We are going to consider the motion of a rigid body about a fixed axis of rotation. x + d dt t

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1 R-1 Rotatonal Moton We ae gong to consde the moton of a gd body about a fxed axs of otaton. The angle of otaton s measued n adans: (ads) s (dmensonless) s s Notce that fo a gen angle, the ato s/ s ndependent of the sze of the ccle. Example: How many adans n 180 o? Ccumfeence C = s = ads ads = 180 o, 1 ad = 57.3 o s = Angle of a gd object s measued elate to some efeence oentaton, just lke 1D poston x s measued elate to some efeence poston (the ogn). Angle s the "otatonal poston". Lke poston x n 1D, otatonal poston has a sgn conenton. Poste angles ae CCW (counteclockwse). x x + x 0 Defnton of angula elocty: ad unts = s d, = (ad/s) (lke dt t dx x, ) dt t In 1D, elocty has a sgn (+ o ) dependng on decton. Lkewse, fo fxed-axs otaton, has a sgn conenton, dependng on the sense of otaton. ( ) : (+) (+) ( ) 3/17/010 Unesty of Coloado at Boulde

2 R- Moe geneally, when axs not fxed, we defne ecto angula elocty wth decton = the decton of the axs + "ght hand ule". Cul fnges of ght hand aound otaton, thumb ponts n decton of ecto. Fo otatonal moton, thee s a elaton between tangental elocty (elocty along the m) and angula elocty. s s =, s = = = t t s n tme t d Defnton of angula acceleaton :, dt t d ( lke a, a ) Unts: dt t ad = s (ad/s ) = ate at whch s changng. = constant = 0 speed along m = constant = Equatons fo constant : dx d Recall fom Chapte : We defned =, a =, dt dt = 0 a t 1 and then showed that, f a = constant, x x0 0t a t 0 a ( x x 0) d d Now, n Chapte 9, we defne =, =. dt dt = 0 t 1 So, f = constant, 0 0t t 0 ( 0) Same equatons, just dffeent symbols. Example: Fast spnnng wheel wth 0 = 50 ad/s ( 0 = f f 8 e/s ). Apply bake and wheel slows at = 10 ad/s. How many eolutons befoe the wheel stops? 3/17/010 Unesty of Coloado at Boulde

3 R-3 Use 0, fnal = 0 1 e 15 ad e ad ad ( 10) Defnton of tangental acceleaton a tan = ate at whch speed along m s changng d d ( ) d a tan = a tan = dt dt dt a tan s dffeent than the adal o centpetal acceleaton a s due to change n decton of elocty a tan s due to change n magntude of elocty, speed a a a a tan a tan and a ae the tangental and adal components of the acceleaton ecto a. a a a a tan Angula elocty also sometmes called angula fequency. Dffeence between angula elocty and fequency f: # adans, sec f # eolutons sec T = peod = tme fo one complete eoluton (o cycle o e) ad, T T f 1 e 1 T T f Unts of fequency f = e/s = hetz (Hz). Unts of angula elocty = ad /s = s -1 Example: An old nyl ecod dsk wth adus = 6 n = 15. cm s spnnng at 33.3 pm (eolutons pe mnute). What s the peod T? e e 60s ( 60/ 33. 3) s 1mn 60 s e 1e s/e peod T = 1.80 s 3/17/010 Unesty of Coloado at Boulde

4 R-4 What s the fequency f? f = 1 / T = 1 e / (1.80 s) = Hz What s the angula elocty? 1 f ( s ) ad / s What s the speed of a bug hangng on to the m of the dsk? = = (15. cm)(3.49 s -1 ) = 53.0 cm/s What s the angula acceleaton of the bug? = 0, snce = constant What s the magntude of the acceleaton of the bug? The acceleaton has only a adal component a, snce the tangental acceleaton a tan = = 0. a = a ( m/s) m/s (about 0. g's) m Fo eey quantty n lnea (1D tanslatonal) moton, thee s coespondng quantty n otatonal moton: Tanslaton x dx dt d a dt Rotaton d = dt d = dt F (?) M (?) F = Ma (?) = (?) KE = (1/) m KE = (1/) (?) The otatonal analogue of foce s toque. Foce F causes acceleaton a F = F sn Toque causes angula acceleaton The toque (ponounced "tok") s a knd of axs F F "otatonal foce". F magntude of toque: F Fsn F m N 3/17/010 Unesty of Coloado at Boulde

