Centroid is the point where the resultant of distributed force system is assumed to act and generate the same dynamic results.
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1 DYNMI ORE NLYSIS If he cceler f g lks echs re rug wh csderble u f ler d/r gulr ccelers, er frces re geered d hese er frces ls us be erce b he drg r s dd he frces eered b he eerl ld r wrk he echs des. S, 0 d τ 0 re lger pplcble. Gerg rules wll be: τ Iα I d re erl (bdl) prperes. hs sge we eed kw he descrp f he erl prperes. erd d Mss ere erd s he p where he resul f dsrbued frce sse s ssued c d geere he se dc resuls. Resul rce erd If he dsrbued frce s gr frce cg ech prcle f ss, he ccered frce self s clled he wegh d he cerd s clled he ceer f gr r ss ceer. Mss es dsce, r, s clled s he frs ss e. Ths ccep f frs ss e s rll used derg he ceer f ss f sse f prcles r rgd bd. I fgure seres f sses re lced le. The ceer f ss r cerd s lced
2 - G - ȳ G The crdes f he sses lced ple c be bed s: d Ths prcedure c be eeded sses ccered lue b spl wrg equ be fr he z s. re geerl fr f ss ceer lc fr hree desl bd c be bed b usg egr sed f su. The rels he bece d, d d zd z Mss Me f Ier Mss e f er s he e ge rl er, he rl lg f ss fr ler. I ppers he relshps fr he dcs f rl. The Mss Me f Ier f sld esures he sld's bl ress chges rl speed bu specfc s. The e f er fr p ss s jus he ss es he squre f perpedculr dsce he r s. The ss e f er fr sgle prcle s ge s:
3 I 00 r where he ss f he prcle, d r he shres dsce fr he s f r he prcle r Whe clculg he ss e f er fr rgd bd, e hks f he bd s su f prcles, ech hg ss f d. Iegr s used su he e f er f ech d ge he ss e f er f bd. The equ fr he ss e f er f he rgd bd s r d I r d r d The egr er ss c be replced b egr er lue, re, r legh. r full hree desl bd usg he des ρ e c rele he elee f ss he elee f lue. llwg re he hecl equs clcule he Mss Me f Ier:
4 r d r rz z I r d ( z )d I r d ( z )d I zz rz d ( )d These hree egrls re clled he prcple ss e f er f he bd. her hree slr egrls re I I d I I z z I zd z I zd z d hese hree egrls re clled he ss prducs f er f he bd. Rdus f gr: See plce f he ss e f er he rdus f gr k s prded. The ss e f er c be clculed fr k usg he rel I k where s he l ss f he bd. Oe c erpre he rdus f gr s he dsce fr he s h e culd pu sgle prcle f ss equl he ss f he rgd bd d he hs prcle he he se ss e f er s he rgl bd.
5 IG, G k k Orgl d Mdel Mdel Prllel-s here: The e f er rud s c be clculed fr he e f er rud prllel s whch psses hrugh he ceer f ss. The equ clcule hs s clled he prllel s here d s ge s I I G d IG, G d Eple ) Wh re he ss es f erl f p ss bu s pssg hrugh self d bu s r dsce w fr? Mss es f erl f p ss bu s pssg hrugh self I r d 0d 0. NS. r Mss es f erl f p ss bu s r dsce w fr. Usg prllel s here I I r 0 r r NS.
6 b) d he ss e f er f sleder rd f legh L (sleder rd es h hs legh, d he reg dess re eglgble sll) bu s perpedculr he rd d pssg hrugh s ss ceer. L L/ I L / 0 d r ser L L/ d Le des f he erl s ρ kg/. The, fesl ss hs be equ, d ρd. Subsug L / L / ρ L L I ρd ρ 0 NS Eple ufr seel br shw he fgure s used s scllg c fllwer. Dre he equ f ss e f er f he fllwer bu s hrugh O. Use he des f seel 7800kg / ρ. H: I ( )d c 5 c 50 c 5 c O 5 c kkn/
7 z r d dz d w l d r I zz where d ρdddz Subsug hese eq. dddz I zz ρ ) ( dddz dddz I zz ρ ρ / 0 / 0 w l zz zz d l d w I d l d w I ρ ρ ρ ρ w l l w I zz ρ ρ w w l l l w I zz ρ ρ sce ρ w l he w l w l I zz ss e f er bu O c be fud b prllel s here, r
8 I 0 l w l 7l 4w I zz d, kg d I kg. 48 D leber s Prcple: D leber s prcple pers he reduc f prble dcs e scs. Ths s ccplshed b rducg fcus frce equl gude he prduc f he ss f he bd d s cceler, d dreced ppse he cceler. The resul s cd f kec equlbru. If we subrc ecr f gude fr bh sdes f he New s secd lw. The eg f he equ;.e. dc f dc cse sll hlds rue, bu equ, hg zer rgh hd sde beces er es sle, lke h sc frce lss prble. 0 d slrl fr e τ α α α 0 Slu f dc prble usg D leber s prcple ) D cceler lss d clcule he ler cceler f he ss ceers f ech g lk. ls clcule he gulr cceler f ech g lk. ) Msses d cerdl ers f ech g lk us be kw befrehd. ) dd e fcus frce ech g bd equl he ss f h bd es he cceler f s ss ceer, drec ppse s cceler, ppled drecl he ceer f gr, pr fr he lred esg rel frces. ) dd fcus rque ech g bd equl he cerdl er f h bd es s gulr cceler, drec r sese ppse h f cceler pr fr he lred esg rel rques. ) Sle scll.
