Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis

Transcription

1 Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta 5.4. Torque, and Newton s Second Law for Rotaton 5.5. Work and Rotatonal Knetc Energy 5.6. Rollng Moton of a Rgd Body 5.7. Angular Momentum of a Rotatng Rgd Body 5.8. Conservaton of Angular Momentum

2 Overvew We have studed the moton of translaton, n whch objects move along a straght or curved lne. In ths chapter, we wll examne the moton of rotaton, n whch objects turn about an axs.

3 5.. Rotatonal varables: We study the rotaton of a rgd body about a fxed axs. Rgd bodes: Bodes can rotate wth all ts parts locked together and wthout any change n ts shape. Fxed axs: A fxed axs means the rotatonal axs does not move. Angular Poston: Reference lne: To determne the angular poston, we must defne a reference lne, whch s fxed n the body, perpendcular to the rotaton axs and rotatng wth the body. The angular poston of ths lne s the angle of the lne relatve to a fxed drecton. s r : radans (rad) where s s the arc length of a crcular path of radus r and angle. rev = = rad

4 Angular Dsplacement Conventon: > 0 n the counterclockwse drecton. < 0 n the clockwse drecton. Angular Velocty Average angular velocty: avg Instantaneous angular velocty: Angular Acceleraton Average angular acceleraton: t t Instantaneous angular acceleraton: lm t 0 avg t t Unt: rad/s or rev/s or rpm; (rev: revoluton) Unt: rad/s or rev/s d dt t t t d lm t 0 t dt Note: Angular dsplacement, velocty, and acceleraton can be treated as vectors (see page 46).

5 5.. Rotaton wth Constant Angular Acceleraton For one dmensonal moton: dt dv a dt dx v ; Let s change varable names: a v x,, ) ( t t t Checkpont (p. 48): In four stuatons, a rotatng body has angular 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 angular equatons above apply? ) ( x x a at t x x at

6 5.3. Knetc Energy of Rotaton a. Lnear and Angular Varable Relatonshp The poston: s where angle measured n rad; s: dstance along a crcular arc; r: radus of the crcle The speed: ds dt r d r dt The perod of revoluton: The Acceleraton: v r (rad) dv dt Tangental acceleraton: r T d v r dt a t r Radal acceleraton: v r a r r

7 b. Knetc Energy of Rotaton: The KE of a rotatng rgd body s calculated by addng up the knetc energes of all the partcles: K K m v I m r Unt for I: kg m m m v ( r) m v mr m v Rotaton Inerta (or moment of nerta) K I (radan) K (J)

8 c. Calculatng the Rotatonal Inerta: If the rgd body conssts of a few partcles: I m r For contnuous bodes: I r dm Parallel-Axs Theorem: calculate I of a body of mass M about a gven axs f we already know I com : I O I com Mh h: the perpendcular dstance between the gven axs at 0 and the axs through the COM of the body.

9 Some Rotatonal Inertas Table 0-

10 Sample problem: In 985,Test Devces, Inc. ( was spn testng a sample of a sold steel rotor (a dsk) of mass M= 7 kg and radus R = 38.0 cm. When the sample reached an angular speed of 4000 rev/mn, the test engneers heard a dull thump from the test system, whch was located one floor down and one room over from them. Investgatng, they found that lead brcks had been thrown out n the hallway leadng to the test room, a door to the room had been hurled nto the adjacent parkng lot, one lead brck had shot from the test ste through the wall of a ktchen, the structural beams of the test buldng had been damaged, the concrete floor beneath the spn chamber had been shoved downward by about 0.5 cm, and the 900 kg ld had been blown upward through the celng and had then crashed back onto the test equpment. The explodng peces had not penetrated the room of the test engneers only by luck. How much energy was released n the exploson of the rotor?

