CHAPTER 1 FUNDAMENTAL CONCEPTS: VECTORS

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1 CHAPTER FUNDAMENTAL CONCEPTS: VECTORS. ( A + B ( iˆ+ ˆj + ( ˆj+ kˆ iˆ+ ˆj+ kˆ A+ B ( (b A B ( iˆ+ ˆj ( ˆj+ kˆ iˆ+ ˆj k ˆ (c AB (( + (( + (( (d iˆ ˆj kˆ A B iˆ( + ˆj( + kˆ( iˆ ˆj+ kˆ A B ( (( + ((4 + (( 6. ( A ( B C ( iˆ ˆj ( iˆ ˆj kˆ A+ B C iˆ+ ˆj+ k 4 ˆj (( + ((4 + (( 4 (b A ( B C ( ˆ 8 4 A B C A B C 8 A B C A C B A B C 4 iˆ+ k 4ˆj 4iˆ 8ˆj+ 4 kˆ (c ( ˆ A B C C A B C B A C A B ( iˆ+ ˆj 4( iˆ+ kˆ 4iˆ+ 4kˆ ( ( ( (

2 . AB ( ( + ( ( + (( 5 csθ AB θ cs ( A iˆ+ ˆj+ kˆ : bdy dignl A A A iˆ iˆ+ ˆj ˆj+ kˆ kˆ (b B iˆ+ ˆj : fce dignl B B B (d (c A B csθ AB C A B θ 9 iˆ ˆj kˆ.5 B B B A C ACsinθ Cy Csinθ A u A C ACcsθ u Cx Ccsθ A A B A u B A B C Cx + Cy A+ A B A A AB A u A B A A + A da ˆ d ˆ d d i αt + j βt + k γt iˆα + ˆ j βt+ kˆγt dt dt dt dt.6 ˆ ˆ j β + kˆ 6γ t dt d A

3 A B q q + q + q q+.7 ( q ( q, q A B A B A B A B A B A + B A + B + AB Since A B ABcsθ AB, A + B A + B A B ABcsθ A B csθ A B B csθ B A B C A C B A B C.9 Shw ( ( ( iˆ ˆj kˆ B B AC + A C + AC B A B + A B + A B C A B ( x y z x x y y z z x x y y z z Cx Cy Cz ( AxBC x x + ABC y x y + ABC z x z ABC x x x ABC y y x ABC z z xˆ i ( A ˆ xbc y x ABC y y y ABC z y z ABC x x y ABC y y y ABC z z y j ( A ˆ xbc z x ABC y z y ABC z z z ABC x x z ABC y y z ABC z z z k ( ABC ˆ y x y + ABC z x z ABC y y x ABC z z x i ( x y x z y z x x y z z y ( x z x y z y x x z y y z + ABC + ABC ABC ABC ˆj + ABC + ABC ABC ABC kˆ

4 iˆ ˆj kˆ A A A x y z B C B C B C B C B C B C y z z y z x x z x y y x ( y x y y y x z z x + z x z ˆj ( ABC z y z ABC z z y ABC x x y ABC x y x kˆ ( Ax BzCx AxBxCz AyByCz AyBzCy iˆ A B C A B C A B C A B C y Asinθ Α xy + y ( B x xy + yb xy AB sinθ Α A B. iˆ ˆj kˆ Ax Ay Az Bx By Bz A B C A B B B B B B A A A B A C x y z x y z x y z ( C C C C C C x y z x y z C C C x y z. x z u A C B y Let A (Ax,Ay,Az, B (,By, nd C (,Cy,Cz C z is the pependicul distnce between the plne A, B nd its ppsite. u B xc is diected lng the x-xis since the ects B, C e in the y,z plne. ux B xc ByC z is the e f the pllelg fed by the ects B, C. Multiply tht e ties the height f plne A, B A x t get the lue f the pllpiped V Au A B C A BxC x x x y z 4

5 . F ttin but the z xis: iˆ iˆ cs φ ˆj ˆj, kˆ k ˆ iˆ ˆj sinφ ˆji ˆ sinφ F ttin but the y xis: iˆ iˆ cs θ kˆ kˆ, ˆj ˆj iˆ kˆ sinθ kˆ iˆ sinθ csθ sinθ csφ sinφ csθ csφ csθ sinφ sinθ t T sinφ csφ sinφ csφ sinθ csθ sinθ csφ sinθ sinφ csθ.4 ˆ ˆ i i cs ˆ ˆ i j sin ˆ ˆ j i sin ˆ ˆ j j cs kˆ iˆ kˆ ˆj iˆ kˆ ˆj kˆ + Ax A y A z A.iˆ ˆj kˆ kˆ kˆ 5

6 .5. Rtte thu φ but z-xis φ 45. Rtte thu θ but x -xis θ 45. Rtte thu ψ but z -xis ψ 45 R φ R θ R ψ R θ R φ R ψ + α R( ψθφ,, RψRθRφ + R ( ψ, θ, φ β γ α Cnditin is: x Rx whee x nd x β γ Since x x we he: ψ + β + α Afte lt f lgeb:.6 ˆ τ ctˆ τ c t & ˆ τ + nˆ cˆ τ + nˆ ρ b α, 4 β +, γ 4 6

7 b t t, ˆ τ bc nd cˆ τ + cnˆ c c bc csθ bc c θ 45.7 ( t ib ˆ ω sin ( ωt + ˆj bωcs( ω t b t+ b t b + t t ib ˆ ω csωt ˆj bω sinω t ( ω sin ω 4 ω cs ω ω( cs ω ( bω + t sin ω t t, bω ; t t ib cs t jb sin t+ kct t ib ˆ ω sinωt ˆjbω csωt+ kc.8 ( ˆ ω ω ˆ ω ω ˆ ( ˆ π t, bω ω b t+ b t+ c b + c ( ω sin ω ω cs ω 4 ( ω 4 kt kt.9 e & ˆ ˆ ˆ + & θeθ bke e + bce êθ kt kt && & θ eˆ + && θ + & & θ eˆ b k c e eˆ + bcke ê θ kt ( + kt bk k c e bcke csφ be k + c be k c + 4c k kt kt k( k + c k ( k + c ( k + c ( k + c csφ. ( & φ ˆR ( & φ && φ R&& R e + R& + R eˆφ + &&ˆ ze bω eˆ + ceˆ R b + c 4 ω 4 z z, cnstnt θ 7

8 t i e + ˆje ( ˆ kt ˆ kt t ike + jke ˆ kt t ik e + ˆjk e. ˆ kt ( ( kt kt. eˆ & + eˆ φ sinθ + eˆ & θ φ θ π π eˆφbω sin + cs 4ωt eθb ωsin 4 t 4 π π eˆ b cs cs( 4 t eˆ φ ω ω θbω sin ( 4 t 8 ω ˆ ( ω π π bω cs cs 4ωt+ sin 4ω t 8 4 Pth is sinusidl scilltin but the equt.. d d + & dt dt & & 8

9 d d d dt dt dt d d ( + + dt dt + + ( & d ( ( & dt.4 ( ( + (.5 τˆ nd ττˆ+ ˆ nn τ, s τ n τ +, s n τ.6 F.4,.7 τ τ 4 ( b ω + 4c t b ω csωt sinωt+ b ω sinωt csωt+ 4c t ct b c n ω + 4 F.5, τ ( b ω cs ωt+ b ω sin ωt+ 4c t 4 6ct b ω + 4c t kt ( + kt be ( k + c bk k c e bcke kt kt bke ( k + c kt kt kt n b e k + c b k e k + c bce k + c ˆn ˆ τ dθ dˆ τ n ˆ dθ ρ 9

10 The figue be shws the unit ects ˆn nd ˆ τ which e nl nd tngent t the cue. The ects e shwn t diffeent pints lng the tjecty. Since the pticle is ing tngentilly t the cue nd in diectin pependicul t ρ, the lcl dius f cutue, we he ˆ τ & ˆ τ + ˆ& τ θ& nˆ n ˆ (since & nd f the be figue ρ ˆ τ nˆ ρ ρ ρ.8 ˆ sin ˆ P ib θ + jbcsθ ˆ cs ˆ el ib & θ θ jb & θ sin θ ˆ ( cs sin ˆ ( sin el ib && θ θ & θ θ jb && θ θ + & θ csθ t the pint θ π, el S, bθ & Nw, P el & θ && b θ & b b el ˆ ˆ ˆ el & elτ + n τ + ρ b + el + el 4 b + nd P el ˆ cs sin ˆ P i + b θ θ jb sinθ + cs θ b b b b ˆn 4 P + csθ + sinθ b b is xiu t θ, i.e., t the tp f the wheel. P

11 sinθ csθ b θ tn b Cents : Nte tht pint n the btt f the wheel is instntneusly t est, i.e., thee is n eltie tin between the gund nd the btt f the wheel ssuing n slipping. x -x x x x.9 %RR x x x x x Theefe, x The tnsftin epesents ttin f 45 but the z-xis (see Exple.8. θ b φ. ( iˆcsθ + ˆj sinθ b iˆcsϕ + ˆj sinϕ ( θ ϕ ( ˆ θ ˆ θ ( ˆ ϕ ˆ b cs ics + jsin ics + jsinϕ cs θ ϕ csθ csϕ+ sinθsinϕ (b b k ˆ sin ( θ ϕ ( i ˆ csθ + ˆ jsinθ ( i ˆ csϕ+ ˆ jsinϕ sin θ ϕ sinθcsϕ csθsinϕ

