2/5/2012 9:01 AM. Chapter 11. Kinematics of Particles. Dr. Mohammad Abuhaiba, P.E.
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1 /5/1 9:1 AM Chper 11 Kinemic of Pricle 1
2 /5/1 9:1 AM Inroducion Mechnic Mechnic i Th cience which decribe nd predic he condiion of re or moion of bodie under he cion of force I i diided ino hree pr 1. Mechnic of rigid bodie. Mechnic of deformble bodie 3. Mechnic of fluid
3 3 /5/1 9:1 AM Inroducion Mechnic of rigid bodie i ubdiided ino: 1. Sic: del wih bodie re. Dynmic: del wih bodie in moion
4 4 /5/1 9:1 AM Inroducion Dynmic i ubdiided ino 1. Kinemic udy of he geomery of moion. I i ued o rele diplcemen, elociy, ccelerion, nd ime wihou reference o he cue of moion. Kineic udy of he relion eiing beween he force cing on body, he m of he body, nd he moion of he body
5 5 /5/1 9:1 AM Inroducion A dynmic udy could be done on wo leel: 1. Pricle n objec whoe ize nd hpe cn be ignored when udying i moion.. Rigid ody collecion of pricle h remin fied dince from ech oher ll ime nd under ll condiion of loding.
6 6 /5/1 9:1 AM Moion of Pricle Moion of Pricle: 1. Reciliner Moion. Curiliner Moion
7 7 /5/1 9:1 AM Reciliner Moion of Pricle Poiion Velociy Aerge elociy Innneou elociy lim
8 8 /5/1 9:1 AM Reciliner Moion of Pricle Accelerion Aerge ccelerion Innneou ccelerion lim
9 9 /5/1 9:1 AM Reciliner Moion of Pricle Conider pricle wih moion gien by 6 3 d d d =, =, =, = 1 m/ =, = 16 m, = m = 1 m/, = = 4, = m = 3 m, =, = -1 m/ = 6, =, = -36 m/, = 4 m/
10 1 /5/1 9:1 AM Deerminion of he Moion of Pricle Typiclly, condiion of moion re pecified by he ype of ccelerion eperienced by he pricle. Deerminion of elociy nd poiion require wo ucceie inegrion. Three cle of moion my be defined for: 1. ccelerion gien funcion of ime, = f(). ccelerion gien funcion of poiion, = f() 3. ccelerion gien funcion of elociy, = f()
11 11 /5/1 9:1 AM Deerminion of he Moion of Pricle 1. Accelerion gien funcion of ime, = f(): d f d f d f d d d f. Accelerion gien funcion of poiion, = f(): d d f or d d d d f or d d f d 1 1 f d
12 1 /5/1 9:1 AM Deerminion of he Moion of Pricle 3. Accelerion gien funcion of elociy, = f(): d d d d f f f d f d f d d f d d f d f
13 13 /5/1 9:1 AM Smple 11. ll oed wih 1 m/ ericl elociy from window m boe ground. Deermine:. elociy nd eleion boe ground ime, b. highe eleion reched by bll nd correponding ime, nd c. ime when bll will hi he ground nd correponding elociy.
14 Smple SOLUTION: Inegre wice o find () nd y(). d 9.81m d m 1 m 9.81 dy y dy y 1 y y y m m /5/1 9:1 AM m
15 Smple /5/1 9:1 AM Sole for which elociy equl zero nd elue correponding liude. m m y y m 1 m 1 m m m 4.95 m y 5.1m
16 Smple /5/1 9:1 AM Sole for which liude equl zero nd elue correponding elociy. y m m m m 1 meningle m 9.81 m m. m
17 17 /5/1 9:1 AM Smple 11.3 rke mechnim ued o reduce gun recoil coni of pion ched o brrel moing in fied cylinder filled wih oil. A brrel recoil wih iniil elociy, pion moe nd oil i forced hrough orifice in pion, cuing pion nd cylinder o decelere re proporionl o heir elociy; h i = -k Deermine (), (), nd ().
