2/2/ :36 AM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E.

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1 //16 1:36 AM Chpter 11 Kinemtics of Prticles 1

2 //16 1:36 AM First Em Wednesdy 4//16

3 3 //16 1:36 AM Introduction Mechnics Mechnics = science which describes nd predicts the conditions of rest or motion of bodies under the ction of forces It is diided into three prts: 1. Mechnics of rigid bodies. Mechnics of deformble bodies 3. Mechnics of fluids

4 4 //16 1:36 AM Introduction Mechnics of rigid bodies is subdiided into: 1. Sttics: dels with bodies t rest. Dynmics: dels with bodies in motion

5 5 //16 1:36 AM Introduction Dynmics is subdiided into: 1. Kinemtics study of geometry of motion. relting displcement, elocity, ccelertion, nd time without reference to the cuse of motion. Kinetics study of the reltion eisting between the forces cting on body, the mss of the body, nd the motion of the body

6 6 //16 1:36 AM Introduction A dynmic study could be done on two leels: 1. Prticle n object whose size nd shpe cn be ignored when studying its motion.. Rigid Body collection of prticles tht remin t fied distnce from ech other t ll times nd under ll conditions of loding.

7 7 //16 1:36 AM Motion of Prticles Motion of Prticles: 1. Rectiliner Motion. Curiliner Motion

8 8 //16 1:36 AM Rectiliner Motion of Prticles Position Velocity Aerge elocity Instntneous elocity t lim t t

9 9 //16 1:36 AM Rectiliner Motion of Prticles Accelertion Aerge ccelertion Instntneous ccelertion t lim t t

10 1 //16 1:36 AM Rectiliner Motion of Prticles Consider prticle with motion gien by 3 d 6t t 1t 3t d d 1 t 6 t t =, =, =, = 1 m/s t t = s, = 16 m, = m = 1 m/s, = t t = 4 s, = m = 3 m, =, = -1 m/s t t = 6 s, =, = -36 m/s, = 4 m/s

11 11 //16 1:36 AM Determintion of Motion of Prticle Three clsses of motion my be defined: 1.Accelertion is function of time, = f(t).accelertion is function of position, = f() 3.Accelertion is function of elocity, = f()

12 1 //16 1:36 AM Determintion of the Motion of Prticle 1. Accelertion is function of time, = f(t) d f t d f t t t d d d t t t f t t f t d t t t t t t

13 13 //16 1:36 AM Determintion of the Motion of Prticle. Accelertion is function of position, = f() d d f or d d d d f or d d d f 1 1 f d

14 14 //16 1:36 AM Determintion of the Motion of Prticle 3. Accelertion is function of elocity, = f() d t t d d d d f t f t f d f t d f t d f d t d f t d f

15 15 //16 1:36 AM Smple 11. Bll tossed with 1 m/s erticl elocity from window m boe ground. Determine:. elocity & eletion boe ground t time t b. highest eletion reched by bll nd corresponding time c. time when bll will hit the ground & corresponding elocity

16 Smple 11.3 Brke mechnism used to reduce gun recoil consists of piston ttched to brrel moing in fied cylinder filled with oil. As brrel recoils with initil elocity, piston moes nd oil is forced through orifices in piston, cusing piston nd cylinder to decelerte t rte proportionl to their elocity; = -k Determine (t), (t), nd (). 16 //16 1:36 AM

17 17 //16 1:36 AM Assignment #11.1 1, 6, 11, 17,, 9 Due Wednesdy 3//16

18 18 //16 1:36 AM Uniform Rectiliner Motion Accelertion is zero nd elocity is constnt d constnt d t t t

19 19 //16 1:36 AM Uniformly Accelerted Rectiliner Motion Accelertion of the prticle is constnt d d d d t constnt t t 1 t constnt d t d d t t d 1 t t 1 t

20 Motion of Seerl Prticles Reltie Motion A B A B reltie position of B wrt A A B A B A B A B reltie elocity of B wrt A A B A B A B A B reltie ccelertion of B wrt A A B A B //16 1:36 AM

21 1 //16 1:36 AM Smple 11.4 Bll thrown erticlly from 1 m leel in eletor shft with initil elocity of 18 m/s. At sme instnt, open-pltform eletor psses 5 m leel moing upwrd t m/s. Determine. when nd where bll hits the eletor b. reltie elocity of bll wrt eletor t contct

22 //16 1:36 AM Motion of Seerl Prticles: Dependent Motion Position of B depends on position of A Rope length = constnt Sum of lengths of segments = constnt A B (one DOF) const

23 Motion of Seerl Prticles: Dependent Motion const C B A ( DOF) C B A C B A 3 //16 1:36 AM

24 4 //16 1:36 AM Smple 11.5 Pulley D is ttched to collr which is pulled down t 3 cm/s. At t =, collr A strts moing down from K with constnt ccelertion nd zero initil elocity. Knowing tht elocity of collr A is 1 cm/s s it psses L, determine the chnge in eletion, elocity, nd ccelertion of block B when block A is t L.

25 5 //16 1:36 AM Assignment #11. 33, 38, 4, 47, 5, 57 Due Mondy 8//16

26 6 //16 1:36 AM Grphicl Solution of Rectiliner-Motion Problems Gien -t cure, -t cure = -t cure slope Gien -t cure, -t cure = -t cure slope

27 Grphicl Solution of 7 //16 1:36 AM Rectiliner-Motion Problems Gien -t cure, chnge in elocity between t 1 & t = re under -t cure between t 1 & t. Gien -t cure, chnge in position between t 1 & t = re under -t cure between t 1 & t.

