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Maps, platas. charts, ate, may ba fllmad at diffarant reduction ratios. Thosa too larga to ba entirely included in one exposure ara filmed beginning in the upper left hand corner, left to right and top to bottom, aa many frames aa required. The following diagrams illustrate tha method: Les cartes, planches, tableaux, etc., peuvent hum filmte A des taux da reduction diff fronts. Lorsque le document est trop grand pour fttra reproduit en un seul cllchi, il est film* 6 partir da ('angle supirieur geuche. de gauche k droite, et de haut mn bes. en prenant le nombre d'images ntcessaira. Les diagrammes suivants lllustrant la mAthode. 1 2 3 1 2 3 4 5 6 \ } N •AN ELEMENTARY TREATISE ON MECHANICS; DESIGNED AS A TEXT-BOOK FOR THE UNIVERSITY EXAMINATIONS FOR THE ORDINARY DEGREE OF B. A. PART I. STATICS. BY J. B. CHERRIMAN, M. A., LATE FELLOW OP ST. JOHN's COLLEGE, CAMBRIDGE, AND PROFESSOR OF NATURAL PHILOSOPHY IN UNIVERSITY COLLEGE, TORONTO. SECOND EDITION. TORONTO: COPP, CLARK & CO., KING STREET EAST. 1870. ^V C T, CLARK k CO., PRINTERS, KINO STREET EAST, TORONTO. FEtTB mi C^r^l * . . ' J:, < > PREFACE TO FIRST £dITION. This Treatise contains the text of the Lectures which I have delivered for some years past in University College. As my design has been only to furnish to Students a text- book for such parts of the subject as are required for the ordinary degree of B, A. in Universities, I have not thought it advisable to burden this work with mere explanation or illustration, or to add examples ; presuming that such, where necessary, will be furnished in the Lecture-room or by the Tutor. University College, Toronto, April r, 1858. CHAPTER I. DEFINITIONS AND nUNt'lPf.ES. 1. A material 2iar tide is a portion of matter occupying ni y^J^'iTi,-.' iDdofinitoly small spnco ; or, a geometrical point endowed with the properties of matter. All bodies may bo gcomctrioally conceived as made up of particles. 2. When the distance lotween two particles remains un- Force, changed during any period of time, they are relatively at rest, and we conceive that they will continue so unless one or both bo acted on by some causo to which we give the name of Fo7'ce. The state of rest or motion of a particle can only be con- ceived of in relation to others, but it is oonyenient to speak of it absolutely as being at rest or in motion, reference being understood to ourselves (or some particles in a known relation to ourselves), and changes of rest or motion are to be consi- dered as produced by forces acting on the particle alone. 3. When a particle at rest is set in motion by a force, it j^^ijfj*""^'"" will begin to move in a particular line, which we may define uitgnitude to be the direction or line of action of the force. The motion might be just prevented and the particle kept at rest, by a suitable force applied in an opposite direction. In this case the two forces are said to balance or counter-balance each other ; and the magnitudes of two forces are said to be equal Eqnai when each would separately counter-balance the same force. 4. Generally, when forces acting on any system of particles statics, keep them at rest, the forces are said to counter-balance, or problem or. 6 il Forro, hfiw ni«Muri'il. Wclb'ht. JJlllt of Kuri'u. niKldity. to bo in or|ullibrium ; ntxl tliu investigation o? tho rdutioni ainon^ tlioni in such oaiio, or the conditions of o<{uilibrium, conHtitutoa tho ^oionoo of Slutla. 6. 8oino convenient forco beinp; nsHunuMl nn n Ntnnilard or ifiiiV, tho uingnitudo of any furco in niciiHured numerically with rcferonco to tho unit l>y tho nuinl)or of nuch uniln (acting hiniultnncouHly at a point and opponito to tho force), which it will counter-balanco. Thus, if tho forco will couutor-balanco V unit forces, its magnitude is Huid to bo n. Tliia Hupposes n to bo a wholo number, nnd wo can nlwayn take a unit-foroe of auoh magnitudo that it Nliall be so ; then, whon any other force is tnkon tin tlio unit, tho mnKnituIo of rur original force will bo cxprcMHcJ by tho ratio (whethor n wliolo number or n fraction) wliicU its niagnituiio bears to tlint of tho forco assumed nn unit. In general, tho term Pmtsnre may bo used for a Force thus statically conaidcred and measured. C. It is found that on all bodies on tho Earth a pressure is exerted downwards, in a vertical direction : that is, in a direc- tion perpendicular to tho surface of still water at tho place; and this pressure (which, for any particular body, is culled tho wciyht of that body) is invariable at tho same place for the same body at all timos, whatever form, size or position tho body bo made to take. Hence the weight of some particular body may conveniently bo assumed as a unit to which other pressures may bo referred for measurement. In the EDglish systom the weight of a certain picco of brass caro- fuUy preserved as a standard, is called one pound Troy, and all other weights are referred to this. If lost, it might be restored from tho knowledge that this pound being divided into ^7 GO ff rains, '• a cubic inch of distilled water, of the temperature of 62° Fahrenheit, when the barometer is at 30 inohee, weighs 252.724 grains " 7. When a system of particles, or a body, is such that the relative distances of the particles undergo no change by the action of the forces applied to them in any manner, the system is said to bo rigidly connected, and the body is called a riyid body, relatively to the forces concerned. 8. Prnm the dcfinitloni^ laid down, it will bo ohsprviMJ fhfit thrco nletnenti enter into ovory Forcp : (I), itii pi)int of np- pHcntion, or tho pnrliolo on which it nctn; (2), \tn ilirpction ; (8), \tn ma^iiitudo. When thr«o arc known, tho forco in fully dolcriJiiimd. Tho fullowinuf is tho fumlaniontiil law, deduced from r;tr"iiMKrj eiporiiuunt, oo whigU tho Sciuuco of 8tatic» is based : *" '"* // two equal forcct act rmpectiiulij on tiro jxirtiiltt, whirh are rijUUij counrctrd, in the line jolninij them but in "pftosite ilii'f>CtioH»f thnj will connterhahinvr. Ileiicc, either of these forces iiiny bo tninHfcrrod to tho other particle, preserving tho hhmio direction, without altera- tion of its Htuticol effect) or: " // force may be tvppoBcil to act at ASY j>oint in its own line of action, the new point of ajtpliation hc!uni- MiiUU tiy ■tr»lKht llhM, 8 11. Sino« the (hreo olcmonti which lorro to dttlcrmine a proMuro aro in thoir naturo iJootioal with thoM which dutor- piino A itraight line — namely, magnitude, direction, and point through which it ia to be drawn — it foilowi that a atrnight lino may properly bo takon aa the reproaentatife of a proaaure. When, however, a line yl Z7 ia ao takon, it will be underatood that the prcaaure ^cU in tho ditootion from A towarda B ; if written // A, thon fr()m Ji towarda A. Frequently alao, the worda " reprcaentod by " will bo omitted, and wo ahall uao " tho force A Ji" to indicnto tho foroo reprcaentod in ningni- tudu and direction by tho line A 11, acting in a iiroction from A towarda Ji. 12. Wo now prncood to atnto tho two problems of Stntica which alone will bo hero touched upon. (I). Tho condition.s of crjuilibrium for any act of Forott acting on tho Hnnio particlo. (2). Tho conditiona of equilibrium when Forcca oct on a rigid ftystcm of particica which hn.s n fixed axia round which it can turn fruoly, tho Forcoa acting perpendioularly to thia axia. (JlIAl'TlUl II. AOTIN(» AT A roINT. in. Whon Forcftn net ninmltancouNly on a pnrliclo at rest, noflniiion of IkMUlUltl. if tho pikrtiolo ho^iri to inovo, itfl tnutiun will coiumcnco in a dcflnito (liroction, and mi^lit bo junt provontoJ by n nin^lo furco of auitablo lna^nitudo applied in an opposite direction. ThiM force would then couiitorbalnnco tho ori^^inal act of Foroei, and a force equal and oppoHito to it would produce tho Hoino Itatical efTcct us tho first not of Forces, and is therefore termed their liraultnnt, 14. Hence, whon any set of Forces acting at a point keep it at rest, sinco any one of thotn may be considered as coun- tcrbiilancinp; nil the rest, a force c(|ual and opposite to any one of them is tho UcsulUnt of all tho others. 15. Ilcnco also tho condition, In order that Forces acting ron.iition of at a point may keep it at rest, is that the magnitude of their '•'i""''*''"'" Resultant shall bo zero, or that their Resultant shall vanish. IG. Whon Forces act in the same line and direction on a ForcMinthn point, their Resultant acts in tho same direction, and its mag- •■'"" "'"-' nitude is equal to tho sum of th'^ir magnitudes. If some of tho Forces bo acting in the opposite direction, tho magnitude of their Resultant will bo the difference between tho sums of tho magnitude of those acting in the one direction and in the other, and it will act in the same direction ns those Fores whose sum is the greater. We can, however, indicate oppo- siteness of direction by attaching to the magnitudes of the Forces the Algebraic signs -|- and — ; so that, any one Forco being considered positive) all Forces in that direction i*^M. 10 will also bo considcrcJ ponili've, but forces in tho opposite diroftion will be considered ncjativc. The above results may then bo combined into the following ; Till ir Resui- fjig Resultant of am/ set of Forces act inn on a point in taiit and ... , the same linCf is the ahjchraic sum of the Forces. ( rnniition of 17. Hence also the condition that tho point may be kept at Iviuilibriuiu Ml 1 1 , _ _ forces act; then, comple- ting tho parallelogram ABCD, the two forces re- presented by ABf B Of will be represented by A B, A D, and their resultant by A C, which is equal and opposite to CA, the third force. 28. Conversely. If three forces acting at a point and keeping it at rest, be represented in direction by the sides of a triangle taken in order, these sides will represent them also in magnitude. 29. Hence, all problems relative to three forces keeping a point at rest are reduced to the solution of a plane triangle. Thus, if P, Q, R, be the forces, and the angle between Pand Q be represented by (P, Q); then the angles of the triangle in the above proposition are the supplements of the angles be. tween the forces ; and, since the sine of an angle is equal to that of its supplement, and the cosine of an angle is the cosine of the supplement with opposite sign, we have (Trig. §34, 40. P Q R Sia(g,i2) Sin (i?, P) Sin (P, Q) ' Lami's For- mulas. ill I*i»lyj{on of 16 and also the equivalent oxprcssioni — » ^ 7?— i" 4- g» -f 2 P (2 cos (P, <2) P«— (^i-f-iiji-l- 2 QR COB (Q,Ji) ^« — i2« + p» 4. 2 /? /» cos (7?, /*) 30. Polygon of Forces. If Forces acting on a point he represented in magnitude and direction by the aides of a polygon, taken in order, they will krep the point at rest. For \f ABCDEF be the polygon, the forces AB,BC, have for their resultant AC ; and the resultant of this and CD is AD; and so on till we come to the last side which is equal and opposite to the resultant of all the previous ones. Hence the proposition, as well as its converse, is established. 