,transcrib]'s ,notes3 ,und]scores 80 be=e & aft] a ^w or phrase indicate italics 9 ! orig9al text4 ,small capitals h be5 conv]t$ to ,,solid capitals4 ,illu/rations h be5 mov$ s !y d n break up p>agraphs4 ,typographical errors h be5 sil5tly correct$4 ,,exp]im5tal ,,me*anics ,illu/ration ,illu/ration3 ,,! ,,pa? ,,( ,a ,,projectile ,,is ,a ,,p>abola4 ,,exp]im5tal ,,me*anics ,a ,,c\rse ,,( ,,lectures ,,deliv]$ ,,at ,,! ,,royal ,,college ,,( ,,sci5ce ,,= ,,irel& ,,by ,,sir ,,rob]t ,,/awell ,,ball1 ,,ll4,d41 ,f4,r4,s4 ,,a/ronom] ,,royal ,,( ,,irel& ,,=m]ly ,,pr(essor ,,( ,,appli$ ,,ma!matics ,,& ,,me*anism ,,9 ,,! ,,royal ,,college ,,( ,,sci5ce ,,= ,,irel& 7,,sci5ce ,,& ,,art ,,dep>tm5t7 ,,) ,,illu/rations ,,second ,,edition ,london ,,macmillan ,,& ,,co4 ,,& ,,new ,,york #ahhh4 ,! ,"r ( ,translation & ,reproduction is res]v$ ,,ri*>d ,,clay ,,& ,,sons1 ,,limit$1 ,,london ,,& ,,bungay4 ,! ,fir/ ,edition 0 pr9t$ 9 #ahga4 ,,preface4 ,i "h pres5t ! revis$ edition ( a c\rse ( lectures on ,exp]im5tal ,me*anics :i* ,i deliv]$ 9 ! ,royal ,college ( ,sci5ce at ,dubl9 eighte5 ye>s ago4 ,! audi5ce 0 a l>ge ev5+ class consi/+ *iefly ( artisans4 ,! tea*] ( ,elem5t>y ,me*anics1 :e!r he be 9 a ,bo>d ,s*ool1 a ,te*nical ,s*ool1 a ,public ,s*ool1 a ,sci5ce ,college1 or a ,univ]sity1 frequ5tly desires to en=ce 8 lessons by exhibit+ work+ app>atus to 8 pupils1 & by mak+ c>eful measurem5ts 9 _! pres5ce4 ,he wants = ?is purpose app>atus ( sub/antial proportions visible f e "p ( 8 lecture room4 ,he wants to h x ( su* a univ]sal "* t he c produce f x day aft] day comb9ations ( an ev]-v>y+ type4 ,he wi%es x to be compos$ ( well-design$ & well-made p>ts t %all be /rong & durable1 & t w n easily get out ( ord]4 ,he wi%es ^? p>ts to be su* t ev5 p]sons n specially tra9$ 9 manual skill %all pres5tly le>n h[ to comb9e !m ) good effect4 ,la/ly1 he desires to economize 8 money 9 ! matt]s ( v>ni%1 mahogany1 & glass cases4 ,i f\nd t ,i 0 able to satisfy all ^! requirem5ts by a suitable adaptation ( ! v +5i\s sy/em ( me*anical app>atus devis$ by ! late ,pr(essor ,willis ( ,cambridge4 ,! elem5ts ( ! sy/em ,i h briefly describ$ 9 an ,app5dix1 & :at adaptations ,i h made ( x are %[n 9 almo/ e page & e figure ( ! book4 ,9 revis+ ! pres5t edition ,i h be5 aid$ by my fri5ds ,mr4 ,g4 ,l4 ,ca?c>t1 ! ,rev4 ,m4 ,h4 ,close1 & ,mr4 ,e4 ,p4 ,culv]well4 ,,rob]t ,s4 ,,ball4 ,,obs]vatory1 ,,co4 ,,dubl91 #crd ,augu/1 #ahhh4 ,,table ,,( ,,cont5ts4 ,,lecture ,i4 ,,! ,,composition ,,( ,,=ces4 ,,page ,introduction4--,! ,def9ition ( ,=ce4--,! ,measurem5t ( ,=ce4--,equilibrium ( ,two ,=ces4--,equilibrium ( ,?ree ,=ces4--,a ,small ,=ce c sometimes balance ,two ,l>g] ,=ces #a ,,lecture ,,ii4 ,,! ,,resolution ,,( ,,=ces4 ,introduction4--,"o ,=ce resolv$ into ,two ,=ces4--,exp]im5tal ,illu/rations4--,sail+4--,"o ,=ce resolv$ into ,?ree ,=ces n 9 ! same ,plane4--,! ,jib & ,tie-rod #af ,,lecture ,,iii4 ,,p>allel ,,=ces4 ,introduction4--,pressure ( a ,load$ ,beam on its ,supports4--,equilibrium ( a ,b> support$ on a ,knife-edge4--,! ,composition ( ,p>allel ,=ces4--,p>allel ,=ces act+ 9 opposite directions4--,! ,c\ple4--,! ,weigh+ ,scales #cd ,,lecture ,,iv4 ,,! ,,=ce ,,( ,,gravity4 ,introduction4--,specific ,gravity4--,! ,plummet & ,_s-,level4--,! ,c5tre ( ,gravity4--,/able & ,un/able ,equilibrium4--,prop]ty ( ! ,c5tre ( ,gravity 9 a ,revolv+ ,:eel #ej ,,lecture ,v4 ,,! ,,=ce ,,( ,,friction4 ,! ,nature ( ,friction4--,! ,mode ( ,exp]im5t+4--,friction is proportional to ! pressure4--,a m accurate =m ( ! ,law4--,! ,coeffici5t v>ies ) ! weights us$4--,! ,angle ( ,friction4--,ano!r ,law ( ,friction4--,conclud+ ,rem>ks #fe ,,lecture ,,vi4 ,,! ,,pulley4 ,introduction4--,friction betwe5 a ,rope & an ,iron ,b>4--,! ,use ( ! ,pulley4--,l>ge & ,small ,pulleys4--,! ,law ( ,friction 9 ! ,pulley4--,:eels4--,en]gy #he ,,lecture ,,vii4 ,,! ,,pulley-,,block4 ,introduction4--,! ,s+le ,movable ,pulley4--,! ,?ree-%eave ,pulley-block4--,! ,diff]5tial ,pulley-block4--,! ,epicycloidal ,pulley-block #ii ,,lecture ,,viii4 ,,! ,,lev]4 ,! ,lev] ( ! ,fir/ ,ord]4--,! ,lev] ( ! ,second ,ord]4--,! ,%e>s4--,! ,lev] ( ! ,?ird ,ord] #aai ,,lecture ,,ix4 ,,! ,,incl9$ ,,plane ,,& ,,! ,,screw4 ,! ,incl9$ ,plane )\t ,friction4--,! ,incl9$ ,plane ) ,friction4--,! ,screw4--,! ,screw-jack4--,! ,bolt & ,nut #aca ,,lecture ,x4 ,,! ,,:eel ,,& ,,axle4 ,introduction4--,exp]im5ts ^u ! ,:eel & ,axle4--,friction ^u ! ,axle4--,! ,:eel & ,b>rel4--,! ,:eel & ,p9ion4--,! ,crane4--,conclusion #adi ,,lecture ,,xi4 ,,! ,,me*anical ,,prop]ties ,,( ,,timb]4 ,introduction4--,! ,g5]al ,prop]ties ( ,timb]4--,resi/ance to ,ext5sion4--,resi/ance to ,compression4--,condition ( a ,beam /ra9$ by a ,transv]se ,=ce #afi ,,lecture ,,xii4 ,,! ,,/r5g? ,,( ,a ,,beam4 ,a ,beam free at ! ,ends & load$ 9 ! ,middle4--,a ,beam uni=mly load$4--,a ,beam load$ 9 ! ,middle1 ^: ,ends are secur$4--,a ,beam support$ at "o end & load$ at ! o!r #ahh ,,lecture ,,xiii4 ,,! ,,pr9ciples ,,( ,,framework4 ,introduction4--,weight su/a9$ by ,tie & ,/rut4--,bridge ) ,two ,/ruts4--,bridge ) ,f\r ,/ruts4--,bridge ) ,two ,ties4--,simple ,=m ( ,truss$ ,bridge #bjc ,,lecture ,,xiv4 ,,! ,,me*anics ,,( ,a ,,bridge4 ,introduction4--,! ,gird]4--,! ,tubul> ,bridge4--,! ,susp5sion ,bridge #bah ,,lecture ,,xv4 ,,! ,,motion ,,( ,a ,,fall+ ,,body4 ,introduction4--,! ,fir/ ,law ( ,motion4--,! ,exp]im5t ( ,galileo f ! ,t[] ( ,pisa4--,! ,space is proportional to ! ,squ>e ( ! ,"t4--,a ,body falls #af' 9 ! ,fir/ ,second4--,! ,action ( ,gravity is indep5d5t ( ! ,motion ( ! ,body4--,h[ ! ,=ce ( ,gravity is def9$4--,! ,pa? ( a ,projectile is a ,p>abola #bcj ,,lecture ,,xvi4 ,,in]tia4 ,in]tia4--,! ,hamm]4--,! ,/or+ ( ,en]gy4--,! ,fly-:eel4--,! ,pun*+ ,ma*9e #bej ,,lecture ,,xvii4 ,,circul> ,,motion4 ,! ,nature ( ,circul> ,motion4--,circul> motion 9 ,liquids4--,! ,applications ( ,circul> ,motion4--,! ,p]man5t ,axes #bfg ,,lecture ,,xviii4 ,,! ,,simple ,,p5dulum4 ,introduction4--,! ,circul> ,p5dulum4--,law connect+ ! ,"t ( ,vibration ) ! ,l5g?4--,! ,=ce ( ,gravity det]m9$ by ! ,p5dulum4--,! ,cycloid #bhd ,,lecture ,,xix4 ,,! ,,comp\nd ,,p5dulum ,,& ,,! ,,composition ,,( ,,vibrations4 ,! ,comp\nd ,p5dulum4--,! ,c5tre ( ,oscillation4--,! ,c5tre ( ,p]cussion4--,! ,conical ,p5dulum4--,! ,composition ( ,vibrations #bii ,,lecture ,,xx4 ,,! ,,me*anical ,,pr9ciples ,,( ,a ,,clock4 ,introduction4--,! ,comp5sat+ ,p5dulum4--,! ,escapem5t4--,! ,tra9 ( ,:eels4--,! ,h&s4--,! ,/rik+ ,p>ts #cah ,,app5dix ,i4 ,! ,me?od ( ,graphical ,con/ruction #cci ,! ,me?od ( ,lea/ ,squ>es #cdb ,,app5dix ,,ii4 ,details ( ! ,willis ,app>atus us$ 9 illu/rat+ ! =ego+ lectures #cde ,,index #cee ,,exp]im5tal ,,me*anics4 ,,lecture ,i4 ,,! ,,composition ,,( ,,=ces4 ,introduction4--,! ,def9ition ( ,=ce4--,! ,measurem5t ( ,=ce4--,equilibrium ( ,two ,=ces4--,equilibrium ( ,?ree ,=ces4--,a ,small ,=ce c sometimes balance ,two ,l>g] ,=ces4 ,,introduction4 #a4 ,i %all endeav\r 9 ?is c\rse ( lectures to illu/rate ! elem5t>y laws ( me*anics by means ( exp]im5ts4 ,9 ord] to und]/& ! subject treat$ 9 ?is mann]1 y ne$ n possess any ma!matical k beyond an acqua9tance ) ! rudim5ts ( algebra & ) a few geometrical t]ms & pr9ciples4 ,b ev5 to ^? :o1 hav+ an acqua9tance ) ma!matics1 h by its means acquir$ a k ( me*anics1 exp]im5tal illu/rations may /ill be useful4 ,by actually see+ ! tru? ( results ) :i* y are !oretically famili>1 cle>] conceptions may be produc$1 & p]haps new l9es ( ?\ght op5$ up4 ,besides1 _m ( ! me*anical pr9ciples :i* lie r beyond ! scope ( elem5t>y works on ! subject are v susceptible ( be+ treat$ exp]im5tally2 & to ! consid]ation ( ^! "s ( ! lectures ( ?is c\rse w be devot$4 ,_m ( our illu/rations w be design$ly drawn f v commonplace s\rces3 by ?is means ,i w\ld try to impress ^u y t me*anics is n a sci5ce t exi/s 9 books m]ely1 b t x is a /udy ( ^? pr9ciples :i* are con/antly 9 action ab\t u4 ,our own bodies1 our h\ses1 our vehicles1 all ! implem5ts & tools :i* are 9 daily use--9 fact all objects1 natural & artificial1 conta9 illu/rations ( me*anical pr9ciples4 ,y %\ld acquire ! habit ( c>efully /udy+ ! v>i\s me*anical contrivances :i* may *ance to come be=e y\r notice4 ,exam9e ! action ( a crane rais+ weights1 ( a canal boat desc5d+ ?r\gh a lock4 ,notice ! way a ro( is made1 or h[ x is t a bridge c su/a9 its load4 ,ev5 a well-con/ruct$ f>m-gate1 ) its po/s & h+es1 w give y admirable illu/rations ( ! me*anical pr9ciples ( framework4 ,take "s opportunity ( exam9+ ! p>ts ( a clock1 ( a sew+-ma*9e1 & ( a lock & key2 visit a saw-mill1 & asc]ta9 ! action ( all ! ma*9es y see !re2 try to famili>ize y\rself ) ! pr9ciples ( ! tools :i* are to be f\nd 9 any work%op4 ,a va/ deal ( int]e/+ & useful k is to be acquir$ 9 ?is way4 ,,! ,,def9ition ,,( ,,=ce4 #b4 ,x is necess>y to "k ! answ] to ?is "q1 ,:at is a =ce8 ,p :o h n /udi$ me*anics occasionally reply1 ,a pu% is a =ce1 a /eam-eng9e is a =ce1 a horse pull+ a c>t is a =ce1 gravitation is a =ce1 a movem5t is a =ce1 c41 c4 ,! true def9ition ( =ce is t :i* t5ds to produce or to de/roy motion4 ,y may probably n fully und]/& ?is until "s fur!r explanations & illu/rations %all h be5 giv52 b1 at all ev5ts1 put any o!r notion ( =ce out ( y\r m9d4 ,:5ev] ,i use ! ^w ,=ce1 d y ?9k ( ! words 8some?+ :i* t5ds to produce or to de/roy motion10 & ,i tru/ be=e ! close ( ! lecture y w und]/& h[ admirably ! def9ition conveys :at =ce really is4 #c4 ,:5 a /r+ is atta*$ to ?is small weight1 ,i c1 by pull+ ! /r+1 move ! weight along ! table4 ,9 ?is case1 !re is some?+ transmitt$ f my h& along ! /r+ to ! weight 9 consequ5ce ( :i* ! weight moves3 t some?+ is a =ce4 ,i c also move ! weight by pu%+ x ) a /ick1 because =ce is transmitt$ along ! /ick1 & makes itself kn[n by produc+ motion4 ,! ar*] :o has b5t 8 b[ & holds ! arr[ betwe5 8 f+] & ?umb feels ! /r+ pull+ until ! impati5t arr[ d>ts (f4 ,"h motion has be5 produc$ by ! =ce ( ela/icity 9 ! b5t b[4 ,be=e he releas$ ! arr[ !re 0 no motion1 yet /ill ! b[ 0 ex]t+ =ce & t5d+ to produce motion4 ,h5ce 9 def9+ =ce we mu/ say 8t :i* t5ds to produce motion10 :e!r motion %all actually result or n4 #d4 ,b =ces may also be recogniz$ by _! capability or t5d5cy to prev5t or to de/roy motion4 ,be=e ,i release ! arr[ ,i am consci\s ( ex]t+ a =ce ^u x 9 ord] to c\nt]act ! pull ( ! /r+4 ,"h my =ce is m]ely manife/$ by de/roy+ ! motion t1 if x 7 abs5t1 ! b[ w\ld produce4 ,s :5 ,i hold a weight 9 my h&1 ! =ce ex]t$ by my h& de/roys ! motion t ! weight w\ld acquire 7 ,i to let x fall2 & if a weight great] ?an ,i c\ld support 7 plac$ 9 my h&1 my ef=ts to su/a9 x w\ld /ill be prop]ly call$ =ce1 because !y t5d$ to de/roy motion1 ?\gh unsuccessfully4 ,we see by ^! simple cases t a =ce may be recogniz$ ei!r by produc+ motion or by try+ to produce x1 by de/roy+ motion or by t5d+ to de/roy x2 & h5ce ! propriety ( ! def9ition ( =ce mu/ be admitt$4 ,,! ,,measurem5t ,,( ,,=ce4 #e4 ,z =ces diff] 9 magnitude1 x becomes necess>y to e/abli% "s conv5i5t means ( express+ _! measurem5ts4 ,! pressure ex]t$ by "o p\nd weight at ,london is ! /&>d ) :i* we %all comp>e o!r =ces4 ,! piece ( iron or o!r sub/ance :i* is attract$ to ! e>? ) ?is =ce 9 ,london1 is attract$ to ! e>? ) a great] =ce at ! pole & a less =ce at ! equator2 h5ce1 9 ord] to def9e ! /&>d =ce1 we h to m5tion ! locality 9 :i* ! pressure ( ! weight is ex]t$4 ,x is easy to conceive h[ ! magnitude ( a pu%+ or a pull+ =ce may be describ$ z equival5t to s _m p\nds4 ,! =ce :i* ! muscles ( a man's arm c ex]t is measur$ by ! weight :i* he c lift4 ,if a weight be susp5d$ f an india-rubb] spr+1 x is evid5t ! spr+ w /ret* s t ! weight pulls ! spr+ & ! spr+ pulls ! weight2 h5ce ! numb] ( p\nds 9 ! weight is ! measure ( ! =ce ! spr+ is ex]t+4 ,9 e case ! magnitude ( a =ce c be describ$ by ! numb] ( p\nds express+ ! weight to :i* x is equival5t4 ,!re is ano!r b mu* m difficult mode ( measur+ =ce occasionally us$ 9 ! high] bran*es ( me*anics 7,art4 #dig71 b ! simpl] me?od is pref]able = our pres5t purpose4 ,illu/ration3 ,,fig4 #a4 #f4 ,! /raight l9e 9 :i* a =ce t5ds to move ! body to :i* x is appli$ is call$ ! direction ( ! =ce4 ,let u suppose1 = example1 t a =ce ( #c lbs4 is appli$ at ! po9t ,a1 ,fig4 #a1 t5d+ to make ,a move 9 ! direction ,,ab4 ,a /&>d l9e ,c ( c]ta9 l5g? is to be tak54 ,x is suppos$ t a l9e ( ?is l5g? repres5ts a =ce ( #a lb4 ,! l9e ,,ab is to be measur$1 equal to ?ree times ,c 9 l5g?1 & an arr[-head is to be plac$ ^u x to %[ ! direction 9 :i* ! =ce acts4 ,h5ce1 by means ( a l9e ( c]ta9 l5g? & direction1 & hav+ an arr[-head atta*$1 we are able completely to repres5t a =ce4 ,,equilibrium ,,( ,,two ,,=ces4 ,illu/ration3 ,,fig4 #b4 #g4 ,9 ,fig4 #b we h repres5t$ two equal weights to :i* /r+s are atta*$2 ^! /r+s1 aft] pass+ ov] pulleys1 are fa/5$ by a knot ,c4 ,! knot is pull$ by equal & opposite =ces4 ,i m>k (f p>ts ,,cd1 ,,ce1 to indicate ! =ces2 & s9ce !re is no reason :y ,c %\ld move to "o side m ?an ! o!r1 x rema9s at re/4 ,h5ce1 we le>n t two equal & directly oppos$ =ces c\nt]act ea* o!r1 & ea* may be reg>d$ z de/roy+ ! motion :i* ! o!r is /riv+ to produce4 ,if ,i make ! weights unequal by add+ to "o ( !m1 ! knot is no long] at re/2 x in/antly beg9s to move 9 ! direction ( ! l>g] =ce4 #h4 ,:5 two equal & opposite =ces act at a po9t1 !y are said to be 9 equilibrium4 ,m g5]ally ?is ^w is us$ ) ref]5ce to any set ( =ces :i* c\nt]act ea* o!r4 ,:5 a =ce acts ^u a body1 at lea/ "o m =ce mu/ be pres5t 9 ord] t ! body %\ld rema9 at re/4 ,if two =ces act+ on a po9t be n opposite1 !y w n be 9 equilibrium2 ?is is easily %[n by pull+ ! knot ,c 9 ,fig4 #b d[nw>ds4 ,:5 releas$1 x flies back aga94 ,?is proves t if two =ces be 9 equilibrium _! directions mu/ be opposite1 = o!rwise !y w produce motion4 ,we h already se5 t ! two =ces mu/ be equal4 ,a book ly+ on ! table is at re/4 ,?is book is act$ ^u by two =ces :i*1 be+ equal & opposite1 de/roy ea* o!r4 ,"o ( ^! =ces is ! gravitation ( ! e>?1 :i* t5ds to draw ! book d[nw>ds1 & :i* w\ld1 9 fact1 make ! book fall if x 7 n su/a9$ by an opposite =ce4 ,! pressure ( ! book on ! table is (t5 call$ ! action1 :ile ! resi/ance (f]$ by ! table is ! =ce ( reaction4 ,we "h see an illu/ration ( an important pr9ciple 9 nature1 :i* says t action & reaction are equal & opposite4 ,,equilibrium ,,( ,,?ree ,,=ces4 ,illu/ration3 ,,fig4 #c4 #i4 ,we n[ come to ! important case :]e ?ree =ces act on a po9t3 ?is is to be /udi$ by ! app>atus repres5t$ 9 ,fig4 #c4 ,x consi/s ess5tially ( two pulleys ,h1 ,h1 ea* ab\t #b diamet]1#a :i* are capable ( turn+ v freely on _! axles2 ! di/ance betwe5 ^! pulleys is ab\t #e'1 & !y are support$ at a height ( #f' by a frame1 :i* w easily be und]/ood f ! figure4 ,ov] ^! pulleys passes a f9e cord1 #i' or #aj' long1 hav+ a light hook at ea* ( ! ends ,e1 ,f4 ,to ! c5tre ( ?is cord ,d a %ort piece is atta*$1 :i* at its free end ,g is also furni%$ ) a hook4 ,a numb] ( iron weights1 #je lb41 #a lb41 #b lbs41 c41 ) r+s at ! top1 are us$2 "o or m ( ^! c easily be susp5d$ f ! hooks z occasion may require4 #a ,we %all (t51 9 ^! lectures1 repres5t feet or in*es 9 ! mann] usual among practical m5--#a' is "o foot1 #a is "o in*4 ,?us1 = example1 #c' #d is to be read 8?ree feet f\r in*es40 ,:5 x is necess>y to use fractions we %all always employ decimals4 ,= example1 #je is ! mode ( express+ a l5g? ( half an in*2 #c' #ai is to be read 8?ree feet "o in* & n9e-t5?s ( an in*40 #aj4 ,we comm5ce by plac+ "o p\nd on ea* ( ! hooks4 ,! cords are fir/ se5 to make a few oscillations & !n to settle into a def9ite position4 ,if we di/urb ! cords & try to move !m into "s new position !y w n rema9 !re2 :5 releas$ !y w return to ! places !y orig9ally occupi$4 ,we n[ conc5trate our att5tion on ! c5tral po9t ,d1 at :i* ! ?ree =ces act4 ,let ?is be repres5t$ by ,o 9 ,fig4 #d1 & ! l9es ,,op1 ,,oq1 & ,,os w be ! directions ( ! ?ree cords4 ,on exam9+ ^! positions we f9d t ! ?ree angles ,p ,o ,s1 ,q ,o ,s1 ,p ,o ,q1 are all equal4 ,?is may v easily be prov$ by hold+ beh9d ! cords a piece ( c>dbo>d on :i* ?ree l9es meet+ at a po9t & mak+ equal angles h be5 drawn2 x w !n be se5 t ! cords co9cide ) ! ?ree l9es on ! c>dbo>d4 ,illu/ration3 ,,fig4 #d4 #aa4 ,a little reflection w\ld h l$ u to anticipate ?is result4 ,= ! ?ree cords be+ ea* /ret*$ by a t5sion ( a p\nd1 x is obvi\s t ! ?ree =ces pull+ at ,o are all equal4 ,z ,o is at re/1 x seems obvi\s t ! ?ree =ces mu/ make ! angles equal1 = suppose t "o ( ! angles1 ,p ,o ,q = in/ance1 0 less ?an ei!r ( ! o!rs1 exp]im5t %[s t ! =ces ,o ,p & ,,oq w\ld be too /rong to be c\nt]act$ by ,o ,s4 ,! ?ree angles mu/ !re=e be equal1 & !n ! =ces are arrang$ symmetrically4 #ab4 ,! =ces be+ ea* #a lb41 m>k (f along ! ?ree l9es 9 ,fig4 #d 7:i* repres5t _! directions7 ?ree equal p>ts ,o ,p1 ,o ,q1 ,o ,s1 & place ! arr[heads to %[ ! direction 9 :i* ea* =ce is act+2 ! =ces are !n completely repres5t$ bo? 9 position & 9 magnitude4 ,s9ce ^! =ces make equilibrium1 ea* ( !m may be consid]$ to be c\nt]act$ by ! o!r two4 ,= example1 ,o ,s is annull$ by ,o ,q & ,o ,p4 ,b ,o ,s c\ld be balanc$ by a =ce ,o ,r equal & opposite to x4 ,h5ce ,,or is capable ( produc+ by itself ! same effect z ! =ces ,o ,p & ,,oq tak5 toge!r4 ,!re=e ,o ,r is equival5t to ,o ,p & ,,oq4 ,"h we le>n ! important tru? t two =ces n 9 ! same direction c be replac$ by a s+le =ce4 ,! process is call$ ! composition ( =ces1 & ! s+le =ce is call$ ! resultant ( ! two =ces4 ,o ,r is only "o p\nd1 yet x is equival5t to ! =ces ,o ,p & ,o ,q toge!r1 ea* ( :i* is also "o p\nd4 ,?is is because ! =ces ,o ,p & ,o ,q p>tly c\nt]act ea* o!r4 #ac4 ,draw ! l9es ,p ,r & ,q ,r2 !n ! angles ,p ,o ,r & ,q ,o ,r are equal1 because !y are ! supplem5ts ( ! equal angles ,p ,o ,s & ,q ,o ,s2 & s9ce ! angles ,p ,o ,r & ,q ,o ,r toge!r make up "o-?ird ( f\r "r angles1 x foll[s t ea* ( !m is two-?irds ( "o "r angle1 & !re=e equal to ! angle ( an equilat]al triangle4 ,also ,o ,p be+ equal to ,o ,q & ,o ,r common1 ! triangles ,o ,p ,r & ,o ,q ,r mu/ be equilat]al4 ,!re=e ! angle ,p ,r ,o is equal to ! angle ,r ,o ,q2 ?us ,p ,r is p>allel to ,o ,q2 simil>ly ,q ,r is p>allel to ,o ,p2 t is1 ,o ,p ,r ,q is a p>allelogram4 ,"h we fir/ p]ceive ! great law t ! resultant ( two =ces act+ at a po9t is ! diagonal ( a p>allelogram1 ( :i* !y are ! two sides4 #ad4 ,?is rem>kable geometrical figure is call$ ! p>allelogram ( =ces4 ,/at$ 9 its g5]al =m1 ! prop]ty we h discov]$ ass]ts t two =ces act+ at a po9t h a resultant1 & t ?is resultant is repres5t$ bo? 9 magnitude & 9 direction by ! diagonal ( ! p>allelogram1 ( :i* two adjac5t sides are ! l9es :i* repres5t ! =ces4 ,illu/ration3 ,,fig4 #e4 #ae4 ,! p>allelogram ( =ces may be illu/rat$ 9 v>i\s ways by means ( ! app>atus ( ,fig4 #c4 ,atta*1 = example1 to ! middle hook ,g #ae lb41 & place #a lb4 on ea* ( ! rema9+ hooks ,e1 ,f4 ,"h ! ?ree weights are n equal1 & symmetry w n enable u1 z x did 9 ! previ\s case1 to =esee ! condition :i* ! cords w assume2 b !y w be obs]v$ to settle 9 a def9ite position1 to :i* !y w inv>iably return if )drawn f x4 ,let ,o ,p1 ,o ,q 7,fig4 #e7 be ! directions ( ! cords2 ,o ,p & ,o ,q be+ ea* ( ! l5g? :i* corresponds to #a lb41 :ile ,o ,s corresponds to #ae lb4 ,"h1 z be=e1 ,o ,p & ,o ,q toge!r may be consid]$ to c\nt]act ,o ,s4 ,b ,o ,s c\ld h be5 c\nt]act$ by an equal & opposite =ce ,o ,r4 ,h5ce ,o ,r may be reg>d$ z ! s+le =ce equival5t to ,o ,p & ,o ,q1 t is1 z _! resultant2 & ?us x is prov$ exp]im5tally t ^! =ces h a resultant4 ,we c fur!r v]ify t ! resultant is ! diagonal ( ! p>allelogram ( :i* ! equal =ces are ! sides4 ,con/ruct a p>allelogram on a piece ( c>dbo>d hav+ its f\r sides equal1 & "o ( ! diagonals half z long aga9 z "o ( ! sides4 ,?is may be done v easily by fir/ draw+ "o ( ! two triangles into :i* ! diagonal divides ! p>allelogram4 ,! diagonal is to be produc$ beyond ! p>allelogram 9 ! direction ,o ,s4 ,:5 ! c>dbo>d is plac$ close aga9/ ! cords1 ! two cords w lie 9 ! directions ,o ,p1 ,o ,q1 :ile ! produc$ diagonal w be 9 ! v]tical ,o ,s4 ,?us ! application ( ! p>allelogram ( =ce is v]ifi$4 ,illu/ration3 ,,fig4 #f4 #af4 ,! same exp]im5t %[s t two unequal =ces may be comp\nd$ into "o resultant4 ,= 9 ,fig4 #e ! two =ces ,o ,p & ,o ,s may be consid]$ to be c\nt]balanc$ by ! =ce ,o ,q2 9 o!r words1 ,o ,q mu/ be equal & opposite to a =ce :i* is ! resultant ( ,o ,p & ,o ,s4 #ag4 ,let u place on ! c5tral hook ,g a weight ( #e lbs41 & weights ( #c lbs4 on ! hook ,e & #d lbs4 on ,f4 ,?is is actually ! case %[n 9 ,fig4 #c4 ,! weights be+ unequal1 we _c imm$iately inf] any?+ ) ref]5ce to ! position ( ! cords1 b /ill we f9d1 z be=e1 t ! cords assume a def9ite position1 to :i* !y return :5 tempor>ily displac$4 ,let ,fig4 #f repres5t ! positions ( ! cords4 ,no two ( ! angles are 9 ?is case equal4 ,/ill ea* ( ! =ces is c\nt]balanc$ by ! o!r two4 ,ea* is !re=e equal & opposite to ! resultant ( ! o!r two4 ,con/ruct ! p>allelogram on c>dbo>d1 z c be easily done by =m+ ! triangle ,o ,p ,r1 ^: sides are #c1 #d1 & #e1 & !n draw+ ,o ,q & ,r ,q p>allel to ,r ,p & ,o ,p4 ,produce ! diagonal ,o ,r to ,s4 ,?is p>allelogram be+ plac$ beh9d ! cords1 y see t ! directions ( ! cords co9cide ) its sides & diagonal1 ?us v]ify+ ! p>allelogram ( =ces 9 a case :]e all ! =ces are ( diff]5t magnitudes4 #ah4 ,x is easy1 by ! application ( a set squ>e1 to prove t 9 ?is case ! cords atta*$ to ! #c lb4 & #d lb4 weights are at "r angles to ea* o!r4 ,we c\ld h inf]r$1 f ! p>allelogram ( =ce1 t ?is mu/ be ! case1 = ! sides ( ! triangle ,o ,p ,r are #c1 #d1 & #e respectively1 & s9ce ! squ>e ( #e is #be1 & ! squ>es ( #c & ( #d are #i & #af respectively1 x foll[s t ! squ>e ( "o side ( ?is triangle is equal to ! sum ( ! squ>es ( ! two opposite sides1 & !re=e ?is is a "r-angl$ triangle 7,euclid1 i4 #dh74 ,h5ce1 s9ce ,p ,r is p>allel to ,o ,q1 ! angle ,p ,o ,q mu/ also be a "r angle4 ,a ,,small ,,=ce ,,sometimes ,,balances ,,two ,,l>g] ,,=ces4 #ai4 ,cases might be multipli$ indef9itely by plac+ v>i\s am\nts ( weight on ! hooks1 con/ruct+ ! p>allelogram on c>dbo>d1 & comp>+ x ) ! cords z be=e4 ,we %all1 h[ev]1 conf9e ourselves to "o m illu/ration1 :i* is capable ( v rem>kable applications4 ,atta* #a lb4 to ea* ( ! hooks ,e & ,f2 ! cord jo9+ !m rema9s /raight until drawn d[n by plac+ a weight on ! c5tre hook4 ,a v small weight w suffice to d ?is4 ,let u put on half-a-p\nd2 ! position ! cords !n assume is indicat$ 9 ,fig4 #g4 ,z be=e1 ea* =ce is equal & opposite to ! resultant ( ! o!r two4 ,h5ce a =ce ( half-a-p\nd is ! resultant ( two =ces ea* ( #a lb4 ,! app>5t p>adox is expla9$ by notic+ t ! =ces ( #a lb4 are v ne>ly opposite1 & !re=e to a l>ge ext5t c\nt]act ea* o!r4 ,con/ruct+ ! c>dbo>d p>allelogram we may easily v]ify t ! pr9ciple ( ! p>allelogram ( =ces holds 9 ?is case also4 ,illu/ration3 ,,fig4 #g4 #bj4 ,no matt] h[ small be ! weight we susp5d f ! middle ( a horizontal cord1 y see t ! cord is deflect$3 & no matt] h[ great a t5sion 7 appli$1 x w\ld be impossible to /raight5 ! cord4 ,! cord c\ld break1 b x c\ld n aga9 become horizontal4 ,look at a telegraph wire2 x is nev] 9 a /raight l9e betwe5 two consecutive poles1 & its curv$ =m is m evid5t ! great] be ! di/ance betwe5 ! poles4 ,b 9 putt+ up a telegraph wire great /ra9+ =ce is us$1 by means ( special ma*9es = ! purpose2 yet ! wires _c be /raight5$3 because ! weight ( ! heavy wire itself acts z a =ce pull+ x d[nw>ds4 ,j z ! cord 9 our exp]im5ts _c be /raight :5 any =ce1 h[ev] small1 is pull+ x d[nw>ds at ! c5tre1 s x is impossible by any ex]tion ( =ce to /raight5 ! long wire4 ,"s fur!r illu/rations ( ?is pr9ciple w be giv5 9 our next lecture1 & ) "o application ( x ! pres5t w be conclud$4 #ba4 ,"o ( ! mo/ important practical problems 9 me*anics is to make a small =ce ov]come a great]4 ,!re are a numb] ( ways 9 :i* ?is may be accompli%$ = diff]5t purposes1 & to ! consid]ation ( !m sev]al lectures ( ?is c\rse w be devot$4 ,p]haps1 h[ev]1 !re is no arrangem5t m simple ?an t :i* is furni%$ by ! pr9ciples we h be5 consid]+4 ,we %all employ x to raise a #bh lb4 weight by means ( a #b lb4 weight4 ,i d n say t ?is p>ticul> application is ( mu* practical use4 ,i %[ x to y r z a rem>kable d$uction f ! p>allelogram ( =ces ?an z a useful ma*9e4 ,illu/ration3 ,,fig4 #h4 ,a rope is atta*$ at "o end ( an upright1 ,a 7,fig4 #h71 & passes ov] a pulley ,b at ! same v]tical height ab\t #af' di/ant4 ,a weight ( #bh lbs4 is fa/5$ to ! free end ( ! rope1 & ! supports mu/ be heavily weight$ or o!rwise secur$ f mov+4 ,! rope ,,ab is app>5tly /raight & horizontal1 9 consequ5ce ( its weight be+ inappreciable 9 comp>ison ) ! /ra9 7#bh lbs47 to :i* x is subject$2 ?is position is indicat$ 9 ! figure by ! dott$ l9e ,,ab4 ,we n[ susp5d f ,c at ! middle ( ! rope a weight ( #b lbs4 ,in/antly ! rope moves to ! position repres5t$ 9 ! figure4 ,b ?is x _c d )\t at ! same mom5t rais+ slightly ! #bh lbs41 =1 s9ce two sides ( a triangle1 ,,cb1 ,,ca1 are great] ?an ! ?ird side1 ,,ab1 m ( ! rope mu/ lie betwe5 ! supports :5 x is b5t d[n by ! #b lb4 weight ?an :5 x 0 /raight4 ,b ?is c only h tak5 place by %ort5+ ! rope betwe5 ! pulley ,b & ! #bh lb4 weight1 = ! rope is firmly secur$ at ! o!r end4 ,! effect on ! heavy weight is s small t x is h>dly visible to y f a di/ance4 ,we c1 h[ev]1 easily %[ by an electrical arrangem5t t ! big weight has be5 rais$ by ! little "o4 #bb4 ,:5 an electric curr5t passes ?r\gh ?is al>um y he> ! bell r+1 & ! mom5t ,i /op ! curr5t ! bell /ops4 ,i h fa/5$ "o piece ( brass to ! #bh lb4 weight1 & ano!r to ! support close above x1 b unless ! weight be rais$ a little ! two w n be 9 contact2 ! electricity is int5d$ to pass f "o ( ^! pieces ( brass to ! o!r1 b x _c pass unless !y are t\*+4 ,:5 ! rope is /raight ! two pieces ( brass are sep>at$1 ! curr5t does n pass1 & our al>um is dumb2 b ! mom5t ,i hang on ! #b lb4 weight to ! middle ( ! rope x raises ! weight a little1 br+s ! pieces ( brass 9 contact1 & n[ y all he> ! al>um4 ,on remov+ ! #b lbs4 ! curr5t is int]rupt$ & ! noise ceases4 #bc4 ,i am sure y mu/ all h notic$ t ! #b lb4 weight desc5d$ ?r\gh a di/ance ( _m in*es1 easily visible to all ! room2 t is to say1 ! small weight mov$ ?r\gh a v consid]able di/ance1 :ile 9 s do+ x only rais$ ! l>g] "o a v small di/ance4 ,?is is a po9t ( ! v greate/ importance2 ,i !re=e take ! fir/ opportunity ( call+ y\r att5tion to x4 ,,lecture ,,ii4 ,,! ,,resolution ,,( ,,=ces4 ,introduction4--,"o ,=ce resolv$ into ,two ,=ces4--,exp]im5tal ,illu/rations4--,sail+4--,"o ,=ce resolv$ into ,?ree ,=ces n 9 ! same ,plane4--,! ,jib & ,tie-rod4 ,,introduction4 ,illu/ration3 ,,fig4 #i4 #bd4 ,z ! la/ lecture 0 pr9cipally conc]n$ ) discuss+ h[ "o =ce c\ld replace two =ces1 s 9 ! pres5t we %all exam9e ! conv]se "q1 ,h[ may two =ces replace "o =ce8 ,s9ce ! diagonal ( a p>allelogram repres5ts a s+le =ce equival5t to ^? repres5t$ by ! sides1 x is obvi\s t "o =ce may be resolv$ into two o!rs1 provid$ x be ! diagonal ( ! p>allelogram =m$ by !m4 #be4 ,we %all frequ5tly employ 9 ! pres5t lecture1 & 9 "s ( ^? t foll[1 ! spr+ balance1 :i* is repres5t$ 9 ,fig4 #i3 ! weight is atta*$ to ! hook1 & :5 ! balance is susp5d$ by ! r+1 a po9t] indicates ! numb] ( p\nds on a scale4 ,?is balance is v conv5i5t = %[+ ! /ra9 along a cord2 = ?is purpose ! balance is held by ! r+ :ile ! cord is atta*$ to ! hook4 ,x w be notic$ t ! balance has two r+s & two correspond+ hooks4 ,! hook & r+ at ! top & bottom w weigh up to #cjj lbs41 correspond+ to ! scale :i* is se54 ,! hook & r+ at ! side correspond to ano!r scale on ! o!r face ( ! plate3 ?is second scale weighs up to ab\t #ej lbs41 consequ5tly = a weight "u #ej lbs4 ! side hook & r+ are employ$1 z !y give a m accurate result ?an w\ld be obta9$ by ! top & bottom hook & r+1 :i* are int5d$ = l>g] weights4 ,^! +5i\s & useful balances are suffici5tly accurate1 & c easily be te/$ by rais+ kn[n weights4 ,besides ! in/rum5t ?us describ$1 we %all sometimes use "o ( a small] size1 & we %all be able ) ?is aid to trace ! exi/5ce & magnitude ( =ces 9 a mo/ conv5i5t mann]4 ,,"o ,,=ce ,,resolv$ ,,into ,,two ,,=ces4 #bf4 ,we %all fir/ illu/rate h[ a s+le =ce may be resolv$ into a pair ( =ces2 = ?is purpose we %all use ! arrangem5t %[n 9 ,fig4 #aj 7see next page74 ,! ends ( a cord are fa/5$ to two small spr+ balances2 to ! c5tre ,e ( ?is cord a weight ( #d lbs4 is atta*$4 ,at ,a & ,b are pegs f :i* ! balances c be susp5d$4 ,let ! di/ances ,,ae1 ,,be be ea* #ab1 & ! di/ance ,,ab #af4 ,:5 ! cord is ?us plac$1 & ! weight all[$ to hang freely1 ea* ( ! cords ,,ea1 ,,eb is /ra9$ by an am\nt ( =ce t is %[n to be v ne>ly #c lbs4 by ! balances4 ,b ! weight ( #d lbs4 is ! only weight act+2 h5ce x mu/ be equival5t to two =ces ( v ne>ly #c lbs4 ea* along ! directions ,,ae & ,,be4 ,"h ! two =ces to :i* #d lbs4 is equival5t are ea* ( !m less ?an #d lbs41 ?\gh tak5 toge!r !y exce$ x4 ,illu/ration3 ,,fig4 #aj4 #bg4 ,b remove ! cords f ,,ab & hang !m on ,,cd1 ! l5g? ,,cd be+ #a' #aj1 !n ! =ces %[n along ,,fc & ,d are ea* #e lbs42 "h1 !re=e1 "o =ce ( #d lbs4 is equival5t to two =ces ea* ( #e lbs4 ,9 ! la/ lecture 7,art4 #ai7 we saw t "o =ce c\ld balance two great] =ces2 "h we see ! analog\s case ( "o =ce be+ *ang$ into two great] =ces4 ,fur!r1 we le>n t ! numb] ( pairs ( =ces into :i* "o =ce may be decompos$ is unlimit$1 = ) e diff]5t di/ance betwe5 ! pegs diff]5t =ces w be indicat$ by ! balances4 ,:5ev] ! weight is susp5d$ f a po9t half-way betwe5 ! balances1 ! =ces along ! cords are equal2 b by plac+ ! weight ne>] "o balance ?an ! o!r1 a great] =ce w be indicat$ on t balance to :i* ! weight is ne>e/4 ,,exp]im5tal ,,illu/rations4 ,illu/ration3 ,,fig4 #aa4 #bh4 ,! resolution or decomposition ( "o =ce into two =ces ea* great] ?an itself is capable ( be+ illu/rat$ 9 a v>iety ( ways1 two ( :i* w be "h expla9$4 ,9 ,fig4 #aa an arrangem5t = ?is purpose is %[n4 ,a piece ( /\t tw9e ,,ab1 able to support f #bj lbs4 to #cj lbs41 is fa/5$ at "o end ,a to a fix$ support1 & at ! o!r end ,b to ! eye ( a wire-/ra9]4 ,a wire-/ra9] consi/s ( an iron rod1 ) an eye at "o end & a screw & a nut at ! o!r2 x is us$ = tight5+ wires 9 wire f5c+2 & is employ$ 9 ?is case = ! purpose ( /ret*+ ! cord4 ,?is be+ done1 ,i take a piece ( ord9>y sew+-?read1 :i* is ( c\rse weak] ?an ! /\t tw9e4 ,i tie ! ?read to ! middle ( ! cord at ,c1 cat* ! o!r end 9 my f+]s1 & pull2 some?+ mu/ break--some?+ has brok53 b :at has brok58 ,n ! slight ?read1 x is /ill :ole2 x is ! cord :i* has snapp$4 ,n[ ?is illu/rates ! po9t on :i* we h be5 dwell+4 ,! =ce :i* ,i transmitt$ along ! ?read 0 insuffici5t to break x2 ! ?read transf]r$ ! =ce to ! cord1 b "u su* circum/ances t ! =ce 0 greatly magnifi$1 & ! consequ5ce 0 t ?is magnifi$ =ce 0 able to break ! cord be=e ! orig9al =ce c\ld break ! ?read4 ,we c also see :y x 0 necess>y to /ret* ! cord4 ,9 ,fig4 #aj ! /ra9s along ! cords are great] :5 ! cords are atta*$ at ,c & ,d ?an :5 !y are atta*$ at ,a & ,b2 t is to say1 ! m ! cord is /ret*$ t[>ds a /raight l9e1 ! great] are ! =ces into :i* ! appli$ =ce is resolv$4 #bi4 ,we give a second example1 9 illu/ration ( ! same pr9ciple4 ,9 ,fig4 #ab is %[n a *a9 #h' long1 "o end ( :i* ,b is atta*$ to a wire-/ra9]1 :ile ! o!r end is fa/5$ to a small piece ( p9e ,a1 :i* is #je squ>e 9 section1 & #e long betwe5 ! two upright irons by :i* x is support$4 ,by means ( ! nut ( ! wire-/ra9] ,i /raight5 ! *a9 z ,i did ! /r+ ( ,fig4 #aa1 & = ! same reason4 ,i !n put a piece ( tw9e r\nd ! *a9 & pull x g5tly4 ,! /ra9 br\ght to be> on ! wood is s great t x breaks across4 ,"h1 ! small =ce ( a few p\nds1 transmitt$ to ! *a9 by pull+ ! sir+1 is magnifi$ to upw>ds ( a hundr$weight1 = less ?an ?is w\ld n break ! wood4 ,! explanation is precisely ! same z :5 ! /r+ 0 brok5 by ! ?read4 ,illu/ration3 ,,fig4 #ab4 ,,sail+4 #cj4 ,! action ( ! w9d ^u ! sails ( a vessel af=ds a v in/ructive & useful example ( ! decomposition ( =ces4 ,by ! p>allelogram ( =ces we are able to expla9 h[ x is t a vessel is able ev5 to sail aga9/ ! w9d4 ,a =ce is t :i* t5ds to produce motion1 & motion g5]ally takes place 9 ! l9e ( ! =ce4 ,9 ! case ( ! action ( w9d on a vessel ?r\gh ! m$ium ( ! sails1 we h motion produc$ :i* is n necess>ily 9 ! direction ( ! w9d1 & :i* may be to a c]ta9 ext5t oppos$ to x4 ,?is app>5t p>adox requires "s elucidation4 ,illu/ration3 ,,fig4 #ac4 #ca4 ,let u fir/ suppose ! w9d to be bl[+ 9 a direction %[n by ! arr[s ( ,fig4 #ac1 p]p5dicul> to ! l9e ,,ab 9 :i* ! %ip's c\rse lies4 ,9 :at direction mu/ ! sail be set8 ,x is cle> t ! sail mu/ n be plac$ along ! l9e ,,ab1 = !n ! only effect ( ! w9d w\ld be to bl[ ! vessel sideways2 nor c\ld ! sail be plac$ ) its edge to ! w9d1 t is1 along ! l9e ,o ,w1 = !n ! w9d w\ld m]ely glide along ! sail )\t produc+ a propell+ =ce4 ,let1 !n1 ! sail be plac$ betwe5 ! two positions1 z 9 ! direction ,p ,q4 ,! l9e ,o ,w repres5ts ! magnitude ( ! =ce ( ! w9d press+ on ! sail4 ,we %all suppose = simplicity t ! sail ext5ds on bo? sides ( ,o4 ,?r\gh ,o draw ,o ,r p]p5dicul> to ,p ,q1 & f ,w let fall ! p]p5dicul> ,w ,x on ,p ,q1 & ,w ,r on ,o ,r4 ,by ! pr9ciple ( ! p>allelogram ( =ces1 ! =ce ,o ,w may be decompos$ into ! two =ces ,o ,x & ,o ,r1 s9ce ^! are ! sides ( ! p>allelogram ( :i* ,o ,w1 ! =ce ( ! w9d1 is ! diagonal4 ,we may !n leave ,o ,w out ( consid]ation1 & imag9e ! =ce ( ! w9d to be replac$ by ! pair ( =ces ,o ,x & ,o ,r2 b ! =ce ,o ,x _c produce an effect1 x m]ely repres5ts a =ce :i* glides along ! surface ( ! sail1 n "o :i* pu%es aga9/ x2 s f> z ?is compon5t goes1 ! sail has its edge t[>ds x1 & !re=e ! =ce produces no effect4 ,on ! o!r h&1 ! sail is p]p5dicul> to ! =ce ,o ,r1 & ?is is !re=e ! effici5t compon5t4 ,! =ce ( ! w9d is ?us measur$ by ,o ,r1 bo? 9 magnitude & direction3 ?is =ce repres5ts ! actual pressure on ! ma/ produc$ by ! sail1 & f ! ma/ communicat$ to ! %ip4 ,/ill ,o ,r is n 9 ! direction 9 :i* ! %ip is sail+3 we mu/ aga9 decompose ! =ce 9 ord] to f9d its useful effect4 ,?is is done by draw+ ?r\gh ,r ! l9es ,r ,l & ,r ,m p>allel to ,o ,a & ,o ,w1 ?us =m+ ! p>allelogram ,o ,m ,r ,l4 ,h5ce1 by ! p>allelogram ( =ces1 ! =ce ,o ,r is equival5t to ! two =ces ,o ,l & ,o ,m4 ,! effect ( ,o ,l ^u ! vessel is to propel x 9 a direction p]p5dicul> to t 9 :i* x is sail+4 ,we mu/1 !re=e1 endeav\r to c\nt]act ?is =ce z f> z possible4 ,?is is accompli%$ by ! keel1 & ! =m ( ! %ip is s design$ z to pres5t ! greate/ possible resi/ance to be+ pu%$ sideways ?r\gh ! wat]3 ! deep] ! keel ! m completely is ! effect ( ,o ,l annull$4 ,/ill ,o ,l w\ld 9 all cases produce "s leeway 7 x n = ! rudd]1 :i*1 by turn+ ! head ( ! vessel a little t[>ds ! w9d1 makes h] sail 9 a direction suffici5tly to w9dw>d to c\nt]act ! small effect ( ,o ,l 9 driv+ h] to leew>d4 ,?us ,o ,l is dispos$ (1 & ! only =ce rema9+ is ,o ,m1 :i* acts directly to pu% ! vessel 9 ! requir$ direction4 ,"h1 !n1 we see h[ ! w9d1 aid$ by ! resi/ance ( ! wat]1 is able to make ! vessel move 9 a direction p]p5dicul> to t 9 :i* ! w9d bl[s4 ,we h se5 t ! sail mu/ be set some:]e betwe5 ! direction ( ! w9d & t ( ! %ip's motion4 ,x c be prov$ t :5 ! direction ( ! sail suppos$ to be flat & v]tical1 is su* z to bisect ! angle ,w ,o ,b1 ! magnitude ( ! =ce ,o ,m is great] ?an :5 ! sail has any o!r position4 #cb4 ,! same pr9ciples %[ h[ a vessel is able to sail aga9/ ! w9d3 %e _c1 ( c\rse1 sail /raight aga9/ x1 b %e c sail )9 half a "r angle ( x1 or p]haps ev5 less4 ,?is c be se5 f ,fig4 #ad4 ,! small arr[s repres5t ! w9d1 z be=e4 ,let ,o ,w be ! l9e p>allel to !m1 :i* measures ! =ce ( ! w9d1 & let ! sail be plac$ along ! l9e ,p ,q2 ,o ,w is decompos$ into ,o ,x & ,o ,y1 ,o ,x m]ely glides along ! sail1 & ,o ,y is ! effective =ce4 ,?is is decompos$ into ,o ,l & ,o ,m2 ,o ,l is c\nt]act$1 z already expla9$1 & ,o ,m is ! =ce t propels ! vessel onw>ds4 ,h5ce we see t !re is a =ce act+ to pu% ! vessel onw>ds1 ev5 ?\gh ! movem5t be p>tly aga9/ ! w9d4 ,illu/ration3 ,,fig4 #ad4 ,x w be notic$ 9 ?is case t ! =ce ,o ,l act+ to leew>ds exce$s ,o ,m pu%+ onw>ds4 ,h5ce x is t vessels ) a v deep keel1 & !re=e oppos+ v great resi/ance to mov+ leew>ds1 c sail m closely to ! w9d ?an o!rs n s con/ruct$2 a vessel %\ld be =m$ s t %e %all move z freely z possible 9 ! direction ( h] l5g?1 = :i* reason %e is %>p5$ at ! b[1 & o!rwise %ap$ = glid+ ?r\gh ! wat] easily2 ?is is 9 ord] t ,o ,m may h to ov]come z little resi/ance z possible4 ,if ! sail 7 flat & v]tical x %\ld bisect ! angle ,a ,,ow = ! w9d to act 9 ! mo/ effici5t mann]4 ,s9ce1 !n1 a vessel c sail t[>ds ! w9d1 x foll[s t1 by tak+ a zigzag c\rse1 %e c proce$ f "o port to ano!r1 ev5 ?\gh ! w9d be bl[+ f ! place to :i* %e w\ld g t[>ds ! place f :i* %e comes4 ,?is well kn[n manuvre is call$ 8tack+40 ,y w und]/& t 9 a sail+-vessel ! rudd] has a m important "p to play ?an 9 a /eam]3 9 ! latt] x is only useful = *ang+ ! direction ( ! vessel's motion1 :ile 9 ! =m] x is n only necess>y = *ang+ ! direction1 b mu/ also be us$ to keep ! vessel to h] c\rse by c\nt]act+ ! effect ( leeway4 ,,"o ,,=ce ,,resolv$ ,,into ,,?ree ,,=ces ,,n ,,9 ,,! ,,same ,,plane4 ,illu/ration3 ,,fig4 #ae4 #cc4 ,up to ! pres5t we h only be5 consid]+ =ces :i* lie 9 ! same plane1 b 9 nature we meet ) =ces act+ 9 all directions1 & !re=e we mu/ n be satisfi$ ) conf9+ our inquiries to ! simpl] case4 ,we proce$ to %[1 9 two diff]5t ways1 h[ a =ce c be decompos$ into ?ree =ces n 9 ! same plane1 ?\gh pass+ ?r\gh ! same po9t4 ,! fir/ mode ( do+ s is z foll[s4 ,to ?ree po9ts ,a1 ,b1 ,c 7,fig4 #ae7 ?ree spr+ balances are atta*$2 ,a1 ,b1 ,c are n 9 ! same /raight l9e1 ?\gh !y are at ! same v]tical height3 to ! spr+ balances cords are atta*$1 :i* unite 9 a po9t ,o1 f :i* a weight ,w is susp5d$4 ,?is weight is support$ by ! ?ree cords1 & ! /ra9s along ^! cords are indicat$ by ! spr+ balances4 ,! greate/ /ra9 is on ! %orte/ cord & ! lea/ /ra9 on ! longe/4 ,"h ! =ce ,w lbs4 produces ?ree =ces :i*1 tak5 toge!r1 exce$ its own am\nt4 ,if ,i add an equal weight ,w1 ,i f9d1 z we might h anticipat$1 t ! /ra9s indicat$ by ! scales are precisely d\ble :at !y 7 be=e4 ,?us we see t ! proportion ( ! =ce to ea* ( ! compon5ts into :i* x is decompos$ does n dep5d on ! actual magnitude ( ! =ce1 b on ! relative direction ( ! =ce & its compon5ts4 ,illu/ration3 ,,fig4 #af4 #cd4 ,ano!r mode ( %[+ ! decomposition ( "o =ce into ?ree =ces n 9 ! same plane is repres5t$ 9 ,fig4 #af4 ,! tripod is =m$ ( ?ree /rips ( p9e1 #d' #je #je1 secur$ by a piece ( wire runn+ ?r\gh ea* at ! top2 "o end ( ?is wire hangs d[n1 & c>ries a hook to :i* is atta*$ a weight ( #bh lbs4 ,?is weight is support$ by ! wire1 b ! /ra9 on ! wire mu/ be borne by ! ?ree wood5 rods3 h5ce !re is a =ce act+ d[nw>ds ?r\gh ! wood5 rods4 ,we _c r5d] ?is manife/ by a contrivance l ! spr+ scales1 because x is a pu% in/ead ( a pull4 ,h[ev]1 by rais+ "o ( ! legs ,i at once become aw>e t !re is a =ce act+ d[nw>ds ?r\gh x4 ,! weight is1 !n1 decompos$ into ?ree =ces1 :i* act d[nw>ds ?r\gh ! legs2 ^! ?ree =ces are n 9 a plane1 & ! ?ree =ces tak5 toge!r are l>g] ?an ! weight4 #ce4 ,! tripod is (t5 us$ = support+ weights2 x is conv5i5t on acc\nt ( its portability1 & x is v /eady4 ,y may judge ( its /r5g? by ! model repres5t$ 9 ! figure1 = ?\gh ! legs are v slight1 yet !y support v securely a consid]able weight4 ,! pulleys by means ( :i* gigantic weights are rais$ are (t5 support$ by colossal tripods4 ,!y possess /ability & /ead9ess 9 addition to great /r5g?4 #cf4 ,an important po9t may be br\ght out by contra/+ ! arrangem5ts ( ,figs4 #ae & #af4 ,9 ! "o case ?ree cords are us$1 & 9 ! o!r ?ree rods4 ,?ree rods w\ld h answ]$ = bo?1 b ?ree cords w\ld n h done = ! tripod4 ,9 "o ! cords are /ra9$1 & ! t5d5cy ( ! /ra9 is to break ! cords1 b 9 ! o!r ! nature ( ! =ce d[n ! rods is entirely diff]5t2 x does n t5d to pull ! rod asund]1 x is try+ to cru% ! rod1 & _h ! weight be5 l>ge en\gh ! rods w\ld b5d & break4 ,i hold "o end ( a p5cil 9 ea* h& & !n try to pull ! p5cil asund]2 ! p5cil is 9 ! condition ( ! cords ( ,fig4 #ae2 b if in/ead ( pull+ ,i pu% my h&s toge!r1 ! p5cil is l ! rods 9 ,fig4 #af4 #cg4 ,?is di/9ction is ( great importance 9 me*anics4 ,a rod or cord 9 a /ate ( t5sion is call$ a 8tie02 :ile a rod 9 a /ate ( compression is call$ a 8/rut40 ,s9ce a rod c resi/ bo? t5sion & compression x c s]ve ei!r z a tie or z a /rut1 b a cord or *a9 c only act z a tie4 ,a pill> is always a /rut1 z ! sup]9cumb5t load makes x to be 9 a /ate ( compression4 ,^! di/9ctions w be v frequ5tly us$ dur+ ?is c\rse ( lectures1 & x is necess>y t !y be ?or\ghly und]/ood4 ,,! ,,jib ,,& ,,tie ,,rod4 #ch4 ,z an illu/ration ( ! nature ( ! 8tie0 & 8/rut10 & also = ! purpose ( giv+ a useful example ( ! decomposition ( =ces1 ,i use ! app>atus ( ,fig4 #ag 7see next page74 ,x repres5ts ! pr9ciple ( ! framework 9 ! common lift+ crane1 & has num]\s applications 9 practical me*anics4 ,a rod ( wood ,b ,c #c' #f long & #a #a section is capable ( turn+ r\nd its support at ! bottom ,b by means ( a jo9t or h+e3 ?is rod is call$ ! 8jib02 x is held at its upp] end by a tie ,a ,c #c' long1 :i* is atta*$ to ! support above ! jo9t4 ,a ,b is "o foot long4 ,f ! po9t ,c a wire desc5ds1 hav+ a hook at ! end on :i* a weight c be hung4 ,! tie is atta*$ to ! spr+ balance1 ! index ( :i* %[s ! /ra94 ,! spr+ balance is secur$ by a wire-/ra9]1 by turn+ ! nut ( :i* ! l5g? ( ! wire c be %ort5$ or l5g!n$ z occasion requires4 ,?is is necess>y1 because :5 diff]5t weights are susp5d$ f ! hook ! spr+ is /ret*$ m or less1 & ! screw is !n employ$ to keep ! entire l5g? ( ! tie at #c'4 ,! rema9d] ( ! tie consi/s ( copp] wire4 #ci4 ,suppose a weight ( #bj lbs4 be susp5d$ f ! hook ,w1 x endeav\rs to pull ! top ( ! jib d[nw>ds2 b ! tie holds x back1 consequ5tly ! tie is put into a /ate ( t5sion1 z inde$ its "n signifies1 & ! magnitude ( t t5sion is %[n to be #fj lbs4 by ! spr+ balance4 ,"h we f9d aga9 :at we h already s (t5 ref]r$ to2 namely1 "o =ce develop+ ano!r =ce t is great] ?an itself1 = ! /ra9 along ! tie is ?ree times z great z ! /ra9 9 ! v]tical wire by :i* x 0 produc$4 ,illu/ration3 ,,fig4 #ag4 #dj4 ,:at is ! condition ( ! jib8 ,x is evid5tly be+ pu%$ d[nw>ds on its jo9t at ,b2 x is !re=e 9 a /ate ( compression2 x is a /rut4 ,?is w be evid5t if we ?9k = a mom5t h[ absurd x w\ld be to endeav\r to replace ! jib by a /r+ or *a93 ! :ole arrangem5t w\ld collapse4 ,! weight ( #bj lbs4 is !re=e decompos$ by ?is contrivance into two o!r =ces1 "o ( :i* is resi/$ by a tie & ! o!r by a /rut4 ,illu/ration3 ,,fig4 #ah4 #da4 ,we h no means ( %[+ ! magnitude ( ! /ra9 along ! /rut1 b we %all prove t x c be comput$ by means ( ! p>allelogram ( =ce2 ?is w also expla9 h[ x is t ! tie is /ra9$ by a =ce ?ree times t ( ! weight :i* is us$4 ,?r\gh ,c 7,fig4 #ah7 draw ,c ,p p>allel to ! tie ,a ,b1 & ,p ,q p>allel to ! /rut ,c ,b !n ,b ,p is ! diagonal ( ! p>allelogram ^: sides are ea* equal to ,b ,c & ,b ,q4 ,if !re=e we consid] ! =ce ( #bj lbs4 to be repres5t$ by ,b ,p1 ! two =ces into :i* x is decompos$ w be %[n by ,b ,q & ,b ,c2 b ,a ,b is equal to ,b ,q1 s9ce ea* ( !m is equal to ,c ,p2 also ,b ,p is equal to ,a ,c4 ,h5ce ! weight ( #bj lbs4 be+ repres5t$ by ,a ,c1 ! /ra9 along ! tie w be repres5t$ by ! l5g? ,a ,b1 & t along ! /rut by ! l5g? ,b ,c4 ,rememb]+ t ,a ,b is #c' long1 ,c ,b #c' #f1 & ,a ,c #a'1 x foll[s t ! /ra9 along ! tie is #fj lbs41 & along ! /rut #gj lbs41 :5 ! weight ( #bj lbs4 is susp5d$ f ! hook4 #db4 ,9 e o!r case ! /ra9s along ! tie & /rut c be det]m9$1 :5 ! susp5d$ weight is kn[n1 by _! proportionality to ! sides ( ! triangle =m$ by ! tie1 ! jib1 & ! upright po/1 respectively4 #dc4 ,9 ?is contrivance y w recognize1 no d\bt1 ! framework ( ! common lift+ crane1 b t v ess5tial portion ( ! crane :i* provides = ! rais+ & l[]+ is n %[n "h4 ,to ?is we %all return aga9 9 a subsequ5t lecture 7,art4 #ccb74 ,y w ( c\rse und]/& t ! tie rod we h be5 consid]+ is entirely diff]5t f ! *a9 = rais+ ! load4 #dd4 ,x is easy to see ( :at importance to ! eng9e] ! in=mation acquir$ by means ( ! decomposition ( =ces may become4 ,?us 9 ! simple case ) :i* we are at pres5t engag$1 suppose an eng9e] 7 requir$ to erect a frame :i* 0 to su/a9 a weight ( #aj tons1 let u see h[ he w\ld be enabl$ to det]m9e ! /r5g? ( ! tie & jib4 ,x is ( importance 9 design+ any /ructure n to make any "p unnecess>ily /rong1 z do+ s involves a wa/e ( valuable mat]ial1 b x is ( /ill m vital importance to make e "p /rong en\gh to avoid ! risk ( accid5t1 n only "u ord9>y circum/ances1 b also "u ! exceptionally great %ocks & /ra9s to :i* e ma*9e is liable4 #de4 ,accord+ to ! num]ical proportions we h employ$ = illu/ration1 ! /ra9 along ! tie rod w\ld be #cj tons :5 ! load 0 #aj tons1 & !re=e ! tie mu/ at lea/ be /rong en\gh to be> a pull ( #cj tons2 b x is cu/om>y1 9 good eng9e]+ practice1 to make ! ma*9e ( ab\t t5 times ! /r5g? t w\ld j be suffici5t to su/a9 ! ord9>y load4 ,h5ce ! crank mu/ be s /rong t ! tie w\ld n break ) a t5sion less ?an #cjj tons1 :i* w\ld be produc$ :5 ! crane 0 lift+ #ajj tons4 ,s great a m>g9 ( safety is necess>y on acc\nt ( ! j]ks & o!r occasional great /ra9s t arise 9 ! rais+ & ! l[]+ ( heavy weights4 ,= a crane int5d$ to raise #aj tons1 ! eng9e] mu/ !re=e design a tie rod :i* n less ?an #cjj tons w\ld te> asund]4 ,x has be5 prov$ by actual trial t a rod ( wr\ght iron ( av]age quality1 "o squ>e in* 9 section1 c j )/& a pull ( tw5ty tons4 ,h5ce fifte5 su* rods1 or "o rod ! section ( :i* 0 equal to fifte5 squ>e in*es1 w\ld be j able to resi/ #cjj tons2 & ?is is !re=e ! prop] area ( section = ! tie rod ( ! crane we h be5 consid]+4 #df4 ,9 ! same way we asc]ta9 ! actual ?ru/ d[n ! jib2 x am\nts to #ce tons1 & ! jib %\ld be t5 times z /rong z a /rut :i* w\ld collapse "u a /ra9 ( #ce tons4 #dg4 ,x is easy to see f ! figure t ! tie rod is pull+ ! upright1 & t5d+1 9 fact1 to make x snap (f ne> ,b4 ,x is !re=e necess>y t ! upright support ,a ,b 7,fig4 #ag7 be secur$ v firmly4 ,,lecture ,,iii4 ,,p>allel ,,=ces4 ,introduction4--,pressure ( a ,load$ ,beam on its ,supports4--,equilibrium ( a ,b> support$ on a ,knife-edge4--,! ,composition ( ,p>allel ,=ces4--,p>allel ,=ces act+ 9 opposite directions4--,! ,c\ple4--,! ,weigh+ ,scales4 ,,introduction4 #dh4 ,! p>allelogram ( =ces enables u to f9d ! resultant ( two =ces :i* int]sect3 b s9ce p>allel =ces d n int]sect1 ! con/ruction does n avail to det]m9e ! resultant ( two p>allel =ces4 ,we c1 h[ev]1 f9d ?is resultant v simply by o!r means4 ,illu/ration3 ,,fig4 #ai4 #di4 ,fig4 #ai repres5ts a wood5 rod #d' long1 su/a9$ by re/+ on two supports ,a & ,b1 & hav+ ! l5g? ,a ,b divid$ into #ad equal p>ts4 ,let a weight ( #ad lbs4 be hung on ! rod at its middle po9t ,c2 ?is weight mu/ be borne by ! supports1 & x is evid5t t !y w be> ! load 9 equal %>es1 = s9ce ! weight is at ! middle ( ! rod !re is no reason :y "o end %\ld be diff]5tly circum/anc$ f ! o!r4 ,h5ce ! total pressure on ea* ( ! supports w be #g lbs41 toge!r ) half ! weight ( ! wood5 b>4 #ej4 ,if ! weight ( #ad lbs4 be plac$1 n at ! c5tre ( ! b>1 b at "s o!r po9t su* z ,d1 x is n !n s easy to see 9 :at proportion ! weight is di/ribut$ betwe5 ! supports4 ,we c easily und]/& t ! support ne> ! weight mu/ be> m ?an ! remote "o1 b h[ mu* m8 ,:5 we are able to answ] ?is "q1 we %all see t x w lead u to a k ( ! composition ( p>allel =ces4 ,,pressure ,,( ,a ,,load$ ,,beam ,,on ,,its ,,supports4 #ea4 ,to /udy ?is "q we %all employ ! app>atus %[n 9 ,fig4 #bj4 ,an iron b> #e' #f long1 weigh+ #aj lbs41 re/s 9 ! hooks ( ! spr+ balances ,a1 ,c1 9 ! mann] %[n 9 ! figure4 ,^! hooks are exactly five feet ap>t1 s t ! b> projects #c beyond ea* end4 ,! space betwe5 ! hooks is divid$ into tw5ty equal portions1 ea* ( c\rse #c long4 ,! b> is suffici5tly /rong to be> ! weight ,b ( #bj lbs4 susp5d$ f x by an ,s hook1 )\t appreciable deflection4 ,be=e ! weight ( #bj lbs4 is susp5d$1 ! spr+ balances ea* %[ a /ra9 ( #e lbs4 ,we w\ld expect ?is1 = x is evid5t t ! :ole weight ( ! b> am\nt+ to #aj lbs4 %\ld be borne equally by ! two supports4 #eb4 ,:5 ,i place ! weight 9 ! middle1 #aj divisions f ea* end1 ,i f9d ! balances ea* indicate #ae lbs4 ,b #e lbs4 is due to ! weight ( ! b>4 ,h5ce ! #bj lbs4 is divid$ equally1 z we h already /at$ t x %\ld be4 ,b let ! #bj lbs4 be mov$ to any o!r position1 suppose #d divisions f ! "r1 & #af f ! left2 !n ! "r-h& scale reads #ba lbs41 & ! left-h& reads #i lbs4 ,to get rid ( ! weight ( ! b> itself1 we mu/ subtract #e lbs4 f ea*4 ,we le>n !re=e t ! #bj lb4 weight pulls ! "r-h& spr+ balance ) a /ra9 ( #af lbs41 & ! left ) a /ra9 ( #d lbs4 ,obs]ve ?is closely2 y see ,i h made ! numb] ( divisions 9 ! b> equal to ! numb] ( p\nds weight susp5d$ f x1 & "h we f9d t :5 ! weight is #af divisions f ! left1 ! /ra9 ( #af lbs4 is %[n on ! "r4 ,at ! same "t ! weight is #d divisions f ! "r1 & #d lbs4 is ! /ra9 %[n to ! left4 ,illu/ration3 ,,fig4 #bj4 #ec4 ,i w /ate ! law ( ! di/ribution ( ! load a little m g5]ally1 & we %all f9d t ! b> w prove ! law to be true 9 all cases4 ,divide ! b> into z _m equal p>ts z !re are p\nds 9 ! load1 !n ! pressure 9 p\nds on "o end is ! numb] ( divisions t ! load is di/ant f ! o!r4 #ed4 ,= example1 suppose ,i place ! load #b divisions f "o end3 ,i read by ! scale at t end #bc lbs42 subtract+ #e lbs4 = ! weight ( ! b>1 ! pressure due to ! load is %[n to be #ah lbs41 b ! weight is !n exactly #ah divisions di/ant f ! o!r end4 ,we c easily v]ify ?is rule :atev] be ! position :i* ! load occupies4 #ee4 ,if ! load be plac$ betwe5 two m>ks1 in/ead ( be+1 z we h hi!rto suppos$1 exactly at "o1 ! p>tition ( ! load is also det]m9$ by ! law4 ,7 x1 = example1 #ce divisions f "o end1 ! /ra9 on ! o!r w\ld be #ce lbs42 & 9 l mann] = o!r cases4 #ef4 ,we h ?us prov$ by actual exp]im5t ?is useful & in/ructive law ( nature2 ! same result c\ld h be5 inf]r$ by reason+ f ! p>allelogram ( =ce1 b ! purely exp]im5tal pro( is m 9 accordance ) our s*eme4 ,! doctr9e ( ! composition ( p>allel =ces is "o ( ! mo/ fundam5tal p>ts ( me*anics1 & we %all h _m occasions to employ x 9 ?is z well z 9 subsequ5t lectures4 #eg4 ,return+ n[ to ,fig4 #ai1 ) :i* we comm5c$1 ! law we h discov]$ w enable u to f9d h[ ! weight is di/ribut$4 ,we divide ! l5g? ( ! b> betwe5 ! supports into #ad equal p>ts because ! weight is #ad lbs42 if1 !n1 ! weight be at ,d1 #aj divisions f "o end ,a1 & #d f ! o!r ,b1 ! pressure at ! correspond+ ends w be #d & #aj4 ,if ! weight 7 #be divisions f "o end1 & !re=e #aae f ! o!r1 ! %>es 9 :i* ?is load w\ld be support$ at ! ends are #aae lbs4 & #be lbs4 ,! actual pressure su/a9$ by ea* end is1 h[ev]1 ab\t #f ounces great] if ! weight ( ! wood5 b> itself be tak5 into acc\nt4 #eh4 ,let u susp5d a second weight f ano!r po9t ( ! b>4 ,we mu/ !n calculate ! pressures at ! ends :i* ea* weight sep>ately w\ld produce1 & ^? at ! same end are to be add$ toge!r1 & to half ! weight ( ! b>1 to f9d ! total pressure4 ,?us1 if "o weight ( #bj lbs4 7 9 ! middle1 & ano!r ( #ad lbs4 at a di/ance ( #aa divisions f "o end1 ! middle weight w\ld produce #aj lbs4 at ea* end & ! #ad lbs4 w\ld produce #c lbs4 & #aa lbs41 & rememb]+ ! weight ( ! b>1 ! total pressures produc$ w\ld be #ac lbs4 #f oz4 & #ba lbs4 #f oz4 ,! same pr9ciples w evid5tly apply to ! case ( sev]al weights3 & ! application ( ! rule becomes especially easy :5 all ! weights are equal1 = !n ! same divisions w s]ve = calculat+ ! effect ( ea* weight4 #ei4 ,! pr9ciples involv$ 9 ^! calculations are ( s mu* importance t we %all fur!r exam9e !m by a diff]5t me?od1 :i* has _m useful applications4 ,,equilibrium ,,( ,a ,,b> ,,support$ ,,on ,a ,,knife-,,edge4 #fj4 ,! weight ( ! b> has hi!rto some:at complicat$ our calculations2 ! results w\ld appe> m simply if we c\ld avoid ?is weight2 b s9ce we want a /rong b>1 its weight is n s small t we c\ld af=d to ov]look x altoge!r4 ,by means ( ! arrangem5t ( ,fig4 #ba1 we c c\nt]poise ! weight ( ! b>4 ,to ! c5tre ( ,a ,b a cord is atta*$1 :i*1 pass+ ov] a fix$ pulley ,d1 c>ries a hook at ! o!r end4 ,! b>1 be+ a p9e rod1 #d feet long & #a in* squ>e1 weighs ab\t #ab ounces2 consequ5tly1 if a weight ( twelve ounces be susp5d$ f ! hook1 ! b> w be c\nt]pois$1 & w rema9 at :atev] height x is plac$4 ,illu/ration3 ,,fig4 #ba4 #fa4 ,a ,b is divid$ by l9es drawn along x at di/ances ( #a ap>t2 !re are ?us #dh ( ^! divisions4 ,! weights employ$ are furni%$ ) r+s l>ge en\gh to enable !m to be slipp$ on ! b> & ?us plac$ 9 any desir$ position4 #fb4 ,und]nea? ! b> lies an important portion ( ! arrangem5t2 namely1 ! knife-edge ,c4 ,?is is a blunt edge ( /eel firmly fa/5$ to ! support :i* c>ries x4 ,?is support c be mov$ along und]nea? ! b> s t ! knife-edge c be plac$ "u any ( ! divisions requir$4 ,! b> be+ c\nt]pois$1 ?