X 1000 = 120000 units. Work of friction in moving 1 lb. over 1000 ft. = 1 X ^ X 1000 = 100 units. Work of gravity in moving 1 lb, over 1000 ft. = 1 X 17 = 17 units. Total work in moving 1 lb. over 1000 ft = work of friction + work of gravity = 100 + 17 = 117 units. . •. Number of lbs. drawn by horse =-L2^l^— 1025*641. Ans, Example 167. — What backward pressure is exerted by a horse in going down a hill which has a rise of 7 feet in 100, with a load whose gross weight is 2000 lbs., assuming ^ to be the co- efficient of friction? 62 WORK ON AN INCLINED PLANE. tAM.llSl. t :;■■■;• ,--;'■! i'. 1^1, 1- '1V SOLUTION. Here on a level plane the friction would be ^ of 2000 lbs. = 5714 lbs. = units of work for each foot. Work of gravity = yg-^y of 2000 = 140 units to each foot. Therefore, the backward pressure is 140— 57*14 = 82*86 lbs. Ans. SXERCISB. 168. What backward pressure will a horse exert in going down a hill which has a rise of 9 feet in 100, with a load whose gross weight is 1200 lbs., assuming the coefficient of fric- tion of the road to be ^ ? ^ns. 68 lbs. 169. What gross load will a horse exerting a traction of 150 lbs. draw up a hill whose inclination is 3 in 100— assuming the coefficient of friction to be 16 J:ns. 1551*72 lbs. 170. What will be the maximum speed attained by a train weighing 200 tons, drawn by a locomotive of 160 H. P. up a gradient having a rise of ^ in 100 — taking friction as usual and neglecting atmospheric resistance ? jins. 29*032 miles per hour. 171. A train weighing 88 tons moves up a gradient having a rise of J in 100 with the uniform velocity of 20 miles per hour — taking friction and atmospheric resistance as usual, what are the H. P. of the locomotive ? ^ws. H. P. = 71-182. 172. A train weighing 95 tons descends a gradient having a fall of ^ in 1000 with the uniform speed of 40 miles per hour — taking friction and atmospheric resistance as usual, what are the H. P. of the locomotive ? Jns. H. P..= 113*742. 173. A train weighing 125 tons moves along a gradient having a rise of ^ in 100 with the uniform speed of 25 miles per hour — taking friction and atmospheric resistance as usual what are the H. P. of the engine, 1st. When the train ascends the gradient? 2nd. When the train descends the gradient ? Jns. Going up, H P.=113*75 ; going down, H.P.=30*416. 151. For finding the H. P., maximum speed, weight of train, X 88 r. h ' ^* . " " to gravity = 2000tt; X -jQQ- X 88 r = 20 Aw X 88 r. Units of work due per min. to attnos. resist. = R I — J x 88ri Units of work per min. in given H. P. = H. P. x 33000. Hence H. P. X 33000 =fw X 88r -f- jR (—) X 88r+ 20hw X 88r, and factoring this, we get : H. P. X 33000 = (/w + J2 Cjj ± 20hw) 88r. / r \ * . 88r 1?herefore H. P. =z{fw^R i-j\ ± 20hw) Or H. P. =z(fw-{'R (y) ± 20hw) 375 33000 * (1.) From this we obtain by transposition and reduction^ and neglecting atmospherib resistance, H.P. X375 W Z=Z (II.) (/+ 20A)r H.P. X375 **- if±20h)w ^^^^'^ Sin6e / is commonly =7, JK=33, and «=10, these formulas become respectively, H. P. = (7W + -asr 2 t 29 hw) j~ (IV.) H.P. X375 ,^, . r = (71 20fc)r H.P.X875 (7t 'M)w (VL) 'iii 64 THE MODULTIS OF A MACHINE. tAiiTS.168,lM. -1 li^6 i Example 174.— A train weighing 140 tons moves along a gradient having a rise of i in 100 with the unifonn speed of 30 miles per hour ; taking friction and atmospheric resistance as usual, what are the H. P. of the locomotive ; 1st, when the train moves up the gradient ? 2nd, when the train moves down the gradient? BOLUTIOir. • Herew=140,r=80, * = i. * H.P.= (7w + -33r« t 20Aw) -g^ = (7 X 140 + -33 X 302 i 20 X i X 140) ^l6 gives isTS. 164-1660 WORK OP STEAM. 65 ' TABLE OF MODULI. MACHINE. MODULUS. Inclined chain pump, | Upright " J Bucket wheel, | Archimedian screw, J^ Pumps for draining mines, a Example 176. — If 7 H. P. be applied to an upright chain pump, how many gallons of water will be raised per hour to the height of 50 feet? SOLUTION. Work applied per hour =33000 X 7 X 60. Work done = 38000 X 7 X 60 X i, since the modulus of t^e ur.right chain pump is \. Work expended in raising 1 gallon of water 50 feet — 10 X 5 j, ,^ ^ , „ 33000X7X60Xi . • , Number of gallons = iox"50 =13860. Atu, Example 177. — ^What must be the H. P. of an er.gme to pump 9000 cubic feet of water per hour from a mine t^ jiose depth is 110 feet? SOLUTION. Work of raising water per hour =9000 X 62J X 110. Effective work of one H. P. per hour — 33000 X 60 X f. 9000 X 62^ X 110 61875000 • • •^•P— 33000X60X1 — ■» "20000 ~~*^^^*** * WORK OP WATER. 154. When water falls from a >r ght upon the float, boards of a wheel, &c., the quantity of work it performs is found by multiplying the wei<:^ht of the water by the height through which it falls. (See Chap. VIII.) STEAM ENGINES AND WORK OF STEAM. 165. A constant power is obtained from the confine- ment and regulated escape of steam in the various kinds of steam engines. 156. Steam engines, though differing very materially from one another in detail, are all modifications of two distinct machines, viz : — 5 .1 ' ' 66 WORK OF STEAM. [Aets. 157-163. . 1st. The high pressure steam engine, or non-condensing engine. 2nd, The low pressure steam engine, or condensing engine. " • 157. The high pressure engine, which is the simpler form of the two, consists essentially of a strong vessel or boiler in which the steam is generated, a cylinder, in which a tightly fitting piston moves backwards and forwards, an arrangement of valves so adjusted as to admit the steam alternately above and below the piston and also alternately open and close a way of escape into the air, and lastly various contrivances by which the oscillations of the piston may be converted into other kinds of motion suited to the work the engine is to perform. 158. In the low pressure engine, the space into which the steam drives the piston is converted, by means of a condensing chamber, into a vacuum, so that the motion of the piston is not resisted by atmospheric pressure, and steam generated at a low temperature can therefore be used, 159. The varieties of the low pressure engine are chiefly two, — the single acting, and the double acting engine. 160. In the single acting engine the piston is driven forward by means of steam acting against a vacuum, and backward by the counterpoising weight of the machinery. The machine is therefore in action only half the time of the movement. 161. In the double acting engine the piston is driven both backward and forward by the steam acting against a vacuum on the opposite side, and the machine therefore acts continuously. 162. In the high pressure engine the piston moves both forwards and backwards against the pressure of the air. 163. The following are the leading ideas that enter into the construction and operation of the steam engine. I. When steam is condensed, a vacuum Is produoca tn»o which the adja- cent bodies liave a tendency tu rush. II. When cold water is placed in contact with steam, i oondonseslt with great rapidity, producing a vacuum ; and this vacuum may be produced without cooling the cylinder containing ih(> steam, if a communication bo kept up between this and a vessel containing water. mk ITS. 157-163, ndensing ndensing 3 simpler vessel or , in which wards, an he steam Iternately ind lastly the piston suited to ito which eans of a motion of sure, and 'e be used, ire chiefly gine, is driven uum, and lachinery. imeof the . is driven against a therefore lovcs both the air. enter into inc. ich the adja- lonsesitwiili be producod mnioationbo AET3.164,ie5.] WORK OF STEAM. 67 III. The vapour of water exerts a considerable pressure even at compara- tively low temperatures ; for example, far below its boiling point. IV. If the pressure exerted by the piston on a quantity of steam confined in a cylinder be less than the elastic force of the steam, the steam will expand and give motion to the piston. v. If a vacuum be produced in a cylinder behind the piston, the atmos- pheric pressure will orive the piston backwards. VI. The same quantity of fuel will convert the same quantity of water into steam whatever may be the pressure on its surface. VII. The higher the pressure under which steam is generated, the smaller its bulk, and tlie greater its elastic force. VIII. The same quantity of water converted into steam at any pressure will produce the same mechanical effect ; i. o., if the pressure below, the steam generated is large in quantity and possessed of comparatively little elastic force ; if the pressure be high, the steam generated is of small quan- tity, but of high elastic force. IX. One cubic inch of water converted into vapour produces 1696 cubic inches of steam, and, since the pressure of steam is, under ordinary circum- stances, equal to that of the atmosphere, the mechanical force produced by the evaporation of one cubic incn of water is sufficient to raise 15 lbs. through 1696 inches or 141^ feet. This is the same, in effect, as raising 1414 times 16 lbs., i. e., 2120 lbs. through one foot. The conversion of one cubic inch of water into steam therefore does work equivalent to raising rather more than one ton weight through one foot. Deducting loss by friction and other causes, about 60 per cent, of this total force is available for use. One cubic foot of water evaporated in one hour will hence do work equal to about 60 per cent, of 1728 times 2120 units, or in other words about 2000000 units, which is about equivalent to the work of one horse for the same space of time. A boiler then of 7, 8, 9, 10, &c., horse powers is a boiler capable of evaporating 7, 8, 9, 10, Ac, cubic feet of water per hour. A. The common allowance of fuel for the steam engine is 10 lbs. of bitu- minous coals for every horse power of the boiler, (i. e., every cubic foot of water it evaporated per hour.) In Cornwall, however, this effect has been E reduced by the consumption of 5 lbs. of coal only. In the American oilers about 6i lbs. of anthracite coal suffice for the evaporation of one cubic foot of water, or in other words the combustion of 1 lb. of coal is sufficient to evaporate 10 lbs. of water. 164. High pressure engines are commonly used where it is desirable to have the engine as simple, cheap, compact, and light as possible, as the condensing apparatus renders the engine more costly and cumbrous. The high pressure engine is, however, far more liable to burst and get other- wise out of repair. 165. The units cf work performed per miy,ute hy a steam engine are found hy multiplying together the pres- sure per square inch on the boiler y the area of the j^iston in inches, the length of the stroke of the piston in feet, and the number of strokes per minute, Thui let iho pressure exerted on each square inch of the piston bo 30 lbs., and let the piston make 40 strokes per minute of 3 ft. each, also let the area of the piston bo 100 square inches : Now if a weight of 30 lbs. be placed on each sciuare incli of the surface of the piston, tho clastic foroo of the steam will bo just sufficient to lift the i 68 WORK OF STEAM. [Mis. 166, 167. ^'1 'I I t^l H IP' loaded piston through the length of the stroke in opposition to gravity, then the work performed on 1 sq. in. of the piston would be 30X3 for each stroke. Work performed on whole piston would be 30X3X100 for each stroke. Work " " " " 30X3X100X40 per minute. 166. In the high pressure engine, the pressure of the atmosphere, about 15 lbs. to the square inoh, acts in op- position to the pressure of the steam ; and in the low-pres- sure or condensing engine a pressure of about 4 lbs. to the square inch of the piston is exerted by the vapour in the condensing chamber. Besides these, a resistance of 1 lb. per square inch is commonly allowed for the friction of the piston. Deducting these allowances from the total pres- sure, we obtain the effective pressure ; and we must further make an allowance of | of this for the friction of the whole engine. Thus in the high pressure engine : Load + 1 load -f- 1 -J- 15 = whole pressure. In the condensing engine : \ Load + \ load + 1 + 4 = whole pressure. For example,— if the whole pressure be 58 lbs. per square inch. Then for the high pressure engine 58—1—15=42 is the working pres- sure on the piston, and 42 is ^ (i. e., load + ^ load) of the useful pressure, and hence useful or eflfective pressure = 42 -f- f = 363. For the low pressure engine68— 1—4 =63= working pressure on the piston, and 53 is ^ of the useful pressure. Therefore usefUl or effective I>ressure is 63 -f- f = 46g . 167. For finding the H. P. of a steam engine, let^ === useful pressure in lbs. on each square inch of the piston, a = area of piston, I = length of piston stroke in feet, and n = number of strokes per minute. pain Then H. P. = P 33000 H.P. X 33000 a = aln H. P. X 33000 . (n.) ,j . (in). pin pan ^ ' ; e of the steam as 28 lbs. per sq. inch, lyhat are theH. P. ofthteii'^ine? SOLUTION. Here p = 28, a = 250, » = 110, and ? = 6. 28X250X110X5 Then (Formula I.) H. P. = 33000 =11G|. Ans. Example 179. — The piston of a high pressure engine has an area of 1200 inches and makes in each minute 30 strokes of 7 feet each — taking the gross pressure of the steam as 48 lbs. per square inch, what are the H. P. of the engine ? SOLTJTIOir. Here48=p + |pH-16 + l, orf jp=32, and hence p = 32 4- f= 28 lbs. Then p = 28, a = 1200, w = 30, and Z = 7. ByPorm»l»I.,H.P. = ?2>L' =213-81. 33000 Ans. Example 180. — The piston of a low pressure engine has a diameter of 20 iu. and makes 60 strokes of 4 ft. each, per minute — the pressure of the steam on the boiler is 45 lbs. to the sq. inch, what are the H. P. of the engine ? SOLUTION. ' • Here45=p + |2>-}- 4-1-1, or ^p = 40, and hencep=40-7-f =35. a* = 102 X 3*1416 = 100 X 31416 = 314*16. Then p = 36, a = 314*16, » = 60, and Z =: 4, 36X314*16X60X4 H.P. 33000 79*968. Ans. Example 181. — In a steam engine of 32 horse power, the area of the piston is 500 inches, the length of the stroke 4 feet, and the useful pressure of the steam 33 Hl. to the sq. inch, how many strokes does the piston make per minute 7 , SOLUTION. Here, H. P. = 32, a = 600, 1 = 4, and p = 33. Then (Formula IV.) w = H.P.X3300a 32X33000 = 16. Ans. pal r00X4X33 Example 182. — In a low pressure steam vingin ^ of 190 H. P. the area of the piston is 1000 inches, the lengtl stroke 6 feet, and the number of strokes per minute 110, what is the useful pressure per square inch on the piston, and also, what is the gross pressure of the steam ? * When the diameter of the piston is given, its area is found by multiply- ing the square of half the diameter by 3*1410. 70 WORK OF STEAM. [Aet. 167, f|i SOLUTION. Here, H. P. = 190, a = 1000, ? = 6, and w = 100. mi- /Ti 1 TT X 190X33000 ^.., , . Then (Formula II.) p = ,nn».x«v.^-.A = 9i l^s. = useful pressure. lUOOXoXllO And pressure on boiler (Art. 166) = 9i+| of 9^+4+1 = 15f lbs. Example 183. — In a high pressure engine the piston has an area of 800 inches, and makes 40 strokes per minute, of 10 feet each, what must be the pressure of the steam on the boiler in order that the engine may pump 120 cubic feet of water per minute from a mine whose depth is 400 feet — making the usual allowance for friction and the modulus of the pump ? SOLUTION. Here, work done per minute = 120 X 62*5 X 400 = 3000000 units. Work applied, i. e., work of engine = 3000000 -^1 = 4500000 units = H. P. X 33000. mu u T. 1 TT H.P.X33000 4500000 ,. , lu ^ % ThenbyFormulaII,p= — ^-^^^ = 800X10X40 =^^^ ^^'^ ^^'"^"^ pressure. And Art. 166, gross pressure = 14-^^+1 of 14^1^+15+1 = S2^ lbs. Ans. ExAMPi^ 1 84. — The piston of a high pressure engine has an area of 600 inches, and makes 20 strokes per minute, each 8 ft. in length, gross pressure of the steam 52 lbs. to the square inch. How many gallons of water per minute will this engine pump from a mine whose depth is 500 feet, making the usual allow- ance for friction and the modulus of the pump ? SOLUTION. Hereo=600, ? = 8, n=20, and since B2=p + i3> + 15 + l; |p = 36, andp=31i. Work of engine per mmntezn pain = 31 J^ X 600 X 8 X 20= 3024000. Useful work per minute =3024000 X 1=2016000. Work of pump'va: 1 gallon of water to height of 500 feet = 10 X 500 = 5000 units. . • . No. of gallons pumped per minute = s ^^,"(,'{,0 " = 403^. Ans. EXERCISB. 185. The piston of a low pressure steam engine is 40 inches in diameter and makes 40 strokes of 5 feet each per minute ; — • the gross pressure of the steam is 37 lbs. per square inch ; what are the H. P. of the engine ? Ms. 213-248. 186. The piston of a high-pressure engine is 20 inches in diame- ter and makes 50 strokes of 4 feet per minute ; taking the gross pressure of the steam as 40 lbs. per square inch and making the usual allowance for friction, what are the H. P. of the engine ? jins. 39'984. 189. 190. 191. AeT. 168.3 WORK OF STEAM. n 187. The piston of an engine has an area of 2400 inches and makes 16 strokes per minute, each 10 feet in length ; the useful pressure of the steam on the piston is 20 lbs. per square inch, what are the H. P. of the engine ? Jns. 232- '72. 188. In a high pressure engine of 140 H. P. the piston has an area of 1000 inches, and makes 20 strokes, of 5 feet each, per minute ; what is the useful pressure of the steam on the piston and also the gross pressure per square inch ? ^ns. Useful pressure = 46*2 lbs. per sq. in. Gross pressure = 68-8 lbs. per sq. in. 189. In a low pressure engine of 100 H. P. the piston has an area of 200 inches and makes 40 strokes per minute ; the gross pressure of the steam is 45 lbs. per square inch. Required the length of the stroke made by the piston. Jns. 11* 78 5 feet. 190. In a high pressure engine of 80 H. P. the piston makes 44 strokes per minute, each 6 feet in length, and the gross pres- sure of the steam is 66 lbs. per square inch. "What is the area of the piston? Ans. 285*714 sq. in. 191. How many cubic feet of water may be pumped per minute from a mine whose depth is 500 feet by an engine in which the piston has an area of 2000 inches, and makes 30 strokes per minute, each 8 feet in length, the useful pressure of tlie steam being 40 lbs. per square inch, and the usual allow- ance being made for the modulus of the pump ? ? ' ., / ./Jws. 409-6 cubic feet. 168. In all the modifications of the steam engine, the real source of work is the evaporating power of the boiler ; the amount of work done by the engine depending not only upon the rapidity with which the water is evaporated, but also upon the temperature, and consequently the pressure under which the steam is produced. The following is a specimen of an experimental table, given by Pambour, showing the relation between the pressure, temperature,and volume of the steam produced by one cubic foot of water. By means of this table, wo are enabled to ascertain the volume of the steam produced by a given quantity of water, when we know the pressure or temperature under which it is formed. Note 1.— The first column gives the pressure in lbs, to the square incli under which the steam is produced ; the second cohimn shows the corres- ponding temperature, as indicated by Fahrcnhoit's thermomoter; and the third column, tlic vohuno of the steam compared with the volume of the wa- 1t^ WORK OF STEAM. CAet. 169. Aet. '^ . , .it ter which produced it. It will be observed that the lower the temperature, or what amounts to the same thing, the less the pressure under which the steam is formed, the greater its volume. Thus under the usual atmospheric pressure of 15 lbs. to the square inch (or at the common temperature of boiling water, 212° or 213° Fahr.),acubic foot of water produces 1669 cubic feet of steam. If, however, the pressure be decreased to 1 lb. to the square inch, the steam is formed at the temperature of 103° Fahr. and occupies 20964 cubic feet ; while if the pressure oe increased to 30 lbs. to the square inch, the temperature required for the production of the steam rises to 251° Fahr. and the steam only occupies 882 cubic feet. Note 2.— It has been shown by numerous experiments that the quantity of fuel requisite for the evaporation of a given quantity of water is in- variably the same, no matter what may be the pressure under which the steam is produced. Hence it is obvious that it is most advantageous to employ steam of a high pressure. TABLE ' • ' "' ' SHOWING THE VOLUME OF STEAM PEODUCED BY ONE CUBIC POOT OF WATE§ AT THE COEEESPONDINa PEE8SUEE AND TEMPEEATUEE. r ai56 B ^ B-^6 ^4 • p5 |5^ Peessue to square in Tempeeat Pahi -nheit's mometer. Volume of s comp'd with of the water ducing it. PEESSUE] to square in Tempeeat Fahrenheit's mometer. Volume of s comp'd with of the water duciiig it. 1 103° 20954 66 288° 606 5 161» 4624 60 294° 467 10 192° 2427 65 299» 434 15 213° 1669 70 304P 406 20 228° 1280 76 309° 381 25 241° 1042 80 313° 369 30 261° 882 86 318° 340 36 260° 765 90 322P 323 40 268° 677 95 326° 307 46 276° 608 100 330° 293 60 282° 652 106 333° 281 169. If we let a: ' ' ' ' ■ n P V area of the piston in square inches, length of stroke made by the piston. : number of strokes made per minute. : effective pressure to each sq. inch of the piston, cubic feet of water evaporated per minute, volume of one cubic foot of water in the form of steam under tbe given pressure p. Now! Cubi the [Aet. 169. Art. 169.3 WORK OP STEAM. n [uperature, which the }mospheric lerature of 1669 cubic the square id occupies the square uu rises to quantity ater is in- nrhich the bageous to ! POOT OF LTUEE. 'I Dches. piston, linute. !• inch id per water ^r tbe Then to find a, /, w, p^ c, or v, when the others are given, we proceed as follows : When p is given, v is found by the table. ''' Now the cubic feet of steam produced per minute = cv. Cubic feet of steam used at each stroke of the piston = ^ .*. cubic feet of steam used in n strokes = :; — = also, 144 the steam evaporated or used per minute. Hence ^-jj = cv, and from this by reduction w^ obtam nal . nal ■,and vz=.- 144 l=z 144cv n 144cv a 144cv na 144c. al ' nl '' 144v' When V is known ^ may be found by the table. Example 192. — The piston of a steam engine has an area of 200 square inches and makes a stroke 4 feet in length, the boiler evaporatmg ^j of a cubic foot of water per minute, under a pressure of 40 lbs. to the square inch. What number of strokes per minute does the piston make ? Here a=200, Z = Then n = — =— = SOLUTIOir. 4, 74 WORK OP STEAM. CArt. 170. SOLUTIOir. '4 .•',,. IS ■'■; I j^ Ma "!' ■•1! r .1 \i -u «E Here a = 80, w = 20, c = '1 and p = 50 and (table) v = 552. „, , lUcv 144 X '1 X 552 , , .„,,,,. ^ Then? 9ia 20X80 - = 4*968 ft. =4 ft. Hi inches. Ans. Example 195. — The boiler of an engine evaporates f of a cubic foot of water per minute under a pressure of 45 lbs. to the square inch ; the piston has an area of 250 inches and makes a stroke 4 feet in length. Required the number of strokes made by the piston per minute. Here a = 250, 1 Then«= — =—• at SOLUTIOir. = 4, c = •4,31) = 45, and hence (table) v = 608. 144 X '4 X 608 : 2^ . — 35-0208, i. e. 35 strokes per minute. Ans. 250 X 4 H^ EXERCISE. 196. The boiler of a steam engine evaporates f of a cubic foot of water per minute under a pressure of 65 lbs. to the square inch. If the piston has an area of 144 square inches and makes strokes 5 feet in length, how many- strokes are made per minute ? *Ans, 69'44. 197. The piston of an engine has an area of 288 inches and makes T strokes per minute. If the boiler evapojates ^ of a cubic foot of water per minute under the pressure of 55 lbs. to the square inch, what is the length of the stroke of the piston ? Ans. 2^^^ feet. 198. The piston of an engine makes 10 strokes of 6 feet each per minute ; the boiler evaporating \ a cubic foot of water per minute under a pressure of 25 lbs. to the square inch, what is the area of the piston ? Ans. 1250*4 inches. 199. In a steam engine the piston having an area of 720 inches makes 20 stroke.?, of 3 feet each, per minute, what volume of water convened intft steam under a pressure of 20 lbs. to the square inch, is evaporated per minute by the boiler ? Ans. ^ of a cubic foot. 200. The piston of a steam engine has an area of 600 inches and makes 12 strokes, of 10 feet each, per minute. Now if the boiler evaporates 1 cubic foot of water per minute, what is the volume of the steam produced per minute and the pressure under which it is generated ? Ans. Volume == 500 cubic feet. Pressure =r nearly 55 lbs. to the square inch. 170. To find the useful H. P. of an engine when a, w, /, c, and V are given we proceed as follows : Find the pressure per square inch of the steam from the TabUy and thence Art. 166 the useful load on each square inch of the piston ; find also when required any of the other quantities^ a, w, or /, and then apply the rules givenin Art. 167. [Art. 170. ABT. 170.] WORK OF STEAM. 75 ins. s f of a 3S. to the makes a es made iiute. Ans. ubic foot 3. to the i square »w many IS. 69-44. jhes and )jates -^ pressure h of the jSt'o" ^eet. feet each foot of le square i inches. inches t volume f 20 lbs. boiler ? ibic foot. inches B. Now minute, minute ire inch. len a, w, B Tablcj h of the iesj a, w, Example 201. — ^What is the useful load per square inch on the piston, and what is the effective horse powers of a high pressure engine in which the area of the piston is 200 inches, the length of stroke 6 feet, the effective evaporation of the boiler f of a cubic foot per minute and the pressure of the steam 70 lbs. to the square inch ? SOLUTION. By Art. 166, 70 = f p + 15 + 1, and hence p = 54 -^ f = 47'25 — Useful load. ByArt.m« = Hf^ = l*t^^*>^=19-488. . - - : al 200X6 Hence we have »= 19*488, i) = 47*25, a = 200, ?= 6. mu A^ ifltT TT T» P^^^'"' 47*25 X 200 X 6X 19*488 _„„..o - ^ Then Art. 167, H. P. = 3^ = 33000 ^3 48. Ans. Example 202. — ^What are the effective horse powers of a low pressure engine in which the piston has an area of 288 inches and makes every minute 16 strokes, the boiler converting ^ of a cubic foot of water per minute into 304 cubic feet of steam ? solution. Since J of a cubic foot of water produces 304 cubic feet of steam, 1 cubic foot of water would produce 608 cubic feet of steam^ and hence (Table) the gross pressure of the steam is 45 lbs. to the square inch. Then (Art. 166) 45 = fi5 + 4 + l, or&p = 40 whence p = 35. Al /A4.-.flA\7 144e confined in a vessel and a pressure to any amount be exerted upon any one square inch of the surface of that water, a pressure to an equal amount will be transmitted to every square inch of the interior surface of the vessel in which the water is confined. Fig. 16. Note. — In the accompanying figure suppose th€! piston P has an area of 1 square inch, and the piston ij' an area of 100 square inches, then if 1 lb. pres- sure be applied to P as weight of 100 lb. must be applied to p' in order to maintain equilibrium. It is this pro- perty of equal and instant transmis- sion of pressure which enables us to make use of hydrostatic pressure as a mechanical power,and it is upon this principle that Braniah's Hydrostatic Press 18 constructed. 183. Bramali's Hydrostatic Press consists of two stron metallic cylinders A and a, one many times as large as th other, connected together by a tube. » The small cylin- der is supplied with a strong forc- ing pump s\ and the larger one with a tightly fit- ting piston Sy at- tached to a firm platform or strong head P. Both the cylinders and the communicating: tube contain wa- ter, and when downward pressure is applied to the water in the smaller cylinder, by means of the attached forcing pump, the pis- ton in the larger is forced upward by a pressure as much greater than the downward pressure in the smaller, as the sectional area of the larger cylinder is greater than that of the smaller. Asa and! pres mtl pa] for for the! I. II. "Aets. 182, 183. pressure to och of the mount will 'ior surface ;wo stron irge as th ,s smaller the pis- much as the that of ASIS.184,185.] HYDROSTATICS. 85 For example, if the smaller cylinder have an area of half a square inch , and the large cylinder an area of 500 square inches then the upward pressure in the latter will be 1000 times as great as the downward pressure m the former. 184. Bramah's Hydrostatic Press is used for pressing paper, cotton, cloth, gunpowder, and other things — also for testing the strength of ropes, for uprooting trees, and for other purposes. 185. To find the relation between the force applied and the pressure obtained in Bramah's Hydrostatic Press. RULE. /. If the power he applied by means of a lever, find the amount of downward pressure in the smaller cylinder by the rule in Art. 11. II. Divide the sectional area of the larger cylinder by that of the smaller cylinder and multiply the quotient by the power applied to the smaller cylinder. Example 238. — In a hydrostatic press the force pump has a sectional area of one square incb ; the large cylinder a sectional area of one square foot, the force pump is worked by means of a lever whose arms are to one another as 21 : 2. If a power of 20 lbs. be applied to the extremity of the lever what will be the upward pressure exerted against the piston in the large cylinder ? SOIjUi^ION. ''0 X 21 Power applied to a fr :'ce pump ~ ^^ — = 210 lbs. Sectional arcaof smaller cylinder -^ 1 inch, and of larger cylinder = 144 inches. Then 144 -M = 144 X 210 =- 30240 lbs. Ans. Ex.'-MPLB 239. — In a hydrostatic press the sectional areas of the cylinders are ^ of an inch and 150 inches, and the power lever is so divided that its arms are to one another as 1 to 43. What pressure will be exerted by a power of 100 lbs. applied at the extremity of the long arm of the lever? SOLUTION. Downward pressure in small cylinder— — -^ = 614? lbs. 160 Upward pressure in largo cylinder— I'Z X 0142 =45u X 6142 = 276428| lbs. Ans, i ^ Example 240. — The aron of the small piston of a hydrostatic 3SS is i an inch and thai of the larger one 300 inches, the lever is 30 inches long and the piston rod is placed 5 inches from the fulcrum (so as to form a lever of the second order) wl'.at power must bo pnplied to the end of the lever in order to produce an upward picssurc iii the cylinder of 1000000 lbs. ? 8ft HYDROSTATICS, [Abt , 186, SOLUTION. 300 Downward pressure in smaller cylinder =; 1000000 lbs. -f -r- = 1000000 lbs. ■T-600 = 1666|lbs. 30 Then power applied =1666§ lbs. -J- -- = 1666§ -r- 6 = 277| lbs. Ans, EXERCISE. 241. In a hydrostatic press the area of the small cylinder is one inch, and that of the large one 300 inches, the force pump is worked by a lever of the second order 30 inches lonpj, havingj the piston rod 2 inches from the fulcrum ; if a pressure of 50 lbs. be applied to the lever what upward pressure will be produced in the large cylinder ? ^ns. 225000 lbs. 242. In a hydrostatic press the force pump las a sectional area of half an inch, the large cylinder a sectional area of 200 inches ; the force pump is worked by means of a lever whose arms are to one another as 1 to 50 ; now suppose a force of 50 lbs. be applied to the ex- tremity of the lever, what will be the upward pressure exerted against the pistoii in the large cylinder ? Ans. 1000000. 243. In a hydrostatic press the small cylinder has an area of one inch, and the large one an area of 500 inches, the pump lever is so divided that its arms are to one another as 1 to 25. What will be the upward pressure against the piston in the large cylinder produced by a force of 100 lbs. acting at i he extremity of the lever ? Ans. 1250000. 244. The area ri' the small piston of a hydrostatic press is J of an hicli and that of the large one 120 inches — the arms A the lever by which the force pump is worked are to r ue another as 40 to 3. Requireu tlie upward pressure exerted against the piston of the large cylinder by a power of 1 T lbs. aj, plicvi at the extremity of the lever. Ans, 3626Cf lbs. 245. The area of the SL\fiil piston of a hydrostatic press is li inches, and that of^ tiie large one 200 inches — the arms of the lever by which t"ie force pump is worked are to one another as 20 to 1^. What power applied at the extre- mity of the levci- will produce a pressure of t500001bs. ? Ans. 421| lbs. 186. Since the pressure of water upon a given base depends upon tbc height of the liquid and not upon ita quantity, it follows that : — [Abt . 186, I Aets. 187, 188.] HYDROSTATICS. 87 = 1000000 lbs. dns. der is one the force 30 inches fulcrum ; ver what cylinder ? 15000 lbs. 3nal area nal area y means I to 50 J the ex- pressure 000000. I area of ties, the another i.inst the brce of 250000. ess is I 3s— the ced are ressure • by a 'er. 3J *lbs. J is li rms of to one extre- )lbs. ? I lbs. base 1 ita Ani/ quantity of water however small^ may be made to balance the 'pressure of any other quantity however greats or to raise any weight however large, IfoTB.— This is what is commonly called the Hydrostatic Paradox. In reality, however, there is nothing at all paradoxical in it ; since, althouKh a pound of water may be made to balance 10 lbs., or 1000 lbs., or 100000 lbs-, it does it upon precisely the same principle that the power balances the weight in the lever and, other mechanical powers. Thus in order to raise 20 lbs., of water by the descending force of 1 lb., the latter must descend 20 inches in order to raise the former 1 inch. Hence what is called the hydrostatic paradox is in strict conformity to the principle of virtual velo' cities. 187. This principle is illustrated by an instrument called the Hydrostatic Bellows, which consists of a pair of boards united together by leather as in the common bellows and made water-tight. From the upper board there rises a long tube, B^ finished with a funnel-shaped termin- ation, C. Note. — When water is poured into the tube an upward pressure is exerted against the upper board as much greater than the weight of the water in the tube as the area of the board is greater than the sectional area of the tube. Fig. 18. For example, if the sectional area of the tube be ^ of an inch, and the area of the board be 2.30 inches, then the area of the board will be 1000 times as great as that of the tube, and consequently 1 lb. of water in the tube will exert a pressure of 1000 lbs. against the upper board of tiie bellows. 188. To find the upward pressure exerted against the board of a hydrostatic bellows by the water contained in the tube. RULE. Divide the sectional area of the hoard by that of the tube^ and multiply the result by the weight of the water in ihe tube. Note. — The weight of water in the tube is found by multiply^ ing the sectional area of the tube by the height of the water in inches and the product, which is cubic inches of water, by 0'03616 Ibs.^ the loeight of one cubic inch of water. Example 24G. — The upper board of a hydrostatic bellows has an area of 1 foot, the tube has a sectional area of i an inch and is tilled with water to the height of 7 feet. What upward pressure is exerted againHt the top board of the bellows ? HYDROSTATICS. [Abts. 189-193. SOLUTION. Cubic inches of water in the tube=i X 84 =42. Weight of water in tube =0'03616 X 42 =1*51872 lbs. 144 Upward pressure against bellows board =1-51872 X— r-= 1'51872X288 =437'391bs. ^ws. ;' . - Example 24*7. — In a Hydrostatic Bellows the board has an area of 200 inches and the tube a sectional area of i of an inch. What upward pressure is exerted on the board by 7 lbs. of water in the tube? _^ SOLUTIOir. Upward pressure=7 X -7-= 7 X 800= 5600 lbs. Ans. EXERCISE. 248. In a hydrostatic bellows the board has an area of 250 inches, the tube has a sectional area of 1^ inches, and ^' contains 11 lbs. of water. What is the amount of upward pressure exerted against the board of the bellows ? Jns. 2200 lbs. 249. The board of a hydrostatic bellows has an area of 300 inches, the tube a sectional area of 1 inch and is filled with water to the height of 10 feet — what pressure will be exerted against the upper board of the bellows ? ^ns. 1301-76 lbs. 250. The tube of a hydrostatic bellows has a sectional area of •72 of an inch and is filled with water to the height of 50 feet — what weight will be sustained on the bellows' board if the latter have an area of 3 feet? — jlns. 9372-672 lbs. 189. A body immersed in any liquid will either float, sink, or rest in equilibrium, according as it is specifically lighter, heavier, or the same as the liquid. 190. A floating body displaces a quantity of liquid equal to its own weight. • • 191. A body immersed in any liquid loses a portion of its weight-equal to the weight of the liquid displaced, and, hence, by weighing a body first in air and then in water, its relative weight or specific gravity may be determined. 192. The specific gravity of a body is its weight as com])ared with the weight of an equal bulk or volume of some Other body assumed as a standard. 193. Pure distilled w;\ter at the temperature of 60° Fahr. is taken as the standard with which to compare all 253. 254. \ rs. 189*193. Aets. 194, 195,] HYDROSTATICS. 89 L872X288 i has an an inch, of water : of 250 iies, and upward 1200 lbs. of 300 is filled are will \.'16 lbs. area of ight of )ellows' 12 lbs. r float, fically liquid ion of , and, vater, ned. t as ne of 3 all solids and liquids, and pure dry atmospheric air at a tempe- rature of 32° Fahr., and a barometric pressure of 30 inches is taken as the standard with which all gases are compared. 194. To find the specific gravity of a solid heavier than water : — RULE. Divide the weight of the body in air by its loss of weight in water J the result will be its specific gravity. Example 251. — A piece of lead weighs 225 grains in air and only 205 grains in water j required its specific gravity. SOLUTION". Loss of weight = 225—205 = 20 grains. Hence specific gravity = 225-r-20 = 11*250. Ans. Example 252. — A piece of sulphur weighs 97 grains in air and but 50*6 grains in water ; what is its specific gravity ? SOLUTION. Loss of weight in water =97— 50"5= 46*5 grains. Then specific gravity = 97 -f- 46*5 = 2*008 Ans. EXERCISE. 253. A piece of silver weighs 200 grains in air and only 180 grains in water ; required its specific gravity. .^ns. 10*000. 254. A piece of platinum weighs 154^ oz. in air and only 147^ oz. in water ; required its specific gravity. Jlns. 22*071. 255. A piece of glass weighs 193 oz. in air and but 130 oz. in .^ns. 3-063. water ; required its specific gravity. 195. To find the specific gravity of a solid not suflficiently heavy to sink in water. RULE. V '*-' To the body whose specific gravity is sought attach some other body sufficiently heavy to sink it, and of which the weight in air and loss of weight in water are known. Then weigh the united mass in water and in air, from its loss of weight deduct the loss of weight of the heavier body in water, and divide the absolute weight of the lighter body by the remainder, the (juotient will be the specific gravity of the lighter body. Example 256. — A piece of wood which weighs 55 oz. in air has attached to it a piece of lead which weighs 45 oz. in air and 41 in water, the united mass weighs 30 oz. in water j required the specific gravity of the piece of wood. t j. 90 t HYDROSTATICS. SOLUTION. [ABT. 196. Wt. of united mass in air =55+45 = 100 oz. ' « " water= 30 " Loss of wt. of united mass in water = 70 *' Loss of wt. of lead in water = 4 " Remainder = 66 " =loss of v, ijight of the wood. Then 55 -4- 66= '833 = specific gravity of the wood. Example 257. — A piece of wood which weighs TO oz. in air has attached to it a piece of copper which weighs 36 oz. in air and 31*5 oz. in water, the united mass weighs only ll*t oz. in water j what is the specific gravity of the wood ? ' ' • - ■'/ SOLUTION. Wt. of united mass in air =70 + 36 =106 oz. " water = 11*7 " fi' .,• Loss of wt. of united mass in water=: 94*3 " • Loss of wt. of copper " = 4*5 " Loss of wt. of wood " = 89*8 " = loss of weight of the wood. Then specific gravity of wood=70-f-89'8= '779. Ans. EXERCISE, 258. A piece of pine wood which weighs 15 lbs. in air has attached to it a piece of copper which weighs 18 lbs. in air and 16 lbs. in water ; the weight of the united mass in water is C lbs. ; required the specific gravity of the pine ? Jlns, -600. 259. A piece of cork which weighs 20 oz. in air has attached to it an iron sinker which weighs 18 oz. in air and 15-73 oz. in water, the united mass weighs 1 oz. in water; required the specific gravity of the cork ? Ans. '575. 260. A piece of wood which weighs 33 oz. in air has attached to it a metal sinker which weighs 21 oz. in air and 18-19 oz. in water, the united mass weighs 2 5 oz. in water ; what is the specific gravity of the wood ? Ans. 'Q*l*J, 196. The specific gravities of liquids may be determined in three different ways. First Method.—.^ small glass Jlaskj which holds precisely 1000 grains of pure distilled water at the temperature of 60? Fahr.j is filled with the liquid in question and accurately weighed^ the result indicates the specific gravity of the liquid. Second Method. — A piece of substance of known specific gra- vity is weighed both in and out of the liquid in question. The difference of weight / ? multiplied by the specific gravity of the solid^ Art. 196.] a?id the the resul That Third ' most C01 strument\ graduate^ beneath u some othe that the g be its spet mcnt sink ated scale For liquic graduated vierj from EXAMPI with sulj the specif ExAMPI weighs 79 ExAMPI 27*4 oz. ii specific g Here w - Then s = EXAMP] weighs 4 is the sp * That ] [Aet. 196. ughtofthc wood. s To oz. in air IS 36 oz. in air nly ll-*? oz. in 3ight of the wood. bs. in air has ; 18 lbs. in air mass in water pine? Arts. '600. as attached to and 15- "73 oz. bter ; required Ans. •575. as attached to and 18-19 oz. ater ; what is Ans. -677. ) determined precisely 1000 60 o Fahr.