Kinetics of a Particle: Impulse and Momentum

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1 Kineics of a Paricle: Impulse and Momenum Linear momenum L mv d Fma ( mv) d dl d Newon s nd law: The resulan of all forces acing on a paricle is equal o is ime rae of change of linear momenum. F d L L m v m v Linear impulse I Fd Chungnam Naional Universiy

2 Principle of Linear Impulse and momenum F d L L m v m v m d m 1 v F v 1 mv Fdmv ( x) ( ) x x 1 1 mv Fdmv ( y) ( ) y y 1 1 mv Fdmv ( z) ( ) z z 1 1 Chungnam Naional Universiy

3 Linear Impulse and momenum: Examples Chungnam Naional Universiy

4 Chungnam Naional Universiy

5 Principle of Linear Impulse and momenum for a sysem of paricles dm ( v i ) Fi fi for paricle i d d i Fi fi v i m d mi i id mi i ( v ) F ( v ) 1 1 For sysem of paricles mr G mv mr m i i v G i i where m m i m d m ( vg) Fi ( vg) 1 1 For rigid body Chungnam Naional Universiy

6 Conservaion of Linear Momenum for a sysem of paricles If he resulan force on a paricle is zero during an inerval of ime, m ( v ) Fd m ( v ) m Chungnam Naional Universiy i i 1 i i i 1 ( v ) m ( v ) i i 1 i i ( v ) ( v ) For rigid body, since mvg miv i G 1 G When is he resulan force on a paricle zero during an inerval of ime? (1) Paricles collide or inerac. m d m 1 v F v A A1 A A m d m 1 v F v B B1 B B () Exernal impulse is negligible, when he ime period is very shor m v m v m v m v A A1 B B1 A A B B

7 Conservaion of Linear Momenum: example Chungnam Naional Universiy

8 Conservaion of Linear Momenum: example mprojvproj 1mBlockvBlock1 mprojv mblockv ( mproj mblock ) v Chungnam Naional Universiy

9 Impac Definiion of impac: collision beween wo bodies is characerized by he generaion of relaively large conac forces ha ac over a very sho inerval of ime. Chungnam Naional Universiy

10 Cenral Impac Law of conservaion of linear momenum is valid. Why? Inernal impulse of deformaion and resiuion cancel, during collision m ( v ) m ( v ) m ( v ) m ( v ) Chungnam Naional Universiy A A 1 B B 1 A A B B How many unknowns in he above equaion?

11 Cenral Impac Deformaion Period For paricle A ma ( va) 1 PdmAv - P d ma ( v A) 1 m v A For paricle B mb( vb) 1 PdmBv P d m v B mb( v B) 1 Chungnam Naional Universiy

12 Cenral Impac Resiuion Period For paricle A mav Rd ma( va) - R d m v A ma( v A) For paricle B mbv Rd mb( vb) R d mb ( v B) m v B Chungnam Naional Universiy

13 Coefficien of Resiuion Coefficien of resiuion: he raio of he resiuion impulse o he deformaion impulse. For paricle A e Rd v ( va) Pd ( v ) v A 1 (1) For paricle B Eliminae v using eqs. 1 & e Rd ( vb ) v Pd v ( v ) e B 1 ( vb) ( v ) ( v ) ( v ) A A 1 B 1 () Chungnam Naional Universiy e relaive velociy of separaion relaive velociy of approach

14 Cenral Impac e=1: resiuion impulse = deformaion impulse No energy loss perfecly elasic ( v ) ( v ) v e=0: plasic impac 100% energy loss A B Summary of Cenral impac problem m ( v ) m ( v ) m ( v ) m ( v ) A A 1 B B 1 A A B B e ( vb) ( v ) ( v ) ( v ) A A 1 B 1 Chungnam Naional Universiy

15 Impac example Chungnam Naional Universiy

16 Impac example Chungnam Naional Universiy

17 Oblique Impac unknowns X direcion ( vax ) 1 ( va) 1cos1 ( v ) ( v ) cos Bx Y direcion 1 B 1 1 ( vax ) ( va) cos ( v ) ( v ) cos Bx ( vay ) 1 ( va) 1sin1 ( v ) ( v ) sin By B 1 B 1 1 ( vay ) ( va) sin ( v ) ( v ) sin By B Chungnam Naional Universiy

18 Oblique Impac X direcion m ( v ) m ( v ) m ( v ) m ( v ) A Ax 1 B Bx 1 A Ax B Bx e ( v ) ( v ) Bx Ax ( v ) ( v ) Ax 1 Bx 1 Y direcion m ( v ) m ( v ) A Ay 1 A Ay m ( v ) m ( v ) B By 1 B By Chungnam Naional Universiy

19 Angular Momenum The momen of he linear momenum L abou O is defined as he angular momenum Ho of paricle P abou O. ( H ) ( d)( mv) o H r mv H o i j k r r r o x y z mv mv mv x y z Uni: kg(m/s)m=kg(m/s )ms=nms Chungnam Naional Universiy

20 Relaion Beween Momen of a Force and Angular Momenum F mv M r Frmv o d H o ( rmv) r mvrmv d M H o o F L The resulan momen abou he fixed poin O of all forces acing on a paricle is equal o is ime rae of change of angular momenum of he paricle abou O. Chungnam Naional Universiy

21 Sysem of Paricles For he paricle i ( r F) ( r f ) ( H ) i i i i i o For sysem of paricles ( r F) ( r f ) ( H ) i i i i i o M o H o The sum of he momens abou poin O of all he exernal forces acing on a sysem of paricles is equal o he ime rae of change of he oal angular momenum of he sysem abou poin O. Chungnam Naional Universiy

22 Angular Impulse and Momenum Principles Principle of angular impulse and momenum o M H o o o 1 M d ( H ) ( H ) o o o 1 dh d ( Ho) Mod ( Ho) 1 1 M d o d H angular impulse M d o ( rf) d 1 1 For a paricle ( Ho) Mod ( Ho) 1 1 For sysem of paricles Chungnam Naional Universiy

23 Impulse and Momenum Principles Vecor formulaion m d m 1 v F v 1 H M d H ( o) ( ) o o 1 1 Scalar formulaion (D case) mv Fd mv ( x) x ( x) 1 1 mv Fd mv ( y) y ( y) 1 1 ( Ho) Mod ( Ho) 1 1 Chungnam Naional Universiy

24 Conservaion of Angular Momenum When he angular impulses acing on a paricle are all zero during he ime 1 o, hen ( H ) ( H ) o 1 o Conservaion of angular momenum of a sysem of paricles ( H ) ( H ) o 1 o From 1 o, he paricles angular momenum remains consan. Chungnam Naional Universiy

25 Conservaion of Angular Momenum: example Chungnam Naional Universiy

26 Conservaion of Angular Momenum: example Chungnam Naional Universiy

Lecture 4 Kinetics of a particle Part 3: Impulse and Momentum

Lecture 4 Kinetics of a particle Part 3: Impulse and Momentum MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an