Dynamics of Rigid Bodies

Transcription

1 Dynamcs of Rgd Bodes A gd body s one n whch the dstances between consttuent patcles s constant thoughout the moton of the body,.e. t keeps ts shape. Thee ae two knds of gd body moton: 1. Tanslatonal Rectlnea foces actng. Patcles moe on staghtlne paths.. Rotatonal Rotatonal foces,.e. toques, actng. Patcles moe on ccula paths Knematcs of Rgd Bodes s Rθ dθ et α be a constant. Then, d θ α α d d θ ntegatng, Angle θ s n adans dθ αt + C et 0 at t 0, ntegatng, et θ 0 at t 0, dθ αt + 0 θ ½αt + 0t + c θ ½αt + 0t COMPARE THESE WTH THE EQUATONS FOR NEAR MOTON Note: θ s n adans

2 Angula Momentum Angula momentum s the otatonal equalent of lnea momentum. t s a conseed quantty A gd body can be thought y of as a lage numbe of patcles, at postons A, wth masses m, at poston ectos fom an ogn on the axs of, at dstances R fom A the axs, and wth eloctes. x z The patcle at the pont A has lnea momentum p m and Angula Momentum about the ogn p m m nˆ whee n ˆ s unt ecto nomal to both and p. Note that s not geneally paallel to the axs of otaton. Moment of neta The component of the angula momentum n the y-decton,.e. along th axs of otaton, s m sn θ Howee, sn θ R, so y m R The magntude of s the same fo all ponts, so we dop the ndex, and R. Then Summng o ntegatng oe all ponts, Ths sum, m R y + y + 1y y K m R y m R s called the Moment of neta, and (c.f. p m)

3 Moment of neta of Potato-Shapes The Moment of neta depends on the axs of otaton. The Angula Momentum s geneally not paallel to the axs of otaton. Fo a body of geneal shape (an asteod, a potato...) thee ae thee mutually pependcula axes fo whch the angula momentum s paallel to the axs. These ae called the Pncpal Axes of nteta and the moments of neta about them ae the Pncpal Moments of neta. Fo bodes of hghe symmety than potatoes, the Pncpal Axes ae geneally Axes of Symmety. Angula Momentum Examples: Two masses m ae gong ound the z-axs, at adus R, n the x-y plane, wth speed R. m fo each mass R R mr fo the pa Decton of s at ght angles to and,.e. n same decton as,.e. mr and mr Now nclne the masses to the axs. m fo each mass R Rsn ϕ mr sn ϕ fo the pa Decton of s at ght angles to and,.e. at ϕ to,.e. t pecesses about z. z mr sn ϕ, and mr sn ϕ about the z-axs.

4 Calculaton of Moments of neta m R R s dstance to axs of otaton f an object s consdeed to consst of elemental patcles of mass dm, then dm ρdv and the sum becomes an ntegal oe the olume. f the densty ρ s constant, t comes out of the ntegal and R dm ρ V V R dv Note that V R dv s a puely geometcal facto Example: Moment of neta of a thn od otated about one end. Coss-sectonal aea S, length, densty ρ. Element dv s dsc of aea S, dstasnce fom axs x, thckness dx. So we hae (. e. dv Sdx, dm ρsdx, d ρsx dx) x x 0 ρs 1 dx ρs Example: Moment of neta of the same thn od otated about ts cente. / 1 1 ρs x dx S ρ M x / M n geneal, Mk. The length k s a chaactestc length of the object, called the Radus of Gyaton compae wth Cente of Gaty.

