CHAPTER- 1: INTRODUCTION TO DYNAMICS

Transcription

1 CHPTER- : INTRODUCTION TO DYNMICS Mechics s he igi f Dics: Mechics is defied s h sciece which descibe d pedics he cdiis f es i f bdies ude he ci f fces. I is he fudi f s egieeig scieces. I c be diided d subdiided s belw: (i Newi Mechics (Egieeig Mechcis (ii Reliisic Mechics (I dels wih he cdiis ilig seed f bdies clse he speed f ligh (iii Quu Mechics (I dels wih he cdiis ilig eeel sll ss d size ie ic disce ( Mechcis f igid bdies (b Mechics f defble bdies (c Mechics f fluids Sics Dics Mechics f Cpessible fluids Mechics f Icpessible fluids Kieics Kieics Dics: I is which f Newi Mechics which dels wih he fces d hei effecs, while cig up he bdies i i. Whe we lk bu he i f he ples i u sl sse, i f spce cf, he ccelei f ubile, he i f chged picle i elecic field, swigig f pedulu, we e lkig bu Dics. Kieics: I is h bch f Dics which dels wih he displcee f picles igid bd e ie wih u efeece he fces h cuse chge he i. I is cceed wih he psii, elci d ccelei f ig bdies s fucis f ie. Kieics: I is h bch f Dics which dels wih he i f picle igid bd, wih he efeece he fces d he fc h cuse ifluece he i. F he sud f i New s Secd Lw is widel used. Dwlded f -

2 Dwlded f Chpe:- Deeii f i f picles: I geel i f picles (psii, elci d ccelei is epessed i es f fuci s, X f(, [ 6 3 ] * u i pcice he eli f i be defied b he equi wih fuci f,,&. f( f( f( ec. s hese gie eli e ieged ge he geel eli f i f(. Cse-I: Whe ccelei is gie s fuci f ie [i.e f( ] [ 6 3 ] We kw, d/d d d, d f( d Nw iegig bh sides kig lii s ie ies f d elci ies f. d f ( d f ( d f ( d...( i gi, elci is gie b, d/d d d gi iegi bh sides f equi siill f ie d psii. We ge, d d f ( d f ( d d Puig lue f f equi (i d...( ii Thus psii is bied f equi f f( # Fid he elci d psii f picles fe is 5 sec f Res, which es wih equi f 6-4. Slui: Gie equi f( 6 4, d 5. We kw, f ( d f ( d (6 4 d [ ] s / gi, - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -

3 X f ( d d 5 [ ] d d [] 5 Theefe, d /s fe 5 secd f i. Cse-II Whe he ccelei is gie fuci f psii [ i.e f( eg. 4] We kw, d/d d/d. d/d.d/d, d d, d f(d [ f(] Nw, Iegig bh sides f be equi, kig lii s elci ies f s psii p ies f. i.e d f ( d f ( d, f ( d (...( f d gi We kw, d/d d d. Iegig bh sides wih liis s ie ies f d psii f. i.e d d...( Puig lue f ies f equi ( we ge, d ( f d f ( d d Cse III : Whe ccelei is gie fuci f elci (i.e f( eg. We kw, d/d f( d/d O, d d/f( Iegig bh sides kig lii s elci ies f d psii ies f d d d f ( f ( d f ( e.g The ccelei f picle is defied s -.5, he picle is gie s iiil elci, fid he disce eled befe is elci dps hlf. Slui: Gie, -.5 i.e f(, Iiil elci, fil elci / F i f(,? Dwlded f -3

4 Dwlded f d f ( d d.5.5 [ l ] l.5.5 O, 4.8 s.. Uif Recilie i: * Uif i es ceig equl disce e equl iels f ie. ie elci cs. We he, d/d [ cs elci f bd] d ud d d [ Iegig bh sides ude liis s psii ies f d ie ] Chge i psii ( displcee is equl uif elci chge i ie [ i.e s ].3 Uif cceleed Recilie i: If cs ccelei be he, d/d cs d d Iegig bh sides wih lii d. We ge, d d...( gi f psii, we he d/d..( f d. d ( d, Iegig bh sides e he liis d d d ls, d O, d d Iegig bh sides ude liis d d ( ( ( ( Iiil psii Fil psii Iiil elci. Fil elci Iiil ie Fil ie.4 Mi f seel picles: Tw e picles ig i sigh lie. Equis f i be wie f ech picles s: - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -4

5 ( Relie i f w picles (b Depede i. ( Relie i f w picles: Cside w picles d ig lg he se sigh lie s fllws: / [ / - ] Psii c-dies f Psii c-dies f Relie psii c-die f w.. - / / ( [ / is igh i psii ] [ - / is lef i psii ] Diffeeiig equi ( w.. ie we ge, / ( [, bslue elci f p. d ] [ / elci f bseed f p. ] Diffeeiig equi ( w.. ie we ge, / (3 [, bslue elci f p d d / ccelei f p w.. p. ] (b Depede Mi: Whe he psii f picles will depede up he psii f he seel he picles, he is e sid be depede. eg :- pulle sses, Ge sse ec. Pulle sse s Depede i: Cside he pulles sse i which he psii f lck depeds up he psii f lck s fllws: F Figue: IH Cs H G J Cs c CD Cs c EF Cs C D D CDEFG Cs Nw, X C IH C Cs. (i X FG J FG Cs.(ii E J F Muliplig equi (ii b d ddig (ii we ge, X CFG Cs C FGFG Cs C FGDED G cs [ Sice FG DED G] CFGDE Cs [ Sice D G Cs ] Dwlded f -5

6 Dwlded f Cs..(i [ CDEFG Cs ] If blck is gie i i will pduce (- / s he i f lck. Diffeeiig equi (i w.. ie, we ge,, ( O, -( / Siill diffeeiig equi ( w.. ie we ge, (i O, - ( / [ Negie sig dees ppsie i dieci ] * I his cse displcee, elci d ccelei f e bd gies he displcee, elci d ccelei f he bd. This gee is clled -degee feed. # Deie he equi f i f he gie pulle sse. Slui: c Cs.( c..( c.(3 G I M N C D E F J K L C.6 Gphicl slui f Recilie i pbles: * Gphicl slui e e helpful sipl d sle he pbles f Dics. * Usig he i gphs (i.e, - d - he issig lue pi c be bied. * If equi f i is kw ll he hee gphs c be bied s fllws. If equi f displcee f(..(i is kw, The, d/d.(ii d/d.(ii i.e elci is slpe f - cue d ccelei is slpe f - cue. f( Sle d/d f( Sle d/d f( - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -6

7 If - cue is gie, he cpuig slpe ech pi f - cue cespdig - cue c be geeed d cpuig slpe ech pi f - cue - cue c be geeed. gi, F equi (ii d d d d d f equi (iii, d d d d d...( i...( d This es chge i psii i gie b he e ude cue - d chge i elci is gie b e ude he cue -d. d - Tuil Eples: 3 The i f picles is defied b he psii ec 6i 4 ˆj kˆ whee i ee d i 4 secd. he is whe 3 sec, fid he ui psii ec, elci d ccelei. Slui: 3 We he, 6i 4 ˆj kˆ 4 ie 3 sec. 8 iˆ 36 ˆj 7 4 kˆ Nw ui psii ec 3 sec. 7 8ˆ i 36 ˆj kˆ ˆ ˆ.44ˆ i.88 ˆj.65kˆ gi, ( 8 ( ( s Dwlded f -7

8 Dwlded f 3 d d iˆ 6 4 ˆj kˆ d d 4 3 i 8j ˆ 6ˆ kˆ 4 ie 3 sec. 4 6ˆ i 4 ˆj kˆ 4 4 elci ( / s /s d d 3 gi, ccelei ( 6ˆ i 8j ˆ k d d ˆj kˆ 3 sec, 8 ˆj 4.5kˆ ccelei, ( ( 4.5 [ ] / s s bll is hw eicll upwd wih elci f 9.5/s. fe s he bll is hw wih he se elci. Fid he heigh which he w bll pss ech he? Slui: Le he iiil elci f bh blls 9.5 /s h be he heigh which w blls pss ech he be he ie elpsed b he fis bll befe pssig secd d be he ie elpsed b secd. F he gie cdii:..(i f s bll, ½ 9..(ii F d bll ½ 9 (iii Subsiuig equi (iii f equi (ii, we ge, 9( ( 9( ( ( 9.5 ( ( ( i ddig equi (i d (ii, we ge.43 sec d.43 sec h h 3.5 Hece, w blls pss ech he 3.5 be he gud. 3 I he fllwig pulle sse, lck hs elci /s upwd d is ccelei is 3/s dwwd while blck 3 hs elci d ccelei /s up wd d 4/s dwwd especiel. Fid he elci d ccelei f blck. - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -8

