Physcs : Mechancs Đào Ngọc Hạnh Tâm Offce: A.503, Emal: dnhtam@hcmu.edu.vn HCMIU, Vetnam Natonal Unvesty Acknowledgment: Sldes ae suppoted by Pof. Phan Bao Ngoc
Contents of Physcs Pat A: Dynamcs of Mass Pont Chapte Bases of Knematcs Chapte Foce and Moton (Newton s Laws) Goup Assgnment Pat B: Laws of Consevaton Chapte 3 Wok and Mechancal Enegy Mdtem exam Chapte 4 Lnea Momentum and Collsons Pat C: Dynamcs and Statcs of Rgd Body Chapte 5 Rotaton of a Rgd Body About a Fxed Axs Goup Assgnment Chapte 6 Equlbum and Elastcty Chapte 7 Gavtaton Fnal exam
Pat C: Dynamcs and Statcs of Rgd Body Chapte 5: Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Vaables 5.. Rotaton wth Constant Angula Acceleaton 5.3. Knetc Enegy of Rotaton, Rotatonal Ineta 5.4. Toque, and Newton s Second Law fo Rotaton 5.5. Wok and Rotatonal Knetc Enegy 5.6. Rollng Moton of a Rgd Body 5.7. Angula Momentum of a Rotatng Rgd Body 5.8. Consevaton of Angula Momentum
Ovevew We have studed the moton of tanslaton, n whch objects move along a staght o cuved lne. In ths chapte, we wll examne the moton of otaton, n whch objects tun about an axs.
5.. Rotatonal vaables: We study the otaton of a gd body about a fxed axs. Rgd bodes: Bodes can otate wth all ts pats locked togethe and wthout any change n ts shape. Fxed axs: A fxed axs means the otatonal axs does not move. Angula Poston: Refeence lne: To detemne the angula poston, we must defne a efeence lne, whch s fxed n the body, pependcula to the otaton axs and otatng wth the body. The angula poston of ths lne s the angle of the lne elatve to a fxed decton. s : adans (ad) whee s s the ac length of a ccula path of adus and angle. ev = 360 0 = ad
Angula Dsplacement Conventon: > 0 n the counteclockwse decton. < 0 n the clockwse decton. Angula Velocty Aveage angula velocty: avg Instantaneous angula velocty: Angula Acceleaton Aveage angula acceleaton: t t Instantaneous angula acceleaton: lm t 0 avg t t Unt: ad/s o ev/s o pm; (ev: evoluton) Unt: ad/s o ev/s d dt t t t d lm t 0 t dt Note: Angula dsplacement, velocty, and acceleaton can be teated as vectos (see page 46).
5.. Rotaton wth Constant Angula Acceleaton Fo one dmensonal moton: dt dv a dt dx v ; Let s change vaable names: a v x,, ) ( 0 0 0 0 0 t t t Checkpont (p. 48): In fou stuatons, a otatng body has angula poston (t) gven by (a) =3t-4, (b) =-5t 3 +4t +6, (c) =/t -4/t, and (d) =5t -3. To whch stuatons do the angula equatons above apply? ) ( 0 0 0 0 0 x x a at t x x at Ans: a) and d)
5.3. Knetc Enegy of Rotaton a. Lnea and Angula Vaable Relatonshp The poston: s whee angle measued n ad; s: dstance along a ccula ac; : adus of the ccle The speed: ds dt d dt v The peod of evoluton: The Acceleaton: dv dt Tangental acceleaton: (ad) T d v dt a t Radal acceleaton: v a
b. Knetc Enegy of Rotaton: The KE of a otatng gd body s calculated by addng up the knetc eneges of all the patcles: K K m v I m Unt fo I: kg m m m v ( ) m v 3 3... m m v Rotaton Ineta (o moment of neta) K I (adan) K (J)
c. Calculatng the Rotatonal Ineta: If the gd body conssts of a few patcles: I m Fo contnuous bodes: I dm Paallel-Axs Theoem: calculate I of a body of mass M about a gven axs f we aleady know I com : I O I com Mh h: the pependcula dstance between the gven axs at 0 and the axs though the COM of the body.
Some Rotatonal Inetas Table 0-
Sample poblem: In 985,Test Devces, Inc. (www.testdevces.com) was spn testng a sample of a sold steel oto (a dsk) of mass M= 7 kg and adus R = 38.0 cm. When the sample eached an angula speed of 4000 ev/mn, the test engnees head a dull thump fom the test system, whch was located one floo down and one oom ove fom them. Investgatng, they found that lead bcks had been thown out n the hallway leadng to the test oom, a doo to the oom had been huled nto the adjacent pakng lot, one lead bck had shot fom the test ste though the wall of a ktchen, the stuctual beams of the test buldng had been damaged, the concete floo beneath the spn chambe had been shoved downwad by about 0.5 cm, and the 900 kg ld had been blown upwad though the celng and had then cashed back onto the test equpment. The explodng peces had not penetated the oom of the test engnees only by luck. How much enegy was eleased n the exploson of the oto?