31. In this way, the Besultant of any number of Forces at a point can be constructed geometrically ; for, having drawn consecutive lines, so that, taken in order, they are parallel to, in the same direction with, and proportional in magnitude to, the forces ; the line drawn to complete the polygon will repre- sent in magnitude and in reversed direction the Resultant required. It may be noticed that the Polygon referred to need not be a plane one, neither are re-entering angles or crossed sides excluded. CHAPTER m. Forces in one plane acting on a system op rigidly connected p0int8, which can turn ftteely about a fixed point in the plane. , , 32. Two intnrsecting forces act on a rigid sgsfem, in the Condition o tame plane with a fixed point round which the »i/slem can wlicnthe turn. two forces in»«t. Lot be the fixed point; P^ Q,t\7o forces in the same plane with 0, their direc- tions intersecting in Af at which point, rigidly connected with 0, they may be supposed to aot. Then if R be the Be- sultant of Ff Q, ia order Q that the point A and the ^ whole system with which it is rigidly oonnected may be kept ai rest, it is necessary and sufficient that the direction of li shall pass through the fixed point : that is, J.0 must be the direction of R. Draw 0J3, perpendicular to the directions of P, Q. Then, resolving the forces at ul in a direction perpendicular 1)0-40, we have (§21, 23): PBrnOAB-^-Q Bin OAC=0,&nd therefore P. OB— Q. OC =0,QTf P.OBz=Q.OO. V 18 MoDtf^nt (le- 33. The prodaot P. B, which is the product of the num* bor eipresBtng the magnitude of the force, and the length of the perpendicular dropped from the point upon its direction, is called the momml of the Force about that point. Ilcnco the above result may be expressed by saying that when the two forces keep the system at rest, their moment* about the Jixed point are equal, the forces tending to turn the system in opposite directions about the point : but if wo indicate this oppositencss of direction by difference of alge- braic sign, so that the moment of one Force which tends to turn the system round in one direction being considered poai- tive, that of another Force tending to turn it in the opposite direction will be considered negative;, we may still more briefly express it in the form : The algebraic turn of the momenta of the two Forces round the point must be zero. , ,, -^. Moment Of 84. The moment of the Resultant of two intersecting two forces Forces round any point in their plane is tqfual to the ahje- •oct iseauti braic Bum of the moments of the Forces. to sum 01 Let P, Q be the two forces intersecting in A ; O, any poiut in their plane; i2, their Resul- tant. Draw the ;>erpendicu- lars OB, C, D. Then (§ 23) the Re- sultant resolved in any direction being equal ><^ the algebraic sum of the resolved Forces in that direction, let them be resolved perpendicularly to OA. Hence, from § 21, iJsin OAD = Pb\q 0AB'{' Q sm AC.', momenta of '•T***—- 10 nn' whioh tho foroM r, b«Iog •qaal tnd oppoiltei iMrium n>iy b« remoTed dtogethor, leaviog the forooa /*, Q lotiag in a direotion parallol to their origioftl direotioo, and oombiniog into a liogle foroe (P -f Q). Again, the moment of this tingle foroe (P -j- Q) ^bout !• equal to the algebraic turn of thoM of ita components R and 8 (§ 84) ; and th« momanl of iZ if equal to (h« ran of tho moments of its oompooenti, namely, P and T; and aq is that of ^ to the sum of the momenta of Q aod T; amoo| whioh momenta thoic of the foroea T deairoy each olher| leaving the algebraio anm of the momenta of the original forces P, Q eqaal to the momsntof the single foro« (P+ Q)% which has been shown to be their equivalent Md WhOM niniiifltit 1« tb« aum uf their uiu- iiianta. If the Forces P, Q had bean taken aoiing in opposit* directions, we should have found by the same procssa that tha single equivalent force had for its magnif.de the difference of those of P and 9> ^^'^ t^cU^ in the direction of the greater foroe, but that its moment was still equal to the algebraio aun^ of their moments. If, therefore, we now extend to parallel forces tbo same method of indicating oppositencss of direction by difference of sign, which was used in tho case of Voroea aoting in tbt same line, we can include the above oases in a aingle Btat*> jnent, as follows : Two parallel foreu acting on a riyid tjfiiim are equivalent to a tingle parallel force which it equal to their algebraio eum, and wfioae moment round any point in the platu of the fortwe, ie equal to the algebraic eumqfthe momenUof the tvoforeee. In one case, however, the above proeess becomes nogatorj, which is when the two forces are parsUel, equal, and oppositei Snob a pair • of forces is called a eoujJe, and the case must b« exduded from onr general statement. Any two 80. If to this slugls equivalent 7oroe in § 86, we give the '■tdtaAiof.' name of Resultant, we can now include the results of the two last articles in one statement : 'Exception. A couple. SI Anjf two Forctt in th« iam« plant acting on a rigid tyttfm {unlttt thty form a coupU) an ^quivalmt to a tingle Jietul- tant fura, tohote monunt round any point in their plane it tqual to the algibraic 9Mmqfth$ir momtnti round thit point, 87. Any Forcet act in one plane on a rigid tyttem. Tak{0(^ aoy two of thene, we find their Resultant, Its mo- ment being the sum of the momentA of tho two round any as- lumcd point in the plune ; combining this Resultant with a third to form a new Resultant, whotio moment will be the sum of those of the throe forocM ; and this again with a fourth ; and ■0 on till we have taken all the forces, we are left at lost with a kingle RcHultnnt only, whoso moment is equal to tho •um of the moments of all the Forces. In thus proceedings we must avoid combining with any one of the partial rosult- antfl a force which would form with it a couple ; and this we ean always do by taking instead of this force another one which will not form a couple, for if it did, there would then bo two equal and parallel forces, not opposite, and those two oould be oombinod into one which would now no longer form a couple with the Resultant spoken of; we can thus always evade forming a couple until we have combined all the forces but one, and it may happen that this one is equal, parallel, and opposite to the Resultant we have obtained from all the re»t, so that we have a couple remaining. Any Kon-r* Hence, any tet of Forces acting in one plane on a rigid are reduci- tysfetn are either reducible to a couple, or else to a tingle Be- iil^iuiuntf^ ndtant Force, tohone moment round any point in the plane it '°**^*"'» equal to the algebraic turn of the moments of the Forces round that point.. 88. To find the conditions of equilibrium when Forces in one plane act on a rigid system which can turn freely about a fixed point in that plane. The forces are reducible either to a couple or to a single andforaqtiu Resultant Force. ^"^^ lo th« fbnntr mm, equilibrium !i notpoiiibtt; In the Ut- ter, equilibrium will ■ubaint if the UeaulUnt either be loro or p«M through the flieout the fixed axit mvat vanith. t ■' "/r^ii M W WP W WH I I M i I ' CflAPTEH TV. ciinriui or parallbl roucui, and op oiuvitt. 40. It hu bo«n ■hown that two ptnllel foroM (not furminff RMniunt of • ooupI«) Aottng on a rigid ■jst«in, hare for KMultaDt » lingU vnrvm iim force in the Mme pUne ; ita direction being parallel to that of the two, i^j magnitude being the algebraio lum of their mag- momani oitudoa, and ita moment about any point in thia plane being rxTnlitculTi'' the algobraio lum of their momenta About thia point. Alao, the moment of a force about a line to which ita diroo- iion ia perpendicular haa boon defined to be the product of the number ezpreeaing the magnitude of the Force, by the perpendicular distance between itM direction and the line. 41. It will now be shown that the moment of thia Resultant of two parallel forcea, about any line perpendicular to their direction, is equal to the algebraio sum of the momenta of the two forcea about this line. Suppose the forces P, Q, to be acting in the same direction perpendicularly to the plane of the figure and meeting this plane , „^, ^^ ..^ in ihn points B, C; and their Resultant R aumof Um (which ^P-f* ^1 (^°<^ '^oii parallel to and in the same plane with them), to meet the plane of the figure in A. Let d a c be any line in this plane, and draw to it the per- pendiculars Aa, Bbf Cc. Then, \t B A C be parallel to 6 a c, Aa^ Mb, Cc, are all equal, and the proposition ia manifestly true, for E. Aaz=(P-\- Q) Aa = P. Bb -\- Q. Cc. Il 'lit! 1 IW ia*'r .1 'i 24 But if BA C bo not parallel to bac, let them meet in 0. Then, being a point in the plane of the forces, the moment of H round is equal to the sum of those of Pand Q round it, and therefore H. OA = P. 0B-\- Q. OC. But by similar triangles Aa Bh Cc OA OB 00 moments of tlio Forces. and therefore R. Aa = P.Bb-{- Q. Cc, taut. which proves the proposition for this case, and the proof holds good also for the cases where the forces or their moments are in opposite directions, having due regard to algebraic sign. Anynurahcr 42. Any parallel Forces, acting on a rigid si/stem, are FonfeThavo C'^^^'* reducible to a couple or else to a single Resultant Force B?n«ieBe8ui- ^^*^^ ^^'* *^ ^ parallel direction, its magnitude being the algebraic sum of the magnitudes of all the Forces, and its moment about any assumed line perpendicular to their direc- tion, being equal to the algebraic sum of the moments of aU the Forces about this line. Taking any two of the forces (which do not form a couple), we find their Resultant, which acts in a parallel direction, its magnitude being the algebraic sum of their magnitudes, and its moment, about any assumed line perpendicular to the direction of the forces, being equal to the algebraic sum of their moments about this line : combining this Resultant with any third force to form a new Resultant, and this again with a fourth, and so on as in § 37, we arrive at last either at a couple or a single Resultant Force acting in a parallel direc- tion, its magnitude being equal to the algebraic sum of the magnitudes of all the Forces, and its moment about the assumed line being equal to the algebraic sum of all their moments about this line. t - ■ L ... ■WP" iMiiiMHMMiaaHWMnHMii 25 43. The centre of Parallel Forces. When ghen parallel Forres, aeting at givai points of a rigidly connected syxtem, are redwAble to a nngle Remltcnt, its direction passes through a point tvhose position is invaria- ble loith regard to the points of the system^ whaiever he the direction of the Forces. Take any two of the Forces P, Q, acting [at the given points 5, 6^ Join J5Cand let their resultant R cut it in A. Then, the moment of R about any point in the plane I being equal to the algebraic sum of the moments of P, Q, about this point; let these moments be taken about A. The moment of R about A is zero; hence, drawing h Ac perpendicular to the direction of the forces, Contro o( imrallcl Forces. P. Ah— $. ^c=0. and, by Bimilar triangles, — == -- ' and therefore, AC Ac * Hence B C, which is given, is cut in the point ^ in a ratio which is independent of the direction of the forces with regard to B C, and the position of A is therefore given with regard to B and G. • i a : . . . y Now, taking any third force, acting at Z>, we may combine it with the resultant of P and Q, and the point in which the new resultant cats A D will be given in position with regard to ^ and Z> or to ^, JB and C. ' And tiitiB we tiia^ go on till We Urtive at the final resultaiit. Henoe, the proposition as enunciated is true. ■ f , This point is called the centre of Parallel Forces^ %' €■ 26 Vm "bo*"" ^'^' ^^ ^^'^^ point — the centre of parallel forces — in a given turaudabout gygtem be rlgidlj connected with the system and supported or fixed, the system will be kept at rest, and will remain so when the forces are turned about their points of action into any other direction. It will also still be at rest if it be turned about this point into any other position, the forces acting always at the same points of the system and being always parallel to each other, though their directions may be varied at pleasure. The pressure supported by this fixed centre is evidently the algebraic sum of the forces, and the algebraic &um of their moments about any line through this point vanishes. Centre of Gravity. Whole weight may be collected At: 45. The Centre of Qraviti/. When the only forces acting on a system are th€ weights of t1ie several particles of that system, if we suppose the vertically-downward directions in which these weights act to be parallel to each other, and the weight of any particle to be independent of its position ; then, since the forces all act in the same direction, they have a single Besultant which is equal to their sum, that is, to the weight of the whole system, and acts vertically downwards through the centre of parallel forceSf which is in this case called the centre of gravity. 46. The statical effect, therefore, of any rigid system will not be altered by supposing it to be without weight, and the whole weight to be collected at its centre of gravity and there to act — this point, however, being rigidly connec^fid with the system. We may also, without alteration of the statical effect, con- ceive the system to be geometrically divided into any number of systems, and the weight of each of these to be collected at its own centre of gravity and there act, these partial centres of gravity being rigidly connected with each other and the system. v ■i mj i yLJ iiii « i lJMJ ' JLI ' I '' ..Jl- 97 47. Also, if the centre of gravity of a system be supported syst«m or fixed, the system will balance about this point in all posi- lilunJunali tions under the sole action of the weights of the parts of the i"^'*'^'"": system, these being rigidly connected with each other and the centre of gravity, and this is sometimes made the definition oi the centre of (jravity. 