CHAPTER 8 MECHANICS OF RIGID BODIES: PLANAR MOTION. m and centered at. ds xdy b y dy 1 b ( ) 1. ρ dy. y b y m ( ) ( ) b y d b y. ycm.

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1 CHAPTER 8 MECHANCS OF RGD BODES: PLANAR MOTON 8. ( For ech portion of the wire hving nd centered t,, (,, nd, ( d d d ρ d ρ d π ρ π Fro etr, π (c (d The center of i on the -i. d d ( d ρ ρ d d d d The center of i on the -i. dv πr d π + d πd

2 ρπ d d ρπd d (e ( The center of i on the -i. α i the hf-nge of the pe of the cone. r i the rdiu of the e t nd r i rdiu of circe t oe ritrr in pne pre to the e. r r tnα, contnt dv π r d π tn αd ρ πr πρ tn ρπ tn αd πρ tn α α ( + d 8. ρd ρd c d cd 8. The center of i on the -i. Conider the phere with the cvit to e de of (i oid phere of rdiu nd M, with it center of t, nd (ii oid phere the ie of the cvit, with M c nd center of t. The ctu phere with the cvit h M Mc nd center of. M c + M M π ρ, M c π ρ +

3 8. ( iri + + i 6 ( d rdθ dr, R rinθ R ρd r θ π r rdr π in d r θ π ρ π in θ dθ ρ θ θ θ in θ in θdθ ρ π ρπ ( π π (c d hd d Where the pro interect the ine, ( ± ρ d ρ d ρ ρ d ρ (d dv π RhdR h

4 R + dr d R ρdv ρ π ( d πρ ( d πρ 6 ρdv ρπ ( πρ ( d πρ d (e α i the hf-nge of the pe of the cone. r i the rdiu of the e t nd r i rdiu of circe t oe ritrr in pne pre to the e. ( R R tnα, contnt R dv πrhdr π dr ( R R Since R, dr d, nd the iit of integrtion for R R correpond to ( ( R R R R ρdv ρπ R πρ + πρ + d ρ πr R d R

5 8. Conider the phere with the cvit to e de of (i oid phere of rdiu nd nd M, with it center of t, nd (ii oid phere the ie of the cvit, with M c nd center of t h M Mc nd center of.. The ctu phere with the cvit 7 M Mc π ρ π ρ π ρ 8 8 M nd M c 7 7 Fro eqn. 8.., c + M M c The oent of inerti out one of the tright edge i R ρdv where R +. Fro Appendi F φ θ R r ρπ π ρ π ρ 8 6 dv r inθ dr dθ dφ R + r in θ Let rdiu of phere π π r θ φ r in r in drd r θ φ θρ θ θ dφ π π r θ ρ r in drd θ θ r θ π ρπ in θd θ co θ in θ dθ coθ

6 6 8.7 For rectngur preepiped: dv hdd h i the ength of the o in the -direction R + R ρdv + ρhdd ρh + d ρh + ρ h ρh + For n eiptic cinder: Agin dv hdd, nd R + On the urfce, + ± R dv + ρhdd ρh + d ρh ( d ( d + Fro te of integr: ( d ( + ( + in 8 8 ( d ( + ( ρh h π + π ρ π ( in 6

7 7 ρh π + For n eipoid: dv hdd, R + nd on the urfce, c + + ± c h c n the pne + ± ρ ρ R dv + c dd d+ Fro te of integr: ρ c d k k ( k d ( k + ( k + in 8 8 k k ( k d ( k + in k ρc π π d + 8 ρπ c + d + ρπ c( + For n eipoid, ρ πc +, o 7

8 8 8.8 (See Figure 8.. Note tht + i the ditnce fro to, defined d Fro eqn. 8.., k k k + d Fro eqn. 8..9, k d Fro eqn. 8..6, T π d g k k + T k π g 8.9 Period of ipe penduu: T π Period of re penduu: T π (eqn. 8.. Mg Where oent of inerti ditnce to CM of phic penduu ditnce to CM of o rdiu of o oction of CM of phic penduu: ( + ( M + ( + + M M M M M Moent of inerti: o ( M ( M ( M + + M + M rod ( M M o + rod M + + M M etting α nd β M M 8

9 9 T M ( α + β + α( β π α αβ Mg ( α ( T + β + α β α to t order inα T α αβ ( g M kg.7 α. β.9 T α.999 T (ctu.999 uing copete epreion 8. The period of the econd penduu i T π Mg The period of the odified penduu i n T π Mg n where, M, refer to preter of penduu with ttched nd n ( 6 6 i the nuer of econd in d. M+ + M where i the ditnce of the ttched π π + π So n Mg M+ g Mg +π ( M+ g ( + M g Thu n Mg + π Or n πα ( α + α M + g Soving for α give the pproite reut fro the pivot point. 9

10 n α M π g Letting.;.; we otin α. 8. ( in ri ( ri + ri T π π g g ( + ( hence ri + ri T π π g g 8. T g g T ω T ω ω g 7 g g g 8. When two en hod the pnk, ech upport When one n et go: Fro te 8.., R 6R ω g. g R nd R ω