12 CHAPTER NEWTONIAN MECHANICS: RECTILINEAR MOTION OF A PARTICLE t F c x& F + ct dt t+ t t F c F c x t t dt t + + t 6. ( && x ( F + ct (b (c F && x sin ct t F F t F x& sin ct dt cs ct cs ct c c t F F x ( csct dt sinct c c c F ct && x e F ct t F ct x & e ( e c c F ct F ct x e t e c c c c ct. ( (b dx& dx& dx dx& && x x& dt dx dt dx dx& x& ( F + cx dx xdx & & ( F + cx dx cx x& Fx + x x& ( F + cx dx& && x x& F e dx cx

13 cx xdx & & F e dx F cx F x& e e c c cx F cx x& ( e c dx& && x x& F cs cx dx F xdx & & cs cx dx F x& sin cx c (c x. ( V ( x F x& sin cx c cx F + cx dx Fx + C x V x x cx F cx Fe dx e + C x c V x x F F cxdx x c cx+ C (b (c cs sin.4 ( F( x dv x kx dx x V ( x kx dx kx (b T T x + V x T x (c E T ka (d tuning T x T V( x k( A x x ± A kx A.5 ( F( x kx+ s (b T( x (c E T T V x T kx + x kx kx V x kx dx kx A 4 A kx 4 A 4 4

14 (d V ( x hs xiu t F( x kx kx x ± A A 4 ka V ( x ka ka 4 A 4 E < V x tuning pints exist. If ( Tuning T x ku E ku+ 4 A sling f u, we btin 4E u A ± ka 4E x ± A ka let u x.6 x& ( x.7 F x α x α x && x α α && x x& x x F Mgsinθ.8 dx& F x && x& dx x& bx dx& 4 bx dx 4 F ( bx ( bx F b x 7

15 V gx.45kg ft.48 54J s ft.9 ( (b g g T t c. D (.45kg 9.8 s T 87J kg ( (. ( (.66 t Fdx c dx c dt c t tnh dt (c τ tnh t t t ct τ + tnh d τ τ τ tnh t t ct τ + ln csh τ τ t Nw tnh f t >> τ τ t t Menwhile x dt t tnh dt τ t ln csh τ τ t x ln csh τ τ x ( 5 ft.48 8 ft t (.45kg 9.8 g s t 4.7 c kg (.(.7 s (.45kg τ.54s cg kg (.( s.8 Fdx (.(.7 ( 4.7 ( J ( 4.7(.54 V T 54J 87J 454J. Ging up: F gsin μgcs x 4

16 ( sin.cs && x g + s +t t the highest pint s tup.74 s xup t up + tup Ging dwn: x.87, xdwn t tdwn.7 s t t + t.8 s ttl up dwn, 9.8( dwn. d d dx d c dt dx dt dx c d dx xx c d dx c xx x x c. Ging up: Fx g c d c g k, k dx d dx g k x (Letting x stting f gund ln ( g k x k g+ k kx e g+ k g kx g + e k k 5

17 Ging dwn: F g+ c x d g dx +k d x dx g+ k ln ( g+ k x x k k kx kx e e g g g kx kx e e k k. At the tp s kxx e g + k Cing dwn x xx nd t the btt x g g g k ( k k g g + + k k t t +, g t k g c g k.4 d k x dx x kdx d b x k x b dx k k b x dt x b b x t x b x b b x dt dx d k b x k x b b b 6

18 Since x b, sy.5 Using x sin θ b ( d b sinθ sinθ csθ θ b t k cs b t π 8k d g c c s t, t dt π π θ k dt d g c c 4 dx cx + b b c ln, c + bx+ cx b 4c cx+ b+ b 4 t c c c + 4gc ln c gc c c c gc ( c gc sin θ dθ ( c + c+ c + 4gc( c c + 4gc ( ( t + 4 ln c c c gc c c gc t c+ c c + 4gc c c g t + + c c c Altentiely, when t, d g c t c t dt t c c c c c g c kg.6 5 c.. kg s s 7

19 t + + t.79 s Using Equtin.4. d dt d f g d dx.7 f ( x g( x dx dx dt F xt, f x g t : By integtin, get ( x If d x d dx dt dt dt 7 ( ( 9.8 g t. 5 c. s f ( x g( t This cnnt, in genel, be sled by integtin. F t, f g t : If d f g dt d g( t dt f Integtin gies ( t dx ( t dt dx t dt ( t A secnd integtin gies x x( t.8 d F kx dt d d kx dt dx kxdx d x k +C At x, t C 8

20 x dx k + dt dx dx dt kx + A + B x tn B t x AB A Sling whee A B k x tn k α x&.9 F x Ae Let u e α Integting du t α &&x αe α d k d A F Ae & dt e α du du d α αe du α A dt αu u A t u u α nd substituting e α A ( ln e α + αt α (b t A α α ln + e αt α A α α e + e αt T e α A d A α d A (c e α dx dx e du gin, let u e α du αud d αu du ln u α αu A dx Integting nd sling u α x ( + α α A e u ln u α. d F + g dt 9

21 but 4 ρ π ρπ s ( πρ+ πρ π πρg ρ Nw ρ s, secnd te is negligible-sll hence g nd gt speed t but ρ ρ & ρ 4π & ρ π & Hence gt nd te f 4 ρ 4 ρ gwth t The exct diffeentil equtin f ( be is: 4 4ρ 4ρ & 4 πρ && + πρ πρ g ρ ρ & ρ which educes t: && + g 4ρ Using Mthcd, sle the be nn-line d.e. letting ρ nd R. (sll indp. Gphs ρ shw tht & t nd t

22 . x.sin π ( 5 s t [ ] CHAPTER OSCILLATIONS x& x (.( π ( s s 4 && xx. π 5.7 s s. x.sinω t [] x&. ω cs ω t s When t, x nd x&.5.ω s π ω 5s T.6 s ω x&. x( t x csω t+ sinω t nd ω π f ω ( π ( π [ ] x.5cs t +.59sin t.4 cs α β csαcs β + sinαsin β x Acs ω t φ Acsφcsω t+ Asinφsin t ω x Α csωt+βsinω t, Α Acsφ, Β Asinφ.5 x& + kx x& + kx k x x x& x& ( ( & k x& x ω x x ka x& + kx xx& xx& A x& + x + k x& x& & xx& x& x& xx A x

23 .6 l T π π s.5 s g F spings tied in pllel: F x k x k x k + k x ω s k + k F spings tied in seies: The upwd fce is keq x. Theefe, the dwnwd fce n sping k is keq x. The upwd fce n the sping k is kx whee x is the displceent f P, the pint t which the spings e tied. Since the sping is in equilibiu, kx k x. k eq Menwhile, The upwd fce t P is kx. k x x. The dwnwd fce t P is Theefe, kx k ( x x And eq kx x k + k kx k x k k+ k ω k eq kk ( k+ k.8 F the syste ( M +, kx ( M + X&& The psitin nd cceletin f e the se s f ( M k && x x M + k k x Acs t+ δ dcs t M + M + The ttl fce n, F x && g F + :

24 F g + k x kd g + cs k t M + M + M + F the blck t just begin t lee the btt f the bx t the tp f the eticl scilltins, F t x d : kd g M + g( M + d k γ t.9 x e Acs( ω t φ d dx γt γt e Aω d sin ( ωdt φ γe Acs( ωdt φ dt dx xi t ωd sin( ωdt φ + γ cs( ωdt φ dt γ tn ( ωdt φ ω d thus the cnditin f eltie xiu ccus eey tie tht t inceses by π ω : d π ti+ ti + ω d γ t F the i th xiu: x e i Acs( ω t φ i d i t γ γ i + ω cs( d i+ ωd i+ φ i x e A t e x xi x i+ π γ ω d γ Td e e π. ( (b (c c γ s ω ω γ 6 s d ω 7 s F 48 Ax. Cω 6.4 d γω γω ω tnφ γ γ ( ω ω k ω 5s ω ω γ 7 d s 7 φ 4.4

25 . ( 7 x && + βx& + β x 7 γ β nd ω β ω ω γ 4β ω β (b A F x γω d A 5β ω ω γ β 4 5 d 5 ωd β. ( S, e γ T d γ ln fd ln T d ω ( ω γ d ω ( ω + γ d γ ln f fd + fd + π π f.6hz (b ω ω γ d γ ln f fd fd π π 99.4Hz f d. Since the plitude diinishes by e γ T γt n d ( e e e γ Tn d ωd γ Tn π n Nw ω ω γ d S ( d d ω ω + γ ω + 4 π n d in ech cplete peid, 4

26 π Td ωd ω T π + ωd 4π n ω T F lge n, d + 8 T π n.4 ( (b (c ω.77ω ω ω γ ω ω ω ( ω γ d 4 Q.866 γ γ ω ω ( ω γω tnφ ω ω ω 4ω φ tn 46. ω ω ω ω + ω ω F F A ω.77 D ω ω (d D.5 A( ω f x Axγ ( ω ω + γ γ A ω A, ( ω ω + γ ω ω + γ 4γ ω ω ± γ ω ω ± γ 5

27 .6 (b (c ω ω γ d Q γ γ ω LC, γ R R L LC 4L L Q R RC 4 L L ω C R Q γ R R i t.7 Fext F sinωt I Fe ω nd x( t is the iginy pt f the slutin t: i t x && + cx& + kx F e ω i( ωt φ i.e. xt ( I Ae Asin ( ωt φ whee, s deied in the text, F A ( k ω + c ω nd γω tnφ ω ω t.8 Using the hint, Fext Re( Fe β, whee β α + iω, nd x(t is the el pt f the slutin t: x && + cx& + kx F e βt. t i Assuing slutin f the f: x Ae β φ ( F c k x xe i φ β + β + A F α i αω ω c α + ic ω + k ( cs isin A φ + φ F ( α ω cα + k csφ A ω ( F α+ c s A inφ 6

