Physics 240: Worksheet 16 Name

Transcription

1 Phyic 4: Workhee 16 Nae Non-unifor circular oion Each of hee proble involve non-unifor circular oion wih a conan α. (1) Obain each of he equaion of oion for non-unifor circular oion under a conan acceleraion, α. Show how hee equaion can be odified o obain he equaion of oion when he angular acceleraion varie (i) a a funcion of ie and (ii) a a funcion of angular velociy. () A wheel i fixed a i cener and ha a radiu of.7. A =, he wheel i a θ=, i no oving bu i ubjec o an acceleraion α=.5 rad /. Anwer he following a =1 : ω 1 (a) Wha i ( ) (b) Wha i θ( 1 ) (c) Wha i he angenial velociy of a poin on he ri a hi ie? (d) Wha i he angenial acceleraion of a poin on he ri? (e) Wha i he cenripeal acceleraion of a poin on he ri? (f) Wha i he agniude of he oal acceleraion of a poin on he ri? (3) A wheel iniially a re i ubjec o a conan angular acceleraion. Afer econd, he wheel ha oved hrough an angle of 8 degree. (a) Wha i he angular acceleraion of he wheel? (b) Wha i he angular velociy a econd? (4) A dik i iniially roaing a 45 revoluion/inue. By placing a finger on he dik, i i oberved ha he dik op in a ie of 1.5. Anwer he following: (a) wha i he average angular acceleraion? (b) wha i he angle ha he dik urn hrough during hi ie? (5) A wheel iniially a re i roaed wih α=3 rad/ for 1. I i hen brough o re in 5 revoluion. Anwer he following: (a) Wha wa he angular acceleraion required? (b) How long did i ake o op he wheel?

2 Phyic 4: Workhee 16 Nae (1) Obain each of he equaion of oion for non-unifor circular oion under a conan acceleraion, α. Show how hee equaion can be odified o obain he equaion of oion when he angular acceleraion varie (i) a a funcion of ie and (ii) a a funcion of angular velociy. Soluion: The equaion decribing ranlaional oion in workhee direcly apply here if you change variable in he following way: x θ : v ω : a α. α i he angular dω acceleraion which i defined by α. Thi urn ou o be a vecor quaniy bu we ll ju worry abou a poiive or negaive direcion. If he objec end o increae ω in a couner-clockwie direcion hen we ll call hi poiive. The oher direcion i negaive. The equaion of oion for non-unifor circular oion direcly ranfer fro hoe for ranlaional oion: α=conan ONLY! Tranlaional Equaion Roaional Equaion X θ v ω a α 1 1 x= x+ v,x+ ax θ=θ +ω + α v = v + a +α x,x x ( ) ω =ω + α( θ) v = v + a x x,x x You igh ay ha here a big difference in ha for ranlaional oion we have hree direcion and here i only one (i ee) for roaional oion. Thi i acually no he cae we can have independen roaion abou hree orhogonal (=perpendicular) axe bu funny hing happen over long ie which would be udied in ore advanced coure. For our coure, we ll worry abou roaion abou only one axi. Now if he angular acceleraion varie a a funcion of ie a: angular velociy by inegraion: ω= 1 + b. In order o obain he angle, we inegrae again: If he angular acceleraion varie a a funcion of velociy a: a he definiion of angular acceleraion: b hu obain: ln( ω ) b e angular velociy: f i α= b, (wih b a conan) we can find he 1 b b = f i. Now if he iniial ie i a =, we obain: θ= ω + b =ω + b = = α= bω (where b i conan), we need o look α bω= = b = b = b. We can dω dω dω dω ω ω ω ω ω =. We can find he angle urned hrough by looking a he definiion of average angular velociy i given by: d b d b b ω b e d e e b e θ θ ω ω = θ=ω θ=ω = 1. Of coure, he ω which i quie differen fro he inananeou θ.

3 Phyic 4: Workhee 16 Nae If we furher have a rigid body (which i defined by a body which ha α he ae hroughou he body), hen here i a connecion beween he wo acceleraion and velociie which coe fro he arc eaureen forula. For =arc lengh, he angenial velociy and he angular velociy are relaed by: d dθ ( ) ( ) = Rθ v= = Rω= R v=ω R and he angenial acceleraion and he angular acceleraion are relaed by: dv dω ( ) ( ) a= = R = Rα a=α R In general, you don necearily have a rigid body. In order o find he vecor relevan o he oal acceleraion, you ll need o do hi vecorially: In general, you don necearily have a rigid body. In order o find he vecor relevan o he oal acceleraion, you ll need o do hi vecorially: a= a + a cenripeal angenial Where he angenial acceleraion i he change in angenial velociy wih repec o ie.

