Capíulo 5 Kieics of Paicles: Eegy ad Momeum Mehods
Mecáica II Coes Ioducio Wok of a Foce Piciple of Wok & Eegy pplicaios of he Piciple of Wok & Eegy Powe ad Efficiecy Sample Poblem 3. Sample Poblem 3. Sample Poblem 3.3 Sample Poblem 3.4 Sample Poblem 3.5 Poeial Eegy Cosevaive Foces Cosevaio of Eegy Moio Ude a Cosevaive Ceal Foce Escuela Técica Supeio de Igeieos Idusiales Sample Poblem 3.6 Sample Poblem 3.7 Sample Poblem 3.9 Piciple of Impulse ad Momeum Impulsive Moio Sample Poblem 3.0 Sample Poblem 3. Sample Poblem 3. Impac Diec Ceal Impac Oblique Ceal Impac Poblems Ivolvig Eegy ad Momeum Sample Poblem 3.4 Sample Poblem 3.5 Sample Poblems 3.6 Sample Poblem!3.7 5 -
Mecáica II Piciple of Impulse ad Momeum Dimesios of he impulse of a foce ae foce*ime. Uis fo he impulse of a foce ae ( kg m s ) s kg m s N s Fom Newo s secod law, d F ( mv ) mv liea momeum d Fd d mv Fd Imp Fd mv ( ) mv + Imp mv impulse of mv he foce F The fial momeum of he paicle ca be obaied by addig vecoially is iiial momeum ad he impulse of he foce duig he ime ieval. Escuela Técica Supeio de Igeieos Idusiales 5-3
Mecáica II Impulsive Moio Foce acig o a paicle duig a vey sho ime ieval ha is lage eough o cause a sigifica chage i momeum is called a impulsive foce. Whe impulsive foces ac o a paicle, mv F m + v Whe a baseball is suck by a ba, coac occus ove a sho ime ieval bu foce is lage eough o chage sese of ball moio. Noimpulsive foces ae foces fo which F is small ad heefoe, may be egleced. Escuela Técica Supeio de Igeieos Idusiales 5-4
Mecáica II Sample Poblem 3.0 SOLUTION: pply he piciple of impulse ad momeum. The impulse is equal o he poduc of he cosa foces ad he ime ieval. auomobile weighig 4000 lb is dive dow a 5 o iclie a a speed of 60 mi/h whe he bakes ae applied, causig a cosa oal bakig foce of 500 lb. Deemie he ime equied fo he auomobile o come o a sop. Escuela Técica Supeio de Igeieos Idusiales 5-5
Mecáica II Sample Poblem 3.0 SOLUTION: pply he piciple of impulse ad momeum. mv Imp m + v Takig compoes paallel o he iclie, mv + 4000 3. ( W si 5 ) F 0 ( 88f s) + ( 4000si 5 ) 500 0 9.49s Escuela Técica Supeio de Igeieos Idusiales 5-6
Mecáica II Sample Poblem 3. SOLUTION: pply he piciple of impulse ad momeum i ems of hoizoal ad veical compoe equaios. 4 oz baseball is piched wih a velociy of 80 f/s. fe he ball is hi by he ba, i has a velociy of 0 f/s i he diecio show. If he ba ad ball ae i coac fo 0.05 s, deemie he aveage impulsive foce exeed o he ball duig he impac. Escuela Técica Supeio de Igeieos Idusiales 5-7
Mecáica II Sample Poblem 3. y x SOLUTION: pply he piciple of impulse ad momeum i ems of hoizoal ad veical compoe equaios. mv Imp m + v x compoe equaio: mv + F 4 6 3. F 89lb x x mv cos40 4 6 3. ( 80) + F ( 0.5) ( 0cos 40 ) y compoe equaio: 0 + F mv si 40 F F y y y x 4 6 3. 39.9lb F 89 lb i + 39.9lb j, F 97.5 ( 0.5) ( 0cos 40 ) ( ) ( ) lb Escuela Técica Supeio de Igeieos Idusiales 5-8
Mecáica II Sample Poblem 3. 0 kg package dops fom a chue io a 4 kg ca wih a velociy of 3 m/s. Kowig ha he ca is iiially a es ad ca oll feely, deemie (a) he fial velociy of he ca, (b) he impulse exeed by he ca o he package, ad (c) he facio of he iiial eegy los i he impac. SOLUTION: pply he piciple of impulse ad momeum o he package-ca sysem o deemie he fial velociy. pply he same piciple o he package aloe o deemie he impulse exeed o i fom he chage i is momeum. Escuela Técica Supeio de Igeieos Idusiales 5-9