48. The position of the centre of gravity relative to a given system iiow found, will be determined from the consideration, that, placing the system so that any given line in it shall bo horizontal, and equating the mo- ment of the whole weight collected at the centre of gravity with the moments of the several weights of the particles about this lino, the distance of the centre of gravity from the vertical plane passing through this lino will be found. Taking thus three planes in succes- sion intersecting in a point, the distances of the centre of gravit}'^ from each of these planes can be found, and its position therefore determined. 49. Since the position of the centre of gravity in the sys- tem depends on the relative and not the absolute weights of its parts, this position will not be affected by increasing or diminishing proportionally these weights. 50. If a rigid body be of uniform density : thru is, if the or aunifom weight a given volume of its substance be the same in ^"^y* every part of the body j then, if there be a line about which the form of the body is symmetrical, the centre of gravity will be in that line ; and if there be two such lines, the centre of gravity will be their intersection. Thus the centre of gravity of a circle or sphere is the centre; of a parallelogram or paral- lelepiped, the intersection of its diagonals; of areguh.. prism or cylinder, the middle poiut of its axis. 51. If a uniform body balance in every position about a line, the centre of gravity lies in that line ; and if about two such lines separately, it will be their intersection. Thus a triangular area will balance about a line drawn from one angle to bisect the opposite side, for the triangle can be generated by a line moving parallel to one side, and the small area gene- rated at any stage of its motion will balance about the line tmm \4 or a trian- V7hich biseots it. Hence the centre of gravity of a triangular «u ar area. ^^^^ .^ ^^^ intersection of lines drawn IVom the anglea to bisect the opposite sides, and this intersection is at a distance from the angle of two-thirds of the bisector drawn from it. For \ei A B C be the triangle, and BD, CE bisect AO, AB, and meet in 0. Then G in the centre of gravity. Join E D, which is parallel to B C (Eucl. B. VI. 2), Thon = by similar triangles BOO, EOD OB ED' ^ C A = , by similar triangles A CB, ABE AB _ %_ 2 Hence B 0, being double of OB, la — BB. o The same point O is also the centre of gravity of three equal bodies placed at the points A,B,C. Of any poly- Cor. In this way can the centre of gravity of any polygonal area be found; for, dividing the figure into triangles, the weight of each of these may be supposed collected at its own centre of gravity, and the centre of gravity of the whole figure will be that of these weights, considered as heavy particles situated in those points. The method of finding this latter will be tr'^'vted in the following article. ofanyheavy 52. To find tlie centre of gravity of a system of particles paiticles in ,, . , neplane. all in one plane. •< . , : . ? ". .; ..^ ' , . •.. Let Ox, Oy be two perpendicular liaes in this plane, with regard to which the positions of the particles are known. < Let P be the place of one of the particles, io its weight. '.,. ■■»WWIT".'HI'.l Draw P Ny i^ jl/ perpendicular to Ox, Oy^ respectively; and denote P M hj x, F N hy i/. Suppose the plane of the figure to be horizontal ; then Ox is a line perpendicular to the direction of the weights, and therefore (§ 36) the moment about Ox of the whole weight collected at the centre of g. tvity is equal to the algcbiuio sum of the separate moments about it. Hence if TV be the whole weight, and the distance of the centre of gravity from Ox be denoted by f, we have W7i = I {w. 1/) y = and W where I denotes the algebraic sum of all the products corrcp ponding to that within the bracket. Aho, if a moment be reckoned positive when Pis above the line Oj\'\i will {lalnly be negative when P is below the lino, and the difiFercnco in sign of the moments will therefore at once bo inuicatcd by considering a y positive when drawn upwards from Ox, and negative when downwards. Similarly, by taking moments round Oy, if oj be the dis- tance of the centre of gravity from Oj/, we have Hi'io.x') X: w where x will be considered positive when drawn to the right of Oy, negative when to the left. „ The distances of the centre of gravity from these two lines being thus found, and the directions in which these distances are drawn being indicated by the signs with which they are aflFeo- ted,the position of the centre of gravity is fully determined. r» p -X +y _ --,P M +y +y N' Hn -y ■ -y p -X +x ^A' p . •- T - V * i . : ■■■ • 'I^- . II Of pnrtlclt'fi n A HtrulKlit liie. 30 Cor. — If the particles all lie in the same line, take this for Ox. Then, every ij being zero, y is so also, and the centre of gravity is in Ox, its distance from being given by — S(wx) W The following independent proof of this may be noted. 6a. Let Oz be the line in which the particles He, O being any point from which the distancea of tlio particlca are known, and lot this line be placed horizontal. Let x bo the distance from of the particle whose weight is w. Let W bo the whole weight, and x the distance of the centre of gravity from 0. Draw another horizontal line from O perpendicular to Ox. This line will then be perpendicular to the direction of the weights, and the moment about it of the whole weight collected at the centre of gravity will be equal to the algebraic sum of the moments of the several weights. Hence we have W~x = 2 [w.x) — 2 (w.x) or X = — ^ ' W where S denotes the algebraic sum of all the- products corresponding to that within the bracket. Also, if a moment be reckoned positive wlien the particle is on one side of 0, that of a particle on the other side of will be negative, and the difference in algebraic sign of the moments will therefore at once be indicated by considering Mie x'a of the particles to be positive or negative according as they lie on one or the other side of 0. I j iioavy body 54. When a rig.id hody rests suspended from or supported freely, its hy a fixed point, and acted on only by its weight, the vertical gravity is line drawn through the centre of gravity will pass through the above^oi^ho- point of Suspension or support ; and, conversely. low the point of suspen- *'""• For the weight of the body may be supposed collected at its centre of gravity, and there to act vertically downwards ; and the necessary and sufficient condition of equilibrium is that its moment about the fixed point must vanish, which requires that its direction shall pass through this point. 10 centre of 81 Cor. 1. — Whon the ^ontre of grovity h vertically under the point of support or suspension, if the body bo slif^htly dis- tutbcd from rest, tho moment of the weight will tend to bring it back again to its original position. Tho equilibrium i^ thorcforo said to bo alable. Whon the centre of gravity is vertically above, tho contrary takes place, and the equilibrium is umtable. Cor. 2. — This affords a practical method of finding the centre of gravity of any plane area. Thus, suspending it freely from any one point, trace on it, when at rest, tho direction of tho vertical passing through this point : thon, taking any other point (not in this line) for a new point of suspension, trace also the vertical through it. The intersec- tion of the two lines thus drawn is the cectre of the gravity required. ' 55. When a rv/id body, having a plane base, is j)^f^<^^^ ""''y ri.u'-'' with (his in contact with a fixed horizontal plane, and is t.ii j)i.inf, acted on mdy uj its men weight, it will stand or /nil according or fall ovU as the vertical through its centre of gravity passes within or without the base. By the base is here meant the figure included by a string stretched completely round the outside of the plane section of the body which is in contact with the horizontal plane. If the body fall over, it must begin to turn round some tangent to the curve formed by this string, and the moment of the weight, supposed collected at the centre of gravity, must tend about this tangent in a direction from the inside towards the outside of the area of the base, and the vertical through the centre of gravity will pass outside the base. A ho, when this vertical passes outside, the body must fall over; but if this vertical pass within the base, the moment of the weight about every tangent to the string tends in direc- tion from the outside towards the inside, and the body cannot fall over. ^ - \l' ,1 CHAPTER V THB MECUANICAL POWERSw do Mftchinet. 56. It is usual to trout of the Simph Machines ^ or ^leihanU cal Powers as thoy arc soniotiincs culled, under six classes, namely — the Lever, the Wheel and Axle, the Pullios, the InoliDed Plane, the Screw, and the Wedge. Of tho^c, the Wedge will not be hero considered, as in its practical appli* cation the investigation on the principles of the fuicgoing chapters would be of small utility. u When a power P sustains on any one of these machines a Mechanical weight TT, the latio W: Pis called the mechanical advantage «ioflned. of the machine; and the machine is said to gain or lose ad* vantage according as this ratio is greater or less than unity. In the following Investigations, bodies will be supposed rigid, snr« faces smooth, string? perfectly flexible and of insenbible size, and the parts of the machine to be without weight, unless otherwise specified. 57. The Lever. ,, straight Lever. Archimedes A straight lever is a rod capable of turning freely in one plane about a point in itself which is fixed. This fixed point nn^iiXd. is called the yw/crum. Case 1. — The weight Tf at one end of the lever supported Pig. 1- by a weight Pat the other end. ^ B A Cthe lever; A the fulcrum. Draw h Ac horizontal^ and therefore perpendicular to the direction of the weights. ■)ro r. An= w. AC. Cor. 1. — Tho prcflfluro on the fulcrum U tho weight (/'p IT) acting vertically downwards. Cur. 2.— Since tho relation P. AD = IF. A (J, doo.s nut involve tho atij^lc at which tlio lover h inclined to tho horizon, it foll(jwa that if tho lover bo at rost in any one position (excopt a vortical one), on being turned into any other posi- tion it will still bo at reat. Cask II. — Tho power Pand tho weight W acting in oppo- p, , ._, fiito (l)ut parallel) directions, and tho weight nearer to tho fulcrum tiiun the power. ILsinti' the same construction and reasoning as in the former case, wo have here also P.AB=W.Aa Cor. 1. — Tho pressure on tho fulcrum is hero IK — 7* acting vertically downwards. Tho second corollary also holds. Ca.se III. — Tho power P and tho weight IF acting in p, ., opposite but parallel directions, and the power nearer to tho fulcrum than the weight. As before, we have P.AB=W.Aa Cor. 1. — The pressure on tho fulcrum is P — IV and acts vertically upwards. The second corollary also holds. 58. In all these cases, the toecbanical advantage ( — i is "^^ — „ . or the ratio of the arms of the power and weight. In Case I. m 34 thiH ratiu tnny bu ciiliur o(|unl to, greater, or Il>m than unity; hut in Cofio H. it in always f^rcator, and in Cmo III. \vnn: honco, advanta^^o iw iilwnyN gained in tho looond caiio nnd lust in tho third, but may bo either gninud or lottt in tho lirnt. [.••v.T diij)- f*'*' If '■'"' wolifht of tlin I«'Tor (w) ho tnkon Into nrconnt, It niny br iKx.aiiciivjr. BH|,|„,,o,i collDctiMl (it tlm ci-rilro of jfrutlty (/ (which will Uu lh« niiiMlo |)oli)t If tho loror bu uniform). T.ct tlu) Turtlcal through (^ meet tliu liori/.ontal A ic In {/, Tlioti, by Ihu vanlHliIng uf niomontH uhont A, P. Ah-{-w. Ag— W.Ae^O but by Bimllar trianglos, AB A a AO ,,, , •-r-r = —. — ==s — : — I ond thcrcforo Ab Aij Ac P. A B + w. AG — ir. AO ~ a or, 1\ A B ■\- w. AO =^ W. AC Slmiliirly, In Cusps II. nnd III. wo should find P.AB=:W.AC^w. AO. Hero nlso tho leYur will balance in nil positions about the fiili-rum. 00, In tho common Bnlnnco, which consiHtsof n heavy boiun. having scalo pans uuHpcmk'd at Its ends, nnd balancin*; about a hoii/.onlul knifo edge, tho pans and arms of tho beams are made pcifcctly cqnnl nnd similar on each side of tho cdjjo, but tho centre of ;;ravity of ilio beam is nmdo to fall vertically below tho knife-ed |Miliit vtliuiico tlio lubittatici', whono wulght U' i« rv«|itlr«(l, U ■iiN|icii<|i'il. t' A /! I)i>tni( hoHxrintnl, lot O im tlio piiliit on th« othor uliln of A wlicro /'woiilil kcoptlio Sti'i'lyiirtl ot nmt, wln-n th« Wtl^lit U'Ih awny. Till) moment of ilio wulght uf thu Htuulyuril about A h therefore «qii*l to /'. A O. Now lot thu wolght 11'!)') iittni'hml, iitnl lot .1/ ho iho plnco of P MrhoD vquUlbrliiiii U ohtuliii'tl. Thuti, Utkitig uioniuiit* about A, W. AC T=t P A Sf f- moment of weight of Stoolyard = /'. ,1 1/-}- r. Ao Ili-nco, ir=--'' OM, AC And, ulnco Pand AC Mfi Invnrinblo, W U proportlonol io M : thorcforo lg thu point from which the (^riuluiition niiirit bu nmdo. ThuH, If /' bo at 1\ when \V U 1 lb, niul wo tftko O li^ .:= /?, Jl.