\gh /ill unload$ ) weights1 may be br\ght d[n till x j t\*es ! knife-edge2 x w !n rema9 horizontal1 & w reta9 ?is position :e!r ! knife-edge be at ei!r end ( ! b> or 9 any int]m$iate position4 ,i %all hang weights at ! extremities ( ! rod1 & we %all f9d t !re is = ea* pair ( weights j "o position at :i*1 if ! knife-edge be plac$1 x w su/a9 ! rod horizontally4 ,we %all !n exam9e ! relations betwe5 ^! di/ances & ! weights t h be5 atta*$1 & we %all trace ! connection betwe5 ! results ( ?is me?od & ^? ( ! arrangem5t t we la/ us$4 #fc4 ,suppos+ t #f lbs4 be hung at ea* end ( ! rod1 we might easily =esee t ! knife-edge %\ld be plac$ 9 ! middle1 & we f9d our anticipations v]ifi$4 ,:5 ! edge is exactly at ! middle1 ! rod rema9s horizontal2 b if x be mov$1 ev5 ?r\gh a v small di/ance1 to ei!r side1 ! rod in/antly desc5ds on ! o!r4 ,! knife-edge is #bd in*es di/ant f ea* end2 & if ,i multiply ?is numb] by ! numb] ( p\nds 9 ! weight1 9 ?is case #f1 ,i f9d #add = ! product1 & ?is product is ! same = bo? ends ( ! b>4 ,! importance ( ?is rem>k w be se5 directly4 #fd4 ,if ,i remove "o ( ! #f lb4 weights & replace x by #b lbs41 leav+ ! o!r weight & ! knife-edge unalt]$1 ! b> in/antly desc5ds on ! side ( ! heavy weight2 b1 by slipp+ ! knife-edge along ! b>1 ,i f9d t :5 ,i h mov$ x to )9 a di/ance ( #ab in*es f ! #f lbs41 & !re=e #cf in*es f ! #b lbs41 ! b> w rema9 horizontal4 ,! edge mu/ be put c>efully at ! "r place2 a qu>t] ( an in* to "o side or ! o!r w\ld upset ! b>4 ,! :ole load borne by ! knife-edge is ( c\rse #h lbs41 be+ ! sum ( ! weights4 ,if we multiply #b1 ! numb] ( p\nds at "o end1 by #cf1 ! di/ance ( t end f ! knife-edge1 we obta9 ! product #gb2 & we f9d precisely ! same product by multiply+ #f1 ! numb] ( p\nds 9 ! o!r weight1 by #ab1 its di/ance f ! knife-edge4 ,to express ?is result concisely we %all introduce ! ^w mom5t1 a t]m ( frequ5t use 9 me*anics4 ,! #b lb4 weight produces a =ce t5d+ to pull its end ( ! b> d[nw>ds by mak+ ! b> turn r\nd ! knife-edge4 ,! magnitude ( ?is =ce1 multipli$ into its di/ance f ! knife-edge1 is call$ ! mom5t ( ! =ce4 ,we c express ! result at :i* we h arriv$ by say+ t1 :5 ! knife-edge has be5 s plac$ t ! b> rema9s horizontal1 ! mom5ts ( ! =ces ab\t ! knife-edge are equal4 #fe4 ,we may fur!r illu/rate ?is law by susp5d+ weights ( #g lbs4 & #e lbs4 respectively f ! ends ( ! b>2 x is f\nd t ! knife-edge mu/ !n be plac$ #bj in*es f ! l>g] weight1 &1 !re=e1 #bh in*es f ! small]1 b #e #bh #adj1 & #g #bj #adj1 ?us aga9 v]ify+ ! law ( equality ( ! mom5ts4 ,f ! equality ( ! mom5ts we c also d$uce ! law = ! di/ribution ( ! load giv5 9 ,art4 #ec4 ,?us1 tak+ ! figures 9 ! la/ exp]im5t1 we h loads ( #g lbs4 & #e lbs4 respectively4 ,^! produce a pressure ( #g #e #ab lbs4 on ! knife-edge4 ,?is edge presses on ! b> ) an equal & opposite reaction4 ,to asc]ta9 ! di/ribution ( ?is pressure on ! ends ( ! beam1 we divide ! :ole beam into #ab equal p>ts ( #d in*es ea*1 & ! #g lb4 weight is #e ( ^! p>ts1 i4e41 #bj in*es di/ant f ! support4 ,h5ce ! edge %\ld be #bj in*es f ! great] weight1 :i* is ! condition also impli$ by ! equality ( ! mom5ts4 ,,! ,,composition ,,( ,,p>allel ,,=ces4 #ff4 ,hav+ n[ exam9$ ! subject exp]im5tally1 we proce$ to inve/igate :at may be le>n$ f ! results we h prov$4 ,illu/ration3 ,,fig4 #bb4 ,! weight ( ! b> be+ all[$ = 9 ! way we h expla9$1 by subtract+ "o-half ( x f ea* ( ! /ra9s indicat$ by ! spr+ balance 7,,fig4 #bj71 we may omit x f consid]ation4 ,z ! balances are pull$ d[nw>ds by ! b> :5 x is load$1 s !y w react to pull ! b> upw>ds4 ,?is w be evid5t if we ?9k ( a weight--say #ad lbs4--susp5d$ f "o ( ^! balances3 x hangs at re/2 !re=e its weight1 :i* is con/antly urg+ x d[nw>ds1 mu/ be c\nt]act$ by an equal =ce pull+ x upw>ds4 ,! balance ( c\rse %[s #ad lbs42 ?us ! spr+ ex]ts 9 an upw>d pull a =ce :i* is precisely equal to t by :i* x is itself pull$ d[nw>ds4 #fg4 ,h5ce ! spr+s are ex]t+ =ces at ! ends ( ! b> 9 pull+ !m upw>ds1 & ! scales indicate _! magnitudes4 ,! b> is ?us subject to ?ree =ces1 viz43 ! susp5d$ weight ( #bj lbs41 :i* acts v]tically d[nw>ds1 & ! two o!r =ces :i* act v]tically upw>ds1 & ! unit$ action ( ! ?ree make equilibrium4 #fh4 ,let l9es be drawn1 repres5t+ ! =ces 9 ! mann] already expla9$4 ,we h !n ?ree p>allel =ces ,,ap1 ,,bq1 ,,cr act+ on a rod 9 equilibrium 7,fig4 #bb74 ,! two =ces ,,ap & ,,bq may be consid]$ z balanc$ by ! =ce ,,cr 9 ! position %[n 9 ! figure1 b ! =ce ,,cr w\ld be balanc$ by ! equal & opposite =ce ,,cs1 repres5t$ by ! dott$ l9e4 ,h5ce ?is la/ =ce is equival5t to ,,ap & ,,bq4 ,9 o!r words1 x mu/ be _! resultant4 ,"h !n we le>n t a pair ( p>allel =ces1 act+ 9 ! same direction1 c be comp\nd$ into a s+le resultant4 #fi4 ,we also see t ! magnitude ( ! resultant is equal to ! sum ( ! magnitudes ( ! =ces1 & fur!r we f9d ! position ( ! resultant by ! foll[+ rule4 ,add ! two =ces toge!r2 divide ! di/ance betwe5 !m into z _m equal p>ts z are conta9$ 9 ! sum1 measure (f f ! great] ( ^! two =ces z _m p>ts z !re are p\nds 9 ! small] =ce1 & t is ! po9t requir$4 ,?is rule is v easily inf]r$ f t :i* we 7 taught by ! exp]im5ts 9 ,art4 #ea4 ,,p>allel ,,=ces ,,act+ ,,9 ,,opposite ,,directions4 #gj4 ,s9ce ! =ces ,,ap1 ,,bq1 ,,cr 7,fig4 #bb7 are 9 equilibrium1 x foll[s t we may look on ,,bq z balanc+ 9 ! position :i* x occupies ! two =ces ( ,,ap & ,,cr 9 _! positions4 ,?is may rem9d u ( ! num]\s in/ances we h already met )1 :]e "o =ce balanc$ two great] =ces3 9 ! pres5t case ,,ap & ,,cr are act+ 9 opposite directions1 & ! =ce ,,bq :i* balances !m is equal to _! diff]5ce4 ,a =ce ,,bt equal & opposite to ,,bq mu/ !n be ! resultant ( ,,cr & ,,ap1 s9ce x is able to produce ! same effect4 ,notice t 9 ?is case ! resultant ( ! two =ces is n betwe5 !m1 b t x lies on ! side ( ! l>g]4 ,:5 ! =ces act 9 ! same direction1 ! resultant is always betwe5 !m4 #ga4 ,! actual position :i* ! resultant ( two opposite p>allel =ces occupies is to be f\nd by ! foll[+ rule4 ,divide ! di/ance betwe5 ! =ces into z _m equal p>ts z !re are p\nds 9 _! diff]5ce1 !n measure outw>ds f ! po9t ( application ( ! l>g] =ce z _m ( ^! p>ts z !re are p\nds 9 ! small] =ce2 ! po9t ?us f\nd det]m9es ! position ( ! resultant4 ,?us1 if ! =ces be #ad & #bj1 ! diff]5ce betwe5 !m is #f1 & !re=e ! di/ance betwe5 _! directions is divid$ into six p>ts2 f ! po9t ( application ( ! =ce ( #bj1 #ad p>ts are measur$ outw>ds1 & ?us ! position ( ! resultant is det]m9$4 ,h5ce we h ! means ( comp\nd+ two p>allel =ces 9 g5]al4 ,,! ,,c\ple4 #gb4 ,9 "o case1 h[ev]1 two p>allel =ces h no resultant2 ?is occurs :5 ! two =ces are equal1 & 9 opposite directions4 ,a pair ( =ces ( ?is k9d is call$ a c\ple2 !re is no s+le =ce :i* c\ld balance a c\ple1--x c only be c\nt]balanc$ by ano!r c\ple act+ 9 an opposite mann]4 ,?is rem>kable case1 may be /udi$ by ! arrangem5t ( ,fig4 #bc4 ,illu/ration3 ,,fig4 #bc4 ,a wood5 rod1 ,a ,b #dh #je #je1 has /r+s atta*$ to x at po9ts ,a & ,d1 "o foot di/ant4 ,! /r+ at ,d passes ov] a fix$ pulley ,e1 & at ! end ,p a hook is atta*$ = ! purpose ( receiv+ weights1 :ile a simil> hook desc5ds f ,a2 ! weight ( ! rod itself1 :i* only am\nts to ?ree ounces1 may be neglect$1 z x is v small comp>$ ) ! weights :i* w be us$4 #gc4 ,suppos+ #b lbs4 to be plac$ at ,p1 & #a lb4 at ,q1 we h two p>allel =ces act+ 9 opposite directions2 & s9ce _! diff]5ce is #a lb41 x foll[s f our rule t ! po9t ,f1 :]e ,d ,f is equal to ,a ,d1 is ! po9t :]e ! resultant is appli$4 ,y see ?is is easily v]ifi$1 = by plac+ my f+] ov] ! rod at ,f x rema9s horizontal & 9 equilibrium2 :]eas1 :5 ,i move my f+] to "o side or ! o!r1 equilibrium is impossible4 ,if ,i move x ne>] to ,b1 ! end ,a asc5ds4 ,if ,i move x t[>ds ,a1 ! end ,b asc5ds4 #gd4 ,to /udy ! case :5 ! two =ces are equal1 a load ( #b lbs4 may be plac$ on ea* ( ! hooks ,p & ,q4 ,x w !n be f\nd t ! f+] _c be plac$ 9 any position along ! rod s z to keep x 9 equilibrium2 t is to say1 no s+le =ce c c\nt]act ! two =ces :i* =m ! c\ple4 ,let ,o be ! po9t midway betwe5 ,a & ,d4 ,! =ces evid5tly t5d to raise ,,ob & turn ! "p ,o ,a d[nw>ds2 b if ,i try to re/ra9 ,o ,b by hold+ my f+] above1 z at ! po9t ,x1 in/antly ! rod beg9s to turn r\nd ,x & ! "p f ,a to ,x desc5ds4 ,i f9d simil>ly t any attempt to prev5t ! motion by hold+ my f+] und]nea? is equally unsuccessful4 ,b if at ! same "t ,i press ! rod d[nw>ds at "o po9t1 & upw>ds at ano!r ) suitable =ce1 ,i c succe$ 9 produc+ equilibrium2 9 ?is case ! two pressures =m a c\ple2 & x is ?is c\ple :i* neutralizes ! c\ple produc$ by ! weights4 ,we le>n1 !n1 ! important result t a c\ple c be balanc$ by a c\ple1 & by a c\ple only4 #ge4 ,! mom5t ( a c\ple is ! product ( "o ( ! two equal =ces into _! p]p5dicul> di/ance4 ,two c\ples t5d+ to turn ! body to :i* !y are appli$ 9 ! same direction w be equival5t if _! mom5ts are equal4 ,two c\ples :i* t5d to turn ! body 9 opposite directions w be 9 equilibrium if _! mom5ts are equal4 ,we c also comp\nd two c\ples 9 ! same or 9 opposite directions into a s+le c\ple ( :i* ! mom5t is respectively ei!r ! sum or ! diff]5ce ( ! orig9al mom5ts4 ,,! ,,weigh+ ,,scales4 #gf4 ,ano!r app>atus by :i* ! nature ( p>allel =ces may be inve/igat$ is %[n 9 ,fig4 #bd2 x consi/s ( a slight frame ( wood ,a ,b ,c1 #d' long4 ,at ,e1 a pair ( /eel knife-edges is clamp$ to ! frame4 ,! knife-edges re/ on two pieces ( /eel1 "o ( :i* is %[n at ,o ,f4 ,:5 ! knife-edges are suitably plac$ ! frame is balanc$1 s t a small piece ( pap] laid at ,a w cause t side to desc5d4 ,illu/ration3 ,,fig4 #bd4 #gg4 ,we susp5d two small hooks f ! po9ts ,a & ,b3 ^! are made ( f9e wire1 s t _! weight may be left out ( consid]ation4 ,) ?is app>atus we c 9 ! fir/ place v]ify ! pr9ciple ( equality ( mom5ts3 = example1 if ,i place ! hook ,a at a di/ance ( #i f ! c5tre ,o & load x ) #a lb41 ,i f9d t :5 ,b is lad5 ) #je lb4 x mu/ be at a di/ance ( #ah f ,o 9 ord] to c\nt]balance ,a2 ! mom5t 9 ! "o case is #i #a1 9 ! o!r #ah #je1 & ^! are obvi\sly equal4 #gh4 ,let a weight ( #a lb4 be plac$ on ea* ( ! hooks1 ! frame w only be 9 equilibrium :5 ! hooks are at precisely ! same di/ance f ! c5tre4 ,a famili> application ( ?is pr9ciple is f\nd 9 ! ord9>y weigh+ scales2 ! frame1 :i* 9 ?is case is call$ a beam1 is su/a9$ by two knife-edges1 small]1 h[ev]1 ?an ^? repres5t$ 9 ! figure4 ,! pans ,p1 ,p are susp5d$ f ! extremities ( ! beam1 & %\ld be at equal di/ances f its c5tre4 ,^! scale-pans mu/ be ( equal weight1 & !n1 :5 equal weights are plac$ 9 !m1 ! beam w rema9 horizontal4 ,if ! weight 9 "o slightly exce$ t 9 ! o!r1 ! pan conta9+ ! heavi] weight w ( c\rse desc5d4 #gi4 ,t a pair ( scales %\ld weigh accurately1 x is necess>y t ! weights be correct2 b ev5 ) correct weights1 a balance ( defective con/ruction w give an inaccurate result4 ,! error frequ5tly arises f "s inequality 9 ! l5g?s ( ! arms ( ! beam4 ,:5 ?is is ! case1 ! two weights :i* really balance are n equal4 ,suppos+1 = in/ance1 t ) an imp]fect balance ,i endeav\r to weigh a p\nd ( %ot4 ,if ,i put ! weight on ! %ort side1 !n ! quantity ( %ot balanc$ is less ?an #a lb42 :ile if ! #a lb4 weight be plac$ at ! long side1 x w require m ?an #a lb4 ( %ot to produce equilibrium4 ,! mode ( te/+ a pair ( scales is !n evid5t4 ,let weights be plac$ 9 ! pans :i* balance ea* o!r2 if ! weights be int]*ang$ & ! balance /ill rema9s horizontal1 x is correct4 #hj4 ,suppose1 = example1 t ! two arms be #aj in*es & #aa in*es long1 !n1 if #a lb4 weight be plac$ 9 ! pan ( ! #aj-;in* end1 its mom5t is #aj2 & if _/ ( #a lb4 be plac$ 9 ! pan belong+ to ! #aa-;in* end1 its mom5t is also #aj3 h5ce #a lb4 at ! %ort end balances _/ ( #a lb4 at ! long end2 & !re=e1 if ! %opkeep] plac$ 8 weight 9 ! %ort arm1 8 cu/om]s w\ld lose _/ "p ( ea* p\nd = :i* !y paid2 on ! o!r h&1 if ! %opkeep] plac$ 8 #a lb4 weight on ! long arm1 !n n less ?an _/ lb4 w\ld be requir$ 9 ! pan belong+ to ! %ort arm4 ,h5ce 9 ?is case ! cu/om] w\ld get _/ lb4 too mu*4 ,x foll[s1 t if a %opman plac$ 8 weights & 8 goods alt]nately 9 ! "o scale & 9 ! o!r he w\ld be a los] on ! :ole2 =1 ?\gh e second cu/om] gets _/ lb4 less ?an he ought1 yet ! o!rs get _/ lb4 m ?an !y h paid =4 ,,lecture ,,iv4 ,,! ,,=ce ,,( ,,gravity4 ,introduction4--,specific ,gravity4--,! ,plummet & ,_s-,level4--,! ,c5tre ( ,gravity4--,/able & ,un/able ,equilibrium4--,prop]ty ( ! ,c5tre ( ,gravity 9 a ,revolv+ ,:eel4 ,,introduction4 #ha4 ,9 ! la/ ?ree lectures we consid]$ =ces 9 ! ab/ract2 we saw h[ !y are to be repres5t$ by /raight l9es1 h[ comp\nd$ toge!r & h[ decompos$ into o!rs2 we h expla9$ :at is meant by =ces be+ 9 equilibrium1 & we h %[n in/ances :]e ! =ces lie 9 ! same plane or 9 diff]5t planes1 & :]e !y int]sect or are p>allel to ea* o!r4 ,^! subjects are ! elem5ts ( me*anics2 !y =m ! framework :i* 9 ?is & subsequ5t lectures we %all try to pres5t 9 a m attractive g>b4 ,we %all comm5ce by /udy+ ! mo/ rem>kable =ce 9 nature1 a =ce con/antly 9 action1 & "o to :i* all bodies are subject1 a =ce :i* di/ance _c annihilate1 & "o ! prop]ties ( :i* h l$ to ! mo/ sublime discov]ies ( human intellect4 ,?is is ! =ce ( gravity4 #hb4 ,if ,i drop a /one f my h&1 x falls to ! gr\nd4 ,t :i* produces motion is a =ce3 h5ce ! /one mu/ h be5 act$ ^u by a =ce :i* drew x to ! gr\nd4 ,on e "p ( ! e>?'s surface exp]i5ce %[s t a body t5ds to fall4 ,?is fact proves t !re is an attractive =ce 9 ! e>? t5d+ to draw all bodies t[>ds x4 ,illu/ration3 ,,fig4 #be4 #hc4 ,let ,a ,b ,c ,d 7,fig4 #be7 be po9ts f :i* /ones are let fall1 & let ! circle repres5t ! section ( ! e>?2 let ,p ,q ,r ,s be ! po9ts at ! surface ( ! e>? ^u :i* ! /ones w drop :5 all[$ to d s4 ,! f\r /ones w move 9 ! directions ( ! arr[s3 f ,a to ,p ! /one moves 9 an opposite direction to ! motion f ,c to ,r2 f ,b to ,q x moves f "r to left1 :ile f ,d to ,s x moves f left to "r4 ,! movem5ts are 9 diff]5t directions2 b if ,i produce ^! directions1 z indicat$ by ! dott$ l9es1 !y ea* pass ?r\gh ! c5tre ,o4 #hd4 ,h5ce ea* /one 9 fall+ moves t[>ds ! c5tre ( ! e>?1 & ?is is consequ5tly ! direction ( ! =ce4 ,we !re=e ass]t t ! e>? has an attraction = ! /one1 9 consequ5ce ( :i* x tries to get z ne> ! e>?'s c5tre z possible1 & ?is attraction is call$ ! =ce ( gravitation4 #he4 ,we are s excessively famili> ) ! ph5om5on ( see+ bodies fall t x does n excite our a/oni%m5t or ar\se our curiosity4 ,a clap ( ?und]1 :i* e "o notices1 because mu* less frequ5t1 is n really m rem>kable4 ,we (t5 look ) att5tion at ! attraction ( a piece ( iron by a magnet1 & ju/ly s1 = ! ph5om5on is v int]e/+1 & yet ! fall+ ( a /one is produc$ by a f> gr&] & m important =ce ?an ! =ce ( magnetism4 #hf4 ,x is gravity :i* causes ! weight ( bodies4 ,i hold a piece ( lead 9 my h&3 gravity t5ds to pull x d[nw>ds1 ?us produc+ a pressure on my h& :i* ,i call weight4 ,gravity acts ) slightly v>y+ int5sity at v>i\s p>ts ( ! e>?'s surface4 ,?is is due to two di/9ct causes1 "o ( :i* may be m5tion$ "h1 :ile ! o!r w be subsequ5tly ref]r$ to4 ,! e>? is n p]fectly sph]ical2 x is flatt5$ a little at ! poles2 consequ5tly a body at ! pole is ne>] ! g5]al mass ( ! e>? ?an a body at ! equator2 !re=e ! body at ! pole is m attract$1 & seems heavi]4 ,a mass :i* weighs #bjj lbs4 at ! equator w\ld weigh "o p\nd m at ! pole3 ab\t "o-?ird ( ?is increase is due to ! cause "h po9t$ out4 7,see ,lecture ,,xvii47 #hg4 ,gravity is a =ce :i* attracts e p>ticle ( matt]2 x acts n m]ely on ^? p>ts ( a body :i* lie on ! surface1 b x equally affects ^? 9 ! int]ior4 ,?is is prov$ by obs]v+ t a body has ! same weight1 h[ev] its %ape be alt]$3 = example1 suppose ,i take a ball ( putty :i* weighs #a lb41 ,i %all f9d t its weight rema9s un*ang$ :5 ! ball is flatt5$ into a ?9 plate1 ?\gh 9 ! latt] case ! surface1 & !re=e ! numb] ( sup]ficial p>ticles1 is l>g] ?an x 0 9 ! =m]4 ,,specific ,,gravity4 #hh4 ,gravity produces diff]5t effects ^u diff]5t sub/ances4 ,?is is commonly express$ by say+ t "s sub/ances are heavi] ?an o!rs2 = example1 ,i h "h a piece ( wood & a piece ( lead ( equal bulk4 ,! lead is drawn to ! e>? ) a great] =ce ?an ! wood4 ,sub/ances are usually t]m$ heavy :5 !y s9k 9 wat]1 & light :5 !y float ^u x4 ,b a body s9ks 9 wat] if x weighs m ?an an equal bulk ( wat]1 & floats if x weigh less4 ,h5ce x is natural to take wat] z a /&>d ) :i* ! weights ( o!r sub/ances may be comp>$4 #hi4 ,i take a c]ta9 volume1 say a cubic in* ( ca/ iron su* z ?is ,i hold 9 my h&1 & :i* has be5 accurately %ap$ = ! purpose4 ,?is cube is heavi] ?an "o cubic in* ( wat]1 b ,i %all f9d t a c]ta9 quantity ( wat] is equal to x 9 weight2 t is to say1 a c]ta9 numb] ( cubic in*es ( wat]1 & x may be fractional p>ts ( a cubic in*1 are precisely ( ! same weight4 ,?is numb] is call$ ! specific gravity ( ca/ iron4 #ij4 ,x w\ld be impossible to c\nt]poise wat] ) ! iron )\t hold+ ! wat] 9 a vessel1 & ! weight ( ! vessel mu/ !n be all[$ =4 ,i adopt ! foll[+ plan4 ,i h "h a numb] ( in* cubes ( wood 7,fig4 #bf71 :i* w\ld ea* be light] ?an a cubic in* ( wat]1 b ,i h weight$ ! wood5 cubes by plac+ gra9s ( %ot into holes bor$ into ! wood4 ,! weight ( ea* cube has ?us be5 accurately adju/$ to be equal to t ( a cubic in* ( wat]4 ,?is may be te/$ by actual weigh+4 ,i weigh "o ( ! cubes & f9d x to be #beb gra9s1 :i* is well kn[n to be ! weight ( a cubic in* ( wat]4 ,illu/ration3 ,,fig4 #bf4 #ia4 ,b ! cubes may be %[n to be id5tical 9 weight ) ! same bulk ( wat] by a simpl] me?od4 ,"o ( !m plac$ 9 wat] %\ld h no t5d5cy to s9k1 s9ce x is n heavi] ?an wat]1 nor on ! o!r h&1 s9ce x is n light]1 %\ld x h any t5d5cy to float4 ,x %\ld !n rema9 9 ! wat] 9 :atev] position x may be plac$4 ,x is difficult to prep>e "o ( ^! cubes s accurately t ?is result %\ld be atta9$1 & x is impossible to ensure its cont9uance = any "t ow+ to *anges ( temp]ature & ! absorption ( wat] by ! wood4 ,we c1 h[ev]1 by a slight modification1 prove t "o ( ^! cubes is at all ev5ts ne>ly equal 9 weight to ! same bulk ( wat]4 ,9 ,fig4 #bf is %[n a tall glass j> fill$ ) a fluid 9 appe>ance l pla9 wat]1 b x is really compos$ 9 ! foll[+ mann]4 ,i fir/ p\r$ into ! j> a v weak solution ( salt & wat]1 :i* p>tially fill$ x2 ,i !n p\r$ g5tly ^u ?is a little pure wat]1 & f9ally fill$ up ! j> ) wat] conta9+ a little spirits ( w9e3 ! salt & wat] is a little heavi] ?an pure wat]1 :ile ! _s & wat] is a little light]4 ,i take "o ( ! cubes & drop x g5tly into ! glass2 x falls ?r\gh ! _s & wat]1 & aft] mak+ a few oscillations settles itself at re/ 9 ! /ratum %[n 9 ! figure4 ,?is %[s t our prep>$ cube is a little heavi] ?an _s & wat]1 & a little light] ?an salt & wat]1 & h5ce we inf] t x mu/ at all ev5ts be v ne> ! weight ( pure wat] :i* lies betwe5 ! two4 ,we h also a numb] ( half cubes1 qu>t] cubes1 & half-qu>t] cubes1 :i* h be5 simil>ly prep>$ to be ( equal weight ) an equal bulk ( wat]4 #ib4 ,we %all n[ be able to measure ! specific gravity ( a sub/ance4 ,9 "o pan ( ! scales ,i place ! in* cube ( ca/ iron1 & ,i f9d t #g ( ! wood5 cubes1 :i* we may call cubes ( wat]1 w balance x4 ,we !re=e say t ! specific gravity ( iron is #g4 ,! exact numb] f\nd by m accurate me?ods is #gb4 ,x is (t5 conv5i5t to rememb] t #bc cubic in*es ( ca/ iron weigh #f lbs41 & t !re=e "o cubic in* weighs v ne>ly lb4 #ic4 ,i h also cubes ( brass1 lead1 & ivory2 by c\nt]pois+ !m ) ! cubes ( wat]1 we c easily f9d _! specific gravities2 !y are %[n toge!r ) t ( ca/ iron 9 ! foll[+ table3-- ,sub/ance4 ,specific ,gravity4 ,ca/ ,iron #gb ,brass #ha ,lead #aac ,ivory #ah #id4 ,! mode "h adopt$ ( f9d+ specific gravities is entirely diff]5t f ! f> m accurate me?ods :i* are commonly us$1 b ! explanation ( ! latt] involve m difficult pr9ciples ?an ^? we h be5 consid]+4 ,our me?od r (f]s an explanation ( ! nature ( specific gravity ?an a good means ( det]m9+ x1 ?\gh1 z we h se51 x gives a result suffici5tly ne> ! tru? = _m purposes4 ,,! ,,plummet ,,& ,,_s-,,level4 #ie4 ,! t5d5cy ( ! e>? to draw all bodies t[>ds x is well illu/rat$ by ! useful 8l9e & plummet40 ,?is consi/s m]ely ( a /r+ to "o end ( :i* a lead5 weight is atta*$4 ,! /r+ :5 at re/ hangs v]tically2 if ! weight be drawn to "o side1 x w1 :5 releas$1 sw+ backw>ds & =w>ds1 until x f9ally settles aga9 9 ! v]tical2 ! reason is t ! weight always tries to get z ne> ! e>? z x c1 & ?is is accompli%$ :5 ! /r+ hangs v]tically d[nw>ds4 #if4 ,! surface ( wat] 9 equilibrium is a horizontal plane2 t is also a consequ5ce ( gravity4 ,all ! p>ticles ( wat] try to get z ne> ! e>? z possible1 & !re=e if any portion ( ! wat] 7 high] ?an ! re/1 x w\ld imm$iately spread1 z by do+ s x c\ld get l[]4 #ig4 ,h5ce ! surface ( a fluid at re/ enables u to f9d a p]fectly horizontal plane1 :ile ! plummet gives u a p]fectly v]tical l9e3 bo? ^! consequ5ces ( gravity are ( ! utmo/ practical importance4 #ih4 ,! _s-level is ano!r common & v useful in/rum5t :i* dep5ds on gravity4 ,x consi/s ( a glass tube slightly curv$1 ) its convex surface upw>ds1 & atta*$ to a /& ) a flat base4 ,?is tube is ne>ly fill$ ) _s1 b a bubble ( air is all[$ to rema94 ,! tube is p]man5tly adju/$ s t1 :5 ! plate is laid on a p]fectly horizontal surface1 ! bubble w /& 9 ! middle3 accord+ly ! position ( ! bubble gives a means ( asc]ta9+ :e!r a surface is level4 ,,! ,,c5tre ,,( ,,gravity4 ,illu/ration3 ,,fig4 #bg4 #ii4 ,we proce$ to an exp]im5t :i* w give an insight into a curi\s prop]ty ( gravity4 ,i h "h a plate ( %eet iron2 x has ! irregul> %ape %[n 9 ,fig4 #bg4 ,five small holes ,a ,b ,c ,d ,e are pun*$ at diff]5t positions on ! m>g94 ,atta*$ to ! framework is a small p9 f :i* ,i c susp5d ! iron plate by "o ( its holes ,a3 ! plate is n support$ 9 any o!r way2 x hangs freely f ! p91 ar\nd :i* x c be easily turn$4 ,i f9d t !re is "o position1 & "o only1 9 :i* ! plate w re/2 if ,i )draw x f t position x returns !re aft] a few oscillations4 ,9 ord] to m>k ?is position1 ,i susp5d a l9e & plummet f ! p91 hav+ rubb$ ! l9e ) *alk4 ,i all[ ! l9e to come to re/ 9 front ( ! plate4 ,i !n flip ! /r+ aga9/ ! plate1 & ?us produce a *alk$ m>k3 ?is ( c\rse traces out a v]tical l9e ,a ,p on ! plate4 ,i n[ remove ! plummet & susp5d ! plate f ano!r ( its holes ,b1 & repeat ! process1 ?us draw+ a second *alk$ l9e ,b ,p across ! plate1 & s on ) ! o!r holes3 ,i ?us obta9 five l9es across ! plate1 repres5t$ by dott$ l9es 9 ! figure4 ,x is a v rem>kable circum/ance t ^! five l9es all int]sect 9 ! same po9t ,p2 & if additional holes 7 bor$ 9 ! plate1 :e!r 9 ! m>g9 or n1 & ! *alk l9e drawn f ea* ( !m 9 ! mann] describ$1 !y w\ld "o & all pass ?r\gh ! same po9t4 ,?is rem>kable po9t is call$ ! c5tre ( gravity ( ! plate1 & ! result at :i* we h arriv$ may be express$ by say+ t ! v]tical l9e f ! po9t ( susp5sion always passes ?r\gh ! c5tre ( gravity4 #ajj4 ,at ! c5tre ( gravity ,p a hole has be5 bor$1 & :5 ,i place ! support+ p9 ?r\gh ?is hole y see t ! plate w re/ indiff]5tly 9 all positions3 ?is is a curi\s prop]ty ( ! c5tre ( gravity4 ,! c5tre ( gravity may 9 ?is respect be contra/$ ) ano!r hole ,q1 :i* is only an in* di/ant3 :5 ,i support ! plate by ?is hole1 x has only "o position ( re/1 viz4 :5 ! c5tre ( gravity ,p is v]tically b5ea? ,q4 ,?us ! c5tre ( gravity diff]s rem>kably f any o!r po9t 9 ! plate4 #aja4 ,we may conceive ! =ce ( gravity on ! plate to act z a =ce appli$ at ,p4 ,x w !n be easily se5 :y ?is po9t rema9s v]tically und]nea? ! po9t ( susp5sion :5 ! body is at re/4 ,if ,i atta*$ a /r+ to ! plate & pull$ x1 ! plate w\ld evid5tly place itself s t ! direction ( ! /r+ w\ld pass ?r\gh ! po9t ( susp5sion2 9 l mann] gravity s places ! plate t ! direction ( its =ce passes ?r\gh ! po9t ( susp5sion4 #ajb4 ,:atev] be ! =m ( ! plate x always conta9s "o po9t possess+ ^! rem>kable prop]ties1 & we may /ate 9 g5]al t 9 e body1 no matt] :at be its %ape1 !re is a po9t call$ ! c5tre ( gravity1 su* t if ! body be susp5d$ f ?is po9t x w rema9 9 equilibrium indiff]5tly 9 any position1 & t if ! body be susp5d$ f any o!r po9t1 !n x w be 9 equilibrium :5 ! c5tre ( gravity is directly und]nea? ! po9t ( susp5sion4 ,9 g5]al1 x w be impossible to support a body exactly at its c5tre ( gravity1 z ?is po9t is )9 ! mass ( ! body1 & x may also sometimes happ5 t ! c5tre ( gravity does n lie 9 ! sub/ance ( ! body at all1 z = example 9 a r+1 9 :i* case ! c5tre ( gravity is at ! c5tre ( ! r+4 ,we ne$ n1 h[ev]1 dwell on ^! exceptional cases1 z suffici5t illu/rations ( ! tru? ( ! laws m5tion$ w pres5t !mselves subsequ5tly4 ,,/able ,,& ,,un/able ,,equilibrium4 ,illu/ration3 ,,fig4 #bh4 #ajc4 ,an iron rod ,a ,b1 capable ( revolv+ r\nd an axis pass+ ?r\gh its c5tre ,p1 is %[n 9 ,fig4 #bh4 ,! c5tre ( gravity lies at ! c5tre ,b1 & consequ5tly1 z is easily se51 ! rod w rema9 at re/ 9 :atev] position x be plac$4 ,b let a weight ,r be atta*$ to ! rod by means ( a b9d+ screw4 ,! c5tre ( gravity ( ! :ole is no long] at ! c5tre ( ! rod2 x has mov$ to a po9t ,s ne>] ! weight2 we may easily asc]ta9 its position by remov+ ! rod f its axle & !n asc]ta9+ ! po9t ab\t :i* x w balance4 ,?is may be done by plac+ ! b> on a knife-edge1 & mov+ x to & fro until ! "r position be secur$2 m>k ?is position on ! rod1 & return x to its axle1 ! weight be+ /ill atta*$4 ,we d n n[ f9d t ! rod w balance 9 e position4 ,y see x w balance if ! po9t ,s be directly und]nea? ! axis1 b n if x lie to "o side or ! o!r4 ,b if ,s be directly ov] ! axis1 z 9 ! figure1 ! rod is 9 a curi\s condition4 ,x w1 :5 c>efully plac$1 rema9 at re/2 b if x receive ! slighte/ displacem5t1 x w tumble ov]4 ,! rod is 9 equilibrium 9 ?is position1 b x is :at is call$ un/able equilibrium4 ,if ! c5tre ( gravity be v]tically bel[ ! po9t ( susp5sion1 ! rod w return aga9 if mov$ away3 ?is position is !re=e call$ "o ( /able equilibrium4 ,x is v important to notice ! di/9ction betwe5 ^! two k9ds ( equilibrium4 #ajd4 ,ano!r way ( /at+ ! case is z foll[s4 ,a body is 9 /able equilibrium :5 its c5tre ( gravity is at ! l[e/ po9t3 un/able :5 x is at ! highe/4 ,?is may be v simply illu/rat$ by an ellipse1 :i* ,i hold 9 my h&4 ,! c5tre ( gravity ( ?is figure is at its c5tre4 ,! ellipse1 :5 re/+ on its side1 is 9 a position ( /able equilibrium & its c5tre ( gravity is !n cle>ly at its l[e/ po9t4 ,b ,i c also balance ! ellipse on its n>r[ end1 ?\gh if ,i d s ! smalle/ t\* suffices to ov]turn x4 ,! ellipse is !n 9 un/able equilibrium2 9 ?is case1 obvi\sly1 ! c5tre ( gravity is at ! highe/ po9t4 ,illu/ration3 ,,fig4 #bi4 #aje4 ,i h "h a sph]e1 ! c5tre ( gravity ( :i* is at its c5tre2 9 :atev] way ! sph]e is plac$ on a plane1 its c5tre is at ! same height1 & !re=e _c be said to h any highe/ or l[e/ po9t2 9 su* a case z ?is ! equilibrium is neutral4 ,if ! body be displac$1 x w n return to its old position1 z x w\ld h done _h t be5 a position ( /able equilibrium1 nor w x deviate fur!r !refrom z if ! equilibrium _h be5 un/able3 x w simply rema9 9 ! new position to :i* x is br\ght4 #ajf4 ,i try to balance an iron r+ ^u ! end ( a /ick ,h1 ,fig4 #bi1 b ,i _c easily succe$ 9 do+ s4 ,?is is because its c5tre ( gravity ,s is above ! po9t ( support2 b if ,i place ! /ick at ,f1 ! r+ is 9 /able equilibrium1 = n[ ! c5tre ( gravity is bel[ ! po9t ( support4 ,,prop]ty ,,( ,,! ,,c5tre ,,( ,,gravity ,,9 ,a ,,revolv+ ,,:eel4 #ajg4 ,!re are o!r curi\s consequ5ces :i* foll[ f ! prop]ties ( ! c5tre ( gravity1 & we %all conclude by illu/rat+ "o ( ! mo/ rem>kable1 :i* is at ! same "t ( ! utmo/ importance 9 ma*9]y4 ,illu/ration3 ,,fig4 #cj4 #ajh4 ,x is g5]ally necess>y t a ma*9e %\ld "w z /eadily z possible1 & t undue vibration & %ak+ ( ! framework %\ld be avoid$3 ?is is p>ticul>ly ! case :5 any p>ts ( ! ma*9e rotate ) great velocity1 z1 if ^! be heavy1 inconv5i5t vibration w be produc$ :5 ! prop] adju/m5ts are n made4 ,! connection betwe5 ?is & ! c5tre ( gravity w be und]/ood by ref]5ce to ! app>atus repres5t$ 9 ! accompany+ figure 7,fig4 #cj74 ,we h "h an arrangem5t consi/+ ( a l>ge cog :eel ,c work+ into a small "o ,b1 :]eby1 :5 ! h&le ,h is turn$1 a velocity ( rotation c be giv5 to ! iron disk ,d1 :i* weighs #ad lbs1 & is #ah 9 diamet]4 ,?is disk be+ uni=m1 & be+ atta*$ to ! axis at its c5tre1 x foll[s t its c5tre ( gravity is also ! c5tre ( rotation4 ,! :eels are atta*$ to a /&1 :i*1 ?\gh massive1 is /ill unconnect$ ) ! floor4 ,by turn+ ! h&le ,i c rotate ! disk v rapidly1 ev5 z mu* z twelve times 9 a second4 ,/ill ! /& rema9s q /eady1 & ev5 ! %utt] bell atta*$ to x at ,e is sil5t4 #aji4 ,?r\gh "o ( ! holes 9 ! disk ,d ,i fa/5 a small iron bolt & a few wa%]s1 altoge!r weigh+ ab\t #a lb42 t is1 only "o-f\rte5? ( ! weight ( ! disk4 ,:5 ,i turn ! h&le sl[ly1 ! ma*9e works z smoo?ly z be=e2 b z ,i increase ! spe$ up to "o revolution e two seconds1 ! bell beg9s to r+ viol5tly1 & :5 ,i increase x /ill m1 ! /& q %akes ab\t on ! floor4 ,:at is ! reason ( ?is8 ,by add+ ! bolt1 ,i slightly alt]$ ! position ( ! c5tre ( gravity ( ! disk1 b ,i made no *ange ( ! axis ab\t :i* ! disk rotat$1 & consequ5tly ! disk 0 n on ?is occasion turn+ r\nd its c5tre ( gravity3 ?is x 0 :i* caus$ ! vibration4 ,x is absolutely necess>y t ! c5tre ( gravity ( any heavy piece1 rotat+ rapidly ab\t an axis1 %\ld lie 9 ! axis ( rotation4 ,! am\nt ( vibration produc$ by a high velocity may be v consid]able1 ev5 :5 a v small mass is ! orig9at+ cause4 #aaj4 ,9 ord] t ! ma*9e may "w smoo?ly aga91 x is n necess>y to remove ! bolt f ! hole4 ,if by any means ,i br+ back ! c5tre ( gravity to ! axis1 ! same end w be atta9$4 ,?is is v simply effect$ by plac+ a second bolt ( ! same size at ! opposite side ( ! disk1 ! two be+ at equal di/ances f ! axis2 on turn+ ! h&le1 ! ma*9e is se5 to "w z smoo?ly z x did 9 ! fir/ in/ance4 #aaa4 ,! mo/ common rotat+ pieces 9 ma*9es are :eels ( v>i\s k9ds1 & 9 ^! ! c5tre ( gravity is evid5tly id5tical ) ! c5tre ( rotation2 b if f any cause a :eel1 :i* is to turn rapidly1 has an extra weight atta*$ to "o "p1 ?is weight mu/ be c\nt]pois$ by "o or m on o!r portions ( ! :eel1 9 ord] to keep ! c5tre ( gravity ( ! :ole 9 its prop] place4 ,?us x is t ! driv+ :eels ( a locomotive are always weight$ s z to c\nt]act ! effect ( ! crank & re/ore ! c5tre ( gravity to ! axis ( rotation4 ,! cause ( ! vibration w be und]/ood aft] ! lecture on c5trifugal =ce 7,lect4 ,,xvii474 ,,lecture ,v4 ,,! ,,=ce ,,( ,,friction4 ,! ,nature ( ,friction4--,! ,mode ( ,exp]im5t+4--,friction is proportional to ! pressure4--,a m accurate =m ( ! ,law4--,! ,coeffici5t v>ies ) ! weights us$4--,! ,angle ( ,friction4--,ano!r ,law ( ,friction4--,conclud+ ,rem>ks4 ,,! ,,nature ,,( ,,friction4 #aab4 ,a discussion ( ! =ce ( friction is a necess>y prelim9>y to ! /udy ( ! me*anical p[]s :i* we %all pres5tly comm5ce4 ,friction r5d]s ! inquiry into ! me*anical p[]s m difficult ?an x w\ld be if ?is =ce 7 abs5t2 b its effects are too important to be ov]look$4 ,illu/ration3 ,,fig4 #ca4 #aac4 ,! nature ( friction may be und]/ood by ,fig4 #ca1 :i* repres5ts a section ( ! top ( a table ( wood or any o!r sub/ance levell$ s t ,c ,d is horizontal2 on ! table re/s a block ,a ( wood or any o!r sub/ance4 ,to ,a a cord is atta*$1 :i*1 aft] pass+ ov] a pulley ,p1 is /ret*$ by a susp5d$ weight ,b4 ,if ! magnitude ( ,b exce$s a c]ta9 limit1 !n ,a is pull$ along ! table & ,b desc5ds2 b if ,b be small] ?an ?is limit1 bo? ,a & ,b rema9 at re/4 ,:5 ,b is n heavy en\gh to produce motion x is support$ by ! t5sion ( ! cord1 :i* is itself neutraliz$ by ! friction produc$ by a c]ta9 coh]5ce betwe5 ,a & ! table4 ,friction is by ?is exp]im5t prov$ to be a =ce1 because x prev5ts ! motion ( ,b4 ,inde$ friction is g5]ally manife/$ z a =ce by de/roy+ motion1 ?\gh sometimes indirectly produc+ x4 #aad4 ,! true s\rce ( ! =ce lies 9 ! inevitable r\ghness ( all kn[n surfaces1 no matt] h[ !y may h be5 wr\ght4 ,! m9ute asp]ities on "o surface are deta9$ 9 correspond+ holl[s 9 ! o!r1 & consequ5tly =ce mu/ be ex]t$ to make "o surface slide ^u ! o!r4 ,by c>e 9 poli%+ ! surfaces ! am\nt ( friction may be dim9i%$2 b x c only be decreas$ to a c]ta9 limit1 beyond :i* no am\nt ( poli%+ seems to produce mu* diff]5ce4 #aae4 ,! law ( friction "u diff]5t conditions mu/ be inquir$ into1 9 ord] t we may make all[ance :5 its effect is ( importance4 ,! discussion ( ! exp]im5ts is sometimes a little difficult1 & ! tru?s arriv$ at are pr9cipally num]ical1 b we %all f9d t "s int]e/+ laws ( nature w appe>4 ,,! ,,mode ,,( ,,exp]im5t+4 #aaf4 ,friction is pres5t betwe5 e pair ( surfaces :i* are 9 contact3 !re is friction betwe5 two pieces ( wood1 & betwe5 a piece ( wood & a piece ( iron2 & ! am\nt ( ! =ce dep5ds ^u ! "* ( bo? surfaces4 ,we %all only exp]im5t ^u ! friction ( wood ^u wood1 z m w be le>n$ by a c>eful /udy ( a special case ?an by a less m9ute exam9ation ( a numb] ( pairs ( diff]5t sub/ances4 #aag4 ,! app>atus us$ is %[n 9 ,fig4 #cb4 ,a plank ( p9e #f' #aa #b is plan$ on its upp] surface1 levell$ by a _s-level1 & firmly secur$ to ! framework at a height ( ab\t #d' f ! gr\nd4 ,on x is a p9e slide #i #i1 ! gra9 ( :i* is crosswise to t ( ! plank2 ^u ! slide ! load ,a is plac$4 ,a rope is atta*$ to ! slide1 :i* passes ov] a v freely m\nt$ ca/ iron pulley ,c1 #ad diamet]1 & c>ries at ! o!r end a hook weigh+ "o p\nd1 f :i* weights ,b c be susp5d$4 #aah4 ,! mode ( exp]im5t+ consi/s 9 plac+ a c]ta9 load ^u ,a1 & !n asc]ta9+ :at weight appli$ to ,b w draw ! load$ slide along ! plane4 ,z sev]al trials are g5]ally necess>y to det]m9e ! p[]1 a rope is atta*$ to ! back ( ! slide1 & passes ov] ! two pulleys ,d2 ?is makes x easy = ! exp]im5t]1 :5 apply+ ! weights at ,b1 to draw back ! slide to ! end ( ! plane by pull+ ! r+ ,e3 ?is rope is ( c\rse left q slack dur+ ! process ( ! exp]im5t1 s9ce ! slide mu/ n be ret>d$4 ,! loads plac$ ^u ,a dur+ ! s]ies ( exp]im5ts rang$ betwe5 "o /one & eight /one4 ,9 ! loads /at$ ! weight ( ! slide itself1 :i* 0 less ?an #a lb41 is always includ$4 ,a v>iety ( small weights 7 provid$ = ! hook ,b2 !y consi/$ ( #ja1 #je1 #a1 #b1 #g1 & #ad lbs4 ,!re is "s friction to be ov]come 9 ! pulley ,c1 b z ! pulley is comp>atively l>ge its friction is small1 ?\gh x 0 always all[$ =4 ,illu/ration3 ,,fig4 #cb4 #aai4 ,an example ( ! exp]im5ts made is ?us describ$4 ,a weight ( #ef lbs4 is plac$ ^u ! slide1 & x is f\nd on trial t #bi lbs4 on ,b 7includ+ ! weight ( ! hook itself7 is suffici5t to />t ! slide2 ?is weight is plac$ ^u ! hook p\nd by p\nd1 c>e be+ tak5 to make ea* addition g5tly4 #abj4 ,exp]im5ts 7 made 9 ?is way ) v>i\s weights ^u ,a1 & ! results are record$ 9 ,table ,i4 ,,table ,i4--,,friction4 ,smoo? horizontal surface ( p9e #gb #aa2 slide also ( p9e #i #i2 gra9 crosswise2 slide is n />t$2 =ce act+ on slide is gradually increas$ until motion comm5ces4 ---------------------------------------- --------------------- ,numb] ( ,load on slide,=ce necess>y,=ce necess>y ,mean ,exp]im5t4 9 lbs41 to move slide4 to move slide4 values4 includ+ weight #a/ ,s]ies4 #bnd ,s]ies4 ( slide4 ---------------------------------------- --------------------- #a #ad #e #h #fe #b #bh #ae #af #aee #c #db #bj #ae #age #d #ef #bi #bd #bfe #e #gj #cc #ca #cbj #f #hd #dc #cc #chj #g #ih #db #ch #djj #h #aab #ej #cc #dae ---------------------------------------- --------------------- ,9 ! fir/ column a numb] is giv5 to ea* exp]im5t = conv5i5ce ( ref]5ce4 ,9 ! second column ! load on ! slide is /at$ 9 lbs4 ,9 ! ?ird column is f\nd ! =ce necess>y to ov]come ! friction4 ,! f\r? column records a second s]ies ( exp]im5ts p]=m$ 9 ! same mann] z ! fir/ s]ies2 :ile ! la/ column %[s ! mean ( ! two frictions4 #aba4 ,! fir/ rem>k to be made ^u ?is table is1 t ! results d n appe> satisfactory or concordant4 ,?us f #f & #g ( ! #a/ s]ies x w\ld appe> t ! friction ( #hd lbs4 0 #dc lbs41 :ile t ( #ih lbs4 0 #db lbs41 s t "h ! great] weight appe>s to h ! less friction1 :i* is evid5tly contr>y to ! :ole t5or ( ! results1 z a glance w %[4 ,moreov] ! frictions 9 ! #a/ & ! #bnd s]ies d n agree1 be+ g5]ally great] 9 ! =m] ?an 9 ! latt]1 ! discordance be+ especially noticeable 9 exp]im5t #h1 :]e ! results 7 #ej lbs4 & #cc lbs4 ,9 ! f9al column ( means ^! irregul>ities are less5$1 = ?is column %[s t ! friction increases ) ! weight1 b x is suffici5t to obs]ve t z ! diff]5ce ( ! #a/ & ! #bnd is #i lbs41 & t ( ! #bnd & ! #crd is only #b lbs41 ! discov]y ( any law f ^! results is hopeless4 #abb4 ,b is friction s caprici\s t x is am5able to no bett] law ?an ^! exp]im5ts appe> to indicate8 ,we mu/ look a little m closely into ! matt]4 ,:5 two pieces ( wood h rema9$ 9 contact & at re/ = "s "t1 a second =ce besides friction resi/s _! sep>ation3 ! wood is compressible1 ! surfaces become closely approximat$1 & ! coh]5ce due to ?is cause mu/ be ov]come be=e motion comm5ces4 ,! initial coh]5ce is unc]ta92 x dep5ds probably on a multitude ( m9ute circum/ances :i* x is impossible to e/imate1 & its pres5ce has vitiat$ ! results :i* we h f\nd s unsatisfactory4 #abc4 ,we c remove ^! irregul>ities by />t+ ! slide at ! comm5cem5t4 ,?is may be conv5i5tly effect$ by ! screw %[n at ,f 9 ,fig4 #cb2 a /r+ atta*$ to its end is fa/5$ to ! slide1 & by giv+ ! h&le ( ! screw a few turns ! slide beg9s to creep4 ,a body once set 9 motion w cont9ue to move ) ! same velocity unless act$ ^u by a =ce2 h5ce ! weight at ,b j ov]comes ! friction :5 ! slide moves uni=mly aft] receiv+ a />t3 ?is velocity 0 9 "o case ( av]age spe$ measur$ to be #af in*es p] m9ute4 #abd4 ,inde$ 9 no case c ! slide comm5ce to move unless ! =ce exce$ ! friction4 ,! am\nt ( ?is excess is indet]m9ate4 ,x is c]ta9ly great] betwe5 wood5 surfaces ?an betwe5 less compressible surfaces l ^? ( metals or glass4 ,9 ! latt] case1 :5 ! =ce exce$s ! friction by a small am\nt1 ! slide />ts (f ) an excessively sl[ motion2 ) wood ! =ce mu/ exce$ ! friction by a l>g] am\nt be=e ! slide comm5ces to move1 b ! motion is !n comp>atively rapid4 #abe4 ,if ! p[] be too small1 ! load ei!r does n cont9ue mov+ aft] ! />t1 or x /ops irregul>ly4 ,if ! p[] be too great1 ! load is drawn ) an accel]at$ velocity4 ,! correct am\nt is easily recognis$ by ! uni=mity ( ! movem5t1 & ev5 :5 ! slide is heavily lad51 a few t5?s ( a p\nd on ! p[] hook cause an appreciable diff]5ce ( velocity4 #abf4 ,! accuracy ) :i* ! friction c be measur$ may be appreciat$ by inspect+ ,table ,,ii4 ,,table ,,ii4--,,friction4 ,smoo? horizontal surface ( p9e #gb #aa2 slide also ( p9e #i #i2 gra9 crosswise2 slide />t$2 =ce appli$ is suffici5t to ma9ta9 uni=m motion ( ! slide4 ---------------------------------------- ------------------- ,load on slide ,=ce ,=ce ,numb] ( 9 lbs41 necess>y necess>y ,mean ,exp]im5t4 includ+ to ma9ta9 to ma9ta9 values4 weight ( motion4 motion4 slide4 #a/ ,s]ies4 #bnd ,s]ies4 ---------------------------------------- ------------------- #a #ad #di #di #di #b #bh #he #hf #he #c #db #abf #abd #abe #d #ef #afc #afb #afb #e #gj #aig #bjj #aih #f #hd #bcg #bcj #bcd #g #ih #bfe #bfa #bfc #h #aab #big #bii #bih ---------------------------------------- ------------------- #abg4 ,two s]ies ( exp]im5ts to det]m9e ! p[] necess>y to ma9ta9 ! motion h be5 record$4 ,?us1 9 exp]im5t #g1 ! load on ! slide be+ #ih lbs41 x 0 f\nd t #bfe lbs4 0 suffici5t to su/a9 ! motion1 & a second trial be+ made indep5d5tly1 ! p[] f\nd 0 #bfa lbs43 a mean ( ! two values1 #bfc lbs41 is adopt$ z be+ ne> ! tru?4 ,! greate/ diff]5ce betwe5 ! two s]ies1 am\nt+ to #jg lb41 is f\nd 9 exp]im5t #f2 a ?ird value 0 !re=e obta9$ = ! friction ( #hd lbs43 ?is am\nt$ to #bce lbs41 :i* is int]m$iate betwe5 ! two =m] results1 & #bcd lbs41 a mean ( ! ?ree1 is adopt$ z ! f9al result4 #abh4 ,! close accordance ( ! exp]im5ts 9 ?is table %[s t ! means ( ! fif? column are probably v ne> ! true values ( ! friction = ! correspond+ loads ^u ! slide4 #abi4 ,! mean frictions mu/1 h[ev]1 be slightly dim9i%$ be=e we c ass]t t !y repres5t only ! friction ( ! wood ^u ! wood4 ,! pulley ov] :i* ! rope passes turns r\nd its axle ) a small am\nt ( friction1 :i* mu/ also be ov]come by ! p[]4 ,! mode ( e/imat+ ?is am\nt1 :i* 9 ^! exp]im5ts nev] exce$s #je lb41 ne$ n n[ be discuss$4 ,! correct$ values ( ! friction are %[n 9 ! ?ird column ( ,table ,,iii4 ,?us1 = example1 ! #di lbs4 ( friction 9 exp]im5t #a consi/s ( #dg1 ! true friction ( ! wood1 & #jb1 :i* is ! friction ( ! pulley2 & #bfc ( exp]im5t #g is simil>ly compos$ ( #beh & #je4 ,x is ! correct$ frictions :i* w be employ$ 9 our subsequ5t calculations4 ,,friction ,,is ,,proportional ,,to ,,! ,,pressure4 #acj4 ,hav+ asc]ta9$ ! values ( ! =ce ( friction = eight diff]5t weights1 we proce$ to inquire into ! laws :i* may be f\nd$ on our results4 ,x is evid5t t ! friction increases ) ! load1 ( :i* x is always great] ?an a f\r?1 & less ?an a ?ird4 ,x is natural to surmise t ! friction 7,f7 is really a con/ant fraction ( ! load 7,r7--9 o!r words1 t ,f k,r1 :]e k is a con/ant numb]4 #aca4 ,to te/ ?is supposition we mu/ try to det]m9e k2 x may be asc]ta9$ by divid+ any value ( ,f by ! correspond+ value ( ,r4 ,if ?is be done1 we %all f9d t ea* ( ! exp]im5ts yields a diff]5t quoti5t2 ! fir/ gives #jccf1 & ! la/ #jbfb1 :ile ! o!r exp]im5ts give results betwe5 ^! extreme values4 ,^! numb]s are tol]ably close toge!r1 b !re is /ill suffici5t discrepancy to %[ t x is n /rictly true to ass]t t ! friction is proportional to ! load4 #acb4 ,b ! law t ! friction v>ies proportionally to ! pressure is s approximately true z to be suffici5t = mo/ practical purposes1 & ! "q !n arises1 :i* ( ! diff]5t values ( k %all we adopt8 ,by a me?od :i* is describ$ 9 ! ,app5dix we c det]m9e a value = k :i*1 :ile x does n repres5t any "o ( ! exp]im5ts precisely1 yet repres5ts !m collectively bett] ?an x is possible = any o!r value to d4 ,! numb] ?us f\nd is #jbg4 ,x is int]m$iate betwe5 ! two values already /at$ to be extreme4 ,! "* ( ?is result is det]m9$ by an inspection ( ,table ,,iii4 ,! f\r? column ( ?is table has be5 calculat$ f ! =mula ,f #jbg ,r4 ,?us1 = example1 9 exp]im5t #e1 ! friction ( a load ( #gj lbs4 is #aid lbs41 & ! product ( #gj & #jbg is #ahi1 :i* is #je lb4 less ?an ! true am\nt4 ,9 ! la/ column ( ?is table ! discrepancies betwe5 ! obs]v$ & ! calculat$ values are record$1 = facility ( comp>ison4 ,x w be obs]v$ t ! greate/ diff]5ce is "u #a lb4 ,,table ,,iii4--,,friction4 ,friction ( p9e ^u p9e2 ! mean values ( ! friction giv5 9 ,table ,,ii4 7correct$ = ! friction ( ! pulley7 comp>$ ) ! =mula ,f #jbg ,r4 ---------------------------------------- ---------------- ,numb] ( ,r4 ,correct$ ,f4 ,discrepancies ,exp]im5t4,total loadmean value,calculat$ betwe5 ! on slide ( value ( obs]v$ & 9 lbs4 friction4 friction4 calculat$ frictions4 ---------------------------------------- ---------------- #a #ad #dg #ch -#ji #b #bh #hb #gf -#jf #c #db #abb #aac -#ji #d #ef #aeh #aea -#jg #e #gj #aid #ahi -#je #f #hd #bcj #bbg -#jc #g #ih #beh #bfe #jg #h #aab #bic #cjb #ji ---------------------------------------- ---------------- #acc4 ,h5ce ! law ,f #jbg ,r repres5ts ! exp]im5ts ) tol]able accuracy2 & ! num]ical ratio ,o#bg is call$ ! coeffici5t ( friction4 ,we may apply ?is law to asc]ta9 ! friction 9 any case :]e ! load lies betwe5 #ad lbs4 & #aab lbs42 = example1 if ! load be #fc lbs41 ! friction is #fc #jbg #agj4 #acd4 ,! coeffici5t ( friction w\ld h be5 slightly diff]5t _h ! gra9 ( ! slide be5 p>allel to t ( ! plank2 & x ( c\rse v>ies ) ! nature ( ! surfaces4 ,exp]im5t]s h giv5 tables ( ! coeffici5ts ( friction ( v>i\s sub/ances1 wood1 /one1 metals1 c4 ,! use ( ^! coeffici5ts dep5ds ^u ! assumption ( ! ord9>y law ( friction1 namely1 t ! friction is proportional to ! pressure3 ?is law is accurate en\gh = mo/ purposes1 especially :5 us$ = loads t lie betwe5 ! extreme weights employ$ 9 calculat+ ! value ( ! coeffici5t :i* is employ$4 ,a ,,m ,,accurate ,,law ,,( ,,friction4 #ace4 ,9 mak+ "o ( our measurem5ts ) c>e1 x is unusual to h an error ( m ?an a few t5?s ( #a lb4 & x is h>dly possible t any ( ! mean frictions we h f\nd %\ld be 9 error to s great an ext5t z #je lb4 ,b ) ! value ( ! coeffici5t ( friction :i* is us$ 9 ,table ,,iii41 ! discrepancies am\nt sometimes to #ji lbs4 ,) any o!r num]ical coeffici5t ?an #jbg1 ! discrepancies w\ld h be5 ev5 /ill m s]i\s4 ,z ^! are too great to be attribut$ to errors ( exp]im5t1 we h to inf] t ! law ( friction :i* has be5 assum$ _c be /rictly true4 ,! signs ( ! discrepancies indicate t ! law gives frictions :i* = small loads are too small1 & = l>ge loads are too great4 #acf4 ,we are !re=e l$ to inquire :e!r "s o!r relation betwe5 ,f & ,r may n repres5t ! exp]im5ts ) great] fidelity ?an ! common law ( friction4 ,if we dim9i%$ ! coeffici5t by a small am\nt1 & !n add$ a con/ant quantity to ! product ( ! coeffici5t & ! load1 ! effect ( ?is *ange w\ld be t = small loads ! calculat$ values w\ld be increas$1 :ile = l>ge loads !y w\ld be dim9i%$4 ,?is is ! k9d ( *ange :i* we h indicat$ to be necess>y = reconciliation betwe5 ! obs]v$ & calculat$ values4 #acg4 ,we !re=e inf] t a relation ( ! =m ,f x y ,r w probably express a m correct law1 provid$ we c f9d x & y4 ,"o equation betwe5 x & y is obta9$ by introduc+ any value ( ,r ) ! correspond+ value ( ,f1 & a second equation c be f\nd by tak+ any o!r simil> pair4 ,f ^! two equations ! values ( x & ( y may be d$uc$ by elem5t>y algebra1 b ! be/ =mula w be obta9$ by comb9+ toge!r all ! pairs ( correspond+ values4 ,= ?is reason ! me?od describ$ 9 ! ,app5dix mu/ be us$1 :i*1 z x is f\nd$ on all ! exp]im5ts1 mu/ give a ?or\ghly repres5tative result4 ,! =mula ?us det]m9$1 is ,f #add #jbeb ,r4 ,?is =mula is comp>$ ) ! exp]im5ts 9 ,table ,,iv4 ,,table ,,iv4--,,friction4 ,friction ( p9e ^u p9e2 ! mean values ( ! friction giv5 9 ,table ,,ii4 7correct$ = ! friction ( ! pulley7 comp>$ ) ! =mula ,f #add #jbeb ,r4 ---------------------------------------- ----------------- ,discrepancies ,numb] ( ,r4 ,correct$ ,f4 betwe5 ! ,exp]im5t4,total load mean ,calculat$obs]v$ & on slide value ( value ( calculat$ 9 lbs4 friction4 friction4 frictions4 ---------------------------------------- ----------------- #a #ad #dg #ej #jc #b #bh #hb #he #jc #c #db #abb #abj -#jb #d #ef #aeh #aef -#jb #e #gj #aid #aia -#jc #f #hd #bcj #bbf -#jd #g #ih #beh #bfa #jc #h #aab #bic #big #jd ---------------------------------------- ----------------- ,! f\r? column conta9s ! calculat$ values3 ?us1 = example1 9 exp]im5t #d1 :]e ! load is #ef lbs41 ! calculat$ value is #add #jbeb #ef #aef2 ! diff]5ce #jb betwe5 ?is & ! obs]v$ value #aeh is %[n 9 ! la/ column4 #ach4 ,x w be notic$ t ! greate/ discrepancy 9 ?is column is #jd lbs41 & t !re=e ! =mula repres5ts ! exp]im5ts ) consid]able accuracy4 ,x is und\bt$ly ne>] ! tru? ?an ! =m] law 7,art4 #acb72 9 fact1 ! diff]5ces are n[ su* z might really belong to errors unavoidable 9 mak+ ! exp]im5ts4 #aci4 ,?is =mula may be us$ = calculat+ ! friction = any load betwe5 #ad lbs4 & #aab lbs4 ,?us1 if ! load be #fc lbs41 ! friction is #add #jbeb #fc #agc lbs41 :i* does n diff] mu* f #agj lbs41 ! value f\nd by ! m ord9>y law4 ,we mu/1 h[ev]1 be cauti\s n to apply ?is =mula to weights :i* d n lie betwe5 ! limits ( ! greate/ & lea/ weight us$ 9 ^? exp]im5ts by :i* ! num]ical values 9 ! =mula h be5 det]m9$2 = example1 to take an extreme case1 if ,r #j1 ! =mula w\ld indicate t ! friction 0 #add1 :i* is evid5tly absurd2 "h ! =mula errs 9 excess1 :ile if ! load 7 v l>ge x is c]ta9 ! =mula w\ld err 9 defect4 ,,! ,,coeffici5t ,,v>ies ,,) ,,! ,,weights ,,us$4 #adj4 ,9 a subsequ5t lecture we %all employ z an incl9$ plane ! plank we h be5 exam9+1 & we %all require to use ! k ( its friction :i* we are n[ acquir+4 ,! weights :i* we %all !n employ range f #g lbs4 to #ef lbs4 ,assum+ ! ord9>y law ( friction1 we h f\nd t #jbg is ! be/ value ( its coeffici5t :5 ! loads range betwe5 #ad lbs4 & #aab lbs4 ,suppose we only consid] loads up to #ef lbs41 we f9d t ! coeffici5t #jbhh w be/ repres5t ! exp]im5ts )9 ?is range1 ?\gh = #aab lbs4 x w\ld give an error ( ne>ly #c lbs4 ,! results calculat$ by ! =mula ,f #jbhh ,r are %[n 9 ,table ,v41 :]e ! greate/ diff]5ce is #jg lb4 ,,table ,v4--,,friction4 ,friction ( p9e ^u p9e2 ! mean values ( ! friction giv5 9 ,table ,,ii4 7correct$ = ! friction ( ! pulley7 comp>$ ) ! =mula ,f #jbhh ,r ---------------------------------------- ------------- ,discrepancies ,numb] ( ,r4 ,correct$ ,f4 betwe5 ! ,exp]im5t4,total load mean ,calculat$ obs]v$ & on slide value ( value ( calculat$ 9 lbs4 friction4 friction4 frictions4 ---------------------------------------- ------------- #a #ad #dg #dj -#jg #b #bh #hb #ha -#ja #c #db #abb #aba -#ja #d #ef #aeh #afa #jc ---------------------------------------- ------------- #ada4 ,b we c replace ! common law ( friction by ! m accurate law ( ,art4 #acg1 & ! =mula comput$ s z to be/ h>monise ! exp]im5ts up to #ef lbs41 disreg>d+ ! high] loads1 is ,f #ji #jbff ,r4 ,?is =mula is obta9$ by ! me?od ref]r$ to 9 ,art4 #acg4 ,we f9d t x repres5ts ! exp]im5ts bett] ?an t us$ 9 ,table ,v4 ,betwe5 ! limits nam$1 ?is =mula is also m accurate ?an t ( ,table ,,iv4 ,x is comp>$ ) ! exp]im5ts 9 ,table ,,vi41 & x w be notic$ t x repres5ts !m ) great precision1 z no discrepancy exce$s #ja4 ,,table ,,vi4--,,friction4 ,friction ( p9e ^u p9e2 ! mean values ( ! friction giv5 9 ,table ,,ii4 7correct$ = ! friction ( ! pulley7 comp>$ ) ! =mula ,f #ji #jbff ,r4 ---------------------------------------- ------------- ,discrepancies ,numb] ( ,r4 ,correct$ ,f4 betwe5 ! ,exp]im5t4,total load mean ,calculat$ obs]v$ & on slide value ( value ( calculat$ 9 lbs4 friction4 friction4 frictions4 ---------------------------------------- ------------- #a #ad #dg #df -#ja #b #bh #hb #hc #ja #c #db #abb #aba -#ja #d #ef #aeh #aeh #jj ---------------------------------------- ------------- ,illu/ration3 ,,fig4 #cc4 ,,! ,,angle ,,( ,,friction4 #adb4 ,!re is ano!r mode ( exam9+ ! action ( friction besides t we h be5 consid]+4 ,! app>atus = ?is purpose is %[n 9 ,fig4 #cc1 9 :i* ,b ,c repres5ts ! plank ( p9e we h already us$4 ,x is n[ m\nt$ s z to be capable ( turn+ ab\t "o end ,b2 ! end ,c is susp5d$ f "o hook ( ! *a9 f ! 8epicycloidal0 pulley-block ,e4 ,?is block is v conv5i5t = ! purpose4 ,by its means ! incl9ation ( ! plank c be adju/$ ) ! greate/ nicety1 z ! rais+ *a9 ,g is held 9 "o h& & ! l[]+ *a9 ,f 9 ! o!r4 ,ano!r great conv5i5ce ( ?is block is t ! load does n run d[n :5 ! lift+ *a9 is left free4 ,! plank is clamp$ to ! h+e ab\t :i* x turns4 ,! frames by :i* bo? ! h+e & ! block are support$ are weight$ 9 ord] to secure /ead9ess4 ,! incl9ation ( ! plane is easily asc]ta9$ by measur+ ! diff]5ce 9 height ( its two ends above ! floor1 & !n mak+ a draw+ on ! prop] scale4 ,! />t+-screw ,d1 ^: use has be5 already m5tion$1 is also fa/5$ to ! framework 9 ! position %[n 9 ! figure4 #adc4 ,suppose ! slide ,a be weight$ & plac$ ^u ! incl9$ plane ,b ,c2 if ! end ,c be only slightly elevat$1 ! slide rema9s at re/2 ! reason be+ t ! friction betwe5 ! slide & ! plane neutralizes ! =ce ( gravity4 ,b suppose1 by means ( ! pulley-block1 ,c be gradually rais$2 an elevation is at la/ rea*$ at :i* ! slide />ts (f1 & runs ) an accel]at+ velocity to ! bottom ( ! plane4 ,! angle ( elevation ( ! plane :5 ?is occurs is call$ ! angle ( /atical friction4 #add4 ,! weights ) :i* ! slide 0 lad5 9 ^! exp]im5ts 7 #ad lbs41 #ef lbs41 & #aab lbs41 & ! results are giv5 9 ,table ,,vii4 ,we see t a load ( #ef lbs4 />t$ :5 ! plane rea*$ an incl9ation ( #bja 9 ! fir/ s]ies1 & ( #agb 9 ! second1 ! mean value #ahf be+ giv5 9 ! fif? column4 ,^! means = ! ?ree diff]5t weights agree s closely t we ass]t ! rem>kable law t ! angle ( friction does n dep5d ^u ! magnitude ( ! load4 ,,table ,,vii4--,,angle ,,( ,,/atical ,,friction4 ,a smoo? plane ( p9e #gb #aa c>ries a load$ slide ( p9e #i #i2 "o end ( ! plane is gradually elevat$ until ! slide />ts (f4 ---------------------------------------- ----------------- ,numb] ( ,total load on ,angle ( ,angle ( ,mean values ,exp]im5t4 ! slide elevation4 elevation4 ( ! 9 lbs4 #a/ ,s]ies4#bnd ,s]ies4 angles4 ---------------------------------------- ----------------- #a #ad #aie ---- #aie #b #ef #bja #agb #ahf #c #aab #bjc #ahi #aif ---------------------------------------- ----------------- #ade4 ,we might1 h[ev]1 proce$ diff]5tly 9 det]m9+ ! angle ( friction1 by giv+ ! load a />t1 & asc]ta9+ if ! motion w cont9ue4 ,to d s requires ! aid ( an assi/ant1 :o w />t ! load ) ! help ( ! screw1 :ile ! elevation ( ! plane is be+ sl[ly increas$4 ,! result ( ^! exp]im5ts is giv5 9 ,table ,,viii4 ,,table ,,viii4--,,angle ,,( ,,friction4 ,a smoo? plane ( p9e #gb #aa c>ries a load$ slide ( p9e #i #i2 "o end ( ! plane is gradually elevat$ until ! slide1 hav+ receiv$ a />t1 moves (f uni=mly4 ---------------------------------------- ,numb] ( ,total load on ,angle ( ,exp]im5t4! slide 9 lbs4incl9ation4 ---------------------------------------- #a #ad #adc #b #ef #acj #c #aab #acj ---------------------------------------- ,we see f ?is table also t ! angle ( friction is indep5d5t ( ! load1 b ! angle is 9 ?is case less by #e or #f ?an 0 f\nd necess>y to imp>t motion :5 a />t 0 n giv54 #adf4 ,x is commonly /at$ t ! coeffici5t ( friction equals ! tang5t ( ! angle ( friction1 & ?is c be prov$ to be true :5 ! ord9>y law ( friction is assum$4 ,b z we h se5 t ! law ( friction is only approximately correct1 we ne$ n expect to f9d ?is o!r law completely v]ifi$4 #adg4 ,:5 ! slide is />t$1 ! mean value ( ! angle ( friction is #acd4 ,! tang5t ( ?is angle is #jbd3 ?is is ab\t #aa p] c5t4 less ?an ! coeffici5t ( friction #jbg1 :i* we h already det]m9$4 ,! mean value ( ! angle ( friction :5 ! slide is n />t$ is #aib1 & its tang5t is #jce4 ,! exp]im5ts ( ,table ,i4 are1 z already po9t$ out1 r unsatisfactory1 b we ref] to !m "h to %[ t1 s f> z !y g1 ! coeffici5t ( friction is 9 no s5se equal to ! tang5t ( ! angle ( friction4 ,if we adopt ! mean values giv5 9 ! la/ column ( ,table ,i41 ! be/ coeffici5t ( friction :i* c be d$uc$ is #jda4 ,:e!r1 !re=e1 ! slide be />t$ or n />t$1 ! tang5t ( ! angle ( friction is small] ?an ! correspond+ coeffici5t ( friction4 ,:5 ! slide is />t$1 ! tang5t is ab\t #aa p] c5t4 less ?an ! coeffici5t2 & :5 ! slide is n />t$1 x is ab\t #ad p] c5t4 less4 ,!re are d\btless _m cases 9 :i* ^! diff]5ces are suffici5tly small to be neglect$1 & 9 :i*1 !re=e1 ! law may be receiv$ z true4 ,,ano!r ,,law ,,( ,,friction4 #adh4 ,! area ( ! wood5 slide is #i #i1 b we w\ld h f\nd t ! friction "u a giv5 load 0 practically ! same :atev] 7 ! area ( ! slide1 s long z its mat]ial rema9$ unalt]$4 ,?is foll[s z a consequ5ce ( ! approximate law t ! friction is proportional to ! pressure4 ,suppose t ! weight 7 #ajj lbs41 & ! area ( ! slide #ajj in*es1 !re w\ld !n be a pressure ( #a lb4 p] squ>e in* ov] ! surface ( ! slide1 & !re=e ! friction to be ov]come on ea* squ>e in* w\ld be #jbg lb41 or = ! :ole slide #bg lbs4 ,if1 h[ev]1 ! slide _h only an area ( #ej squ>e in*es1 ! load w\ld produce a pressure ( #b lbs41 p] squ>e in*2 ! friction w\ld !re=e be #b #jbg #jed lb4 = ea* squ>e in*1 & ! total friction w\ld be #ej #jed #bg lbs41 ! same z be=e3 h5ce ! total friction is indep5d5t ( ! ext5t ( surface4 ,?is w\ld rema9 equally true ev5 ?\gh ! weight 7 n1 z we h suppos$1 uni=mly di/ribut$ ov] ! surface ( ! slide4 ,,conclud+ ,,rem>ks4 #adi4 ,! importance ( friction 9 me*anics arises f its univ]sal pres5ce4 ,we (t5 recognize x z a de/roy] or imp$] ( motion1 z a wa/] ( our en]gy1 & z a s\rce ( loss or inconv5i5ce4 ,b1 on ! o!r h&1 friction is (t5 indirectly ! means ( produc+ motion1 & ( ?is we h a spl5did example 9 ! locomotive eng9e4 ,! eng9e be+ v heavy1 ! :eels are press$ closely to ! rails2 !re is friction en\gh to prev5t ! :eels slipp+1 consequ5tly :5 ! eng9es =ce ! :eels to turn r\nd !y mu/ roll onw>ds4 ,! coeffici5t ( friction ( wr\ght iron ^u wr\ght iron is ab\t #jb4 ,suppose a locomotive weigh #cj tons1 & ! %>e ( ?is weight borne by ! driv+ :eels be #aj tons1 ! friction betwe5 ! driv+ :eels & ! rails is #b tons4 ,?is is ! greate/ =ce ! eng9e c ex]t on a level l9e4 ,a =ce ( #aj lbs4 = e ton weight ( ! tra9 is kn[n to be suffici5t to su/a9 ! motion1 consequ5tly ! eng9e we h suppos$ %\ld draw along ! level a load ( #ddh tons4 #aej4 ,b we ne$ n invoke ! /eam-eng9e to %[ ! use ( friction4 ,we c\ld n exi/ )\t x4 ,9 ! fir/ place we c\ld n move ab\t1 = walk+ is only possible on acc\nt ( ! friction betwe5 ! soles ( our boots & ! gr\nd2 nor if we 7 once 9 motion c\ld we /op )\t com+ into collision ) "s o!r object1 or grasp+ some?+ to hold on by4 ,objects c\ld only be h&l$ ) difficulty1 nails w\ld n rema9 9 wood1 & screws w\ld be equally useless4 ,build+s c\ld h>dly be erect$1 nay1 ev5 hills & m\nta9s w\ld gradually disappe>1 & f9ally dry l& w\ld be imm]s$ b5ea? ! level ( ! sea4 ,friction is1 s f> z we are conc]n$1 q z ess5tial a law ( nature z ! law ( gravitation4 ,we mu/ n seek to evade x 9 our me*anical discussions because x makes !m a little m difficult4 ,friction obeys laws2 its action is n vague or unc]ta94 ,:5 inconv5i5t x c be dim9i%$1 :5 useful x c be increas$2 & 9 our lectures on ! me*anical p[]s1 to :i* we n[ proce$1 we %all h opportunities ( describ+ ma*9es :i* h be5 devis$ 9 ob$i5ce to its laws4 ,,lecture ,,vi4 ,,! ,,pulley4 ,introduction4--,friction betwe5 a ,rope & an ,iron ,b>4--,! use ( ! ,pulley4--,l>ge & ,small ,pulleys4--,! ,law ( ,friction 9 ! ,pulley4--,:eels4--,en]gy4 ,,introduction4 #aea4 ,! pulley =ms a good introduction to ! important subject ( ! me*anical p[]s4 ,b be=e ent]+ on ! discussions ( ! next few *apt]s1 x w be necess>y = u to expla9 :at is meant 9 me*anics by 8"w10 & by 8en]gy10 :i* is ! capacity = p]=m+ "w1 & we %all !re=e include a %ort outl9e ( ?is subject 9 ! pres5t lecture4 #aeb4 ,! pulley is a ma*9e :i* is employ$ = ! purpose ( *ang+ ! direction ( a =ce4 ,we frequ5tly wi% to apply a =ce 9 a diff]5t direction f t 9 :i* x is conv5i5t to ex]t x1 & ! pulley enables u to d s4 ,we are n n[ speak+ ( ^! arrangem5ts = increas+ p[] 9 :i* pulleys play an important "p2 ^! w be consid]$ 9 ! next lecture3 we at pres5t ref] only to *ange ( direction4 ,9 fact1 z we %all %ortly see1 "s =ce is ev5 wa/$ :5 ! s+le fix$ pulley is us$1 s t ?is ma*9e c]ta9ly _c be call$ a me*anical p[]4 #aec4 ,! occasions ^u :i* a s+le fix$ pulley is us$ are num]\s & famili>4 ,let u suppose a sack ( corn has to be elevat$ f ! l[] to "o ( ! upp] /ories ( a build+4 ,x may ( c\rse be rais$ by a man :o c>ries x1 b he has to lift 8 own weight 9 addition to t ( ! sack1 & !re=e ! quantity ( ex]tion us$ is great] ?an absolutely necess>y4 ,b suppos+ !re be a pulley at ! top ( ! build+ ov] :i* a rope passes2 !n1 if a man atta* "o end ( ! rope to ! sack & pull ! o!r1 he raises ! sack )\t rais+ 8 own weight4 ,! pulley has ?us provid$ ! means by :i* ! d[nw>d =ce has be5 *ang$ 9 direction to an upw>d =ce4 #aed4 ,! weights1 ropes1 & pulleys :i* are us$ 9 our w9d[s = c\nt]pois+ ! weight ( ! sa% af=d a v famili> in/ance ( h[ a pulley *anges ! direction ( a =ce4 ,"h ! d[nw>d =ce ( ! weight is *ang$ by means ( ! pulley into an upw>d =ce1 :i* ne>ly c\nt]balances ! weight ( ! sa%4 ,,friction ,,betwe5 ,a ,,rope ,,& ,,an ,,iron ,,b>4 #aee4 ,e "o is famili> ) ! ord9>y =m ( ! pulley2 x consi/s ( a :eel capable ( turn+ freely on its axle1 & x has a groove 9 its circumf]5ce 9 :i* ! rope lies4 ,b :y is x necess>y to give ! pulley ?is =m8 ,:y c\ld n ! direction ( ! rope be *ang$ by simply pass+ x ov] a b>1 z well z by ! m complicat$ pulley8 ,we %all be/ answ] ?is "q by actually try+ ! exp]im5t1 :i* we c d by means ( ! app>atus ( ,fig4 #cd 7see page #ij74 ,9 ?is are %[n two iron /uds1 ,g1 ,h1 #jf diamet]1 & ab\t #h ap>t2 ov] ^! passes a rope1 :i* has a hook at ea* end4 ,if ,i susp5d a weight ( #ad lbs4 f "o hook ,a1 & pull ! hook ,b1 ,i c by ex]t+ suffici5t =ce raise ! weight on ,a1 b ) ?is arrangem5t ,i am consci\s ( hav+ to ex]t a v mu* l>g] =ce ?an w\ld h be5 necess>y to raise #ad lbs4 by m]ely lift+ x4 #aef4 ,9 ord] to /udy ! "q exactly1 we %all asc]ta9 :at weight susp5d$ f ! hook ,b w suffice to raise ,a4 ,i f9d t 9 ord] to raise #ad lbs4 on ,a no less ?an #dg lbs4 is necess>y on ,b1 consequ5tly !re is an enorm\s loss ( =ce3 m ?an two-?irds ( ! =ce :i* is ex]t$ is exp5d$ uselessly4 ,if in/ead ( ! #ad lbs4 weight ,i sub/itute any o!r weight1 ,i f9d ! same result1 viz4 t m ?an ?ree times its am\nt is necess>y to raise x by means ( ! rope pass+ ov] ! /uds4 ,if a lab\r]1 9 rais+ a sack1 7 to pass a rope ov] two b>s su* z ^!1 !n = e /one ! sack weigh$ he w\ld h to ex]t a =ce ( m ?an ?ree /ones1 & !re w\ld be a v extravagant loss ( p[]4 #aeg4 ,:5ce arises ?is loss8 ,! rope 9 mov+ slides ov] ! surface ( ! iron /uds4 ,al?