^ is hed^ the result specific gra- lestion. The ^y of the solid J Abt. 196.] HYDROSTATICS. 91 ajid the product divided by the absolute weight of the solid^ and the result is the specific gravity of the liquid, w — v/ That is s = • X s'; where w = absolute weight of solid, w/ = weight in the liquid. Therefore w — v/ = loss of weight. " s = specific gravity of the liquid. s' = specific gravity of the solid. Third Method. — This specific gravity of liquids is most commonly found in practice by means of an in- ^^S- !"• strument called the Hydrometer, which consists of a graduated scale rising from a glass silver bulb^ beneath which is a small appendage lo<. d with shot or some other heavy substance. It acts upon the principle that the greater the density of a liquid the greater will be its specific gravity. The depth to which the instru- ment sinks in different liquids is shown by the gradu- ated scalcj which thus indicates their specific gravities. For liquids specifically lighter than water j the scale is graduated from the bottom upwards ; for those hea- vier, from the top downwards. Example 261. — The Thousand-grain Bottle filled with sulphuric acid weighs 1841 grains.* What is the specific gravity of the sulphuric acid ? 1841- SOLUTION. 1000 = 1*841. Ans. Example 262. — The Thousand-grain Bottle filled with i^lcohol weighs 792 grains, required the specific gravity of alcohol. SOLUTION: 792-M000 = -792. ^w*. Example 263. — A piece of zinc (specific gravity 7*190) weighs 27*4 oz. in a certain liquid and 32-7 oz. out of it, required the specific gravity of the liquid. SOLUTION. Here w — 32*7, w' = 27'4, s' = 7190. Then s = w — w w Xs' = 32-7 — 27'4 827 X 7*190 = 6-8 X 7*1 90 32*7 = 1-165 Ans. Example 264. — A piece of silver (specific gravity 10-500) weighs 47-8 g»-ains in a liquid and 58-2 grains out of it — what is the specific gravity of the liquid ? * That is not including the weight of the bottle itself. IMAGE EVALUATION TEST TARGET (MT-3) K^ e ^^i^ 4^ ^ 4^ 1.0 1.1 ■u Uii |22 £f L& 12.0 1^25 |Jj4 |L6 Photographic Sciences Corporalion 4^ <^ V ia WfST MAM STIHT WIUTII.N.Y. 14510 (7U) 173.4101 4^ rf^ V ^r\\ ,^ K<^ n^^ ^ % t f <> ^ ^ ►^ 92 HYDROSTATICS. BOLUTIOir. [Aets. 107. 108. Here to = 58*2, to' = 47'8 and af = 10*6. w — w',, 68-2 — 4/7-8^ ,^. 10'4X10*5 -.-^- .^ Then s== — tz — X «' = — zx::: — X 10*6 = — =7::; — = 1 876. Ans. to 68*2 68*2 EXEROISB. 265. A piece of copper (specific gravity 8-850) weighs 446*3 grains in liquid, and 490 grains out of it, required the specific gravity of the liquid. -Ans. *789. 266. The Thousand-grain Bottle filled with olive oil weighs 915 grains — what is the specific gravity of olive oil ? Ans. •916, 2C7. The Thousand-grain Bottle filled with mercury weighs 13596 grains — ^what is the specific gravity of mercury ? Ans. 13*596. 268. A piece of cast-iron (specific gravity 7*425) weighs 34*61 oz. in a liquid and 40 oz. out of it— what is the specific gravity of the liquid ? Ans. 1*000 nearly. 269. A piece of gold (specific gravity 19*360) weighs 139*85 grains in a liquid and 159*7 grains in the air, required the specific gravity of the liquid ? Ans. 2*406. 270. A piece of marble (specific gravity 2*850) weighs 30 lbs. in a certain liquid, and 35*9 lbs. in the air, required the spe- cific gravity of the liquid? Ans. -468. 197. The specific gravity of gases is found by exhaust- ing a fiask of atmospheric air and filling it with the gas in question previously well dried. This is accurately weighed and its weight compared with the weight of the same volume of dry atmospheric air at the temperature of 60^ Fahr. and under a barometric pressure of 30 inches. 198. The following table gives the specific gravities of the most common substances ; — ^I3J AbTS. 199, 200.] T8. 197, 198. HYDROSTATICS. TABLE OK SPECIFIC TORAVITIES. 176. Ana. ghs 446*3 quired the Ms, -789. »il weighs eoil? AuB, -DIS. ry weighs lercurjr ? 18. 13-596. ighs 34-61 le specifio )0 nearly. hs 139-85 , required 'n«. 2-406. 30 lbs. in d the spe- dm. -468. czhaust- the gas icuratcly it of the ature of inches. vitios of OASES. Atmospheric air, 1 Hydrogen, Oxygen, 1 Nitrogen, ,. AmmoHiacal gas, Carbonic acid gas, .... 1 Sulphurous acid gas, . . 2 Chlorine, 2 LIQUIDS. Distilled water, 1 Mercury, 13 Sulphuric acid, 1 Nitric acid,.... 1 Milk, 1 Sea water, 1 Wine, Olive oil, ^ Spirits of turpentine,... Pure alcohol, Ether, Prussic acid, SOLIDS. Platinum, 22 Gold, 19 Silver, 10 Lead, 11 000 069 106 972 596 629 234 470 000 596 841 220 030 026 993 915 869 792 715 696 050 360 500 250 Copper, 8 Brass, 8 Iron, 7 Tin, 7 Zinc, 7 Diamond, 3 Flint glass, 3 Sulphur 2 Slate, 2 Brick, 2 Common stone, 2 Marble, 2 Ivory, Phosphorus, Lignum vitie, Boxwood, Potassium, Sodium, Pumice stone, Dry pine, • Dry poplar, Ice, Living man, Cork, Graphite, Bituminous coal, Anthracite coal, 93 •850 •300 •788 •293 •190 •530 •330 -080 -840 •000 •400 •850 •825 •770 •350 •320 •875 •972 •914 •657 •383 •865 •891 •240 •500 •250 •800 199. A cubic foot of pure distilled water at the tem- perature of 60^ Fahr. weighs exactly 1000 ounces. Hence if the specific gravity of any substance be known the weight of a cubic foot, &c., may be easily found. For example.— The specifio frravity of mercury is 13*696 water, beinR 1*000, and a cubic foot of water weighing 1000 ounces it follows that a cubic foot of mercury weighs 13590 ounces. 200. To find the solid contents of a body from its weight : — RULE. w Contents in feet = -r; where w = whole weighty and v/ = weight of a cubic foot as ascertained from its specific gravity. Example 271. — How many cubic feet are there in 2240 lbs. of dry oak (specific gravity -926.) ? ill 94 HYDROSTATICS. [Abt. 201. 801UTI0N. W 2240 lbs. 35840 „„ , , „ , • - i. Here —r = -rr- = ■— ;-- = 3H3 ^ cubic feet. w' 926 oz. 925 ^'^o Example 2T2. — How many cubic feet are there in a mass of iron which weighs 17829 lbs. ? SOLUTION. Specific gravity of iron — 7788. Therefore 1 cu. ft. weighs 7788 oz. Then cubic feet in mass = 17829 lbs. -r 7788 oz. = 36-628. Ana, 201. To find the weight of a body from its solid con- tents : — LULE. w = contents in ft, X "w', ' Where w and w' are same as in last rule. Example 273. — What is ihe weight f)f a block of dry oalt 10 ft. long) 3 ft. thick, 2^ ft. wide. Here 10 x 3 X 2^ = 75 cubic feet. . % Then Mr=M)'x 75=925 oz.X 75=69375 oz.=4335|§ lbs. Jns. Example 274. — What is the weight of a block of marble 8 ft. long, 2 ft. wide, and U ft. thick. SOLUTION. Cubic feet of marble = 8 X 2 X 1 J = 24. Spec. grav. of marble = 2*860. Therefore one cubic foot weighs 2850 oz. Then weight of block = 2860 X 24 = 68400 oz. = 4276 lbs. Ans. EXERCISE. 275. What is the weight of a mass of copper which contains 20 cubic feet? Jins. 16040 lbs. 10 oz. 276. How many cubic feet are there in a mass of lead which weighs seven million pounds ? Ans. 9955-55 cub. ft. 277. How many cubic feet of sulphuric acid are there in 78124732 lbs. ? Jlns. 678976-48 cub. ft. 278. What is the weight of the mercury contained in a rectan- gular cistern 6 feet long, 4 feet wide and 10 feet dec]*, the mercury filling it ? Jns. 203940 lbs. 279. If a block of zinc bo 11 feet long by 3 feet wide and 2 feet thick, how much does it weigh ? Jins. 29658 i lbs. 280. What is the weight of a squared log of dry pine 44 feet long and 18 inches square ? /ins, 4065 lbs. 3 oz. CAet. 201. in a mass of • 1 A" 1 7788 oz. m. 3 solid con- ' dry oak 10 lbs. jins, marble 8 ft. 3iG;1is 2850 07.. contains 20 lbs. 10 oz. lead which 55 cub. ft. e there in 48 cub. ft. 1 a rectan- feet decj), 203940 lbs. and 2 feet 29658i lbs. ne 44 feet 5 lbs. 3 oz. »;?;. ABTS.2Q2-208.] PNEUMATICS. 95 202. In order that a floating body may be in equilibrium it is requisite that :— 1st. The weight of the water displaced shall be equal to tlie weight of the floating body ; and 2nd. The resultant of all the upward pressures of the liquid shall act in the line of direction of the centre of gravity of the body. 203. The centre of buoyancy of a floating body is the point upon which the resultant of all the upward pressures of the liquid acts. Note.— The centre of buoyancy coincides not with the centre of gravity of the floating body, but with the centre of gravity of the fluid displaced. While the body floats the centre of buoyancy is always below the centre of gravity, but the two coincide when the body sinks. In a shii), however, or other noUow body, containing much heavy ballast in the hold, the centre of gravity is below the centre of buoyancy. 204. A floating body is in equilibrium when the centre of gravity and the centre of buoyancy are in the same vertical line and the equilibrium is :— Stable when the centre of gravity is below the centre of buoy- ancy. Neuj^ when the centre of gravity coincides with the centre of buoyancy. UmtabU when the centre of gravity is above t^e centre of buoyancy. ., CnAPTER VI. PifEUMATICS. 206. Pneumatics treats of the mechanical properties of permanently elastic fluids^ of which atmospheric air may be taken as the type. 206. The atmosphere (Greek attnoi " gases ") or sphere of gases is the name applied to the gaseous envelope which surrounds the earth. 207. It is supposed, from certain astronomical con- siderations that the atmosphere extends to the height of only about 45 miles above the surface of the earth. NoTB.— The height of the atmosphere is only about ^L of the radius of the earth, ao that upon an artiflcial globe 24 inches in diameter the atmosphere would be represented by a covering ^ of an inch in thickuost. 208. Atmospheric air ' > a mechanical mixture chiefly of two gases, oxygen and nitrogen in the proportion of 96 PNEUMATICS. [Aet.200. v.\ 1 gallon of the former to 4 gallons of the latter. Its exact composition, omitting the aqueous vapour, is as follows : — COMPOSITION BY VOLUME. Nitrogen, Oxygen, Carbonic acid, Carburetted Hydrogen, Ammonia, 79*12 per cent. 20-80 " •04 " •04 " Trace. y OTB.— Ox-yrrcn is the sustaining principle of animal life and of ordinary combustion. When an animal is placed in a vessel of pure oxygen its heart boats with, increased energy and rapidity and it very soon dies from excess of vital action. Many substances, also, that are not all combustible under ordinary circumstances, burn, when placed in pure oxygen, with extraor- dinary brilliancy and vigour. Nitrogen, on the other hand, ^.upports i leithcr respiration nor combustion . In its chemical nature it is distinguished chiefly by its negative properties. In the atmosphere it serves the important purpose of diluting the oxygen and thus fltting it for the function it is designed to perform in the animal economy. Carbonic acid is a highly poisonous gas, formed by the union of oxygen and carbon (charcoal). It is produced in large quantities during the pro- cesses of animal respiration, common combustion, fermentntion, volcanic action and the decay of animal and vegetable substances. Altbf^h when inhaled, it rapidly destroys animal life it constitutes the chief source of food to the plant. Animals take into the lungs air loaded with oxyiofen and throw it otf so charged with carbonic acid as to bo incapable of again serving for the purposes of respiration. The green parts of plants, on the contrary, absorb air, decompose the carboni ' acid it contains, retain the carbon and ^ve off air containing no carbonic acid but a large amount of oxygen. This IS a most beautiful illustration of the mutual dependence of the different orders of created beings upon one another. Were it not for plants the air would rapidly become so vitiated as to cause the total extinction of animal life ; were it not for animals, plants would not thrive for want of the food now supplied in the form of carbonic acid by the ii.ing animal. As it is, the one order of beings prepar«s the air for the sustenance ant! support of the other, and so admirably is the matter adjusted that the con\positionof the air is, within very narrow limits, invariaoly the same. The amount of carbonic acid varies from 37 as a minimum to 6'2 as a maximum in 10000 volumes. Carburetted Hydrogen is produced during the decay of animal and vege- table substances. It is one of*the chief ingredients of common illuminat- ing gas, aud is poisonous to animals when present iu the air in large quantities. 209. One of the most remarkable characteristics of gases, is the property they possess of diffusing themselves among one another. Thus if a light gas and a heavy one are once mixed they exhibit no tendency to sepaj'ate again, and no matter how long they may be allowed to stand at rest, they are found upon examination intimately mingled with each other. Moreover if two vessels 6e placed one upon »« [Aet.209. tcr. Its )ur, is as nt. ABTS.21(h212.] PNEUMATICS. 9T I of ordinary geii its heart < from excess istible under irith extraor- combustion. e properties. g the oxygen II the animal in of oxygen ring the pro- ion, volcanic >b«^h when ourceof food m and throw \ serving for he contrary, carbon and xygen. This he ditferent lants the air 3n of animal t of the food il. As it is, } support of mposition of n to 6'2 as a lalandvegc- I illuminat- air in largo ristics of iemselvcs eavy one lie again, nd at rest, gled with one upon the other, the upper being filled with any light gas (hydro- gen) and the lower with any heavy gas (carbonic acid) and if the two gases be allowed to communicate with one another by a narrow tube, or a porous membrane, a remarkable interchange rapidly takes place, ue,^ in direct opposition to the attraction of gravity the heavy gas ascends and the light gas descends until they become perfectly mixed in both vessels. Note.— The property of gaseous di^j^usion has a very intimate bearing upon the composition of the air. If either of the constituents of the air were to separate firom the mass, the extinction of life would soon follow. Beside^ were it not for the existence of this property, various vapours would accumulate in certain localities, as large cities, manufacturing dis- tricts, volcanic regions, &c., in such quantities as to render them totally uninhabitable. 210. In addition to the gases already mentioned, atmospheric air always contains more or less water in the form of invisible vapour. This is derived partly from combustion, respiration and decay, but chiefly from spon- taneous evaporation from the surface of the earth. The amoint of invisible vapour thus held in solution depends upon the temperature of the air being as high as ?V of the weight of the air in very hot weather, and as low as rb in cold, 211. The blue color of the sky is due to light that has suflfered polarization, and which is, therefore, reflected light, like the white light of the clouds. The air appears to absorb to a certain extent the red rays and yellow rays of solar light and to reflect the blue rays. In the higher regions the blue becomes deeper in color and is mixed with black. The golden tints of sunset depend upon the large amount of aqueous vapour held in solution by the air. 212. Air like all other material bodies possesses the properties of impenetrability, extension, inertia, porosity, compressibility, elasticity, t motion, imbrella. rfeotly calm ilatform of a )ythe body am, when it atmosphere, ns a greater leather, and natter, is sequcntly » of pneu- inches, and 1 air pump, ts weight is grains. t « « »pearstobe nosphere is irth to the :Iobe to the •ressible, )8sure of that the in the netrical imetical ) atmoB- height ith, ^0. preMure height at reoareAil AeT8. Slih210.] PNEUMATICS. 99 collection, by Biot and Arago, of the observations made on the Andes and in balloons, respecting the upward decrease of pressure and temperature, has led to the adoption of 8^ miles as the point at which we may aay that one- iK.rlf of the atmosphere is beneath us. HBIOHTIVMILB8. DBKBITT. UKIOHT, Iir IN., OF COLUMir OV MEBCU&Y. PBESSUKE IN LBS. TO THB 8Q. INCH. 2 '7 \ 1» 7-6 5-4 i 7'5 3-76 8 1 k 3 '75 1 -875 10-8 iV 1'876 •937 13-6 ■h •937 '468 16-2 ^ '468 '234 18 '9 ritt '234 -117 21 '6 th •317 •058 24 '3 Trti 058 '029 27 10^84' 029 014 29 '7 ro'48 014 '007 216. The pressure of the air is a necessary consequence of its weight, and is equal, at the level of the sea, to about 15 lbs. to the square inch. NoTB.— By saying that the pressure of the atmosphere is equal to 15 lbs. to the sq. inch, we mean that it is capable of balancing a column of mercury 30 inches in height, and a colmim of mercury 30 inches in height and having a sectional area of 1 sq. inch 'r^ighs 15 lbs. Or in other words, that a col- umn of air having a section&ii '^7ea of 1 sq. inch, and extending fh>m the level of the sea to the top of the atmosphere weighs 15 lbs. 217. Air at 60® F. is 810 times as light as water, and 10466 times as light as mercury. It follows that the pressure of the atmosphere is equal to that of a column of air of the same density as that at the surface of the earth 810 times 82 feet or 10466 times 30 inches in height. That is, if the air were throughout of the same density that it is at the level of the sea, it would extend to the height of about 5 miles* 218. The particles of elastic gases, unlike those of solids or liquids^ possess no cohesive attraction, but on the contrary a powerful repulsion, by means of which they tend to separate from one another as far as possible. 818* Pennanently elasUc fluids such as atmospheric ^ 100 PNEUMATICS. [ABTS. 220-221. '. ? ";■ ;||-; If ..I Fig. 20. air, and certain gases, are chiefly distinguished from non elastic fluids, such as water, by the possession of almost perfect elasticity and compressibility. Note.— Air and certain gases as Oxygen, Hydrogen, Nitrogen, &c., arc called permanently elastic to distinguish them from a number uf others as Carbonic Acid, Nitrous Oxide, &c., which under great pressure and intense cold pass first into the liquid and finally into the solid state. 220. If a liquid be placed in a cylinder under the piston it will remain at the same level, no matter to what height the piston may be raised above it, but if a portion of air or any other elastic gas be thus placed in the cylinder, and the piston be air tight, the confined air will expand upon raising the piston and will always fill the space beneath it, however great this may become. This expansibility or tendency to enlarge its volume so as to entirely fill the space in which it is inclosed is termed elasticity. Note.— It is obvious that the elasticity of air is due to the repulsive power possessed by the particles. 221. The law determining the density j^ and elasticity of gases under diflerent pres- sures was investigated by Boyle in 1660, and afterwards by Mariotte. Note.— To illustrate this law we take a bent glass tube Fig. 20, having one limb AC much longer than t ie other. The longer limb is open and the shorter fu*- nished with a stop-cock. Both ends being open a quantity of mercury is poured into the tube and of course rises to the same ' level in both legs— the surface of the mercury at A a, sustaining the weight of a column of air extending to the top of the atmosphere. We now close the stop* cock and thus shut off the pressure of the atmosphere above that point, so that the surface a, cannot be affected by the weight of the atmosphere— i. e., cannot be influenced by atmospheric pressure. We find, however, that the mercury in both limbs remains at the same level, fh)m which we infer that the elastic force of the air confined above a is equal to the weight of the whole column on a before the stop-cock was closed. Hence the elasticity of the air is equal to its weight, which is equal to a column of mercury SO inches h^h. If now we pour mercury into the tube until the air confined above a is compressed into half its former volume, i. «., until the mercury rises to 6 in the shorter tube, we shall find that the dolumn of mercury oBiM exactly 80 inoliei in length, or in other words, wo h«ve doubled the prenure on the 'H Ai»rs. 222-225.] PNEUMATICS. 101 <5 air confined in the shorter tube and have decreased its volume to one*half its former dimensions, and at the same time doubled its elastic force since it now reacts against the surface of the mercury with a force equal to 30 lbs. to the square inch. If we increase the height of the mercury in the longer leg to 60 inches above its height in the shorter leg, we shall compress the air into one-third its original volume and at the same time treble its elasticity, and so on. Henoe the law of Mariotte. 222. Mariotte's law may be thus enunciated. /. The density and elasticity of a gas vary directly as the pressure to which it is subjected. II, The volume which a gas occupies under different pressures varies inversely as the force of compression. NoTB.— Recent researches tend to prove that Marietta's law is true only within certain limits, and that all gases vary ft'om the law when subjected to very greatpressures, their density increasing in a greater ratio than their elasticiv^. With atmospheric air the law holds good to a far greater extent than with any other gas, the correspondence being found to be rigidly exact when the air is expanded to 300 volumes, and also when it is compressed into ^ of its primary volume. Mariotte's law would require the air to be indefinitely expansible while we know that there is, beyond all doubt, an upward limit to the atmos- phere. Dr. WoUaston imagines that when the particles of air are driven a certain distance apart by their mutual repulsive power, the weight of the individual particles comes at last to balance tnis repulsive force and thus prevent their further divergence. If this be the case as is probable fh>m various considerations, there is a limit »to the rarefac- tion of a gas, arriving at which the gas ceases to expand further and oomes to have a true upper surface like a liquid. As has oeen already remarked this exact limit and upper surfiu:e of the atmosphere is supposed to be at an elevation certainly not greater than 45 miles— Biot fixes it at 30 miles ; Bunsen and others place it at 200 miles. 223. The air-pump, as its name implies, is an instrument used for pumping out or exhausting the air from any closed vessel. 224. The bell-shaped glass vessel usually attached to the air-pump is called a Receivety and when the air is exhausted as far as practicable from this a vacuum is said to have been produced. NoTB.— The air pump was invented by Otto Guericke, a celebrated Burgomaster of Magdeburg, in the year 1560. At the close of the Imperial Diet in 1564. he exhibited nis first public experiments with it before the emperor and assembled princes and nobles of Germany. On this occasion he exhausted the air Arom two 12-inch hemispheres fittea together by ground edges, and greatlv astonished his noble audience by showing that the com- bined strength or 12 horses was insufficient to pull them asunder. The exhausting syringe of Otto Guericke was so imperfect in its action that while using it he was compelled to keep it immeraed in water to pre- vent the inward leakage of the air. Since his time, however, the attention of many eminent men has been directed to the subject, and the form and construction of the air-pump have been greatly improved. 226. The exhausting syringe "vthioXx is the essential part of an air-pump, consists of a brass cylinder ahcd^ supplied 102 PNEUMATICS. [ASTS.220.227. •*;;■! with an air-tiglit piston ef^ and an arrangement of valves h!k^ by means of which the air is permitted to pass out from the receiver q and through the piston ef^ but not in the contrary direction. NoTB.— Whon the piston 0/is raised the valve A closes, and as the piston in its ascent produces a partial vacuum be neath it, the air contained in the receiver ens the valve h by its expansive power and us refills the cylinder ahcd. Now when the piston is forced down again, the air contained Fig. 21. op th tin ^f / r\ '^ XL \m \\\ the cylinder tends to rush back into the re- ceiver but in doing so closes the valve A;, and has therefore no other mode of escape than through A, thus passing above the piston to be lifted out at the next stroke. In this manner the air continues to be exhausted until what remains in the receiver has not sufficient ex- pansive power to open the valve A;, when the exhaustion is said to be complete. 226. The principle upon which the air-pump acts is the elasticity or expansibility of the air, and since in order to enable the pump to act, the air contained in the receiver roust possess sufficient elastic force to raise the valve, it follows thm the receiver ; the remaining half completely filling it, but having only half as much density and elastidty as before. The second stroke of the puton will reduce the quantity, density, and elasticity, to one-fourth, the tmrd to one-eighth, and so on as exhibited by the following table:— Elastic force of the remainder. in. of mercury, or 7 'S5 lbs. per so. in* Stroke, Croes out. 1st, of 1 £nd. of * 3rd, of I 4th, of i 6th, of ^ 6th, of S 7th, of ^ 8th, of lU 0th, of ITS Left in Vuttl. E i 16 i 7* \ 3f tV 1*878 •h 0-987 ^ 0*468 rhv 234 £ 6 117 yif 068 «( «( H •« <( (( « <« 3*676 1*837 '918 '469 •229 '114 •067 028 «« is ] wit cii i i 1.220.227. Art. 228.] PNEUMATICS. 103 valves >as8 out not in lio piston •J //I acts is I order Bceiver alve, it by the ^ill not at the lave the I at each dng the bottom, ing half tioityas density. OUbited )r84.in. «« .« «( M «( M M >rcDgi m an exhausting syringe only in the fact that its valves open inward towards the chamber instead of outward. 228. The Air-pump is chiefly employed to illustrate the pressure and elasticity of the air. NoTB 1.— The pretture of the atmosphere may be shoMm by innumerable experiments among which are the following :-> I. When the air is ezhaosted fh)m the receiver of an air-pump the re oeiver is firmly fastened to the plate and cannot be removed until the air is re-admitted. II. The handjplaoed on the open end of the receiver is pressed inward with a force sufficiently great to cause pain. III. Thin square glais-tubes are crushed when the air is exhausted Arom them. IV. In the sui^cal operation of cuppina, the air is removed fh)m a.small cup which is then placed over an opened vein ; the pressure of the air on the surrounding parts causes the blood to flow rapidly into the cup. V. When a cask of beer is tapped, the beer does not run until a small hole called the vent'hole has been made in the upper part of the cask. TIirouj[h this the atmospheric air enters and pressing on the surfStoe of the beer with a force of 15 lbs. to the square inch, forces it through the tap. ^ VI. The usefiil small glass instruments called pipettes act upon the priu- ciple of atmospheric pressure. vll. A hole 18 usually made in the lid of a tea-pot so as to bring into play the pressure of the atmosphere and thus cause the beverage to flow more rapidly. VIII. Flies walk on glass or on the ceiling by producing a vacuum under each foot which is thus pressed against the surrace with a force sufficient to sustain the weight of the insect. The Greoko, a South American lizard, has a similar apparatus attached to each foot. And within the past few years a man has succeeded in walking across a ceiling with his head down- wards, by alternately withdrawing and admitting the air between his feet and the ceiling. IX. Pneumatic chemistry, i. «., the mode of collecting gases over water depends upon the principle of atmospheric pressure. X. If a tumbler or other glass vessel be filled with water and covered with a piece of paper, and the hand be then placed firmly on the paper and the whole suddenly and carefully inverted, the water does not now out of the vessel upon removing the hand— being held by the upward pressure of the atmosphere. XI. Suction is the effect of atmospheric pressure, as illustrated by drato- ing^liquids into the mouth, also by the leather sucker used by boys. XII. The pressure of the air is shown by the fwst that it supports or ba- lances a column of mercury 30 inches or a column of water 32 feet in height. XIII. The pressure of the atmosphere retards ebullition or boiling. Thus if some boiling water be partially cooled and then placed under the receiver of an air-pump and the air exhausted, the water recommences to boil, owing to the decreased pressure. Or if a flask containing boiling water be corked and the water be allowed to cool partially, upon plunging the flask in a large vessel of cold water, the water in the flask again begins to boil ; the reason is. the cold water condenses the vapour in the upper part of the flask and thus produces a partial vacuum. Note 2.--The elasticity of the air may be shown by various experiments among which are the following ^- I. The exhaustion of the receiver of the air-pump is a proof of the elasti- city of the idr. II. The elaitioity of the air is shown by placing a thin square bottle with its mouth closed, under the receiver, and exhausting the surrounding air, the bottle is brokMi by the elastic fero9 of the contuned air. 104 PNEUMATICS. [Abtb. 220-2S2. m. When some withered fruit, as apples, figs, or raisins, with unbroken skins are placed under the receiver, and the surrounding air exhausted, they become plump flrom the elasticity of the included air. Iv. The eUutieily of the air is shown by the operation of the air-gun. y . The elasticity of the air is taken advantage of in applying air as a stuflSng material for cushions, pillows, and beds. 229. The barometer (Greek haros " weight" and me^rco '^ I measure" is an instrument designed to measure the variations in the amount of atmospheric pressure. NoTB.— The barometer was invented about the middle of the seven' teenth century by Torricelli, a pupil of the celebrated Gkdileo. 230. The essential parts of a barometer are : — Ist. A well formed glass tube 33 or 34 inches long, closed at one end and having a bore equal throughout, of two or three lines in diameter. The tube contains pure mercury only, and is so arranged that the mercury is sup- ported in the tube by the pressure of the atmosphere ; and 2nd. An attached graduated scale and various appliances for protecting the tube and ascertaining the exact height of the column of mercury. NoTB.— The vacant space between the top of the column of mercury and the top of the tube is called the TorricelUan «acwMm. in honour of the inventor of the barometer, and in a Kood instrument is the most perfect vacuum that can be produced by mechanical means. 231. The excellency of a barometer depends principally upon the purity of the mercury in the tube aud the per- fectness of the Torricellian vacuum. The value of the instrument may be tested :— Ist. By the brightness of the column of mercury, and the absence of any Bpedc, &w, or dullness on its surface. 2nd. By the barometric light; i. e., flashes of electric light produced in the dark in the Torricellian vacuum by the firiction of the mercury against the glass. 8rd. By the clearness of the ring or clicking sound produced by making the mercury strike the top of the tube, and which is greatly modified when any jwrticles of air are present above the column. 232. The cause of all the oscillations in the barometer is to be found in the unequal and constantly varying dis- tribution of heat over the earth's surface. If tLe air is much heated at any spot it expands, rises above the mass of air, and rests upon the colder portions surrounding it. The ascended air consequently flows off laterally from above, the pressure of the air is decreased in the warmer place and the barometer falls. In the colder surrounding TS. 229-2S2. Arts. 283, 234.] PNEUMATICS. 105 1 unbroken exhausted, air-gun. ngidr as a d metreo sure the the seven* es long, hout, of OS pure Y is sup- re ; and pliances height wnuyand ur of the It perfect icipally be peiv Be of any ducedin y against T making modified Dmeter Dg dis- I air is B mass ing it. ' from armer mding places, however, the barometer rises, because the air that ascended in the warmer regions is diffused over and presses upon the atmosphere of these cooler parts. NoTB.— It is found that the fluctuations in the height of the barometer vary greatly in extent in different latitudes— being so small in tropical regions as almost to escape notice, and comparatively so fitftd and extreme in the temperate and frigid zones as to defy all attempts at reducing them to any system. In our climate the column varies in height from a little over 30 Inches as a maximum to a little over 27 inches as a minimum. Within the torrid zone the column of mercury scarcely ever exhibits any disturbance greater t^n what would occur in Canada before a slight thun- der storm— but such a disturbance is there the sure and rapid precursor of one of those mighty atmospheric convulsions which sometimes desolate vast regions and which are frequently as disastrous in their effects as the most violent earthquakes. 233. Besides the irregular fluctuations depending upon the weather,the barometer hsahjeot to Tegular semi-diurnal oscillations depending upon atmospheric tides, caused by the heat of the sun — the two maxima of pressure always occurring at about a.m. and 9 p.m. and the two minima at about 3 a.m. and 3 p.m. NoTB.— The semi-diuiTial oscillLtion is greatest at the equator, where it averages one-tenth of an inch— diminishing to Hx-htindrMtht of an inch in lat. 80°, beyond which it still decreases, and in our climate becomes completely masked by the irregular fluctuations peculiar to the temperate and flrigid zones. 234. USE OF THE BAROMETER AS A WEATHER-QLASS. I. The state of the weather to be expected depends not so much upon the absolute height of the column of mercury as upon the RAPIDITY AND BXTBNT OF ITS MOTION whether rising or falling, NoTB.— If themeroiuryhavea convex surfttce the column is rising; if the surfice is concave the column is falling ; when the surfue is flat the column is usually changing from one of these states to the other. II. Jifall in the barometer generally indicates approaching rain^ high windSf or a thunder storm. III. A rise in the mercury commonly indicates the approach of fine weaiher ; sometimes^ however^ it indicates the approach of a mow storm. IV. A rapid rise or fall in the mercury indicates a sudden change of weather. V. Ji steady rise in the column^ continued for two or three days, is generally followed by a long contintumce of fine settled weather, VI. A steady fall in the columny continued for two or three days^ is commonly followed by along continuance of rainy weather. VII. A fluctuating state in the height of the mercury coincides with unsettled weather, NoTB.— The barometer is flu* more valuable as a means of asoertaiuing approaching changes in the state of the wind than in foretelling the ap- proach of wet or dry weaker. 106 PNEUMATICS. i? CAbT.285. ABI.! ^'•1: 4 h '\ :!■■ 235. To ascertain the height of mountains, ^c, by the barometer. hallbt's bulb. I. Find the logarithm corresporulir^ to the number which ex- presses the height in inches of the column of mercury in the baro- meter at the level of the sea. II. Find also the logarithm corresponding to the number which expresses in inches the height of the column in the barometer at the top of the mountain or other given elevation. III. Subtract the latter of these logarithms from the former^ multiply the remainder by the constant number ^ 621*70, and the result will be the elevation in English feet. NOTB.— Xh^ number 08170 in this rule, and 63946 in the following, were selected by Halley for certain mathematical reasons into which it is unnecessary to enter. ExAMPLB 281. — On the top of a certain mountain the barometer stands at the height of 21*793 inches, while on the surface of the earth it stands at 29*780 inches ; required the height of the mountain. SOLUTIOir. Logarithm of 29*780 = 1*473985 and logarithm of 21*793 = 1*328317. Then from 1*473926 Subtract 1*828317 Bemainderrs *146608X 62170 =9052 feet. Ans. BULB WITH COBBBOTION FOB TBMPBBATUBB. I. Obtain^ as before, the difference between the logarithms of the numbers expressing the heights at tphich the mercury stands at the surface of the earth and on the summit of the mountain. II. MtUtytly this difference by the constant number, 63946 — the result is the elevation in feet, \fthe mean temperature of the surface of the earth and the elevation is 69*68^ Fahr. III. If the mean temperature of the two elevations be not 69*68° Fahr., add ^^ of the whole weight found for eacn degree above 69*68° or subtract the same quantity }f the mean temperature be below. ExAMPLB 282. — Humboldt found that at the level of the sea, near the foot of Ohimborazo,«the mercury stood at the height of 30 inches, while at the summit of the mountain it was only 14*85 inches. At the same time the temperature at the base of the mountain was 8*7° Fahr., and at the top 50*40° Fahr. What is the height of Ohimboraao ? BOLUTIOH. Log. of 80 = 1*477121, log. of 14*85 =1*171724 and mean temperature :^ 87lf60;4^_l37:£_gg^^ Then 1*477121 —:i*l7l724=*305397. And '305397 X 63946 = 19639 feetf TTff 286. [Abt. 286. fc, by the which eX' I the baro' iber which leter at the le former J 0, and the owing, were wrhioh it is barometer surface of ifht of the J17. t. Atu, 'ritkms of itands at lin. 3946— Me he surface iot 69*68° ree above \rature be ' the sea, height of nly 14-85 le of the What is erature=-; ABT. 386.3 PNEUMATICS. 107 Since the mean temperature of the two stationa is V* lees than 60 68, what is the eleyation of the balloon ? Jilts. 12703 feet. 236. The common pump consists of a barrel SB, a tube ASy which descends into the water reservoir, a piston erf, moving air-tight in the barrel and two valves, v and x^ which act in the same manner as in the exhausting syringe of the air pump. NoTB 1.— When the machine begins to act the piston is raised and pro* duces a vacuum below it in the barrel, and the atmospheric pressure on the water hi the reservoir forces it up the tube and through the valve x into the lower part of the barrel. As the piston descends the valve x closes and the water contained in the barrel passes through the valve v above the piston, to be lifted out at the next stroke. Hence the common pump is sometimes oaUed a lifting pump. Note 2.~Since the specific gravity of mercury is 13*696 and the pressure of the atmosphere sustains a column of mercury, 30 inches in height— it follows that atmospheric pressure will sustain a column of water 30 X 13*596 inches, or jS4 ft. in height. Hence the vertical distance of the valve x above thesurfaceofthewaterinthereservoirmustbeless Fig. 23 than 34 feet,or takingthe variations inatmospheric pressure into account, about 32 feet. 237. The forcing pump consists of a suction pump A, in which the piston P is a solid plug without a valve. When the piston P descends the valve V closes and the water is forced through the valve v' into the chamber MN, The upper part of this chamber is filled with compressed air, which, by the pressure it exerts against the surface of the water, ww^ drives it with considerable force through the pipe or tube HG, NoTB.— Sometimes the forcing pump is used without the air chamber, ^fiV. Fig. 23 eznibits the arrangement of the valves, Ac., in a common fire engine with the exception that .there is another similar forcing pump on the other side of the air chamber. MO represents the tube leading to the hose. 238. The Syphon is a bent tube of glass or other material having one leg somewhat longer than the other and is used for transferring liquids from one vessel to another. AATs.: NOTl ingthe sucking the atn thebei suokini be set and, w audim ;A&T6. 236« 238. nperatiire of the balloon ? s, 12703 feet. SB, a tube ft piston erf, 3, V and a;, ing syringe aised and pro* ic pressure on ;h the valve x I valve a? closes valve V above K)mmon pump i the pressure in height— it ber 80 X 13'596 I valve a; above ;, 23.. A&TS. 239-241.] DYNAMICS* or other the other vessel to fig. 24. 109 Note.— The machine is set in operation by imme'st ing the shorter leg in the liquid to be decanted* a>nd sucking the air out of the tube, when the pressui'^ of the atmosphere force i the liquid into the syphon over the bend and down through the longer leg. Instead of sucking the air out of the syphon, the instrument may be set in operation by first filling it with the liquid and, while thus full, placing the finger over each end, and immersing the shorter leg in the liquid. NoTB 2.— In order to understand why one limb must be shorter than the other, it is only necessary to remember that the pressure of the atmosphere acts as much at one extremity as at the other. If we raise the column of liquid as fkr as J&, by sucking at the ex- tremity C, and then withdraw the mouth, the water falls back into the vessel !F. The column will likewise run back if we get it no farther than L, which is the level of the water in the vessel F, because at that point the upward pressure of the atmosphere prevails over the downwarapreasare of the limiid, but if we get the column below L, the downward pressure of the liquid exceeds the upward pressure of the atmosphere and the liquid will flow. Thus the motion of the fluid in the syphon is similar to the motion of a chain hanging over a pulley.