28 φ tn ω ( c α Using sin φ + cs φ, F A α ω cα +k ( α ω cα + k + ω ( c α A F { ( α ω cα k + + ω ( c α } αt nd x( t Ae cs( ωt φ +the tnsient te..9 ( (b (c l A T π g 8 f A π l, T π.4 4 g 4π l g.84 T l 4π l Using T π gies g, ppxitely 8% t sll. g T λ A B ω ω nd λ 6 B A A 9 f A π B,. 4 A in t. f ( t c e ω n, ±, ±,... n n n n,,,, n n f ( t c cs nω t+ c isin nω t T in t ω nd cn T f ( t e dt T, n, ±, ±,... T T i T cs ω T T n ± ±... cn f t n t dt f t n t T ( ( sin ( ω dt The fist te n cn is the se f n nd n ; the secnd te chnges sign f n s. n. The se hlds tue f the tignetic tes in f t. Theefe, when tes tht cncel in the sutins e discded: 7

29 T f ( t c + T f ( t cs( nω t dtcs nωt n T T + T f ( t sin ( nω t dtsin nωt n T, T T T n ±, ±,..., nd c f ( t dt Nw, due t the equlity f tes in ± n : T f ( t c + T f ( t cs( nω t dtcs nω t n T T + T f ( t sin ( nω t dtsin nωt n T n,,,... Equtins.9.9 nd.9. fllw diectly., in t. f ( t c e ω, c ( n n n T inω t T f t e T π ω ω inω t π f t e π ω π T s cn ( ω dt dt, nd n, ±, ±, π ω inωt ω inωt π ( e dt e dt π + ω π ω in t ω ω inωt e e π inω π inω ω + inπ inπ e e + π in inπ inπ F n een, e e nd the te in bckets is ze. inπ inπ F n dd, e e 4 c n πin, n ±, ±,... 4 inω t f ( t e, n ±, ±,... n π in 4 inωt inωt ( e e, n,, 5,... π n i n 4 sin ( nω t, n,, 5,... n π n f 4 ( t sinω t sin ω sin 5 t ω π 5 t K 8

30 in t n. In stedy stte, xt ( Ae ω φ n n Fn An ( ω n ω + 4γ n ω 4F Nw Fn, n,, 5,... nd ω ω nπ ω Q 9ω s γ γ 4, 4F A π 9ω ω ( 9ω ω F A πω 4F A π 9ω ( 9ω 9ω + 4 4F A 7πω 4F A5 5π ω ( 9ω 5ω + 4 ( 5ω F A5 πω i.e., A : : A : 9.6 :. A 5. ( && x+ ω x y x& Thus y& ω x x& y diide these tw equtins: y& dy ω x x& dx y (b Sling ydy xdx ω + nd Integting Let C A y x + ω A A n ellipse y x ω + C 9

31 .4 The equtin f tin is F ( x x x x &&. F siplicity, let. Then ` ( (b (c && x x x. This is equilent t the tw fist de equtins x& y nd y& x x The equilibiu pints e defined by x x x x + x Thus, the pints e: (-,, (, nd (+,. We cn tell whethe nt the pints epesent stble unstble pints f equilibiu by exining the phse spce plts in the neighbhd f the equilibiu pints. We ll d this in pt (c. dy y& x x The enegy cn be fund by integting dx x& y ydy x x dx+ C 4 y x x + C 4 4 y x x In the wds E T V + + C 4. The ttl enegy C is cnstnt. The phse spce tjecties e gien by slutins t the be equtin 4 x y ± x + C. The uppe ight qudnt f the tjecties is shwn in the figue belw. The tjecties e syeticlly dispsed but the x nd y xes. They f clsed pths f enegies C< but the tw pints (-, nd (+,. Thus, these e pints f stble equilibiu f sll excusins wy f these pints. The tjecty psses thu the pint (, f C nd is sddle pint. Tjecties nee pss thu the pint (, f psitie enegies C>. Thus, (, is pint f unstble equilibiu

32 .5 && θ + sinθ Integting: T 4 & θ & θ d θ csθ dt csθ θ θ dθ csθ csθ θ & θ ( csθ csθ T Tie f pendulu t swing f θ t θ θ is 4 Nw substitute θ sin sinφ θ sin s π φ t θ θ θ dθ nd use the identity csθ sin T 4 θ θ 4 sin sin dφ dθ nd fte se lgeb θ θ θ sin 4 sin sin ( π dφ θ T 4 whee α sin α sin φ 4 αsin φ + αsin φ + α sin φ + K 8 (b π 4 T 4 dφ + αsin φ + α sin φ + K 8 (c θ θ θ θ sin 48 4 α + K... T θ α 9 π α θ T π + + K K

33 CHAPTER 4 GENERAL MOTION OF A PARTICLE IN THREE DIMENSIONS Nte t instucts thee is typ in equtin The nge f the pjectile is sin α R x NOT... sin α g g ( ˆ V ˆ V ˆ V F V i j k x y z F c( iyz ˆ + ˆjxz + kxy ˆ (b F V iˆα x ˆjβy kˆ γz ( αx+ βy+ γz (c F V ce ( iˆα + ˆjβ + kˆ γ V V V (d F V eˆ ˆ ˆ eθ eφ θ sinθ φ n F eˆ cn 4. ( (b (c (d iˆ ˆj kˆ F x y z x y z iˆ ˆj kˆ F x y z kˆ ( y x z iˆ ˆj kˆ F kˆ ( x y z y x z cnsetie ScnsetieS cnsetie

34 eˆ ˆ ˆ eθ eφsinθ F sinθ θ φ k n cnsetie 4. ( (b iˆ ˆj kˆ F k cx x x y z xy cx z cx x c ( iˆ ˆj kˆ F x y z z cxz x y y y ˆ x cx ˆ ˆ cz z i j k y y y y y y x cx y y c ls cz z + y y iplies tht c s it ust V x, y, z + t the igin E ( E cnstnt t (,, E α + β + γ +

35 (b α + β + γ α β γ + + α + β + γ α β γ + + (c V x && Fx x x && α V y && βy y V z && γ z z 4.5 ( F ix ˆ + ˆjy n the pth x y : d idx ˆ + ˆjdy (, F d (, F xdx + F ydy xdx + ydy n the pth lng the x-xis: d idx ˆ nd n the line x : d ˆjdy (, F d F dx+ F dy, x y F is cnsetie. (b F iy ˆ ˆjx n the pth x y : (, F d (, F xdx + F ydy ydx xdy (, nd, with x y F d xdx ydy n the x-xis: nd, with y n the x-xis n the line x :, (,,, x (, F d F dx ydx F d F d F dy xdy, y,

36 nd, with x n this pth (, (, F is nt cnsetie. (, F d dy (, F d F Exple.., e V z g z ( + F Appendix D, e z V( z ge + e + x x+ x +K z z V ( z ge + + K e e gz V ( z ge + gz +K With ge n dditie cnstnt, z V ( z gz e F V kˆ V( z z ˆ z kg + z e e ˆ z F kg e x && F x, y && F y dz& z z& g dz e h z zdz & g dz z e h g h z e e z h h e + g e z z && g e 4

37 h g e e z h g e e z e ( hz F Appendix D, ( x h + g + 4g + e, e x x + + +K 8 4 e e z z K z z h + g ge F Exple.., h And with ( x + x, e g g e h + g g e 4.7 F pint n the i esued f the cente f the wheel: ib ˆ csθ ˆjbsin θ t θ ωt, s & i ˆ sinθ ˆ jcsθ b i ˆ sinθ ˆj csθ Reltie t the gund, F pticle f ud leing the i: y bsinθ nd y csθ S y y gt csθ gt nd y bsinθ t csθ gt At xiu height, y : csθ t g csθ csθ h bsinθ csθ g g g h bsinθ + Mxiu h ccus f cs θ g dh b csθ dθ csθ sin g θ 5

38 gb sinθ 4 g b cs θ sin θ 4 4 gb g b gb hx + + g g Mesued f the gund, gb h x b+ + g gb The ud lees the wheel t θ sin 4.8 x Rcsφ R csφ s t csα nd ( csα y Rsinφ x t t x y t gt t gt nd ( sinα y Rcsφ Rcsφ Rsinφ ( sinα g φ csα csα gr cs φ sinφ tnαcsφ cs α cs cs R α α ( tnα csφ sinφ ( sinαcsφ csαsinφ gcs φ gcs φ sin θ + φ sinθ csφ + csθ sinφ csα R sin ( α φ g cs φ dr sinα sin α φ csαcs α φ dα gcs φ + csαcs α φ sinαsin α φ cs θ + φ csθ csφ sinθ sinφ cs α φ F Appendix B, R is xiu f Iplies tht F ppendix B, s R π π φ α φ α + 4 π φ π φ cs + sin g cs φ 4 4 x α 6