Mecáica II Sample Poblem 3. SOLUTION: pply he piciple of impulse ad momeum o he package-ca sysem o deemie he fial velociy. y x m p v + ( mp + mc ) Imp v x compoes: m v cos30 + 0 ( m + m ) p ( 0 kg)( 3 m/s) cos30 ( 0 kg + 5 kg) v p c v v 0.74 m/s Escuela Técica Supeio de Igeieos Idusiales 5-0
Mecáica II Sample Poblem 3. pply he same piciple o he package aloe o deemie he impulse exeed o i fom he chage i is momeum. y x m p v + Imp mpv x compoes: m p v cos30 + F x m ( 0 kg)( 3 m/s) cos30 + Fx ( 0 kg) v p v F x 8.56 N s y compoes: m p v si30 + F 0 ( 0 kg)( 3 m/s) si30 + F 0 y y F y 5N s ( 8.56 N s) i + ( 5 N s) j F 3.9 N s Imp F Escuela Técica Supeio de Igeieos Idusiales 5 -
Mecáica II Sample Poblem 3. To deemie he facio of eegy los, T T m p v ( 0 kg)( 3m s) 45 J ( m + m ) v ( 0 kg + 5 kg)( 0.74m s) 9.63 J p c T T 45 J 9.63 J 45 J T 0.786 Escuela Técica Supeio de Igeieos Idusiales 5 -
Mecáica II Impac Impac: Collisio bewee wo bodies which occus duig a small ime ieval ad duig which he bodies exe lage foces o each ohe. Lie of Impac: Commo omal o he sufaces i coac duig impac. Diec Ceal Impac Ceal Impac: Impac fo which he mass cees of he wo bodies lie o he lie of impac; ohewise, i is a ecceic impac.. Diec Impac: Impac fo which he velociies of he wo bodies ae dieced alog he lie of impac. Oblique Ceal Impac Oblique Impac: Impac fo which oe o boh of he bodies move alog a lie ohe ha he lie of impac. Escuela Técica Supeio de Igeieos Idusiales 5-3
Mecáica II Diec Ceal Impac Escuela Técica Supeio de Igeieos Idusiales odies movig i he same saigh lie, v > v. Upo impac he bodies udego a peiod of defomaio, a he ed of which, hey ae i coac ad movig a a commo velociy. peiod of esiuio follows duig which he bodies eihe egai hei oigial shape o emai pemaely defomed. Wish o deemie he fial velociies of he wo bodies. The oal momeum of he wo body sysem is peseved, m v + m v m v + m v secod elaio bewee he fial velociies is equied. 5-4
Mecáica II Diec Ceal Impac Peiod of defomaio: Peiod of esiuio: m v Pd m m u Rd m v simila aalysis of paicle yields Combiig he elaios leads o he desied secod elaio bewee he fial velociies. u e coefficie of Rd Pd 0 e v u e u v v v u v v u e ( v v ) esiuio Pefecly plasic impac, e 0: Pefecly elasic impac, e : Toal eegy ad oal momeum coseved. v v v m v + mv ( m + m )v v v v v Escuela Técica Supeio de Igeieos Idusiales 5-5
Mecáica II Oblique Ceal Impac Fial velociies ae ukow i magiude ad diecio. Fou equaios ae equied. No ageial impulse compoe; ageial compoe of momeum fo each paicle is coseved. Nomal compoe of oal momeum of he wo paicles is coseved. Nomal compoes of elaive velociies befoe ad afe impac ae elaed by he coefficie of esiuio. Escuela Técica Supeio de Igeieos Idusiales ( v ) ( v ) ( v ) ( v ) ( v ) + m ( v ) m( v ) + m ( v ) m [ ] ( v ) ( v ) e ( v ) ( v ) 5-6
Mecáica II Oblique Ceal Impac lock cosaied o move alog hoizoal suface. Impulses fom ieal foces F ad F alog he axis ad fom exeal foce F ex exeed by hoizoal suface ad dieced alog he veical o he suface. Fial velociy of ball ukow i diecio ad magiude ad ukow fial block velociy magiude. Thee equaios equied. Escuela Técica Supeio de Igeieos Idusiales 5-7
Mecáica II Oblique Ceal Impac Tageial momeum of ball is coseved. Toal hoizoal momeum of block ad ball is coseved. Nomal compoe of elaive velociies of block ad ball ae elaed by coefficie of esiuio. ( v ) ( v ) ( v ) + m ( v ) x m( v ) + m ( v ) x m [ ] ( v ) ( v ) e ( v ) ( v ) Noe: Validiy of las expessio does o follow fom pevious elaio fo he coefficie of esiuio. simila bu sepaae deivaio is equied. Escuela Técica Supeio de Igeieos Idusiales 5-8
Mecáica II Poblems Ivolvig Eegy ad Momeum Thee mehods fo he aalysis of kieics poblems: - Diec applicaio of Newo s secod law - Mehod of wok ad eegy - Mehod of impulse ad momeum Selec he mehod bes suied fo he poblem o pa of a poblem ude cosideaio. Escuela Técica Supeio de Igeieos Idusiales 5-9
Mecáica II Sample Poblem 3.4 SOLUTION: Resolve ball velociy io compoes omal ad ageial o wall. Impulse exeed by he wall is omal o he wall. Compoe of ball momeum ageial o wall is coseved. ball is how agais a ficioless, veical wall. Immediaely befoe he ball sikes he wall, is velociy has a magiude v ad foms agle of 30 o wih he hoizoal. Kowig ha e 0.90, deemie he magiude ad diecio of he velociy of he ball as i ebouds fom he wall. ssume ha he wall has ifiie mass so ha wall velociy befoe ad afe impac is zeo. pply coefficie of esiuio elaio o fid chage i omal elaive velociy bewee wall ad ball, i.e., he omal ball velociy. Escuela Técica Supeio de Igeieos Idusiales 5-0
Mecáica II Sample Poblem 3.4 SOLUTION: Resolve ball velociy io compoes paallel ad pepedicula o wall. v vcos 30 0.866v v vsi 30 0. 500v Compoe of ball momeum ageial o wall is coseved. v v 0. 500v pply coefficie of esiuio elaio wih zeo wall velociy. 0 v v e 0.9 ( v 0) ( 0.866v) 0.779v v 0.779vλ + 0.500vλ v 0.96v a 0.779 3.7 0.500 Escuela Técica Supeio de Igeieos Idusiales 5 -
Mecáica II Sample Poblem 3.5 SOLUTION: Resolve he ball velociies io compoes omal ad ageial o he coac plae. Tageial compoe of momeum fo each ball is coseved. The magiude ad diecio of he velociies of wo ideical ficioless balls befoe hey sike each ohe ae as show. ssumig e 0.9, deemie he magiude ad diecio of he velociy of each ball afe he impac. Toal omal compoe of he momeum of he wo ball sysem is coseved. The omal elaive velociies of he balls ae elaed by he coefficie of esiuio. Solve he las wo equaios simulaeously fo he omal velociies of he balls afe he impac. Escuela Técica Supeio de Igeieos Idusiales 5 -
Mecáica II Sample Poblem 3.5 SOLUTION: Resolve he ball velociies io compoes omal ad ageial o he coac plae. ( v ) v cos 30 6.0f s ( v ) v si 30 5.0f s ( v ) v cos60 0.0f s ( v ) v si 60 34.6f s Tageial compoe of momeum fo each ball is coseved. ( v ) ( v ) 5.0f s ( v ) ( v ) 34.6f s Toal omal compoe of he momeum of he wo ball sysem is coseved. m( v) ( ) ( ) ( + m v m v + m v ) m( 6.0) + m( 0.0) m( v ) + ( m v ) ( v ) + ( v ) 6. 0 Escuela Técica Supeio de Igeieos Idusiales 5-3
Mecáica II Sample Poblem 3.5 The omal elaive velociies of he balls ae elaed by he coefficie of esiuio. v v e v v [ ] ( ) ( ) ( ) ( ) 0.90[ 6.0 ( 0.0) ] 4. 4 Solve he las wo equaios simulaeously fo he omal velociies of he balls afe he impac. ( v ) 7.7f s ( ) 3.7f s v v v v v 7.7λ + 5.0λ 5.0 3.f s a 40.3 7.7 3.7λ + 34.6λ 4.9f s a 34.6 3.7 55.6 Escuela Técica Supeio de Igeieos Idusiales 5-4