— if the two parts of the chain be equal, the fluid remains at rest, but if one end be longer than the other, it moves in the direction of the longer, and flresh linu, so to speak, are added con- tinuously to the fluid chain by the atmospheric pressure exerted on the surface of the water. CHAPTER VII. t\.. DYNAMICS. 239. When the forces which are the subject of inves- tigation are balanced^ the consideration of them properly comes under the science of Statics^ but when they cease to be balanced, and the body acted upon is set in motion other principles become involved, and the investigation of these constitutes the more complex science of Dynamics, 240. Statics is a deductive science, since all its facts are deducible, like those of Arithmetic and Geometry from abstract truths ; dynamics is an inductive, experimental or physical science, many of its principles being capable of proof only by an appeal to the laws of nature. 241. Force may be defined to be the cause of the change of motion, i. e., force is required : — 1st. To change the state of a body from rest to motion or from motion to rest* 2nd. To change the velocity of motion. 3rd, To change the direction of motion, r-f-..- ■I ■: I: V::- ii 110 DYNAMICS. tAilTS. 24^250* 242. Fol^ces are either instantaneous or continued, and continued forces are either accelerating, constant or retard- ing, 243. Motion may be defined to be the opposite of rest or a continuous changing of place. 244. Motion has two qualities, direction and velocity, and is of three kinds. 1st. Direct, 2nd. Rotatory or Circular; and 3rd. Vibratory or Oscillatory. 246. An accelerating, constant or retarding force pro- duces an accelerated, uniform or retarded motion. 246. Velocity is the degree of speed in the motion of a body and may be either uniform or varied. It is w»(/brm when all equal spaces great or small, are passed over in equal times. 247. The principles of the composition and resolution oi force are equally applicable to motion. 248. Momentum or Motal Force or Quantity of Motion is the force exerted by a mass of matter in motion. 249. The momenta of a body depends upon its weight and velocity, thus : I. When the velocities of two moving bodies are equal, their momenta are proportional to their masses. II. When the masses of two moving bodies are equal, their momenta are proportional to their velocities. III. When neither the masses nor velocities of two moving bodies are equal, their momenta are in proportion to the products of their weights by their velocities. NoTB.—'WIien we speak of multiplying a velocity by a weight, we refer to multiplying the number of units of weight by the number qf unite of velooity* and it makes no difference what units of each kind are employed for the product, thus obtained, means nothing by itself, but only by com- parison with other products similarly obtained by the use of the same units. Forexample, when we say that a weight of U lbs. moring 6 feet per second, has a momentum of 66, all we mean is, that in this case the weight strikes a body at rest with 66 times the force that a body weighing one lb. and moving only one foot per second would exert. 250. If a moving body M, having a velocity V, strike another m at rest, so that the two masses shall coalesce, and move on together with a velocity v, then M X V=» 21 anot velo< toge (M- have impi 2 [Arts. 24^260* AllTS.251,253.] DYNAMICS. Ill \tinuedj and t or retard- »site of rest nd velocity^ force pro- motion of is uniform ied over in resolution tantity of in motion. ts weight are equal, re equal, s. 8 of two •roportion es. ht,we refer qfwnita of re employed ily by com- if the same ■ per second, ight strikes one lb. and F, strike coalesce, X F« (M + f?i) X w ; or whatever momentum may be acquired by the body m must be lost by JIT. 251. If a moving body M^ having a velocity Vy strike another body m moving in the same direction with a velocity v, so that the two may coalesce, and move on together with a velocity vel, — then My, V-^-my. v= (M-\'m)yvel, or in other words, the two bodies united have the same momentum that they separately had before impact. 262. If a moving body M having a velocity F, strike another body m moving with a velocity v, in the opposite direction, so that the two masses shall coalesce and move on together with a velocity vel — then M X Vr^ fn X f =» (if -+• m) X vel or in other words the body moving with least force will destroy as much of the monientum of the other as is equal to its own momentum. 263. If a moving body M^ having a velocity F, strike another body m moving obliquely towards it with a velocity v,80 that the two masses shall coalesce and move on together, then by representing their momenta, just before impact by lines in the direction of their motion and completing the parallelogram, the diagonal will represent the quanti- ty and direction of the momentum of the combined mass. ExAMPLi 239. — ^What ia the momentum of a body weighing 78 lbs. and moving with a velocity of 20 feet per second ? SOLUTIOir. Momentum =^7SX 20 =16604 Am, That is, the momentum of such a body is 1660 times as great as the mo* mentum of a body weighing only 1 lb., and moving only 1 ft. per second. Example 290.— If a body weighing 67 lbs. be moving with the velocity of 11 feet per second and strike a second body at rest weighing 33 lbs., so that the two bodies may coalesce, and more on together, what will be the velocity of the united mass ? BOLUTIOir. • Art. 260.— If If be the movins body, V its yelooity, m the body at rest and V the velocity of the uuitea mass ;— Then(ir+f») X f> = if XF and therefore i>= ^ In this example, ilf = 67* F= 11 and m =: 33. M-i-m Then v — JlfXF_ 67X11 787 if+m"* 67 + 83 = ,-— =:7*37 feet per second. -An*. IvO fc*l MH. ■!^ 112 DWAMlCS* [AET.26S. ExAMPL«291.— If a body weighing 50 lbs., and moving with a velocity of 100 ft. per second, come in contact with another body weighing 40 lbs. and moving in the same direction with a velocity of 20 feet per second, so that the two bodies coalesce and move on together, what will be the velocity and momen- tum of the united mass ? ' ; • ^ - . y. SOLVTIOir. Art.251.— If 3fand mhe the two bodies, and V aud« their separate velocities and vel the velocity of the united mass :—• • Then (3f + m)Xvel=MXV+mXv, Hence vel = j^t^T^ — In this example M= 50, m = 40, V= 100 and v — 20. T7- ?_ MXV+mXv __ 60X100-f40X20 _ 50004-800 __ 6800 men Vel-~ lf+« 60 + 40 90 "" lo =: 04^ fb. per second, and momentum := (50+40) X 64^ = 5800. Ans. ExAUPLB 292.— If a body weighing 120 lbs., and moving to the east with a velocity of 40 feet per second, come into contact with a second body weighing 90 lbs., and moving to the west, with a speed of 80 feet per second, so that the two bodies coal- esce and move onward together, in what direction will they move, with what velocity, and what will be their momentum 1 80LTTTI0K. Prom Art. 252, if M and m be the bodies, and Fand v their respective velocities, and'wl the velocity of the united mass after impact :— Then (If +m) X vel = Jf X Vn^rnXv and hence , _ MX F/-'»X« vel = =>-T In this example Jf=: 120, «»= 90, Fs 40 and «= 80. MxVr-'tnXv ^ (120 X 40) /^ (90 X 80) _ 4800/^7200 *^ 120 + 90 "" Then vel = Jkf + m 210 2400 = ^ =aif feet per second = the velocity, llf X (120+90) = llf X 210 = 2400 = momentum. And since 90 X 80, the momentum of the body moving to the west is greater than 120 X 40, the momentum of the body moving to the east, the united mass moves to the west. IXSBOISI. 293. What is the momentum of a body weighing "79 lbs. moving with a velocity of 64 feet per second ? Jina, 5056. 294. Which would strike an object with greatest force, a bullet weighing one ounce and propelled with a velocity of 2000 feet per second, or a ball weighing 6 lbs. and thrown with a velocity of 28 feet per second 7 jim. Momentum of bullet = 125. .• ^ . " of ball =140. Therefore the ball would exert most force of impact. Abt. 295. CAbt.263. Art. 253.] DYNAMICS. 113 oving with ith another iction with les coalesce ad momen- leir separate M-\-m -800 6800 90 0. Ans, moving to ito contact ;o the west| odies coal- 1 will they mentam 1 ir respectiVd t:— 4800 /^yaoo 210 BO) = llf X the west is ihe east, the }S. moving ^ns. 5056. e, a ballet relocity of nd thrown llet=125. 1 = 140. luld exert 295. Which has the greatest momentum, a train of cars weighing 1*70 tons and moving at the rate of 40 miles per hour, or a steamer weighing 790 tons and moving at the rate of 9 miles per hour ? ^ns. Momentum of train =6800, of steamers 7110, and therefore the latter has most mo- mentum. 296. If a body weighing 60 lbs. and moving at the rate of 86 feet per second, come in contact with another body weighing 400 lbs., and moving in the same direction at the rate of 12 feet per second, so that the two bodies coalesce and move on together ; what will be the velocity and momentum of the united mass ? ./f?n«o Velocity = 21^ feet per second; momentum r= 9960. 297. If a body weighing 56 lbs. and moving with a velocity of 80 feet per second come in contact with a body at rest, weighing 70 lbs., so that the two bodies coalesce and move on together; what will be the velocity of the united mass ? OP SECONDS. SPACB PASSED OYER BACH SECOND. TERMINAL VELOCITIES. TOTAL SPACB. 1 1 a 1 2 8 4 4 3 5 6 9 4 1 8 16 5 9 10 26 6 11 12 86 7 13 14 49 8 15 16 64 9 17 18 81 10 19 20 100 Note.— The numbers iu the second, third and fourth columns means so many times 16 feet. From this it is evident that : — I. The spaces through which the body descends in equally succeS' sive portions of time increase as the oddnumhersy 1, 3, 5, 7, 9, ^c, and hence the space through which the body falls during any second of its flight f is found by multiplying 16 feet by the odd number which corresponds to that second ; i» e., one less than twice the number of the second, II. The final velocity acquired by a falling body at th\. end of successive equal portions of time^ varies as the even numberSy 2, 4, 6, 8, Sfc.j and hence the final velocity acquired by a body at the end of any second of its fall ^ is found by multiplying 16 feet by twice the number of seconds, III. 7%e whole space passed over by a body falling during equal successive portions of time^ varies as the square of the numbers ly 2, 3, 4| ifc.y and hence the whole space parsed over during any given number of seconds^ is found by multiplying 16 feet by the square of the number of seconds, 266. Let f= the time of descent in seconds, v = the terminal velocity, i.0., the velocity acquired at the end of the last second of its fall,«= whole space passed over, and g = 82, i.0., the measure of the attraction of gravity. Then Art. 263, the time is equal to the space divided by half the terminal velocity, or t=-r— = — , ^ V ITS. 265» 266. DT. Abts. 267, 268.] DYNAMICS. 117 1 SPAOS. 1 4 9 16 25 36 19 S4 31 )0 a means so ly succes' ny second d number twice the iL end of ers, 2, 4, ty at the 5 feet by ^fig equal nuinhers ring any t by the I velocity, I fall,tf= raction of I terminal Affain (Art. 265, III) the whole space passed over is equal to 16. t, «., half of thu gravity, g, multiplied by the square of the time or »=\gt*. Also (Art. 265, 1) the terminal velocity is equal to 16, «. e., }ai multiplied by twice the time or v =:^ X 2^ ^=gU These three formulas, viz : « = igt*, v=:gt and < =— are fundamental V and the remaining six of the following table are derived fh>m them by transposition and substitution :— TABLE OF FORMULAS FOR DESCENT OF BODIES FALLING FREELY THROUGH SPACE. iro. QITBV, TO PIND. 70BMULAS. WHENCE DEBITED. I II III IV V VI VII VIII IX ^ g Vi g t, V S s = \gt^ Jlrt. 265, IIL 1 From formula V. | From formula VII. ST, t fir, < », t V i>=V2gr». _2s ''"■ t Art, 265, J. From IV and VII by sub- stituting the value of t. From formula VII. ■0, g », g t 2s V t = V 2s 'sr Art. 263. * From formula IV. From formula I. 267. When a body is thrown vertically upward it rises with a regularly retarded motion, losing 32 feet of its original velocity every second, and it occupies as much time in rising as it would have required in falling to ac- quire its initial velocity. If a body be projected upwards or downwards with a given initial velocity F, and is at the same time acted ui)on by the force of gravity, then when the body descends, in t seconds the initial velocity alone would carry it through Vt feet, and gravity alone would carry it tihrough ^gt feet, therefore together they carry it tmrough Vt+hfft* f^U and the terminal velocity willevidently be V+tg, 118 DYNAMICS. [Abt. ^68. I AbtJ t ,f 'fr — 5f< = 32 X 9 = 288 feet per second. Ans. ' • Example 304. — What will be the terminal velocity of a fall- ing body at the end of the 25th second of its fall, also at the end of the 33rd second ? * \ SOLUTION. Formula IV. « = flr^ = 32 X 26 = 800 feet per second at end of 26th second. « = gr^ = 32 X 33 = 1056 feet per second at end of 83rd " Example 305. — Through how many feet will a body fall during 5 seconds 7 ' SOLUTION. Formula I. 8 = ^gti = i X 32 X S* =16 X 25 = 400 feet. Ans. Example 306. — Through how many feet will a body fall in 12 seconds ? SOLUTION. Formula I. « = yt* = i X 82 X 12* = 16 X 144 = 2304 feet. Ans. Example 307. — If a body has fallen untilit has acquired a terminal velocity of 400 feet per second, what is the whole space through which it has descended ? molution. 160000 V* 400* Formula II. •=- = 2^ = E a tei E a tei Fc 64 = 2500 feet. Ans. ■ W CAET.268. I Abt.266.] DYNAMICS. 119 ild carry it in WiVity would e ascent will y fall dur- 16=21X16 y fall dur- mt? fa falling of a fall- so at the >th second. I3rd « ^odj fall yfall in quired a le space Example 308.— -How long must a body fall in order to acquire a terminal velocity of 1000 feet ? il '!,,' i '. .■Kij V SOLUTION. 31i seconds. Ans. Formula VIII. t = ~= ^^ g 82 Example 309. — How long must a body fall in order to acquire a terminal Telocity of 8000 feet per second ? V 8000 SOLUTION. Formula VIII. t = — = -^^= 260 seconds. Ans. g 82 Example 310.— What time does a body require to fall through 11200 feet? SOLUTION. Formula IX. <=^ ^ =^ ?>^=^^00" = 26*45 seconds. Ans. Example Sll.—When a body has descended through 4400 feet, what velocity has it acquired ? SOLUTION. Formula V. tr = J 2g8 = V2X82X4400= a/ 281600 = ^'® '®®* P®' second. » » v Example 312. — ^If an arrow be shot vertically upwa»rda and reach the ground again after the lapse of 20 seconds, to what height did it rise ? SOLUTION. From Art. 267 it apnears that the arrow will be as long ascending as de- scending, and hence the problem is reduced to finding the distance through which tne arrow will fall in half of 20 seconds, i. e., in 10 seconds. Then formula I. » = igt* = iX32X10* = 16X100 = 1600 feet. Ans. Example 313. — If a cannon ball be fired vertically with an in- itial velocity of 1600 feet per second to what height will it rise ? SOLUTION. First, the time it ascends is equal to the time it would require if descend- ing to acquire a terminal velocity of 1600 feet. 1) 1600 By formula VIII. t = — = = 60 seconds = time of ascent. g '62, Then formula XI. s^Vt — ^gt^ = 1600x50 — iX32X602 = 80000 — 16 X 2500 = 80UUO — 40000 = 40000 feet. Ans. Ejl&mplb 314. — If a bovly be shot upward with an initial velocity of 1200 feet per second, at what height will it be at the end of the 10th second, and also at the end of the 70th second of its flight ? SOLUTION. Formula XI. « = Fi? — * gt» = 1200 X 10 — iX32X10« = 12000 — 1600 = 1040O feet = elevation at end of 10th second. Also 1200X70 — iX32X70« = 84000 — 16X4900=84000— 784000 = 5600 feet = elevation at end of the 70th second. 120 DYNAMICS. Cast. ^. I AktI m r; ■ * ■ if Example 315. — If a cannon ball be fired vertically with an initial velocity of 2400 feet per second : — 1st. In how many seconds will it again reach the ground ? 2nd. How far will it rise? 3rd. Where will it be at the end of the 40th second ? 4th. What will be its terminal velocity ? 5th. In what other moment of its flight will it have the same velocity as at the end of the 19th second of its ascent ? SOLUTION. Since the initial velocity =: terminal velocity =-. 2400 ft. I. Formula VIII. time of ascent =: — = ——-=76 seconds, and since fl' 82 it is as long ascending as descending, it again reaches the ground m 160 sec. II. Formula I. « = i fl^^* = i X 82 X 75* = 16 X 5625 = 90000 ft. — height to which it rises. III. Formula XI. «= Vt — igt^ = 24C0X40 — iX32X402 = 96000— 16X 1600 = 96000 — 25600 = 70400 ft. = elevation at end of 40th second. IV. Terminal velocity = initial velocity — 2400 feet per second. V. Since the whole time of flight = 160 seconds, and! since at all equal spaces of time from the moment it ceases to ascend and begins to descend, the velocity is the same in rising as in falling, it follows that the moment in which the body has the same velocity as at the end of the 19th second of its ascent is 19 full seconds before it again reaches the ground, or in 150^ 19 = 181st second, i. e., in the end of the 131st second. ExAMPLB 316. — If a body is thrown downwards from an eleva- tion with an initial velocity of *I0 feet per second, how far will it descend in 27 seconds 7 SOLUTIOir. Formula X. s =rt + ifl^* = 70 X 27 + i X 32 X 27« = 1890 + 16 X 729 =^ 1890 + 11664 = 18554 ft. Ans. ExAMPLB 31 7. — If a body is thrown down from an elevation with an initial velocity of 140 ft. per second, what will be its velocity at the end of the 30th second ? SOLUTION. V = r+tg = 140+30X32 = 140+960= 1100 feet per second. Ans. Example 318.— If a body be projected vertically with an ini- tial velocity of 400 feet per second, what will be its velocity at the end of the 12th second? SOLUTION. Formula XIII. v = F—^flc=400— 12X82=400— 384=16 feet per second. Ana* Example 319. — If a cannon ball be flred vertically upwards with an initial velocity of 1800 feet per second : — 1st. In how many seconds will it again reach the ground? 2nd. What will be its terminal velocity ? Srd. How far will it rise ? 4th. Where will it be at the end of the 90th second ? 6th. In what other moment of its flight will it have the same velocity as at the end of the 27th second of its ascent ? 11 III E is 4 trai reql tAsT. 268. llj with an le ground ? nd? ' "■e the same its ascent ? s, and since din 160 sec. ft. = height ►6000 —lex ond. id. t all equal to descend, le moment h second of or in 160— an eleva- r how far f 16X elevation 11 be its h an ini' locity at ^nd. Anst upwards und? AM. 268.1 V. DYNAMICS; SOLUTION. 121 V I. <=— = ff 1800 32 : 66i = time of ascent or descent, hence whole time of flight = 66J X 2 = 112i seconds. II. Terminal velocity = initial velocity = 1800 feet per second. III. Formula 1. S = i flr <« = i X 32 X (56i) 2 = 16 X 3164-0626 = 50625 ft. IV. Formula XI. S= Vt-^ t* =1800 X 90— i X 32 X 902 == 162000 — 16 X 8100=162000—129600 =32400 ft. = elevation at end of the 90th second. V. 112i— 27 =854 = middle of 86th second of flight. ExAMPLB 320. — A stone is dropt into the shaft of a mine and is heard to strike the bottom in 9 seconds ; allowing sound to travel at the rate of 1142 ft. per second, and taking gr= 32^ ; required the depth of the shaft. SOLUTIOir. Let X = time stone takes to fall. Then (9— ;r) =time sound takes to reach the top and a?2 x 16i^ = depth of shaft =(9— ar) X 1142 feet. „, , 193a?2 Therefore — ^^ — =1028 — 11420?. 193ar« + 13704a? = 128336. , 148996aj« + 10679488a? + 187799616 = 95216392 + 187799616 = 283016008. *' '886»-i- 13704 = 16823 + 386aj = 3119 x = 8*0803 = number of seconds body was falling. 9— « = 9—8*0803 = '9197 = time sound travelled. And 1142 X '9197 = 1060-2974 feet = depth of shaft. EXA.MPLB 321. — A body has fallen through m feet when another body begins to fall at a point n feet below it ; required the distance the latter body will fall before it is passed by the former? PIEST SOLUTION. Atendofmft.f= 1^ jj.mdv=:gt=^gj^ =V2^andsince n = distance to be traversed t = ,p=, hence S = \gt^ = y X '\J2mg 7=)2=ifyx-f— = ^. Am. le same nt? 122 DYNAMICS. [AST. 268. AST. SBCOITD SOLUTION. fife \2S |2r U2m-\rX-{-x) Let a? = distance. Then (of 2nd = body) ^ = |— ■ = | — and I ~ = entire time taken by the first body to pass through whole space. Then l^SEi- 1^=1^ and multiplying all by V^ N N9 N9 V2(M+n+a;)— Viw =V2^ V2(i»+w+a?)=V2jp +V2w,and squaring. .1 2(w+M+a?) = 2a? + 2m + 2,'\l\ynx, ,. .' 2m* + 2» + 2a?=2a? + 2i» + %'4'kinx. .• , .; » = 2V«M?. .. ' • ^ • ^^ m8 =:4mx, ■!■-':_ »2 *=4^- ^'»*' t .u ' » M . ir\.' ■.■:.< ♦exercise. 322. Through how many feet will a body fall during the STth second of its descent? ^^ns. 1168 ft. 323. Through what space will a body descend in 25 seconds ? Jlns. 10000 ft. 324. With what velocity does a body more at the close of the 20th second of its fall ? jins. 640 ft. per sec. 325. During how many seconds must a body fall in order to ac- quire a terminal velocity of 1100 ft. per sec. ? Ans, 34| sec. 326. Through what space must a falling body pass before it ac- quires a terminal velocity of 1700 ft. per sec. ? Jlns. 45156i ft. 327. "What will be the terminal velocity of a body thai has fallen through 25000 ft. ? Jns. 1264-8 ft. 328. If a body is projected upwards with an initial velocity of 6000 ft. per second, where will it be at the end of the 40th second? ^n«. At an elevation of 214400ft. 329. If a body be thrown downward with an initial velocity of 120 ft. per second, through how many feet will it fall in 32 seconds? ^n«. 20224ft. 330. A cannon ball is fired vertically, with an initial velocity of 1936 per second : — * In all cases, when not otiierwise dlroctodt use d^ = 32 ft. Art. 268.3 DYNAMICS* 123 1st. How far will it rise ? i .; 2nd. Where will it be at the end of the 6th second? 3rd. In how many seconds will it again reach the ground ? 4th. What will be its terminal velocity ? 5th: In what other moment of its flight will it have the same velocity as at the end o^the 13th second of its ascent ? Ans. 1st. 58564 ft. -' 2nd. At an elevation of 11040 ft. 3rd. 121 seconds. 4th. 1936 ft. per second. 5th. Atendof 108th sec. of flight. 331. If a body be projected vertically with an initial velocity of 4000 feet per second, taking gravity to 32^ feet : — 1st. How high will the body rise ? 2nd. Where will it be at the end of the 50th second ? 3rd. Where will it be at the end of the 100th second ? 4th. Where will it be at the end of the 200th second ? 5th. In what time will it again reach the gound? Ans, 1st. 248704-66 ft. 2nd. At an elevation of 159791-66 ft. 3rd. « 239166-66 ft. 4th, " 156666-66 ft. 5th. 248*70 seconds. 332. If a cannon ball be fired vertically With an initial velocity of 1100 feet per second, what will be its velocity at the end of the 7th second, at the end of the 20th second, and at the end of the 33rd second ? Ans, End of tth sec. vel. r= 876 ft. ', « 20th " =460 ft. ' " 33rd " = 44 ft. 333. If a stone be dropped into a well and is seen to strike the water after the lapse of 5 seconds how deep is the well ? Ans. 400 ft. 334. If a stone be thrown downward with an initial velocity of 250 ft. per second, what will be its velocity at the end of the 3rd, the 9th, the 30th, and the 90th seconds of its des- cent ? Ans. End of 3rd sec. vel. = 346 ft. per sec. " 9th " = 538 ft. « " 80th " =1210 ft. " « 90th " = 3130 ft. " 335. A stone is dropt into the shaft of a mine and is hewrd to strike the bottom in 12-76 seconds, assuming that sound travels at the rate of 1100 ft. per second, what is the depth of the mine ? Ans, 1936 ft. 124 DieBCENT ON INCLINED PLANES. CAet8. 209-271. I Aets 336. A body has fallen through 400 feet, when another body- begins to fall at a point 2500 feet below it ; through what space will the latter body fall before the former overtakes it ? Jins. 3906i feet. 337. A body Jl has fallen during m seconds, when another body B begins to fall,/ feet below it ; in what time will *A over- take 5? / 2| dra^ whal in t| DESCENT ON INCLINED PLANES. 269. When a body is descending an inclined plane a portion of the gravity of the body is expended in pressure on the plane and the remainder in accelerating the motion of the descending body. . n . : • 270. The following are the laws of the descent of bodies on inclined planes : — J. The pressure 07i the inclined plane is to the weight of the body as the base of the plav€ is to its length, II. The terminal velodity of the descending body is that which it would have acquired in falling freely through a distance equal to the height of the plane, III. The space parsed through by a body falling freely ^ is to that gone over an inclined plane j in equal times, as the length of the plane is to its height. IV. If a body which has descended an inclined plane meets at the foot of it another inclined plane of equal altitude, it will ascend this plane with the velocity acquired in coming dovm, the former, it will then descend the second and re-ascend the former plane, and will thus continue oscillating down one plane and up the other. NoTB.— The same takes place if the motion be made in a curve instead of on an incliued plane. In practice, however, the resistance of the atmos- phere and ft'iction retard the motion \'c I V an^eatly at each oscillation and very soon bring the body to a state oi xi:ii^. 271. The final velocity, neglecting friction, on arriving at the bottom of the plane is dependent solely on the height of the plane, and will be the same for all planes of equal height, however various may be their lengths and the times of descent are exactly proportional to the lengths of the planes. h- BT8. 269-271. ther body »ugh what overtakes {906i feet. )ther body 11 A over- Aets. 272-275.] DESCENT ON INCLINED PLANES. 125 s. f 32m plane a pressure 3 motion )f bodies ^t of the which it equal to is to that thofthe its at the U ascend former^ unCf and ier. > instead le atmos- tiou and rriving >n the mes of tis and )ngths 272. If in a vertical semicircle any number of cords be drawn from any points Fig. 25. whatever and A\ meeting in the lowest point of the semicircle, and a number of bodies be allowed to start along these cords at the same instant they will all arrive at the bottom at the same instant, and at every instant of their descent they will all be in the cir- cumference of a smaller circle. Thus in the accompanying fifnire if ADP be a semicircle and BP, CP, DP, EP, FP, any cords, and balls be allowed to start simultaneously firom A, B, C, 2>, E, and F, they will all arrive at P at the same instant. At the end of one-fourth the entire time they take to fall to P, A will have arrived at g, and the other bodies vrill be in the circumference gP: at the end of one-half the time of descent all will be in the circum- ference h, &o. 273. Bodies descending curves are subject to the same law as regards velocity as those on inclined planes, i. e., the terminal velocity is due only to the perpendicular fall. 274. The Brachystochrone (Greek brachistoSy "shortest," and chronoSy " time,") or curve of quickest descent, is a curve somewhat greater than a circular curve, being what mathematicians denominate a cycloid, or that which is de- scribed by a point in the circumference of a carriage- wheel rolling along a plane. 275. Since Art. 270, the effect of gravity as an accelerating force on a body descending an inclined plane is to the effect of gravity on a body freely falling through the air as the height of the plane is to its length ; we have accelerating force of gravity on inclined plane :g::h:li and hence accelerating force of gravity on inclined planes = j * where h = height of plane. I = length. a == effect of gravity = 32. Substituting this value of the effect of gravity In the formulas in Art. 266» WQ gqt the foUQwing formulas for the descent of bod^M on inclined ^la^eg. 126 DESCENT ON INCLINED PLANES. LAet. 276. 'if .V iJi 1'' 'i t i . r FORMULAS FOR DESCENT OF BODIES ON INCLINED PLANES. NO. 2 3 4 5 6 8 'GIVEN. Si hj Ij t Sj t Si /», li t Si ^1 h « 9, V Si A, li V Si ^1 h « TO PIND. V POBMULAS. 21 2gh i'! J!, i; = i; = T I 2ghs V 28 V gh COBBESPONDINQ FORMULA IN ABT. 266. Ill VI IV V J 2ls = V gh VII VIII 276. When the body is projected down an inclined plane with a given initial velocity V; « = F< -}" ^-—j- (10.) (;ht 21 and v ■= F+ -y- (II') When the body is projected up an inclined plane with a given initial velocity V; s=Vt -?|!(12.)and.= r-^4^(13.) Note.— MThen » body is throwu up an inclined plane, the attraction of gravity acts as a unirormly retarding force as when a body is projected itwil Itwi veloc veloci E3 15 sc LAbt. 276. INED SPONDING [ULA IN r. 266. Aet. 276 J DESCENT ON INCLINED PLANES. 127 II ' [II VI [V V II % [II X inclined 3cted up raction of projected vertically into the air. In the case of the inclined plane the body will continue to rise with a constantly retarded motion until Vt— ght i 21 when it will remain stationary for an instL and then commence to descend. It will occupy the same time in coming down as in going up : its terminal velocity will be the same as its initial velocity, and it will have the samo velocity at any given point of the plane both m ascending and descending. Example 338. — Through how many feet will a body fall in 15 seconds on an inclined plane which rises 1 feet in 40 ? SOLUTION. Here ^ = 16, % = 7, 2 =• 40, and ^ = 32. Then , = »|''=»i4pli»-' =630 feet. ^«,. .:.: , Example 339. — Through how many feet must a body have fallen on an inclined plane, having a rise of 3 feet in 32, in order to acquire a terminal velocity qf 1700 feet per second ? solution. f; 'i Here ^r = 32, « = 1700, A = 3, ? =32. Then . = ^. =|^;=48ie66i feet. ^„,. ^ Example 340. — What will be the velocity at the end of the 20th second, of a body falling down an inclined plane, having an inclination of 7 feet in 60 feet ? SOLUTION. - '■ i .;.:.. '< • ■ ' y ' ;^ ''''^ ■--' " Hereflr = 32, * = 20,fe=7,and? = 60. ' ™^ i, ^ m ^ ff^* 32X7X20_^^„, , , . Then formula 6. « = -^= — = — = 7*2. feet per second. Ans, f 60 ** Example 341. — On an inclined plane rising 3 ft. in 17, a body has fallen through one mile, what velocity has it then acquired ? SOLUTION. Here « = 1 mile = 6280 ft. ft=3 , 1 = l7andflr=32. 12 X 32 X 3 X 5280__ , = J 17 V59632'94i = Then formula VI. v = 2gh8 V i 244'17 feet per second. Ans. Example 342. — ^In what time will a body falling down an inclined plane, having a rise of 7 feet in 16, acquire a terminal velocity of 777 feet per second ? SOLUTION. Here flr = 32, A = 7, ? = 16, and V =: 777. J/it 1 ft b^ "T^T Then formula 8. < = —- = „^ ^^ ■■ = 55i seconds. Ans. •'■'^■ hg 32 X 7 Example 343. — In what time will a body fall through 4780 feet on an inclined plane, having a rise of 3 feet in 4? SOLUTION. Herefir=32,A=3, ^=4» and g =: 4780. a,v * 1 « a His 2 X 4X4780 _, Then formula 9. « = ^ f^ = J ■~32X~3 ^*®^'^ = ^®'^ seconds. ■:.i..pL ■■■•J 1 1 II ft ' ■ ''- ■') It ■■ |,_j; -ivMi:!,' 4-m 128 DESCENT ON INCLINED PLANES. [Abt. 276. ExAMPLB 344. — If a body be projected down an inclined plane, having a rise of 8 feet in 15, with an initial velocity of 80 feet per second, throngh what space will it pass in 40 seconds ? SOLUTION, Here v=: 80, 9 =32, A = 8. 2=16, and i^=40. 32X8X402 Then formula 10. »=Fif+ ^^ ^^^^^^^"^ Tv l7 =^^^ +18653j^ = 168531 ft. Am. Example 345. — If a body be projected up an inclined plane having a rise of 5 feet in 16, with an initial velocity of 2000 ft. per second ;; — -• *■ - 1st. How far will it rise? 2nd. "When will it again reach the bottom of the plane ? 3rd. "What will be its terminal velocity ? 4th. Where will it be at the end of the 100th second? ' '- 5th. In what other moment nf its flight will it have the same velocity as at the end of the 11th second of its ascent ? SOLTTTIOir. * J .: . . Here hzzz 6, Z =16, flf = 32 and v =2000. Then formula 8, ^ = -r = -r --=200 seconds. flffc 5X32 1st. Formula 12. «=F^-^ =200X2000- ^4^^^!^* = 400000 21 2 X 16 —200000 = aV'^mo ft. Ans. 2nd. Ascent =:200 se<^ -}• descent 200 sec. = 400 sec. Ans. -. ,«^ 3rd. Terminal velocity = initial velocity = 2000 feet per sec. Ans, * '" " ^ 4th. Formula 12. « = FiJ - ^ = 100X20000 - ^^ 1l.^,T' = aOOMO av « X lo — 50000 = 150000 = elevation at end of 100th sec. Ans, 5th. 400 — 11 = 389th second. Ans. EXBRCISB. 346. On an inclined plane rising 5 feet in 19 through what space will'a body descend in half a minute ? .Ans. ai89 -^^ ft. 34'7. On an inclined plane rising 3 feet in 13, what velocity will a descending body acquire in 39 seconds ? »dns. 288 ft. per second. 348. What time does a body require to descend through 3800 feet on a plane rising 19 feet in 32 ? .^ns. 20 seconds. 349. If a body be projected down an inclined plane, having a fall of 7 in 11 with an initial velocity of 50 feet per second, what will be its velocity at the end of the 44th second ? .dns. 946 feet per second. .350. If a body be thrown down an inclined plane having a rise of 13 feet in 32 with an initial velocity of 100 feet per second, througti bov many feet will it descend in 130 sec. ? Jins. 122850 feet. AKTB 351. [Am. 276. .Q inclined locity of 80 ) seconds ? 3200 +13653J ined plane of 2000 ft. ane? id? "^ e the same its ascent ? — = 400000 Ans, mfi -=200000 ',*»,j^*. i»^4f«. Ugh what 3k'789-^^ft. locity will w. 288 ft. )ugh 3800 seconds, having a feet per fthe 44th 3r second, 'ing a rise ) feet per 130 sec. ? 2850 feet. ASTB. 277, 278.] ILES. ^&^ havii ^ a fall det per of 800 351. If a body be projected up an inclined plr of 5 feet in 8, with an initial veloGit second: — .1st. How far will it rise? 2nd. In how many seconds will it again reach the bottom of the plane ? 3rd. What will be its terminal velocity ? 4th. Where will it be at the end of the 68th second ? 5th. In what other moment of its flight will it have the same velocity as at the end of the 37th second of its ascent ? Jns, 1st. Rise= 16000 ft. ; 2nd. Time of flight= 80 seconds ; 3rd. Terminal velocity = 800 feet per second; 4th. Ele- vation at end of 68th sec. = 8160 feet. ; 5th. At the end of the 43d second. 352. A body rolls down an inclined plane, being a rise of 7 ft. ,, . .* in 20 — when it has descended through / feet, another body commences to descend at a point m feet beneath it. Through how many feet will the second body descend before the first body passes it ? jins. 4? PROJECTILES. 277. A projectile is a solid body to which a motion has been communicated near the surface of the earth, by any force, as muscular exertion, the action of a spring, the explosive effects of gunpowder, <&;c., which ceases to act the moment the impulse has been given. 278. A projectile is at once acted upon by two forces : — - 1st. The projectile force which tends to make the body move over equal spaces in equal times ; and 2nd. The force of gravity, which tends to make the body move towards the centre of the earth over spaces which are proportional to the squares of the times. Under the joint influences of these two forces the pro^ jectile describes a curve, which in theory is the parabola, but which in practice departs very materially from that figure. Note l.— The »ara&o2a is that curve which is produced by cutting a cone parallel to its side. Note 2.— The parabolic theory is based upon three suppositions, all of which »re more or less inaccurate. Ist. That the force of gpravity is the same in every part of the curve described by the projectile. 9 -: ^ 13a PROJECTILES, [Abts. 279, 280. ,1 ,;- .>|i . )• .; 2nd. That the forc« of gravity acts in parallel lines. 3rd. That the projectile moves through a non-resisting medium. The first and second of these suppositions differ so insensibly trom truth that they may be assumed to be absolutely correct, but the resistance of the atmosphere so materially affects the motions of aHl bodies, especially when their velocities are considerable, that it renders the paraooUc theory practically useless. 279. When a body is projected liurizdntally forward, the horizontal motion does not interfere with the action of gravity, — the projectile descending with the same rapMity while moving forward, that it would if acted upon by gravity alone. NoTB.-~The accompanying figure represents a tower 144 feet in height. Now if three balls a, 5, and c, be made to start simultaneously from P, one dropping vertically, one being projected forward with sufflcient force to carry it hori«ootalfy half a mile, and tbe third with suffident force to carry ' it horizontally to any otber distance, say one mU^all three balls will reach the ground, provide it be a horizontal plane, at the same instant. Thus each ball will have fkllen 16 feet at the end of the 1st second, and they will simultaoeoudy cross the line d^. At the wd of tl^e ^d second they have ea^ descended 64 feet, and are.reepectively at g, A, and «, ^.. Fig. 26. ;.„ 280. According to the parabolic theory : — 1st. The projectile rises to the greatest height, and remains longest before it again reaches the ground, when -vl thrown vertically upwards. 2nd. The distance or range over a horizontal plane is greatest, when the angle of elevation is 45**. 3rd. With an initial velocity of 2000 per second, the pro- jectile should go about 24 miles. NoTB.— The first of these laws is found by experiment to be absolutely correct, and the second is not fiur trom the truth, the greatest range taking place at an angle of elevation somewhat leas ttaiia 481^ BTS. 279, 280. lum. yfroin truth resistanoe of , especially )Uc theory ibol ^ forward, he action .he same i if acted )et in height, from P, one lent force to 'oroetocarnr Is will reach itant. Thus md they will sd they have d remains id, when plane is the pro- » aibiolately ing* taking Abts. 281, 282.3 PKOJECXILES. 131 The difference between the third law and the result of experiment is pro- digious ; for no projectile, however great its initial velocity may have been, has ever been thrown from the surface of the earth to a horizontal distance of 5 miles. 281. Whatever may be the initial velocity of projection, it is speedily reduced by atniospheric pressure to a velocity not exceeding 1280 feet per second. Note 1.— This arises flrom the tact that atmospheric air flows into a vacuum with a velocily of only 1280 feet per second, so that when a ball moves with a greater velocity than this, it leaves a vacuum behind it into which the strongly compressed air in front tends powerfully to force it. Note 2.— From experiments made with great care, it has been ascertained that when the velocity of a ball or other projectile is 2000 feet per sc^^nd, the ball meets with an atmospheric resistance equal to 100 times its own weight. Note 3.— Another great irregularity in the firing of balls arises firom the fact that the ball deviates more or less to the right or left, sometimes crossing the direct line several times in a very short course. This deflection sometimes amounts flrom ito i of the whole range, or. as much as 30O or 400 yards in a mile when there is considerable windage ; i. e., when the ball is too small for the calibre of the gun. 282. The motion of projectiles has recently been inves- tigated with much care, with the view of deducing a new theory in which the resistance of the air should be taken into account. The following are the most important results: — • „ ,^ WHSN THE BODY IS THBOWN YKBTICALLY UPWARDS INTO THE AIR. I. The time of ascent is less than the time of descent. II. TTie velocity of descent is less than that of ascent. ] III. The terminal velocity is less than the initial velocity. [^ r IV. The velocity of descent is not infinitely a/:celeratedy since when the velocity becomes very great ^ the resistance of the atmos- phere becomes so great as to counterbalance the accelerating force of gravity f and the velocity of the descending body is thenceforth uniform. , WHEN the PROJECTILE IS THROWN AT AN ANGLE OF ELEVATION. I. The ascending branch of the curve is longer than the descend- ing branch. II. The time of describing the ascending branch is less than that of describing the descending branch. III. The descending velocity is less than the ascending. lY . The terminal velocity is less than the iniHai. y . The direction of the descending branch is constantly approart- matin^ to a vertical.linef ifiMch it never reaches. j I ;.., 132 PROJECTILES. [Abts. 28^-288. !lv*. '. VI. The descending velocity is not infinitely accelerated^ butj as in case of a body falling vertically ^ becomes constant after reaching a certain limit. VII. 2%c limit of the velocity of descent is different in different bodies J being greatest when they are dense j and increasing with the diameter of spherical bodies. 283. The explosive force of gunpowder, fired in apiece of ordnance, is equal to 2000 atmospheres, or 30000 lbs. to the square inch, and it tends to expand itself with a velocity of 5000 feet per second. Note.— Gunpowder is an intimate mixture of 6 parts saltpetre, 1 par charcoal, and 1 part sulphur. In firing good perfectly dry gunpowder, th* ignition takes place in a space of time so short as to appear instantaneous. 1 cubic inch of powder produces 800 cubic inches of cold gas, and, as at the moment of ezpUraion the gas is red hot, we may safely reckon the expansion as about 1 into 2000. 284. The greatest initial velocity that can be given to a cannon ball is little mov'e than 2000 feet per second, and that only at the moment, it leaves the gun. - »- f NoTB.— The velocity is ^atest in the longest pieces ; thus Hutton found the velocity of a ball of given weight, fired with a given charge of powder to be in proportion to the fifth root of the length of the piece. 