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 0 0 5 N m ma0 N m m N m 0 4 4 5 5 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 4 4 5. (a Fcen m ( Fo, F m e ˆ ( F cen cen 500 s 000π s ( π 6 0 000 5eˆ 5π dynes ouwad Fcen m F ( π 000 5 5.04 0 m 980 5. 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 5.7 0 m T.005m cosθ 5.4 The non-ineia oseve hinks ha poins downwad in he diecion of he hanin pum o Thus 4
ˆ A j ˆ i 0 Fo sma osciaions of a simpe penduum: T π + 0.005 T π.995π.005 5.5 (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 5..4. 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
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,
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
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
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 (.7 0.866 o cosh T 0.866 5. 400 ˆj mph 586.67 ˆj f s 5 7.7 0 ( cos 4 ˆ j + sin 4 kˆ s ( 5 7.7 0 ( 586.67( sin 4 i ˆ F m co F m F F av co av 5 ( ( ( 7.7 0 586.67 sin 4 0.007 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
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 5.4. 8h x h cos λ hoiz z and y h 0. 5 8 50 f x h ( 7.7 0 s cos 4 f s x 0.404 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 ( 5 7.7 0 s ( 00 f sin 4 0.069 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
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
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 5.4.5 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
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 0 5.9 m qe + q ( v B Equaion 5..4 + + 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
Coecin ems and neecin ems in : x+ x cos + y+ y sin 0 x + x sin y+ y cos 0 5. 4 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
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 5... --------------------------------------------------------------------------------------------------