9 Slui: Gie, /s ( 3/s ( 3 /s ( 3 4/s ( Hee, CF cs GI HJ Cs DE Cs Pi f pe ud he pulle is ls cs. Nw, X cs..(i X GI CF cs.(ii X 3 HJCF Cs (iii E C 3 Muliplig equi (i b ( d ddig (i, (ii d (iii, we ge 3 CFCFGIHJCs. 3 cs...(i G I F H Diffeeiig equi (i w.. ie. 3 ( 3..(i J 3 F equi 3 / s Theefe, elci f blck ( /s ( F equi (i / s Theefe, ccelei f blck ( 3.5 /s ( Dwlded f -9

10 Dwlded f Chpe 3 Cuilie Mi f Picles 3. Psii ec, elci d ccelei: Whe picle es lg cue ph he h sigh lie, i is sid be i cuilie i. ec lsis is used lze he chge i psii d dieci f i f picles. Y O p s p Y p s p ( X O (b X (c X Y p s Le, ie he psii ec f picle be d he ie ( he picle kes ew psii p d is psii be!. The epeses he chge i dieci s well s giude f he psii ec. (fig. The ege elci f he picles ie iel s ( i giude d dieci f Iseus elci, li d beces she, d d P & P ges clse d is ge he ph f he picle. (fig c d, s deceses, legh f P P ( equls legh f c PP s ds ν li li d Chge i psii ( c be esled i w cpes, i Oe pllel -is d ( P P ii Ohe pllel -is ( P P PP P P PP P P, li li li Y s (fig b P - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ - O α X Y O P p p α Y X

11 iˆ, ν li li ˆj The, d, ν iˆ d, ν iˆ, ν iˆ j ˆ d d ˆj ˆj [Mgiude f elci] α α [Dieci f elci] Psiie lue f Psiie lue f d d ; d d Righ Dieci Upwd Dieci Siill, ccelei b cuilie i c be cpued s: Y Y Y s Q Q Q q β q q O ( X O (b X O (c X If d be he elciies ie 4 ( i.e. ges P d P, he he ege ccelei f he picle e he ie iel is gie b ( d, li d gi, c be esled i Q Q & Q Q pllel & -es especiel. The, QQ Q Q QQ Q Q, li li li, li iˆ li ˆj d d iˆ ˆj d d, iˆ ˆj, iˆ j ˆ d d, ( ( ccelei i giude d d Dwlded f -

12 β β Dwlded f ( i dieci 3. Deiies f ec fuci: Y p Y dp du O ( X O (b X Ζ Ζ pf(uu 4u3 Le, P(u be ec fuci f scl ible u. If lue f u is ied, P will ce cue i spce. Csideig chge f ec P cespdig he lues u 4 ( u u s shw i figue(. The p p( u u p( u i. e. d p du p li u u p( u u p( u li u u - ( s u, d p p beces ge he cue. Thus du is ge he cue s shw i figue(b. gi, Csideig he su f w ec fucis p( u& Q( u he deiie f he ec ( P Q is gie b: d du ( P Q { P { Q P Q ( P Q li li li li u u u u u u u u u f he se scl ible u. The d ( P Q du d P du dq du - ( gi, pduc f scl fuci f(u d pf ec fuci P(u f he se scl ible u. The, deiie f f P is gie b: df. p du li u ( f f ( P P f li P u u u P f u f P df P du df du P f d P du - (3 - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -

13 Siill, scl pduc d ec pduc f w ec fucis P( u d Q( u be bied s: d du d P dq ( P. Q. Q P. du du - (4 [Scl Pduc] gi, d du ( P Q d P du P P iˆ P ˆj P kˆ Q P z dq du - (6 - (5 [ec Pduc] whee, P, P & P e he ecgul scl cpes f ec P & iˆ, ˆ, j kˆ e he ui ec. z d P du dp du dp dp iˆ ˆj kˆ du du - (7 [whee, P f (u] d, d P d dp d dp dp iˆ ˆj kˆ d d - (8 [whee, P f (] 3.3 Recgul Cpes f elci d ccelei: Whe, he psii f picle is defied b is b is ecgul c-dies,, z s: i & ˆ j & ˆ zk & ˆ - (i The, diffeeiig bh sides w... ie, we ge, d ν d & î && ĵ & zkˆ whee, S, &, & &&, && & z& z & && z z iˆ ˆj kˆ ( z z ( iˆ ˆj kˆ z z Y Z z X Dwlded f -3

14 Dwlded f Whe he i i ech is c be epeseed idepede wih ech he he he use f ecgul cpes descibe he psii, elci d ccelei f picle is effecie i.e. i i ech is c be csideed sepel. F e.g. f pjecile i, eglecig i esisce, he cpes f ccelei e:, d d g g, d d d gd O Iegig bh sides ude he liis, d d d d g d g, & d & g g d d, d g d d, d d d d d gd Iegig bh sides ude liis csideig i ss f igi b c-dies i.e. ; ; d, d d d d d d g d d g ( i Thus i ude pjecile c be epeseed b -idepede ecilie i. Y X u d ( O u X Pbles:. bulle is fied upwd gle f 3 he hizl f pi P hill d i skies ge which is 8 lwe h he leel f pjeci s shw i figue. The iiil elci f he bulle is /s. Clcule:. The iu heigh which he bulle will ise be he hizl. b. The cul elci wih which i will sike he ge /s c. The l ie equied f he fligh f he bulle. Slui: P / s cs / s si 3 5 / s - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -4 Hill 8

15 si 3 h 7.4 g h 7.4( s b Le, eicl cpe f elci highes pi eicl cpe f elci sikig ge H eicl disce bewee pi d ge The, gh [ ], / s 86.6 / s [ cs ] 7.55 / s (s & z P c h ( g, , Slig,.6sec( s The i f ibig picle is defied b he equi siπ d 5csπ, whee & e epded i & i sec. Deeie he elci d ccelei whe 5 b Shw h he ph f he picle is pblic. Slui: We he, siπ & π csπ & π siπ gi, 5csπ & 5π si π & π cs π The, f sec, [π csπ ]: [ 5π siπ ] 8 Tl ie f fligh is he su f ie ech f & C f T siα & PRge siα g g ( π csπ ( 5si π π / s α ( Dwlded f -5

16 d, F sec Dwlded f [( π siπ ]: π cs π β ( b Sice, siπ siπ 7 π / s ( si π ( i gi, 5cs π, cs π 5, cs π 5 5 cs π ( ii 5 ddig equis (i d (ii, we ge ; 5 si π cs π 5, 5, 5, which is he equi f pbl [ b c] 3 The i f picle is gie b he eli cs & si. I is kw h iiill bh & c-dies e ze. Deeie Tl ccelei he is f 5 b Equi f he pbl Hee, cs & si The, d, d d d d iˆ si ˆj cs si iˆ cs ˆj ( sec -siz-.8 cs-.4 α.865 / s, cs cs ( ii F sec ddig ( i d ( ii, 4 ( - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -6, 4 8, which is he equied equi f pbl β 93

17 si ˆ i cs ˆ i Q.8ˆ i.4 ˆj α α α Tl ccelei (.8 [ whee,is i di] ( / s.4 β 93 ( s.8 d b cs d d d cs cs, si, 4 gi, si ( i d d si d si d 3.4 Mi Relie Fe i Tsli: Le d be he picles ig is se ple wih & be hei psii wih espec XY is. Csideig New es (X-Y ceeed d pllel igil es X-Y, he i f picle c be defied wih espec i f picle such h: F ec igle O / ( i whee, d siill, X,Y & X & Y e c-dies f & w... XY es X X X / X /, Y / e c-dies f... X-Y is - (ii Y Y Y / Diffeeiig equ(i w... ie, we ge: (iii I scl f: X& X& X& / Y& Y& Y& / - (i Equi f i i bh d es X Y Equi f i i - is Equi f i i - is ( OR ( ( ( / ( ( / X&, Y& e X & Y cpes f X& / Y& / e X & Y cpes f gi, diffeeiig (iii wih espec ie, we ge: I scl, / ( whee, X&, Y& e X & Y cpes f / β ( ( Dwlded f -7

18 X&& Y&& X&& X&& Y&& Y&& / / OR Dwlded f ( ( ( / - (i ( ( ( / 3.4 Tgeil d Nl Cpes: The elci f picle is ec ge he ph f picle. u ccelei be ge i cuilie i. The ccelei ec be esled i w cpes pepedicul wih ech he i diecis i Fis cpe lg he ge f ph f picle ( ii Secd cpe lg he l f ph f picle ( Le eˆ d eˆ be he ui ecs dieced lg he ge d l f he ph especiel. The, i cuilie i, es f e pi he. / s eˆ d eˆ wuld chge he dieci s picle ed ds p p p ( (b F fig ( The, & eˆ ˆ li s s e d ds [ li s ] s d ds ( i s gi, e ˆ li [ li s] Theefe, s s s s e ˆ d ds is he ui ec lg he ge ph. Le, ρ be he dius f cuue f he ph he pi P d P d P. ê be he chge i ui ec while he picle es f P P. eˆ & eˆ be he ge ui ecs - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -8

