1. Kinematics of Particles

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1 1. Kinemics o Picles 1.1 Inoducion o Dnmics Dnmics - Kinemics: he sud o he geome o moion; ele displcemen, eloci, cceleion, nd ime, wihou eeence o he cuse o he moion. - Kineics: he sud o he elion eising beween he oces cing on bod, he mss o he bod, nd he moion o he bod; pedic he moion cused b gien oces o deemine he oces equied o poduce gien moion. 1. Posiion, Veloci, nd cceleion Reciline moion: moion long sigh line Posiion coodine: SI Unis: m, sec ege eloci Δ Δ Insnneous eloci (m/s) lim Δ Δ Δ Speed: mgniude o Δ ege cceleion (m/s ) Δ Insnneous cceleion lim Δ Δ Δ d d o d d d 1-1

2 1.3 Deeminion o he Moion o Picle In genel, he cceleion o picle cn be epessed s uncion o one o moe o ibles,, nd. Le us conside hee common clsses o moion: () d d d () () () () () () () ( ) d 1 d d ( ) ( ) () d ( ) 1 d d () d () ( ) ( ) d ( ) ( ) () 1-

3 Emple 1) The posiion o picle which moes long sigh line is deined b he elion , whee is epessed in mees nd in seconds. Deemine () he ime which he eloci will be zeo, (b) he posiion nd disnce eled b he picle he ime, (c) he cceleion o he picle h ime, (d) he disnce eled b he picle om 4s o 6s. ns) The equions o moion e d 6 1 (1) () (3) ) Time which. 4 5 ( 5)( + 1) 5s b) Posiion nd Disnce Teled when. Posiion 5s: ( m) Disnce 5s: 1m 4( m d) Disnce eled om 4s o 6s. The picle moes in he negie diecion om 4s o 5s nd in he posiie diecion om 5s o 6s. ) c) cceleion when. ( m ) 6 s / ( ) + dis( ) ( m) ol dis n ce dis Emple ) bll is ossed wih eloci o 1m/s dieced eicll upwd om window loced m boe he gound. Knowing h he cceleion o he bll is consn nd equl o 9.81m/s 1-3

downwd, deemine () he eloci nd eleion o he bll boe he gound n ime, (b) he highes eleion eched b he bll nd he coesponding lue o, (c) he ime when he bll will hi he gound nd he coesponding eloci.

4 downwd, deemine () he eloci nd eleion o he bll boe he gound n ime, (b) he highes eleion eched b he bll nd he coesponding lue o, (c) he ime when he bll will hi he gound nd he coesponding eloci. ns) ) eloci nd eleion d ( m / ) g, g 9.81 s g 1, ( m s) d b) highes eleion , ( m) d s c) bll his he gound highes s o 3.8s ( m s) ( m) 1-4

5 1.4 Uniom Reciline Moion When he eloci o picle is consn, cons n Unioml cceleed Reciline Moion When he cceleion is consn, d cons n d o d d d cons n + ( ) 1.6 Moion o Seel Picles Relie Moion o Two Picles The elie posiion coodine o wih espec o is denoed b o + The elie eloci o wih espec o is denoed b o + 1-5

6 The elie cceleion o wih espec o is denoed b o + Dependen Moions In he cse o hee blocks o Fig. 11.9, + consn : DOF + C 1 linel dependen equion 3 se ibles hen C + C Emple 4) bll is hown eicll om he 1-m leel in n eleo sh wih n iniil eloci o 18 m/s. he sme insn n open-plom eleo psses he 5-m leel, moing upwd wih consn eloci o m/s. Deemine () when nd whee he bll will hi he eleo, (b) he elie eloci o he bll wih espec o he eleo when he bll his he eleo. ns) Moion o he ll : unioml cceleed moion Moion o he Eleo : uniom moion E E + m s 1 + E

ll His he Eleo When he bll his he eleo, 5 + 1 + 18 4.95.39 o 3.65 E E E 5 + 3.65 1.3 E ( m) 16 9.81 3.65 19.

Pulles C nd E e ied, while D is ched o coll which is pulled downwd wih consn eloci o 75m/s.

