# Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations

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1 Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1

2 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy dq ds dq w d d d w slide

3 Veco angula elociy Define eco ang. elociy w d q magniude w d diecion along oaion axis sense using igh-hand ule Examples Recod a lef has w along z axis body o igh has w along +z axis w slide 3

4 eiew Veco Poduc (ch 3) Conenion fo w ecalls he definiion of he eco poduc (no a coincidence) The eco poduc of chape 3 is a conenien way o use he eco angula elociy (along wih ohe oaion quaniies) c a b b a c ab sinf slide 4

5 Veco poduc gies angenial elociy w w w Noe el pa of is he ^ disance o he axis of oaion w ' Aleady seen he angenial speed in ems of dis fom axis and ang elociy Veco o coss-poduc gies diecion of ^ w & Tue fo any oaing eco slide 5

6 One Sep Fuhe w Any oaing eco of fixed lengh simple ime ae of change Was ue fo adius eco Also ue fo elociy eco a c / Dieced inwad The elociy eco Has fixed magniude Roaes wih angula elociy, w Fo c any eco of cons. lengh oaing w, hen dc w c d slide 6

7 Cenipeal Acceleaion (eisied as a Veco) ac a c a c w w w w ( w ) w Rae of change of posiion aound axis gien by coss-poduc Also, ae of change of elociy aound axis of oaion gien by same cosspoduc Veco Ideniy (appx E) a ( b c) ba ( c) - ca ( b) slide 7

8 a d w d a Angula acceleaion If ang elociy,w, changes, hee is an angula acceleaion, a. Poin a adius,, has linea acceleaion, a a, angen o cicle a. a d w a a d angula acceleaion.mov slide 8

9 Relaionships fo consan angula acceleaion simila o hose of consan linea acceleaion See able 11-1 Noe ha p moing on cicle, in geneal, will hae wo componens of acceleaion The wo add ecoially o gie oal acceleaion eco Angula and Cenipeal Acceleaion a a c a d d a a c a a poining o oaion axis along diecion of oaion slide 9

10 Hin: HW poblem 11-3 w f c f Waemelon seed on ecod playe Sas a es w 0 0 Reaches 33 1/3 pm Cone o sandad unis Two componens (a igh angles) of acceleaion Find ne acceleaion ne ficion foce causes i N mg w a a a c a w w w 0 Coefficien of saic ficion mus be big enough o wihsand his oal foce slide 10

11 Kineic Enegy of Roaion: Momen of Ineia w i m i i y x M Roaion of a igid body abou an axis implies Kineic Enegy... Look down fom +z axis a mass pieces in body K K i i I w m m i i i i w I m discee m i i i I dm coninuous m M enegy in oaional moion.mov i slide 11

12 Poin masses: sample pob Rigid body of wo masses (m) conneced by igid massless ba. Find I fo wo axes I m discee m i i i I dm coninuous m M i 1 1 I m( L) + m( L) I com com 1 ml I m() 0 + ml ( ) I end end ml slide 1

13 I M dm Momens of Ineia coninuous igid bodies R Solid cylinde Rod of mass, M, adius, R, and lengh, L R<<L Momens of ineia abou wo axes, (a) and (b) L Thin od Deemined by geomey of body Chaaceisic dimension of body abou oaion axis I I a b MR ML slide 13

14 I M dm Typical Shapes able 11- slide 14

15 P Paallel Axis Theoem Simple echnique fo finding I abou a diffeen axis (when know i abou axis hough he cn of mss) Pick oigin a c of m wih z-axis along w So x com y com 0 {( ) ( ) } I dm x a + y b dm x - ax + a + y - by + b ( x + y ) dm axdm bydm + h dm I P I + com Mh paallel axis heoem.mov slide 15

16 Toque: Can change Roaional Kineic Enegy Definiion of Toque F F sin f in diecion F plane F F Toque poduces Angula Acceleaion Body s oaional enegy changes as esul Wok done by oque wok fom oque.mov Wok Fs Fq Powe q d q w d slide 16

17 Uiliy of Toque- oaion abou fixed axis Massless od of lengh,, wih p-mass, m, a end apply foce, F, as shown F componen can' moe F ma ma F I a wih I m Noe diecions of and Ia wih I m ma i i m a When ne oque on body is zeo, hee can be no angula acceleaion If w0, and no oque, w says ha way slide 17

18 Specifying Toque I I m i i a wih F F sinf plafom- oque.mov Noe he equialence of he wo pespecies disance imes ansese foce componen lee, o momen, am imes foce Impoance of all seen in pix slide 18

19 I lage, a smalle I I m i i a wih Noe ha his fom is simila o Newon s nd Law: Fma Moion die (foce) poduces linea acceleaion smalle if ineia (mass) lage Toque dies angula acceleaion I, popey of he body, is a measue of oaional ineia plafom- I w bodies1-sndrs.mov plafom- I w bodies-sndrs.mov slide 19

20 Roaional Mechanical Enegy Toques do Wok and Make Enegy Kineic Enegy esul of angula acceleaion o Wok done Enegy o pix Analogy beween enegies anslaional oaional Wok Fs Fq Powe q w enegy-powe in oaional moion.mov slide 0

21 Cene of Gaiy Rigid body feels gaiy as if all he body mass concenaed a cene of mass See poof in ex (sec 13-3) Consequence Pio igid body and cog mus lie below i Demo Pio body a cog and sable (no oque) a pio poin momen am slide 1

22 Finding C of M Empiically N 1 x 1 x N f 1 f mg Why does his wok?? l l Allow od o slide acoss hands Woks because he foces and oques on he od mus sum o zeo dm m com 1 M 1 D -- like sick dm dl M L L L 1 1 com M lmdl L 0 0 L ldl slide

23 Conclude Remembe ha HW 6 on chapes due on Wednesday Read hough chapes 1-13 will finish in he nex wo lecues slide 3

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