\gh ^! are q smoo? & poli%$1 yet :5 !re is a /ra9 on ! rope x presses closely ^u !m1 & !re is a c]ta9 am\nt ( =ce necess>y to make ! rope slide along ! iron4 ,9 o!r words1 :5 ,i am try+ to raise up #ad lbs4 ) ?is contrivance1 ,i n only h its weight oppos$ to me1 b also ano!r =ce due to ! slid+ ( ! rope on ! iron3 ?is =ce is due to friction4 ,7 x n = friction1 a =ce ( #ad lbs4 on "o hook w\ld exactly balance #ad lbs4 on ! o!r1 & ! slighte/ addition to ei!r weight w\ld make x desc5d & raise ! o!r4 ,if1 !n1 we are oblig$ to *ange ! direction ( a =ce1 we mu/ devise "s means ( do+ s :i* does n require s great a sacrifice z ! arrangem5t we h j us$4 ,,! ,,use ,,( ,,! ,,pulley4 #aeh4 ,we %all next inquire h[ x is t we are enabl$ to obviate friction by means ( a pulley4 ,x is evid5t we mu/ provide an arrangem5t 9 :i* ! rope %all n be requir$ to slide ^u an iron surface4 ,?is end is atta9$ by ! pulley1 ( :i* we may take ,i1 ,fig4 #cd1 z an example4 ,?is repres5ts a ca/ iron :eel #ad 9 diamet]1 ) a ,v-%ap$ groove 9 its circumf]5ce to receive ! rope3 ?is :eel turns on a in* wr\ght iron axle1 :i* is well oil$4 ,! rope us$ is ab\t #jbe 9 diamet]4 #aei4 ,f ! hooks ,e1 ,f at ea* end ( ! rope a #ad lb4 weight is susp5d$4 ,^! equal weights balance ea* o!r4 ,accord+ to our =m] exp]im5t ) ! /uds1 x w\ld be necess>y = me to treble ! weight on "o ( ^! hooks 9 ord] to raise ! o!r1 b n[ ,i f9d t an additional #je lb4 plac$ on ei!r hook causes x to desc5d & make ! o!r asc5d4 ,?is is a great improvem5t2 #je lb4 n[ accompli%es :at #cc lbs4 0 be=e requir$ =4 ,we h avoid$ a great deal ( friction1 b we h n got rid ( x altoge!r1 = #jbe lb4 is incompet5t1 :5 add$ to ei!r weight1 to make t weight desc5d4 #afj4 ,to :at is ! improvem5t due8 ,:5 ! weight desc5ds ! rope does n slide ^u ! :eel1 b x causes ! :eel to revolve ) x1 consequ5tly !re is little or no friction at ! circumf]5ce ( ! pulley2 ! friction is transf]r$ to ! axle4 ,we /ill h "s resi/ance to ov]come1 b = smoo? oil$ iron axles ! friction is v small1 h5ce ! advantage ( ! pulley4 ,!re is 9 e pulley a small loss ( p[] f ! =ce exp5d$ 9 b5d+ ! rope2 ?is ne$ n conc]n u at pres5t1 = ) ! pliable plait$ rope t we h employ$ ! effect is inappreciable1 b ) l>ge /rong ropes ! loss becomes ( importance4 ,! am\nt ( loss by us+ diff]5t k9ds ( ropes has be5 det]m9$ by c>eful exp]im5ts4 ,,l>ge ,,& ,,small ,,pulleys4 #afa4 ,!re is (t5 a consid]able advantage obta9$ by us+ l>ge r ?an small pulleys4 ,! am\nt ( =ce necess>y to ov]come friction v>ies inv]sely z ! size ( ! pulley4 ,we %all demon/rate ?is by actual exp]im5t ) ! app>atus ( ,fig4 #cd4 ,a small pulley ,k is atta*$ to ! l>ge pulley ,i2 !y are 9 fact "o piece1 & turn toge!r on ! same axle4 ,h5ce if we fir/ det]m9e ! friction ) ! rope ov] ! l>ge pulley1 & !n ) ! rope ov] ! small pulley1 any diff]5ce c only be due to ! diff]5ce 9 size1 z all ! o!r circum/ances are ! same4 #afb4 ,9 mak+ ! exp]im5ts we mu/ att5d to ! foll[+ po9t4 ,! pulleys & ! socket on :i* !y are m\nt$ weigh sev]al p\nds1 & consequ5tly !re is friction on ! axle aris+ f ! weight ( ! pulleys1 q indep5d5tly ( any weights t may be plac$ on ! hooks4 ,we mu/ !n1 if possible1 evade ! friction ( ! pulley itself1 s t ! am\nt ( friction :i* is obs]v$ w be entirely due to ! weights rais$4 ,?is c be easily done4 ,! rope & hooks be+ on ! l>ge pulley ,i1 ,i f9d t #jaf lb4 atta*$ to "o ( ! hooks1 ,e1 is suffici5t to ov]come ! friction ( ! pulley1 & to make t hook desc5d & raise ,f4 ,if !re=e we leave #jaf lb4 on ,e1 we may consid] ! friction due to ! weight ( ! pulley1 rope1 & hooks z neutraliz$4 ,illu/ration3 ,,fig4 #cd4 #afc4 ,i n[ place a /one weight on ea* ( ! hooks ,e & ,f4 ,! am\nt necess>y to make ! hook ,e & its load desc5d is #jbh lb4 ,?is does n ( c\rse include ! weight ( #jaf lb4 already ref]r$ to4 ,we see !re=e t ) ! l>ge pulley ! am\nt ( friction to be ov]come 9 rais+ "o /one is #jbh lb4 #afd4 ,let u n[ p]=m precisely ! same exp]im5t ) ! small pulley4 ,i transf] ! same rope & hooks to ,k1 & ,i f9d t #jaf lb4 is n n[ suffici5t to ov]come ! friction ( ! pulley1 b ,i add on weights until ,c w j desc5d1 :i* occurs :5 ! load rea*es #jie lb4 ,?is weight is to be left on ,c z a c\nt]poise1 = ! reasons already po9t$ out4 ,i place a /one weight on ,c & ano!r on ,d1 & y see t ,c w desc5d :5 x receives an additional load ( #ace lbs42 ?is is !re=e ! am\nt ( friction to be ov]come :5 a /one weight is rais$ ov] ! pulley ,k4 #afe4 ,let u comp>e ^! results ) ! dim5sions ( ! pulleys4 ,! prop] way to measure ! effective circumf]5ce ( a pulley :5 c>ry+ a c]ta9 rope is to measure ! l5g? ( t rope :i* w j embrace x4 ,! l5g? measur$ 9 ?is way w ( c\rse dep5d to a c]ta9 ext5t ^u ! size ( ! rope4 ,i f9d t ! circumf]5ces ( ! two pulleys are #dcj & #ie4 ,! ratio ( ^! is #de2 ! correspond+ resi/ances f friction we h se5 to be #jbh lb4 & #ace lbs4 ,! l>g] ( ^! quantities is #dh times ! small]4 ,?is numb] is v close to #de2 we mu/ n1 z already expla9$1 expect p]fect accuracy 9 exp]im5ts 9 friction4 ,9 ! pres5t case ! agreem5t is )9 ! _/? ( ! :ole1 & we may reg>d x z a pro( ( ! law t ! friction ( a pulley is inv]sely proportional to its circumf]5ce4 #aff4 ,x is easy to see ! reason :y friction %\ld dim9i% :5 ! size ( ! pulley is increas$4 ,! friction acts at ! circumf]5ce ( ! axle ab\t :i* ! :eel turns2 x is !re pres5t z a =ce t5d+ to ret>d motion4 ,n[ ! l>g] ! :eel ! great] w be ! di/ance f ! axis at :i* ! =ce acts :i* ov]comes ! friction1 & !re=e ! less ne$ be ! magnitude ( ! =ce4 ,y w p]haps und]/& ?is bett] aft] ! pr9ciple ( ! lev] has be5 discuss$4 #afg4 ,we may d$uce f ^! consid]ations ! practical maxim t l>ge pulleys are economical ( p[]4 ,?is rule is well kn[n to eng9e]s2 l>ge pulleys %\ld be us$1 n only = dim9i%+ friction1 b also to avoid loss ( p[] by excessive b5d+ ( ! rope4 ,a rope is b5t gradually ar\nd ! circumf]5ce ( a l>ge pulley ) f> less =ce ?an is necess>y to accommodate x to a small] pulley3 ! rope also is apt to become injur$ by excessive b5d+4 ,9 coal pits ! trucks lad5 ) coal are hoi/$ to ! surface by means ( wire ropes :i* pass f ! pit ov] a pulley into ! eng9e-h\se3 ?is pulley is ( v l>ge dim5sions1 = ! reasons we h po9t$ out4 ,,! ,,law ,,( ,,friction ,,9 ,,! ,,pulley4 #afh4 ,i h "h a wood5 pulley #ce 9 diamet]2 ! hole is l9$ ) brass1 & ! pulley turns v freely on an iron sp9dle4 ,i place ! rope & hooks ^u ! groove4 ,brass rubb+ on iron has b little friction1 & :5 #g lbs4 is plac$ on ea* hook1 #je lb4 add$ to ei!r w make x desc5d & raise up ! o!r4 ,let #ad lbs4 be plac$ on ea* hook1 #je lb4 is no long] suffici5t2 #a lb4 is requir$3 h5ce :5 ! weight is d\bl$ ! friction is also d\bl$4 ,repeat+ ! exp]im5t ) #ba lbs4 & #bh lbs4 on ea* side1 ! correspond+ weights necess>y to ov]come friction are #ae lbs4 & #b lbs4 ,9 ! f\r exp]im5ts ! weights us$ are 9 ! proportion #a1 #b1 #c1 #d2 & ! =ces necess>y to ov]come friction1 #je lb41 #a lb41 #ae lbs41 & #b lbs41 are 9 ! same proportion4 ,h5ce ! friction is proportional to ! load4 ,,:eels4 #afi4 ,! :eel is "o ( ! mo/ simple & effective devices = ov]com+ friction4 ,a sleigh is an admirable vehicle on a smoo? surface su* z ice1 b x is totally unadapt$ = use on common roads2 ! reason be+ t ! am\nt ( friction betwe5 ! sleigh & ! road is s great t to move ! sleigh ! horse w\ld h to ex]t a =ce :i* w\ld be v great comp>$ ) ! load he 0 draw+4 ,b a vehicle prop]ly m\nt$ on :eels moves ) ! greate/ ease along ! road1 = ! circumf]5ce ( ! :eel does n slide1 & consequ5tly !re is no friction betwe5 ! :eel & ! road2 ! :eel h[ev] turns on its axle1 !re=e !re is slid+1 & consequ5tly friction1 at ! axle1 b ! axle & ! :eel are prop]ly fitt$ to ea* o!r1 & ! surfaces are lubricat$ ) oil1 s t ! friction is extremely small4 #agj4 ,) l>ge :eels ! am\nt ( friction on ! axle is less ?an ) small :eels2 o!r advantages ( l>ge :eels are t !y d n s9k mu* into depressions 9 ! roads1 & t !y h also an increas$ facility 9 surm\nt+ ! innum]able small ob/acles f :i* ev5 ! be/ road is n free4 #aga4 ,:5 x is desir$ to make a pulley turn ) extremely small friction1 its axle1 in/ead ( revolv+ 9 fix$ be>+s1 is m\nt$ ^u :at are call$ friction :eels4 ,a set ( friction :eels is %[n 9 ! app>atus ( ,fig4 #ff3 z ! axle revolves1 ! friction betwe5 ! axles & ! :eels causes ! latt] to turn r\nd ) a comp>atively sl[ motion2 ?us all ! friction is transf]r$ to ! axles ( ! f\r friction :eels2 ^! revolve 9 _! be>+s ) extreme sl[ness1 & consequ5tly ! pulley is b little affect$ by friction4 ,! am\nt ( friction 9 a pulley s m\nt$ may be und]/ood f ! foll[+ exp]im5t4 ,a silk cord is plac$ on ! pulley1 & #a lb4 weight is atta*$ to ea* ( its ends3 ^! ( c\rse balance4 ,a numb] ( f9e wire hooks1 ea* weigh+ #jjja lb41 are prep>$1 & x is f\nd t :5 a weight ( #jjjd lb4 is atta*$ to ei!r side x is suffici5t to ov]come friction & set ! weights 9 motion4 ,,en]gy4 #agb4 ,9 connection ) ! subject ( friction1 & also z introductory to ! me*anical p[]s1 ! notion ( 8"w10 or z x is m prop]ly call$ 8en]gy10 is ( great importance4 ,! mean+ ( ?is ^w z employ$ 9 me*anics w require a little consid]ation4 #agc4 ,9 ord9>y language1 :atev] a man does t c cause fatigue1 :e!r ( body or m9d1 is call$ "w4 ,9 me*anics1 we mean by en]gy t p>ticul> k9d ( "w :i* is directly or indirectly equival5t to rais+ weights4 #agd4 ,suppose a weight is ly+ on ! floor & a /ool is /&+ beside x3 if a man raise ! weight & place x ^u ! /ool1 ! ex]tion t he exp5ds is en]gy 9 ! s5se 9 :i* ! ^w is us$ 9 me*anics4 ,! am\nt ( ex]tion necess>y to place ! weight ^u ! /ool dep5ds ^u two ?+s1 ! magnitude ( ! weight & ! height ( ! /ool4 ,x is cle> t bo? ^! ?+s mu/ be tak5 into acc\nt1 = al?\gh we "k ! weight :i* is rais$1 we _c tell ! am\nt ( ex]tion t w be requir$ until we "k ! height ?r\gh :i* x is to be rais$2 & if we "k ! height1 we _c appreciate ! quantity ( ex]tion until we "k ! weight4 #age4 ,! foll[+ plan has be5 adopt$ = express+ quantities ( en]gy4 ,! small am\nt ( ex]tion necess>y to raise #a lb4 avoirdupois ?r\gh "o ,briti% foot is tak5 z a /&>d1 comp>$ ) :i* all o!r quantities ( en]gy are e/imat$4 ,?is quantity ( ex]tion is call$ 9 me*anics ! unit ( en]gy1 & sometimes also ! 8foot-p\nd40 #agf4 ,if a weight ( #a lb4 has to be rais$ ?r\gh a height ( #b feet1 or a weight ( #b lbs4 ?r\gh a height ( #a foot1 x w be necess>y to exp5d twice z mu* en]gy z w\ld h rais$ a weight ( #a lb4 ?r\gh #a foot1 t is1 #b foot-p\nds4 ,if a weight ( #e lbs4 _h to be rais$ f ! floor up to a /ool #c feet high1 h[ _m units ( en]gy w\ld be requir$8 ,to raise #e lbs4 ?r\gh #a foot requires #e foot-p\nds1 & ! process mu/ be aga9 repeat$ twice be=e ! weight arrive at ! top ( ! /ool4 ,= ! :ole op]ation #ae foot-p\nds w !re=e be necess>y4 ,if #ajj lbs4 be rais$ ?r\gh #bj feet1 #ajj foot-p\nds ( en]gy is requir$ = ! fir/ foot1 ! same = ! second1 ?ird1 c41 up to ! tw5tie?1 mak+ a total ( #b1jjj foot-p\nds4 ,"h is a practical "q = ! sake ( illu/ration4 ,:i* w\ld x be pref]able to hoi/1 by a rope pass+ ov] a s+le fix$ pulley1 a trunk weigh+ #dj lbs4 to a height ( #bj feet1 or a trunk weigh+ #ej lbs4 to a height ( #ae feet8 ,we %all f9d h[ mu* en]gy w\ld be necess>y 9 ea* case3 #dj times #bj is #hjj2 !re=e 9 ! fir/ case ! en]gy w\ld be #hjj foot-p\nds4 ,b #ej times #ae is #gej2 !re=e ! am\nt ( "w1 9 ! second case1 is only #gej lbs4 ,h5ce x is less ex]tion to c>ry #ej lbs4 up #ae feet ?an #dj lbs4 up #bj feet4 #agg4 ,! rate ( work+ ( e s\rce ( en]gy1 :e!r x lie 9 ! muscles ( m5 or o!r animals1 9 wat]-:eels1 /eam-eng9es1 or o!r prime mov]s1 is to be measur$ by ! numb] ( foot-p\nds produc$ 9 ! unit ( "t4 ,! p[] ( a /eam-eng9e is def9$ by its equival5t 9 horse-p[]4 ,= example1 x is meant t a /eam-eng9e ( #c horse-p[]1 c\ld1 :5 work+ = an h\r1 d z mu* "w z #c horses c\ld d :5 work+ = ! same "t4 ,! p[] ( a horse is1 h[ev]1 an unc]ta9 quantity1 diff]+ 9 diff]5t animals & n q uni=m 9 ! same individual2 accord+ly ! selection ( ?is measure = ! effici5cy ( ! /eam-eng9e is inconv5i5t4 ,we replace x by a conv5i5t /&>d horse-p[]1 :i* is1 h[ev]1 a good deal l>g] ?an t cont9u\sly obta9able f any ord9>y horse4 ,a "o horse-p[] /eam-eng9e is capable ( accompli%+ #cc1jjj foot-p\nds p] m9ute4 #agh4 ,we %all illu/rate ! num]ical calculation ( horse-p[] by an example3 if a m9e be #a1jjj feet deep1 h[ mu* wat] p] m9ute w\ld a #ej horse-p[] eng9e be capable ( rais+ to ! surface8 ,! eng9e w\ld yield #ej #cc1jjj units ( "w p] m9ute1 b ! weight has to be rais$ #a1jjj feet1 consequ5tly ! numb] ( p\nds ( wat] rais$ p] m9ute is #ej #cc1jjj ------------------------ #a1fej4 #a1jjj #agi4 ,we %all apply ! pr9ciple ( "w to ! consid]ation ( ! pulley already describ$ 7p4 #ij74 ,9 ord] to raise ! weight ( #ad lbs41 x is necess>y t ! rope to :i* ! p[] is appli$ %\ld be pull$ d[nw>ds by a =ce ( #ae lbs41 ! extra p\nd be+ on acc\nt ( ! friction4 ,to fix our ideas1 we %all suppose ! #ad lbs4 to be rais$ #a foot2 to lift ?is load directly1 )\t ! int]v5tion ( ! pulley1 #ad foot-p\nds w\ld be necess>y1 b :5 x is rais$ by means ( ! pulley1 #ae1 foot-p\nds are necess>y4 ,h5ce !re is an absolute loss ( _/? ( ! en]gy :5 ! pulley is us$4 ,if a /eam-eng9e ( #a horse-p[] 7 employ$ 9 rais+ weights by a rope pass+ ov] a pulley simil> to t on :i* we h exp]im5t$1 only _/?s ( ! "w w\ld be usefully employ$2 b we f9d #cc1jjj #ad #cj1hjj4 -------- #ae ,! eng9e w\ld !re=e p]=m #cj1hjj foot-p\nds ( useful "w p] m9ute4 #ahj4 ,! effect ( friction on a pulley1 or on any o!r ma*9e1 is always to wa/e en]gy4 ,to p]=m a piece ( "w directly requires a c]ta9 numb] ( foot-p\nds1 :ile to d x by a ma*9e requires m1 on acc\nt ( ! loss by friction4 ,?is may at fir/ sight appe> some:at p>adoxical1 z x is well kn[n t1 by lev]s1 pulleys1 c41 an enorm\s me*anical advantage may be ga9$4 ,?is subject w be fully expla9$ 9 ! next & foll[+ lectures1 :i* relate to ! me*anical p[]s4 #aha4 ,we %all conclude ) a few obs]vations on a po9t ( ! greate/ importance4 ,we h se5 a case :]e #ae foot-p\nds ( en]gy only accompli%$ #ad foot-p\nds ( "w1 & ?us #a foot-p\nd appe>$ to be lo/4 ,we say t ?is 0 exp5d$ ^u ! friction2 b :at is ! friction8 ,! axle is gradually worn away by rubb+ 9 its be>+s1 &1 if x be n prop]ly oil$1 x becomes heat$4 ,! am\nt ( en]gy t seems to disappe> is p>tly exp5d$ 9 gr9d+ d[n ! axle1 & is p>tly trans=m$ into heat2 x is ?us n really lo/1 b un=tunately assumes a =m :i* we d n require & 9 :i* x is r injuri\s ?an o!rwise4 ,inde$ we "k t en]gy _c be de/roy$1 h[ev] x may be trans=m$2 if x disappe> 9 "o %ape1 x is only to reappe> 9 ano!r4 ,a s-call$ loss ( en]gy by friction only means a div]sion ( a "p ( ! "w to "s purpose o!r ?an t :i* we wi% to accompli%4 ,x has long be5 kn[n t matt] is inde/ructible3 x is n[ equally c]ta9 t ! same may be ass]t$ ( en]gy4 ,,lecture ,,vii4 ,,! ,,pulley-,,block4 ,introduction4--,! ,s+le ,moveable ,pulley4--,! ,?ree-%eave ,pulley-block4--,! ,diff]5tial ,pulley-block4--,! ,epicycloidal ,pulley-block4 ,,introduction4 #ahb4 ,9 ! fir/ lecture ,i %[$ h[ a l>ge weight c\ld be rais$ by a small] weight1 & ,i /at$ t ?is subject w\ld aga9 occupy our att5tion4 ,i n[ fulfil ?is promise4 ,! que/ions to be discuss$ involve ! mo/ advantage\s me?ods ( employ+ a small =ce to ov]come a great]4 ,"h is a subject ( practical importance4 ,a man ( av]age /r5g? _c raise m ?an a hundr$weight )\t great ex]tion1 yet ! weights :i* x is necess>y to lift & move ab\t (t5 weigh _m hundr$weights1 or ev5 _m tons4 ,x is n always practicable to employ num]\s h&s = ! purpose1 nor is a /eam-eng9e or o!r great s\rce ( p[] at all times available4 ,b :at are call$ ! me*anical p[]s enable ! =ces at our disposal to be greatly increas$4 ,"o man1 by _! aid1 c ex]t z mu* =ce z sev]al c\ld )\t su* assi/ance2 & :5 !y are employ$ to augm5t ! p[] ( sev]al m5 or ( a /eam-eng9e1 gigantic weights1 am\nt+ sometimes to hundr$s ( tons1 c be manag$ ) facility4 #ahc4 ,9 ! v>i\s arts we f9d innum]able cases :]e great resi/ances h to be ov]come2 we also f9d a correspond+ numb] & v>iety ( devices contriv$ by human skill to conqu] !m4 ,! gird]s ( an iron bridge h to be lift$ up to _! pi]s2 ! boil]s & eng9es ( an ocean /eam] h to be plac$ 9 position2 a great ca/+ has to be rais$ f its m\ld2 a railway locomotive has to be plac$ on ! deck ( a vessel = transit2 a weighty an*or has to be lift$ f ! bottom ( ! sea2 an iron plate has to be roll$ or cut or pun*$3 = all ( ^! cases suitable arrangem5ts mu/ be devis$ 9 ord] t ! requisite p[] may be obta9$4 #ahd4 ,we "k b little ( ! means :i* ! anci5ts employ$ 9 rais+ ! va/ /ones ( ^? build+s :i* travell]s 9 ! ,ea/ h describ$ to u4 ,x is sometimes ?\ght t a l>ge numb] ( m5 c\ld h transport$ ^! /ones )\t ! aid ( appliances :i* we w\ld n[ use = a simil> purpose4 ,b x is m likely t "s ( ! me*anical p[]s 7 us$1 z1 ) a multitude ( m51 x is difficult to ensure ! prop] application ( _! unit$ /r5g?4 ,9 ,ea/] ,isl&1 hundr$s ( miles di/ant f civilis$ l&1 & n[ inhabit$ by savages1 va/ idols ( /one h be5 f\nd 9 ! hills :i* mu/ h be5 rais$ by human lab\r4 ,x is useless to speculate on ! ext9ct race by :om ?is "w 0 a*iev$1 or on ! means !y employ$4 #ahe4 ,! me*anical p[]s are usually enum]at$ z foll[s3--,! pulley1 ! lev]1 ! :eel & axle1 ! w$ge1 ! incl9$ plane1 ! screw4 ,^! diff]5t p[]s are s frequ5tly us$ 9 comb9ation t ! di/9ctions _c be always ma9ta9$4 ,! classification w1 h[ev]1 suffice to give a g5]al notion ( ! subject4 #ahf4 ,_m ( ! mo/ valuable me*anical p[]s are ma*9es 9 :i* ropes or *a9s play an important "p4 ,pulleys are usually employ$ :]ev] x is necess>y to *ange ! direction ( a rope or *a9 :i* is transmitt+ p[]4 ,9 ! pres5t lecture we %all exam9e ! mo/ important me*anical p[]s t are produc$ by ! comb9ation ( pulleys4 ,,! ,,s+le ,,moveable ,,pulley4 #ahg4 ,we comm5ce ) ! mo/ simple case1 t ( ! s+le moveable pulley 7,fig4 #ce74 ,! rope is firmly secur$ at "o end ,a2 x !n passes d[n "u ! moveable pulley ,b1 & upw>ds ov] a fix$ pulley4 ,to ! free end1 :i* dep5ds f ! fix$ pulley1 ! p[] is appli$ :ile ! load to be rais$ is susp5d$ f ! moveable pulley4 ,we %all fir/ /udy ! relation betwe5 ! p[] & ! load 9 a simple way1 & !n we %all describe a few exact exp]im5ts4 #ahh4 ,:5 ! load is rais$ ! moveable pulley itself mu/ ( c\rse be also rais$1 & a "p ( ! p[] is exp5d$ = ?is purpose4 ,b we c elim9ate ! weight ( ! moveable pulley1 s f> z our calculations are conc]n$1 by fir/ atta*+ to ! p[] end ( ! rope a suffici5t weight to lift up ! moveable pulley :5 n c>ry+ a load4 ,! weight necess>y = do+ ?is is f\nd by trial to be a little ov] #ae lbs4 ,s t :5 a load is be+ rais$ we mu/ r$uce ! app>5t p[] by #ae lbs4 to obta9 ! p[] really effective4 #ahi4 ,let u susp5d #ad lbs4 f ! load hook at ,b1 & asc]ta9 :at p[] w raise ! load4 ,we leave ! weight ( ! moveable pulley & #ae lbs4 ( ! p[] at ,c out ( consid]ation4 ,i !n f9d by exp]im5t t #g lbs4 ( effective p[] is n suffici5t to raise ! load1 b if "o p\nd m be add$1 ! p[] desc5ds1 & ! load is rais$4 ,"h1 !n1 is a rem>kable result2 a weight ( #h lbs4 has ov]come #ad lbs4 ,9 ?is we h ! fir/ application ( ! me*anical p[]s to increase our available =ces4 ,illu/ration3 ,,fig4 #ce4 #aij4 ,let u exam9e ! reason ( ?is me*anical advantage4 ,if ! load be rais$ "o foot1 x is pla9 t ! p[] mu/ desc5d two feet3 = 9 ord] to raise ! load ! two p>ts ( ! rope desc5d+ to ! moveable pulley mu/ ea* be %ort5$ "o foot1 & ?is c only be done by ! p[] desc5d+ two feet4 ,h5ce :5 ! load ( #ad lbs4 is lift$ by ! ma*9e1 = e foot x is rais$ ! p[] mu/ desc5d two feet3 ?is simple po9t leads to a conception ( ! greate/ importance1 on :i* dep5ds ! effici5cy ( ! pulley4 ,9 ! /udy ( ! me*anical p[]s x is ess5tial to exam9e ! numb] ( feet ?r\gh :i* ! p[] mu/ act 9 ord] to raise ! load "o foot3 ?is numb] we %all always call ! velocity ratio4 #aia4 ,to raise #ad lbs4 "o foot requires #ad foot-p\nds ( en]gy4 ,h5ce1 7 !re no su* ?+ z friction1 #g lbs4 on ! p[] hook w\ld be suffici5t to raise ! load2 because #g lbs4 desc5d+ ?r\gh two feet yields #ad foot-p\nds4 ,b !re is a loss ( en]gy on acc\nt ( friction1 & a p[] ( #g lbs4 is n suffici5t3 #h lbs4 are necess>y4 ,eight lbs4 9 desc5d+ two feet p]=ms #af foot-p\nds2 ( ^! only #ad are utilis$ on ! load1 ! rema9d] be+ ! quantity ( en]gy t has be5 div]t$ by friction4 ,we le>n1 !n1 t 9 ! moveable pulley ! quantity ( en]gy employ$ is really great] ?an t :i* w\ld lift ! weight directly1 b t ! actual =ce :i* has to be ex]t$ is less4 #aib4 ,suppose t #bh lbs4 be plac$ on ! load hook1 a few trials assure u t a p[] ( #af lbs4 7b n less7 w be suffici5t = motion2 t is to say1 :5 ! load is d\bl$1 we f9d1 z we might h expect$1 t ! p[] mu/ be d\bl$ also4 ,x is easily se5 t ! loss ( en]gy by friction !n am\nts to #d foot-p\nds4 ,we ?us v]ify1 9 ! case ( ! moveable pulley1 ! approximate law t ! friction is proportional to ! load4 #aic4 ,by means ( a moveable pulley a man is able to raise a weight ne>ly d\ble z great z he c\ld lift directly4 ,f a s]ies ( c>eful exp]im5ts x has be5 f\nd t :5 a man is employ$ 9 ! p>ticul> ex]tion necess>y = rais+ weights ov] a pulley1 he is able to "w mo/ effici5tly :5 ! pull he is requir$ to make is ab\t #dj lbs4 ,a man c\ld1 ( c\rse1 ex]t great] =ce ?an ?is1 b 9 an ord9>y day's "w he is able to p]=m m foot-p\nds :5 ! pull is #dj lbs4 ?an :5 x is l>g] or small]4 ,if !re=e ! weights to be lift$ am\nt to ab\t #hj lbs41 en]gy may really be economiz$ by ! use ( ! s+le moveable pulley1 al?\gh by s do+ a great] quantity ( en]gy w\ld be actually exp5d$ ?an w\ld h be5 necess>y to raise ! weights directly4 #aid4 ,"s exp]im5ts on l>g] loads h be5 tri$ ) ! moveable pulley we h j describ$2 ! results are record$ 9 ,table ,,ix4 ,,table ,,ix4--,,s+le ,,moveable ,,pulley4 ,moveable pulley ( ca/ iron #cbe diamet]1 groove #jf wide1 wr\ght iron axle #jf diamet]2 fix$ pulley ( ca/ iron #e diamet]1 groove #jd wide1 wr\ght iron axle #jf diamet]1 axles oil$2 flexible plait$ rope #jbe diamet]2 velocity ratio #b1 me*anical effici5cy #ah1 useful effect #ij p] c5t42 =mula ,p #bba #jedec ,r4 ---------------------------------------- ------------------ ,p4 ,discrepancies ,numb] ( ,r4 ,obs]v$ ,calculat$ betwe5 obs]v$ ,exp]im5t4 ,load p[] p[] & calculat$ 9 lbs4 9 lbs4 9 lbs4 p[]s4 ---------------------------------------- ------------------ #a #bh #age #age #jj #b #eg #cce #ccc - #jb #c #he #dhe #dhf #ja #d #aac #fdj #fch - #jb #e #adb #hjj #gif - #jd #f #agj #ide #idi #jd #g #aih #aaje #aajb - #jc #h #bbf #abee #abee #jj ---------------------------------------- ------------------ ,! dim5sions ( ! pulleys are precisely /at$ because1 = pulleys ( diff]5t con/ruction1 ! num]ical coeffici5ts w\ld n necess>ily be ! same4 ,an att5tive /udy ( ?is table w1 h[ev]1 %[ ! g5]al "* ( ! relation betwe5 ! p[] & ! load 9 all arrangem5ts ( ?is class4 ,! table consi/s ( five columns4 ,! fir/ conta9s m]ely ! numb]s ( ! exp]im5ts = conv5i5ce ( ref]5ce4 ,9 ! second column1 head$ ,r1 ! loads1 express$ 9 p\nds1 :i* are rais$ 9 ea* exp]im5t1 are giv52 t is1 ! weight atta*$ to ! hook1 n includ+ ! weight ( ! l[] pulley4 ,! weight ( ?is pulley is n includ$ 9 ! /at$ loads4 ,9 ! ?ird column ! p[]s are record$1 :i* 7 f\nd to be suffici5t to raise ! correspond+ loads 9 ! second column4 ,?us1 9 exp]im5t #g1 x is f\nd t a p[] ( #aaje lbs4 w be suffici5t to raise a load ( #aih lbs4 ,! ?ird column has ?us be5 det]m9$ by gradually increas+ ! p[] until motion beg9s4 #aie4 ,f an exam9ation ( ! columns %[+ ! p[] & ! load1 we see t ! p[] always am\nts to m ?an half ! load4 ,! excess is p>tly due to a small portion ( ! p[] 7ab\t #ae lbs47 be+ employ$ 9 rais+ ! l[] block1 & p>tly to friction4 ,= example1 9 exp]im5t #g1 if !re _h be5 no friction & if ! l[] block 7 )\t weight1 a p[] ( #ii lbs4 w\ld h be5 suffici5t2 b1 ow+ to ! pres5ce ( ^! di/urb+ causes1 #aaje lbs4 are necess>y3 ( ?is am\nt #ae lbs4 is due to ! weight ( ! pulley1 #aj lbs4 is ! =ce ( friction1 & ! rema9+ #ii lbs4 raises ! load4 #aif4 ,by a calculation bas$ on ?is table we h asc]ta9$ a c]ta9 relation betwe5 ! p[] & ! load2 !y are connect$ by ! =mula :i* may be enunciat$ z foll[s3 ,! p[] is f\nd by multiply+ ! weight ( ! load into #jedec1 & add+ #bba to ! product4 ,call+ ,p ! p[] & ,r ! load1 we may express ! relation ?us3 ,p #bba #jedec ,r4 ,= example1 9 exp]im5t #e1 ! product ( #adb & #jedec is #ggdc1 to :i*1 :5 #bba is add$1 we f9d = ,p #gifd1 v ne>ly ! same z #hj lbs41 ! obs]v$ value ( ! p[]4 ,9 ! f\r? column ! values ( ,p calculat$ by means ( ?is =mula are giv51 & 9 ! la/ we exhibit ! discrepancies betwe5 ! obs]v$ & ! calculat$ values = ! sake ( comp>ison4 ,x w be se5 t ! discrepancy 9 no case am\nts to #je lb41 consequ5tly ! =mula expresses ! exp]im5ts v well4 ,! mode ( d$uc+ x is giv5 9 ! ,app5dix4 #aig4 ,! quantity #bba is p>tly t portion ( ! p[] exp5d$ 9 ov]com+ ! weight ( ! moveable pulley1 & p>tly arises f friction4 #aih4 ,we c readily calculate f ! =mula h[ mu* p[] w be requir$ to raise a giv5 weight2 = example1 suppose #bjj lbs4 be atta*$ to ! moveable pulley1 we f9d t #aaa lbs4 mu/ be appli$ z ! p[]4 ,b 9 ord] to raise #bjj lbs4 "o foot1 ! p[] ex]t$ mu/ act ov] two feet2 h5ce ! numb] ( foot-p\nds requir$ is #b #aaa #bbb4 ,! quantity ( en]gy t is lo/ is #bb foot-p\nds4 ,out ( e #bbb foot-p\nds appli$1 #bjj are usefully employ$2 t is to say1 ab\t #ij p] c5t4 ( ! appli$ en]gy is utiliz$1 :ile ! rema9+ #aj p] c5t4 is lo/ by friction4 ,,! ,,?ree-,,%eave ,,pulley-,,block4 #aii4 ,! next arrangem5t we %all employ is a pair ( pulley-blocks ,s ,t1 ,fig4 #ce1 ea* conta9+ ?ree %eaves1 z ! small pulleys are t]m$4 ,a rope is fa/5$ to ! upp] block1 ,s2 x !n passes d[n to ! l[] block ,t "u "o %eave1 up aga9 to ! upp] block & ov] a %eave1 & s on1 z %[n 9 ! figure4 ,to ! end ( ! rope f ! la/ ( ! upp] %eaves ! p[] ,h is appli$1 & ! load ,g is susp5d$ f ! hook atta*$ to ! l[] block4 ,:5 ! rope is pull$1 x gradually raises ! l[] block2 & to raise ! load "o foot1 ea* ( ! six p>ts ( ! rope f ! upp] block to ! l[] block mu/ be %ort5$ "o foot1 & !re=e ! p[] mu/ h pull$ out six feet ( rope4 ,h5ce1 = e foot t ! load is rais$ ! p[] mu/ h act$ ?r\gh six feet2 t is to say1 ! velocity ratio is #f4 #bjj4 ,if !re 7 no friction1 ! p[] w\ld only be "o-six? ( ! load4 ,?is foll[s at once f ! pr9ciples already expla9$4 ,suppose ! load be #fj lbs41 !n to raise x "o foot w\ld require #fj foot-p\nds2 & ! p[] mu/ !re=e ex]t #fj foot-p\nds2 b ! p[] moves ov] six feet1 !re=e a p[] ( #aj lbs4 w\ld be suffici5t4 ,ow+1 h[ev]1 to friction1 "s en]gy is lo/1 & we mu/ h rec\rse to exp]im5t 9 ord] to te/ ! real effici5cy ( ! ma*9e4 ,! s+le moveable pulley ne>ly d\bl$ our p[]2 we %all prove t ! ?ree-%eave pulley-block w quadruple x4 ,9 ?is case we deal ) l>g] weights1 ) ref]5ce to :i* we may leave ! weight ( ! l[] block out ( consid]ation4 #bja4 ,let u fir/ atta* #a cwt4 to ! load hook2 we f9d t #bi lbs4 on ! p[] hook is ! smalle/ weight t c produce motion3 ?is is only #a lb4 m ?an "o-qu>t] ( ! load rais$4 ,if #b cwt4 be ! load1 we f9d t #ef lbs4 w j raise x3 ?is "t ! p[] is exactly "o-qu>t] ( ! load4 ,! exp]im5t has be5 tri$ ( plac+ #d cwt4 on ! hook2 x is !n f\nd t #aji lbs4 w raise x1 :i* is only #c lbs4 %ort ( #a cwt4 ,^! exp]im5ts demon/rate t = a ?ree-%eave pulley-block ( ?is con/ruction we may safely apply ! rule1 t ! p[] is "o-qu>t] ( ! load4 #bjb4 ,we are ?us enabl$ to see h[ mu* ( our ex]tion 9 rais+ weights mu/ be exp5d$ 9 m]ely ov]com+ friction1 & h[ mu* may be utiliz$4 ,suppose = example t we h to raise a weight ( #ajj lbs4 "o foot by means ( ! pulley-block2 ! p[] we mu/ apply is #be lbs41 & six feet ( rope mu/ be drawn out f betwe5 ! pulleys3 !re=e ! p[] ex]ts #aej foot-p\nds ( en]gy4 ,( ^! only #ajj foot-p\nds are usefully employ$1 & ?us #ej foot-p\nds1 "o-?ird ( ! :ole1 h be5 exp5d$ on friction4 ,"h we see t not)/&+ a small =ce ov]comes a l>ge "o1 !re is an actual loss ( en]gy 9 ! ma*9e4 ,! real advantage ( c\rse is t by ! pulley-block ,i c raise a great] weight ?an ,i c\ld move )\t assi/ance1 b ,i d n create en]gy2 ,i m]ely modify x1 & lose by ! process4 #bjc4 ,! result ( ano!r s]ies ( exp]im5ts made ) ?is pair ( pulley-blocks is giv5 9 ,table ,x4 ,,table ,x4--,,?ree-,,%eave ,,pulley-,,blocks4 ,%eaves ca/ iron #be diamet]2 plait$ rope #jbe diamet]2 velocity ratio #f2 me*anical advantage #d2 useful effect #fg p] c5t42 =mula ,p #bcf #jbch ,r4 ---------------------------------------- ------------------ ,p4 ,discrepancies ,numb] ( ,r4 ,obs]v$ ,calculat$ betwe5 obs]v$ ,exp]im5t4 ,load p[] p[] & calculat$ 9 lbs4 9 lbs4 9 lbs4 p[]s4 ---------------------------------------- ------------------ #a #eg #aee #aei #jd #b #aad #bie #bie #jj #c #aga #dce #dca - #jd #d #bbh #efj #eff #jf #e #bha #gjj #fib - #jh #f #cch #hcj #hbh - #jb #g #cie #igj #ifd - #jf #h #deb #ajij #ajii #ji ---------------------------------------- ------------------ #bjd4 ,?is table conta9s five columns2 ! weights rais$ 7%[n 9 ! second column7 range up to some:at ov] #d cwt4 ,! obs]v$ values ( ! p[] are giv5 9 ! ?ird column2 ea* ( ^! is g5]ally ab\t "o-qu>t] ( ! correspond+ value ( ! load4 ,!re is1 h[ev]1 a m accurate rule = f9d+ ! p[]2 x is z foll[s4 #bje4 ,to f9d ! p[] necess>y to raise a giv5 load1 multiply ! loads 9 lbs4 by #jbch1 & add #bcf lbs4 to ! product4 ,we may express ! rule by ! =mula ,p #bcf #jbch ,r4 #bjf4 ,to f9d ! p[] :i* w\ld raise #bbh lbs42 ! product ( #bbh & #jbch is #edbf2 add+ #bcf1 we f9d #eff lbs4 = ! p[] requir$2 ! actual obs]v$ p[] is #ef lbs41 s t ! rule is accurate to )9 ab\t half a p\nd4 ,9 ! f\r? column w be f\nd ! values ( ,p calculat$ by means ( ?is rule4 ,9 ! fif? column1 ! discrepancies betwe5 ! obs]v$ & ! calculat$ values ( ! p[]s are giv51 & x w be se5 t ! diff]5ce 9 no case rea*es #a lb4 ,( c\rse x w be und]/ood t ?is =mula is only reliable = loads :i* lie betwe5 ^? employ$ 9 ! fir/ & la/ ( ! exp]im5ts4 ,we c calculate ! p[] = any load betwe5 #eg lbs4 & #deb lbs41 b = loads mu* l>g] ?an #deb or less ?an #eg x w\ld probably be bett] to use ! simple f\r? ( ! load r ?an ! p[] comput$ by ! =mula4 #bjg4 ,i w next p]=m an exp]im5t ) ! ?ree-%eave pulley-block1 :i* w give an insight into ! exact am\nt ( friction )\t calculation by ! help ( ! velocity ratio4 ,we fir/ c\nt]poise ! weight ( ! l[] block by atta*+ weights to ! p[]4 ,x is f\nd t ab\t #af lbs4 is suffici5t = ?is purpose4 ,i atta* a #ef lb4 weight z a load1 & f9d t #aca lbs4 is suffici5t p[] = motion4 ,?is am\nt is p>tly compos$ ( ! =ce necess>y to raise ! load if !re 7 no friction1 & ! re/ is due to ! friction4 ,i next gradually remove ! p[] weights3 :5 ,i h tak5 (f a p\nd1 y see ! p[] & ! load balance ea* o!r2 b :5 ,i h r$uc$ ! p[] s l[ z #ee lbs4 7n includ+ ! c\nt]poise = ! l[] block71 ! load is j able to ov]haul ! p[]1 & run d[n4 ,we h !re=e prov$ t a p[] ( #aca lbs4 or great] raises #ef lbs41 t any p[] betwe5 #aca lbs4 & #ee lbs4 balances #ef lbs41 & t any p[] less ?an #ee lbs4 is rais$ by #ef lbs4 ,:5 ! p[] is rais$1 ! =ce ( friction1 toge!r ) ! p[]1 mu/ be ov]come by ! load4 ,let u call ,x ! real p[] t w\ld be necess>y to balance #ef lbs4 9 a p]fectly frictionless ma*9e1 & ,y ! =ce ( friction4 ,we %all be able to det]m9e ,x & ,y by ! exp]im5ts j p]=m$4 ,:5 ! load is rais$ a p[] equal to ,x ,y mu/ be appli$1 & !re=e ,x ,y #aca4 ,on ! o!r h&1 :5 ! p[] is rais$1 ! =ce ,x is j suffici5t to ov]come bo? ! friction ,y & ! weight #ee2 !re=e ,x ,y #ee4 ,solv+ ?is pair ( equations1 we f9d t ,x #ic & ,y #ch4 ,h5ce we inf] t ! p[] 9 ! frictionless ma*9e w\ld be #ic2 b ?is is exactly :at w\ld h be5 d$uc$ f ! velocity ratio1 = #ef #f #ic lbs4 ,9 ?is result we f9d a p]fect accordance betwe5 !ory & exp]im5t4 ,,! ,,diff]5tial ,,pulley-,,block4 #bjh4 ,by increas+ ! numb] ( %eaves 9 a pair ( pulley-blocks ! p[] may be increas$2 b ! l5g? ( rope 7or *a97 requisite = sev]al %eaves becomes a practical inconv5i5ce4 ,!re are also o!r reasons :i* make ! diff]5tial pulley-block1 :i* we %all n[ consid]1 m conv5i5t = _m purposes ?an ! common pulley blocks :5 a consid]able augm5tation ( p[] is requir$4 #bji4 ,! pr9ciple ( ! diff]5tial pulley is v anci5t1 & 9 mod]n times x has be5 embodi$ 9 a ma*9e ( practical utility4 ,! object is to secure1 t :ile ! p[] moves ov] a consid]able di/ance1 ! load %all only be rais$ a %ort di/ance4 ,:5 ?is has be5 atta9$1 we !n "k by ! pr9ciple ( en]gy t we h ga9$ a me*anical advantage4 #baj4 ,let u consid] ! means by :i* ?is is effect$ 9 t +5i\s contrivance1 ,we/on's diff]5tial pulley-block4 ,! pr9ciple ( ?is ma*9e w be und]/ood f ,fig4 #cf & ,fig4 #cg4 ,illu/ration3 ,,fig4 #cf4 ,x consi/s ( ?ree p>ts1--an upp] pulley-block1 a moveable pulley1 & an endless *a94 ,we %all briefly describe !m4 ,! upp] block ,p is furni%$ ) a hook = atta*m5t to a support4 ,! %eave x conta9s resembles two %eaves1 "o a little small] ?an ! o!r1 fa/5$ toge!r3 !y are 9 fact "o piece4 ,! grooves are provid$ ) ridges1 adapt$ to prev5t ! *a9 f slipp+4 ,! l[] pulley ,q consi/s ( "o %eave1 :i* is also furni%$ ) a groove2 x c>ries a hook1 to :i* ! load is atta*$4 ,! endless *a9 p]=ms a "p t w be und]/ood f ! sket* ( ! pr9ciple 9 ,fig4 #cf4 ,! *a9 passes f ! h& at ,a up to ,l ov] ! l>g] groove 9 ! upp] pulley1 !n d[nw>ds at ,b1 "u ! l[] pulley1 up aga9 at ,c1 ov] ! small] groove 9 ! upp] pulley at ,a1 & !n back aga9 by ,d to ! h& at ,a4 ,:5 ! h& pulls ! *a9 d[nw>ds1 ! two grooves ( ! upp] pulley beg9 to turn toge!r 9 ! direction %[n by ! arr[s on ! *a94 ,! l>ge groove is !re=e w9d+ up ! *a91 :ile ! small] groove is l[]+4 #baa4 ,9 ! pulley :i* has be5 employ$ 9 ! exp]im5ts to be describ$1 ! effective circumf]5ce ( ! l>ge groove is f\nd to be #aahd1 :ile t ( ! small groove is #ajcf4 ,:5 ! upp] pulley has made "o revolution1 ! l>ge groove mu/ h drawn up #aahd ( *a91 s9ce ! *a9 _c slip on acc\nt ( ! ridges2 b 9 ! same "t ! small groove has l[]$ #ajcf ( *a93 h5ce :5 ! upp] pulley has revolv$ once1 ! *a9 betwe5 ! two mu/ h be5 %ort5$ by ! diff]5ce betwe5 #aahd & #ajcf1 t is by #adh2 b ?is c only h tak5 place by rais+ ! moveable pulley ?r\gh half #adh1 t is1 ?r\gh a space #jgd4 ,! p[] has !n act$ ?r\gh #aahd1 & has rais$ ! resi/ance #jgd4 ,! p[] has !re=e mov$ ?r\gh a space #af times great] ?an t ?r\gh :i* ! load moves4 ,9 fact1 x is easy to v]ify by actual trial t ! p[] mu/ be mov$ ?r\gh #af feet 9 ord] t ! load may be rais$ #a foot4 ,we express ?is by say+ t ! velocity ratio is #af4 #bab4 ,by apply+ p[] to ! *a9 at ,d proce$+ f ! small] groove1 ! *a9 is l[]$ by ! l>ge groove fa/] ?an x is rais$ by ! small "o1 & ! l[] pulley desc5ds4 ,! load is ?us rais$ or l[]$ by simply pull+ "o *a9 ,a or ! o!r ,d4 ,illu/ration3 ,,fig4 #cg4 #bac4 ,we %all next consid] ! me*anical effici5cy ( ! diff]5tial pulley-block4 ,! block 7,fig4 #cg7 :i* we %all use is int5d$ to be work$ by "o man1 & w raise any weight n exce$+ a qu>t] ( a ton4 ,we h already le>n$ t ) ?is block ! p[] mu/ act ?r\gh sixte5 feet = ! load to be rais$ "o foot4 ,h5ce1 7 x n = friction1 ! p[] ne$ only be ! sixte5? "p ( ! load4 ,a few trials w %[ u t ! real effici5cy is n s l>ge1 & t 9 fact m ?an half ! "w ex]t$ is m]ely exp5d$ ^u ov]com+ friction4 ,?is w lead aft]w>ds to a result ( consid]able practical importance4 #bad4 ,plac+ ^u ! load hook a weight ( #bjj lbs41 ,i f9d t #ch lbs4 atta*$ to a hook fa/5$ on ! p[] *a9 is suffici5t to raise ! load2 t is to say1 ! p[] is ab\t "o-six? ( ! load4 ,if ,i make ! load #djj lbs4 ,i f9d ! requisite p[] to be #fd lbs41 :i* is only ab\t #c lbs4 less ?an "o-six? ( #djj lbs4 ,we may safely adopt ! practical rule1 t ) ?is diff]5tial pulley-block a man w\ld be able to raise a weight six times z great z he c\ld raise )\t su* assi/ance4 #bae4 ,a s]ies ( exp]im5ts c>efully tri$ ) diff]5t loads h giv5 ! results %[n 9 ,table ,,xi4 ,,table ,,xi4--,,! ,,diff]5tial ,,pulley-,,block4 ,circumf]5ce ( l>ge groove #aahd1 ( small groove #ajcf2 velocity ratio #af2 me*anical effici5cy #fjg2 useful effect #ch p] c5t42 =mula ,p #chg #jaejh ,r4 ---------------------------------------- --------------- ,p4 ,diff]5ce ,numb] ( ,r4 ,obs]v$ ,calculat$( ! obs]v$ ,exp]im5t4 ,load p[] p[] & calculat$ 9 lbs4 9 lbs4 9 lbs4 values4 ---------------------------------------- --------------- #a #ef #aj #abc #bc #b #aab #bj #bjh #jh #c #afh #ca #bib - #ah #d #bbd #ch #cgg - #jc #e #bhj #dh #dfa - #ai #f #ccf #ed #edf #jf #g #cib #fd #fca - #ji #h #ddh #gb #gae - #je #i #ejd #hj #hjj #jj #aj #efj #hf #hhd #bd ---------------------------------------- --------------- ,! fir/ column conta9s ! numb]s ( ! exp]im5ts1 ! second ! weights rais$1 ! ?ird ! obs]v$ values ( ! correspond+ p[]s4 ,f ^! ! foll[+ rule = f9d+ ! p[] has be5 obta9$3-- #baf4 ,to f9d ! p[]1 multiply ! load by #jaejh1 & add #chg lbs4 to ! product2 ?is rule may be express$ by ! =mula ,p #chg #jaejh ,r4 7,see ,app5dix47 #bag4 ,! calculat$ values ( ! p[]s are giv5 9 ! f\r? column1 & ! diff]5ces betwe5 ! obs]v$ & calculat$ values 9 ! la/ column4 ,! diff]5ces d n 9 any case am\nt to #be lbs41 & consid]+ t ! loads rais$ are up to a qu>t] ( a ton1 ! =mula repres5ts ! exp]im5ts ) satisfactory precision4 #bah4 ,suppose = example #bhj lbs4 is to be rais$2 ! product ( #bhj & #jaejh is #dbbb1 to :i*1 :5 #chg is add$1 we f9d #dfji to be ! requisite p[]4 ,! me*anical effici5cy f\nd by divid+ #dfji into #bhj is #fjg4 #bai4 ,to raise #bhj lbs4 "o foot #bhj foot-p\nds ( en]gy w\ld be necess>y1 b 9 ! diff]5tial pulley-block #dfji lbs4 mu/ be ex]t$ = a di/ance ( #af feet 9 ord] to accompli% ?is object4 ,! product ( #dfji & #af is #gcgd4 ,h5ce ! diff]5tial pulley-block requires #gcgd foot-p\nds ( en]gy to be appli$ 9 ord] to yield #bhj useful foot-p\nds2 b #bhj is only #ch p] c5t4 ( #gcgd1 & !re=e ) a load ( #bhj lbs4 only #ch p] c5t4 ( ! en]gy appli$ to a diff]5tial pulley-block is utiliz$4 ,9 g5]al1 we may /ate t n m ?an ab\t #dj p] c5t4 is pr(itably us$1 & t ! rema9d] is exp5d$ 9 ov]com+ friction4 #bbj4 ,x is a rem>kable & useful prop]ty ( ! diff]5tial pulley1 t a weight :i* has be5 hoi/$ w rema9 susp5d$ :5 ! h& is remov$1 ev5 ?\gh ! *a9 be n secur$ 9 any mann]4 ,! pulleys we h previ\sly consid]$ d n possess ?is conv5i5t prop]ty4 ,! weight rais$ by ! ?ree-%eave pulley-block1 = example1 w run d[n unless ! free end ( ! rope be prop]ly secur$4 ,! diff]5ce 9 ?is respect betwe5 ^! two me*anical p[]s is n a consequ5ce ( any special me*anism2 x is simply caus$ by ! excessive friction 9 ! diff]5tial pulley-block4 #bba4 ,! reason :y ! load does n run d[n 9 ! diff]5tial pulley may be ?us expla9$4 ,let u suppose t a weight ( #djj lbs4 is to be rais$ "o foot by ! diff]5tial pulley-block2 #djj units ( "w are necess>y1 & !re=e #a1jjj units ( "w mu/ be appli$ to ! p[] *a9 to produce ! #djj units 7s9ce only #dj p] c5t4 is utiliz$74 ,! friction w ?us h consum$ #fjj units ( "w :5 ! load has be5 rais$ "o foot4 ,if ! p[]-weight be remov$1 ! pressure support$ by ! upp] pulley-block is dim9i%$4 ,9 fact1 s9ce ! p[]-weight is ab\t _/? ( ! load1 ! pressure on ! axle :5 ! p[]-weight has be5 remov$ is only _/?s ( its previ\s value4 ,! friction is ne>ly proportional to t pressure3 h5ce :5 ! p[] has be5 remov$ ! friction on ! upp] axle is _/?s ( its previ\s value1 :ile ! friction on ! l[] pulley rema9s unalt]$4 ,we may !re=e assume t ! total friction is at lea/ _/?s ( :at x 0 be=e ! p[]-weight 0 remov$4 ,w friction all[ ! load to desc5d8 #fjj foot-p\nds ( "w 7 requir$ to ov]come ! friction 9 ! asc5t3 at lea/ _/ #fjj #ead foot-p\nds w\ld be necess>y to ov]come friction 9 ! desc5t4 ,b :]e is ?is en]gy to come f8 ,! load 9 its desc5t c\ld only yield #djj units1 & ?us desc5t by ! m]e weight ( ! load is impossible4 ,to enable ! load to desc5d we h actually to aid ! movem5t by pull+ ! *a9 ,d 7,figs4 #cf & #cg71 :i* proce$s f ! small groove 9 ! upp] pulley4 #bbb4 ,! pr9ciple :i* we h "h e/abli%$ ext5ds to o!r me*anical p[]s1 & may be /at$ g5]ally4 ,:5ev] m ?an half ! appli$ en]gy is consum$ by friction1 ! load w rema9 )\t runn+ d[n :5 ! ma*9e is left free4 ,,! ,,epicycloidal ,,pulley-,,block4 #bbc4 ,we %all conclude ?is lecture ) "s exp]im5ts ^u a useful me*anical p[] introduc$ by ,mr4 ,eade "u ! "n ( ! epicycloidal pulley-block4 ,x is %[n 9 ,fig4 #cc1 & also 9 ,fig4 #di4 ,9 ?is ma*9e !re are two *a9s3 "o a slight endless *a9 to :i* ! p[] is appli$2 ! o!r a /\t *a9 :i* has a hook at ea* end1 f ei!r ( :i* ! load may be susp5d$4 ,ea* ( ^! *a9s passes ov] a %eave 9 ! block3 ^! %eaves are connect$ by an +5i\s piece ( me*anism :i* we ne$ n "h describe4 ,?is me*anism is s contriv$ t1 :5 ! p[] causes ! %eave to revolve ov] :i* ! slight *a9 passes1 ! %eave :i* c>ries ! l>ge *a9 is also made to revolve1 b v sl[ly4 #bbd4 ,by actual trial x is asc]ta9$ t ! p[] mu/ be ex]t$ ?r\gh twelve feet & a half 9 ord] to raise ! load "o foot2 ! velocity ratio ( ! ma*9e is !re=e #abe4 #bbe4 ,if ! ma*9e 7 frictionless1 its me*anical effici5cy w\ld be ( c\rse equal to its velocity ratio2 ow+ to ! pres5ce ( friction ! me*anical effici5cy is less ?an ! velocity ratio1 & x w be necess>y to make exp]im5ts to det]m9e ! exact value4 ,i atta* to ! load hook a weight ( #bhj lbs41 & ins]t a few small hooks into ! l9ks ( ! p[] *a9 9 ord] to receive weights3 #ef lbs4 is suffici5t to produce motion1 h5ce ! me*anical effici5cy is #e4 ,_h !re be5 no friction a p[] ( #ef lbs4 w\ld h be5 capable ( ov]com+ a load ( #abe #ef #gjj lbs4 ,?us #gjj units ( en]gy mu/ be appli$ to ! ma*9e 9 ord] to p]=m #bhj units ( "w4 ,9 o!r words1 only #dj p] c5t4 ( ! appli$ en]gy is utiliz$4 #bbf4 ,an ext5d$ s]ies ( exp]im5ts ^u ! epicycloidal pulley-block is record$ 9 ,table ,,xii4 ,,table ,,xii4--,,! ,,epicycloidal ,,pulley-,,block4 ,size adapt$ = lift+ weights up to #e cwt42 velocity ratio #abe2 me*anical effici5cy #e2 useful effect #dj p] c5t42 calculat$ =mula ,p #eh #jahe ,r4 ---------------------------------------- --------------- ,p4 ,diff]5ce ,numb] ( ,r4 ,obs]v$ ,calculat$( ! obs]v$ ,exp]im5t4 ,load p[] p[] & calculat$ 9 lbs4 9 lbs4 9 lbs4 values4 ---------------------------------------- --------------- #a #ef #ae #afb #ab #b #aab #bg #bfe - #je #c #afh #dj #cfi - #ca #d #bbd #dg #dgb #jb #e #bhj #ef #egf #af #f #ccf #ff #fhj #bj #g #cib #gh #ghc #jc #h #ddh #hh #hhf #jf #i #ejd #ajj #iij - #aj #aj #efj #aaj #ajid - #jf ---------------------------------------- --------------- ,! f\r? column %[s ! calculat$ values ( ! p[]s d]iv$ f ! =mula4 ,x w be se5 by ! la/ column t ! =mula repres5ts ! exp]im5ts ) b little error4 #bbg4 ,s9ce #fj p] c5t4 ( en]gy is consum$ by friction1 ?is ma*9e1 l ! diff]5tial pulley-block1 su/a9s its load :5 ! *a9s are free4 ,! diff]5tial pulley-block gives a me*anical effici5cy ( #f1 :ile ! epicycloidal pulley-block has only a me*anical effici5cy ( #e1 & s f> ! =m] ma*9e has ! advantage2 on ! o!r h&1 t ! epicycloidal pulley conta9s b "o block1 & t its lift+ *a9 has two hooks1 are practical conv5i5ces /rongly 9 its fav\r4 ,,lecture ,,viii4 ,,! ,,lev]4 ,! ,lev] ( ! ,fir/ ,ord]4--,! ,lev] ( ! ,second ,ord]4--,! ,%e>s4--,! ,lev] ( ! ,?ird ,ord]4 ,,! ,,lev] ,,( ,,! ,,fir/ ,,ord]4 #bbh4 ,!re are _m cases 9 :i* a ma*9e = ov]com+ great resi/ance is necess>y :]e pulleys w\ld be q inapplicable4 ,to meet ^! v>i\s dem&s a correspond+ly v>i\s numb] ( contrivances has be5 devis$4 ,among/ ^! ! lev] 9 sev]al diff]5t =ms holds an important place4 #bbi4 ,! lev] ( ! fir/ ord] w be und]/ood by ref]5ce to ,fig4 #ch4 ,x consi/s ( a /raight rod1 to "o end ( :i* ! p[] is appli$ by means ( ! weight ,c4 ,at ano!r po9t ,b ! load is rais$1 :ile at ,a ! rod is support$ by :at is call$ ! fulcrum4 ,9 ! case repres5t$ 9 ! figure ! rod is ( iron1 #a #a 9 section & #f' long2 x weighs #ai lbs4 ,! p[] is produc$ by a #ef lb4 weight3 ! fulcrum consi/s ( a mod]ately %>p /eel edge firmly secur$ to ! framework4 ,! load 9 ?is case is replac$ by a spr+ balance ,h1 & ! hook ( ! balance is atta*$ to ! frame4 ,! spr+ is /ra9$ by ! action ( ! lev]1 & ! index records ! magnitude ( ! =ce produc$ at ! %ort end4 ,?is is ! lev] ) :i* we %all comm5ce our exp]im5ts4 ,illu/ration3 ,,fig4 #ch4 #bcj4 ,9 exam9+ ! relation betwe5 ! p[] & ! load1 ! "q is a little complicat$ by ! weight ( ! lev] itself 7#ai lbs471 b we %all be able to evade ! difficulty by means simil> to ^? employ$ on a =m] occasion 7,art4 #fj72 we c c\nt]poise ! weight ( ! iron b>4 ,?is is easily done by apply+ a hook to ! middle ( ! b> at ,d1 !nce c>ry+ a rope ov] a pulley ,f1 & susp5d+ a weight ,g ( #ai lbs4 f its free extremity4 ,?us ! b> is balanc$1 & we may leave its weight out ( consid]ation4 #bca4 ,we might also adopt ano!r plan analog\s to t ( ,art4 #ea1 :i* is h[ev] n s conv5i5t4 ,! weight ( ! b> produces a c]ta9 /ra9 ^u ! spr+ balance4 ,i may fir/ read (f ! /ra9 produc$ by ! b> alone1 & !n apply ! weight ,c & read aga94 ,! obs]v$ /ra9 is due bo? to ! weight ,c & to ! weight ( ! b>4 ,if ,i subtract ! kn[n effect ( ! b>1 ! rema9d] is ! effect ( ,c4 ,x is1 h[ev]1 less complicat$ to c\nt]poise ! b>1 & !n ! /ra9s indicat$ by ! balance are entirely due to ! p[]4 #bcb4 ,! lev] is #f' long2 ! po9t ,b is #f f ! end1 & ,b ,c is #e' long4 ,b ,c is divid$ into #e equal portions ( #a'2 ,a is at "o ( ^! divisions1 #a' di/ant f ,b1 & ,c is #e' di/ant f ,b 9 ! figure2 b ,c is capable ( be+ plac$ at any position1 by simply slid+ its r+ along ! b>4 #bcc4 ,! mode ( exp]im5t+ is z foll[s3--,! weight is plac$ on ! b> at ! position ,c3 a /ra9 is imm$iately produc$ ^u ,h2 ! spr+ /ret*es a little1 & ! b> becomes incl9$4 ,x may be notic$ t ! hook ( ! spr+ balance passes ?r\gh ! eye ( a wire-/ra9]1 s t by a few turns ( ! nut ^u ! /ra9] ! lev] c be re/or$ to ! horizontal position4 #bcd4 ,! p[] ( #ef lbs4 be+ #d' f ! fulcrum1 :ile ! load is #a' f ! fulcrum1 x is f\nd t ! /ra9 indicat$ by ! balance is #bbd lbs42 t is1 f\r times ! am\nt ( ! p[]4 ,if ! weight be mov$1 s z to be #c' f ! fulcrum1 ! /ra9 is obs]v$ to be #afh lbs42 & :atev] be ! di/ance ( ! p[] f ! fulcrum1 we f9d t ! /ra9 produc$ is obta9$ by multiply+ ! magnitude ( ! p[] 9 p\nds by ! di/ance express$ 9 feet1 & fractional p>ts ( a foot4 ,?is law may be express$ m g5]ally by /at+ t ! p[] is to ! load z ! di/ance ( ! load f ! fulcrum is to ! di/ance ( ! p[] f ! fulcrum4 #bce4 ,we c v]ify ?is law "u v>i$ circum/ances4 ,i move ! /eel edge :i* =ms ! fulcrum ( ! lev] until ! edge is #b' f ,b1 & secure x 9 t position4 ,i place ! weight ,c at a di/ance ( #c' f ! fulcrum4 ,i n[ f9d t ! /ra9 on ! balance is #hd lbs42 b #hd is to #ef z #c is to #b1 & !re=e ! law is also v]ifi$ 9 ?is in/ance4 #bcf4 ,!re is ano!r aspect 9 :i* we may express ! relation betwe5 ! p[] & ! load4 ,! law 9 ?is =m is ?us /at$3 ,! p[] multipli$ by its di/ance f ! fulcrum is equal to ! load multipli$ by its di/ance f ! fulcrum4 ,?us1 9 ! case we h j consid]$1 ! product ( #ef & #c is #afh1 & ?is is equal to ! product ( #hd & #b4 ,! di/ances f ! fulcrum are commonly call$ ! arms ( ! lev]1 & ! rule is express$ by /at+ t ,! p[] multipli$ into its arm is equal to ! load multipli$ into its arm3 h5ce ! load may be f\nd by divid+ ! product ( ! p[] & ! p[] arm by ! load arm4 ,?is simple law gives a v conv5i5t me?od ( calculat+ ! load1 :5 we "k ! p[] & ! di/ances ( ! p[] & ! load f ! fulcrum4 #bcg4 ,:5 ! p[] arm is long] ?an ! load arm1 ! load is great] ?an ! p[]2 b :5 ! p[] arm is %ort] ?an ! load arm1 ! p[] is great] ?an ! load4 ,we may reg>d ! /ra9 on ! balance z a p[] :i* supports ! weight1 j z we reg>d ! weight to be a p[] produc+ ! /ra9 on ! balance4 ,we see1 !n1 t = ! lev] ( ! fir/ ord] to be effici5t z a me*anical p[] x is necess>y t ! p[] arm be long] ?an ! load arm4 #bch4 ,! lev] is an extremely simple me*anical p[]2 x has only "o mov+ "p4 ,friction produces b little effect ^u x1 s t ! laws :i* we h giv5 may be actually appli$ 9 practice1 )\t mak+ any all[ance = friction4 ,9 ?is we notice a m>k$ diff]5ce betwe5 ! lev] & ! pulley-blocks already describ$4 #bci4 ,9 ! lev] ( ! fir/ ord] we f9d an excell5t ma*9e = augm5t+ p[]4 ,a p[] ( #ad lbs4 c by its means ov]come a resi/ance ( a hundr$weight1 if ! p[] be eight times z f> f ! fulcrum z ! load is f ! fulcrum4 ,?is pr9ciple x is :i* gives utility to ! cr[b>4 ,! end ( ! b> is plac$ "u a heavy /one1 :i* x is requir$ to raise2 a support ne> t end s]ves z a fulcrum1 & !n a comp>atively small =ce ex]t$ at ! p[] end w suffice to elevate ! /one4 #bdj4 ,! applications ( ! lev] are innum]able4 ,x is us$ n only = increas+ p[]1 b = modify+ & trans=m+ x 9 v>i\s ways4 ,! lev] is also us$ 9 weigh+ ma*9es1 ! pr9ciples ( :i* w be readily und]/ood1 = !y are consequ5ces ( ! law we h expla9$4 ,into ^! v>i\s appliances x is n our int5tion to ent] at pres5t2 ! great majority ( !m may1 :5 met )1 be easily und]/ood by ! pr9ciple we h laid d[n4 ,,! ,,lev] ,,( ,,! ,,second ,,ord]4 #bda4 ,9 ! lev] ( ! second ord] ! p[] is at "o end1 ! fulcrum at ! o!r end1 & ! load lies betwe5 ! two3 ?is lev] !re=e diff]s f ! lev] ( ! fir/ ord]1 9 :i* ! fulcrum lies betwe5 ! two =ces4 ,! relation betwe5 ! p[] & ! load 9 ! lev] ( ! second ord] may be /udi$ by ! arrangem5t 9 ,fig4 #ci4 #bdb4 ,! b> ,a ,c is ! same rod ( iron #gb #a #a :i* 0 us$ 9 ! =m] exp]im5t4 ,! fulcrum ,a is a /eel edge on :i* ! b> re/s2 ! p[] consi/s ( a spr+ balance ,h1 9 ! hook ( :i* ! end ,c ( ! b> re/s2 ! spr+ balance is su/a9$ by a wire-/ra9]1 by turn+ ! nut ( :i* ! b> may be adju/$ horizontally4 ,! "p ( ! b> betwe5 ! fulcrum ,a & ! p[] ,c is divid$ into five portions1 ea* #a' long1 & ! po9ts ,a & ,c are ea* #f di/ant f ! extremities ( ! b>4 ,! load employ$ is #ef lbs42 ?r\gh ! r+ ( ?is weight ! b> passes1 & ?us ! b> supports ! load4 ,! b> is c\nt]pois$ by ! weight ( #ai lbs4 at ,g1 9 ! mann] already expla9$ 7,art4 #bca74 #bdc4 ,! mode ( exp]im5t+ is z foll[s3--,let ! weight ,b be plac$ #a' f ! fulcrum2 ! /ra9 %[n by ! spr+ balance is ab\t #aa lbs4 ,if we calculate ! value ( ! p[] by ! rule already giv51 we %\ld h f\nd ! same result4 ,! product ( ! load by its di/ance f ! fulcrum is #ef1 ! di/ance ( ! p[] f ! fulcrum is #e2 h5ce ! value ( ! p[] %\ld be #ef #e #aab4 ,illu/ration3 ,,fig4 #ci4 #bdd4 ,if ! weight be plac$ #b' f ! fulcrum ! /ra9 is ab\t #bbe lbs4 & x is easy to asc]ta9 t ?is is ! same am\nt z w\ld h be5 f\nd by ! application ( ! rule4 ,a simil> result w\ld h be5 obta9$ if ! #ef lb4 weight _h be5 plac$ ^u any o!r "p ( ! b>2 & h5ce we may reg>d ! rule prov$ = ! lev] ( ! second ord] z well z = ! lev] ( ! fir/ ord]3 t1 ! p[] multipli$ by its di/ance f ! fulcrum is equal to ! load multipli$ by its di/ance f ! fulcrum4 ,9 ! pres5t case ! load is uni=mly #ef lbs41 :ile ! p[] by :i* x is su/a9$ is always less ?an #ef lbs4 ,illu/ration3 ,,fig4 #dj4 #bde4 ,! lev] ( ! second ord] is frequ5tly appli$ to practical purposes2 "o ( ! mo/ in/ructive ( ^! applications is illu/rat$ 9 ! %e>s %[n 9 ,fig4 #dj4 ,! %e>s consi/ ( two lev]s ( ! second ord]1 :i* by _! unit$ action enable a man to ex]t a greatly increas$ =ce1 suffici5t1 = example1 to cut ) ease a rod ( iron #jbe squ>e4 ,! mode ( action is simple4 ,! fir/ lev] ,a ,f has a h&le at "o end ,f1 :i* is #bb di/ant f ! o!r end ,a1 :]e ! fulcrum is plac$4 ,at a po9t ,b on ?is lev]1 #ah di/ant f ! fulcrum ,a1 a %ort l9k ,b ,c is atta*$2 ! end ( ! l9k ,c is jo9t$ to a second lev] ,c ,d2 ?is second lev] is #h long2 x =ms "o edge ( ! cutt+ %e>s1 ! o!r edge be+ fix$ to ! framework4 #bdf4 ,i place a rod ( iron #jbe squ>e betwe5 ! jaws ( ! %e>s 9 ! position ,e1 ! di/ance ,d ,e be+ #ce1 & proce$ to cut ! iron by apply+ pressure to ! h&le4 ,let u calculate ! am\nt by :i* ! lev]s increase ! p[] ex]t$ ^u ,f4 ,suppose = example t ,i press d[nw>ds on ! h&le ) a =ce ( #aj lbs41 :at is ! magnitude ( ! pressure ^u ! piece ( iron8 ,! effect ( ea* lev] is to be calculat$ sep>ately4 ,we may asc]ta9 ! p[] ex]t$ at ,b by ! rule ( mom5ts already expla9$2 ! product ( ! p[] & its arm is #bb #aj #bbj2 ?is divid$ by ! numb] ( in*es1 #ah 9 ! l9e ,a ,b1 gives a quoti5t #abb1 & ?is quoti5t is ! numb] ( p\nds pressure :i* is ex]t$ by means ( ! l9k ^u ! second lev]4 ,we proce$ 9 ! same mann] to f9d ! magnitude ( ! pressure ^u ! iron at ,e4 ,! product ( #abb & #h is #igf4 ,?is is divid$ by #ce1 & ! quoti5t f\nd is #bgi4 ,h5ce ! ex]tion ( a pressure ( #aj lbs4 at ,f produces a pressure ( #bgi lbs4 at ,e4 ,9 r\nd numb]s1 we may say t ! pressure is magnifi$ #bh-;fold by means ( ?is comb9ation ( lev]s ( ! second ord]4 #bdg4 ,a pressure ( #aj lbs4 is n suffici5t to %e> across ! b> ( iron1 ev5 ?