Capítulo Knematcs of Rgd Bodes n Thee Dmensons
Mecánca Contents ntoducton Rgd Bod Angula Momentum n Thee Dmensons Pncple of mpulse and Momentum Knetc Eneg Sample Poblem 8. Sample Poblem 8. Moton of a Rgd Bod n Thee Dmensons Eule s Equatons of Moton and D Alembet s Pncple Moton About a Fed Pont o a Fed As Sample Poblem 8.3 Moton of a Goscope. Eulean Angles Stead Pecesson of a Goscope Moton of an Asmmetcal Bod Unde No Foce Escuela Técnca Supeo de ngeneos ndustales 5 -
Mecánca ntoducton F ma M G & G The fundamental elatons developed fo the plane moton of gd bodes ma also be appled to the geneal moton of thee dmensonal bodes. The elaton G whch was used to detemne the angula momentum of a gd slab s not vald fo geneal thee dmensonal bodes and moton. The cuent chapte s concened wth evaluaton of the angula momentum and ts ate of change fo thee dmensonal moton and applcaton to effectve foces, the mpulse-momentum and the wokeneg pncples. Escuela Técnca Supeo de ngeneos ndustales 5-3
Escuela Técnca Supeo de ngeneos ndustales Mecánca 5-4 Rgd Bod Angula Momentum n Thee Dmensons Angula momentum of a bod about ts mass cente, ( ) ( ) [ ] n n G m m v The component of the angula momentum, ( ) ( ) [ ] ( ) ( ) [ ] ( ) n n n n n m m m m m ( ) dm dm dm
Mecánca Rgd Bod Angula Momentum n Thee Dmensons Tansfomaton of nto G s chaacteed b the neta tenso fo the bod, Wth espect to the pncpal aes of neta, 0 0 0 0 0 0 The angula momentum of a gd bod and ts angula veloct G have the same decton f, and onl f, s dected along a pncpal as of neta. Escuela Técnca Supeo de ngeneos ndustales 5-5
Mecánca Rgd Bod Angula Momentum n Thee Dmensons The momenta of the patcles of a gd bod can be educed to: L lnea momentum mv G angula momentum about G The angula momentum about an othe gven pont O s mv O G Escuela Técnca Supeo de ngeneos ndustales 5-6
Mecánca Rgd Bod Angula Momentum n Thee Dmensons The angula momentum of a bod constaned to otate about a fed pont ma be calculated fom mv O G O, the angula momentum ma be computed dectl fom the moments and poducts of neta wth espect to the O fame. n v m O ( ) n [ ( ) m ] Escuela Técnca Supeo de ngeneos ndustales 5-7
Mecánca Pncple of mpulse and Momentum The pncple of mpulse and momentum can be appled dectl to the thee-dmensonal moton of a gd bod, Sst Momenta Sst Et mp - Sst Momenta The fee-bod dagam equaton s used to develop component and moment equatons. Fo bodes otatng about a fed pont, elmnate the mpulse of the eactons at O b wtng equaton fo moments of momenta and mpulses about O. Escuela Técnca Supeo de ngeneos ndustales 5-8
Escuela Técnca Supeo de ngeneos ndustales Mecánca 5-9 Knetc Eneg Knetc eneg of patcles fomng gd bod, ) ( n n mv m mv m v mv T f the aes coespond nstantaneousl wth the pncple aes, ) ( mv T Wth these esults, the pncples of wok and eneg and consevaton of eneg ma be appled to the thee-dmensonal moton of a gd bod.