11 Energy was released n the exploson of the rotor K I () Rotatonal Inerta of the dsk I MR Angular speed (7kg)(0.38m) 9.64kg. m =4000 rev/mn = /60= rad/s () K I (9.64kg. m =. 0 7 J )( rad / s)

12 Homework:, 3, 7, 4, 0, 6, 39, 43 (p )

13 Part C Dynamcs and Statcs of Rgd Body Chapter: 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta 5.4. Torque, and Newton s Second Law for Rotaton 5.5. Work and Rotatonal Knetc Energy 5.6. Rollng Moton of a Rgd Body 5.7. Angular Momentum of a Rotatng Rgd Body 5.8. Conservaton of Angular Momentum

14 5.4. Torque, and Newton s Second Law for Rotaton a. Torque ( to twst ): A force F appled at pont P of a body that s free to rotate about an axs through O. The force has no component parallel to the rotaton axs. F F f Rotaton axs

15 Resolve nto components: F t (tangental) and F r (radal). The ablty of to rotate the body s defned by torque F F r F (Unt: N.m) r ( F sn ) rft ( r sn ) F r F r : the moment arm of F

16 Drecton of torque: Use the rght hand rule to determne F f F

17 Important Note: Torque s a vector quantty, however, because we consder only rotaton around a sngle axs, we therefore do not need vector notaton. Instead, a torque s a postve value f t would produce a counterclockwse rotaton and a negatve value for a clockwse rotaton. counterclockwse f f t net =t -t = F r snf -F r snf clockwse

18 Checkpont 6 (p. 59): A meter stck can pvot about the dot at the poston marked 0 (cm). All fve forces on the stck are horzontal and have the same magntude. Rank the forces accordng to ther torque magntude, greatest frst F F F F F Lne of acton F-F3, F4, F-F5 (0)

19 b. Newton s Second Law for Rotaton We consder a smple problem, the rotaton of a rgd body consstng of a partcle of mass m on one end of a massless rod. Ft ma t The torque actng on the partcle: F r ma r m( r) r ( mr ) t (a t : tangental acceleraton) t I (radan measure) : angular acceleraton f more than one force appled to the partcle: I net Ths s vald for any rgd body rotatng about a fxed axs.

20 5.5. Work and Rotatonal Knetc Energy We consder a torque of a force F accelerates a rgd body n rotaton about a fxed axs: The work-knetc theorem appled for rotaton of a rgd body: K K f K I f I W The work done by the torque: If s a constant: W K ) ( f f d The power: P dw dt

21 Problem 50 (p. 70): If a 40.0 N.m torque on a wheel causes angular acceleraton 5.0 rad/s, what s the wheel s rotatonal nerta? I 40.0N m; 5.0 rad/s I (kg m )

22 Problem 6 (p. 7): A 3.0 kg wheel, essentally a thn hoop wth radus. m, s rotatng at 80 rev/mn. It must be brought to a stop n 5.0 s. (a) How much work must be done to stop t? (b) What s the requred average power? The rotatonal nerta of a wheel (a thn hoop) about central axs: I MR To stop the wheel, =0: (kg m 80 (rad) 0 80 rev/mn 60 (s) (a) The work s needed to stop the wheel: W K f K 0 ) 9.3 (rad/s) I (J) or 9.8 (kj) (W<0: energy transferred from the wheel) (b) The average power: P W t (W) or.3 ( kw)

23 5.6. Rollng Moton of a Rgd Body We consder a rgd body smoothly rollng along a surface. Its moton conssts of two motons: translaton of the center of mass and rotaton of the rest of the body around that center. Example: A bke wheel s rollng along a street. Durng a tme nterval t, both O (the center of mass) and P (the contact pont between the wheel and the street) move by a dstance s: s R where R s the radus of the wheel.

24 The speed of the center of the wheel: v com R The lnear acceleraton of the center of the wheel: a com R The Knetc Energy of Rollng K I com Mv com Rotatonal KE Translatonal KE

25 Examples:. Rollng on a horzontal surface: A.0 kg wheel, rollng smoothly on a horzontal surface, has a rotatonal nerta about ts axs I = MR /, where M s ts mass and R s ts radus. A horzontal force s appled to the axle so that the center of mass has an acceleraton of 4.0 m/s. What s the magntude of the frctonal force of the surface actng on the wheel? Applyng Newton s second law for rotatonal moton: a com, x Rf Icom () net 0 (n the 0 (clockwse) a (), (): f s I com R s postvedrecton of com MR, x R () a com, x R Ma (for ) + f s the x axs) com, x 4.0 F appled + (for a com,x ) 4.0 (N)

26 Examples:. Rollng Down a Ramp: We consder a rgd body smoothly rollng down a ramp. Force analyss: F g, F N, and f s (opposng the sldng of the body, so the force s up the ramp) Applyng Newton s second law for translatonal moton: f Mg sn,x s Ma com () Applyng Newton s second law for rotatonal moton: Rf Icom () a com, x net 0 (n the 0 (counterclockwse) a (), (), (3): s negatve com drecton of, x R (3) a com, x the x axs) g sn Icom MR