286. The velocities communicated to balls of equal weights, from the same piece of ordnai^ce, by unequal weights of powder, are as the square roots of the quantities of powder. 286. The velocities communicated to balls of different weights and of the same dimensions, by equal quantities of powder, are inversely proportional to the square roots of tne weights of the balls. 287. The depth to which a ball penetrates into an obstacle is in proportion to the density and diameter of the ball and the square root of the velocity with which it enters. Note 1.— An 18-pound ball with a velocity of 1200 feet per second pene> trates 34 inches into dry oak, and a 24cpouud ball with a velocity of 1300 ft. per second penetrates 13 feet into dry earth. Note 2.— The length of guns has been much reduced in all possible caaes. Field pieces are now seldom made of greater length than 12 or 14 calibres (diameter of the ball). The maximum charge of powder has also been diminished very greatly—now seldom exceeding one-third, and often being as low as one-tweltlh of the weight of the ball. 288. The following rule, obtained from experiment, has been given, to find the velocity of any i^hot or Bhell, when ! ITS. 283-288. edj but J as ir reaching n different gwitk the n apiece )0 lbs. to I velocity letre, 1 par M)waer, th% tautaneous. id, as at the e expansion given to ;ond, and itton found of powder of equal unequal uantities different uantities re roots into an erofthe vbich it ond pene- of 1300 ft. ibie oases. L4 oalibroM also been ften being ent, has 11, when AST. 288.] PROJECTILES. 133 the weight of the charge of powder and also that of the shot are known : — . , ,, RULE. Divide three times the weight of powder 6j/ the weight of the ahotf multiply the square root of the quotient by 1600, and ihepro- duct will be the velocity per second in feet. Or ifp = charge of powder in Ibs.j w = weight of ball in lbs., .( 3/> \ and V = velocity per second in feet ; then v = 1600 x v I ) Example 353. — What is the velocity of a ball weighing 48 lbs., fired by a charge of 4 lbs. of powder? SOLUTION. 3X4 Here 2)= 4 and ti; = 48. Thenv = 1600X V(-J^) =1600 X V( 48 •" <*' I . , !^s, 1 ,*»> ■;* *■» ' ■j=1600X V(-J j =: 1600 X i = 800 feet per second. Ans, Example 354. — With what velocity will a charge of 7 lbs. of powder throw a ball weighing 32 lbs. ? •1-' u Here p =7 and w =32. solutiow. ,'j- Then v = 1600 X J-^ = 1600 X J-^^ = 1600 X V 'QMiH =1600 V w V 82 X '81 = 1296 feet per second. Ans. Example 355. — If 4 lbs. of powder throw a ball 16 lbs. in weight with a velocity of 1200 ft. per second, what amount of powder would throw the same ball with a velocity of 600 feet per second ? solution. Art. 285. vel. : vel . : : V( weight of powder) • V(weight of powder ); or 1200 : 600 : : V 4 : V a;, and hence « = 1 lb. Ans. Example 366. — If 3 lbs. of powder throw a ball 6 inches in diameter and weighing 32 lbs., with a velocity of 850 feet per second, with what velocity will the same charge throw another ball of the same dimensions but weighing only 9 lbs. ? solution. Art. 286. V'9: VsS": : 850 : ar, or 3 : 5'66 : : 850 : x. And hence x = 1600 feet. Ans. EXERCISE. 357. With what velocity will a charge of 11 lbs. of powder throw a cannon ball weighing 24 lbs. ? Ans. 1876 feet per second. I • t: I 134 CIRCULAR MOTION. [Aets. 289-292. Abts. !:r.v ll. < 358. With what velocity will a charge of 9 lbs. powder throw a ball weighing 36 lbs ? Ans. 1385 feet per second. 359. If n lbs. of powder throw a ball with a velocity of 1000 feet per second, what charge will throw the same ball with a velocity of 1500 feet per second ? Ans. lb\ lbs. 360. If a certain charge of powder throw a 10-inch ball weighing 20 lbs. with a velocity 973 feet per second, with what velocity will the same charge throw a ball of the same . dimensions weighing only 25 lbs. ? Am. 870 feet per second. CIRCULAR MOTION. 289. Centrifugal force (Lat. centrum^ " the centre," and fugioy ** I flee ") is that force by which a body mov- ing in H circle tends to fly oft' from the centre. NoTB.— Slncea body moving in a circle would, if not restrained by other forces, fly off in a tangent to ^^ at circle, centrifugal force is sometimes called tai^ential force. 290. Centripetal force (Lat. centrum, *' the centre,'* and petOy " I seek or rush to ") is that force by which a body moving in a circle is held or attracted to the centre. 291. When a body is at once acted upon by both cen- trifugal and centripetal force, it moves in a curve, and the form of this curve depends upon the relative intensities of the two forces : i. e., if the two be equal at all points, the curve will be a circle, and the velocity of the body will be uniform ; but if the centrifugal force, at different points of the body's orbit, be inversely as the square of the distance from the centre of gravity, the curve will be an ellipse, and the velocity of the body will be variable. 292. When a body rotates upon an axis, all its parts revolve in equal times; hence the velocity of each particle increases with its perpendicular distance from the axis, and so also does its centrifugal force. NoTB 1.— As long as the oentriftigml force is lesis than the cohesive force by which the particles are held together, the body can preserve itself ; but, as soon as the centrifugal fo^.tw exceeds the oohesivoi the parts of the rotating mass fly off in directions which are tangents to the circles in which they were moving. NoTB 2.— We have examples of the effects of centrifugal force in the destructive violence with which rapidly revolving grindstones burst and fly to pieces— the expulsion of water firom a rotating mop, the projection of a stone from a sling, the action of the conical pendulum or governor in regulating the supply of steam in an engine, &c., &c. ITS. 289-292. er throw a er second. Y of 1000 same ball s. 159 lbs. weighing viih what the same er second. centre," dy mov- ed by other sometimes ;re," and I a body re. ►oth cen- rve, and itensities I points, be body different •e of the II be an >le, ts parts particle tie axis, >8ive force self; but, e rotating hioh they ce in the rnnit ami >roJeetioa •vernoriii Aets. 293-297.] CIRCULAR MOTION. 135 -1 :• 293. When the velocity and radius are constant, the centrifugal force is proportional to the weight. 294. When the radius is constant, the centrifugal force varies as the square of the yeldcity. Note.— At the equator the centrifugal fbrc^ of a i)article is ?i? of its gravity or weight and from the equator it diminishes as we approach tlie poles where it becomes C It follows tliat if the earth were to revolve 17 times faster than it does, the centrifugal force at the equator would be equal to gravity, and a body would not foil there at all. If the earth revolved still more rapidly, the water, inhabitants, &c., would be whirled away into space, and the equatorial regions would constitute an impassable zone of sterility. 295. When the velocity is constant, the centrifugal force is inversely proportional to the radius. 296. When the number of revolutions is constant, the centrifugal force is directly proportional to the radius. 297. Let c = centrifiigal force, v = the velocity per second in feet, r =: radius in feet, g- = 32, w = weight, and n = the number of revolutions per second. -!^ > ttw* wv^ cgr ./cgr\ Then c = — (I), r = — (II), w = j,- (III), v = y/{j^j (IV). Also, since v = r X 2 X 3-1416 X w, v* = r* X 4 X (3'1416)* X »*, WX»'*X4X(31416)«X»«, and hence formula I. : c = gr and re- c - ducing this we get c = wm* x 1-2345 (V), w = yy^g^ 1.23 45 (^^)» urn* X 1*2346 (VII), «.d » = V(„TXT2346) <^"'>- BxAMPLi 361. — ^What is the centrifugal force exerted by a body weighing 10 lbs. revolving with a velocity of 20 feet per second in a circle 8 feet in diameter ? Here «0 = 10, « Then c = — = soLirrioir. 80,r = 4,andp=a8. 10 X 20* 10 X 400 Slilbs. Ant. gr 82 X 4 S2 X 4 ExAifpLi 362. — What centrifugal force is exerted by a body weighing 15 lbs. revolving in a circle 3 feet in diameter and making 100 revolutions per minute ? : :m 136 If U': '^< CIRCULAR MOTION. SOLUTION. [Abt. 297. Here w = 15, r =1-5, » = W = 1|. ThenfonnulaV.: c=wm^ X 1*2345 =16X1-6X (If)* X 1*2345 =77* 15625 lbs. Ant, Example 363. — A body weighing 40 lbs. revolves in a circle 4 feet in diameter ; in order that its centrifugal force maj be 1847 lbs., what must be its velocity and number of revolutions per second ? SOLUTION. Here to — 40 lbs., r = 2, and c = 1847. c Then formula YIII. [.:» = V(; V ~'^\*0X2Xl*a845| ^wrxr2846j y 18*7019 = 4*32 = number of revolutions per second, and hence revolu- tions per minute =£ 266'8. Also V = 4X3*1416X4*32 = 64.28 feet per second. EXA.MPLB 364. — The diameter of a grindstone is 4 feet, its Weight half a ton, and the centrifugal force required to burst it is 45 tons : with what velocity must it revolve, and how many revolutions must it make per minute in order to burst ? ^^ ' SOLUTION. wrX 1*2345 7 ='^(4X2X 1*2346 j — 1 ittere w = i, c — 45, and r = 2. Then formula VIII. : « = Vf; V 36*462 = 6.03 = revolutions per second, and hence 6*08X60=361*8 = the revolutions per minute. Alip velocity = 4X3*1416X6*03 = 76*775 feet per second. BXEROISB. 365. If a ball weighing 4 lbs. be attached to a string 2| feet long and whirled round in a circle so as to to make 120 revolutions per minute, — what must be the strength of the string in order to just keep the ball from flying off? '- Jint, 49*38 lbs. 366. A ball weighing 2 lbs. is attached to a string 3) feet long and capable of resisting a strain of 200 lbs. ; if the ball be whirled in a circle with the whole length of the string as radius, how many revolutions per minute must it make in order to break the string? ^tm. 2 88|^ revolutions. 367. A ball is whirled in a circle, with a velocity of 64 feet per second, by means of a string 4 feet in length and capa- ble of resisting a strain of 840 lbs. ; what must be the weight of the ball in order to break the string ? jfnx. 261 lbs. ABTi 368. [Abt. 297. 5 =77-15625 i a circle e maj be iTolutions ice reyolu- feet, its 3 burst it )w many 'uT — sv ■;• = the 2| feet ake 120 sngth of igoflf? ^38 lbs. iet long the ball e string it make Intioni. feet per d capa- ;be the :6i lbs. Abts. ms, 299.] ACCUMULATED WORK. 13T 368. What is the centrifugal force exerted by a body weighing 20 lbs. revolving in a circle 10 feet in diameter and mak- ing 2-8 revolutions per second? Ans. 967*848 lbs. 369. What is the centrifugal force exerted by a body weighing 8 lbs. and revolving in a circle 20 feet in diameter with a velocity of 100 feet per second ? »Ang. 250 lbs. AOCfUMULATED WORK. :IS\ 298. Work is required to set a body in motion or to bring a moving body to a state of rest. For example, when a common engine is first set in action a considerable portion of the work of the engine goes to give motion to the fly-wheel and other parts of the machinery ; and before the engine can come to a state of rest, all of this accumu- lated work must be destroyed by friction, atmospheric resistance, &c. 299. To find the work accumulated in a moving body : — *-V ','v/>U.|>; BULB. /. Find the height in feet from which the body must have fallen to have acquired the given velocity, II. Multiply the number thus found by the weight of the body in pounds. Or let U=z units of work accumulatedj v = velocity ^ w = the weight in Ibs.f and g = 32. 2fir Then Art. 266, since s=:h:=z V* v^w — '*««' — 2g — 2g ExAMPLB 370. — A ball weighing 10 lbs. is projected on smooth ice with a velocity of 100 feet per second : assuming the friction to be ^V ^'f the weight of the ball, and neglecting atmospheric resistance ; over what space will it pass before coming to a state ofredt? SOLUTIOSr. Here v=rl00, 10=10, and g = 82. v*w 100* X 10 100000 Then 17= -— = 64 = 16624 = units of work accu- iff 2 XS2 mulated in the ball. Alio ^X10xl=| = units of work destroyed by fWction in moving the ball through 1 foot. Therefore the nu^er of feet = 1662i-f-f = 23433 ^ns. 5A •'!':• 114* .''••". Bu.> (I 138 ACCUMULATED WORK. [Art. 299. EzAMPLB 371. — A train weighs 100 tons and has a Yelocify of 40 miles per hour when the steam is turned off : how far will it ascend a plane having an inclination of | in 100, taking fric- tion as 11 lbs. per ton, and neglecting the resistance of the atmosphere ? SOLUTION. Here v = 40 nr'les per hour = 40 X 5280 : d8 f feet x)er second, to =100 60X60 tons = 200000 lbs. and g = 32. Then F=H!!? = M)!x?««»2? = ?*«l2ody to be destroyed, so that, at the moment of impact, the body would have no tendency to move in any direction. In a rod of inappreciable thickness the centre of percussion is two-thirds of the length of the rod A'om the axis about which it moves. 309. The centres of suspension and oscillation in the pendulum are interchangeable, i. e., if the pendulum be inverted and suspended by its centre of oscillation, the former point of suspension will become the centre of oscil- lation and the pendulum will vibrate in precisely the same time. E'« -\ m LAWS OF THE OSCILLATION OF THE PENDULUM. 810. The duration of an oscillation is independent of its amplitude, provided it does not exceed 4® or 6^. Note 1.— This fact is commonly stated by saying that the vibrations of the pendililum are isochronous ; i. e., equal-timed. Thus, a pendulum of a given length will oscillate through an arc of 6° in the same time it would have required to vibrate through an arc of 0*1^ although the amplitude of the vibration is in the one case 50 times as great as in the other. This arises fh)m the fkct that the pendulum in moving through the larger arc ftills through a greater vertical distance, and hence acqiures a greater velocity. 1 ?"rt ETS. 306-310. distance of oscil la- nd which vibrating 5ted, the led by one pendulum, separately, kpidly than r, all of its m of those •ded, while in the rod, respect to the acce- ;ed by the lis particle if liberated I, and were the centre Rides with ion is that mmovable revolviuK »dy would ppreoiable of the rod I in the [lum be on, the [)f oscil- le same lent of rations of lulum of i it would tlitude of bis arises arc fills relocity. Aets. 311-316.] THE PENDULUM. 141 Note 2.— Strictly speaking, the oscillations of the pendulum are isochro- nous only when the curve in which they move is a cycloid. When, however, the common pendulum vibrates in very small arcs, as of 2" or 3^, the oscillations are, for all practical purposes, isochronous. 311. The duration of the vibration is independent of the weight of the ball and the nature of its substance. 312. Two pendulums of equal lengths perform an equal number of vibrations in the same period of time. 313. Two pendulums of unequal lengths perform an unequal number of vibrations in the same period of time — the longest pendulum performing the smallest number of oscillations. 314. Pendulums of unequal lengths vibrate in times which are to one another as the square roots of their lengths. 315. A seconds pendulum is one that performs exactly sixty vibrations in a minute, or one vibration in one second. 316. The time occupied by a vibration depends : — 1st. Upon the length of the pendulum ; and 2nd. Upon the intensity of the force of gravity. Note.— Since the earth is not an exact sphere, being flattened at the poles* the surface of the earth at the poles is nearer to the centre than at the equator. Hence the intensity of the force of gravity is less at the equator tmm at the poles, and a pendulum that beats seconds at the equator must be lengthened in order to beat seconds as it is carried towards the poles. In point of fact, a seconds pendulum at the poles is about one-fifth of an incQ longer than a seconds pendulum at the equator. The following table shows the length of the seconds ^. tndulum at different parts of the earth's surface, and also the magnitude of the force of gravity ; i. e., the velocity which the force of gravity will impart to a dense body in falling for one entire second. Phice. Latitude. Length of Seconds Pen- dulum. Velocity acquired by a body falling one second. St. Thomas 0" 24' 7«66' 40» 42' 48'' 60' 61» 31' 70» 60* 39 01 inches 39 02 " 39 10 •• 39 12 " 39 13 " 39*21 *' 3849 86 inches 3842 86 385-978 386 076 " 386 174 886-984 •• Ascension New York Paris London Spitzbencen 142 THE PENDULUM. [Arts. 317-320. AST. >* Hot' s I ^, 'I'it, m- in h Note.— In Canad the seconds pendulum is about 3911 in. in length. 317. The pendulum is applied to three purposes : — 1st. As a measure of time : 2nd. As a measure of the force of gravity : and :*^^ 3rd. As a standard of measure. a, Note.— The pendulum is used as a measure of time by attaching it to clock-work, which serves the double purpose of registering its oscillations and restoring to the pendulum the motion lost in its vibration by friction and atmospheric resistance. The use of the pendulum as a standard of measure will be seen from the following statements, viz : 1st. A. pound pressure metms that amount of pressure which is exerted towards the earth, in the latitude of London and at the level of the sea, by the quantity of matter called a pound. '2nd. A pound of matter means a quantity equal to that quantity of pure water which, at the temperature of 62 deg. Fahrenheit, would occupy a7'727 cubic inches. Srd. AcudttftncAisthat cube whose side, taken 39*1393 times, would measure the effective length of a London seconds pendulum. 4th. A S6con(29 pendulum is that which, by the unassisted and unopposed effect of its own gravity, would make 864(00 vibrations in an artificial solar day, or 86163*09 in a natural sidereal day. 318. Ift=^the time of oscillationj I = the length of the pendu- lum, g =: the force of gravity ; i. c, the velocity which the farce of gravity would impart to a dense body faHling through one entire second, and = 3*1416; i. e,, the ratio between the diameter of a circle and its circumference. Then t =7r x \ (1\ t^g \fi:/(i.)Z = -.(ii.) andg= -j^ (m.) T' When t = (me si:ond,formulas (ii.) and (iii.) respectively become 8 I =— J (rv.) and g = Zir' (v.) IT 319. To find the time in which a pendulum of given length will vibrate, or the length of a pendulum that vibrates in a given time : — Let I = the length and t = the time. Then since {Art. 314) the times are as the square roots of the lengths, and in Canada the seconds pendulum is 39*11 inches in length — we have t'.\'.: VZ:V(39'11) ; and hence f = V\"39Tr/ C^i-), o-nd I = ^«X39*11. (vii.) 320. To find the number of vibrations which a pendulum of given length will lose by decreasing the force of gravity, 1. e., or Ia face sain the »' = ;TS. 317-320. in length. 368 : — : and 3hinK it to oscillations by Motion tandard of L is exerted the sea, Ijy lityofpure lid occupy nes, would unopposed ficial solar le pendu- e/jrceof »e entire meter of II.) ly become ►f given im that rt. 314) nada the Abt. 321.] THE PENDULUM. 143 odulum gravity, i. e., by carrying the pendulum to the top of a mountain or other elevation. Let n = the nwnber of vi^raiions performed at the eartKs sur- face in the given time^ n' = the number of vibrations lost in the sante time^ r = the radius of the earthy = 4000 miles^ and h = th^ height of the mountain in miles or fraction of a mile ; then nh n'r » = —- (viii.), and hence A =r — (ix.) 321. To find the number of vibrations which a pendu' lum of given length will gain in a given time by shortening the pendulum. Let I = the given length ot the pendulum^ and V = the decrease in length: also let nz;i the number ofvibrtUions performed in the given time, and n' =5 the number of vibrations gained in the same time; then n' = rr- (x.) and V = 2 n' l n (XI.) ExAMPLB 382. — ^How many vibrations will a pendulum 36 inches long make in one minute ? 8OLUT101?. Formula TI: t= Vlgg^j") =Vl3a:ij"y = y*9204=: •969 seconds. Hence the number of vibrations = 60-4- *959 = 62*56. ExAMPLS 383.— Required the length of a pendulum that makes 80 vibrations in a minute. SOLUTION. Here< = f^ = }. Then foromla VII. Z ^ *« x 8911 ^ (i) « X 39*11 =* ^ X 3911 = 21*999 inches. ExAMPLs 384.— In what time will a pendulum 60 inches long vibrate ? € SOLUTION. Formula VI. : e = V ( 39^ ) =V l^n ) =Vr6341 = 1*289 seconds. Ant. ExAMPLB 385.-^A pendulum which beats seconds is taken to the top of a mountain one mile high : how many seconds will it lose in 6 hours f 144 THE PENDULUM. [Am. S21. ▲kt M.t ' I,} .: ■ ;/:•^• ■,{'"■■ 10 ;ty' }'.; h-/ ''•:■'',■:'■.. \ n:^- -. 1 '.'-■■r ' :■ ;W ^ ii 'V ■-,*':■■■ •1 i ■'■ -.- ' '/•^ ■ ; 1 1 „; \ ■ . fl .'1 1 Itf' ' ■ ^1 ■ ' , I4.,,^.y. ■■■Ft ,^.£ --. ■-■. r. '■ :'.' ;!! 'jti .. :, ■it 7--- h ^^ '■'■' :_ : ,' . ',S.'ii' !;'"^! r^ fi .1L. SOLUTION. Here » = 6XflOX60,A = l, and r=4000. "H mi- , 1 /TTfTTX » *^ 6 X 60 X 60 X 1 _ 21600 ,.^ . Then formula (VIII.): »' = — = ^^ = looo" = **' ^*'- ' ExAMPLX 386. — ^If a clock lose 1 minute in 24 hours, how much must the pendulum be shortened to make it keep true time? SOLUTION. , Here»=24X60X60,»'=60,andi=39"ll. mi. * 1 ITT »i 2 n7 2 X 60 X S9'll ...,., , * , i.u • Then formula XI. : V = — :— = ^ ^ = 0'0i43 or about ^i^th of an inch. Ans. n 24X60X60 ^ Example 387. — Through what distance will a heavy body fall in Canada during one entire second, and what will* be its ter- minal velocity 7 solution. u I ;:^> .: Here ^ = 1, and 2 = 89*11. Then formula V. p = l^' — 39*11 X (3.U16) « = 39*11 X 9*86966066 = 386*002 inches = terminal velocity. :i^,*,i^--:. H«nce the space passed through = ^ = 193*001 inches = '^'* ' ^ 16*0835 feet. Ant, 2 ExAMPLs 388. — ^What must be the length of a pendulum in order to vibrate ten times in a minute 7 solution. Here * = Tjr = ^ seconds. ,stcr Then formula VIL ? = *» X 39*11 = 10« X 39*11 = 100 X 39*11 = 3911 in. = 326 feet nearly. Ans, Example 389. — A pendulum which vibrates seconds at the surface of the earth is taken to the top of the mountain and is there found to lose 18 seconds in a day of 24 hours : required the height of the fountain. SOLUTION. Here »' = 24 X 60 X 60, n = 18, and r = 4000. 18X4000 Then A = "^-S^l^ = I mn« = «00 fert. A»,. Example 390. — If a seconds pendulum be shortened 1^ inekts how many vibrations will it make in one minute 7 [Abt. S31> = 5*4. Ant. ours, how keep true L . ^ k., i » >out ^th of jr body fall !)e! its ter- '86965066=5 dulum in = 3911 in. is at the in and is required i inekts ▲it. 8SL] THE PENDULUM. SOLUTION. 145 Here » = 60, 2 = SfU, and i' = l-a. Then formuht X. : «' = — = "V;*'° = 0-968 = the number of vibra- tious gained ; hence the number of vibrations made == 60*968. Atu. ExAMPLS 391. — ^What will be the velocity acquired by a heavy body falling during one entire second in the latitude of Spitzbergen? SOLUTION. ' ' -■ ' - "J' '; Here ^ = 1. and by the table Art. 316, 2 = 39*21. Then ^= iw* = 39*21 X (3*1416) « = 39*21 X 9'86966 = 386*988 inches. Ana. BXBRCISB. ». *>*^ 392. What must be the length of a pendulum in the latitude ^^ ^;: of Canada in order that it shall vibrate once in 3 seconds ? ^ns. 351*997 inches. 393. A pendulum that vibrates seconds at the surface of the earth is carried to the summit of a mountain 3 miles in ! height: how many seconds will it lose in 24 hours? .^# tp -tfn». 64*8. 394. In what time will a pendulum 10 inches in length vibrate ? ,dns, '505 seconds. 395. What velocity will a heavy body falling in the latitude of New York acquire in one entire second ? Jlns. 385*903. 396. If a clock lose 10 minutes in 24 hours, how much must the pendulum be shortened in order that it shall keep correct timb ? ^ns. '543 or over i of an inch. 397. If a seconds pendulum be shortened 6 incheS; how many vibrations will it make in a minute ? Jins. 63*83. 398. A pendulum which vibrates seconds at the surface of the earth is carried to the summit of a mountain, where it is observed to lose 30 seconds in 24 hours : required the height of the mountain. Jtns. 7333*3 feet. 399. In what time will a pendulum 100 inches long vibrate ? jins. 1*59 seconds. 400. Required the length of a pendulum which makes 120 vibra- tions per minute. •^m. 9*77 inches. 40 1\ Through how many feet will a body fall in one second, and what will be its terminal velocity at the end of that portion of time in the latitude of Paris ? V e^iw. Terminal velocity =r 386*1 in. Space passed over = 16*0876 ft. 10 I > i' 146 HYDROPTNAMICS. [AbT8. 1'.> ; ; CHAPTER Vm. ' :\ - ' .^)' :Jf^.i HYDRODYNAMICS. 322. Hydrodynamics treats of the motions of liquids and of the forces which they exert upon the bodies when their action is applied. 323. The particles of a fluid on escaping from an orifice possess the same velocity as if they had fallen freely in vacuo from a height equal to that of the fluid surfiEice above the centre of the orifice. This is known as TorricelWs theorem. 324. The principal deductions from the Torricellian theorem are— '"^' 1st. The velocity of an escaping fluid depends upon the depth of the orifice beneath the surface and is independent of the density of the liquid. 2nd. The velocity of efflux from an orifice is as the square root of the height of the fluid surface above the centre of the orifice. NoTX.— Since all bodies fUling in vacuo firom the sune height acquire the same velodty» density has no effect in increasing the velooity of a liquid escaping firom an wifloe in the side or in the bottom ot a Teasel. Thus water, alcohol, and mercury will all flow with the same rapidity ; for though the pressure of the mercury is IS^ times greater than that of water, it has l&k times as much matter to move. , 326. When a liquid flows from an orifice in a vessel which is not replenished but the level of which continually descends, the velocity of the escaping liquid is uniformly retarded, being as the decreasing series of odd numbers 9, 7, 5, 9, &c., so that an unreplenished reservoir empties itself through a given aperture in twice the time the same quantity of water would have required to flow through the same aperture had the level been maintained constantly at the same point. 826^ The (]^uantity of fluid discharged from a given aperture in a given time is found by multiplying the area of the aperture by the velocity of the escaping liquid. NoTB.— Experiments do not agree with this theory as regards the quan- tity of liquid discharged. The whole sutject has been carefully investi- gated by Bossut. and ne has shown that Actual dtsohargt i Theoretical ditchargt i : '62 : 1 or as 6 : 8. of liquids dies when i an orifice freely in hce above ^orricellVs >rricellian upon the lependent is as the Lbove the Sht acquire relodty of a of a yesMl. mpidity: for tan that of a vessel ntinuallj iniformly mbers 9, empties the same ough the instantly a given the area uid. I the quan- lly inveati- ARTS.8W-8S1.] HtI^R0DY3^A]Jii;CS. m Hence the theoretical dischai^ must be multiplied by 4 to obtain the true quantity. " Fig. 27. This discrepancy arises from the fact that the escaping jet diminishes in diameter just after leaving tJiie Tesselj formiuff what is known as the vena conifrv>cto or oontraoted vain. The minimum diameter of the vein is found at a dufeanee about equal to half the diameter of the aperture 9i»e«f Fig. 27. This effect arises from the ftet that just above the orifice the lateral particles of fluid move as well as the descending portions. If the Jet of liquid be thrown upwarda at an angle of from 26(> to 46^ the vein retuns thediameter of the aperture, but iT thrown at an angle greater than 4Bfi its section increases. 327. Let Qz± the quantity discharged in 1 second^ azz area of aperturey h =: height ofjluid lenel above the centre of the orifice^ g = accelerating force of gravity ^ and v=: velocity. _ Q Then Art, 266 v =z *j2gh^ (i) Q == a^lgh^ (ii) a — ^—h > 0") «», t «* m' i },' ~ Ml f 332. As the velocity of a liquid escaping through an orifice is the same as it would have acquir- -pig. 28. ed in filling freely in vacuo through a space equal to the distance of the orifice below the level of the liquid, it follows that a jet of water spouting upwards should rise to the level of the liquid in the reservoir.In practice I however the spouting jet never reaches thii$ ' height owing to certain disturbing forces, nanoiely : 1st. Friction in the conducting tube in part destroys the velocity. 2nd. Atmospheric resistance. 3rd. The returning water falls upon that which is rising and thus tends to stop its ascent. NoTB.— The height to which the liquid spouts is inoreaseU by Ist. Having the orifice very bmall in comparison with th '> acting tube. 2nd. Piercing the orifice in a very thin wall ; and Srd. Inclining the jet a little so as to avoid the retu^^ning water. ExuiPLi 402. — ^With what velocity does water issue from a small aperture at the bottom of a vessel filled to the height of 100 feet ? SOLUTION. Formula 1 « = sV* = sV 100 = 8 X 10 = 80 feet per second. Ans. Example 403. — What quantity of water will be discharged in one minute firom an aperture of half an inch in area-^the height of the water in the vessel being kept constant at 10 feet above the centre of the orifice ? SOLUTIOir. Here a = i square inch = jir of a square foot. The cubic feet discharged in 1 second = Bayh. Cubic feet discharged in 1 minute = 60 X 8axVA=60XifTX VlO =60 X ■^ X 8-168 = 5*27 cubic feet = the theoretical quantity, and 6*27 X i =^ S '29 cubic feet =^ true quantity. Example 404. — What must be the area of an orifice in the side of a vessel in order that 40 cubic feet of water may issue per hour — the water in the reservoir being kept constantly at the level of 20 feet above the centre of the aperture 7 •x^- HereQ = 40 60 X60 = -A SOLUTIOir. of a cubic foot, and since this is only fof th« [ART.SSt. brough an ig. 28. I IS rising . .Vs,T;ucting ig water. sue from a e height of i. An«. icharged in -the height feet above kVio=60x 27X4=^3'29 i in the side ^ issue per Qtly at the mly f of the A9T. 332.J HYDRODYNAMICS. 149 theoretical quantity, Q = |. of ^ = -^ of a cubic foot. Also A = 20 Q Then formula III.a=-^ = -^ = ^^ = ^^ofafoot=r W^ of an inch. Ant, BxAMPLB 405.— An upright vessel 16 feet deep is filled with Water and just contains 15 cubic feet. Now if a small aperture i of an inch in area be made in the bottom, in what time will the vessel empty itself? SOLUTION. Hire fc = 16 ft., a = J of an inch, and Q = 15 cubic feet. Hence the theoretical quantity = 15 X |- = 24 cubic feet. Then velocity at commencement = sV* = sVlB = 82 ft. Quantity discharged in 1 second = 82 X shs = ^W = tV of a cubic foot. Time required to discharge 24 cubic feet = 24 -f- -j^g = 432 seconds. But, Art. 324, when a vessel empties itself, the time required to discharge a given quantity of water is double that requisite for discharging the same quantity when the level is maintained. Hence time = 482 X 2 = 864 seconds = 16*4 minutes. Ant. EXBROISB. 406. With what velocity does water issue from a small aperture in the side of a vessel filled to the height of 25 feet above the centre of the orifice ? Jlns. 40 feet per second. 407. With what velocity does water flow from a small aperture in the side of a vessel filled with water to the height of 1*7 feet above the centre of the orifice ? ^ns. 32*984 feet per second. 408. In the last example, if the water flows into a vacuum, what is its velocity ? .Ans. 56 feet per second. Note.— Since the pressure of the atmosphere is equal to that of a column of water 32 feet high, the effective height of the column of water is 17 + 32 = 40 feet. 409. How much water is discharged per minute from an aperture having an area of ^ of an inch — the surface of the fluid being kept constant at 36 feet ? •Ans. 2^ cubic feet. 410. What must be the area of the aperture in the bottom of a vessel in order that 90 cubic feet of water may issue per hour — the level of the water in the vessel being constantly kept at 20 feet above the centre of the orifice ? j^n«. '161 of about ^ of an inch. 411. A vessel contains 20 cubic feet of water, which fills it to the depth of 30 feet — now if an aperture having an area of } of an inch be made in the bottom of the vessel, in what time will it empty itself? Jn$, 2 min. 30} sec. ■m '^ 150 HYDRODYNAMICS. [AST8. 389-^80. 333. When water spouts from several apertures in the side of a vessel, it is thrown with the greatest random from the orifice nearest the centre, the jet issuing from the centre will reach a horizontal distance equal to the entire height of the liciuid, and all jets equally distant from the centre will be thtown to an equal horizontal distance. Fig. 29. NoTB.— Let VA he tk vessel filled with water, having its side AB perpendicular to the hori^ntal pume £M, On AB de8m B wil' spout to twice BI? =:AB=BM .- the liquid from CorC will spout to JEr= twice CCarCiy and that tcoin JJ or D will reach iC=twio e DD or jyjff, 334. The horizontal distance to which the liquid spouts under these circumstances may be found as follows : Let if = height of water above horizontal plane, (2 = per- pendicular let fall to the orifice from the circumference A E' B, and h z= height of orifice above the horizontal plane. Then (tiuclid iii. 35) d*=h (H— h) and hence d = ^/h{H — h) Thus if the reservoir in Fig. 29 be 20 feet in height and be filled with water and the apertures C, B, and D, be respectively 4, 10 and 15 feet above the plane B M; then the segments of A B are respectively 4 and 16, 10 and 10 , and 1 5 and 6 feet and the rando ms will be respectively 2 X V^ X 16, 2 X V^O X 10 and 2 X Vl6 X 6 i.e. 2 X V^ or 16 ft. 2 X '^Jl^ov 20 ft. and 2 X yl*lh or lT-32 ft. 386. When water flows in any bed, as in the channel of a river or in a pijpe, the velocity becomes constant when the length of the bed bears a large proportion to its sec- tional area. Thus in pipes of more than 100 feet in len^h or in rivers whose course is unopposed by natural obstacles, the velocity of the body of the stream is the same through- out. When this occurs the liquid is said to be in train. iBTS. 389-^85. ures in the idom from the centre ire height the centre ..^- v/\ Inr ill iL=twioe [uid spouts >ws : 5, d =r per- nce A E' B, me. Then ) i^htand be sspectively gments of 6 feet and VlO X 10 20 ft. and B channel ant when ;o its sec- in leo^h obBtacTesy through- I train. ABT8.83ft-839.] WATER WHEELS. 151 336. The velocity of the liquid flowing in a pipe or channel is not the same in every part of its section^ being greatest in the centre of the section of the pipe or in the middle of the sur&ce of the stream. Kom l.— Thig arises flrom the friction exerted against the fluid by the interior sorflMM of the pijpe or the banks of the stream. In a stream, on accoant of the middle piurt having the greatest velodl^, the surface is always more or leiw oonyex. NoTS St— The vdodty of a stream may be determined in three different ways:— 1st. An open tiibe bent at right angles is placed in a stream with one of its legs opposed to the current and the other branch vertical^the velocity of the stream is measured by the height to which the water rises in the vertical leg. 2nd. A float is thrown into the stroam and the time occupied by it in passing over a known distanoe observed. Srd. The convexifiy of the surface may be measured by a levelling instru- ment, and itst^'-slooity thus determined. 837. To find the velocity of efflux^ and hence the quantity of water discharged in a given time from a reservoir of given height through a pipe of given length and diameter ;-— Let d = diameter of pipe, I =: lengthy h =: height ^ and v = vefoei^y. Then^ all the dimensions being in feet ^ v = 48V \ ( NoTB.— This is the formula of M. Foncelet and is regarded as strictly accurate. WATER WHEELS. 838. Water is frequently made to drive machinery by its weight or momentum exerted on a vertical water-wheel. 839. There are three varieties of vertical water-wheels, viz : the underskotj the overshot^ and the breast wheel. Fig. 30. 9MABT WHIII). ]^ ^ 152 UPRIGHT WATER WHEELS. [Aet. 840. 841. Fig. 31. Fig. 32. CNDBRSHOT WHBBL. OTBRSHOT WHIBL. Note. — The mode in which the water is made to act on these is represen- ted in Figs. 29, 80, 31. It will be observed that the undershot wheel is moved by the momentum of the water— the bruast wheel and overshot wheel by its weight aided by its momentum. An overshot wheel will produce twice the eftiect of an undershot wheel, the dimensions, fall, and quantity of water being the same. The breast wheel is found to consume twice the quantity of water required by an overshot wheel to do the same work. 340. In all water-wheels the greatest mechanical effect 18 produced when the velocity of the water is 2^ times that of the wheel. 341 To jfind the horse powers of a vertical water- wheel : — Let b = breadth of stream in feet^ d = depth of stream^ V = mean velocity in feet of stream per minute j h = height of fatly s = weight of one cubic foot of water j and m = modulus of the wheel. mbdvsh Then horse powers =r ,.q - ExAMPLB 412. — A water-wheel is worked by a stream 6 feet wide and 3 feet deep, the velocity of the water is 22 feet per minute, and the height of the fall 30 feet, required the horso powers of the wheel, the modulus being *7. H.P.=: SOLUTION. mbdvsh 6X8X22X30X62'6X '7 = Wl6.Ans, 33000 83000 ExAMPLB 413. — ^What is the horse powers of a water-wheel worked by a stream 2 feet deep, 1 feet wide and having a velocity of 33 feet per minute — the fall being 10 feet and modulus of the wheel '6 ? SOLUTION. ir.P.s: mbdvsh 'B X7 X2 X 88 X BTS X 10 _ 88000 88000 4*25 Ans, KT. 840, 841. 18 represen- 1 is moved by wheel by its oduce twice quantity of ne twice the e work. ical effect 2i times ;al watcr- of stream f ght of folly iiltu of the earn 6 feet 22 feet per I the horse 6. An», ater-wheel : a velocity ulus of the 4*26 Am, Art. 342.3 TURBINE WHEELS, 158 ExAKPLi 414. — A water reservoir is 100 feet in height, supplies water to a city by a pipe 10000 feet in length and 6 inches in diameter, what is the velocity per second and what quantity will be =48V) 100 Xi Z + 54d ( — "^^V J ib0OO4-64X i -i- 48 '-52- V10027 =3*36 feet ])er second =velooity. Quantity discharged in 1 second =8*1416 X (i)« X 3'36. Quantity discharged la i hours = 3*1416 X ^ X 8*36 X 60 X 60 X 34= 57001*1904 cubic feet. il?w. • BXBROISB. 415. A water-wheel is worked by a stream 4 feet wide and 3 feet deep, the velocity of the water is 29 feet per minute, the fall 20 feet ; required the horse powers of the wheel, its modulus being -56 ? Ans. hzB. 416. A water-wheel is worked by a stream 2 feet deep and 4 feet wide, and having a velocity of 50 feet per minute, the fall is 33 feet and the modulus *84, how many cubic feet of water per hour vrill this wheel raise from the depth of 44 feet?- ^n«. 15120. 417. A water-wheel is worked by a stream 4 feet wide and 3 feet deep, the velocity of the water being 16 feet per minute, and the fall 27 feet, how many gallons of water per hour will this wheel raise to a height of 80 feet, the modulus being *8 7 Am, 18225 gallons. 418. A water reservoir 80 feet in height supplies water to a city through a pipe 1742 feet in length and 4 inches in diameter what is the velocity of the water per second and how many gallons will it deliver in 10 hours ? Ana. 115925*04 gallons. 342. The turbine is a horizontal water wheel having a vertical axle. It revolves entirely submerged, and is of all varieties of water wheels the most economical and pow- erful. It was invented by M. Fourneyron in 1827, but has since been much modified. in form and greatly im- proved. The water enters at the centre of the wheel, descends in its vertical axis and is delivered by a great number of curved guides so as to strike the buckets in a direction nearly tangential to the circumference of the 164 TUKBINE WHBiBLS. [ASTS. 849-846. ;(' ~ >.i'->i wheel. The buckets are also curved in the direction re- quired to give the tuachine the greatest possible amount of efficiency. The water having expended its force es- capes from the wheel in a direction corresponding very nearly with the radii. 343. Turbine wheels may be divided into high pres- sure and low pressure machines. 344. High pressure turbines are such as are worked by a high fadl of water, and iwe adapted to hilly countries and deep mines, where the height of the fall may be made to compensate for the smallness of the volun^e of water. ^ 345. Low pressure turbines are employed where a large stream of water possesses but little fall ; they are said to produce powerful effects with a head of water of but nine inches. 346. A committee of investigation appointed by the French Academy of Sciences, and consisting of Arago, Frony and others, gave the following report on these wheels : — I. Turbines are equally applicable to high or low falls of water. _. II. Their effective work is from 70 to 78 per cent, of the \:, work applied. (Turbines made by Boyden of Boston, have given 88 per cent, of the work ap- plied). III. They work without much loss of power at velocities both above and below that required to produce the maximum effect. IV. They will work without appreciable loss at a depth of from 4 to 6 feet beneath the surface of water. NoTii.— In another modification of thete horicontal wheeli the water is made to apply at the periphery of the wheel. Many varieties are patented and highly spoken of ais to their elQ^eotite performance, TS. JMa-846. I Abts. 8*7-3610 THEORY OP UNDULATIONS. 155 iction re- amount force es- ling very [gli pres- ) worked countries be made water. where a V are said er of but d by the >f Arago, on these w falls of at. of the oydeu of work ap- velocities ) produce a depth of rater. the water is wepfttentecl * ■*nr-?:-^ CHAPTER IX. THEORY OP UNDULATIONS. 