MÜHENDİSLİK MEKANİĞİ. HAFTA İMPULS- MMENTUM-ÇARPIŞMA Linea oenu of a paicle: The sybol L denoes he linea oenu and is defined as he ass ies he elociy of a paicle. L ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN Calculae he linea oenu of a paicle of ass 0 kg which has a elociy of 3i + 4j + 3 k /s. Çözü: L 0(3i + 4j + 3k) 30i + 40j + 30k kg s Ipulse of a foce fo ie o : The inegal of he foce oe he ie ineal of concen is is ipulse. The ipulse of a foce is a eco gien by he inegal F
I F d ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN Calculae he ipulse of he foce s. Çözü: I Fd ( i + 3 0 F i + 3 j) d ( j N fo ie 0 s o ie i + 4i + 8j N s Newon s nd law: Newon s second law saes ha he esulan of all foces applied on a paicle is equal o he ae of change of linea oenu of he paicle. F L & This educes o he oe failia saeen of his law if one noes ha in Newonian echanics i is assued ha ass is consan, and, heefoe, L& & + & a. The linea ipulse-oenu elaion: Inegaion of Newon s nd law oe he ie ineal fo o esuls in 3 j) 0 F d L& d F d L( ) L( ) L + F d L ÖRNEK 3: THE LINEAR IMPULSE-MMENTUM RELATIN
3 w ind foce (N ) w ind 0 i e (s) 0 A pojecile of ass 0 kg haing an iniial elociy of 3 i + 4j /s is subjeced o a hoizonal wind foce as shown in he figue. Calculae he pojecile s elociy jus afe he wind has ended. F w Çözü: Linea ipulse-oenu elaion: L + Fd L() L 0(3i + 4j) 30i + 40j Fd Fw i g j) d Subsiuion ino () gies: g kg s ( F di g d j 0i 0(9.8)() j 0i 96. jn s L 0 30i + 40j 0i 96.j 0 0 w 0 i 5. 6j s
4 Theefo, he linea oenu of a paicle is changed by he ipulse of he esulan foce on he paicle. Thee will be conseaion of linea oenu only if he ipulse of he esulan foce is zeo. Fo a syse of paicles: Conside he syse of paicles shown below. Each paicle in he syse has a ass i and a ie has a elociy i, and fo ie o ie is aced upon by he esulan exenal foce F i and he inenal foces of ineacion beween he paicles of f ij. As a esul of he ipulse of he inenal and exenal foces on each paicle, a ie he paicle wih ass i has a elociy i. Tie Fo o Tie F f f 3 f 3 f 3 f 3 3 3 3 f 3 F 3 F 3 Wiing he ipulse oenu equaion fo each paicle and adding he up, consideing ha Newon s 3 d law equies ha f ij f ji, we ge fo n paicles fo : + n i i + i ( fo : +.. fo n : n n + ( F + f + f 3 +...) d ( F + f + f +...) d ( Fn + f n + f n +...) d n n n n Fi ) d i i i i
5 Theefoe, he oal linea oenu of he syse is changed by he ipulse of he exenal foces. Recalling he elaion CM, we can ewie his equaion as CM + Fex d CM ÖRNEK 4: THE LINEAR IMPULSE-MMENTUM RELATIN 5kg 0kg The wo paicles ae oing a elociy 0i / s and 5i / s jus befoe hey collide and becoe conneced. Calculae he elociy of he syse afe he collision. Soluion: Tie Tie 5kg 0kg + Fexd () (5)(0i ) + (0)(5i ) 00i F ex d 0 kg s (No exenal foces)
6 (5 + 0) i (Boh hae he sae elociy) Subsiuion ino () gies 00i 5 i 6. 67 ÇARPIŞMA Ficionless ipac: Ipac efes o he ineacion of wo paicles when he ineacion ineal is ey sho. Ficionless ipac efes o sho ineacions when hee is no noiceable effec due o ficion. s The fee-body diaga of he wo paicles duing he collision is
7 The ipulse oenu fo each paicle is Whee and ae he exenally applied foces. The geneal ipulse oenu elaion fo his syse will be An epiical elaion is used o accoun fo he loss of echanical enegy in he ipac. This elaion is beween he elaie speed ha he paicles appoach each ohe and he elaie speed ha hey sepaae fo each ohe. This elaion is