18 18 /5/1 9:1 AM Smple 11.3 SOLUTION: Inegre = d/ = -k o find (). d d k k ln Inegre () = d/ o find (). d d e e k k 1 e k k k e k k k 1 e
19 19 /5/1 9:1 AM Smple 11.3 Inegre = d/d = -k o find (). d d k k d k d d k k d Alerniely, wih nd hen k k 1 e k e or 1 k e k k
20 /5/1 9:1 AM Home Work Aignmen #1 1, 6, 11, 17,, 9 Due Tuedy 7//1
21 1 /5/1 9:1 AM Uniform Reciliner Moion Accelerion i zero nd elociy i conn d d conn
22 /5/1 9:1 AM Uniformly Accelered Reciliner Moion Accelerion of he pricle i conn. d d d d conn 1 conn d d d d 1 1
23 Moion of Seerl Pricle Relie Moion 3 For pricle moing long he me line, ime hould be recorded from he me ring inn nd diplcemen hould be meured from he me origin in he me direcion. A A relie poiion of wih repec o A A A A A relie elociy of wih repec o A A A A A relie ccelerion of wih repec o A A A /5/1 9:1 AM
24 4 /5/1 9:1 AM Smple 11.4 ll hrown ericlly from 1 m leel in eleor hf wih iniil elociy of 18 m/. A me inn, open-plform eleor pe 5 m leel moing upwrd m/. Deermine. when nd where bll hi eleor nd b. relie elociy of bll nd eleor conc.
25 5 SOLUTION: Subiue iniil poiion nd elociy nd conn ccelerion of bll ino generl equion for uniformly ccelered reciliner moion. 1 m 4.95 m 18 1m m 9.81 m 18 y y Subiue iniil poiion nd conn elociy of eleor ino equion for uniform reciliner moion. y y E E E m 5m m Smple 11.4 /5/1 9:1 AM
26 Smple 11.4 Wrie equion for relie poiion of bll wih repec o eleor nd ole for zero relie poiion, i.e., impc. y E.39 meningle 3.65 Subiue impc ime ino equion for poiion of eleor nd relie elociy of bll wih repec o eleor. 6 /5/1 9:1 AM ye y E 1.3m E E m 19.81
27 Moion of Seerl Pricle: Dependen Moion Poiion of pricle my depend on poiion of one or more oher pricle. Poiion of block depend on poiion of block A. Since rope i of conn lengh, i follow h um of lengh of egmen mu be conn. conn (one degree of freedom) A Poiion of hree block re dependen. conn (wo degree of freedom) A 7 /5/1 9:1 AM C For linerly reled poiion, imilr relion hold beween elociie nd ccelerion. d A d d C or A C d A d d C or A C
28 8 /5/1 9:1 AM Smple 11.5 Pulley D i ched o collr which i pulled down 3 cm/. A =, collr A r moing down from K wih conn ccelerion nd zero iniil elociy. Knowing h elociy of collr A i 1 cm/ i pe L, deermine he chnge in eleion, elociy, nd ccelerion of block when block A i L.
29 Smple /5/1 9:1 AM Differenie moion relion wice o deelop equion for elociy nd ccelerion of block. A A 1 cm D D conn cm cm cm A D cm 9 cm 9 cm 9
30 Smple /5/1 9:1 AM SOLUTION: Define origin upper horizonl urfce wih poiie diplcemen downwrd. Collr A h uniformly ccelered reciliner moion. Sole for ccelerion nd ime o rech L. A A cm 1 A cm 1 A cm 9 A A A A A 8cm A cm
31 Smple 11.4 D D D cm cm D D lock moion i dependen on moion of collr A nd pulley D. Wrie moion relionhip nd ole for chnge of block poiion ime. A D A D A A D D 8cm 4cm 16cm 16cm 31 /5/1 9:1 AM Pulley D h uniform reciliner moion. Clcule chnge of poiion ime.
32 3 /5/1 9:1 AM Home Work Aignmen # 33, 38, 4, 47, 5, 57 Due Sundy 1//1
33 33 /5/1 9:1 AM Grphicl Soluion of Reciliner-Moion Problem Gien he - cure, he - cure i equl o he - cure lope. Gien he - cure, he - cure i equl o he - cure lope.