19 e e ( e p p e (b e e (c p e e de d Nw, f fig, s PP ρ eˆ eˆ eˆ eˆ eˆ eˆ ˆ de eˆ li eˆ s d ( ii Siill, d deˆ d eˆ ds ρ d (iii ls, ds s ρ d li li ρ d d d ρ ρ d (i d d d ds. eˆ eˆ ds d [ eˆ ] ( d, d d d deˆ ( eˆ eˆ d d d d, e & ˆ (ˆ e ( ρ, e & ˆ eˆ ρ [ s s ê eˆ i giude] d deˆ d ds eˆ. d d ds d. which c be epeseed s i fig(c. eˆ eˆ, whee ( i d Tgeil cpe f ccelei & d Nl cpe f ccelei ρ& ρ Nes: F icesig elci will be i he dieci f elci d f decesig elci will be i ppsie he dieci f elci. If he speed is cs bu. [ f Reci ρ α ] is lws dieced wds he cee f cuue F highe elci d slle dius highe is. 3.6 Rdil d Tsese Cpes: Dwlded f -9

20 Dwlded f F he i descibed b pl c-dies. Psii f picles P is defied b he c-dies &, whee is he legh d is he gle i dis. The ui ecs i dil d sese dieci e deed b dius d 9 clckwise he dius i dieci. fig ( e e e ˆ e d eˆ especiel lg e P(, e e e ( s he picle es f e d e especiel. Hee, eˆ & (b P P. The ui ecs e, & e chge eˆ & eˆ b deˆ ˆ de d dê. eˆ & (i dieci f is i dieci f ê d d d d & deˆ ˆ de d dê e ˆ ˆ. e & (ii dieci f is i dieci f - ê d d d d Nw, eˆ d d d deˆ The, ( eˆ eˆ e & ˆ e& ˆ d d d d [ e & ˆ & e ] [& eˆ & ˆ - (iii ê ] which c epessed s eˆ eˆ, whee Rdil cpe f elci & d, Tsese cpe f elci & Siill, ( e & & e d d ˆ d d & eˆ e & ˆ & & & eˆ && eˆ && eˆ & & & ϑ && & & eˆ & eˆ eˆ eˆ eˆ [ (& & eˆ ( eˆ && & & ] ( which c be epeseed s, eˆ eˆ whee, d, [ e& eˆ & & eˆ & eˆ & ] Rdil cpe f ccelei (& & Tsese cpe f ccelei ( & & & - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -

21 I cse f picle ig lg cicul ph wih is cee he igi O, we he cs, & & & The, eˆ ( i & eˆ eˆ & ( ii Pbles: The i f picle is defied b he psii ec, ˆ 3 4 ˆ 4 3 i j 5 kˆ, whee is i d is i sec. is whe 4 sec, fid he l d geil cpe f ccelei d he dius f cuue. Slui, we he 3 iˆ 4 & gi, d d d d 3 ˆ 4 j 5 kˆ 6iˆ ˆ 3 j kˆ i 4j ˆ 6ˆ 6 kˆ 4 6 ( ( i [ 36 ( 4 ( 6 ] ( ii Nw, 4sec 94.54/s [puig 4 i equ-(i] 964.8/s [puig 4 i equ-(ii] gi, Tgeil cpe f ccelei, d d. d d 4 6 ( ( ie 4 sec, /s (s Nw, 3 5 ( ( ( / s ( s gi, ( ρ ( s 49.. c is elig cued seci f he d f dius 95 he speed f 5k/h. kes e suddel pplied cusig he c slw dw he 3 k/h fe 6 sec. Clcule he ccelei f he c iediel fe he bke he bee pplied. Dwlded f -

22 Dwlded f - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ - Slui: Gie, 8.88 /sec 3k/h 3.88 /sec 5k/h 95 ρ he is whe he bke is pplied, ( e e e e s s.833ˆ. ˆ ˆ ˆ /.833 / ρ ( ( ( ( / s s s β 3. The ple cuilie i f he picle is defied i pl c-dies b 3/4 3 d.5 whee is i, is i di d is i secd. he is whe 4 sec, deeie he giude f elci, ccelei d dius f cuue f he ph. Slui: We he, / & & & gi,.5 & & & Nw, we he ( ˆ 3 4 ˆ ˆ ˆ 3 i e e e e & & gi, 4 sec ( ( ( / 33 5 ˆ 5ˆ s s e e gi, ( e e e e ˆ ˆ ˆ ( ˆ 3 3 & & && & && 4 sec, ( ( ( s s e e / ˆ 44ˆ gi, f equ(i [f ρ ]

23 Dwlded f -3 d d d d Q Q 4 sec, ( [ ] ( ( [ ] / 6.55 s Q ( s / ρ Hece, Rdius f cuue 7.4 (s

24 Dwlded f Chpe 4 KINETICS OF PRTICLES NEWTONS SECOND LW 4. New s Secd Lw f Mi: New hs gie his udesdig f i f picles d hei cuses d effecs i 3 lws. The fis d hid lw f i dels wih he bdies es ig wih uif elci i.e. wihu ccelei. F he bdies ude he i wih ccelei he lsis f i d fces pducig i is de b he pplici f New s Secd Lw. See f New s d Lw: If he esul fce cig picle is ze, he picle will he ccelei ppil he giude f he esul d i he dieci f his esul fce. F F F If F, F, F3, ec be he esul fces f diffee giude d dieci cig he picle. Ech ie he picle es i he dieci f he fce cig i d if,, 3, ec be he giude f he cceleis pduced b he esul fces. The, F, F F F 3 3 F, F cs ss f picle ( 3 S, whe picle f ss is ced up b fce F d ccelei, he us sisf he eli, F ( i - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -4 ec [whee dieci f F & e se] i.e. F iˆ F ˆj F kˆ ( iˆ ˆj kˆ z which is New s Secd Lw. Whe picle is subjeced siuleusl seel fces equi(i is dified s: F i. e. ( F iˆ F ˆj F kˆ ( iˆ ˆ j zkˆ z whee, F su f esul f ll fces cig. F, F, z z z F,,, e, d z cpe f he fces d ccelei cig he picle especiel. Nes: (i Whe he esul fce is ze, he ccelei f he picle is ze. (ii Whe d F, The picle wuld ei es.

25 (iiiwhe d F, The picle wuld e wih cs elci, lg he sigh lie. (i ll he be cses defies he fis lw, hece he New s s Lw f Mi is picul cse f New s d Lw f Mi. 4. Lie Meu d Re f Chge [Ipulse Meu Thee}: F New s d Lw, d, F d F d F ( ( i d Muliplig bh sides b d d iegig ude he liis, we ge: Fd d I Ippe Ph Fuci The e Fd is clled he ipulse ( I f he fce duig ie iel ( - whees is he lie eu ec f he picle. S, equi (ii ses h The ipulse ( I e he ie iel ( - equl he chge i lie eu f picle duig h iel. [Ipulse Meu Thee] The ipulse f fce is kw ee whe he fce iself be kw. gi, f equi(ii I ( iii i.e. Fil eu ( f he picle be bied b ddig ecicll is iiil eu d he ipulse f he fce F duig he ie iel csideed. O, shwig i ec f. ( ii I- I- Whe seel fces c picle, he ipulse pduced b ech f he fces shuld be csideed. i.e. I ( i Fd... whee, I ( F d ( F F F3... d F d Dwlded f -5

26 F F Dwlded f F F F (e 4.3 Sse f Ui: Uis f esuee shuld be csise d e f he sdds shuld be fllwed. Geell sdd uis e ke: Sse de Ieil Ui (SI ui b U.S. Cus Uis (used b eic Egiees SI Uis: SI sds f Sse de Ieil. SI uis e he wld-wide sdds f he esuig sse. SI uis e fudel deied. Fudel d Deied Uis: Fudel d Deied uis e he SI uis. Fudel uis e idepede f he esuig uis d e he bsic uis f ll he sse whees Deied uis e he uis which e epessed i es f pwes f e e fudel uis. Fudel Uis Deied Uis Legh ee ( elci L/T /s Mss kilg (kg ccelei /T L/T /s Tie secd (s Fce ML/T kg/s (N SI uis e he bslue sse f uis d esuls e idepede up he lci f esuee. US Cus Uis: This sse is bslue sse f ui. The e giil sse f uis. se Uis legh f(f fce pud (lb ie secd (s Cesi f US Cus Uis SI Uis: legh : f.348 fce : lb N ss : slug 4.59 kg : pud.4536 kg 4.4 Equis f Mi d Dic Equilibiu - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -6

27 Csideig picle ss ced up b seel fces. The f secd lw, F i F ( i j kˆ ( ˆ ˆ z Usig ecgul cpes, he equi f is e F, F, Fz z ( ii (i d (ii gies he equi f i f picle ude he fce F, F &, F &&, Fz & z Iegig hese equi s de i 3.3, he equi f i c be bied. gi, he equi(i be epessed s F i.e., if we dd ec he esul fce i ppsie dieci, he sse ces ude he equilibiu se. This fce ( ppsie he esul fce is clled Ieil Fce Iei ec. This equilibiu se f picle ude he gie fces d he iei ec is sid be dic equilibiu. he dic equilibiu, F, F & Fz Iei ec esue he esisce h picles ffe whe we se he i i whe we chge he cdii f hei i. gul Meu d Re f Chge (gul Meu Thee See The e f chge f gul eu f he picle bu pi is is equl he e f he fce ( F cig h picle bu he se pi. Le picle f ss ig i he XY-ple d he lie eu f he picle is equl he ec. Y The e bu O f he ec. (lie eu is clled gul eu f he picle h is d is deed b H Nw, d e cpes f. i & dieci. The, f defiii, ( H ( -( Tl eu bu is su f e bu he pi - igs Thee X pi bu O [ H ( ] ( i Diffeeiig equ(i wih espec ie, we H& d d ( Q & & & & & & ge: Dwlded f -7