7 ll His he Eleo When he bll his he eleo, o 3.65 E E E E ( m) ( m s) Emple 5) Coll nd block e conneced b cble pssing oe hee pulles C, D, nd E s shown. Pulles C nd E e ied, while D is ched o coll which is pulled downwd wih consn eloci o 75m/s., coll ss moing downwd om posiion K wih consn cceleion nd no iniil eloci. Knowing h he eloci o coll is 3 mm/s s i psses hough poin L, 1-7

deemine he chnge in eleion, he eloci, nd he cceleion o block when coll psses hough L. ns) Moion o Coll We plce he oigin O he uppe hoizonl suce nd choose he posiie diecion downwd.

8 deemine he chnge in eleion, he eloci, nd he cceleion o block when coll psses hough L. ns) Moion o Coll We plce he oigin O he uppe hoizonl suce nd choose he posiie diecion downwd. ( ) ( ) When L 3mm / s 3 ( ) + [ ( ) ] + 5mm / s ( ) s Moion o Pulle D D D 75mm / s ( ) + ( ) D D D D s ( ) ( mm) D D 1 + D + ( ) + ( D ) + ( ) [ ( ) ] + [ D ( D ) ] + [ ( ) ] [ ( ) ] ( ) 4mm Moion o lock The ol lengh o he cble is consn. Consideing he imes nd 1.333s, D + ( ) + ( ) ( ) D + [ ( ) ] + [ ( ) ] + [ ( ) ] D D

9 mm s D D + 5mm s Homewok 1 Poblems 11.34, 11.41, 11.45, 11.47, Posiion Veco, Veloci, nd cceleion Cuilie moion: moion long cue ohe hn sigh line Posiion eco: posiion o he picle w... he eeence me, ege eloci: Δ Δ Insnneous eloci: Insnneous cceleion: () Δ () Δ lim Δ Δ () Δ lim Δ ( ) d( ) ( ) d( ) 1.1 Deiies o Veco Funcions Le P(u) nd Q(u) be eco uncions o he scl ible u. Then Sum o wo eco uncions o he sme scl ible u: d ( P + Q) du dp dq + du du Poduc o scl uncion (u) nd eco uncion P(u): ( P) d du d du P + dp du 1-9

10 Scl poduc nd eco poduc: d ( P Q) du ( ) dp dq Q + P du du d P Q dp dq Q + P du du du Recngul componens o he deiie o eco uncion P(u): P P i + P j + P k whee i, j, k e uni ecos coesponding o he,, z es. dp dp du du Y Z dp i + du dpz j + k du Re o Chnge o Veco: dp dp dp dpz i + j + k The e o chnge o eco is he sme wih espec o ied me nd wih espec o me in nslion Recngul Componens o Veloci nd cceleion The posiion eco o picle in ecngul es cn be wien i + j + zk whee he coodines,, z e uncions o. Dieeniing wice, we obin d d d dz i + j + k d d d z i + j + k 1.1 Moion Relie o Fme in Tnslion Fied me o Reeence: he eeence me ched o he eh Moing Fmes o Reeence: he eeence me in eihe nslion o oion n me cn be designed s ied; ll ohe mes no igidl ched o his me will be descibed s moing. 1-1

11 + + + : The bsolue moion o cn be obined b combining he moion o nd he elie moion o wih espec o he moing me ched o. Emple 1.8) pojecile is ied wih n iniil eloci o 4m/s ge loced 6m boe he gun nd hoizonl disnce o 36m. Neglecing i esisnce, deemine he lue o he iing ngle α. ns) Hoizonl Moion 36m Veicl Moion Pojecile His Tge ( ) ( ) 4 cosα 4 cosα 36 4 cosα 15 cosα ( ) 4sinα 9.81m s 1 ( ) + 4 sinα cos nα.574 sinα 4.9 α nd cosα 6 α 9.9 o nd 69.6 o 1-11

Emple1.9) uomobile is eling es he consn speed o 36km/h. s uomobile cosses he inesecion shown, uomobile ss om es 35m noh o he inesecion nd moes souh wih consn cceleion o 1.m/s.