39 4.9 π φ π π φ π φ Nw sin cs cs π φ Rx cs + g cs φ 4 Agin using Appendix B, cs θ cs θ sin θ cs θ π Rx cs φ π + + g cs φ cs + φ + g cs φ Using cs π + θ sin θ, Rx sinφ g sin φ R x g ( + sinφ ( Hee we nte tht the pjectile is lunched dwnhill twds the tget, which is lcted distnce h belw the cnnn lng line t n ngle φ belw the hizn. α is the ngle f pjectin tht yields xiu nge, RBxB. We cn use the α esults f pble 4.8 f this pble. We siply he t eplce the ngle φ in the be φ esult with the ngle -φ, t ccunt f the dwnhill h slpe. Thus, we get f the dwnhill nge RBxB csα sin ( α + ϕ R g cs ϕ The xiu nge nd the ngle is α e btined f the pble be gin by ( + sinϕ π eplcing φ with the ngle -φ Rx nd α ϕ. g cs ϕ h ( + sinϕ We cn nw clculte α Rx sinϕ g cs ϕ g sinϕ gh gh Sling f sinϕ sinϕ + π But, f the be sinϕ sin α cs α sin α gh gh Thus sin α + 7

40 Finlly gh gh α + gh + sin csc α csc α + gh (b h h h Sling f RBxB Rx sinϕ sin α csc α Substituting f R csc α nd sling gh x + g 4. We cn gin use the esults f pble 4.8. The xiu slpe nge f pble 4.8 is gien by h Rx g ( + sinϕ sinϕ Sling f sinϕ gh gh sinϕ Thus csϕ xx Rx csϕ h sinϕ We cn clculte csϕ f the be eltin f sinϕ gh gh csϕ ( sin ϕ Inseting the esults f sinϕ nd csϕ int the be x csϕ gh x h sinϕ g 4. We cn siplify this pbles sewht by nting tht the tjecty is syetic but eticl line tht psses thugh the highest pint f the tjecty. Thus we he the fllwing pictue 8

41 xbb hbb (8 ssuing xbb ( hbb, is BB (.8 z zbxb α δ x We he eesed the tjecty s tht hbb 9.8 ft, nd xbb the height nd nge within which Mickey cn ctch the bll epesent the stting pint f the tjecty. hbb ft is the height f the bll when Mickey stikes it t he plte. δ is the distnce behind he plte whee the bll wuld be hyptheticlly lunched t se ngle α t chiee the ttl nge R. xbb ft is the distnce the bll ctully wuld tel f he plte if nt cught by Mickey. (Nte, becuse f the syety, BB the speed f the bll when it stikes the gund ls t the se ngle α t which ws lunched. We will clculte the lue f xbb tie-eesed tjecty! sin α sinαcsα ( The nge f the bll R g g ( The xiu height R x tn g R z α cs α g ( The height t xbb h xtnα x cs α g tnα F ( nd inseting this int ( gies cs α R R R R zx tnα tnα tnα 4 4 4zx Thus, R nd inseting this expessin nd the fist peiusly deied int ( tnα x tnα (4 h xtnα 4zx Let u x tnα nd we btin the fllwing qudtic u 4z u+ 4z h nd sling f u x x u z x ± h zx nd letting h ε z R x, we get 9

42 using u z ε h x ( ε u z z.475.9z. This esult is the cect ne x x x.9zx Thus, tnα.8 α 9.4 x Nw sle f xbb eltin identicl t (4 h x tnα ( x tnα 4z x Agin we btin qudtic expessin f u x tnα which we sle s befe. This tie, thugh, the fist esult f u is the cect ne t use u z ε h nd we btin x x h tnα.9 ft 4. The x nd z psitins f the bll s. tie e x t cs θ z t cs θsinθ gt Since x cs θ The hizntl nge is R g dr The xiu nge dθ cs θ sin θ dr cs θcsθ cs θsin θsinθ dθ g Thus, cs θ cs θ cs θ sin θ sin θ Using the identities: cs θ + cs θ nd sin θ sinθcsθ We get: + csθ cs θ sinθ sinθ csθ cs θ csθ ( + csθ ( cs θ csθ Thus csθ, csθ ( ± 6 Only the psitie t pplies f the θ -nge: θ csθ θ Thus (b f 5s Rx θ 9 5 π

43 dz ( The xiu height ccus t dt cs θ sinθ cs θsinθ gt t T g H cs θsin θ g xiu t fixed θ dh The xiu UpssibleU height dα dh cs θsinθcsθ cs θsin θsin θ dα g Using the be tignetic identities, we get ( + csθ sin cs sin sin sin ( cs θ θ θ θ θ θ sinθ( + csθ( csθ Thee e -ts: sinθ, csθ, csθ The fist tw ts gie iniu heights; the lst gies the xiu Thus, Hx θ cs 7 4. The tjecty f the shell is gien by Eq. 4.. with eplcing x z& g z whee & csθ z& sinθ & & g Thus, z tnθ sec θ θ + Since sec tn θ We he: g g tn θ tn θ + z+ (,z e tget cdintes. The be equtin yields tw pssible ts: 4 tnθ gz g ± g The ts e nly el if 4 gz g The UciticlU sufce is theefe: 4 gz g 4.4 If the elcity ect, f gnitude s&, kes n ngle θ with the z-xis, nd its

44 pjectin n the xy-plne ke n ngle φ with the x-xis: x& s& sinθ csφ, nd F F sinθ csφ x && y& s& sinθ sinφ, nd F F sinθ sinφ y && z& s& csθ, nd F g + F csθ z && Since F cs c( x + y + z x y z & & & &, the diffeentil equtins f tin e nt sepble. x && cs& sinθ csφ csx && dx& dx& ds dx& s& csx && dt ds dt ds dx & c c ds γ ds, whee γ x& ln x ln x ln x & & & γ s x& Siilly x& xe & γ s y& ye & γ s z θ s & φ x y z& g γxx g x 4.5 F eqn 4..6, ln γx + + γ γ x& γ x& u u F Appendix D: ln ( u u K f u < γx x γxx γ xx γ xx 4 ln + tes in γ x& x& x& x& z& xx gxx gxx gxx g xx x + γ γx γx x x + tes in γ & & & & & x x z x + & x xx γ & & gγ x x& 9x& x& z& x ± + 4γ 6γ gγ x & x & 6γ xx z & ± + 4γ 4γ g Since x x >, the + sign is used. F Appendix D: 6γz& 8γz& 6γz& tes in g g 8 g x x x z 8x z x & γ x 4γ + & 4γ + & & g & & g + tes in γ γ

45 F xz 8xz x & & x g & & g γ + K z& sinα nd xz & & sinα : sin α 4 sin αsinα x γ + K x g g 4.6 y x Acs( ωt+ α, x& Aω sin ( ωt+ α f x &, α f x A, x Acsωt x y Bcs( ωt+ β, y& ωbsin ( ωt+ β kb ky + y& with k y 4A, y& ω A nd ω : B 6A + ( 9ω A 5A ω B 5A Then 4A 5Acsβ nd ω A 5ωAsinβ 4 β cs sin y 5Acs( ωt 6.9 Since xiu x nd y displceents e ± A nd ± 5A, espectiely, the tin tkes plce entiely within ectngle f diensin A nd A. Δ β α AB csδ F eqn 4.4.5, tn ψ A B 4 ( A( 5A cs ( tn ψ A 5A 4 ψ tn V && x k x Acs t Acs( t α & + π + α x Fx kx π x

46 V y && 4π x y y Bcs( π t+ β V z && 9π z z z Ccs( π t+ γ π Since x y z t t, α β γ π x Acs πt Asinπt x& Aπ csπt Since x& + y& + z& nd x& y& z&, x& Aπ A π x sinπt π y Bsin π t, y& Bπ csπ t y& π B B π y sin π t π z Csin π t, z& Cπ csπ t z& Cπ C π z sin π t π Since ωx π, ω y π, nd ω z π the bll des etce its pth. π n π n π n tin ω ω ω x y z The iniu tie ccus t n, n, n. 4

47 t in π π AB csδ 4.8 Equtin is tn ψ A B Tnsfing the cdinte xes xyz t the new xes x yz by ttin but the z-xis thugh n ngleψ gien, f Sectin.8: x xcsψ + ysinψ, y xsinψ + ycsψ, x x csψ y sinψ, nd y x sinψ + y csψ x csδ y F eqn. 4.4.: xy + sin Δ A AB B Substituting: ( x cs ψ xy csψsinψ + y sin ψ A csδ x csψ sinψ + xy ( cs ψ sin ψ y csψsinψ AB + ( x sin ψ + xy csψsinψ + y cs ψ sin Δ B F x t be j in xis f the ellipse, the cefficient f x y ust nish. cs ψ sin ψ cs Δ cs sin ( cs ψ sin ψ + ψ ψ A AB B F Appendix B, csψ sinψ sinψ nd cs ψ sin ψ cs ψ sin ψ cs Δ cs ψ sin ψ + A AB B csδ tn ψ B A AB AB csδ tn ψ A B 4.9 Shwn belw is fce-centeed cubic lttice. Ech t in the lttice is centeed within cube n whse 6 fces lies nthe djcent t. Thus ech t is suunded by 6 neest neighbs t distnce d. We neglect the influence f ts tht lie t futhe distnces. Thus, the ptentil enegy f the centl t cn be ppxited s 6 V c α i i d 5

48 d x + y + z α α α x x y z α + + ( d dx x y z d d d n F Appendix D, ( + x + nx+ n( n x +K 4. α α α x x + y + z α α d + + d d x x x + y + z x + y + z + + tes in d d d d α α αx α α α 4x d + x + y + z d d 4 tes in d α α αx α α α d + ( x + y + z + x + d d d ( d x y z d dx x y z α α α x x + y + z d + + d d α α αx α α α d ( x + y + z + x + d d d α α α α α + d ( x + y + z + ( α + x d d Siilly: α α α α α + 4 d x + y + z + y α + d d α α α α α d ( x + y + z + ( α + z d d α α α α V cd 6 ( x + y + z + ( x y z d d d α α 6cd + cd α α x + y + z V A+ B x + y + z x d x d 6