28 ( & & & ( Q H & H& F F H& e f Fce bu O H& ii [ ] ( Dwlded f & Thus, he e f chge f gul eu f he picle bu pi is is equl he e f fce ( F cig h picle bu he se pi. i.5 Equi f Mi ( Recilie i f picles: If picle f ss is ig i sigh lie ude he ci f cpl fces F, F, F3, ec The he i f picle c be wie s F i, whee F F F ( F3 F Recilie i, i is l lg he sigle ci.e. & Equ -(i be wie s F - (ii F These e he equi f i f he picle ig i he sigh lie. F F F3 p die, (b Cuilie i f picles: i Recgul cpes ii Tgeil d Nl cpes iii Rdil d Tsese Cpes i. Recgul cpes F New s secd lw, F ; F F Pjecile i, eglecig i esisce F F F w g g, g (i g These e he equis f i. W F F - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -8

29 ii. Tgeil d Nl cpes: F New s d lw, F F d d ρ F F F These e he equis f i. F FT d d iii. Rdil d Tsese cpes: F New s d lw, F ( && & F ( && & & ` F F F F These e he equis f i. Ne: I cse f Dic Equilibiu ll he cpes f fces e blced b Ieil ec Iei fce. S, f dic equilibiu cdii, he equi f i beces F F F F F F - (i - (ii - (iii F i.6 Mi due Cel Fce-Csei f gul Meu Whe he fce F cig picle P is dieced wds w f he fied pi O, he picle is sid be ig ude cel fce. The fied pi O is clled he cee f fce. s shw i he figue, picle P es lg he cue ph. O igi f c-dies Nw, F Rdil cpe f fce F F Tsese cpe f fce F F cel i F QF && & &, d, d ( && & & ( & d( & Dwlded f -9

30 Dwlded f Iegig bh sides we ge & csh - (i Nw, if elee seci e swep i ie d be d Q d. d [ S / R; S f d ], d d d d, & d d d Re f chge f d Q.. & h h.. ( ii [ Diidig bh sides b d] Hee, sweepig e eelci (.. Thus, whe picle es ude he cel fce, he el elci is cs. This is ls clled Keple s Lw gi, gul eu eu f lie eu bu he fied pi. H Nw, & H & H & ( iii, H, H H h (.. cs [ Q & h] [ Qh..] [ si ce,.. cs & cs ] Hece, whe picle is ig ude cel fce, gul eu is lws cseed. i.7 New s Uiesl Lw f Gii: See: Ee picle i he uiese cs ee he picle wih fce, which is diecl ppil he pduc f hei sses d iesel ppil he sque f he disce bewee hei cees. Y Mheicll, M F d GM F d whee, d M e sses i kg ( i d is he disce i ls, G is he Uiesl Gii cs wih is lue N /kg d Z F is he fce f ci bewee he. d d F F P P d X - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -3

31 F bd f ss lced e he sufce f eh, fce eeed b he eh bd equls he weigh f he bd i.e. F g d d R (dius f he eh. GM F g R GM Q g ( ii R whee, g is he ccelei due gi wih is sdd lue 9.8 /s he se leel. Sice, eh is pefecl spheicl s he lue f R is diffee d hece g ies ccdig he ii f liude d liude. i.8 pplici i spce echics: Eh sellie d spce ehicles e subjeced l he giil pull f he eh fe cssig he sphee. The gii fce cs s cel fce he d hece hei is c be pediced s fllws: F cel fce i, & h ( i Tjec f picle ude cel fce: Csideig picle P ude cel fce F (i.e. dieced wds cee O The we he Rdil cpe f fce, (& & F ( i F d, Tsese cpe f fce F ( & & & ( ii F equ(ii sice & & & d, ( & d O iegig, cs h ( iii d h d gi, d d d h d d &.. h d d d d d ( i d& d& d h d& h d d &.. h d d d d d d h d & ( d Puig lues f & & & i equ( i we ge Dwlded f -3

32 Puig h d d Dwlded f h F 4, we ge h u d u h u F d d u F u ( i d h u This is secd de diffeeil equi, which is he jec fllwd b he picle which is ig ude cel fce F. Ne: i. F is dieced wds O ii. Mgiude f F is e if F is cull wds O (i.e. cie fce iii. F shuld be -e if F is dieced w f O. The jec f picle ude cel fce is d u d gi, F u h u GM F GMu whee, ( ii ( iii M ss f eh ss f he spce ehicle disce f he cee f eh he spce ehicle, u F equ (ii & (iii d u GMu u d h u GM h cs ( i This equ GM (i is secd de diffeeil equi wih cs c-efficie. h slui f he diffeeil equi is equl he su f he cplee i.e. The geel U U c U p whee, U c si cs GM U p h gi, U c si cs C (cs cs si si C cs(- GM U C cs( h U c cplee slui i.e. f gie cdii U p picul slui i.e. f sed se cdii - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -3

33 Nw, we, ge: GM c cs ( h gi, we he he equi f cic seci, l l ecs e cs ( i l l Cpig equ ( & (i, we ge: e c e cl l gi, chsig d U [ i. e.iil lie is is f se] GM l h h l GM ch e which is ecceici f he cic seci. GM S, hee cses ise: If e> (i.e. cic is hpebl e> e< e> e< sellie ch GM i.e. >, c > GM h b If e (i.e. cic is pbl ch GM i.e., c GM h c If e< (i.e. cic is ellipse eh jec f i ch GM i.e. <, c < GM h Specil Cses: Whe ec he legh f dius ec is cs d he cic seci educes cicle. he ls sge f luchig sellie i bi, i hs he elci pllel he sufce f he eh d he sellie begis is fee fligh he ee. Le,, be he dius ec d elci he begiig f fee fligh. Hee, elci educes sese cpes l. & gi, & ω h & & ( ii h ω ω The f equ GM ( c cs h Dwlded f -33

34 Dwlded f ee,, d c, GM ( GM GM ( iii [ cs ] GM F pblic jec, c h GM c F equ (iii & (i GM GM ( ( i fee fligh pweed fligh This elci is clled he escpe elci. Sice, his is he iiu elci equied f he ehicle s h i des eu is sig pi. esc esc GM gr gr ( i Ne: If > esc, jec will be hpeblic esc, jec will be pblic < esc, jec will be ellipicl GM gr GM g R g he ellipicl bi, if c he he ellipse educes cicle. i.e. puig c i equ (iii, we ge cic GM gr cic ( ii GM Peigee d pgee The clses pi f he bi f he eh is clled peigee d he fhes pi f bi f he eh is clled pgee. (ii F cic < < esc, peigee d pgee (iii F < cic, pgee d "peigee (i F << cic, he ehicle des g bi. - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -34

35 Tie Peid ( Peidic ie f Spce ehicle I is he ie equied f he sellie cplee is bi d is deed b τ. eisidebi ( i.e.ef ellipse τ eelci πb τ h b, τ π h ( iii Dwlded f b O ese ci<ese ci <ci <<ese whee, sei-j is f ellipse b sei-i is f ellipse h Tuils: sellie is luched i dieci pllel he sufce f he eh wih elci f 37 k/h f liude f 5 k. Deeie he liude ied b i whe i ces he gul disce equl 35. ls clcule he peidic ie f he sellie. Tke dius f eh, R637 k. Sl :- Hee, Luchig elci ( 37 k/h 77 /s 5k Rdius f eh (R 637 k liude f luchig (h 5 k 5 5 The, h GM gr 9.8 ( /s GM h ( 7.6 GM We kw, c cs ( i h pi d Q c gi, c cs k O

36 Dwlded f liude gied b sellie (H -R k gi, clcule ie peid: Whe he sellie ces 8, i will ke k The, k b. 49k 3 πb π QTie peid f he sellie, τ h 7.6 τ sec τ hs8 i6sec 4 6. The w blcks shw i he figue s f es. The hizl ple d he pulles e ficiless, d he pulle is ssued be f egligible ss. Deeie he ccelei f ech blck d he esi i ech cd. T Sl : Le, esi i he cd CD be T & cd C be T. F 7kg figue, if blck es hugh disce S he blck es hugh S /. Q S S Usig New s d lw f lck, lck d Pulle C lck : F T lck : W T F T T T 3 ( ii 3 ( iii / Pulles Sice ss f pulle is csideed ze, we he: T F T c c ( i Puig lues f T & T i equ (i, we ge: Q 8.4 / s 45 / s ( i - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -36 T W W T F..D f 4kg 3kg D T 3kg c