5 R-5 = "lee am" = dstance fom axs to pont of applcaton of foce F = component of foce pependcula to lee am Example: Wheel on a fxed axs: Notce that only the pependcula component of the foce F wll otate the wheel. The component of the foce paallel to the lee am (F ) has no effect on the otaton of the wheel. If you want to easly otate an object about an axs, you want a lage lee am and a lage pependcula foce F : ( = 0) Example: Pull on a doo handle a dstance = 0.8 m fom the hnge wth a foce of magntude F = 0 N at an angle = 30 o fom the plane of the doo, lke so: hnge axs no good! bad bette best F F no good! (F = 0) = F = F sn = (0.8 m)(0 N)(sn 30 o ) = 8.0 mn Anothe example: a Pulley = F F Fo fxed axs, toque has a sgn (+ o ) : Poste toque causes counte-clockwse CCW otaton. Negate toque causes clockwse (CW) otaton. + If seeal toques ae appled, the net toque causes angula acceleaton: net Asde: Toque, lke foce, s a ecto quantty. Toque has a decton. Defnton of ecto toque : F = coss poduct of and F: " coss F" Vecto Math ntelude: The coss-poduct of two ectos s a thd ecto AB C defned lke ths: The magntude of AB s A B sn. The decton of AB s the decton pependcula to the plane defned by the ectos A and B plus ght-hand-ule. (Cul fnges fom fst ecto A to second ecto B, thumb ponts n decton of A B 3/17/010 Unesty of Coloado at Boulde

6 R-6 AB A B To see the elaton between toque and angula acceleaton, consde a mass m at the end of lght od of length, potng on an axs lke so: axs m Apply a foce F to the mass, keepng the foce pependcula to the lee am. axs F F acceleaton a tan = Apply F net = m a, along the tangental decton: F = m a tan = m Multply both sdes by ( to get toque n the game ): F = (m ) Defne "moment of neta" = I = m = I ( lke F = m a ) Can genealze defnton of I: Defnton of moment of neta of an extended object about an axs of otaton: I m m m axs m Examples: small masses on ods of length : m axs m 3/17/010 Unesty of Coloado at Boulde

7 R-7 I = m m A hoop of total mass cente, has I hoop = M R M, adus R, wth axs though the R I m m R M R (snce = R fo all ) In detal: I m m m m R m R m R ( m m m ) R MR 1 3 A sold dsk of mass M, adus R, wth axs though the cente: I dsk = (1/) MR (need to do ntegal to poe ths) See Appendx fo I s of aous shapes. Moment of neta I s "otatonal mass". Bg I had to get otatng ( lke Bg M had to get mong ) If I s bg, need a bg toque to poduce angula acceleaton accodng to net = I ( lke F net = m a ) R mass M Example: Apply a foce F to a pulley consstng of sold dsk of adus R, mass M. =? R F R F I 1 MR F MR 3/17/010 Unesty of Coloado at Boulde

8 R-8 Paallel Axs Theoem Relates I cm (axs though cente-of-mass) to I w..t. some othe axs: I = I cm + M d (See poof n text.) Example: Rod of length L, mass M 1 ICM M R, d = L/ 1 d od mass M length L Iend axs ICM M d M L M L M L axs hee ( I ) axs hee ( I cm ) Rotatonal Knetc Enegy How much KE n a otatng object? Answe: KE 1 ot I ( lke KE 1 tans m ) Poof: KE ( m ) 1 tot, KE ( m ) m I axs m How much KE n a ollng wheel? The fomula = s tue fo a wheel spnnng about a fxed axs, whee s speed of ponts on m. A smla fomulas CM = woks fo a wheel ollng on the gound. Two ey dffeent stuatons, dffeent s: = speed of m s. cm = speed of axs. But = tue fo both. cm = cente-of-mass elocty = axs statonay: = pont touchng gound nstantaneously at est 3/17/010 Unesty of Coloado at Boulde