^ sa "^a ^a = ••• ' '*»''" ^''«'* ^ '" "^t ^j • ^^3 ' - ^^' ^'''' '- ^ '^' ^' - "^''■ ('2. Tho prcccdinj; cases of tho lover oro only special appli- "Prinrij.io cations of tho pcnoral invcHtij^iition in Chap. III. In I'uct, any body moveable about a fixed point and acted on by forces in the piano of that point may bo considered ft lever, and tho principle of § IJS is often quoted as tho principle of the lever. C3. The Mhed and Axle. This trachinc consists of a circular drum or wheel, which is attached to a cylinder or axle, its centre lying in the axis of tho cylinder and its piano being perpendicular to this axis. The whole system runs freely on this axis, which is fixed; and the power P acts by a string coiled round tho wheel, and sup- ports a weight W which bangs from a string coiled round tho axle. Tho strings being perpendicular to the axis, and also to the radii of the wheel and axle respectively at the points where thoy becomo uncoiled, wo have for tho condition of equilibrium, by § 39, taking moments about the axis, P X radius of wheel — W X radius of axlo = 0, Wheel and nxlu i'ig. 5. nn I . Putltoi. ti. idv, ITvnoo tho ntiu'hanical ndvnntngo ( ., I i* 'n'"^^ ^^ ^^ ratio of iho rtulii of tho wliucl iuhI iixU. Cor. — Any niimbor of whocN ntnl axlt^N mny run on tlio ■aino Axiii, uiul tho contiition uf «<(|uilihritim will hu thut tho Muni uf thu products of each power into tho rmliuN of itft wheel la cqunl to tlie corrcHpotuliii^ nam for tho wci^hta and radii of tho axluR, tho powcra hoin^ nil niippoMMl to turn in tho tam% dirootion and thu woi^htii ull in tho oppoaito. C4. ThtPuWrn. A ptili)' is n whocl runninj^ frcoly on an axiw, which, pn«« ^in^; through ita centre, in flxod in a blt)ck by which thu pully .8 auMpcndud and to which n weight mny bo nttnched. Tho circunifercnco of tho pully in grooved to admit of a Mtring passing over or under it. Tho pully i.s miid t') bo fued or tnoveahle according as itH block ia ao. Single fixed PuUy. Lot /*, Q bo iho ft rcoa, applied nt the cndn of tho string passing over tho pully. Tho whole Hystem being stnooth, tho ten.sion of tho atring ia tho aamo throughout (§ lU), and, thcrtforo, No mechanical advantage is g.'iincd or lost. C5. Single moveable pully supported Itj a stritifj passin(/ under it, the free portions of the strinj being parallel, and a wci'jht attached to the block. Let J* be the force applied to the string on one side of tho pully; then, tho whole system being smooth, tho tension of the string is the same throughout, and P is therefore also tho force applied to tho string on tho other side. There are then two parullel forces, each equal to P, supporting a weight W which acts vertically. Hence the strings must be vertical, and 37 Ilvro iho mochani'^nl n(lvaninp;o U 2. Cnr. I. If nn« «ncl of thn atrlni; hfl iittiich«d to • fliA^t |>olnt, th« Mmo rvault I111I1I4 Kooii, fur tlit< tttnnlon tniiat !*« th« iiariio throiiu^hoQt. Cur. 1. If tlio wi'l^lit w of titit piilly, lni'ltilul, li.* takim luto account, It may ba luppuacd Mttachau to Uio block, and wu h«To IK -f n -I 9 A CO. Firtt ijf»ttm of pulliet, A nuitibor of puliion aro nttnchod to tho iiAmo block, which pupporta a weight, ami tho aatno Htriiig paa^cN round all tho pulliea. Tho portiona of tho Mtririj^H botweon tho pullics aro Huppoaod to bo pnriiliul, and will thoruforo aNo bo vortical as in § OS- Lot /'bi! tho forco applied to tho atrinj;; /*will then bo tho tcnHiou thr()Uf,:huut, and tho weight W is aupportod by na many purfillol forces, each e(|ual to /*, nn thero aro pnriillul portions of tho string at tho lower block; and tho number of thcNU pitrtiona is OTidently double tho number uf pullics at this block. Ilonce, if n bo the number of movcublo pullic.>«, W=2nP; and tho mcchaniciil advantage is 2 n. Cor. If the weight of all the pnlliua within tho lower block bo w, tiiu wcigltt of the block Also being iucludud, we may luppotio this wcijjht ulUchtid tu thu blurk, and ir 4- !• = 2 n /». 07. Second tyntem of pulliet. Each pully hongs by a separate string, tho last pully sup porting tho weight; tho frco portions of all tho strings are parallel, and thorcforo vertical. Let yli, ^j, vlj,... bo tho pullies, n being tho number of them ; W, tho weight supported at tho last ono; /*, tho power applied at tho first string. Number tho strings 1, 2, 3,... according to tho pully under which each passes. The tension of each separate string is tho same throughout. flr»tiy»lo;u BprouJ 2 P, and this is the tension of (2^. The weight supported at A.^ is double the tension of (2) and therefore = 2 (2 P) = 2' P, and this is the tension of (3). The weif!;ht supported at A3 is double thfr tension of (3), and therefore = 2 (2^^ P) = 2^ P, and this is the tension of Proceeding in this way, we como at last to the weight sup- ported at An == 2" P, and this is ihe attached weight, ftence, Tr=2"P, and the mechanical advantage is 2". Cor. 1. The mechanical advantage is cioublod by every additional pully. Piillioa sup- Cor. 2. The weight of the pullies may be readily taken into account by observing that, from the preceding, the force required to support a W weight W on n moveable pullies is — -. o •'2" Let Wj, Wj, Wg, ... be the weights of th" several pullies, blocks in- cluded. Each of these weights may be supposed a weight attached to its block, and supported on the system of pullie» above ii w The power required to support w on one moveable pully is -J. w w on> two " 2 w on three "• 3 to on n n pullies is —f. Also iron n 2 w a 2 w n w " • n ill 9^ And tho wliolo power roqn'ircd will be the sum of tlioso ; therefore, w W- 9 P-= J + -^ +... + -+ -.or 2 2 2 2 IF / ,«-l 71-2 \ Tilt' WL'if^lit of tho pullied therefore lessens tho advantage of the rnnchiiie. Cor. If tho weight of each puUy bo tho same (w), then G8. Third system of pullies. Each pully hangs by a separate string which is attached to a bar or block carrying the weight, and the free portions of all the strings are parallel, and therefore vertical. Third sy.s- teiii, Fig. 9. This is the second system turned upside down, the weight becomiiiw a fixture, and tlio beam to which the strings are attached becoming a moveable bar carrying a woiglit, and the mechanical advantage njight be inferred from the preceding. The pressure supported by the beam in the second system is the sum of the tensions of the strings, that is, P + 2 P + 22 P 4- ... to 71 terms, == (2» — 1) P, and this becomes the weight IF in tho third system. Therefore, Tr=(2''— 1)P The last puUy ( J„), however, becomes fixed, so that the number of moveable pullies is only {n — 1). Making n the number of moveable pullies we have lF=(2»+»— 1)P. The following is an independent investigation for this case. Let ^1, ^2» ^3) ••• ^nt he the pullies, n being their num- ber exclusive of the last one A, which is fixed, and n -\- 1 the number of strinjis. 'I n 40 ^1, 7>a, B3, ... Bn +1, tho points at which the respective strinj^s are attached to tho strai}j;ht bar which carries the wcijzht W. Number the strings 1, 2, 3, ... accordinj; to the pully over which each passes. The tenison of each separate string is the same throughout. The weight supported is tho sum of tiie pressures of tho strings at i/j, B.^, By ... The tension of (1) is P, and this is tho pressure at B^. The weight supported at yI,^ is double the tension of (1) and = 2 P, and this is therefore the tension of (2) and tho pressure at i)^ The weight supported at ^3 is double the tension of (2) and = 2 (2 P) = 2^^ P; and this is therefore the tension of (3) and the pressure at By Proceeding in tliis way wc obtain the tension of the (?i +l)th string and pressure at B^^i = 2"^ P. Taldng the sura of all these pressures, pr=P+2P+2^P+ + 2T =(^r + ^-l-)P, ' and the mechanical advantaere is 2'*"^^ — 1. P lilies Slip- Cor. The weights of the pullies may be taken into account by P .e( K'a\y. Q^^ggj-ving that each maybe considered as a power ' jting by means of the string from which it hangs, and supporting a weight on the sys- tem of moveable pullies above it. Let u' , w , w , ... w , be the weights of the pullies, blocks included. 12 3 n The weight supported by w on (m — 1) moveable pullies is (2 — 1) w . ■>i-l " (2 —\)w . -Also w on (n — 2) 2 M on n P on n " (2— l)w n' " (2 — IJP. 41 Tlio whole weight ]V (including tlint of tho bar) is the sum of theeo ; therefore, Tlio woij;ht of tho pullics therefore incrcdsea tho fiilvftntage of tlio nincliine. Cor. 1. If tho weight of oftch pully bo tho same (w), then, ir= P (2" + ^ — 1) -i- w (2'* 4- 2**~ ^ + ... + 2 — «) = 7^(2" + 1_ 1) + „, (2" + ^ _ 2 -n). If we put i' = 0, we have whicli is the weiglit that would bo supported by tho puUies alone. Cor. 2. The point of tho bar to which the weight should be attached in order tiiat tlio bar may bo horizontal will be tho centre of pdrallcl forces for tho tensions of the strings and the weight of tho bar. If wo neglect tho weight of tlie puUies ond the bar, this point will remain the same in a system, whatever bo the power; if, however, tho weight of bar and pulliea be considered, it will be different for different powers. 69. Taking the same number n of moveable pr.i.Ic? in each sys- systems tern, the respective mechanical advantages are 2n, 2", 2' — 1, '^'^"'i''^'" and these numbers are in ascending order of magnitude. Hence the mechanical advantage of the third system is greater than that of the second, and of the second than of the first when there is more ihan one puUj. 70. Tho following combination of puUies may bo noticed. Spanisii It is called the Spanish Bartun. Fig. lu! The tension of the string to whicli F is attached is the s.'vme throughout and = P. That of the '. ther string is also the same throughout and = 22. Therefore IF == 4 P. If we take the weights of the pullies A, B into account, we have W-\- B=^ P-^ A. 42 I ilK'lilii'd I>Iimt'. Fi«. 11. 71. Tho, Inclined Plane. This is a plane fixed at a certain angle (called its huiiiuttion') to tiio horizon, and on it a heavy particle is supported by a force applied and the reaction of the plane. Since the piano is smooth, its reaction is exerted in a normal direction ; also the weij^ht of the particle acts vertically : therefore if a verti- cal plane be drawn through the particle and the normal to tho inclined plane, since the plane thus drawn contains the direc- tions of those two forces acting on the particle, tho third force or Power must also act in this plane. Let tl figure represent this plane ; AB, the section of the inclined ine; yl C, horizontal. The angle BAC\s the inclination, a (suppose). Let /*, the power, act at an angle to AB, and let li be the reaction of the plane exerted perpendicularly to AB. W the weight of the particle acting vertically downwards. The particle is then kept at rest by the three forces P, R, W. Taking the resolved parts of these along AB, that of P is P cos ^; of i? is 0; of W^is TF cos (90°— a) = W sin a. Hence by the " vanishing of the Resultant," ,.