Kinetics of Particles. Chapter 3

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1 Kinetics f Particles Chapter 3 1

2 Kinetics f Particles It is the study f the relatins existing between the frces acting n bdy, the mass f the bdy, and the mtin f the bdy. It is the study f the relatin between unbalanced frces and the resulting mtin. The three general appraches t the slutin f kinetics prblems Direct applicatin f Newtn s secnd law (called the frcemassacceleratin methd), Use f wrk and energy principles, and Slutin by impulse and mmentum methds. 2

3 Frce, Mass and Acceleratin Newtn s secnd law states that F=ma The mass m is used as a quantitative measure f inertia (The resistance f any physical bject t change in its state f mtin r rest) F and a are tw vectrs having identical directin with respect t an internal frame f reference. When a particle f mass m acted upn by several frces. The Newtn s secnd law can be expressed by the equatin. F = ma 3

4 T determine the acceleratin we must use the analysis used in kinematics, i.e Rectilinear mtin Curvilinear mtin Rectilinear Mtin F = ma If we chse the x-directin, as the directin f the rectilinear mtin f a particle f mass m, the acceleratin in the y and z directin will be zer, i.e F = ma Generally, Resultant frce are given by F F x z y = 0 = 0 x F F F x y Z = ma = ma x y = ma Z ( ) F = Fx + ( Fy) + ( Fz) 4

5 Curvilinear mtin In applying Newtn's secnd law, we shall make use f the three crdinate descriptins f acceleratin in curvilinear mtin. Rectangular crdinates F F = ma = ma Where Nrmal and tangentialcrdinate Plar crdinates x y F F F n t r = ma x y = ma t = ma n r F θ = ma θ Where Where ar a a a x a y t = v = x = y 2 2 v n = ρβ =, ρ r = rθ 2 an = rθ + 2 rθ 5

6 Example 6

7 1. 7

8 2. 8

9 3. 9

10 4. 10

11 6. 11

12 7. 12

13 8. 13

14 Wrk and Energy 14

15 Wrk and Kinetic Energy There are tw general classes f prblems in which the cumulative effects f unbalanced frces acting n a particle ver an interval f mtin. These requires integratin f the frces with respect t the displacement. OR integratin f the frces with respect t the time they are applied. The first ne will result in Wrk-Energy equatin The secnd will give Impulse-mmentum equatin 15

16 Wrk Cnsider the fllwing figure du The wrk dne by the frce ver the displacement is du dr = F dr = ds The magnitude f the dt prduct is du = Fds cs α where dr = ds ds csα and α is the angle between F & Or du = ( F cs α) ds - the cmpnent f resultant f resultant displacement in the directin f frce. F csα - the cmpnent f resultant frce in the directin f displacement F - cmpnent in the displacement = directin f displacement F dr 16 dr

17 F n - has n wrk because it can nt displace the particles alng its directin, it is called reactive frce. F t - has wrk besause there is displacement in its directin and its called active frce. If the frce and displacement are in the same directin the wrk dne by is +ve, therwise it is ve. Example f negative wrk F Wrk by spring frce Wrk by gravitatinal frce du = F ds t 17

18 Cnsider the wrk dne n a particle f mass m, mving alng a curved path under the actin frce F as shwn belw. 2 2 s2 U = du = Fdr = F ds 1,2 1,2 1 1 U F 1, U1,2 = mv ( 2 v1 ) 2 1 t t a ds = s s 2 1 ma = matds t = vdv 2 v2 U = Fdr = mvdv v 1 s 1 t 18

19 Kinetic Energy f a Particle (T) K.E f a particle is the wrk dne n the particle t bring the particle frm state f rest t a velcity V. 1 2 T = mv 2 U1 2 K.E is a scalar quantity with the unit f Nm r Jule (J). K.E is always psitive. -the ttal wrk dne by all external acting frces n the particle during an interval f its mtin frm cnditin 1 t 2. Pwer U = 1 2 T 2 T = 1 T du F dr P= = = F v Time rate f ding wrk. dt dt Scalar quantity and units f Nm/s=J/s=watt(W) Efficiency:- the rati f wrk dne by a machine t the wrk dne n the machine during the same f interval Wrk dne by the machine Pwer ut put em = = wrk dne n the machine Pwer in put 19

20 Ptential Energy Gravitatinal ptential energy U = mg( h h ) Elastic ptential energy 2 1 Wrk energy equatin U g = mgh Ue = kx ( 2 x1 ) 2 g If U1 2stands wrks dne f all external frce ther than gravitatinal ptential energy and elastic ptential energy. U 1 2 U e U = g T If there is n external frces U1 2 =0 0 = Ue + Ug + T = 1 k( x 1 2 x1) + mg( h2 h1) + m( v2 v1) 2 2 = U U + U U + T T 0 e2 e1 g2 g1 2 1 U + U + T = U + U + T 20 e g 1 e g

21 Example 21

22 1. 22

23 2. 23

24 3. 24

25 4. 25

26 5. 26

27 Impulse and Mmentum 27

28 Integrating the equatin f mtin F=ma with respect t time leads t the equatins f Impulse and Mmentum. Linear impulse and mmentum Cnsider the general curvilinear mtin in space f a particle f mass m, F = ma where dv a = where mv = G Linear mmentum f the Particle. d F = G Fdt = dg dt t2 Fdt = G = G G = mv mv t 1 dt dv d F = m = ( mv) dt dt 2 1 Mmentum:-capacity fr prgressive develpment. The pwer t increase r develp at an every ging Frward mvement