\gh x be magnifi$ to #bgi lbs4 ,i !re=e susp5d weights f ,f1 & gradually increase ! load until ! b> is cut4 ,i f9d at ! fir/ trial t #aab lbs4 is suffici5t1 & a second trial ) ! same b> gives #aad lbs42 #aac lbs41 ! mean betwe5 ^! results1 may be consid]$ an adequate =ce4 ,?is is ! load on ,f2 ! real pressure on ! b> is #aac #bgi #caec lbs43 ?us ! actual pressure :i* 0 necess>y to cut ! b> am\nt$ to m ?an a ton4 #bdh4 ,we c calculate f ?is exp]im5t ! am\nt ( =ce necess>y to %e> across a b> "o squ>e in* 9 section4 ,we may reasonably suppose t ! necess>y p[] is proportional to ! section1 & !re=e ! p[] w be> to #caec lbs4 ! proportion :i* a squ>e ( "o in* be>s to ! squ>e ( a qu>t] in*2 b ?is ratio is #af3 h5ce ! =ce is #af #caec lbs41 equal to ab\t #bbe tons4 #bdi4 ,x is noticeable t #bbe tons is ne>ly ! =ce :i* w\ld suffice to te> ! b> 9 sund] by actual t5sion4 ,we %all subsequ5tly return to ! subject ( %e>+ iron 9 ! lecture ^u ,in]tia 7,lecture ,,xvi474 ,,! ,,lev] ,,( ,,! ,,?ird ,,ord]4 #bej4 ,! lev] ( ! ?ird ord] may be easily und]/ood f ,fig4 #ci1 ( :i* we h already made use4 ,9 ! lev] ( ! ?ird ord] ! fulcrum is at "o end1 ! load is at ! o!r end1 :ile ! p[] lies betwe5 ! two4 ,9 ?is case1 !n1 ! p[] is repres5t$ by ! #ef lb4 weight1 :ile ! load is indicat$ by ! spr+ balance4 ,! p[] always exce$s ! load1 & consequ5tly ?is lev] is to be us$ :]e spe$ is to be ga9$ in/ead ( p[]4 ,?us1 = example2 :5 ! p[]1 #ef lbs41 is #b' di/ant f ! fulcrum1 ! load indicat$ by ! spr+ balance is ab\t #bc lbs4 #bea4 ,! treadle ( a gr9d/one is (t5 a lev] ( ! ?ird ord]4 ,! fulcrum is at "o end1 ! load is at ! o!r end1 & ! foot has only to move ?r\gh a small di/ance4 #beb4 ,! pr9ciples :i* h be5 discuss$ 9 ,lecture ,,iii4 ) respect to p>allel =ces expla9 ! laws n[ laid d[n = lev]s ( diff]5t ord]s1 & w also enable u to express ^! laws m concisely4 #bec4 ,a comp>ison betwe5 ,figs4 #bj & #ci %[s t ! only diff]5ce betwe5 ! contrivances is t 9 ,fig4 #bj we h a spr+ balance ,c 9 ! same place z ! /eel edge ,a 9 ,fig #ci4 ,we may 9 ,fig4 #bj reg>d "o spr+ balance z ! p[]1 ! o!r z ! fulcrum1 & ! weight z ! load4 ,nor is !re any ess5tial diff]5ce betwe5 ! app>atus ( ,fig4 #ch & t ( ,fig4 #bj4 ,9 ,fig4 #ch ! b> is pull$ d[n by a =ce at ea* end1 "o a weight1 ! o!r a spr+ balance1 :ile x is support$ by ! upw>d pressure ( ! /eel edge4 ,9 ,fig4 #bj ! b> is be+ pull$ upw>ds by a =ce at ea* end1 & d[nw>ds by ! weight4 ,! two cases are sub/antially ! same4 ,9 ea* ( !m we f9d a b> act$ ^u by a pair ( p>allel =ces appli$ at its extremities1 & reta9$ 9 equilibrium by a ?ird =ce4 #bed4 ,we may !re=e apply to ! lev] ! pr9ciples ( p>allel =ces already expla9$4 ,we %[$ t two p>allel =ces act+ ^u a b> c\ld be comp\nd$ into a resultant1 appli$ at a c]ta9 po9t ( ! b>4 ,we h def9$ ! mom5t ( a =ce 7,art4 #fd71 & prov$ t ! mom5ts ( two p>allel =ces ab\t ! po9t ( application ( _! resultant are equal4 #bee4 ,9 ! lev] ( ! fir/ ord] !re are two p>allel =ces1 "o at ea* end2 ^! are comp\nd$ into a resultant1 & x is necess>y t ?is resultant be appli$ exactly ov] ! /eel edge or fulcrum 9 ord] t ! b> may be ma9ta9$ at re/4 ,9 ! lev]s ( ! second & ?ird ord]s1 ! p[] & ! load are two p>allel =ces act+ 9 opposite directions2 _! resultant1 !re=e1 does n lie betwe5 ! =ces1 b is appli$ on ! side ( ! great]1 & at ! po9t :]e ! /eel edge supports ! b>4 ,9 all cases ! mom5t ( "o ( ! =ces ab\t ! fulcrum mu/ be equal to t ( ! o!r4 ,f ! equality ( mom5ts x foll[s t ! product ( ! p[] & ! di/ance ( ! p[] f ! fulcrum equals ! product ( ! load1 & ! di/ance ( ! load f ! fulcrum3 ?is pr9ciple suffices to demon/rate ! rules already giv54 #bef4 ,! laws gov]n+ ! lev] may be d$uc$ f ! pr9ciple ( "w2 ! load1 if ne>] ?an ! p[] to ! fulcrum1 is mov$ ?r\gh a small] di/ance ?an ! p[]4 ,?us1 = example1 9 ! lev] ( ! fir/ ord]3 if ! load be #ab times z f> z ! p[] f ! fulcrum1 !n = e in* ! load moves x c be demon/rat$ t ! p[] mu/ move #ab in*es4 ,! numb] ( units ( "w appli$ at "o end ( a ma*9e is equal to ! numb] yield$ at ! o!r1 always except+ ! loss due to friction1 :i* is1 h[ev]1 s small 9 ! lev] t we may neglect x4 ,if !n a p[] ( #a lb4 be appli$ to move ! p[] end ?r\gh #ab in*es1 "o unit ( "w w h be5 put into ! ma*9e4 ,h5ce "o unit ( "w mu/ be done on ! load1 b ! load only moves ?r\gh _/ ( a foot1 & !re=e a load ( #ab lbs4 c\ld be ov]come3 ?is is ! same result z w\ld be giv5 by ! rule 7,art4 #bcf74 #beg4 ,to conclude3 we h fir/ det]m9$ by actual exp]im5t ! relation betwe5 ! p[] & ! load 9 ! lev]2 we h se5 t ! law ?us obta9$ h>monizes ) ! pr9ciple ( ! composition ( p>allel =ces2 &1 f9ally1 we h %[n h[ ! same result c be d$uc$ f ! f]tile & important pr9ciple ( "w4 ,,lecture ,,ix4 ,,! ,,incl9$ ,,plane ,,& ,,! ,,screw4 ,! ,incl9$ ,plane )\t ,friction4--,! ,incl9$ ,plane ) ,friction4--,! ,screw4--,! ,screw-jack4--,! ,bolt & ,nut4 ,,! ,,incl9$ ,,plane ,,)\t ,,friction4 #beh4 ,! me*anical p[]s n[ to be consid]$ are (t5 us$ = o!r purposes beside ^? ( rais+ great weights4 ,= example3 ! p>ts ( a /ructure h to be =cibly drawn toge!r1 a p[]ful compression has to be ex]t$1 a mass ( timb] or o!r mat]ial has to be riv5 asund] by splitt+4 ,= purposes ( ?is k9d ! incl9$ plane 9 its v>i\s =ms1 & ! screw1 are ( ! greate/ use4 ,! screw also1 9 ! =m ( ! screw-jack1 is sometimes us$ 9 rais+ weights4 ,x is pr9cipally conv5i5t :5 ! weight is enorm\sly great1 & ! di/ance ?r\gh :i* x has to be rais$ comp>atively small4 ,illu/ration3 ,,fig4 #da4 #bei4 ,we %all comm5ce ) ! /udy ( ! incl9$ plane4 ,! app>atus us$ is %[n 9 ,fig4 #da4 ,a ,b is a plate ( glass #d' long1 m\nt$ on a frame & turn+ r\nd a h+e at ,a2 ,b ,d is a circul> arc1 ) its c5tre at ,a1 by :i* ! glass may be support$2 ,d ,c is a v]tical rod1 to :i* ! pulley ,c is clamp$4 ,?is pulley c be mov$ up & d[n1 to be accommodat$ to ! position ( ,a ,b2 ! pulley is made ( brass1 & turns v freely4 ,a little truck ,r is adapt$ to run on ! plane ( glass4 ,! truck is lad5 to weigh #a lb41 & ?is weight is unalt]$ ?r\gh\t ! exp]im5ts2 ! :eels are v free1 s t ! truck runs ) b little friction4 #bfj4 ,b ! friction1 ?\gh small1 is appreciable1 & x w be necess>y to measure ! am\nt & !n endeav\r to c\nt]act its effect ^u ! motion4 ,! silk cord atta*$ to ! truck is v f9e1 & its weight is neglect$4 ,a s]ies ( weights is provid$2 !y are made f pieces ( brass wire1 & weigh #ja lb4 & #jja lb43 ^! c easily be hook$ into ! loop on ! cord at ,p4 ,we fir/ make ! plane ,a ,b horizontal1 & br+ d[n ! pulley ,c s t ! cord %all be p>allel to ! plane2 we f9d t a =ce mu/ be appli$ by ! cord 9 ord] to draw ! truck along ! plane3 ?is =ce is ( c\rse ! friction1 & by a suitable weight at ,p ! friction may be said to be c\nt]balanc$4 ,b we _c expect t ! friction w be ! same :5 ! plane is horizontal z :5 ! plane is incl9$4 ,we mu/ !re=e exam9e ?is "q by a me?od analog\s to t us$ 9 ,art4 #bjg4 #bfa4 ,let ! plane be elevat$ until ,b ,e1 ! elevation ( ,b above ,a ,d1 is #bj2 let ,c be prop]ly adju/$3 x is f\nd t :5 ,p is ,o#de lb4 ,r is j pull$ up2 & on ! o!r h&1 :5 ,p is only #jdj lb4 ! truck desc5ds & raises ,p2 & :5 ,p has any value int]m$iate betwe5 ^! two1 ! truck rema9s 9 equilibrium4 ,let u d5ote ! =ce ( gravity act+ d[n ! plane by ,r1 & x foll[s t ,r mu/ be #jdbe lb41 & ! friction #jjbe lb4 ,= :5 ,p raises ,r1 x mu/ ov]come friction z well z ,r2 !re=e ! p[] mu/ be #jjbe #jdbe #jde4 ,on ! o!r h&1 :5 ,r raises ,p1 x mu/ also ov]come ! friction #jjbe1 !re=e ,p c only be #jdbe - #jjbe #jdj2 & ,r is ?us f\nd to be a mean betwe5 ! greate/ & lea/ values ( ,p consi/5t ) equilibrium4 ,if ! plane be rais$ s t ! height ,b ,e is #cc1 ! greate/ & lea/ values ( ,p are #jff & #jga2 !re=e ,r is #jfhe friction #jjbe1 ! same z be=e4 ,f9ally1 mak+ ! height ,b ,e only #b1 ! friction is f\nd to be #jjbj1 :i* is n mu* less ?an ! previ\s det]m9ations4 ,^! exp]im5ts %[ t we may consid] ?is v small friction to be practically con/ant at ^! incl9ations4 7,7 ! friction l>ge1 o!r me?ods are necess>y1 see ,art4 #bfe47 ,z 9 ! exp]im5ts ,r is always rais$ we %all give ,p ! p]man5t load ( #jjbe lb41 ?us suffici5tly c\nt]act+ friction1 :i* we may !re=e dismiss f consid]ation4 ,x is h>dly necess>y to rem>k t1 9 aft]w>ds record+ ! weights plac$ at ,p1 ?is c\nt]poise is n to be includ$4 #bfb4 ,we h n[ ! means ( /udy+ ! relation betwe5 ! p[] & ! load 9 ! frictionless incl9$ plane4 ,! incl9e be+ set at diff]5t elevations1 we %all obs]ve ! =ce necess>y to draw up ! con/ant load ( #a lb4 ,our c\rse w be guid$ by fir/ mak+ use ( ! pr9ciple ( en]gy4 ,suppose ,b ,e to be #b'2 :5 ! truck has be5 mov$ f ! bottom ( ! plane to ! top1 x w h be5 rais$ v]tically ?r\gh a height ( #b'1 & two units ( en]gy mu/ h be5 consum$4 ,b ! plane be+ #d' long1 ! =ce :i* draws up ! truck ne$ only be #je lb41 = #je lb4 act+ ov] #d' produces two units ( "w4 ,9 g5]al1 if l be ! l5g? ( ! plane & h its height1 ,r ! load1 & ,p ! p[]1 ! numb] ( units ( en]gy necess>y to raise ! load is ,r h1 & ! numb] ( units exp5d$ 9 pull+ x up ! plane is ,p l3 h5ce ,r h ,p l1 & consequ5tly ,p 3 h 33 ,r 3 l2 t is1 ! p[] is to ! height ( ! plane z ! load is to its l5g?4 ,9 ! pres5t case ,r #a lb41 l #dh2 !re=e ,p #jjbjh h1 :]e h is ! height ( ! plane 9 in*es1 & ,p ! p[] 9 p\nds4 #bfc4 ,we comp>e ! p[]s calculat$ by ?is =mula ) ! actual obs]v$ values3 ! result is giv5 9 ,table ,,xiii4 ,,table ,,xiii4--,,incl9$ ,,plane4 ,glass ,plane #dh long1 truck #a lb4 9 weight1 friction c\nt]pois$2 =mula ,p #jjbjh h4 ---------------------------------------- --------------------- ,numb] ( ,height,obs]v$ ,p4 ,diff]5ce ( ! ,exp]im5t4 ( p[] ,calculat$ obs]v$ & calculat$ plane4 9 lbs4p[] 9 lbs4 p[]s ---------------------------------------- --------------------- #a #b #jjd #jjd #jjj #b #d #jjh #jjh #jjj #c #f #jac #jab -#jja #d #h #jaf #jag #jja #e #aj #jba #jba #jjj #f #ae #jca #jca #jjj #g #bj #jdb #jdb #jjj #h #cc #jga #jfi -#jjb ---------------------------------------- --------------------- ,?us = example1 9 exp]im5t #f1 :]e ! height ,b ,e is #ae1 x is obs]v$ t ! p[] necess>y to draw ! truck is #jca lb4 ,! truck is plac$ 9 ! middle ( ! plane1 & ! p[] is adju/$ s z to be suffici5t to draw ! truck to ! top ) c]ta9ty2 ! necess>y p[] calculat$ by ! =mula is also #jca lbs41 s t ! !ory is v]ifi$4 #bfd4 ,! fif? column ( ! table %[s ! diff]5ce betwe5 ! obs]v$ & ! calculat$ p[]s4 ,! v slight diff]5ces1 9 no case exce$+ ! fiftie? "p ( a p\nd1 may be ref]r$ to ! inevitable errors ( exp]im5t4 ,,! ,,incl9$ ,,plane ,,) ,,friction4 #bfe4 ,! friction ( ! truck ^u ! glass plate is always v small1 & is %[n to h b little v>iation at ^? incl9ations ( ! plane :i* we us$4 ,b :5 ! friction is l>ge1 we %all n be ju/ifi$ 9 neglect+ its *anges at diff]5t elevations1 & we mu/ adopt m rigor\s me?ods4 ,= ?is inquiry we %all use ! p9e plank & slide already describ$ 9 ,art4 #aag4 ,we d n 9 ?is case attempt to dim9i% friction by ! aid ( :eels1 & consequ5tly x w be ( consid]able am\nt4 #bff4 ,9 ano!r respect ! exp]im5ts ( ,table ,,xiii4 are also 9 contra/ ) ^? n[ to be describ$4 ,9 ! =m] ! load 0 con/ant1 :ile ! elevation 0 *ang$4 ,9 ! latt] ! elevation rema9s con/ant :ile a succession ( diff]5t loads are tri$4 ,we %all f9d 9 ?is inquiry also t :5 ! prop] all[ance has be5 made = friction1 ! !oretical law connect+ ! p[] & ! load is fully v]ifi$4 #bfg4 ,! app>atus us$ is %[n 9 ,fig4 #cc2 ! plane1 is1 h[ev]1 secur$ at "o incl9ation1 & ! pulley ,c %[n 9 ,fig4 #cb is adju/$ to ! app>atus1 s t ! rope f ! pulley to ! slide is p>allel to ! incl9e4 ,! elevation ( ! plane 9 ! position adopt$ is #agb1 s t its l5g?1 base1 & height are 9 ! proportions ( ! numb]s #a1 #jiee1 & #jbif4 ,weights rang+ f #g lbs4 to #ef lbs4 are plac$ ^u ! slide1 & ! p[] is f\nd :i*1 :5 ! slide is />t$ by ! screw1 w draw x /eadily up ! plane4 ,! requisite p[] consi/s ( two p>ts1 t :i* is necess>y to ov]come gravity act+ d[n ! plane1 & t :i* is necess>y to ov]come friction4 ,illu/ration3 ,,fig4 #db4 #bfh4 ,! =ces are %[n 9 ,fig4 #db4 ,r ,g1 ! =ce ( gravity1 is resolv$ into ,r ,l & ,r ,m2 ,r ,l is evid5tly ! compon5t act+ d[n ! plane1 & ,r ,m ! pressure aga9/ ! plane2 ! triangle ,g ,l ,r is simil> to ,a ,b ,c1 h5ce if ,r be ! load1 ! =ce ,r ,l act+ d[n ! plane mu/ be #jbif ,r1 & ! pressure ^u ! plane #jiee ,r4 #bfi4 ,we %all fir/ make a calculation ) ! ord9>y law t ! friction is proportional to ! pressure4 ,! pressure ^u ! plane ,a ,b1 to :i* ! friction is proportional1 is n ! weight ( ! load4 ,! pressure is t compon5t 7,r ,m7 ( ! load :i* is p]p5dicul> to ! plane ,a ,b4 ,:5 ! weights d n ext5d beyond #ef lbs41 ! be/ value = ! coeffici5t ( friction is #jbhh 7,art4 #ada73 h5ce ! am\nt ( friction ^u ! plane is #jbhh #jiee ,r #jbge ,r4 ,?is =ce mu/ be ov]come 9 addition to #jbif ,r 7! compon5t ( gravity act+ d[n ! plane73 h5ce ! expression = ! p[] is #jbge ,r #jbif ,r #jega ,r4 #bgj4 ,! values ( ! obs]v$ p[]s comp>$ ) ! p[]s calculat$ f ! expression #jega ,r are %[n 9 ,table ,,xiv4 ,,table ,,xiv4--,,incl9$ ,,plane4 ,smoo? plane ( p9e #gb #aa2 angle ( incl9ation #agb2 slide ( p9e1 gra9 crosswise2 slide />t$2 =mula ,p #jega ,r4 ---------------------------------------- --------------- ,r4 ,p[] 9 ,p4 ,diff]5ce ( ,numb] ( ,total loadlbs4 :i*,calculat$ ! obs]v$ ,exp]im5t4 on slide j draws value ( & calculat$ 9 lbs4 up slide4! p[]4 p[]s4 ---------------------------------------- --------------- #a #g #df #dj -#jf #b #ad #hc #hj -#jc #c #ba #abc #abj -#jc #d #bh #afe #afj -#je #e #ce #bjj #bjj #jj #f #db #bdb #bdj -#jb #g #di #bhj #bhj #jj #h #ef #cah #cbj #jb ---------------------------------------- --------------- #bga4 ,?us = example1 9 exp]im5t #f1 a load ( #db lbs4 0 rais$ by a =ce ( #bdb lbs41 :ile ! calculat$ value is #bdj lbs42 ! diff]5ce1 #jb lbs41 is %[n 9 ! la/ column4 #bgb4 ,! calculat$ values are f\nd to agree tol]ably well ) ! obs]v$ values1 b ! pres5ce ( ! l>ge diff]5ces 9 ,no4 #a & ,no4 #d leads u to inquire :e!r by employ+ ! m accurate law ( friction 7,art4 #ada7 a bett] result may n be obta9$4 ,9 ,table ,,vi4 we h %[n t ! friction = weights n exce$+ #ef lbs4 is express$ by ! =mula ,f #ji #jbff pressure1 b ! pressure is 9 ?is case #jiee ,r1 & h5ce ! friction is #ji #jbed ,r4 ,to ?is mu/ be add$ #jbif ,r1 ! compon5t ( ! =ce ( gravity :i* mu/ be ov]come1 & h5ce ! total =ce necess>y is #ji #jee ,r4 ,! p[]s calculat$ f ?is expression are comp>$ ) ^? actually obs]v$ 9 ,table ,,xv4 ,,table ,,xv4--,,incl9$ ,,plane4 ,smoo? plane ( p9e #gb #aa2 angle ( incl9ation #agb2 slide ( p9e1 gra9 crosswise2 slide />t$2 =mula ,p #ji #jee ,r4 ---------------------------------------- --------------- ,r4 ,p[] 9 ,p4 ,diff]5ce ( ,numb] ( ,total loadlbs4 :i*,calculat$ ! obs]v$ ,exp]im5t4 on slide j draws value ( & calculat$ 9 lbs4 up slide4! p[]4 p[]s4 ---------------------------------------- --------------- #a #g #df #dg #ja #b #ad #hc #hf #jc #c #ba #abc #abe #jb #d #bh #afe #afc -#jb #e #ce #bjj #bja #ja #f #db #bdb #bdj -#jb #g #di #bhj #bgh -#jb #h #ef #cah #cag -#ja ---------------------------------------- --------------- ,= example3 9 exp]im5t #e1 a load ( #ce lbs4 is f\nd to be rais$ by a p[] ( #bjj lbs41 :ile ! calculat$ p[] is #ji #jee #ce #bja lbs4 #bgc4 ,! calculat$ values ( ! p[]s are %[n by ?is table to agree extremely well ) ! obs]v$ values1 ! greate/ diff]5ce be+ only ,o#c lb4 ,h5ce !re c be no d\bt t ! pr9ciples on :i* ! =mula has be5 calculat$ are correct4 ,?is table may !re=e be reg>d$ z v]ify+ bo? ! law ( friction1 & ! rule laid d[n = ! relation betwe5 ! p[] & ! load 9 ! incl9$ plane4 #bgd4 ,! incl9$ plane is prop]ly /yl$ a me*anical p[]4 ,= let ! weight be #cj lbs41 we calculate by ! =mula t #agd lbs4 w\ld be suffici5t to raise x1 s t1 not)/&+ ! loss by friction1 we h "h a small] =ce ov]com+ a l>g] "o1 :i* is ! ess5tial feature ( a me*anical p[]4 ,! me*anical effici5cy is #cj #agd #agb4 #bge4 ,! velocity ratio 9 ! incl9$ plane is ! ratio ( ! di/ance ?r\gh :i* ! p[] moves to ! height ?r\gh :i* ! weight is rais$1 t is #a #jbif #cch4 ,to raise #cj lbs4 "o foot1 a =ce ( #agd lbs4 mu/ !re=e be ex]t$ ?r\gh #cch feet4 ,! numb] ( units ( "w exp5d$ is ?us #agd #cch #ehh4 ,( ?is #cj units1 equival5t to #ea p] c5t41 are utiliz$4 ,! rema9+ #bhh units1 or #di p] c5t41 are absorb$ by friction4 #bgf4 ,we h po9t$ out 9 ,art4 #bbb t a ma*9e 9 :i* less ?an half ! en]gy is lo/ by friction w p]mit ! load to run d[n :5 free3 ?is is ! case 9 ! pres5t in/ance2 h5ce ! weight w run d[n ! plane unless specially re/ra9$4 ,t x %\ld d s agrees ) ,art4 #adg1 = x 0 !re %[n t at ab\t #acd1 & /ill m at any great] incl9ation1 ! slide w\ld desc5d :5 />t$4 ,,! ,,screw4 #bgg4 ,! incl9$ plane z a me*anical p[] is (t5 us$ 9 ! =m ( a w$ge or 9 ! /ill m disguis$ =m ( a screw4 ,a w$ge is an incl9$ plane :i* is =c$ "u ! load2 x is usually mov$ by means ( a hamm]1 s t ! effici5cy ( ! w$ge is augm5t$ by ! dynamical effect ( a bl[4 #bgh4 ,! screw is "o ( ! mo/ useful me*anical p[]s :i* we possess4 ,its =m may be trac$ by wrapp+ a w$ge-%ap$ piece ( pap] ar\nd a cyl9d]1 & !n cutt+ a groove 9 ! cyl9d] along ! spiral l9e indicat$ by ! m>g9 ( ! pap]4 ,su* a groove is a screw4 ,9 ord] t ! screw may be us$ x mu/ revolve 9 a nut :i* is made f a holl[ cyl9d]1 ! int]nal diamet] ( :i* is equal to t ( ! cyl9d] f :i* ! screw is cut4 ,! nut conta9s a spiral ridge1 :i* fits into ! correspond+ ?read 9 ! screw2 :5 ! nut is turn$ r\nd1 x moves backw>ds or =w>ds accord+ to ! direction ( ! rotation4 ,l>ge screws ( ! bett] class1 su* z ^? ^u :i* we %all fir/ make exp]im5ts1 are always turn$ 9 a la!1 & are ?us =m$ ) extreme accuracy4 ,small screws are made 9 a simpl] mann] by means ( dies & o!r contrivances4 #bgi4 ,a *>act]i/ic feature ( a screw is ! incl9ation ( ! ?read to ! axis4 ,?is is mo/ conv5i5tly describ$ by ! numb] ( complete turns :i* ! ?read makes 9 a specifi$ l5g? ( ! screw1 usually an in*4 ,= example3 a screw is said to h t5 ?reads to ! in* :5 x requires #aj revolutions ( ! nut 9 ord] to move x "o in*4 ,! %ape ( ! ?read itself v>ies ) ! purposes = :i* ! screw is int5d$2 ! section may be squ>e or triangul>1 or1 z is g5]ally ! case 9 small screws1 ( a r\nd$ =m4 #bhj4 ,!re is s mu* friction 9 ! screw t exp]im5ts are necess>y = ! det]m9ation ( ! law connect+ ! p[] & ! load4 #bha4 ,we %all comm5ce ) an exam9ation ( ! screw by ! app>atus %[n 9 ,fig4 #dc4 ,! nut ,a is secur$ ^u a /\t frame2 to ! end ( ! screw hooks are atta*$1 9 ord] to receive ! load1 :i* 9 ?is app>atus does n exce$ #bbd lbs42 at ! top ( ! screw is an arm ,e by :i* ! screw is turn$2 to ! end ( ! arm a rope is atta*$1 :i* pass+ ov] a pulley ,d1 c>ries a hook = receiv+ ! p[] ,c4 ,illu/ration3 ,,fig4 #dc4 #bhb4 ,we fir/ apply ! pr9ciple ( "w to ?is screw1 & calculate ! relation betwe5 ! p[] & ! load z x w\ld be f\nd if friction 7 abs5t4 ,! diamet] ( ! circle describ$ by ! end ( ! arm is #bje2 its circumf]5ce is !re=e #fdd4 ,! screw conta9s ?ree ?reads 9 ! in*1 h5ce 9 ord] to raise ! load #a ! p[] moves #c #fdd #aic v ne>ly2 !re=e ! velocity ratio is #aic1 & 7 ! screw capable ( work+ )\t friction1 #aic w\ld repres5t ! me*anical effici5cy4 ,9 actually p]=m+ ! exp]im5ts ! arm ,e is plac$ at "r angles to ! rope lead+ to ! pulley1 & ! p[] hook is weight$ until1 ) a slight />t1 ! arm is /eadily drawn2 b ! p[] w only move ! arm a few in*es1 = :5 ! cord ceases to be p]p5dicul> to ! arm ! p[] acts ) dim9i%$ effici5cy2 consequ5tly ! load is only rais$ 9 ea* exp]im5t ?r\gh a small fraction ( an in*1 b q suffici5t = our purpose4 ,,table ,,xvi4--,,! ,,screw4 ,wr\ght iron screw1 squ>e ?read1 diamet] #abe1 ) #c ?reads to ! in*1 l5g? ( arm #ajbe2 nut ( ca/ iron1 be>+ surfaces oil$1 velocity ratio #aic1 useful effect #cf p] c5t41 me*anical effici5cy #gj2 =mula ,p #jjadc ,r4 ---------------------------------------- -------------------- ,p4 ,diff]5ce ( ,numb] ( ,r4 ,obs]v$ ,calculat$ ! obs]v$ & ,exp]im5t4 ,load p[] p[] calculat$ 9 lbs4 9 lbs4 9 lbs4 p[]s4 ---------------------------------------- -------------------- #a #bh #jd #jd #jj #b #ef #jh #jh #jj #c #hd #ab #ab #jj #d #aab #af #af #jj #e #adj #bj #bj #jj #f #afh #bd #bd #jj #g #aif #bg #bh #ja #h #bbd #cc #cb -#ja ---------------------------------------- -------------------- #bhc4 ,! results ( ! exp]im5ts are %[n 9 ,table ,,xvi4 ,if ! motion _h n be5 aid$ by a />t ! p[]s w\ld h be5 great]4 ,?us 9 exp]im5t #f1 #bd lbs4 is ! p[] ) a />t1 :5 )\t a />t #cb lbs4 0 f\nd to be necess>y4 ,! exp]im5ts h all be5 aid$ by a />t1 & ! results record$ h be5 correct$ = ! friction ( ! pulley ov] :i* ! rope passes3 ?is correction is v small1 9 no case exce$+ #jb lb4 ,! f\r? column conta9s ! values ( ! p[]s comput$ by ! =mula ,p #jjadc ,r4 ,?is =mula has be5 d$uc$ f ! obs]vations 9 ! mann] describ$ 9 ! ,app5dix4 ,! fif? column proves t ! exp]im5ts are truly repres5t$ by ! =mula3 9 ea* ( ! exp]im5ts #g & #h1 ! diff]5ce betwe5 ! calculat$ & obs]v$ values am\nts to #ja lb41 b ?is is q inconsid]able 9 comp>ison ) ! weights we are employ+4 #bhd4 ,9 ord] to lift #ajj lbs4 ! expression #jjadc ,r %[s t #adc lbs4 w\ld be necess>y3 h5ce ! me*anical effici5cy ( ! screw is #ajj #adc #gj4 ,?us ?is screw is va/ly m p[]ful ?an any ( ! pulley sy/ems :i* we h discuss$4 ,a ma*9e s capable1 s compact1 & s /rong z ! screw1 is invaluable = innum]able purposes 9 ! ,arts1 z well z 9 multitudes ( appliances 9 daily use4 #bhe4 ,x is evid5t1 h[ev]1 t ! di/ance ?r\gh :i* ! screw c raise a weight mu/ be limit$ by ! l5g? ( ! screw itself1 & t 9 ! l5g? ( lift ! screw _c compete ) _m ( ! o!r contrivances us$ 9 rais+ weights4 #bhf4 ,we h se5 t ! velocity ratio is #aic2 !re=e1 to raise #ajj lbs4 #a foot1 we f9d t #adc #aic #bgf units ( en]gy mu/ be exp5d$3 ( ?is only #ajj units1 or #cf p] c5t41 is usefully employ$2 ! re/ be+ consum$ 9 ov]com+ ! friction ( ! screw4 ,?us ne>ly two-?irds ( ! en]gy appli$ to su* a screw is wa/$4 ,h5ce we f9d t friction does n p]mit ! load to run d[n1 s9ce less ?an fifty p] c5t4 ( ! appli$ en]gy is usefully employ$ 7,art4 #bbb74 ,?is is "o ( ! valuable prop]ties :i* ! screw possesses4 #bhg4 ,we may contra/ ! screw ) ! pulley-block 7,art4 #aii74 ,!y are bo? p[]ful ma*9es3 ! latt] is bulky & economical ( p[]1 ! =m] is compact & wa/eful ( p[]2 ! latt] is adapt$ = rais+ weights ?r\gh consid]able di/ances1 & ! =m] = ex]t+ pressures ?r\gh %ort di/ances4 ,illu/ration3 ,,fig4 #dd4 ,,! ,,screw-,,jack4 #bhh4 ,! importance ( ! screw z a me*anical p[] ju/ifies u 9 exam9+ ano!r ( its useful =ms1 ! screw-jack4 ,?is ma*9e is us$ = ex]t+ great pressures1 su* = example z />t+ a %ip :i* is reluctant to be laun*$1 or replac+ a locomotive ^u ! l9e f :i* its :eels h slipp$4 ,^! ma*9es v>y 9 =m1 z well z 9 ! weights = :i* !y are adapt$2 "o ( !m is %[n at ,d 9 ,fig4 #dd1 & a description ( its details is giv5 9 ,table ,,xvii4 ,we %all det]m9e ! p[]s to be appli$ to ?is ma*9e = ov]com+ resi/ances n exce$+ half a ton4 #bhi4 ,to employ weights s l>ge z half a ton w\ld be inconv5i5t if n actually impossible 9 ! lecture room1 b ! requir$ pressures c be produc$ by means ( a lev]4 ,9 ,fig4 #dd is %[n a /\t wood5 b> #af' long4 ,x is prev5t$ f b5d+ by means ( a *a92 at ,e ! lev] is atta*$ to a h+e1 ab\t :i* x turns freely2 at ,a a tray is plac$ = ! purpose ( receiv+ weights4 ,! screw-jack is #b' di/ant f ,e1 consequ5tly ! b> is a lev] ( ! second ord]1 & any weight plac$ 9 ! tray ex]ts a pressure eightfold great] ^u ! top ( ! screw-jack4 ,?us ea* /one 9 ! tray produces a pressure ( #a cwt4 at ! po9t ,d4 ,! weight ( ! lev] & ! tray is c\nt]pois$ by ! weight ,c1 s t until ! tray receives a load !re is no pressure ^u ! top ( ! screw-jack1 & ?us we may omit ! lev] itself f consid]ation4 ,! screw-jack is furni%$ ) an arm ,d ,g2 at ! extremity ,g ( ?is arm a rope is atta*$1 :i* passes ov] a pulley & supports ! p[] ,b4 #bij4 ,! velocity ratio = ?is screw-jack ) an arm ( #cc1 is f\nd to be #dad1 by ! me?od already describ$ 7,art4 #bhc74 #bia4 ,to det]m9e its me*anical effici5cy we mu/ resort to exp]im5t4 ,! result is giv5 9 ,table ,,xvii4 ,,table ,,xvii4--,,! ,,screw-,,jack4 ,wr\ght iron screw1 squ>e ?read1 diamet] #b1 pit* #b ?reads to ! in*1 arm #cc2 nut brass1 be>+ surfaces oil$2 velocity ratio #dad2 useful effect1 #bh p] c5t42 me*anical effici5cy #aaf2 =mula ,p #jff #jjjge ,r4 ---------------------------------------- --------------- ,p4 ,diff]5ce ( ! ,numb] ( ,r4 ,obs]v$ ,calculat$ obs]v$ & ,exp]im5t4 ,load p[] p[] calculat$ 9 lbs4 9 lbs4 9 lbs4 p[]s4 ---------------------------------------- --------------- #a #aab #ad #ae #ja #b #bbd #bb #bc #ja #c #ccf #cc #cb -#ja #d #ddh #da #dj -#ja #e #efj #ej #di -#ja #f #fgb #eg #eg #jj #g #ghd #fe #fe #jj #h #hif #gd #gd #jj #i #ajjh #ha #hb #ja #aj #aabj #ij #ia #ja ---------------------------------------- --------------- #bib4 ,x may be se5 f ! column ( diff]5ces h[ closely ! exp]im5ts are repres5t$ by ! =mula4 ,! p[] :i* is requir$ to raise a giv5 weight1 say #fjj lbs41 may be calculat$ by ?is =mula2 x is #jff #jjjge #fjj #eaf4 ,h5ce ! me*anical effici5cy ( ! screw-jack is #fjj #eaf #aaf4 ,?us ! screw is v p[]ful1 increas+ ! =ce appli$ to x m ?an a hundr$fold4 ,9 ord] to raise #fjj lbs4 "o foot1 a quantity ( "w repres5t$ by #eaf #dad #bacf units mu/ be exp5d$2 ( ?is only #fjj1 or #bh p] c5t41 is utiliz$1 s t ne>ly ?ree-qu>t]s ( ! en]gy appli$ is exp5d$ ^u friction4 #bic4 ,?is screw does n let ! load run d[n1 s9ce less ?an #ej p] c5t4 ( en]gy is utilis$2 to l[] ! weight ! lev] has actually to be press$ backw>ds4 #bid4 ,! details ( an exp]im5t on ?is subject w be in/ructive1 & af=d a confirmation ( ! pr9ciples laid d[n4 ,9 exp]im5t #aj we f9d t #ij lbs4 suffice to raise #a1abj lbs42 n[ by mov+ ! pulley to ! o!r side ( ! lev]1 & plac+ ! rope p]p5dicul>ly to ! lev]1 ,i f9d t to produce motion ! o!r way--t is1 ( c\rse to l[] ! screw--a =ce ( #cd lbs4 mu/ be appli$4 ,h5ce1 ev5 ) ! assi/ance ( ! load1 a =ce ( #cd lbs4 is necess>y to ov]come friction4 ,?is w enable u to det]m9e ! am\nt ( friction 9 ! same mann] z we det]m9$ ! friction 9 ! pulley-block 7,art4 #bjg74 ,let ,x be ! =ce usefully employ$ 9 rais+1 & ,y ! =ce ( friction1 :i* acts equally 9 ei!r direction aga9/ ! production ( motion2 !n to raise ! load ! p[] appli$ mu/ be suffici5t to ov]come bo? ,x & ,y1 & !re=e we h ,x ,y #ij4 ,:5 ! weight is to be l[]$ ! =ce ,x ( c\rse aids 9 ! l[]+1 b ,x alone is n suffici5t to ov]come ! friction2 x requires ! addition ( #cd lbs41 & we h !re=e ,x #cd ,y1 & h5ce ,x #bh1 ,y #fb4 ,t is1 #bh is ! am\nt ( =ce :i* ) a frictionless screw w\ld h be5 suffici5t to raise half a ton4 ,b 9 ! frictionless screw ! p[] is f\nd by divid+ ! load by ! velocity ratio4 ,9 ?is case #aabj #dad #bg1 :i* is )9 #ja lb4 ( ! value ( ,x4 ,! agreem5t ( ^! results is satisfactory4 ,,! ,,screw ,,bolt ,,& ,,nut4 ,illu/ration3 ,,fig4 #de4 #bie4 ,"o ( ! mo/ useful applications ( ! screw is met ) 9 ! common bolt & nut1 %[n 9 ,fig4 #de4 ,x consi/s ( a wr\ght iron rod ) a head at "o end & a screw on ! o!r1 ^u :i* ! nut works4 ,bolts 9 _m diff]5t sizes & =ms repres5t ! /it*es by :i* ma*9es & frames are mo/ readily unit$4 ,!re are sev]al reasons :y ! bolt is s conv5i5t4 ,x draws ! p>ts into close contact ) trem5d\s =ce2 x is itself s /rong t ! p>ts unit$ practically =m "o piece4 ,x c be adju/$ quickly1 & remov$ z readily4 ,! same bolt by ! use ( wa%]s c be appli$ to pieces ( v diff]5t sizes4 ,no skill$ h& is requir$ to use ! simple tool t turns ! nut4 ,add+ to ?is t bolts are *eap & durable1 we %all easily und]/& :y !y are s ext5sively us$4 #bif4 ,we mu/ rem>k 9 conclusion t ! bolt owes its utility to friction2 screws ( ?is k9d d n ov]haul1 h5ce :5 ! nut is screw$ home x does n recoil4 ,if x 7 n t m ?an half ! p[] appli$ to a screw is consum$ 9 friction1 ! bolt & ! nut w\ld ei!r be r5d]$ useless1 or at lea/ w\ld require to be furni%$ ) "s complicat$ app>atus = prev5t+ ! motion ( ! nut4 ,,lecture ,x4 ,,! ,,:eel ,,& ,,axle4 ,introduction4--,exp]im5ts ^u ! ,:eel & ,axle4--,friction ^u ! ,axle4--,! ,:eel & ,b>rel4--,! ,:eel & ,p9ion4--,! ,crane4--,conclusion4 ,,introduction4 #big4 ,! me*anical p[]s discuss$ 9 ^! lectures may be gr\p$ into two classes1--! fir/ :]e ropes or *a9s are us$1 & ! second :]e ropes or *a9s are abs5t4 ,belong+ to t class 9 :i* ropes are n employ$1 we h ! screw discuss$ 9 ! la/ lecture2 & ! lev] discuss$ 9 ,lecture ,,viii42 :ile among ^? ma*9es 9 :i* ropes or *a9s =m an ess5tial "p ( ! app>atus1 ! pulley & ! :eel & axle hold a prom95t place4 ,we h already exam9$ sev]al =ms ( ! pulley1 & we n[ proce$ to ! n less important subject ( ! :eel & axle4 #bih4 ,:]e great resi/ances h to be ov]come1 b :]e ! di/ance ?r\gh :i* ! resi/ance mu/ be urg$ is %ort1 ! lev] or ! screw is g5]ally f\nd to be ! mo/ appropriate means ( increas+ p[]4 ,:51 h[ev]1 ! resi/ance has to be mov$ a consid]able di/ance1 ! aid ( ! pulley1 or ! :eel & axle1 or sometimes ( bo? comb9$1 is call$ 94 ,! :eel & axle is ! =m ( me*anical p[] :i* is g5]ally us$ :5 ! di/ance is consid]able ?r\gh :i* a weight mu/ be rais$1 or ?r\gh :i* "s resi/ance mu/ be ov]come4 ,illu/ration3 ,,fig4 #df4 #bii4 ,! :eel & axle assumes v _m =ms correspond+ to ! v>i\s purposes to :i* x is appli$4 ,! g5]al =m ( ! arrangem5t w be und]/ood f ,fig4 #df4 ,x consi/s ( an iron axle ,b1 m\nt$ 9 be>+s1 s z to be capable ( turn+ freely2 to ?is axle a rope is fa/5$1 & at ! extremity ( ! rope is a weight ,d1 :i* is gradually rais$ z ! axle revolves4 ,atta*$ to ! axle1 & turn+ ) x1 is a :eel ,a ) hooks 9 its circumf]5ce1 ^u :i* lies a rope2 "o end ( ?is rope is atta*$ to ! circumf]5ce ( ! :eel1 & ! o!r supports a weight ,e4 ,?is latt] weight may be call$ ! p[]1 :ile ! weight ,d susp5d$ f ! axle is ! load4 ,:5 ! p[] is suffici5tly l>ge1 ,e desc5ds1 mak+ ! :eel to revolve2 ! :eel causes ! axle to revolve1 & ?us ! rope is w\nd up & ! load ,d is rais$4 #cjj4 ,:5 comp>$ ) ! diff]5tial pulley z a means ( rais+ a weight1 ?is arrangem5t appe>s r bulky & o!rwise inconv5i5t1 b1 z we %all pres5tly le>n1 x is a f> m economical means ( apply+ en]gy4 ,9 its practical application1 moreov]1 ! arrangem5t is simplifi$ 9 v>i\s ways1 two ( :i* may be m5tion$4 #cja4 ,! cap/an is ess5tially a :eel & axle2 ! p[] is n 9 ?is case appli$ by means ( a rope1 b by direct pressure on ! "p ( ! m5 work+ x2 nor is !re actually a :eel employ$1 = ! pressure is appli$ to :at w\ld be ! extremities ( ! spokes ( ! :eel if a :eel exi/$4 #cjb4 ,9 ! ord9>y w9*1 ! p[] ( ! lab\r] is directly appli$ to ! h&le :i* moves r\nd 9 ! circumf]5ce ( a circle4 #cjc4 ,!re are innum]able o!r applications ( ! pr9ciple :i* are con/antly met )1 & :i* c be easily und]/ood ) a little att5tion4 ,^! we %all n /op to describe1 b we pass on at once to ! important "q ( ! relation betwe5 ! p[] & ! load4 ,,exp]im5ts ,,^u ,,! ,,:eel ,,& ,,axle4 #cjd4 ,we %all comm5ce a s]ies ( exp]im5ts ^u ! :eel ,a & axle ,b ( ,fig4 #df4 ,we %all fir/ det]m9e ! velocity ratio1 & !n asc]ta9 ! me*anical effici5cy by actual exp]im5t4 ,! :eel is ( wood2 x is ab\t #cj 9 diamet]4 ,! /r+ to :i* ! p[] is atta*$ is coil$ r\nd a s]ies ( hooks1 plac$ ne> ! m>g9 ( ! :eel2 ! effective circumf]5ce is ?us a little less ?an ! real circumf]5ce4 ,i measure a s+le coil ( ! /r+ & f9d ! l5g? to be #hhe4 ,?is l5g?1 !re=e1 we %all adopt = ! effective circumf]5ce ( ! :eel4 ,! axle is #jge 9 diamet]1 b its effective circumf]5ce is l>g] ?an ! circle ( :i* ?is l5g? is ! diamet]4 #cje4 ,! prop] mode ( f9d+ ! effective circumf]5ce ( ! axle 9 a case :]e ! rope be>s a consid]able proportion to ! axle is z foll[s4 ,atta* a weight to ! extremity ( ! rope suffici5t to /ret* x ?or\ghly4 ,make ! :eel & axle revolve suppose #bj times1 & measure ! height ?r\gh :i* ! weight is lift$2 !n ! "o-tw5tie? "p ( t height is ! effective circumf]5ce ( ! axle4 ,by ?is means ,i f9d ! circumf]5ce ( ! axle we are us+ to be #bhg4 #cjf4 ,we c n[ asc]ta9 ! velocity ratio 9 ?is ma*9e4 ,:5 ! :eel & axle h made "o complete revolution ! p[] has be5 l[]$ ?r\gh a di/ance ( #hhe1 & ! load has be5 rais$ ?r\gh #bhg4 ,?is is evid5t because ! :eel & axle are atta*$ toge!r1 & !re=e ea* completes "o revolution 9 ! same "t2 h5ce ! ratio ( ! di/ance :i* ! p[] moves ov] to t ?r\gh :i* ! load is rais$ is #hhe #bhg #ca v ne>ly4 ,we %all !re=e suppose ! velocity ratio to be #ca4 ,?us ?is :eel & axle has a f> high] velocity ratio ?an any ( ! sy/ems ( pulleys :i* we h be5 consid]+4 #cjg4 ,7 friction abs5t ! velocity ratio ( #ca w\ld necess>ily express ! me*anical effici5cy ( ?is :eel & axle2 ow+ to ! pres5ce ( friction ! real effici5cy is less ?an ?is--h[ mu* less1 we mu/ asc]ta9 by exp]im5t4 ,i atta* a load ( #ef lbs4 to ! hook :i* is borne by ! rope desc5d+ f ! axle3 ?is load is %[n at ,d 9 ,fig4 #df4 ,i f9d t a p[] ( #bf lbs4 appli$ at ,e is j suffici5t to raise ,d4 ,we inf] f ?is result t ! me*anical effici5cy ( ?is ma*9e is #ef #bf #bae4 ,i add a second #ef lb4 weight to ! load1 & ,i f9d t a p[] ( #ej lbs4 raises ! load ( #aab lbs4 ,! me*anical effici5cy 9 ?is case is #aab #ee #bbe4 ,we adopt ! mean value #bb4 ,h5ce ! me*anical effici5cy is r$uc$ by friction f #ca to #bb4 #cjh4 ,we may compute f ?is result ! numb] ( units ( en]gy :i* are utiliz$ out ( e #ajj units appli$4 ,let u suppose a load ( #ajj lbs4 is to be rais$ "o foot2 a =ce ( #ajj #bb #df lbs4 w suffice to raise ?is load4 ,?is =ce mu/ be ex]t$ ?r\gh a space ( #ca'1 & consequ5tly #ca #df #adc units ( en]gy mu/ be exp5d$2 ( ?is am\nt #ajj units are usefully employ$1 & !re=e ! p]c5tage ( en]gy utiliz$ is #ajj #adc #ajj #gj4 ,x foll[s t #cj p] c5t4 ( ! appli$ en]gy is consum$ 9 ov]com+ friction4 #cji4 ,we c see ! reason :y ! :eel & axle ov]hauls--t is1 runs d[n ( its own accord--:5 all[$ to d s2 x is because less ?an half ! appli$ en]gy is exp5d$ ^u friction4 #caj4 ,a s]ies ( exp]im5ts :i* h be5 c>efully made ) ?is :eel & axle are record$ 9 ,table ,,xviii4 ,,table ,,xviii4--,,:eel ,,& ,,axle4 ,:eel ( wood2 axle ( iron1 9 oil$ brass be>+s2 weight ( :eel & axle toge!r1 #afe lbs42 effective circumf]5ce ( :eel1 #hhe2 effective circumf]5ce ( axle1 #bhg2 velocity ratio1 #ca2 me*anical effici5cy1 #bb2 useful effect1 #gj p] c5t42 =mula1 ,p #jbjd #jjdbf ,r4 ---------------------------------------- -------------- ,p4 ,diff]5ce ( ,numb] ( ,r4 ,obs]v$ ,calculat$ ! obs]v$ ,exp]im5t4 ,load p[] p[] & calculat$ 9 lbs4 9 lbs4 9 lbs4 values4 ---------------------------------------- -------------- #a #bh #ad #ad #jj #b #db #bj #bj #jj #c #ef #bf #bf #jj #d #gj #cb #cb #jj #e #hd #cg #ch #ja #f #ih #dd #dd #jj #g #aab #ej #ej #jj ---------------------------------------- -------------- ,by ! me?od ( ! ,app5dix a relation connect+ ! p[] & ! load has be5 det]m9$2 x is express$ 9 ! =m-- ,p #jbjd #jjdbf ,r4 #caa4 ,?us = example 9 exp]im5t #e a load ( #hd lbs4 0 f\nd to be rais$ by a p[] ( #cg lbs4 ,! value calculat$ by ! =mula is #jbjd #jjdbf #hd #ch4 ,! calculat$ value only diff]s f ! obs]v$ value by #ja lb41 :i* is %[n 9 ! fif? column4 ,x w be se5 f ?is column t ! values calculat$ f ! =mula repres5ts ! exp]im5ts ) fidelity4 #cab4 ,we h d$uc$ ! relation betwe5 ! p[] & ! load f ! pr9ciple ( en]gy1 b we might h obta9$ x f ! pr9ciple ( ! lev]4 ,! :eel & axle bo? revolve ab\t ! c5tre ( ! axle2 we may !re=e reg>d ! c5tre z ! fulcrum ( a lev]1 & ! po9ts :]e ! cords meet ! :eel & axle z ! po9ts ( application ( ! p[] & ! load respectively4 #cac4 ,by ! pr9ciple ( ! lev] ( ! fir/ ord] 7,art4 #bcg71 ! p[] is to ! load 9 ! inv]se proportion ( ! arms2 9 ?is case1 !re=e1 ! p[] is to ! load 9 ! inv]se proportion ( ! radii ( ! :eel & ! axle4 ,b ! circumf]5ces ( circles are 9 proportion to _! radii1 & !re=e ! p[] mu/ be to ! load z ! circumf]5ce ( ! axle is to ! circumf]5ce ( ! :eel4 #cad4 ,?is mode ( arriv+ at ! result is a little artificial2 x is m natural to d$uce ! law directly f ! pr9ciple ( en]gy4 ,9 a me*anical p[] ( any complexity x w\ld be difficult to trace exactly ! transmission ( p[] f "o "p to ! next1 b ! pr9ciple ( en]gy evades ?is difficulty2 no matt] :at be ! me*anical arrangem5t1 simple or complex1 ( few p>ts or ( _m1 we h only to asc]ta9 by trial h[ _m feet ! p[] mu/ trav]se 9 ord] to raise ! load "o foot2 ! numb] ?us obta9$ is ! !oretical effici5cy ( ! ma*9e4 ,,friction ,,^u ,,! ,,axle4 #cae4 ,9 ! :eel & axle ^u :i* we h be5 exp]im5t+1 we h f\nd t ab\t #cj p] c5t4 ( ! p[] is consum$ by friction4 ,we %all be able to asc]ta9 to :at ?is loss is due1 & !n 9 "s degree to remove its cause4 ,f ! exp]im5ts ( ,art4 #afe we le>n$ t ! friction ( a small pulley 0 v mu* great] ?an t ( a l>ge pulley--9 fact1 ! friction is inv]sely proportional to ! diamet] ( ! pulley4 ,we inf] f ?is t by w9d+ ! rope ^u a b>rel in/ead ( ^u ! axle1 ! friction may be dim9i%$4 ,illu/ration3 ,,fig4 #dg4 #caf4 ,we c exam9e exp]im5tally ! effect ( friction on ! axle by ! app>atus ( ,fig4 #dg4 ,b is a %aft #jge diamet]1 ab\t :i* a rope is coil$ sev]al times2 ! ends ( ?is rope hang d[n freely1 & to ea* ( !m hooks ,e1 ,f are atta*$4 ,?is %aft revolves 9 brass be>+s1 :i* are oil$4 ,9 ord] to inve/igate ! am\nt ( p[] lo/ by w9d+ ! rope ^u an axle ( ?is size1 ,i %all place a c]ta9 weight--suppose #ef lbs4--^u "o hook ,f1 & !n ,i %all asc]ta9 :at am\nt ( p[] hung ^u ! o!r hook ,e w be suffici5t to raise ,f4 ,!re is "h no me*anical advantage1 s t ! excess ( load :i* ,e mu/ receive 9 ord] to raise ,f is ! true measure ( ! friction4 #cag4 ,i add on weights at ,e until ! p[] rea*es #he lbs41 :5 ,e desc5ds4 ,we ?us see t to raise #ef lbs4 an excess ( #bi lbs4 0 necess>y to ov]come ! friction4 ,we may r\ghly enunciate ! result by /at+ t to raise a load 9 ?is way1 half z mu* aga9 is requir$ = ! p[]4 ,?is law is v]ifi$ by susp5d+ #bh lbs4 at ,f1 :5 x is f\nd t a p[] ( #dc lbs4 at ,e is requir$ to lift x4 ,_h ! p[] be5 #db lbs41 x w\ld h be5 exactly half z mu* aga9 z ! load4 #cah4 ,h5ce 9 rais+ ,f ^u ?is axle1 ab\t "o-?ird ( ! p[] :i* mu/ be appli$ at ! circumf]5ce ( ! axle is wa/$4 ,?is exp]im5t tea*es u :]e ! loss lies 9 ! :eel & axle ( ,art4 #cjd1 & expla9s h[ x is t ab\t a ?ird ( its effici5cy is lo/4 #he lbs4 0 only able to raise two-?irds ( its own weight1 ow+ to ! friction2 & h5ce we %\ld expect to f9d1 z we actually h f\nd1 t ! p[] appli$ at ! circumf]5ce ( ! :eel has an effect :i* is only two-?irds ( its !oretical effici5cy4 #cai4 ,f ?is exp]im5t we %\ld inf] t ! prop] mode ( avoid+ ! loss by friction is to w9d ! rope ^u a b>rel ( consid]able diamet] r ?an ^u ! axle itself4 ,i place ^u a simil> axle to t on :i* we h be5 already exp]im5t+ a b>rel ( ab\t #ae circumf]5ce4 ,i coil ! rope two or ?ree times ab\t ! b>rel1 & let ! ends hang d[n z be=e4 ,i !n atta* to ea* end #ef lbs4 weight1 & ,i f9d t #aj lbs4 add$ to ei!r ( ! weights is suffici5t to ov]come friction1 to make x desc5d1 & raise ! o!r weight4 ,! app>atus is %[n 9 ,fig4 #dg4 ,a is ! b>rel1 ,c & ,d are ! weights4 ,9 ?is arrangem5t #aj lbs4 is suffici5t to ov]come ! friction :i* requir$ #bi lbs4 :5 ! rope 0 simply coil$ ar\nd ! axle4 ,9 o!r words1 by ! b>rel ! loss by friction is r$uc$ to "o-?ird ( its am\nt4 ,,! ,,:eel ,,& ,,b>rel4 #cbj4 ,we next place ! b>rel ^u ! axis already exp]im5t$ ^u & %[n 9 ,fig4 #df at ,b4 ,! circumf]5ce ( ! :eel is #hhe2 ! circumf]5ce ( ! b>rel is #adi4 ,! prop] mode ( f9d+ ! circumf]5ce ( ! b>rel is to susp5d a weight f ! rope1 !n raise ?is weight by mak+ "o revolution ( ! :eel1 & ! di/ance ?r\gh :i* ! weight is rais$ is ! effective circumf]5ce ( ! b>rel4 ,! velocity ratio ( ! :eel & b>rel is !n f\nd1 by divid+ #adi into #hhe1 to be #eid4 #cba4 ,! me*anical effici5cy ( ?is ma*9e is det]m9$ by exp]im5t4 ,i susp5d a weight ( #ef lbs4 f ! hook1 & apply p[] to ! :eel4 ,i f9d t #aja lbs4 is j suffici5t to raise ! load4 #cbb4 ,! me*anical effici5cy is to be f\nd by divid+ #aja into #ef2 ! quoti5t ?us obta9$ is #eed4 ,! me*anical effici5cy does n diff] mu* f #eid1 ! velocity ratio2 & consequ5tly 9 ?is ma*9e b little p[] is exp5d$ ^u friction4 #cbc4 ,we c asc]ta9 ! loss by comput+ ! p]c5tage ( appli$ en]gy :i* is utiliz$4 ,let u suppose a weight ( #ajj lbs4 has to be rais$ "o foot3 = ?is purpose a =ce ( #ajj #eed #aha lbs4 mu/ be appli$4 ,?is is evid5t f ! def9ition ( ! me*anical effici5cy2 b s9ce ! load has to be rais$ "o foot1 x is cle> f ! mean+ ( ! velocity ratio t ! p[] mu/ move ov] #e'id3 h5ce ! numb] ( units ( "w to be appli$ is to be measur$ by ! product ( #eid & #aha1 t is1 by #ajge2 9 ord] !re=e to accompli% #ajj units ( "w #ajge units ( "w mu/ be appli$4 ,! p]c5tage ( en]gy usefully employ$ is #ajj #ajge #ajj #ic4 ,?is is f> m ?an #gj1 :i* is ! p]c5tage utiliz$ :5 ! axle 0 us$ )\t ! b>rel 7,art4 #cji74 #cbd4 ,a s]ies ( exp]im5ts made ) c>e ^u ! :eel & b>rel are record$ 9 ,table ,,xix4 ,,table ,,xix4--,,! ,,:eel ,,& ,,b>rel4 ,:eel ( wood1 #hhe 9 circumf]5ce1 on ! same axle z a ca/ iron b>rel ( #adi circumf]5ce2 axle is ( wr\ght iron1 #jge 9 diamet]1 m\nt$ 9 oil$ brass be>+s2 p[] is appli$ to ! circumf]5ce ( ! :eel1 load rais$ by rope r\nd b>rel2 velocity ratio1 #eid2 me*anical effici5cy1 #eed2 useful effect1 #ic p] c5t42 =mula1 ,p #je #jafi ,r4 ---------------------------------------- ----------------- ,p4 ,diff]5ce ( ,numb] ( ,r4 ,obs]v$ ,calculat$ ! obs]v$ ,exp]im5t4 ,load p[] p[] & calculat$ 9 lbs4 9 lbs4 9 lbs4 values4 ---------------------------------------- ----------------- #a #ad #bg #bi #jb #b #bh #ec #eb - #ja #c #db #gg #gf - #ja #d #ef #aja #ajj - #ja #e #gj #abd #abd #jj #f #hd #adg #adg #jj #g #ih #aga #aga #jj #h #aab #aid #aie #ja ---------------------------------------- ----------------- ,! =mula :i* repres5ts ! exp]im5ts ) ! greate/ am\nt ( accuracy is ,p #je #jafi ,r4 ,?is =mula is comp>$ ) ! exp]im5ts1 & ! column ( diff]5ces %[s t ! calculat$ & ! obs]v$ values agree v closely4 ,! con/ant "p #je is p>tly due to ! con/ant friction ( ! heavy b>rel & :eel1 & p>tly1 x may be1 to small irregul>ities :i* h prev5t$ ! c5tre ( gravity ( ! :ole mass f be+ /rictly 9 ! axle4 #cbe4 ,?\gh ?is ma*9e is m economical ( p[] ?an ! :eel & axle ( ,art #cje1 yet x is less p[]ful2 9 fact1 ! me*anical effici5cy1 #eed1 is only ab\t "o-f\r? ( t ( ! :eel & axle4 ,x is !re=e necess>y to inquire :e!r we _c devise "s me?od by :i* to secure ! advantages ( b little friction1 & at ! same "t h a l>ge me*anical effici5cy3 ?is we %all proce$ to inve/igate4 ,,! ,,:eel ,,& ,,p9ion4 #cbf4 ,by means ( :at are call$ cog-:eels or too!d :eels1 we are enabl$ to comb9e two or m :eels & axles toge!r1 & ?us greatly to increase ! p[] :i* c be produc$ by a s+le :eel & axle4 ,too!d :eels are us$ = a great v>iety ( purposes 9 me*anics2 we h already _h "s illu/ration ( _! use dur+ ^! lectures 7,fig4 #cj74 ,! :eels :i* we %all employ are ^? (t5 us$ 9 la!s & o!r small ma*9es2 !y are :at are call$ #aj-pit* :eels1--t is to say1 a :eel ( ?is class conta9s t5 times z _m tee? 9 its circumf]5ce z !re are in*es 9 its diamet]4 ,i h "h a :eel #bj diamet]1 & consequ5tly x has #bjj tee?2 "h is ano!r :i* is #be diamet]1 & :i* consequ5tly conta9s #be tee?4 ,we %all m\nt ^! :eels ^u two p>allel %afts1 s t !y ge> "o into ! o!r 9 ! mann] %[n 9 ,fig4 #df3 ,f is ! l>ge :eel conta9+ #bjj tee?1 & ,g ! p9ion ( #be tee?4 ,! axles are #jge diamet]2 ar\nd ea* ( !m a rope is w\nd1 by :i* a hook is susp5d$4 #cbg4 ,a small weight at ,k is suffici5t to raise a mu* l>g] weight on ! o!r %aft2 b be=e exp]im5t+ on ! me*anical effici5cy ( ?is arrangem5t1 we %all z usual calculate ! velocity ratio4 ,! :eel conta9s eight times z _m tee? z ! p9ion2 x is !re=e evid5t t :5 ! :eel has made "o revolution1 ! p9ion w h made eight revolutions1 & conv]sely ! p9ion mu/ turn r\nd eight times to turn ! :eel r\nd once3 h5ce ! p[] :i* is turn+ ! p9ion r\nd mu/ be l[]$ ?r\gh eight times ! circumf]5ce ( ! axle1 :ile ! load is rais$ ?r\gh a l5g? equal to "o circumf]5ce ( ! axle4 ,we ?us f9d ! velocity ratio ( ! ma*9e to be #h4 #cbh4 ,we det]m9e ! me*anical effici5cy by trial4 ,atta*+ a load ( #ef lbs4 to ! axle ( ! l>ge :eel1 x is obs]v$ t a p[] ( #acg lbs4 at ,k w raise x2 ! me*anical effici5cy ( ! ma*9e is !re=e ab\t #da1 :i* is almo/ exactly half ! velocity ratio4 ,we note t ! load w only j run d[n :5 ! p[] is remov$2 f ?is we might h inf]r$1 by ,art4 #bbb1 t ne>ly half ! p[] is exp5d$ on friction1 & t !re=e ! me*anical effici5cy is ab\t half ! velocity ratio4 ,! actual p]c5tage ( en]gy t is utilis$ ) ?is p>ticul> load is #ea4 ,if we susp5d #aab lbs4 f ! load hook1 #bf lbs4 is j en\gh to raise x2 ! me*anical effici5cy t w\ld be d$uc$ f ?is result is #aab #bf #dc1 :i* is slightly 9 excess ( ! am\nt obta9$ by ! =m] exp]im5t4 ,x is (t5 f\nd to be a prop]ty ( ! me*anical p[]s1 t z ! load increases ! me*anical effici5cy slightly improves4 #cbi4 ,9 ,table ,,xx4 w be f\nd a record ( exp]im5ts ^u ! relation betwe5 ! p[] & ! load ) ! :eel & p9ion2 ! table w suffici5tly expla9 itself1 aft] ! description ( simil> tables already giv5 7,arts4 #caj1 #cbd74 ,,table ,,xx4--,,! ,,:eel ,,& ,,p9ion4 ,:eel 7#aj-pit*71 #bjj tee?2 p9ion1 #be tee?2 axles equal1 effective circumf]5ce ( ea* be+ #bhg2 oil$ brass be>+s2 velocity ratio1 #h2 me*anical effici5cy1 #da2 useful effect1 #ea p] c5t42 =mula1 ,p #bdf #jba ,r4 ---------------------------------------- --------------- ,p4 ,diff]5ce ( ,numb] ( ,r4 ,obs]v$ ,calculat$ ! obs]v$ ,exp]im5t4 ,load p[] p[] & calculat$ 9 lbs4 9 lbs4 9 lbs4 p[]s4 ---------------------------------------- --------------- #a #ad #ed #ed #jj #b #bh #hg #hc - #jd #c #db #aaj #aac #jc #d #ef #acg #adb #je #e #gj #age #agb - #jc #f #hd #bjj #bja #ja #g #ih #bcj #bcj #jj #h #aab #bfj #bfj #jj ---------------------------------------- --------------- #ccj4 ,! l>ge am\nt ( friction pres5t 9 ?is contrivance is ! consequ5ce ( w9d+ ! rope directly ^u ! axle in/ead ( ^u a b>rel1 z already po9t$ out 9 ,art4 #cai4 ,we might place b>rels ^u ^! axles & demon/rate ! tru? ( ?is /atem5t2 b we ne$ n delay to d s1 z we use ! b>rel 9 ! ma*9es :i* we %all next describe4 ,,! ,,crane4 #cca4 ,we h already expla9$ 7,art4 #ch7 ! con/ruction ( ! lift+ crane1 s f> z its framework is conc]n$4 ,we n[ exam9e ! me*anism by :i* ! load is rais$4 ,we %all employ = ?is purpose ! model :i* is repres5t$ 9 ,fig4 #dh4 ,! jib is support$ by a wood5 b> z a tie1 & ! crane is /eadi$ by means ( ! weights plac$ at ,h3 "s su* c\nt]poise is necess>y1 = o!rwise ! ma*9e w\ld tumble ov] :5 a load is susp5d$ f ! hook4 #ccb4 ,! load is support$ by a rope or *a9 :i* passes ov] ! pulley ,e & !nce to ! b>rel ,d1 ^u :i* x is to be w\nd4 ,?is b>rel receives its motion f a l>ge :eel ,a1 :i* conta9s #bjj tee?4 ,! :eel ,a is turn$ by ! p9ion ,b :i* conta9s #be tee?4 ,9 ! actual use ( ! crane1 ! axle :i* c>ries ?is p9ion w\ld be turn$ r\nd by means ( a h&le2 b = ! purpose ( exp]im5ts ^u ! relation ( ! p[] to ! load1 ! h&le w\ld be inconv5i5t1 & !re=e we h plac$ ^u ! axle ( ! p9ion a :eel ,c conta9+ a groove 9 its circumf]5ce4 ,ar\nd ?is groove a /r+ is wrapp$1 s t :5 a weight ,g is susp5d$ f ! /r+ x w cause ! :eel to revolve4 ,?is weight ,g w con/itute ! p[] by :i* ! load may be rais$4 #ccc4 ,let u compute ! velocity ratio ( ?is ma*9e be=e comm5c+ exp]im5ts ^u its me*anical effici5cy4 ,! effective circumf]5ce ( ! b>rel ,d is f\nd by trial to be #adi4 ,s9ce !re are #bjj tee? on ,a & #be on ,b1 x foll[s t ! p9ion ,b mu/ revolve eight times to produce "o revolution ( ! b>rel4 ,h5ce ! :eel ,c at ! circumf]5ce ( :i* ! p[] is appli$ mu/ also revolve eight times = "o revolution ( ! b>rel4 ,! effective circumf]5ce ( ,c is #dc2 ! p[] mu/ !re=e h be5 appli$ ?r\gh #h #dc #cdd1 9 ord] to raise ! load #aei4 ,! velocity ratio is #cdd #adi #bc v ne>ly4 ,we c easily v]ify ?is value ( ! velocity ratio by actually rais+ ! load #a'1 :5 x appe>s t ! numb] ( revolutions ( ! :eel ,b is su* t ! p[] mu/ h mov$ #bc'4 ,illu/ration3 ,,fig4 #dh4 #ccd4 ,! me*anical effici5cy is to be f\nd z usual by trial4 #ef lbs4 plac$ at ,f is rais$ by #ca lbs4 at ,g2 h5ce ! me*anical effici5cy d$uc$ f ?is exp]im5t is #ef #ca #ah4 ,! p]c5tage ( useful effect is easily %[n to be #gh by ! me?od ( ,art4 #cbc4 ,"h1 !n1 we h a ma*9e possess+ v consid]able effici5cy1 & be+ at ! same "t economical ( en]gy4 ,,table ,,xxi4--,,! ,,crane4 ,circumf]5ce ( :eel to :i* ! p[] is appli$1 #dc2 tra9 ( :eels1 #be #bjj2 circumf]5ce ( drum on :i* rope is w\nd1 #adi2 velocity ratio1 #bc2 me*anical effici5cy1 #ah2 useful effect1 #gh p] c5t42 =mula1 ,p #jjeef ,r4 ---------------------------------------- ----------- ,p4 ,diff]5ce ( ,numb] ( ,r4 ,obs]v$,calculat$ ! obs]v$ ,exp]im5t4 ,load p[] p[] & calculat$ 9 lbs4 9 lbs4 9 lbs4 values4 ---------------------------------------- ----------- #a #ad #ji #jh -#ja #b #bh #af #af #jj #c #db #bd #bc -#ja #d #ef #ca #ca #jj #e #gj #ch #ci #ja #f #hd #de #dg #jb #g #ih #ec #ee #jb #h #aab #fb #fb #jj ---------------------------------------- ----------- #cce4 ,a s]ies ( exp]im5ts made ) ?is crane is record$ 9 ,table ,,xxi41 & a comp>ison ( ! calculat$ & obs]v$ values w %[ t ! =mula ,p #jjeef ,r repres5ts ! exp]im5ts ) consid]able accuracy4 #ccf4 ,x may be notic$ t 9 ?is =mula ! t]m indep5d5t ( ,r1 :i* we frequ5tly meet ) 9 ! expression ( ! relation betwe5 ! p[] & ! load1 is abs5t4 ,! probable explanation is to be f\nd 9 ! fact t "s m9ute irregul>ity 9 ! =m ( ! b>rel or ( ! :eel has be5 con/antly act+ l a small weight 9 fav\r ( ! p[]4 ,9 ea* exp]im5t ! motion is always />t$ f ! same position ( ! :eels1 & h5ce any irregul>ity w be con/antly act+ 9 fav\r ( ! p[] or aga9/ x2 "h ! =m] appe>s to h happ5$4 ,9 o!r cases d\btless ! latt] has occurr$2 ! diff]5ce is1 h[ev]1 ( extremely small am\nt4 ,! friction ( ! ma*9e itself :5 )\t a load is ano!r s\rce = ! production ( ! con/ant t]m2 x has happ5$ 9 ! pres5t case t ?is friction has be5 almo/ exactly balanc$ by ! accid5tal influ5ce ref]r$ to4 #ccg4 ,9 cranes x is usual to provide means ( add+ a second tra9 ( :eels1 :5 ! load is v heavy4 ,9 ano!r model we appli$ ! p[] to an axle ) a p9ion ( #be tee?1 ge>+ into a :eel ( #bjj tee?2 on ! axle ( ! :eel ) #bjj tee? is a p9ion ( #cj tee?1 :i* ge>s into a :eel ( #ahj tee?2 ! b>rel is on ! axle ( ! la/ :eel4 ,a s]ies ( exp]im5ts ) ?is crane is %[n 9 ,table ,,xxii4 ,,table ,,xxii4--,,! ,,crane ,,= ,,heavy ,,loads4 ,circumf]5ce ( :eel to :i* p[] is appli$1 #dc2 tra9 ( :eels1 #be #bjj #cj #ahj2 circumf]5ce ( drum on :i* rope is w\nd1 #adi2 velocity ratio1 #acg2 me*anical effici5cy1 #hg2 useful effect1 #fc p] c5t42 =mula1 ,p #jahe #jjjghb ,r4 ---------------------------------------- ----------- ,p4 ,diff]5ce ( ,numb] ( ,r4 ,obs]v$,calculat$ ! obs]v$ ,exp]im5t4 ,load p[] p[] & calculat$ 9 lbs4 9 lbs4 9 lbs4 values4 ---------------------------------------- ----------- #a #ad #jcj #jbi -#jja #b #bh #jdj #jdj #jjj #c #db #jej #jea #jja #d #ef #jfj #jfb #jjb #e #gj #jge #jgc -#jjb #f #hd #jhe #jhd -#jja #g #ih #jie #jie #jjj #h #aab #aje #ajf #jja ---------------------------------------- ----------- ,! velocity ratio is n[ #acg1 & ! me*anical effici5cy is #hg2 "o man c\ld !re=e raise a ton ) ease by apply+ a p[] ( #bf lbs4 to a crane ( ?is k9d4 ,,conclusion4 #cch4 ,x w be useful to contra/ ! :eel & axle on :i* we h exp]im5t$ 7,art4 #cjd7 ) ! diff]5tial pulley 7,art4 #bji74 ,! velocity ratio ( ! =m] ma*9e is ne>ly d\ble t ( ! latt]1 & its me*anical effici5cy is ne>ly f\r times z great4 ,less ?an half ! appli$ p[] is wa/$ 9 ! :eel & axle1 :ile m ?an half is wa/$ 9 ! diff]5tial pulley4 ,?is makes ! :eel & axle bo? a m p[]ful ma*9e1 & a m economical ma*9e ?an ! diff]5tial pulley4 ,on ! o!r h&1 ! great] compactness ( ! latt]1 its facility ( application1 & ! practical conv5i5ces aris+ f ! prop]ty ( n all[+ ! load to run d[n1 d (t5 m ?an comp5sate = ! sup]ior me*anical advantage ( ! :eel & axle4 #cci4 ,we may also contra/ ! :eel & axle ) ! screw 7,art4 #bgg74 ,! screw is rem>kable among ! me*anical p[]s = its v high velocity ratio1 & its excessive friction4 ,?us we h se5 9 ,art4 #bia h[ ! velocity ratio ( a screw-jack ) an arm atta*$ am\nt$ to #dad1 :ile its me*anical effici5cy 0 little m ?an "o-f\r? z great4 ,no s+le :eel & axle c\ld conv5i5tly be made to give a me*anical effici5cy ( #aaf2 b f ,art4 #ccg we c\ld easily design a comb9ation ( :eels & axles to yield an effici5cy ( q ?is am\nt4 ,! friction 9 ! :eel & axle is v mu* less ?an 9 ! screw1 & consequ5tly en]gy is sav$ by ! use ( ! =m] ma*9e4 #cdj4 ,9 practice1 h[ev]1 x g5]ally happ5s t economy ( en]gy does n weigh mu* 9 ! selection ( a me*anical p[] = any purpose1 z !re are always o!r consid]ations ( great] consequ5ce4 #cda4 ,= example1 let u take ! case ( a lift+ crane employ$ 9 load+ or unload+ a vessel1 & inquire :y x is t a tra9 ( :eels is g5]ally us$ = ! purpose ( produc+ ! requisite p[]4 ,! answ] is simple1 ! tra9 ( :eels is conv5i5t1 = by _! aid any l5g? ( *a9 c be w\nd ^u ! b>rel2 :]eas if a screw 7 us$1 we %\ld require a screw z long z ! greate/ height ( lift4 ,?is screw w\ld be inconv5i5t1 & inde$ impracticable1 & ! additional circum/ance t a tra9 ( :eels is m economical ( en]gy ?an a screw has no influ5ce 9 ! matt]4 #cdb4 ,on ! o!r h&1 suppose t a v heavy load has to be ov]come = a %ort di/ance1 z = example 9 />t+ a %ip laun*1 a screw-jack is evid5tly ! prop] ma*9e to employ2 x is easily appli$1 & has a high me*anical effici5cy4 ,! want ( economy ( en]gy is ( no consequ5ce 9 su* an op]ation4 ,,lecture ,,xi4 ,,! ,,me*anical ,,prop]ties ,,( ,,timb]4 ,introduction4--,! ,g5]al ,prop]ties ( ,timb]4--,resi/ance to ,ext5sion4--,resi/ance to ,compression4--,condition ( a ,beam /ra9$ by a ,transv]se ,=ce4 ,,introduction #cdc4 ,9 ! lectures on ! me*anical p[]s :i* h be5 j complet$1 we h se5 h[ great weights may be rais$ or o!r l>ge resi/ances ov]come4 ,we are n[ to consid] ! important subject ( ! application ( me*anical pr9ciples to /ructures4 ,^! are fixtures1 :ile ma*9es are adapt$ = motion2 a ro( or a bridge is a /ructure1 b a crane or a screw-jack is a ma*9e4 ,/ructures are employ$ = support+ weights1 & ! me*anical p[]s give ! means ( rais+ !m4 #cdd4 ,a /ructure has to support bo? its own weight & also any load t is to be plac$ ^u x4 ,?us a railway bridge mu/ at all times su/a9 :at is call$ ! p]man5t load1 & frequ5tly1 ( c\rse1 ! weight ( "o or m tra9s4 ,! problem :i* ! eng9e] solves is to design a bridge :i* %all be suffici5tly /rong1 &1 at ! same "t1 economical2 8 skill is %[n by ! mann] 9 :i* he c atta9 ^! two ends 9 ! same /ructure4 #cde4 ,9 ! f\r lectures ( ! c\rse :i* w be devot$ to ?is subject x w only be possible to give a slight sket*1 & !re=e b few details c be introduc$4 ,an ext5d$ acc\nt ( ! prop]ties ( diff]5t mat]ials us$ 9 /ructures w\ld be beyond our scope1 b !re are "s g5]al pr9ciples relat+ to ! /r5g? ( mat]ials :i* may be discuss$4 ,timb]1 z a build+ mat]ial1 has1 9 mod]n times1 be5 replac$ to a great ext5t by iron 9 l>ge /ructures1 b timb] is m capable ?an iron ( be+ exp]im5t$ ^u 9 ! lecture room4 ,! elem5t>y laws :i* we %all demon/rate ) ref]5ce to ! /r5g? ( timb]1 are also1 sub/antially ! same z ! correspond+ laws = ! /r5g? ( iron or any o!r mat]ial4 ,h5ce we %all comm5ce ! /udy ( /ructures by two lectures on timb]4 ,! laws :i* we %all prove exp]im5tally w aft]w>ds be appli$ to a few simple cases ( bridges & o!r actual /ructures4 ,,! ,,g5]al ,,prop]ties ,,( ,,timb]4 #cdf4 ,! uses ( timb] 9 ! arts are z v>i\s z its qualities4 ,"s woods are useful = _! beauty1 & o!rs = _! /r5g? or durability "u diff]5t circum/ances4 ,we %all only employ 8p9e0 9 our exp]im5tal inquiries4 ,?is wood is select$ because x is s well kn[n & s mu* us$4 ,a k ( ! prop]ties ( p9e w\ld probably be m useful ?an a k ( ! prop]ties ( any o!r wood1 & at ! same "t x mu/ be rememb]$ t ! laws :i* we %all e/abli% by means ( slips ( p9e may be g5]ally appli$4 #cdg4 ,a transv]se section ( a tree %[s a numb] ( r+s1 ea* ( :i* repres5ts ! gr[? ( wood 9 "o ye>4 ,! age ( ! tree may sometimes be approximately f\nd by c\nt+ ! numb] ( di/+ui%able r+s4 ,! out] r+s are ! new] portions ( ! wood4 #cdh4 ,:5 a tree is fell$ x conta9s a l>ge quantity ( sap1 :i* mu/ be all[$ to evaporate be=e ! wood is fit = use4 ,) ?is object ! timb] is /or$ 9 suitable y>ds = two or m ye>s accord+ to ! purposes = :i* x is int5d$2 sometimes ! process ( season+1 z x is call$1 is ha/5$ by o!r means4 ,wood1 :5 season+1 contracts2 h5ce blocks ( timb] are (t5 f\nd split f ! circumf]5ce to ! c5tre1 = ! out] r+s1 be+ new] & conta9+ m sap1 contract m ?an ! inn] r+s4 ,= ! same reason a plank is f\nd to w>p :5 ! wood is n ?or\ghly season$4 ,! side ( ! plank :i* 0 f>!