12 Emple1.9) uomobile is eling es he consn speed o 36km/h. s uomobile cosses he inesecion shown, uomobile ss om es 35m noh o he inesecion nd moes souh wih consn cceleion o 1.m/s. Deemine he posiion, eloci, nd cceleion o elie o 4s e cosses he inesecion. ns) We choose nd es wih oigin he inesecion o he wo sees nd wih posiie senses dieced especiel es nd noh. Moion o uomobile 1 m s Fo n ime 1m s ( ) Fo 5s 1m s ( m) Moion o uomobile Fo n ime 1.m s ( ) ( ) + ( ) Fo 5s 1. m s ( m s) ( m) 1-1

13 Moion o Relie o We dw he ingle coesponding o he eco equion + / nd obin he mgniude nd diecion o he posiion eco o elie o. / 53.9m α 1.8 / 11.66m/s β 31. Homewok Poblems 11.97, 11.13, 11.1,

14 1.13 Tngenil nd Noml Componens Plne Moion o Picle See Fig nd Fig

15 Le us conside picle which moes long cue conined in he plne o he igue. Le P be he posiion o he picle gien insn. We ch P uni eco e ngen o he ph o he picle nd poining in he diecion o moion. Le e / be he uni eco coesponding o he posiion P / o he picle le insn. Dwing boh ecos om he sme oigin O /, we deine he eco Δe e / - e. Since e / nd e e o uni lengh, hei ips lie on cicle o dius 1. Denoing b Δθ he ngle omed b e / nd e, we ind h he mgniude o Δe is sin(δθ/). Consideing now he eco Δe / Δθ, we noe h s Δθ ppoches zeo, his eco becomes ngen o he uni cicle, pependicul o e, nd h is mgniude ppoches Denoing his eco b e n, we wie sin Δθ ( Δθ ) sin( Δθ ) lim lim Δθ e e n n Δlim θ de Δe Δθ Δθ Δθ Since he eloci o he picle is ngen o he ph, i cn be epessed s he poduc o he scl nd he uni eco e : e d d e de + u de de ds ds 1 e n e n ρ ρ whee ρ is he dius o cuue o he ph P. Thus, 1 d d e + e ρ n The scl componens o he cceleion e n d ρ 1-15

16 1.14. Rdil nd nsese Componens See Fig de eθ deθ e In cein poblems o moion, he posiion o he picle P is deined b is pol coodines nd θ. I is hen conenien o esole he eloci nd cceleion o he picle ino componens pllel nd pependicul, especiel, o he line OP. These componens e clled dil nd nsese componens. deiion simil o he one we used o deemine he deiie o he uni eco e leds o he elions We epess he ime deiies o he uni ecos s ollows: de deθ de eθ deθ e Using dos o indice dieeniion wih espec o, The posiion eco o P is epessed s e. Thus, e& e& θ & θeθ & θe Dieeniing gin w, d & e && e d ( e ) + & θ e && & e d θ + e& ( & e + & θe ) + e + θ + θ + & & θe θ && θe && θe θ 1-16

( & & θ ) e + ( && θ + & & θ ) e θ In he cse o picle moing long cicle o cene O, we he consn: & θ e θ & θ e & + θ e θ Eension o he Moion o Picle in Spce: Clindicl Coodines See ig. 11.

17 ( & & θ ) e + ( && θ + & & θ ) e θ In he cse o picle moing long cicle o cene O, we he consn: & θ e θ & θ e & + θ e θ Eension o he Moion o Picle in Spce: Clindicl Coodines See ig The posiion o picle P in spce is someimes deined b is clindicl coodines R, θ, nd z; uni ecos e, e θ, nd k. Then he posiion eco o he picle P cn be wien s Thus d R& e d R e R + zk R + R & θe θ + z& k ( R&& R & θ ) e + ( R && θ + R& & θ ) e + && zk R θ Emple 1.1) moois is eling on cued secion o highw o dius 5 he speed o 6 mi/h. The moois suddenl pplies he bkes, cusing he uomobile o slow down consn e. Knowing h e 8s he speed hs been educed o 45mi/h, deemine he cceleion o he uomobile immediel e he bkes he been pplied. ns) Tngenil Componen o cceleion 58 6mi / h 6 36s 45mi / h 66 / s 88 / s Δ Δ 66 / s 88 / s 8s.75 / s 1-17

18 Noml Componen o cceleion Mgniude nd Diecion o cceleion Poblems , 11.14, , , 11.C3 ( 88 / s) n 3.1 / s ρ 5 nα α / s.75 / s n 4.14 / s sinα Homewok n o 1-18

ME 141. Engineering Mechanics

ME 141. Engineering Mechanics ME 141 Engineeing Mechnics Lecue 13: Kinemics of igid bodies hmd Shhedi Shkil Lecue, ep. of Mechnicl Engg, UET E-mil: sshkil@me.bue.c.bd, shkil6791@gmil.com Websie: eche.bue.c.bd/sshkil Couesy: Veco Mechnics