49 î x qb y && Fy qe qxb & qe qb x& + y & qe qbx& qb ee ebx eb && y y + y ee eb && y+ ω y + ωx&, ω ee y x Acs ( t ω + + ω ω + θ y& Aω sin ( ωt+ θ y &, s θ ee y, s A x ω ω + & ee y ( csωt, x ω ω + & qb x& x& + y x& ω y x& ω csωt x& ( x& ω + ωcsωt x ( x& ω t+ sinωt x sinωt+ bt, b x& ω z && F z z z& t+ z z ˆkB ĵe y F q( E+ B B ( ix ˆ& + ˆjy & + kz ˆ& kb ˆ iyb ˆ& ˆjxB & F iqyb ˆ & + ˆjq( E xb & x && Fx qyb & qb x& x& y 4. y b h x b + qh g g b h F gcsθ + R b 7

50 h csθ b h g g R g h b h h b b b b b the pticle lees the side f the sphee when R b b h, i.e., be the centl plne 4. gh + t the btt f the lp, h b b h s gb, gb F g+ R b R g + g + g g b 4. F the equtin f the enegy s functin f s in Exple 4.6., g E s& + s, 4A s is undeging hnic tin with: " k" g g ω 4A A Since s 4Asinφ, φ inceses by π dins duing the tie intel: π T π A ω g F cyclidl tin, x nd z e functins f φ s they undeg cplete cycle eey tie φ chnges by π. Theefe, the peid f the cyclidl tin is ne-hlf the peid f s. A T T π g

51 CHAPTER 5 NONINERTIAL REFERENCE SYSTEMS 5. ( The nn-inetil bsee beliees tht he is in equilibiu nd tht the net fce cting n hi is ze. The scle exets n upwd fce, N, whse lue is equl t the scle eding --- the weight, W, f the bsee in the cceleted fe. Thus N ( A (b N+ g A g 5 N g A N g N g W N g W 4 4 W 5 lb. g W 9 lb. (b The cceletin is dwnwd, in the se diectin s g g N g W + W W W ( Fcent ω ( ω F ω, F eˆ (b F cent cent ω ω 5 s π s ( π 5eˆ 5π dynes utwd Fcent F g 6 ( π ω g g + T A (See Figue 5.. ˆ ˆ ˆ g g j+ Tcsθ j+ Tsinθi i ˆ g Tcsθ g, nd T sinθ tnθ, θ 5.7 g T.5g csθ 5.4 The nn-inetil bsee thinks tht g pints dwnwd in the diectin f the hnging plub bb Thus 4

52 ˆ g g g A g j ˆ i F sll scilltins f siple pendulu: T π g g g g +.5g T π.995π.5g g 5.5 ( f μg is the fictinl fce cting n the A f (b ( bx, s f A ( is the cceletin f the bx eltie t the tuck. See Equtin 5..4b. Nw, f the nly el fce cting hizntlly, s the ccetin eltie t the d is f μ g μg g (F + in the diectin f the ing tuck, the indictes tht fictin ppses the fwd sliding f the bx. g A (The tuck is deceleting. f be, A s g g g A + 6 i x R t ˆ + Ω + jrsin Ωt & iˆωrsin Ω t+ ˆjΩRcsΩ t & & Ω R Ω R cicul tin f dius R 5.6 ( ˆ( cs (b & & ω whee ix ˆ + ˆjy iˆωrsin Ω t+ ˆjΩRcs Ωt ωkˆ ix ˆ + jy ( ˆ

53 iˆωrsin Ω t+ ˆjΩRcsΩt ˆjω x + iˆω y x& ω y ΩRsin Ωt y& ωx +ΩRcsΩt (c Let u x + iy hee i! u& x& + iy& ω y ΩRsin Ωt iω x + iωrcs Ωt iω u ω y + iω x sin cs Re i Ω u& + iωu ΩR Ω t+ iωr Ω t iω t Ty slutin f the f i t i t u Ae ω Ω + Be ω u& i Ae + iωbe ω iωu iωae + iωbe i t i t ω Ω i t iωt i t ( ω u& + iωu i +Ω Be ω s ΩR B ω + Ω Als t t the cdinte systes cincide s u A+ B x + iy x + R Ω A x + R B x + R s, ω + Ω ωr iωt ΩR iωt Thus, u x + e + e ω +Ω ω+ω R ωr A x + ω + Ω 5.7 The x, y fe f efeence is ttched t the Eth, but the x-xis lwys pints wy f the Sun. Thus, it ttes nce eey ye eltie t the fixed sts. The X,Y fe f efeence is fixed inetil fe ttched t the Sun. ( In the x, y tting fe f efeence x( t Rcs( Ω ω t R ε (b x& ( t ( Ω ω Rsin ( Ω ω t t t ( ( Ω ω cs( Ω ω ( Ω sin ( Ω ω y t R t whee R is the dius f the steid s bit nd R E is the dius f the Eth s bit. Ω is the ngul fequency f the Eth s elutin but the Sun nd ω is the ngul fequency f the steid s bit. y& t R t ω R t t (c A A Ω & Ω & Ω Ω ε

54 Whee is the cceletin f the steid in the x, y fe f efeence, A, A ε e the cceletins f the steid nd the Eth in the fixed, inetil fe f efeence. st : exine: A Aε Ω Ω ω ω R Ω Ω Rε Ω Ω R R ω ω Ω Ω R ω Ω R ( ε Thus: Ω ω R Ω nte: ω ω ˆk, Ω Ω ˆk Theefe: ( ix ˆ&& + ˆjy && ( Ω ω Ω Ω ir ˆ cs( ω t ˆjR sin ( ω t ˆ ˆ jω x& + iωy& Thus: && x Ω ω Rcs Ω ω t+ Ω&y ( ω ( ω && y Ω Rsin Ω t Ωx & Let && x ( Ω ω y& nd && y ( Ω ω Then, we he Ω y& ( Ω ω Rcs( Ω ω t+ Ω y & which educes t ( ω cs( ω y& Ω R Ω t Integting ( ω ( ω y Rsin Ω t t t Als, x& Ω ω Ω ω Rsin Ω ω t Ωx& Ω+ sin Ω + Ω&x x& ( ω R ( ω t x& ( Ω ω Rsin ( Ω ω t Integting cs( ω x R Ω t+cnst x Rcs( Ω ω t R R R t t ε ε 5.8 Reltie t efeence fe fixed t the tuntble the cckch tels t cnstnt speed in cicle. Thus y ω eˆ. b Since the cente f the tuntble is fixed. b x A x& 4

55 The ngul elcity, ω, f the tuntble is cnstnt, s ω ω ˆk, with & ω be ˆ, s ω ( ω bω e ˆ e ˆ θ, s ω ω e ˆ F eqn 5..4, + ω + ω ( ω nd putting in tes f be ω bω b F n slipping F μsg, s μsg + ω + bω μsg b + ωb + b ω bμ g s ± + ωb ω b b ω bμsg Since ws defined psitie, the +sque t is used. ωb+ bμ g s (b e ˆ θ ω + ω e ˆ + ω bω b ω + bω μsg b ωb+ bμ g s 5.9 As in Exple 5.., V ω ˆ j nd A ρ F the pint t the fnt f the wheel: V && ˆ j b nd Vk ˆ & ω V ˆ ( ˆ Vb ω k bj i ˆ ρ ρ V ˆ V ˆ Vb ˆ V b ω ω k i ˆj ρ ρ ρ V ω kˆ ( Vkˆ ρ i ρ V ˆ V V b && + ω ω + A i + + ˆ j ρ b ρ 5

56 5. (See Exple 5.. ω x x && ω x t Ae + Be t ωt ( ωt ωt ω ω x& t Ae Be Bundy Cnditins: l x A+ B x& ω A B l l A B 4 4 l l ( x ( t cshωt x& ( t ω sinhωt l l l (b x ( T + cshωt when the bed eches the end f the d.7 cshωt T csh ω ω l (c x& ( T ω sinhωt l l ω sinh csh ω (.7.866ωl l ωl ω csh ωt.866ωl 5. 4 ˆj ph ˆj ft s ( cs 4 ˆ ω j + sin 4 kˆ s ( 5 ω 7.7 ( ( sin 4 i ˆ F ω c F g F F g c g 5 ( sin 4.7 The Cilis fce is in the 5. (See Figue 5.4. ω ω ˆj + ω kˆ y z ft s ω diectin, i.e., + î est. 6

57 ˆ ˆ x i + y j ω ω ˆ ˆ z y i + ωz x j ωy x kˆ ω ω i + ω j ˆ ˆ hiz z y z x ( ω ( ω hiz z y + ωz x ωz ( x + y ω z Fc ω F ω ω, independent f the diectin f. ( c hiz 5. F Exple h x h ω cs λ g hiz z nd y h ft x h ( 7.7 s cs 4 ft s x.44 ft t the est. h 5.4 F Exple 5.4.: ωh Δ sin λ is the deflectin f the bsebll twds the suth since it ws stuck due Est t Ynkee Stdiu t ltitude λ 4 N (pble 5.. is the initil speed f the bsebll whse nge is H. F eqn 4..8b, withut i esistnce in n inetil efeence fe, the hizntl nge is sin α H g Sling f gh sin α ft s ft f sin ( s ( ft t s Δ sin 4.69 ft. in ft s A deflectin f. inches shuld nt cuse the utfielde ny difficulty. 5.5 Equtin 5.. gies the eltinship between the tie deitie f ny ect in fixed nd tting fe f efeence. Thus &&& d d + ω dt dt fixed t 7