347. All undulations or waves have their origin in vibratory or oscillatory movements imparted to the mole- cules of the solid, liquid, or gaseous body in which tho UDdulations occur. d48. Undulations are of two kinds, 1st. Progressive undulations. ; \ ^ , . ■,. > 2nd. Stationary undulations. '" 349. Progressive undulations are such as are produced by the vibratory movement passing from the particles first affected to those next them, and the oscillation being thus communicated successively from particle to particle, the wave advances with a progressive movement. As fkmiliar illustrations of this kind of undulatory movement, we may mention the waves produced on water by the wind, or by casting a stone on its surface, and those produced in a cord made fast at one end, by smartly shaking the other end up and down. In the latter case, a wave- like movement u observed to pass firom the hand to the fast end of the cord, and then a similar wave returns to the hand. NoTB.— It must be carefully remembered that although the wave ad- vances, the particles by whose vibration it is produced have themselves no progressive motion, but a mere oscillatory movement up and down like that of a penduluir . Thus in the case of the cord, the particles of matter that compose the curd do not themselves recede firom the hand and advance to it. And that there is no progressive forward movement in the particles of water producing water-waves is evidenced by the fi&ct, that a~float placed on the surface of the water simply rises and falls with the wave but does not move forward with it. 360. Stationary undulations are such as are produced when all the particles of a body are made to assume and to complete these vibrations at the same times. Thus when a cord or a wire is stretched between two fixed points, and is made to vibrate by drawing it at the middle from its rectilinear position, it recovers its normal concution after performing a series of unaiUations in which all the particles of the cord or wire take part. 361. In every undula- tion certain parts are to be Pig. 33. distinguished as follows: — * The curve a i 6 e c, is cal led an undulation wave. The part adb^ is the Z. phase of elevation. The part h e c^ is the phase of depression. •5 156 THEORY OF UNDULATIONS. [Abis. 362-864. m"^' The distance a c, is the length of the wave. ^ 5.^- V The distance d g, is its height. The distance /«, is its depth. Twice d g^ op fe, is its amplitude, 362. The vibration of solid bodies may be conveniently considered under the heads of cords, rods, planes and masses. Stretched strings, wires or other linear Fig. 34. solids, are susceptible of three kinds of vi- bration, viz. : 1st. Transverse vibrations. ; 2nd. Longitudinal vibrations^ Srd. Torsional vibrations. Thus if a cord be secured at one end and held stretch- ed by a weight attached to the other as in Fig. 34, then Ist. Upon drawing the string to one side and suddenly letting it go the vibrations which it makes and which are represented by the dotted lines are at right angles to the axis of the cord and are called trans- verse vibrations. 2nd. If the ball B be raised a little and suddenly dropped, it will continue for some time advancing and receding from its original position, the cord performing a series of longU/itdinal vibrations. 3rd. If the ball be turned round its vertical axis several times, and then let go. the cord vrill for some time continue to twist and untwist, thus performing a series of ^f>r«i , I. The vibratier of rods then we may consider the no- dal lines will be made up of their nodal points. The plate is divided by the nodal lines into intemodal spaces^ the acy&' cent spaces being always in opposite phases as shewn in the Fig. 86. where the sign -^ indicates the phauae of elevation and the sign — the pbase of depression. 369. The nodal lines vary in number and position ac-^ cording to the form of the plate, its size, its elasticity, the rapidity of the vibrations, the mode in which they are produced, the point by which the plate is fixed, &c. Their position may be determined by scattering sand or colored powder on the plate and vibrating by means of a violin- bow j — the sand is thrown off the internodes and arranges itself along the nodal lines forming the so called nodal figures or acoustic figures^ 360. Nodal figures have a gfeat variety in their form but are generally very symmetrical. Several hundred have been figured. The accompanying illustration repre- sents a few of those obtained on square and circular plates. The plates are supposed to be fastened in a vice at the point a, and the violin-bow drawn over the edge at the point h. In figure III the finger is placed on the edge of the plate at a point 450 from 6, in IV at a point 60<* or 30° or 90^ from 6. In Y the finger is placed at w, . '■'•.-t'' V'^ -A-iV TS. 857-369< material^ and in- >y fasten - ntre and a B }»» the acya- .36. where le phase of »sition ac-^ iicity, the they are 3. Their T colored a violin- arranges ed nodal leit form hundred on repre- clrcular the point nt b. In it a point b. InY Aets.1360-362.1 undulations IN LIQUIDS. Fig. 37. 159 S^[111SS 4- a. a t f M + 861. The vibrations c^ elastic plat*^ v are performed ac- cording to the following lav/s : — I. The number of vibratums is independent ./* the breadth of the plate, II, The number of vibrations is proportional to the thick- ness of the plate, III. The thickness being the same, the number of vibra- tions varies inversely as the square of its length, NoTB.— The phtte is supposed to be, in each case, composed of the same lubstance. UNDULATIONS IN LIQUIDS. 862. Undulations in a liquid are caused by the vibra- tory morement of its molecules in such a manner that each particle describes a vertical circle, about the spot in which it may chance to be, revolving in the direction of the ad- vancing wave. This rotating movement among the par- ticles is progressively carried to the contiguous particles, so that different atoms will be describing different parts of their circular path at the same moment. Thus some will be at the point of highest elevation, forming the crest of the wave, others at the point of lowest depression forming the trough, and others at intermediate points. ■1 ■ it I I 160 UNDULATIONS IN LIQUIDS^ AsHs. 363,904. 1«l ■ I 1 The diameter of the vertical cirole described by a single particle is called the amplitude of the wave and is, in the case of ocean waves, often as much as 20 feet. It has been ascertained by experiment that a liquid is not disturbed by the undulations on its surface, to a depth greater than about 176 times the amplitude of the wave. 363. Progressive undulations striking against a solid surface are reflected and the angle of reflection is always equal to the angle of incidence. It follows from this law that : — 1st. If the wave be linear, i. e., if its crest is at right angles to its course and it meets a plane burface perpendicularly it will be reflected back in the same path; and if it meet the plane surface at an angle of 80", 40«>, 30*», &c., degrees it will be reflected on the other side of the perpendicular at the same angle. ^-^ 2nd. The rays of a wave originating in one focus of an el- liptical vessel are all reflected to the other xbcus. . 3rd. The rays of a wave propagated in the focus of a par- abola are all reflected in parallel lines. 4th. A line or wave impinging on a parabola has all its rays reflected to the focus of the parabola. 5th. If two parabolas face each other with their axes coincident, a system of circular waves originat- ing in one focus will be followed by a correspond- ing system having the other focus for their centre. 6th. When the rays of a circular wave impinge at right anglei< upon a plane surface they are reflected so as to form a circular wave having the same degree of curvature but in the opposite direction. 364. When two systems of waves originating in diffe- rent centres meet, ihey either combine or interfere and their interference may be either complete or partial. I. When two waves meet in the same phase, i. e. so that their elevations and depressions coincide they com- , bine and form a new wave having an amplitude equal to the sum of the amplitudes of the com- bining waves. II. If the two waves of equal amplitude meet in oppo- site phases, i. e. so tliat the depression of the one Bi?s.363,9d4. ABT8. 866-868.] ACOUSTICS. 161 ;iole is called ves, often m at a liquid is greater than ist a solid is always Q this law \ at right Qe surface 1 the same b an angle e reflected ^ the same is of an el- 3r I'ocus. . s of a par- has all its a. their axes 3 originat- orrespond- leir centre. ;e at right eflected so me degree n. ig in diflfe- erfere and rtial. . e. so that they corn- amplitude the com- t in oppo- of the one 'mllifj coincides with the elevation of the other they in- terfere, both waves disappear, and the liquid sur- face becomes perfectly horizontal. III. If two waves of imequal amplitude meet in opposite ._^ phases they partially interfere and the resulting ^ wave has an amplitude equal only to the difference between the amplitude of the meeting waves. UNDULATIONS IN ELASTIC FLUIDS. 366. All elastic fluids, such as atmospheric air, are sub- ject to surface undulations such as occur in liquids and these surface undulations are governed by the same laws. S66. When an elastic fluid is compressed and the com- pressing force is suddenly removed, the fluid expands be- yond its uormal dimensions, it then contracts, a second time expands, and thus continues, for some time, to pscillate alternately on each side of its original volume. The pul- sations or waves which are thus engendered in elastic fluids differ from the surface waves produced in the same fluid, and also from the waves that are peculiar to water and other non-elastic fluids in the following particulars. Ist. Aerial waves or undulations consist in the alternate rarefaction and condensation of the air or other gas and are hence called waves of rarefaction and waves of condensation ; and 2nd. Aerial waves are always spherical in form. 367. Aerial waves are influenced with respect to in- terference and combination by the same general laws as govern the surface wave of liquids (See Art. 364.) but we must bear in mind that the term rarefaction corresponds to phase of elevation, and condensation to phase of depression. CHAPTER X. ACOUSTICS. 368. Acoustics (Greek " Akoud" " to hear,") treats of sounds, their cause, production and nature, and the laws by which they are governed. * 4 ,•.-■' 162 ACOUSTICS. [Abts. 86^874. ?:>. 869. Sounds are sensations arising from impressions made upon the auditory nerve by waves or undulations in the surrounding medium, 370. All bodies producing sound are in a state of more or less rapid vibration and these vibrations impinging upon the atmosphere or other elastic medium, produce in it a scries of undulations of condensation and rarefaction. The vibrations of a stretched cord producing sound may be perceived by placing the finger ou it ; the vibrations of a sonorous plate by scatter- ing sand upon it. &c. 871. The intensity of the sound produced by a vibrat- ing sonorous body depends chiefly upon two circum* stances : — ""• Ist. The density of the surrounding medium, and 2nd. The rapidity of the vibration of the sonorous body. . ^ 872. Sound is not propagated at all in a vacuum, and the sound produced in atmospheric air by a vibrating sonorous body is much more intense than that produced in hydrogen and other gases of less density than air. On the other hand solid bodies, vapours, water and other liquids of greater density than air, transmit sound with increased energy. Sounds are not only much louder but can be heard to a much greater distance in water and solids than in air. Thus if the ear be applied to one end of a long beam of wood and the least tapping noise or even the scratch of a pin be applied to the other— the sound is distinctly perceived by the ear. The report of cannon is said to have been distinctly heard to the distance of 250 miles by applying the ear to the solid earth. If the ear be placed under the surface of water, and two pebbl 3s be knocked together, the sound conveyed to the ear is very loud and it .s said that if a cannon be fired close to a body of water in which a person has his head immersed, the report is sufficient to destroy his sense of hearing. 878. All sounds travel, in the same medium, with the same velocity, whatever may be their pitch or their strength. Were it not for this property of sound— the notes produced by the musi- cal instruments of an orchestra would be discordant, except to those in the immediate neighbourhood of the performers. Note.— It has lately been satisfkotorily shown that in the case of sounds differing very widely in intensity this is not strictly true,— very lAlense sounds travel rather more rapidly than others. 874. The velocity of sound in atmospheric air varies : lit With the temperature, decreasing about 1^^ ft. per LTS. 869-374. pressions iations in a of more ;ing upon 36 in it a 5tion. B perceived ) by scatter- a vibrat- circum- ), and sonorous uum, and vibrating produced I air. On ind other >und with mch greater >e applied to I or even the sly perceived Dtly heard to rth. If the be knocked Raid that if a has his head ing. I, with the A»TS. 376-378.] ACOUSTICS. 163 or their by the musi' those in the the case of f true,— very air varies: ^ ft. per second, for every degree, Fahr. the temperature is lowered. 2nd. With the velocity and direction of the wind. Note.— The intensity jof a sound as heard at a distance is much modified but its velocity is not affected by the condition of the air as to its being clear or foggy, the barometric pressure great or small, the sky clear or cloudy. 375. Accurate experiments have determined the velo- city of sound in atmospheric air at a temperature of 60° F., to be 1118 feet per second. 376. The velocity of sound in vapours and gases at 32^ 1^., has been determined from, calculation by Dulong to be as follows : — 860 feet per second. tt it u II u CT» Carbonic acid, ^ . , Alcohol vapour, 862 Oxygen, 1040 Olefiantgas, 1030 Air, 1092 Carbonic oxide, 1105 Water vapour, , . . 1347 Hydrogen, 4163 377. The following table gives the results of experi- ments made upon the velocity of sound in liquids and solids : — In Water, sound travels at rate of 4708 ft. per second. " « « 8386 " «« . « " " 11609 " " •' Tin, Cast Iron, u Copper, " " " 13416 Wood, " " " 16770 NoTB.— That is, in .^ater sound travels 44 times as fast as in air; in wood about 16 times, and in metals from 7 to 12 times as fast. 378. The distance to which sound may be propagated depends upon the following circumstances : — I. The greater the intensity of the sound the greater the distance to which it will travel. II. The denser the f •. or other conducting medium the greater the distance to which the sound will travel. III. In atmospheric air the distance to which the sound will travel is much influenced by the condition of the air as regards winds, &c. 164 ACOUSTICS. [Abts. 379*882. ■■' ' '\. " *& MIIjI I"' I 879. It has been experimentally ascertained that the following sounds may, under ordinary circumstances, be heard at the annexed distances. The human voice in the open air, 700 ft.\^ The marching of a company of soldiers at night, 2500 ft.^' The marching of a company or squadron of cavalry, 3000 ft. The report of a musket, 3000 ft. Note.— Lieut. Foster conversed with a man, in frosty weather, across the harbor of Port Bowen, a distance of li miles. Dr. Young states that the watchword " all's well " has been heard from Old to New Gibraltar, a dis- tance of 10 miles. The cannonading of a sea fight between the English and Dutch in 1672 was heard at Shrewsbury, a distance of 200 miles. The cannonading at the sie^eof Antwerp in 1832 was heard in the mines of Saxony, a distance of 320 miles. The noise produced by the volcanic eruption in Tombers in Sumbawawas heard at a distance of 900 miles. 380. When two series of sonorous undulations en- counter each other in opposite phases of vibration, they interfere, and, if the sound produced by each separately are equal, the interference will be complete, they will des- troy each other and produce silence. 381. The phenomenon of interference of sonorous waves so as to produce silence may be conveniently shown in the following manner : — Fig. 38. Take two tuning-forks of the same note, fasten to one prong of each a small disc of card board half an inch in diameter and make one fork rather heavier than the other by loading it with a little sealing wax at the end. Also take a glass jar about ten indhesin height and two inches in diameter. Then make one of the forks vibrate, and holding it lust above the mouth of the glass vessel as seen at a, Fig. 38 ; care- fully pour in water till the air in the jar vibrates in unison with the fork and the result will be the produc- tion of a prolonged uniform and clear sound without stop or cessation. When either fork is made to vibrate and is held alone over the jar, we obtain a uniform sound, but when both are made to vibrate and are at the same time held over the mouth of the jar there . t arise a series of sounds alternating with a series of si- lences, this alternation continuing as long as the forks are vibrating. The explanation is simply that tne long waves arising from one fork over- take the shorter waves produced by the other and alternately interfere and combine with them. The destruction of sonorous waves by interference may also be observed by holding a vibrating tuning*fork about a foot from the ear and gradually turning it round. When the prongs are equally distant firom the ear a note is heard, but when one is more distant thau the other partial or com- plete interference takes places and the sound dies out m part or alto- gether. 382. Soundwaves are reflected upon striking" any solid S. 379*882. that the ices, be rooft. 500 ft/'" aoo ft. }00 ft. across the i that the tar, a dis- e English iles. the mines ibawawas M18 en- )n, they paralely v^ill des- 18 waves Q in the SftI Arts. 38^385.] ACOUSTICS. 165 Injr. fork over- r interfere > observed gradually the ear a a or oom- rt or alto- ny solid or liquid surface according to the laws enumerated in Art. 363. Note.— A certain portion of the sound enters the second medium and undergoes refhiction. ^ 883. An echo is a sound reflected by a surface suffi- ciently distant to allow a short space of time to intervene between the direct and the reflected soitnds in order that these may not be confounded. 884. The ^ar cannot distinguish one sound from an other unless there be an interval between the two at least one-ninth of a second. In one-ninth of a second sound travels 124 feet (1118 -r- 9) so that a perfect echo cannot exist unless there be at least 62 feet (half of 124) between the ear and the reflecting surface. If a sentence be repeated in a loud voice at the distance of 62 feet from a reflecting wall the last syllable will be distinctly echoed ; if at the dis- tance of 124 ft. the last two syllables ; if at the distance of 186 ft. (62X3) the last three syllables, &c. If the reflecting wall is at a less distance than 62 ft. from the speaker, the reflected sounds blend with the emitted so as to prolong and strength- en them. This is expressed by the term resonance. Hangings, draperies, carpets, &c., about a room tend to smother or stifle the sounds as they are bad reflectors. A crowded audience has a similar effect— increasing the difficulty of speaking by presenting non-reflecting surfaces. If .i person stands 1118 feet fh)m a reflecting surface ^and articulates loudly at the rate of four syllables per second the echo will repeat the last eight syllables clearly ; because the sound will require two seconds to tra- vel to the reflecting surface and back to the ear and in two seconds he gives utterance to eight syllables. 385. Repeating or multiplying echoes are those that repeat the same sound several times. Such echoes com- monly occur where parallel walls or other obstacles are . placed opposite each other at a suflScient distance apart to reflect the sonorous undulations alternately from side to side. In a multiplying echo each repetition is less loud because the reflected wave is always more feeble than the direct wave, so that intensity is lost by each reflection until the sonorous undulations become incapable of con- veying any impression to the ear. 386. Betnarkable Echoes.— There is an echo at Verch^res between two towers that repeats a word thirteen times. At Ademach in Bohemia there is an echo which repeats seven syllables three times distinctly. At Lurleyfels on the Rhine there is an echo which repeats seventeen times. At Belvidere, Allegheny County, N, T., there is an echo between two bams which repeats distinctly a word'of one, two or three syllables eleven times. ' ' ' ".■ H*i! ; * < 166 ACOUSTICS. [ASTS. 387-38{^. j-r At Woodstock in Eugland there is an echo which repeats seventeen times during the day and twenty times during the night. In the Villa Simoiietta, near Milan, there is an echo which repeats a sharp sound thirty times. The celebrated ancient echo of Metelli at Rome is reputed to have been capable of repeating the first line of the ^neid containing fifteen syllables eight times distinctly. 387. Whispering galleries are so called because a whis- per uttered at one point may be distinctly heard in some other remote locality although quite inaudible in all other positions. They are generally domed or are of an ellipsoi- dal form — the point of utterance and the focus of reflec- tion being tl.^ two foci of the ellipse (compare Art. 363). The most remarkable whispering galleries in the world are the follow- ing :— The gallery beneath the dome of St. Paul's Cathedral in London. The Gothic vault of the Cathedral at Gloucester. A Church at Girgeuti in Sicily in which a whisper near the door is dis- tinctly heard at the remote end of the Church 200 feet distant. The Grotto di Favella, at Syracuse, (supposed to be the celebrated Ear of Dionvsius.) The dome of the rotunda of the Capitol at Washington, &c 388. The speaking trumpet is an instrument designed to enable the human voice to be heard to a great distance. Its efficacy is due to the fact that the confined column of air is made to vibrate in unison with the voice and hence the pulsations that impinge upon the exterior air, have a greater energy and give rise to sonorous waves of greater intensity. It has been satisfactorily shown by Hassenfratz that the old explanation by reflection of the rays of sound is inadmissible. This is proved b;^ the fact that the power or the instrument is not impaired by lining its inte- rior with linen, a very bad reflector, or by making the trumpet in the form of a cylinder provided with a belUshaped extremity. The shape of the ex- tremity exerts an unexplained influence upon the action of the instrument. The sound emitted through the trumpet is increased in all directions, i. e., not merely in the quarter to which it is pointed. 389. The ear trumpet is designed to enable partially deat people to distinguish sounds more distinctly. It acts upon the principle that the portion of the sonorous wave that enters the large end of the instrument imparts its energy to portions of air smaller and smaller and conse- quently causes it to vibrate or pulsate with more intensity as it approaches the ear. We have an illustration of something of the same kind of concentration when we attach a weight to a string and cause it to wind rapidly round the finger \ the revolutioni become more npid m the ttring shortens. t-i 887-3881. in timeft a sharp ivebeen yllables whis- 8ome other lipsoi- reflec- 363). e follow- or is dis* kted Ear jsigned stance, umn of hence have a greater )1anation id by the ; its inte- the form >f the ex- trument. ions, i. e., artially It acts 18 wave arts its conse- I tensity miration lly round ^ns. Aii*B. 860-366.3 MJEJCflAiTlCAL THilORt OF MtJSiC. 161 It was formerly customai^ to explain the action of the ear trumpet upon the principle of reflection of the rays or waves of sound. This explanation is diiiproved by the fact that so long as the extremity remote from the ear is much larger than that applied to that organ, it makes but little or no difrerence what may be the shape of the trumpet. It likewise makes no difference whether the interior surface is rough or polished. ■y : •» ■&<* CHAPTER XI. MECHANICAL THEORY OP MUSIC. 890. Noise is the effect of a series of sonorous undula- tions produced by unequal or irregular vibrations. The report of a gun, the crack of a whip, the rumble of a train of cars or of a carriage on a stone road, &c., are familiar examples of noises. 391. Musical sounds are the result of sonorous waves produced by equal or regular vibrations. 1 892. Every sound has thr^e distinct qualities distin- guishable by the ear, viz. : — ; T. The pitch or tone. ' ' / . - II. The intensity. * ' III. The quality or timbre. ' '''" - 898. The tone or pitch of a sound is high or low and depends upon the rapidity of the vibratory movement pro- ducing the sonorous undulation. The more rapid the vib- rations are, the higher will be the pitch of the note. '"■ 894. The intensity or loudness of the isound depends upon the amplitude of the vibrations which produce the sonorous wave, or what amounts to the same thing, upon the degree of condensation produced in the middle of the sonorous undulation. NoTB.~A sound may maintain the same pitch, i. e.^ the same length of wave and yet vary in intensity. 895. The quality or timbre of a souadis that property or peculiarity which enables us to distinguish it from all other sounds of the same pitch and intensity. Thus if a flute, a piano, a violin, and a clarionet, all sound a note of the same pitch and with the same intensity, we can readily distibguish the sound produced by each. 896. Sounds produced by the sRme number of vibra- tions per Bocond, are said to be in unison. ** . %w - »»♦ 168 MECHANICAL THEORY OF MUSIC. [AETa.a»7-406. rr 'M I? m^ 897. A melody is a succession of single mnsical sounds which bear to each other such simple relations afi are readily perceived by the ear, and which consequently pro- duce an agreeable impression. 388. A chord consists of two or more melodious sounds produced simultaneously. -— - 899. A harmonized passage consists of a succession of chords wllowing one another in melodious order. To a cultivated ear a riue of bells js musical or noiqr according as its tones are musical or unmusical iiit6mls; ft i^ hturmonious or discordant according as the intervids are concords or dissonances ; it will be " cheer- fill " or '* sad " according as the intervals producing the concordances are major or minor. 400. The instruments used for determining the number of vibrations performed by a sonorous body giving a tone of definite pitch, are the Siren and Savarfs toothed wheeU The essential parts of the Sliren are a brass tube about 4 inches in dia> meter, terminatmg in a smooth brass plate which has about twenty small holes pierced obliquely near its circumference. A sdQOnd thick plate having the same number of equidistant holes, but pierced obliqudy in the reverse direction, is supported just above the first plate ih such a man- ner as to revolve with extreme ease. At the upper extremity of the verti- cal axis which bears the second plate, there is an endless screw, which acts upon a counter, like that on a gas meter. The lower part of the tube bearing the first plate, terminates in au air chamber whi6h ii kept filled with uniformly compressed air by a double acting bellows. When a current of air arrives flrom the bellows it passes through the holes in the first plate, and in escaping through the second plate impart to the latter a rotary motion. As the upper plate revolves the avenues of escape for tlie compressed air are rapimy cut off and renewed, and consequently wiien the plate revolves regularly and with sufficient rapidity, sonorous undulations are ])roduced in the exterior air by the minuite puffs of wind that escape at uniform intervals through the plates,— the sound increasing in acuteness as the velocity of the revolving disc becomes greater. The rapidity of the revolution is governed by the degree of pressure to which the air in the chamber is subjected. Savart's toothed wheel consists of a large wheel connected by means of an endless band with the axle of a smaller toothed wheel, the cogs of which are made to touch in succession a small tongue or slip of metal, thus causing it to vibrate. The number of revolutions made oy the toothed wheel IS recorded by an attached s^'stem of clock-work; and thd muttlwr of vibrations made by the tongue is found of course by simply multiplying the number of revolutions by the number of teeth in the wheel. It Is, per- haps, unnecessary to remark that the more rapid the revolution of the wheel the more rapid is the vibration of the tongue, and consequently the higher the pitch of the note produced. Each tooth causes the tongue to make two movements, i. e., one down and the other up. each of these is called a single vibration, and the two together a double vibration. Both the Siren and Savart's toothed wheel act upon the recognized prin- ciple that two sounds are in unison when they are produced by the ciame number of vibrations per second. The instrument is made to.revolVe more or less rapidly till it is brought in unison with the sound experi- mented on when the rate of vibration is at once obtained flromtlle dM ftee. ^i (.897-400* AiitB.401-403.3 MECHANICAL THEORY OP MUSIC. 169' sounds as are ;ly pro- 'I fA sounds (cession ng as its scordant " cheer- knees are lumber a tone toothed 3S in dia- twenl^ icktilata qudbr in laman- he verti- hichacts the tube i) kept When a 38 in the lie latter icape for equently sonorous of wind creasing er. The o which oeans of cogs of tal,thus toothed uakerof tiidying kfs,p6r- 3 of the iitiy the ngneto these is sdprin- le same revolVe eypeii- ihedfal 401. The Monochord or Sonometer is an instrument; used to study the taransverse vibrations of cords and, hence, the relation that subsists as regards number of vibrations, &c^ between the several notes of the musical scale. tig. 39. •jp c^s jt=sx: T =35= w \*ir v: ■ V ■ :■» 4- -. The monochord consists of a thin wooden case SS' above which a metallic wire or a cord of catgut FTF' is stretched over the pulley M by the weight P. A movable bridge HH' can be placed at any desired point between the filed biKdges F and F. The weight P is commonly adjustied so that the string or wire when vibfating its whole length shall give the note G. 402. If the whole length of cord vibrating produces the note C it is found by eicperiment that when f of its length vibrate the note D is produced ; f of its length vi- brating give the note Ej &Ci, and since (Art. ) the number of vibrations varies inversely as the length of the string these' fractions inverted give the number of vibrations necessary to produce the notes I), E, &c., as compared with C. The following table gives the rela- tive letigths of cord producing the notes of the common diatonic scale and the relative numbers of vibrations per second belonging to them. ^. Relative lengths of cords, Relative number of vibrations. C do I D re t E mi n 5 4 P fa Q sol ^ i A la il B C do i 403. It is common to indicate the different scales in use by means of indices attached to the various notes. Thus the fundamental C which corresponds to the highest sound of the base, is represented by C*, the successive higher octaves by C*, C», C*, &c., and the successive lower octaves by C-" , C"^, C-». l: 170 MECHANICAL THEOET OF MUSICi [Astg. Ml-MA. m pi:' I', n§ ^ i' ■ c. 404. The absolute number of vibrations corresponding to any given note can easily be determined by setting the Siren or Savart's wheel in unison with it. It has been thus ascertained that the fundamental C is produced by 128 simple vibrations per second, and by multiplying this successively by f , {, f , f , f , &c., we obtain the number of vibrations for the other notes as given in the following table :— Notes CDBPGABO Absolute number of ) . simple vibrations > 128, 144, 160, 170§, 192, 213^, 240, 256. per second. ) 405. The number of vibrations corresponding to the several notes of any superior gamut, is found by multi- plying the above numbers by 2, 4, 8, &c., and for the inferior gamut by dividing by 2, 4, 8, &c. Thus A3 = 213^ X 4= 853^ simple vibrations =426| complete vibrations. C3=128 X 4=512 " " =266 A-2 = 213^-7-4= 63^ " " = 26§ 0-2 = 128 -r4= 32 " " = 16 " " Note.— There is a slight difference in the actual number of vibrations producing a particular note as performed in different cities. Thus the number of viorations required to produce A3 varies as follows :— ^^ .^i Theoretical number, 426|. Orchestra of Berlin Opera, 437^. Opera Comique, Paris, 427^. Academic de la musique, Paris 431 j. * ^ • '^ | Italian Opera, (1855), 449. The General Musical Congress which met in London last year (1860) to consider the propriety of adopting a uniform musical pitch, fixed upon the number 628 complete vibrations for C3, 440 for A3. The commission recently appointed in France have recommended C3 r= 522 ; = A3 = 435. In the report submitted by this committee the folloW' ing pitches were discussed :— Handel's Tuning Fork (c. 1740) A at 416 = C at 499^. Theoretical Pitch, A at 426^ = C at 512. Philharmonic Society (1812-42) A at 433 =0 at 518^. Diapason Normal (Paris. 1859), A at 435 = C at 522. Stuttgard Congress (1834), A at 440 = C at 528. Italian Opera, London, (1859) A at 455 = C at 546. Piano-fortes for private purposes are usually tuiied somewhat below concert pitch, so that A3 is produced by about 420 complete vibrations per second. 406. The length of a sonorous vibration is found by dividing 1120 feet, the velocity of sound per second, by the number of vibrations made per second, in order to pro- bs. 40^-406. ■ ■'— ■ -^ ': z jsponding Itting the (has been luced by lying this number [following B - i 1240, 256. ig to the by multi- d for the te vibrations. <« • -r ' M - i tTf* -^ of vibrations es. Thus the rs:— year (1860) to ixed upon the landed 03 = !ethe folloW' Ants. 407-409.3 MECHANICAL THEORY OF MUSIC. 171 f^-^Vf*i('i'n, mn ewhat below ibrations per found by econd, by ier to pro- duce the note. The following table gives the wave-length of the C of diflferent scales : — Notes. C-3.. C-2.. d .. c« .. c* .. c« .. C6 .. Simple Vibrations per second. 16 32 64 128 256 512 1024 2048 4096 "Wave-lengfths in feet. 70. 36. 17i. 407. Interval indicates how much one sound is higher than another in pitch, and is of course greater or less as the difference in the number of vibrations, producing the two sounds, is greater or less. 408. Musical intervals are named thirds, fourths, fifths, &c., from the position of the higher note counting up- wards from the lower, according to the following table, in which the first line gives the name of the note ; the second line, the number of its vibrations, as compared with the first note; the third line, the name of the inter- val ; and the fourth line, the interval obtained by dividing each note by that which precedes it. C" c D E F G A B C D' E' F' G' A' B' 1 g f i f ^ Y 2 ¥ K"- f f ¥• ¥ Ist. 2nd. 3rd. 4th. 6th. 6th. 7th. 8th. 9th. 10th. 11th. 12th. 13th 14th. 16th. f ^9^ i^ f ¥ I if § Y if f ¥ f it Note.— The second line of this table must be interpreted thusr—In order to produce the second note, D, 9 vibrations n^^ist be made in the same time required by 8 vibrsitions giving the first no^/C C ; in order to obtain the third note E, 5 vibrations must be in time required by 4 of the first note C and so on ; or, taking 24 the least common denommators of the fractions, while the vibrations producing the first note G number 24, those required to produce the successive following notes will be 27* 30, 32, 36, 40, 45. 48, 60. 72. 80, 90, and 96. 409. In examining the foregoing table two points must be carefully noted. I. There are but three different intervals between the successive notes of the scale, viz., f, y and if. II. These intervals occur in the same order in each successive octave. I 172 MEOHAMICAT^ THEORY OF MUSIC. [Asis. 410, 411. 1% * i-r- J The interval f, bti; *? iLe largest interval found in the scale, is called a major tone; y is called a minor tone, and |f is called a semitone^ although it is greater than one-half of either a major or a minor tone. Note.— The internal-]-^ Uf^oqueiMyaipokenot as tk diatonic semitone; the difference between a major tone and the diatonic semitbne, i. e., f — if ^^ TsTT ^^ coiled a chromatic semitone; the difference between a minor tone and the diatonic semitone, i. e., J^ — -^ = -^^ is called a grave chromatic semitone ; the difference between a major tone and a minor tone, i. e., |^ •— l^ = .^t. is called a comma, 410. The following table exhibits all the intervals that occur in comparing the notes of the common scale two and two.-' -:-•>■■<> '?-.-^ -.i- ■f^.-^r- -", ■•'••-"' '••■■.'-• . (O..D = F..Gt=A..B • }d..b = g..a ^B..F=B..C CC..B = F..A=rG..B >B..G = A..C'=B..D' = f , a major tone. = ^, a minor tone = ^\ ef f . = ^y diatonic semitone = |^ of ^ori^off^off. = f , a major third. = f , a minor third = if of | D;.F = f^ of a minor third = f? of ^ = Mof ftoff. CO.. F=D . . Gr=E . . A=r G . . C = f , a perfect fourth. ^A..D^ F..B C..G=B..B=F..C'=G..D' = A..B' D..A B..F = 1^, a sharp fourth = f^ of f . = f« = itf 0^ a perfect fourth = If of U of 'J. = f, a perfect fifth: ■^. = ^ = f ? of a perfedt fifih; =zf^, an inharmonious interval. \ . . A = D. . B=F . . D'=!G. . B = f , a perfect sixth. A..F' = B..G' F..D^ O..B=P..B^ D..C' = G..F'=B..A^ B..D' = A..G' CO' = f , a minor fifth = f f of f . rsfl, an inharmonious interval. =:-^, a seventh, an inharmo- nious interval. 3s -^g^, a flattened seventh, more harmonious than the perr feet seventh. = ^,a minor seventh = || of ^. = f , an octave. 411. Compound chords consist of three or four notes whose vibrations have a simple numerical relation to one another, ^n:d which tak< ducehannony,"* "-*"" ITS. 410, 411. found in SL minor is greater (C aeimione; nitbne, i. e., B bcjtween a p- is called a tone and a rvala that scale two ones: 1^ of off. = ifof| •d = If of )f i. ih. = Moff feet fourth t. .■■.-."/:,.■ Ctfifthr ''' a interval. = iioff. 8 interval. I inharmo- enth| more ,n the per- = It of ^. bur notes lation to two^ pro- AETs. 412-415.3 MECHANICAL THEORY OP MUSIC. 173 The Perfect Major Record consists of the three notes C, E and G, whose vibrations are to each other as the num- bers 4, 5 and 6, and which compared together two and two give the relations {, f and f . The Perfect Minor Accord consists of the three notes E, G and B, whose vibra- tions are as the numbers 10, 12 and 15, and which give the relations |, J and |. < Note. —The Intervals of the perfect minor differ firom those of the per- fect major accord only hi their order. 41:2. Any tone whatever in the common scale or any pitch whatever, may be adopted as the basis of another similar scale, provided means be employed to preserve. the s^me relative intervals between the successive notes. When a piece of music is thus changed from one scale into another it is said to be transposed, and the process is called the transposition of scales, ■ ^ ' 413. In the transposition ofscales.it is found necessary to introduce additional notes, in order to paaintain the rela- tive intervals between the successive notes. Such addi- •tional notes are called sharps (Jf) And flats (]>) according as the tone oorresponding to any given note is raised or lowered. - « ^ 414. When these new notes are interpolated in every full tone (major or minor), of the diatonic. scale, the re- . suit is a series of twelve intervals in the octavQ, forming \ what is known as the chromatic scale, 415. Temperament is an artifice by means of which the ; introduction of an inconveniently large number of addi- { tional notes into the scale is prevented. lu the transfor- mation of scales it IS assumed that. every note may be-raised or lowered by a diatonic semitone |f, but in order actu- ally to raise and lower each tone by that amount, we- would require a very great number of new notes. To pre- ' vent this, such notes as Cj( and D\) are regarded as iden- tical, though in reality they differ from one anothei slightly, and are played differently on stringed instruments, as the harp and violin by Skilful players. For practical purposes musical instruments such as pianos, organs, the fifths. we iiave to ibove or be- affering the per tono is iveone, and L to any of 1 be so uapea- almost to pie- produced at nbleness are :r at regular le vibrations between ibe Dccurs; aod ABTs. 417-^21.] MECHANICAL THEORY OP MUSIC. 175 when the vibrations differ widely in rapidity, they pro- duce merely an unpleasant rattle. 417. The tuning fork or diapason is a two-pronged steel fork of peculiar form, by means of which we can produce an invariable note. It is commonly formed to give A^, corresponding to 428 vibrations per second, but may be made so as to give any other note of the gamut. It is much used as a standard in tuning instruments, or striking the key note in vocal music, &c. NoTB.