Impulse & Momenum
Objecies To deelop he principle of linear impulse and momenum for a paricle. To sudy he conseraion of linear momenum for paricles. To analyze he mechanics of impac. To inroduce he concep of angular impulse and momenum. To sole problems inoling seady fluid sreams and propulsion wih ariable mass. 5/3/58 ME ดร. พ ภ ทร
Principle of linear impulse and momenum ΣF Σ ma Fd m m d d d Σ Fd m m Linear Impulse Linear Momenum 5/3/58 ME ดร. พ ภ ทร 3
Principle of linear impulse and momenum m + F d m m ( m ( m ( x y z ) ) ) + + + F F F x y z d d d m ( m ( m ( x z y ) ) ) 5/3/58 ME ดร. พ ภ ทร 4
Procedure for Analysis Selec he inerial coordinae sysem Draw FBD o accoun for all forces Esablish direcion and sense of acceleraion Sole for unknowns 5/3/58 ME ดร. พ ภ ทร 5
Problem 5- A man kicks he -g ball such ha i leaes he ground a an angle of 3⁰ wih he horizonal and srikes he ground a he same eleaion a disance of 5 m away. Deermine he impulse of his foo F on he ball. 5/3/58 ME ดร. พ ภ ทร 6
Problem 5- The paricle P is aced upon by is weigh of 3 N and forces F and F, where is in seconds. If he paricle originally has a elociy of (3i + j + 6k) m/s, deermine is speed afer s. 5/3/58 ME ดร. พ ภ ทร 7
Problem 5-7 Block A weighs N and block B weighs 3 N. If B is moing downward wih a elociy ( B ) m/s a, deermine he elociy of A when s. Assume ha he horizonal plane is smooh. 5/3/58 ME ดร. พ ภ ทร 8
Principle of Linear Impulse and Momenum for a Sysem of Paricles F i d d i m i mi ( ) + ( ) i Fi d mi i 5/3/58 ME ดร. พ ภ ทร 9
Conseraion of Linear Momenum for a Sysem of Paricles When he sum of he exernal impulsesacing on a sysem of paricles is zero, he equaion is m( ) m( ) i i i i This equaion is referred o as he conseraion of linear momenum. 5/3/58 ME ดร. พ ภ ทร
Example The 5-Mg boxcar A is coasing a.5 m/s on he horizonal rack when i encouners a -Mg ank B coasing a.75 m/s oward i. If he cars mee and couple ogeher, deermine (a) he speed of boh cars jus afer he coupling, and (b) he aerage force beween hem if he coupling akes place in.8 s. 5/3/58 ME ดร. พ ภ ทร
Problem 5-5 The boy B jumps off he canoe a A wih a elociy of 5 m/s relaie o he canoe as shown. If he lands in he second canoe C, deermine he final speed of boh canoes afer he moion. Each canoe has a mass of 4 kg. The boy s mass is 3 kg, and he girl has a mass of 5 kg. Boh canoes are originally a res. 5/3/58 ME ดร. พ ภ ทร
Impac Cenral Impac Velociies are colinear(a) Deforms during collision (b) Resore afer deformaion (b) Coefficien of resiuion I resiuion /I deform 5/3/58 ME ดร. พ ภ ทร 3
Coefficien of Resiuion Consider m )] ( [ )] ( [ m m d F d F e d r Consider m 5/3/58 ME ดร. พ ภ ทร 4 ] [ ] [ m m d F d F e d r e
Impac Oblique Impac e ( ) ( ) n n ( ) ( ) n n 5/3/58 ME ดร. พ ภ ทร 5
Problem 5-64 If he girl hrows he ball wih a horizonal elociy of 3 m/s, deermine he disance dso ha he ball bounces once on he smooh surface and hen lands in he cup a C. Take e.8. 5/3/58 ME ดร. พ ภ ทร 6
Angular Momenum If he paricle is moing along a space cure, he ecor cross produc can be used o deermine he angular momenum abou O. H r m O H mrsinθ r r i+ r j+ r x x y y z i+ j+ k z k 5/3/58 ME ดร. พ ภ ทร 7
Angular Momenum Recall d Σ F ( m) d L& d Σ ( r F ) ( r m ) d ΣM & O H O H & H O The momen abou he fixed poin O of all forces acing on m equals o he rae of change of angular momenum. 5/3/58 ME ดร. พ ภ ทร 8
Principle of Angular Momenum H O ΣMO H & O ΣM d d Σ M O dh d O H O d O H O O H + Σ MOd H ΣHO +Σ M OdΣHO O 5/3/58 ME ดร. พ ภ ทร 9
Conseraion of Angular Momenum Finally, if all angular impulse is zero. H H O O For a sysem of paricles ΣH ΣH O O 5/3/58 ME ดร. พ ภ ทร
Example The kg disk ress on a smooh horizonal surface and is aached o an elasic cord ha has a siffness k c N/m and is iniially unsreched. If he disk is gien a elociy ( D ).5 m/s, perpendicular o he cord, deermine he rae a which he cord is being sreched and he speed of he disk a he insan he cord is sreched. m. 5/3/58 ME ดร. พ ภ ทร
Meriam3-6 The small spheres, which hae he masses and iniial elociies shown in he figure, srike and become aached o he spiked ends of he rod, which is freely pioed a O and is iniially a res. Deermine he angular elociy ωof he assembly afer impac. Neglec he mass of he rod. 5/3/58 ME ดร. พ ภ ทร
Meriam3-3 The 6-kg sphere and 4-kg block are secured o he arm of negligible mass which roaes in he erical plane abou a horizonal axis a O. The -kg plug is released from res a A and falls ino he recess in he block when he arm has reached he horizonal posiion. An insan before engagemen, he arm has an angular elociy ω rad/s. Deermine he angular elociy ωof he arm immediaely afer he plug has wedged iself in he block. 5/3/58 ME ดร. พ ภ ทร 3