Escuela Técnca Supeo de ngeneos ndustales Mecánca 5-0 Knetc Eneg Knetc eneg of a gd bod wth a fed pont, ) ( T f the aes O coespond nstantaneousl wth the pncple aes O, ) ( T
Mecánca Sample Poblem 8. Rectangula plate of mass m that s suspended fom two wes s ht at D n a decton pependcula to the plate. mmedatel afte the mpact, detemne a) the veloct of the mass cente G, and b) the angula veloct of the plate. Escuela Técnca Supeo de ngeneos ndustales SOLUTON: Appl the pncple of mpulse and momentum. Snce the ntal momenta s eo, the sstem of mpulses must be equvalent to the fnal sstem of momenta. Assume that the suppotng cables eman taut such that the vetcal veloct and the otaton about an as nomal to the plate s eo. Pncple of mpulse and momentum elds to two equatons fo lnea momentum and two equatons fo angula momentum. Solve fo the two hoontal components of the lnea and angula veloct vectos. 5 -
Mecánca Sample Poblem 8. SOLUTON: Appl the pncple of mpulse and momentum. Snce the ntal momenta s eo, the sstem of mpulses must be equvalent to the fnal sstem of momenta. Assume that the suppotng cables eman taut such that the vetcal veloct and the otaton about an as nomal to the plate s eo. v v v k j Snce the,, and aes ae pncpal aes of neta, G j mb ma j Escuela Técnca Supeo de ngeneos ndustales 5 -
Mecánca Sample Poblem 8. Pncple of mpulse and momentum elds two equatons fo lnea momentum and two equatons fo angula momentum. Solve fo the two hoontal components of the lnea and angula veloct vectos. 0 mv 0 v F t mv F t v v ( F t m)k m bf t mb 6F t mb 6F t mab af t ( a b j ) ma ( 6F t ma) Escuela Técnca Supeo de ngeneos ndustales 5-3
Mecánca Sample Poblem 8. v ( F t m)k G 6F t mab mb ( a b j ) ma j Escuela Técnca Supeo de ngeneos ndustales 5-4
Mecánca Sample Poblem 8. A homogeneous dsk of mass m s mounted on an ale OG of neglgble mass. The dsk otates counteclockwse at the ate about OG. Detemne: a) the angula veloct of the dsk, b) ts angula momentum about O, c) ts knetc eneg, and d) the vecto and couple at G equvalent to the momenta of the patcles of the dsk. SOLUTON: The dsk otates about the vetcal as though O as well as about OG. Combne the otaton components fo the angula veloct of the dsk. Compute the angula momentum of the dsk usng pncple aes of neta and notng that O s a fed pont. The knetc eneg s computed fom the angula veloct and moments of neta. The vecto and couple at G ae also computed fom the angula veloct and moments of neta. Escuela Técnca Supeo de ngeneos ndustales 5-5
Mecánca Sample Poblem 8. SOLUTON: The dsk otates about the vetcal as though O as well as about OG. Combne the otaton components fo the angula veloct of the dsk. j Notng that the veloct at C s eo, vc C 0 0 ( j ) ( L j ) L k ( ) L ( L) j Escuela Técnca Supeo de ngeneos ndustales 5-6
Escuela Técnca Supeo de ngeneos ndustales Mecánca 5-7 Sample Poblem 8. Compute the angula momentum of the dsk usng pncple aes of neta and notng that O s a fed pont. ( ) j L k j O ( ) ( )( ) ( ) 0 0 4 4 m ml L m ml m ( )( ) j L m L m O 4 The knetc eneg s computed fom the angula veloct and moments of neta. ( ) ( )( ) [ ] 4 L m L m T 8 6 L m T
Mecánca Sample Poblem 8. ( L) j The vecto and couple at G ae also computed fom the angula veloct and moments of neta. mv m k G j k m m L 4 ( ) j G m j L Escuela Técnca Supeo de ngeneos ndustales 5-8
Mecánca Moton of a Rgd Bod n Thee Dmensons F ma M & G Angula momentum and ts ate of change ae taken wth espect to centodal aes GX Y Z of fed oentaton. Tansfomaton of nto G s ndependent of the sstem of coodnate aes. Convenent to use bod fed aes G whee moments and poducts of neta ae not tme dependent. Defne ate of change of change of Gwth espect to the otatng fame, & & & j & ( ) k & G G Then, & G ( ) Ω Ω G G G Escuela Técnca Supeo de ngeneos ndustales 5-9
Mecánca Eule s Eqs of Moton & D Alembet s Pncple Wth Ω and G chosen to coespond to the pncpal aes of neta, M & M M M G ( ) G Ω G & & G Eule s Equatons: & ( ) ( ) ( ) Sstem of etenal foces and effectve foces ae equvalent fo geneal thee dmensonal moton. Sstem of etenal foces ae equvalent to the vecto and couple, ma and &. G Escuela Técnca Supeo de ngeneos ndustales 5-0
Mecánca Moton About a Fed Pont o a Fed As Escuela Técnca Supeo de ngeneos ndustales Fo a gd bod otaton aound a fed pont, M O O & Ω ( ) O O O Fo a gd bod otaton aound a fed as, ( ) M & O O O O ( j k )& k ( j k ) j k α j M M M ( ) ( ) α α α 5 -