58 && + & ω + ω & + ω ( ω d &&& + && ω + & ω & + & ω & + ω && dt t + & ω ( ω + ω ( & ω + ω ( ω & ω ω && + ω ( & ω + ω ( ω & + ω ω ( ω Nw ( ω is t ω nd. Let this define diectin ˆn : ω ω nˆ Since ω ˆn, ω ( ω is in the plne defined by ω nd nd ω ( ω ω nˆ ω ω ω. Since ω ω ( ω ω ω ( ω ω ω And ω ω ( ω is in the diectin ˆn Thus ω ω ( ω ω ( ω ω ω && + ω ( & ω + ω ( ω & ω ( ω &&& &&& + && ω + & ω & + ω && + & ω ( ω + ω & ω + ω ω & ω ω 5.6 With x y z x& y&, nd z& x ( t ωgt cs λ ωt cs λ y t ( z ( t gt + t, s When the bullet hits the gund z ( t t g Equtins c bece: 8 4 x g cs cs ω g λ ω g λ 4ω x cs λ g x is negtie nd theefe is the distnce the bullet lnds t the west. 5.7 With x y z nd x& csα y & z& sinα we cn sle equtin 5.4.5c t find the tie it tkes the pjectile 8

59 t stike the gund z ( t gt + tsinα + ωt csαcsλ sinα sinα t g ω csαcsλ g We he igned the secnd te in the denint since wuld he t be ipssibly lge f the lue f tht te t ppch the gnitude g F exple, f λ 4 nd α 45 g ω csαcsλ g ω g k 44! ω s Substituting t int equtin 5.4.5b t find the ltel deflectin gies 4ω y& ( t [ ω csαsin λ] t sin λsin αcsα g 5.8 Let cceletin f bject eltie t Eth ω ω ˆk its ngul speed y x A cceletin f stellite R ω ω ˆk its ngul speed R + ω + ω ( ω + A (Equtin 5..4 A ω ω ω As in pble 5.7 Elute the te Δ A ω ω ω ω R ω ω R ω ω R R Δ ω ω R ( gien the cnditin tht R Δ ω R R R R R+ R+ R + + Rcsθ but Letting x csθ ω R ω R x f sll R R R Rx R x nd R R + R R x + R R x R ˆ ω R Δ + ω x ω xi f sll R R R 9

60 Hence: Δ ˆ ˆ ( ˆ ω ω xi ωk ix& + ĵy& S ˆ ˆ ix && + jy && ω xiˆ+ ωyi & ˆ ωxĵ & && x ωy& ω x && y+ ω x& 5.9 && qe + q ( B Equtin 5..4 && && + & ω + ω + ω ( ω Equtin ω q ω B s & ω q && q ( B B ( ω qe + q ( + ω B q && + q ( B + ( ω B qe + q ( B + q ( ω B q && qe + ( ω B q q qb ( ω B ( ( sinθ( B B Neglecting tes in B, && qe 5. F x xcsω t+ ysinω t y xsinω t+ ycsω t x& x& csω t xω sinω t+ y& sinω t+ yω csω t y& x& sinω t xω csω t+ y& csω t yω sinω t x& x& csω t+ y& sinω t+ ω y y& x& sinω t+ y& csω t ω x && x && xcsω t x& ω sinω t+ && ysinω t+ y& ω csω t+ ω y& && y && xsinω t x& ω csω t+ && ycsω t y& ω sinω t ω x& && && && & x xcsω t+ ysinω t+ ω y + ω x && y && xsinω t+ && ycsω t ω x& + ω y Substituting int Eqns 5.6.: && x csω t+ && ysinω t+ ω y& + ω x g g x csω t ysinω t+ ω y& l l && xsinω t+ && ycsω t ω x& + ω y

61 g g + xsinω t ycsω t ω x& l l Cllecting tes nd neglecting tes in ω : g g x+ xcsω t+ y+ ysinω t && l && l g g x + xsinω t y+ ycsω t && l && l 5. 4 T sin λ hus 4 T 7.7 hus sin9 5. Chse cdinte syste with the igin t the cente f the wheel, the x nd y xes pinting twd fixed pints n the i f the wheel, nd the z xis pinting twd the cente f cutue f the tck. Tke the initil psitin f the x xis t be hizntl in the V diectin, s the initil psitin f the y xis is eticl. The bicycle wheel is tting with ngul elcity V but its xis, s b ˆ V ω l k b A unit ect in the eticl diectin is: ˆ Vt ˆ sin ˆ Vt n i + j cs b b At the instnt pint n the i f the wheel eches its highest pint: ˆ ˆ Vt sin ˆ Vt bn b i + j cs b b Since the cdinte syste is ing with the wheel, eey pint n the i is fixed in tht cdinte syste. & nd && The x yz cdinte syste ls ttes s the bicycle wheel cpletes cicle und the tck: ˆ ˆ ˆ n V V i sin Vt j cs Vt ω + ρ ρ b b The ttl ttin f the cdinte xes is epesented by: ω ω V ˆ ˆ ˆ ω i sin Vt j cs Vt k V ρ b b b & ω V iˆcs Vt ˆj sin V t ρ b b b

62 & V ˆ ˆ ω k cs Vt k sin Vt V + ˆk ρ b b ρ ω & Vb ˆ sin Vt cs Vt ˆ sin Vt cs Vt ω k k V ˆj sin Vt iˆcs Vt + ρ b b b b b b V ˆ sin Vt ˆ cs Vt V ω ω k k iˆ sin Vt ˆj cs Vt + + ρ b b b b b ˆ V V ω ω k nˆ ρ b Since the igin f the cdinte syste is teling in cicle f dius ρ : A kˆ V ρ && && + & ω + ω & ++ ω ( ω + A V V V V && kˆ kˆ nˆ kˆ + + ρ ρ b ρ && V kˆ V ρ b n ˆ With pppite chnge in cdinte nttin, this is the se esult s btined in Exple

63 CHAPTER 6 GRAVITATIONAL AND CENTRAL FORCES 6. 4 ρv ρ π s s 4πρ ( G G πρ G πρ 4 4 F 4 4 s F F G 4πρ W g 4g 4 6 π ( kg F 6.67 N kg 4.5 g c kg c W s g F W ( The deitin f the fce is identicl t tht in Sectin 6. except hee < R. This ens tht in the lst integl equtin, (6..7, the liits n u e R t R +. GM R+ R Q F ds 4R + R s R s GM R R R+ ( R + θ ψ 4R R+ R P GM F + R ( R+ 4R (b Agin the deitin f the gittinl ptentil enegy is identicl t tht in Exple 6.7., R R+. except tht the liits f integtin n s e πρr R+ φ G d R s R πρr G R+ ( R R 4πR ρ M φ G G R R F < R, φ is independent f. It is cnstnt inside the spheicl shell. GM F e 6. ˆ

64 The gittinl fce n the pticle is due nly t the ss f the eth tht is inside the pticle s instntneus displceent f the cente f the eth,. The net effect f the ss f the eth utside is ze (See Pble 6.. F 4 M π ρ 4 F Gπρeˆ ˆ ke The fce is line esting fce nd induces siple hnic tin. T π π π ω k 4Gπρ The peid depends n the eth s density but is independent f its size. At the sufce f the eth, GM G 4 g π R e ρ Re Re 4Gπρ g R e 6 Re 6.8 h T π π.4h g 9.8 s 6 s GM 6.4 F ˆ g e, whee M 4 π ρ The cpnent f the gittinl fce pependicul t the tube is blnced by the nl fce ising f the side f the tube. The cpnent f fce lng the tube is Fx Fg csθ The net fce n the pticle is ˆ 4 F i Gπρcsθ csθ x ˆ 4 F i Gπρx ik ˆ x As in pble 6., the tin is siple hnic with peid f.4 hus.

65 6.5 GM GM s f cicul bit, is cnstnt. π T 4π 4π T GM π 6.6 ( T F Exple 6.5., the speed f stellite in cicul bit is gr e π T g R e T gr e 4π T g 4 h 6 s h 9.8 s 6 Re 4π Re 4π R e (b π ( R e e e g Re π π 6 6 R T π g g R 9.8 s 6 s h 4 h dy T 7.7 dy 7 dy 6.7 F Exple 6.5., the speed f stellite in cicul bit just be the eth s sufce is gr e π Re T π Re g

66 This is the se expessin s deied in Pble 6. f pticle dpped int hle dilled thugh the eth. T.4 hus. 6.8 The Eth s bit but the Sun is cunte-clckwise s seen f, sy, the nth xy, ε, α. st. It s cdintes n ppch t the ltus ectu e The esiest wy t sle this pble is t nte tht ε is sll. The 6 bit is lst cicul! GM S nd GM S with α b when ε GM S 4 α s Me exctly l αcs β l, but α (equtin k l Since k GMS l α cs β αgms GM GM O S α cs β The ngle β cn be clculted s fllws: s hee nd α, hence S y x + (see ppendix C b dy b x xy, ε, α dx y nd t dy b b dx ε ε ε α b since ε ( ε dy tn β ε β ε (sll ε dx GM S GM S α csε α 6.9 F Fs + Fd s befe. GM Fs GM d Fd The net effect f the dust utside the plnet s dius is ze (Pble 6.. The ss f the dust inside the plnet s dius is: 4