9 R-9 To see why same fomula woks fo both, look at stuaton fom the bcyclst's pont of ew: axs statonay, gound mong Rollng KE: Rollng wheel s smultaneously tanslatng and otatng: = + KE tot = KE tans + KE ot KE M I ( = V cm ) (See poof n text.) 1 1 tot Conseaton of enegy poblem wth ollng moton: A sphee, a hoop, and a cylnde, each wth mass M and adus R, all stat fom est at the top of an nclned plane and oll down to the bottom. Whch object eaches the bottom fst? M R ( = 0) h f = =? Apply Conseaton of Enegy to detemne fnal. Lagest fnal wll be the wnne. KE PE KE PE f f 0 Mgh M I KEtans KEot Value of moment of neta I depends on the shape of the ollng thng: I dsk = (1/)M R, I hoop = M R, I sphee = ( /5)M R (Computng coeffcent eques ntegal.) Let's consde a dsk, wth I = (1/)MR. Fo the dsk, the otatonal KE s 3/17/010 Unesty of Coloado at Boulde

10 R I ( M R ) M [used / ] R M g h M M ( )M M g h, g h 1.16 g h Notce that fnal speed does not depend on M o R. Let's compae to fnal speed of a mass M, sldng down the amp (no ollng, no fcton). h M ( = 0) f = =? 1 M g h M (M's cancel) = g h 1.4 g h Sldng mass goes faste than ollng dsk. Why? As the mass descends, PE s coneted nto KE. Wth a ollng object, KE tot = KE tans + KE ot, so some of the PE s coneted nto KE ot and less enegy s left oe fo KE tans. A smalle KE tans means slowe speed (snce KE tans = (1/) M ). So ollng object goes slowe than sldng object, because wth ollng object some of the enegy gets "ted up" n otaton, and less s aalable fo tanslaton. Compang ollng objects: I hoop > I dsk > I sphee Hoop has bggest KE ot = (1/) I, hoop ends up wth smallest KE tans hoop olls down slowest, sphee olls down fastest. Anothe conseaton of otatonal enegy poblem: Rod of mass M, length L, one end statonay on gound, stats fom est at angle and falls. What s speed of end of stck, when stck hts gound? L I 1 3 M L (axs at end) fxed axs CM h CM =? Plan: Use conseaton of enegy to get, then = = L E E M g h I f CM 1 3/17/010 Unesty of Coloado at Boulde

11 R-11 Impotant pont: PE ga = Mgh whee h = heght of cente-of-mass, ndependent of the oentaton of the stck. PE m g h g m h g M Y M g h Poof: ga CM CM ) ( Hae used defnton of cente-ofmass: MYCM m y same h CM, same PE ga Back to the poblem: M g hcm M g L sn M L 1 I, gsn h Lsn, L I 1 M L 3 L/ h Use / / L to get: 3g Lsn Done. 3gsn L L L The tp of the stck stats at heght h tp = L sn, but ts fnal speed s faste than the speed of an object that falls fom that heght h [ g h 1 m mg h ]. The tp of the stck falls faste than t would n fee-fall, because the cental pat of the od pulls t down. Ths s why tall chmneys always beak apat when toppled: : Let's Reew: Tanslaton Rotaton x dx dt d a dt F M I d = dt d = dt F = Ma = I KE = (1/) m KE = (1/) I 3/17/010 Unesty of Coloado at Boulde

12 R-1 Appendx: Moments of Ineta fo some shapes: Hoop I = M R R Dsk I = (1/) M R Sold sphee I = (/5) M R Thn sphecal shell I = (/3) M R L L Thn od, axs thu cente I = (1/1) M L Thn od, axs thu end I = (1/3) M L 3/17/010 Unesty of Coloado at Boulde