34 34 /5/1 9:1 AM Grphicl Soluion of Reciliner-Moion Problem Gien he - cure, he chnge in elociy beween 1 nd i equl o he re under he - cure beween 1 nd. Gien he - cure, he chnge in poiion beween 1 nd i equl o he re under he - cure beween 1 nd.
35 Grphicl Soluion of Reciliner-Moion Problem 35 Momen-re mehod o deermine pricle poiion ime direcly from he - cure: 1 re under cure 1 uing d =, d bci of cenroid C fir momen of re under - cure wih repec o = 1 line. under - cure re 1 /5/1 9:1 AM
36 36 /5/1 9:1 AM Grphicl Soluion of Reciliner-Moion Problem Mehod o deermine pricle ccelerion from - cure: d d A n C ubnorml o - cure
37 37 /5/1 9:1 AM SAMPLE PROLEM 11.6 A ubwy cr lee ion A; i gin peed he re of 4 f/ for 6 nd hen he re of 6 f/ unil i h reched he peed of 48 f/. The cr minin he me peed unil i pproche ion ; brke re hen pplied, giing he cr conn decelerion nd bringing i o op in 6. The ol running ime from A o i 4. Drw he,, nd cure, nd deermine he dince beween ion A nd.
38 38 /5/1 9:1 AM Homework Aignmen #3 61, 67, 73, 79, 87 Due Tuedy 14//1
39 Curiliner Moion: Poiion, Velociy & Accelerion Pricle moing long cure oher hn righ line i in curiliner moion. Poiion ecor of pricle ime i defined by ecor beween origin O of fied reference frme nd he poiion occupied by pricle. Conider pricle which occupie poiion P defined r by ime nd P defined r by +, r dr lim 39 /5/1 9:1 AM innneou elociy (ecor) lim d innneou peed (clr)
40 4 /5/1 9:1 AM Curiliner Moion: Poiion, Velociy & Accelerion Conider elociy of pricle ime nd elociy +, d lim innneou ccelerion (ecor) In generl, ccelerion ecor i no ngen o pricle ph nd elociy ecor.
41 Deriie of Vecor Funcion /5/1 9:1 AM Le P u, Q u be ecor funcion of clr rible u, Le f dp du u be clr funcion of clr rible u P Pu u Pu lim lim u u u u Deriie of ecor um, dp Q dp dq du du du Deriie of produc of clr nd ecor funcion, d f P df dp P f du du du Deriie of clr produc nd ecor produc, dp Q dp dq Q P du du du dp Q dp dq Q P du du du 41
42 Recngulr Componen of Velociy & Accelerion When poiion ecor of pricle P i gien by i recngulr componen, r i y j zk Velociy ecor, d dy dz i j k i j k Accelerion ecor, y y z z i d d y d i j i j k z k y j zk i y j zk /5/1 9:1 AM 4
43 Recngulr Componen of Velociy & Accelerion Recngulr componen priculrly effecie when componen ccelerion cn be inegred /5/1 9:1 AM independenly, e.g., moion of projecile, y g z wih iniil condiion, y z y,, y z Inegring wice yield y y g z y y 1 g z Moion in horizonl direcion i uniform. y Moion in ericl direcion i uniformly ccelered. Moion of projecile could be replced by wo independen reciliner moion. 43 z
44 Moion Relie o Frme in Trnlion Deigne one frme he fied frme of reference. All oher frme no rigidly ched o he fied reference frme re moing frme of reference. Poiion ecor for pricle A nd wih repec o he fied frme of reference Oyz re r A nd r. Vecor r A joining A nd define he poiion of wih repec o he moing frme A y z nd r ra r A A A A elociy of relie o A. A A A ccelerion of relie o A. /5/1 9:1 AM Abolue moion of cn be obined by combining moion of A wih relie moion of wih repec o moing reference frme ched o A. 44
45 45 /5/1 9:1 AM Homework Aignmen #4 89, 95, 11, 17, 113, 1, 16 Due 19//1