67 4 π ρ GM 4 F πρg M d k 6. u e du k e k dθ du k k e θ k dθ F equtin 6.5. du + u k u + u f u du l u f u l k + u θ θ u l ( k + f The fce ies s the inese cube f. F equtin 6.5.4, & θ l dθ l k e θ dt kθ l e dθ dt k lt e θ + C k klt θ ln C + k θ ies lgithiclly with t. k 6. f ku F equtin 6.5. du ku u ku dθ + l u l du k + u + dθ l 5

68 k If + <, l k If +, l du C dθ u c θ + c c θ + c k If + >, l u Acs( cθ + δ du cu dθ du dθ, c >, f which du cu dθ +, c > u b e θ is slutin. k Acs l θ δ u csθ du sinθ dθ cs θ du sin θ cs θ + + dθ csθ cs θ csθ cs θ cs θ cs θ du u ( u u u dθ Substituting int equtin 6.5. u u+ u f ( u l u 5 f u l u f l 5 6. F Chpte, the tnsese cpnent f the cceletin is θ && θ + & & θ If this te is nnze, then thee ust be tnsese fce gien by f ( θ ( && θ + & & θ F θ, nd θ bt f θ b Since f ( θ, the fce is nt centl field. 6

69 F θ, nd the fce t be centl, ty θ bt n n f θ b n t + b n( n t F centl field f ( θ ( n n+ n θ bt n 6.4 ( Clculting the ptentil enegy d 4 f k + 5 d Thus, The ttl enegy is 9k 9k E T + V k Its ngul entu is 9k 4 l cnstnt & θ Its KE is d d l T ( & + & θ + & θ + dθ dθ The enegy equtin f the bit is d l T + V + k dθ 4 d 9 k dθ 4 4 d ( dθ 9 Letting csφ then d dφ sinφ dθ d θ S dφ φ θ dθ 9 4 7

70 Thus (b t Since cs θ θ π θ the igin f the fce. T find hw lng it tkes & l θ cs θ cs θ dt cs θ dθ π π π T cs θ dθ cs φdφ 4 T 9k π π 4 9k 4 k (c Since the pticle flls int the cente f the fce (since l cnst 6.5 F Exple c + Letting V we he V c + dv S: + d V dv Thus V d

71 dv V ( d dv (b ( 6 % %! d V The ppxitin f diffeentil hs bken dwn cect esult cn be btined by clculting finite diffeences, but the iplictin is cle % e in bst cuses cket t iss the n by huge fct ---! F sectin 6.5, ε.967 nd 55 i. + ε F equtins 6.5.&b, ε 6 55 i AU ( + 6 ε i 7.9 AU F equtin 6.6.5, τ c τ y AU 7.9 AU τ 75.9 y F equtin 6.5. nd α l ε k ε nd k GM k GM ( ε + F Exple 6.5. we cn tnslte the fct GM int the e cnenient GM with e the dius f cicul bit nd e the bitl speed e e 6 e e 9 i ( ε 6 ( i + e.84 e Since l is cnstnt ε e.6 + ε.967 e 6 π e π 9 i 66, 75 ph τ y 65dy y 4 h dy 5. ph nd 6.7 F Exple ph e 9

72 ε + q ( qdsinφ d whee q nd d e e e diensinless tis f the cet s speed nd distnce f the Sun in tes f the Eth s bitl speed nd dius, espectiely (q nd d e the se s the fcts V nd R in Exple 6... φ is the ngle between the cet s bitl elcity nd diectin ect twds the Sun (see Figue 6... The bit is hypeblic, pblic, elliptic s ε is >,, < i.e., s q q is >,, <. R is >,, < s qd is >,, <. R 6.8 Since l is cnstnt, ccus t nd ccus t, i.e. nd x in in nd f the cnstncy f l inx k ε + (See Exple F equtin k π GM τ ( ε + x π inx τ + ε F equtin 6.5.&b. With + : ε ( ε + ( + ( ε + ε ( ε ( ε + ε + in x π τ 6.9 As esult f the ipulse, the speed f the plnet instntneusly chnges; its bitl dius des nt, s thee is n chnge in its ptentil enegy V. The instntneus chnge in its ttl bitl enegy E is due t the chnge in its kinetic enegy, T, nly, s δ δ δe δt δ δ T δ E δ T But the ttl bitl enegy is

73 k k E S δ E δ Since plnety bits e nely cicul V ~ k k nd T ~ δ δ E δ Thus, δ E T nd T δ δ We btin 6. ( V τ V τ dt k V F equtin 6.5.4, l & θ dθ l dθ dt dt l τ π k Vdt dθ l F equtin ( ε + ε csθ ( τ k ε π dθ Vdt l + ε csθ π F equtin τ ε l k ε π dθ V π + εcsθ π d θ π k, ε < V + εcsθ ε (b This pble is n exple f the iil thee which, f bunded, peidic syste, eltes the tie ege f the quntity deie it f plnety tin s fllws: τ τ τ p dt dt F dt τ & τ && τ Integte LHS by pts τ p& i t its kinetic enegy T. We will i

74 τ τ τ & & dt F dt τ τ τ The fist te is ze since the quntity hs the se lue t nd τ. Thus T F whee dente tie ege f the quntity within bckets. but hence T dv k F V V d V V V but E T + V + V k hence V E but E cnstnt τ k k nd E Edt E τ s E Thus: V k s befe nd theefe T k V 6. The enegy f the initil bit is k k E k ( + ε t pgee, the speed, t pgee is Since ( ε ( ε k k ( ε + + T plce stellite in cicul bit, we need t bst its speed t k c k since the dius f the bit is k k c + ε Thus, the bst in speed Δ c c such tht k ( Δ ( ε ( + ε Nw we sle f the sei-j xis nd the eccenticity ε f the fist bit. F ( be, t lunch t R, s k RE E

75 nd sling f RE nting tht RE k k GME ( gre RE RE RE RE 4.49 k.46 gre The eccenticity ε cn be fund f the ngul entu pe unit ss, l, equtin 6.5.9, nd the dt n ellipses defined in figue 6.5. kα k & θ ( E sinθ ε l R whee, θ e the lunch elcity, ngle Sling f ε (using ( be ε sin θ.795 gre gre ε.89 Inseting these lues f, ε int ( nd using ( gies E ( ( ε R Δ gre 4.6 k s ε + h + ε R.9 k {ltitude be the Eth t peigee} (b 6. E b b b be e e f k k b+ f ( b + f F equtin 6.4., ψ π f + ( f ( π b+ π ψ b 6. F Pble 6.9, GM 4 f πρg GM 4 f πρg

76 4 4πρ f ( M f ( 4 GM πρg 4πρ + M GM πρg + f F equtin 6.4., ψ π + f ( ( 4πρ 4πρ M M ψ π π + 4πρ 4πρ + + M M + c ψ π + 4c, 4πρ c M 6.4 We diffeentite equtin 6..b t btin F cicul bit t, && s du d F sll displceents x f, x+ nd && && x F Appendix D x f ( x+ f ( + xf ( + f ( +K Tking f ( t be du d, du f d Ne du du d U + x +K d d d du x && x d && du d This epesents esting fce, i.e., stble tin, s lng s du d > t. k 4ε f f f F equtin 6..7, the cnditin f stbility is + < 4

77 k ε k 4ε < k ε + < 4 ε k < ε > k f e k 6.6 ( b b b b e f ke k b + f f F equtin 6..7, the cnditin f stbility is + < b b e e k + k b + < b b e be k + k < b < < b k f k f 4 k k f ( + f ( + 4 f + f is nt less thn ze, the bit is nt stble. (b Since ( ε 6.7 (See Figue 6.. F equtin ε csθ nd the dt n ellipses in Figue 6.5. p ( ε s ( + ε p + ε csθ 5

78 F pblic bit, ε The cet intesects eth s bit t. p + csθ csθ + p 6.8 (See Figue 6.. T d t lng the cet s tjecty inside eth s bit F equtin 6.5.4, & θ dθ θ l s dt dt l dθ T l ( ε F equtin ε csθ nd the dt n ellipses in Figue 6.5. p ( ε s ( + ε p + ε csθ F equtin 6.5.8b, with ε f pblic bit: p + csθ π p At θ the distnce t the cet is α p π + cs l l F equtin 6.5.9, α, whee k GM, s p k GM As shwn in Exple 6.5., GM e π F cicul bit, e y whee d e ( ( ( l GMp p p π y d 4 p T θ l + θ + θ ( π θ θ p d y θ θ cs + T p π p ( + csθ + θ dθ θ F tble f integls, ( + csθ f Pble 6.7 y ( + csx dx x x tn + tn 6 6

79 p θ θ T tn + tn y π x csx tn + cs x p θ p tn p p p p p T + y π p p p p p + y π p p p p T + π T is xiu when ( d dp y p+ p is xiu. ( p+ ( p ( p+ ( p + ( p+ ( T is xiu when p. ( p ( 6p + T y 77.5 dy π π When p.6 T ( y 76. dy π k kε V dv k kε k f ( + ε d k kε k f 5 5 ( + 6ε f ( + 6ε f ( + ε 6.9 7

80 f F equtin 6.4., ψ π + f ( ( + 6ε + ε ψ π π + ε ε F ε 5 RΔ R, R 4i, Δ R i 4 ε ( 4 (.8 i 5 7 F R,.6 i ψ.9π k k Vel E + c dv k k k f + E + d c k k + E+ c f k k k k f E + + c c k k f E + + c k + E + f ( c f ( k E + + c k k + E+ E+ f ( c c + f ( k + E + c E + c k + E + c 8