Mecáica II Sample Poblem 3.6 SOLUTION: Deemie oieaio of impac lie of acio. The momeum compoe of ball ageial o he coac plae is coseved. all is hagig fom a iexesible cod. ideical ball is eleased fom es whe i is jus ouchig he cod ad acquies a velociy v 0 befoe sikig ball. ssumig pefecly elasic impac (e ) ad o ficio, deemie he velociy of each ball immediaely afe impac. Escuela Técica Supeio de Igeieos Idusiales The oal hoizoal momeum of he wo ball sysem is coseved. The elaive velociies alog he lie of acio befoe ad afe he impac ae elaed by he coefficie of esiuio. Solve he las wo expessios fo he velociy of ball alog he lie of acio ad he velociy of ball which is hoizoal. 5-5
Mecáica II Sample Poblem 3.6 siθ 0.5 θ 30 SOLUTION: Deemie oieaio of impac lie of acio. The momeum compoe of ball ageial o he coac plae is coseved. mv + F mv mv 0 ( v ) 0.5v0 ( ) si30 + 0 m v The oal hoizoal (x compoe) momeum of he wo ball sysem is coseved. mv + T mv + mv 0 0 0.5 m( v ) cos30 m( v ) ( 0.5v0 ) cos30 ( v ) ( v ) + v 0.433v0 si30 mv si 30 v Escuela Técica Supeio de Igeieos Idusiales 5-6
Mecáica II Sample Poblem 3.6 The elaive velociies alog he lie of acio befoe ad afe he impac ae elaed by he coefficie of esiuio. [ ] ( v ) ( v ) e ( v) ( v ) v si 30 ( v ) v0 cos30 0.5v ( v ) 0.866v0 0 Solve he las wo expessios fo he velociy of ball alog he lie of acio ad he velociy of ball which is hoizoal. ( v ) 0.50v0 v 0. 693v0 v v v 0 0.7v α 46. 30 6. 0.5v λ 0.50v λ 0.693v 0 0 0 β a 0.5 46. 0.5 Escuela Técica Supeio de Igeieos Idusiales 5-7
Mecáica II Sample Poblem 3.7 30 kg block is dopped fom a heigh of m oo he he 0 kg pa of a spig scale. ssumig he impac o be pefecly plasic, deemie he maximum deflecio of he pa. The cosa of he spig is k 0 kn/m. SOLUTION: pply he piciple of cosevaio of eegy o deemie he velociy of he block a he isa of impac. Sice he impac is pefecly plasic, he block ad pa move ogehe a he same velociy afe impac. Deemie ha velociy fom he equieme ha he oal momeum of he block ad pa is coseved. pply he piciple of cosevaio of eegy o deemie he maximum deflecio of he spig. Escuela Técica Supeio de Igeieos Idusiales 5-8
Mecáica II Sample Poblem 3.7 SOLUTION: pply piciple of cosevaio of eegy o deemie velociy of he block a isa of impac. T T T 0 + V m T 0 + 588 J V ( v ) ( 30)( v ) W + V y ( 30)( 9.8)( ) 588 J 0 ( 30)( v ) + 0 ( v ) 6.6m s V Deemie velociy afe impac fom equieme ha oal momeum of he block ad pa is coseved. m( v) + m( v ) ( m + m ) v3 ( )( 6.6) + 0 ( 30 + 0) v v 4.70m s 30 3 3 Escuela Técica Supeio de Igeieos Idusiales 5-9
Mecáica II Sample Poblem 3.7 x Iiial spig deflecio due o pa weigh: 3 W k ( )( 9.8) 0 3 4.9 0 m 3 0 0 pply he piciple of cosevaio of eegy o deemie he maximum deflecio of he spig. T V T V T 3 3 4 4 3 V 0 + 0 V ( m + m ) v ( 30 + 0)( 4.7) g g + V + V 39 39 e kx e 3 0.30 m 3 3 3 ( 0 0 )( 4.9 0 ) ( W + W )( h) 3 ( x ) 4 x3 + ( 0 0 ) x 4 3 3 ( x ) 4 4.9 0 + ( 0 0 ) x4 44 + 0.4 0 39 x 4 + V 3 T 4 + V 4 + kx 4 44 J 0.4 J 3 3 ( x 4.9 0 ) + ( 0 0 ) 4 x 4 h x x 4 3 0.30 m 4.9 0 3 m h 0.5 m Escuela Técica Supeio de Igeieos Idusiales 5-30