5 Paallel Axs Theoem Consde an object otatng aound ts Cente of Mass, wth the z-axs as the axs of otaton. We know the moment of neta CM m R m ( x + y ). What would be the moment of neta about the axs P, paallel to the z-axs but offset at x a, y b? P m m [( x a) + ( y b) ] [( x + y ) ax by + ( a + b )] Recall that the defnton of the Cente of Mass ges x CM m x we hae placed the Cente of mass at the ogn, ou tem n x anshes. Smlaly fo y. So, P ( x + y ) + ( a + b ) m m CM + Md m. Snce Compae ths esult wth the thn od n the eale example. end CM + Md, and d ½. CM M M M end 4 4 whch s what we found by dect calculaton. M 1

6 Toque To cause a body to otate we need to apply foces whch do not pass though the axs of otaton. The effect of such a foce depends on ts magntude and decton and also on how fa fom the axs t s. n ecto tems, ths s called the Toque. ts scala magnstude s often called the Moment. t s gen by τ F τ RF Toque s the otatonal analogue of foce n lnea mechancs. As foce s ate of change of lnea momentum, so toque s ate of change of angula momentum, d τ d d d θ α τ α s to be compaed wth F ma Example: The Yo-Yo T Vetcal acceleaton a Angula acceleaton α τ α ½ MR * T ½ MRα Mg * How would you calculate ths? Note that a αr and that Mg T Ma MRα So Mg ½MRα MRα Mg ½Ma Ma g a a g 6.5 m s 1 Note that a < g and s ndependent of R and M.

7 Gyoscopes and Pecesson z y x F Consde a gyoscope whch s not spnnng. t s eleased n the poston shown. ts cente of mass s at. ntally s zeo Then d τ d τ The toque causes the gyoscope to stat otatng about the pot, ganng angula elocty n the y-decton. Now let the gyoscope be spnnng ntally. ts angula momentum s not zeo but les along the axs of otaton, along the x-decton. Now, afte a tme, + τ whch s stll n the x-y plane, whch has the same magntude, but has changed decton by the amount τ dϕ dϕ τ mg Ω Ths ges us an expesson fo the well-known otaton n the hozontal plane of a gyoscope, ts PRECESSON. Fo a smple dsc, the moment of neta ½ mr and g Ω ½R Astonomcal Example: The Eath s spnnng and so acts as a gyoscope. ts axs of otaton makes an angle of º7 to the nomal to the plane of ts obt. Thee s a toque due to (tdal) gatatonal effects of the Sun and Moon. As a consequence, the decton of the Eath s axs pecesses (the Pecesson of the Equnoxes) wth a peod of 775 yeas. Consequently, whle sambad Bunel bult the Box Tunnel (between Bath and Bstol on the Geat Westen Ralway) so that the sun would shne though t at sunse on hs bthday, that conon wll not be met fo ey long.

8 Knetc Enegy of Rotaton. Fo a body consdeed as a lage numbe of pont masses, E K ½ ½ ½ m R m ½ m R Compae E ½m n the geneal case of lnea tanslaton moton togethe wth otaton about the cente of mass, E K ½m + ½ Example: A ng olls* down an nclned plane, heght h 0, angle ϕ * Defnton of Rollng: Fcton peents slppage. No wok s done aganst fcton as the bottom of the ng touchng the amp s nstantaneously at est. f the ng stats at heght h 0 at zeo lnea and angula elocty, and olls down to a heght h, then at h, ½ E K ½m + ½ mg( h0 Now, fo a ng of mass m, adus R s mr, and R, so E ½m g( h 0 + ½mR R mg( h 0 Compae fctonless sldng: E K ½m g( h 0 mg( h 0 The ntal Potental Enegy s used to supply both the otatonal and tanslatonal knetc enegy, so the ollng moton slows down the tanlatonal.

9 Compason of Dynamcal Quanttes n Tanslaton and Rotaton. Tanslatonal Rotatonal Poston s Angle θ ds Angula dθ Velocty elocty d s Angula d θ Acceleaton a α acceleaton Moment of Mass m neta Momentum p m Angula Momentum p Knetc Enegy E ½m..... E ½ m dp d Foce F Toque τ F Powe P F P τ. Second aw F ma..... α τ