/ f ! c5tre ( ! tree contracts m ?an ! o!r side1 & becomes concave4 ,?is c be easily v]ifi$ by look+ at ! edge ( ! plank1 = we !re see ! r+s ( :i* x is compos$4 #cdi4 ,timb] may be s(t5$ by /eam+4 ,i h "h a rod ( p9e1 #bd #je #je1 & "h a second rod cut f ! same piece & ( ! same size1 :i* has be5 expos$ to /eam ( boil+ wat] = m ?an an h\r3 secur+ ^! at "o end to a firm /&1 ,i b5d !m d[n toge!r1 & y see t aft] ! dry rod has brok5 ! /eam$ rod c be b5t mu* f>!r be=e x gives way4 ,?is prop]ty ( wood is utiliz$ 9 %ap+ ! timb]s ( wood5 %ips4 ,we %all be able to und]/& ! action ( /eam if we reflect t wood is compos$ ( a numb] ( fibres rang$ side by side & unit$ toge!r4 ,a rope is compos$ ( a numb] ( fibres laid toge!r & twi/$1 b ! fibres are n coh]5t z !y are 9 wood4 ,h5ce we f9d t a rod ( wood is /iff1 :ile a rope is flexible4 ,! /eam f9ds its way into ! int]/ices betwe5 ! fibres ( ! wood2 x s(t5s _! connections1 & increases ! pliability ( ! fibres !mselves1 & ?us1 ! op]ation ( /eam+ t5ds to s(t5 a piece ( timb] & r5d] x tractable4 #cej4 ,! /ructure ( wood is exhibit$ by ! foll[+ simple exp]im5t3--,"h are two pieces ( p9e1 ea* #i #a #a4 ,"o ( !m ,i easily snap across ) a bl[1 :ile my bl[s are unable to break ! o!r4 ,! diff]5ce is m]ely t "o ( ^! pieces is cut aga9/ ! gra91 :ile ! o!r is ) x4 ,9 ! fir/ case ,i h only to sep>ate ! connection betwe5 ! fibres1 :i* is q easy4 ,9 ! o!r case ,i w\ld h to te> asund] ! fibres !mselves1 :i* is va/ly m difficult4 ,to a c]ta9 ext5t ! gra9$ /ructure is also f\nd 9 wr\ght iron1 b ! contra/ betwe5 ! /r5g? ( iron ) ! gra9 & aga9/ ! gra9 is n s m>k$ z x is 9 wood4 ,,resi/ance ,,to ,,ext5sion4 #cea4 ,x w be necess>y to expla9 a little m def9itely :at is meant by ! /r5g? ( timb]4 ,we may conceive a rod to be brok5 9 ?ree diff]5t ways4 ,9 ! fir/ place ! rod may be tak5 by a =ce at ea* end & torn asund] by pull+1 z a ?read may be brok54 ,to d ?is requires v great p[]1 & ! /r5g? ( ! mat]ial ) ref]5ce to su* a mode ( de/roy+ x is call$ its resi/ance to ext5sion4 ,9 ! second place1 x may be brok5 by longitud9al pressure at ea* end1 z a pill> may be cru%$ by ! sup]9cumb5t weight be+ too l>ge2 ! /r5g? t relates to ?is =m ( =ce is call$ resi/ance to compression3 f9ally1 ! rod may be brok5 by a =ce appli$ transv]sely4 ,! /r5g? ( p9e ) ref]5ce to ^! diff]5t applications ( =ce w be consid]$ successively4 ,! rods t are to be us$ h be5 cut f ! same piece ( timb]1 :i* has be5 select$ on acc\nt ( its /raightness ( gra9 & fre$om f knots4 ,!y are ( diff]5t rectangul> sections1 #a #je & #je #je be+ g5]ally us$1 b sometimes #a #a is employ$4 ,illu/ration3 ,,fig4 #di4 #ceb4 ,) ref]5ce to ! /r5g? ( timb] 9 its capacity to resi/ ext5sion1 we c d b little 9 ! lecture room4 ,i h "h a p9e rod ,a ,b1 ( dim5sions #dh #j4e #je1 ,fig4 #di4 ,ea* end ( ?is rod is firmly secur$ betwe5 two *eeks ( iron1 :i* are bolt$ toge!r3 ! rod is susp5d$ by its upp] extremity f ! hook ( ! epicycloidal pulley-block 7,art4 #bac71 :i* is itself support$ by a tripod2 hooks are atta*$ to ! l[] end ( ! rod = c>ry+ ! weights4 ,by plac+ #c cwt4 on ^! hooks & pull+ ! h& *a9 ( ! pulley-block1 ,i f9d t ,i c raise ! weight safely1 & !re=e ! rod w resi/ at all ev5ts a t5sion ( #c cwt4 ,f exp]im5ts :i* h be5 made on ! subject1 x is asc]ta9$ t ab\t a ton w\ld be necess>y to te> su* a rod asund]2 h5ce we see t p9e is enorm\sly /rong 9 resi/+ a =ce ( ext5sion4 ,! t5sile /r5g? ( ! rod does n dep5d ^u its l5g?1 b ^u ! area ( ! cross section4 ,t ( ! rod we h us$ is "o-f\r? ( a squ>e in*1 & ! break+ weight ( a rod "o squ>e in* 9 section is ab\t f\r tons4 #cec4 ,a rod ( any mat]ial g5]ally elongates to "s ext5t "u ! action ( a susp5d$ weight2 & we %all asc]ta9 :e!r ?is occurs p]ceptibly 9 wood4 ,be=e ! rod 0 /ra9$ ,i _h m>k$ two po9ts ^u x exactly #b feet ap>t4 ,:5 ! rod supports #c cwt4 ,i f9d t ! di/ance betwe5 ! two po9ts has n appreciably alt]$1 ?\gh by m delicate measurem5t ,i h no d\bt we %\ld f9d t ! di/ance _h elongat$ to an insignificant ext5t4 #ced4 ,let u contra/ ! resi/ance ( a rod ( timb] to ext5sion ) ! effect ^u a rope "u ! same circum/ances4 ,i h "h a rope ab\t #j4be diamet]2 x is susp5d$ f a po9t1 & be>s a #ad lb4 weight 9 ord] to be completely /ret*$4 ,i m>k po9ts ^u ! rope #b' ap>t4 ,i n[ *ange ! /one weight = a weight ( #a cwt41 & on measurem5t ,i f9d t ! two po9ts :i* be=e 7 #b' ap>t1 are n[ #b' #b2 ?us ! rope has /ret*$ at ! rate ( an in* p] foot = a /ra9 ( #a cwt41 :ile ! timb] did n /ret* p]ceptibly = a /ra9 ( #c cwt4 #cee4 ,we h already expla9$ 9 ,art4 #cg ! mean+ ( ! ^w 8tie40 ,! mat]ial suitable = a tie %\ld be capable ( (f]+ great resi/ance1 n only to actual rupture by t5sion1 b ev5 to appreciable elongation4 ,^! qualities we h f\nd to be possess$ by wood4 ,!y are1 h[ev]1 possess$ 9 a mu* high] degree by wr\ght iron1 :i* possesses o!r advantages 9 durability & facility ( atta*m5t4 ,,resi/ance ,,to ,,compression4 #cef4 ,we proce$ to exam9e into ! capability ( timb] to resi/ =ces ( longitud9al compression1 ei!r z a pill> or 9 any o!r =m ( 8/rut10 su* = in/ance1 z ! jib ( ! crane repres5t$ 9 ,fig4 #ag4 ,! use ( timb] z a /rut dep5ds 9 a great degree ^u ! coh]5ce ( ! fibres to ea* o!r1 z well z ^u _! actual rigidity4 ,! action ( timb] 9 resi/+ =ces ( compression is ?us v diff]5t f its action :5 resi/+ =ces ( ext5sion2 we c exam9e1 by actual exp]im5t1 ! /r5g? ( timb] "u ! =m] conditions1 z ! weights :i* x w be necess>y to employ are )9 ! capabilities ( our lecture room app>atus4 #ceg4 ,! app>atus is %[n 9 ,fig4 #ej4 ,x consi/s ( a lev] ( ! second ord]1 #aj' long1 ! me*anical advantage ( :i* is ?reefold2 ! resi/ance ( ! pill> ,d ,e to cru%+ is ! load to be ov]come1 & ! p[] consi/s ( weights1 to receive :i* ! tray ,b is us$2 e p\nd plac$ 9 ! tray produces a compressive =ce ( #c lbs4 on ! pill> at ,d4 ,! fulcrum is at ,a & guides at ,g4 ,! lev] & ! tray w\ld some:at complicate our calculations unless _! weights 7 c\nt]pois$4 ,a cord atta*$ to ! extremity ( ! lev] passes ov] a pulley ,f2 at ! o!r end ( ?is cord1 suffici5t weights ,c are atta*$ to neutralize ! weight ( ! app>atus4 ,9 fact1 ! lev] & tray n[ sw+ z if !y _h no weight1 & we may !re=e leave !m out ( consid]ation4 ,! pill> to be exp]im5t$ ^u is fitt$ at its l[] end ,e into a hole 9 a ca/ iron bracket3 ?is bracket c be adju/$ s z to take 9 pieces ( diff]5t l5g?s2 ! upp] end ( ! pill> passes ?r\gh a hole 9 a second piece ( ca/ iron1 :i* is bolt$ to ! lev]3 ?us our little exp]im5tal column is secur$ at ea* end1 & ! risk ( slipp+ is avoid$4 ,! /&s are heavily weight$ to secure ! /ability ( ! arrangem5t ,illu/ration3 ,,fig4 #ej4 #ceh4 ,! fir/ exp]im5t we %all make ) ?is app>atus is ^u a p9e rod #dj long & #je squ>e2 ! l[] bracket is s plac$ t ! lev] is horizontal :5 j re/+ ^u ! top ( ! rod4 ,weights plac$ 9 ! tray produce a pressure ?ree times z great d[n ! rod1 ! effect ( :i* w fir/ be to b5d ! rod1 &1 :5 ! deflection has rea*$ a c]ta9 am\nt1 to break x across4 ,i place #bh lbs4 9 ! tray3 ?is produces a pressure ( #hd lbs4 ^u ! rod1 b ! rod /ill rema9s p]fectly /raight1 s t x be>s ?is pressure easily4 ,:5 ! pressure is increas$ to #if lbs4 a v slight am\nt ( deflection may be se54 ,:5 ! /ra9 rea*es #aad lbs4 ! rod beg9s to b5d into a curv$ =m1 ?\gh ! deflection ( ! middle ( ! rod f its orig9al position is /ill less ?an #jbe4 ,gradually augm5t+ ! pressure1 ,i f9d t :5 x rea*es #acb lbs4 ! deviation has rea*$ #je2 & f9ally1 :5 #dh lbs4 is plac$ 9 ! tray1 t is1 :5 ! rod is subject$ to #add lbs41 x breaks across ! middle4 ,h5ce we see t ?is rod su/a9$ a load ( #if lbs4 )\t s5sibly b5d+1 b t fracture ensu$ :5 ! load 0 increas$ ab\t half z mu* aga94 ,ano!r exp]im5t ) a simil> rod gave a slightly less value 7#acb lbs47 = ! break+ load4 ,if ,i add ^! results toge!r1 & divide ! sum by #b1 ,i f9d #ach lbs4 z ! mean value ( ! break+ load1 & ?is is a suffici5tly exact det]m9ation4 #cei4 ,let u next try ! resi/ance ( a %ort] rod ( ! same section4 ,i place a piece ( p9e #bj long & #je squ>e 9 ! app>atus1 firmly secur+ ea* end z 9 ! =m] case4 ,! l[] bracket is adju/$ s z to make ! lev] horizontal2 ! c\nt]poise1 ( c\rse1 rema9s ! same1 & weights are plac$ 9 ! tray z be=e4 ,no deflection is notic$ :5 ! rod supports #abf lbs42 a v slight am\nt ( b5d+ is noticeable ) #ahf lbs42 ) #bbh lbs41 ! am\nt by :i* ! c5tre ( ! rod has deviat$ lat]ally f its orig9al position is ab\t #jb2 & f9ally1 :5 ! load rea*es #bid lbs41 ! rod breaks4 ,fracture fir/ occurs 9 ! middle1 b is imm$iately foll[$ by o!r fractures ne> :]e ! ends ( ! rod are secur$4 #cfj4 ,h5ce ! break+ load ( a rod #bj long is m ?an d\ble ! break+ load ( a rod ( #dj long ! same section2 f ?is we le>n t ! sections be+ equal1 %ort pill>s are /rong] ?an long pill>s4 ,x has be5 asc]ta9$ by exp]im5t t ! /r5g? ( a squ>e pill> to resi/ compression is proportional to ! squ>e ( its sectional area4 ,h5ce a rod ( p9e1 #dj long & #a squ>e1 hav+ f\r times ! section ( ! rod ( ! same l5g? we h exp]im5t$ on1 w\ld be sixte5 times z /rong1 & consequ5tly its break+ weight w\ld am\nt to ne>ly a ton4 ,! /r5g? ( a rod us$ z a tie dep5ds only on its section1 :ile ! /r5g? ( a rod us$ z a /rut dep5ds on its l5g? z well z on its section4 ,,condition ,,( ,a ,,beam ,,/ra9$ ,,by ,a ,,transv]se ,,=ce4 #cfa4 ,we next come to ! important practical subject ( ! /r5g? ( timb] :5 support+ a transv]se /ra92 t is1 :5 us$ z a beam4 ,! nature ( a transv]se /ra9 may be und]/ood f ,fig4 #ea1 :i* repres5ts a small beam1 /ra9$ by a load at its c5tre4 ,fig4 #eb %[s two supports #dj ap>t1 across :i* a rod ( p9e #dh #a #a is laid2 at ! middle ( ?is rod a hook is plac$1 f :i* a tray = ! reception ( weights is susp5d$4 ,a rod ?us support$1 & be>+ weights1 is said to be /ra9$ transv]sely4 ,a raft] ( a ro(1 ! floor+ ( a room1 a gangway f ! :>f to a %ip1 _m =ms ( bridge1 & innum]able o!r examples1 might be giv5 ( beams /ra9$ 9 ?is mann]4 ,to ?is important subject we %all devote ! rema9d] ( ?is lecture & ! :ole ( ! next4 #cfb4 ,! fir/ po9t to be notic$ is ! deflection ( ! beam f :i* a weight is susp5d$4 ,! beam is at fir/ horizontal2 b z ! weight 9 ! tray is augm5t$1 ! beam gradually curves d[nw>ds until1 :5 ! weight rea*es a c]ta9 am\nt1 ! beam breaks across 9 ! middle & ! tray falls4 ,illu/ration3 ,,fig4 #ea4 ,= conv5i5ce 9 record+ ! exp]im5ts ! tray *a9 & hooks h be5 adju/$ to weigh exactly #ad lbs4 7,fig4 #eb74 ,a ,b is a cord :i* is kept /ret*$ by ! little weights ,d3 ?is cord gives a r\gh measure ( ! deflection ( ! beam f its horizontal position :5 /ra9$ by a load 9 ! tray4 ,9 ord] to obs]ve ! deflection accurately an in/rum5t is us$ call$ ! ca!tomet] 7,g74 ,x consi/s ( a small telescope1 always direct$ horizontally1 ?\gh capable ( be+ mov$ up & d[n a v]tical triangul> pill>2 on "o ( ! sides ( ! pill> a scale is engrav$1 s t ! height ( ! telescope 9 any position c be accurately det]m9$4 ,! ca!tomet] is levell$ by means ( ! screws ,h ,h1 s t ! triangul> pill> on :i* ! telescope slides is accurately v]tical3 ! dott$ l9e %[s ! direction ( ! visual ray :5 ! c5tre ,c ( ! beam is se5 by ! obs]v] ?r\gh ! telescope4 ,illu/ration3 ,,fig4 #eb4 ,inside ! telescope & at its focus a l9e ( spid]'s web is fix$ horizontally2 on ! b> to be obs]v$1 & ne> its middle po9t ,c1 a cross ( two f9e l9es is m>k$4 ,! tray be+ remov$1 ! beam becomes horizontal2 ! telescope ( ! ca!tomet] is !n direct$ t[>ds ! beam1 s t ! l9es m>k$ ^u x c be se5 di/9ctly4 ,by means ( a screw ! telescope may be rais$ or l[]$ until ! spid]'s web inside ! telescope is obs]v$ to pass ?r\gh ! image ( ! int]section ( ! l9es4 ,! scale !n indicates precisely h[ high ! telescope is on ! pill>4 #cfc4 ,:ile ,i look ?r\gh ! telescope my assi/ant susp5ds ! tray f ! beam4 ,in/antly ,i see ! cross desc5d 9 ! field ( view4 ,i l[] ! telescope until ! spid]'s web aga9 passes ?r\gh ! image ( ! int]section ( ! l9es1 & !n by look+ at ! scale ,i see t ! telescope has be5 mov$ d[n #jai1 t is1 ab\t "o-fif? ( an in*3 ?is is1 !re=e1 ! di/ance by :i* ! cross l9es on ! beam1 & !re=e ! c5tre ( ! beam itself1 mu/ h desc5d$4 ,inde$1 ev5 a simpl] app>atus w\ld be compet5t to measure ! am\nt ( deflection ) "s degree ( precision4 ,by plac+ successively "o /one aft] ano!r ^u ! tray1 ! beam is se5 to deflect m & m1 until ev5 )\t ! telescope y see ! beam has deviat$ f ! horizontal4 #cfd4 ,by c>efully obs]v+ ) ! telescope1 & measur+ 9 ! way already describ$1 ! deflections %[n 9 ,table ,,xxiii4 7 det]m9$4 ,! scale along ! v]tical pill> 0 read aft] ! spid]'s web _h be5 adju/$ = ea* increase 9 ! weight4 ,! movem5t f ! orig9al position is record$ z ! deflection = ea* load4 ,,table ,,xxiii4--,,deflection ,,( ,a ,,beam4 ,a rod ( p9e #dh #a #a2 re/+ freely on supports #dj ap>t2 & lad5 9 ! middle4 ------------------------------------- ,numb] ( ,magnitude ,deflection4 ,exp]im5t4 ( load4 ------------------------------------- #a #ad #jai #b #bh #jcg #c #db #jee #d #ef #jgd #e #gj #jid #f #hd #aac #g #ih #ace #h #aab #afa #i #abf #aie #aj #adj #bcg ------------------------------------- #cfe4 ,! fir/ column records ! numb] ( ! exp]im5t4 ,! second repres5ts ! load1 & ! ?ird conta9s ! correspond+ deflections4 ,x w be se5 t up to #ih lbs4 ! deflection is ab\t #jb = e /one weight1 b aft]w>ds ! deflection increases m rapidly4 ,:5 ! weight rea*es #adj lbs4 ! deflection at fir/ indicat$ is #bcg2 b gradually ! cross l9es are se5 to desc5d 9 ! field ( ! telescope1 %[+ t ! beam is yield+ & f9ally x breaks across4 ,?is exp]im5t tea*es u t a beam is at fir/ deflect$ by an am\nt proportional to ! weight x supports2 b t :5 two-?irds ( ! break+ weight is rea*$1 ! beam is deflect$ m rapidly4 #cff4 ,x is a "q ( ! utmo/ importance to asc]ta9 ! greate/ load a beam c su/a9 )\t injury to its /r5g?4 ,?is subject is to be /udi$ by exam9+ ! effect ( diff]5t deflections ^u ! fibres ( a beam4 ,a beam is always deflect$ :atev] be ! load x supports2 ?us by look+ ?r\gh ! telescope ( ! ca!tomet] ,i c detect an increase ( deflection :5 a s+le p\nd is plac$ 9 ! tray3 h5ce :5ev] a beam is load$ we mu/ h "s deflection4 ,an exp]im5t w %[ :at am\nt ( deflection may be exp]i5c$ )\t produc+ any p]man5tly injuri\s effect4 #cfg4 ,a p9e rod #dj #a #a is freely support$ at ea* end1 ! di/ances betwe5 ! supports be+ #ch1 & ! tray is susp5d$ f its middle po9t4 ,a f9e pair ( cross l9es is m>k$ ^u ! beam1 & ! telescope ( ! ca!tomet] is adju/$ s t ! spid]'s l9e exactly passes ?r\gh ! image ( ! int]section4 #ad lbs4 be+ plac$ 9 ! tray1 ! cross is se5 to desc5d2 ! weight be+ remov$1 ! cross returns precisely to its orig9al position ) ref]5ce to ! spid]'s l9e3 h5ce1 aft] ?is am\nt ( deflection1 ! beam has cle>ly return$ to its initial condition1 & is evid5tly j z good z x 0 be=e4 ,! tray next receiv$ #ef lbs42 ! beam 01 ( c\rse1 consid]ably deflect$1 b :5 ! weight 0 remov$ ! cross aga9 return$1--at all ev5ts1 to )9 #jja ( :]e ! spid]'s l9e 0 left to indicate its =m] position4 ,we may consid] t ! beam is 9 ?is case also re/or$ to its orig9al condition1 ev5 ?\gh x has borne a /ra9 :i*1 includ+ ! tray1 am\nt$ to #gj lbs4 ,b :5 ! beam has be5 made to c>ry #hd lbs4 = a few seconds1 ! cross does n completely return on ! removal ( ! load f ! tray1 b x %[s t ! beam has n[ receiv$ a p]man5t deflection ( #jjc4 ,?is is /ill m app>5t aft] ! beam has c>ri$ #ih lbs41 = :5 ?is load is remov$ ! c5tre ( ! beam is p]man5tly deflect$ by #jac4 ,"h1 !n1 we may inf] t ! fibres ( ! beam are beg9n+ to be /ra9$ beyond _! p[]s ( resi/ance1 & ?is is v]ifi$ :5 we f9d t ) #bh additional p\nds 9 ! tray a collapse ensues4 #cfh4 ,reason+ f ?is exp]im5t1 we might inf] t ! ela/icity ( a beam is n affect$ by a weight :i* is less ?an half t :i* w\ld break x1 & t1 !re=e1 x may be> )\t injury a weight n exce$+ ?is am\nt4 ,z1 h[ev]1 9 our exp]im5ts ! weight 0 only appli$ once1 & !n b = a %ort "t1 we _c be sure t a long]-cont9u$ or m frequ5t application ( ! same load might n prove injuri\s2 h5ce1 to be on ! safe side1 we assume t "o-?ird ( ! break+ weight ( a beam is ! greate/ load x %\ld be made to be> 9 any /ructure4 ,9 _m cases x is f\nd desirable to make ! beam mu* /rong] ?an ?is ratio w\ld indicate4 #cfi4 ,we next consid] ! condition ( ! fibres ( a beam :5 /ra9$ by a transv]se =ce4 ,x is evid5t t s9ce ! fracture comm5ces at ! l[] surface ( ! beam1 ! fibres !re mu/ be 9 a /ate ( t5sion1 :ile ^? at ! concave upp] surface ( ! beam are compress$ toge!r4 ,?is condition ( ! fibres may be prov$ by ! foll[+ exp]im5t4 #cgj4 ,i take two p9e rods1 ea* #dh #a #a1 p]fectly simil> 9 all respects1 cut f ! same piece ( timb]1 & !re=e probably ( v ne>ly id5tical /r5g?4 ,) a f9e t5on saw ,i cut ea* ( ! rods half ?r\gh at its middle po9t4 ,i n[ place "o ( ^! beams on ! supports #dj ap>t1 ) ! cut side ( ! beam upw>ds4 ,i susp5d f x ! tray1 :i* ,i gradually load ) weights until ! beam breaks1 :i* x does :5 ! total weight is #ha lbs4 ,if ,i 7 to place ! second beam on ! same supports ) ! cut upw>ds1 !n !re c be no d\bt t x w\ld require z ne>ly z possible ! same weight to break x4 ,i place x1 h[ev]1 ) ! cut d[nw>ds1 ,i susp5d ! tray1 & f9d t ! beam breaks ) a load ( #ca lbs4 ,?is is less ?an half ! weight t w\ld h be5 requir$ if ! cut _h be5 upw>ds4 #cga4 ,:at is ! cause ( ?is diff]5ce8 ,! fibres be+ compress$ toge!r on ! upp] surface1 a cut has no t5d5cy to op5 !re2 & if ! cut c\ld be made ) an extremely f9e saw1 s z to remove b little mat]ial1 ! beam w\ld be sub/antially ! same z if x _h n be5 tamp]$ )4 ,on ! o!r h&1 ! fibres at ! l[] surface are 9 a /ate ( t5sion2 !re=e :5 ! cut is bel[ x yawns op51 & ! beam is greatly weak5$4 ,x is1 9 fact1 no /rong] ?an a beam ( #dh #je #a1 plac$ ) its %orte/ dim5sion v]tical4 ,if we rememb] t an entire beam ( ! same size requir$ ab\t #adj lbs4 to break x 7,art4 #cff71 we see t ! /r5g? ( a beam is r$uc$ to "o-f\r? by be+ cut half-way ?r\gh & hav+ ! cut und]nea?4 #cgb4 ,we may le>n f ?is ! practical consequ5ce t ! s\nd] side ( a beam %\ld always be plac$ d[nw>ds4 ,any flaw on ! l[] surface w s]i\sly weak5 ! beam3 s t ! mo/ knotty face ( ! wood %\ld c]ta9ly be plac$ upp]mo/4 ,if a portion ( ! actual sub/ance ( a beam be remov$--= example1 if a not* be cut out ( x--?is w be almo/ equally injuri\s on ei!r side ( ! beam4 #cgc4 ,we may illu/rate ! condition ( ! upp] surface ( ! beam by a fur!r exp]im5t4 ,i make two cuts #je deep 9 ! middle ( a p9e rod #dh #a #a4 ,^! cuts are #je ap>t1 & slightly incl9$2 ! piece betwe5 !m be+ remov$1 a w$ge is %ap$ to fit tightly into ! space2 ! w$ge is long en\gh to project a little on "o side4 ,if ! w$ge be upp]mo/ :5 ! beam is plac$ on ! supports1 ! beam w be 9 ! same condition z if x _h two f9e cuts on ! upp] surface4 ,i n[ load ! beam ) ! tray 9 ! usual mann]1 & ,i f9d x to be> #gj lbs4 securely4 ,on exam9+ ! beam1 :i* has curv$ d[n consid]ably1 ,i f9d t ! w$ge is held 9 v tightly by ! pressure ( ! fibres ^u x1 b1 by a %>p tap at ! end1 ,i knock out ! w$ge1 & in/antly ! load ( #gj lbs4 breaks ! beam2 ! reason is simple--! piece be+ remov$1 !re is no long] any resi/ance to ! compressive /ra9 ( ! upp] fibres1 & consequ5tly ! beam gives way4 #cgd4 ,! collapse ( a beam by a transv]se /ra9 comm5ces by fracture ( ! fibres on ! l[] surface1 foll[$ by a rupture ( all fibres up to a consid]able dep?4 ,"h we see t by a transv]se =ce ! fibres 9 a beam ( #dh #a #a h be5 brok5 by a /ra9 ( #adj lbs4 7,art4 #cff72 b we h already /at$ 7,art4 #cec7 t to te> su* a rod across by a direct pull at ea* end a =ce ( ab\t f\r tons is necess>y4 ,! break+ /ra9 ( ! fibres mu/ be a c]ta9 def9ite quantity1 yet we f9d t to ov]come x 9 "o way f\r tons is necess>y1 :ile by ano!r mode ( apply+ ! /ra9 #adj lbs4 is suffici5t4 #cge4 ,to expla9 ?is discrepancy we may ref] to ! exp]im5t ( ,art4 #bh1 :]e9 a piece ( /r+ 0 brok5 by ! transv]se pull ( a piece ( ?read 9 illu/ration ( ! fact t "o =ce may be resolv$ into two o!rs1 ea* ( !m v mu* great] ?an itself4 ,a simil> resolution ( =ce occurs 9 ! transv]se deflection ( ! beam1 & ! =ce ( #adj lbs4 is *ang$ into two o!r =ces1 ea* ( !m enorm\sly great] & suffici5tly /rong to rupture ! fibres4 ,we ne$ n suppose t ! =ce ?us develop$ is s great z f\r tons1 because t is ! am\nt requir$ to te> across a squ>e in* ( fibres simultane\sly1 :]eas 9 ! transv]se fracture ! fibres appe> to be brok5 r[ aft] r[2 ! fracture is ?us only gradual1 nor does x ext5d ?r\gh ! entire dep? ( ! beam4 #cgf4 ,we %all conclude ?is lecture ) "o m rem>k1 on ! condition ( a beam :5 /ra9$ by a transv]se =ce4 ,we h se5 t ! fibres on ! upp] surface are compress$1 :ile ^? on ! l[] surface are ext5d$2 b :at is ! condition ( ! fibres 9 ! int]ior8 ,!re c be no d\bt t ! foll[+ is ! /ate ( ! case3--,! fibres imm$iately b5ea? ! upp] surface are 9 compression2 at a great] dep? ! am\nt ( compression dim9i%es until at ! middle ( ! beam ! fibres are 9 _! natural condition2 on approa*+ ! l[] surface ! fibres comm5ce to be /ra9$ 9 ext5sion1 & ! am\nt ( ! ext5sion gradually increases until x rea*es a maximum at ! l[] surface4 ,,lecture ,,xii4 ,,! ,,/r5g? ,,( ,a ,,beam4 ,a ,beam free at ! ,ends & load$ 9 ! ,middle4--,a ,beam uni=mly load$4--,a ,beam load$ 9 ! ,middle1 ^: ,ends are secur$4--,a ,beam support$ at "o end & load$ at ! o!r4 ,a ,,beam ,,free ,,at ,,! ,,ends ,,& ,,load$ ,,9 ,,! ,,middle4 #cgg4 ,9 ! prec$+ lecture we h exam9$ "s g5]al circum/ances 9 connection ) ! condition ( a beam act$ on by a transv]se =ce2 we proce$ 9 ! pres5t to inquire m p>ticul>ly into ! /r5g? "u ^! conditions4 ,we %all1 z be=e1 use = our exp]im5ts rods ( p9e only1 z we wi% r to illu/rate ! g5]al laws ?an to det]m9e ! /r5g? ( diff]5t mat]ials4 ,! /r5g? ( a beam dep5ds ^u its l5g?1 bread?1 & ?ickness2 we mu/ endeav\r to di/+ui% ! effects ( ea* ( ^! elem5ts on ! capacity ( ! beam to su/a9 its load4 ,we %all only employ beams ( rectangul> section2 ?is be+ g5]ally ! =m 9 :i* beams ( wood are us$4 ,beams ( iron1 :5 l>ge1 are usually n rectangul>1 z ! mat]ial c be m effectively dispos$ 9 sections ( a diff]5t =m4 ,x is important to di/+ui% betwe5 ! /iffness ( a beam 9 its capacity to resi/ flexure1 & ! /r5g? ( a beam 9 its capacity to resi/ fracture4 ,?us ! /iffe/ beam :i* c be made f ! cyl9drical trunk ( a tree #a' 9 diamet] is #f broad & #aje deep1 :ile ! /ronge/ beam is #g broad & #ige deep4 ,we are n[ discuss+ ! /r5g? 7n ! /iffness7 ( beams4 #cgh4 ,we %all comm5ce ! inquiry by mak+ a numb] ( exp]im5ts3 ^! we %all record 9 a table1 & !n we %all endeav\r to see :at we c le>n f an exam9ation ( ?is table4 ,i h "h t5 pieces ( p9e1 ( l5g?s v>y+ f #a' to #d'1 & ( ?ree diff]5t sections1 viz4 #a #a1 #a #je1 & #je #je4 ,i h arrang$ f\r diff]5t /&s1 on :i* we c break ^! pieces3 on ! fir/ /& ! di/ance betwe5 ! po9ts ( support is #dj1 & on ! o!r /&s ! di/ances are #cj1 #bj1 & #aj respectively2 ! pieces be+ #d'1 #c'1 #b'1 & #a' long1 w j be conv5i5tly held on ! supports4 #cgi4 ,! mode ( break+ is z foll[s3--,! beam be+ laid ^u ! supports1 an ,s hook is plac$ at its middle po9t1 & f ?is ,s hook ! tray is susp5d$4 ,weights are !n c>efully add$ to ! tray until ! beam breaks2 ! load 9 ! tray1 toge!r ) ! weight ( ! tray1 is record$ 9 ! table z ! break+ load4 #chj4 ,9 ord] to gu>d z mu* z possible aga9/ error1 ,i h "h ano!r set ( t5 pieces ( p9e1 duplicates ( ! =m]4 ,i %all also break ^!2 & :5ev] ,i f9d any diff]5ce betwe5 ! break+ loads ( two simil> beams1 ,i %all record 9 ! table ! mean betwe5 ! two loads4 ,! results are %[n 9 ,table ,,xxiv4 ,,table ,,xxiv4--,,/r5g? ,,( ,a ,,beam4 ,slips ( p9e 7cut f ! same piece7 support$ freely at ea* end2 ! l5g? record$ is ! di/ance betwe5 ! po9ts ( support2 ! load is susp5d$ f ! c5tre ( ! beam1 & gradually increas$ until ! beam breaks2 area ( section dep? ,=mula1 ,p #fjhj ---------------------------------------- -------- span ---------------------------------------- ---------------------- ,mean ( ! ,p4 ,diff]5ce ,dim5sions4 obs]vations,calculat$ ( ! ,no4 ( ------------------- ( ! break+ obs]v$ & ,exp]i- break+ load calculat$ m5t4 ,span4,bread?4,dep?4load 9 lbs4 9 lbs4 values4 ---------------------------------------- -------------------- #a #djj #aj #aj #aeb #aeb #jj #b #djj #je #aj #gg #gf -#aj #c #djj #aj #je #ch #ch #jj #d #djj #je #je #ai #ai #jj #e #cjj #aj #je #ei #ea -#hj #f #cjj #je #je #be #be #jj #g #bjj #aj #je #gd #gf #bj #h #bjj #je #je #cf #ch #bj #i #ajj #aj #je #aed #aeb -#bj #aj #ajj #je #je #fh #gf #hj ---------------------------------------- -------------------- #cha4 ,9 ! fir/ column is a s]ies ( figures = conv5i5ce ( ref]5ce4 ,! next ?ree columns are occupi$ ) ! dim5sions ( ! beams4 ,by span is meant ! di/ance betwe5 ! po9ts ( support2 ! real l5g? is ( c\rse great]2 ! dep? is t dim5sion ( ! beam :i* is v]tical4 ,! fif? column gives ! mean ( two obs]vations ( ! break+ load4 ,?us = example1 9 exp]im5t ,no4 #e ! two beams us$ 7 ea* #cf #a #je1 !y 7 plac$ on po9ts ( support #cj di/ant1 s ! span record$ is #cj3 "o ( ! beams 0 brok5 by a load ( #eh lbs41 & ! second by a load ( #fj lbs42 ! mean betwe5 ! two1 #ei lbs41 is record$ z ! mean break+ load4 ,9 ?is mann] ! column ( break+ loads has be5 f\nd4 ,! mean+ ( ! two la/ columns ( ! table w be expla9$ pres5tly4 #chb4 ,we %all endeav\r to elicit f ^! obs]vations ! laws :i* connect ! break+ load ) ! span1 bread?1 & dep? ( ! beam4 #chc4 ,let u fir/ exam9e ! effect ( ! span2 = ?is purpose we br+ toge!r ! obs]vations ^u beams ( ! same section1 b ( diff]5t spans4 ,sections ( #je #je w be conv5i5t = ?is purpose2 ,nos4 #d1 #f1 #h1 & #aj are exp]im5ts ^u beams ( ?is section4 ,let u fir/ comp>e #d & #h4 ,"h we h two beams ( ! same section1 & ! span ( "o 7#dj7 is d\ble t ( ! o!r 7#bj74 ,:5 we exam9e ! break+ weights we f9d t !y are #ai lbs4 & #cf lbs42 ! =m] ( ^! numb]s is r m ?an half ( ! latt]4 ,9 fact1 _h ! break+ load ( #dj be5 lb4 less1 #ahbe lbs41 & _h t ( #bj be5 lb4 m1 #cfe lbs41 "o ( ! break+ loads w\ld h be5 exactly half ! o!r4 #chd4 ,we mu/ n look = p]fect num]ical accuracy 9 ^! exp]im5ts2 we mu/ only expect to meet ) approximation1 because ! laws = :i* we are 9 se>* are 9 reality only approximate laws4 ,wood itself is v>iable 9 quality1 ev5 :5 cut f ! same piece3 p>ts ne> ! circumf]5ce are diff]5t 9 /r5g? f ^? ne>] ! c5tre2 9 a "y tree !y are g5]ally weak]1 & 9 an old tree g5]ally /rong]4 ,m9ute diff]5ces 9 ! gra91 great] or less p]fectness 9 ! season+1 ^! are also among ! circum/ances :i* prev5t "o piece ( timb] f be+ id5tical ) ano!r4 ,we %all1 h[ev]1 g5]ally f9d t ! effect ( ^! diff]5ces is small1 b occasionally ?is is n ! case1 & 9 try+ _m exp]im5ts ^u ! break+ ( timb]1 discrepancies occasionally appe> = :i* x is difficult to acc\nt4 #che4 ,b y w f9d1 ,i ?9k1 t1 mak+ reasonable all[ances = su* difficulties z d occur1 ! laws on ! :ole repres5t ! exp]im5ts v closely4 #chf4 ,we %all1 !n1 assume t ! break+ weight ( a b> ( #dj is half t ( a b> ( #bj ( ! same section1 & we ask1 ,is ?is g5]ally true8 is x true t ! break+ weight is inv]sely proportional to ! span8 ,9 ord] to te/ ?is hypo!sis1 we c calculate ! break+ weight ( a b> ( #cj 7,no4 #f71 & !n comp>e ! result ) ! obs]v$ value2 if ! supposition be true1 ! break+ weight %\ld be giv5 by ! proportion-- #cj 3 #dj 33 #ai 3 ,answ]4 ,! answ] is #bec lbs42 on ref]5ce to ! table we f9d #be lbs4 to be ! obs]v$ value1 h5ce our hypo!sis is v]ifi$ = ?is b>4 #chg4 ,let u te/ ! law also = ! #aj b>1 ,no4 #aj-- #aj 3 #dj 33 #ai 3 ,answ]4 ,! answ] 9 ?is case is #gf1 :]eas ! obs]v$ value is #fh1 or #h lbs4 less2 ?is does n agree v well ) ! !ory1 b /ill ! diff]5ce1 ?\gh #h lbs41 is only ab\t #aa or #ab p] c5t4 ( ! :ole1 & we %all /ill reta9 ! law1 = c]ta9ly !re is no o!r t c express ! result z well4 #chh4 ,b ! table w supply ano!r v]ification4 ,9 exp]im5t ,no4 #c a #dj b>1 #a broad1 & #je deep1 broke ) #ch lbs42 & 9 exp]im5t ,no4 #g a #bj b> ( ! same section broke ) #gd lbs42 b ?is is s ne>ly d\ble ! break+ weight ( ! #dj b>1 z to be an additional illu/ration ( ! law1 t = a giv5 section ! break+ load v>ies inv]sely z ! span4 #chi4 ,we next inquire z to ! effect ( ! bread? ( ! beam ^u its /r5g?8 ,= ?is purpose we comp>e exp]im5ts ,nos4 #c & #d3 we !re f9d t a b> #dj #a #je is brok5 by a load ( #ch lbs41 :ile a b> j half ! bread? is brok5 by #ai lbs4 ,we might h anticipat$ ?is result1 = x is evid5t t ! b> ( ,no4 #c mu/ h ! same /r5g? z two b>s simil> to t ( ,no4 #d plac$ side by side4 #cij4 ,?is view is confirm$ by a comp>ison ( ,nos4 #g & #h1 :]e we f9d t a #bj b> takes twice ! load to break x t is requir$ = a b> ( half its bread?4 ,! law is n q s well v]ifi$ by ,nos4 #e & #f1 = half ! break+ weight ( ,no4 #e1 namely #bie lbs41 is m ?an #be1 ! obs]v$ break+ weight ( ,no4 #f3 a simil> rem>k may be made ab\t ,nos4 #i & #aj4 #cia4 ,suppos+ we _h a beam ( #dj span1 #b broad1 & #je deep1 we c easily see t x is equival5t to two b>s l t ( ,no4 #c plac$ side by side2 & we inf] g5]ally t ! /r5g? ( a b> is proportional to its bread?2 or to speak-m def9itely1 if two beams h ! same span & dep?1 ! ratio ( _! break+ loads is ! same z ! ratio ( _! bread?s4 #cib4 ,we next exam9e ! effect ( ! dep? ( a beam ^u its /r5g?4 ,9 exp]im5t+ ^u a beam plac$ edgewise1 a precaution mu/ be obs]v$1 :i* w\ld n be necess>y if ! same beam 7 to be brok5 flatwise4 ,:5 ! load is susp5d$1 ! beam1 if m]ely laid edgewise on ! supports1 w\ld almo/ c]ta9ly turn ov]2 x is !re=e necess>y to place its extremities 9 recesses 9 ! supports1 :i* w obviate ! possibility ( ?is occurr5ce2 at ! same "t ! ends mu/ n be prev5t$ f b5d+ upw>ds1 = we are at pres5t discuss+ a beam free at ea* end1 & ! case :]e ! ends are n free w be subsequ5tly consid]$4 #cic4 ,let u fir/ comp>e toge!r exp]im5ts ,nos4 #b & #c2 "h we h two b>s ( ! same dim5sions1 ! section 9 ea* be+ #aj #je1 b ! fir/ b> is brok5 edgewise1 & ! second flatwise4 ,! fir/ breaks ) #gg lbs41 & ! second ) #ch lbs42 h5ce ! same b> is twice z /rong plac$ edgewise z flatwise :5 "o dim5sion ( ! section is twice z great z ! o!r4 ,we may g5]alize ?is law1 & ass]t t ! /r5g? ( a rectangul> beam brok5 edgewise is to ! /r5g? ( a beam ( l span & section brok5 flatwise1 z ! great] dim5sion ( ! section is to ! less] dim5sion4 #cid4 ,! /r5g? ( a beam #dj #je #a is f\r times z great z ! /r5g? ( #dj #je #je1 ?\gh ! quantity ( wood is only twice z great 9 "o z 9 ! o!r4 ,9 g5]al we may /ate t if a beam 7 bisect$ by a longitud9al cut1 ! /r5g? ( ! beam w\ld be halv$ :5 ! cut 0 horizontal1 & unalt]$ :5 ! cut 0 v]tical2 ?us1 = example1 two beams ( exp]im5t ,no4 #d1 plac$ "o on ! top ( ! o!r1 w\ld break ) ab\t #dj lbs41 :]eas if ! same rods 7 9 "o piece1 ! break+ load w\ld be ne>ly #hj lbs4 #cie4 ,?is may be illu/rat$ 9 a diff]5t mann]4 ,i h "h two beams ( #dj #a #je sup]pos$2 !y =m "o beam1 equival5t to t ( ,no4 #a 9 bulk1 b ,i f9d t !y break ) #hj lbs41 ?us %[+ t ! two are only twice z /rong z "o4 #cif4 ,i take two simil> b>s1 &1 in/ead ( lay+ !m loosely "o on ! o!r1 ,i unite !m tightly ) iron clamps l ^? repres5t$ 9 ,fig4 #ef4 ,i n[ f9d t ! b>s ?us fa/5$ toge!r require #ajd lbs4 = fracture4 ,we c readily und]/& ?is increase ( /r5g?4 ,z soon z ! b>s beg9 to b5d "u ! action ( ! weight1 ! surfaces :i* are 9 contact move slightly "o ^u ! o!r 9 ord] to accommodate !mselves to ! *ange ( =m4 ,by clamp+ ,i greatly imp$e ?is motion h5ce ! beams deflect less1 & require a great] load be=e !y collapse2 ! case is !re=e to "s ext5t approximat$ to ! /ate ( ?+s :5 ! two rods =m "o solid piece1 9 :i* case a load ( #aeb lbs4 w\ld be requir$ to produce fracture4 #cig4 ,we %all be able by a little consid]ation to und]/& ! reason :y a b> is /rong] edgewise ?an flatwise4 ,suppose ,i try to break a b> across my knee by pull+ ! ends held "o 9 ea* h&1 :at is x t resi/s ! break+8 ,x is *iefly ! t5acity ( ! fibres on ! convex surface ( ! b>4 ,if ! b> be edgewise1 ^! fibres are fur!r away f my knee & !re=e resi/ ) a great] mom5t ?an :5 ! b> is flatwise3 nor is ! case diff]5t :5 ! b> is support$ at ea* end1 & ! load plac$ 9 ! c5tre2 = !n ! reactions ( ! supports correspond to ! =ces ) :i* ,i pull$ ! ends ( ! b>4 #cih4 ,we c n[ calculate ! /r5g? ( any rectangul> beam ( p9e3 ,let u suppose x to be #ab' long1 #e broad1 & #g deep4 ,?is is five times z /rong z a beam #a broad & #g deep = we may conceive ! orig9al beam to consi/ ( #e ( ^! beams plac$ side by side 7,art #cia72 ! beam #a broad & #g deep1 is #g times z /rong z a beam #g broad1 #a deep 7,art4 #cic74 ,h5ce ! orig9al beam mu/ be #ce times z /rong z a beam #g broad1 #a deep2 b ! beam #g broad & #a deep is sev5 times /rong] ?an a beam ! section ( :i* is #a #a1 h5ce ! orig9al beam is #bde times z /rong z a beam #ab' long & #a #a 9 section2 ( :i* we c calculate ! /r5g?1 by ,art4 #chh1 f ! proportion-- #add 3 #dj 33 #aeb 3 ,answ]4 ,! answ] is #dbb lbs41 & ?us ! break+ load ( ! orig9al beam is ab\t #aj1cjj lbs4 #cii4 ,x w be useful to d$uce ! g5]al expression = ! break+ load ( a beam l span1 b broad1 & d deep1 support$ freely at ! ends & lad5 9 ! c5tre4 ,let u suppose a b> l long1 & #a #a 9 section4 ,! break+ load is f\nd by ! proportion-- l 3 #dj 33 #aeb 3 ,answ]2 & ! result obta9$ is #fjhj_/l4 ,a beam :i* is d broad1 l span1 & #a deep1 w\ld be j z /rong z d ( ! beams l #a #a plac$ side by side2 ( :i* ! collective /r5g? w\ld be-- #fjhj ---------------- d4 l ,if su* a beam1 in/ead ( re/+ flatwise1 7 plac$ edgewise1 its /r5g? w\ld be increas$ 9 ! ratio ( its dep? to its bread?--t is1 x w\ld be increas$ d-fold--& w\ld !re=e am\nt to #fjhj ------------------ d4 l ,we ?us le>n ! /r5g? ( a beam #a broad1 d deep1 & l span4 ,! /r5g? ( b ( ^! beams plac$ side by side1 w\ld be ! same z ! /r5g? ( "o beam b broad1 d deep1 & l span1 & ?us we f9ally obta9 #fjhj ---------------- d b4 l ,s9ce b d is ! area ( ! section1 we c express ?is result conv5i5tly by say+ t ! break+ load 9 lbs4 ( a rectangul> p9e beam is equal to area ( section dep?2 #fjhj ---------------------------------------- ---------- span ! dep? & span be+ express$ 9 in*es l9e> measure1 & ! section 9 squ>e in*es4 #djj4 ,9 ord] to te/ ?is =mula1 we h calculat$ f x ! break+ loads ( all ! t5 beams giv5 9 ,table ,,xxiv4 & ! results are giv5 9 ! six? column4 ,! diff]5ce betwe5 ! am\nt calculat$ & ! obs]v$ mean break+ weight is %[n 9 ! la/ column4 #dja4 ,?us1 = example1 9 exp]im5t ,no4 #g ! span is #bj1 bread?1 #a1 dep? #je2 ! =mula gives1 s9ce ! area is #je1 #je #je ,p #fjhj -------------------- #gf #bj ,?is agrees suffici5tly ) #gd lbs41 ! mean ( two obs]v$ values4 #djb4 ,except 9 exp]im5ts ,nos4 #e & #aj1 ! diff]5ces are v small1 & ev5 9 ^! two cases ! diff]5ces are n suffici5t to make u d\bt t we h discov]$ ! correct expression = ! load g5]ally suffici5t to produce fracture4 #djc4 ,we h already po9t$ out t a beam beg9s to su/a9 p]man5t injury :5 x is subject$ to a load great] ?an half t :i* w\ld break x 7,art4 #cfh71 & we may inf] t x is n 9 g5]al prud5t to load a beam :i* is "p ( a p]man5t /ructure ) m ?an ab\t a ?ird or a f\r? ( ! break+ weight4 ,h5ce if we want$ to calculate a fair work+ load 9 lbs4 = a beam ( p9e1 we might obta9 x f ! =mula4 area ( section dep?4 #aejj ---------------------------------------- ---------- span ,probably a small] coeffici5t ?an #aejj w\ld (t5 be us$ by ! cauti\s build]1 especially :5 ! beam 0 liable to sudd5 bl[s or %ocks4 ,! coeffici5t obta9$ f small select$ rods su* z we h us$ w\ld also be great] ?an t f\nd f l>ge beams 9 :i* imp]fections are inevitable4 #djd4 ,_h we adopt$ any o!r k9d ( wood we %\ld h f\nd a simil> =mula = ! break+ weight1 b ) a diff]5t num]ical coeffici5t4 ,= example1 _h ! beams be5 made ( oak ! numb] #fjhj mu/ be replac$ by a l>g] figure4 ,a ,,beam ,,uni=mly ,,load$4 #dje4 ,we h up to ! pres5t only consid]$ ! case :]e ! load is susp5d$ f ! c5tre ( ! beam4 ,b 9 ! actual employm5t ( beams ! load is n g5]ally appli$ 9 ?is mann]4 ,see 9 ! raft]s :i* support a ro( h[ e in* 9 ! entire l5g? has its burd5 ( slates to be>4 ,! beams :i* support a w>eh\se floor h to c>ry _! load 9 :atev] mann] ! goods are dispos$3 sometimes1 z = example 9 a gra9-/ore1 ! pressure w be tol]ably uni=m along ! beams1 :ile if ! weights be irregul>ly scatt]$ on ! floor1 !re w be correspond+ inequalities 9 ! mode 9 :i* ! loads are di/ribut$ ov] ! beams4 ,x w !re=e be useful = u to exam9e ! /r5g? ( a beam :5 its load is appli$ o!rwise ?an at ! c5tre4 #djf4 ,we %all employ1 9 ! fir/ place1 a beam #dj span1 #je broad1 & #a deep2 & we %all break x by apply+ a load simultane\sly at two po9ts1 z may be mo/ conv5i5tly done by ! contrivance %[n 9 ! diagram1 ,fig4 #ec4 ,a ,b is ! beam re/+ on two supports2 ,c & ,d are ! po9ts ( trisection ( ! span2 f :5ce loops desc5d1 :i* c>ry an iron b> ,p ,q2 at ! c5tre ,r ( :i* a weight ,w is susp5d$4 ,! load is ?us divid$ equally betwe5 ! two po9ts ,c & ,d1 & we may reg>d ,a ,b z a beam load$ at its two po9ts ( trisection4 ,! tray & weights are employ$ :i* we h us$ 9 ! app>atus repres5t$ 9 ,fig4 #eh4 ,illu/ration3 ,,fig4 #ec4 #djg4 ,we proce$ to break ?is beam4 ,add+ weights to ! tray1 we see t x yields ) #aag lbs41 & cracks across betwe5 ,c & ,d4 ,on ref]5ce to ,table ,,xxiv4 we f9d f exp]im5t ,no4 #b t a simil> b> 0 brok5 by #gg lbs4 at ! c5tre2 n[ _/ #gg #aaee2 h5ce we may /ate ) suffici5t approximation t ! b> is half z /rong aga9 :5 ! load is susp5d$ f ! two po9ts ( trisection z x is :5 susp5d$ f ! c5tre4 ,x is rem>kable t 9 break+ ! beam 9 ?is mann] ! fracture is equally likely to occur at any po9t betwe5 ,c & ,d4 #djh4 ,a beam uni=mly load$ requires twice z mu* load to break x z w\ld be suffici5t if ! load 7 m]ely susp5d$ f ! c5tre4 ,! mode ( apply+ a load uni=mly is %[n 9 ,fig4 #ed4 ,illu/ration3 ,,fig4 #ed4 ,9 an exp]im5t actually tri$1 a beam #dj #je #a plac$ edgewise 0 f\nd to support t5 #ad lb4 weights rang$ z 9 ! figure2 "o or two /one m w\ld1 h[ev]1 d\btless produce fracture4 #dji4 ,we inf] f ^! consid]ations t beams load$ 9 ! mann] 9 :i* !y are usually employ$ are consid]ably /rong] ?an w\ld be indicat$ by ! results 9 ,table ,,xxiv4 ,,effect ,,( ,,secur+ ,,! ,,ends ,,( ,a ,,beam ,,^u ,,its ,,/r5g?4 #daj4 ,x has be5 notic$ dur+ ! exp]im5ts t :5 ! weights are susp5d$ f a beam & ! beam beg9s to deflect1 ! ends curve upw>ds f ! supports4 ,?is b5d+ ( ! ends is = example %[n 9 ,fig4 #ed4 ,if we re/ra9 ! ends ( ! beam f b5d+ up 9 ?is mann]1 we %all add v consid]ably to its /r5g?4 ,?is we c d by clamp+ !m d[n to ! supports4 #daa4 ,let u exp]im5t ^u a beam #dj #a #a4 ,we clamp ea* ( ! ends & !n break ! beam by a weight susp5d$ f ! c5tre4 ,x requires #bch lbs4 to accompli% fracture4 ,?is is a little m ?an half z mu* aga9 z #aeb lbs41 :i* we f9d f ,table ,,xxiv4 0 ! weight requir$ to break ?is b> :5 its ends 7 free4 ,calculation %[s t ! /r5g? ( a beam may be ev5 d\bl$ :5 ! ends are kept horizontal by m p]fect me?ods ?an we h us$4 #dab4 ,:5 ! beam gives way "u ^! circum/ances1 !re is n only a fracture 9 ! c5tre1 b ea* ( ! halves are also f\nd to be brok5 across ne> ! po9ts ( support2 ! necessity = ?ree fractures in/ead ( "o expla9s ! increase ( /r5g? obta9$ by re/ra9+ ! ends to ! horizontal direction4 #dac4 ,9 /ructures ! beams are g5]ally m or less secur$ at ea* end1 & are !re=e m capable ( be>+ resi/ance ?an w\ld be indicat$ by ,table ,,xxiv4 ,f ! consid]ation ( ,arts4 #djh & #daa1 we c inf] t a beam secur$ at ea* end & uni=mly load$ w\ld require ?ree or f\r times z mu* load to break x z w\ld be suffici5t if ! ends 7 free & if ! load 7 appli$ at ! c5tre4 ,,beams ,,secur$ ,,at ,,"o ,,end ,,& ,,load$ ,,at ,,! ,,o!r4 #dad4 ,a beam1 "o end ( :i* is firmly imb$d$ 9 masonry or o!rwise secur$1 is occasionally call$ ^u to support a weight susp5d$ f its extremity4 ,su* a beam is %[n 9 ,fig4 #ee4 ,9 ! case we %all exam9e1 ,a ,b is a p9e beam ( dim5sions #bj #je #je1 & we f9d t1 :5 ,w rea*es #aj lbs41 ! beam breaks4 ,9 exp]im5t ,no4 #h1 ,table ,,xxiv41 a simil> beam requir$ #cf lbs42 h5ce we see t ! beam is brok5 9 ! mann] ( ,fig4 #ee1 by ab\t "o-f\r? ( ! load :i* w\ld h be5 requir$ if ! beam _h be5 support$ at ea* end & lad5 9 ! c5tre4 ,illu/ration3 ,,fig4 #ee4 ,we %all pres5tly h occasion to apply "s ( ! results obta9$ by ! exp]im5ts made 9 ! lecture n[ t]m9at$4 ,,lecture ,,xiii4 ,,! ,,pr9ciples ,,( ,,framework4 ,introduction4--,weight su/a9$ by ,tie & ,/rut4--,bridge ) ,two ,/ruts4--,bridge ) ,f\r ,/ruts4--,bridge ) ,two ,ties4--,simple ,=m ( ,truss$ ,bridge4 ,,introduction4 #dae4 ,9 ?is lecture & ! next we %all exp]im5t ^u "s ( ! arts ( con/ruction4 ,we %all employ slips ( p9e #je #je 9 section = ! purpose ( mak+ models ( simple framework3 ^! slips c be atta*$ to ea* o!r by means ( ! small clamps ab\t #c long1 %[n 9 ,fig4 #ef1 & ! g5]al appe>ance ( ! models ?us produc$ may be se5 f ,figs4 #eh & #fb4 ,illu/ration3 ,,fig4 #ef4 #daf4 ,! foll[+ exp]im5t %[s ! t5acity ) :i* ^! clamps hold4 ,two slips ( p9e1 ea* #ab #je #je1 are clamp$ toge!r1 s t !y ov]lap ab\t #b1 ?us =m+ a l5g? ( #bb3 ?is composite rod is rais$ by a pulley-block z 9 ,fig4 #di1 :ile a load ( #b cwt4 is susp5d$ f x4 ,?us ! clamp$ rods be> a direct t5sion ( #b cwt4 ,! effici5cy ( ! clamps dep5ds pr9cipally ^u friction1 aid$ d\btless by a slight cru%+ ( ! wood1 :i* br+s ! surfaces into p]fect contact4 #dag4 ,^! slips ( p9e unit$ by ! clamps are possess$ ( /r5g? q suffici5t = ! exp]im5ts n[ to be describ$4 ,models ?us con/ruct$ h ! great advantage ( be+ erect$1 v>i$ or pull$ d[n1 ) ! utmo/ facility4 ,we h le>n$ t ! compressive /r5g?1 &1 /ill m1 ! t5sile /r5g? ( timb]1 is mu* great] ?an its transv]se /r5g?4 ,?is pr9ciple is l>gely us$ 9 ! arts ( con/ruction4 ,we endeav\r by means ( suitable comb9ations to turn transv]se =ces into =ces ( t5sion or compression1 & ?us /r5g!n our con/ructions4 ,we %all illu/rate ! mode ( do+ s by simple =ms ( framework4 ,,weight ,,su/a9$ ,,by ,,tie ,,& ,,/rut4 #dah4 ,we beg9 ) ! /udy ( a v simple contrivance1 repres5t$ 9 ,fig4 #eg4 ,a ,b is a rod ( p9e #bj long4 ,9 ! diagram x is repres5t$1 = simplicity1 imb$d$ at ! end ,a 9 ! support4 ,9 reality1 h[ev]1 x is clamp$ to ! support1 & ! same rem>k may be made ab\t "s o!r diagrams us$ 9 ?is lecture4 ,7 ,a ,b unsupport$ except at its end ,a1 x w\ld ( c\rse break :5 a weight ( #aj lbs4 0 susp5d$ at ,b1 z we h already f\nd 9 ,art4 #dad4 #dai4 ,we mu/ asc]ta9 :e!r ! transv]se =ce on ,a ,b _c be *ang$ into =ces ( t5sion & compression4 ,! tie ,b ,c is atta*$ by means ( clamps2 ,a ,b is su/a9$ by ?is tie2 x _c b5d d[nw>ds "u ! action ( ! weight ,w1 because we %\ld !n require to h on ! same base & on ! same side ( x two triangles hav+ _! cont]m9\s sides equal1 b ?is we "k f ,euclid 7,i4 #g7 is impossible4 ,h5ce ,b is support$1 & we f9d t #aab lbs4 may be safely susp5d$1 s t ! /r5g? is enorm\sly increas$4 ,9 fact ! transv]se =ce is *ang$ into a compressive =ce or ?ru/ d[n ,a ,b1 & a t5sile =ce on ,b ,c4 ,illu/ration3 ,,fig4 #eg4 #dbj4 ,! actual magnitudes ( ^! c be comput$4 ,draw ! p>allelogram ,c ,d ,e ,b2 if ,b ,d repres5t ! weight ,w1 x may be resolv$ into two =ces1--"o1 ,b ,c1 a =ce ( ext5sion on ! tie2 ! o!r1 ,b ,e1 a compressive =ce on ,a ,b1 :i* is !re=e a /rut4 ,h5ce ! =ces are proportional to ! sides ( ! triangle1 ,a ,b ,c4 ,9 ! pres5t case ,a ,b #bj1 ,a ,c #ah1 ,b ,c #bg2 !re=e1 :5 ,w is #aab lbs41 we calculate t ! =ce on ,a ,b is #abd lbs41 & on ,c ,b #afh lbs4 ,a ,b w\ld require ab\t #cjj lbs4 to cru% x1 & ,c ,b ab\t #b1jjj lbs4 to te> x asund]1 consequ5tly ! tie & /rut c support #a cwt4 ) ease4 ,if1 h[ev]1 ,w 7 increas$ to ab\t #bgj lbs41 ! =ce on ,a ,b w\ld become too great1 & fracture w\ld arise f ! collapse ( ?is /rut4 #dba4 ,:5 a /ructure is load$ up to ! break+ po9t ( "o "p1 x is prop] = economy t all ! o!r p>ts %\ld be s design$ t !y %all be z ne> z possible to _! break+ po9ts4 ,9 fact1 s9ce no?+ is /rong] ?an its weake/ "p1 any additional /r5g? :i* ! rema9+ p>ts may possess adds no /r5g? to ! :ole1 & is only s mu* mat]ial wa/$4 ,h5ce our /ructure w\ld be j z /rong1 & w\ld be m prop]ly design$ if ! section ( ,b ,c 7 r$uc$ to "o-fif?1 = ! tie w\ld !n break :5 ! t5sion ^u x am\nt$ to #djj lbs4 ,:5 ,w is #bgj lbs4 ! compression on ,a ,b is #cjj lbs41 & ! t5sion on ,b ,c is #dje lbs41 s t bo? tie & /rut atta9 _! break+ loads toge!r4 ,! pr9ciple ( duly apportion+ ! /r5g? ( ea* piece to ! load x has to c>ry1 involves ! ess5ce ( s\nd eng9e]+4 ,9 t greate/ ( me*anical feats1 ! con/ruction ( a mighty railway bridge across a wide span1 att5tion to ?is pr9ciple is ( vital importance4 ,su* a bridge has to be> ! occasional load ( a pass+ tra91 b x has always to support ! f> great] load ( ! bridge mat]ials4 ,!re is ?us e inducem5t to make ! weight ( ea* "p ( ! bridge z light z may be consi/5t ) safety4 ,a ,,bridge ,,) ,,two ,,/ruts4 #dbb4 ,we %all next exam9e ! /ructure ( a type ( bridge1 %[n 9 ,fig4 #eh4 ,illu/ration3 ,,fig4 #eh4 ,x consi/s ( two beams1 ,a ,b1 #d' long1 plac$ p>allel to ea* o!r at a di/ance ( #ce1 & support$ at ea* end2 !y are firmly clamp$ to ! supports1 & a roadway ( %ort pieces is laid ^u !m4 ,at ! po9ts ( trisection ( ! beams ,c1 ,d1 /ruts ,c ,f & ,d ,e are clamp$1 _! l[] ends be+ support$ by ! framework3 ^! /ruts are #b' long1 & !re are two ( !m support+ ea* ( ! beams4 ,! tray ,g is atta*$ by a *a9 to a /\t piece ( wood1 :i* re/s ^u ! roadway at ! c5tre ( ! bridge4 #dbc4 ,we %all fir/ det]m9e ! /r5g? ( ?is bridge by actual exp]im5t1 & !n we %all endeav\r to expla9 ! results 9 accordance ) me*anical pr9ciples4 ,we c obs]ve ! deflection ( ! bridge by ! ca!tomet] 9 ! mann] already describ$ 7,art4 #cfb74 ,by ?is means we %all asc]ta9 :e!r ! load has p]man5tly injur$ ! ela/icity ( ! /ructure 7,art4 #cfg74 ,we beg9 by te/+ ! deflection :5 a load is di/ribut$ uni=mly1 z ! weights are dispos$ 9 ! case ( ,fig4 #fb4 ,a cross is m>k$ ^u "o ( ! beams1 & is view$ 9 ! ca!tomet]4 ,we arrange #aa /one weights along ! bridge1 & ! ca!tomet] %[s t ! deflection is only #jji3 ! ela/icity ( ! bridge rema9s unalt]$1 = :5 ! weights are remov$ ! cross on ! beam returns to its orig9al position2 h5ce ! bridge is well able to be> ?is load4 #dbd4 ,we remove ! r[ ( weights f ! bridge & susp5d ! tray f ! roadway4 ,i take my place at ! ca!tomet] to note ! deflection1 :ile my assi/ant places weights ,h ,h on ! tray4 #a cwt4 be+ ! load1 ,i see t ! deflection am\nts to #jb2 ) #b cwt4 ! deflection rea*es #jdc2 & ! bridge breaks ) #bch lbs4 #dbe4 ,let u endeav\r to calculate ! additional /r5g? :i* ! /ruts h imp>t$ to ! bridge4 ,by ,table ,,xxiv4 we see t a rod #dj #je #je is brok5 by a load ( #ai lbs43 h5ce ! beams ( ! bridge w\ld h be5 brok5 by a load ( #ch lbs4 if _! ends _h be5 free4 ,z1 h[ev]1 ! ends ( ! beams _h be5 clamp$ d[n1 we le>n f ,art4 #daa t a d\ble load w\ld be necess>y4 ,we may1 h[ev]1 be confid5t t ab\t #hj lbs4 w\ld h brok5 ! unsupport$ bridge4 ,! /r5g? is1 !re=e1 increas$ ?reefold by ! /ruts1 = a load ( #bch lbs4 0 requir$ to produce fracture4 #dbf4 ,we might h anticipat$ ?is result1 because ! po9ts ,c & ,d be+ support$ by ! /ruts may be consid]$ z almo/ fix$ po9ts2 9 fact1 we see t ,c _c desc5d1 because ! triangle ,a ,c ,f is unalt]able1 & = a simil> reason ,d rema9s fix$3 ! beam breaks betwe5 ,c & ,d1 & ! =ce requir$ mu/ !re=e be suffici5t to break a beam support$ at ! po9ts ,c & ,d1 ^: ends are secur$4 ,b ,c ,d is "o-?ird ( ,a ,b1 & we h already se5 t ! /r5g? ( a beam is inv]sely z its l5g? 7,art4 #chh72 h5ce ! =ce requir$ to break ! beam :5 support$ by ! /ruts is ?ree times z l>ge z w\ld h be5 necess>y to break ! unsupport$ beam4 ,?us ! /r5g? ( ! bridge is expla9$4 #dbg4 ,z a load ( #bch lbs4 appli$ ne> ! c5tre is necess>y to break ?is bridge1 x foll[s f ! pr9ciple ( ,art4 #djh t a load ( ab\t d\ble ?is am\nt mu/ be plac$ uni=mly on ! roadway be=e x succumbs2 we c1 !re=e1 und]/& h[ a load ( #aa /one 0 easily borne 7,art #dbc7 )\t p]man5t injury to ! ela/icity ( ! /ructure4 ,if we take ! factor ( safety z #c1 we see t a bridge ( ! =m we h be5 consid]+ may c>ry1 z its ord9>y work+ load1 a f> great] weight ?an w\ld h cru%$ x if unsupport$ by ! /ruts & ) free ends4 #dbh4 ,! /r5g? ( ! bridge 9 ,fig4 #eh is great] 9 "s p>ts ?an 9 o!rs4 ,at ! po9ts ,c & ,d a maximum load c\ld be borne2 ! weake/ places on ! bridge are 9 ! middle po9ts ( ! segm5ts ,a ,c1 ,d ,c1 & ,d ,b4 ,! load appli$ by ! tray 0 pr9cipally borne at ! middle ( ,d ,c1 b ow+ to ! piece ( wood :i* su/a9$ ! *a9 be+ ab\t #ah long1 ! load 0 to "s ext5t di/ribut$4 ,! ?ru/ ^u ! /ruts is n s easy to calculate accurately4 ,t d[n ,c ,f = example mu/ be less ?an if ! "p ,c ,d 7 remov$1 & half ! load 7 susp5d$ f ,c4 ,! =ce 9 ?is case c be det]m9$ by pr9ciples already expla9$ 7,art4 #dbj74 ,a ,,bridge ,,) ,,f\r ,,/ruts4 #dbi4 ,! same pr9ciples t we h employ$ 9 ! con/ruction ( ! bridge ( ,fig4 #eh may be ext5d$ fur!r1 z %[n 9 ! diagram ( ,fig4 #ei4 ,illu/ration3 ,,fig4 #ei4 ,we h "h two horizontal rods1 #dh #je #je1 ea* end be+ secur$ to ! supports2 "o ( ^! rods is %[n 9 ! figure4 ,x is divid$ into five equal p>ts 9 ! po9ts ,b1 ,c1 ,c1 ,b4 ,we support ! rod 9 ^! f\r po9ts by /ruts1 ! o!r extremities ( :i* are fa/5$ to ! framework4 ,! po9ts ,b1 ,c1 ,c1 ,b are fix$1 z !y are su/a9$ by ! /ruts3 h5ce a weight susp5d$ f ,p1 :i* is to break ! bridge1 mu/ be suffici5tly /rong to break a piece ,c ,c1 :i* is secur$ at ! ends2 ! rod ,a ,a w\ld h be5 brok5 ) #ch lbs41 h5ce #aij lbs4 w\ld be necess>y to break ,c ,c4 ,!re is a simil> beam on ! o!r side ( ! bridge1 & !re=e to break ! bridge #chj lbs4 w\ld be necess>y1 b ?is =ce mu/ be appli$ exactly at ! c5tre ( ,c ,c2 & if ! weights be spread ov] any consid]able l5g?1 a heavi] load w be necess>y4 ,9 fact1 if ,i 7 to di/ribute ! weight uni=mly ov] ! di/ance ,c ,c1 x appe>s f ,art4 #djh t d\ble ! load w\ld be necess>y to produce fracture4 #dcj4 ,we %all n[ break ?is model4 ,i place #ah /one ^u x rang$ uni=mly1 & ! ca!tomet] tells me t ! bridge only deflects #ja1 & t its ela/icity is n injur$4 ,plac+ ! tray 9 position1 & load+ ! bridge by ?is means1 ,i f9d ) a weight ( #b cwt4 t !re is a deflection ( #jae2 ) #d cwt4 ! deflection am\nts to #j'gb4 ,we !re=e inf] t ! bridge is beg9n+ to yield1 & ! clamps give way :5 ! load is increas$ to #ejj lbs4 ,a ,,bridge ,,) ,,two ,,ties4 #dca4 ,x might happ5 t circum/ances w\ld n make x conv5i5t to obta9 po9ts ( support bel[ ! bridge on :i* to erect ! /ruts4 ,9 su* a case1 if suitable positions = ties c be obta9$1 a bridge ( ! =m repres5t$ 9 ,fig4 #fj may be us$4 ,a ,d is a horizontal rod ( p9e #dj #je #je2 x is trisect$ 9 ! po9ts ,b & ,c1 f :i* po9ts ! ties ,b ,e & ,c ,f are secur$ to ! upp] p>ts ( ! framework4 ,a ,d is !n support$ 9 ! po9ts ,b & ,c1 :i* may !re=e be reg>d$ z fix$ po9ts4 ,h5ce1 = ! reasons we h already expla9$1 ! /r5g? ( ! bridge %\ld be increas$ ne>ly ?reefold4 ,rememb]+ t ! bridge has two beams we "k x w\ld require ab\t #gj lbs4 or #hj lbs4 to produce fracture )\t ! ties1 & !re=e we might expect t ov] #bjj lbs4 w\ld be necess>y :5 ! beams 7 support$ by ! ties4 ,i p]=m ! exp]im5t1 & y see ! bridge yields :5 ! load rea*es #aid lbs43 ?is is some:at less ?an ! am\nt we _h calculat$2 ! reason be+1 ,i ?9k1 t "o ( ! clamps slipp$ be=e fracture4 ,illu/ration3 ,,fig4 #fj4 ,a ,,simple ,,=m ,,( ,,truss4 #dcb4 ,x is (t5 n conv5i5t1 or ev5 possible1 to su/a9 a bridge by ! me?ods we h be5 consid]+4 ,x is desirable !re=e to inquire :e!r we _c arrange "s plan ( /r5g!n+ a beam1 by giv+ to x :at %all be equival5t to an increase ( dep?4 #dcc4 ,we %all only be able to describe "h "s v simple me?ods = do+ ?is4 ,sup]b examples are to be f\nd 9 railway bridges all ov] ! c\ntry1 b ! full inve/igation ( ^! complex /ructures is a problem ( no little difficulty1 & "o into :i* x w\ld be q beyond our prov9ce to ent]4 ,we %all1 h[ev]1 %[ h[ by a judici\s comb9ation ( sev]al p>ts a /ructure c (f] suffici5t resi/ance4 ,! mo/ complex lattice gird] is little m ?an a network ( ties & /ruts4 ,illu/ration3 ,,fig4 #fa4 #dcd4 ,let ,a ,b 7,fig4 #fa7 be a rod ( p9e #dj #je #je1 secur$ at ea* end4 ,we %all suppose t ! load is appli$ at ! two po9ts ,g & ,h1 9 ! mann] %[n 9 ! figure4 ,! load :i* a bridge mu/ be> :5 a tra9 passes ov] x is di/ribut$ ov] a di/ance equal to ! l5g? ( ! tra91 & ! weight ( ! bridge itself is ( c\rse arrang$ along ! entire span2 h5ce ! load :i* a bridge be>s is at all times m or less di/ribut$ & nev] entirely conc5trat$ at ! c5tre 9 ! mann] we h be5 consid]+4 ,9 ! pres5t exp]im5t we %all apply ! break+ load at ! two po9ts ,g & ,h1 z ?is w be a v>iation f ! mode we h latt]ly us$4 ,e ,f is an iron b> support$ 9 ! loops ,e ,g & ,f ,h4 ,let u fir/ try :at weight w break ! beam4 ,susp5d+ ! tray f ,e ,f1 ,i f9d t a load ( #dh lbs4 is suffici5t2 mu* less w\ld h done _h n ! ends be5 clamp$4 ,we h already appli$ a load 9 ?is mann] 9 ,art4 #djf4 #dce4 ,y obs]v$ t ! beam1 z usual1 deflect$ be=e x broke2 if we c\ld prev5t deflection we might reasonably expect to increase ! /r5g?4 ,?us if we support ! c5tre ( ! beam ,c1 deflection w\ld be prev5t$4 ,?is c be done v simply4 ,we clamp ! pieces ,d ,a1 ,d ,b1 ,d ,c1 on a simil> beam1 & x is evid5t t ,c _c desc5d s long z ! jo9ts at ,a1 ,b1 ,d1 ,c rema9 firmly secur$4 ,we n[ f9d t ev5 ) a weight ( #aab lbs4 9 ! tray1 ! b> is unbrok54 ,an arrangem5t ( ?is k9d is frequ5tly employ$ 9 eng9e]+1 = x seems to be able to be> m ?an d\ble ! load :i* is suffici5t to break ! unsupport$ beam4 ,illu/ration3 ,,fig4 #fb4 #dcf4 ,two frames ( ?is k9d1 ) a roadway laid betwe5 !m1 w\ld =m a bridge1 or if ! frames 7 turn$ upside d[n !y w\ld answ] equally well1 ?\gh ( c\rse 9 ?is case ,d ,a & ,d ,b w\ld become ties1 & ,d ,c a /rut1 b a bett] arrangem5t = a bridge w be next describ$4 ,,! ,,wye ,,bridge4 #dcg4 ,an in/ructive bridge 0 erect$ by ! late ,sir ,i4 ,brunel ov] ! ,wye1 = ! purpose ( c>ry+ a railway4 ,! ess5tial p>ts ( ! bridge are repres5t$ 9 ! model %[n 9 ,fig4 #fb1 :i* z be=e is made ( slips ( p9e clamp$ toge!r4 #dch4 ,our model is compos$ ( two simil> frames1 "o ( :i* we %all describe1 ,a ,b is a rod ( p9e #dh #je #je1 support$ at ea* extremity4 ,?is rod is su/a9$ at its po9ts ( trisection ,d1 ,c by ! uprights ,d ,e & ,c ,f1 :ile ,e & ,f are support$ by ! rods ,b ,e1 ,f ,e1 & ,a ,f2 ! rectangle ,d ,e ,f ,c is /iff5$ by ! piece ,c ,e1 & x w\ld be prop] 9 an actual /ructure to h a piece connect+ ,d & ,f1 b x has n be5 introduc$ into ! model4 ,illu/ration3 ,,fig4 #fc4 #dci4 ,we %all und]/& ! use ( ! diagonal ,c ,e by an inspection ( ,fig4 #fc4 ,suppose ! quadrilat]al ,a ,b ,c ,d be =m$ ( f\r pieces ( wood h+$ at ! corn]s4 ,x is evid5t t ?is quadrilat]al c be de=m$ by press+ ,a & ,c toge!r1 or by pull+ !m asund]4 ,ev5 if !re 7 actual jo9ts at ! corn]s1 x w\ld be almo/ impossible to make ! quadrilat]al /iff by ! /r5g? ( ! jo9ts4 ,y see ?is by ! frame :i* ,i hold 9 my h&2 ! pieces are clamp$ toge!r at ! corn]s1 b no matt] h[ tightly ,i compress ! clamps1 ,i am able ) ! slighte/ ex]tion to de=m ! figure4 #ddj4 ,we mu/ !re=e look = "s me?od ( /iff5+ ! frame4 ,i h "h a triangle ( ?ree pieces1 :i* h be5 simply clamp$ toge!r at ! corn]s2 ?is triangle is unalt]able 9 =m2 9 fact1 s9ce x is impossible to make two diff]5t triangles ) ! same ?ree sides1 x is evid5t ! triangle _c be de=m$4 ,?is po9ts to a guid+ pr9ciple 9 all bridgework4 ,! quadrilat]al is n /iff because innum]able diff]5t quadrilat]als c be made ) ! same f\r sides4 ,b if we draw ! diagonal ,a ,c ( ! quadrilat]al x is divid$ into two triangles1 & h5ce :5 we atta* to ! quadrilat]al1 :i* has be5 clamp$ at ! f\r corn]s1 an additional piece 9 ! direction ( "o ( ! diagonals1 x becomes unalt]able 9 %ape4 #dda4 ,9 ,fig4 #fc we h drawn ! two diagonals ,a ,c & ,b ,d3 "o w\ld be !oretically suffici5t1 b x is desirable to h bo?1 & = ! foll[+ reason4 ,if ,i pull ,a & ,c ap>t1 ,i /ret* ! diagonal ,a ,c & compress ,b ,d4 ,if ,i compress ,a & ,c toge!r1 ,i compress ! l9e ,a ,c & ext5d ,b ,d2 h5ce 9 "o ( ^! cases ,a ,c is a tie1 & 9 ! o!r x is a /rut4 ,x !re=e foll[s t 9 all cases "o ( ! diagonals is a tie1 & ! o!r a /rut4 ,if !n we h only "o diagonal1 x is call$ ^u to p]=m alt]nately ! functions ( a tie & ( a /rut4 ,?is is n desirable1 because x is evid5t t a piece :i* may act p]fectly z a tie may be v unsuitable = a /rut1 & vice v]s4 ,b if we ins]t bo? diagonals we may make bo? ( !m ties1 or bo? ( !m /ruts1 & ! frame mu/ be rigid4 ,?us = example1 ,i might make ,a ,c & ,b ,d sl5d] b>s ( wr\ght iron1 :i* =m admirable ties1 ?\gh q incapable ( act+ z /ruts4 #ddb4 ,:at we h said ) ref]5ce to ! necessity = divid+ a quadrilat]al figure into triangles applies /ill m to a polygon ) a l>ge numb] ( sides1 & we may lay d[n ! g5]al pr9ciple t e su* piece ( framework %\ld be compos$ ( triangles4 #ddc4 ,return+ to ,fig4 #fb1 we see ! reason :y ! rectangle ,e ,d ,c ,f %\ld h "o or bo? ( its diagonals introduc$4 ,a load plac$1 = example1 at ,d w\ld t5d to depress ! piece ,d ,e1 & ?us de=m ! rectangle1 b :5 ! diagonals are introduc$ ?is de=mation is impossible4 #ddd4 ,h5ce "o ( ^! frames is almo/ z /rong z a beam support$ at ! po9ts ,c & ,d1 & !re=e1 f ! pr9ciples ( ,art4 #chh1 its /r5g? is ?ree times z great z t ( an unsupport$ beam4 #dde4 ,! two frames plac$ side by side & c>ry+ a roadway =m an admirable bridge1 q indep5d5t ( any ext]nal support1 except t giv5 by ! pi]s ^u :i* ! extremities ( ! frames re/4 ,x w\ld be prop] to connect ! frames toge!r by means ( braces1 :i* are n1 h[ev]1 %[n 9 ! figure4 ,! model is repres5t$ z c>ry+ a uni=m load 9 contradi/9ction to ,fig4 #eh1 :]e ! weight is appli$ at a s+le po9t4 #ddf4 ,) eight /one rang$ along x1 ! bridge ( ,fig4 #fb did n indicate an appreciable deflection4 ,,lecture ,,xiv4 ,,! ,,me*anics ,,( ,a ,,bridge4 ,introduction4--,! ,gird]4--,! ,tubul> ,bridge4--,! ,susp5sion ,bridge4 ,,introduction4 #ddg4 ,p]haps x may be ?