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PHY205 Lecture 17 Ch. 8.4, 8.6 (kip 8.5) Rtatinal Equilibrium, Rtatinal rmulatin f ewtn Law
Linear v Rtatinal Variable Linear Variable Tranfrmatin Table Rtatinal Variable ditance velcity acceleratin ma [inertia] mmentum p = mv frce, = ma angle angular velcity angular acceleratin mment f inertia angular mmentum, L = ω trque, τ = α Ktran = ½mv 2 Krtatin = ½ω 2 PHY205, Lecture 17 Rtatinal rm f ewtn Law
Right Hand Rule fr rtatinal ytem change f angle ( θ) i a vectr alng the axi f θ rtatin, and it directin i determined by the right hand rule: θ rient yur right hand that the finger pint in the directin in which the angle i changing; yur thumb pint in the directin f the rtatinal vectr PHY205, Lecture 17 Rtatinal rm f ewtn Law 4
Reminder: frm angle t angular acceleratin intantaneu angular velcity: intantaneu angular acceleratin: PHY205, Lecture 17 Rtatinal rm f ewtn Law 5
Trque The equivalent f frce, fr rtatinal ytem (mewhat hand-waving derivatin f equivalence) trque = mmentum f inertia angular acceleratin nt quite trivial, need t accunt fr the rientatin f frce wrt lever arm f bject i, ueful frce PHY205, Lecture 17 Rtatinal rm f ewtn Law 6
= inθ Perpendicular frce θ θ r r r = r inθ lever arm θ fr the ame frce, the trque i directly prprtinal t the lever arm (prjectin f r perpendicular t ) PHY205, Lecture 17 Rtatinal rm f ewtn Law 7
Wrk dne by trque Start frm definitin f wrk, fr rtatinal ytem: m Hke law, fr rtatinal ytem (trin pring): r θ PHY205, Lecture 17 Rtatinal rm f ewtn Law 8
Unuual Atwd Machine n the machine belw, the mall weight (ma m) i cnnected via an ideal pulley t the male hrizntal wheel f radiu r. gnre the mment f inertia f the cnnecting rd; the tw mae M are bth R away frm the axi f rtatin. What i the velcity f ma m a it fall h frm it initial pitin, where it wa at ret? R M M r m h PHY205, Lecture 16, Rtatinal Energy and nertia 9
Dicuin: Unuual Atwd Machine, via energy cnervatin PHY205, Lecture 16, Rtatinal Energy and nertia 10
l -q l l l b rl L} Jl, lr'{b L} / f Jr l, p Pfrl Tl P /Pr_lp 1P ll p r ll nv ; l p rlnv J Jl q il PHY205, Lecture 16, Rtatinal Energy and nertia lr'{ rl q l.t' il R )1^ R R )1^ r R lil e rle.t' q l l / f / f lr l 'rla,llr'{pp 'rla,l pp n tt J! n tt J! R )1^ R /_1 / _r l 1 rl r l e.t' l? 'rl)a,l pp? n) t TJ! ;l rlnv t.4 t.4 b - l t r - 1 i' ll p' tr " ' rt- l1f i' lqp' r " ' A-a ; l p l ll A-a? ) J L} J, < - tl. t d < 5 J - J * r -fvl -, (-.-.-l 6'' 1 6'' 1 6'' 1 )- b,?' b,?' (-.-.-l T ) b T fvl b * r--, i E i E t r l r l t. r -G - r G Gb,? L G l 5 L q7). < t d q'(-.-.-l t <. * T 7b fvl r--, i E L L l l : : -t -t H B H B a a L cn- L Ln c /f /f e ir0 6' c eic6 ' r0 G cng / / / e c i6' r0 G / / / /f Dicuin: Unuual Atwd Machine, via frce, tenin and trque 11
Dicuin: Unuual Atwd Machine, via frce, tenin and trque 2 c n,^) -*. tl ) r0,"? c rtl^ -r- CS.' _t ) rdp P - t r4 q n 5 v r ( A % $ e r b 6-5 Av - $ u- - ck "$,rc ) 6 *0 X.--.S' 5li ilpn, f) *^l,n* ll 'J'l* ll ' l, lb' tl P X t, a 6- f.j ll E {0 ṯ l- { t ) 5u fi-.9- D c) n 6 C D PHY205, Lecture 16, Rtatinal Energy and nertia 12