Mecánca Rotaton About a Fed As Fo a gd bod otaton aound a fed as, M α M M α α f smmetcal wth espect to the plane, M 0 M 0 M α f not smmetcal, the sum of etenal moments wll not be eo, even f α 0, M M M A otatng shaft eques both statc 0 and dnamc ( 0balancng ) to avod ecessve vbaton and beang eactons. ( ) 0 Escuela Técnca Supeo de ngeneos ndustales 5 -
Mecánca Sample Poblem 8.3 SOLUTON: Evaluate the sstem of effectve foces b educng them to a vecto ma attached at G and couple G. Rod AB wth weght W 40 lb s pnned at A to a vetcal ale whch otates wth constant angula veloct 5 ad/s. The od poston s mantaned b a hoontal we BC. Detemne the tenson n the we and the eacton at A. Escuela Técnca Supeo de ngeneos ndustales Epessng that the sstem of etenal foces s equvalent to the sstem of effectve foces, wte vecto epessons fo the sum of moments about A and the summaton of foces. Solve fo the we tenson and the eactons at A. 5-3
Mecánca Sample Poblem 8.3 SOLUTON: Escuela Técnca Supeo de ngeneos ndustales Evaluate the sstem of effectve foces b educng them to a vecto ma attached at G and couple G. a an ( Lcos β ) ( 450ft s ) 40 ma ( 450) ( 559lb) g j k G G & G ml cos β sn β 0 ml cos β ( & G ) G G 0 cos β sn β j ml cos β ml sn β cos β k ( 645lb ft)k 0 ml ( ) ( ) 5-4
Mecánca Sample Poblem 8.3 Epessng that the sstem of etenal foces s equvalent to the sstem of effectve foces, wte vecto epessons fo the sum of moments about A and the summaton of foces. M 6.93J ( ) eff ( T ) ( 40J ) 3.46J ( 559 ) A M A ( 6.93T 80) K ( 934 645)K 645K F F A A X ( ) eff Y J A Z K T 384lb 384 40J 559 A 75 ( lb) ( 40 lb)j Escuela Técnca Supeo de ngeneos ndustales 5-5
Mecánca Moton of a Goscope. Eulean Angles A goscope conssts of a oto wth ts mass cente fed n space but whch can spn feel about ts geometc as and assume an oentaton. Fom a efeence poston wth gmbals and a efeence damete of the oto algned, the goscope ma be bought to an oentaton though a successon of thee steps: ) otaton of oute gmbal though ϕ about ) AA, otaton of nne gmbal though θ about 3) BB, otaton of the oto though ψ about CC. ϕ, θ, and ψ ae called the Eulean Angles and & φ ate of pecesson & θ ate of nutaton Ψ& ate of spn Escuela Técnca Supeo de ngeneos ndustales 5-6
Mecánca Moton of a Goscope. Eulean Angles The angula veloct of the goscope, & φk & θ j Ψ& k wth K snθ cosθ j & φ snθ & θ j ( Ψ& & φ cosθ )k Equaton of moton, M & M M M O ( ) O Ω O O & φ snθ & θ j cos Ω & φk & θ j O ( Ψ& & φ θ ) k (&& φ snθ && θφ cosθ ) & θ ( Ψ& & φ cosθ ) (&& ) θ & φ snθ cosθ & φ snθ ( Ψ& & φ cosθ ) d dt ( Ψ& & φ cosθ ) Escuela Técnca Supeo de ngeneos ndustales 5-7
Mecánca Stead Pecesson of a Goscope Stead pecesson, θ, & φ, ψ& ae constant & φ snθ k O & φ snθ k Ω & φ snθ & φ cosθ k M Ω Escuela Técnca Supeo de ngeneos ndustales O ( & φ cosθ )& φ snθ j Couple s appled about an as pependcula to the pecesson and spn aes O When the pecesson and spn as ae at a ght angle, θ 90 Ψφ && j M O Goscope wll pecess about an as pependcula to both the spn as and couple as. 5-8
Mecánca Moton of an Asmmetcal Bod Unde No Foce Consde moton about ts mass cente of an asmmetcal bod unde no foce but ts own weght, e.g., pojectles, satelltes, and space caft. & G 0 constant G Defne the Z as to be algned wth Gand n a otatng aes sstem along the as of smmet. The as s chosen to le n the Z plane. G snθ G snθ 0 0 G cosθ G cosθ θ constant and bod s n stead pecesson. Note: tanγ tanθ Escuela Técnca Supeo de ngeneos ndustales 5-9
Mecánca Moton of an Asmmetcal Bod Unde No Foce Two cases of moton of an asmmetcal bod whch unde no foce whch nvolve no pecesson: Bod set to spn about ts as of smmet, 0 and ae algned G and bod keeps spnnng about ts as of smmet. Bod s set to spn about ts tansvese as, 0 and ae algned G and bod keeps spnnng about the gven tansvese as. Escuela Técnca Supeo de ngeneos ndustales 5-30
Mecánca Moton of an Asmmetcal Bod Unde No Foce The moton of a bod about a fed pont (o ts mass cente) can be epesented b the moton of a bod cone ollng on a space cone. n the case of stead pecesson the two cones ae ccula. <. Case of an elongated bod. γ < θ and the vecto les nsde the angle ZG. The space cone and bod cone ae tangent etenall; the spn and pecesson ae both counteclockwse fom the postve as. The pecesson s sad to be dect. >. Case of a flattened bod. γ > θ and the vecto les outsde the angle ZG. The space cone s nsde the bod cone; the spn and pecesson have opposte senses. The pecesson s sad to be etogade. Escuela Técnca Supeo de ngeneos ndustales 5-3