\ght t ! /ructures we h be5 lately consid]+ are n ^? :i* are mo/ univ]sally us$1 & t ! bridges :i* are g5]ally ref]r$ to z monum5ts ( eng9e]+ skill are ( q a diff]5t con/ruction4 ,e "o is famili> ) ! ar*1 & mo/ ( u h se5 susp5sion bridges & ! celebrat$ ,m5ai tube4 ,we mu/ !re=e allude fur!r to "s ( ^! /ructures1 & ?is we propose to d 9 ! pres5t lecture4 ,x w only be possible to take a v slight survey ( an ext5sive subject to :i* elaborate treatises h be5 devot$4 ,we %all fir/ give a brief acc\nt ( ! use ( iron 9 ! arts ( con/ruction4 ,we %all !n expla9 simply ! pr9ciple ( ! tubul> bridge1 & also ( ! susp5sion bridge4 ,! m complex =ms are beyond our scope4 ,,! ,,gird]4 #ddh4 ,a horizontal beam support$ at ea* end1 & p]haps at int]m$iate po9ts1 & design$ to support a heavy load is call$ a gird]4 ,^? rods ^u :i* we h p]=m$ exp]im5ts1 ! results ( :i* h be5 giv5 9 ,table ,,xxiv41 are small gird]s2 b ! t]m is g5]ally und]/ood to relate to /ructures ( iron3 ! greate/ gird]s = railway bridges are made ( b>s or plates ( iron rivet$ toge!r4 #ddi4 ,we %all fir/ consid] ! application ( ca/ iron to gird]s1 & %[ :at =m !y %\ld assume4 #dej4 ,a beam ( ca/ iron1 suppos+ its section to be rectangul>1 has its /r5g? det]m9$ by ! same laws z ! beams ( p9e4 ,?us1 suppos+ ! section ( two beams to be ! same1 _! /r5g?s are inv]sely proportional to _! l5g?s1 & ! /r5g? ( a beam plac$ edgewise is to its /r5g? plac$ flatwise 9 ! proportion ( ! great] dim5sion ( its section to ! less dim5sion4 ,^! laws det]m9e ! /r5g? ( e rectangul> beam ( ca/ iron :5 t ( "o beam is kn[n1 & we mu/ p]=m an exp]im5t 9 ord] to f9d ! break+ load 9 a p>ticul> case4 #dea4 ,i take "h a piece ( ca/ iron1 :i* is #b' long1 & #je #je 9 section4 ,i support ?is beam at ea* end ^u a frame2 ! di/ance betwe5 ! supports is #bj4 ,i atta* ! tray to ! c5tre ( ! beam & load x ) weights4 ,! ends ( ! beam re/ freely ^u ! supports1 b ,i h tak5 ! precaution ( ty+ ea* end by a piece ( wire1 s t !y may n fly ab\t :5 ! fracture occurs4 ,load+ ! tray1 ,i f9d t ) #bhj lbs4 ! cra% comes4 #deb4 ,let u comp>e ?is result ) ,no4 #h ( ,table ,,xxiv4 7p4 #aij74 ,!re we f9d t a piece ( p9e1 ! same size z ! ca/ iron1 0 brok5 ) #cf lbs43 ! ratio ( #bhj to #cf is ne>ly #h1 s t ! beam ( ca/ iron is ab\t #h times z /rong z ! piece ( p9e ( ! same size4 ,?is result is a little l>g] ?an we w\ld h inf]r$ f an exam9ation ( tables ( ! /r5g? ( l>ge b>s ( ca/ iron2 ! reason may be t a v small ca/+1 su* z ?is b>1 is /rong] 9 proportion ?an a l>g] "o1 ow+ to ! iron n be+ s uni=m ?r\gh\t ! l>g] mass4 #dec4 ,i hold "h a b> ( ca/ iron #ab long & #a #a 9 section4 ,i h n suffici5t weights at h& to break x1 b we c compute h[ mu* w\ld be necess>y by our =m] exp]im5t4 #ded4 ,9 ! fir/ place a b> #ab long1 & #je #je ( section1 w\ld require #bj #bhj #ab #dfg lbs4 by ! law t ! /r5g? is inv]sely z ! l5g?4 ,we also "k t "o beam #ab #a #a is j z /rong z two beams #ab #a #je1 ea* plac$ edgewise2 ea* ( ^! latt] beams is twice z /rong z #ab #a #je plac$ flatwise1 because ! /r5g? :5 plac$ edgewise is to ! /r5g? :5 plac$ flatwise1 z ! dep? to ! bread?1 t is z #b to #a3 h5ce ! orig9al beam is f\r times z /rong z "o beam #ab #a #je plac$ flatwise3 b ?is la/ beam is twice z /rong z a beam #ab #je #je1 & h5ce we see t a beam #ab #a #a mu/ be #h times z /rong z a beam ( #ab #je #je1 b ?is la/ beam w\ld require a load ( #dfg lbs4 to break x1 & h5ce ! beam ( #ab #a #a w\ld require #dfg #h #cgcf lbs4 to produce fracture4 ,?is am\nts to m ?an a ton & a half4 #dee4 ,x is a rule sometimes useful to practical m5 t a ca/ iron b> "o foot long by "o in* squ>e w\ld break ) ab\t a ton weight4 ,if ! iron be ( ! same quality z t :i* we h us$1 ?is result is too small1 b ! error is on ! safe side2 ! real /r5g? w !n be g5]ally a little great] ?an ! /r5g? calculat$ f ?is rule4 ,:at we h said 7,art4 #djc7 ) ref]5ce to ! precaution = safety 9 b>s ( wood applies also to ca/ iron4 ,! load :i* ! beam has to be> 9 ord9>y practice %\ld only be a small fraction ( t :i* w\ld break x4 #def4 ,9 mak+ any description ( gird] x is desirable on v special gr\nds t z little mat]ial z possible be uselessly employ$4 ,x w ( c\rse be rememb]$ t a gird] has to support its own weight1 besides :atev] may be plac$ ^u x3 & if ! gird] be massive1 its own weight is a s]i\s item4 ,( two gird]s1 ea* capable ( be>+ ! same total load1 ! light]1 besides employ+ less mat]ial1 w be able to be> a great] weight plac$ ^u x4 ,x is !re=e = a d\ble reason desirable to dim9i% ! weight4 ,?is rem>k applies especially to su* a mat]ial z ca/ iron1 :i* c be at once giv5 ! =m 9 :i* x %all be capable ( (f]+ ! greate/ resi/ance4 #deg4 ,! pr9ciples :i* w guide u 9 asc]ta9+ ! prop] =m to give a ca/ iron gird]1 are easily d$uc$ f :at we h laid d[n 9 ,lectures ,,xi4 & ,,xii4 ,we h se5 t dep? is v desirable = a /rong beam4 ,if !re=e we /rive to atta9 great dep? 9 a light beam1 ! beam mu/ be v ?94 ,n[ an extremely ?9 beam w n be safe4 ,9 ! fir/ place x w\ld be flexible & liable to displacem5t sideways2 &1 9 ! second place1 !re is a /ill m fatal difficulty4 ,we h %[n t :5 a beam ( wood is support+ a weight1 ! fibres at ! bottom ( ! beam are ext5d$1 ! t5d5cy be+ to te> !m 7,art4 #cgf74 ,! fibres on ! top ( ! beam are compress$1 :ile ! c5tre ( ! beam is 9 its natural /ate4 ,! condition ( /ra9 9 a ca/ iron beam is precisely simil>2 ! bottom portions are 9 a /ate ( ext5sion1 :ile ! top is compress$4 ,if !re=e a beam be v ?91 ! mat]ial at ! l[] "p may n be suffici5t to )/& ! =ces ( ext5sion1 & fracture is produc$4 ,to obviate ?is1 we /r5g!n ! bottom ( ! beam by plac+ extra mat]ial !re4 ,?us we are l$ to ! idea ( a ?9 beam ) an excess ( iron at ! bottom4 ,illu/ration3 ,,fig4 #fd4 #deh4 ,e ,f 7,fig4 #fd7 is ! ?9 iron beam along ! bottom ( :i* is ! /\t flange %[n at ,c ,d2 rupture _c comm5ce at ! bottom unless ?is flange be torn asund]2 = until ?is happ5s x is cle> t fracture _c beg9 to attack ! upp] & sl5d] "p ( ! beam ,e ,f4 #dei4 ,b ! beam is 9 a /ate ( compression along its upp] side1 j z 9 ! wood5 beams :i* we h already consid]$4 ,if !re=e ! upp] p>ts 7 n p[]ful en\gh to resi/ ?is compression1 !y w\ld be cru%$1 & ! beam w\ld give way4 ,! rem$y = ?is s\rce ( weakness is obvi\s2 a second flange runs along ! top ( ! beam1 z %[n at ,a ,b4 ,if ?is be /rong en\gh to resi/ ! compression1 ! /ability ( ! beam is ensur$4 #dfj4 ,! upp] flange is made v mu* small] ?an ! l[] "o1 9 consequ5ce ( a prop]ty ( ca/ iron4 ,?is metal is m capable ( resi/+ =ces ( compression ?an =ces ( ext5sion1 & x is only necess>y to use "o-six? ( ! iron on ! upp] flange t is requir$ = ! l[]4 ,:5 ! section has be5 ?us proportion$1 ! beam is equally /rong at bo? top & bottom2 add+ mat]ial to ei!r flange )\t /r5g!n+ ! o!r1 w n b5efit ! gird]1 b w r prove a s\rce ( weakness1 by increas+ ! weight :i* has to be support$4 #dfa4 ,i h "h a small gird] made ( :at we are famili> ) "u ! "n ( 8t910 b :i* is ( c\rse %eet iron ?9ly cov]$ ov] ) t94 ,x has ! %ape %[n 9 ,fig4 #fd1 & x is #ab long4 ,i support x at ea* end1 & y see x be>s two hundr$ weight )\t app>5t deflection4 ,,! ,,tubul> ,,bridge4 #dfb4 ,i %all comm5ce ! description ( ! pr9ciple ( ?is bridge by p]=m+ "s exp]im5ts ^u a tube1 :i* ,i hold 9 my h&4 ,! tube is squ>e1 #a #a 9 section1 & #ch long4 ,x is made ( 8t910 & weighs r less ?an a p\nd4 #dfc4 ,"h is a solid rod ( iron ( ! same l5g? z ! tube1 b conta9+ consid]ably m metal4 ,?is is easily v]ifi$ by weigh+ ! tube & ! rod "o aga9/ ! o!r4 ,i %all reg>d !m z two gird]s1 & exp]im5t ^u _! /r5g?1 & we %all f9d t1 ?\gh ! tube conta9s less sub/ance ?an ! rod1 x is mu* ! /rong]4 #dfd4 ,i place ! rod on a pair ( supports ab\t #c' ap>t2 ,i !n atta* ! tray to ! middle ( ! rod3 #ad lbs4 produce a deflection ( #jea1 & #db lbs4 b5ds d[n ! rod ?r\gh #cah4 ,?is is a l>ge deflection2 & :5 ,i remove ! load1 ! rod only returns ?r\gh #agh1 ?us %[+ t a p]man5t deflection ( #adj is produc$4 ,?is proves t ! rod is greatly injur$1 & demon/rates its unsuitability = a gird]4 #dfe4 ,next we place ! tube ^u ! same supports1 & treat x 9 ! same mann]4 ,a load ( #ef lbs4 only produces a deflection ( #jji1 &1 :5 ?is load is remov$1 ! tube returns to its orig9al position3 ?is is %[n by ! ca!tomet]1 = a cross is m>k$ on ! tube1 & ,i br+ ! image ( x on ! horizontal wire ( ! telescope be=e ! load ( #ef lbs4 is plac$ 9 ! tray4 ,:5 ! load is remov$1 ,i see t ! cross returns exactly to :]e x 0 be=e1 ?us prov+ t ! ela/icity ( ! tube is unimpair$4 ,we d\ble ! load1 ?us plac+ #a cwt4 9 ! tray1 ! deflection only rea*es #jbf1 &1 :5 ! load is remov$1 ! tube is f\nd to be p]man5tly deflect$ by a quantity1 at all ev5ts n great] ?an #jjjd2 h5ce we le>n t ! tube be>s easily & )\t injury a load m ?an twice z great z t :i* practically de/roy$ a rod ( wr\ght iron1 conta9+ m iron ?an ! tube4 ,we load ! tube /ill fur!r by plac+ additional weights 9 ! tray1 & ) #adj lbs4 ! tube breaks2 ! fracture has occurr$ at a jo9t :i* 0 sold]$1 & ! real break+ /r5g? ( ! tube1 _h x be5 cont9u\s1 is d\btless f> great]4 ,en\gh1 h[ev]1 has be5 borne to %[ ! increase ( /r5g? obta9$ by ! tubul> =m4 #dff4 ,we c expla9 ! reason ( ?is rem>kable result by means ( ,fig4 #fd4 ,7 ! ?9 portion ( ! gird] ,e ,f made ( two p>ts plac$ side by side1 ! /r5g? w\ld n be alt]$4 ,if we !n imag9e ! flange ,a ,b wid5$ to ! wid? ( ,c ,d1 & ! two p>ts :i* =m ,e ,f op5$ out s z to =m a tube1 ! /r5g? ( ! gird] is /ill reta9$ 9 its modifi$ =m4 #dfg4 ,a tube ( rectangul> section has ! advantage ( great] dep? ?an a solid rod ( ! same weight2 & if ! bottom ( ! tube be /rong en\gh to resi/ ! ext5sion1 & ! top /rong en\gh to resi/ ! compression1 ! gird] w be /iff & /rong4 #dfh4 ,9 ! ,m5ai ,tubul> ,bridge1 :]e a gigantic tube support$ at ea* end bridges ov] a span ( f\r hundr$ & sixty feet1 special arrangem5ts h be5 made = /r5g!n+ ! top4 ,x is =m$ ( cells1 z wr\ght iron dispos$ 9 ?is way is especially adapt$ = resi/+ compression4 #dfi4 ,we h only spok5 ( rectangul> tubes1 b x is equally true = tubes ( circul> or o!r sections t :5 suitably con/ruct$ !y are /rong] ?an ! same quantity ( mat]ial1 if made into a solid rod4 #dgj4 ,we f9d ?is pr9ciple 9 nature2 bones & quills are (t5 f\nd to be holl[ 9 ord] to comb9e lightness ) /r5g?1 & ! /alks ( :eat & o!r plants are tubul> = ! same reason4 ,,! ,,susp5sion ,,bridge4 #dga4 ,:]e a great span is requir$1 ! susp5sion bridge possesses _m advantages4 ,x is light] ?an a gird] bridge ( ! same span1 & consequ5tly *eap]1 :ile its s+ul> elegance contra/s v fav\rably ) ! appe>ance ( m solid /ructures4 ,on ! o!r h&1 a susp5sion bridge is n s well suit$ = railway traffic z ! lattice gird]4 #dgb4 ,! me*anical "* ( ! susp5sion bridge is simple4 ,if a rope or a *a9 be susp5d$ f two po9ts to :i* its ends are atta*$1 ! *a9 hangs 9 a c]ta9 curve kn[n to ma!maticians z ! cat5>y4 ,! =m ( ! cat5>y v>ies ) ! l5g? ( ! rope1 b x w\ld n be possible to make ! *a9 lie 9 a /raight l9e betwe5 ! two po9ts ( support1 = reasons po9t$ out 9 ,art4 #bj4 ,no matt] h[ great be ! =ce appli$1 x w /ill be concave4 ,:5 ! *a9 is /ret*$ until ! depression 9 ! middle is small comp>$ ) ! di/ance betwe5 ! po9ts ( support1 ! curve ?\gh always a cat5>y1 has a v close resemblance to ! p>abola4 #dgc4 ,9 ,fig4 #fe a model ( a susp5sion bridge is %[n4 ,! two *a9s are fix$ "o on ea* side at ! po9ts ,e & ,f2 !y !n pass ov] ! pi]s ,a1 ,d1 & bridge a span ( n9e feet4 ,! v]tical l9e at ! c5tre ,b ,c %[s ! greate/ am\nt by :i* ! *a9 has deflect$ f ! horizontal ,a ,d4 ,:5 ! deflection ( ! middle ( ! *a9 is ab\t "o-t5? "p ( ,a ,d1 ! curve ,a ,c ,d becomes = all practical purposes a p>abola4 ,! roadway is susp5d$ by sl5d] iron rods f ! *a9s1 ! l5g?s ( ! susp5sion rods be+ s regulat$ z to make x ne>ly horizontal4 #dgd4 ,! roadway 9 ! model is lad5 ) #h /one weights4 ,we h di/ribut$ !m 9 ?is mann] 9 ord] to repres5t ! p]man5t load :i* a great susp5sion bridge has to c>ry4 ,! s]ies ( weights ?us arrang$ produces sub/antially ! same effect z if x 7 actually di/ribut$ uni=mly along ! l5g?4 ,9 a real susp5sion bridge ! weight ( ! *a9 itself adds greatly to ! t5sion4 #dge4 ,we assume t ! *a9 hangs 9 ! =m ( a p>abola1 & t ! load is uni=mly rang$ along ! bridge4 ,! t5sion ^u ! *a9s is greate/ at _! highe/ po9ts1 & lea/ at _! l[e/ po9ts1 ?\gh ! diff]5ce is small4 ,! am\nt ( ! t5sion c be calculat$ :5 ! load1 span1 & deflection are kn[n4 ,we _c give ! /eps ( ! calculation1 b we %all enunciate ! result4 ,illu/ration3 ,,fig4 #fe4 #dgf4 ,! magnitude ( ! t5sion at ! l[e/ po9t ,c ( ea* *a9 is f\nd by multiply+ ! total weight 7includ+ *a9s1 susp5sion rods1 & roadway7 by ! span1 & divid+ ! product by sixte5 times ! deflection4 ,! t5sion ( ! *a9 at ! highe/ po9t ,a exce$s t at ! l[e/ po9t ,c1 by a weight f\nd by multiply+ ! total load by ! deflection1 & divid+ ! product by twice ! span4 #dgg4 ,! total weight ( roadway1 *a9s1 & load 9 ! model is #abj lbs42 ! deflection is #aj1 ! span #ajh2 ! product ( ! weight & span is #ab1ifj2 sixte5 times ! deflection is #afj2 &1 !re=e1 ! t5sion at ! po9t ,c is f\nd1 by divid+ #ab1ifj by #afj1 to be #ha lbs4 ,to f9d ! t5sion at ! po9t ,a1 we multiply #abj by #aj1 & divide ! product by #baf2 ! quoti5t is ne>ly #f4 ,?is add$ to #ha lbs4 gives #hg lbs4 = ! t5sion on ! *a9 at ,a4 #dgh4 ,"o *a9 ( ! model is atta*$ to a spr+ balance at ,a2 by ref]5ce to ! scale we see ! t5sion indicat$ to be #ij lbs43 a suffici5tly close approximation to ! calculat$ t5sion ( #hg lbs4 #dgi4 ,a l>ge susp5sion bridge has its *a9s /ra9$ by an enorm\s =ce4 ,x is !re=e necess>y t ! ends ( ^! *a9s be v firmly secur$4 ,a good atta*m5t is obta9$ by an*or+ ! *a9 to a l>ge iron an*or imb$d$ 9 solid rock4 #dhj4 ,9 ,art4 #de we h po9t$ out h[ ! dim5sions ( ! tie rod c\ld be det]m9$ :5 ! t5sion 0 kn[n4 ,simil> consid]ations w enable u to calculate ! size ( ! *a9 necess>y = a susp5sion bridge :5 we h asc]ta9$ ! t5sion to :i* x w be subject$4 #dha4 ,we c easily det]m9e by trial :at effect is produc$ on ! t5sion ( ! *a91 by plac+ a weight ^u ! bridge 9 addition to ! p]man5t load4 ,?us an additional /one weight 9 ! c5tre raises ! t5sion ( ! spr+ balance to #ajj lbs42 ( c\rse ! t5sion 9 ! o!r *a9 is ! same3 & ?us we f9d a weight ( #ad lbs4 has produc$ additional t5sions ( #aj lbs4 ea* 9 ! two *a9s4 ,) a weight ( #bh lbs4 at ! c5tre we f9d a /ra9 ( #aaj lbs4 on ! *a94 #dhb4 ,^! additional weights may be reg>d$ z analog\s to ! weights ( ! vehicles :i* ! susp5sion bridge is requir$ to c>ry4 ,9 a l>ge susp5sion bridge ! t5sion produc$ by ! pass+ loads is only a small fraction ( ! p]man5t load4 ,,lecture ,,xv4 ,,! ,,motion ,,( ,a ,,fall+ ,,body4 ,introduction4--,! ,fir/ ,law ( ,motion4--,! ,exp]im5t ( ,galileo f ! ,t[] ( ,pisa4--,! ,space is proportional to ! ,squ>e ( ! ,"t4--,a ,body falls #af' 9 ! ,fir/ ,second4--,! ,action ( ,gravity is indep5d5t ( ! ,motion ( ! ,body4--,h[ ! ,=ce ( ,gravity is def9$4--,! ,pa? ( a ,projectile is a ,p>abola4 ,,introduction4 #dhc4 ,k9etics is t bran* ( me*anics :i* treats ( ! action ( =ces 9 ! production ( motion4 ,we %all f9d x r m difficult ?an ! subjects ) :i* we h be5 hi!rto occupi$2 ! difficulties 9 k9etics arise f ! introduction ( ! elem5t ( "t1 into our calculations4 ,! pr9ciples ( k9etics 7 unkn[n to ! anci5ts4 ,galileo discov]$ "s ( its tru?s 9 ! sev5te5? c5tury2 &1 s9ce 8 "t1 ! sci5ce has gr[n rapidly4 ,! motion ( a fall+ body 0 fir/ correctly appreh5d$ by ,galileo2 & ) ?is subject we c appropriately comm5ce4 ,,! ,,fir/ ,,law ,,( ,,motion4 #dhd4 ,velocity1 9 ord9>y language1 is suppos$ to convey a notion ( rapid motion4 ,su* is n precisely ! mean+ ( ! ^w 9 me*anics4 ,by velocity is m]ely meant ! rate at :i* a body moves1 :e!r ! rate be fa/ or be sl[4 ,?is rate is mo/ conv5i5tly measur$ by ! numb] ( feet mov$ ov] 9 "o second4 ,h5ce :5 x is said ! velocity ( a body is #be1 x is meant t if ! body cont9u$ to move = "o second ) its velocity unalt]$1 x w\ld 9 t "t h mov$ ov] #be feet4 ,illu/ration3 ,,fig4 #ff4 #dhe4 ,! fir/ law ( motion may be /at$ ?us4 ,if no =ce act ^u a body1 x w1 if at re/1 rema9 = ev] at re/2 or if 9 motion1 x w cont9ue = ev] to move ) a uni=m velocity4 ,we "k ?is law to be true1 & yet no "o has ev] se5 x to be true = ! simple reason t we _c realise ! condition :i* x requires4 ,we _c place a body 9 ! condition ( be+ unact$ ^u by any =ces4 ,b we may conv9ce ourselves ( ! tru? ( ! law by "s su* reason+ z ! foll[+4 ,if a /one be ?r[n along ! road1 x soon comes to re/4 ,! body leaves ! h& ) a c]ta9 initial velocity & is n fur!r act$ ^u by x4 ,h5ce1 if no o!r =ce act$ on ! /one1 we %\ld expect1 if ! fir/ law be true1 t x w\ld cont9ue to run on = ev] ) ! orig9al velocity at ! mom5t ( leav+ ! h&4 ,b o!r =ces d act ^u ! /one2 ! attraction ( ! e>? pulls x d[n2 &1 :5 x beg9s to b\nd & roll ^u ! gr\nd1 friction comes into op]ation1 deprives ! /one ( its velocity1 & br+s x to re/4 ,b let ! /one be ?r[n ^u a surface ( smoo? ice2 :5 x beg9s to slide1 ! =ce ( gravity is c\nt]act$ by ! reaction ( ! ice3 !re is no o!r =ce act+ ^u ! /one except friction1 :i* is small4 ,h5ce we f9d t ! /one w run on = a consid]able di/ance4 ,x requires b little ef=t ( ! imag9ation to suppose a lake ^: surface is an inf9ite plane1 p]fectly smoo?1 & t ! /one is p]fectly smoo? also4 ,9 su* a case z ?is ! fir/ law ( motion am\nts to ! ass]tion t ! /one w\ld nev] /op4 #dhf4 ,we may1 9 ! lecture room1 see ! tru? ( ?is law v]ifi$ to a c]ta9 ext5t by ,atwood's ma*9e 7,fig1 #ff74 ,?is ma*9e has be5 devis$ = ! purpose ( inve/igat+ ! laws ( motion by actual exp]im5t4 ,x consi/s pr9cipally ( a pulley ,c1 m\nt$ s t its axle re/s ^u two pairs ( :eels1 z %[n 9 ! figure2 x be+ ! object ( ?is contrivance to enable ! :eel to revolve ) ! utmo/ fre$om4 ,a pair ( equal weights ,a1 ,b1 are atta*$ by a silk5 ?read1 :i* passes ov] ! pulley2 ea* ( ! weights is c\nt]balanc$ by ! o!r3 s t :5 ! two are 9 motion1 we may consid] ei!r z a body n act$ ^u by any =ces1 & x w be f\nd t x moves uni=mly1 z f> z ! size ( ! app>atus w p]mit4 #dhg4 ,if we try to conceive a body free 9 space1 & n act$ ^u by any =ce1 x is m natural to suppose t su* a body1 :5 once />t$1 %\ld g on mov+ uni=mly = ev]1 ?an t its velocity %\ld be alt]$4 ,! true pro( ( ! fir/ law ( motion is1 t all consequ5ces prop]ly d$uc$ f x1 9 comb9ation ) o!r pr9ciples1 are f\nd to be v]ifi$4 ,a/ronomy pres5ts u ) ! be/ examples4 ,! calculation ( ! "t ( an eclipse is bas$ ^u laws :i* 9 !mselves assume ! fir/ law ( motion2 h5ce1 :5 we inv>iably f9d t an eclipse occurs precisely at ! mom5t = :i* x has be5 pr$ict$1 we h a spl5did pro( ( ! sublime tru? :i* ! fir/ law ( motion expresses4 ,,! ,,exp]im5t ,,( ,,galileo ,,f ,,! ,,t[] ,,( ,,pisa4 #dhh4 ,! contra/ betwe5 heavy bodies & light bodies is s m>k$ t )\t trial we h>dly believe t a heavy body & a light body w fall f ! same height 9 ! same "t4 ,t !y d s ,galileo prov$ by dropp+ a heavy ball & a light ball toge!r f ! top ( ! ,lean+ ,t[] at ,pisa4 ,!y 7 f\nd to rea* ! gr\nd simultane\sly4 ,we %all repeat ?is exp]im5t on a scale suffici5tly r$uc$ to correspond ) ! dim5sions ( ! lecture room4 ,illu/ration3 ,,fig4 #fg4 #dhi4 ,! app>atus us$ is %[n 9 ,fig4 #fg4 ,x consi/s ( a /\t framework support+ a pulley ,h at a height ( ab\t #bj feet above ! gr\nd4 ,?is pulley c>ries a rope2 "o end ( ! rope is atta*$ to a triangul> piece ( wood1 to :i* two electro-magnets ,g are fa/5$4 ,! electro-magnet is a piece ( iron 9 ! =m ( a horse-%oe1 ar\nd :i* is coil$ a long wire4 ,! horse-%oe becomes a magnet imm$iately an electric curr5t passes ?r\gh ! wire2 x rema9s a magnet z long z ! curr5t passes1 & returns to its orig9al condition ! mom5t ! curr5t ceases4 ,h5ce1 if ,i h ! means ( controll+ ! curr5t1 ,i h complete control ( ! magnet2 y see ?is ball ( iron rema9s atta*$ to ! magnet z long z ! curr5t passes1 b drops ! in/ant ,i break ! curr5t4 ,! same electric circuit includes bo? ! magnets2 ea* ( !m w hold up an iron ball ,f :5 ! curr5t passes1 b ! mom5t ! curr5t is brok5 bo? balls w be releas$4 ,electricity travels along a wire ) prodigi\s velocity4 ,x w\ld pass ov] _m ?\s&s ( miles 9 a second2 h5ce ! "t t x takes to pass ?r\gh ! wires we are employ+ is q inappreciable4 ,a piece ( ?9 pap] int]pos$ betwe5 ! magnets & ! balls w ensure t !y are dropp$ simultane\sly2 :5 ?is precaution is n tak5 "o or bo? balls may hesitate a little be=e comm5c+ to desc5d4 ,a long pair ( wires ,e1 ,b1 mu/ be atta*$ to ! magnets1 ! o!r ends ( ! wires communicat+ ) ! batt]y ,d2 ! triangle & its load is hoi/$ up by means ( ! rope & pulley & ! magnets ?us c>ry ! balls to a height ( #bj feet3 ! balls we are us+ weigh ab\t #jbe lb4 & #a lb4 respectively4 #dij4 ,we are n[ ready to p]=m ! exp]im5t4 ,i break ! circuit2 ! two balls are dis5gag$ simultane\sly2 !y fall side by side ! :ole way1 & rea* ! gr\nd toge!r1 :]e x is well to place a cu%ion to receive !m4 ,?us y see ! heavy ball & ! light "o ea* require ! same am\nt ( "t to fall f ! same height4 #dia4 ,b ^! balls are bo? ( iron2 let u comp>e toge!r balls made ( diff]5t sub/ances1 iron & wood = example4 ,a flat-head$ nail is driv5 into a wood5 ball ( ab\t #be 9 diamet]1 & by means ( ! iron 9 ! nail ,i c susp5d ?is ball f "o ( ! magnets2 :ile ei!r ( ! iron balls we h already us$ hangs f ! o!r4 ,i repeat ! exp]im5t 9 ! same mann]1 & y see !y also fall toge!r4 ,f9ally1 :5 an iron ball & a cork ball are dropp$1 ! latt] is )9 two or ?ree in*es ( its weighty companion :5 ! cu%ion is rea*$3 ?is small diff]5ce is due to ! great] effect ( ! resi/ance ( ! air on ! light] ( ! two bodies4 ,!re c be no d\bt t 9 a vacuum all bodies ( :atev] size or mat]ial w\ld fall 9 precisely ! same "t4 #dib4 ,h[ is ! fact t all bodies fall 9 ! same "t to be expla9$8 ,let u fir/ consid] two iron balls4 ,take two equal p>ticles ( iron2 x is evid5t t ^! fall 9 ! same "t2 !y w\ld d s if !y 7 v close toge!r1 ev5 if !y 7 t\*+1 b !n !y might z well be 9 "o piece3 & ?us we %\ld f9d t a body consi/+ ( two or m iron p>ticles takes ! same "t to fall z "o 7omitt+ ( c\rse ! resi/ance ( ! air74 ,?us x appe>s mo/ reasonable t two balls ( iron1 ev5 ?\gh unequal 9 size1 %\ld fall 9 ! same "t4 #dic4 ,! case ( ! wood5 ball & ! iron ball w require m consid]ation be=e we realise ?or\ghly h[ mu* ,galileo's exp]im5t proves4 ,we mu/ fir/ expla9 ! mean+ ( ! ^w mass 9 me*anics4 #did4 ,x is n correct to def9e mass by ! introduction ( ! idea ( weight1 because ! mass ( a body is some?+ indep5d5t ( ! exi/5ce ( ! e>?1 :]eas weight is produc$ by ! attraction ( ! e>?4 ,x is true t weight is a conv5i5t means ( measur+ mass1 b ?is is only a consequ5ce ( ! prop]ty ( gravity :i* ! exp]im5t proves1 namely1 t ! attraction ( gravity = a body is proportional to its mass4 #die4 ,let u select z ! unit ( mass ! mass ( a piece ( plat9um :i* weighs #a lb4 at ,london2 x is !n evid5t t ! mass ( any o!r piece ( plat9um %\ld be express$ by ! numb] ( p\nds x conta9s3 b h[ are we to det]m9e ! mass ( "s o!r sub/ance1 su* z iron8 ,a piece ( iron is def9$ to h ! same mass z a piece ( plat9um1 if ! same =ce act+ on ei!r ( ! bodies = ! same "t produces ! same velocity4 ,?is is ! prop] te/ ( ! equality ( masses4 ,! mass ( any o!r piece ( iron w be repres5t$ by ! numb] ( times x conta9s a piece equal to t :i* we h j comp>$ ) ! plat9um2 simil>ly ( c\rse = o!r sub/ances4 #dif4 ,! magnitude ( a =ce act+ = ! "t unit is measur$ by ! product ( ! mass set 9 motion & ! velocity :i* x has acquir$4 ,?is is a tru? e/abli%$1 l ! fir/ law ( motion1 by indirect evid5ce4 #dig4 ,let u apply ^! pr9ciples to expla9 ! exp]im5t :i* demon/rat$ t a ball ( wood & a ball ( iron fall 9 ! same "t4 ,=ces act ^u ! two bodies = ! same "t1 b ! magnitudes ( ! =ces are proportional to ! mass ( ea* body multipli$ into its velocity1 &1 s9ce ! bodies fall simultane\sly1 _! velocities are equal4 ,! =ces act+ ^u ! bodies are !re=e proportional to _! masses2 b ! =ce act+ on ea* body is ! attraction ( ! e>?1 !re=e1 ! gravitation to ! e>? ( diff]5t bodies is proportional to _! masses4 #dih4 ,we may "h note ! contra/ betwe5 ! attraction ( gravitation & t ( a magnet4 ,a magnet attracts iron p[]fully & wood n at all2 b ! e>? draws all bodies ) =ces dep5d+ on _! masses & _! di/ances1 & n on _! *emical composition4 ,,! ,,space ,,describ$ ,,by ,a ,,fall+ ,,body ,,is ,,proportional ,,to ,,! ,,squ>e ,,( ,,! ,,"t4 #dii4 ,we h next to discov] ! law by :i* we asc]ta9 ! di/ance a body fall+ f re/ w move 9 a giv5 "t2 x is n possible to exp]im5t directly ^u ?is subject1 z 9 two seconds a body w drop #fd feet & acquire an inconv5i5tly l>ge velocity2 we c1 h[ev]1 resort to ,atwood's ma*9e 7,fig4 #ff7 z a means ( dim9i%+ ! motion4 ,= ?is purpose we require a clock ) a seconds p5dulum4 #ejj4 ,on "o ( ! equal cyl9d]s ,a ,i place a slight brass rod1 ^: weight gives a prepond]ance to ,a1 :i* w consequ5tly desc5d4 ,i hold ! load$ weight 9 my h&1 & release x simultane\sly ) ! tick ( ! p5dulum4 ,i obs]ve t x desc5ds #e be=e ! next tick4 ,return+ ! weight to ! place f :5ce x />t$1 ,i release x aga91 & ,i f9d t at ! second tick ( ! p5dulum x has travell$ #bj4 ,simil>ly we f9d t 9 ?ree seconds x desc5ds #de4 ,x greatly facilitates ^! exp]im5ts to use a little /age :i* is capable ( be+ slipp$ up & d[n ! scale1 & :i* c be clamp$ to ! scale 9 any position4 ,by actually plac+ ! /age at ! di/ance ( #e1 #bj1 or #de bel[ ! po9t f :i* ! weight />ts1 ! co9cid5ce ( ! tick ( ! p5dulum ) ! tap ( ! weight on its arrival at ! /age is v m>k$4 #eja4 ,^! ?ree di/ances are 9 ! proportion ( #a1 #d1 #i2 t is1 z ! squ>es ( ! numb]s ( seconds #a1 #b1 #c4 ,h5ce we may inf] t ! di/ance trav]s$ by a body fall+ f re/ is proportional to ! squ>e ( ! "t4 #ejb4 ,! motion ( ! bodies 9 ,atwood's ma*9e is mu* sl[] ?an ! motion ( a body fall+ freely1 b ! law j /at$ is equally true 9 bo? cases s t 9 a free fall ! di/ance trav]s$ is proportional to ! squ>e ( ! "t4 ,atwood's ma*9e _c directly tell u ! di/ance ?r\gh :i* a body falls 9 "o second4 ,if we c f9d ?is by o!r means1 we %all easily be able to calculate ! di/ance ?r\gh :i* a body w fall 9 any numb] ( seconds4 ,a ,,body ,,falls #af' ,,9 ,,! ,,fir/ ,,second4 #ejc4 ,! app>atus by :i* ?is important tru? maybe demon/rat$ is %[n 9 ,fig4 #fg4 ,a "p ( x has be5 already employ$ 9 p]=m+ ! exp]im5t ( ,galileo1 b two o!r p>ts mu/ n[ be us$ :i* w be briefly expla9$4 #ejd4 ,at ,a a p5dulum is %[n :i* vibrates once e second2 x ne$ n be connect$ ) any clockwork to su/a9 ! motion1 = :5 once set vibrat+ x w cont9ue to sw+ "s hundr$s ( times4 ,:5 ?is p5dulum is at ! middle ( its sw+1 ! bob j t\*es a sl5d] spr+1 & presses x slightly d[nw>ds4 ,! electric curr5t :i* circulates ab\t ! magnets ,g 7,art4 #dhi7 passes ?r\gh ?is spr+ :5 9 its natural position2 b :5 ! spr+ is press$ d[n by ! p5dulum1 ! curr5t is int]rupt$4 ,! consequ5ce is t1 z ! p5dulum sw+s backw>ds & =w>ds1 ! curr5t is brok5 once e second4 ,!re is also 9 ! circuit an electric al>m bell ,c1 :i* is s arrang$ t1 :5 ! curr5t passes1 ! hamm] is drawn f ! bell2 b1 :5 ! curr5t ceases1 a spr+ =ces ! hamm] to /rike ! bell4 ,:5 ! circuit is clos$1 ! hamm] is aga9 drawn back4 ,! p5dulum & ! bell are 9 ! same circuit1 & ?us e vibration ( ! p5dulum produces a /roke ( ! bell4 ,we may reg>d ! /rokes f ! bell z ! ticks ( ! p5dulum r5d]$ audible to ! :ole room4 #eje4 ,y w n[ und]/& ! mode ( exp]im5t+4 ,i draw ! p5dulum aside s t ! curr5t passes un9t]rupt$ly4 ,an iron ball is atta*$ to "o ( ! electro-magnets1 & x is !n g5tly hoi/$ up until ! height ( ! ball f ! gr\nd is ab\t #af'4 ,a cu%ion is plac$ on ! floor 9 ord] to receive ! fall+ body4 ,y are to look /eadily at ! cu%ion :ile y li/5 = ! bell4 ,all be+ ready1 ! p5dulum1 :i* has be5 held at a slight incl9ation1 is releas$4 ,! mom5t ! p5dulum rea*es ! middle ( its sw+ x t\*es ! spr+1 r+s ! bell1 breaks ! curr5t :i* circulat$ ar\nd ! magnet1 & z !re is n[ no?+ to su/a9 ! ball1 x drops d[n to ! cu%ion2 b j z x arrives !re1 ! p5dulum has a second "t brok5 ! electric circuit1 & y obs]ve ! fall+ ( ! ball ^u ! cu%ion to be id5tical ) ! second /roke ( ! bell4 ,z ^! /rokes are repeat$ at int]vals ( a second1 x foll[s t ! ball has fall5 #af' 9 "o second4 ,if ! magnet be rais$ a few feet high]1 ! ball may be se5 to rea* ! cu%ion aft] ! bell is he>d4 ,if ! magnet be l[]$ a few feet1 ! ball rea*es ! cu%ion be=e ! bell is he>d4 #ejf4 ,we h previ\sly %[n t ! space is proportional to ! squ>e ( ! "t4 ,we n[ see t :5 ! "t is "o second1 ! space is #af feet4 ,h5ce if ! "t 7 two seconds1 ! space w\ld be #d #af #fd feet2 & 9 g5]al ! space 9 feet is equal to #af multipli$ by ! squ>e ( ! "t 9 seconds4 #ejg4 ,by ! help ( ?is rule we are sometimes enabl$ to asc]ta9 ! height ( a p]p5dicul> cliff1 or ! dep? ( a well4 ,= ?is purpose x is conv5i5t to use a /op-wat*1 :i* w enable u to measure a %ort int]val ( "t accurately4 ,b an ord9>y wat* w d ne>ly z well1 = ) a little practice x is easy to c\nt ! beats1 :i* are usually at ! rate ( five a second4 ,by obs]v+ ! numb] ( beats f ! mom5t ! /one is releas$ till we see or he> its arrival at ! bottom1 we det]m9e ! "t occupi$ 9 ! act ( fall+4 ,! squ>e ( ! numb] ( seconds 7tak+ acc\nt ( fractional p>ts7 multipli$ by #af gives ! dep? ( ! well or ! height ( ! cliff 9 feet1 provid$ x be n high4 ,,! ,,action ,,( ,,gravity ,,is ,,indep5d5t ,,( ,,! ,,motion ,,( ,,! ,,body4 #ejh4 ,we h already le>n$ t ! effect ( gravity does n dep5d ^u ! actual *emical composition ( ! body4 ,we h n[ to le>n t its effect is un9flu5c$ by any motion :i* ! body may possess4 ,gravity pulls a body d[n #af' p] second1 if ! body />ts f re/4 ,b suppose a /one be ?r[n upw>ds ) a velocity ( #bj feet1 :]e w x be at ! end ( a second8 ,did gravity n act ^u ! /one1 x w\ld be at a height ( #bj feet4 ,! pr9ciple we h /at$ tells u t gravity w draw ?is /one t[>ds ! e>? ?r\gh a di/ance ( #af'1 j z x w\ld h done if ! /one _h />t$ f re/4 ,s9ce ! /one asc5ds #bj' 9 consequ5ce ( its own velocity1 & is pull$ back #af' by gravity1 x w1 at ! end ( a second1 be f\nd at ! height ( #d'4 ,if1 in/ead ( be+ %ot up v]tically1 ! body _h be5 project$ 9 any o!r direction1 ! result w\ld h be5 ! same2 gravity w\ld h br\ght ! body at ! end ( "o second #af' ne>] ! e>? ?an x w\ld h be5 _h gravity n act$4 ,= example1 if a body _h be5 %ot v]tically d[nw>ds ) a velocity ( #bj'1 x w\ld 9 "o second h mov$ ?r\gh a space ( #cf'4 #eji4 ,we %all illu/rate ?is rem>kable prop]ty by an exp]im5t4 ,! pr9ciple ( do+ s is z foll[s3--,suppose we take two bodies1 ,a & ,b4 ,if ^! be held at ! same height1 & releas$ toge!r1 ( c\rse !y rea* ! gr\nd at ! same in/ant2 b if ,a1 in/ead ( be+ m]ely dropp$1 be project$ ) a horizontal velocity at ! same mom5t t ,b is releas$1 x is /ill f\nd t a & ,b /rike ! floor simultane\sly4 #eaj4 ,y may v simply try ?is )\t special app>atus4 ,9 y\r left h& hold a m>ble1 & drop x at ! same in/ant t y\r "r h& ?r[s ano!r m>ble horizontally4 ,x w be se5 t ! two m>bles rea* ! gr\nd toge!r4 #eaa4 ,a m accurate mode ( mak+ ! exp]im5t is %[n by ! contrivance ( ,fig4 #fh4 ,illu/ration3 ,,fig4 #fh4 ,9 ?is we h an arrangem5t by :i* we ensure t "o ball %all be releas$ j z ! o!r is project$4 ,at ,a ,b is %[n a piece ( wood ab\t #b ?ick2 ! circul> portion 7#b' radius7 on :i* ! ball re/s is groov$1 s t ! ball only t\*es ! two edges & n ! bottom ( ! groove4 ,ea* edge ( ! groove is cov]$ ) t9foil ,c1 b ! pieces ( t9foil on ! two sides mu/ n communicate4 ,"o edge is connect$ ) "o pole ( ! batt]y ,k1 & ! o!r edge ) ! o!r pole1 b ! curr5t is unable to pass until a communication by a conductor is op5$ betwe5 ! two edges4 ,! ball ,g supplies ! bridge2 x is cov]$ ) t9foil1 & !re=e1 z long z x re/s ^u ! edges1 ! circuit is complete2 ! groove is s plac$ t ! tang5t to x at ! l[e/ po9t ,b is horizontal1 & !re=e1 :5 ! ball rolls d[n ! curve1 x is project$ f ! bottom 9 a horizontal direction4 ,an india-rubb] spr+ is us$ to propel ! ball2 & by draw+ x back :5 embrac$ by ! spr+1 ,i c communicate to ! missile a velocity :i* c be v>i$ at pleasure4 ,at ,h we h an electro-magnet1 ! wire ar\nd :i* =ms "p ( ! circuit we h be5 consid]+4 ,?is magnet is s plac$ t a ball susp5d$ f x is precisely at ! same height above ! floor z ! t9n$ ball is at ! mom5t :5 x leaves ! groove4 #eab4 ,we n[ und]/& ! mode ( exp]im5t+4 ,s long z ! t9n$ ball ,g rema9s on ! curve ! bridge is complete1 ! curr5t passes1 & ! electro-magnet w su/a9 ,h1 b ! mom5t ,g leaves ! curve1 ,h is all[$ to fall4 ,we inv>iably f9d t :atev] be ! velocity ) :i* ,g is project$1 x rea*es ! gr\nd at ! same in/ant z ,h arrives !re4 ,v>i\s dott$ l9es 9 ! figure %[ ! diff]5t pa?s :i* ,g may trav]se2 b :e!r x fall at ,d1 at ,e1 or at ,f1 ! "t ( desc5t is ! same z t tak5 by ,h4 ,( c\rse1 if ,g 7 n project$ horizontally1 we %\ld n h arriv$ at ?is result2 all we ass]t is t :atev] be ! motion ( a body1 x w 7:5 possible7 be at ! end ( a second1 sixte5 feet ne>] ! e>? ?an x w\ld h be5 if gravity _h n act$4 ,if ! body be project$ horizontally1 its desc5t is due to gravity alone1 & is nei!r accel]at$ nor ret>d$ by ! horizontal velocity4 ,:at ?is exp]im5t proves is1 t ! m]e fact ( a body hav+ velocity does n affect ! action ( gravity !reon4 #eac4 ,?\gh we h only %[n t a horizontal velocity does n affect ! action ( gravity1 yet nei!r does a velocity 9 any direction4 ,?is is v]ifi$1 l ! fir/ law ( motion1 by ! accordance betwe5 ! consequ5ces d$uc$ f x & ! facts ( obs]vation4 #ead4 ,we may summ>ize ^! results by say+ t no matt] :at be ! mat]ial ( :i* a p>ticle is compos$1 :e!r x be heavy or light1 mov+ or at re/1 if no =ce b gravity act ^u ! p>ticle = t seconds1 x w !n be #aft feet ne>] ! e>? ?an x w\ld h be5 _h gravity n act$4 ,illu/ration3 ,,fig4 #fi4 #eae4 ,a proposition :i* is ( "s importance may be introduc$ "h4 ,let u suppose a c]ta9 velocity & a c]ta9 =ce4 ,let ! velocity be su* t a po9t />t+ f ,a1 ,fig4 #fi1 w\ld 9 "o second move uni=mly to ,b4 ,let ! =ce be su* t if x act$ on a p>ticle orig9ally at re/ at ,a1 x w\ld 9 "o second draw ! p>ticle to ,d2 if !n ! =ce act on a p>ticle hav+ ?is velocity :]e w x be at ! end ( ! second8 ,complete ! p>allelogram ,a ,b ,c ,d1 & ! p>ticle w be f\nd at ,c4 ,by :at we h /at$ ! =ce w equally dis*>ge its duty :atev] be ! initial velocity4 ,! =ce w !re=e make ! p>ticle move to a di/ance equal & p>allel to ,a ,d f :atev] position ! p>ticle w\ld h assum$1 _h ! =ce n act$2 b _h ! =ce n act$1 ! p>ticle w\ld h be5 f\nd at ,b3 h5ce1 :5 ! =ce does act1 ! p>ticle mu/ be f\nd at ,c1 s9ce ,b ,c is equal & p>allel to ,a ,d4 ,,h[ ,,! ,,=ce ,,( ,,gravity ,,is ,,def9$4 #eaf4 ,f ! =mula ,di/ance #aft1 we le>n t a body falls ?r\gh #fd' 9 #b seconds2 & z we "k t x falls #af' 9 ! fir/ second1 x mu/ fall #dh' 9 ! next second4 ,let u exam9e ?is4 ,aft] fall+ = "o second1 ! body acquires a c]ta9 velocity1 & ) t velocity x comm5ces ! next second4 ,n[1 accord+ to :at we h j se51 gravity w act dur+ ! next second q indep5d5tly ( :atev] velocity ! body may h previ\sly _h4 ,h5ce 9 ! second second gravity pulls ! body d[n #af'1 b ! body moves altoge!r ?r\gh #dh'2 !re=e x mu/ move ?r\gh #cb' 9 consequ5ce ( ! velocity :i* has be5 impress$ ^u x by gravity dur+ ! fir/ second4 ,we le>n by ?is t :5 gravity acts = a second1 x produces a velocity su* t1 if ! body be conceiv$ to move uni=mly ) ! velocity acquir$1 ! body w\ld 9 "o second move ov] #cb'4 #eag4 ,9 ?ree seconds ! body falls #add'1 !re=e 9 ! ?ird second x mu/ h fall5 #add' - #fd' #hj'2 b ( ?is #hj' only #af' c\ld be due to ! action ( gravity impress$ dur+ t second2 ! re/1 #hj' - #af' #fd'1 is due to ! velocity ) :i* ! body comm5c$ ! ?ird second4 #eah4 ,we see !re=e t aft] ! lapse ( two seconds gravity has communicat$ to ! body a velocity ( #fd' p] second2 we %\ld simil>ly f9d1 t at ! end ( ! ?ird second1 ! body has a velocity ( #if'1 & 9 g5]al at ! end ( t seconds a velocity ( #cbt4 ,?us we illu/rate ! rem>kable law t ! velocity develop$ by gravity is proportional to ! "t4 #eai4 ,?is law po9ts out t ! mo/ suitable way ( measur+ gravity is by ! velocity acquir$ by a fall+ body at ! end ( "o second4 ,h5ce we are accu/om$ to say t g 7z gravity is g5]ally designat$7 is #cb4 ,we %all aft]w>ds %[ 9 ! lecture on ! p5dulum 7,,xviii47 h[ ! value ( g c be obta9$ accurately4 ,f ! two equations1 v #cbt & s #aft x is easy to inf] ano!r v well kn[n =mula1 namely1 v #fds4 ,,! ,,pa? ,,( ,a ,,projectile ,,is ,a ,,p>abola4 #ebj4 ,we h already se51 9 ! exp]im5ts ( ,fig4 #fh1 t a body project$ horizontally describes a curv$ pa? on its way to ! gr\nd1 & we h to det]m9e ! geometrical nature ( ! curve4 ,z ! movem5t is rapid1 x is impossible to foll[ ! projectile ) ! eye s z to asc]ta9 ! %ape ( its pa? ) accuracy2 we mu/ !re=e adopt a special contrivance1 su* z t repres5t$ 9 ,fig4 #gj4 ,b ,c is a quadrant ( wood #b ?ick2 x conta9s a groove1 along :i* ! ball ,b w run :5 releas$4 ,a s]ies ( c>dbo>d hoops are prop]ly plac$ on a black bo>d1 & ! ball1 :5 x leaves ! quadrant1 w pass ?r\gh all ^! hoops )\t t\*+ any1 & f9ally fall into a basket plac$ to receive x4 ,! quadrant mu/ be secur$ firmly1 & ! ball mu/ always />t f precisely ! same place4 ,! hoops are easily adju/$ by trial4 ,lett+ ! ball run d[n ! quadrant two or ?ree times1 we c see h[ to place ! fir/ hoop 9 its "r position1 & secure x by draw+ p9s2 !n by a few m trials ! next hoop is to be adju/$1 & s on = ! :ole eight4 #eba4 ,! curv$ l9e f ! bottom ( ! quadrant1 :i* passes ?r\gh ! c5tres ( ! hoops1 is ! pa? 9 :i* ! ball moves2 ?is curve is a p>abola1 ( :i* ,f is ! focus & ! l9e ,a ,a ! directrix4 ,illu/ration3 ,,fig4 #gj4 ,x is a prop]ty ( ! p>abola t ! di/ance ( any po9t on ! curve f ! focus is equal to its p]p5dicul> di/ance f ! directrix4 ,?is is %[n 9 ! figure4 ,= example1 ! dott$ l9e ,f ,d1 drawn f ,f to ! c5tre ( ! l[e/ hoop ,d1 is equal 9 l5g? to ! p]p5dicul> ,d ,p let fall f ,d on ! directrix ,a ,a4 #ebb4 ,! direction 9 :i* ! ball is project$ is 9 ?is case horizontal1 b1 :atev] be ! direction ( projection1 ! pa? is a p>abola4 ,?is c be prov$ ma!matically z a d$uction f ! !orem ( ,art4 #eae4 ,,lecture ,,xvi4 ,,in]tia4 ,in]tia4--,! ,hamm]4--,! ,/or+ ( ,en]gy4--,! ,fly-:eel4--,! ,pun*+ ,ma*9e4 ,,in]tia4 #ebc4 ,a body unact$ ^u by =ce w cont9ue = ev] at re/1 or = ev] mov+ uni=mly 9 a /raight l9e4 ,?is is ass]t$ by ! fir/ law ( motion 7,art4 #dhe74 ,x is usual to say t ,in]tia is a prop]ty ( all matt]1 by :i* x is meant t matt] _c ( itself *ange its /ate ( re/ or ( motion4 ,=ce is accord+ly requir$ = ?is purpose4 ,9 ! pres5t *apt] we %all discuss "s important me*anical consid]ations connect$ ) ! application ( =ce 9 *ang+ ! /ate ( a body f re/ or 9 alt]+ its velocity :5 9 motion4 ,9 ! next *apt] we %all /udy ! application ( =ce 9 compell+ a body to sw]ve f its motion 9 a /raight l9e4 #ebd4 ,we h 9 e>li] lectures be5 conc]n$ ) ! application ( =ce ei!r to raise a weight or to ov]come friction4 ,we h n[ to consid] ! application ( =ce to a body1 n = ! purpose ( rais+ x1 nor = pu%+ x along aga9/ a frictional resi/ance1 b m]ely = ! purpose ( g5]at+ a velocity4 ,un=tunately !re is a practical difficulty 9 ! way ( mak+ ! exp]im5ts precisely 9 ! mann] we %\ld wi%4 ,we want to get a mass isolat$ bo? f gravitation & f friction1 b ?is is j :at we _c d--t is1 we _c d x p]fectly4 ,we h1 h[ev]1 a simple appliance :i* w be suffici5tly isolat$ = our pres5t purpose4 ,"h is a heavy weight ( iron1 ab\t #be lbs41 susp5d$ by an iron wire f ! ceil+ ab\t #cb feet above ! floor 7see ,fig4 #hb74 ,?is weight may be mov$ to & fro by ! h&4 ,x is q free f friction1 = we ne$ n at pres5t rememb] ! small resi/ance :i* ! air (f]s4 ,we may also reg>d ! gravity ( ! weight z neutraliz$ by ! su/a9+ =ce ( ! wire1 & accord+ly z ! body n[ hangs at re/ x may = our purposes be reg>d$ z a body unact$ ^u by any =ce4 #ebe4 ,to give ?is ball a horizontal velocity ,i feel t ,i mu/ apply =ce to x4 ,?is w be manife/ to y all :5 ,i apply ! =ce ?r\gh ! m$ium ( an india-rubb] spr+4 ,if ,i pull ! spr+ %>ply y notice h[ mu* x /ret*es2 y see !re=e t ! body w n move quickly unless a consid]able =ce is appli$ to x4 ,x ?us foll[s t m]ely to g5]ate motion 9 ?is mass =ce has be5 requir$4 #ebf4 ,s1 too1 :5 ! body is 9 motion z x n[ is ,i _c /op x )\t ! ex]tion ( =ce4 ,see h[ ! spr+ is /ret*$ & h[ /rong a pull has ?us be5 ex]t$ to deprive ! body ( motion4 ,notice also t :ile a small =ce appli$ suffici5tly long w always re/ore ! body to re/1 yet t to produce re/ quickly a l>ge =ce w be requir$4 #ebg4 ,x is an univ]sal law ( nature t action & reaction are equal & opposite4 ,h5ce :5 any ag5t acts to set a body 9 motion1 or to modify its motion 9 any way1 ! body reacts on ! ag5t1 & ?is =ce has be5 call$ ! ,k9etic reaction4 #ebh4 ,= example4 ,:5 a railway tra9 />ts1 ! locomotive applies =ce to ! c>riages1 & ! spe$ g5]at$ dur+ "o second is add$ to t produc$ dur+ ! next1 & ! pace improves4 ,! k9etic reaction ( ! tra9 ret>ds ! eng9e f atta9+ ! spe$ x w\ld acquire if free f ! tra94 ,,! ,,hamm]4 #ebi4 ,! hamm] & o!r tools :i* give a bl[ dep5d = _! action ^u in]tia4 ,a gigantic hamm] might =ce 9 a nail by ! m]e weight ( ! head re/+ on ! nail1 b ) ! help ( in]tia we drive ! nail by bl[s f a small hamm]4 ,we h "h in]tia aid+ 9 ! production ( a me*anical p[] to ov]come ! consid]able resi/ance :i* ! wood opposes to ! entrance ( ! nail4 ,to drive 9 ! nail usually requires a direct =ce ( "s hundr$s ( p\nds1 & ?is we are able conv5i5tly to produce by sudd5ly *eck+ ! velocity ( a small mov+ body4 ,illu/ration3 ,,fig4 #ga4 #ecj4 ,! !ory ( ! hamm] is illu/rat$ by ! app>atus 9 ,fig4 #ga4 ,x is a tripod1 at ! top ( :i*1 ab\t #i' f ! gr\nd1 is a /\t pulley ,c2 ! rope is ab\t #ae' long1 & to ea* end ( x ,a & ,b are weights atta*$4 ,^! weights are at fir/ ea* #ad lbs4 ,i raise ,a up to ! pulley1 leav+ ,b ^u ! gr\nd2 ,i !n let g ! rope1 & d[n falls ,a3 x fir/ pulls ! slack rope ?r\gh1 & !n1 :5 ,a is ab\t #c' f ! gr\nd1 ! rope becomes tight1 ,b gets a viol5t *uck & is lift$ into ! air4 ,:at has rais$ ,b8 ,x _c be ! m]e weight ( ,a1 because t be+ equal to ,b1 c\ld only j balance ,b1 & is insuffici5t to raise x4 ,x mu/ h be5 a =ce :i* rais$ ,b2 t =ce mu/ h be5 some?+ m ?an ! weight ( ,a1 :i* 0 produc$ :5 ! motion 0 *eck$4 ,a 0 n /opp$ completely2 x only lo/ "s ( its velocity1 b x c\ld n lose any velocity )\t be+ act$ ^u by a =ce4 ,?is =ce mu/ h be5 appli$ by ! rope by :i* ,a 0 held back1 & ! t5sion ?us aris+ 0 suffici5t to pull up ,b4 #eca4 ,let u remove ! #ad lb4 weight f ,b1 & atta* !re a weight ( #bh lbs41 ,a rema9+ ! same z be=e 7#ad lbs474 ,i raise ,a to ! pulley2 ,i all[ x to fall4 ,y obs]ve t ,b1 ?\gh d\ble ! weight ( ,a1 is aga9 *uck$ up aft] ! rope has become tight4 ,we c only expla9 ?is by ! supposition t ! t5sion 9 ! rope ex]t$ 9 *eck+ ! motion ( ,a is at lea/ #bh lbs4 #ecb4 ,f9ally1 let u remove ! #bh lbs4 f ,b1 put on #ef lbs41 & p]=m ! exp]im5t aga92 y see t ev5 ! #ef lbs4 is rais$ up sev]al in*es4 ,"h a t5sion 9 ! rope has be5 g5]at$ suffici5t to ov]come a weight f\r times z heavy z ,a4 ,we h !n1 by ! help ( in]tia1 be5 able to produce a me*anical p[]1 = a small =ce has ov]come a great]4 #ecc4 ,aft] ,b is rais$ by ! *uck to a c]ta9 height x desc5ds aga91 if heavi] ?an ,a1 & raises ,a4 ,! height to :i* ,b is rais$ is ( c\rse ! same z ! height ?r\gh :i* ,a desc5ds4 ,y notic$ t ! height ?r\gh :i* #bh lbs4 0 rais$ 0 consid]ably great] ?an t ?r\gh :i* ! #ef lbs4 0 rais$4 ,h5ce we may draw ! inf]5ce1 t :5 ,a 0 depriv$ ( its velocity :ile pass+ ?r\gh a %ort space1 x requir$ to be oppos$ by a great] =ce ?an :5 x 0 gradually depriv$ ( its velocity ?r\gh a long] space4 ,?is is a mo/ important po9t4 ,suppos+ ,i 7 to put a hundr$weight at ,b1 ,i h little d\bt1 if ! rope 7 /rong en\gh to be> ! /ra91 t ?\gh ,a only weighs #ad lbs41 ,b w\ld yet be rais$ a little3 "h ,a w\ld be depriv$ ( its motion 9 a v %ort space1 b ! =ce requir$ to arre/ x w\ld be v great4 #ecd4 ,x is cle> t matt]s w\ld n be mu* alt]$ if ,a 7 to be /opp$ by "s =ce1 ex]t$ f bel[ r ?an above2 9 fact1 we may conceive ! rope omitt$1 & suppose ,a to be a hamm]-head fall+ ^u a nail 9 a piece ( wood4 ,! bl[ w\ld =ce ! nail to p5etrate a small di/ance1 & ! entire velocity ( ,a w\ld h to be de/roy$ :ile mov+ ?r\gh t small di/ance3 consequ5tly ! =ce betwe5 ! head ( ! nail & ! hamm] w\ld be a v l>ge "o4 ,?is expla9s ! effect ( a bl[4 #ece4 ,9 ! case t we h suppos$1 ! weight m]ely drops ^u ! nail3 ?is is actually ! pr9ciple ( ! hamm] us$ 9 pile-driv+ ma*9es4 ,a pile is a l>ge piece ( timb]1 po9t$ & %od ) iron at "o end3 ?is end is driv5 d[n into ! gr\nd4 ,piles are requir$ = v>i\s purposes 9 eng9e]+ op]ations4 ,!y are (t5 int5d$ to support ! f\ndations ( build+s2 !y are !re=e driv5 until ! resi/ance ) :i* ! gr\nd opposes _! fur!r entrance af=ds a gu>antee t !y %all be able to be> :at is requir$4 #ecf4 ,! ma*9e = driv+ piles consi/s ess5tially ( a heavy mass ( iron1 :i* is rais$ to a height1 & all[$ to fall ^u ! pile4 ,! resi/ance to be ov]come dep5ds ^u ! dep? & nature ( ! soil3 a pile may be driv5 two or ?ree in*es ) ea* bl[1 b ! less ! di/ance ! pile ent]s ea* "t1 ! great] is ! actual pressure ) :i* ! bl[ =ces x d[nw>ds4 ,9 ! ord9>y hamm]1 ! p[] ( ! arm imp>ts velocity to ! hamm]-head1 9 addition to t :i* is due to ! fall2 ! effect produc$ is m]ely ! same z if ! hamm] _h fall5 f a great] height4 #ecg4 ,ano!r po9t may be m5tion$ "h4 ,a nail w only ent] a piece ( wood :5 ! nail & ! wood are press$ toge!r ) suffici5t =ce4 ,! nail is urg$ by ! hamm]4 ,if ! wood be ly+ on ! gr\nd1 ! reaction ( ! gr\nd prev5ts ! wood f gett+ away & ! nail w ent]4 ,9 o!r cases ! elem5t ( "t is all-important4 ,if ! wood be massive less =ce w make ! nail p5etrate ?an w\ld suffice to move ! wood quickly en\gh4 ,if ! wood be ?9 & unsupport$1 less =ce may be requir$ to make x yield ?an to make ! nail p5etrate4 ,! usual rem$y is obvi\s4 ,hold a heavy mass close at ! back ( ! wood4 ,! nail w !n ent] because ! augm5t$ mass _c n[ escape z rapidly z be=e4 ,,! ,,/or+ ,,( ,,en]gy4 #ech4 ,our /udy ( ! subject w be facilitat$ by "s consid]ations f\nd$ on ! pr9ciples ( en]gy4 ,9 ! exp]im5t ( ,fig4 #ga let ,a be #ad lbs41 & ,b1 on ! gr\nd1 be #ef lbs4 ,s9ce ! rope is #ae' long1 ,a is #c' f ! gr\nd1 & !re=e #f' f ! pulley4 ,i raise ,a to ! pulley1 &1 9 do+ s1 exp5d #f #ad #hd units ( en]gy4 ,en]gy is nev] lo/1 & !re=e ,i %all expect to recov] ?is am\nt4 ,i all[ ,a to fall2 :5 x has fall5 #f'1 x is !n precisely 9 ! same condition z x 0 be=e be+ rais$1 except t x has a consid]able velocity ( desc5t4 ,9 fact1 ! #hd units ( en]gy h be5 exp5d$ 9 giv+ velocity to ,a4 ,b is !n lift$ to a maximum height ,x1 9 :i* #ef ,x units ( en]gy h be5 consum$4 ,at ! in/ant :5 ,b is at ! summit ,x1 ,a mu/ be at a di/ance ( #f ,x feet f ! pulley2 h5ce ! quantity ( "w p]=m$ by ,a is #ad 7#f ,x74 ,b ! "w done by ,a mu/ be equal to t done ^u ,b1 & !re=e #ad7f ,x7 #ef ,x1 :5ce ,x #b4 ,if !re 7 no loss by friction1 ,b w\ld !re=e be rais$ #b'2 b ow+ to friction1 & d\btless also to ! imp]fect flexibility ( ! rope1 ! effect is n s great4 ,we may reg>d ! "w done 9 rais+ ,a z s mu* en]gy /or$ up1 & :5 ,a is all[$ to fall1 ! en]gy is reproduc$ 9 a modifi$ =m4 #eci4 ,let u apply ! pr9ciple ( en]gy to ! pile-driv+ eng9e to :i* we h ref]r$ 7,art4 #ecf72 we %all !n be able to f9d ! magnitude ( ! =ce develop$ 9 produc+ ! bl[4 ,suppose ! 8monkey10 t is ! heavy hamm]-head1 weighs #efj lbs4 7a qu>t] ( a ton74 ,a c\ple ( m5 raise ?is by means ( a small w9* to a height ( #ae'4 ,x takes !m a few m9utes to d s2 _! en]gy is !n sav$ up1 & !y h accumulat$ a /ore ( #efj #ae #h1djj units4 ,:5 ! monkey falls ^u ! top ( ! pile x transf]s !reto ne>ly ! :ole ( ! #h1djj units ( en]gy1 & ?is is exp5d$ 9 =c+ ! pile into ! gr\nd4 ,suppose ! pile to ent] "o in*1 ! reaction ( ! pile ^u ! monkey mu/ be s great t ! numb] ( units ( en]gy consum$ 9 "o in* is #h1djj4 ,h5ce ?is reaction mu/ be #h1djj #ab #ajj1hjj lbs4 ,if ! reaction did n rea* ?is am\nt1 ! monkey c\ld n be br\ght to re/ 9 s %ort a di/ance4 ,! reaction ( ! pile ^u ! monkey1 & !re=e ! action ( ! monkey ^u ! pile1 is ab\t #de tons4 ,?is is ! actual pressure ex]t$4 #edj4 ,if ! soil :i* ! pile is p5etrat+ be m resi/+ ?an t :i* we h suppos$1--= example1 if ! pile require a direct pressure ( #ajj tons to =ce x 91--! same monkey ) ! same fall w\ld /ill be suffici5t1 b ! pile w\ld n be driv5 s f> ) ea* bl[4 ,! pressure requir$ is #bbd1jjj lbs43 ?is ex]t$ ov] a space ( #jde w\ld be #h1djj units ( en]gy2 h5ce ! pile w\ld be driv5 #jde4 ,! m ! resi/ance1 ! less ! p5etration produc$ by ea* bl[4 ,a pile int5d$ to be> a v heavy load p]man5tly mu/ be driv5 until x ent]s b little ) ea* bl[4 #eda4 ,we may comp>e ! pile-driv] ) ! me*anical p[]s 9 "o respect1 & contra/ x 9 ano!r4 ,9 ea*1 we h ma*9es :i* receive en]gy & re/ore x modifi$ into a great] p[] ex]t$ ?r\gh a small] di/ance2 b :ile ! me*anical p[]s re/ore ! en]gy at "o end ( ! ma*9e1 simultane\sly ) _! reception ( x at ! o!r1 ! pile-driv] is a res]voir = keep+ en]gy :i* w re/ore x 9 ! =m want$4 #edb4 ,we h1 !n1 a class ( me*anical p[]s1 ( :i* a hamm] may be tak5 z ! type1 :i* dep5d ^u ! /orage ( en]gy2 ! p[] ( ! arm is accumulat$ 9 ! hamm] ?r\gh\t its desc5t1 to be in/antly transf]r$ to ! nail 9 ! bl[4 ,in]tia is ! prop]ty ( matt] :i* qualifies x = ?is purpose4 ,en]gy is develop$ by ! explosion ( gunp[d] 9 a cannon4 ,?is en]gy is transf]r$ to ! ball1 f :i* x is aga9 9 l>ge "p pass$ on to d "w aga9/ ! object :i* is /ruck4 ,"h we see en]gy /or$ 9 a rapidly mov+ body1 a case to :i* we %all pres5tly return4 #edc4 ,b en]gy c be /or$ 9 _m ways2 we might almo/ say t gunp[d] is itself en]gy 9 a compact & /orable =m4 ,! ef=ts :i* we make 9 =c+ air into an air-cane are pres]v$ z en]gy !re /or$ to be reproduc$ 9 ! dis*>ge ( a numb] ( bullets4 ,dur+ ! few seconds occupi$ 9 w9d+ a wat*1 a small *>ge ( en]gy is giv5 to ! spr+ :i* x exp5ds economically ov] ! next tw5ty-f\r h\rs4 ,9 us+ a b[ my en]gy is /or$ up f ! mom5t ,i beg9 to pull ! /r+ until ,i release ! arr[4 #edd4 ,_m ma*9es 9 ext5sive use dep5d ^u ^! pr9ciples4 ,9 ! clock or wat* ! dem& = en]gy to su/a9 ! motion is con/ant1 :ile ! supply is only occasional2 9 o!r cases ! supply is con/ant1 :ile ! dem& is only int]mitt5t4 ,we may m5tion an illu/ration ( ! latt]4 ,suppose x be requir$ occasionally to hoi/ heavy weights rapidly up to a height4 ,if an eng9e suffici5tly p[]ful to raise ! weights be employ$1 ! eng9e w be idle except :5 ! weights are be+ rais$2 & if ! ma*9]y 7 to h mu* idle "t1 ! wa/e ( fuel 9 keep+ up ! fire dur+ ! int]vals w\ld (t5 make ! arrangem5t uneconomical4 ,x w\ld be a f> bett] plan to h a small] eng9e2 & ev5 ?\gh ?is 7 n able to raise ! weight directly ) suffici5t spe$1 yet by keep+ ! eng9e cont9ually work+ & /or+ up its en]gy1 we might produce en\gh 9 ! tw5ty-f\r h\rs to raise all ! weights :i* x w\ld be necess>y to lift 9 ! same "t4 #ede4 ,let u suppose we want to raise slates f ! bottom ( a qu>ry to ! surface4 ,a l>ge pulley is m\nt$ at ! top ( ! qu>ry1 & ov] ?is a rope is pass$3 to ea* end ( ! rope a bucket is atta*$1 s t :5 "o ( ^! is at ! bottom ! o!r is at ! top1 & _! sizes & t ( ! pulley are s arrang$ t !y pass ea* o!r ) safety4 ,a res]voir is e/abli%$ at ! top ( ! qu>ry on a level ) ! pulley1 & an eng9e is set to "w con/antly pump+ up wat] f ! bottom ( ! qu>ry into ! res]voir4 ,ea* ( ! buckets is p>tly compos$ ( a l>ge tank1 :i* c be quickly fill$ or empti$4 ,! l[] bucket is load$ ) slates1 & :5 ready = "w1 ! man at ! top fills ! tank ( ! upp] bucket ) wat]3 ?is accord+ly becomes s heavy t x desc5ds & raises ! slates4 ,:5 ! heavi] "o rea*es ! bottom1 ! wat] f its tank is let out into ! l[] res]voir1 f :i* ! eng9e pumps1 & ! slates are remov$ f ! bucket :i* has be5 rais$4 ,all is !n ready = a repetition ( ! same op]ation4 ,if ! slates be rais$ at int]vals ( t5 m9utes1 ! en]gy ( ! eng9e w be suffici5t :5 9 t5 m9utes' "w x c pump up en\gh wat] to fill "o tank2 by ! aid ( ?is contrivance we are !re=e able to accumulate = "o ef=t ! :ole p[] ( ! eng9e = t5 m9utes4 ,,! ,,fly-,,:eel4 #edf4 ,"o ( ! be/ means ( /or+ en]gy is by sett+ a heavy body 9 rapid motion4 ,?is has already be5 ref]r$ to 9 ! case ( ! cannon-ball4 ,9 ord] to r5d] ?is me?od practically available = ! purposes ( ma*9]y1 ! heavy body we use is a fly-:eel1 & ! rapid motion imp>t$ to x is t ( rotation ab\t its axis4 ,a v l>ge am\nt ( en]gy c by ?is means be /or$ 9 a manageable =m4 #edg4 ,we %all illu/rate ! pr9ciple by ! app>atus ( ,fig4 #gb4 ,?is repres5ts an iron fly-:eel ,b3 its diamet] is #ah1 & its weight is #bf lbs42 ! fly is c>ri$ ^u a %aft 7,a7 ( wr\ght iron 9 diamet]4 ,we %all /ore up a quantity ( en]gy 9 ?is :eel1 by sett+ x 9 rapid motion1 & !n we %all see h[ we c recov] f x ! en]gy we h imp>t$4 #edh4 ,a rope is coil$ ar\nd ! %aft2 by pull+ ?is rope ! :eel is made to turn r\nd3 ?us ! rope is ! m$ium by :i* my en]gy %all be imp>t$ to ! :eel4 ,to measure ! op]ation accurately1 ,i atta* ! rope to ! hook ( ! spr+ balance 7,fig4 #i72 & by tak+ ! r+ ( ! balance 9 my h&1 ,i le>n f ! index ! am\nt ( ! =ce ,i am ex]t+4 ,i f9d t :5 ,i walk backw>ds z quickly z is conv5i5t1 pull+ ! rope all ! "t1 ! scale %[s a t5sion ( ab\t #ej lbs4 ,to set ! :eel rapidly 9 motion1 ,i pull ab\t #bj' ( rope f ! axle1 s t ,i h imp>t$ to ! :eel some:]e ab\t #ej #bj #a1jjj units ( en]gy4 ,! rope is fa/5$ to ! %aft1 s t1 aft] x has be5 all unw\nd1 ! :eel n[ rapidly rotat+ w9ds x 94 ,by measur+ ! "t 9 :i* ! :eel made a c]ta9 numb] ( coils ( ! rope ar\nd ! %aft1 ,i f9d t x makes ab\t #fjj revolutions p] m9ute4 ,illu/ration3 ,,fig4 #gb4 #edi4 ,let u see h[ ! /or$-up en]gy c be exhibit$4 ,a piece ( p9e #bd #a #a ( :i* bo? ends are support$1 requires a =ce ( #cjj lbs4 appli$ to its c5tre to produce fracture 7,see p4 #aij74 ,i arrange su* a piece ( p9e ne> ! :eel4 ,z ! %aft is w9d+ 9 ! rope1 a trem5d\s *uck w\ld be giv5 to any?+ :i* tri$ to /op ! motion4 ,if ,i ti$ ! end ( ! rope to ! piece ( p9e1 ! *uck w\ld break ! rope2 !re=e ,i h fa/5$ "o end ( a #aj' l5g? ( *a9 to ! rope1 & ! o!r has be5 ti$ r\nd ! middle ( ! wood5 b>4 ,! :eel fir/ w9ds 9 ! rope1 !n ! *a9 takes a few turns be=e x tight5s1 & crack goes ! rod ( p9e4 ,! :eel _h no *oice2 x mu/ ei!r /op or break ! rod3 b nature =bids x to be /opp$1 unless by a great =ce1 :i* ! rod 0 n /rong en\gh to apply4 ,"h ,i nev] ex]t$ a =ce great] ?an #ej lbs4 9 sett+ ! :eel 9 motion4 ,! :eel /or$ up & modifi$ my en]gy s z to produce a =ce ( #cjj lbs41 :i* _h1 h[ev]1 only to be ex]t$ ov] a v small di/ance4 #eej4 ,b we may %[ ! exp]im5t 9 ano!r way1 :i* is t repres5t$ 9 ! figure 7#gb74 ,we see ! *a9 is !re atta*$ to two #ef lb4 weights4 ,! mode ( proce$+ is t already describ$4 ,! rope is fir/ w\nd r\nd ! %aft1 !n by pull+ ! rope ! :eel is made to revolve2 ! :eel !n beg9s to w9d 9 ! rope aga91 & :5 ! *a9 tight5s ! two #ef lbs4 are rais$ up to a height ( #c or #d feet4 ,"h1 aga91 ! en]gy has be5 /or$ & recov]$4 ,b ?