s = rθ Chapter 10: Rotation 10.1: What is physics?

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Chape : oaon Angula poson, velocy, acceleaon Consan angula acceleaon Angula and lnea quanes oaonal knec enegy oaonal nea Toque Newon s nd law o oaon Wok and oaonal knec enegy.: Wha s physcs?

We wll Dscuss he quanes n angula vaables (we wll ocus on he angle when objec oang) nd he angula poson, velocy, and acceleaon.

1 Chape : oaon Angula poson, velocy, acceleaon Consan angula acceleaon Angula and lnea quanes oaonal knec enegy oaonal nea Toque Newon s nd law o oaon Wok and oaonal knec enegy.: Wha s physcs? In pevous chapes we have dscussed he anslaonal moon In hs chape we wll dscuss he moon when objec un abou an axs (oaonal Moon) Vaables n oaonal moon ae analogous o hose o anslaonal moon wh ew changes We wll Dscuss he quanes n angula vaables (we wll ocus on he angle when objec oang) nd he angula poson, velocy, and acceleaon. Apply Newon second law bu nsead o oce and mass we wll use oque and oaonal nea. Apply enegy conceps o angula quanes lke wok knec enegy heoem.: oaonal vaables We wll ocus on oaon o a gd body abou a xed axs gd body: body ha can oae wh all s pas packed ogehe whou any change n s shape Fxed axs: oaon abou an axs ha does no move Fgue shows a gd body o abay shape n pue oaon abou he z- axs o a coodnae Sysem evey pon o he body moves n a ccle whose cene les on he axs o oaon (see he abay eeence lne), and evey pon moves hough he same angle dung a pacula me neval.: oaonal vaables: Consde a pacle on gd objec a pon p oaes hough an angle θ As objec oaes, he pon P make an Ac lengh s Angula poson θ s s θ n adans (ad.) o evoluon adans: ao beween he wo lengh s and dmensonless quany When s θ ad (دورة) Full ccle 36 π ad. evoluon whee π /7 3. we can conve beween ad. and degee om he coelaon: Degee ad. 36 π π θ (ad) θ (deg) θ?? 36

2 .: oaonal vaables Angula dsplacemen: Aveage angula speed: θ θ θ θ ωavg (ad/s) s - (ad.) As pacle moves om angula poson θ o θ.: oaonal vaables: Example Insananeous angula speed: dθ ω lm d (ad/s) s - Aveage angula ω ω ω α avg acceleaon: (ad/s²) s - Insananeous ω dω angula α lm (ad/s²) s acceleaon: d - All pacles o a gd objec oae a he same angula dsplacemen, speed and acceleaon. θ ω 7 ev/mn 9 6 cm θ.: oaonal vaables: Example A chld s op s spun wh angula acceleaon A, he op has angula velocy 5 ad/s, and a eeence lne on s a angula poson (θ ) θ ad. Fnd (a) angula velocy ω a he op a any me (b) angula poson θ a any me a) ω αd b) ω ω (5 ) d 5 ω 5 5 ω ωd θ θ 5 ( 5 3 θ θ ) d.3: ae angula quanes vecos? ω Angula velocy ω, s a veco can be wen as Fo oaon abou a xed axs, he decon o he angula velocy s along he axs o oaon. Use he gh hand ule o deemne decon. Also angula acceleaon α s a veco quany can be wen α havng same ules o decon and same ules o speedng up oaon o slowng down oaon Fo oaon couneclockwse (بعكس عقارب الساعة ( +ve ω Fo oaon clockwse (مع عقارب الساعة ( -ve ω

.: oaon wh consan angula acceleaon Fo oaonal moon wh consan oaonal acceleaon α The equaons o moon ae smla n o he equaon o moon n one dmenson (D); Only do he ollowng symbol eplacemen x θ v ω a α.