81 f ( c E + + f ( k c + E+ f ψ π + f ( ( k ψ π + c + E ( ε 6. F equtin (Hee θ is the pl ngle f cnic + ε csθ sectin tjecties s illustted by the cdintes in Figue 6.5. nd the dt n ellipses in Figue 6.5. ( ε s + ε c + ε csθ F equtin 6.5.8b α + ε csθ nd t θ α + ε l l And f equtin α s k k( + ε k GM GM F Exple 6.5., GM nd l φ c c sin φ ( + ε ( + ε ( + εcsθ c e e RV sin φ + ε csθ csθ ( RV sin φ ε e e c c sin sinθ cs θ sin φ ε ( RV sinθ ε ( RV sin φ + ( RV sin φ ε Agin f Exple 6.5. ε + V si R ( RV n φ 9

82 ( RV sin ε + φ RV sin φ sinθ sin sin ε sinθ RV sinφ csφ ε csθ RV sin φ sinθ RV sinφcsφ ctθ tnφ RV sin φ θ ct tnφ csc φ RV F V.5, R 4, φ : ( RV φ ( RV φ θ ct tn csc ct ct θ 6. The pictue t left shws the bitl tnsfe nd the psitin f the tw stellites t the ent the tnsfe is initited. Stellite B is hed f stellite A by the ngle θ + is the sei-j xis f the ellipticl tnsfe bit. F Keple s d lw (Equtin pplied t bjects in θ bit but Eth 4π τ GM E The tie t intecept is τ π GM E Tt π since g GM E RE g RE Letting RE + h nd RE + h whee h nd h e the heights f the stellites be the gund. Inseting these int the be gies π h+ h π h+ h Tt RE + + R R E g E R R Eg E

83 F Exple 6.6., R E 67 k, h i 4 k nd h - R E 4,4 k - 6,9 k. Putting in the nubes T t 4.79 h T (b Thus, 8 t θ 8 6. θ S θ θ θ b The ptentil f the inese-cube fce lw k is V Letting u, we he (Equtin 6.9. du l + u + V ( u E dθ ( du E V u dθ l l dθ du ( E V ( E V l u u l du Nw, integting f θ ux ( ( u up t ( u u l E V l u du But E, l b, s ux b θ ku b u θ ux b ku b u Befe eluting this integl, we need t find u x ( in distnce f clsest ppch t the sctteing cente. k E T( in + V( in + in But, the ngul entu pe unit ss l is l b nd substituting f int the be gies in in du du x, in the wds, the

84 l Sling f k + s in in u x ux k b + Nw we elute the integl f θ θ Sling f b b u l k in + x u u x x b b u b du sin k k ux k b u b b + + π + k θ ( θ π 4θ But θ ( π θ S. Thus, we he k π θs b( θs θ π θ S S We cn nw cpute the diffeentil css sectin b db kπ ( π θs σ ( θs sinθs dθs θs ( π θs sinθs Since dω π sinθ s dθ s we get kπ ( π θ S σ ( θs dω π bdb d θ S E ( π θs θs

85 CHAPTER 7 DYNAMICS OF SYSTEMS AND PARTICLES 7. F eqn. 7.., c i i i ( ( ˆ ˆ ˆ ˆ ˆ c + + i + j+ j+ k+ k ( ˆ ˆ ˆ c i + j+ k d ˆ ˆ ˆ ˆ ˆ c c + + i + j+ i + j+ k dt ( ˆ ˆ ˆ c i + j+ k F eqn 7.., p i i + + p iˆ+ ˆj+ k ˆ i z y x 7. ( F eqn. 7..5, T i i i 4 T ˆ ˆ ˆ c i + j+ k c ( (c F eqn.7..8, L i i (b F Pb. 7., i L i + j i + j+ k j + k i + j+ k L k + i + j i i + j ( ˆ ˆ ˆ ( ˆ ˆ ˆ ˆ ( ˆ ˆ ˆ ( ˆ ( ˆ ( ˆ ˆ ˆ ˆ kˆ 7. b g Since entu is cnseed nd the bullet nd gun wee initilly t est: + M b g g γb, γ M ( +γ b b + γ

86 γ g + γ 7.4 Mentu is cnseed: + M blk γ γ M Ti Tf M γ + γ M 4 4 Ti Tf γ T 4 4 i 7.5 At the tp f the tjecty: ˆ cs 6 ˆ i i Mentu is cnseed: ˆ ˆ i j + ˆ ˆ i j Diectin: θ tn 6.6 Speed: When bll eches the fl, ε. blk belw the hizntl. gh. As esult f the bunce, The height f the fist bunce: gh ε h ε h g g Siilly, the height f the secnd bunce, h ε h ε 4 h 6 / θ

87 4 Ttl distnce h+ ε h+ ε h+ K h + ε n n n Nw, <. n + ε + + n ε n ε ε + ε ttl distnce h ε F the fist fll, gt h, s t h g h h F the fll f height h : t ε g g Accunting f equl ise nd fll ties: h h Ttl tie ( + ε + ε + K ε n g g + n h + ε Ttl tie g ε 7.7 F eqn : ( ε x& + ( + ε x& x& + x& ( ε & ( + x + ε x& c t Bth c nd tuck e teling in the initil diectin f the tuck with speeds nd 8, espectiely. - / 4 c t

88 7.8 F eqn. 7..5, T i i + i Menwhile: + c + μ + ( Theefe, T c + μ 7.9 F Pb. 7.8, T c + μ Q T T nd since c c : Q μ μ F eqn , ε Q μ ( ε 7. Cnsetin f entu: + + Cnsetin f enegy: T T ( + 4

89 T T T μ 4μ 7. F eqn. 7..4, L c c + i i i i i i i + i F eqn. 7.., + R + μ Since f eqn. 7.., R μ μ μ i i i R + R i μr ( R μ L + R μ c c 7. Let s ss f Sun nd e sei-j xis f Eth s bit then f eqn. 7..9c, τ + s s e τ + e s s y y 5.6 d 65d 6. e e 5 ( + y 6 s s 7. (Igne pies in nttin The cdintes f the tw piies, P nd P, e shwn t left lng with the cdintes f L 4 nd. L 5 5

90 b V( x, y ( α α x + y ( x α + y ( x+ α + y V ( α( x α α( x+ α + x x [ ] [ ] Nw x α.5 t L 4 nd L5 ls, ech bcket te in the denint equls t L 4, L5 V α α.5 α + α α.5 + α α.5 x.5 +.5α +.5α α +.5 V ( α y α y + y y [ ] [ ] Agin, the denint in bckets L 4, L5 V S, ( α ± + α ± ± y ± α ± α V V V x, y i + j x y t L, L. 4 5 Thus ˆ ˆ ( Cnsetin f entu: p p p + 4 p α cs α csφ p 4 cs p α φ sin 45 4 α sinφ p 4 sinφ α p 6 + p p α Cnsetin f enegy: 45 φ p Subtcting: p p p + 4 p α 6 4 α 4 p + 5 p p 6

91 ( 6 ± + 6 p ± >, s the psitie sque t is used. p p.988 p px py.657 α ( p (.988 α.85 p p.988 tnφ p p.988 φ tn αx α csφ.86 y sinφ.64 α α 7.5 Cnsetin f enegy: p p p + 4 p α + p 4 6 α 4 p F the cnsetin f entu eqn f Pb. 7.4: 6 α + p p Subtcting: + 5 p p ( 4 ± + 4 p ± Using the psitie sque t, since p > : p.7895 p px py.558 p α ( α.78 F the cnsetin f entu eqns f Pb. 7.4: p.7895 tnφ.7895 p 7

92 φ tn αx α csφ. y α α sinφ.4 sinθ 7.6 F eqn , tnφ γ + csθ φ nd θ e the sctteing ngles in the Lb nd C.M. fes espectiely. F eqn , f Q, γ sinθ tn 45 + cs θ 4 + cs θ sin θ nd squing 4 cs cs cs + θ + θ θ 6 5 cs θ + csθ 6 5 ± + cs θ 4.5 ± π Since < θ <, θ cs sinθ 7.7 F eqn , tnφ γ + csθ F eqn , Q γ + T γ sinθ tn csθ.5 + csθ sinθ (since sinθ > csθ, θ > 45.5 sin θ sinθcsθ + cs θ Using the identity sinθ csθ sin θ sin θ.5.99 Since θ > 45, θ > 9 : θ sin θ 57. 8

93 P 7.8 Cnsetin f entu: P P csφ + P cs ψ φ P sinφ P sin( ψ φ F Appendix B f sin ( α + β nd cs( α β + : P P csφ + P csψ csφ + sinψ sinφ P sinφ P sinψ csφ csψ sinφ ( φ P P cs φ + P cs ψ cs φ + csψ csφsinφsinψ + sin ψsin φ + PP cs φ csψ + csφsinψsin P sin φ + P sin ψ cs φ sinψ csφcsψsinφ + cs ψsin φ PP sinφ sinψ csφ csψ sin φ Adding: P P + P + P P csψ Cnsetin f enegy: P P P + + Q Q ( P P P ( P P cs ψ PP csψ Q P φ ψ P 7.9 T T T let ti f sctteed pticle t incident pticle enegy T Lking t Figue 7.6. c c + c c csφ hence c + cγ whee γ csφ c cγ + but the cente f ss speeds f the incident nd sctteed pticle e the se. α + + α f equtin 7.6. wheeα 9