\gh ! fly-:eel w ?us pres]ve en]gy1 x does s at "s co/3 ! /ore is cont9ually be+ fritt]$ away by friction & ! resi/ance ( ! air2 9 fact1 ! en]gy w\ld altoge!r disappe> 9 a little "t1 & ! :eel w\ld come to re/2 x is !re=e economical to make ! :eel yield up :at x has receiv$ z soon z possible4 #eea4 ,^! pr9ciples are illu/rat$ by ! function ( ! fly-:eel 9 a /eam-eng9e4 ,! pressure ( ! /eam ^u ! pi/on v>ies accord+ to ! diff]5t p>ts ( ! /roke3 & ! fly-:eel obviates ! inconv5i5ce :i* w\ld arise f su* irregul>ity4 ,its great in]tia makes its velocity b little augm5t$ by ! exub]ant action ( ! pi/on :5 ! pressure is greate/1 :ile x also su/a9s ! motion :5 ! pi/on is giv+ no assi/ance4 ,! fly-:eel is a va/ res]voir into :i* ! eng9e p\rs its en]gy1 sudd5 floods alt]nat+ ) dr\ghts2 b ^! succe$ ea* o!r s rapidly1 & ! area ( ! res]voir is s va/1 t its level rema9s s5sibly uni=m1 & ! supplies s5t out to ! consum]s are regul> & unv>i$4 ,! consum]s ( ! en]gy /or$ 9 ! fly-:eel ( an eng9e are ! ma*9es 9 ! mill2 !y are suppli$ by %afts :i* trav]se ! build+1 convey+1 by _! rotation1 ! en]gy orig9ally cond5s$ )9 ! coal f :i* combu/ion has set x free4 ,,! ,,pun*+ ,,ma*9e4 #eeb4 ,:5 en]gy has be5 /or$ 9 a fly-:eel1 x c be )drawn ei!r z a small =ce act+ ov] a great di/ance1 or z a l>ge =ce ov] a small di/ance4 ,9 ! latt] case ! fly-:eel acts z a me*anical p[]1 & 9 ?is =m x is us$ 9 ! v important ma*9e to be next describ$4 ,a model ( ! pun*+ ma*9e is %[n 9 ,fig4 #gc4 ,! pun*+ ma*9e is usually work$ by a /eam-eng9e1 a h&le w move our small model4 ,! h&le turns a %aft on :i* ! fly-:eel ,f is m\nt$4 ,on ! %aft is a small p9ion ,d ( #dj tee?3 ?is works into a l>ge :eel ,e ( #bjj tee?1 s t1 :5 ! fly & ! p9ion h turn$ r\nd #e times1 ,e w h turn$ r\nd once1 ,c is a circul> piece ( wood call$ a cam1 :i* has a hole bor$ ?r\gh x1 betwe5 ! c5tre & circumf]5ce2 by means ( ?is hole1 ! cam is m\nt$ on ! same axle z ,e1 to :i* x is rigidly fa/5$1 s t ! two mu/ revolve toge!r4 ,a is a lev] ( ! fir/ ord]1 ^: fulcrum is at ,a3 ! remote end ( ?is lev] re/s ^u ! cam ,c2 ! o!r end ,b conta9s ! pun*4 ,z ! :eel ,e revolves x c>ries ) x ! cam3 ?is raises ! lev] & =ces ! pun* d[n a hole 9 a die1 into :i* x fits exactly4 ,! metal to be op]at$ on is plac$ "u ! pun* be=e x is depress$ by ! cam1 & ! pressure drives ! pun* ?r\gh1 cutt+ out a cyl9drical piece ( metal f ! plate3 ?is model w1 z y see1 pun* ?r\gh ord9>y t94 ,illu/ration3 ,,fig4 #gc4 #eec4 ,let u exam9e ! mode ( action4 ,! ma*9e be+ made to rotate rapidly1 ! pun* is depress$ once = e #e revolutions ( ! fly2 ! resi/ance :i* ! metal opposes to be+ pun*$ is no d\bt v great1 b ! lev] acts at a twelve-fold advantage4 ,:5 ! pun* comes d[n on ! surface ( ! metal1 "o ( ?ree ?+s mu/ happ53 ei!r ! motion mu/ /op sudd5ly1 or ! ma*9e mu/ be /ra9$ & injur$1 or ! metal mu/ be pun*$4 ,b ! motion _c be /opp$ sudd5ly1 because1 be=e ?is c\ld happ51 an inf9ite =ce w\ld be develop$1 :i* mu/ make some?+ yield4 ,if !re=e we make ! /ructure suffici5tly massive to prev5t yield+1 ! metal mu/ be pun*$4 ,su* ma*9es are necess>ily built /rong en\gh to make ! pun*+ ( ! metal easi] ?an break+ ! ma*9e4 #eed4 ,we %all be able to calculate1 f :at we h already se5 9 ,art4 #bdh1 ! magnitude ( ! =ce requir$ = pun*+4 ,we !re le>n$ t ab\t #bbe tons ( pressure 0 necess>y to %e> a b> ( iron "o squ>e in* 9 section4 ,pun*+ is s f> analog\s to %e>+ t 9 ea* case a c]ta9 area ( surface has to be cut2 ! area 9 pun*+ is measur$ by ! cyl9d] ( iron :i* is remov$4 #eee4 ,suppose x be requir$ to pun* a hole #je 9 diamet] ?r\gh a plate #jh ?ick1 ! area ( iron t has to be cut across is _/ _/ #abf squ>e in*es3 & z #bbe tons p] squ>e in* are requir$ = %e>+1 ?is hole w require #bbe #abf #bhd tons4 ,a =ce ( ?is am\nt mu/ !re=e be ex]t$ ^u ! pun*3 :i* w require f ! cam a =ce ( m ?an #b tons ^u its end ( ! lev]4 ,?\gh ! iron mu/ be pi]c$ to a dep? ( #jh1 yet x is obvi\s t almo/ imm$iately aft] ! pun* has p5etrat$ ! surface ( ! iron1 ! cyl9d] mu/ be entirely cut & beg9 to em]ge f ! o!r side ( ! plate4 ,we %all c]ta9ly be correct 9 suppos+ t ! pun*+ is complet$ be=e ! pun* has p5etrat$ to a dep? ( #jb1 & t = n m ?an ?is di/ance has ! great pressure ( #bh tons be5 ex]t$2 = a small pressure is aft]w>ds suffici5t to ov]come ! friction :i* opposes ! motion ( ! cyl9d] ( iron4 ,h5ce1 ?\gh s great a pressure has be5 requir$1 yet ! numb] ( units ( en]gy consum$ is n v l>ge2 x is _/ #bbdj #bhd #ajfb4 ,! en]gy actually requir$ to pun* a hole ( half an in* diamet] ?r\gh a plate eight-t5?s ( an in* ?ick is !re=e less ?an t :i* w\ld be exp5d$ 9 rais+ #a cwt4 to a height ( t5 feet4 #eef4 ,! fly-:eel may be lik5$ to ! res]voir 9 ,art4 #ede4 ,! "t t is actually occupi$ 9 ! pun*+ is extremely small1 & ! sudd5 exp5diture ( #ajfb units is gradually reimburs$ by ! eng9e4 ,if ! rotat+ fly-:eel conta9 #ej1jjj units ( en]gy1 ! ab/raction ( #acfb units w n p]ceptibly affect its velocity4 ,!re is !re=e an advantage 9 hav+ a heavy fly su/a9$ at a high spe$ = ! work+ ( a pun*+ ma*9e4 ,,lecture ,,xvii4 ,,circul> ,,motion4 ,! ,nature ( ,circul> ,motion4--,circul> motion 9 ,liquids4--,! ,applications ( ,circul> ,motion4--,! ,p]man5t ,axes4 ,,! ,,nature ,,( ,,circul> ,,motion4 #eeg4 ,to compel a body to sw]ve f motion 9 a /raight l9e =ce mu/ be ex]cis$4 ,9 ?is *apt] we %all /udy ! comp>atively simple case ( a body revolv+ 9 a circle4 #eeh4 ,:5 a body moves r\nd uni=mly 9 a circle =ce mu/ be cont9u\sly appli$1 & ! fir/ "q = u to exam9e is1 z to ! direction ( t =ce4 ,we h to demon/rate ! important fact1 t x con/antly t5ds t[>ds ! c5tre4 #eei4 ,! direction ( ! =ce c be exhibit$ by actual exp]im5t1 & its magnitude w be at ! same "t cle>ly indicat$ by ! ext5t to :i* a spr+ is /ret*$4 ,! app>atus we use is %[n 9 ,fig4 #gd4 ,! ess5tial p>ts ( ! ma*9e consi/ ( two balls ,a1 ,b1 ea* #b 9 diamet]3 ^! are ?9 holl[ sph]es ( silv]$ brass4 ,! balls are support$ on arms ,p ,a1 ,q ,b1 :i* are atta*$ to a piece ( wood1 ,p ,q1 capable ( turn+ on a socket at ,c4 ,! arm ,a ,p is rigidly fix$ to ,p ,q2 ! o!r arm1 ,b ,q1 is capable ( turn+ r\nd a p9 at ,q4 ,an india-rubb] door-spr+ is %[n at ,f2 "o end ( ?is is secur$ to ,p ,q1 ! o!r end to ! movable arm1 ,q ,b4 ,if ! arm ,q ,b be turn$ s z to move ,b away f ,c1 ! spr+ ,f mu/ be /ret*$4 ,illu/ration3 ,,fig4 #gd4 ,a small too!d :eel is m\nt$ on ! same socket z ,c2 ?is is beh9d ,p ,q1 & is !re=e n se5 9 ! figure3 ! :ole is made to revolve rapidly by ! l>ge :eel ,e1 :i* is turn$ by ! h&le ,d4 #efj4 ,! room be+ d>k5$1 a beam f ! lime-light is all[$ to fall on ! app>atus3 ! reflections ( ! light are se5 9 ! two silv]$ balls z two bright po9ts4 ,:5 ,d is turn$1 ! balls move r\nd rapidly1 & y see ! po9ts ( light reflect$ f !m describe circles4 ,! ball ,b :5 at re/ is #d f ,c1 :ile ,a is #h f ,c2 h5ce ! circle describ$ by ,b is small] ?an t describ$ by ,a4 ,! appe>ance pres5t$ is t ( two conc5tric lum9\s circles4 ,z ! spe$ increases1 ! inn] circle enl>ges till ! two circles bl5d into "o4 ,by increas+ ! spe$ /ill m1 y see ! circle ^: diamet] is enl>g+ actually exce$+ ! fix$ circle1 & its size cont9ues to increase until ! highe/ velocity :i* x is safe to employ has be5 communicat$ to ! ma*9e4 #efa4 ,:at is ! explanation ( ?is8 ,! arm ,a is fix$ & ! di/ance ,a ,c _c alt]1 h5ce ,a describes ! fix$ circle4 ,b1 on ! o!r h&1 is n fix$2 x c rec$e f ,c1 & we f9d t ! quick] ! spe$ ! fur!r x rec$es4 ,! l>g] ! circle describ$ by ,b ! m is ! spr+ /ret*$1 & ! great] is ! =ce ) :i* ,b is attract$ t[>ds ! c5tre4 ,?is exp]im5t proves t ! =ce necess>y to reta9 a body 9 a circul> pa? mu/ be increas$ :5 ! spe$ is increas$4 #efb4 ,?us we see t uni=m motion ( a body 9 a circle c only be produc$ by an uni=m =ce direct$ to ! c5tre4 ,if ! motion1 ev5 ?\gh circul>1 h v>iable spe$ ! law ( ! =ce is n s simple4 #efc4 ,we c measure ! magnitude ( ?is =ce by ! same app>atus4 ,! ball ,b weighs #ja lb4 ,i f9d t ,i mu/ pull x ) a =ce ( #c lbs4 9 ord] to draw x to a di/ance ( #h f ,c2 t is1 to ! same di/ance z ,a is f ,c4 ,h5ce1 :5 ! diamet]s ( ! circles 9 :i* ! balls move are equal1 ! c5tral =ce mu/ be #c lbs42 t is1 x mu/ be ne>ly ?irty times z great z gravity4 #efd4 ,! necessity = ! c5tral =ce is ?us %[n3 ,let u conceive a weight atta*$ to a /r+ to be swung r\nd 9 a circle1 a portion ( :i* is %[n 9 ,fig4 #ge4 ,illu/ration3 ,,fig4 #ge4 ,suppose ! weight be at ,s & mov+ t[>ds ,p1 & let a tang5t to ! circle be drawn at ,p4 ,take two po9ts on ! circle1 ,a & ,b1 v ne> ,p2 ! small arc ,a ,b does n diff] p]ceptibly f ! "p ,a ,b on ! tang5t l9e2 h5ce1 :5 ! p>ticle arrives at ,a1 x is a matt] ( indiff]5ce :e!r x travels 9 ! arc ,a ,b1 or along ! l9e ,a ,b4 ,let u suppose x to move along ! l9e4 ,by ! fir/ law ( motion1 a p>ticle mov+ 9 ! l9e ,a ,b w\ld cont9ue to d s2 h5ce1 if ! p>ticle be all[$1 x w move on to ,q3 b ! p>ticle is n all[$ to move to ,q2 x is f\nd at ,r4 ,h5ce x mu/ h be5 )drawn by "s =ce4 #efe4 ,?is =ce is suppli$ by ! /r+ to :i* ! weight is atta*$4 ,! incessant *ange f ! rectil9e> motion ( ! weight requires a con/antly appli$ =ce1 & ?is is always direct$ to ! c5tre4 ,%\ld ! /r+ be releas$1 ! body flies (f 9 ! direction ( ! tang5t to ! circle at ! po9t :i* ! body occupi$ at ! in/ant ( release4 #eff4 ,! c5tral =ce increases 9 proportion to ! squ>e ( ! velocity4 ,if ,i d\ble ! spe$ ) :i* ! weight is :irl$ r\nd 9 ! circle1 ,i quadruple ! =ce :i* ! /r+ mu/ ex]t on ! body4 ,if ! spe$ be trebl$1 ! =ce is increas$ n9efold1 & s on4 ,:5 ! spe$s ) :i* two equal masses are revolv+ 9 two circles are equal1 ! c5tral =ce 9 ! small] circle is great] ?an t ( ! l>g] circle1 9 ! proportion ( ! radius ( ! l>g] circle to t ( ! small]4 ,,! ,,action ,,( ,,circul> ,,motion ,,^u ,,liquids4 #efg4 ,i h "h a small bucket ne>ly fill$ ) wat]3 to ! h&le a piece ( /r+ is atta*$4 ,if ,i :irl ! bucket r\nd 9 a v]tical plane suffici5tly fa/1 y see no wat] escapes1 al?\gh ! bucket is turn$ upside d[n once 9 e revolution4 ,?is is because ! wat] has n "t to fall out dur+ su* a brief int]val4 ,a body w\ld n fall half an in* f re/ 9 ! tw5tie? ( a second4 #efh4 ,! action ( circul> motion ^u liquids is illu/rat$ by ! exp]im5t :i* is repres5t$ 9 ,fig4 #gf4 ,illu/ration3 ,,fig4 #gf4 ,a glass beak] ab\t half full ( wat] is m\nt$ s t x c be spun r\nd rapidly4 ,! motion is giv5 by means ( a l>ge :eel turn$ by a h&le1 z %[n 9 ! figure4 ,:5 ! rotation comm5ces1 ! wat] is se5 to rise up aga9/ ! glass sides & =m a holl[ 9 ! c5tre4 #efi4 ,9 ord] to demon/rate ?is cle>ly1 ,i turn ^u ! vessel a beam f ! lime-light4 ,i h previ\sly dissolv$ a little qu99e 9 ! wat]4 ,! light f ! lamp is transmitt$ ?r\gh a piece ( d5se blue glass4 ,:5 ! light ?us col\r$ falls on ! wat]1 ! pres5ce ( ! qu99e makes ! entire liquid gl[ ) a blui% hue4 ,?is rem>kable prop]ty ( qu99e1 :i* is kn[n z fluoresc5ce1 enables y to see di/9ctly ! holl[ =m caus$ by ! rotation4 #egj4 ,y obs]ve t z ! spe$ becomes great] ! dep? ( ! holl[ increases1 & t if ,i turn ! :eel suffici5tly fa/ ! wat] is actually driv5 out ( ! glass4 ,! %ape ( ! curve :i* ! wat] assumes is t :i* w\ld be produc$ by ! revolution ( a p>abola ab\t its axis4 #ega4 ,! explanation is simple4 ,z soon z ! glass beg9s to revolve1 ! friction ( its sides spe$ily imp>ts a revolv+ motion to ! wat]2 b 9 ?is case !re is no?+ to keep ! p>ticles ne> ! c5tre l ! /r+ 9 ! revolv+ weight1 s ! liquid rises at ! sides ( ! glass4 #egb4 ,b y may ask :y all ! p>ticles ( ! wat] %\ld n g to ! circumf]5ce1 & ?us l9e ! inside ( ! glass ) a holl[ cyl9d] ( wat] in/ead ( ! p>abola4 ,su* an arrangem5t c\ld n exi/ 9 a liquid act$ on by gravity4 ,! l[] p>ts ( ! cyl9d] mu/ be> ! pressure ( ! wat] above1 & !re=e h m t5d5cy to flatt5 out ?an ! upp] portions4 ,?is t5d5cy c\ld n be ov]come by any consequ5ces ( ! movem5t1 = su* mu/ be alike on all p>ts at ! same di/ance f ! axis4 #egc4 ,a v beautiful exp]im5t 0 devis$ by ,plateau = ! purpose ( /udy+ ! revolution ( a liquid remov$ f ! action ( gravity4 ,! app>atus employ$ is repres5t$ 9 ,fig4 #gg4 ,a glass vessel #i cube is fill$ ) a mixture ( alcohol & wat]4 ,! relative quantities ought to be s proportion$ t ! fluid has ! same specific gravity z olive oil1 :i* is heavi] ?an alcohol & light] ?an wat]4 ,9 practice1 h[ev]1 x is f\nd s difficult to adju/ ! composition exactly t ! be/ plan is to make two alcoholic mixtures s t olive oil w j float on "o ( !m1 & j s9k 9 ! o!r4 ,! l[] half ( ! glass is to be fill$ ) ! d5s] mixture1 & ! upp] half ) ! light]4 ,if1 !n1 ! oil be c>efully introduc$ ) a funnel x w =m a beautiful sph]e 9 ! middle ( ! vessel1 z %[n 9 ! figure4 ,we ?us see t a liquid mass fre$ f ! action ( t]re/rial gravity1 =ms a sph]e by ! mutual attraction ( its p>ticles4 ,illu/ration3 ,,fig4 #gg4 ,?r\gh ! liquid a v]tical sp9dle passes4 ,on ?is !re is a small disk at ! middle ( its l5g?1 ab\t :i* ! sph]e ( oil arranges itself symmetrically4 ,to ! end ( ! sp9dle a h&le is atta*$4 ,:5 ! h&le is turn$ r\nd sl[ly1 ! friction ( ! disk & sp9dle communicates a motion ( rotation to ! sph]e ( oil4 ,we h ?us a liquid sph]oidal mass end[$ ) a movem5t ( rotation2 & we c /udy ! effect ( ! motion ^u its =m4 ,we fir/ see ! sph]e flatt5 d[n at its poles1 & bulge at its equator4 ,9 ord] to %[ ! ph5om5on to ^? :o may n be ne> to ! table1 ! sph]e c be project$ on ! scre5 by ! help ( ! lime-light lamp & a l5s4 ,x fir/ appe>s z a yell[ circle1 & !n1 z ! rotation beg9s1 ! circle gradually trans=ms into an ellipse4 ,b a v rem>kable modification takes place :5 ! h&le is turn$ some:at rapidly4 ,! ellipsoid gradually flatt5s d[n until1 :5 a c]ta9 velocity has be5 atta9$1 ! surface actually becomes ind5t$ at ! poles1 & flies f ! axis altoge!r4 ,ultimately ! liquid assumes ! =m ( a beautiful r+1 & ! appe>ance on ! scre5 is %[n 9 ,fig4 #gh4 ,illu/ration3 ,,fig4 #gh4 #egd4 ,! explanation ( ! developm5t ( ! r+ involves "s additional pr9ciples3 z ! sph]e ( oil sp9s r\nd 9 ! liquid1 its surface is ret>d$ by friction2 s t :5 ! velocity atta9s a c]ta9 am\nt1 ! int]nal portions ( ! sph]e1 :i* are 9 ! neighb\rhood ( ! sp9dle1 are driv5 f ! c5tre into ! out] portions1 b ! full acc\nt ( ! ph5om5on _c be giv5 "h4 #ege4 ,! e>? 01 we believe1 orig9ally 9 a fluid condition4 ,x _h !n1 z x has n[1 a diurnal rotation1 & "o ( ! consequ5ces ( ?is rotation has be5 to cause ! =m to be slightly protub]ant at ! equator1 j z we h se5 ! sph]e ( oil to bulge out "u simil> conditions4 #egf4 ,bodies ly+ on ! e>? are :irl$ ar\nd 9 a great circle e day4 ,h5ce1 if !re 7 n "s =ce draw+ !m to ! c5tre1 !y w\ld fly (f at a tang5t4 ,a "p ( ! e>?'s attraction goes = ?is purpose1 & ! rema9d]1 :i* is ! app>5t weight1 is ?us dim9i%$ by a quantity increas+ f ! pole to ! equator 7,art4 #hf74 ,,! ,,applications ,,( ,,circul> ,,motion4 #egg4 ,^! pr9ciples h _m applications 9 ! me*anical arts2 we %all m5tion two ( !m4 ,! fir/ is to ! gov]nor-balls ( a /eam-eng9e2 ! second is to ! process ( sug> ref9+4 ,an eng9e :i* turns a numb] ( ma*9es 9 a factory %\ld "w uni=mly4 ,irregul>ities ( motion may be productive ( loss & v>i\s inconv5i5ces4 ,an eng9e w\ld "w irregul>ly ei!r f v>iation 9 ! production ( /eam1 or f ! dem&s ^u ! p[] be+ less5$ or increas$4 ,ev5 if ! fir/ ( ^! s\rces ( irregul>ity c\ld be avoid$ by c>e1 x is cle> t ! second c\ld n4 ,"s ma*9es 9 ! mill are occasionally /opp$1 o!rs occasionally set 9 motion1 & ! eng9e g5]ally t5ds to g fa/] ! less x has to d4 ,x is !re=e necess>y to provide means by :i* ! spe$ %all be re/ra9$ )9 n>r[ limits1 & x is obvi\sly desirable t ! contrivance us$ = ?is purpose %\ld be self-act+4 ,we mu/1 !re=e1 h "s arrangem5t :i* %all admit m /eam to ! cyl9d] :5 ! eng9e is mov+ too sl[ly1 & less /eam :5 x is mov+ too quickly4 ,! valve :i* is to regulate ?is mu/ be work$ by "s ag5t :i* dep5ds ^u ! velocity ( ! eng9e2 ?is at once po9ts to circul> motion because ! =ce act+ on ! revolv+ body dep5ds ^u its velocity4 ,su* 0 ! tra9 ( reason+ :i* l$ to ! happy inv5tion ( ! gov]nor-balls3 ^! are %[n 9 ,fig4 #gi4 ,illu/ration3 ,,fig4 #gi4 ,a ,b is a v]tical sp9dle :i* is turn$ by ! eng9e4 ,p ,p is a piece firmly atta*$ to ! sp9dle & turn+ ) x4 ,p ,w1 ,p ,w are arms t]m9at+ 9 weights ,w ,w2 ^! are balls ( iron1 g5]ally v massive3 ! arms are free to turn r\nd p9s at ,p ,p4 ,at ,q ,q l9ks are plac$1 atta*$ to ano!r piece ,r ,r1 :i* is free to slide up & d[n ! sp9dle4 ,:5 ,a ,b rotates1 ,w & ,w are c>ri$ r\nd1 & !re=e fly outw>ds f ! sp9dle2 to d ?is !y mu/ evid5tly pull ! piece ,r ,r up ! %aft4 ,we c easily imag9e an arrangem5t by :i* ,r ,r %all be made to %ut or op5 ! /eam-valve accord+ z x asc5ds or desc5ds4 ,! problem is !n solv$1 = if ! eng9e beg9 to g too rapidly1 ! balls fly out fur!r j z !y did 9 ,fig4 #gd3 ?is movem5t raises ! piece ,r ,r1 :i* dim9i%es ! supply ( /eam1 & consequ5tly *ecks ! spe$4 ,on ! o!r h&1 :5 ! eng9e works too sl[ly1 ! balls fall 9 t[>ds ! sp9dle1 ! piece ,r ,r desc5ds1 ! valve is op5$1 & a great] supply ( /eam is admitt$4 ,! objection to ?is gov]nor is t ?\gh x mod]ates1 x does n completely *eck irregul>ity4 ,!re are o!r gov]nors occasionally employ$ :i* dep5d also on circul> motion2 "s ( ^! are m s5sitive ?an ! gov]nor-balls2 b !y are elaborate ma*9es1 only to be employ$ "u exceptional circum/ances4 #egh4 ,! application ( circul> motion to sug> ref9+ is a v beautiful inv5tion4 ,to expla9 x ,i mu/ briefly describe ! process ( ref9+4 ,! raw sug> is dissolv$ 9 wat]1 & ! solution is purifi$ by /ra9+ & by filtration ?r\gh animal *>coal4 ,! syrup is !n boil$4 ,9 ord] to pres]ve ! col\r ( ! sug>1 & to prev5t loss1 ?is boil+ is conduct$ 9 vacuo1 z by ?is means ! temp]ature requir$ is mu* less ?an w\ld be necess>y ) ! ord9>y atmosph]ic pressure4 ,! evaporation hav+ be5 complet$1 cry/als ( sug> =m ?r\gh\t ! mass ( syrup4 ,to sep>ate ^! cry/als f ! liquor :i* surr\nds !m1 ! aid ( circul> motion =ce is call$ 94 ,a mass ( ! mixture is plac$ 9 a l>ge iron tub1 ! sides ( :i* are p]=at$ ) small holes4 ,! tub is !n made to rotate ) prodigi\s velocity2 its cont5ts in/antly fly (f to ! circumf]5ce1 ! liquid portions f9d an exit ?r\gh ! p]=ations 9 ! sides1 b ! cry/als are left beh9d4 ,a little cle> syrup is !n spr9kl$ ov] ! sug> :ile /ill rotat+3 ?is wa%es f ! cry/als ! la/ traces ( ! col\r$ liquid1 & passes out ?r\gh ! holes2 :5 ! motion ceases1 ! inside ( ! tub conta9s a lay] ( p]fectly pure :ite sug>1 "s in*es 9 ?ickness1 ready = ! m>ket4 #egi4 ,circul> motion is peculi>ly fitt$ = ?is purpose2 ea* p>ticle ( liquid /rives to get z f> away f ! axis z possible4 ,! action on ! sug> is v diff]5t f :at x w\ld h be5 _h ! mass be5 subject$ to pressure by a screw-press or simil> contrivance2 ! p>ticles imm$iately act$ on w\ld !n h to transmit ! pressure to ^? )92 & ! consequ5ce w\ld be t :ile ! cry/als ( sug> on ! outside w\ld be cru%$ & de/roy$1 ! wat] w\ld only be v imp]fectly driv5 f ! int]ior3 = x c\ld lurk 9 ! int]/ices ( ! sug>1 :i* rema9 not)/&+ ! pressure4 #ehj4 ,b ) circul> motion ! wat] mu/ g1 n because x is pu%$ by ! cry/als1 b because ( its own in]tia2 & x c be p]fectly expell$ by a velocity ( rotation less ?an t :i* w\ld be necess>y to produce suffici5t pressure to make ! cry/als injure ea* o!r4 ,,! ,,p]man5t ,,axes4 #eha4 ,!re are "s curi\s prop]ties ( circul> motion :i* rema9 to be consid]$4 ,^! we %all inve/igate by means ( ! app>atus ( ,fig4 #hj4 ,?is consi/s ( a pair ( :eels ,b ,c1 by :i* a consid]able velocity c be giv5 to a horizontal %aft4 ,?is %aft is connect$ by a pair ( bevell$ :eels ,d ) a v]tical sp9dle ,f4 ,! ma*9e is work$ by a h&le ,a1 & ! object to be exp]im5t$ ^u is susp5d$ f ! sp9dle4 #ehb4 ,i fir/ take a disk ( wood #ah 9 diamet]2 a hole is bor$ 9 ! m>g9 ( ?is disk2 ?r\gh ?is hole a rope is fa/5$1 by means ( :i* ! disk is susp5d$ f ! sp9dle4 ,! disk hangs ( c\rse 9 a v]tical plane4 ,illu/ration3 ,,fig4 #hj4 #ehc4 ,i n[ beg9 to turn ! h&le r\nd g5tly1 & y see ! disk beg9s to rotate ab\t ! v]tical diamet]2 b1 z ! spe$ increases1 ! motion becomes a little un/eady2 & f9ally1 :5 ,i turn ! h&le v rapidly1 ! disk spr+s up into a horizontal plane1 & y see x l ! surface ( a small table3 ! rope sw+s r\nd & r\nd 9 a cone1 s rapidly t x is h>dly se54 #ehd4 ,we may repeat ! exp]im5t 9 a diff]5t mann]4 ,i take a piece ( iron *a9 ab\t #b' long1 ,g2 ,i pass ! rope ?r\gh ! two la/ l9ks ( its extremities1 & susp5d ! rope f ! sp9dle4 ,:5 ,i comm5ce to turn ! h&le1 y see ! *a9 gradually op5s out into a loop ,h2 & z ! spe$ increases1 ! loop becomes a complete r+4 ,/ill increas+ ! spe$1 ,i f9d ! r+ becomes un/eady1 till f9ally x rises into a horizontal plane4 ,! r+ ( *a9 9 ! horizontal plane is %[n at ,i4 ,:5 ! motion is fur!r increas$1 ! r+ sw+s ab\t viol5tly1 & s ,i cease turn+ ! h&le4 ,illu/ration3 ,,fig4 #ha4 #ehe4 ,! pr9ciples already enunciat$ w expla9 ^! rem>kable results2 we %all only describe t ( ! *a91 z ! same explanation w include t ( ! disk ( wood4 ,we %all beg9 ) ! *a9 hang+ v]tically f ! sp9dle3 ! mom5t rotation comm5ces1 ! *a9 beg9s to sp9 ab\t a v]tical axis2 ! p>ts ( ! *a9 fly outw>ds f ?is axis j z ! ball flies outw>ds 9 ,fig4 #gd2 ?is is ! cause ( ! loop$ =m ,h :i* ! *a9 assumes4 ,z ! spe$ is increas$ ! loop gradually op5s m & m1 j z ! diamet] ( ! circle ,fig4 #gd increases ) ! velocity4 ,b we h also to inquire into ! cause ( ! rem>kable *ange ( position :i* ! r+ und]goes2 in/ead ( cont9u+ to rotate ab\t a v]tical diamet]1 x comes into a horizontal plane4 ,?is w be easily und]/ood ) ! help ( ,fig4 #ha4 ,let ,o ,p repres5t ! rope atta*$ to ! r+1 & ,o ,c be ! v]tical axis4 ,suppose ! r+ to be sp9n+ ab\t ! axis ,o ,c1 :5 ,o ,c 0 a diamet]2 if !n1 f any cause1 ! r+ be slightly displac$1 we c %[ t ! circul> motion w t5d to drive ! r+ fur!r f ! v]tical plane1 & =ce x into ! horizontal plane4 ,let ! r+ be 9 ! position repres5t$ 9 ! figure2 !n1 s9ce x revolves ab\t ! v]tical l9e ,o ,c1 ! t5d5cy ( ,p ,p & ,q ,q is to move outw>ds 9 ! directions ( ! arr[s1 ?us evid5tly t5d+ to br+ ! r+ into ! horizontal plane4 #ehf4 ,9 ,art4 #ajc we h expla9$ :at is meant by /able & un/able equilibrium2 we h "h f\nd a precisely analog\s ph5om5on 9 motion4 ,! rotation ( ! r+ ab\t its diamet] is un/able1 = ! m9ute/ deviation ( ! r+ f ?is position is fatal2 circum/ances imm$iately comb9e to augm5t ! deviation m & m1 until f9ally ! r+ is rais$ into ! horizontal plane4 ,once 9 ! horizontal plane1 ! motion !re is /able1 = if ! r+ be displac$ ! t5d5cy is to re/ore x to ! horizontal4 #ehg4 ,! r+1 :5 9 a horizontal plane1 rotates p]man5tly ab\t ! v]tical axis ?r\gh its c5tre2 ?is axis is call$ p]man5t1 to di/+ui% x f all o!r directions1 z be+ ! only axis ab\t :i* ! motion is /able4 #ehh4 ,we may %[ ano!r exp]im5t ) ! *a93 if in/ead ( pass+ ! rope ?r\gh ! l9ks at its ends1 ,i pass ! rope ?r\gh ! c5tre ( ! *a91 & all[ ! ends ( ! *a9 to hang d[nw>ds4 ,i n[ turn ! h&le2 in/antly ! p>ts ( ! *a9 fly outw>ds 9 a curv$ =m2 & by increas+ ! velocity1 ! p>ts ( ! *a9 at l5g? come to lie almo/ 9 a /raight l9e4 ,,lecture ,,xviii4 ,,! ,,simple ,,p5dulum4 ,introduction4--,! ,circul> ,p5dulum4--,law connect+ ! ,"t ( ,vibration ) ! ,l5g?4--,! ,=ce ( ,gravity det]m9$ by ! ,p5dulum4--,! ,cycloid4 ,,introduction4 #ehi4 ,if a weight be atta*$ to a piece ( /r+1 ! o!r end ( :i* hangs f a fix$ po9t1 we h :at is call$ a simple p5dulum4 ,! p5dulum is ( ! utmo/ importance 9 sci5ce1 z well z = its practical applications z a "t-keep]4 ,9 ?is lecture & ! next we %all treat ( its g5]al prop]ties2 & ! la/ w be devot$ to ! practical applications4 ,we %all comm5ce ) ! simple p5dulum1 z already def9$1 & prove1 by exp]im5t1 ! rem>kable prop]ty :i* 0 discov]$ by ,galileo4 ,! simple p5dulum is (t5 call$ ! circul> p5dulum4 ,,! ,,circul> ,,p5dulum4 ,illu/ration3 ,,fig4 #hb4 #eij4 ,we fir/ exp]im5t ) a p5dulum on a l>ge scale4 ,our lecture !atre is #cb feet high1 & !re is a wire susp5d$ f ! ceil+ #bg' long2 to ! end ( ?is a ball ( ca/ iron weigh+ #be lbs4 is atta*$4 ,?is wire :5 at re/ hangs v]tically 9 ! direction ,o ,c 7,fig4 #hb74 ,i draw ! ball f its position ( re/ to ,a2 :5 releas$1 x sl[ly returns to ,c1 its orig9al position2 x !n moves on ! o!r side to ,b1 & back aga9 to my h& at ,a4 ,! ball--or to speak m precisely1 ! c5tre ( ! ball--moves 9 a circle1 ! c5tre be+ ! po9t ,o 9 ! ceil+ f :i* ! wire is susp5d$4 #eia4 ,:at causes ! motion ( ! p5dulum :5 ! weight is releas$8 ,x is ! =ce ( gravity2 = by mov+ ! ball to ,a ,i raise x a little1 & !re=e1 :5 ,i release x gravity compels x to return to ,c x be+ ! only mann] 9 :i* ! mode ( susp5sion w all[ x to fall4 ,b :5 x has rea*$ its orig9al position at ,c1 :y does x cont9ue its motion8--= gravity mu/ be act+ aga9/ ! ball dur+ ! j\rney f ,c to ,b4 ,! fir/ law ( motion expla9s ?is4 7,art4 #dhe74 ,9 travell+ f ,a to ,c ! ball has acquir$ a c]ta9 velocity1 h5ce x has a t5d5cy to g on1 & only by ! "t x has arriv$ at ,b w gravity h arre/$ ! velocity1 & beg9 to make x desc5d4 #eib4 ,y see1 ! ball cont9ues mov+ to & fro--oscillat+1 z x is call$--= a long "t4 ,! fact is t x w\ld oscillate = ev]1 7 x n = ! resi/ance ( ! air1 & = "s loss ( en]gy at ! po9t ( susp5sion4 #eic4 ,by ! "t ( an oscillation is meant ! "t ( go+ f ,a to ,b1 b n back aga94 ,! "t ( our long p5dulum is ne>ly ?ree seconds4 #eid4 ,) ref]5ce to ! "t ( oscillation ,galileo made a great discov]y4 ,he f\nd t :e!r ! p5dulum 7 sw++ ?r\gh ! arc ,a ,b1 or :e!r x _h be5 br\ght to ! po9t ,a1 & 0 ?us describ+ ! arc ,a ,b1 ! "t ( oscillation rema9$ ne>ly ! same4 ,! arc ?r\gh :i* ! p5dulum oscillates is call$ its amplitude1 s t we may enunciate ?is tru? by say+ t ! "t ( oscillation is ne>ly indep5d5t ( ! amplitude4 ,! means by :i* ,galileo prov$ ?is w\ld h>dly be adopt$ 9 mod]n days4 ,he all[$ a p5dulum to p]=m a c]ta9 numb] ( vibrations1 say #ajj1 ?r\gh ! arc ,a ,b1 & he c\nt$ 8 pulse dur+ ! "t2 he !n c\nt$ ! numb] ( pulsations :ile ! p5dulum vibrat$ #ajj times 9 ! arc ,a ,b1 & he f\nd ! numb] ( pulsations 9 ! two cases to be equal4 ,assum+1 :at is probably true1 t ,galileo's pulse rema9$ uni=m ?r\gh\t ! exp]im5t1 ?is result %[$ t ! p5dulum took ! same "t to p]=m #ajj vibrations1 :e!r x swung ?r\gh ! arc ,a ,b1 or ?r\gh ! arc ,a ,b4 ,?is discov]y x 0 :i* fir/ sugge/$ ! employm5t ( ! p5dulum z a means ( keep+ "t4 #eie4 ,we %all adopt a diff]5t me?od to %[ t ! "t does n dep5d ^u ! amplitude4 ,i h "h an arrangem5t :i* is repres5t$ 9 ,fig4 #hc4 ,x consi/s ( two p5dulums ,a ,d & ,b ,c1 ea* #ab' long1 & susp5d$ f two po9ts ,a ,b1 ab\t #a' ap>t1 9 ! same horizontal l9e4 ,ea* ( ^! p5dulums c>ries a weight ( ! same size3 !y are 9 fact id5tical4 ,illu/ration3 ,,fig4 #hc4 #eif4 ,i take "o ( ! balls 9 ea* h&4 ,if ,i )draw ea* ( !m f its position ( re/ ?r\gh equal di/ances & !n release !m1 bo? balls return to my h&s at ! same in/ant4 ,?is might h be5 expect$ f ! id5tity ( ! circum/ances4 #eig4 ,i next )draw ! weight ,c 9 my "r h& to a di/ance ( #a'1 & ! weight ,d 9 my left h& to a di/ance ( #b'1 & release !m simultane\sly4 ,:at happ5s8 ,i keep my h&s /eadily 9 ! same position1 & ,i f9d t ! two weights return to !m at ! same in/ant4 ,h5ce1 ?\gh "o ( ! weights mov$ ?r\gh an amplitude ( #b' 7,c ,e7 :ile ! o!r mov$ ?r\gh an amplitude ( #d' 7,d ,f71 ! times occupi$ by ea* 9 mak+ two oscillations are id5tical4 ,if ,i draw ! "r-h& ball away #c'1 :ile ,i draw ! left h& only #a' f _! respective positions ( re/1 ,i /ill obs]ve ! same result4 #eih4 ,9 two oscillations we c see no effect on ! "t produc$ by ! amplitude1 & we are correct 9 say+ t1 :5 ! amplitude is only a small fraction ( ! l5g? ( ! p5dulum1 its effect is inappreciable4 ,b if ! amplitude ( "o p5dulum 7 v l>ge1 we %\ld f9d t its "t ( oscillation is slightly great] ?an t ( ! o!r1 ?\gh to detect ! diff]5ce w\ld require a delicate te/4 ,"o consequ5ce ( :at is "h rem>k$ w be notic$ at a lat] page4 7,art4 #fee47 #eii4 ,we next inquire :e!r ! weight :i* is atta*$ to ! p5dulum has any influ5ce ^u ! "t ( vibration4 ,us+ ! #ab' p5dulums ( ,fig4 #hc1 ,i place a weight ( #ab lbs4 on "o hook & "o ( #f lbs4 on ! o!r4 ,i )draw "o 9 ea* h&2 ,i release !m2 !y return to my h& at ! same mom5t4 ,:e!r ,i )draw ! weights ?r\gh long arcs or %ort arcs1 equal or unequal1 !y inv>iably return toge!r1 & bo? !re=e h ! same "t ( vibration4 ,) o!r iron weights ! same law is confirm$1 & h5ce we le>n t1 besides be+ indep5d5t ( ! amplitude1 ! "t ( vibration is also indep5d5t ( ! weight4 #fjj4 ,f9ally1 let u see if ! mat]ial ( ! p5dulum c influ5ce its "t ( vibration4 ,i place a ball ( wood on "o wire & a ball ( iron on ! o!r2 ,i sw+ !m z be=e3 ! vibrations are /ill p]=m$ 9 equal times4 ,a ball ( lead is f\nd to sw+ 9 ! same "t z a ball ( brass1 & bo? 9 ! same "t z a ball ( iron or ( wood4 #fja4 ,9 ?is we may be rem9d$ ( ! exp]im5ts on gravity 7,art4 #dia71 :]e we %[$ t all bodies fall to ! gr\nd 9 equal times1 :atev] be _! sizes or _! mat]ials4 ,f bo? cases ! inf]5ce is drawn t ! =ce ( gravity ^u diff]5t bodies is proportional to _! masses1 ?\gh ! bodies be made ( v>i\s sub/ances4 ,x 0 inde$ by means ( exp]im5ts ) ! p5dulum t ,newton prov$ t gravity _h ?is prop]ty1 :i* is "o ( ! mo/ rem>kable tru?s 9 nature4 ,,law ,,connect+ ,,! ,,"t ,,( ,,vibration ,,) ,,! ,,l5g?4 #fjb4 ,we h se5 t ! "t ( vibration ( a p5dulum dep5ds nei!r ^u its amplitude1 mat]ial1 nor weight2 we h n[ to le>n on :at ! "t does dep5d4 ,x dep5ds ^u ! l5g? ( ! p5dulum4 ,! %ort] a p5dulum ! less is its "t ( vibration4 ,we %all f9d by exp]im5t ! relation betwe5 ! "t & ! l5g? ( ! cord by :i* ! weight is susp5d$4 ,illu/ration3 ,,fig4 #hd4 #fjc4 ,i h "h 7,fig4 #hd7 two p5dulums ,a ,d1 ,b ,c1 "o ( :i* is #ab' long & ! o!r #c'2 !y are m\nt$ side by side1 & ! weights are at ! same di/ance f ! floor4 ,i take "o ( ! weights 9 ea* h&1 & )draw !m to ! same di/ance f ! position ( re/4 ,i release ! balls simultane\sly2 ,c moves (f rapidly1 arrives at ! end ,c :ile ,d has only rea*$ ,d1 & returns to my h& j z ,d has complet$ "o oscillation4 ,i d n seize ,c3 x goes (f aga91 only to return at ! same mom5t :5 ,d rea*es my h&4 ,?us ,c has p]=m$ f\r oscillations :ile ,d has made no m ?an two4 ,?is proves t :5 "o ( two p5dulums is a qu>t] ! l5g? ( ! o!r1 ! "t ( vibration ( ! %ort] "o is half t ( ! o!r4 #fjd4 ,we %all repeat ! exp]im5t ) ano!r p5dulum #bg' long1 susp5d$ f ! ceil+1 & comp>e its vibrations ) ^? ( a p5dulum #c' long4 ,i draw ! weights to "o side & release !m z be=e2 & y see t ! small p5dulum returns twice to my h& :ile ! long p5dulum is /ill abs5t2 b t1 keep+ my h&s /eadily 9 ! same place ?r\gh\t ! exp]im5t1 ! long p5dulum at la/ returns simultane\sly ) ! ?ird arrival ( ! %ort "o4 ,h5ce we le>n t a p5dulum #bg' long takes ?ree times z mu* "t = a s+le vibration z a #c' p5dulum4 #fje4 ,! l5g?s ( ! ?ree p5dulums on :i* we h exp]im5t$ 7#bg'1 #ab'1 #c'71 are 9 ! proportions ( ! numb]s #i1 #d1 #a2 & ! times ( ! oscillations are proportional to #c1 #b1 #a3 h5ce we le>n t ! p]iod ( oscillation ( a p5dulum is proportional to ! squ>e root ( its l5g?4 #fjf4 ,b ! "t ( vibration mu/ also dep5d ^u gravity2 = x is only ow+ to gravity t ! p5dulum vibrates at all4 ,x is evid5t t1 if gravity 7 increas$1 all bodies w\ld fall to ! e>? m ?an #af' 9 ! fir/ second4 ,! effect on ! p5dulum w\ld be to draw ! ball m quickly f ,d to ,d 7,fig4 #hd71 & ?us ! "t ( vibration w\ld be dim9i%$4 ,x is f\nd by calculation1 & ! result is confirm$ by exp]im5t1 t ! "t ( vibration is repres5t$ by ! expression1 #cadaf 7 ,l5g? _/ ,=ce ( gravity74 #fjg4 ,! accurate value ( ! =ce ( gravity 9 ,london is #cbaijh1 s t ! "t ( vibration ( a p5dulum !re is #jeecg l5g?3 ! l5g? ( ! seconds p5dulum is #c'bfaf4 ,,! ,,=ce ,,( ,,gravity ,,det]m9$ ,,by ,,! ,,p5dulum4 #fjh4 ,! p5dulum af=ds ! prop] means ( measur+ ! =ce ( gravity at any place on ! e>?4 ,we h se5 t ! "t ( vibration c be express$ 9 t]ms ( ! l5g? & ! =ce ( gravity2 s conv]sely1 :5 ! l5g? & ! "t ( vibration are kn[n1 ! =ce ( gravity c be det]m9$ & ! expression = x is-- ,l5g? 7#cadaf _/ ,"t74 #fji4 ,x is impossible to obs]ve ! "t ( a s+le vibration ) ! necess>y degree ( accuracy2 b suppos+ we consid] a l>ge numb] ( vibrations1 say #ajj1 & f9d ! "t tak5 to p]=m !m1 we %all !n le>n ! "t ( "o oscillation by divid+ ! entire p]iod by #ajj4 ,! amplitudes ( ! oscillations may dim9i%1 b !y are /ill p]=m$ 9 ! same "t2 & h5ce1 if we are sure t we h n made a mi/ake ( m ?an "o second 9 ! :ole "t1 !re _c be an error ( m ?an #jja second1 9 ! "t ( "o oscillation4 ,by tak+ a /ill l>g] numb] ! "t may be det]m9$ ) ! utmo/ precision1 s t ?is "p ( ! inquiry pres5ts little difficulty4 #faj4 ,b ! l5g? ( ! p5dulum has also to be asc]ta9$1 & ?is is a r baffl+ problem4 ,! ideal p5dulum ^: l5g? is requir$1 is suppos$ to be compos$ ( a v f9e1 p]fectly flexible cord1 at ! end ( :i* a p>ticle )\t appreciable size is atta*$2 b ?is is v diff]5t f ! p5dulum :i* we mu/ employ4 ,we are n sure ( ! exact position ( ! po9t ( susp5sion1 &1 ev5 if we _h a p]fect sph]e = ! weight1 ! di/ance betwe5 its c5tre & ! po9t ( susp5sion is n ! same ?+ z ! l5g? ( ! simple p5dulum t w\ld vibrate 9 ! same "t4 ,ow+ to ^! circum/ances1 ! measurem5t ( ! p5dulum is emb>rass$ by consid]able difficulties1 :i* h h[ev] be5 ov]come by +5i\s contrivances to be describ$ 9 ! next *apt]4 #faa4 ,b we %all p]=m1 9 a v simple way1 an exp]im5t = det]m9+ ! =ce ( gravity4 ,i h "h a silk5 ?read :i* is fa/5$ by be+ clamp$ betwe5 two pieces ( wood4 ,a ca/ iron ball #bed 9 diamet] is susp5d$ by ?is piece ( silk4 ,! di/ance f ! po9t ( susp5sion ( ! silk to ! ball is #bdjg1 z well z x c be measur$4 ,! l5g? ( ! ideal p5dulum :i* w\ld vibrate iso*ron\sly ) ?is p5dulum is #becg1 be+ ab\t #jjc great] ?an ! di/ance f ! po9t ( susp5sion to ! c5tre ( ! sph]e4 #fab4 ,! l5g? hav+ be5 asc]ta9$1 ! next elem5t to be det]m9$ is ! "t ( vibration4 ,= ?is purpose ,i use a /op-wat*1 :i* c be />t$ or /opp$ in/antane\sly by t\*+ a little /ud3 ?is wat* w indicate "t accurately to "o-fif? ( a second4 ,x is necess>y t ! p5dulum %\ld sw+ 9 a small arc1 z o!rwise ! oscillations are n /rictly iso*ron\s4 ,q suffici5t amplitude is obta9$ by all[+ ! ball to rotate to & fro ?r\gh a few t5?s ( an in*4 #fac4 ,9 ord] to obs]ve ! movem5t easily1 ,i h m\nt$ a little telescope1 ?r\gh :i* ,i c view ! top ( ! ball4 ,9 ! eye-piece ( ! telescope a v]tical wire is fa/5$1 & ,i c\nt ea* vibration j z ! silk5 ?read passes ! v]tical wire4 ,tak+ my seat ) ! /op-wat* 9 my h&1 ,i write d[n ! position ( ! h&s ( ! /op-wat*1 & !n look ?r\gh ! telescope4 ,i see ! p5dulum sl[ly mov+ to & fro1 cross+ ! v]tical wire at e vibration2 on "o occasion1 j z x passes ! wire1 ,i t\* ! /ud & />t ! wat*4 ,i all[ ! p5dulum to make #cjj vibrations1 & z ! silk arrives at ! v]tical wire = ! #cjj? "t1 ,i promptly /op ! wat*2 on ref]5ce ,i f9d t #bdaf seconds h elaps$ s9ce ! "t ! wat* 0 />t$4 ,to avoid error1 ,i repeat ?is exp]im5t1 ) precisely ! same result3 #bdaf seconds are aga9 requir$ = ! completion ( #cjj vibrations4 #fad4 ,x is desirable to reckon ! vibrations f ! in/ant :5 ! p5dulum is at ! middle ( its /roke1 r ?an :5 x arrives at ! end ( ! sw+4 ,9 ! =m] case ! p5dulum is mov+ ) ! greate/ rapidity1 & !re=e ! "t ( co9cid5ce betwe5 ! ?read & ! v]tical wire c be obs]v$ ) especial def9it5ess4 #fae4 ,! "t ( a s+le vibration is f\nd1 by divid+ #bdaf by #cjj1 to be #jhje second4 ,?is is c]ta9ly correct to )9 a ?\s&? "p ( a second4 ,we conclude t a p5dulum ^: l5g? is #becg #baad1 vibrates 9 #jhje second2 & f ?is we f9d t gravity at ,dubl9 is #b'aad 7#cadaf _/ #j4hje7 #cbaif4 ,?is result agrees ) "o :i* has be5 det]m9$ by measurem5t made ) e precaution4 ,ano!r me?od ( measur+ gravity by ! p5dulum w be describ$ 9 ! next lecture 7,art4 #fcg74 ,,! ,,cycloid4 #faf4 ,if ! amplitude ( ! vibration ( a circul> p5dulum be> a l>ge proportion to ! radius1 ! "t ( oscillation is slightly great] ?an if ! amplitude be v small4 ,! iso*ronism ( ! p5dulum is only true = small arcs4 #fag4 ,b !re is a curve 9 :i* a weight may be made to move :]e ! "t ( vibration is precisely ! same1 :atev] be ! amplitude4 ,?is curve is call$ a cycloid4 ,x is ! pa? describ$ by a nail 9 ! circumf]5ce ( a :eel1 z ! :eel rolls along ! gr\nd4 ,?us1 if a circle be roll$ und]nea? ! l9e ,a ,b 7,fig4 #he71 a po9t on its circumf]5ce describes ! cycloid ,a ,d ,c ,p ,b4 ,! l[] "p ( ?is curve does n diff] v mu* f a circle ^: c5tre is a c]ta9 po9t ,o above ! curve4 #fah4 ,suppose we _h a piece ( wire c>efully %ap$ to ! cycloidal curve ,a ,d ,c ,p ,b1 & t a r+ c\ld slide along x )\t friction1 x w\ld be f\nd t1 :e!r ! r+ be all[$ to drop f ,c1 ,p or ,b1 x w\ld fall to ,d precisely 9 ! same "t1 & w\ld !n run up ! wire to a di/ance f ,d on ! o!r side equal to t f :i* x _h orig9ally />t$4 ,9 ! oscillations on ! cycloid1 ! amplitude is absolutely )\t effect ^u ! "t4 #fai4 ,z a frictionless wire is impossible1 we _c adopt ?is me?od1 b we c nev]!less con/ruct a cycloidal p5dulum 9 ano!r way1 by utiliz+ a prop]ty ( ! curve1 ,o ,a 7,fig4 #he7 z a half cycloid2 9 fact1 ,o ,a is j ! same curve z ,b ,d1 b plac$ 9 a diff]5t position1 s also is ,o ,b4 ,if a /r+ ( l5g? ,o ,d be susp5d$ f ! po9t ,o1 & h a weight atta*$ to x1 ! weight w describe ! cycloid1 provid$ t ! /r+ wrap itself along ! arcs ,o ,a & ,o ,b2 ?us :5 ! weight has mov$ f ,d to ,p1 ! /r+ is wrapp$ along ! curve ?r\gh ! space ,o ,t1 ! "p ,t ,p only be+ free4 ,?is arrangem5t w always =ce ! po9t ,p to move 9 ! cycloidal arc4 ,illu/ration3 ,,fig4 #he4 #fbj4 ,we are n[ 9 a condition to asc]ta9 exp]im5tally1 :e!r ! "t ( oscillation 9 ! cycloid is indep5d5t ( ! amplitude4 ,we use = ?is purpose ! app>atus %[n 9 ,fig4 #hf4 ,d ,c ,e is ! arc ( ! cycloid2 two /r+s are atta*$ at ,o1 & equal weights ,a1 ,b are susp5d$ f !m2 ,c is ! l[e/ po9t ( ! curve4 ,! "t ,a w take to fall ?r\gh ! arc ,a ,c is ( c\rse half ! "t ( its oscillation4 ,if1 !re=e1 ,i c %[ t ,a & ,b bo? take ! same "t to fall d[n to ,c1 ,i %all h prov$ t ! vibrations are iso*ron\s4 ,illu/ration3 ,,fig4 #hf4 #fba4 ,hold+1 z %[n 9 ! figure1 ,a 9 "o h& & ,b 9 ! o!r1 ,i release !m simultane\sly1 & y see ! result1--!y bo? meet at ,c3 ev5 if ,i br+ ,a up to ,e1 & br+ ,b d[n close to ,c1 ! result is ! same4 ,! motion ( ,a is s rapid t x arrives at ,c j at ! same in/ant z ,b4 ,:5 ,i br+ ! two balls on ! same side ( ,c1 & release !m simultane\sly1 ,a ov]takes ,b j at ! mom5t :5 x is pass+ ,c4 ,h5ce1 "u all circum/ances1 ! times ( desc5t are equal4 #fbb4 ,x w be notic$ t ! /r+ atta*$ to ! ball ,b1 9 ! position %[n 9 ! figure1 is almo/ z free z if x 7 m]ely susp5d$ f ,o1 = x is only :5 ! ball is "s di/ance f ! l[e/ po9t t ! side arcs produce any appreciable effect 9 curv+ ! /r+4 ,! ball sw+s f ,b to ,c ne>ly 9 a circle ( :i* ! c5tre is at ,o4 ,h5ce1 9 ! circul> p5dulum1 ! vibrations :5 small are iso*ron\s1 = 9 t case ! cycloid & ! circle become indi/+ui%able4 ,,lecture ,,xix4 ,,! ,,comp\nd ,,p5dulum ,,& ,,! ,,composition ,,( ,,vibrations4 ,! ,comp\nd ,p5dulum4--,! ,c5tre ( ,oscillation4--,! ,c5tre ( ,p]cussion4--,! ,conical ,p5dulum4--,! ,composition ( ,vibrations4 ,,! ,,comp\nd ,,p5dulum4 ,illu/ration3 ,,fig4 #hg4 #fbc4 ,p5dul\s motion mu/ n[ be /udi$ 9 o!r =ms besides t ( ! simple p5dulum1 :i* consi/s ( a weight & a cord4 ,any body rotat+ ab\t an axis may be made to oscillate by gravity4 ,a body ?us vibrat+ is call$ a comp\nd p5dulum4 ,! ideal =m1 :i* consi/s ( an indef9itely small weight atta*$ to a p]fectly flexible & impond]able /r+1 is an ab/raction :i* c only be approximately imitat$ 9 nature4 ,x foll[s t e p5dulum us$ 9 our exp]im5ts is /rictly speak+ comp\nd4 #fbd4 ,! fir/ p5dulum ( ?is class :i* we %all notice is t us$ 9 ! common clock 7,fig4 #hg74 ,?is consi/s ( a wood5 or /eel rod ,a ,e1 to :i* a brass or lead5 bob ,b is atta*$4 ,?is p5dulum is susp5d$ by means ( a /eel spr+ ,c ,a1 :i* be+ v flexible1 all[s ! vibration to be p]=m$ ) consid]able fre$om4 ,! use ( ! screw at ! end w be expla9$ 9 ,art4 #ffd4 ,a p5dulum l ?is vibrates iso*ron\sly1 :5 ! amplitude is small1 b x is n easy to see precisely :at is ! l5g? ( ! simple p5dulum :i* w\ld oscillate 9 ! same "t4 ,9 ! fir/ place1 we are unc]ta9 z to ! virtual position ( ! po9t ( susp5sion1 = ! spr+1 ?\gh flexible1 w n yield at ! po9t ,c to ! same ext5t z a /r+2 ?us ! effective po9t ( susp5sion mu/ be some:at l[] ?an ,c4 ,! o!r extremity is /ill m unc]ta91 = ! weight1 s f> f be+ a s+le po9t1 is n exclusively 9 ! neighb\rhood ( ! bob1 inasmu* z ! rod ( ! p5dulum has a mass t is appreciable4 ,?is =m ( p5dulum _c !re=e be us$ :]e x is necess>y to det]m9e ! l5g? ) accuracy4 #fbe4 ,:5 ! l5g? ( a p5dulum is to be measur$1 we mu/ adopt o!r means ( support+ x ?an t ( susp5sion by a spr+1 z o!rwise we _c h a def9ite po9t f :i* to measure4 ,to illu/rate ! mode t is to be adopt$1 ,i take "h an iron b> #f' long & #a squ>e1 :i* weighs #ai lbs4 ,i wi% to support ?is at "o end s t x c vibrate freely1 & at ! same "t h a def9ite po9t ( susp5sion4 ,i h "h two small prisms ( /eel ,e 7,fig4 #hh7 fa/5$ to a brass frame2 ! faces ( ! prisms meet at ab\t an angle ( #fj & =m ! edges ab\t :i* ! oscillation takes place3 ?is frame & ! edges c be plac$ on ! end ( ! b>1 & c be fix$ !re by tight5+ two nuts4 ,! object ( hav+ ! edges on a slid+ frame is t !y may be applicable to diff]5t p>ts ( ! b> ) facility4 ,9 "s in/rum5ts us$ 9 exp]im5ts requir+ extreme delicacy1 ! edges :i* are atta*$ to ! p5dulum are support$ ^u plates ( agate2 !y are to be adju/$ on ! same horizontal l9e1 & ! p5dulum really vibrates ab\t ?is l9e1 z ab\t an axis4 ,= our purpose x w be suffici5t to support ! edges ^u small pieces ( /eel4 ,a ,b1 ,fig4 #hh1 repres5ts "o side ( ! top ( ! iron b>2 ,e is ! edge project+ f x1 ) its edge p]p5dicul> to ! b>4 ,c ,d is a /eel plate be>+ a knife edge on its upp] surface2 ?is piece ( /eel is firmly secur$ to ! framework4 ,!re is ( c\rse a simil> piece on ! o!r side1 support+ ! o!r edge4 ,! b>1 ?us delicately pois$1 w1 :5 once />t$1 vibrate backw>ds & =w>ds = an h\r1 z !re is v little friction betwe5 ! edges & ! pieces :i* support !m4 ,illu/ration3 ,,fig4 #hh4 #fbf4 ,! g5]al appe>ance ( ! app>atus1 :5 m\nt$1 is %[n 9 ,fig4 #hi4 ,a ,b is ! b>3 at ,a ! two edges are %[n1 & also ! pieces ( /eel :i* support !m4 ,! :ole is c>ri$ by a horizontal beam bolt$ to two uprights2 & a glance at ! figure w expla9 ! arrangem5ts made to secure ! /ead9ess ( ! app>atus2 ! second pair ( edges %[n at ,b w be ref]r$ to pres5tly 7,art4 #fce74 #fbg4 ,?is b>1 z y see1 vibrates to & fro2 & we %all det]m9e ! l5g? ( a simple p5dulum :i* w\ld vibrate 9 ! same p]iod ( "t4 ,! l5g? might be d$uc$ by f9d+ ! "t ( vibration1 & !n calculat+ f ,art4 #fjf4 ,?is w\ld be ! mo/ accurate mode ( proce$+1 b ,i h pref]r$ to adopt a direct me?od :i* does n require calculation4 ,a simple p5dulum1 consi/+ ( a f9e cord & a small iron sph]e ,c1 is m\nt$ beh9d ! edge1 ,fig4 #hi4 ,! po9t f :i* ! cord is susp5d$ lies exactly 9 ! l9e ( ! two edges1 & !re is an adju/m5t = l5g!n+ or %ort5+ ! cord at pleasure4 ,illu/ration3 ,,fig4 #hi4 #fbh4 ,we fir/ try ) #f' ( cord1 s t ! simple p5dulum %all h ! same l5g? z ! b>4 ,tak+ ! ball 9 "o h& & ! b> 9 ! o!r1 ,i draw !m aside1 & y see1 :5 !y are releas$1 t ! b> p]=ms two vibrations & returns to my h& be=e ! ball4 ,h5ce ! l5g? ( ! iso*ron\s simple p5dulum is c]ta9ly less ?an ! l5g? ( ! b>2 = a p5dulum ( t l5g? is too sl[4 #fbi4 ,i n[ %ort5 ! cord until x is only half ! l5g? ( ! b>2 &1 repeat+ ! exp]im5t1 we f9d t ! ball returns be=e ! b>1 & !re=e ! simple p5dulum is too %ort4 ,h5ce we le>n t ! iso*ron\s p5dulum is great] ?an half ! l5g? ( ! b>1 & less ?an ! :ole l5g?4 #fcj4 ,let u f9ally try a simple p5dulum two-?irds ( ! l5g? ( ! b>4 ,i make ! exp]im5t1 & f9d t ! ball & ! b> return to my h& precisely at ! same in/ant4 ,!re=e two-?irds ( ! l5g? ( ! b> is ! l5g? ( ! iso*ron\s simple p5dulum4 ,we may /ate g5]ally t ! "t ( vibration ( a uni=m b> ab\t "o end equals t ( a simple p5dulum ^: l5g? is two-?irds ( ! b>2 no d\bt ! b> we h us$ is n /rictly uni=m1 because ( ! edges2 b 9 ! positions !y occupy1 _! influ5ce on ! "t ( vibrations is imp]ceptible4 #fcb4 ,= ?is rule to be v]ifi$1 x is ess5tially necess>y t ! edges be prop]ly situat$ on ! b>2 to illu/rate ?is we may exam9e ! oscillations ( ! small rod1 %[n at ,d 7,fig4 #hi74 ,?is rod is also ( iron #bd #je #je1 & x is susp5d$ f a po9t ne> ! c5tre by a pair ( edges2 if ! edges c\ld be plac$ s t ! c5tre ( gravity ( ! :ole lay 9 ! l9e ( ! edges1 x is evid5t t ! b> w\ld re/ indiff]5tly h[ev] x 7 plac$1 & w\ld n oscillate4 ,if !n ! edges be v ne> ! c5tre ( gravity1 we c easily und]/& t ! oscillations may be v sl[1 & ?is is actually ! case 9 ! b> ,d4 ,by ! aid ( ! /op-wat*1 ,i f9d t "o hundr$ vibrations are p]=m$ 9 #bdh seconds1 & t !re=e ea* vibration occupies #bdh seconds4 ,! l5g? ( ! simple p5dulum :i* has #bdh seconds = its p]iod ( oscillation1 is ab\t #bj'4 ,_h ! edges be5 at "o end1 ! l5g? ( ! simple p5dulum w\ld h be5 #bd #af4 ,a b> #gb long w vibrate 9 a %ort] "t :5 ! edge is #aeb f "o end ?an :5 x has any o!r position4 ,! l5g? ( ! correspond+ simple p5dulum is #daf4 ,,! ,,c5tre ,,( ,,oscillation4 #fcc4 ,x appe>s t correspond+ to ea* comp\nd p5dulum we h a specific l5g? equal to t ( ! iso*ron\s simple p5dulum4 ,to take z example ! #f' b> already describ$ 7,art4 #fbe71 ?is l5g? is #d'4 ,if ,i measure (f f ! edges a di/ance ( #d'1 & m>k ?is po9t ^u ! b>1 ! po9t is call$ ! c5tre ( oscillation4 ,m g5]ally ! c5tre ( oscillation is f\nd by draw+ a l9e equal to ! iso*ron\s simple p5dulum f ! c5tre ( oscillation ?r\gh ! c5tre ( gravity4 #fcd4 ,9 ! b> ,d ! c5tre ( oscillation w\ld be at a di/ance ( #bj' bel[ ! edges2 & 9 g5]al ! position w v>y ) ! position ( ! edges4 #fce4 ,9 ! #f' b> ,b is ! c5tre ( oscillation4 ,i take ano!r pair ( edges & place !m on ! b>1 s t ! l9e ( ! edges passes ?r\gh ,b4 ,i n[ lift ! b> c>efully & turn x upside d[n1 s t ! edges ,b re/ ^u ! /eel plates4 ,9 ?is position "o-?ird ( ! b> is above ! axis ( susp5sion1 & ! rema9+ two-?irds bel[4 ,a is ( c\rse n[ at ! bottom ( ! b>1 & is on a level ) ! ball1 ,c3 ! p5dulum is made to oscillate ab\t ! edges ,b1 & ! "t ( its vibration may be approximately det]m9$ by direct comp>ison ) ,c1 z already expla9$4 ,i f9d t1 :5 ,i all[ ,c & ! b> to sw+ toge!r1 !y bo? vibrate precisely 9 ! same "t4 ,y w rememb]1 t :5 ! ball 0 susp5d$ by a /r+ #d' long1 its vibrations 7 iso*ron\s ) ^? ( ! b> :5 susp5d$ f ! edges ,a4 ,)\t hav+ alt]$ ,c1 b hav+ made ! b> to vibrate ab\t ,b1 ,i f9d t ! "t ( oscillation ( ! b> is /ill equal to t ( ,c4 ,!re=e1 ! p]iod ( oscillation ab\t ,a is equal to t ab\t ,b4 ,h5ce1 :5 ! b> is vibrat+ ab\t ,b1 its c5tre ( oscillation mu/ be #d' f ,b1 t is1 x mu/ be at ,a3 s t :5 ! b> is susp5d$ f ,a1 ! c5tre ( oscillation is ,b2 :ile1 :5 ! b> is susp5d$ f ,b1 ! c5tre ( oscillation is ,a4 ,?is is an int]e/+ dynamical !orem4 ,x may be m concisely express$ by say+ t ! c5tre ( oscillation & ! c5tre ( susp5sion are reciprocal4 #fcf4 ,?\gh ! pro( t we h giv5 ( ?is curi\s law applies only to a uni=m b>1 yet ! law is itself true 9 g5]al1 :atev] be ! nature ( ! comp\nd p5dulum4 #fcg4 ,we allud$ 9 ! la/ lecture 7,art4 #faj7 to ! difficulty ( measur+ ) accuracy ! l5g? ( a simple p5dulum2 b ! reciprocity ( ! c5tres ( oscillation & susp5sion1 sugge/$ to ! +5i\s ,capta9 ,kat] a me?od by :i* ?is difficulty c\ld be evad$4 ,we %all expla9 ! pr9ciple4 ,let "o pair ( edges be at ,a4 ,let ! o!r pair ( edges1 ,b1 be mov$ z ne> z possible to ! c5tre ( oscillation4 ,we c te/ :e!r ,b has be5 plac$ correctly3 = ! "t tak5 by ! p5dulum to p]=m #ajj vibrations ab\t ,a %\ld be equal to ! "t tak5 to p]=m #ajj vibrations ab\t ,b4 ,if ! times are n q equal1 ,b mu/ be mov$ slightly until ! times are prop]ly br\ght to equality4 ,! l5g? ( ! iso*ron\s simple p5dulum is !n equal to ! di/ance betwe5 ! edges ,a & ,b2 & ?is di/ance1 f "o edge to ! o!r edge1 pres5ts none ( ! difficulties 9 its exact measurem5t :i* we _h be=e to cont5d )3 x c be f\nd ) precision4 ,h5ce1 kn[+ ! l5g? ( ! p5dulum & its "t ( oscillation1 gravity c be f\nd 9 ! mann] already expla9$ 7,art4 #fjh74 #fch4 ,i h adju/$ ! two edges ( ! #f' b> z ne>ly z ,i c\ld at ! c5tres ( oscillation & susp5sion1 & we %all proce$ to te/ ! correctness ( ! positions4 ,m\nt+ ! b> fir/ by ! edges at ,a1 ,i set x vibrat+4 ,i take ! /op-wat* already ref]r$ to 7,art4 #fab71 & record ! position ( its h&s4 ,i !n place my f+] on ! /ud1 &1 j at ! mom5t :5 ! b> is at ! middle ( "o ( its vibrations1 ,i />t ! wat*4 ,i c\nt a hundr$ vibrations2 & :5 ! p5dulum is aga9 at ! middle ( its /roke1 ,i /op ! wat*1 & f9d x records an int]val ( #aajd seconds4 ,?us ! "t ( "o vibration is #aajd seconds4 ,rev]s+ ! b>1 s t x vibrates ab\t its c5tre ( oscillation ,b1 ,i n[ f9d t #aajj is ! "t occupi$ by "o hundr$ vibrations c\nt$ 9 ! same mann] z be=e2 h5ce #aajj seconds is ! "t ( "o vibration ab\t ,b3 ?us1 ! p]iods ( ! vibrations are v ne>ly equal1 z !y diff] only by _/? "p ( a second4 #fci4 ,x w\ld be difficult to r5d] ! times ( oscillation exactly equal by m]ely alt]+ ! position ( ,b4 ,9 ,kat]'s p5dulum ! two knife-edges are fir/ plac$ s t ! p]iods are z ne>ly equal z possible4 ,! f9al adju/m5ts are giv5 by mov+ a small slid+-piece on ! b> until x is f\nd t ! times ( vibration ab\t ! two edges are id5tical4 ,we %all n1 h[ev]1 use ?is ref9em5t 9 a lecture exp]im5t2 ,i %all adopt ! mean value ( #aajb seconds4 ,! di/ance ( ! knife-edges is ab\t #c'iib2 h5ce gravity may be f\nd f ! expression 7,art4 #fjh7 #c'iib 7#cadaf _/ #a4ajb74 ,! value ?us d$uc$ is #cb'd1 :i* is )9 a small fraction ( ! true value4 #fdj4 ,) suitable precautions ,kat]'s p5dulum c be made to give a v accurate result4 ,x is to be adju/$ s t !re %all be no p]ceptible diff]5ce 9 ! numb] ( vibrations 9 tw5ty-f\r h\rs1 :i*ev] edge be ! axis ( susp5sion3 ! di/ance betwe5 ! edges is !n to be measur$ ) ! la/ degree ( precision by comp>ison ) a prop] /&>d4 ,,! ,,c5tre ,,( ,,p]cussion4 ,illu/ration3 ,,fig4 #ij4 #fda4 ,! c5tre ( oscillation 9 a body free to rotate ab\t a fix$ axis is id5tical ) ano!r rem>kable po9t1 call$ ! c5tre ( p]cussion4 ,we proce$ to exam9e "s ( ! prop]ties ( a body ?us susp5d$ ) ref]5ce to ! effects ( a bl[4 ,= ! purposes ( ^! exp]im5ts ! me?od ( susp5sion by edges is h[ev] q unsuit$4 #fdb4 ,we %all fir/ use a rod susp5d$ f a p9 ab\t :i* ! rod c rotate4 ,a ,b1 ,fig4 #ij1 is a p9e rod #dh #a #a1 free to turn r\nd ,b4 ,suppose ?is rod be hang+ v]tically at re/4 ,i take a /ick 9 my h&1 &1 giv+ ! rod a bl[1 an impulsive %ock w in/antly be communicat$ to ! p9 at ,b2 b ! actual effect ^u ,b w be v diff]5t accord+ to ! position at :i* ! bl[ is giv54 ,if ,i /rike ! upp] "p ( ! rod at ,d1 ! action ( ,a ,b ^u ! p9 is a pressure to ! left4 ,if ,i /rike ! l[] "p at ,a1 ! pressure is to ! "r4 ,b if ,i /rike ! po9t ,c1 :i* is di/ant f ,b by two-?irds ( ! l5g? ( ! rod1 !re is no pressure ^u ! p94 ,concisely1 = a bl[ bel[ ,c1 ! pressure is to ! "r2 = "o above ,c1 x is to ! left2 = "o at ,c x is no?+4 #fdc4 ,we c easily v]ify ?is by hold+ "o extremity ( a rod betwe5 ! f+] & ?umb ( ! left h&1 & /rik+ x 9 diff]5t places ) a /ick held 9 ! "r h&2 ! pressure ( ! rod1 :5 /ruck1 w be s felt t ! circum/ances already /at$ c be v]ifi$4 ,illu/ration3 ,,fig4 #ia4 #fdd4 ,a m visible way ( inve/igat+ ! subject is %[n 9 ,fig4 #ia4 ,f ,b is a rod ( wood1 susp5d$ f a beam by ! /r+ ,f ,g4 ,a piece ( pap] is fa/5$ to ! rod at ,f by means ( a small slip ( wood clamp$ firmly to ! rod2 ! o!r ends ( ?is piece ( pap] are simil>ly clamp$ at ,p & ,q4 #fde4 ,:5 ! rod receives a bl[ on ! "r-h& side at ,a1 we f9d t ! piece ( pap] is brok5 across at ,e1 because ! end ,f has be5 driv5 by ! bl[ t[>ds ,q1 & consequ5tly caus$ ! fracture ( ! pap] at a place1 ,e1 :]e x _h be5 specially n>r[$4 ,i remove ! pieces ( pap]1 & replace !m by a new piece precisely simil>4 ,i n[ /rike ! rod at ,b1--a sm>t tap is all t is necess>y1--& ! piece ( pap] breaks at ,d4 ,f9ally replac+ ! pieces ( pap] by a ?ird piece1 ,i f9d t :5 ,i give ! rod a tap 7n a viol5t bl[7 at ,c1 nei!r ,d nor ,e are brok54 #fdf4 ,?is po9t ,c1 :]e ! rod c receive a bl[ )\t produc+ a %ock ^u ! axis ( susp5sion is ! c5tre ( p]cussion4 ,we see1 f its be+ two-?irds ( ! l5g? ( ! rod di/ant f ,f1 t x is id5tical ) ! c5tre ( oscillation ( ! rod1 if vibrat+ ab\t knife-edges at ,f4 ,x is true 9 g5]al1 :atev] be ! %ape ( ! body1 t ! c5tre ( oscillation is id5tical ) ! c5tre ( p]cussion4 #fdg4 ,! pr9ciple embodi$ 9 ! prop]ty ( ! c5tre ( p]cussion has _m practical applications4 ,e cricket] well kn[s t !re is a "p ( 8 bat f :i* ! ball flies )\t giv+ 8 h&s any unpleasant feel+4 ,! explanation is simple4 ,! bat is a body susp5d$ f ! h&s ( ! batman2 & if ! ball be /ruck ) ! c5tre ( p]cussion ( ! bat1 !re is no %ock exp]i5c$4 ,! c5tre ( p]cussion 9 a hamm] lies 9 its head1 consequ5tly a nail c receive a viol5t bl[ ) p]fect com=t to ! h& :i* holds ! h&le4 ,,! ,,conical ,,p5dulum4 #fdh4 ,i h "h a tripod 7,fig4 #ib7 :i* supports a heavy ball ( ca/ iron by a /r+ #f' long4 ,if ,i )draw ! ball f its position ( re/1 & m]ely release x1 ! ball vibrates to & fro1 ! /r+ cont9ues 9 ! same plane1 & ! motion is t already discuss$ 9 ! circul> p5dulum4 ,if at ! same in/ant t ,i release ! ball1 ,i imp>t to x a slight pu% 9 a direction n pass+ ?r\gh ! position ( re/1 ! ball describes a curv$ pa?1 return+ to ! po9t f :i* x />t$4 ,?is motion is t ( ! conical p5dulum1 because ! /r+ support+ ! ball describes a cone4 #fdi4 ,9 ord] to exam9e ! nature ( ! motion1 we c make ! ball depict its own pa?4 ,at ! opposite po9t ( ! ball to t f :i* x is susp5d$1 a hole is drill$1 & 9 ?is ,i h fitt$ a camel's hair pa9t-bru% fill$ ) ink4 ,i br+ a %eet ( pap] on a draw+-bo>d "u ! vibrat+ ball2 & y see ! bru% traces an ellipse ^u ! pap]1 :i* ,i quickly )draw4 #fej4 ,by />t+ ! ball 9 diff]5t ways1 ,i c make x describe v diff]5t ellipses3 "h is "o t is extremely long & n>r[1 & "h ano!r almo/ circul>4 ,:5 ! magnitude ( ! initial velocity is prop]ly adju/$1 & its direction is p]p5dicul> to ! radius1 ,i c make ! /r+ describe a "r cone1 & ! ball a horizontal circle1 b x requires "s c>e & sev]al trials 9 ord] to succe$ 9 ?is4 ,! ellipse may also become v n>r[1 s t we pass by ins5sible gradations to ! circul> p5dulum1 9 :i* ! bru% traces a /raight l9e4 ,illu/ration3 ,,fig4 #ib4 #fea4 ,:5 ! ball is mov+ 9 a circle1 its velocity is uni=m2 :5 mov+ 9 an ellipse1 its velocity is greate/ at ! extremities ( ! lea/ axes ( ?is ellipse1 & lea/ at ! extremities ( ! greate/ axes2 b1 :5 ! ball is vibrat+ to & fro1 z 9 ! ord9>y circul> p5dulum1 ! velocity is greate/ at ! middle ( ea* vibration1 & vani%es ( c\rse ea* "t ! p5dulum atta9s ! extremity ( its sw+4 ,x is wor?y ( notice t "u all circum/ances ! bru% traces an ellipse ^u ! pap]2 = ! circle & ! /raight l9e are only extreme cases1 ! "o be+ a v r\nd ellipse & ! o!r a v ?9 "o4 ,if1 h[ev]1 ! arc ( vibration be l>ge ! movem5t is by no means s simple4 #feb4 ,h[ are we to acc\nt = ! elliptic movem5t8 ,to d s fully w\ld require m calculation ?an c be admitt$ "h1 b we may give a g5]al acc\nt ( ! ph5om5on4 ,let u suppose t ! ellipse ,a ,c ,b ,d1 ,fig4 #ic1 is ! pa? describ$ by a p>ticle :5 susp5d$ by a /r+ f a po9t v]tically above ,o1 ! c5tre ( ! ellipse4 ,to produce ?is motion ,i )draw ! p>ticle f its position ( re/ at ,o to ,a4 ,if m]ely releas$1 ! p>ticle w\ld sw+ ov] to ,b1 & back aga9 to ,a2 b ,i d n simply release x1 ,i imp>t a velocity impell+ x 9 ! direction ,a ,t4 ,?r\gh ,o draw ,c ,d p>allel to ,a ,t4 ,if ,i _h tak5 ! p>ticle at ,o1 &1 )\t )draw+ x f its position ( re/1 _h />t$ x (f 9 ! direction ,o ,d1 ! p>ticle w\ld cont9ue = ev] to vibrate backw>ds & =w>ds f ,c to ,d4 ,h5ce1 :5 ,i release ! p>ticle at ,a1 & give x a velocity 9 ! direction ,a ,t1 ! p>ticle comm5ces to move "u ! action ( two di/9ct vibrations1 "o p>allel to ,a ,b1 ! o!r p>allel to ,c ,d1 & we h to f9d ! effect ( ^! two vibrations impress$ simultane\sly ^u ! same p>ticle4 ,!y are p]=m$ 9 ! same "t1 s9ce all vibrations are iso*ron\s4 ,we mu/ conceive "o motion />t+ f ,a ,a t[>ds ,o at ! same mom5t t ! o!r comm5ces to />t f ,o t[>ds ,d4 ,aft] ! lapse ( a %ort "t1 ! body has mov$ ?r\gh ,a ,y 9 its oscillation t[>ds ,o1 & 9 ! same "t ?r\gh ,o ,z 9 its oscillation t[>ds ,d2 x is !re=e f\nd at ,x4 ,n[1 :5 ! p>ticle has mov$ ?r\gh a di/ance equal & p>allel to ,a ,o1 x mu/ be f\nd at ! po9t ,d1 because ! motion f ,o to ,d takes ! same "t z f ,a to ,o4 ,simil>ly ! body mu/ pass ?r\gh ,b1 because ! "t occupi$ by go+ f ,a to ,b1 w\ld h be5 suffici5t = ! j\rney f ,o to ,d1 & back aga94 ,! p>ticle is f\nd at ,p1 because1 aft] ! vibration return+ f ,b has arriv$ at ,q1 ! movem5t f ,d to ,o has travell$ on to ,r4 ,9 ?is way ! p>ticle may be trac$ completely r\nd its pa? by ! composition ( ! two motions4 ,x c be prov$ t = small motions ! pa? is an ellipse1 by reason+ f\nd$ ^u ! fact t ! vibrations are iso*ron\s4 ,illu/ration3 ,,fig4 #ic4 #fec4 ,close exam9ation reveals a v int]e/+ circum/ance connect$ ) ?is exp]im5t4 ,x may be obs]v$ t ! ellipse describ$ by ! body is n q fix$ 9 position1 b t x gradually moves r\nd 9 its plane4 ,?us1 9 ,fig4 #ib1 ! ellipse :i* is be+ trac$ out by ! bru% w gradually *ange its position to ! dott$ l9e %[n on ! bo>d4 ,! axis ( ! ellipse revolves 9 ! same direction z t 9 :i* ! ball is mov+4 ,?is ph5om5on is m m>k$ ) an ellipse ^: dim5sions are consid]able 9 proportion to ! l5g? ( ! /r+4 ,9 fact1 if ! ellipse be v small1 ! *ange ( position is imp]ceptible4 ,! cause ( ?is *ange is to be f\nd 9 ! fact already m5tion$ 7,art4 #eih71 t ?