3 .: oaon wh consan angula acceleaon Fo oaonal moon wh consan oaonal acceleaon α The equaons o moon ae smla n o he equaon o moon n one dmenson (D); Only do he ollowng symbol eplacemen x θ v ω a α.: oaon wh consan angula acceleaon Lnea (D) Moon wh consan lnea acceleaon, a v v + a x ) x x x ( v + ) x + ( v + v x ( v + x x v) x v + a v v + a x oaonal Moon wh consan oaonal acceleaon, α ω ω + α θ θ + + ( ω ω θ θ + ω + α ( ω ω ) ω ω + α ).: oaon wh consan angula acceleaon: Example α.: oaon wh consan angula acceleaon: Example: connued om pevous slde α 3.5ad, s, ω. ad s,?, evoluons?, ω?.s ev. ad.75 ev. π ad. α 3.5ad, s, ω. ad?, evoluons?, ω? s,.s 3

.5: elang The Lnea And Angula Vaables Fo a oang objec, boh lnea and angula quanes ae smulaneously exs hee mus be a elaon beween hem Ac lengh s: Tangenal speed o a pon P: Tangenal acceleaon o a pon P:

4 .: oaon wh consan angula acceleaon: Example Whle you ae opeang a oo cylnde, he angula velocy o he cylnde om 3. ad/s o. ad/s n ev., a consan angula acceleaon. Fnd (a) he angula acceleaon (b) me o decease he angula speed. We have ω 3. ad/s ω ad/s ev. a) π ad. ω + α wh ev.( ) 5.7 ad ev. ω ω α b) ω ω + α.56 ω ω α.3ad/s² 5. α s. 3 ω..5: elang The Lnea And Angula Vaables Fo a oang objec, boh lnea and angula quanes ae smulaneously exs hee mus be a elaon beween hem Ac lengh s: Tangenal speed o a pon P: Tangenal acceleaon o a pon P: Cenpeal acceleaon o oaon objec: Magnude o oal acceleaon s θ v ω a α v a ω a a + a oaonal axs - ou o page oaonal axs - ou o page A ace ca acceleaes consanly om a speed o m/s o 6 m/s n 5 s aound a ccula ack o adus m. When he ca eaches a speed o 5 m/s nd he (a) Angula speed (b) Cenpeal acceleaon, Tangenal acceleaon, and angula acceleaon (d) The magnude o he oal acceleaon. v We have 5s, v m/s ω.ad v 6 m/s ω.5ad a v 5 m/s nd ω?, a c?, a?, a o? v 5 a) v ω ω.5 ad/s b).5: elang The Lnea And Angula Vaables: Example v (5) a c 6.5 m/s o ac ω (.5) 6.5 m/s.5: elang The Lnea And Angula Vaables: Example: connued om pevous slde We have 5s, v m/s ω.ad, v 6 m/s ω.5ad a v 5 m/s a?, a o? c) Fom lnea quanes we can nd he lnea acceleaon a v v 6 v v + a a m/s 5 a a α α.ad./ s O om angula quanes we can nd angula acceleaon α ω ω.5. ω ω + α α.ad 5 a α (.) m c) The oal acceleaon a a c + a m/s

.6: Knec Enegy O oaon a collecon o n pacles oang abou a xed axs has a oaonal knec enegy o: n n K mv m ω v s he lnea speed o pacle K Iω whee I s he momen o nea o oaonal nea: (kg.

5 .6: Knec Enegy O oaon a collecon o n pacles oang abou a xed axs has a oaonal knec enegy o: n n K mv m ω v s he lnea speed o pacle K Iω whee I s he momen o nea o oaonal nea: (kg.m (o collecon o pacles) ) s he dsance om oaonal axs I m (J) Mahemacally, Smla n shape o lnea K wh he ollowng eplacemens I m, ω v.6: Knec Enegy O oaon: Example Sphees o mass m has I because, hey le on y-axs.6: Knec Enegy O oaon: Example: connued om pevous slde.7: Calculang he oaonal nea o collecon o pacles, we had: Fo an exended, gd objec: I I m dm o m mρv dm ρdv mσa dmσda mλl dmλdl I dm ρdv I m I ρ dv 5