9 Eple I he fgure, duble- slder echs wrkg hrzl ple s shw. The slder s g rghwrd wh cs elc f /sec. lcule he u f frce he slder he ge kec se. 0 c, G G 5 c, θ kg, 0.8 kg, I 0.0 kg / s? b? G / Scle: 4 c. sds fr /sec 4 / s / s? b G / s fr G? 5.96 / s 7,698 / s
10 G / / s α α 78.98rd / s 0. W D leber frces d es N 90 G N 90 Iα N W N 6. G 584 N Iα N G θ 4
11 7.698 N N N G θ 4 4 0; 0 4 0; N M 0; cs cs N 5. N s 60 0
12 Eple rb pulr wrkg he ercl ple s shw fgure. The pld p s he elc f G /sec wh 0 d cceler f 5 /sec wh G 0. Mss f he pld s 0 kg. d he frces h he curs eer. 0 kg, 0 kg.,, G 0.5. G 0.5 G G Lk s rslg, s p hs lk hs he se elc d cceler,,g, /4. /s /.7 /s 60 /s O 60,G.5 /s G4.4 /s O g9.8 /s 0.0
13 cceler f d G cludg he effec f gr beces: G.4 / 4 s D leber frces P G G N N N 0.0 G τ 4.4 N 44. N ercl cur crres hs frce ( ) s N NS τ ( ) S N G 75 N g τc c c Hrzl cur crres hs frce N NS 5
14 Eple Sple sprg ss sse s shw he fgure, whch hs ss f kg, d sprg sffess s 500 N/. ) Wre he kec eerg f hs sse b) Wre he pel eerg f hs sse c) Dre he equ f f hs sse b usg Lgrge equ. d) Ge slu he equ f. d) k 500N/ If he sse s ll dsplced b 0. d relesed fr res, clcule he plude f he resulg br. ) Kec eerg, K & b) Pel eerg, k c) d d L K & δl δl 0 δ& δ δl d δl δl &, &&, k δ& d δ& δ k Subsug be eq & ( k ) 0 & k 0 d) Sl, csω sω e) r 0 d csω 0 sω 0 r 0 d & 0 & ω sω ω csω 0 ω sω 0 ω csω 0 0 plude
15 Eple rk f he echs shw s blced such h he ss ceer s. Mss ceer f he lk D s s d p. he ge s, lk 4 s rslg rghwrd wh cs elc f 5 /sec. lcule he u f r rque requred crk keep he ge kecs se. D θ c, D4 c, c G 5 c, 4 5 kg, I I I kg-? 5 / sec?.85 /sec /5.05 /sec
16 / sec ?? / sec fr fr / 776 /sec /48 /sec 800 / sec α α 0000 rd / sec W 776 α α 0078 rd / sec D leber frces d es N 8 W I α N I α N W W
17 D 4000N 000N 504 N T θ45 4 D 4000N 000N θ N T 4 T4 4 r D f he lk 0 ; 0 () 0 ; 0 () M 0 ; 504 T s cs 0 0 ()
18 r D f he lk M 0 ; 0 ; 0 ; cs s s 45 cs (4) (5) (6) r D f he lk 4 0 ; 0 (7) 0 ; 4 0 (8) M 0 ;T4 0 (9) r eq N r eq N r eq N r eq ( ) T 6. 9 N W
19 Eple I he fgure r gered shf whch flwheel s ued s shw. The es f er f he prs re s fllws: flwheel, I f kg. ; flwheel shf, I fs 0.0 kg. ; ger, I g 0. kg. ; p, I p kg. ; r, I 0.05 kg.. If he r hs srg rque f 75 N., wh s he gulr cceler he flwheel shf he s he r swch s ured? Tgel ccelers f he gers he cc p re equl ech her. Tgel cceler f he ger d flwheel ssebl s αg g r α d gel cceler f he g g r d p ssebl s g r α. The, p p IIfsIfIg r α r α () g g p p G αp G r he freebd dgr f he r d p ssebl; T IIpI T r I α () G p p r he freebd dgr f he ger d flwheel ssebl; G r g I α () g Slg hese hree equs fr α g ; T α I rp I rg rg rp g 7. 7 rd / sec