37 QT 84N T T 68 N 3. The bb f 3 pedulu descibes c f cicle, i eicl ple. If he esi is wice f he weigh f he bb f he psii whe i is displced hugh gle f 3 f is e psii, he fid he elci d ccelei f he bb. Sl : pplig New s Secd Lw, F, g si 3 g si / s gi, F g g cs3 Q g g cs3. / s [ ].5 / s β 36. elci f b ( ρ ρ Q / sec pep. hechd 4. The i f 5 g lck i hizl ple is defied b he eli (csπ d π, whee is epessed i ees, i secds d i dis. Deeie he dil d sese cpe f he fce eeed he blck whe &.75 sec. Sl : Hee, 5 g.5 kg (csπ - (i π - (ii Diffeeiig wih espec ie, we ge & -4π siπ & π & 8 π cs π & Nw, ( & F && F ( && & & Whe,, 4, & & & 8π & π & & Dwlded f -37

38 Q F F F QF.5.5 Dwlded f 4 4π ( 8π 8.43 N ( 4 π Siill f.75 sec, F 39.5N, F 79. N & 4π &&.5π & π && - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -38

39 Chpe 5 Kieics f Picle : Eeg d Meu Mehd 5. Wk de b Fce: Whe picle es b he pplici f fce F pducig he displcee ds, he he wk de b he fce duig he displcee ds is defied b: du cpe f fce lg he dieci f i disce elled. whee, Q du F csα. ds du Fds.csα ds d ( i F d Fsi F Fcs ds [ α is he gle bewee he fce d dieci f i] Picul cses: ( Whe F is lg he dieci f d, he du Fds cs α cs [ ] (b If F is pepedicul he dieci f du [ csα cs9 ] (c F fiie wk de f s s, Iegig (i, we ge: U ( F cs ds ( ii s α s d, he Fcs U- U e ude he cue Fcsα - s S S 5.. Wk f cs Fce i Recilie Mi U ( F csα Displcee f [ Recilie Mi] 5.. Wk f weigh ( Fce f gi The wk du f he weigh is equl he pduc f weigh (w d he eicl displcee f he cee f gi G f he bd. i.e. du -wd du U U w w wd ( U - is -e whe wk is de he bd U - is e whe wk is de b he bd. Y w Y F Y Dwlded f -39

40 Dwlded f Wk f fce eeed b spig Whe spig is defed, he giude f fce F eeed b i he bd is ppil he elgi f he spig. i.e. F k - (i whee, k spig cs elgi legh gi, Elee wk du -Fd -k d fiie wk de duig elgi f U k d ( U k wk is psiie, whe <, i.e. spig is euig Wk f he Giil Fces: We he, M F G ( i Nw, he elee wk GM du Fd d ce S, he wk f fiie displcee U GM d GM U wr Hee, > R ( ii [ si F is dieced ppsie i dieci f i] ( iii Q GM gr wr ( i 5. Kieic Eeg f Picle: F ss ced up b fce F d ig lg he cue ph, he cpe f fce lg he dieci f i is gie b: F d, F csα Q ds, F csαds d Iegig bh sides, kig liis we ge s s F csαds d d d d ds ds d d ds - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -4

41 , U, U, U T T ( i whee, T d T is fil d iiil K.E. f he picle. Hece, he wk f he fce F is equl he chge f K.E. f he picle. This is ls clled s piciple f wk d eeg. 5.3 pplicis f Piciple f wk d eeg: Wih he help f wk eeg piciple, slui f pbles, ilig fce, displcee d elci c be bied i siple f, e.g. lsis f Pedulu T deeie he elci f bb s i flls feel f, we e U wl gi, KE KE T [ Q ] T Nw, usig piciple f wk d eeg, T T U, T U gl wl gl [ L is he eicl heigh f bb f efeece pi] dges f his ehd: T fid i is ecess fid Equi is i he f f scl, hece i is es hdle. Fces which d wk (e.g. Tesi sigs, ec e eliied. 5.4 Pwe d Efficiec: Pwe is defied s e f chge f wk. u P g, Tkig lii s we ge du P ( i, Puig du ( F csα ds d ds P F csα d ds, P F csα d ( ii whee, is giude f he elci he pi f pplici f fce F. Dwlded f -4

42 Dwlded f Oupu wk Efficiec ( η Ipu wk Pwe upu, η < Pwe ipu [ due lsses due fici] 5.5 Peil Eeg: Cside bd f weigh W, which es lg cue ph f. Qwk de due weigh duig he displcee, U, W W The, wk psii, U W ( i The wk de b gi is idepede f ph d is ppil psii, wk de b gi psii is deed b g. U ii ( ( ( g g d, U ( iii whee W g g If ( g >( g he wk is e (i.e. PE iceses If ( g < ( g he wk is e (i.e. PE deceses ( Peil Eeg f Giil Fce: We kw h wk f Giil Fce, whe he bd is displed f, GM GM i.e. U GM The, wk de psii, U ( i U g ii gi, we kw h wk de, ( F (i d (ii g wr GM GM F g w R GM wr g ( i ( iii is he disce f he bd f he cee f he eh. F lge lue f, (b Peil Eeg due spig: The wk f fce eeed b he spig he bd f he elgi f U k k ( i, U [( ( ] e e - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -4 g

43 Whe, ( e d ( e is he PE due elsic fce, he peil eeg elgi f spig e k ( ii Duig he elgi he peil eeg f he spig iceses. The wk f he fce is idepede f he ph fllwed d is equl ius chge f peil eeg. i.e. U ( ( iii The fce which sisf he equi is clled s he csei fce. The gi fce d he elsic fce e eples f cseie fce. 5.6 Csei f Eeg: We kw, wk f fce is equl U ( [ be he PE] U ( i gi, wk f fce is equl chge i KE U T T ( ii T F (i d (ii T E T E T T ( iii [ be he KE] whee, E T echicl eeg f he sse. Hece, csei f eeg ses h echicl eeg f he sse lws eis cs. Eples f Csei f Eeg: lsis f Pedulu F fee fll f pedulu f l, The F psii, KE, T [ ] PE, wl T wl - (i psii, gl w g PE ( [ L, du] KE (T ( gl wl T wl F psii KE, (T ( ii w glsi wlsi g PE ( w( L L wl wl si Dwlded f -43

44 Dwlded f T wl - (iii F equi (i, (ii d (iii, he l echicl eeg f pedulu psii is se d is equl wl., Tl eeg is eiel due PE, Tl eeg is eiel due KE, Tl eeg is su f PE KE Ne: F he sse iecig wih he fs f eeg s elecicl, ficil, ec ll he fs f eeg shuld be csideed. I h cse s well he l eeg f sse is lws cseed. Hece, eeg is cseed i ll he cses. 5.7 Piciple f Ipulse d Meu: Csideig picle f ss ced up b fce F. The f New s d Lw, F I & cpes, F d F & d Sice ss is cs & F F d d d d F d d ecicll, we he F d d ( & F ( ( i ( ( ii This equi ses, Fce F cig he picle is equl he e f chge f eu ( f he picle. Muliplig equi (i b d d iegig bh sides, we ge: Fd d ( ( ( F d ( ( iii Siill, ( F d ( ( i I ec f, ( Fd F F F whee, - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -44

45 F d Ipulse f fce I Gie, b e ude he cue F- Hece, I p ( i ( F ec dig, p ( Whe seel fces e cig picle, I p ( ii ( 5.8 Ipulsie i d Ipc ( Ipulsie Mi: Whe e lge fce is ced duig e sh ie iel picle d pduce defiie chge i eu, such fce is clled s ipulsie fce d he esulig i is clled ipulsie i. Eple f Ipulsie i: Sikig he bll wih cicke b, lge fce ( ipulse F is pplied i sll ie ( F is lge eugh chge he dieci f i f bll., he esulig F F ipulse eu piciple, ( i F Hee -ipulsie fces (like weigh f bll, b, ec e icluded.. Ipc cllisi bewee w bdies, which ccus i e sh iel f ie d duig which he w bdies ee eliel lge fces ech he is clled Ipc. The c l he sufces i cc duig he ipc is clled he lie f ipc lie f ci. Tpes f Ipc: If he ss cees f he w cllidig bdies e lced his lie f ipc, he ipc is cel ipc hewise ecceic ipc. If he elciies f he w picles e dieced lg he lie f ipc, i is sid be diec ipc. If eihe bh picle es lg he lie he h he lie f ipc, he ipc is Lie f cc sid be blique ipc. Hece, fu pes f ipc ccu. The e: Diec Cel Ipc b Oblique Cel Ipc c Diec Ecceic Ipc C Dwlded f C ( dieci cel ipc Ipc/ci Lie f C C (b blique cel ipc Lie f ipc