\gh ! vibrations ( a p5dulum are v ne>ly iso*ron\s1 yet !y are n absolutely s2 ! vibrations ?r\gh a long arc tak+ a m9ute portion ( "t long] ?an ^? ?r\gh a %ort arc4 ,?is diff]5ce only becomes appreciable :5 ! l>g] arc is ( consid]able magnitude ) ref]5ce to ! l5g? ( ! p5dulum4 ,illu/ration3 ,,fig4 #id4 #fed4 ,h[ ?is causes displacem5t ( ! ellipse may be expla9$ by ,fig4 #id4 ,! p>ticle is describ+ ! figure ,a ,d ,c ,b 9 ! direction %[n by ! arr[s4 ,?is motion may be conceiv$ to be comp\nd$ ( vibrations ,a ,c & ,b ,d1 if we imag9e ! p>ticle to h be5 />t$ f ,a ) ! "r velocity p]p5dicul> to ,o ,a4 ,at ! po9t ,a1 ! entire motion is = ! in/ant p]p5dicul> to ,o ,a2 9 fact1 ! motion is !n exclusively due to ! vibration ,b ,d1 & !re is no movem5t p>allel to ,o ,a4 ,we may def9e ! extremity ( ! major axis ( ! ellipse to be ! position ( ! p>ticle1 :5 ! motion p>allel to t axis vani%es4 ,( c\rse ?is applies equally to ! o!r extremity ( ! axis ,c1 & simil>ly at ! po9ts ,b or ,d !re is no motion ( ! p>ticle p>allel to ,b ,d4 #fee4 ,let u foll[ ! p>ticle1 />t+ f ,a until x returns !re aga94 ,! movem5t is comp\nd$ ( two vibrations1 "o f ,a to ,c & back aga91 ! o!r along ,b ,d2 f ,o to ,d1 !n f ,d to ,b1 !n f ,b to ,o1 tak+ exactly d\ble ! "t ( "o vibration f ,d to ,b4 ,if ! "t ( vibration along ,a ,c 7 exactly equal to t along ,b ,d1 ^! two vibrations w\ld br+ ! p>ticle back to ,a precisely "u ! orig9al circum/ances4 ,b !y d n take place 9 ! same "t2 ! motion along ,a ,c takes a %ade long]1 s t1 :5 ! motion p>allel to ,a ,c has ceas$1 ! motion along ,d ,b has gone pa/ ,o to a po9t ,q1 v ne> ,o4 ,let ,a ,p ,o ,q1 & :5 ! motion p>allel to ,a ,c has vani%$1 ! p>ticle w be f\nd at ,p2 h5ce ,p mu/ be ! extremity ( ! major axis ( ! ellipse4 ,9 ! next revolution1 ! extremity ( ! axis w advance a little m1 & ?us ! ellipse moves r\nd gradually4 ,,! ,,composition ,,( ,,vibrations4 #fef4 ,we h le>n$ to reg>d ! elliptic motion 9 ! conical p5dulum z comp\nd$ ( two vibrations4 ,! importance ( ! composition ( vibrations ju/ifies u 9 exam9+ ?is subject exp]im5tally 9 ano!r way4 ,! app>atus :i* we %all employ is repres5t$ 9 ,fig4 #ie4 ,a is a ball ( ca/ iron weigh+ #be lbs41 susp5d$ f ! tripod by a cord3 ?is ball itself =ms ! support ( ano!r p5dulum1 ,b4 ,! second p5dulum is v light1 be+ m]ely a globe ( glass fill$ ) s&4 ,?r\gh a hole at ! bottom ( ! glass ! s& runs out ^u a draw+-bo>d plac$ und]nea? to receive x4 ,illu/ration3 ,,fig4 #ie4 ,?us ! little /ream ( s& depicts its own j\rney ^u ! draw+-bo>d1 & ! curves trac$ out ?us indicate ! pa? 9 :i* ! bob ( ! second p5dulum has mov$4 ,illu/ration3 ,,fig4 #if4 #feg4 ,if ! l5g?s ( ! two p5dulums be equal1 & _! vibrations be 9 diff]5t planes1 ! curve describ$ is an ellipse1 pass+ at "o extreme into a circle1 & at ! o!r into a /raight l9e4 ,?is is :at we might h expect$1 = ! two vibrations are ea* p]=m$ 9 ! same "t1 & !re=e ! case is analog\s to t ( ! conical p5dulum ( ,art4 #fdh4 #feh4 ,b ! curve is ( a v diff]5t "* :5 ! cords are unequal4 ,let u /udy 9 p>ticul> ! case 9 :i* ! second p5dulum is only "o-f\r? ! l5g? ( ! cord support+ ! iron ball4 ,?is is ! exp]im5t repres5t$ 9 ,fig4 #ie4 ,! =m ( ! pa? del9eat$ by ! s& is %[n 9 ,fig4 #if4 ,! arr[heads plac$ ^u ! curve %[ ! direction 9 :i* x is trac$4 ,let u suppose t ! =mation ( ! figure comm5ces at ,a2 x !n goes on to ,b1 to ,o1 to ,c1 to ,d1 & back to ,a3 ?is %[s u t ! bob ( ! l[] p5dulum mu/ h p]=m$ two vibrations up & d[n1 :ile t ( ! upp] has made "o "r & left4 ,! motion is ?us comp\nd$ ( two vibrations at "r angles1 & ! "t ( "o is half t ( ! o!r4 ,! "t ( vibration is proportional to ! squ>e root ( ! l5g?2 &1 s9ce ! l[] p5dulum is "o-f\r? ! l5g? ( ! upp]1 its "t ( vibration is "o-half t ( ! upp]4 ,9 ?is exp]im5t1 !re=e1 we h a confirmation ( ! law ( ,art #fje4 ,,lecture ,,xx4 ,,! ,,me*anical ,,pr9ciples ,,( ,a ,,clock4 ,introduction4--,! ,comp5sat+ ,p5dulum4--,! ,escapem5t4--,! ,tra9 ( ,:eels4--,! ,h&s4--,! ,/rik+ ,p>ts4 ,,introduction4 #fei4 ,we come n[ to ! mo/ important practical application ( ! p5dulum4 ,! vibrations be+ always iso*ron\s1 x foll[s t1 if we c\nt ! numb] ( vibrations 9 a c]ta9 "t1 we %all asc]ta9 ! duration ( t "t4 ,x is simply ! product ( ! numb] ( vibrations ) ! p]iod ( a s+le "o4 ,let u take a p5dulum #ciaci in*es long2 :i* w vibrate exactly once a second 9 ,london1 & is !re=e call$ a seconds p5dulum 7,see ,art4 #fjg74 ,if ,i set "o ( ^! p5dulums vibrat+1 & contrive me*anism by :i* ! numb] ( its vibrations %all be record$1 ,i h a means ( measur+ "t4 ,?is is ( c\rse ! pr9ciple ( ! common clock3 ! p5dulum vibrates once a second & ! numb] ( vibrations made f "o epo* to ano!r epo* is %[n by ! h&s ( ! clock4 ,= example1 :5 ! clock tells me t #ae m9utes h elaps$1 :at x really %[s is t ! p5dulum has made #fj #ae #ijj vibrations1 ea* ( :i* has occupi$ "o second4 #ffj4 ,"o duty ( ! clock is !re=e to c\nt & record ! numb] ( vibrations1 b ! :eels & works h ano!r "p to dis*>ge1 & t is to su/a9 ! motion ( ! p5dulum4 ,! friction ( ! air & ! resi/ance exp]i5c$ at ! po9t ( susp5sion are =ces t5d+ to br+ ! p5dulum to re/2 & to c\nt]act ! effect ( ^! =ces1 ! ma*9e mu/ be cont9ually invigorat$ ) fre% en]gy4 ,?is supply is communicat$ by ! works ( ! clock1 :i* w be suffici5tly describ$ pres5tly4 #ffa4 ,:5 ! weight driv+ ! clock is w\nd up1 a /ore ( en]gy is communicat$ :i* is dol$ out to ! p5dulum 9 a v small impulse1 at e vibration4 ,! clock-weight is j l>ge en\gh to be able to c\nt]balance ! ret>d+ =ces :5 ! p5dulum has a prop] amplitude ( vibration4 ,9 all ma*9es !re is "s en]gy lo/ 9 ma9ta9+ ! p>ts 9 motion 9 opposition to friction & o!r resi/ances2 9 clocks ?is repres5ts ! :ole am\nt ( ! =ce1 z !re is no ext]nal "w to be p]=m$4 ,,! ,,comp5sat+ ,,p5dulum4 #ffb4 ,! actual l5g? ( ! p5dulum us$1 dep5ds ^u ! purposes = :i* ! clock is int5d$1 b x is ess5tial = correct p]=mance t ! p5dulum %\ld vibrate at a con/ant rate2 a small irregul>ity 9 ?is respect may appreciably affect ! indications ( ! clock4 ,if ! p5dulum vibrates 9 #ajja seconds in/ead ( 9 "o second1 ! clock loses "o ?\s&? ( a second at ea* beat2 &1 s9ce !re are #hf1djj seconds 9 a day1 x foll[s t ! p5dulum w make only #hf1djj - #hfc vibrations 9 a day1 & !re=e ! clock w lose #hfc seconds1 or ne>ly a m9ute & a half daily4 #ffc4 ,= accurate "t-keep+ x is !re=e ess5tial t ! "t ( vibration %all rema9 con/ant4 ,n[ ! "t ( vibration dep5ds ^u ! l5g?1 & !re=e x is necess>y t ! l5g? ( ! p5dulum be absolutely unalt]able4 ,if ! l5g? ( ! p5dulum be *ang$ ev5 by "o-t5? ( an in*1 ! clock w lose or ga9 ne>ly two m9utes daily1 accord+ to :e!r ! p5dulum has be5 made long] or %ort]4 ,9 g5]al we may say t1 if ! alt]ation 9 ! l5g? am\nt to k ?\s&?s ( an in*1 ! numb] ( seconds ga9$ or lo/ p] day is #aajc k ) a seconds p5dulum4 #ffd4 ,?is expla9s ! practice ( rais+ ! bob ( ! p5dulum :5 ! clock is go+ too sl[ or l[]+ x :5 go+ too fa/4 ,if ! ?read ( ! screw us$ 9 do+ ?is h tw5ty ?reads to ! in*2 !n "o complete revolution ( ! screw w raise ! bob ?r\gh #ej ?\s&?s ( an in*1 & !re=e ! effect on ! rate w be #aajc #ej #ee ne>ly4 ,?us1 ! rate ( ! clock w be alt]$ by ab\t #ee seconds daily4 ,:atev] be ! screw1 its effect c be calculat$ by ! simple rule express$ z foll[s4 ,divide #aajc by ! numb] ( ?reads to ! in*2 ! quoti5t is ! numb] ( seconds t ! clock c be made to ga9 or lose daily by "o revolution ( ! screw on ! bob ( ! p5dulum4 #ffe4 ,let u suppose t ! l5g? ( ! p5dulum has be5 prop]ly adju/$ s t ! clock keeps accurate "t4 ,x is necess>y t ! p5dulum %\ld n alt] 9 l5g?4 ,b !re is an ev]-pres5t cause t5d+ to *ange x4 ,t cause is ! v>iation ( temp]ature4 ,we %all fir/ illu/rate by actual exp]im5t ! well kn[n law t bodies exp& "u ! action ( heat2 !n we %all consid] ! irregul>ities ?us introduc$ into ! motion ( ! p5dulum2 &1 f9ally1 we %all po9t out means by :i* ^! irregul>ities may be effectually c\nt]act$4 ,illu/ration3 ,,fig4 #ig4 #fff4 ,we h "h a brass b> a y>d long2 x is at pres5t at ! temp]ature ( ! room4 ,if we heat ! b> ov] a lamp1 x becomes long]2 b ^u cool+1 x returns to its orig9al dim5sions4 ,^! alt]ations ( l5g? are v small1 inde$ too small to be p]ceiv$ except by c>eful measurem5t2 b we %all be able to demon/rate 9 a simple way t elongation is ! consequ5ce ( increas$ temp]ature4 ,i place ! b> ,a ,d 9 ! supports %[n 9 ,fig4 #ig4 ,x is firmly secur$ at ,b by means ( a b9d+ screw1 & passes q freely ?r\gh ,c2 if ! b> elongate :5 x is heat$ by ! lamp1 ! po9t ,d mu/ approa* ne>] to ,e4 ,at ,h is an electric batt]y1 & ,g is a bell rung by an electric curr5t4 ,"o wire ( ! batt]y connects ,h & ,g1 ano!r connects ,g ) ,e1 & a ?ird connects ,h ) ! end ( ! brass rod ,b4 ,until ! electric curr5t becomes complet$1 ! bell rema9s dumb1 ! curr5t is n clos$ until ! po9t t\*es ,e3 :5 ?is is ! case1 ! curr5t ru%es f ! batt]y along ! b>1 !n f ,d to ,e1 f t ?r\gh ! bell1 & s back to ! batt]y4 ,at pres5t ! po9t is n t\*+ ,e1 ?\gh extremely close !reto4 ,inde$ if ,i press ,e t[>ds ! po9t1 y he> ! bell1 %[+ t ! circuit is complete2 remov+ my f+]1 ! bell aga9 becomes sil5t1 because ,e spr+s back1 & ! curr5t is int]rupt$4 #ffg4 ,i place ! lamp "u ! b>3 :i* beg9s to heat & to elongate2 & z x is firmly held at ,b1 ! po9t gradually approa*es ,e3 x has n[ t\*$ ,e2 ! circuit is complete1 & ! bell r+s4 ,if ,i )draw ! lamp1 ! b> cools4 ,i c accel]ate ! cool+ by t\*+ ! b> ) a damp sponge2 ! b> contracts1 breaks ! circuit1 & ! bell /ops3 heat+ ! b> aga9 ) ! lamp1 ! bell aga9 r+s1 to be aga9 /opp$ by an application ( ! sponge4 ,?\gh y h n be5 able to see ! process1 y\r e>s h in=m$ y t heat mu/ h elongat$ ! b>1 & t cold has produc$ contraction4 #ffh4 ,:at we h prov$ ) respect to a b> ( brass1 is true = a b> ( any mat]ial2 & ?us1 :atev] be ! sub/ance ( :i* a p5dulum is made1 a simple uncomp5sat$ rod mu/ be long] 9 hot wea!r ?an 9 cold wea!r3 h5ce a clock w g5]ally h a t5d5cy to g fa/] 9 w9t] ?an 9 summ]4 #ffi4 ,! am\nt ( *ange ?us produc$ is1 x is true1 v small4 ,= a p5dulum ) a /eel rod1 ! diff]5ce ( temp]ature betwe5 summ] & w9t] w\ld cause a v>iation 9 ! rate ( five seconds daily1 or ab\t half a m9ute 9 ! week4 ,! am\nt ( error ?us introduc$ is ( no great consequ5ce 9 clocks :i* are only int5d$ = ord9>y use2 b 9 a/ronomical clocks1 :]e seconds or ev5 portions ( a second are ( importance1 inaccuracies ( ?is magnitude w\ld be q inadmissible4 ,illu/ration3 ,,fig4 #ih4 #fgj4 ,!re are1 x is true1 "s sub/ances--= example1 ord9>y timb]--9 :i* ! rate ( expansion is less ?an t ( /eel2 consequ5tly1 ! irregul>ities introduc$ by employ+ a p5dulum ) a wood5 rod are less ?an ^? ( ! /eel p5dulum we h m5tion$2 b no sub/ance is kn[n :i* w\ld n orig9ate great] v>iations ?an are admissible 9 ! p]=mance ( an a/ronomical clock4 ,we mu/1 !re=e1 devise "s means by :i* ! effect ( temp]ature on ! l5g? ( a p5dulum c be avoid$4 ,v>i\s means h be5 propos$1 & we %all describe "o ( ! be/ & simple/4 #fga4 ,! m]curial p5dulum 7,fig4 #ih7 is frequ5tly us$ 9 clocks int5d$ to s]ve z /&>d "t-keep]s4 ,! rod by :i* ! p5dulum is susp5d$ is made ( /eel2 & ! bob consi/s ( a glass j> ( m]cury4 ,! di/ance ( ! c5tre ( gravity ( ! m]cury f ! po9t ( susp5sion may practically be consid]$ z ! l5g? ( ! p5dulum4 ,! rate ( expansion ( m]cury is ab\t sixte5 times t ( /eel3 h5ce1 if ! bob be =m$ ( a column ( m]cury "o-eigh? "p ( ! l5g? ( ! /eel rod1 ! comp5sation w\ld be complete4 ,=1 suppose ! temp]ature ( ! p5dulum be rais$1 ! /eel rod w\ld be l5g!n$1 & !re=e ! vase ( m]cury w\ld be l[]$2 on ! o!r h&1 ! column ( m]cury w\ld exp& by an am\nt d\ble t ( ! /eel rod3 ?us ! c5tre ( ! column ( m]cury w\ld be elevat$ z mu* z ! /eel 0 elongat$2 h5ce ! c5tre ( ! m]cury is rais$ by its own expansion z mu* z x is l[]$ by ! expansion ( ! /eel1 & !re=e ! effective l5g? ( ! p5dulum rema9s unalt]$4 ,by ?is contrivance ! "t ( oscillation ( ! p5dulum is r5d]$ indep5d5t ( ! temp]ature4 ,! bob ( ! m]curial p5dulum is %[n 9 ,fig4 #ih4 ,! screw is = ! purpose ( rais+ or l[]+ ! entire vessel ( m]cury 9 ord] to make ! rate correct 9 ! fir/ in/ance4 ,,! ,,escapem5t4 #fgb4 ,practical skill z well z "s !oretical inve/igation has be5 exp5d$ ^u t "p ( a clock :i* is call$ ! escapem5t1 ! excell5ce ( :i* is ess5tial to ! correct p]=mance ( a timepiece4 ,! p5dulum mu/ h its motion su/a9$ by receiv+ an impulse at e vibration3 at ! same "t x is desirable t ! vibration %\ld be hamp]$ z little z possible by me*anical connection4 ,! iso*ronism on :i* ! "t-keep+ dep5ds is 9 /rictness only a *>act]i/ic ( oscillations p]=m$ ) a total fre$om f con/ra9t ( e description2 h5ce we mu/ endeav\r to approximate ! clock p5dulum z ne>ly z possible to "o :i* is sw++ q freely4 ,to effect ?is1 & at ! same "t to ma9ta9 ! arc ( vibration tol]ably con/ant1 is ! prop]ty ( a good escapem5t4 ,illu/ration3 ,,fig4 #ii4 #fgc4 ,a common =m ( escapem5t is %[n 9 ,fig4 #ii4 ,! arrangem5t is no d\bt diff]5t f t actually f\nd 9 a clock2 b ,i h con/ruct$ ! ma*9e 9 ?is way 9 ord] to %[ cle>ly ! action ( ! diff]5t p>ts4 ,g is call$ ! escapem5t-:eel3 x is surr\nd$ by ?irty tee?1 & turns r\nd once :5 ! p5dulum has p]=m$ sixty vibrations1--t is1 once a m9ute4 ,i repres5ts ! escapem5t2 x vibrates ab\t an axis & c>ries a =k at ,k :i* projects beh9d1 & ! rod ( ! p5dulum hangs betwe5 its prongs4 ,! p5dulum is itself susp5d$ f a po9t ,o4 ,at ,n1 ,h are a pair ( poli%$ surfaces call$ ! pallets3 ^! fulfil a v important function4 #fgd4 ,! escapem5t-:eel is con/antly urg$ to turn r\nd by ! action ( ! weight & tra9 ( :eels1 ( :i* we %all speak pres5tly2 b ! action ( ! pallets regulates ! rate at :i* ! :eel c revolve4 ,:5 a too? ( ! :eel falls ^u ! pallet ,n1 ! latt] is g5tly press$ away3 ?is pressure is transmitt$ by ! =k to ! p5dulum2 z ,n moves away f ! :eel1 ! o!r pallet ,h approa*es ! :eel2 & by ! "t ,n has rec$$ s f> t ! too? slips f x1 ,h has advanc$ suffici5tly f> to cat* ! too? :i* imm$iately drops ^u ,h4 ,9 fact1 ! mom5t ! too? is free f ,n1 ! :eel beg9s to revolve 9 consequ5ce ( ! driv+ weight2 b x is quickly /opp$ by ano!r too? fall+ on ,h3 & ! noise ( ?is collision is ! well kn[n tick ( ! clock4 ,! p5dulum is /ill sw++ to ! left :5 ! too? falls on ,h4 ,! pressure ( ! too? !n t5ds to pu% ,h outw>ds1 b ! in]tia ( ! p5dulum 9 =c+ ,h inw>ds is at fir/ suffici5t to ov]come ! outw>d pressure aris+ f ! :eel2 ! consequ5ce is t1 aft] ! too? has dropp$1 ! escapem5t-:eel moves back a little1 or 8recoils10 z x is call$4 ,if y look at any ord9>y clock1 :i* has a second-h&1 y w notice t aft] ea* second is complet$ ! h& recoils be=e />t+ = ! next second4 ,! reason ( ?is is1 t ! second-h& is turn$ directly by ! escapem5t-:eel1 & t ! in]tia ( ! p5dulum causes ! escapem5t-:eel to recoil4 ,b ! con/ant pressure ( ! too? soon ov]comes ! in]tia ( ! p5dulum1 & ,h is gradually pu%$ out until ! too? is able to 8escape02 ! mom5t x does s ! :eel beg9s to turn r\nd1 b is quickly br\ght up by ano!r too? fall+ on ,n1 :i* has mov$ suffici5tly inw>ds4 ,! process we h j describ$ !n recurs ov] aga94 ,ea* too? escapes at ea* pallet1 & ! escapem5ts take place once a second2 h5ce ! escapem5t-:eel ) ?irty tee? w turn r\nd once 9 a m9ute4 #fge4 ,:5 ! too? is pu%+ ,n1 ! p5dulum is be+ urg$ to ! left2 ! in/ant ?is too? escapes1 ano!r too? falls on ,h1 & ! p5dulum1 ere x has accompli%$ its sw+ to ! left1 has a =ce ex]t$ ^u x to br+ x to ! "r4 ,:5 ?is =ce & gravity comb9$ h /opp$ ! p5dulum1 & caus$ x to move to ! "r1 ! too? soon escapes at ,h1 & ano!r too? falls on ,n1 !n ret>d+ ! p5dulum4 ,h5ce1 except dur+ ! v m9ute portion ( "t t ! :eel turns aft] "o escapem5t1 & be=e ! next tick1 ! p5dulum is nev] free2 x is urg$ =w>ds :5 its velocity is great1 b be=e x comes to ! end ( its vibration x is urg$ backw>ds2 ?is escapem5t does n !re=e possess ! *>act]i/ics :i* we po9t$ out 7,art4 #fgb7 z necess>y = a really good in/rum5t4 ,b = ord9>y purposes ( "t-keep+1 ! recoil escapem5t works suffici5tly well1 z ! =ce :i* acts ^u ! p5dulum is 9 reality extremely small4 ,= ! ref9$ applications ( ! a/ronomical clock1 ! p]=mance ( a recoil escapem5t is inadequate4 ,! obvi\s defect 9 ! recoil is t ! p5dulum is ret>d$ dur+ a portion ( its vibration2 ! impulse =w>d is ( c\rse necess>y1 b ! ret>d+ =ce is useless & injuri\s4 #fgf4 ,! 8dead-beat0 escapem5t 0 devis$ by ! celebrat$ clockmak] ,graham1 9 ord] to avoid ?is difficulty4 ,if y obs]ve ! second-h& ( a clock1 controll$ by ?is escapem5t1 y w und]/& :y x is call$ ! dead beat3 !re is no recoil2 ! second-h& moves quickly ov] ea* second1 & rema9s !re fix$ until x />ts = ! next second4 ,! :eel & escapem5t by :i* ?is effect is produc$ is %[n 9 ,fig4 #ajj4 ,a & ,b are ! pallets1 by ! action ( ! tee? on :i* ! motion is giv5 to ! crut*1 :i* turns ab\t ! c5tre ,o2 f ! axis ?r\gh ?is c5tre ! =k desc5ds1 s t z ! crut* is made to vibrate to & fro by ! :eel1 ! =k is also made to vibrate1 & ?us su/a9 ! motion ( ! p5dulum4 ,! ess5tial feature 9 :i* ! dead-beat escapem5t diff]s f ! recoil escapem5t is t :5 ! too? escapes f ! pallet ,a1 ! :eel turns3 b ! too? :i* 9 ! recoil escapem5t w\ld h fall5 on ! o!r pallet1 n[ falls on a surface ,d1 & n on ! pallet ,b4 ,d is "p ( a circle ) its c5tre at ,o1 ! c5tre ( motion2 consequ5tly1 ! too? rema9s almo/ entirely in]t s long z x rema9s on ! circul> arc ,d4 #fgg4 ,!re is ?us no recoil1 & ! p5dulum is all[$ to rea* ! extremity ( its sw+ to ! "r unret>d$2 b :5 ! p5dulum is return+1 ! crut* moves until ! too? passes f ! circul> arc ,d on to ! pallet ,b3 in/antly ! too? slides d[n ! pallet1 giv+ ! crut* an impulse1 & escap+ :5 ! po9t has trav]s$ ,b4 ,! next too? t comes into action falls ^u ! circul> arc ,c1 ( :i* ! c5tre is also at ,o2 ?is too? likewise rema9s at re/ until ! p5dulum has f9i%$ its sw+1 & has comm5c$ its return2 !n ! too? slides d[n ,a1 & ! process recomm5ces z be=e4 ,illu/ration3 ,,fig4 #ajj4 #fgh4 ,! op]ations are s tim$ t ! p5dulum receives its impulse 7:i* takes place :5 a too? slides d[n a pallet7 precisely :5 ! oscillation is at ! po9t ( greate/ velocity2 ! p5dulum is !n unact$ ^u till x rea*es a simil> position 9 ! next vibration4 ,?is impulse at ! middle ( ! sw+ does n affect ! "t ( vibration4 #fgi4 ,!re is /ill a small frictional =ce act+ to ret>d ! p5dulum4 ,?is arises f ! pressure ( ! tee? ^u ! circul> arcs1 = !re is a c]ta9 am\nt ( friction1 no matt] h[ c>efully ! surfaces may be poli%$4 ,x is n h[ev] f\nd practically to be a s\rce ( appreciable irregul>ity4 ,9 a clock furni%$ ) a dead-beat escapem5t & a m]curial p5dulum1 we h a sup]b "t-keep]4 ,,! ,,tra9 ,,( ,,:eels4 #fhj4 ,we h next to consid] ! mann] 9 :i* ! supply ( en]gy is communicat$ to ! escapem5t-:eel1 & also ! mode 9 :i* ! vibrations ( ! p5dulum are c\nt$4 ,a tra9 ( :eels = ?is purpose is %[n 9 ,fig4 #ii4 ,! same rem>k may be made ab\t ?is tra9 t we h already made ab\t ! escapem5t1--namely1 t x is m design$ to expla9 ! pr9ciple cle>ly ?an to %[ ! actual con/ruction ( a clock4 #fha4 ,! weight ,a :i* animates ! :ole ma*9e is atta*$ to a rope1 :i* is w\nd ar\nd a b>rel ,b2 ! process ( w9d+ up ! clock consi/s 9 rais+ ?is weight4 ,on ! same axle z ! b>rel ,b is a l>ge too?-:eel ,c2 ?is :eel conta9s #bjj tee?4 ,! :eel ,c works into a p9ion ,d1 conta9+ #bj tee?2 consequ5tly1 :5 ! :eel ,c has turn$ r\nd once1 ! p9ion ,d has turn$ r\nd t5 times4 ,! l>ge :eel ,e is on ! same axle ) ! p9ion ,d1 & turns ) ,d2 ! :eel ,e conta9s #ahj tee?1 & works into ! p9ion ,f1 conta9+ #cj tee?3 consequ5tly :5 ,e has gone r\nd once1 ,f w h turn$ r\nd six times2 & !re=e1 :5 ! :eel ,c & ! b>rel ,b h made "o revolution1 ! p9ion ,f w h gone r\nd sixty times2 b ! :eel ,g is on ! same %aft z ! p9ion ,f1 & !re=e1 = e sixty revolutions ( ! escapem5t-:eel1 ! :eel ,c w h gone r\nd once4 ,we h already %[n t ! escapem5t-:eel goes r\nd once a m9ute1 & h5ce ! :eel ,c mu/ g r\nd once 9 an h\r4 ,if !re=e a h& be plac$ on ! same axle ) ,c1 9 front ( a clock dial1 ! h& w g completely r\nd once an h\r2 t is1 x w be ! m9ute-h& ( ! clock4 #fhb4 ,! tra9 ( :eels s]ves to transmit ! p[] ( ! desc5d+ weight & ?us supply en]gy to ! p5dulum4 ,9 ! clock model y see be=e y1 ! weight su/a9+ ! motion is #ef lbs4 ,! diamet] ( ! escapem5t-:eel is ab\t d\ble t ( ! b>rel1 & ! :eel turns r\nd sixty times z fa/ z ! b>rel2 !re=e = e in* ! weight desc5ds1 ! circumf]5ce ( ! escapem5t-:eel mu/ move ?r\gh #abj in*es4 ,f ! pr9ciple ( "w x foll[s t ! en]gy appli$ at "o end ( a ma*9e equals t obta9$ f ! o!r1 friction be+ neglect$4 ,! =ce ( #ef lbs4 is !re=e1 r$uc$ to ! "o hundr$-&-tw5tie? "p ( its am\nt at ! circumf]5ce ( ! escapem5t-:eel4 ,& z ! friction is consid]able2 ! actual =ce ) :i* ea* too? acts ^u ! pallet is only a few ounces4 #fhc4 ,9 a good clock an extremely m9ute =ce ne$ only be suppli$ to ! p5dulum1 s t1 not)/&+ #hf1djj vibrations h to be p]=m$ daily1 "o w9d+ ( ! clock w supply suffici5t en]gy to su/a9 ! motion = a week4 ,,! ,,h&s4 #fhd4 ,we %all expla9 by ! model %[n 9 ,fig4 #aja1 h[ ! h\r-h& & ! m9ute-h& are made to revolve ) diff]5t velocities ab\t ! same dial4 ,illu/ration3 ,,fig4 #aja4 ,g is a h&le by :i* ,i c turn r\nd ! %aft :i* c>ries ! :eel ,f1 & ! h& ,b4 ,!re are #bj tee? 9 ,f1 & x ge>s into ano!r :eel1 ,e1 conta9+ #hj tee?2 ! %aft :i* is turn$ by ,e c>ries a ?ird :eel ,d1 conta9+ #be tee?1 & ,d works ) a f\r? ,c1 conta9+ #ge tee?1 ,c is capable ( turn+ freely r\nd ! %aft1 s t ! motion ( ! %aft does n affect x1 except ?r\gh ! int]v5tion ( ! :eels ,e1 ,f1 & ,d4 ,to ,c ano!r h& ,a is atta*$1 :i* !re=e turns r\nd simultane\sly ) ,c4 ,let u comp>e ! motion ( ! two h&s ,a & ,b4 ,we suppose t ! h&le ,g is turn$ twelve times2 !n1 ( c\rse1 ! h& ,b1 s9ce x is on ! %aft1 w turn twelve times4 ,! :eel ,f also turns twelve times1 b ,e has f\r times ! numb] ( tee? t ,a has1 & !re=e1 :5 ,f has gone r\nd f\r times1 ,e w only h gone r\nd once3 h5ce1 :5 ,f has revolv$ twelve times1 ,e w h gone r\nd ?ree times4 ,d turns ) ,e1 & !re=e ! twelve revolutions ( ! h&le w h turn$ ,d r\nd ?ree times1 b s9ce ,c has #ge tee? & ,d #be tee?1 ,c w h only made "o revolution1 :ile ,d has made ?ree revolutions2 h5ce ! h& ,a w h made only "o revolution1 :ile ! h& ,b has made twelve revolutions4 ,we h already se5 7,art4 #fha7 h[1 by a tra9 ( :eels1 "o :eel c be made to revolve once 9 an h\r4 ,if t :eel be ^u ! %aft in/ead ( ! h&le ,g1 ! h& ,b w be ! m9ute-h& ( ! clock1 & ! h& ,a ! h\r-h&4 #fhe4 ,! adju/m5t ( ! numb]s ( tee? is important1 & ! *oice ( :eels :i* w\ld answ] is limit$4 ,= s9ce ! %afts are p>allel1 ! di/ance betwe5 ! c5tres ( ,f & ,e mu/ equal t betwe5 ! c5tres ( ,c & ( ,d4 ,b x is evid5t t ! di/ance f ! c5tre ( ,f to ! c5tre ( ,e is equal to ! sum ( ! radii ( ! :eels ,f & ,e4 ,h5ce ! sum ( ! radii ( ! :eels ,f & ,e mu/ be equal to ! sum ( ! radii ( ,c & ,d4 ,b ! circumf]5ces ( circles are proportional to _! radii1 & h5ce ! sum ( ! circumf]5ces ( ,f & ,e mu/ equal t ( ,c & ,d2 x foll[s t ! sum ( ! tee? 9 ,e & ,f mu/ be equal to ! sum ( ! tee? 9 ,c & ,d4 ,9 ! pres5t case ea* ( ^! sums is "o hundr$4 #fhf4 ,o!r arrangem5ts ( :eels might h be5 devis$1 :i* w\ld give ! requir$ motion2 = example1 if ,f 7 #bj1 z be=e1 & ,e #bdj1 & if ,c & ,d 7 ea* equal to #acj1 ! sum ( ! tee? 9 ea* pair w\ld be #bfj4 ,e w\ld only turn once = e twelve revolutions ( ,f1 & ,c & ,d w\ld turn ) ! same velocity z ,e2 h5ce ! motion ( ! h& ,a w\ld be "o-twelf? t ( ,b4 ,?is plan requires l>g] :eels ?an ! tra9 already propos$4 ,,! ,,/rik+ ,,p>ts4 #fhg4 ,we h exam9$ ! ess5tial features ( ! go+ p>ts ( ! clock2 to complete our sket* ( ?is in/rum5t we %all describe ! beautiful me*anism by :i* ! /rik+ is arrang$4 ,! model :i* we repres5t 9 ,fig4 #ajb is1 z usual1 r int5d$ to illu/rate ! pr9ciples ( ! contrivance ?an to be an exact c\nt]-"p ( ! arrangem5t f\nd 9 clocks4 ,"s ( ! details are n reproduc$ 9 ! model2 b en\gh is %[n to expla9 ! pr9ciple1 & to enable ! model to "w4 #fhh4 ,:5 ! h\r-h& rea*es c]ta9 po9ts on ! dial1 ! /rik+ is to comm5ce2 & a c]ta9 numb] ( /rokes mu/ be deliv]$4 ,! /rik+ app>atus has bo? to initiate ! /rik+ & to control ! numb] ( /rokes2 ! latt] is by f> ! m difficult duty4 ,two contrivances are 9 common use2 we %all describe t :i* is us$ 9 ! be/ clocks4 #fhi4 ,an ess5tial feature ( ! /rik+ me*anism 9 ! repeat+ clock is ! snail1 :i* is %[n at ,b4 ,?is piece mu/ revolve once 9 twelve h\rs1 & is1 !re=e1 atta*$ to an axle :i* p]=ms its revolution 9 exactly ! same "t z ! h\r-h& ( ! clock4 ,9 ! model1 ! /rik+ ge> is %[n deta*$ f ! go+ p>ts1 b x is easy to imag9e h[ ! snail c receive ?is motion4 ,! m>g9 ( ! snail is m>k$ ) twelve /eps1 numb]$ f "o to twelve4 ,! portions ( ! m>g9 betwe5 ea* pair ( /eps is a "p ( ! circumf]5ce ( a circle1 ( :i* ! axis ( ! snail is ! c5tre4 ,! correct figur+ ( ! snail is ( ! utmo/ importance to ! correct p]=mance ( ! clock4 ,above ! snail is a portion ( a too!d :eel1 ,f1 call$ ! rack2 ?is conta9s ab\t f\rte5 or fifte5 tee?4 ,:5 ?is :eel is free1 x falls d[n until a p9 comes 9 contact ) ! snail at ,b4 ,illu/ration3 ,,fig4 #ajb4 #fij4 ,! di/ance ?r\gh :i* ! rack falls dep5ds ^u ! position ( ! snail2 if ! p9 come 9 contact ) ! "p m>k$ ,i41 z x does 9 ! figure1 ! rack w desc5d b a small di/ance1 :ile1 if ! p9 fall on ! "p m>k$ ,,vii41 ! rack w h a long] fall3 h5ce z ! snail *anges its position ) ! successive h\rs1 s ! di/ance ?r\gh :i* ! rack falls *anges also4 ,! snail is s contriv$ t at ea* h\r ! rack falls on a l[] /ep ?an x does 9 ! prec$+ h\r2 = example1 dur+ ! h\r ( ?ree o'clock1 ! rack w\ld1 if all[$ to fall1 always drop ^u ! "p ( ! snail m>k$ ,,iii41 b1 :5 f\r o'clock has arriv$1 ! rack w\ld fall on ! "p m>k$ ,,iv42 x is to insure t ?is %all happ5 correctly t su* att5tion mu/ be paid to ! =m ( ! snail4 #fia4 ,a is a small piece call$ ! 8ga!r+ pallet02 x is s plac$ ) ref]5ce to ! rack t1 at ea* revolution ( ,a1 ! pallet raises ! rack "o too?4 ,?us1 aft] ! rack has fall51 ! ga!r+ pallet gradually raises x4 #fib4 ,on ! same axle z ! ga!r+ pallet1 & turn+ ) x1 is ano!r piece ,c1 ! object ( :i* is to arre/ ! motion :5 ! rack has be5 rais$1 suffici5tly4 ,on ! rack is a project+ p92 ! piece ,c passes free ( ?is p9 until ! rack has be5 lift$ to its orig9al height1 :5 ,c is caught by ! p91 & ! me*anism is /opp$4 ,! magnitude ( ! tee? 9 ! rack is s arrang$ ) ref]5ce to ! snail1 t ! numb] ( lifts :i* ! pallet mu/ make 9 rais+ ! rack is equal to ! numb] m>k$ ^u ! /ep ( ! snail ^u :i* ! rack _h fall52 h5ce ! snail has ! effect ( controll+ ! numb] ( revolutions :i* ! ga!r+ pallet c make4 ,! rack is reta9$ by a det5t ,f1 aft] be+ rais$ ea* too?4 #fic4 ,! ga!r+ pallet is turn$ by a small p9ion ( #bg tee?1 & ! p9ion is work$ by ! :eel ,c1 ( #ahj tee?4 ,?is :eel c>ries a b>rel1 to :i* a movem5t ( rotation is giv5 by a weight1 ! arrangem5t ( :i* is evid5t3 a second p9ion ( #bg tee? on ! same axle ) ,d is also turn$ by ! l>ge :eel ,c4 ,s9ce ^! p9ions are equal1 !y revolve ) equal velocities4 ,ov] ,d ! bell ,i is plac$2 its hamm] ,e is s arrang$ t a p9 atta*$ to ,d /rikes ! bell once 9 e revolution ( ,d4 ,! action w n[ be easily und]/ood4 ,:5 ! h\r-h& rea*es ! h\r1 a simple arrangem5t raises ! det5t ,f2 ! rack !n drops2 ! mom5t ! rack drops1 ! ga!r+ pallet comm5ces to revolve & raises up ! rack2 z ea* too? is rais$ a /roke is giv5 to ! bell1 & ?us ! bell /rikes until ! piece ,c is br\ght to re/ aga9/ ! p94 #fid4 ,! object ( ! fan ,h is to control ! rapidity ( ! motion3 :5 its blades are plac$ m or less obliquely1 ! velocity is less5$ or increas$4 ,,app5dix ,i4 ,! =mul 9 ! tables on p4 #gc & aft] c be d$uc$ by two me?ods1--"o t ( graphical con/ruction1 ! o!r t ( lea/ squ>es4 ,! fir/ me?od is ! m simple & requires b little calculation2 ?\gh neatness & c>e are necess>y 9 con/ruct+ ! diagrams4 ,! second me?od w be describ$ = ! b5efit ( ^? :o possess ! requisite ma!matical k4 ,! =mul us$ 9 ! prep>ation ( ! tables h be5 g5]ally d$uc$ f ! me?od ( lea/ squ>es1 z ! results are to a slight1 ?\gh insignificant1 ext5t m accurate ?an ^? ( ! me?od ( graphical con/ruction4 ,?is rem>k w expla9 :y ! figures 9 "s ( ! =mul are c>ri$ to a great] numb] ( places ( decimals ?an c\ld be obta9$ by ! o!r me?od4 ,we %all conf9e ! num]ical examples to ,tables ,,iii4 & ,,iv41 & %[ h[ ! =mul ( ^! tables h be5 d$uc$ by ! two diff]5t me?ods4 ,tables ,v41 ,,xiv41 ,,xvi41 ,,xxi41 are to be f\nd 9 ! same mann] z ,table ,,iii42 & ,tables ,,vi41 ,,ix41 ,x41 ,,xi41 ,,xv41 ,,xvii41 ,,xviii41 ,,xix41 ,,xx41 ,,xxi41 ,,xxii41 9 ! same mann] z ,table ,,iv4 ,,! ,,me?od ,,( ,,graphical ,,con/ruction4 ,,table ,,iii4 ,a horizontal l9e ,,aps1 %[n on a dim9i%$ scale 9 ,fig4 #ajc1 is to be neatly drawn ^u a piece ( c>dbo>d ab\t #ad #f4 ,a scale :i* reads to ! hundr$? ( an in* is to be us$ 9 ! con/ruction ( ! figure4 ,a pocket l5s w be f\nd conv5i5t 9 read+ ! small divisions4 ,by means ( a pair ( compasses & ! scale1 po9ts are to be m>k$ ^u ! l9e ,,aps1 at di/ances #ad1 #bh1 #db1 #ef1 #gj1 #hd1 #ih1 #aab f ! orig9 ,a4 ,^! di/ances correspond to ! magnitudes ( ! loads plac$ ^u ! slide on ! scale ( #ja to #a lb4 ,p]p5dicul>s to ,,aps are to be erect$ at ! po9ts m>k$1 & di/ances ,f1 ,f1 ,f1 c4 set (f ^u ^! p]p5dicul>s4 ,^! di/ances are to be equal1 on ! adopt$ scale1 to ! frictions = ! correspond+ loads4 ,= example1 we see f ,table ,,iii41 ,exp]im5t #c1 t :5 ! load ^u ! slide is #db lbs41 ! friction is #abb lbs42 h5ce ! po9t ,f is f\nd by measur+ a di/ance #db f ,a1 & erect+ a p]p5dicul> #abb4 ,?us1 = ea* ( ! loads a po9t is det]m9$4 ,! positions ( ^! po9ts %\ld be indicat$ by mak+ ea* ( !m ! c5tre ( a small circle #ja diamet]4 ,^! circles1 besides neatly def9+ ! po9ts1 w be useful 9 a subsequ5t "p ( ! process4 ,illu/ration3 ,,fig4 #ajc4 ,x w be f\nd t ! po9ts ,f1 ,f1 c4 are v ne>ly 9 a /raight l9e4 ,we assume t1 if ! app>atus & obs]vations 7 p]fect1 ! po9ts w\ld lie exactly 9 a /raight l9e4 ,! object ( ! con/ruction is to det]m9e ! /raight l9e1 :i* on ! :ole is mo/ close to all ! po9ts4 ,if x be true t ! friction is proportional to ! pressure1 ?is l9e %\ld pass ?r\gh ! orig9 ,a1 = !n ! p]p5dicul> :i* repres5ts ! friction is proportional to ! l9e cut (f f ,a1 :i* repres5ts ! load4 ,x w be f\nd t a l9e ,,at c be drawn ?r\gh ! orig9 ,a1 s t all ! po9ts are 9 ! imm$iate vic9ity ( ?is l9e1 if n actually ^u x4 ,a /r+ ( f9e black silk ab\t #ae long1 /ret*$ by a b[ ( wire or :alebone1 is a conv5i5t /raight-edge = f9d+ ! requir$ l9e4 ,! circles describ$ ab\t ! po9ts ,f1 ,f1 c4 w facilitate ! plac+ ( ! silk l9e z ne>ly z possible ?r\gh all ! po9ts4 ,x w n be f\nd possible to draw a l9e ?r\gh ,a1 :i* %all int]sect all ! circles2 ! be/ l9e passes bel[ b v ne> to ! circles r\nd ,f1 ,f1 ,f1 ,f1 t\*es ! circle ab\t ,f1 int]sects ! circles ab\t ,f & ,f1 & passes above ! circle r\nd ,f4 ,! l9e %\ld be s plac$ t its dep? bel[ ! po9t :i* is mo/ above x1 is equal to ! height at :i* x passes above ! po9t :i* is mo/ bel[ x4 ,f ,a measure ,,z1 a l5g? ( #aj1 & erect ! p]p5dicul> ,,/4 ,we f9d by measurem5t t ,,/ is #bg4 ,if1 !n1 we suppose t ! friction = any load is really repres5t$ by ! di/ance cut (f by ! l9e ,,at ^u ! p]p5dicul>1 x foll[s t ,f 3 ,r 33 #bg 3 #aj4 or ,f #jbg ,r4 ,?is is ! =mula f :i* ,table ,,iii4 has be5 con/ruct$4 ,,table ,,iv4 ,by a c>eful application ( ! silk b[-/r+1 ,x ,y ,q c be drawn1 :i*1 itself 9 close proximity to ,a1 passes m ne>ly ?r\gh ,f1 ,f1 c4 ?an is possible = any l9e :i* passes exactly ?r\gh ,a4 ,x ,y ,q w be f\nd n only to int]sect all ! small circles1 b to cut (f a consid]able arc f ea*4 ,measure (f ,x ,p a di/ance ( #aj1 & erect ! p]p5dicul> ,p ,q2 !n1 if ,r be ! load1 & ,f ! correspond+ friction1 we mu/ h f simil> triangles-- ,,ay ,f - -------- #a lb4 #ja ,,pq ---------------------------------- ------------4 ,r ,,px ,by measurem5t x is f\nd t ,,ay #jad1 & ,,pq #bec4 ,we h1 !re=e1 ,f #ad #jbec ,r4 ,?is is practically ! same =mula z ,f #add #jbeb ,r1 f :i* ! table has be5 con/ruct$4 ,9 fact1 ! column ( calculat$ values ( ! friction might h be5 comput$ f ! =m]1 )\t appreciably diff]+ f :at is f\nd 9 ! table4 ,,! ,,me?od ,,( ,,lea/ ,,squ>es4 ,,table ,,iii4 ,let k be ! coeffici5t ( friction4 ,x is impossible to f9d any value = k :i* w satisfy ! equation1 ,f - k ,r #j1 = all ! obs]v$ pairs ( values ( ,f & ,r4 ,we h !n to f9d ! value = k :i*1 ^u ! :ole1 be/ repres5ts ! exp]im5ts4 ,f - k ,r is to be z ne> z]o z possible = ea* pair ( values ( ,f & ,r4 ,9 accordance ) ! pr9ciple ( lea/ squ>es1 x is well kn[n to ma!maticians1 ! be/ value ( k is t :i* makes 7,f - k ,r7 7,f - k ,r7 c4 7,f - k ,r7 a m9imum :]e ,f & ,r1 ,f & ,r c4 are ! simultane\s values ( ,f & ,r 9 ! sev]al exp]im5ts4 ,9 fact1 x is easy to see t1 if ?is quantity be small1 ea* ( ! ess5tially positive elem5ts1 7,f - k ,r71 c4 ( :i* x is compos$1 mu/ be small also1 & t !re=e ,f - k ,r mu/ always be ne>ly z]o4 ,diff]5tiat+ ! sum ( squ>es & equat+ ! diff]5tial coeffici5t to z]o1 we h accord+ to ! usual notation1 ,r 7,f - k ,r7 #j2 ,r ,f :5ce k --------------------------------4 ,r ,! calculation ( k becomes simplifi$ :5 7z is g5]ally ! case 9 ! tables7 ! loads ,r1 ,r1 c41 ,r are ( ! =m1 ,n1 #b,n1 #c,n c41 m ,n4 ,9 ?is case1 ,r ,f ,n 7,f #b;,f #c;,f c4 m,f74 ,r ,n 7#a #b c4 m7 m7m #a7 7#bm #a7 ,n ---------------------------------------- ------ #f 7,f #b;,f c4 m,f7 k #f ---------------------------------------- ------------------ 4 ,nm7m #a7 7#bm #a7 ,9 ! case ( ,table ,,iii4 m #h1 ,n #ad1 ,f #b;,f #c;,f m,f #ggji2 :5ce k #jbg4 ,?us ! =mula ,f #jbg ,r is d$uc$ bo? by ! me?od ( lea/ squ>es1 & by ! me?od ( graphical con/ruction4 ,,table ,,iv4 ,! =mula = ?is table is to be d$uc$ f ! foll[+ consid]ations4 ,no values exi/ = x & y1 s t ! equation ,f x y ,r %all be satisfi$ = all pairs ( values ( ,f & ,r1 b ! be/ values = x & y are ^? :i* make 7,f - x - y ,r7 7,f - x - y ,r7 c4 7,f - x - y ,r7 a m9imum4 ,diff]5tiat+ ) respect to x & y1 & equat+ ! diff]5tial coeffici5ts to z]o1 we h 7,f - x - y ,r7 #j1 ,r7,f - x - y ,r7 #j4 ,?is gives two equations = ! det]m9ation ( x & y4 ,suppose1 z is usually ! case1 ! loads be ( ! =m1 ,n1 #b,n1 #c,n1 #d,n c4 m,n1 & mak+ ,a ,f ,f ,f c4 ,f ,b ,f #b;,f #c;,f c4 m,f1 we h ! equations m7m #a7 ,a - mx - -------------------------- ,n y #j1 #b m7m #a7 m7m #a7 7#bm #a7 ,b - -------------------------- x - ---------------------------------------- ------ ,n y #j4 #b #f ,solv+ ^!1 we f9d #b #dm #f x ---------------------- ,a - ------------------ ,b1 m - m m - m #ab ,b #f ,a y ------------------ ---------- - ------------------- ---------- 4 m - m ,n m - m ,n ,9 ! pres5t case1 m #h1 ,n #ad1 ,a #achd1 ,b #ggji2 :5ce x #add y #jbeb1 & we h ! =mula1 ,f #add #jbeb ,r4 ,,app5dix ,,ii4 ,,details ,,( ,,! ,,willis ,,app>atus ,,us$ ,,9 ,,illu/rat+ ,,! ,,=ego+ ,,lectures4 ,! ultimate p>ts ( ! v>i\s contrivances figur$ 9 ?is volume are ma9ly ^? inv5t$ by ! late ,pr(essor ,willis ( ,cambridge4 ,!y are m9utely describ$ & illu/rat$ 9 a "w writt5 by him = ! purpose "u ! title ,sy/em ( ,app>atus = ! use ( ,lectur]s & ,exp]im5t]s 9 ,me*anical ,philosophy1 ,london1 ,weale ,co41 #ahea4 ,?is "w has long be5 out ( pr9t4 ,x may !re=e be conv5i5t if ,i give "h a brief acc\nt ( ^? p>ts ( ?is admirable app>atus t ,i h f\nd especially useful ,! illu/rations h be5 copi$ f ! plates 9 ,pr(essor ,willis' book4#b #b ,i ought to ackn[l$ge ! k9dness ) :i* ,mr4 ,j4 ,willis ,cl>k1 ( ,cambridge1 ! lit]>y executor ( ,pr(essor ,willis1 has respond$ to my qu]ies1 :ile ,i am also "u obligations to ! c\rtesy ( ,messrs4 ,crosby1 ,lockwood1 ,co4 ,! ,willis sy/em provides ! means = putt+ v]satile framework toge!r ) or )\t revolv+ ge> = ! purpose ( me*anical illu/ration4 ,_m p>ts :i* ent] into ! con/ruction ( ! ma*9e us$ at ! lecture to-day w reappe> to-morr[ z ess5tial p>ts ( "s totally diff]5t contrivance4 ,! p>ts are suffici5tly sub/antial to "w ?or\ghly well4 ,! scantl+s & dim5sions g5]ally h be5 s *os5 z to produce models readily visible to a l>ge class4 ,x w ( c\rse be und]/ood t e model conta9s "s "o or m special p>ts su* z ! pun* & die 9 ,fig4 #gc1 or ! spr+ balance 9 ,fig4 #ag1 or ! pulley-block 9 ,fig4 #cc4 ,b = ! due exhibition ( ! op]ation ( ! ma*9e a fur!r quantity ( ord9>y framework & ( mov+ me*anism is usually necess>y4 ,?is mat]ial1 :i* may be reg>d$ z ( a g5]al type1 x is ! function ( ! ,willis sy/em to provide4 ,,! ,,bolts4--,! sy/em ma9ly owes its v]satility & its /ead9ess to ! use ( ! iron screw bolt = all atta*m5ts4 ,! bolts us$ are diamet]2 ! %ape ( ! head is hemisph]ical & ! %ank mu/ be squ>e = a %ort di/ance f ! head s t ! bolt _c turn r\nd :5 pass$ ?r\gh ! slits ( ! brackets or rectangles4 ,:5 ! head ( ! bolt be>s on a slit 9 "o ( ! wood5 pieces a circul> iron wa%] #b 9 diamet]1 or a squ>e wa%] #b on ea* side1 is necess>y to protect ! wood f cru%+4 ,!re is to be a squ>e hole1 9 ! wa%] to receive ! squ>e %ank ( ! bolt & ! ?ickness ( ! wa%]s %\ld be 4 ,! nut is squ>e or hexagonal1 & %\ld always h a wa%] und]nea? :5 screw$ home ) a spann] or screw-wr5*4 ,! mo/ useful l5g?s are #b1 #d1 #f4 ,! prop] k9d are kn[n comm]cially z coa*-bolts1 & !y %\ld be *os5 ) easy screws1 = facility 9 erect+ or modify+ app>atus4 ,at lea/ two doz5 ( ! int]m$iate size & a doz5 ( ea* ( ! o!rs are requir$4 ,= elaborate contrivances _m m w be necess>y4 ,illu/ration3 ,,! ,,b$4 ,,fig4 #ajd4 ,,! ,,b$s4--,! simple/ z well z ! longe/ p>ts ( ! framework are call$ 8b$s0 7,fig4 #ajd74 ,ea* b$ is made ( two wood5 b>s4 ,^! b>s are unit$ by /rong screws pass+ ?r\gh small blocks ( h>d wood s z to keep ! b>s full asund]1 & ?us all[ ! %anks ( ! bolts to pass freely ?r\gh ! slit ,! scantl+ ( ea* b> is #b #a1 & ! b$s are ( v>i\s l5g?s f #a' to #aj' or ev5 long]4 ,! b$s c be atta*$ toge!r 9 any requir$ position by bolts #f long4 ,! rectangles & ! brackets are atta*$ to ! b$s by #d bolts4 ,9 "o conjunction or ano!r ! b$s w be f\nd repres5t$ 9 almo/ e figure 9 ! book4 ,we may specially ref] to ,figs4 #bj1 #dd1 #dh1 #di1 #ej1 #fe1 #hc4 ,illu/ration3 ,,! ,,/ool4 ,,fig4 #aje4 ,,! ,,/ool4--,mo/ ( ! l>g] pieces ( app>atus h ! /ool z _! f\ndation 7see ,figs4 #aa1 #ci1 #ajb74 ,x is (t5 conv5i5t z 9 ,fig4 #fe to employ a pair ( /ools1 :ile "o /ool sup]pos$ on ano!r gives ! conv5i5t /& 9 ,fig4 #hj4 ,! /ool is a /\t wood5 frame1 provid+ a *oice ( slits to :i* b$s or o!r pieces may be atta*$ by bolts4 ,! /ructure ( ! frame is %[n 9 ,fig4 #aje4 ,x is #b' #f high & its extreme horizontal dim5sions are #b' #f #a' #i ( :i* ! great] is ,a ,e4 ,9 o!r words1 ! long] sides ( ! /ool are ^? op5 at ! top4 ,ea* top corn] is /r5g!n$ by an iron plate ( :i* a sep>ate sket* is %[n4 ,! scantl+s ( ! p>ts ( ! /ool are z foll[s3--,! legs & horizontal top rails are #c #b4 ,two ( ^! rails ) ! int]v5+ slit make ! top & legs to be #d wide4 ,! bottom front rail ,i is #c wide & #d deep4 ,! d\ble side rails ,d1 ,h are #a wide & #b deep1 be+ made ?9n] ?an ! legs into :i* !y are mortis$ 9 ord] to all[ ! wa%]s ( ! bolts to pass beh9d !m4 ,! slits are to be full wide ?r\gh\t4 ,bee* or bir* are v suitable mat]ials1 b s(t] woods w answ] if l>ge wa%]s are inv>iably us$4 ,illu/ration3 ,,! ,,rectangle4 ,,fig4 #ajf4 ,,! ,,rectangle4--,! useful elem5t ( ! ,willis sy/em kn[n by ?is "n is ( iron ca/ 9 "o piece 7,fig4 #ajf74 ,! rectangles are us$ 9 ! atta*m5t ( b$s to ea* o!r "u special conditions1 or !y are (t5 atta*$ to ! /ools or to brackets4 ,inde$ _! uses are multif>i\s1 see = examples ,figs4 #ab1 #eh1 #fb1 #hi1 #ig1 #ajb & _m o!rs4 ,! faces ( ! rectangle are #b broad4 ,! outside dim5sions are #f & #i1 & ! ?ickness ( metal is 4 ,ea* side ( ! rectangle has ! usual bolt slit cle>4 ,rectangles ( a l>g] size are (t5 f\nd useful1 _! weight makes !m effective /&s 7see ,figs4 #ce1 #dc1 #eb1 #fe74 ,,! ,,too!d ,,:eel4--,! mo/ conv5i5t type ( too!d :eel = our pres5t purpose is t kn[n z ! ca/ iron t5-pit*4 ,9 all su* :eels ! numb] ( tee? is simply t5 times ! numb] ( in*es 9 ! diamet]4 ,= example a :eel ) #abj tee? is #ab in*es 9 diamet]4 ,a numb] ( t5-pit* :eels l>ge & small mu/ be provid$4 ,! actual assortm5t t w be necess>y dep5ds ^u circum/ances4 ,= mo/ purposes x w be suffici5t to h ! multiples ( #e f #be upw>ds to #abj1 & !n a few l>g] sizes su* z #aej1 #ahj1 #bjj4 ,duplicates ( ! con/antly recurr+ numb]s su* z #cj1 #fj1 #abj are conv5i5t4 ,arm :eels are always pref]able to plate :eels 9 lightness & appe>ance z well z 9 price4 ,all :eels are to be #a ?ick at ! boss :i* is fac$ 9 ! latt] at ea* side1 & bor$ ) a hole full #a diamet]1 9 :i* a key groove is cut4 ,a pair ( mitre :eels su* z are us$ 9 ,fig4 #hj are sometimes useful4 ,illu/ration3 ,,! ,,/ud ,,socket4 ,,fig4 #ajg4 ,,! ,,pulley4--,we h frequ5t occasion to use ! pulley = convey+ a cord1 & a some:at v>i$ /ock is conv5i5t ,?us light brass pulleys are us$ 9 ! app>atus %[n 9 ,fig4 #c1 & a /\t pulley 9 ,fig4 #ga4 ,a ca/ iron pulley ab\t #aj 9 diamet] is se5 9 ,figs4 #cb & #cd4 ,x is bor$ #a 9 diamet] ) a key groove1 & ! boss is #a ?ick4 ,"s small pulley-blocks simil> to ^? us$ on ya*ts are (t5 v useful4 ,,! ,,/ud ,,socket4--,= m\nt+ too!d :eels on ! l>g] pulleys or = almo/ any rotat+ or oscillat+ pieces ! /ud socket is us$ 7see ,fig4 #ajg74 ,! socket ,a ,b may be made ( brass or ( ca/ iron4 ,x is #a 9 diamet] s z to pass ?r\gh ! bosses ( ! :eels t h be5 bor$ to #a ) ?is object3--,! socket is provid$ ) a %\ld] at "o end 7,a7 :i* is #a diamet]1 & ) a /rong screw ,b & octagonal nut at ! o!r end4 ,! extreme l5g? ( ! socket is #c1 & ! pla9 "p ( ! #a cyl9d] is #a long4 ,:5 two :eels are plac$ on ! socket ea* ( :i* has a boss #a ?ick1 ! tight5+ ( ! nut w secure ! :eels aga9/ ! %\ld]4 ,a fea!r is screw$ on ! pla9 "p :i* ent]s ! key grooves 9 ! :eels1 & ?us ensures t ! :eels %all turn toge!r4 ,?is fea!r %\ld be small en\gh to slip easily into ! key groove4 ,if only a s+le :eel or if any peculi> piece su* z a wood5 cam or a disk ( %eet iron has to be m\nt$1 !n coll>s or l>ge ?ick wa%]s mu/ be plac$ on ! socket s z p]mit ! screw to b9d ! :ole toge!r4 ,! socket revolves ^u a /\t iron /ud ,c ,d1 :i* is 9 diamet]4 ,x be>s a %\ld] or flange ,c at ! back ( ! same diamet] z ! base ( ! socket ,! /ud be>s on ! o!r side ( ! %\ld] a /rong screw & nut :i* project #a s z to all[ ! /ud to be secur$ 9 a hole #a deep 9 "o ( ! brackets 7to be pres5tly describ$74 ,! pla9 "p ( ?is screw ne> ! %\ld] mu/ be diamet]4 ,! front end ( ! /ud is pi]c$ ) a hole to receive a spr+ p9 to keep ! socket f slid+ (f ! /ud4 ,among ! _m applications ( ! /ud socket we may m5tion ^? %[n 9 ,figs4 #cj1 #gc1 #gd4 ,illu/ration3 ,,bracket ,,no4 ,i4 ,,fig4 #ajh4 ,illu/ration3 ,,bracket ,,no4 ,,ii4 ,,fig4 #aji4 ,illu/ration3 ,,bracket ,,no4 ,,iii4 ,,fig4 #aaj4 ,,! ,,bracket4--,!re are six diff]5t =ms ( ca/ iron brackets repres5t$ 9 ! adjo9+ figures 7,figs4 #ajh-aac74 ,! brackets are prim>ily int5d$ z ! supports ( ! /ud sockets4 ,= ?is purpose ea* has a head #a ?ick bor$ ) a hole diamet]1 & ?us fitt$ to receive ! screw on any ( ! /uds4 ,ea* bracket /&s on a base or sole ) a slit full wide = ! bolts4 ,! ?ickness ( ! sole is 4 ,! l>g] ( ! brackets ,i41 ,,ii41 & ,,iv4 h also slits 9 _! v]tical faces4 ,brackets c be fa/5$ ei!r to ! /ool or to ! b$s or rectangles1 & ! v>iety ( _! =ms enables ! :eel-"w c>ri$ on ! /ud sockets to be dispos$ 9 any desir$ fa%ion4 ,brackets avail = _m o!r purposes besides ^? ( support+ rotat+ me*anism4 7,look at ,figs4 #aa1 #ab1 #ag1 #bj1 #cc1 #ch1 #ci1 #gc & _m o!rs47 ,illu/ration3 ,,bracket ,,no4 ,,iv4 ,,fig4 #aaa4 ,illu/ration3 ,,bracket ,,no4 ,v4 ,,fig4 #aab4 ,illu/ration3 ,,bracket ,,no4 ,,vi4 ,,fig4 #aac4 ,illu/ration3 ,,! ,,tube ,,fitt+4 ,,fig4 #aad4 ,,! ,,%afts ,,& ,,tube-,,fitt+s4--,! /ud sockets w n provide = e case 9 :i* :eels h to be m\nt$ & driv54 ,we mu/ (t5 employ %afts 7see = in/ance ,figs4 #cj1 #dg1 #aja74 ,! %afts we use are turn$ iron rods 9 diamet]1 & ( v>i\s l5g?s f #f up to #d'4 ,to support ! %afts we use = be>+s ! tube fitt+ 7,fig4 #aad74 ,?is is a brass ca/+ :i* consi/s ( a tube ,m ,n #b long1 & #a 9 ext]nal diamet]1 bor$ s z to fit ! %aft4 ,! back ( ?is tube is a flat surface p>allel to ! bore1 & f x projects a screw diamet]1 & #a long ) a nut :i* is h[ev] omitt$ 9 ! draw+4 ,?is screw may be ( ! same size z t ( ! /uds1 & x is int5d$ = ! same purpose1 namely to atta* ! be>+ to ! hole 9 a bracket4 ,! tube may ( c\rse be fix$ at any desir$ angle 9 ! plane p>allel to ! face ( ! bracket4 ,to prev5t ! endlong motion ( ! %aft ca/ iron or brass r+s are employ$ 7,fig4 #aae74 ,^! are bor$ 1 & furni%$ ) a b9d+ screw by :i* !y may be tight5$ on ! %aft 9 any position4 ,to avoid injury to ! %aft x is well to h a n>r[ flat surface fil$ along x to receive ! end ( ! b9d+ screw4 ,! use ( ! r+s is %[n 9 ,fig4 #dg4 ,if z (t5 happ5s 7see = example ,fig4 #ajb7 a b>rel has to be set 9 motion by a %aft ! requir$ atta*m5t c be made by simply slipp+ on ! b>rel1 & !n putt+ at ea* end ( x two ( ! p9n$ r+s 7,fig4 #aae74 ,! p9s ent] holes bor$ into ! b>rel = _! reception s t :5 ! r+s are b\nd to ! %aft by _! screws ! b>rel mu/ revolve ) ! %aft4 ,illu/ration3 ,,! ,,p9n$ ,,r+4 ,,fig4 #aae4 ,illu/ration3 ,,! ,,adapt]4 ,,fig4 #aaf4 ,,! ,,adapt]4--,= ! atta*m5t ( :eels or o!r rotat+ pieces to ! %aft an adapt] is employ$ 7,fig4 #aaf74 ,x is bor$ ) a hole to fit ! %aft1 & ! ext]nal diamet] is #a4 ,at "o end is a %\ld] ?r\gh :i* ! b9d+ screw is tapp$1 & !re is a nut & screw at ! opposite end4 ,a fea!r w prev5t ! :eel f turn+ r\nd on ! adapt] :i* is itself made to revolve ) ! %aft by screw+ ! b9d+ screw d[n on ! %aft4 ,"s adapt]s are only l>ge en\gh = a s+le :eel #a ?ick 9 ! boss1 b x is useful to h o!rs t w take two :eels4 ,adapt]s are %[n 9 use 9 ,figs4 #df & #aja4 ,illu/ration3 ,,! ,,lev] ,,arm4 ,fig4 #aag4 ,,! ,,lev] ,,arm4--,to give motion to ! me*anism a lev] arm ) a h&le is frequ5tly requir$ 7,fig4 #aag74 ,x is bor$ #a & has a key groove1 & ! hole is #a long1 s t ! lev] arm c be fix$ on a /ud socket l a :eel4 ,by ! aid ( an adapt] ! lev] arm is atta*$ to a %aft4 ,= ! use ( ! h&le see ,figs4 #cj & #aja4 ,!re are h[ev] _m o!r uses to :i* ! lev] arm is occasionally put4 ,x c be us$ z a crank1 & 9 l9kage arrangem5ts a pair ( lev] arms are v conv5i5t4 ,/uds ,a or ,c c replace ! h&le :5 necess>y4 ,su* are ! p>ts ( ! ,willis app>atus :i* are adapt$ = our pres5t purpose4 ,x rema9s to add t ! fits %\ld be v easy1 & ! p>ts %\ld be readily int]*angeable4 ,,index4 ,a4 ,accid5t1 risk (1 #cb ,action1 #f ,adapt]1 ,willis app>atus1 #ceb ,angle ( friction1 #gh ( /atical friction1 #hj ,app>atus = c5tre ( gravity1 #fb = equilibrium ( ?ree =ces1 #g to %[ friction1 #fe1 #gh ! ,willis1 #cde ,app5dix ,i41 #cci ,atwood's ma*9e1 #bcb ,axes1 p]man5t1 #bgi ,b4 ,balance1 defective1 #dh spr+1 #af ,b>1 equilibrium ( a1 #ch ,bat1 cricket1 #cji ,beam1 bread? (1 #aic break+ load (1 #aic1 #aif ca/ iron1 #bbb collapse (1 #ahf deflection (1 #agi ela/icity (1 #ahd load on1 #aig plac$ edgewise1 #aic /ra9$1 #agh /r5g? (1 #aij uni=mly load$1 #aih ) bo? ends secur$1 #bjj ) "o end secur$1 #bja ,b$s 9 ,willis app>atus1 #cdf ,bob1 rais+ or l[]+ !1 #cbj ,bolts1 use (1 9 ,willis app>atus1 #cdf ,bracket1 ,willis app>atus1 #cej ,brass1 specific gravity (1 #ef ,break+ load1 #agg ,bridge1 deflection (1 #bjh me*anics (1 #bah ,m5ai1 #bah susp5sion1 #bbe ! ,wye1 #bae tubul>1 #bbc ) f\r /ruts1 #baj two /ruts1 #bjf two ties1 #baa ,brunei1 ,sir ,j41 ! ,wye bridge1 #bae ,c4 ,cap/an1 #aea ,ca/ iron beam1 #bbb ,cat5>y1 #bbf ,ca!tomet]1 #ahj ,c5tre ( gravity1 #eg ( a :eel1 #fa position (1 #ei oscillation1 #cjd p]cussion1 #cjg ,circul> motion1 #bfg action (1 #bga applications (1 #bgf cause (1 #bgj 9 gov]nor-balls1 #bgf 9 sug> ref9+1 #bgf nature (1 #bfg on liquids1 #bga on ! e>?1 #bgf ,circul> p5dulum1 #bhd ,clamps1 #bjc ,clock p5dulum1 #bii pr9ciples (1 #cah rate (1 #cbb ,coeffici5t ( friction1 #gd1 #hb ,collapse ( a beam1 #ahf ,comp5sat+ p5dulum1 #cai ,composition ( =ces1 #a1 #i p>allel =ces1 #ce1 #cg1 #db vibrations1 #bii1 #cae ,conical p5dulum1 #caj ,c\ple1 #dd ,crane1 #bi1 #afb friction 91 #aff me*anical effici5cy (1 #afe ,table ,,xxi4 #afe ,,xxii4 #aff velocity1 ratio (1 #afc ,cricket bat1 #cji ,cr[b>1 #abc ,cycloid1 #bie ,d4 ,dead-beat escapem5t1 #cbh ,def9ition ( =ce1 #b ,deflection ( a beam1 ,table ,,xxiii4 #ahb ,diff]5tial pulley1 #aab ,table ,,xi4 #aad ,direction ( a =ce1 #e ,e4 ,eade1 ,mr41 epicycloidal pulley-block1 #aaf ,ea/] ,isl&1 #ajj ,ela/icity ( beam1 #ahd ,en]gy1 #he1 #id /orage (1 #bef1 #beh unit (1 #ie ,eng9e1 locomotive1 #hc ,epicycloidal pulley-block1 #hj1 #aaf ,table ,,xii4 #aah ,equilibrium1 neutral1 #fa ( a b>1 #ch1 #da ?ree =ces1 #f two =ces1 #f /able1 #ei un/able1 #ei ,escapem5t1 #cbd dead-beat1 #cbh recoil1 #cbh ,expansion ( bodies1 #cba ,exp]im5t by ,m4 ,plateau1 #bgc ,f4 ,fall 9 a second1 #bci ,fall+ body1 motion (1 #bcj ,feet1 h[ repres5t$1 #g ,fibres 9 /ate ( compression1 #ahd t5sion1 #ahd ,fir/ law ( motion1 #bcj ,fly-:eel1 #bfj 9 /eam-eng9e1 #bfb ,foot-p\nd1 #ie ,=ce1 a small1 & two l>g]1 #ab def9ition (1 #b de/roy+ motion1 #c direction ( a1 #e magnitude ( a1 #d measurem5t (1 #d ( friction1 #fe gravity1 #ej "o1 resolv$ into ?ree1 #bf two1 #ag repres5tation (1 #e /&>d (1 #d ,=ces1 composition (1 #a1 #i equilibrium ( ?ree1 #f two1 #f illu/rations (1 #c 9 incl9$ plane1 #acf p>allel1 #cd p>allelogram (1 #aj resolution (1 #af ,=mula = pulley-block1 #aji1 #aad ,framework1 #bjc1 #cde ,friction1 #fe accurate law (1 #ge a =ce1 #ff & pressure1 #gb angle (1 #gh angle ( /atical1 #hj app>atus to %[1 #fe1 #fh1 #gh caus$ by r\ghness1 #ff coeffici5t (1 #gd1 #hb dim9i%$1 #ff excessive1 #aae exp]im5t+ on1 #ff 9 crane1 #aff diff]5tial pulley-block1 #aac incl9$ plane1 #acb lev]1 #abc pulleys1 #hi law (1 #ia rope & iron b>1 #hg :eel & axle1 #aec :eel & b>rel1 #aeh laws (1 #gc1 #ha1 #hb mean1 #ge motion imp$$ by1 #gj nature (1 #fe ov]com+1 #ic ,table ,i4 #fi ,,ii4 #ga ,,iii4 #gd ,,iv4 #gf ,v4 #gh ,,vi4 #ha ,,vii4 #ha ,,viii4 #ha ^u axle1 #aee :eels1 #ic ,g4 ,galileo & fall+ bodies1 #bce k9etics1 #bcj ! p5dulum1 #bhd t[] ( ,pisa1 #bcc ,ga!r+ pallet1 #ccf ,gird]1 #bai z slight z possible1 #bba ,gov]nor-balls1 #bgf ,graham1 dead-beat escapem5t1 #cbh ,graphical con/ruction1 #cci ,gravity1 #ej action (1 #bdc & ! p5dulum1 #bib & weight1 #eb c5tre (1 #eg def9$1 #bdf diff]5t effects (1 #ec indep5d5t ( motion1 #bda 9 ,london1 #bib specific1 #ec ,gr9d/one1 treadle (1 #abh ,h4 ,hamm]1 #beb !ory ( !1 #beb ,h&s ( a clock1 #cca ,horse-p[]1 #if ,i4 ,illu/ration ( p>allelogram ( =ces1 #aj ,illu/rations ( =ces1 #aj resolution1 #ai ,in*es1 h[ repres5t$1 #g ,incl9ation ( ?read1 #adj ,incl9$ plane1 #aca =ces on1 #acf friction 91 #acb me*anical effici5cy (1 #aci ,table ,,xiii4 #acd ,,xiv4 #acg ,,xv4 #ach velocity1 ratio (1 #aci ,in]tia1 #bej inh]5t 9 matt]1 #beb ,iron gird]s1 #bai specific gravity (1 #ee ,iso*ron\s simple p5dulum1 #cjc ,ivory1 specific gravity (1 #ef ,j4 ,jib1 #bi1 #afc ,k4 ,kat]1 ,capta91 #cje ,k9etics1 #bcj ,l4 ,l>ge :eels1 advantages (1 #ic ,law ( fall+ bodies1 #bch friction 9 pulleys1 #ia lev] ( fir/ ord]1 #abb pressure1 #cg ,laws ( friction1 #gc1 #ha1 #hb ,lead1 specific gravity (1 #ef ,lean+ t[] ( ,pisa1 #bcc ,level1 #ef ,lev]1 #aai & friction1 #abc applications (1 #abc arm1 ,willis app>atus1 #ceb laws (1 #acj ( fir/ ord]1 #aai law (1 #abb ( second ord]1 #abd ( ?ird ord]1 #abh weight (1 #aba ,lift+ crane1 #bi ,l9e & plummet1 #ef ,load1 break+1 #agg ,locomotive eng9e1 #hc ,m4 ,ma*9e1 ,atwood's1 #bcb pun*+1 #bfc ,ma*9es1 pile-driv+1 #bee ,magnitude ( a =ce1 #d ,m>g9 ( safety1 #cc ,mass1 #bcf ,mean frictions1 #ge ,measurem5t ( =ce1 #d ,me*anical p[]s1 #he1 #ajj app>atus1 ,willis1 #cde ,m5ai ,bridge1 #bah ,me?od ( lea/ squ>es1 #cdb ,mom5t1 #acj ,monkey1 #beg ,motion1 fir/ law (1 #bcj ( fall+ body1 #bcj ,n4 ,neutral equilibrium1 #fa ,newton & gravity1 #bhi ,nut1 #adj ,o4 ,oscillation1 c5tre (1 #cjd ,p4 ,pair ( scales1 #dh te/+1 #dh ,p>abola1 #bbf ,p>allel =ces1 #cd composition (1 #ce1 #cg1 #db opposite1 #dd resultant (1 #dc ,p>allelogram ( =ces1 #aj ,pa? ( a projectile1 #bdg ,p5dulum & gravity1 #bib circul>1 #bhd comp5sat+1 #cai comp\nd1 #bii1 #cja conical1 #caj =mula =1 #bib ,galileo & !1 #bhf iso*ron\s simple1 #cjc l5g? ( ! seconds1 #bib1 #cah motion ( !1 #bhe ( a clock1 #bii simple1 #bhd "t ( oscillation1 #bhf1 #bhi ,p]cussion1 c5tre (1 #cjg1 #cji ,p]man5t axes1 #bgi ,pile-driv+ ma*9es1 #bee ,plateau1 ,m41 exp]im5t by1 #bgc ,plummet1 #ef ,p[]s1 me*anical1 #he ,pressure & friction1 #gb law (1 #cg ( a load$ beam1 #ce1 #cg ,pr9ciples ( framework1 #bjc ,projectile1 pa? (1 #bdg ,pulley-block1 #ii diff]5tial1 #aaj epicycloidal1 #hj ?ree-%eave1 #ajf velocity1 ratio (1 #aab ,pulley1 ord9>y =m (1 #hf s+le movable1 #aja fix$1 #hf use (1 #hh velocity1 ratio (1 #ajc ,pulleys1 friction 91 #hi 9 w9d[s1 #hf 9 ,willis app>atus1 #cdi ,pun*+ ma*9e1 #bfc =ce (1 #bfe ,r4 ,rack1 #ccd ,reaction1 #f ,recoil escapem5t1 #cbh ,rectangle 9 ,willis app>atus1 #cdh ,repres5tation ( a =ce1 #e ,resi/ance to compression1 #agb1 #age ext5sion1 #agb ,resolution ( =ces1 #af "o =ce into ?ree1 #bf two1 #ag ,resultant1 #i ( p>allel =ces1 #dc ,r+s 9 ,willis app>atus1 #ceb ,risk ( accid5t1 #cb ,s4 ,safety1 m>g9 (1 #cc ,sail+1 #ba aga9/ ! w9d1 #bd ,scales1 #df ,screw1 #aci & :eel & axle1 #afg =m (1 #aci ,table ,,xvi4 #adb velocity1 ratio (1 #adc ,screw bolt & nut1 #adh jack1 #aca1 #ade ,table ,,xvii4 #adf ,second1 fall 9 a1 #bci ,seconds1 p5dulum1 #cah ,%afts1 ,willis app>atus1 #cea ,%e>s1 #abf ,simple p5dulum1 #bhd ,s+le movable pulley1 ,table ,,ix4 #ajd ,snail1 #ccd ,specific gravity1 #ec ( brass1 #ef iron1 #ee ivory1 #ef lead1 #ef ,_s-level1 #ef ,spr+ balance1 #af ,/able equilibrium1 #ei1 #bhb ,/&>d ( =ce1 #d ,/atical friction1 angle (1 #hj ,/ool 9 ,willis app>atus #cdg ,/orage ( en]gy1 #bef1 #beh ,/or$-up en]gy exhibit$1 #bfa ,/r5g? ( a beam1 #aij ,/rik+ p>ts1 #ccc ,/ructures1 #afi ,/rut1 #bh ,/ud socket 9 ,willis app>atus1 #cdi ,sug> ref9+1 #bgf ,susp5sion bridge1 #bbe me*anics (1 #bbe t5sion 91 #bbh ,t4 ,table ,i4 #fi ,,ii4 #ga ,,iii4 #gd ,,iv4 #gf ,v4 #gh ,,vi4 #gh ,,vii4 #ha ,,viii4 #ha ,,ix4 #ajd ,x4 #ajh ,,xi4 #aad ,,xii4 #aah ,,xiii4 #acd ,,xiv4 #acg ,,xv4 #ach ,,xvi4 #adb ,,xvii4 #adf ,,xviii4 #aed ,,xix4 #aei ,,xx4 #afb ,,xxi4 #afe ,,xxii4 #aff ,,xxiii4 #ahb ,,xxiv4 #aij ,tack+1 #be ,t5sion along a cord1 #ag ,?ree-%eave pulley-block1 #ajf ,tie1 #bh1 #age rod1 #bi1 #cb ,timb]1 b5d+1 #aga compression (1 #agb ext5sion (1 #agb prop]ties (1 #agj r+s 91 #aga season+1 #aga w>p+1 #aga ,t91 #bbc ,too!d :eels1 #afj ,t[] ( ,pisa1 #bcc ,tra9 ( :eels1 #ccj ,transv]se /ra91 #aha ,treadle ( a gr9d/one1 #abh ,tripod1 #bh /r5g? (1 #bh ,truss1 simple =m (1 #bab ,tube fitt+1 ,willis app>atus1 #cea ,tubul> bridge1 #bbc ,u4 ,un/able equilibrium1 #ei1 #bhb ,v4 ,velocity1 #bca ratio ( incl9$ plane1 #aci pulley1 #ajc pulley-block1 #aab screw1 #adc :eel & axle1 #aeb :eel & p9ion1 #afa ,vibrations1 composition (1 #bii1 #cae ,w4 ,w$ge1 #aci ,weigh+ ma*9es1 #abc scales1 #df1 #dh ,weight caus$ by gravity1 #eb ( wat]1 #ed ,:eel & axle1 #adi & diff]5tial pulley1 #afg screw1 #afg exp]im5ts on1 #aeb =mula =1 #aed friction 91 #aec ,table ,,xviii4 #aed velocity1 ratio (1 #aeb ,:eel & b>rel1 #aeh =mula =1 #afj friction 91 #aeh ,table ,,xix4 #aei ,:eel & p9ion1 #afj effici5cy (1 #afa ,table ,,xx4 #afb velocity1 ratio (1 #afa ,:eel1 c5tre ( gravity (1 #fa ,:eels1 #ib friction1 #ic ,:eels 9 ,willis app>atus1 #cdh ,willis sy/em ( app>atus1 #cde ,w9*1 #aea ,w9d1 direction (1 #bb ,"w1 #he1 #id ,wye ,bridge1 #bae ,,! ,,end4 ,,ri*>d ,,clay ,,& ,,sons1 ,,limit$1 ,,london ,,& ,,bungay4