Static, Rtatinal Equilibrium Cnditin 1: ewtn 1t law: a tatinary bject that ha n net frce acting n it remain tatinary Cnditin 2: r the bject nt t tart rtating abut an axi, the net trque n the bject mut be zer mprtant: Cnditin f tatic equilibrium, nce etablihed fr ne axi, apply t any arbitrary axi PHY205, Lecture 17 Rtatinal rm f ewtn Law 1
Example: Springbard A diver weighing 80 kg i tanding n the end f a 2 m lng pringbard which weigh 240 kg. The pringbard i upprted by tw pring n the ther end. One pring i at the very end f the bard, and the ther i 0.5 m tward the center f the bard. Cmpute the frce generated by the pring if the ytem i in tatic equilibrium. Are the pring cmpreed r tretched? PHY205, Lecture 17 Rtatinal rm f ewtn Law 14
-l lt l -, n a_ -f q, a- xlr q-, r ca l - X 1r-. l' t c r V l - u1 l - E r' tl t. J' r,9.'l r J- 9 5 n t?'t- ( b f /t t- ll r - ' J- e r J c r- )' Y /*l-l X tl l a_5 c E" --{- t-) 5 T E,. Elfl v 9 -. AQ lr-. q{) f- tl O - f-6 t/ 9 c X rl ) Q.1 t- rvl.?t, 6.a tl - - C 1 ^, t - ^.0 AQ u A x t e. t^ qp e R $ v U e q. c -lr-l T ) g,t- =r -*," A. * * 5.-4 9 ( t (.b E S (,.. Ji a*r-- : c6 a_ u Jp * i J- Ḏ J-.( A V - Dicuin: Springbard PHY205, Lecture 16, Rtatinal Energy and nertia 15
Dicuin: Springbard 2 -r ilr r if>,,rb XJ rf ->r ' CJ t t. ll -:-,.n f, -J,g ^l & S { L J) -ttr. ') SD rr S, 1 -St a.) -S p,["rt t l -, J tl Lt.{ {l O 00 b E / SD O {9 tf Crb 5 SD >. -f- n E t J! ) l J x. u l l rf u' a) - : * n 1/1 n T(.) tn a- t B _ ^P {, l*v tn.-l r - 9 J. l-r- lf!' (-7 -'l -u "g -}.l t ar b 0 O _v R r t l l il.? h-. crd J C 6 5 P AJ _ a- Y) l/r CtD H := q.).4 [*v b q) -.S -19 * u E a,) f ẖ r _ v Pq E (J 6 PHY205, Lecture 16, Rtatinal Energy and nertia 16
Angular Mmentum Mmentum-impule therem: by analgy with p = mv, define angular mmentum: angular mmentum-impule therem: PHY205, Lecture 17 Rtatinal rm f ewtn Law 17
Example: Angular Clliin A dik f ma m1 and radiu R i pinning with an angular frequency ωi. A ecnd dik, f ma m1 i drpped nt dik 1 that they hare the ame axi f rtatin. After a brief perid f frictin, the tw dik are rtating with the ame angular frequency ωf. What i the rati ωf/ωi? What fractin f the initial kinetic energy wa wated thrugh frictin? PHY205, Lecture 17 Rtatinal rm f ewtn Law 18
Dicuin: Angular Clliin ^5 tl J J H? t -...r<. ^) j, L $ fl, -J-.l t- t--l p l lq e..$ 1 p <$ J- c6 - a c fbtal *[ }}q. c.l q ^{ t - ll.h -i l,t J Jr ll > P l^ J.q, - l-.l t'j f-l H. H - J r0 t1 l--'. t (r R -) tl ' ii " -' l-- -"l' h C.f rl rj.d J^ ns ll UJ t H ) L-t h- tl fllr * h f- )l t J: n *H t (r r f--).j f-j n) L_J -l 7 rd 5 CQ e - ) T CO c 5 0 h -t 5 6 vl v1 -L J q- Y c) r q,0 r0 ọ* PHY205, Lecture 16, Rtatinal Energy and nertia 19
5' d -T-? -)- )-L- 6 : V1 e* 6. cb. E R c 6 y 6 t$ 1* d' $ (tl. -, " ={-.' p e* 5- q - a ^ <. u - { r t l-1 L (,1) t) c dl- r t l-r il -cl* n L.) ;a -rln t -. a l H r T S. t t d t p l-j t-j H P J > e {.J L rr' r rl r, -,. r L qt -t -, Plt rlaj r p { r l* {.-' fr l 5 t 6 ll *l T n P-l ' to h h ui. lry, A Jr J,n H e e. d t -1 ti h l-!,*e" -r h t-r H l*r- A H ' lp qb Dicuin: Angular Clliin 2 PHY205, Lecture 16, Rtatinal Energy and nertia 20
The vectr nature f trque and angular mmentum PHY205, Lecture 17 Rtatinal rm f ewtn Law 21
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