— The note ^ven bv the diapason is much strengthened by mount* ing it on a box of thm wood open at one end. MUSICAL INSTRUMENTS. 418. Musical instruments may be for the most part divided into I. Wind instruments. n. Stringed instruments. in. Instruments of which the essential part is a stretch- ed membrane. 419. Wind instruments are sounded either by an em- bouchure like the flute, organ, pipe, flageolet, &c., or by reeds as in the Jew's-harp, clarionet, vnelodeon, horns, trumpets, trombones, &c. 420. Stringed instruments are all compound — ^the sounds produced by the vibrating string being strengthened by elastic plates of wood or meial and inclosed portions of air to which the cords transmit their own vibrationn. Stringed instraments are i)layed I. By a bow as in the violin. II. By percussion as in the piano, or III. By twanging as in the harp. 421. The third class of musical instruments includes drums, tamborines, Src Drums are of three kinds ; the small drum or common regimental drum which is a brass cylinder having both heads covered with membrane but beaten only at one end ; the base or double drum which is much larger aad which is beaten at both heads; and the kettlo drum which is a hemispherical copper vessel supported c u a iripod and having its head covered with ^4 1J6 MECHANICAL THEORY OF MUSIC. CAbt8.^A2.4M vellam. The kettle drum has an opening in the metalKc case to equalize the vibrations. 422. In all wind instruments the sounds are produced by throwing the column of air contained in tubes into vi- bration. The pitch of the sound produced depends upon: — .. 1st. The length of the tube containing the air. 2nd. The position and dimensions of the embouchure. 3rd. The manner of imparting the primary motion to tiie air. The diflference of quality belonging to the notes given by pipes of different materials is due most likely to a feeble vibration of the sides of the tube. 423. Sonorous vibrations are produced in tubes I. By blowing obliquely into the open end of the tube, lis In the Pandean pipe. II. By casting a current of air into an embouchure near the closed end of the tube as in mouth pipes. III. By thin laminae of metal or wood placed at the end of the tube and which vibrate as the current of air sweeps past. These lamina) are called reeds, IV. By the lips acting as reeds. - v^ V. By a small burning jet of hydrogen gas. r. %r 424. The laws that govern the vibration of air in tubes were investigated by Bernoulli in 17.82. He divides all tubes into two classes : Ist. Tubes having the extremity opposite the mouth closed. 2nd. Tubes open at both extremities. For tubes with the end remote from the mouth closed he gives the following laws : — I. The same tube may produce different sounds and in this case the number of vibrations will be to each other as the odd numbers 1, 3, 6, 7, 9, roduced into vi- depends uchure. LOtion to es given cely to a )e3 the tube, Ibouchure id at the 3nt of air :T -^ : . fv *"* ' ii ■. - :. Ti' ili'i, !«* [r in tubes divides all le iDouth th closed ds and in ich other Eime order )so are in is divided AElB.425-4a7.J THE OEGANS OF VOICE. 17T into equal parts which vibrate separately and in unison — the open orifice being always in the middle of a vibrating part and the length of a vibrating part equal to the length of a wave corresponding to the sound produced. For tubes open at both ends the foregoing laws prevail with the following modifications : I. The sounds produced are represented by the natural numbers 1, 2, 3, 4, 6, 6, 7, &c. / ^^ ., II. The fundamental sound of a tube open at both extremities is always the acute octave of the same sound in a tube closed at one extremity. III. The extremities of the tube are in the middle of a vibrating part. *!': 1,1 -,'»jV' ■:'-U- CHAPTER XII. ,,i,f^. THE ORGANS OP VOICE AND HEARING. ^hfi'V i THB ORGANS OF VOICE. 426* Many animals have the power of producing sounds and as a general rule those that are endowed with a voice have also the organ of hearing well developed. Man alone, howeverj possesses the gift of speech, i. e. the power of giving to the tones he utters a variety of definite articulate sounds. ' » • 426. The vocal apparatus of maa consists of the fol- lowing parts :-^ I. The Thorax which, by means of the intercostal muscles and the diaphragm, acts as a beliows ia producing a current of air for the production of sounds. II. The Windpipe which is a long tube carrying the air from the lungs to the organs more immediately con- cerned in forming the voice. III. The Larynx (Adam's apple) which is the musical organ of the voice and corresponds to the mouth-piece of a musical instrument. IV. The Pharynx — a large funnel-shuped cavity at the top of the larynz or at the back of the mouth, which 12 fei 178 THE OEGANS OF VOICE. AbT8. 428— 4S0. f • by varying in form and tension modifies the tones of the voice. --,>7/ir ■ iifi' .- Wi'. V. The mouth and nasal passages which correspond to the upper part of an organ tube and throw the vibrations into the air. - ^ I - ■ 'i n, * '\:i : i ">^iv^ - r' %t^Kr-^^> 427. The larynx is composed of the hi/oid bone, and its attached cartilages, viz. the two thyroid, which form the sides and front of the larynx and which constitute the prominence known as the pomum Adami — the cricoid which is ring-shaped and rests upon the top of the trachea, and the two arytenoid at the back of the larynx and between the two thyroid cartilages. The arytenoid cartil- ages are movable to a small extent by means of several muscles attached to them, 428. The Cord>i5 vocales or vocal cords are two ligaments, of elastic fibrous substance, which extend from the arytenoid cartilages behind to the thyroid cartilages in front. The ligaments meet in front but are somewhat separated behind so that when at rest they form an open- ing in the interior of the larynx shaped like a V ; but by the drawing together of the arytenoid cartilages the open end may be closed in such a manner that the two vocal cordfi shall toucli one another along their entire length and the aperture be completely closed. The opening between the vocal cords is called the rima glottir^h or fissure of the glottis. . - 429. The membrane which lines the interior of the larynx doubles bo as to form a second pair of folds just above the vocal cords. The space between these is much wider than that between the vocal (;ords and is covered during the act of deglutition by a valvo-like flap called the epiglottis. The space between the upper and lower pair of ligaments \^ called the glottis or the vc '^les of the larynx, 430. Except during the production of vocal sounds the arytenoid cartilages are wide apart «d the vocal cords wrinkled and plicated, but while the organs of voice are in action tho rrrna ghttidifi ia po narrowed that the 428--480. ABIS.431,432. TH^ ORGANS OF VOICE. 179 of the >ond to rations le, and jh form ute the cricoid •achea, DX and cartil- several lire two d from irtilagbS mewhat ,n open- but bv he open vocal 1 length opening tif^is or r of the Ids just is mucli covered > called d lower *les oj aounds e vocal of voice hat the H sides rather than the edges of the vocal cords are in con- tact and while the ligaments are thus in contact the air passing through the larynx sets them in vibration, some- what like the reed of a clarionet or the tongufe of a trumpet, and the result is the production of a sound. The pitch of the sound depends of course on the rapidity of the vibra- tions and this is governed by the length and the degree of tension of the vocal cords. The vocal cords are tightened or relaxed by means of the muscles that act on the thyroid and arytenoid cartilages. Note.— Some physiologists regard the return of the glottis in producing sound as analogous to that of a bird call. 431. One of the most remarkable circumstances in connection with the organs of voice and their action is the perfect precision with which the will can determine the degree of tension of these ligaments. Their average length while in repose is in the adult male about yVo ^^ ^" incn and in the adult female -{^-^ and when stretched to their utmost capacity their length is only //^ ^^ ^^® male and -j^ in the female. The extreme difference of length is therefore about \ of an inch in the male and about -J- of an inch in the female. The average compass of the cultivated voice is about two octaves or 24 semi- tones, a id as a practised singer can produce at least 10 distinct intervals within each semitone, the range of the voicQ may be said to be 240 note?. Each of these 240 notes corresponds to a different degree of tension of the vocal cords and as the utmost limits of tension are \ of an inch in the male and \ of an inch in the female it follows that in man the difference in length of the vocal cords required to pass from one interval to another will not be more than |y\,„ of an inch and in woman not more than ^oV? °^ ^" inch. NoTB. —It is said that the celebrated vocalist Madame Mara, M'as able to sound 100 different notes within each interval of the diatonic scale, and as the compass of hel* voice was 20 tones, the whole number of notos she could sountt was 2000, all cf course comprised within the extreme variation of \ of an inch. It may hence be said that she was capable of determining with precision the contraction of the vocal cords to the 1 6 l^o o of an inch. 482. The larynx is about the same size and conse- quently the vocal cords are about the same length m both 86X03 up to the age of 14 or 16 years, however from that 180 THE ORGANS Of VOICE. fAKls. 4^, 436. time it rapidly increases in size in tbe male, but remains stationary in the female. It is owing to this greater length of the vocal cords that the pitch of a man's voice is lower than that of a woman, or of a girl, or of a boy. ^^ 433. The difference of timbre or quality in different voices, appears to be chiefly due to the difference of flexi- bility and smoothness in the cartilages of the larynx. Wo- men and children have these cartilages smooth and flex- ible, and hcjnce their voice is smooth, men on the contrary, have cartilages which are harder, and sometimes com- pletely ossified, and hence the roughness — the want of flexibility of their voices. 434. The loud«ies8 of the voice depends principally upon the force with which the air is expelled from the chest, but in part also the resonance produced by the other parts of the larynx and the neighbouring cavities. Note. — In the howling monkeys of South America there are several hollow pouches which open from the larynx, and one in the hyoid hone (which is greatly enlarged). The voice of this variety of monkey is said to be louder than tne roar of the lion. 435. Voices are divided by musicians into the follow- ing classes : — . < . - Female voices^ ( Soprano Mftl e voices. Mezzo-Soprano. ( Contralto. Tenor. Barytone. Base. Double vibrations per second made by vocal cords. From 106fi to 264. 930 « 220* '704 " 628 « 862 "* 220 (( u 176. 132. 110. 82j. I Note. --Ir» speaking, the range of the voice is limited to about half an c.ctave, in sinK'O?, to about two octaves. Occasionally a person may be met with who has ci-Itivated his voice so as to reach through three out an to the hairs, Ire is a >ng its imal ears I, and it is irard with directed n behind, ont. fmpanic alf-inch lending \\T. The iross the val, and looking [ leading into the ,ns of a ^ves exit also per- iy, when ^ Abt. 440.3 THE ORGANS OF HEABINa. 183 the OS orbiculare or round bone o, (the smallest bone in the body), and the stapes or stirrup, s. The handle h of the malleus is fastened to the membrana tympani, and the base of the stapes to the membrane covering the fenestra oxalis. The bones are joined to one another in the position represented in the figure, and are capable of slight move* ment by means of attached muscles. "^ ^ 440. The labyrinth or internal ear has its channels excavated in the petreous bone, the hardest of any in the body. It consists of the followinsf parts : — I. The Vestibule. II. The Semi-circular Canals. ' * ^ ' III. The Cochlea. .' • The vestibule (I) is a chamber formed in the petreous bone. Various branches of the auditory nerve and of ar- teries pass into it, and it is connected with the tympanic cavity by means of two orifices which are covered with membranes, viz., the fenestra ovalis or oval window (o. Fig. 42.) and the fenestra rotunda or round window (r. Fig. 42.) The semicircular canals (aj, y and z\ are three in num- ber, passing from and returning into the vestibule in the upper posterior part. They _^,^^^^_ Fig. 42. are placed at right angles to one another, and are each filled ^J by a membraneous canal of the same shape, containing fluid. The cochlea (snail shell), n Fig. 40 and k Fig. 42 is a spi- ral cavity, having the exact form of a snail's shell, the con- volutions making just two turns and a half around a central pil- lar. The canal is divided into two passages by a partition (the lamina spiralis), which runs its entire length. These passages do not communicate except at the top, where there is a small opening through the partition ; at the lower 184 THE OPiQANS OF HEARING. [Aets. 441, 443. 11 m m I I! li end, (corresponding to the mouth of the snail shell), they terminate separately, one with the tympanic cavity by means oHhQ fenestra rotunda^ and the other opens freely into the vestibule. . . v ... .4 ^ 441. The whole interior of the labyrinth is lined by a delicate membrane, on which the auditory nerve is mi- nutely distributed. Small looped fibrils of this nerve depend from the lamina spiralis, and float in the watery liquid which fills the cochlea as well as the other parts of the labyrinth. . , 'u - \' 442. The functions of the different parts of the ear are as follows:—^ ^ ; I. The waves of sound are collected in the pinna or ex- ternal ear, are directed through the auditory canal, and striking upon the membrana tympani throw it into vibra- tion. ' - . II. The chain of small bones connecting the membrana ^ tympani with the membrane that covers ihQ fenestra ovdlis receives the vibrations from the drum or membrana tym- pani, and transmits them across the tympanic cavity through the fenestra ovalis into the vestibule. . i -.i. III. The vibrations which are thus excited in the fluid which fills the vestibule, semi-circular canals and cochlea, are received by the expanded filaments of the auditory nerve, and the sensation of sound is thus transmitted to the brain. , i. . ^ r., ,»•; 'i'ti.?'; >».; ■ 44.3. Careful experiments have determined the follow- ing principles with regard to the transmission of vibra- tions from one medium to another, and a due considera- tion of these will explain the arrangement of membranes, and solids, and fluids in the ear. -a >it imn;f: I. Atmospheric vibrations lose much of their intensity when transmitted directly either to solids or liquids. ,■ II. The intervention of a membrane greatly facilitates the communications of vibrations from air to liquids. III. Atmospheric vibrations are readily communicated to a solid, if the latter be attached to a membrane so placed th? the vibrations of the air act upon it. +.,«. 441, 443. I), they ^ity by freely jcl by a is mi- nerve watery )arts of ear are a or ex- iial, and to vibra- ]emhrana ra ovalis ana tym- 3 cavity the fluid cochlea, auditory aitted to 3 follow- of vibra- >nsidera- mbranes, intensity ds. acilitates lids. unicated brane so ABT8. 444-446.] THE ORGANS OF HEARING. 185 IV. A solid body fixed in an opening by a border mem- brane so as to be movable, communicates sonorous vibra- tions from air on one side to water or other similar fluids on the other, much better than solid media not so con- structed. . , .. . 444. The peculiar functions of the semi-circular canals and of the cochlea, are not very well known. As the for- n^er are always placed at right angles to each other, occupying the position of the bottom and two sides of a cube, it has been supposed that they enable us to judge of the direction of sound, it is also deemed highly probable by physiolor' ts that the cochlea serves to give us the idea ofthepitc. i sounds. 445. According to Savart the most grave note the ear is capable of appreciating, is formed by from seven to eight complete vibrations per second. When fewer vibrations are made per second, they are heard as distinct sounds, i. e., do not produce a note. The most acute note appre- ciated by the ear is produced by 36500 complete vibra- tions per second. ^l a* < -^ * .. NoTE.—The interval la is said to be heard by rapidly moving the head, from side to side owing to the motions of the small bones of the ear. 446. The mechanism of hearing is not equally compli- cated in all classes of animals. Birds have the internal an 1 middle ear constructed on the same general plan as man, but the external ear is merely a circlet of feathers. ; ';^ Reptiles have no external ear, and in many cases no middle ear. The fluid in the vestibule is rendered milky in color, owing to the abundance of minute crystals of phosphate of lime. ^1 Fishes have no external or middle ear, but simply a membraneous vestibule situated in the skull, and sur- mounted by semi-circular canals from one to three in number. The ear of the moUusk is simply a sac filled with liquid, and having the auditory nerve expanded upon its inner surface. ^ IMAGE EVALUATION TEST TARGET (MT-3) // ^/ ^ 1.0 JJL 1.25 ■u U^ 12.2 2f Hit "^ '/ FhotogFa[Jiic Sciences CarporatiQn 4^ :\ ^V ^. ^ ;\ MwnT WltSTIR, (7U) MAIN ITMIT ,N.Y. USM •7a-4»0S A% • 45. If a stone be thrown vertically upwards and again reaches the ground after a lapse of 16 seconds^ to what height did it rise ? 46. In a composition of levers the arms of the power are 8, 4, 2 and 7, the arms of the weight are 3, 1, |^ and 4 \ what weight will be sustained by a power of 29 lbs. ? 47. A piece of wood which weighs 17 oz. has attached to it A metal sinker which weighs 13-7 ot. in air and 8*6 dz. in water-->the united mass weighs oilly -5 of an ounce ic water ; What is the specific gravity of the wood ? 48. What must be the area of an aperture in the bottom of a Vessel of water 18 feet deep and kept constantly full in order that 27 cubic feet may be discharged per hour^ 190 MISCBLLANEOUS PROBLEMS. I ( 49. How many tons of coal will be raised per day of ten hourii from a mine whose depth is 400 feet, bj a low pressure engine in which the piston has an area of 1200 inches and makes 20 strokes of 6 feet each per minute, the pressure of the steam on the boiler being 46 lbs. to the sq. inch ? 50. What power will support a weight of 70000 by means of a screw having a pitch of ^ of an inch and a power leyer 9 feet two inches in length ? 61. In what time will a pendulum 60 inches long vibrate in the latitude of Canada ? 62. In a leyer whose power arm is 8^ times as long as the arm of the weight, what p«wer will sustain a weight of 729 lbs. 7 63. A train weighing 130 tons is drawn along an incline of ^ in 100 with a uniform velocity of 26 miles per hoar ; taking friction and atmospheric resistance as usual, what is the horse powers of the locomotive : — 1st. If the train is ascending the incline ? 2nd. If the train is descending the incline? 64. A seconds pendulum is observed to lose 40 seconds in 24 ' hours on the summit of a mountain ; required its height. 65. A body is fired vertically with an initial velocity of 2000 feet por second ; it is required to find :— 1st. Where it will be at the end of the 120th seeond. 2nd. How for it will rise. 3rd. In what space of time it will again reach the ground. 4th. Its terminal velocity. 6th. In what other moment of its flight its velocity will be the same as at the end of the 49th second. 66. In a wheel and axle the radius of the axle is 3 inches and a weight of 247 lbs. is sustained by a power of 17 lbs.; what is the radius of the wheel ? 67. With what velocity does water flow from a small aperture in the side or bottom of a vessel, the fluid level being kept constant at 40 feet above the centre of the orifice ? 68. In a train of wheel work there aie four wheels and four axles, the first wheel and last axle being plain, i.e., without cogs, and having radii respectively of 12 and 2 feet — the second wheel has 70, the third 80, and the fourth 100 teeth : the first axle has 8, the second 7, and the third 11 leaves ; with this machine what weight will be sustained by a power of 130 lbs. 7 69. To what depth may a closed empty glass vessel capable of sustaining a pressure of 200 lbs. to the square inch be sunk in water before it breaks 7 f ten hourd ow pressure ) inches and the pressure ! sq. inch ? means of a power lever brate in the as the arm weight of cline of ^ in our ; taking what is the conds in 24 1 its height. city of 2000 Bcond. the ground, ocity will be inches and a • of 17 lbs.; all aperture level being the orifice ? i\a and four i.e., without I 2 feet— the i fourth 100 the third 11 be sustained i\ capable of lare inch be MISCELLANEOUS t>HOBLEMS. 191 60. In a differential wheel and axle the radii of the axles are If and 1| inches : a power of 2 lbs. sustains a weight of 749, what is the radius of the wheel ? 61. How many units of work are expended in raising 247 tons of coal from a depth of 478 feet ? 62. What are the horse powers of an upright water wheel worked by a stream five feet wide and 2| feet deep, the velocity of the water being 110 feet per minute, the fall 6 feet, and * the modulus of the wheel f ? 63. A train weighing 140 tons ascends a gradient having a rise of i in 100 ; taking friction as usual and neglecting at- mospheric resistance, what is the maximum speed the train will attain if the horse powers of the locomotive be 160 7 64. A barometer at the summit of a mountain indicates a pres- sure of 21*4 inches while at the base the pressure is 30*2 inches, what is the height of the mountain ? 65. On an incline of 7 in 100 what power acting parallel to the plane will sustain a weight of 947 lbs. ? 66. What centrifugal force is exerted by a ball weighing 40 lbs. revolving in a circle 20 feet in diameter and making 140 revolutions per minute ? 67. What is the specific gravity of a piece of metal which weighs 23-49 oz. in air and only 18*12 Oz. in water? 68. If a b6dy be thrown vertically upwards and again reaches the ground in 22 seconds— , 1st. With what velocity was it projected ? 2nd. How far did it rise ? 69. In A screw the pitch is -iV of an inch, the power lever is 40 inches long ; what power will sustain a weight of 95000 ^ lbs. ? 70. In what time will an engine of 120 horse powers, moving a train Whose gross weight is 100 tons, complete a journey of 300 miles, taking fi-iction as usual, neglecting atmos- pheric resistance, and assuming the rail to ascend regu- larly i in 100 ? 71. An engine of 60 horse powers raises 50 tons of coal per hour from the bottom of a mine 200 feet deep, and et the same time causes a forge hammer to make forty lifts per minute of 3 feet each : required the weight of the hammer. 72. In a hydrostatic press the sectional areas of the cylinders are to one another as 1411 to 3, the force pump is worked by a lever whose arms are to one another as 28 to 3, the upward pressure required is 9000 lbs ; what must be the power applied ? 192 MISCELLANEOUS PEOBLEMS. ■!■: 73. In a differential screw the pitch of the exterior BcreW is ■^g and that of the inner screw ^ of an inch, the power lever is 6 feet 8 inches in length ; what pressure will be exerted by a power of 19 Ibs.T 74. A piece of nickel (spec. gray. 7*816) weighs 24 grains in air and only 16'4 grains in a certain liquid; required the specific gravity of the liquid. 75. tna differential wheel and axle the radii of the axles are 1^ and IyV inches, the radius of the wheel is 42 inches ; what weight may be sustained by a power of 23' 7 lbs. ? 76. What gross load will a horse draw when travelling at the rate of 3^ miles per hour on a road whose coefficient of friction is iV ? 77. A body has descended through a-^-x feet when a second body commences to fkll at a point 2m feet beneath it ; what distance will the latter body fall before the fbrmer passes it? 78. On an incline of | in 70 what power acting parallel to the plane will sustain a weight of 4790 lbs. ? 79. When a body has fallen 7000 feet down an incline of 7 in 20 what velocity per second has it acquired 7 80. A body weighing 100 lbs. and moving from south to north ' with a velocity of 60 feet per second comes into contact with another body which weighs 430 lbs. and is moving from north to south with a velocity of 20 feet per second, and the two bodies coalesce and move on together ; re^ quired the direction and velocity of the motion of the united mass. 81. An engine of 21 horse powers pulnps 40 cubic fbet of Water per hour from the bottom of a mine whose depth is 200 feet and at the same time draws coals from the bottom of the mine ; required the tons of coals drawn up per hour. 82. In a system of pulleys worked by several cords, each attached by both ends to the pulleys, there are 8 movable pulleys and as many separate cords ; what weight will be sustain- ed by a power of 73 lbs. ? 8di A body weighing 20 lbs. and moving at the rate of 47 feet per second comes in contact with another body weighing 270 lbs. and moving in the same direction with the vel- ocity of 30 feet per second ; required the velocity and momentutn of the united mass. 84. In what time will an engine of 150 horse powers draw a train whose gross weight is 90 tons through a journey of 220 miles, taking friction as usual and neglecting atmos- pheric resistance, one half of the journey to be on a level plane and the other half up an incline of) in 100 1 ■ icrew li the power '6 will be ains in air quired the :les are 1^ les : what ng at the ifficient of a second )eneath it ; the former Lllel to the ) of 7 in 20 th to north Ito contact is moving per second, ^ether ; re* tion of the et of water pth is 200 ) bottom of )er hour. ih attached 3le pulleys be sustain- I of 4T feet f weighing :h the yel- ilocitj and srs draw a journey of ing atmoB- on a lerel 00? MISCELLANEOUS PROBLEMS. 193 85. In a common wheel and axle a power of *l lbs. sustains a weight of 9T4 ; the radius of the wheel is 51 inches, what is the radius of the axle ? 86. At what height would the mercury stand in a barometer placed at an elevation of 43*2 miles above the level of the earth ? 87. If a body be projected down an incline of 7 in 12 with an initial velocity of 40 feet per second, through how many feet will it move during the tenth second and over what space will it have passed in 23 seconds ? 88. In a high pressure engine the piston has an area of 360 inches and makes 17 strokes of 5 feet each per minute ; taking the pressure of the steam on the boiler as equal to 66 lbs. to the square inch, what are the horse powers of the engine? 89. If a body weighing 111 lbs. moving to the east with a velocity of 90 feet per second come in contact with another body '^ • which weighs 70 lbs. and is moving to the west with a velocity of 40 feet per second, and after the two have coalesced they come in contact with a third which weighs • 80 lbs. and is moving to the east with a velocity of 20 feet » per second and the three then coalesce and move on to- gether ; what will be the direction, velocity, and momen- tum of the united mass ? 90. What must be the length of a pendulum in the latitude of Canada in order that it may make 40 vibrations in 1 minute ? 91. What pressure will be exeorted upon the body of a man at the depth of 97 feet beneath the si ;T.ce of the water, the man's body having a surface equal to ' 4 square feet ? 92. A piece of cork which weighs 27'42 grains has attached to it a sinker which weighs 34*71 grains in air and only 30*12 grains in water, the united mass weighs nothing in water i. e., is of the same specific gravity as water j required the specific gravity of the cork. 93. What is the weight of a mass of slate which contains 27 cubic feet? 94. How many cubic feet of iron are there in 87 tons ? 95. What backward pressure is exerted by ahorse in going down a hill which has a rise of 3 in 40 with a load whose gross weight is 2100 lbs. assuming friction to be equal to ^jof the load ? 96. What pressure is exerted against one square yard of an embankment if the upper edge of the yard be 17 feet and the lower edge 18 feet beneath the surface of the water ? 13 ^'^" ' 194 MISCELLANEOUS PROBLEMS. 9*7. The length of a wedge is 2Y inches and the thickness of the back 2)1 inches ; what weight maybe raised by a pressure of nibs. ? 98. What are the effective horse powers of a high pressure engine in which the piston has an area of 420 inches and makes 30 strokes per minute, the boiler evaporating ^ of a cubic foot of water per minute under a pressure of 60 lbs. to the square inch ? 99. A train drawn by a locomotive of 100 H. P. moves along an incline of ^ in 100 with a uniform velocity of 25 miles per hour ; taking friction as usual and neglecting atmos- pheric resistance, what is the weight of the train : — 1st. If it is ascending the incline ? 2nd. If it is descending the incline ? 100. A lightning flash is seen 9} seconds before the report is heard, at what distance did the discharge occur ? 101. A body YOOO miles from the surface of the earth weighs 500 lbs., what would be its weight at the distance of 4000 miles ? 102. How long would sound require to travel from Toronto to Markham a distance of 21 miles, the thermometer indi- cating a temperature of 82^ F. ? 103. At what distance from the source of sound must the reflect- ing surface be in order that the last 20 syllables uttered may be distinctly repeated by the echo ? 104. On a perfectly calm day the report of a cannon fired on the northern shore of Lake Ontario is heard on the southern shore a distance' of 40 miles. How much sooner will the report' arrive at the southern shore through the water of the lake than through the overlying air ? 105. A metallic wire placed on the monochord vibrates 800 times in a second — ^by how much must its length be increased in order that with the same degree of tension, &c.f it shall vibrate only 550 times in a second ? 106. What are the relative number of vibrations per second required to produce the notes E and D sharp ? 107. What is the length of a wave of air producing P* of the Italian Opera (1856) ? 108. A cord of certain length and diameter makes 90 vibrations per second when stretched over the sonometer by a weight of 100 lbs., by what weight must it be stretched in order to make 135 vibrations per second? 109. In the year 1*783, the report of a meteor was heard at Windsor Castle 10 minutes after the flash of the meteor was seen, what was its distance assuming the tempera- ture of the air to be 52® F. ? EXAMINATION PAPERS. 196 110. An upright vessel is filled with water and is pierced in the side at the heights of 2, 5, 9, and 16 feet from the ground, taking the whole height of the water as 24 feet, what in each case will be the random of the jet ? HI. A person supposes himself to be in the range of a distant cannon, the report of which he hears 23 seconds after seeing the flash, how soon may he apprehend danger from the ball assuming that it travels with the uniform velocity of i^ of a mile per second ? r / ■ EXAMINATION PAPERS. ; 1. A railway train weighing 110 tons is drawn along an incline of \ in 100 with a uniform velocity of 42 miles per hour, -^' taking friction as usual and atmospheric resistance equal to 20 lbs. when the train is moving at the rate of 1 miles per hour, what are the horse powers of the locomotive ? * 1st. If the train is ascending the gradient ? 2nd. If the train is descending the gradient ?^ 2. Enunciate the principle of virtual velocities and calculate through how many feet a weight of 89**7 lbs. will be car- ried by a power of 11*7 lbs. moving through 123 feet? 3. In a differential wheel and axle the radii of the axles are 3| and 3^ inches j the radius of the wheel is 42 inches, what power will sustain a weight of 444*4 lbs. ? 4. Describe the Barometer and explain the principles on which it acts. 5. What is the weight of a log of boxwood (spec. grav. 1*320) 1*7 feet long, 1 foot 9 inches wide, and 2 feet 3 inches thick? 6. The upright gate of a canal is 12 feet wide and 16 feet deep, the water being on one side only and level with the top j required the pressure : — 1st. On the whole gate : 2nd. On the lowest five-eighths of the gate ; and, 3rd. On the middle seventh of the gate. 7. Give the composition of atmospheric air, and state what are the chief sources of the carbonic acid. 8. The piston of a high pressure engine has an area of 400 inches and makes 12 strokes of 6 feet each per minute, the pressure of the steam on the boiler is 64 lbs. per square inch ; how many tons of coal per hour will this engine raise from a mine whose depth is 240 feet ? i ;'U 196 EXAMINATION PAPERS. \i) ! I ( w\ { 8. Distinguish between the essential and the accessory proper- ties of matter and enumerate the former. 10. An upright vessel 1 T feet in height is filled with water and holds just 290 gallons ; in what time will it empty itself through an aperture in the bottom two-fifths of an inch in area ? II. - 1. A cannon ball is fired vertically with an initial velocity of 800 feet per second ; required— , ^ 1st. How far it will ascend. 2nd. In what space of time it will again reach the ground. 3rd. Where it will be at the end of the 31st second. 4th. Its terminal velocity. 5th. In what other moment of its flight it will have the same velocity as at the close of the 13 th second. 2. Enumerate the different kinds of attraction, define what is meant by the attraction of gravity, and state by what law its intensity varies. 3. A piece of stone weighs 13 grains in air and only 35 grains in water ; required its specific gravity. 4. In a hydrostatic press the areas of the cylinders are to one another as 1347 : 2, the force pump is worked by means of a lever whose arms are to one another as 23 : 2, the power applied is 120 lbs. ; required the upward pressure exerted against the piston in the larger cylinder. 5. In a lever the power arm is 7 feet long, the arm of the weight is 5 inches, the power is 11 lbs. ; required the weight. 6. Enunciate the principle of the parallelogram of forces and explain how it is that a force may be more advantageous- ly represented by a line of given length than by saying it is equal to a given number of lbs., &c. 7. Name the dififerent kinds of upright water wheels, explain the difference between them, and give the rule for finding the horse powers of a water wheel. 8. If a closed empty vessel be sunk in water to the depth of 143 feet before it breaks, what was the extreme pressure to the square inch it was capable of sustaining ? 9. Describe what is meant by the vena contraeta of escaping fluids, indicate its position with reference to the orifice of escape, and give the proportion between the area of the aperture and the sectional area of the vena contraeta. 1>( proper- rater and pty itaelf { an inch relocity of le ground, id. e the same ine what is te bj what y 35 grains } are to one t>7 means of {, the power ore exerted arm of the equired the f forces and rantageous- II bj saying els, explain ) for finding he depth of ne pressure of escaping he orifice of area of the miracta. EXAMINATION PAPERS. 197 10. An engine of 60 horse powers draws a train weighing 60 tons up an incline of i in 100 with a uniform velocity of ' 20 miles per hour; taking atmospheric resistance as usual, what is the friction per ton ? III. 1. By means of a lever a certain number of lbs. Troy attached to the arm of the weight balances the same number of ounces Avoirdupois attached to the arm of the power ; required the ratio between the arms of the lever, a pound Troy being to a pound Avoirdupois as 5760 : 7000. 2. Enunciate Torricdlis theorem and calculate the velocity with which a liquid spouts from a small orifice in the side of a vessel when the level of the fluid is 24 feet above the centre of the orifice. 3. In a hydrostatic bellows the sectional area of the tube is three-sevenths of an inch and it contains 12 lbs. of water, ;j; 7/ the area of the board of the bellows is 3' 7 square feet ; '.All « '<; "^hat is the upward pressure exerted against the board of the bellows 7 4. Through how many feet will a body fall during the 22nd second of its descent ? 6. Define what is meant by specific gravity. Give the rule for calculating the specific gravity of a solid not sufficiently heavy to sink in water and calculate the specific gravity of cork from the following data : — A piece of cork which weighs 20 oz. in air has attached to it an iron sinker which weighs 18 oz. in air and only 15*73 oz. in water ; the united mass weighs 1 oz. in water, required the specific gravity of the cork. 6. What weight would be carried through a space of 7 feet by a power of 5 lbs. moving through 40 feet? t< 7. Define what is meant by the centre of gravity of a body and explain how it may be experimentally determined in a solid. 8. How many tons of coal per day of ten hours may be raised from a mine of 660 feet in depth by a low pressure engine vr< having a piston which has an area of 600 inches, and makes 20 strokes of 11 feet each per minute, the gross pressure of the steam on the boiler being 37 lbs. per square inch ? 9. The power arm of a lever is 32 times as long as the arm of the weight, the power is 97 oz. •, required the weight. i 198 EXAMINATION PAPERS. 10. A city is supplied with water through a pipe 8 inches in diameter and 1 mile in length, leading to a reservoir whose height is 140 feet above the remote end of the pipe ; what will be the velocity of the water per second and how much will be discharged in one hour ? ii.| B-. i * I ' ■ IV. 1. Enunciate the law of decrease in the pressure and density of the air as we ascend into the higher regioQS of the atmosphere ? 2. In a hydrostatic presF; the sectional areas of the cylinders are to one another as 943 : 2, the force pump is worked by means of u lever whose arms are to one another as 19 : 3 ; if the power applied be 87 lbs., what is the upward pres- sure exerted against the piston in the larger cylinder ? 3. The power arm of a lever is 9 feet long, the arm of the weight is 17 feet long and the weight is 6^ lbs. ; required the power. 4. Explain when a body is said to be in a condition of stable^ unstable f or indifferent equilibrium. 6. A train weighing 90 tons is drawn along an incline of 2 in 900 with a uniform velocity of 30 miles per hour ; taking friction and atmospheric resistance as usual, what are the . horse powers of the locomotive : — . ^ 1st. If the train is ascending the gradient ? ,^ 2nd. If the train is descending the gradient ? 6. A stone is dropt into a mine and is heard to strike the bot* tom in 11| seconds; required the depth of the mine, if sound travels at the rate of 1066^ feet per second. 7. State the condition of equilibrium in the differential wheel and axle. 8. What is the weight of the sulphuric acid (specific gravity 1'841) contained in a rectangular vat 7 feet 4 inches long, 2 feet 5 inches deep, and 3 feet 7 inches wide ? 9. At the top of a mountain a barometer indicates a pressure of 21 inches while at the base the pressure is 29*78 inches — the temperature at the top is 40* "JO" F?hr. and that at the base is 70- 70^ Fahr. ; required the height of the mountain. 10. A high pressure steam engine raises 200 cubic feet of water per minute from a depth of 80 feet, the piston has an area * ' of 800 inches and makes 10 strokes per minute of 8 feet ^ each, what is the pressure of the steam on the boiler ? iches in ir whose e; what ind how density of 8 of the • cylinders (Torked by as 19 : 3 ; ^ard pres- linder ? he weight uired the of stable^ ine of 2 in ur; taking hat are the [e the bot- le mine, if 3nd. itial wheel fie gravity iches long, pressare of 78 inches — that at the I mountain. et of water lias an area ta of 8 feet boiler ? EXAMINATION PAPERS. V. i9d 1. The flood gate of a canal is 10 feet long and 1 feet deep, the water being on one side and level with the top ; what is the pressure: — 1st. On the whole gate ? 2nd. On the lowest two-sevenths of the gate ? 3rd. On the middle three-sevenths of the gate ? 4th. On the lowest one-ninth of the gate ? 2. In a compound lever the arms of the power are 6, 7, and 11 feet, the^arms of the weight are 2, 3, and 5 feet ; by means of this combination what power will sustain a weight of 1000 lbs.? ' ^ 3. Enunciate Mariotte's law and ascertain what will be the density, volume, and elasticity oi that amount of atmos- pheric air, which, under ordinary circumstances, i. e., at the level of the sea or under a pressure of 15 lbs. to the square inch, fills a gallon measure, if it be placed in a piston and subjected to a pressure of 60 lbs. to the square inch. 4. What power moving through 29 feet will carry a weight of 7 lbs. through 70 feet? 5. An engine of 12 horse powers gives motion to a forge ham- mer which weighs 400 lbs. and makes 40 lifts of 3 feet '' each per minute and at the same time pumps water from '^ a mine 100 feet deep ; required the number of cubic feet of water it pumps per hour from the mine. 6. On an inclined plane a power of 341 lbs. acting parallel to the base sustains a weight of 27300 lbs. ; what must be the length of the base in order that the height maybe 11 feet? 7. Enunciate the three laws of motion commonly known as Newton's laws, and state to whom they respectively belong. 