MEE 214 (Dynamics) Tuesday Dr. Soratos Tantideeravit (สรทศ ต นต ธ รว ทย )

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1 MEE 14 (Dynmics) Tuesdy Dr. Sortos Tntideerit (สรทศ ต นต ธ รว ทย ) sortos@oep.go.th Lecture Notes, Course updtes, Extr problems, etc No Homework Finl Exm (Dte & Time TBD) 1/03/58 MEE14 Dynmics 1

2 Course Oeriew Kinemtics of Prticle Rectiliner nd Curiliner Motion Kinetics of Prticle Force nd Accelertion Work nd Energy Impulse nd Momentum Kinetics of System of Prticles 1/03/58 MEE14 Dynmics

3 Introduction to Dynmics Sttics Engineering Mechnics Dynmics Kinemtics Kinetics 1/03/58 MEE14 Dynmics 3

4 Kinemtics of prticle

5 Objecties Concepts of position, displcement, elocity, nd ccelertion. Prticle motion long stright line Prticle motion long cured pth using different coordinte systems. Anlysis of dependent motion of two prticles. Principles of reltie motion of two prticles using trnslting xes. 1/03/58 MEE14 Dynmics 5

6 Rectiliner Kinemtics Origin Define fixed point in spce Position Defined by position ector r or n lgebric sclr s 1/03/58 MEE14 Dynmics 6

7 Rectiliner Kinemtics Displcement Chnge in position s Velocity s g t ds dt 1/03/58 MEE14 Dynmics 7

8 Rectiliner Kinemtics Accelertion g t d dt d s dt ds d 1/03/58 MEE14 Dynmics 8

9 Constnt Accelertion c t c /03/58 MEE14 Dynmics t t s s c + + ) ( 0 0 s s c +

10 Problem 1-31 The ccelertion of prticle long stright line is defined by (t-9) m/s, where t is in seconds. At t0, s1m nd 10m/s. When t9s, determine () the prticle s position, (b) the totl distnce treled nd (c) the elocity. 1/03/58 MEE14 Dynmics 10

11 Generl Curiliner Motion Curiliner motionoccurs when the prticle moes long cured pth Position.The position of the prticle, mesured from fixed point O, is designted by the position ector r r(t). 1/03/58 MEE14 Dynmics 11

12 Generl Curiliner Motion Displcement. Suppose during smll time interl Δtthe prticle moes distnce Δslong the cure to new position P`,defined by r` r+ Δr. The displcement Δrrepresents the chnge in the prticle s position. t 0, r s 1/03/58 MEE14 Dynmics 1

13 Generl Curiliner Motion Velocity r g t dr ins dt ds dt 1/03/58 MEE14 Dynmics 13

14 Generl Curiliner Motion Accelertion. g t d d r dt dt d s dt 1/03/58 MEE14 Dynmics 14

15 Position. Curiliner Motion: Rectngulr Components Position ector is defined by r xiˆ + yˆj + zkˆ The mgnitude ofr is lwys positie nd defined s r x + y + z The directionof r is specified by the components of the unit ector uˆ r r/ r 1/03/58 MEE14 Dynmics 15

16 Curiliner Motion: Rectngulr Components Velocity. z y x k j i dt dr z y x z y x & & & + + ˆ ˆ ˆ 1/03/58 MEE14 Dynmics 16 u / ˆ The elocity hs mgnitude defined s the positie lue of z y x + +

17 Curiliner Motion: Rectngulr Components Accelertion. x k j i dt d x x z y x && & + + ˆ ˆ ˆ 1/03/58 MEE14 Dynmics 17 z y z z y y && & && & The ccelertion hs mgnitude defined s the positie lue of z y x + +

18 Curiliner Motion: Rectngulr Components The ccelertion hs direction specified by the components of the unit ector uˆ /. Since represents the time rte of chngein elocity, will notbe tngent to the pth. 1/03/58 MEE14 Dynmics 18

19 Motion of Projectile Constnt downwrd ccelertion, no ir resistnce Mthem cl expressions, [] +, [] + && y y& y g y 0y 0 + gt 0y t 1 gt && x x& x 0 x 0x 0 + 0x t 1/03/58 MEE14 Dynmics 19

20 Exmple 1.1 The chipping mchine is designed to eject wood chips t O 5 ft/s. If the tube is oriented t 30 from the horizontl, determine how high, h, the chips strike the pile if they lnd on the pile 0 ft from the tube. 1/03/58 MEE14 Dynmics 0

21 Curiliner Motion: Norml nd Tngentil Components When the pth of motion of prticle is known, describe the pth using nnd tcoordintes which ct norml nd tngent to the pth Consider origin locted t the prticle û t Tngentil direction û n Norml direction 1/03/58 MEE14 Dynmics 1

22 Curiliner Motion: Norml nd Tngentil Components Velocity. Since the prticle is moing, sis function of time Prticle s elocity hs direction tht is lwys tngent to the pthnd mgnitudetht is determined by tking the time deritie of the pth function s s(t) uˆ t s& 1/03/58 MEE14 Dynmics

