CHAPTER 5 NONINERTIAL REFERENCE SYSTEMS

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1 CHAPTER 5 NONINERTIAL REFERENCE SYSTEMS 5. (a The non-ineia oseve eieves ha he is in equiiium and ha he ne foce acin on him is zeo. The scae exes an upwad foce, N, whose vaue is equa o he scae eadin --- he weih, W, of he oseve in he acceeaed fame. Thus N (a A 0 ( N+ m ma N m ma0 N m m N m W N m W 4 4 W 50. m W 90. ( The acceeaion is downwad, in he same diecion as N m+ m 0 4 W W W W (a Fcen m ( Fo, F m e ˆ ( F cen cen 500 s 000π s ( π eˆ 5π dynes ouwad Fcen m F ( π m m + T ma 0 (See Fiue 5.. ˆ ˆ ˆ m j+ Tcosθ j+ Tsinθi m i ˆ 0 0 m Tcosθ m, and T sinθ 0 anθ, θ m T.005m cosθ 5.4 The non-ineia oseve hinks ha poins downwad in he diecion of he hanin pum o Thus 4

2 ˆ A j ˆ i 0 Fo sma osciaions of a simpe penduum: T π T π.995π (a f µ m is he ficiona foce acin on he A 0 f ( (a ox, so f ma ma (a is he acceeaion of he ox eaive o he uck. See 0 Equaion Now, f he ony ea foce acin hoizonay, so he acceaion eaive o he oad is f µ a m µ m m (Fo + in he diecion of he movin uck, he indicaes ha ficion opposes he fowad sidin of he ox. A (The uck is deceeain. fom aove, ma ma0 ma so a a A + 6 i x R ˆ + Ω + jrsin Ω iˆωrsin Ω + ˆjΩRcosΩ v Ω R vω R cicua moion of adius R 5.6 (a ˆ( cos ( whee ix ˆ + ˆjy iˆωrsin Ω + ˆjΩRcos Ω kˆ ix ˆ + jy ( ˆ

3 iˆωrsin Ω + ˆjΩRcosΩ ˆj x + iˆ y x y ΩRsin Ω y x +ΩRcos Ω (c Le u x + iy hee i! u x + iy y ΩRsin Ω i x + iωrcosω i u y + i x sin cos Re i Ω u + iu ΩR Ω + iωr Ω iω Ty a souion of he fom i i u Ae Ω + Be u i Ae + iωbe iu iae + ibe i i Ω i iω i ( u + iu i +Ω Be so ΩR B + Ω Aso a 0 he coodinae sysems coincide so u A+ B x 0 + iy 0 x + R ( ( Ω A x + R B x + R so, + Ω R i ΩR iω Thus, u x + e + e +Ω +Ω R R A x + + Ω 5.7 The x, y fame of efeence is aached o he Eah, u he x-axis aways poins away fom he Sun. Thus, i oaes once evey yea eaive o he fixed sas. The X,Y fame of efeence is fixed ineia fame aached o he Sun. (a In he x, y oain fame of efeence x( Rcos( Ω R ε ( x ( ( Ω Rsin ( Ω 0 a 0 ( ( Ω cos( Ω ( Ω ( sin ( Ω y R whee R is he adius of he aseoid s oi and R E is he adius of he Eah s oi. Ω is he anua fequency of he Eah s evouion aou he Sun and is he anua fequency of he aseoid s oi. y R R a 0 (c a A Aε Ω Ω Ω Ω Whee a is he acceeaion of he aseoid in he x, y fame of efeence,

