Page 1. Motion in a Circle... Dynamics of Circular Motion. Motion in a Circle... Motion in a Circle... Discussion Problem 21: Motion in a Circle

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1 Dynics of Circulr Motion A boy ties rock of ss to the end of strin nd twirls it in the erticl plne. he distnce fro his hnd to the rock is. he speed of the rock t the top of its trectory is. Wht is the tension in the strin t the top of the rock s trectory? Drw Free Body Dir. Motion in Circle... We will use F O = (surprise) First find F O in x direction: F O = + x Lecture 5, P Lecture 5, P F O = + Motion in Circle... Now reeber tht = / (in x direction) Motion in Circle... Wht is the iniu speed of the ss t the top of the trectory such tht the strin does not o lip? i.e. find such tht = 0. / = + / = = 0 + = / x = / - Notice tht this does not depend on. Lecture 5, P 3 Lecture 5, P 4 Discussion Proble : Motion in Circle A skier of ss oes oer oul hin rdius of curture. How fst cn she o without lein the round? N / = - N For N = 0: Discussion Proble : N () (b) (c) Lecture 5, P 5 Lecture 5, P 6 Pe

2 Dr Forces When n obect oes throuh fluid, like ir or wter, the ediu exerts dr force D tht opposes the otion of the obect. D points in the direction in which the fluid flows reltie to the body. D W = Dr Forces he dr force is relted to the reltie speed of the obect by n experientlly deterined dr coefficient C. = ir density = 000 k/ 3 A = effectie cross-sectionl re of body D = CA W = Lecture 5, P 7 Lecture 5, P 8 Dr force: erinl Speed D CA When n obect flls fro rest throuh ir, D increses fro zero s increses D eentully equls W net erticl force = 0 Body then flls t constnt terinl speed t. D D C A W = t C A Lecture 5, P 9 Fllin Ct Why would ct hurt itself ore if it fell out of buildin which is 5 floors tll rther thn buildin which is 7 floors tll? Durin initil fll ct is ccelertin» fer cuses it to curl up When ct reches terinl elocity, = 0» relxes nd stretches out little» A increses D increses As the dr force increses the ct slows down» D > W D CA Ct hits round with slower elocity!!!» Ct ust fll bout 6 floors to rech terinl elocity Lecture 5, P 0 Mny-Body Dynics Atwood s Mchine: Systes de up of ore thn one obect. Obects re typiclly connected: By ropes & pulleys. By rods, sprins etc... tody lter on Msses nd re ttched to n idel ssless strin nd hun s shown round the pulley. Find the ccelertions, nd, of the sses. Wht is the tension in the strin? Wht is the force on the pulley support? Fixed Pulley Lecture 5, P Lecture 5, P Pe

3 Atwood s Mchine... Drw free-body dirs for ech obect Applyin Newton s Second Lw: ( -coponents) - = - = But = = since pulley is idel nd = - = -. since the sses re connected by the strin Free Body Dirs Atwood s Mchine... wo equtions & unknowns - = - () we cn sole for both unknowns ( nd ). - = (b) subtrct (b) - (): ( - ) = ( + ) ( - ) = ( ) dd (b) + (): ( - - ( + ) = -( - ) = - = / ( + ) ) Lecture 5, P 3 Lecture 5, P 4 So we find: ( ) ( ) ( ) Atwood s Mchine... Specil cses: Is the esult esonble? Check Liitin Cses! i.) = = = 0 nd =. OK! ii.) or = 0 = nd = 0. OK! Atwood s chine cn be used to deterine (by esurin the ccelertion for ien sses). ( ) ( - ) Lecture 5, P 5 Lecture 5, P 6 Attched Bodies on wo Inclined Plnes sooth pe How Will the Bodies Moe? Fro the free body dirs for ech body, nd the chosen coordinte syste for ech block, we cn pply Newton s Second Lw: kin x coponents: ) - sin = x y x x y N ) - sin = x N All surfces frictionless But = nd - x = x (constrints) Lecture 5, P 7 Lecture 5, P 8 Pe 3

4 Solin the Equtions Usin the constrints, sole the equtions. - sin = () - sin = - (b) Subtrctin () fro (b) ies: sin - sin = ( + ) Specil Cse : ( sin sin) ( ) Borin!! So: ( sin sin) ( ) If = 0 nd = 0, = 0. Lecture 5, P 9 Lecture 5, P 0 ( sin sin) ( ) ( sin sin) ( ) Specil Cse : Specil Cse 3: Atwoods Mchine If = 90 nd = 90, ( ) ( ) If = 0 nd = 90, ( ) Lecture 5, P Lecture 5, P Discussion Proble : wo-body Dynics In which cse does block experience lrer ccelertion? In () there is 0 k ss hnin fro rope. In () hnd is proidin constnt downwrd force of 98 N. In both cses the ropes nd pulleys re ssless. 0k () Cse () (b) Cse () (c) se F = 98 N Cse () Cse () Add () nd (b): = ( + 0k) 0k Discussion Proble : For cse () drw FBD nd write F NE = for ech block: = () () (0k) - = (0k) (b) Note: 0k 0k (b) Lecture 5, P 3 Lecture 5, P 4 Pe 4

5 Discussion Proble : For cse () = 98 N = 0k 0k he nswer is (b) Cse () F = 98 N Cse () Cse () Lecture 5, P 5 Proble: wo Strins & wo Msses on Horizontl Frictionless Floor Gien, nd, wht is nd? - = = Add () + (b): () (b) = ( + ) Pluin solution into (b): = + = + i Lecture 5, P 6 Discussion Proble 3: Mny-Body Dynics Discussion Proble 3: hree blocks of ss 3,, nd re connected by strins nd pulled with constnt ccelertion. Wht is the reltionship between the tension in ech of the strins? 3 3 () > > 3 (b) 3 > > (c) = = 3 Drw free body dirs!! 3 = 3-3 = = + 3 > 3 - = = + > () > > 3 Lecture 5, P 7 Lecture 5, P 8 Alterntie solution: Consider to be pullin ll the boxes is pullin only the boxes of ss 3 nd 3 is pullin only the box of ss 3 Discussion Proble 3: > > 3 Lecture 5, P 9 Proble: ottin Puck & Weiht A ss slides in circulr pth with speed on horizontl frictionless tble. It is held t rdius by strin threded throuh frictionless hole t the center of the tble. At the other end of the strin hns second ss. Wht is the tension in the strin? Wht is the speed of the slidin ss? Lecture 5, P 30 Pe 5

6 Proble: ottin Puck & Weiht... = Proble: ottin Puck & Weiht... Drw FBD of hnin ss: Since is constnt, = 0. so = Drw FBD of slidin ss: Use F = = where = / N = = / Lecture 5, P 3 Lecture 5, P 3 Pe 6

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