8. A piece of sulphur weighs 19 oz. in air and 10 oz. in water ; required its specific gravity. 0. A ball is thrown up an incline of 11 in 16 with an initial velocity of 1100 feet per second ; required—, 1st. To what height it will rise. ' * 2nd. Where H will be at the end of the 20th second. • 3rd. In what time it will again reach the ground. 4th. Its terminal velocity. 6th. In what other moment of its flight it will have the same velocity as at the end of the 17th second of its ascent. 10. Required the pressure exerted against a mill-dam 170 feet long and 16 feet wide, the perpendicular depth of the water being 12 feet. X 200 EXAMINATION PAPEBS. VI. rl < 11 -r fi f 20 miles sistance as Dtive ? elocity has show the omposition is 11 inches r will, with water being squired the ic gravity of axles are 2f Qches ; what ? water to the ) has an area gainst it? ilum 9 yards IS on inclined mother body 1 it ; through he first over- [0 inches and »et in length, r per minute; r minute and ited? ANSWERS TO EXAMINATION PAPERS. 208 7. Give the most important consequences that result from the fact that each atom of a liquid is separately drawn towards the centre of the earth by the force of gravity. 8. What gross load will a horse exerting a traction of 1A lbs. draw on a road whose coefficient of friction is i^ ? 9. What are the conditions of equilibrium between the power and weight in the inclined plane 7 10. Through how many feet must a body fall in order to acquire a velocity of 250 feet per second ? ANSWERS AND REFERENCES TO EXAMINATION PAPERS. I. 1. H.P.=:228-48orl05'28. 2. Art. 66. 3. -1679 lbs. 4. Arts. 227, 229. 5. 5522*34375 lbs. 1. H.P. = 161-28or38-08. 2. Arts. 25, 27. 3. 1-921. 4. 929430 lbs. 5. 184f lbs. 6. 96000 lbs., 82500 lbs., and 13714^ lbs. 7. Art. 205. 8. 151*2 tons. 9. Arts. 9, 19, and 20. 10. 18 min., 45 sec. II. 6. Art. 44. 7. Arts. 339, 341. 8. 62*05 lbs. 341. 9. Arts. 9, 19, and 10. 10. 8*425 lbs. per ton. III. 1. Power arm 13^ times as great as the arm of the weight. 2. Arts. 25, 26, and 27. 3. 14918*4 lbs. 4. 688 feet. 6. Arts. 192, 195, and '57584. 6. 28f lbs. 7. Arts. 57, 58. 8. 1400. 9. 194 lbs. 10. Velocity = 6*336 feet per second. Quantity = 7962*071 cubic feet per hour. IV. 1. Art. 212. 2. 2597961 lbs. 3. 12i»ir. 4. Art. 62. 6. H. P. = 106'16 or 42*16. 6. 1600 feet. 7. Art. 88. 8. 730700144 lbs. 9. 9721*2 feet. 10. 33f to the square inch. 204 ANSWERS 1?0 EXAMINATION PAPEHS. T. 1. 15312ilbs., 7500 lbs., 6562| lbs., aad 3213^1 lbs. 2. 64Hlbs. 3. Art. 219, density 4 times as great, Yolume 1 qt. and elasticity 60 lbs. to the sq. inch. 4. 16$f lbs. 6. 3340^ cubic feet. tla It'c ' VO-V^J 6. 900 feet. 7. Arts. 255, 256,257. '" 8. 2-111. 9. 27500 feet. At elevation of 17600 feet. 100 seconds. 1100 feet per second. *' "* At the end of the 83rd sec 10. 1020000 lbs. ■■ ■ - :* ♦• .' V * *i - • • .:<>•■• VI. 1. 19724 feet. • 6. Arts. 183 and 182, Note. 1 2. Art. 282. 7. -618. 1 8. 24336 feet. v 4. 45-36 H. P. ^i 8. 3990 lbs. 9. Arts. 233 and 234. i-y;i 5. j^^ of an inch. . i ■ ^ 1. 10406i lbs. 10. 2163-4 feet. VII. ■9- ;% .-7 i>^ \i '■'1 ' 6. Arts. 129, 126. K5 2. 1*303 inches. 7. 454 feet. 3. Art. 235. 4. 594 lbs. 8. 1920 lbs. 9. 13791|fV Kftllons. 6. 626933^ lbs. 10. H. P. = 67-87. . .'■' "''.. '■ , -' ■ VIII. 1. Arts. 301, 302, and 308. 8. 185ff lbs. 2. 386*17 inches. 9. 275000 lbs. _ 3. 3022-68672. 185204A lbs. 64320t4 lbs. 4. 14660166-6 lbs. ft. 133-262. 75000 lbs. 6. 400 feet per second. 231000 lbs. ' 7. Art. 206. 10. Art. 196. 1. 8085 lbs. 2. 21479-04 lbs. 3. 20-8. 4. Art. 270. 6. midfeet 6. Volume = 339-5 cub. feet. Pres. = 85 lbs. the sq. inch. 7. Art. 175. 8. 1776 lbs. 9. Art. 116. 10. 976^ feet. Is. 17600 feet. icond. Ihe 83rd sec 182, Note. 234. >r \. )n8. ■ ^7 ' r* :\:^-i-lT 39*6 cub. feet. )S. the sq. inch. EXAMINATION QUESTIONS. 205 QUESTIONS TO BE ANSWERED ORALLY BY THE PUPIL. "SoTE.— The numbera follotoing the questione r^er to the numbered articles in the work, where the answers may be found. 1. What is Natural Science? (1) 2. Into what classes are all natural objects divided and how are these distinguished from each other? (2) 3. How are animals distinguished from vegetables P (3) 4. What is Zoology? (4) 5. What is Botany? (4) ,>'..,■.. 6. What is Mineralogy? (4) 7. What is Astronomy ? (4) t ' ., 8. What is Geology? (4) jmistry? (t) 10. What is the object of Natural Philosophy ? (4) 9. What is Ghemi 11. What are the subdivisions of Natural Philosophy ? (6) 12. In what separate forms does matter exist ? (6/ 13. Define what is meant by the essential properties of matter ? (9) 14 Enumeratetheessential properties of matter. (10) 15. What is extension ? (11) ^ t: 16. What is impenetrability ? Oive some illustrations. (12.) 17. What is divisibility ? (13) 18. Does the property of divisibility belong to masses or to particles of matter or to both? (13) 19. Give some illustrations of the extrome divisibility of matter ? (13,Note) 20. Whatis Indfistruotability? (14) 21. What is Porosity ? (16) 22. What is Compressibility? (16) 23. Whatis Inertia? (17) 24. If bodies cannot bring themselves to a state of rest, how is it that all bodies moving upon or near the earth soon come to rest ? (17, Note) 25. What iselastl^y ? (18) 26. Name the differont kinds of elasticity as applied to solids. (18, Note) 27. 28. 29. 30. 81. 32. 33. 34. 35. 36. 87. 38. 39. 40. 41. 42. 4S. 44. 46. 46. 47. 48. 49. 50. 61. 62. What ara the aoc«Mory properties of matter? (19) Enumerate some of the most important of the accessory properties of matter. (20) What is malleability ? Which are the most malleable of the metals f (21) Whatis ductility? Name the most ductile metals. (22) What is tenacity? (23) What is attraction? (24) Enumerate the difTerent kinds jof attraction. (26) What is the attraction of gravity ? (26) What is the law of variation in the intensity of gravity ? (27) Explain whatis meant bv saying the force of gravity varies inversely as the square of the distance ? (28) What is the attraction of cohesion ? (29) What is the attraction of adhesion ? (30) What is capillary attraction ? Give some examples. (31) What is electrical attraction ? What is magnetic attraction ? What is chemical attraction ? What is the derivation of the word Statics f (86) What is the object of the science of Statics ? (36) What is the derivation of the word HydroslaUes ? (86) What is the object of the science of Hydrostatics ? (36) What is the derivation of the word Dynamics 't (36) What is the object of the science of i^rnamics ? (36) What is the derivation of the word Hydrodynamics? (86) What is the object of the science of Hydrodynamics? (36) What is the derivation of the word Pneumatics i (36) What is the object of the science of Pneumatios ? (36) Yfawiff 206 EXAMINATION QUESTIONS. 53. When is a body said to be in equilibrium ? (37) . . ? / 64. What are statical forces or pressures ? (38) .^ 55. What are the elements of a force ? (39) • ' 56. What are the different modes of representing a force ? (40) 57. When several forces act upon the same point of a body, how many motions can they e^ve it ? (41) 58. Distinguish between component and resultant forces. (42) 59. If several forces act upon a point in the same straight line and in thu same direction, to what is their resultant equal? (48) 60. When several forces act upon a point in the same straight line but in opposite directions, to what is their resultant equal P (43) 61. Enunciate the principle of the i)arallelogram of forces. (44) 62. When several forces act on a point in any direction whatever, state how the resultant may be found. (45) 63. What is the distinction between the parallelogram offerees and the parallelopiped of forces? (46) 64. What is the resultant of two parallel forces which act on different points of a body, but in the same direction ? (47) 65. What is the resultant of two parallel forces which act on different points of a body and in opposite directions? {4&) 66. How do we find the resultant of any number of parallel forces P (49) 67. What is a couple ? (50) 68. Distinguish between the composition of forces and the resolution of 69. What is the centre of gravity of a body ? (67) 70. Why is the centre ofgravity called also the centre of parallel forces? (55) 71. How may the centre of gravity of a solid body be experiment^ly determined? (68) 72. If a body be free to move in any direction, how will it finally rest with reference to its centre of gravity ? (60) 73. How is the stability of a body estimated? (61) 74. When is a body said to be in a condition of stable, unstable, or in- different equilihrivani (62) 76. How may the centre of gravity of two separate bodies be found? (63) 76. 77. 78. 79. 80. 81. 82. 83. 84. 86. 86. 87. 88. 89. 90. 91. 92. 93. 94. What is the object of all mechanical contrivances ? (64) By what law or principle in philosophy is the relaiive gain or lo«s of power and velocity Ui a machine determined ? (65) Enunciate the principle of virtual velocities ? (66) What is a machine ? (67) How many mechanical elements enter into the composition of ma* chineryP (68) Name the primary mechanical elements. (68) Name the seoondiury mechanical elements. (68) From what mechanical element is the wheel and axle formed? (69) Of what mechanical element are the wedge and screw modinca- tions? (69) How are levers, cords, &c., regarded in theoretical mechanics? (70) What is a lever? (71) Of how many kinds are levers? (^) » Of simple straight levers how many kinds are there ? (73) Upon what does the distinction Detween the three kinds of levers depend? (73) Give examples of levers of the first class. (74) How are the ftdorum, power, and weight placed in levers of the first class? (75) How are ttie fulcrum, power, and weight placed in levers of the second class? (75) Give some examples of levers of the second class ? (76) How are the fulcrum, power, and weight placed in levers of the third class? (76) U)) ly, how many 2) ine and in thu ht line but in [43) (44) natever, stato broes and the lifferent points b on different forces? (49) 9 resolution of lei forces? (55) aperiment^ly Ally rest with tstdblOt or in- e found ? (63) gainorIo«sof )8ition of ina- tued? (69) ew modifioa- )chanic8? (70) ads of levers rs of the first of the second ) of the third EXAMINATION QUESTIONS. 207 95. Give some examples of levers of the third class? (76) 96. In levers of the first class which must be greatest, the power or the weight? (76, Note) 97. In levers of the second class, which must be greatest, the power or the weight ? (76, Note) 98. In levers of the third class, which must be the greatest, the power or the weight? (76, Note) 99. What is the arm of the weight ? What is the arm of the power ? (77) 100. What are the conditions of equilibrium between the power and the weight in the lever ? (77) 101 Deduce formulas for finding the power, the weight, the arm of the, power or the arm of the weight when the other three are given. (77) 102. When the arms of the lever are curved or bent, how must their effiective lengths be determined? (79) 103. What is a compound lever or composition of levers? (80) 104. Deduce rules for finding the power or the weight in a compound lever? (81) 105 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. Describe the wheel and axle. (82) Why is the wheel and axle sometimes called a perpetual lever ? (84) What are the conditions of equilibrium in the wheel and axle? (85) Deduce a set of rules for finding the power, the weight, the radius of the axle or the radius of the wheel when the other three are given. (86) Describe the differential wheel and axle ? (87) To what is it, in effect, equivalent ? (87) Deduce a set of rules for the differential wheel and axle. In toothed gear how is the ratio between the power and the weight determined ? (89) How are axles commonly made to act on wheels? (90) When is wheel work used to concentrate power ? Give an example. (92) When is wheel work used to diffuse power ? Give an example. (92) What are the conditions of equilibrium in a system of toothed wheels and pinions? (98) What is a pinion 7 what are leaves 1 (91) Deduce formulas for finding the power and the weight, in a system of wheels and axles? (94) Explain what is meant by the Imntir^ cog ? (95) Name the different kinds of wheels? (96) Explain the difference between crovmt spur, and bevelled gear ? (97) Explain for what purpose crown, spur, or bevelled gear.is used ? (98), Whenbevelled wheels of different diameters are to|beused together show how the sections of the cones of which they are to be frusta are found ? (99) 124. What is a pulley ? (100) 125. Show that from the pulley itself no mechanical advantage is detived P 126. Wherein consists the real advantage of the pulley and cord as a me- chanical power? (101) 127. When ia a pulley said to be'fixed ? (102) 128. What is a single movable pulley called? (108) 129. What are Spanish Bartons ? (103) 130. Explain the meaning of the words sheaf. Mock, and tackle ? (104) 131. What is the only mechanical advantage derived fjrom the use of a fixed pulley? (106) 132. In a system of pulleys worked by a single cord, what are the conditions of equilibrium? (106) 133. Deduce a set of rules for a system of pulleys worked by a single cord ? L107) , ,^ 184. What are the conditions of equilibrium in a Spanish Barton when the separate cords are attached directly to the beam ? (108) ■*»^4i^i&sLj^^at^h 208 EXAMINATION QTJESTIONS. .( ; ' t 135. What are the oonditions of equilibrium when the separate cords are attached to the movable pulleys ? (100) 136. Deduce in each of these last two cases a set of rules for finding the ratio between the power and the weight ? (110 and 111) 137. If the lines of direction of the power and weight make with one another an angle greater than 120°, what is the relation between the IK)wer and the weight? (.12) 138. In theoretical mechanics how is the inclined plane regarded P (113) 139. What are the modes of indicating the inclination of the plane? (114) 140. In the inclined plane how may the power be applied? (115) 141. What are the conditions of equillibrium in the mclined plane ? (116) 142. Deduce a set of rules for the inclined plane ? (117) 143. What is the wedge? (118) 14 K How is the wedge worked? (119) 145. What are the conditions of equilibrium in the wedge when it is worked by pressure? (120) 146. In what important particular does the wedge differ from all the other mechanical powers? C120, Note 1) 147. Give some examples of the application of the wedge to practical pur> poses? (120, Note 2) 148. Dieduce a set of rules for the wedge ? (121) ♦".n ■>> 149. Describe the screw ? (122) 150. How is the screw related to an ordinary inclined plane ? (122, Note.) 151. What is the pitch of the screw ? (123) 152. How is the screw commonly worked ? (124 and 126) 153. What are the conditions of equilibrium in the screw ? (126) 154. How may the efficiency of the screw as a mechanical power be increa- sed? (127) 155. DeduQeaset of rules for the common screw? (128) «[in< 156. By whom was the di^erential screw invented ? (129) i;«><; 157. Upon what principle does the differential screw act ? (129) 158. To what is the differential screw, in effect, equivalent ? (129) 159. Deduce a set of rules for the differential screw ? (130) 160. Describe the endless screw? (131) 161. What are the oonditions of equilibrium in the endless screw ? (132) 162. Deduce a set of rules for the endless screw? (133) 163. How does friction affect the relation between the •po\rer and the weight in the mechanical elements ? (135) 164. What are the different kinds of friction ? (136) 165. What is meant by the coefficient of friction ? (137) 166. What is the coencient of sl.ding flriction ? (138) 167. What is the coeficient of friction on railways ? (138) 168. What is the coefficient of friction on good macadamized roads? (138) 169. What is meant by the force of traction? (\i%) 170. Enumerate the different expedients in common use for diminishing fHction ? (139) 171. Give Coulomb's conclusions as regards sliding friction ? (139) 172. Give Coulomb's conclusions as regards rolling friction ? (139) ^'u 2 S 2 2 2 2 173. What is the unit qf work 1 (140) 174. How are the units of work expended in raising a body found ? (141) 175. What are the most important sources of laboniig forces ? (142) 176. How many units of work are there in one horse power ? (142) 177. What is meant by the Table in Art. 142 ? 178. What is the true work of the horse per minute ? (142. Note) 179. In moving a Carriage along a horizontal plane, for what purpose is work expended ? 180. In the case of railway trains what is the amount of friction ? (143) 181. In the case of railway trains when does the velocity become unifonn? EXAMINATION QUESTIONS, 209 irate cords are ne? (122. Note.) power be increa- 182. Upon what does the traction of force with which an animal pulls de- pend? (146) 183. At what rate per hour must a horse travel to do most work? (146) 184. Upon what does the amount of atmospheric resistance experienced by a moving body depend? (147) 186. Explain what is meant by this? (147) 186. What is the amount of atmospheric resistance experienced by a train of medium length moving; at the rate of 10 miles per hour ? (148) 187. If a body be moved along a surface without friction or atmospheric resistance, how may the units of work performed be found ? (149) 188. When a train is moved along an inclined plane, how is the work per- formed by the locomotive found ? (150) 189. Deduce a set of formulas for finding the liorse power, weight, maximum speed, &c., of trains ? (151) 190. What is meant by the modulus of a machine ? (152) 191. Of machines for raising water which has the greatest modulus ? (158) 192. How may the work performed by water failing from a height be found? (154) 193. How is steam converted into a source of laboring force ? (155) 194. What ar3 the two principal varieties of the steam engine ? (156) 195. What are the essential parts of the high pressure engine ? (157) 196. How does the low pressure differ from the high pressure engine ? (158) 197. What are the varieties of the low pressure engine? (169) 198. How do these differ from each other ? (160, 161) 199. Inthehighpressurecngine, at what part of the stroke does atmos- pheric pressure act against the piston ? (162) 200. Give the leading ideas that enter into the construction of the steam engine ? (163) 201. In what respects is the low pressure engine preferable to the non-con- jv,tj densing engine? (164) . . ., . -. • ■■ 202. How are the units of work performed by an engine found P (165) 203. Knowing the pressure of the steam on the boiler, how do we obtain the useful pressure on the piston ? (166) 204. Give themlesfor finding the H. P. &c.. of engines? (167) 205. What is the real source of work in the steam engine ? (168) 206. Why is it most advantageous to employ steam of high pressure ? (168) 207. Give formulas for finding the area of the piston, length of stroke, pres- sure, effective evaporation, &o.; in the steam engine ? (169) 208. Define what is meant by k fluid / (171) 209. How is the term fluid commonly applied? (172) 210. Into what classes are fluids divided ? Name the type of each ? (173) 211. To what extent is water compressible? Alcohol ? (173, Note) 212. How do liquids chiefly differ from gases? (174) 213. In what respects do liquids chiefly differ ftrom solids? (175) 214. Give the most important consequences that flow from this fact ? (175) 215. How would you illustrate the upward and lateral pressure of liauids ? (176, Note)) What relationexists between the respective heights of two liquids of different densities placed in an inverted syphon? (176) What is the amount of downward pressure exerted oy a liquid confined in any vessel ? (177) How would you illustrate this fact ? (177, Note J Show that weight and pressure are not to be confounded with one another? (177, Note 2.) What are the weights respectively of a cubic inch, a cubic foot, and a gallon of water, at the temperature of 60*' Fahr. ? (178) To what is the pressure exerted by water on a vertical or inclined surface equal? (179) Give a rule for finding the lateral pressure exerted by water P (179) How do you find the pressure exerted by water against a vertical or inoUned lorfaoe »t a given depth beneath the water ? (180) U 216. 217. 218. 219. 220. 221. 282* 223. 210 EXAMINATION QUESTIONS. I 224. How do you find the pressure exerted against any fkvction of aTertical surface when the upper edge is level with the surflsoe of tiie water f (181) Explain what is meant by transmission of pressure by liquids f (18S) Describe Bramah's Hydrostatic Press, and illustrate ay a figure f (188) Explain the principle upon which Bramah's Press acts ? (182, Note) For what purposes is Bramah's Press used ? (184) How do we find the relation between the power applied and the pres* sure obtained by Bramah's Press ? (185) »vi Describe what is meant by the hydrostatic paradox. (186) 1 .'. I'i Show that it is not in reality a paradox? (186, Note) Describe the hydrostatic bellows. (187) Give the rule for finding the upward pressure against the board of a hydrostatic bellows. (188) When will a body float, sink, or rest in eauilibrium in % fluid ? (189) What weight of liquid does a floating body displace? (190) What portion of its weight is lost by a body immersed in a liquid ?(191) What IS the specific gravity of a body ? (192) What is the standard of comparison for solids and liquids ? (198) ; What is the standard of comparison for all gases ? (193) How do we find the specific gravity of a solid heavier tluui water ? (194) How do we find the specific gravity of a solid not suflSciently heavy to sink in water ? (195) What is the first method [of finding the specific gravity of a liquid? (196) What is the second method given for finding the specific gravity of a liquid? (196) How is the specific gravity of a liquid determined by means of the Hydrometer? (196) * Describe the Hydrometer? (196) What difference is there between hydrometers designed for deter> mining the specific gravity of liquids specificidly lighter than water, and those for ascertaining the specific gravity of liquida spedfipaly heavier than water ? (196) How is the specific gravity of gases found ? (197) How ma^ the weight of a cubic foot of any substance be fiound when its specific gravity is known ? (199) How may the solid contents of a body be found ftrom its weight ? (200) How may the weight of a body be found Arom its solid contents f (201) 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 236. 286. 237. 238. 239. 240. 241, 242. 243. 244. 245. 246. 347. 248. 249. 260. 251. 252. 253. 254. 256. 256. 257. 258. 259. 260. 261. 262. 263. 264. 265. 266. 267. ••-:ji What is Pneumatics ? (202) What is the derivation of the word atmosphere ? (208) ^ : What is the atmosphere? (208) To what height does the atmosphere extend ? (204) Give the exact composition of atmospheric air ? (205) What purpose is served by the oxygen in the air ? (205, Note) What purpose is served by the nitrogen ? (205, Note) -^^ Describe the principal properties of carbonic acid ? (205, Note) - ii'f. What are the chief sources of carbonic acid ? (205, Note) '^■^"■ What is the maximum and what the minimum amount of carbonic acid in the air? (206, Note) Describe the mode by which the air is kept sufficiently pure to sustain animal life, (206, Note) « -J^ Describe the property of gaseous diffusion. (206) Explain how the property of gaseous diffusion affects the composition of the atmosphere? (206, Note) Upon whatdoes the amount of aqueous vapor present in the atmos* phere depend ? (207) What is the maximum amount ? What its minimum amoant? (S57) To what is the blue colour of the sky due ? To what the golden tints of sunset? (208) Which of the essential properties of matter belong to tiie air f (SOO) nofftTeiiieal le water f (181) luidsf (18S> figure r (188) (182. Note) indthe pres* 8) .:»3 he board of a Inid? (189) 0) aUquidf(191) B? (198) I water? (194) sntly heavy to ^ of a liquid? [c grmTityofa means of the ,ed for deter- r thau water, idsspedfipaly e found when reight? (200) ntents ? (201) bte) Note) of carbonic ire to sustain composition L the atmoB- int? (257) dden tints of iir?(S09) EXAMINATION QUESTIONS. 211 268. How would you illustrate the impenetrability of the air? (209 Note) 269. How would you illustrate the inertia of the air ? (209, Note 2) 370. Why does air possess weight ? (210) 271. What may be taken as the fundamental fact of Pneumatics ? (210 Note) 272. What if tho weight of 100 cubic inches of each of the following gases, viz., oxygen, hydrogen, nitrogen, atmospheric air, carbonic air ? Give some illuatrutions of the aggr^ate weight of the atmosphere? (210,Kote2) How is it that the lower strata of sir are denser than the upper ? (211) 276. By what law does the density of the atmasphere decrease as we as* cend ? (212) 278. 274. 276. From what does the pressure of the air result ? (213) 277. Whut do we mean by sayii^ the pressure of the air is equal to 15 lbs. to the square inch ? (213, Note) 278. If the air were of the same density throughout to what height would it extend? (214) 279. How is this known ? (214) 280. How are permanently elastic gases chiefly distinguished from non- elastic gases? (216) 281. What is meant by permanently elastic ^es ? (216, Note) 282. Illustrate what is meant by the tlasticity of a gas. (217) 283. To what is the elasticity of gases due ? (217, Note) 234. Enunciate Marriotte's law? (219) 285. Illustrate it by a bent tube as in Art. 218. 286. To whttt extent is Mariotte's law true ? (219, Note) 287. What is the air-pump ? (220) 288. By whom and when was it invented? (221, Note) 289. Describe the exhausting syringe. (222) 290. Draw a sketch of the air-pump and describe its mode of action. (222) 291. Upon what principle does the air-pump act? (223) 292. How perfect a vacuum can be secured by the air-pump ? (223, Note) 293. Describe the condensing syringe. (224) 294. For what purpose is the air-pump chiefly used ? (225) 295. Give some illustrations of the pressure of the air. (225, Note) 296. Give some illustrations of the elasticity of the air. (225, Note) 297. What is the barometer? (226) 298. By whom and when was it invented ? (226, Note) « 299. What are the essential parts of a barometer ? (227) "i 800. What is meant by the Torricellian vacuum ? (227, Note) 301. How may the excellenoy of a barometer be tested? (228) 302. What is the cause of the oscillations of the barometer? (229) 803. In what regions of the earth are the oscillations of the barometer most fitful and extensive ? (2i9. Note) 304. To what regular oscillations is the barometer subject ? (230) 305. At what hours are the two maxima of pressure? (230) 306. At what hours are the two minima of pressure ? (230) 807. In what region are the semi-diurnal oscillations greatest? (230, Note) 308. Give some idea of their extent in tropical countries and explain why they are not observed in our climate. (230, Note) 309. How may the weather to be expected bd foretold by the oscillations in the height of the barometric column ? (251) 810. What does a fall in the barometer denote? (231, IE.) 811. What does a rise in the barometer indicate? (231, III.) 812. What does a sudden change in the height of the mercury in the barometer denote ? (231, IV.) 318, What does a steady nse in the column denote ? (231, V.) 814. What does a steady fall in the column denote? (281, VI.) ^5. What does a fluotuating state in the height of the column of morcunr denote? (231, VIL) 21^ EXAMINATION QtJESTIOlfS. t .1 n 316. Give Halley's rule for ascertaining the height of mountains, &c., by the barometer. (232) » -t; 817. Give Halley's rule with correction for temperature. (232) 318. Give Leslie's rule. (232) 819. Describe the essential parts of a common pump and illustrate by a diagram. (233) 820. Explain why the common pump is sometimes called a lifting pump. (233, Note) 821. Explain the principle upon which the common pump acts. (233, Note 2) 322. Explain why the lower valve must be within 32 feet of the water in the reservoir in order that the pump may act at all times. (234, Note 2) 323. Describe the forcing pump. (234) 324. Describe the essential parts of a fire engine. (234, Note) ^ ^ " -' 325. Describe the syphon. (235) ',; '•"^ 326. How is the syphon set in operation ? (235, Note 1) . 3:27. Explain upon what principle the syphon acts. (235, Note 2) .. . 328. 329. 330. 331. 832. 333. 834. 335. 336. 337. 334. 339. 840. 341. 842. 843. 344. 845. 346. 847. 848. 349. 350. 851. 352. 353. 854. (237) ->^8 When does the consideration of forces -come under the science of statics? (236) What kind of forces are considered in dynamics ? (236) ' '• Why is statics called a deductive science ? (237) Why is dynamics called an inductive or experimental science ? What may force be defined to beP (238) For what purposes is force required? (23S) What are the different kinds of forces as regards duration? (239) What are the different kinds of continued forces ? (239) What may motion be defined to be ? (240) What are the qualities of motion? (241) ' ' " ' ^ •" ** '^^^ What are *^,he different kinds of motion? (241) What kind of a motion is produced by an accelerating, constant, or retarding force ? (242) What is velocity ? (243) , ,. Of how many kinds is velocity? (243) " " " "' ' ' ^ ^ ' '^'^ V .+v»^ When is velocity said to be uniform ? (243) What is momentum or mocal force ? (245) To what are the momenta of bodies proportional ? (246) When the velocities of two moving bodies are equal, to what are their momenta proportional ? (247) When the masses of two moving bodies are equal, to what are their momenta proportional ? (248) When we speak of multiplying a velocity by a weight, what do we mean? (249, Note) When force is communicated by impact to a body at rest, how long will the body remain at rest ? (254) Give the first general law of motion. (255) Whose law is this? (257. Note) i"^- , Give the second law of motion. (266) ' ■ , '";' i v J*^*''^ Whose law is this P (257, Note) . , . -"V ^. Give the third law of motion. (257) -■ •'-'. - '--j* <'■>- MZ Whose law is this? (257, Note) itt'.n' . »<»;n m 365. What is reflected motion? (258) i vf 356. What is the angle of incidence ? (258) ' 357. What is the angle of refle. f.,%»r'JO'. a'e;? lat are their Lat are their what do we st, liow long • ■lu^ nd the angle ring through Uing from ft 862. What velocity is acquired by a heavy body in falling through one second? (264) 863. Through how many feet does a body fall during the first second of its descent? (265) 864. Deduce a set of formulas for the descent of bodies freely through space. (266) 865. When a body is projected upwards what is the nature of its motion? (267) 366. Give the formulas for the motion of a body projected upwards or downwards? (268) 867. When a body is descending an incline how is the gravity expended? (269) 868. What are the laws of descent on inclined planes? (270) 869. Upon what is the final velocity of a body falling down an incline dependent? (271) 870. What are the laws of descent in curves ? (273) "' • • 371. What is the brachystochrone ? '■'•- ♦ - 872. What is a cycloid ? (274) 873. Deduce a set of formulas for descent on inclines. (275, 276) 874. What is a projectile? (277) '^ . ^ 375. What forces influence projectiles ? (278) > •'• ^ - 376. What is the theoretical path of a projectile ? (278) ^ 377. What is a parabola? (278, Note 1) 878. Upon what erroneous suppositions is the parabolic theory based P (278, Note 2) 879. Show that when a body is projected horizontally forward, the horizontal motion does not interfere with the action of gravity. (279, Note) 880. What are the three conclusions of the parabolic theory ? (280) 381. What is the greatest horizontal range of a projectile P (280, Note) 382. To what is the velocity of projection speedily reduced, no matter what it may have been originally? (281) 883. How do you explain this? (281, Note 1) 884. What is the atmospheric resistance encountered by a ball or other projectile having a velocity of 2000 feet per second ? (281, Note 2) 385. When a ball has considerable windage, what is the amount of deflec- tion in its course ? (281, Note 3) 386. What are the roost important laws regarding the motion of projectiles thrown vertically into the air ? (282) 387. What are the most important laws regarding the motion of projectiles thrown at an angle of elevation ? (282) 888. To what is the explosive force of gunpowder exploded in a cannon equal? (283) 389. With what velocity does exploded gunpowder tend to expand? (283) 390. What is the composition of gunpowder ? (283, Note) 301. What is the greatest initial velocity that can be given to a cannon ball ? (284) 892. To what is the velocity of a ball of given weight fired with a given charge of powder proportional ? (284, Note) 393. To what are the velocities of balls of equal weight fired by the same charge of powder proportional ? (285) 894. To what are the velocities of balls of different weight but of the same dimensions fired by equal quantities of powder proportional ? (286) 395. To what is the depth which a oall penetrates into an obstacle propor- tional? (287) 896. Give the rule for finding the velocity of any shot or shell when its weight and also that of the chaise of puwder are known ? (288) 807. What is centrifugal force ? (289) 398. Why is it sometimes called tangential force ? (289, Note) 899. What is centripetal force? (290) 400. When does a body move in a circle? (291) , ^* 401. When does a body move in an ellipse ? (291) 'l» Hh 214 KX A MTNATIOK QUESTIONS. 403. 403. 404. 405. 406. 407. 408. 409. 410. 411. 412. 413. 414. 415. 416. 417. 418. 419. 420. 421. 422. 423. 424. 425. 426. 427. 428. 429. 430. 431. 482. 483. 434. 485. 436. 437. 488. 489. 440. 441. 442. 448. How long can a rotating mans preserve itself ? (292, Note 1) '• Give some examples of the effects of centrifugal force. (292» Note 2) If the velocity and radius are constant, to what is the centrifugal force proportional? (293) When the radius is constant how does the centriftigal force varyP (294) What is the amount of centrifugal force at the equator 9 (294, Note) How rapidly must the earth revolve in order that the centrifi^l force at the equator may equal gravity P (294, Note) When the velocity is constant how does the centrifugal force vary? (295) When the number of revolutions is constant to what is the centrifugal force proportional ? (296) Give a set of formulas for calculating centrifugal force. (297) Give a rule for finding the work accumulated in a moving body. (299) What is a pendulum ? (300) What is a simple pendulum ? (301) What is a compound or material pendulum ? (302) What is an oscillation or vibration ? (303) What is meant by the amplitude of the arc of vibration ? (301) ^ , What is meant by the duration of a vibration ? (305) Whatismeant by the length of a pendulum? (306) .. :. f r What is the centre of suspension? (307) . ,. What is the centre of oscillation ? (308) What is the centre of percussion ? (308, Note) What is meant by saying the centres of oscillation and suspension are interchangeable ? (369) How is the duration of a vibration affected by its amplitude ? (310) What is meant by saying the vibration of the pendulum is isochronous T (310, Note) What relation exists between the lengths and times of vibrations of pendulums? i814) - 'V ■ *A- *? •?.'- centrifugal 597) dy. (299y • (304) pension lure le? (310) ochronous ? ibrations of (Sll-316) t latitudes? 16^ Note 2) r«te) ^^■ 17, Note) I in a givm um of given endnlum of 'iftee In the of given size Itheactua idiacharged bn a4iutage I the bottom 447. When water spouts flrom an aperture in the side of a vessel how is the horizontal distance to which it is thrown found P (833) 448. When a liquid flows through a pipe or cluuiuel, which pm has the greatest velocity? (335) 4i9. How is the velocity of a stream determined ? (336, Note 2) 450. What are the principal varieties of water wheels P (339) 451. In water wheels, when is the greatest mechanical effect produced P (340) 462. Give the rule for finding the horse powers of upright water wheels. (341) 453. What is a turbine wheel ? How does it act ? (342) 454. For what purposes are high and low pressure turbines respectively used? (343-6) 455. What are the principal advantages of the turbine over the upright water wheel? (346) 456. What is the ori^n of all waves or undulations ? (347) 457. Of how many kinds are undulations ? (348) 458. What are progressive undulations ? (349) 459. What are stationary undulations? (360) 460. What kinds of vibration may be imparted to a stretched string ? (352) 461. What is meant by the time of vibration? (353) 462. What are the chief laws of the transverse vibrations of cords ? (354) 463. What are nodal points ? (355): 464. What are the pnncipal laws that govern the transverse vibrations of rods? (366) 465. How may an elastic plate be made to vibrate ? (357) 466. What are nodal lines and nodal figures ? (368-60) ~ *^ " ' ^ ^*^ 467. What are the laws of vibration of elastic plates ? (361) ' , . 468. Explain the cause and mode of undulation in liquids. (362) ' ' " ' i^ 469. Give the law of reflection of progressive undulations. (363) 470. Explain what is meant by the interference of waves and the pheno* mena resulting. (364) 471. Describe carefully the phenomenon of undulations in an elastic fluid ,^ ^ like the air. (366) 472. What are the objects of the science of acoustics ? (368) ^ ' " ' ^ 473. What are sounds? (369) ^,-'. 474. Upon what does the intensity of a sound depend ? (371) 476. How is sound affected by the density of the medium in which it is produced? (372) , . 476. How does the pitch of a sound affect its velocity ? (373) * ""•' 477. How does the velocity of sound in atmospheric air vary ? (374) -.- .-, 478. What is the velocity of sound in atmospheric air ? (375) 479. Give the velocity of sound in several other media. (376, 377.) 480. Upon what does the distance to which sound may be propagated de- pend? (378,879) 481. What is the result of the interference, partial or complete, of sonorous waves? (380,81) 482. What laws govern the reflection of sound waves ? (382, 3S3) 483. What is an echo? (383) 484. What must be the least distance of the reflecting surface in order to produce a perfect echo ? (384) — 485. What are repeating echoes. (386) 486. Give some examples of remarkable echoes ' (386) 487. Explain the construction of the so called whispering galleries. (887) 468. Name some of the best whispering galleries in the world. (387) 489. Describe the speaking trumpet and explain its mode of action. (388) 490. Describe the ear trumpet and explain the principle on which it Mti. (389) 216 EXAMINATION QUESTIONS. 481. 492. , 403. !■ 494. i 495. . 496. i 497. » 498. I » 499. t j 600. f*^ »- 501. '■:m 1 602. ^ 503. i i ' : I 504. 505. 506. 607. 508. 509. 510. 511. 612. 513. 614. 515. 616. 617. 518. 619. 620. 621. 522. 523. 524. Wliat is noise? (390) ■^Lmn^ika. Ml What are musical sounds ? (391) What are the three elements of a sound ? (392) /:• Jtii- What is tone or pitch P Upon what does it depend ? (393) q What is intensity? Upon what does it depend? (394) - "" - What is the quality or timbre of a sound ? (395) When are sounds said to be in unison? (396) .. <:ur What is a melody? (397) What is a chord? (398) >3[ .W. What is a harmony? (399) Describe the siren and Savart's toothed wheel and explain their use. (400) Describe the monochord and explain its use. (401) " " Give the relative length of cords and the number of vibrations required to produce each note of the gamut. (402) How is the absolute number of vibrations required to produce any given note determined^ (403) Give the number required for each of the notes of the common scale. (404) How do we determine the number of vibrations required for the cor- responding notes of higher or lower scales ? (406) How do we determine the len^h of a sonorous vibration ? (406) Give the lengths of the vibrations producing the C of different scales (406) What are intervals? (407) How are musical intervals named ? (408) Give the fractional length of the interval between each two successive notes in the diatonic scale. (408 and 409) What is a major tone 7 a minor tone 1 a semitone 7 (409) \ JHUFi What are diatonic and chromatic semitones? What is a grave chro- matio semitone ? What is a comma 7 (409 Note) What are compound chords ? (411) What is the perfect major accord 7 (4,11) What is the perfect minor accord 7 (411 ) What is the difiference between these as regards intervals ? (411 Note) Explain what is meant by the transposition of scales ? (412) What are sharps Kud flats and for what purpose are they employed ? (418) What is the chromatic scale ? (414) What is temperament ? (416) Eiplain the use of temperament in music. (416) Explain the phenomenon of beating in musical sounds. (416) Describe the diapason or tuning fork. (417) 525. Classify musical instruments ? (418) 526. Describe the mode in which wind instruments are sounded and name the most itnportant wind instruments. (419) 627. Describe the mode in which stringed instruments are sounded. (420) 528. Describe the difTerent varieties of the drum. (421) 529. Upon what circumstances does the pitch of the sound produced by a wind instrument deiiend ? (522) 530. What causes the difference of timbre in wind instruments? (422) 631. What are the different modes of producing sounds in tubes ? (426) 532. Give Bernoulli's lavps governing the vibration of air in tubes. (424) 533. Name the several parts that constitute the oi^n of voice in man. (426) 634. Give the position and common name of the larynx. (427) 535. Describe the structure of the larynx. (427) 636. What cartilages form the prominence known as Adam's apple in the fh)nt part of the throat ? (427) ^7. Where wre the m^t^noid cartilages placed ? What is their use ? (487) EXAMINATION QUESTIONS. 