4 AA, ε ae he acceeaions of he aseoid and he Eah in he fixed, ineia fame of efeence. s : examine: A Aε Ω Ω R Ω Ω Rε Ω Ω ( R Rε Ω Ω R Ω R noe: ˆk ( ( Thus: a Ω R Ω v (, Ω Ω ˆk Theefoe: ( ix ˆ + ˆjy ( Ω ir ˆ cos( ˆ Ω jrsin ( Ω ˆ jω x+ i ˆ Ωy Thus: x Ω Rcos Ω + Ωy ( ( ( ( y Ω Rsin Ω Ωx Le x ( Ω y and y ( Ω x Then, we have Ω y ( Ω Rcos( Ω + Ω y which educes o ( cos( y Ω R Ω Ineain ( ( y Rsin Ω 0 a 0 Aso, x Ω Ω Rsin Ω Ωx ( ( ( o x ( R ( x ( Ω Rsin ( Ω Ineain cos( Ω+ sin Ω + Ωx x R Ω + cons x Rcos( Ω R R R a 0 ε ε 5.8 Reaive o a efeence fame fixed o he unae he cockoach aves a a consan speed v in a cice. Thus y a eˆ. Since he cene of he unae is fixed. x A 0 The anua veociy,, of he unae is consan, so 4

5 ˆk, wih 0 e ˆ, so ( e ˆ vê θ, so ve ˆ Fom eqn 5..4, a a + + ( and puin in ems fom aove a Fo no sippin F µ sm, so a µ s + + µ s + + µ 0 m m s ± + vm µ s Since v was defined posiive, he +squae oo is used. + µ m ( vê θ + ve ˆ a + + µ s + µ m 5.9 As in Exampe 5.., V ˆ j and A ρ Fo he poin a he fon of he whee: V ˆ j and v V ˆk 0 V ˆ ˆ V k j i ˆ ρ ρ s s ( V ˆ V ( ˆ V ˆ V k i ˆj ρ ρ ρ V kˆ ( Vkˆ 0 ρ i ρ V ( ˆ V V a + + A i + + ˆ j ρ ρ 5

6 5.0 (See Exampe 5.. m x mx x Ae + Be ( ( x Ae Be Bounday Condiions: x ( 0 A+ B x 0 0 A B ( ( A 4 B 4 (a x ( cosh x ( sinh ( x ( T + cosht when he ead eaches he end of he od.7 cosht o T cosh (c x ( T sinht sinh cosh ( o cosh T ˆj mph ˆj f s ( cos 4 ˆ j + sin 4 kˆ s ( ( ( sin 4 i ˆ F m co F m F F av co av 5 ( ( ( sin The Coiois foce is in he 5. (See Fiue 5.4. ˆ ˆ y j + z k v iˆ + v ˆj x y f s diecion, i.e., + î o eas. 6

7 ˆ ˆ ˆ z vy i + z vx j y vx k i + v j ( ˆ ˆ hoiz z y z x ( ( hoiz z vy + z vx z ( vx + vy z Fco m v F m v m, independen of he diecion of v. ( co ( hoiz 5. Fom Exampe h x h cos λ hoiz z and y h f x h ( s cos 4 f s x f o he eas. h 5.4 Fom Exampe 5.4.: H sin λ is he defecion of he asea owads he souh since i v was suck due Eas a Yankee Sadium a aiude λ 4 N (poem 5.. v0 is he iniia speed of he asea whose ane is H. Fom eqn 4..8, wihou ai esisance in an ineia efeence fame, he hoizona ane is v sin α H Sovin fo v 0 v H v sin α f s 00 f sin 0 f s ( s ( 00 f sin f 0. in f s A defecion of 0. inches shoud no cause he oufiede any difficuy. 5.5 Equaion 5..0 ives he eaionship eween he ime deivaive of any veco in a fixed and oain fame of efeence. Thus da da + a d fixed d o a ( 7

8 da d o + ( + ( + ( a + ( + ( + ( Now ( is o and. Le his define a diecion ˆn : n ˆ Since ˆn, ( is in he pane defined y and and ( nˆ. Since ( ( And ( is in he diecion ˆn Thus ( ( a + ( + ( ( ( + + ( ( ( 5.6 Wih x y z x y 0, and z Equaions 5.4.5a 5.4.5c ecome: x ( cos λ cos λ y ( 0 z ( + v, so When he ue his he ound z ( 0 8v 4v x cos v cos λ λ 4 x cos λ x is neaive and heefoe is he disance he ue ands o he wes. 5.7 Wih x y z 0 and x vcosα y 0 z vsinα we can sove equaion 5.4.5c o find he ime i akes he pojecie o sike he ound 8