46 Tngenil nd Norml Componen /5/1 9:1 AM Velociy ecor of pricle i ngen o ph of pricle. In generl, ccelerion ecor i no. Wih o epre ccelerion ecor in erm of ngenil nd norml componen. e nd e re ngenil uni ecor for he pricle ph P nd P. When drwn wih repec o he me origin, e e e nd i he ngle beween hem. e in e lim de e n d lim in e n e 46 n
47 Tngenil nd Norml Componen Wih he elociy ecor epreed he pricle ccelerion my be wrien d d de d de d d e e bu d d de d en d d d /5/1 9:1 AM Afer ubiuing, d d e en n Tngenil componen of ccelerion reflec chnge of peed nd norml componen reflec chnge of direcion. Tngenil componen my be poiie or negie. Norml componen lwy poin owrd cener of ph curure. 47
48 Tngenil nd Norml Componen Relion for ngenil nd norml ccelerion lo pply for pricle moing long pce cure. d e e n d n Plne conining ngenil nd norml uni ecor i clled he oculing plne. Norml o he oculing plne i found from eb e en e principl norml e n b binorml /5/1 9:1 AM Accelerion h no componen long binorml. 48
49 Rdil nd Trnere Componen r re r de de d d der der d e d de de d e d r e e r /5/1 9:1 AM r When pricle poiion i gien in polr coordine, i i conenien o epre elociy nd ccelerion wih componen prllel nd perpendiculr o OP. The pricle elociy ecor i d dr de re e r r r r r e r e r dr d r e Similrly, he pricle ccelerion ecor i d dr d er r e d d r dr de dr d d e r r e r e r d r r er r r e e r d de 49
50 Rdil nd Trnere Componen When pricle poiion i gien in cylindricl coordine, i i conenien o epre he elociy nd ccelerion ecor uing he uni ecor, e, nd k. e R Poiion ecor, r Re z k R Velociy ecor, dr R er R e z k /5/1 9:1 AM Accelerion ecor, d R R er R R e z k 5
51 Smple 11.1 A moori i reling on cured ecion of highwy 88 m/. The moori pplie brke cuing conn decelerion re. Knowing h fer 8 he peed h been reduced o 66 m/, deermine he ccelerion of he uomobile immediely fer he brke re pplied. /5/1 9:1 AM SOLUTION: Clcule ngenil nd norml componen of ccelerion. Deermine ccelerion mgniude nd direcion wih repec o ngen o cure. 51
52 Smple 11.1 SOLUTION: Clcule ngenil nd norml componen of ccelerion m m.75 8 n 88m 5m m 3.1 Deermine ccelerion mgniude nd direcion wih repec o ngen o cure m 4.14 n /5/1 9:1 AM n 1 n 1 n
53 Smple 11.1 The roion of he.9 m rm OA bou O i defined by he relion.15 where i epreed in rdin nd in econd. Collr lide long he rm in uch wy h i dince from O i r =.9-.1, where r i epreed in meer nd in econd. Afer he rm OA h roed hrough 3 o, deermine () he ol elociy of he collr, (b) he ol ccelerion of he collr, (c) he relie ccelerion of he collr wih repec o he rm /5/1 9:1 AM 53
54 Smple 11.1 SOLUTION: Elue ime for = 3 o rd Elue rdil nd ngulr poiion, nd fir nd econd deriie ime. r.9.1 r.4 r.4m.481 m.449m /5/1 9:1 AM rd.54 rd.561rd 54
55 Smple 11.1 Clcule elociy nd ccelerion. r r.449 m r.481m.561rd.7m 1 r n r.54m 31. /5/1 9:1 AM r r r.4 m.391m r r.481m.561rd.481m.3rd.449m.561rd.359 m 1 r n.531m 4. 6 r 55
56 Smple 11.1 Elue ccelerion wih repec o rm. Moion of collr wih repec o rm i reciliner nd defined by coordine r. OA r.4m /5/1 9:1 AM 56
57 57 /5/1 9:1 AM Howe Work Aignmen #4 133, 14, 146, 153, 16, 167, 173 Due Tuedy 1//1
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