217 ixplain their f yibrations )roduce any amon scale, for tlieoor- ferent scales ro suocessiYO 538. Describe the cordse vocales. What is their position and their attach- ments? (428) 539. What is the ritna glottidia? What is its shape except during the production of sound ? (428) 540. WhatisthcRlottis? What the «pifir^^^{«? (429) 541. Explain the production of sound in the lai^nx. (430) 542. Illustrate the extreme precision with which the will can determine the exact amount of tension of the vocal cords. (431) 543. Explain why a man sings base or tenor while women, girls and boys sing treble. (432) 544. How do you account for the difference of timbre in voices ? (433) '■■ 545. Upon what does the loudness of the voice depend ? (434) 546. How are voices divided by musicians? (435) 647. Give the extreme number of vibrations of each class of voices. (436) 548. What is the range of the voice in speaking? What is the range in singing? {4S5Noie) 549. Describe the production of sound in the inferior animals ? (436) 550. What are the principal parts of the organ of hearing ? (487) 651. Name and describe the two parts of the external ear. (438) 552. Name and describe the three parts of the middle ear. (439) 553. Name and describe the three parts of the internal oar. (440) 554. What is the fenestra ovalis ? What the fenestra rotunda 7 (440) 565. Describe the position and probable use of the semi-circular canals ? (440, 443) 656. How and where is the auditory nerve distributed in the ear ? (441) 657. Describe the functions of the different parts of the ear. (442) 658. What are the most grave and acute notes that are perceptible to the ear? (444) 559. Describe the mechanism of hearing in the different tribes of animals. (445) !':>/> ? (411 Note) 12) jr employed ? THB BND OF PART I. 416) (" sd and name nded. (420) 'oduced by a ■ •.- 1,»- •■/,>. t H '.■ i, ■■ .M.(i ti ♦•'«♦»■• '■ vt » t, -•■ B? (422) 3S? (426) bes. (424) >ioe in man. •.,-W . f'i ■vi apple in the ruse? (427) ,*rf.* LOVELL'S SERIES OF SCHOOL BOOKS. .r:-c 11^ rpHE undersigned having long felt that it would be highly *** desirable to have a Series of Educational Works prepared and written in Canada and adapted for the purpose of Ca- nadian Education, begs to call attention to the Books with which he has already commenced this Series. These works have met with a very general welcome throughout the Province ; and the Publisher feels confident that the eulo- giums bestowed upon them are fully merited, as considera- ble talent and care have been enlisted in their preparation, Lovbll's Gbneral Geoobaphy will, it is hoped, fbrm a very valuable addition to this Series. While it has the ad- vantage of being prepared in Canada, and fully represents its geographical features, at the same time it embraces a sketch of every other country; and thus, while it contains all the information embraced in other works of the same kind relating to older countries, the different British Colon- ies, in those works but indifferently pourtrayed, are here delineated with due regard to their extent and position and to the importance of the acquisition of a correct knowledge of those Colonies, not only to the children educated in them but to every student of Geography. Tjie Maps illustrating this work have been prepared with the greatest care by draughtsmen in Canada, and will be found to have been brought down to the latest dates. *• JOHN LOVELL, Publisher. Canada DiBEOTOET Offiob, * Montreal, August, 1861. , ^^ '^ 90ES. )e highly prepared le of Ga- wks with ise works hoQt the the eulo- ionsidera- aration, I, fbrm a IS the ad- epresehts ibraces a contains the same sh Colon- are here iition and nowledge d in them ustrating ; care by are been H :hh blUhw, U vs r,'^> OPmrONS ON LOVELL'^ GENERAl. QEOQRAPHY. f" IN view of the promises held owt in the Prospectus of this Work and of its pretensions as a standard Educational Text-Book, it appeared to the Publisher desirable that, be- fore actual publication, the Author's labors might hare the benefit of the independent opinion of those best qualified to judge how far the object had been attained. Actuated by these considerations, the Publisher, with the Author's consent, sent out advance or proof sheets to com- petent persons in yarious parts of the Prorinces, who re- sponded by enclosing in many cases some very valuable suggestions, which were forwarded to the Author, and for which the Publisher tenders his thanks. Attention is re- quettdd to the following extracts from. Opinions upon the Work :— »#• >nyi. ♦ .-t'ifS- As an elementary work on a subject 90 e^tiensive, I con- sider the plan excellent, the matter judiciously selected, and for a text-book surprisingly full and complete.— £t OPmiONS ON LOVELL'S GENERAL 6E0OBAFHT. Autant qu'il m'a 4i6 permis d'en juger par I'aper^u rapide que j'en ai fait, elle m'a paru pleine de connaissances varices, int^ressanteSj et tres utiles & la jeunesse pour laquelle elle a 6t6 faite. — Bishop of Ottawa, »..■;. I have carefully perused it, and have no hesitation in pro- nouncing it as a most useful improvement on the Geogra- phies now used, and I wish you all success in your spirited undertaking.-^JisAop o/" Ontario. > ?rc...i I think Mr, Hodgins will be admitted to have executed his part with much judgment and ability, r . d that the work will give general satisfaction.-— CAie/'Jushce Robinson, The system Mr. Hodgins has adopted is one which, of all others, is altogether e£icient, and no doubt conducive to a clear, easy, and practical teaching of Geography, and in all probability will in most cases ensure success.— Juc^ge Mon- dekt, * V. „ ,, _^ It gives me much pleasure to state that the book is one which is worthy of Canada, and that, both as a scientific production as well as a work of art, it is deserving of all praise.-Wudgre Jiylwin. I trust you will find its sale to be as remunerative, as I am persuaded it will be found to be extremely useful not only to our youth but to ourselves, children of larger growth.— Judgr« Badgley, . , ^...,.,„.,j^^ Je recommande avec plaisir la nouvelle Giographie en langue anglaise, que vous vous proposez de publier, la con- sid^rant comme tres ntile, et comme ^tendue et compacte a la fois. — Judge Morin, As regards ourselves^ it is the first work of the kind in which the magnificent Colonies of Britain have had justice done them, and we should therefore testify our appreciation of such justice by a liberal patronage.-^u^c McCord, PHY. '9U rapide s varices, quelle elle on in pro- e Geogra- ir spirited executed that the Robinson, .- I .- . J « • -J -f—tmat — • ich, of all iicive to a and in all udge Mon- ook is one > scientific ing of all ative, as I useful not of larger raphie en r, la con- compacte e kind in td justice ireoiatioii :!ord. OPINIONS ON LOYEIL'S GENERAL OEOORAPHT. It is a vast improvement upon such works as have here- tofore been in circulation in the countrv, and it is pleasing to observe that you have given to GaUv .a and the British North American possessions generally, of which so little is said in other Geographies, that just degree of notice to which by their importance they are entitled. — Sir W. E. Logan, I think the work a very important one as a standard edu- cational book. It reflects very great credit on the Author and Publisher, and certainly deserves support, in such a very expensive enterprise, from every person who feels an interest in the progress of Canadian educational literature. — Dean of Montreal. Pour moi, je souhaite voir au plus tot votre conscientieux travail livr^ au public, qui lui fera, je n'en doute point, en Canada surtout, un bienveillant accueil. — Superior of the Seminary of St, Sulpice. Apres en avoir pris connaissance, aussi bien que de tons les 41oges flatteurs avec lesquels il a d4ja ^t^ accueilli, je ne puis, pour ma part, que vous exprimer ma parfaite satis- faction et vous f^Uciter pour la publication d'un ouvrage qui fait autant d'honneur a votre presse qu'il doit procurer d'avantages au pays. — Director of the Montreal College. ' ' * It is a work of prodigious labor, and of conscientious eiiort at accuracy of statement ; and therefore well merits the patronage of the classes of students for whom it is intended. I shall consequently introduce your book into my Schools, and shall, without hesitation or reserve, recom- mend it to my Brothers in Canada.— Director of the Chris^ tian Brothers^ Schools in Canada. The most prominent facts seem to have been carefully gleaned, with an arrangement that appears to be very simple and lucid. The illustrations and maps are also highly creditable for their variety and execution ; and the work in general appears to evince a large amount of industry and hhilitj. ^^r chdeacon Bethune. - ^- ~ - - - fi'- I sincerely hope that it maj meet with general adoption in schools and private families, not onlj in order that en- couragement maj thereby be given to the production of books of this class in our own country) which is much to be desired, but also because its general tone is such as to pro- mote a loyal attachment to the QueeU) under whose rule we have the happiness to live, and to the Empire of which we have the honor to form a part. — jirchdec^on Gihon. . s^ We have examined it, and we are conscious that we shall be consulting the best interests of the Schools of the Society by endeayouring to introduce the book into every part of our charge.-^General Superintendent in B. N. A, of the Colo- nial Church and School Society ; and the Superintendent for the JHocete of Montreal, I am glad to perceive that while general information respecting every section of the globe hf^s been equally dis- tributed throughout the General Geography, the resources and commercial importance of the Province of Canada have not been overlooked,~~>a feature which, with the style in which it has been got up and the lowness of the price, cannot fail to recommend it as a text-book for the use of Schools, and especially of Canada. — Rev. Dr. Mathieson. It contains an immense amount of information, and yet the style and arrangement are so natural and easy as to pre- vent any appearance of tediousness and dryness, and greatly to aid the memory.—- ijcv. Wellington Jeffers. I confidently anticipate for this |tnd your Qther sqhool books that large demand that will indicSite the high appre- ciation of the profession. — Rev, Dr. Wilkes* I have no hesitation in pronouncing it siiperior to any work of the same character and size extant.<— i2«v. Dr. Wood. Mr. Hodgins has displayed much ability in his wqrk. It is brief but comprehensive : "without overflowing, full,"— Rev, Dr. Shortt, adoption ' that an- iuotioii of lueh to be as to pro> se rule we which we I. j-«> it we shall le Society ery part of fthe ColO" endent for aformation quallj dis- e resources of Canada ;h the style the price, the use of 'Jathieson. m, and yet ' as to pre- nd greatly her sql^ool igh appr^- ior to any Dr, Wood. work. It ft fuU."- (ffZmOiro ON L0YZIJ.*8 GEN£«AL GBD^dAFHT. tWDUt As a Text-book for Schools, your Geography is, question, greatly in adyance of all others that have > oeen' presented for public use in this country, and canno ail to prove a great boon to both teachers and scholars. — Hev. J. F. Kemp. .., . , Your work on "Geography" supplies a want which teach- ers, and all, I believe, who in Canada take an interest in the education of the young, have long felt and complained of. — Rev. Dr. Leach. 'H r - . ■"TT The whole plan, order, and execution of the work, as well as the low price at which it is proposed to offer it, render it a most excellent and in all respects suitable school-book. — Rev. Dr. Irvine. Such portions as I have paid particular attention to, ap- pear to me to be very accurate, considering the diversity and fulness of the information furnished, and the vast amount of labor which must have been incurred by the selection and arrangement of it.— i2er. W. Snodgrass. I believe the work to be better adapted for use in our Schools than any publication of the kind with which I am apquainted. It will become a necessity in our Seminaries of Education.— i?€r. Dr. Bancroft. An enterprise of this nature, undertaken to meet what may be considered a great national want, deserves to have extended to it such encouragement as its importance merits, and in this case both Author and Publisher are entitled to a large meed of praise, the one for his enterprise and patri- otic spirit, the other for the care bestowed upon its compil- ation and arrangement. The work under review seems to merit the highetit commendation.— l^ev. /. EUegood. The plan and manner of execution, are both admirable. The amount of information given, and mode of its arrange- ment, evince great research and good XoAi^.'^Rev. W. Scott. I I « I OPINIONS ON LOYELL'S OENEBAL GEOQKAFHT. The fairness and impartiality with which the different * countries are described will commend it to general usej and I believe that its introduction into the schools of this con- tinent will greatly promote the acquirement of sonnd and correct information in this branch of education. — Rev. E. J. Rogers, It seems to me just what is needed, suited to the require- ments of the country in its matter, form, and price. It is decidedly superior to the Geographies found in general use in the Schools of Canada. — Rev. J, B. Bonar. I think the publication of the work ought to be regarded as a matter of sincere congratulation to the country at large. The arrangement of the book appears to me to be excellent; the information conveyed is well selected and condensed. — Rev, W, S. Darling, Where all is excellent it is difficult to particularise, but I may state that I consider the introductory part deserving of especial commendation.-^i2ev. John M. Brooke. I have not only looked through the whole work, but I have carefully read large portions of it : and to say I am very much pleased with it would very faintly convey my sense of its excellence. I am really delighted that at last a School Geography, almost perfect, is provided for the youth of the British North American Provinces.— i2er. John Carry, itour book is all that can be desired^ and after a thorough examination I am convinced that, from its merits, it will at once be adopted in all our schools. I have been teaching for fifteen years in Canada, and have found such a text- book to be the great desideratum. Your Geography is a marvel of cheapne3S,->-^admirable in plan^ — and a fine speci- men of what can be done by an enterprising and liberal publisher. We shall at once introduce it into our school, as its want has been long felt. — Rev^ H. J, Borthwick^ Prin- cipal of the County of Carleton Senior Chrammar School, different usei and this con- ound and lev. E. J. e require- ce. It is ineral use regarded gantry at » me to be icted and rise, but I serving of '^ork, but I say I am jonvey my kt at last a d for the ces.— 12er. i thorough its, it will n. teaching ih a tezt- ^aphy is a fine epeci- Buad liberal >ur school, oickf Prin- School, OPINIONS OH LOTELL'S GENERAL GEOGBAPHT. It is my intention to adopt at once this Geography as a text-book in the Grammar School department of this Insti- tution, and I have no doubt tba; it will soon win its way into general use in all our Schools. — Rev. S. S. NelleSj President of Victoria College. The classification appears to be faultless, the definitions .concise and lucid, and the information given in regard to the derivation and pronunciation of proper names is very valuable. It is indeed multum in parvOj and will doubtless become the standard Geography of our schools. — Rev. I. B. Howard. It displays no ordinary degree of ability, industry, taste, and perseverance. A book of this kind is very much re- quired in this country ; and affords information regarding the Colonies which no doubt will be appreciated by old country residents. I shall be most happy to recommend it to the schools in my superintendency, as well as to heads vof families, and hope it will be patronized as extensively as it deserves.— -iiei;. /. Gilbert Armstrong. • " ■ When it comes to be known by the public, I should think it must command a very extensive, if not universal, circula- tion in the Schools of British North America. — Rev. John Gardner. . .; The plan of your School Geography is excellent and I hope it will meet with the success it deserves. — Rev. Dr. Leitchf Principal of Queen^s College. So far as I can judge, Lovell's General Geography is well adapted to our Canadian Schools. — Rev. A. J. Parker. I admire iU arrangement very much. With such brevity as was necessary to the plan pursued, its fulness on all the subjects connected with Geographical study is remarkable. It is most gratifying that Canada is not only preparing her own school books, but that, as in the case of the Geography they are of so high an order of merit. — Rev. S. D. Rite. ■■4fih- OPINIONS ON LOVELL'S GENERAL OEOGBAPHT. I must sincerely congratulate you on the mechanical as well as the literary execution of the book. No existing work can be held to excel it. — Rev, A. de Sola. By the prominence given to our own, and the other Colo- nial possessions of Great Britain and the due proportion of space assigned to other countries it is much more suitable for the use of our Canadian youths than Morse's and other similar Geographies which give such undue proportions of space to the United States. — Rev. Henry Patton. Until your book shall be in the hands of our youth, the only notice so far as I am aware, that our growing country, one of the finest in the world, and likely soon to be one of the most important, has obtained in works on General Geography, is only what can be crowded into some half dozen pages of some small book. — Rev. C. P. Reid, This Geography — without controversy the best yet given to the British American public — will do much toward ex- alting the popular estimate of this branch of study, and fostering the patriotism and loyalty of our people.— iZev. A. Carman, And while it does credit to your enterprise, and to the skill and talent of the accomplished Author, I doubt not but that it will be hailed by every intelligent teacher of youth, as well as by a grateful community, as a boon much- needed and well-timed, — calculated at once to save the minds of our youth from improper associations, and to lead them to cherish national and patriotic feelings.'—iZcv. Dr. Urquhart. The plan is most excellent, inasmuch as it contains mtd' turn in parvOf and brings into one view an immense mass of useful informations, abridging the labors both of teacher and taught in no ordinary degree. — Rev. David Black. I am much pleased with the plan and style of the work. It cannot fail of being useful in the schools for which it is intended— i{ev. /. Ooadby. nical as existing i.v'* sr Colo- )rtioii of suitable id other )ortion3 )ath| the country, )e one of General )me half ret given )ward ex- tudy, and -Rev. Jt, nd to the oubt not eacher of on much- save the id to lead 'Rev. Dr. ains tnul' e mass of acherand he work, rhich it is It is certainly the best and most impartial Geography for the use of Schools which, to my knowledge, has issued from the press on the North American continent, and will, I trust, receive from the public all the encouragement it so eminent- ly deserves. — Rev. Dr. AdavMon, : '■ : :- The work is well planned and executed, comprising in remarkably moderate bounds a vast amount of information. It is an improvement on every other School Geography I am acquainted with, and is likely to take a chief place in Canadian Schools. — Rev. Dr. Willis. The General Geography will, no doubt, become a valuable national work, and take its place as a standard book in ovir schools.— iHrs. Susanna Moodie. I have examined with some care the General Geography you were so kind as to send me, and am very much pleased with it, especially with the portion relating to Canada. The want of a correct description of the British Provinces has long been felt in our schools, and I am sure you will find a hearty appreciation of your efforts to supply that need. — Miss Lyman. The plan is excellent and answers all the requirements of an intelligent work on the subject ; the facts, (so far as I am able to judge of them) are correct and well chosen ; and the pretty and truthful engravings, by which the book is illustrated, considerably enhance its value and usefulness. I believe you have satisfied a want long felt in Canadian schools ; therefore, as soon as it is ready, I shall gladly place the General Geography in the hands of my pupils, as a text-book. — Mrs. Simpson^ Principal of Ladies^ Mademy, 4 Inkermann Terrace^ Montreal. 1 have long desired to see a Geography which would give Canada, and the other British Provinces, a proper share of attention ; and in issuing your new work you have supplied the schools with a valuable auxiliary for conducting the education of our youth. — Mrs. E. H. Lay^ Principal of Young Ladies^ Institute^ Beaver Hallj Montreal. .I t The Author's naniQ (to say nothing of the Publisher's) was sufficient to insure my respectful attention to the admirably got up volume now before me, and I rise from its perusal convinced that I shall be able to use it in my Seminary with considerable advantage to all concerned.-— Mrs. Gwdorij Principal of Ladies^ Seminary^ 5 Jirgyle Terraciy Montreal. , . The work is well adapted to meet the requirements of the schools in our own Province, and will do good service should it find a place in the schools of other lands. — Riv. Dr. Ormiston. , ; . ...... \ '-0 . ■ I have carefully examined the advance sheets of your General Geography, which I think is a great improvement over any other book of the kind now used in Canada. — Hon. John Young. ' f ^5/. ' Its complete description of the British Colonies fills a vacuum not supplied heretofore by either Foreign or Britis|h Geographies, while the style in which it is got up, and its low price, cannot fail to recommend it for general purposes. —Hon. A. A. Dorionj M. P.P. It is a work well calculated to attain the end which you have in view, and will undoubtedly prove invaluable, as a text-book in the hands of our Canadian youth. — J, B. Meil- lew, M.D. I have carefully perused your valuable work on General Geography with much pleasure, and am convinced that it will attain the patriotic ends you aim at. — Wolfred Nelson^ M.D. I think that your Geography forms an exception to other works of the kind, ^s you have dealt in equality of fairness with all countries, thus rendering the volume one which might with the greatest propriety be placed in the hands of a pupil here, in England, the United States, or Australia. In fact I think you have made it as cosmopolitan as such a work can well he. 'Archibald Hall^ M.D, )lisher's) to the rise from it in my erned.^- jSrgyle s of the seryice s. — Rev. of your ovement I. — Hon, s fills a r British and its lurposes. tiich you 3le, as a B. Meil- General that it NeUon^ » -» ^ ^T- **^ i . .; l-.»» to other fairness > which ands of istralia. such a OFnniirs on ixttELvn general geogsapet. The Editorial department has been carried out with a talent and perseverance worthy of the highest encomiums^ and has left nothing to be desired. As an Educational book of the first class, I feel confident that it will supersede any work on the same subject at present in use. — Charles Smallvjoodf M.D. I have much pleasure in saying that I conceive it to be compiled with much care and judgment ; at the same time the admirable engravings and maps add greatly to its value, and make it in my opinion the best School Geography I have ever met with. — Dr, T. Sterry Hunt» I believe that the Geography will prove a boon to the country, and will have a most happy effect in training the youth of the British Provinces to right views of the great extent of their country, and of the variety of its resources, and will largely contribute to the development of a national sentiment. I trust that the Geography will obtain the widest and most general circulation, and that you will thereby be rewarded for your public-spirited enterprise.— Mexander Morris^ M. P. P. The work appears to be well adapted to the purpose of instruction as well as of reference, and I trust that the en- terprise and zeal which yon have shown in thus providing a work more particularly adapted to the Canadian standing point, though by no means confined to it, will meet with the success that it merits. — Colonel Wilmotf R. Jt, I cannot wish you better success than your excellent work so richly merits, and I trust the people of Canada, at least, will show their appreciation of it by its general adoption.— Dun6ar Rott^ M. P. P. I think your Geography better adapted for Schools than any one I have seen used in the Province.— Andrew Robertson, It was high time we should have a School Geography which would give due prominence to our own and the sister ColonieS) as yours does.— 2^oma« D^Jrcy MeOee^ M, P, P, 'rmsas^e^. t OPINIOKS ON LOVELL'S GENERAL OEOOSAPHT. A School Geography giving more ample information to our youth concerning British America, has long been a desideratum in this and our sister Colonies, and I rejoice to find that the work under my notice so fully meets this want. Mr. Hodgins and you have, in this Tolumei made a very valuable addition to our series of School Books, and I have no doubt that your enterprise will be appreciated by every friend of education. — B. Workman^ M.D, Le moins que j'en puisse dire d'apres le specimen que j'en ai devant moi, c'est qu'd mon avis il devra surpasser I'attente, tant dans son ensemble que dans ses details, de ceux qui d^siraient voir remplir la lacune qui existait pour la langue anglaise au moins, dans les livres & I'usage des ^coles.— Etienne Parent. Tj ai admir^ I'ordre et I'arrangement des matieres comme de leurs lucides et classiques dispositions, qui accusent de savantes recherches et d'heureuses combinaisons. — Joseph G. Barthe. J'e ne hasarde rien, en disant qu'il n'y a pas, en g^ographie, de volume qui pour un priz aussi modique, offre la reunion d'un au3si grand nombre de notions pratiques. En un mot, rien n'a 6t^ n^glig^ pour rendre cet ouvrage aussi complet qu'il ^tait possible, en se renfermant dans les limites de ce qui est r^ellement utile auz enfants.— P. R. Lafrenaye. Altogether the volume reflects the highest credit upon its learned Author, Mr. Hodgins, already favourably known by his previous labours in the same field .-^/pAeus Todd, I look upon Lovell's General Geography, to be a great improvement upon the books on the same subject now gener- ally used in this Province.— JFVee/ericX: Griffin. The arrangement of the work is good. Its aim is not to be a history but to fix localities and the prominent character- istics of nations, provinces and peoples, in mind ; to give land-marks to guide the voyager on the ocean of knowledge. — John S, Sanborn, ation to been a . rejoice sets this made a , and I ated by que j'en attente, ;eux qui langue coles.— I comme isent de -Joseph B^raphie, reunion un mot| complet BS de ce lye. ipon its own by A great r gener- ! not to tractor- to give p^ledge. OPINIONS ON LOYEIX'S GENERAL GEOGBAPHT. I have never seen one arranged upon a better system, or more profusely and judiciously illustrated. I have no doubt it will immediately become the standard work in our schools, where it will supply a very great want. — Thomas C. Keefer. I have examined the work with considerable attention and very great pleasure, and think it highly creditable to Mr. Hodgins and to yourself, as well as to the Province. It seems to me to be a very excellent school book. — G. W. Wicksteed, I may mention that your Geography is well adapted to supply a want that has been much spoken of, and occupy a place in our school literature, which hitherto has been but indifferently filled. — Fennings Taylor. I think I am justified in entertaining the confident expec- tation that your "General Geography," through an enlight- ened appreciation of its varied intrinsic merits, is destined very shortly to supersede most of the Geographies now in use in British North America. — T. A. Gibson^ First Assis- tant Master J High School, Montreal. I am sure the Teachers of Canada will feel grateful to you for publishing the new Geography, a specimen copy of which I have just beep looking over with much plesaure. Such a work has been long needed in this country, where the in- structors of youth have been obliged to use books either badly arranged, or very scantily furnished with information connected with the British Provinces of North America.— Professor Hicks, McGill Normal School. I have just been perusing your General Geography, edited by J. George Hodgins, LL.B.,F.R.G.S., and I must say that it is an excellent work, and I make no doubt will soon super- sede all other Geographies in the Schools of Canada. — Charles Nichols, Principal of Collegiate School, Montreal. We have a true representative of our magnificent Provin- ces. The plan of the work is excellent, and the definitions are accurate. — John Smith, Head Master, High School, St. Johns, C. E. \' Its general plan is good. The prominence given to phy- sical phenomena, and natural and artificial products, as well as to history and statistics, is a distinctive feature that will commend the work to those who have enlarged views as to the real nature and objects of geographical science. — Dr. LawioUj Queen's Colleg 3, Kingston. . ^ The whole work is marked by learning, ability, and taste. The arrangement is natural, and therefore excellent. The information supplied is very great and very good, just what is wanted for the school-room, and suited for the studio. The labor and care bestowed on it have been immense, and reflect much credit on all concerned. In making this valu- able addition to ihe school books already published, you have laid all connected with the education of youth under renew- ed obligation. This work should, as I trust it will shortly, be in the hands of every teacher and school officer in Cana- da. — Archibald Macallunij Principal of the Hamilton Central School. It supplies a want which has long been felt in Canadian Schools, and is, I conceive, specially adapted to the youth of British North America. I have no hesitation in saying that the work must come into general use in our Schools. — William TassiCj Principal of the Gait Grammar School, In general terms, I would express the opinion that you have hit upon the just medium between the prolixity of his- tory and the conciseness of mere tabular statistics. It con- tains the general principles of Geography, and enough of des- cription to suit the requirements of Schools ; and the promi' nence given to our oum country is a feature that specially commends it for use in Canadian Schools.— /{o^ut Parmalee. The completeness, with conciseness, of the information it affords must commend it ; and your avoidance of the too com- mon mistake of giving too much space to particular sections of the earth, to ^^4 equal neglect of others just as important, should secure for it general confidence and acceptance.— T%oma8 M. Tayl r« ■'. -s r ; I ?c PHY^ people will re in, and ;ht to con- mpiling a lent on all but is as eneral use. I Globe, a ressed and ^Montreal ret issued 'Ommercial This is a very valuable work, which we warmly commend to the notice of Teachers and all persons engaged in the i&sk ofEdvLC&tion.-^True WitnesSi Montreal. Merely to say that Mr. Hodgins, the able and accom- plished author of the volume, has executed his work well, is, we think, but paying him a poor compliment. He has undertaken and discharged a duty which we think few could have achieved with equal success. — British American Journal^ Montreal^ ^-^ We think the rising generation in these Provinces should have a geographical text-book for themselves, giving a true history and correct description compatible with their politi- cal and social importance, and such a text-book we have in that now before us. — Canada Temperance Advocate^ Montreal, No work of the kind could be more complete.— Toronto Daily Leader. The introductory Chapter on Mathematical, Physical, and Political Geography, is a manual of concise simplicity, which will at the outset enlist the approbation of the thinking Teacher. — Home Journal. Toronto, i,^ ,- ^. n.^ i., ^i It is correct and most explicit wi*h regard to every por- tion of the globe. — Hamilton Daily Spectator, Such a work was needed in the British Provinces, and we feel proud that we now have one every way worthy of the country. — Canada Christian Advocate^ Hamilton, It is exceedingly well got \ip, -^Kingston Daily British Whig, In Canada, we feel assured, it will find its way into every household. — Kingston Daily News, '" *" ^ " LovelFs General Geography is the very thing that is re- quired for our Schools— most ably and correctly got up, handsomely printed, and in a national point of view, it is a boon to the country.— £cra2(2 and jidvertiser^ Kingston, ■ 5.1 OPHnom^ OH LOVSiL'8 OEHEBAL CfflOGOUIPHt. The infoifmatioii is derived from tlie most Approved sources, and is arranged in a manner so systematic as to afibM the greatest facility for both teacher and scholar. — Quebee Gazette, .. ^ It is a valnabl^ contribution to the cause of ediicaiioh.-- > Loiutoh (C,W,) Daily Free Press. ^ . ■:^i- m^""i'h^ i^^.^ isrf^sr.f? One of the most useful works ever issued from the Cana- dian Press. — Ottawa Gazette. ■ ^ AT . Mt-^^ We rejoice in the appearance of this new and excellent compendium of Geography. — Cohourg Star. A4 i>4 It is the most complete and ihte'restiiig work of the kind ever published.— Cofrotfrg- £fe}^ineZ. To Ganadiaas this is an invaluable work, as it is the only Qeography that has ever done justice to Oanada and the other British Provinces. — Belleville Intelligencer, We have examined it carefally and find that it is Superior ti) any Geography now in Mie.-^Perth Couriir. ^^■, ^«« We consider the Geography one of the best extJEuit, and hope, it may soon supersede, in the schools throughout the Province, the use of all similar publications.— £^^. J(^ns News. Lovell's General Geography is a Canadian wonder. In fact, it is just such a manual as we would wish to see intro- duced into every School in Canada.— jRtcAmoni Guardian. Cette G^ographie est destined & rendre un grand service si I'education primaire des enfants. — Courier de St. Bya- cinthe. No other Geography contains duch a store of information respecting the Eritiih North Amisrican possessions, ahd none other doeid lequlil Justice i6 th^ t^irHtorial extent and boundaries of ttie tJAited ProViiced cif ITppWr And Lo^er Canada. — Bwttingdon Herald* n the Cana- • It 18 full of valuable information, is beautifully printed, elegantly illustrated, and is well worth the small price claimed for it— one dollar. — Niagara Mail. ■-.v uhv a After a careful inspection of this Canadian work, we un- hesitatingly pronounce it to be a valuable boon conferred upon the youth of the British American Provinces. — British Constitution f Fergus. 1-: The work is certainly one of inestimable value.— TTAi/iy Prfisf, ':)i.'^:mif\ ^■'»'^iy: '5r:*\i ">«^ 'sAHPk'ti'^iH -v''** i' ii»>;}j 4^«f Its plan and arrangement are both admirable, and while it has the recommendation of brevity, it is a full and complete geographical work. In these respects, as well as in mechan- ical execution and literary ability, it excels all works of the kind hitherto produced.-rJFiiY&y Chronicle, We have closely examined this new School Book, and we are led to the conclusion that it is the most valuable and comprehensive work of the Jdnd, for the use of schools, that could be put into the hands of our students. It must at once become a standard School Bo6k.»^-Whitby Watchinan, Wltile it by no melons neglects ^he Geography of the other Qountri^s of the world, that of Canada occupies the most prominent position. — Paris Star, Mr. Hodgins* work is free from dwarfing the interests of any people, but large attention is ^iven where most needed —to Canada and the sister Colonies. — Argus, Chatham, It metpts a want which nothing has hitherto supplied; and we are convinced that it will work its way into the houses as well as the Schools of our land.— TTe^^Zy Dispatch, St, llioTnat, This is fi.very bej^utlful end laseful Geography, just issued at the ^w yxlfip Qt.qi^^ Af^i^T.~-^Grfi^ donia. OFimOHS ON LOVELL'S OENEBAL GEOGSAFHT. We may safely predict its being adopted as a text-book in all the Schools and Colleges throughout the Proyince.— Gananoque Reporter, It is the best publication of the kind ever issued. — Omemee Warder, We highly commend this Geography, being excellent beyond all competitors. — Cayuga Sentinel, Not only as an exhibition of Canadian literary progress, but as a beautiful and appropriate sample of Canadian art, we must congratulate the Publisher on this yery opportune and praiseworthy donation to the teachers of youth in Canada. — British Flag^ Brighton, The explanatory and descriptiye matter is of the most use- fal and comprehensiye order. — Welland Reporter ^ Drum' mondville. The present work commends itself at once to the atten- tion of parents and teachers. — Waterloo Chronicle, --<-" ^^^ After much care and attention in the examination of this Geography, we haye come to the conclusion that it is the best Greography published, and we can conscientiously re- commend it to the attention of teachers of schools in Canada. '—Maple Leaff Sandwich, ^'We earnestly recommend its general adoption in our Schools. — Essex Journal, Sandwich. ^..^ We earnestly trust that no time will be lost in introduc- ing it to our Common Schools— no Canadian youth can un- derstand the Geography of his country without having studied Loyell's General Geography.— TToodc/ocA; Sentinel, As a complete Geography and Atlas, this new work is superior to any other extant, and is just what is yery neces- sary in our Canadian Schools, into which we hope to see it at once introduced.— Per^A Standard^ St, Marjfs. . ag excellent to the atten- tion in our OPINIONS ON L0YELL*8 OENEBAL OEOOBAFHT. The arrangement of the maps and matter is admirable, and well calculated to make the study attractive to the learner. — St. Mary's Argus. It is in every respect a most excellent elementary work, and admirably adapted for the use of schools, and we hope to see it universally adopted as the School Geography of Canada.— JSramp^on Times. It begins, as it ought to do, with Canada, and is in matter, illustration, execution and general comeliness, a credit to the covmix J. '■^Norfolk Messenger, Simcoe. To those engaged in educational pursuits, we commend Lovell's General Geography. — Northern Advance, Barrie. It is with no ordinary feelings of pleasure we hail the appearance of this work.— OsAatca Vindicator. We hope to see this Geography, introduced into our Com- mon Schools, and generallyadopted by Teachers and Instruc- tors in the Canadas. — Berlin Telegraph. This excellent book, which is creditable to any printing establishment, is well adapted to the use of our Canadian Schools. — Markham Economist. We are fully convinced that it will prove to be of great utility in our schools. It should be highly prized by Cana- dians, not only because it is a Canadian work, but because, in addition to its giving a satisfactory knowledge of all parts of the world, it gives a fair portion of prominence to the British Colonies. — Brantford Courier. vi* We have no hesitation in recommending it to the favour- able notice of teachers and friends of education generally. — Cobourg Sun, ta^^'asfSEj' J 1i! ■').i LOVELL'S IV GENERAL GEOfiBAFHT, BY J. GEORGE HODGINS, LL.B., F.R.G.S., EMBELLISHED WITH 51 SUPERIOR OOLGURED MAPS, AND A TABLE OF CLOCKS OF THE WORLP. rilHIS GEOGRAPHY Js designed to furnish a J- satisfactory resunU of Geographical knowledge of all parts of the World,, and to give equal prominence to the BRITISH OQLONIES, concerning wh^ch such meagre information is generally found in works of this kindi It will be found a suitable Text-Book for chil- dren in CANADA, NOVA SCOTIA, NEW BRUNS- WICK, NEWFOUNDLAND, PRINCE EDWARD ISLAND, the EAST and WEST INDIES, AUS- TRALIA, &c. R. & A. MtLLER are the General Agents for the Sale of this Book throughout Canada, and will supply the trade on advantageous terms. Tl^e Geography is also on Sale at the Bookstores in the pripcipal Cities in ENGLAND, IRELAND, and SCOTLAND— In NOVA SCOTIA— NEW BRUNSWICK — NEWFOUNDLAND — PRINCE EDWARD ISLAND— The EAST and WEST INPIES—AUSTRALIA, &5. JOHN LOVELL, Puhlisher, Montreal, August, 1861. 1 7 1 725$If" OHY, D MAPS, Ib'tti* T ; #-*'| f : : WORLP. to furnish a knowledge of )rominence to 5 whiph such works of this Jook for chil- 3W BRUNS- ; EDWARD )IES, AUS- ^ents for the id will supply ,6 Bookstores IRELAND, riA— NEW — PRINCE and WEST Publisher,