46 Dwlded f d Oblique Ecceic Ipc C C Lie f ipc C C (c diec eceic ipc (d blique eceic ipc 5.9 Diec Cel Ipc: Tw picles d f ss d e ig i sigh lie wih elciies &. If > he picle sikes. Ude he ipc, he def d he ed f peid f defi he will e wih he se elci u. fe he ipc he picles gi hei igil shpe e peel defed, depedig up he giude f ipc d eil iled which is clled esiui. fe he ipc d sepi he picles e wih d elciies. The dui f ie f ipc whe he picles ces ude he defi d esiui duig ipc is clled defi peid d esiui peid especiel. U Csideig h l ipulsie fces e cig, he l eu f he sse is cseed. i.e. ( i I scl f, ( ii e lue is f e is d e lue is f e is. lsis duig Ipc Fllwig phee will ccu f he picle. u pd Rd u ( iii ( i pd Rd whee, p d d Rd e he ipulses duig he peid f defi d esiui especiel. The he c-efficie f esiui is defied s: e Rd Pd ( - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -46

47 Dwlded f lue f e depeds up Meils f picles Ipc elci Shpe & size f cllidig bdies Geell, <e< F pefecl elsic cllisi, e F pefecl plsic cllisi, e F equ(iii, u Pd F equ(i, u Rd u u e u u u u Pd Rd e Q Siill f picle, ( ii u u e ddig especiel he ues d deis f equ (i d (ii, we ge: u u u u e ( ( iii e i.e. Relie elci fe ipc e Relie elci befe ipc F Pefecl Plsic Ipc: e, i.e. hee is peid f esiui. f equ(iii, ( c elci The f equ(ii ( ( ( i ci C el F Pefecl Elsic Ipc: Sice e, f equ (iii, ( f equ (ii, ( ( ( i f equ ( ( ii Muliplig LHS d RHS f equ(i & (ii especiel d diidig bh sides b we ge: ( ( [ ] ( ( [ ],

48 Dwlded f ( ( i.e. Iiil KE f sse Fil KE f sse Hece, f pefecl elsic cdii, KE f he sse is cseed. Whe e, hee is lss f KE d his ls eeg is ceed i he, sud d he fs f eeg. 5. Oblique Cel Ipc: Whe he elciies f he w cllidig bdies e dieced lg he lie f ipc, he i is clled blique ipc s shw i he figue. Lie f cc X Hee, lie f ipc is lg -is d lie f cc is lg -is. The he fllwig phee ccu. ( -cpe f he eu f he picle s is cseed i.e. ( i (b -cpe f he eu f d picle is cseed ii i.e. ( F ( d (b [ ] [ ] (c F ( d (b, he l eu f he picles i -dieci is ls cseed i.e. ( iii (d -cpe f l eu f he picle is cseed. i i.e. ( (e -cpe f elie elci fe ipc is bied b uliplig -cpe f elie elci befe ipc b c-efficie f esiui. ( ( e i.e. ( e( ( The be fie equi e pplied f he lsis f he pbles eled blique ipc. Reebe: ( lg he lie f cc, eu f ech picle is cseed. (b lg he lie f ipc, he l eu f picles is cseed. Tuils: - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -48

49 ( kg cll slides wihu fici lg eicl d. The spig ched he cll hs udefed legh f d cs f 5 N/. If he cll is elesed f es i psii, deeie is elci fe i hs ed 5 psii. Slui: Gie, K 5 N/ Du Udefed legh f spig. We he f csei f eeg, KE PE 5 KE PE i.e. T T ( T gi, i e [ ] 5.5N Y g e g k 5 9.9N w k (. 5 (.5 9.8(.5 Puig ll he lues i equ(, we ge / s g elgi f spig... du pi, he l legh f spig is (. ( picle hig ss.5 kg is elesed f es d sikes. The si picle f ss.4 kg s shw i he figue. ssue he ipc is diec d elsic. If he hizl sufce hs dic c-efficie f fici µ. 3, lce he fil psii f ech ss f he igi f he is. Slui: pplig csei f eeg P. & Ls f eeg wk de gis fici 3 Nw f ss, KE (T [ ] M.5kg PE ( gh (.5-.5si3 M (.63 J.4kg KE (T.5. 5 PE ( [ is du] Nw, f csei f eeg, Dwlded f -49

50 T T.63.5 Dwlded f elci M p. Nw, he pi f ipc (.56 / s.56 / s elci f befe ipc (.56 /s elci f fe ipc ( elci f befe ipc ( elci f fe ipc ( Nw, f diec ipc, , ( i gi, we he: e ( ii Slig equ(i d (ii, we ge:.73 / s.733 / s Nw, usig wk eeg eli fid he disce elled b he picle F M : Wk de due fici T ( iii (Eeg ls due fici T Fil KE - Iiil KE.5 [ h ] ( J wk de due fici f ss F M : T.4 ( J T. 748J T.665J Nw, Wk de due fici µ g disce elled wk de due fici disce elled ( - µg F ss, - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -5

51 ( ( The giude d dieci f he elciies f w ideicl ficiless blls befe he sike ech he s shw. ssuig e.9 deeie he giude d dieci f he elci f ech bll fe he ipc. Slui: cs3 7.8 / s si / s cs6 6. / s si 6.6 / s Nw, i blique ipc F i lg he lie f cc, 4.5 / s d,.6 / s F i lg he lie f ipc, gi, e Q.7 ( i.9.5 Slig ( d (, we ge: ( 7.8 ( 6. (.5 9/s [ ] 7. & 5.4 Resul Mi: ddig cpes f elciies fe ipc, we ge: ( ( 7 / s lie f cc 3 6 9/s ( (.8 / s Dwlded f -5

52 Dwlded f Chpe 6 Sse f Picles 6. Sse f Picles: F New s lw, equi f i f ech picle is F F, F & Fz z i d f Dic Equilibiu, F, F & Fz ( ii u, f he sse ilig lge. f picles, d if he sse s whle is csideed, he ech picle f sse is subjeced w pes f fces: ( Eel Fces: Fces eeed b he bd uside he sse d weigh f picle. (b Iel Fces: [dewl s Mlecul ci] Fces eeed b he he picle f he se sse. Fi Pi fik fij Pk fjk Pj Pi Pj ii Fi ( z z If P be picle i sse f picles P j, P k, ec d F i be he esul f eel fces i h picle d F ij, F ik be he iel fces i h picle f j h d k h picle. The, The su l f fces i h picle su f eel d iel fces cig he picle. Nw, f New s d lw, he esul f fces cig he i h picle is equl i i. Whe ll he picles e csideed siuleusl he iel fces ccel u d he l eel fces cs he sse. Hece, f he whle sse Q ( F ( F e e ( Fz e - (iii 6. New s Lw d Sse f Picles: pplig New s d lw f he i h picle, we he: i i d d d d i Eel fce Iel fce i Fi Fij j i j ( i - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -5 Fi fjk fik fij

53 F ll picles he equi (i c be wie s: d i i Fi Fij i d i i i j j ( ii F csideig ll picles he sui f iel fces is ze. Hece, equ(ii difies s : d i F i Q F F i i d i d, F i i d i ( iii If, c be psii ec f ss cee f sse f picles d M is he l ss f picles, he f piciple f fis e f iei (e due eie ss su f es due idiidul ss, M c M i i F (iii d (i, we ge d F d ( M c d F M d c ( i ( 6. Lie d gul Meu f sse f picles ( Lie Meu f sse f picles: F sse f picles, pplig New s d lw j h picle, we he: d j F M j ( i j d Muliplig (i b d d iegig f, we ge: Fd I e M j j M j j j j This shws, The ipulse f he l eel fce he sse f picles duig ie iel equls he su f he chges f he lie eu ec f he picles duig he se ie iel. F he ccep f ss cee, M c j M j j ( iii Diffeeiig wih espec ie, we ge M c j M F (ii d (i, I e j j ( i Fd M ( c M ( c ( Thus, he l eel ipulse f sse picles is equl he chge i lie eu f he picles, ig wih he ss cee elci. ( ii Dwlded f -53

54 Dwlded f gul Meu F sse f picles: gul Meu f i h picle, d Η ii i i d d Η & ii i d ( P i ( i Pi i i F sse f picles he gul eu equi f he i h picle bu igi O is gie b: [ i Pi ] ( i d i Fi i Fij j d whee, P i Lie eu f picle F he sse f picles, equ (i beces: i j d ( i Fij i Fi µ& i Fi i j d i Sice, iel fce ishes f ll picles, he e ls bece ze f ll picles. Hece, [ M µ ] ( ii & µ i Fi M & i Siill f he fied pi, M µ& ( iii Thus, he l e f eel fces cig ggege f picles bu fied pi i ieil efeece equls he e f chge f he gul eu elie he se pi d se ieil efeece. gi, csideig cee f ss f he ggege f picles F i h picle, [ ρ psii f i w... CG ] ( i ci i c ρ ci Nw, he gul eu f ggege picle bu O is he, Η i pi, Η i [ Q pi lie eu] pi ii i& i [( c ρci { i ( & i & ρ ci }] ( & & & & ρ i i c ci Cig u he css-pduc d ecig c f he sui we ge: Η i c [ c i c c & i ρ ci ρci i & c ρci & i ρci ] i ( i & c c ( & i ρci ( i ρci & c ( ρci & i ρci i i We kw h he su f he fis e f ss bu he ceid is ze i.e. i ρ ci d hece i ρ& ci, The i z fij c M - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -54