20 Eple Sple pedulu s shw he fgure, whch hs ss f kg, d legh f 0.5. ) Wre he kec eerg f hs sse b) Wre he pel eerg f hs sse c) Dre he equ f f hs sse b usg Lgrge equ. d) Ge slu he equ f. d) If he sse s ll dsplced b 0. rd d relesed fr res, clcule he plude f he resulg br. ) Kec eerg, & θ K I,where I b) Pel eerg, g( l l csθ ) & c) L K l θ gl( csθ ) d d δl δl 0 δθ& δθ δ L & d δ L l θ δθ&, l & θ δl d δθ&, gl sθ δ l Subsug be eq l && θ ( gl sθ ) 0 l && θ gl sθ 0 r sll gles s θ θ & θ ω θ 0 where ω && θ g l d) Sl, θ csω sω e) r 0 d θ θ g l θ csω 0 sω 0 θ r 0 d & θ 0 sθ 0 & θ ω sω ω csω 0 ω sω 0 ω csω 0 0 plude rd
21 Eple duble slder echs wrkg hrzl ple s shw fgure. The slder s g upwrd wh cs elc f 4 /sec. lcule he u f frces requred lk he ge kec se. 7 c, G c, 4 0, 4 0 kg, I 0.05 kg- 60 G 4 0? 4 / sec? 0 r 00 /5.85/sec 4/sec./sec 0
22 ?? 0 r / sec fr /58/s / G 758 /s G / s α rd / s D leber frces d es G W N 0 I α N W 60 G 54.4 N 7580 N 4 0
23 N d 60 d G 48.5 N 54.4 N 4 N 0 r he D; 0 ; N cs 0 N cs70 0 () 0 ; s 0 N s70 0 () M 0 ; d N d N fr eq s s N upwrd () N
24 Eple Ech f he w dru d ceced hubs f 0c rdus hs ss e f er f kg-. lcule he gulr cceler f ech dru. se r0 c se r0 c 40 N 40/9.8 kg se r0 c 40 N τ Iα τ T r T r I α 0 40 α 00 α 4rd / sec W se r0 c T T 40/9.8 kg g τ Iα τ T r T r I α () T g T g () r α () Subsug eq d slg fr T T rα g Subsug T eq d slg fr α gr r αr gr Iα α. 84 rd / sec W I r
25 - (5%) I he fgure sldercrk echs s shw wh he pprpre dess. rk f he 4 echs shw s G blced such h he 00 N ss ceer s. Mss D(6c, c) ceer f he lk 4 s. he ge s, lk 4 s rslg rghwrd wh cs elc f 5 /sec. lcule he 0 u f r rque requred crk keep he ge kec se. Mechs s wrkg hrzl ple. ssue frc bewee he lk d 4. Mechs s ge s scle. kg, 6 kg, 4 4 kg, I 0.0 kg, I 0. 0 kg? 5 / sec? / sec 0.05?.6 5. / sec 0.05? fr fr 57. / sec 6
26 774.4 α α 5488rd / sec α α 5408rd / sec 0.05 D leber frces d es N G 5 W W I α N I α N W W X Ig Y X X Y N 4 00 N Y X b c c/b b/c 0 I α r D f he lk 0 ; 0 0 ; 0 M 0; 54,08 0,0 0,0 T 54, , ,6 T T 0,N. ( W ) r D f he lk
27 0; 45,8 N 0; 498,6N M 96,8 N 476,66cs , ,66cs 45 96,8 5, ; 0,044 0,05 I α g 0,05 0 r D f he lk 4 0 ; 00N 0; N 0 N M 0; T4 0,0 0,07 0
28 - (0%) er cupler lk f he slder crk echs ge ques fr where l w p sses d es, ssfg he crer f equlec f sses d lc f ss ceers. The, usg ur w ss del, clcule he u f r rque requred crk keep he ge kec se s ques. pre ur resul wh he resul f ques. Mechs s ge s scle. 4 G D(6c, c) 0 00 N D 4 00 N 0 kg l l 47kg l l 0,05 Lk s w frce eber s 5,47 00/cs(0) r lk 0,005 0,05I α T T
29 Q- (0%) rk f he echs shw s blced such h he ss ceer s. Mss ceer f he lk P s P. Mss f he lk s eglgble sll. he ge s; lk s rg W drec wh cs elc f 5 rd/sec, lk P s rg W drec wh zer gulr elc d gulr cceler f 5 rd/sec. lcule he u f r rque requred crk P keep he ge kecs se. Mechs s wrkg hrzl ple ge scle. 4 5 kg, I I kg- 4 P ω 0, α 5 rd / sec ω 4 5rd / sec P P α / sec D leber frces d rques bece; P N I α ( ) N W Lk 4 d s w frce eber, 8.75 N P.86 N T r freebd dgr f he lk ; ; () 0 M 0; T 0 () M 0; T 0 () Slg hese hree equ fr T 5. 9 N W
KINEMATICS OF RIGID BODIES RELATIVE VELOCITY RELATIVE ACCELERATION PROBLEMS
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