• PcoaO — W sin a=0, which gives the mechanical advantage ( — | = (-; — ). ° ViV Kama/ Cor. 1. For a given inclination, the mechanical advantage is greatest when cos is greatest; that is, when = 0, and the force acts parallel to the plane. For a force acting at a given angle to planes of different inclinations, the mechanical advantage increases as tho incli- nation diminishes. ' '' 43 Coa 2. Resolving the forces horizontally we have P cos («>-[-«) — /? sin a = 0. Also resolving them perpendicularly to the direction of /*, IV cos {0 -f- a) — R cos ^ = 0, These two equations give li in terms o^ P or TF. Or these relations might at once have been asserted fVoiu the " triangle of forces" (§ 29) : for this gives P _ W li sin a cos a cos (lane."' passing over a pully and supporting a weight P hanging freely, we have from the above by putting ^ = 0, or at once by ^''^^■"""' resolving the forces along AB, observing that i-^is the tension of the string, P— Trsina=0 W 1 and the mechanical advantage — = , and is the P sin a cosecant of the inclination. If BC (vertical) be called the height of the plane, AB its length : then, since sin a = , we have AB AB or P W height length Screw. Fii!. M. Fig. 13. Again, resolving furccs perpendicularly to the plane, we have 2i — W 008 a — or JR = Trcos a AJJ and J? W base length Hence the power, weight and prcs.suro on the plane are proportional to the height, length and base of the piano. Cor. This latter result is at once seen from the " triangle of forces;" for, drawing C'A'^ perpendicular to AB, the sides of the triangle BCJV, taken in order, are parallel to the direc- tions of the forces, and therefore represent them ia magnitude; and the triangle ABO ia similar to BOX, 73. The Screw. The Screw is a circular cylinder, on the surface of which runs a protuberant spiral thread, whose inclination to the axis of the cylinder is everywhere the same. This thread works freely in a fixed block, wherein has been cut a corresponding groove. The power is applied perpendicularly to a rigid arm which passes perpendicularly through the axis of the cylinder and is rigidly attached to it, and the weight is supported on the cylinder (whose axis is here supposed to be vertical), and may be supposed to act in the direction of this axis. 74. The complement of the invariable inclination of the thread to the axis, or (the axis being vertical) the inclination to the horizontal line which touches the cylinder at the point, is called the pitch of the screw. If a right angled triangle BAC be drawn, having the base AG equal to the circumfer- ence of the cylinder, and the angle BA G equal to the pitch of the screw (a), and this triangle be wrapped smoothly on the cylinder, its hypothenuse will mark on the cylinder the course of the thread, and by superposing similar triangles the whole ^-T" "■"^fl^^WWBp* 45 course uf tlio thread may bo continued. liU is thou tlio dlitaiico bctwoou two coiitiL'Uou.s thnads, and wo havo tan BA C = - BC or tan a= AC dLtiincu between two conti^^uons threads oircuuit'urence of o^liudur 75. The Screw ia kept at rest by the weij^ht ( W) which ijj,. i j acts vertically, by the power i-* which nets horizoi' !y, and by the reactions of the groove on the thread at tlie various points in contact. Since the thread is smooth, the reaction at each point of it is normal to the thread ; and the anj^le between the directions of this normal and tlie axis, being the same as that between the thread and the horizontal tangent which are respectively p>erpendicular to them, is a, tho^'iVc/t. If then we resolve this reaction at any point, R (suppose) pjg. i,-, into two forces ; one, vertical, and the other, horizontal and touching the cylinder, the former will be R cos «, and the latter R sin a. All these vertical portions being parallel, will form a single vertical resultant whose magnitude is cos a 2' (7t*), and this must counterbalance the weight W^ since all the other forces are horizontal. Hence cos a 2' {R) = W. (1) Again the horizontal portions of the 7i's tend to turn the cylinder about its axis, and since each acts in a horizontal direction touching the cylinder, the radius of the cylinder is itself the perpendicular distance between the asis and the direction in each case. Hence the moment of one of these (i2 sin a) is i? sin a X radius of cylinder, 46 Qud tlio 8UIU of tliciu nil 18 Bin a X rmlius of cyli.idcr X - (^)» and thi.s must bo C(|ual luid opposito to tho moDiutit of tho power, namely, /* X urin of l\ Ilonoc sin a X radius of cylinder i' (^R) z=r P y^ vrm of P. Dividing tlio siile.s of this ccjuality by those of tho C(juality (1) 8in a 008 a r r !• J ^* X «»'"1 of P X radius of cylinder = — W g]p Q disttnoo botwooD threads llencc, since = tan a = ;; — — 7~ ;r., — c^** " circumference or cylMider 1-^1.^, 1 ,,, radius ... P X arm of i* dist.het. threads X( , of cylinder )= — circuin. W but radius 1 circum. 2s and 2rr X fi^m of Pis tho circumference of the circle which the end of i^'s arm would describo, and may be briefly called the circumforeneo of /*; hence P distance between contiguous threads W circumference of tho power Mcjh. adv. and this ratio inverted is the mechanical advantage. Cor. 1. The mechanical advantage is increased by dimin- ishing the distance between the threads or increasing the arm of the power. Cou. 2. If the cylinder be heavy, its A^eight must be inoludcd in W. Instead of supporting a weight, th j screw may bo producing a pressure at its lower end, and in tliis case the pressure produced will bo increased by the woiglit of tho screw. It may also be producing a pressure at its upper end, and theu the pressare produced must be diminished by this weight. Xf''^llgia^l.mrm>«ttmlt)l$0>llmmf^-»-< 4ft Sonn'ilmcn tlio hcppw Im flxod ntul tho block nioToablo, nn In thf pmo of A common fiM^; or thu (^ruoTo may bo cut In liio ecruw ItHi-lf, itnd thu thrcaii project in tho block. To nil IhoHO casc'B tho |>rncf'(lin{; inTCBtljjiitJon Appliofi. 70. The Wrdije. Th. WcIk.-. Tliis is n solid, whose bouiitlinj^ purfaccs arc two iritrrMrcrtiiif^ pliincH. Most outtinjr iristruuietits conio under this clii^s. It is u.^ed ij;encriiily to separate tho parts of bodies, either hy hliiws or ft inoviti}; pressure, and in this mode of use its invos- titration beh)nf;s to Dynamics. When used to keep open a rift in a body, it ucts generally by means of friction, and not by a weij»lit applied to it; it is thcireforo useless to proceed with its examination on tl , principles employed hitherto. VIRTUAL VELOCITIES. 77. If a machine at rest under a power Pand a weight irvirtimi be put in motion, so however that its geometrical relafion.s arc ^'' "*'***• unaltered, the space described by tho point of application o** tho power, estimated always in the direction of the Pov^er, is called the virtual velocity of tho power : and similarly fo'' the weiuht. Tho principle of virtual velocities asserts that the product „ • , ^ of P by P's virtual velocity is equal to that of W by IK's virtual velocity. 78. This principle is only a special application of a far more general one, which it Is not heu Becessarj to exaniine. We Htrnlitht li-vir. P\H. 10. Wi. ol and Fit;. .',. PulUcs. 48 •li.tll thoruforo only oxt!ibIi>vo nttitoJ, in n I'l'W of llio iiioro hiiMpIo Qixnan, by pr.tviii^ thut it Io.kIh to tbo rvhttioiiM of (M|uilibrium nirondy iDurid. Tbu ^noiiictrical rclatiunN in n macbiiio boiii}^ xiioh ilint tho lipneoi (lu!4cribo(l in difTrrcnt ilinplnrcMnontH nro iilwayM propur- tiotiul, it wi" bo only ncccssjiry to prove tho priiu-iplo lor ii pMiticulur di.spiuceniont, und wo mny Huloot thin um convenient- 7U. T/it! »tr tiijht lever under wiit/hia at it» ends. Tjct tho lover liAC bo horizontnl, und diaplaco it round A into thu pohition //,.'! 6\ tho diroctions of 7^ und \V niooling JIAC in I, c. Then /),A is I*'a virtual vehvitt/, und C'lC is irs. Tho prlMciplo then ns.sorts thut n,A but by similar triii»)j^los, nj» CjA li nico P. Ali = W\ A(\ the condiiion found in § 57. 80. Wiirrl aud AAc. 8ii|'po.se the nmchins to make one coiupletc turn ; then, tho jspneo descended by F is the circumfercnco of tlio wheel; and by ir, that of the axle. The principle then asserts that P X ciroumfereuco of wheel = IF* X circuiufereuco of ttxie, and the circunifercnces are as the r;.ilii : therefore, P X radius of wheel = W X radius of axle, the eondiliou fouud in § G3. 81. Pxdlit's. In tho single fixed pully, the principle is obviously true. siiiirK'imiiy. In the single movoal)le pully (6g. G), let the pully be raised through one inch, thou W is raised through one inch, and '0 Htiitoi], in l(^'l(lH to thl) U'\\ iliat tho frt^H prcpor- lu-Iplu ior H c()r)votii«jiit> it rumitl A W III outing t/, niiJ C\c 40 ono iuuli of cnch portion uf tlio ntrin;; irt oct fruo ; iherofurtfi /'Moendu tUruugb two inuhuit, auU tho prinuiplo luiitirtit that i» X 2 - ir X 1 which ii tho condition in § G5. R'J. In tho fifHt »y«tcm of pullicn, lot tho lower hhick bo Kir»tHyNf.i, ruiHod ono inoh : then Win rained ono inch, nnd ono inch of ("ikt- 7 each portion of tho string at tho blocic i.s Mut free ; therefore, on tho whole, 2n inchon aro Bct froo, and tiiin i.s the hpuco through which J* dcflcondfl. Tho principle then asserts that tho condition found in § GO. Hi'cntiil 157. ; then tho vhci'l; and 1 that Toreuco of fore, 83. In tho second system of pullics, lot H'bo rai.sed 1 inch : then A,^ rises through 1 inoh, and each pully rises through Hysuiiu. twice as much as tho ono below it, and J* rises through twice Fig. 8. as much as tho top-pully ; therefore on tho whole, P'» ascent will bo 2" inches ; and the priuciplo a.sserts that PxiJ^^Trxi, the condition found in § G7. 84. In tho third system of pullles let TV and tho bar bo Tiiini raised 1 inch : then each pully descends through this 1 inch '^*'*''^"' and twice as much as tho pully abo 'c it, and P descends ^^^'^' through 1 inch and twice as much as Ai. The last pully A,^ descends through 1 : therefore, the last but one descends through 1 -|- 2 X I true. ■ <' two be raised ■ " three ich, and H and so oo. " ' " 1 + 2 (1 + 2) or 1 + 2 + 2* '< I " l-f-2(l+2-f22) or 1+2+2^+23 [Li jl tl I M: 1. J llM'HtDltl l>Uut). Hiri'W. KlR IS. Kolwrvart liilluilUU. Fix 17. I i. ! V 60 Oo the wbulo, r " l-l-2-f 2M-2*+...+2", or 2-*'-l. A 11*1 tho prinuiplu an^crtN that i»x (2-+'— i)=»rx I tho oonditioii fuuud in § 08. fi5. Tn tho inclinotJ piano, tho power aotlnp; purnllcl to tho piano, (fig. 12), lot H^ bu nt tho bottom of tho plnno niid bo drawn up to tho top. Thon W'n vorticnl diMplncomcnt h tho height o( tho piano, and /"h doHcont i« its length. Tho prio* ciplti ODHortii that P X length = TF X height, tho ooudition found in § 72. 80. In the Bcrow, lot ono ooinplcto turn bo luado. Then the di.stanoo moved through by the end of P'h arm, cti- matod always in tho direction of /* is tho circumfcronco of i'; ond tho apaco doHconded by Win tho diatanco bctwocn two threads. Tho prinoiplo thon assorts that P X circumforcnco of P = IF X distanco botwcoa two threads, tho condition found in § 75. 87. Assuming tho truth of this principle of virtual vrlu' Cities, it may bo conveniently employed to lind the mechanical advantage in many machines — as examples, let us take liober- vol's Balance, The Differential Axle, and Ilunter'a Screio, 88. In RohervaVs Balance tho sides of a parallelogram are connected by free joints with each ot' -r and with a vertical axis passing through tho middle points of opposite sides ; so that the figure is symmetrical about this axis, and tho other opposite sides are always vertical. Tho weights P, W are car- ried by arms fixed perpendicularly to these latter sides, whioh arms are therefore always horizontal. If tho machine, when at rest, be displaced, one of the weights ascends as much as the other descends, and they are therefore equal. 51 Thin roiult U indopendont (if tho particular polntii of tho oriiiN frDiii which tho vr«M^hti4 tlcpoml, and in thia IIom tho oon- voiiiuDoo uf tho iimchino an n Hulance. nade. Thon 's arm, cti- lotwcori two HI). Iti tho Difftrrntltd Axle (tlj?. 18), two axloii of dilToront rn(T.f»ut..» nixiiN run fixod tou'cthor on thu Runio uxin, and tho woi^ht ia *''"' Hiip|)<»rtcd on tho«o by n pnlly, whono Btrin;^ i« coilod round *''* '" thoHO uxhit ill opponito diroctiona. If P bo ruiavd by a com* pliito turn of tho ninchiiio, IK duNcondrt through a Mpiico i'(|uiil to half tho r|(itintity of atrin;; Mot frvo from the azlca; that in, throned) half tho diflforonoo ot tho circutiirorotiouN of tho axica; iind, the oiruuiuferenooH buing an tho riulii, wo havo P X rudiuH of wheul =3 \V. \ (difToronoo of radii of axica). In tho oon)riion whoci nnd oxio, tho powor ond whocl boin;» givon, tho moohrinicitl ndv:intu^o is incronHcd by diiniriiMhin^ tho rndiiiH of tho axlo, but this diuiitiutioii in practically limited by ru^ard to tho Htron;: ii of tho axlo. In tho abovo mrichino, tho mechanical ndvantn;^o mn^ bo inoroa«cd indefinitely, by mnkinj; tho axles more nearly of equal «i/.o, without too much weukonin^ them. If tho Mxles wcro Abnoluloly ociuiil, tho moclinnical adTiititn^o wcild bo ititinlto, uikI it U obvioiirt thai uiiy woljjht would bo liuru supportuil without II powiT nt all. 90. In FIuntrr'H Srrew (fii,'. 19), tho weight is supported niint.r'H on a nmallor screw, which run.s in a couipaniun in tho interior **"''"**'• of a larger screw, tho latter passing? through a fixed block u!id ^''^ *'' being acted on by a power as usual. When tho power makes a complete revolution, nu-J ruincs tho largo screw through tho di.stanco between its threads, tho smaller screw at the samo time descends in tho large 0110 through tho distance between its own threads, and tL j weight thoroforo on tho whole rises through tho difTerencc between tho *' distances of tho threads " in tho two screws. Hence, i* X circumference of P =:^ IF X difference between di.s- tances of contiguous threads in the two screws. 