Rotary motion

Transcription

1 ectue 8 RTARY TN F THE RGD BDY Notes: ectue 8 - Rgd bod Rgd bod: j const numbe of degees of feedom 6 3 tanslatonal + 3 ota motons m j m j Constants educe numbe of degees of feedom non-fee object: 6-p < 6 j Tanslatonal moton Rota moton Combned moton Ke tems: gd bod, pont mass

2 ectue 8 - Toque P F Fee bod P F Toque about an as leve am:pependcula dstance fom the as of otaton to the lne of acton of the foce poston vecto F F F sn (, F ) F 0 0 P F equlbum conons F 0 0 w et w et w Ke tems: Rght hand ule

3 ectue 8-3 Equlbum of the gd bod stable F > 0 ; 0 E p > 0 Fg E p mn F g F g neutal E p 0 0 F g F g F g unstable 0 F g E p ma F g F g E p < 0 F g

4 ectue 8-4 Rota moton v ω Angula momentum: c p m v p toques Total toque ω ω F F F F dp F dp v p 0 d d dp ( p ) p + dp d d ( p ) Rate of change of the angula momentum equals the total toque due to the etenal foces. law of dnamcs m ( ω ) ω dp v P, m d - 0 -

5 ectue 8-5 d d d d Consevaton of the angula momentum dc Fet 0 const c 0 Thetotalmomentumof thesolatedsstem stas unchaged. oton equatons: d. Tanslaton of the cente of mass: m sm F. Rota moton: d Equlbum conons: F et et 0 0 W.H.Feeman &Co - -

6 ectue 8-6 oment of neta Rotaton about pemanent as. As goes though the cente of mass.. As has pemanent decton. p ; p p m v m ω m ω ω m m ω oment of neta (about as) p ω m dm V V ρdv R R R R R R l R 4 R l l l R 5 R - -

7 ectue 8-7 oment of neta aula theoems oment of neta about cente of mass as - C as `// as C ` C + h Paallel aes theoem h ` tene s theoem oment of neta about as plane - as X` as Y as oment of neta about as : + ` Pependcula aes theoem Plana object - 3 -

8 ectue 8-8 ω?? // ω ω ω ω// + ω ; ω// // // ω// ω // + //ω // nl n case of: object wth otatonal smmet pemanent as of otaton as of smmet and not movng): Angula momentum //ω ω // ω // ω ω ω n case of nonsmmetcal object, the component of the total angula momentum on the pemanent as satsfes: ω ; // ω - 4 -

9 ectue 8-9 oment of neta tenso () n geneal case ω + ω + ω + ω + cente of mass sstem ω + ω + ω ω ω // ω m ` ω m ( ) m ( + ) m ( + ) dagonal elements m.neta. about moment about m ( + ) moment about m m m V V V dm dm dm offdagonal elements j - devaton moments - 5 -

10 ectue 8-0 oment of neta tenso () Î ω ω ω ˆ ω ean moe: Rattleback, Celtc stone Tenso s smmetcal ; ; ne can fnd coodnates sstem n whch: ω,, ˆ 0 0 ω Pncpal moments of the object 0 0 ω stem aes pncpal aes of neta An as of the mamal moment of neta stable as of otaton - 6 -

11 ectue 8 - Rotaton about movng as mmetc spnnng top R mg Pecesson: etenal toque nduces otaton ω θ ω p of angula momentum vecto. ω 0 mg R snθ R ω 0 ` + // ' mg ω p φ angula veloct of pecesson t φ sn φ snθ ω p φ mgr snθ t tsnθ snθ snθ ω mgr p ω p ω psnθ ω p //ω snθ φ snθ + θ nutaton - 7 -

12 ectue 8 - Goscope Goscopes ean moe: netal Navgaton stem (N) stem W.H.Feeman &Co No otaton: goscope falls ove W.H.Feeman &Co d τ Rotaton: As of goscope pma otaton otates due to the etenal toque Goscopc effect - mantanng oentaton of the pma as of otaton, based on the pncple of consevaton of angula momentum. goscopes

13 ectue 8-3 Rota moton - law of dnamcs ean moe: d fo the pemanent as of otaton ω //ω d dω ( ω) ε ε flwheel Flwheel eneg stoage: gobus EK EK m v m ω ω ω mv m ω ω m Combned moton: E ω K EK v C + ω ω ω v P, m Ke tems: Newton s second law fo otaton - 9 -

14 ectue 8-4 E K Fo an abta as of oaton ω ˆ ω ω ω o +,, ω, ω, ω Knetc eneg n the ota moton ( ω + ω ω ) About the pncpal aes ˆ oment of neta tenso + + ean moe: Flwheel eneg stoage (FE) woks b acceleatng a oto (flwheel) to a ve hgh speed and mantanng the eneg n the sstem as otatonal eneg. When eneg s etacted fom the sstem, the flwheel's otatonal speed s educed as a consequence of the pncple of consevaton of eneg; addng eneg to the sstem coespondngl esults n an ncease n the speed of the flwheel. moe at: age About the sngle pncpal as X ω ω 0 E K ω Ke tems: tenso quantt - 0 -

15 ectue 8-5 F d s dθ dw F o ds F cos φds F cosφdθ ds dθ π F cosφ F sn( φ) dw dθ Wok and powe n the ota moton F ( t + ) dθ θ F φ ds (t) P W θ θ dθ Wok n the ota moton of the gd bod dw d dθ P dθ ω P ω Powe - -

16 ectue 8-6 Descpton of the gd bod moton Rectlnea moton d v d a m F ma W Fd mv P Fv p mv EK θ dθ ω d θ ε ε W dθ EK ω P ω ω Rota moton Rotaton about pemanent as of smmet passng though the cente of mass. - -

17 ectue 8-7 lde?? Rollng wthout slde Contact pont wth the suface completes the same path as the cente of mass. Combnaton of the tanslatonal and ota moton Rφ v C Notes: Rollng s a combnaton of otaton and tanslaton of the object wth espect to a suface (ethe one o the othe moves), such that the two ae n contact wth each othe wthout sldng. n ollng sldng fcton s not pesent. ean moe: Rollng fcton No slde ollng conons: ds dφ v C R Rω dvc drω dω a R C E K mvc + Cω v C Rω Rε a C Rε ω 0 v C 0 W.H.Feeman &Co slde: v C ωr v C - 3 -

18 ectue 8-8 Rollng () No slde ollng tanslatonal moton of the C + ota moton about C v v v v v v v v v v v v 0 Rota moton C tanslatonal moton Combnaton - ollng F ε F ma v P 0 v Rollng as pue ota moton ω ω ω nstantenous as of otaton