55 µ c M & c ( ρci & i ρci [ i M ] i [ µ c M & c µ c ] ( [ Qµ c ρ cu & i ρ ci ] whee, µ c be he su f he gul eu bu he cee f ss.siill f pi, we ge: [ µ C M & C c ] ( i µ whee, & C is he elci f cee f ss elie fied pi C Nw, diffeeiig equ (i wih espec ie, we ge: & µ M& & µ C C c [ M C M C & µ c ] ( ii [ & µ M ; C ], C 6.3 Mi f ss cee f sse f picle: Cee f ss f picles is he cee f ss f sse. We kw h (f 6. f sse f picles ( F ( e ( F ( e ( Fz ( z e (i If ss cee f he sse f picles is csideed wih c-dies G(,, z, The we he: ( ( ( ( ( z ( z (ii Diffeeiig equ(ii wice wih espec ie, we ge: && && whee, ( ( ( && ( && ( && z ( z && z ( ( ( ( ( z ( z - (iii, & e he cpes f ccelei f G (i.e. cee f ss f he sse, F equ(i & equ(iii, we he: ( F ( e ( F ( e ( Fz ( z e - (i i.e. cpe f e due eie ss cpe f su f he e due idiidul ss which defies he i f cee f ss f sse. I shws h he cee f sse f picles e s if he eie ss f he sse d ll he eel fces wee cceed h pi G. 6.4 Csei f Meu: We kw h he fil eu f he picle is bied b ddig ecicll is iiil eu d he ipulse f he fce F duig he ie iel csideed i.e. Dwlded f -55

56 I p Csideig sse f picles, I p ii ( ( i Dwlded f ( ( sice he iel fces ish csideig he ll picles l ipulse due eel fce eiss i.e. ( I p( ( iii e Whe he ipulse f eel fces is ze, he equ(iii beces, ( i Thus, whe he su f he ipulses f he eel fces cig sse f picles is ze, he l eu f he sse eis cs. G, f he sse, we he, gi, csideig ss cee ( ( ( ( ( - ( Diffeeiig equ( wih espec ie, we ge: ( ( ( ( We he [f 6.3] Nw f equ(i we ge: & ( ii ( ( (i Thus, Whe he su f ipulses f he eel fces cig sse f picles is ze, he ss cee f he sse es wih cs elci. 6.4 Kieic Eeg f sse f picles: Csideig sse f picles, he l K.E. elie z es f sse f picles c be bied s KE i i ( i i ci C.ci gi, i i c ρ ci ( ii Diffeeiig equ(ii wih espec ie, we ge, i c ρ ci F (i & (iii u, KE ( iii ( c & ρ ci i ( c & ρ ci (. & c ρci i i i KE ic i c. & ρ ci i i i & ρ i ci - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -56 z i c [ Q( b(. b. b b ] X

57 M d i i i ρ& ci S, we ge, KE M c i ρ& ci i Hece, KE f he sse f picles KE f he l ss ig wih he ss cee elci su f he KE f idiidul picles hig elci elie he cee f ss. 6.5 Wk-Eeg Piciple: Csei f eeg f sse f picles: Wk f he fce F eeed he picle duig he displcee chge i KE i.e. u ( i F he eie sse, U T T ( ii Siill wk f he cseie fce is idepede f ph fllwed d is equl ius f he chge i peil eeg. i.e. u ( ( iii F he eie sse, U ( ( i F (i & (i, we ge T T T T ( Thus f he picle ig ude he cseie fce, he su f KE d PE eis cs. KE PE Tl Eeg (E cs E E Hece, Eeg is cseed. 6.7 Piciple f Ipulse d Meu f sse f picles: The gul eu f picle bu igi O e f he fis eu f picle bu he igi. Q h ( The f he l sse f picle H ( Diffeeiig wih espec ie, we ge, whee, ( M P ed i 6. equ ( iii [ ] H& e M e is he e f eel fce cig he sse. This eel fce c be cegized i w pes: ( Fces eeig igi These pide lie ipulse C CC Dwlded f Y Y Y (Fed (Me.d

58 i.e. ( Fe d ( ( (b Fces which e w f he igi These pide gul ipulse. i.e. ( F e d ( M Dwlded f e d [Ne: - Eel Ipulse chges he lie eu f sse Eel Me chges he gul eu f sse. Nw, f lie ipulse eu piciple, we he ( ( I ( ( i e gi, f gul eu piciple ( H ( M d ( H ( ii e Cpig hese w equ i shws h su f e (lie eu gul eu f picles ie d he ipulse f he eel fces e equiplle (N cull equile f picles bu equile f igid bd. Whe hee is eel fces, (Sse e (Sse e Tpes f sse f picles: ( Cs Sse f Picles: The sse which eihe gi lse picles duig hei i e clled he cs sse f picles. e.g. Rigid bdies (b ible Sse f Picles: The sse which e ciuusl giig lsig picles dig bh he se ie e clled ible sse f picles. e.g. Hdulic ubies, Rckes, ec. 6.8 Sed Se f Picles wih ible ss I sed se f picles he flw f picles pi eis cs. [ll he be equ deied s f e f sed ses]. ` Le us cside sed se f picle(s such s se f we dieed b fied e flw f i hugh blwe. ( S ii F S ii ( - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -58

59 I his sse i ciuusl gis picles flwig i d lses equl. f picles flwig u, s his pe f ible sse f picles is educed uili cs sse f picles, which eis cs f sh ie If u f ss ees he sse s, ie. The, The eu f he picles eeig he sse ( The eu f he picles leig he sse ( d, he ipulses f he fces eeed s F. d d i i (eu f ech picles ccels f bh sides. Hece, we cclude h, The sse fed b he eu ( f he picles eeig he sse s i he ie d ipulses f he fces eeed s duig h ie is equiplle eu f ( f he picles leig s i he se ie. i.e. ( F ( ( i Diidig bh sides b F li d F d d kig lii, we ge ( R ( ( ii This equi gies he esul f he fces eeed b he e he se. This piciple is pplicble he fllwig ss s: ( Fluid dieed b e/ hd-ubies/ ppeies (b Fluid flwig hugh pipe (c Je egie (d F 6.9 Sse wih ible ss: sse which gis ss b ciuusl bsbig picles lses ss b ciuusl epellig picles is he sse wih ible ss. Cside ie, he sse s bsbs he picles f ss duig he ie iel. S Hee, elci ie ( S elci ie ( bslue elci f he bsbig ss F he piciple f ipulse d eu, ( F ( ( ( ( ( i, F Relie elci wih espec s f he picle Dwlded f -59

60 Dwlded f ( u, si ce < F ( u [ eglecig( u due beig e e sll],, F li li u d d F u d d d d, F u d d d F u d ( ii I he be equ d u is he hus which eds slw dw he i f s. This is i cse f d giig ss. If sse s lses ss (s ppulsi f cke he hus geeed wuld icese he i f s. Tuils: ( Tw picles shw i fig scille he sh ple i he -dieci. ( Wie he diffeeil equi f i f ech ss. (b Fid equi f i f he cee f ss (c Wie he epessi f KE & PE f he sse f picles. g K (X-X g X X KX F F F K K F R R ( Diffeeil equi f i f ss : F && ( & k k Diffeeil equi f i f ss : F F && k ( & (b Equi f i f he cee f ss Hee he esul iel fces due K ccles u s effec is due eel fces l. If c is he c.. f he eie sse he F ( & c [& c ccelei f c.. ] ( &, F k c Tkig M Tl ss - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -6

61 F k M & c (c K.E./P.E. f he sse f picles: KE (T & & PE( g e O e g, ( ( k k ( k k [ beig du] ( zzle dischges se f we f css-secil e wih speed f 5/s, d he se is defleced b fied e s shw i he figue. The desi f we is 3 kg/ 3. Deeie he esul fce F eeed he se b he fied e. Slui: We e f he piciple f Ipulse-Meu f picles Hee, 6 5 /s e, ss flw-e Nw, Ccellig Slig f -is, i i d d ( F ( F F d d 5 ( cs6 gi, slig f -is F si 6 F ( F 5N ρ 5kg / s f bh sides, we he d si d F F iˆ F 6 cs6 ˆj 5ˆ i 6.5 ˆj s ii ( F F F 5/s s ii (3 sse csiss f hee picles, d C s shw i he figue d he elci, ( 5iˆ 8kˆ / s, ( 4ˆ i ˆj kˆ / s d ( 6ˆ i 3 ˆj kˆ / s especiel. If he sses z f hese picles e 3 kg, kg d c kg especiel d he esul gul eu µ f he sse bu igi O is pllel he z-is, deeie he lue f µ. c Dwlded f -6