28 8 //16 1:36 AM Smple Problem 11.6 A subwy cr lees sttion A; it gins speed t the rte of 4 ft/s for 6 s nd then t the rte of 6 ft/s until it hs reched the speed of 48 ft/s. The cr mintins the sme speed until it pproches (cr does not rech B yet) sttion B; brkes re then pplied, giing the cr constnt decelertion nd bringing it to stop in 6 s. The totl running time from A to B is 4 s. Drw the t, t, nd t cures, nd determine the distnce between sttions A nd B.

29 9 //16 1:36 AM Assignment # , 67, 73, 79, 87 Due Sturdy13//16

30 3 //16 1:36 AM Curiliner Motion: Position, Velocity & Accelertion Curiliner motion: Prticle moing long cure other thn stright line Position ector of prticle t time t

31 31 //16 1:36 AM Curiliner Motion: Position, Velocity & Accelertion lim t r t dr instntneous elocity (ector) lim t s t ds instntneous speed (sclr)

32 3 //16 1:36 AM Curiliner Motion: Position, Velocity & Accelertion lim t t d instntneous ccelertion (ector) In generl, ccelertion ector is not tngent to prticle pth

33 Rectngulr Components of Velocity & Accelertion r i y j zk d dy dz i j k i j k y z i y j zk d d y d i j i j k y z z k i y j zk //16 1:36 AM 33

34 Rectngulr Components of Velocity & Accelertion Motion of projectile y g z initil conditions: y z y Integrting twice: 1 t y t gt z Motion in horizontl direction is uniform Motion in erticl direction is uniformly ccelerted //16 1:36 AM 34 y y y gt z z

35 Motion Reltie to Frme in Trnsltion yz = fied frme of reference moing frmes of reference: frmes not rigidly ttched to the fied reference frme Position ectors for prticles A nd B wrt to the fied frme of reference Oyz re r A nd r B. r B A : position of B wrt moing frme A y z r r r //16 1:36 AM 35 B A B A

36 Motion Reltie to Frme in Trnsltion B B A A B B A A B B A A elocity of B wrt A ccelertion of B wrt A Absolute motion of B = combined motion of A nd reltie motion of B wrt moing reference frme ttched to A. //16 1:36 AM 36

37 37 //16 1:36 AM Smple Problem 11.7 A projectile is fired from edge of 15-m cliff with n initil elocity of 18 m/s t n ngle of 3 with the horizontl. Neglecting ir resistnce, find:. horizontl distnce from the gun to the point where the projectile strikes the ground, b. gretest eletion boe the ground reched by the projectile.

38 38 //16 1:36 AM Smple Problem 11.9 Automobile A is treling est t the constnt speed of 36 km/h. As utomobile A crosses the intersection shown, utomobile B strts from rest 35 m north of the intersection nd moes south with constnt ccelertion of 1. m/s. Determine the position, elocity, nd ccelertion of B reltie to A 5 s fter A crosses the intersection.

39 39 //16 1:36 AM Assignment # , 95, 11, 17, 113, 1, 16 Due Mondy 17//16

40 Tngentil nd Norml Components Velocity ector is tngent to pth. = tngentil unit ectors for prticle pth t P & P e t e t nd t t t e e e d de e e e e e t n n n t t sin lim lim sin //16 1:36 AM 4

41 Tngentil nd Norml Components d d e t de d e t de d ds d ds det d e n d ds ds d e t e n t d n //16 1:36 AM 41

42 Tngentil nd Norml Components Tngentil component of ccelertion reflects chnge of speed Norml component reflects chnge of direction Tngentil component my be +e or -e Norml component lwys points towrd center of pth curture //16 1:36 AM 4

43 Tngentil nd Norml Components 3D Pth d e t e n t d n Osculting plne: Plne contining tngentil & norml unit ectors e en e b b e t e n principl binorml norml No Accelertion component long binorml //16 1:36 AM 43

44 Smple 11.1 A motorist is treling on cured section of highwy t 88 m/s. The motorist pplies brkes cusing constnt decelertion rte. Knowing tht fter 8 s the speed hs been reduced to 66 m/s, determine the ccelertion of the utomobile immeditely fter the brkes re pplied. //16 1:36 AM 44

45 Rdil nd Trnserse Components r re r d r e r de d dr rer e r e de d r e e r r de r r dr e r d r e de de r der d d de d e d //16 1:36 AM e r d d

46 Rdil nd Trnserse Components d d dr r e r d er r e dr de r r r e r r r e dr d e r d e r d de //16 1:36 AM 46

47 Rdil nd Trnserse Components 3D r Re z k dr R Re R R e z k d R R e R R e z k R //16 1:36 AM 47

48 Smple 11.1 The rottion of the.9 m rm OA bout O is defined by the reltion.15t where is epressed in rdins nd t in seconds. Collr B slides long the rm in such wy tht its distnce from O is r =.9-.1t, where r is epressed in meters nd t in seconds. After the rm OA hs rotted through 3 o, determine. totl elocity of the collr b. totl ccelertion of the collr c. reltie ccelertion of the collr wrt the rm //16 1:36 AM 48

49 49 //16 1:36 AM Assignment # , 14, 146, 153, 167 Due Sturdy //16

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