8 whee e is he coefficien of esiuion, an epiical consan beween 0 and. The collision is consideed elasic if e. In an elasic collision he agniude of he noal coponen of he elaie elociy befoe and afe he collision is he sae. The collision is consideed fully plasic if e0. In a plasic collision he noal coponen of he elaie elociy of he wo paicles becoes zeo (i.e., he coponen of he elociy of he wo paicles in he noal diecion becoes he sae)., Noe ha (3) is a linea cobinaion of () and (), and, heefoe, is no an independen equaion. Ipac pobles ae soled using equaion (4) and a cobinaion of (), (), and (3). When hee is negligible ipulse due o exenal foces: The ie of he ipac is assued o be sall. If and ae bounded (us be less han infiniy), when one les go o zeo, he ipulse of hese foces go o zeo and one can siplify he aboe se of equaions o ge The subscips and n efe o he coponens of he ecos along he and diecions. Exaples of bounded exenal foces ae consan foces, such as he foce of gaiy, and foces esuling fo oion, such as sping foces. In hese cases, if he ipac ie is sall, we can ignoe he ipulse of hese foces and use equaions (5)-(8). The angula ipulse-oenu elaion
9 Angula oenu of a paicle: The sybol H denoes he angula oenu and is defined as he oen of linea oenu aound he poin, and gien by he equaion H L L ÖRNEK : THE ANGULAR IMPULSE-MMENTUM RELATIN A paicle of ass 0 kg is a posiion (,,3) and has a elociy 3i + j + k /s. Calculae he angula oenu of his paicle abou poin wih coodinaes (0,,). The coodinaes ae gien in ees. Çözü: ( 0) i + ( ) j + (3 ) k i + k kg L 0(3i + j + k) 30i + 0j + 0k s Ho ( i + k) (30i + 0j + 0k) 0k 0j + ()(30) j ()(0) i kg 40i + 50j + 0k s ÖRNEK : THE ANGULAR IMPULSE-MMENTUM RELATIN
0 y P x A paicle of ass 5 kg is a he posiion shown and has a elociy 3i + 4j /s. Calculae he angula oenu of his paicle abou poin P. y y (5)( 4) 0 P x (5)( 3) 5 x 3 Soluion: Assuing coune clockwise o be posiie H p (3)(0) ()(5) 45kg / s Ipulse of a oen fo ie o : The inegal of he oen oe he ie ineal of concen is is ipulse. The ipulse of a oen is a eco gien by he inegal
M d Moen and ae-of-change-of angula oenu elaion: Saing fo Newon s second law, calculaing he oen of boh sided of he equaion wih espec o a poin on an ineial fae esuls in H M L L L L H L F L F & & & & & & + 0 This esuling elaion beween he oen and he ae of change of angula oenu is called he oen-ae of change of angula oenu elaion (MRCAM). ÖRNEK 3: THE ANGULAR IMPULSE-MMENTUM RELATIN F M F
e θ e z e 0. M The oo M applies a consan oen M o o oae he paicle. If he paicle of ass 0 kg sas fo es and he oo applies a oen M o 0 N-, calculae he speed of he paicle afe 0 seconds. Assue he paicle is fixed o he ba a a adius of 0.. Soluion: Ho + M d H o o () Ho 0 (sas fo es, 0) Ho (0.e ) (0) e e θ z subsiuion ino () gies M d M e d 0(0 0) e 00e o 0 0 o z 00 e z e z 50 / s The angula ipulse-oenu elaion: Inegaion of Newon s nd law oe he ie ineal fo o esuls in z z
3 M d H& d Md H ( ) H ( ) + H M d H Theefoe, he angula oenu of a paicle is changed by he ipulse of he esulan oen on he paicle. Thee will be conseaion of angula oenu only if he ipulse of he esulan oen is zeo. ÖRNEK 4: THE ANGULAR IMPULSE-MMENTUM RELATIN 0 / s sooh hole 0.5 A B Paicle A of ass kg oes on a sooh hoizonal suface and is conneced by a sing o paicle B. B has a ass of kg. Paicle B is eleased fo es a ie 0 when paicle A has a cicufeenial elociy of 0 /s and a adius of 0.5. Wie he equaion of oion of he syse. Soluion: e z e θ e θ A Paicle A:
4 g e z T A e N e θ Fz az N Ag 0 ( ) M o 0 Ho consan Ho e A & e & ( ) ( + θ e θ ) A & θ e z H o consan & θ & θ & θ 5 o o (0.5) () 0 ( ) 5 0.5 F a T (&& & θ ) T && & θ (3) A Paicle B: e z T g B B a B Kineaics elaion: Fz az T B g BaB T ( )(9.8) ab (4)
5 l + l consan && && l && a B (5) Add (3) and (4) and subsiue (5) o ge 9. 6 && & θ + && Subsiuion fo () gies 5 9. 6 3&& ( 6) 3 Equaion () and (6) ae he equaion of oion fo he syse.