29 The scalar cmpnents f equatin G = mv are: The ttal linear impulse n a mass m equals the crrespnding change in linear mmentum f m. If F = 0 then F G x, Fy = G y, Fz = = G x z t 2 t1 t 2 t1 t 2 t1 G = 0 G2 G1 = 0 G = G ( ) ( ) F dt = mv mv x x 2 x 1 ( ) ( ) F dt = mv mv y y 2 y 1 ( ) ( ) F dt = mv mv 2 1 z z 2 z 1 The SI units f linear mmentum is Kgm/s r Ns. 29

30 Angular Impulse and Angular Mmentum Cnsider the fllwing H = ( r v) m r = xi + yj + zk v= vi+ v j+ vk x y z H = Angular mmentum H is defined as the mment f linear mmentum vectr ( mv ) abut the rigin. And it is given by:- H0 = r mv = r G The angular mmentum is a vectr perpendicular t plane A defined by r and v. The directin H is clearly defined by the right hand rule fr crss prduct. The scalar cmpnent f angular mmentum btained by crss prduct. i j k m x y z v v v x y z H = r mv= mvy ( z vzi y ) + mvz ( x vx z ) j+ mvx ( y vyk x ) Hx = mvy ( z vz y ), Hy = mvz ( x vx z ) Hz = mvx ( y vy x ) 30

31 In rder t analyze the cmpnents f angular mmentum cnsider fllwing figure The magnitude f Where θ is the angle between r and v SI unit f angular mmentum is kg(m/s)m=kgm 2 /s=nm.s H H = r mv sinθ H = m( r sin θ) v = mr sinθ H = mr( vsin θ) = mr sinθ 31

32 If F represents the resultant f all frces acting n the particles P, the mment M abut the rigin O is the vectr crss prduct. M = r F = r mv (1) Angular mmentum = Differentiate with respect t time d dt Frm equatin (1) and (2) H ( r mv) d H = ( r mv) dt dr d H = mv + r ( mv) dt dt d r ( mv ) = r F dt H r F = Since but (2) d M = H = H dt M dt = dh dr mv v mv 0 dt = = Since the crss prduct f parallel vectr is zer. d F = ( mv) dt 32

33 The mment abut the fixed pint O f all frces acting n M equals the time rate f change f angular mmentum f M abut O. The prduct f mmentum and time is defined as angular impulse. Where d M = H = H dt t 2 t 1 M dt = dh M dt = H H = H H = r mv 2 H = r1 mv If the resultant mment abut a fixed pint O f all frces acting n a particle is zer t2 M dt = 0 = H H 2 1 t 1 H = 0 r H = H

34 Example 34

35 1. 35

36 2. 36

37 3. 37

38 4. 38

39 Impact 39

40 Impact refers t the cllisin between tw bdies and is characterized by the generatin f relatively large cntact frces that act ver a very shrt interval f time. Direct central impact Cnsider the cllinear mtin f tw spheres f masses m 1 and m 2 travelling with velcities V 1 & V 2. If V 1 is greater than V 2, cllisin ccurs with the cntact frces directed alng the line f centers. a. Befre impact b. Maximum defrmatin during impact c. After impact 40

41 As far as the cntact frces are equal and ppsite during impact, the linear mmentum f the system is cnserved. ' ' In this equatin we have tw unknwns v 1 and v 2 s we need anther relatin r equatin. During cllisin, there are tw kinds f frces experienced by the particles Frce f defrmatin: fr Frce f restratin: fr Particle 1 mv + mv = mv + mv ' ' Befre cllisin t t t t t 0 ' 0 t t t t t After cllisin ' During cllisin 41

42 Particle 1 Befre cllisin During cllisin After cllisin 0 t t Applying the principle f linear mmentum and impulse t t (state f defrmatin) t t t t 0 F dt = m ( v ( v )) d t 1 1 t t t ' (State f restratin) ' t Fddt = m1( v1 v) ' Frdt = m1( v v1) Cefficient f restratin (e) t It is the rati f magnitude f the restratin impulse t the magnitude f the defrmatin ' impulse. t Fr dt ' t m1( v 1) v e = = ' t ( v v1 ) m1( v1 v ) e = (1) F dt ( v v ) d t t 1 ' ' F dt = m ( v ( v )) r '

43 Particle 2 Befre cllisin 0 t t t t t ' During cllisin After cllisin Applying the principle f linear mmentum and impulse 0 t t (state f defrmatin) t t t ' (State f restratin) t F dt = m ( v v ) d 2 2 t t ' F dt = m ( v v ) r ' 1 2 Cefficient f restratin (e) It is the rati f magnitude f the restratin impulse t the magnitude f the defrmatin impulse. e t t ' r = = t F dt F dt d ' m1( v2 v ) m ( v v ) 1 2 e = ' ( v2 v ) ( v v ) 2 (2) 43

44 Frm equatin (1) and (2) e = v v ' ' 2 1 v v 2 If e=1 then the cllisin is perfectly elastic cllisin (n energy lss) If e=0 then the cllisin is perfectly plastic cllisin (maximum energy lss) 44

45 Oblique Central Impact Occurs when the directin f mtin f the mass centers f the clliding particles are nt the line f impact. mv + mv = mv + mv ' ' 1 1n 2 2n 1 1n 2 2n mv mv ' 1 1t 1 1t mv = = mv ' 2 2t 2 2t e = v v v ' ' 2n 1n v 1n 2n 45

46 Example 46

47 1. 47

48 2. 48

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