11 R ω R g R R g R 6R 6 g end ω end g 8. For oid phere: M π ρ nd M ( k For ucript c repreenting oid phere the ie of the cvit, fro eqn. 8..: + c π ρ π ρ π ρ 7 π ρ π ρ π ρ 8 8 k 7 7 Fro eqn. 8.6., for phere roing down rough incined pne: g inθ k + ( k + k

12 8. Energ i conerved: g g E g g ( g Whie the cinder re in contct: v fr g coθ R r R v r +, o g coθ R v + Fro conervtion of energ: g ( + v + ω + g ( + coθ g Fro te 8.., v ω v v + g ( + ( coθ v g ( coθ + When the roing cinder eve, R : g coθ g ( coθ 7 coθ θ co N N g

13 8.8 R v + ω + g g coθ, θ inθ inθ, θ coθ θ v + θinθ + θ coθ, nd ω θ θ + θ + g inθ g inθ θ g ( inθ inθ g θ ( inθ inθ, ( θ coθ + θ inθ, ( θ inθ + θ coθ g g g θ ( inθ inθ ( coθ θ co θ g g N coθ ( inθ inθ inθ co + θ Seprtion occur when N : inθ inθ inθ, θ in inθ R g inθ, θ coθ, ( θ inθ + θ coθ coθ, θ inθ, ( θ coθ + θ inθ v + θ + g coθ g v θ,

14 θ + θ g θ g ( coθ g θ ( coθ ( coθ g g g θ ( coθ inθθ in θ g g R ( in ( co co in θ θ θ θ + g R inθ( coθ g g R g co ( co in in θ θ θ θ + g in θ R g coθ co θ + g R ( co θ The rection force contrin the ti of the rocket fro iding ckwrd for R > : coθ > θ < co The rocket i contrined fro iding forwrd for R < : θ > co 8.9 g inθ µ g coθ f g θ µ g coθ ω t ( in co g θ + µ θ Since cceertion i contnt, t+ t gt vt ( inθ + µ coθ Menwhie ( µ g coθ ω ω µ g coθ ω :

15 The egin pure roing when v ω µ g coθ v v + t v g( inθ + µ coθ t t v t g inθ + 7µ coθ At tht tie: v g v ( inθ + µ coθ g inθ + 7µ coθ g inθ + 7µ coθ v g ( inθ + 6µ coθ ( inθ + 7µ coθ 8. µ g µ g µ gt, nd µ gt ω ω µ g µ g ω µ g ω ω t Sipping cee to occur when v ω µ gt ω µ gt ω t 7 g ω µ g 7 µ g ω 9 µ g 8. Let the oent of inerti of A nd B e M nd M. The ngur veocit of A i α whie tht of B i β α + φ (reeer tht in two dienion, ngur veocit i the rte of chnge of n nge etween n ine or direction fied to the od nd one fied in pce. For roing contct, ength trveed ong the perieter of the dik A nd B ut e equ to the rc ength trveed ong the trck C. φ β + α φ

16 + α + α o tht φ nd β + + After oe ger the ngur veocit of B i found to e α ωb β α + φ For A, we tke oent out O nd for B we tke oent out it center. C T A nd T B the coponent of the rection force tngent to A nd B (the upwrd-going T A ct on dik B. The downwrd-going T A ct on dik A 6 Thu K T A A α (Torque on Dik A α T A T B B( β α + φ B (Torque on Dik B TA TB MB( + ( α φ MB α (Force on Dik B Eiinte T A nd TB α K ( A + MB + B ntegrting thi eqution give : Kt α + M + Putting ω A ω A ( A B B α t t t give Kt ( A + MB + B Putting in vue for A nd B give Kt ω A M A + MB α Since the ngur veocit of B i ωb β α + φ, we hve Kt ωb ωa M A + M B 6

17 7 8. Fro ection 8.7 (ee Figure 8.7., the intntneou center of rottion i the point O. f i the ditnce fro the center of to O nd i the ditnce fro the center of to the center of percuion O, then fro eqn M M M 6 8. n order tht no rection occur etween the te urfce nd the, the ut pproch nd recede fro it coiion with the cuhion roing without ipping. Uing prie to denote veocit nd rottion veocit fter the coiion: Pˆ v v Pˆ ω ω ( h The condition for no rection re v ω nd v ω. Pˆ Pˆ ω ω ( h ( h 7 h + d 7 h d 8. During the coiion, ngur oentu out the point i conerved: v θ v θ After the coiion, energ i conerved: θ g g g coθ g coθ v g( coθ + 7

18 8 + v g( coθ The effect of rod BC cting on rod AB i ipue + ˆP. The effect of AB on BC i ˆP. v ˆ ˆ P + P v ˆ P ( ˆ ˆ ω P P ω ˆ P 6 ( Pˆ Pˆ 6 ω ω ˆP vb v ω vb v + ω Pˆ + Pˆ 6 v ( ˆ ˆ B P P Pˆ 6 v ˆ B P + Pˆ + Pˆ Pˆ Pˆ Pˆ Pˆ 8P ˆ Pˆ ˆ ˆ P P Pˆ Pˆ v v 9Pˆ Pˆ ω ω Pˆ vb