69. Ill (weiyma- iliiue, whrxt is j;nincil in is lost in liiiif. The incchmical advantnge can thcrcforo bo indefinitely increased by making the distance between the threads more nearly the same in each screw. In the common screw, the advantage is increased by diminishing the distance between the threads, but the diminution is practically limited by regard to the strength of the thread. If onch screw bad tho same distance of throacis, tho advantage would bo infinite, and it ia obvious that any weight would bo sup- ported without a power at all, the outer screw rising just as much aa tiio inner screw descends within it, so that the weight would bo sta- tionary. 91. When a power P is supporting a weight W on any machine, if the machine be set in motion, it will continue to move uniformly so long as its geometrical relations with the power and weight are unaltered; and if s, S be i. e spaces j,'one through by the power and weight in any time (that is, ihcir virtual velocities) we have i* X * = ^^X *^- Hence a given force acting through a given space for any time will lift the same weight only through a given space, whatever be the machine through which it acts ; and if the weight lifted be increased, in the same proportion will the space through which it is lifted be diminished. Also when a given power lifts a weight through a given space, the greater the weight, tho greater in the same proportion is the space through which tho power must act, and (the motion being uniform) the longer is the time employed. Hence the principle of virtual velocities is sometimes stated in the form, that *' in every machine what is gained in power is lost in time." , v.'oikdono 0^- Hence also this product F X ^ or W X S may be i'itLy!' considered the work clone by the machine, and is sometimes termed its dutj/ ; while with reference to the power, the names of mechanical efficiency and laboring force have been given. In this sense, although advantage may be gained by a machine, no efficiency is gained or {theoretically) lost, but it remains the same as if the power were applied directly without the intervention of the machine. ... 68 rractlcttHy, efficiency is always lost, owing to tho various resiatancea due to the parts of the raacliine. 93. Among ongineors the standard of effictenci/ in tho com- norse parison of machines has usually been taken to be a horse power, i'"**""* which is represented by 33000, a lb. and foot being the units employed, and the power being exerted for one minute of time. Thus a horse in one minute is supposed to lift 33000 lbs. through 1 foot, or 3300 lbs. through 10 feet, or 330 lbs. through 100 feet, and so on. A machine is then said to be of so many horse-powers, whence the work done by it in any time can be ' julated. FRICTION. ^. Hitherto the surfaces of bodies in contact have been considered nmooth, and exerting on each other no pressure except in a normal direction. In nature, however, all surfaces are more or less rough, and when one surface is pressing or moving upon another a force is called into play which acts in a diroction contrary to that of the motion, or to that in which motion would occur if the surfaces were smooth. This force is called Friction. Friction. In machines, when a power is supporting a given weight, the mag- p«- * <• ■ nitude of the power, determined on the supposition of the smoothness macliines. of the machine, may be increased beyond this value without disturbing the equilibrium, until it is great enough to overcome the friction together with the weight; and on the other hand, may be diminished till it is so small as with the aid of friction just to prevent the weight overcoming it. So also, with a given power, the weight may be increased or diminished within certain limits without disturbing the equilibrium. Generally, when the power is on the point of raising the weight, friction acts to the disadvantage of the power ; but, when the power is just preventing the weight from descending, fric. tion acts advantageously. When the equilibrium of a system depends on position, this position may with the aid of friction be varied within certain limits of the position determined on the supposition of smooth- ness, and the equilibrium be still maintained. 95. Tho motioD of one surface upon another may be of the nature of sliding or roHivg, or both these. The former will HiidingFiic- ho tho caso when two plane surfaces are in contact, and the laws of the friction in this case (denominated sliding friction) have been determined by experiment, the two surfaces, how* over, only tending to slide and not in actual motion. They Hot are, 1^,1 W8 of. I. Between plane surfaces of given substances, the amount of friction is independent of the extent of area in contact, and depends only on the mutual pressure between them. II. The amount of friction is, for the same two substances, proportional to this normal pressure. Hence, by the second of these laws, if F be the friction, — : -* F . and E the normal pressure, — is a constant quantity for two Ji cneffloient given substanccs. It is called the coeficient of friction for these substances, and may be determined experimentally as follows : — Found by I xperimeut. Fig. 20. 96. Let one of the substances form an inclined plane (fig.20) and a block of the other, of known weight W, and having a plane base, be placed upon it ; and, by varying the inclina- tion of the plane, let that inclination (^a) be found at which TTis just on the point of sliding down the plane. Then F acts upwards along the plane, and we have (§ 72.) M = IF cos a. (j<'\ Bin a _ I = tan a ' cos a The values of this coefficient for various substances have been found by experiment. ^.''■SSBB«>^,..^ AN ELEMENTARY TREATISE )s, the aDiount rea in contact, en them. ON MECHANICS; DESIGNED AS A TEXT-BOOK FOR THE UNIVERSITY EXAMINATIONS FOR THE ORDINARY DEGREE OF B. A. tntity for two PART II. DYNAMICS OP A PARTICLE. BY J. B. CHERRIMAN, M: A., LATE FELLOW OF ST. JOHN's COLLEGE, CAMBRIDGE, AND PROFESSOR OF NATURAL PHILOSOPHY IN UNIVERSITY COLLEGE, TORONTO. SECOND EDITION. in a — = tan a OS a stances have TORONTO : COPP, CLARK & CO., KING STREET EAST:' I 8 70. us PREFACE TO FIRST EDITION. The arrangement of this elementary work differs from that of most of the recent English writers on the subject, und is in the main the same as that employed by Professor Sandeman in his *' Treatise on the motion of a Particle." Adopting in full the principles and method of that admirable treatise, I have attempted little more than to translate out of the language of the Calculus into ordinary algebra the investiga- tions there given of the simpler cases of particle-motion. For the reason stated in Part I., I have not added any examples^ and have endeavored to be as concise as possible in any explanations jr illustrations that have appeared necessary. University College, Toronto, April I, 1858. i CHAPTER I. TUB MOTION OF &. PARTIOLK OEOMKTUICALLY C0N8IDERKD. 1. When the distance between two particles changes con- Motion of n tinuously during an interval of time, they are relatively jn motion. The position, and consequently the motion, of one particle can only be conceived in relation to other particles, but it is convenient to speak of a particle ahsolutely as being at rest or in motion, reference being made to ourselves or to some points in known relation to ourselves, considering these as Jixed^ and referring all motion and change of motion to the particle itself. By a particle is here to be understood only a geometrical point. Uniform motion. 2. When a particle is moving in a fixed straight line, its in a straight motion is measured by the change of its distance from a fixed point in this line, and the rate of this change of distance at any instant of time is called the velocity of the particle at that velocity. instant. « The change of distance in any time is here the linear space described by the particle in that time. If equal spaces are described in equal times, the change of distance in any given Uniform, time is always the same, and the rate of this change, or the mcusurcd. velocity, is said to be uniform^ and is measured by the space described in a given time. Taking a foot and a second as the units of linear space and time, the velocity t> of a particle moving uniformly will be measured by the number of feet described in one second. 60 ili'Hcrlbod in uuy timu. Velocity, + and — Dlstttii'-o of a iiioviii); ]>i>int from ii llxt'd point ill itH lint! of motion after any timo. Tho space doscribod in 1 sooond boing v, that in 2 socotids will bo '2v; in 'i seconda, '.in', and, Ronorally, in t Hcconds lu : henoo, if k bo tho Hpuco doaoribod in tiiuo t, with a uiiit'urm velocity v, a = vt. Apimrontly this formula is proToil only for tho caso wliere t \h n whole number of 8ocom]« ; but, If t bo fractional, wo can alwayw as- Bumo a unit of timo such that tho interval of tinif oxproHHOil by I nIuiII contain a wholu numbor of thoHo unita, and tho formula can thou bo shown to apply. ThuH lot n bo a wholo numbor Huch that nt in uUu a wholo numbor T; and let -tb of a socoud b«. > tkon as tho unit, and V n be tho velocity roforrod to this unit. Then tho timo t being oxpro88ed in this unit by a wholo number T, wo have a — T. V = nl r-= vl; for V being tho Hpace in one aoconrl or n units, is n times the space la one unit, that is, = n V. Uonc^ the formula is general. 3. Assuming some fixed point in tho line of motion, if a bo the distance of the particle from it at one instant, and s be the distance, estimated in the same direction, after the time t during which the particle has been moving uniformly with the velocity v, we shall have s z= a -\- vt, or 8 = a — vt, according as the particle has been moving in the direction towards which s has been estimated positivcli/, or in the oppo- site direction. Both these cases can be included in one formula by indicating oppositoncss of direction of velocity by the ai^^e- braic signs -\- and — . Thus, fixing on one direction from the fixed point towards which when measured tho distances are to be considered positive, a velocity in this direction will be positive, and in the opposite direction, negative. Hence, if a particle move during successive intervals of time with dififerent uniform velocities, and a be the distance from the fixed point at the beginning of the time, x its distance at the end, thea 8 = a -{- S (yi) . < • where S denotes the algebraic sum of all the products cor- responding to that within the brackets ; and the particle will be on one or the other side of the fixed point at the end of the time according as 8 comes out from this expression positive or vegative. «1 . Tlio wliolo .npitre tfesrri/ieil will, however, ho the nunioriciil Huui ut' thuHu pruduulfl, UUrogardiiig ul^tbruic HigiiH. 4. Wlit'ii ft piirliolo niovcH from a fixc'd point in a Htrnight '.<'t>itnii, lino with difTcront vcU)citio.s 'luriiif^ succcBsivo equal interviils of imm'HUrt- time, oacli velocity continuing uniform throu^'hout its interval, !,'n,.r','i?.*n. the distunco of the particle from the point nt the end of the lI.','II,'H!!fu„|, tinje is the product of the time by the arithmetic mean of ull [il'i'iVi'tiUM. the velocities. "'»»"♦' "«"• By tho arithuutie mean of n nutiibcr of quiintltlos Is monnt tlit-ir nl^'obrnii! huiii lUvidud by tho niiiiibur of tbuiii. / For let t be the whole time : the duration of each interval ; *'ii *'.') ^'3> ••• t^o succesMvo velocitioa during tho first, second, third ... intervals. Then tho required di.stunco will be the alf^ebraic sum of tho f^paces described with these velocities ; l.iUt irt, by § 2 ; ; or, I'l 'f- ^2 + '3 + n '-.t Q. E. D. \fK the i)ro- "liict (if till- llll'llll Vtilo- fity and the time*. The Ci\3o of iiiiy of flio velocities bein;^ in tiio opposite tlirocti'iu (and tlicreforo ftfcountocl iie sured by tho space which it would describe in a unit of time, ]|[|^^^„rc(i if it wore to move uniformly during that unit with this velocity. The rate of change of velocity at any instant (provided it Acceleration be continuous) is called the acceleration. If the change of velocity in a given time be always tho same uniform, throughout the moticn, the acceleration is said to bo uniform^ Illeasurcd. and it i.s measured by this change of velocity in a given time. ( Thtf olian/o of Ti>li)city may ho •lltivr an InorpAHo or docroaiio, nml In tho latter "Auo the atictUratinn Ifl In vfToot • rc/arr/'ifjon. Tin* imo of both tt'rniB In, however, rcndcri'd unneccasary by introducing' ilio iil(;obraio iiiifiih -f- and — ; for n ducruait') in al^obrAli^nlly n niytUn>« innrAAAo, und tituf a retardation in a npi^ativo acculoratlon ; and \\\wn wo Rpoak of ^oli)city bclnjf incrtsiiMed, odd'd, or i;i'nornl«!