9 z ( + sinα + v cosαcosλ 0 sinα sinα o cosαcosλ We have inoed he second em in he denominao since woud have o e impossiy ae fo he vaue of ha em o appoach he maniude o o Fo exampe, fo λ 4 and α 45 cosαcosλ km o v 44! s Susiuin ino equaion o find he aea defecion ives 4 y ( [ v cosαsin λ] sin λsin αcosα 5.8 Le a acceeaion of ojec eaive o Eah ˆk is anua speed y x A acceeaion of saeie R ˆk is anua speed 0 R a a+ v+ ( + A (Equaion 5..4 a a A v As in po em 5.7 Evauae he em a a A R R R R R a ( iven he condiion ha R a R R R R R+ R+ R + + Rcosθ u ( ( Lein x cosθ R R x fo sma R R R Rx R o R x + R R x and R R + R a + R R x R R x R xiˆ fo sma ( 9

10 Hence: a ˆ ˆ ( ˆ a v xi k ix+ ĵy a iˆ x+ ˆjy xiˆ+ yi ˆ xĵ So x y x 0 y+ x m qe + q ( v B Equaion v + ( Equaion 5.. v + q B so 0 m q m q ( B B ( qe + q ( + B q m + q ( B + ( B qe + q ( B + q ( B q m qe + ( B q ( q qb B ( ( sinθ( B B m Neecin ems in B, m qe 5.0 Fo x xcos + ysin y xsin + ycos x xcos x sin + y sin + y cos y xsin x cos + y cos y sin x xcos + y sin + y y xsin + y cos x x xcos x sin + ysin + y cos + y y xsin x cos + ycos y sin x x xcos + ysin + y + x y xsin + ycos x + y Susiuin ino Eqns 5.6.: x cos + ysin + y + x x cos ysin + y xsin + ycos x + y + xsin ycos x 0

11 Coecin ems and neecin ems in : x+ x cos + y+ y sin 0 x + x sin y+ y cos T sin λ hous 4 T 7.7 hous sin9 5. Choose a coodinae sysem wih he oiin a he cene of he whee, he x and y axes poinin owad fixed poins on he im of he whee, and he z axis poinin owad he cene of cuvaue of he ack. Take he iniia posiion of he x axis o e hoizona in he V diecion, so he iniia posiion of he y axis is veica. The icyce whee is oain wih anua veociy V aou is axis, so ˆ V k A uni veco in he veica diecion is: ˆ V ˆ sin ˆ V n i + j cos A he insan a poin on he im of he whee eaches is hihes poin: ˆ ˆ V sin ˆ V n i + j cos Since he coodinae sysem is movin wih he whee, evey poin on he im is fixed in ha coodinae sysem. 0 and 0 The x yz coodinae sysem aso oaes as he icyce whee compees a cice aound he ack: ˆ ˆ ˆ n V V i sin V j cos V + ρ ρ The oa oaion of he coodinae axes is epesened y: V ˆ ˆ ˆ i sin V j cos V k V ρ V iˆcos V ˆj sin V ρ V ˆ ˆ k cos V k sin V V + ˆk ρ ρ

12 0 V ˆ sin V cos V ˆ sin V cos V k k V ˆj sin V iˆcos V + ρ ( V ˆ sin V ˆ cos V V k k iˆ sin V ˆj cos V + + ρ ( k ˆ V n V ˆ ρ Since he oiin of he coodinae sysem is avein in a cice of adius ρ : A kˆ V ρ ( + A V V V V kˆ kˆ nˆ kˆ + + ρ ρ ρ V kˆ V ρ n ˆ Wih appopiae chane in coodinae noaion, his is he same esu as oained in Exampe

Circular Motion. Radians. One revolution is equivalent to which is also equivalent to 2π radians. Therefore we can.

Circular Motion. Radians. One revolution is equivalent to which is also equivalent to 2π radians. Therefore we can. 1 Cicula Moion Radians One evoluion is equivalen o 360 0 which is also equivalen o 2π adians. Theefoe we can say ha 360 = 2π adians, 180 = π adians, 90 = π 2 adians. Hence 1 adian = 360 2π Convesions Rule