Enegy was eleased n the exploson of the oto K I () Rotatonal Ineta of the dsk I MR (7kg)(0.38m) 9.64kg. m Angula speed =4000 ev/mn = 4000. /60=.466 0 3 ad/s () K I (9.64kg. m =. 0 7 J )(.466.0 3 ad / s)
5.4. Toque, and Newton s Second Law fo Rotaton a. Toque ( to twst ): F A foce appled at pont P of a body that s fee to otate about an axs though O. The foce has no component paallel to the otaton axs. F F f Rotaton axs
The ablty of Resolve F F to otate the body s defned by toque F (Unt: N.m) nto components: F t (tangental) and F (adal). ( F sn) F t ( sn) F : the moment F am of F
Decton of toque: Use the ght hand ule to detemne F f F
Impotant Note: Toque s a vecto quantty, howeve, because we consde only otaton aound a sngle axs, we theefoe do not need vecto notaton. Instead, a toque s a postve value f t would poduce a counteclockwse otaton and a negatve value fo a clockwse otaton. counteclockwse f f t net =t -t = F snf -F snf clockwse
b. Newton s Second Law fo Rotaton We consde a smple poblem, the otaton of a gd body consstng of a patcle of mass m on one end of a massless od. Ft ma t The toque actng on the patcle: F ma m( ) ( m ) t (a t : tangental acceleaton) t I (adan measue) : angula acceleaton f moe than one foce appled to the patcle: I net Ths s vald fo any gd body otatng about a fxed axs.
53/70. The fgue below shows a unfom dsk that can otate aound ts cente lke a mey-go-ound. The dsk has a adus of.0 cm and a mass of 0.0 gams and s ntally at est. Statng at tme t=0, two foces ae to be appled tangentally to the m as ndcated, so that at tme t=.5 s the dsk has an angula velocty of 50 ad/s counteclockwse. Foce F has a magntude of 0. N. What s magntude F? F net I net F R F R I R F MR Fo a unfom dsk: Fo otaton: 0 I t I t t 3 MR (0.00 )(.00 )(50) F F 0. t.5 + 0.4 (N)
5.5. Wok and Rotatonal Knetc Enegy We consde a toque of a foce F acceleates a gd body n otaton about a fxed axs: The wok-knetc theoem appled fo otaton of a gd body: K K f K I f I W The wok done by the toque: If s a constant: W K ) ( f f d The powe: P dw dt
Poblem 6 (p. 7): A 3.0 kg wheel, essentally a thn hoop wth adus. m, s otatng at 80 ev/mn. It must be bought to a stop n 5.0 s. (a) How much wok must be done to stop t? (b) What s the equed aveage powe? The otatonal neta of a wheel (a thn hoop) about cental axs: I MR To stop the wheel, =0: 80 ev/mn 3.0. 80 (ad) 60 (s) 46.(kg m 0 (a) The wok s needed to stop the wheel: W K f K 0 I 46. 9.3 ) 9.3 (ad/s) 9788 (J) o 0 (W<0: enegy tansfeed fom the wheel) (b) The aveage powe: P W t 9788 5 39 (W) o 9.8 (kj).3 ( kw)
5.6. Rollng Moton of a Rgd Body We consde a gd body smoothly ollng along a suface. Its moton conssts of two motons: tanslaton of the cente of mass and otaton of the est of the body aound that cente. Example: A bke wheel s ollng along a steet. Dung a tme nteval t, both O (the cente of mass) and P (the contact pont between the wheel and the steet) move by a dstance s: s R whee R s the adus of the wheel.
The speed of the cente of the wheel: v com R The lnea acceleaton of the cente of the wheel: a com R The Knetc Enegy of Rollng K I com Mv com Rotatonal KE Tanslatonal KE
Examples:. Rollng on a hozontal suface: A.0 kg wheel, ollng smoothly on a hozontal suface, has a otatonal neta about ts axs I = MR /, whee M s ts mass and R s ts adus. A hozontal foce s appled to the axle so that the cente of mass has an acceleaton of 4.0 m/s. What s the magntude of the fctonal foce of the suface actng on the wheel? Applyng Newton s second law fo otatonal moton: a com, x Rf Icom () net 0 (n the 0 (clockwse) a (), (): f s I com R s postvedecton of com MR, R () x a com, x R Ma (fo ) + f s the x axs) com, x 4.0 F appled + (fo a com,x ) - 4.0 (N)
Examples:. Rollng Down a Ramp: We consde a gd body smoothly ollng down a amp. Foce analyss: F g, F N, and f s (opposng the sldng of the body, so the foce s up the amp) Applyng Newton s second law fo tanslatonal moton: f Mg sn,x s Ma com () Applyng Newton s second law fo otatonal moton: Rf Icom () a com, x net 0 (n the 0 (counteclockwse) a (), (), (3): com s negatve a decton of, R (3) x com, x the g sn Icom MR x axs)
Examples: 3. The Yo-Yo: Fo tanslaton moton: T Mg Ma com, y R0T Icom () () Fo otatonal moton: a com, y R 0 (3) y a com, y g I MR com 0