<), Wf uUo Inc'udo tho caao of Tuloolty hcln{j dlnilninitod, Hubtrnctcd, or doa* troy»d. Tlie vi'lmity Tukirij; a Bccotid ns tho ui it of tliuc, th ) n'rchration /, whon iorNfof tho motion is nnljormli/ nccdurntod, Ih tho chim^o of volHi'ity amkruiion •" ono Bccjotid. Then 2/' is tho chnn^'o in 2 seconds, 8/ in 3; and j^(!fiunilly tf in t Becoiids. Hotjoc), if u ho tho veh)citv ufc tho hi;;:ifiiiir)}^ of tlui timo ^, and V bu tho velocity at tho cud of this tiuic, wo havu V — u -r=.fi^ or ithiy tho nroa of t)io Innor pnrfttlflo^mtii Cif^ \ no a\n(\ \t U tnovoci ii«i/(f)rw/// tluriniir/) with till) viilucity //./', wliirh it ImN nt tlin onj of thin iiiturval, tho npaou lioMcrihotl wmiM ho rcpn'««»nto(l l»y tho nrni of lh«' oiWcr piir!ill.l<»_'riuii Ti/'. If, thcri'loro, the piirtichi mnviMl unifortnly thrttu^hiMit rtu'h In. tcrviil on thi) rofiner supposition, tho whnlo Hpiico (IfNcrihni wouM b« th«< sum of tho innor prirtinolo^nunx ; ami If on lint Inttor nupponltion, it would ho tho huiu of tho outi^r pariiihto f^niinH ; un*l thu npnco («) notually (loMorilu'tl Uoh iiuinoiiiiilly between tho ^pa<'CN dcticrihoil oti thoMO two HUppoHiiionH. Hut as tho nunihii* uf iritcrvulit \n incroaftcd, nml tiio ii>:i^'riitU)lo of onch cliiiiiulMht'il, tho two HcrloM uf parulli ]o|;rauiH hoth npproucli nvat'or anil nonrer to thu <|Uiiilrihitoral www AKL'tt^ wxA \\\'\n must thiToforo bo tho value of «. Ifciit'o a \a ropro.scntoti by tho parullolojj;ram Ale and triangle dH*', tltut i.s nuniuriciilly by Kk X AK -\ , /.^' X <'/.-, !in;le3 ALa and LPp\ that is, by twice tho triangle ALa and tho quadrilateral Nii'p'P. This result is identical with that in § 8, Cor. 3. 10. When the particle moves from rest and its motion is uniformly accelerated, we have seen that tho velocity and space described at any time from the beginning of motion are given by the formulas, 1 v—ft; s f^\ and these are sufficient to determine all the circumstances of the motion in any case. When any two of the quantities /, v, s, t^ are given, the remaining two can be found from the above equations. The following cases may be noticed : 11. Given the acceleration and space described, to find the velocity acquired. 1 1 / 1^ \ 2 Here s = -fC' = o/i 7 j ? ^^^^ therefore, and conversely, to find the space through which the particle must move to acquire a given velocity, we have ■ ^ = v- ■ ■ ■ .■> -.. 12. The equation s = - ffi becomes, by putting v for //, s = - vt. Hence the space described in acquiring any velo- city is half the space which would be described with that velocity continued uniform through the same time. «t 13. Putting f = 1, we have « = - /, or / = 2s. Hence, twice the space described in the first second from rest measures the acceleration. 14, The qmcei described from rest in successive equal intervals of time, are as the odd numbers, 1, 3, 5, 7, For, taking any interval as the unit of time, let 2^ bo the acceleration referred to it. Then the space described in n — 1 intervals from rest is - F (ii — 1)", and the space described in n intervals from rest is iFn'. ■2 The difference between these is the space described in the 71 t)i interval, and = Fn — - F = - F (2n — 1). Giving to n the successive values 1, 2, 3, this becomes -F . 1. -F . ^) 7.F. 5, which was to be proved. 15. The initial velocity being u, and this being uniformly cnum- accelerated during the time t, the velocity v at the end of this m'otio'ii wi. n time and the distance s of the particle from its initial point; Av!',s\'il!t 'It" 1,1 , • rest -dt tho IS given by the equations i V = « + /U; s :r= ut + - /e, and these are sufficient to determine all the civcumstances of the motion in any case. When any three of the quantities «, /, t, v, s, are given, the Remaining two can be found from the above equations. (■OlilllU'llri'- lllf lit lit tll'i Iierio'l. The following cases may be noted : 16. We have s = ut-i--ft' = lt{2u-\-ft) GS or, the distance is that which would bo described in 'iie same time with a uniform velocity equal to the mean of the initial and terminal velocities. This result might at onco have boon iuferred from ^ 8. 17. Given the initial velocity, the acceleration and the distance, to find the velocity acquired. Here n, /, s are given to find v, and t must be eliminated from the two equations. t; = w -1- //, s = M< -f - /(2. Squaring the first, we have = w2 + 2/s. If the velocity were retarded, we should have Cor. This result might have been obtained without finding the second equation, for we have directly, from § 5, V — U —ft, :■ , : .. ; and from § 1 G or 8, , multiplying these equalities, we have The following geometrical proof may also be noticed : Le'. B be the initial point, where the velocity is ?<; i?C the space described (s) when the velocity is v. C9 Lot A bo the point from winch the pavticlo, proceeding from rest nndor the same acceleration, would ncqnire the velocity w at li. Then (§ 11). uS =r 2 /. yl B. Also, since the wholo motion may bo taken to proceed from rest at At we have (§11) t)2 = 2/. yl C = 2f,{A B + AC) = 2/. A B -{-2/. A By a proper adaptation of the figure, this proof may bo extended to oil the cases included in the algebraic formula. 18. Given the initial velocity and the acceleration, to find the time when the particle will bo at a given distance from the initial point. Here u,f, s are given to find t. Solving as a quadratic in t the equation s =z u t -\- -ft'-, we have t = — M + i/u' + 2ys / The significance of the double sign is here note-wortliy. If/ be positive, or the velocity be numerically accelerated, one of the values of t is positive, and the other negative. The former is the solution required, but the latter can he inter- preted thus : Suppose A the initial point, AP the distance s, and the velocity m at J. to be in the direction AF. Then the positive value of t in the above giv^s the time of moving from .^ to P; the negative value gives the time that would have elapsed if the particle had moved from P towards -4, with a retarded motion, passed through A to the other side of it, been reduced to rest and again returned to A. If/ be negative, then, writing —/ for /, the values of t become M -f- i/w* — 2/s. 70 If then m' > 2 /s, both values of t aro real and positive^ anil the particle will twice be at the same di8^anco from the initial point, once during the recess from and again during the return towards it. If M^ = 2/8, the two values become the same, and the dis- tance in question is that where the particle momentarily comes to rest. If it^ <; 2/s, both values of t are imaginary, and the par- ticle can never reach that distance. If, however, s bo negative,, both values are real, and one positive, the other negative, the latter referring to a time pre- vious to the epoch from which wj aro reckoning, when the particle, if it had been moving towards the initial point from the negative side, would have been at the assumed distance. Comivisition nrvt:lui;ilii;b. Fi^ Component Velocities. 19. The position and motion of a particle moving uniformly in a straight line have been determined by the distance of the particle from a fixed point in the line, and by the change of this distance in a given time. Its position, however, might have been defined by its distances from two fixed lines, mea- sured parallel to these lines. Thus : let Ox^ Oy be two fixed lines, B the place of a particle moving in the line ABC, and A c fixed point in this line, the distance from which deter- mines the place of the particle. Let C be the place at which the particle would arrive after any time if it moved uniformly with the velocity it had at B, and complete the figure by draw, ing lines parallel to Ox, Oy:. The position of B is knowc. when B P, B Q, its distances from thcie fixed lines, are given; and CB, CD, or their equals^ B D, BE, would be the changes of these distances if the par- ticle arrived at C by moving uniformly. Now 5(7, which would be the change of distance in a given time from the fixed point A, measures tlffe velocity of the par- 71 tide : and J57>, BE aro always proportional to li(\ and there- fore measure wliat wo may cull the component velocities of the particle in the directions of the fixed lines. Iloneo, 1/ a stnii'jht line he tuken'to rrpreSfmt in mnyniliule and I'lnii'ii" direction the veliciti/ of a pariiclr, the adjacent sides of any <'i>iii|'"ii'iit parallelogram constructed on this line as diarjonal ivill repre- sent the COMPONENT VELOCITIES in he directions of those ^ides. Conversely. If the COMPONENT VELOCITIES in tico dircrtions be (jiven, the actual velocity loill be found ir .nafjnitude and direction by drawing the diagonal of the parallelogram of which the components form adjacent sides. These two statements constitute the "parallelogram of com- ponent velocities." 20. When the two components are in perpendicular direc- v.io,iiy tions, it will be convenient to call them the resolved parti of Inauy'^ the velocity in these directions; and the rule for finding these ^"'^*^*''"^' resolved parts will be Ihe same as that for the resolved parts of a Force (STATICS, § 21), namely : To find the resolved part of a velocity in any direction, uuietor. midtipJy it by the cosine of the angle between this direction and that of the velocity ; and to find the resolved part perpen- dicular to this direction, multiply by the sine of the aforesaid angle. CIIAl'TKIl TI. TIIK MOTION OF A MATRIIIAL PAHTICLK ACTED ON BY UNIFOUM FUUCKS. vti: Apiilir'ilii'll .if iM-ffi'iliii;; It'SUltS til t!ic nctuil MliitiiPllH nf lll.lt.Tiill |p;irti.'li'S. i;xiiciiiiiciit- ,»1 l.awrt. 21. In the foref^olnrr cliiipter the j^comctrical conditions of the tnotinn of a point have been examined. It now remains to exhibit the connection of tlieso results with the actual motions of material particles, and the relation between thcso niDtions and the forces acting on the particles, and this inves- tigation constitutes the science of Di/namics. For this purpose it is necessary to appeal to experiment and observation, and it appears that all the phenomena of the n)otions of material panicles can bo referred to three elementary principles or laws, which are commonly known as " Newton's Laws of Motion." These laws, from their nature, are incapable of being demonstrated by direct experiment, for it is impossible to make experiments under the precise circum. stances conditioned by the Laws, and which would not involve other phenomena besides those rhioh it is desired to test. Direct experiments may, however, afford a presumption in favor of these laws by showing that the more nearly do the circumstances of the experiment approach to the exact condi- tions required, the more nearly are the results of the experi- ment in accord with those indicated by the Laws; and also that whenever a discrepancy is found between these results, there can always be traced some disturbing cause which ought to have been excluded by the conditions postulated. The ultimate ground on which these and all other laws in Natural Philosophy rest, is the entire and universal concord- ance of the results of experiment or observation with those calculated on the assumption of the truth of the laws. lO 22. Allhouf^h tho motion of u pnrticlo nnJ tlio forces nctin;^' on it can only bo conceived in rolution to other particles, if is convenient to speak uholiitch/ of a particle as beiii<; at rest oi* in motion, rcferenco beinp; made to ourwelves or to some spaco in a known relation to ouiselves which wo consider /j:k/, and then to regard tho phenomena exhibited by the particle as duo to forces acting only on itself, these forces being delined by tho measures of them already cuiployed in Statics. 23. First Law op Motion.