62 Dwlded f Slui: We he, Resul gul eu Y 4 C C 3 5 X Z µ i i i c c c ( 3 ˆj 5kˆ 3( 5ˆ i 8kˆ ( 4ˆ i 5kˆ ( 4ˆ i ˆj kˆ ( 4ˆ i 3kˆ ( 6ˆ i 3 ˆj kˆ ( 5 84 iˆ ( 4 ˆj ( 4 57kˆ Sice, µ is pllel he z-is, µ µ & ( 5 84 ( ii ( 4 ( iii z Slig hese w equis, we ge: z 6.8 & 7.75 The, µ ( 4 57 k 4.k ˆ ˆ z z µ 4. kg / s i dieci e z is - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -6

63 Chpe 7 Kieics f Rigid dies 7. Iduci: Picle: I is eil bd which is s sll h is diesi c be eed s egligible i cpis he diesis iled. Rigid bdies: I is cbii f w e picles, which e ceced i such w h he d chge hei elie psiis due pplici f eel fces. The ius pis picles i igid bdies he diffee is bu hei i e s eled such h hei elie psii eis uchged. I eli ll he igid bdies def i pplici f eel fces bu i egligible u. 7. Tsli: i is sid be sli if sigh lie dw he bd bes he se dieci. If ll he picles e pllel lg sigh lie, i is sid be ecilie sli d if he ph e cued, he i is sid be cuilie sli. If Siill, d be he displcee ecs f he picle & duig, he f sli ( i ( ii Thus, f bd i sli, ll he pis he he se elci d ccelei gie is. F cuilie sli, hee is chge i dieci d giude f elci d ccelei ee is. F ecilie sli elci d ccelei fllw se dieci duig eie i. 7. Ri: i is sid be i whe he picles i he igid bdies (slb es i cceic cicles, wih c fied cee O. If he picle i he slb es d ie d, he he gul elci is gie b: Dwlded f -63

64 Dwlded f d d, he gul ccelei is gie b: F (i dω d α d d d d ω dω dω α ω d d ω dω α ω d ( iii ( ii d ω d i-clckwise dieci is ke e d clckwise dieci is ke e. ( F Uif Ri: α, ω is cs, d ω... ( F Uifl ccelei Ri: α cs ω ω α α ω ω ω α ( Lie d gul elci, Lie d gul ccelei ds d ; ω ; ω d d d dω Tgeil cpe f ccelei, α d d d Nl cpe f ccelei, F he equi f lie i ( ω ω 7.3 Geel Ple Mi: ple i which is su f slil d il i is clled ple i. p f i is sli d he p is il. d ( i - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -64

65 7.4 bslue d Relie elci i ple i bslue i es i wih espec fied es d elie i es i f e wih espec he es i i. F ple i f igid bd, whe eplced b su f sli d i bu, he bslue elci f picle is gie b: ( i whee, bslue elci f is sli f slb wih / elie elci wih espec i.e. i f slb esued wih espec Ple i sli wih Ri f bu / / / The ple i f Rd c be eplied s: ( chsig s efeece \ Tsli wih & i bu i.e. F figue d & / cs ω l [ l ω] ω l cs Hece, gul elci f Rigid bd i ple i is idepede f he efeece pi. ( ii [ se esul c be bied b chsig s efeece pi] 7.5 Iseus Cee f Ri F geel ple i, hee eis cee wih efeece which he elciies f ll he picles i he igid bd e se is f ie. This cee is clled s iseus cee f i d he is is clled iseus is f i. Dwlded f -65

66 Dwlded f W U W ple i Ri sli U If will be he iseus cee f i If ω N iseus cee f i u, If & ω, The If he slb is ed wih gul elci ω bu pi c lced he pepedicul disce ω f he ll he picles will ppe e bu c wih se elci ω. W ple i i ses f i Mehds lce iseus cee f i ( If dieci f elci f picles & e kw bu e pllel, he c is he ieseci f pepedicul dw & C (b If giude d dieci f elci f picles & e kw d e pepedicul lie, he c is lced s C W C W (c If giude, he c wuld be ifiie d he bd wuld sle. Ne: Iseus cee f i be lced eihe he slb uside. If c is lced he slb he c d C. The pi c is diffee diffee ie iel ccelei f ius picle he slb c be deeied b his ehd. Hece, i is used cpue l he elci f picles is f ie. 7.7 bslue d Relie Fe: Cilis ccelei i ple i - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -66

67 Defiii: picle wih i sse f igid bd he i wih espec ig fe iside he bd. I such cse ddiil cp. f ccelei f he picle wih espec he ig fe ces i he eisece, his cplee ccelei i cse f ig fe is clled cilis ccelei, c which is pepedicul he dieci f elie elci f picle wih espec fe. Y u Y w Y u O bslue fe Si fe Reli fe ig fe is es Wih g. elci w Y s Y c el.elci W p p bs. elci u Clculi: Cside he i f picle P, which es lg ph slb s. The slb es lg fied pi O. Mi f P is gie b & wih espec fied is (XOY is d b &, wih espec is ched slb s d ig wih i. I is equied deeie he bslue i f P d elie i f P wih espec S. We kw f dil d sese cpe f elci & & & & & i ( ( ( ( [ ] p p Hee, p is psii wih espec - is p is psii wih espec is. Cse I: Whe P is fied slb s d he slb llwed e wih espec XOY, he P cicides wih P cs cs p & & & S f equi (i p, ( & ( & ( ii p p X X X Cse-II: Whe slb is fied d P is llwed e, he cs & p p s The f equi (i ( & & ( & ( iii p s p s Dwlded f -67

68 Nw, he l elci is gie b: f he ec cpes, we he: Dwlded f ( i p p p s The Rdil d Tsese cpes f elci ( p ( p ( p s ( p ( p ( p s / & / [ls f ccelei] gi f ccelei, we he: ( p & & && ( & && ( & ( & & & ( & & & ( && && & ( & & ( i p Clculig he ccelei f pi p whe he slb is es s P ls es wih espec slb i is bseed h, ised f beig p p p p p / s p / s, we ge c, whee c hs he ec cpe s & & ω ii ( c ( p s ( d, ( ω( ( iii c p / s ( 9 he i f slb c ( c lg he i f slb The cilis ccelei c is hus ec pepedicul he elie elci p / s d f giude z. p / s ω d dieci f c is pepedicul he ec p / s. 7. Geel Mi: Csideig w picles & f he igid bd, he, we he ( i / If he bd es wih gul elci, he ω / / ( ii [ ] ω Fuhe, we he ω bu he pi / α / ω ( / ( iii d d [ Q / ] ( / ( ω /, ω / d d ω d α / / / d ω ω ω d d / / ( ω ω / - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -68 Y /

69 whee α is he gul ccelei f he bd is csideed. The equ (ii d (iii shw h he s geel i f igid bd is equile he su f sli (i which ll picles f he bd he he se elci d ccelei d f i i which picle is ssued be fied. i.e. The i f picle wih espec wuld be chceized b he se ecs ω & α. Thusω & α e idepede f he chice f efeece pi bu he ig fe shuld ei pllel he fied fe f efeece. Tuils:. The ed f he d s shw i he figue es wih cs elci,.9 /s ( Deeie: Y ( gul elci d gul ccelei f he d (b elci d ccelei f ed Slui: 5 Hee, Y 3 & X ( gi, X&.9 / s & ( Nw f O, X 5cs Q X& 5 ω &.3d [ kig e f clckwise dieci] ( si &.9 &.3 / sec gi, & & 5 ( cs & ( si 3 [ && 4 3 ] 5 (.3 &. 5 5 &. α &.d / sec (b We he Y 5si Y& 5 cs & & Y&& 5 cs && si & ( ( ( 3 4, 5 (.3 ( , Y 5.3. / s 5. / s &.75 / s Dwlded f [ ] ( I he egie sse shw, he ck hs cs clckwise gul elci f p. F he ck psii idiced, deeie: ( he gul elci f he cecig d D (b he elci f psii P. Slui: Hee, he ck es bu pi, 4./s X

70 Dwlded f πn π ω p d / sec 9.4d /sec 6 6 ( ω / s Usig sie lw, si 9 si si 4 si 3.94 D 3 8 Nw, f he i f ceci d D, which is ple 4 i: D P he elci D is hizl d is s bied be. Reslig i f D i sli wih d i bu, we ge: 5 D 5 D D WD Ple i Tsli Ri D/ 5 D/ Nw, we he D D / gi, usig sie lw, D D / 5.68 si 53.9 si 5 si 76. si 76. Slig he equi sepel, we he, D /.37 / s D p 3.5 / s gi, D / ω D ω D D ω 6.9d / s D D / D.37. (3 duble ge lls he si lwe ck. The elci f is cee is. /s dieced he igh. Deeie: R ( he gul elci f he ge (b he elciies f he uppe ck R d f pi D f ge. Slui: D /s ( gul elci f he ge: Disce ed b he cee f ge f ech C elui π R - E. ij Sigh Thp (Lecue, Ese Cllege f Egieeig, ig/ -7 ( i 6.9d / s

71 The, π π Diffeeiig, &,..5ω ω ω 8d / sec ( i [ π π ] lzig he ple i: D/ D / D/ D C C/ C Nw, elci f uppe ck R /. ω.. 8 / s / s elci f pi D,. ω..5 8 D.69 / s. φ. φ 45 D D / Dwlded f -7