4 Phyic 4: Workhee 16 Nae () A wheel i fixed a i cener and ha a radiu of.7. A =, he wheel i a θ=, i no oving bu i ubjec o an acceleraion α=.5 rad /. Anwer he following a =1 : (a) Wha i ω( 1 ) (b) Wha i θ( 1 ) (c) Wha i he angenial velociy of a poin on he ri a hi ie? (d) Wha i he angenial acceleraion of a poin on he ri? (e) Wha i he cenripeal acceleraion of a poin on he ri? (f) Wha i he agniude of he oal acceleraion of a poin on he ri? Soluion: (a) +α ω ( 1) = ( ) =. 5 1 (b) θ=θ +ω + α 1 θ ( 1) = + + (. 5)( 1) =. 5 rad (c) v=ωr v( 1) =ω ( 1)R =. 7 (. 5) =. 35 (d) Since he wheel i a rigid body (I auing i no ade of jelly) we can ue he connecion beween he acceleraion: a =αr a = (. 5)(. 7) =. 35 v. (e) a c= 35 = =. 175 R. 7 (f) Iagine you ake a napho of he iuaion a he inan ha he deired poin i exacly along he x-axi. Then, a plo of he vecor look like he picure below: The wo vecor are a righ angle o each oher. We can hu find he agniude of he acceleraion by direc applicaion of a= a + a = = ( ) ( ) c We can even find ou he angle which he reulan acceleraion i a o poined in: an( φ ) = = =. φ= an (. ) = A hi inan a c. 175 in ie, he oal acceleraion i poined a an angle of 11.3 above he (-x) axi a i hown in he econd figure below.

5 Phyic 4: Workhee 16 Nae (3) A wheel iniially a re i ubjec o a conan angular acceleraion. Afer econd, he wheel ha oved hrough an angle of 8 degree. (a) Wha i he angular acceleraion of he wheel? (b) Wha i he angular velociy a econd? Soluion: Probably he fir proble you ll noe wih hi i ha he oal angle i given in degree bu we ve go o work in radian. I did hi inenionally o ha you can be ure you know exacly how o ake hi converion correcly: π π θ radian= oθdegree θ= 368= Now we can anwer he proble. (a) θ=ω + α α= = = θ 13 (. 96) Noice ha he proble did ay he angle oved hrough. (b) Now ha we have he angular acceleraion, finding he angular velociy i eay. I ll do i way: ( ) ( ) (. )(. ). ω =ω + α θ ω=± α θ =± = 1 4 I choe he poiive oluion here ince he angle oved hrough wa poiive and he angular acceleraion wa alo poiive. Here i anoher way o ge he ae reul: +α ω=. 7 ( ) = 1. 4 (4) A dik i iniially roaing a 45 revoluion/inue. By placing a finger on he dik, i i oberved ha he dik op in a ie of 1.5. Anwer he following: (a) wha i he average angular acceleraion? (b) wha i he angle ha he dik urn hrough during hi ie? Soluion: You ll noice ha he angular velociy i again no given in radian/econd. You will need o conver o hee er o ay wihin he SI ye. Thu: revoluion 1 inue π radian rad 45 inue x 6 econd x 1 revoluion= =ω. (a) ω ω α ω ω =α α= = 1 5 = (b) Now ha we have he angular acceleraion, we can find he angle in differen way. I ll do i way: ω ω. ( ) 4 71 ω =ω + α θ θ= α = ( 3. 14) = θ=ω + α = ( 4. 71)( 1. 5) + ( 3. 14)( 1. 5 ) = = 3. 53

6 Phyic 4: Workhee 16 Nae (5) A wheel iniially a re i roaed wih α=3 rad/ for 1. I i hen brough o re in 5 revoluion. Anwer he following: (a) Wha wa he angular acceleraion required o op he wheel? (b) How long did i ake o op he wheel? I i iporan in hi proble o diinguih beween he wo angular acceleraion! Soluion: We need o find fir how fa i wa oving a he end of he iniial 1. Thi i given by: +α1 ω= 31 ( ) = 3 where he ubcrip 1 indicae he fir acceleraion. We can now anwer everyhing ele bu fir, we ough o conver 5 revoluion o radian (a) ( ) ( θ) 5 revoluion x radian. π radian 1 revoluion= ω =ω + α θ α= = = ω ω 3 31 (. 4) +α = = 14 33=. 9 ω ω 3 (b) α.