* A riuiteriul j^article, when not advil on lij nni/ fotce, if it rest, irill so remain ; aml^ if in motion, v:!U move in a »tra!ijht line ioi/h nni/orm vdocitj/. The first part of this law has been already assumed (Statics § 2) as tho basis of our conception of a force. Experience shows that whenever a quiescent body is set in motion, we can trace the action of some cause external to the body ; thus, when a body is suffered to drop to tho earth, wo assign its motion to a pressure exerted on it duo to the earth itself, and which would have no existence if tho earth did not exist. Also, there seems no reason why a particle, apart from any external fljrce, should begin to move in one direction rather than another. til ire I,;l\VH III' Miitliiii. First Law. No forces iii'tiiii,', thi' particle t'itlK'l' ri'iiiiiitm at rest, or Again, when a particle is in motion there seems no reason m.ivpsina why it should change the direction of its motion in one way liuo. rather than another, unless some force bo acting upon it to determine such change; and in all cases of any such change, we can always trace tho action of some external force; as, for instance, when a stone is projected from the earth in any direction, the deflection of its motion from a straight line is produced by the aforesaid pressure due to the earth, whicl? we know is always acting vertically downwards. If this pressure be counteracted by projecting the stone horizontally along a * Lkx. I. Corptis omne perseverare in statu sito qniesccndi vel movendi wuformiter in directiim, riisi guaientcs a viribus impressis coffiiur siatum iilum mutare. — Princ. Leff, Mot. 74 \rlt!i iini''iinn vi'Ioclty, Onllloo. 11x0(1 plnno, the path approaches to a 8trnight line, with onl7 siu.'h (loviutions um niny bo accounted for by friction or irrcgu- huiticfi in the piano, or from the stono not being nuiull enough to bo oonsiilorcd a particle. 80 also with rogurj to the velocity of the particle, it docs not seorn possible to conceive any way in which its velocity could increase or decrease unless by the action of some external cause and iu act lal cases of variation of velocity wo con alwa}. t a.' u existence of such causes. Thus when astono is thru.Vii hi-i, itally along the ground, it gradually loses its velocity I' i .1 aloj." ')ut here the friction of the ground and the resi.stanco of the mt ct as retarding causes, and wo see that in proportion as the surface on which the stone moves is smoother, as on a sheet of ico, the longer and more uniform docs the motion continue. Thia law is sometimes tormod the Law 0/ Inertia, bein;^ understood to express that a material particlo ia inert, and has no tendency of itself to change its state of rest or uniform motion. 24. It follows that tho motion of a material particle when not acted on by any forces, or acted on only by forces which counterbalance, is determined by tho formula of uniform mo- tion, s = vt, investigated in § 2. Unitonn -•'• ^^^ ^^'^ procced to considcr the motion of a particlo ftl" cm a* acted on by any uniform forces, of which the following are i.i»rtiuic. ^jjQ observed laws : — (1.) When a uniform force acts contiimoudi/ upon a par- ticle in the line of its motion, the velocity is uniformly accele- rated. A sinple fort^e ill tho line of motion. The investigations of § 5 e< seg., therefore, apply to this case, (noticing also that retardation is included in the term accele- ration), and we can compare the results there calculated with those of experiment. Thus when a body is permitted to drop freely to the earth, or is projected vertically /^downwards or upwards with any assigned velocity, its path is a vertical line, and the force acting on it is its weight which always acts ver- 7ft tically, and (for not grcnt hci^htH nbuvo the; Hurfuco) in flonHibly ui)it'(irtii. Ilvro then the rcr|uirc(l cuiulitiunH iiro fulfilled, :itiil tbti result of experiment, when duo allowiince is nmdo fur the rc8i.sttinco of the nir, in that tiie motion is uniforiuly accelerated, the ninount of this acceleration being about 2)2.2 feet a second, but vuryinp; slij^htly for different latitudt-M and elevations ubovo the 8ca-level. This acceleration is usually denoted by formforcct are. avtinff Aimuftaneonuli/ in the line oj motion, the reuniting accrlcration I's the ahjdiraic tuni of the accelerations which would he produced hy each forca art in J aeparatdi/. llcnco it follows that n equal forces nctiiij^ Hiniuitn -nsly on a particle in its lino of motion will produce n tip. ^s ltd acceleration which ono of the forces alouo would r. '^^X'^\. i the same particle; and, consequently, the accelerati /r^ 'acod in a given particle i« proportional to the mngni*ud» of tlio force acting. It will bo hereafter shown how this may be tested by com- paring the accelerations of a particle down inclined planes of dilTorent inclinations. Hence also the change of velocity in a given time is pro- portional to the magnitude of the Force, the particle acted on being the same. (3.) When a moving particle is acted on continuously by a uniform force which acts always in the same direction and obliquely to the direction of the particle's motion, its velocity after any time is found to have for components — first, the original velocity, unaltered, in its own direction — second, a velocity in the direction of the Force, the magnitude of which is the same as if the Forcp bad acted on the particle origiually at rest. So that the velocity and direction of the motion may be found at any time by calculating the velocity which would be produced by the Force acting for that time on the particle originally at rest, and then compounding this with the original velocity according to the principle of the " parallelogram of velocities." Any fiiri'i'H ill till' lliii' llf lliotlnli A sin«LL forci' 1)1)- Vu\\w to till- (liiT''tiou (it luutiou. W^ 7n Or, this mtij ho oxproMod moro simply thu.^ : if vro rcnolvc tho original velocity of the prirfiolo into two coinpoiu'ntH, ono in ilircction of tho force, tho other perpendicular to it, iho latter rciniiinH unnitored nnd tlio former is changed hy tlio F<,rco prccinoly an if it uloiio wore tho actual volooity of tlio p.'trticlo. 80 thut T/te change of velocity/ prnJncerl ly th", force in a given time iH !n direction of the force and it proportional to it in tnoynitude. In tliiH cnHo tho pdtli of Uio piirtiolu in no longer a Htrni^lit linu, but a ctirvo, tliu ton^ont to wliicli nt any point U tliu dlroctlun of tho par- ticlu'd ni()th)n tliuro. It: Other vvordH, tho ;.bove oxprosacH tliat the dynamical effect of a force on a particle in icholli/ independent of any motion which the particle mag have, and is the tame a» if it were exerted on th-^ particle orijinalli/ at rent. Thus, tho vertical dc.^cont of a body let full from tho mast- head of a ship in motion is precisely tho samo in all its cir- cumstances as if tho ship wero at rest. Tho principle can nl.Ho bo tested by comparing tho results of calculation with obi^crvations on tho motion of a body projected obli«iucly to the horizon and acted on by gravity, duo allowauco being made for tho resistance of the air. . , (^^' When several Forces act .siniultancously, retaining Any forres ^ •' , . . . a-tin^'in always the samo magnitudes and directions, on a particle iiiiy dirco ..m, . ...«, . 1*1 tionona originally at rest, tho motion la uniformly accelerated in the . ithur origi- direction of the Resultant of tho Forces, and tho acceleration is that duo to this Resultant acting singly. Also tho velocity generated after any time, being that due to this Resultant, is also that which is compounded of the velocities duo to tho Forces acting singly on the particle from rest. orinmotion, -^^^° ^^ ^^'^ particle be in motion when the Forces begin to act, its velocity and direction of motion after any time will 77 to (li'tcrminod by cftmpoumlinjr Un ori^inul Vi'loclty with ihe TuliK'ity (Itio to llio UcMultiitit of tho l''orci>ii, ur with nil thitHC da« to tho Forc'-'i acpuratrly. Ur, Whm forctt act on the iiimf pnrh'rlo uwler any circuiti' ttanrex pruvltl d each forte fm unt/urm and aliiun/H ptesrrre tht t'ime dirertioHj the chanije of rclncitif in a f/iven time d in- to each /one in in direcliitn of thut forcr, and is proportionnl to it in mnjni'.udc. (J).) It follow!* from llio pi-t'(;c*'0s in tlio motion of a partiek' by actiiif; on it for a finite timo. There is, however, a certain class of forces, Huch as those nianifestud in explosioDs or the collision of bodios, which produce Lnite effects in chanp;ing tho velocity or momentum inatantanrouith/. Such forcoH are called itnpul' aiie, and must bo carefully distin':;uishcd from forces of tho former class, with which they do not in any way admit of comparison. These impulsive forces are measured by tho momentum which each would instantaneously communicate to n particle at rest, and tho second Law of Motion applies to them, stated under tho form : HiMond Law Wficn impufshe Forces act on material particles, each li o< u». pi'QfiiK'cg 1,1 {ig fjjtvi direction an instantaneous chamje of momentum proportional to itself in mafjnitude, • Lkx. 11. — Mutatloneiamotm proportionalein exse vi molrid imprmsn, «t fieri secundum lineam rcctam qua vU ilia imprimitur. — Puino. Lko Mot. . 70 Thuii iho moUon of n pftrUcto when noled on hy >(iiiul(a* noous itii|)ulii(>(i will bu (lotcriiilixtl by culculntiriK tint wlucity iiiNtuitt iiiiiouNly Kuncratod by unoh in its own ilirt;i!tion, nn*l couipournlin^ (hoNO with thu uri^itial vulocity of thu |wirti(lo nccorillri;^ to thu parnllulo^roin of vulocitlm. For Inntsinou, if I'lruii.i-^ A pnrticlo ut rout bo notuil on by two impulMCH whioli, mpitratcly xIhuiv* coiniiiuiiicatoil, wouM ^ivo tHiu particle ro^poctivi'ly Huch volo. .vrutotl citioH ns would cnuMO it to iloMcribu unifnrrnly tlio Aden All, AC o( li p(iralli!lti^ra?n JLiCJJ, tlio parliulo will acquiro from thu itnpiilM(vs itiniultaiu'oUMly cottiiiiunioatod a velocity whioli will cauNu it to duHuribo utiifuriiily thu diaj^onal vt/>in that tiuiu. APi'UCATIUNH AND TKSTS Ut' TIIK 8KLUM) LAW Ol' MoTIo.N. V.rtl.ul tiintloit by 27. Thi vertical motion nf a part uh under the ac/iou o/ ""^ "* "^ Untlleo. gravity. The nccolcration of gravity (/e directly wUh given vdoci- tief, fo find their velocity afte - inqtdct. Let A, Ji ho tho masses of tho two balls, and u, v their ve- loe.Lics estimated in tho same direction; then, after impact, thoy proceed with a common velocity, F" (suppose). Tho al- gebraic sum of the momenta before impact is Au -{- Bv ; and, after impact, it is {A + B) V. IIcucc, by the law of equality of momenta, {A -\- B) V=Au-\- ^y, and „ An -\- Bo '^ a + b'' If the sec (id ball were moving in an opposite direction to the first, v would be taken negative, and the direction of V will be indicated by its resulting sign. Coa. 1, If 5 were at rest, then v = 0, and A V: u. A-r B €011. 2. If the balls be brought to rest by the impact, then F= 0, and, therefore, Au -{- Bv = 0, — V A or Or the balls must have been moving in opposite directions with velocities inversely proportional to their respective masses. 89 88. Two clastic balls impinge directly with given vdocilieSf Tw'o ciaiiiit to ditermint their velocitiet o/ter impact, Let A, B, bo tho masnes of tho two balls : lMi|>tni(in(; •llreftly. n, V, their velocities bofuru impact, estimated iu eamo diroctiun, w', v/, " after " " " e, tho elasticity. A is supposed to ovcrtuko B, Then the sum of their momenta before impact is An -\- Bo. and " " after " Au^+Bv'. Hence, by the law of equality of momenta, Au^ + ^vf = Au -{- Bv. Again, by the law of relativo velocities, t;/ — u' = e (u — v). From these two equations, finding m' and v', we have yl « 4" ^^ — -^^ (" — '0 «' = d'= A -\-B Art -{- Bu -{- Ae {u — v) AA-Ji ' If B wore moving before impact in an opposite direction, V would bo taken negative, and the directions of m', v^ will be indicated by their algebraic signs. Cor. 1. In no case can both balls be brought to rest by the impact. Cor. 2. If the balls be perfectly elastic, or e = 1 ; and Two equal if also their masses be equal, or A = B; then we have "yeiasu" W V, V = u. balls exchange velocities. and the two balls exchange velociiies. Thus, if the second were at reit, the first after impact would remain at rest, and the second would go on with the velocity of the first before impact. •«l ss:: 90 fkr. 3. Honcp, If A row of «qiial, pflrfoctly clnstlo bnlln bo rnn^^ixl In contofit In a HtralKiit lino, and nnutliur bull, oUo |>orfootly claiitlo and crjiml tu each of tlium, Impint^u In cncli lino on the (Imt of tlumo bnlU, tlin itiipln^dn^ ( v^ cos 0' V sin = v^ sin 0' Ititt>«''t nt i» from which wc obtain V ]/ ^ siu2 tf + t^ cos^ I , and tan ^ C^/ ^ lis " < ^^^ * \^ _<^. 1.0 I.I l^lii 125 ■U lU 12.2 Z Uo |2.0 m 1 l'-2^lil'-p»'m-fK»TR ■: -m *!«R». «««■• «at'»a»M».>/ ■If/ J ! --■»■ ■ tf - ■ - ■ V-. 1* '' ■ - V-J ' W "wai mmfmmmmmmtf'Mmm \ J i , .M . . - ■ .-.« «■ 1 .^ . , W=j % .' B V f/.7. ) } . w L-l N= Ill»i % .' I /}yj l*T^' V V.7 />'«/. -7 S - ij^py I' I 5*5 — f "*p*i », ..•"!'• i :. t,:H. -ct.'.-c