27 Examples: 3. The Yo-Yo: For translaton moton: T Mg R0T Icom () Ma com, y For rotatonal moton: a com, y R 0 (3) () y a com, y g I MR com 0

28 Examples: 4. Pulley: A 0.0 kg block hangs from a cord whch s wrapped around the rm of a frctonless pulley. Calculate the acceleraton, a, of the block as t moves down? (The rotatonal nerta of the pulley s 0.50 kg m and ts radus s 0.0 m). For translaton moton: mg For rotatonal moton: T ma () + a R m mg I pulley R 0T Ipulley a R 0 (3) () (Note: a s the acceleraton of the block, s the angular acceleraton of the pulley) (m/s ) T T a mg

29 Example: (The Knetc Energy of Rollng) A 0 kg cylnder rolls wthout slppng. When ts translatonal speed s 0 m/s, What s ts translatonal knetc energy, ts rotatonal knetc energy, and ts total knetc energy? The rotatonal nerta of a cylnder about central axs: Translatonal knetc energy: K k I MR 0 0 Mv Rotatonal knetc energy: K r I The Knetc Energy of Rollng MR v R 500 (J) 4 Mv Total knetc energy: K K r 50 (J) K k K r 750 (J)

30 Homework: 53, 56, 6 (p. 70-7);, 5, 9, 7, 6, 35, 38, 4 (p )

31 Part C Dynamcs and Statcs of Rgd Body Chapter: 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta 5.4. Torque, and Newton s Second Law for Rotaton 5.5. Work and Rotatonal Knetc Energy 5.6. Rollng Moton of a Rgd Body 5.7. Angular Momentum of a Rotatng Rgd Body 5.8. Conservaton of Angular Momentum

32 5.7. Angular Momentum of a Rotatng Rgd Body a. Angular Momentum of a Partcle: l r p m( r v) r : p : the lnear momentum of the partcle l the poston vector of the partcle wth respect to O The drecton of determned by the rght-hand rule l

33 l rmvsn (Unt: kg m s - ) or l r p rp l

34 Newton s Second Law n Angular Form for a Partcle: For translatonal motons: net m( r r F net F dv dt net net net dp dt dp r dt dv m( r v v) dt dr d( r v) v) m dt dt So, Newton s Second Law n Angular Form: dl dt dl dt Note: The torques and the angular momentum must be defned wth respect to the same orgn O. l

35 b. Angular Momentum of a System of Partcles: L n l l l3... l n l Newton s Second Law n Angular Form: net dl dt c. Angular Momentum of a Rotatng Rgd Body: Method: To calculate the angular momentum of a body rotatng about a fxed axs (here the z axs), we evaluate the angular momentum of a system of partcles (mass elements) that form the body.

36 l l z l s the angular momentum of element of mass m : v m r p r l 0 sn 90 z v m r m v r l l ) )( sn ( sn, n n n z z r r m m v r l L, ) (

37 L z n m r I z : rotatonal nerta of the body about the z axs L I z z We drop the subscrpt z: L I Note: L and I are the angular momentum and the rotatonal nerta of a body rotatng about the same axs.

38 5.8. Conservaton of Angular Momentum If no net external torque acts on the system: L constant L L f net dl dt t net = 0 I I f f

39 Example: You stand on a frctonless platform that s rotatng at an angular speed of.5 rev/s. Your arms are outstretched, and you hold a heavy weght n each hand. The moment of nerta of you, the extended weghts, and the platform s 6.0 kg.m. When you pull the weghts n toward your body, the moment of nerta decreases to.8 kg.m. (a) What s the resultng angular speed of the platform? (b) What s the change n knetc energy of the system? (c) Where dd ths ncrease n energy come from? (a) No net torque actng on the rotatng system (platform + you + weghts): I constant I I I 5 (rev/s) I.8 (b) The change n knetc energy: K (J) I I (c) Because no external agent does work on the system, so the ncrease n knetc energy comes from your nternal energy (bochemcal energy n your muscles).

40 More Correspondng Varables and Relatons for Translatonal and Rotatonal

41 Homework: 43, 54, 60 (p )