Examples: 4. Pulley: A 0.0 kg block hangs fom a cod whch s wapped aound the m of a fctonless pulley. Calculate the acceleaton, a, of the block as t moves down? (The otatonal neta of the pulley s 0.50 kg m and ts adus s 0.0 m). Fo tanslaton moton: mg Fo otatonal moton: T ma () + a R T m mg I pulley R I 0 pulley a R 0 (3) () (Note: a s the acceleaton of the block, s the angula acceleaton of the pulley) 0 9.8.63 (m/s ) 0.5 0 0. T T a mg
Example: (The Knetc Enegy of Rollng) A 0 kg cylnde olls wthout slppng. When ts tanslatonal speed s 0 m/s, What s ts tanslatonal knetc enegy, ts otatonal knetc enegy, and ts total knetc enegy? The otatonal neta of a cylnde about cental axs: Tanslatonal knetc enegy: K k I MR 0 0 Mv Rotatonal knetc enegy: K I The Knetc Enegy of Rollng MR v R 500 (J) 4 Mv Total knetc enegy: K K 50 (J) K k K 750 (J)
5.7. Angula Momentum of a Rotatng Rgd Body a. Angula Momentum of a Patcle: l p m( v) : p : the lnea momentum of the patcle the poston vecto of the patcle wth espect to O The decton of l detemned by the ght-hand ule l
l p m( v) l mvsn (Unt: kg m s - ) l o p p l
Newton s Second Law n Angula Fom fo a Patcle: Fo tanslatonal motons: dp F net dt dp Fnet dt dv net m( dt d( v) m dt So, Newton s Second Law n Angula Fom: net dl dt d dt dl dt v) Note: The toques and the angula momentum must be defned wth espect to the same ogn O. l
b. Angula Momentum of a System of Patcles: L n l l l3... l n l Newton s Second Law n Angula Fom: net dl dt c. Angula Momentum of a Rotatng Rgd Body: Method: To calculate the angula momentum of a body otatng about a fxed axs (hee the z axs), we evaluate the angula momentum of a system of patcles (mass elements) that fom the body. l p 0 sn 90 m v
l p 0 sn 90 m v l l L s the angula momentum of element of mass m : l sn ( sn )( m v, z ) n n n z l z mv m, ( ) m v l z l L z n m L I z z We dop the subscpt z: I z : otatonal neta of the body about the z axs L I Note: L and I ae the angula momentum and the otatonal neta of a body otatng about the same axs.
Example: At the nstant of the fgue below, a.0 kg patcle P has a poston vecto of magntude 5.0 m and angle =45 0 and a velocty vecto of magntude 4.0 m/s and angle =30 0. Foce of magntude.0 N and angle 3 =30 0, acts on P. All thee vectos le n the xy plane. About the ogn, what ae the (a) magntude and (b) decton of the angula momentum of P and the (c) magntude and (d) decton of the toque actng on P? (a) l v l mvsn p 5.0.0 4.0sn(30) 0 (kg m l / s) (b) Usng the ght-hand ule, ponts out of the page and t s pependcula to the fgue plane. (c) (d) F F sn 3 5.0.0sn(30) 5 (N m) F l ponts out of the page and t s pependcula to the fgue plane.
5.8. Consevaton of Angula Momentum If no net extenal toque acts on the system: L constant L L f net dl dt t net = 0 I I f f
Example: You stand on a fctonless platfom that s otatng at an angula speed of.5 ev/s. You ams ae outstetched, and you hold a heavy weght n each hand. The moment of neta of you, the extended weghts, and the platfom s 6.0 kg.m. When you pull the weghts n towad you body, the moment of neta deceases to.8 kg.m. (a) What s the esultng angula speed of the platfom? (b) What s the change n knetc enegy of the system? (c) Whee dd ths ncease n enegy come fom? (a) No net toque actng on the otatng system (platfom + you + weghts): I 6.0.5 constant I I I 5 (ev/s) I.8 (b) The change n knetc enegy: K.8 5 6.0.5 6 (J) I I (c) Because no extenal agent does wok on the system, so the ncease n knetc enegy comes fom you ntenal enegy (bochemcal enegy n you muscles).
Moe Coespondng Vaables and Relatons fo Tanslatonal and Rotatonal
Homewok:, 3, 7, 4, 0, 6, 39, 43, 53, 56, 6 (p. 67-7), 5, 9, 7, 35, 38, 4, 43, 54, 60 (p. 97-30)
Summay: Constant Lnea vaables Toque: s v F Newton s second law fo otaton: Fo a pont mass: t 0 0 t 0 a t I 0 v a m t (kg.m ) Rotatonal KE: K I net I Fo a gd body: I depends on the object shape (table 0-) ( ) 0 Angula poston: (ad) Angula dsplacement: (ad) Angula velocty: (ad/s) Angula acceleaton: (ad/s ) ev = 360 0 = ad I: Rotaton Ineta (o moment of neta) Wok KE Theoem: K K f K I f I W
Rollng moton: v com R a com R K I com Mv com Angula Momentum: l p Rotatonal KE Tanslatonal KE l mv sn Newton s Second Law n Angula Fom: Angula Momentum of a Rgd Body: L I net dl dt Consevaton of Angula Momentum: If no net extenal toque acts on the system: L constant L L f I I f f