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

Consequence Pio igid body and cog mus lie below i Demo

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

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body. Descipion of he moion of igid