23 Curiliner Motion: Norml nd Tngentil Components Accelertion Accelertion of the prticle is the time rte of chnge of elocity & d( uˆ) dt d & dt d u & ˆ dt s + u& ˆ t t 1/03/58 MEE14 Dynmics 3

24 Curiliner Motion: Norml nd Tngentil Components Accelertion Find u &ˆ t dθ û n ût ' uˆt dˆ ut t t ˆ ' t u ˆ + duˆ u duˆ uˆ & t du duˆ dθ ˆ t t uˆ t u n s θ & & uˆ uˆ n n ρ du t (1) dθ dθ n ρ uˆ n 1/03/58 MEE14 Dynmics 4

25 Curiliner Motion: Norml nd Tngentil Components uˆ t t + n uˆ n t & n ρ ds t d ρ [1+ ( dy/ dx) d y/ dx ] 3/ 1/03/58 MEE14 Dynmics 5

26 Problem 1-10 The utomobile is originlly t rest t s0.if it then strts to increse its speed t & (0.05t ) ft/s where t is in seconds, determine the mgnitudes of its elocity nd ccelertion t s 550 ft. 1/03/58 MEE14 Dynmics 6

27 Problem At gien instnt the trin engine t E hs speed of 0 m/s nd n ccelertion of 14 m/s cting in the direction shown. Determine the rte of increse in the trin s speed nd the rdius of curture of the pth. 1/03/58 MEE14 Dynmics 7

28 Problem 1-15 If the speed of the box t point on the trck is 30ft/s which is incresing t the rte of & 5 ft/s, determine the mgnitude of the ccelertion of the box t this instnt. 1/03/58 MEE14 Dynmics 8

29 Fixed origin Curiliner Motion: Cylindricl Components û r û θ Rdil direction Trnserse direction 1/03/58 MEE14 Dynmics 9

30 Position Curiliner Motion: Cylindricl Components r ruˆ r 1/03/58 MEE14 Dynmics 30

31 Velocity Curiliner Motion: Cylindricl Components or Polr & r r ru & ˆ + ruˆ & r r u & ˆ θu & ˆ θ u r ru & ˆ + rθu & ˆ θ r u ˆ + r θ u ˆθ 1/03/58 MEE14 Dynmics 31

32 Curiliner Motion: Cylindricl Components Accelertion & && ruˆ + ru && ˆ + r& & θuˆ + r&& θuˆ + r r uˆ & θ & ˆ θ u r θ θ r & θu& ˆ θ r u ˆ + r & r r & r θ r& + r& θ θ θ u ˆθ & θ 1/03/58 MEE14 Dynmics 3

33 Exmple 1-19 The serchlight csts spot of light long the fce of wll tht is locted 100m from the serchlight. Determine the mgnitudes of the elocity nd ccelertion t which the spot ppers to trel cross the wll t the instnt θ 45. The serchlight is rotting t constnt rte of 4 rd/s 1/03/58 MEE14 Dynmics 33

34 Problem The slotted rm AB dries pin C through the spirl grooe described by the eqution r (1.5Ө) ft, where Ө is in rdins. If the rmstrts from restwhen Ө 60 nd is drien t n ngulr elocity of Ө (4t) rd/s, where t is in seconds, determine the rdil nd trnserse components of elocity nd ccelertion of thepincwhent1s. 1/03/58 MEE14 Dynmics 34

35 Absolute Dependent Motion Dependent motions of two prticles re normlly ssocited with systems of connected msses i inextensible cords nd pulleys. 1/03/58 MEE14 Dynmics 35

36 Absolute Dependent Motion l s + 3s A B A B A B 1/03/58 MEE14 Dynmics 36

37 Problem 1-06 If the hydrulic cylinder t H drws in rod BC by 00 mm t ft/s, determine how fr the slider A moes nd the speed of the slider. 1/03/58 MEE14 Dynmics 37

38 Exmple 3 A mn t Ais hoisting sfe S by wlking to the right with constnt elocity A 0.5m/s. Determine the elocity nd ccelertion of the sfe when it reches the eletion t E. The rope is 30m long nd psses oer smll pulley t D. 1/03/58 MEE14 Dynmics 38

39 Problem 1-08 If block A is moing downwrd with speed of 6 ft/s while C is moing down t 18 m/s, determine the speed of block B. 1/03/58 MEE14 Dynmics 39

40 Reltie Motion Anlysis The reltie position of B with respect to A is gien by A B A B r r r / 1/03/58 MEE14 Dynmics 40 The reltie elocity nd ccelertion of B with respect to A re gien by A B A B / A B A B /

41 Exmple 4 A trin, treling t constnt speed of 60 mi/h, crosses oer rod. If utomobile A is treling t 45 mi/h long the rod, determine the mgnitude nd direction of reltie elocity of the trin with respect to the utomobile. 1/03/58 MEE14 Dynmics 41

42 Problem The two prticles A nd B strt t the origin O nd trel in opposite directions long the circulr pth t constnt speeds A 0.7m/s nd B 1.5m/s respectiely. Determine t t s, () the displcement long the pth of ech prticle, (b) the position ector to ech prticle, nd (c) the mgnitude of the ccelertion of prticle B. 1/03/58 MEE14 Dynmics 4