nea abou an axs o oaon hough he cene o mass Exended objec ; λdlλdx (L s

7: Calculang he oaonal nea: Paallel axs heoem I I CM s known, he momen

6 .7: Calculang he oaonal nea: Example.7: Calculang he oaonal nea: Momens o nea o vaous objecs I CM : Momen o nea abou an axs o oaon hough he cene o mass Exended objec ; λdlλdx (L s on x-axs) and.7: Calculang he oaonal nea: Paallel axs heoem I I CM s known, he momen o nea hough a paallel axs o oaon a dsance h away om he cene o mass s: I ICM + Mh I I I CM + Mh L ML + M ML 3.8: Toque Fsnφ φ F Fcosφ φ Is angle beween F and decons.(نقطة ارتكاز) Consde a gd objec abou a pvo pon A oce s appled o he objec. Ths oce causes he objec o oae havng wha s called Toque τ. τ F snφ 6

F does no oque snce (an-paallel o ) he angen componen F has a oque τ F (sn 9 ) bu F ma and a α τ F

8: Toque: Example Two oces T and T ae appled as shown Fo oaon couneclockwse (بعكس عقارب الساعة)

5 kg and adus cm, mouned on a xed hozonal axle. A block wh mass m.

Fnd he acceleaon o he allng block, he angula acceleaon o he dsk, and he enson n he cod.

8: Example: A unom od o lengh L and mass M s aached as shown.

7 Toque and Angula Acceleaon.8: Toque consde a pacle o mass m oang a bou a xed axs unde an nluence o appled oce F The componen F does no oque snce (an-paallel o ) he angen componen F has a oque τ F (sn 9 ) bu F ma and a α τ F m α τ Iα I moe han one oce appled o he objec τ τ ne Iα Newon s second law n oaon.8: Toque: Example Two oces T and T ae appled as shown Fo oaon couneclockwse (بعكس عقارب الساعة) +ve α Fo oaon مع عقارب ( clockwse -ve α (الساعة α? a? T?.8: Example: a unom dsk, wh mass M.5 kg and adus cm, mouned on a xed hozonal axle. A block wh mass m. kg hangs om a massless cod ha s wapped aound he m o he dsk. Fnd he acceleaon o he allng block, he angula acceleaon o he dsk, and he enson n he cod. The cod does no slp, and hee s no con a he axle. a wh I M (.5)(.).5kg. m a.8m, T 6N, and α ad.8: Example: A unom od o lengh L and mass M s aached as shown. The od s eleased om es n he hozonal poson. Wha ae he nal angula acceleaon o he od and he nal anslaonal acceleaon o s gh end? α? and a? * The ode wll move lke pendulum unde he eec o F g Mg Exended objec look a he CM soluon L τ F sn φ F ( ) Mg L bu τ Iα ( ) Mg τ ( L / ) Mg 3g α I / 3ML L a The anslaonal acceleaon s 7

8 .8: Wok and oaonal Knec Enegy Wok n lnea moon dw F ds W F s dw P F. v d Wok n oaonal moon dw F ds dw τdθ W τ P τω.8: Wok and oaonal Knec Enegy: Wok Knec Enegy heoem The wok-knec enegy heoem o lnea moon: W mv mv Exenal wok done on an objec changes s knec enegy and o oaonal moon: W Iω Iω K Exenal oaonal wok done on an objec changes s oaonal knec enegy.8: Wok and oaonal Knec Enegy: Example In pevous example o dsk, he dsk sa om es a me. Wha s s oaonal knec enegy K a.5 s? Fom pevous example we have I M (.5)(.).5kg. m a.8m, T 6N, and α ad K Iω We need o nd ω a.5s ω ω + α + (.5) 6 ad/s K Iω M.5 kg,adus cm, and m. kg (.5)(6) 9J.8: Wok and oaonal Knec Enegy: Example: connued om pevous slde K K W K ω + α K τ ( ) We need o nd τ and τ T (. )( 6 ). N.m K + ()(. 5) τ (.)(75) 9J M.5 kg,adus cm, and m. kg o 75ad 8

9 evew Lnea quanes have analogous angula counepas. evew Objec oang make boh lnea and angula quanes a same nsan hee s a elaon wh angula and lnea quanes Toque s he endency o a oce o oae an objec. The oal knec enegy o a oang objec has o nclude s oaonal knec enegy. 9

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( ) 5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma