Analytical Mechanics - الميكانيك التحليلي

.I.-"'~j~wl~ I•..,1;..1 .J~':"".l,.

..~..P " "'"

~~)~\~./}))

~

I.,'

r..r.J 1

L:~ ~;

L.;,. ~. ~..,JII:.- ':i~l

...1... 111:.- ':i~1 vi

J

&f.l.-..t ~

J w...l,::., ~~ ~~II:.-I..s,"J-

ve

• 4...u:JI vl~I~r=-:,~JJ'L:tJ..,JII:.-~' ~rL:J1 c.L::iJ':i1 • ~I.;.:;JI ifl:....JI u~ ~~~I:.-L..:.J UJ...;,I 1

....,..,> •

~ I;;

.J,J I

J

~ ~.,li.:- ~U..~

~ I~.".,..: t l...:i; ':i I c!.lU I:.- 1.;1':

'" ve ,

..:....ul I..iJ • ~.rJ I L:tJ..,J I J..s L-) I V'- ,....J~ v I ., ~ ~.rJ ll...t:{.,JI vi, c.SJL..:::>I~ vIS',

.. ,1

•

~I ~~I~ ~,,:,,~II.a.. ~;: ~

. . 14.-)L., ......::)1+

~,

•• 1 1 .11......,;;.,

~!.u~ ":-~I

1.lJJ

~I;.,...wl ":"~ v-,JL.:,~IV'-~~ L,~~J~,

--.kb.1 J1WL.;1 • ;..sU ",,:4- I c.S.ieUJ""-

J

~,~UJIvJ'J

~L-l l""..s ~ c.S~ r.l~; .iI..:J1 ~~I ~ 'c.SJli1I,.1&

• 'L.j::iI1 ~ ~, ':il • :. :"".uJl ~1=_H'~ J

:........ill I, ~#'S I u.!',J I ~ ~ ~ 1..j U',.l ~ ~,b. .J",..i ,.

J-J

eY"·

.11 ).,,; L>.. U. - ~L.. V'- cd) J J L f J - L..u-.J 1 .;. I I ~ J L...,.;.JI 1..i..A ~ ~ ..u vrS1v I~T,,,,,,,)1...;..1, W I.~

-:,...... J.i: ~

•

~IJ,....uI,......-":,,,....JJ.J1

r.!,

_

J'~

-1;;,. ~ I j....iJ I o.S~

~~J4111..uv.x>'iI·~I~· ~~Wl ~\:;, ~~..::.~

•

t..-ro..::.lul;,.;

~~

~LiJI ;'1

• J-i ...lS'~Y

~UJ'v.~u.J,J'vILS· • ":-' ~ I J

-II L.)~I t.-IJ ", ,j..::.1.i,.i-l1 rl",..>.:-I

,- .&':.:.1 v=:J I..::. ~ '" W

~v-,Jl.;llv.~~~~La

b.,J:: V")"wl~vl ~. Ya~~UJI ~

1~ rliJ I ~ v. ~ ~ • ~ '" WI )OJ,,.;

J r-" ~ vI

V"J..\J 1 ~,;::lv,..s-:- h;ILS· ,,:-,~'~y J~",~Iv-,JI.:l' ~I ~ILS

• ~I ~o.S;>~1 0--',J1.:l14'-:

,r---J,.: •

v' v,'" I.ri.i> v4c ~ ~ L....u:J I ~ I ~ ~ I L..)is. ~ ..s-;

I~~I u-,J..s.:l ~I..::..;,JI vb' 1.11

u--t..ul J~

•• _

t....,.....:;. • ~I lr,J..s; ~ ~ • ~ I• .i.A ~~ I ~.bl qr...ic v I J-..it.ic • J b. ~ j J.s,

J, ~I,,:-,~l ~..ri ~,.I5L. u--t..ul~ ~I ; I'W"..s"f .< 1.i.A . '.I5L.

~ --...-

o.S~ r..s.il Ix>~

• ..1..s.>.::..1 ..s.....-- 'l...:,.- I", Li::; I.. -J ..L-i;; I ·4....;IW'~I~

v-i, l; •

J ..::.-:

Ij

f;.

,.;

fa!

Is-

'"1

-'"I

l-

•

*

E!

~.

I

'"'1

t

L

r

~I ~ '"'1 " . :- ~,..

~

t. ~.... f'

. &'"\

I.

•

r.

:

c[

'i.:..

-

I ,

{::

'r. 0. ..,...

....

f·..

[

~

.-

I

~

~ yo-

~'

'

-

t

....

t

I

I

:

1

J

'-

·

_I

-

);.

I

_

't. .~

·r .f."

I

f.

fa -

I

•

r- -. .

1.1::: 't_o J.

I

b-

or

-r- [~ r- f . 't-

I

-

'I p.:.

.- :- ;.

1

.-·r -e

C.'t,C\·

•

I

:I.r

~

....-

,~:_ .~

t . . 'l

~

J

I....

Po'

-

C-

t ==.r

~",~··f J-

f·

•

I

~

I"

..r.

~

I

i

'"l.1'

{"

f"--

I

.

0

(i'

I

.-

l

't-

'"'1

I

o

0

I . '

..,

•

I

....•

~

0

-

•

-

•

. •

{

fa

~

. 4

-

-

r'

I.

I.'

~.

C :E Ci l:' - l Ci ~. T r :[ t. ("'" :[ ~ .p. f' .~ ~r~ t.1:: 1.- l ~ ~: tf· Of:.

-

(0

~

LL._ . .

t' t'

~ ~

: :C':[ • •

•

oL'

~ ~ - v -

.[ "

.-

~~~~~~F~~t{L.~L~~)t~_Ll L F

I•

'"'1

...

-

~

r' L -{, ~. .: - -, - r ;.[ 1:-I ~::::-'"ll' ;'t. [ I - l ~ 1 .~ t t G.:; t l. 'I. { .~ '.. ~"I, -: ~ ~ ~

r.

~ t~..J \ ~;;.-J}i..

;;_-.5p- • A_ Y ~..J' • ,

'0

r-~ vi;- U

y_ y

0

.,_y

L~

I

~

r

~;;...spJl_4..J..Jlr~~I~~ -A

I 0) ,,.::.\ 1 • ,,-"\ ..J...,. ~ ~ I

J,

'0+" .

J;

:\.1.1 I p)O -" b..,

~)WIUp-· ~t~

rr- . :. .: ~ ;'':>'" J,>

•

~~ I Js. ~.,» I ; " ~

~I, ~UI r~l..:.;ll

J~~ 1 I

..

J:.;.. Ll.i..I", - Y

F)O • '-A

..J....: ~ 0 I." ~ • T'-A ~

0_.' v.;I.iJ1 J,...,.iJ1 rr'":--I....> • t-A ~UH

Y-" _'\ d..-...;: I..i.1I -::..1),-iJ1 '":--~ - IS, l;.l' ..J..~ 1 ~ 1

r>j •,-\

J L_--::5,rll

~';LbJI

• r _,

~)I

..u..J1

~I."b...

• y~,

r-~u=I.i.lI~lrr·t_'l .u....~L.::l.,,~1

(""" -:-\ I· 0 _

'\

.,. ;", 11 ~ I 4SpJ ~ II.:- ~ '" ~ • ,-,

.1.-..... \

-!

J}> ..u...,

rj-oJ.l ueJl:.l1 ~.J I· ~J:, ~ 1~

,..c~ ~},:. 'J

...J .J...J..~ ~ pJl

rr-ll vJW I

I. ~ ..u.,... ~ pJ I v I." ..J 1• y_,

u 1J.'.J1

~~Iu...,.ll

);I.,..;:J1 ~WI _).1;.;: ~

wl,--hJl .,_, . ; ~,~~I 1.1L..:-I.,

·A_' W.r-1l

._,

u,).j~I~;>- ~~I

• ~;'.i.l';,...,..tLl;..s.::.J I ~ I • ..J-.l1 ~ I ~ ~ '"

ro r

~ l"....s~ ,,;;.;~

. r_ \ •

~ t 1oS,.il1

• '( - \ • ;

H

)

\of,\II.:-I;I.J>~I.;

•

~ Ij'J I.:- 'J .,l ~

,,~lrr.>jJl.o_\.

•••,:)1 ; ",.di • y_ \. L....Ai I"J '1oS,.ill

• ,- \ • v::J,..A..::.. 'j '" t.-.. •

~ 1r-5 'J..::.. ~

~ Wi", • A_ \.

;~1

L"l.4 - \ •

.J 1.,:.. 1; 'oJ> ~, • \ _ \ •

~ 1r---5'j I.:- ~ oJ L... I.:- ~ ~ • t_ \.

. 1_'·.

,.I

U,p-l\

,J

~

u-:l.-r-l

~lrS'J..;:... 'J.,I~

p - \\

'7'~.,i:;.l1 L_....

;.JI", ~ • '(_ \ \ )I;i:- 'JI_ vit,.;ll, w.~1 L;U:JI • \_\ \ •

o,J=..;..'7'~.i:

·r_\ \ 4-LI.

!, \.~o .......... ~IL;Ll:J1

u~"'jeu~t,;u~~.i:.· t_,) 4.l>o~ a..-.,)>-~J'" ..::..L..; cl.~ ~W, ...~, Joe

'l_' \ ,,-.,)L:.-JI

~l;loJ>

'11 • 0_\ \

,,,\.:-~..i.;.j.._\\ ~M\.:-~,j.:: .y_\\

L-:.i:J..:J1

•

~--::,;>~.J I ~~..;:... ~ • '1_ \ \

trt ~;,......

-

~-.\!. ...~

""->,,J I 4J ,J L... • L--~ L-.Wl~~ILpl_\\

"_\,,

~~..;:...U2>)L..1_\"

~~

,.,; • t_ \ ~

~w I ~ I U"~ ~ I ..:- ~;

~.......:,

J,...JJ I vJ.i; - ,;:.:",J

f' ~

., ~ \.J.LII

t,,;.~

46.Jol I •

J..~ ~ ~

y-, , v-.jJ I, •

.

0-

"

~ I • 1- 1 \

.,._ 1 'i

,;-:;),J

v--.Y I

.: 'A_' , u)-~&J I JI";1;.:l\ 4~.J =.. .w;, '\_" v---Jl

~.;.J~ I ;S~l =.~

eo-- ~l ~ •

~~l

J l ... -.:...1 • \ ,_ \ , wUJ ~ ~ I L.;~ • \ • - 1, ~;---ll · _.1 .. 11J~./~I- ~~~~t.;~1

J=",..JJ I

J,~ I

c.:.L~1

~L-I ~I.i.

_

Fundamental Concepts-Vectors

:; vi ~ .dc:~1 ~'"

1.\

I ,-" •• ,

~L.::J~;j~1 J

~L....~Iv-~ ~~ ~u"",rtJIv-dJ.LS. L..:J,I~ ~~ .~~I

'L..i.iJI -I..a vL,..,J. ~L..I ~liJl ?I~ vi

Space

~ UpJl

W;~llo.:.-IJ~ ~ I~~l,

T1:me

v-ot}lI,

" lc~~I-; j..L...:J1 ~WJI 'Wai.!l I,)l.;.;a....- ~~I

, •." .•

,.LJ

.., loiA, ~~'il i..-Yl...:)W1 ~~'ilu'; ..,..-Vl 'L...dl~ ;. v-;JI r~ ~~

"';'J";

.j--j"", ~ "'" li:

LI •• v ~I ~ 'Li:S'ile.:l..; ...:

v I ~ JU I ~l=J I;,

p~ I ~ 1.J->. ~ I V- ..I _1_

~jJI, 'LdJI V- ~ .L-~. I

.:: --~~

,.

vi """,;:i:-- tdI.i JI wL;.'i~ • ~ r~ ..;11 L..~.la.iJ1 ~ ~ L..A4 l:.:1 'il. 1~,--~~ • t '\.I:. 1.1"

; l...A ~ u,.- ~.l-1J I o--jJ I, 'LAiJ I ~.'" ~ I ~.; _,

cJl"J.s-J

vi

""""c

~I ..,.-.. u,.-

#..It- ~, • ~ 'i, \Ali.

· ':f:: - ~l ~ l!:J I '" L

... ,,~I cS..o,rAI1 ~ • 'Li.AJ1 '" r

':t

ie).

-

IJ"'-'....' I

u..~ ~ 'i

..,.--L-'Jltr'" d,.:~,H .j~lrl~~' rlJ;.; J..--,: .; ". ~JO ~J~.aJ I ~ I; Iol.4> 'i I t LJi,.; -,.. L....:..; Ij ~ ~ cS .iJ I ,.:.. ~ l.boo 'i I r I..!:.4J

~~,

;

3..l......L...:JI

~l;I.u..~1

tLI.;.;,I Oartesian Ooordina+.;e

~L.;I.u.. ~I • .lA ~ • ..l......w. (...,L...:,... ,I ) ~.I'':;••• ~)I: ~ &: ~

.;...~, z,y,x

;'.

~"",L...,I oJI~IU~

L.b..i: ~~

r-=J I ~ I

J ~ oJ

~ L..:; 1.1> ~ I~,:..: loS I • u--j.J1 ""'~ L.S

I.:- L; 1.1> I

• ~;JI L-~ ~ L_liJl t ~ 'di~l" ;~I ~l.tJI~ ~I ~1,1rr-~lul

~ • ~ 1 • loS ~

...Jcr:J ~,

oJ I.J.:"I

(l) L1:S .J •~

JI:. r,....s. • ~Jl

--J ~,;!l~1 ~;; a;:I..I\" -.I oJ~ ~ oJ~

.

~,

Ii

I

.

.r::--'

:J .

~ I,

:0..e ~

~ ~Jlh; ~ ~rr-.# ~I.SJ~ JJI

• clUJ I ~~" ~J>J~IL1L-..)L:. ~ Physical Quantities &: Units ~I.J>,JI, Y~II.:-~l \_\ ~I.;,

JJ I ~ L;,r..i.II

Sa ~ ~ ~ ;.u L.:.J I ~

;,....JJ I , v--;J I , Jr-W I J.:.

~ Ii>J I ~ ~

~~.r.:iJ Il.:-l...:oiI l ~

L.::.: ~

~ l-,I

'J!.s.-i. t.r~ ~1oS.il1 ·~I 10,(-" ~~I ~_;~.!J1, • .o I~,u,

, ~ ~ 0.\.:

1,p"s:J1("... W' u I cill ~ ~

. ( r- \ ) ; ""->,.1 I J~ ,

...--~II

~~I"~L~II.:..~I ~u..~ ~I~"".....,;I ~.f..L-';, • ~"j)l, ~I,

J,hJl...".

J..i; ~l-I I.:..l..L>, ,~)l~ Ll"Y..1.:

J,JJ I ; ..L>, ~I

• j-J I ~ is'

..u, • ;.J I ~ .. 11.'1

c.;;.,J l;iJ

I

_~II

v--"

~ ~I

Jr-IJ t ~ ""->-, U I

...t••~

r~

1J'1j

•

U

• Io.

( \ )

t. i --do

!:. rB

l:

v,-

~..r

L

rt

-

'1w£r[E -

t

..

.F

r

G fs'-: CO 'E!

-f. CD ~~t, a·

a

~.lo'-1~

-

• f.' (i

I-

~

r-,. ~ '-

~ l::.:.

t:.

'IF

.1'"

r

'!::

fs'

_,

_

'C

r- 'G...

rrt,

2f'.~

f~' 'L 'G...

(i

t~ 1~ t:. ,'L

-~

Co.

c·

I

'G... ~

_

v,

_

•

~_

-

I;:-

{l -

~'-

~. '~B

~

;. f' 1 ~ ~- r

~'\

': 1;

;:- 'to.

....E

~!

-

E

(

"f~-;

~.

~l,J t'L j·t: c. ''''''

f :; ~ ~ ~ I

'-" I

;,.....iI', J.,~...".:Jl , I..t-=..o~ l..:4'1 ~

• 1.:.1 • ~1.r..1!'" J:;~ 1-Ir tJ Notation ;"'-J T' - , L~

(A) ,j.:. ~I ~11J1~,~:; • =:-:J'I.:.~I ~

• ~oJ~'j1 ~u.Jlu,~ ~.s.A.AJII.:.~I ~ IS

.i...-JI r-. ......J is'

- ~ «-~ L- oJ ~I J-

- ~

-

-

J... . ':i 1-l::!J I 'oF.t L1 •

.A j:.

t t~ ~I~I ~J~

...... ~I, ..}..u.rS~ L.l.ts';~ -; J.,..:. ~cJI ~ ,-_)1 j .)~.

"\,!

io

~,

· c----:::-~ ~ ~JJL.:... JI;'1

to .......J,.~~ __ ~.:..

~I .~I JI ~ f'

:--.II.;;I~..JI,.. ~1.;,..1.iI~

'j

~, .~I

t,J ~ J l:J'

.;r:=;, oJ .:; I Vt ~n--: ~ .; , 'IS; lill ~, ~ ~ J}> ~ l..~..i.IloJ;.w' .:;1l'..Jlfrl::u..J; .;b~~~...,b...J}> ~b. ~

I\,J

r""; ~u,-·J6..

~I ~t)l,;,~llSjl,:. .;,;lil,)t ~

• erJl

'.iA V- I.:.~ ~bJi

-:J~ w:

(f)

,

L....:=J~..aJI...,b-1,1

J....

'4'

&'J..';"o

J

11 ",,6.J1 J,.i2

••

· . ~,r-->' ~

a

z2

- l il

L.4.~

Js

..L-> ~

......::~". j~

~I~~". ;-J

.....:4S". LJ~~

.........

A

~I ~

(~..uL.;)l>j.it~vl,;.,,~I~~J~..aJ1

, Ay-Y2-Y1 ' Ax

· ,~,u., ;,iH ~". ...,b.J1Jl:::>1

4-,WL_ ,I

4.l."WI~.. ~~iu.r-~Jli

~'J' P l (xl'Yl' z l)

P2 (x2 'Y2,1i 2 '

Az

Js

[Ax,Ay,AzJ

A = [Ax, Ay' Az ] ...,6.Jb l ..jJJ4-) L.~""b.. J ..... A vifIJ1)l:.;.

( \ - , ) ~I

~~~".

Lhi:J11,)4 L>lj~1 ~

= x2

- Xl

Ax v,s; ~......... ;,......; ~

~

A V if

'.;1 ,

~ ~ ~~~".J ~."..t..oJ1 ~Ivl c:-"~'

,.:--~; L..a.i~ 1:, ~ ~ ue l> ~ ~I I..i 1

~~.LQ; ~vWl~, v-LJ~, .6...... ~l:,s".".wbJl,.w.J

(A!' A2 , A"....

"n'

,;.) I '"'~ V J 1 .." ~~ 11,)4

." ~" ~

v ~I ~ ,j~1~11.i.Jl~,.

•

.t,j~1 ~

n

• ,jl..L..sl: ~

.1......S-\,; , L..::>)lJ..." 1u".J lA:

Jlormal Def'initiona and Rules

,.Li.! 1~)IJa.,.~ 1u...J L..:J I ~ ~ ~ I ~~

;;

t _,

I),j 1~

_:.~~~ ~I

Equality of' Veotors

.....

~

t ~II.S,L-::

A 1:1 B

(Ax, Ay' Ax •

Ay • ~L.::J1

-'"'

L),jWI

Az ) ... (B~, By' Bz ) _: ~l:JI~)WI..:..~ ,jW,

Bx

":v,A.z )

\,.;\1:

Y ~.r ~ L...:: I.H ~ lr·~":;,,.h loS, L...::: loS I ~I 1~1~

: Wl:J14.I,jW~ V':.

A + B =(Ax,

,I

Az = B.

By

Veotor Addition

-

_,

_~

t ~ loS 1 ~un

+ (Bx,By,B z ) ... (Ax+Bx ' Ay+B y ' AIt+B z )

_......::~.r~I~,.. ~1oS1~vl ~ I..;..,

:i..o'

• ~~I v:' .~I~4S'.r~ Multiplioa.tion bY' a. soalar .-

.... oJ. ...

A.

vii ~ ~

o{.Ax'

~I.r-II

ve

Ay' 0

Az }

1:1

(ClAx'

Y~.r I.$~I ;

~,j.u;;~,,:-~I_t

oAy, oAI;) t

=-

'~,j.u~

~~

.....

• l.

or

0

1o:.JlS'1.i1

.ACt

,,:-~I J....b.1.$1

L_~"::•.J1 ~ l,S'.r

ve r,S"1

A

~I

Veotor Subtraction

: ~ IS ~I ~

~

-

-0.

A - B == A +(-1) B

= (A--x

f

A-A

clJ.i

I

~

_t

- Bx , L_ - BY , AZ - Bz ) J ~I ~

--"-

• ~I~ ~

=0

=:--:JI

=:-:JI ~u~

The Bull Veotor

--

t

-"

~I

0 == (0,0,0)

.)s. ....L -: (1:) v-' u~ rJ.;.-J1 ~ _~I, ~;...J I ~

k:..J-

vo

~ ~ }-J I J~'

--'"

• 0 = O,;.} I

u ~ W,

J..:.i:..-..J l.j J..-,: ,,; ..

The Conm1utative Law o:f Addition ~I

J

...,..\.->JI

J.J4: u,.:U -,

v IlSl cl.r:-:JI ~ J v,.:liJll.u. ~ -..

-..=..

--.:a..

~

A+B=B+A z , y..:.~~ ~~cill.i5

~ + Bx

III

Bx +

Ax

.J,..bJI'.:':;:v,.:li

The Associative Law

v ~ c~1 oj ...,.b...I1 '. ••~ v,.:li cill.i..S

A+(13 + c)= [.Ax +(B:x:+ Cx ) ,

u~

~,

Ay+(By+Cy )' AJI+(Bz+C,,) ""

_

_

_Y

J --.1lo

=(Ax+Bx ) + Ox' {Ay+By)+Cy , (A,,+Bs)+Cs)=(A + B) + a The Distributive Law .J,..w.JI(:-'.;,.:v,.:li-A ~ .J ..ts ~

4

-""

.j

C

L,.~ I,--:,.rP ~.

-.J,..bJ It',J,.: v":!.;

- ~ (~) , ('0 V-

-....

J-,..d;.;~

c(A + B) = c(Ax + Bx ' A.y + By' As + Bs > ==

c(~+Bx)' c(Ay+By )' o(AII+B s )

= (cAx + cB x ' cAy + oBy ' oAs + o~s>

-

-

== cA + cB

c--=

.:.1.+-oJ1 ~ ~ ~.J~'SI

&' ~1,.iJ~ ..:-~!

v' u'

., .r-S.iJ' LJ,: I i

~I )1..\...4

Magnitude of: a. Vector

J.i-~I.:u~

A

~,I .,

111

;.)1.:

~IJI.J..iJje.r-

A

~1~l:Sr~..,. ~ j..-bJ ~.;:JI

A= lAI = (~ + ~ + A;)*

(' _. )

)..a..!_ _.. . . . vA6L-4IJ~~ ....

..,.J;4'.)L..;1

._,

.:.~ ISj I,.:. ~

f~ ~IJ~I.l>I,

J,» IS 1 • • As·

,.... ~I

Ay • Ax

...,b-lJ ~I c.:.1..\>,JI (,

Unit Coordinate Veotors (~I~.b-,JI) ~ ~

.J~

"'0.1....&

_1

.J....>~ "JI..u.IS.iJI~1

c)WI ~L.:J.J....>,JI,

~

I:

[l,O,OJ

t

j

= [O,l,OJ,

k = [0.0.1J

('f_t>

l-'j'~~I) ,I ( Jjb...U~Ic.:.I..b-,JI) ~

0.:

-

Jl:l~t,l~~ c.:.1 t~I,,~LJ'j~~1S1

~~"

A = (Ax,Ay.As)=(!x'O.O) + (O,Ay,O + (O,O,A s )

(\"-t>

= Ax(l,O,O) + Ay(O~l~O) + As(O.O,l) c,

:I:

lJ,

iAx

..... 'r

~

jAy + kA s

,,' ,~

t.)..':"-

'4

".# "'pI;' 'j

.~I~

13k

r)t." ~I"j.;.:: vi _~~

».'tob-

cl

~I rU;.: Lh-IJoo: ~I ~1..l>,J1 cu.~hj~, Ji-~I)

( , -,

,.. k

o J----:r--J----- y j

/

x .i.j-k

cl:;I..l>)lI~L:_I.1>,J1 (

; ~ till,j

t; ~"

~I ~l

,_ \) jitJI

u.;,.;, ISri l , I ~ 1.»b... r .1.';"';

.,;:

~ I ..;~

u,x. ..T' ,

~..t~l, ... ~.-JI.»L>.J1

• ( ' _ \)

~l.j

L:.:HJ1,»b-'I

cLt~l& ~.,l;-.JI ~I

(y_\

Geometrio Meaning of Vector Operations • u L.."L-.

4.-1' 1.::.;:1 ,

~~~

l;j,j

w,.J.. ~~

JJJ

~ 4.f,~_J1 u 1 1...,.:..;) l..i 1

u,..)WI u l V4

J-..b..,::ll··tl,~ ~

: W l:J I u l;~ ~ ~

: J, r.:-.... t;.:j ,

c4-~I15,L.-: _\

Equality of Veotors

J,..JJ' v-.&: l..rJ, I>.t'J t,.':.

tl.

y

l; (,

u ~ IS, L.-: l..i I

( '" - \ ~~ · &1' r..,.d; L.rJ u~ u l tS.»~1 V4 ~ ~, L

••..:.

J1I

.J.J.t u L.:.;S".r ..::.....-J ~.... 1>.t'.J L...:. ~~. ~ ·tt~

11

y

....

------..-A--~

Vector Addition .1

t

.1 ...

..w:J

~'-----x

I:,.L~'I~

_ '(

~WIeL-J1 \S,~ ~ 'tI'~ 'JI~I

.~ ~ (t_') J!:JI. ~I~_~I u~~ u1r->'Jl ~dJ..iS"

y

~

..... _""" .....

C=A+13=13+A

-'ort-~:p:;1I;i:;r----

1 ~~I (to_It) jUJl

x

•

I:,.~I

~...;.r'"'t)'. JUH ~

r

. __ I,J ..;.:-

'vi

(n

~,.. LS ~ 'i i ~j ~ ;..s...s1ic ~~H

( U;-:" L.j,J: ~ I Mult~plioation of

.A

u--.,r.S";;-

c

....J,J.,

~~

f

.-r

A

7: . ~I(J~'" -t- . 6.-;1 vi

I::,.~..:~. .

.~.).u~ ~Ie.:-"';

a Vaotor by a

Soalar

JJ

;,; l: ....-.,.. b V-

(t_l) u..~ ~

~I ~j~ . ~

0

III

-1

-,.

~,

y~ L~

~lJ~"'LS

.. ~i.!I.,.JWI (

I» -

\ ..)

J!.tJ1 ~.).J.AJIe.:-,riJ1 (A_'

The Soalar Product -..:a...

~

"1\ :B·

r--)~ ~

-'---.

:B

,

A. J:.:~ u'J ~ .J.,.,J Ie.:-.,.eJ1

: "41WI ~.)W~ u~ 1:.J~~"", (A dot B) ..... .A..• B Ax Bx + Ay By + A. B.

1~,

_

III

.......

........:..-...,

vi 1&:1 ~..wl ~1 ~ ~,

~

A.:B • B...A - • ~__ L...a.. ~Js" rMJ ~..L... -................-.: ~ -a. .1.(13 + 0) • .l.B + .1.0

,J

-=

( .. -H (. _ , )

A 'R_ .. --r""•

:Br-x .1

(J .,

(1-0

~.j....; ~l:(t_\ ~1u.~IIJ,n:.:~

........... A.(B + C) .. ~(Bx + ex> + Ay(B1 + 1 > + A.(B. + 0.>

°

-"'

• Yx + Yl

+

-~....:.-

Ill.A .• B + J. • 0

A,J3. + -'x0x + A.,G1 + A.a.

,.. ~. ~",.wl"r".,..u ~il...i..~ li~L..JIU)l"JIJ~I~

I.:

-.a..

J~

• B

A

,JI u....; 1...: ~

., 4J~, •. .:Jl.:JII,!.."'~I~~~1 ........ fOI,... 1.i = j.j == k.k == 1 '''' A.... .. ... ' i.j '= i.k PI j.k III 0 (A...;\)

/It,..

----»~ «

•

J;.i:

o.l,.'\

'it"; ......--'!1"})l:. 0";

~

~

-:>0.

Pn , •••••••••••••••• , J!l

3

~ I..I~ U I

ub;...!

~

\,;~

, F 2 , F1

~ ~,i-J I

vi l,..:J I .b~ vA. ~..l.:$ ('J-~ ~ p,:; v~ vi ~ • 1S,..iII • .u fr l::..:..:>.:: LoS'L- ~ I ~ 1$.iJ I..~ I loS I r ='

-u l • 1)..0 1oS,~ ~~~, Lr+~ ~ ~ ~

~t.""";fr~

':'~ ~ b.

IS, ~

6.

-.... s

6. W

,

~ c-,JI..,.; ~v'~}1

J..u.J L;

-....

(,-,) l.ol.L:J I

J

~,.,.

-"'"

IJ'

6. S ~li~1 414-:~ F .,.iH ~)O

;£~~-7 Some Applications of Vectors Equilibrium of a Partiole

r::---::- vj~

-,

\c

6W

= (p

oos e) 68

v----.J~ ~~luJ~H~,· ~

-.Jao

6 S

~

6 W &:I P. 6 S

--"0

J ~~!;JI

,

1....> Ij ~ I

'" I 6 S

J

--a.

J

..,.,.

~

C

• B

A ~~I u~1

•

tl.;~1

~ J..-..; ".) ~ --.. .....a..

C.O

-.:..

--=--

.....a..

........&;

J:.:

e ~

.,.w '" .)..\-A.l1 ~,,-J I tl.;~1

Law of Cosines

~

L>lj~IJI~J

",I 68 .....

lili. (Y_l )

u,.:li

_f

~I ~

u,.:li ( Y_') J!:JI

J

~

0

~ I ~".a.u

-...

=-a........ (A + B).(A + B) -a.. ...............

~

= A.A + 2A.B + B.B

( • - , )~~.)lAJl

e u""""; li,.. I ~ , AB cos

J

ui\,.ill ~v- ~ ~l~H.~I"

40~

0

2

-'--

t.-.

= A2

-l·B

J I~I,

( 1_ 1 ) ,

+ 2.A:B oos e + »2

'" I

• u,,,.-I' tW~' -"-

---.

~he Vector Product

.IL AxB r-y.

"/---1, ..... ""'': -.; l.tS,It

-

B"

~~) I ~;.All (\. - \)

--'"

A ~ ",~~I,,:,~I~

1S.il1 ~ I.e U;"t." ( -A cros s -"B)

1~ ,

: 4-.tl b.i1 4i oJ ~ ~ -....

~

'" x B =

... j x t III

W,

( ,t.... ,

)

-. "" "" A '" kx:1r:1j .....1:x:k

r:x:

r

II

(0-0, 0-0, 1-0)

= (0,0,1)

. ¥~~, ~ ~~""w,~ ~~~',;-~

=

k

w;.,.JoC ;J.H-""C~'

~~I ~I

. ( , ,_,)

IOQmot:r:l.c Intel'J;ll'etat1on. ot t1\..: OroBa Product

1jk .... ;iO~ ..,..~~I ,;-~I ~vl

y. ..A.xB-:L

....a.

1\.

B,jBx

T :BXS '

~ ~·I :81

:Bx

--

I\;

+k

+~

BTD.

.,~I"""".""

~I

Arh,

...a.

~t,

( 1t _ 1 )

:8•

c~~~I~oJ~·I~,

......"'~~. ~~~ly.,..,1 ~J.a.: ~ l;~L.....;.......,)I.~loJl B

t.:..U

•.

Ici~

...... J.

~ ~~~I~ti..ll ~ l.~tdJ.iJ

• 'i

. •..;1

•

~

-

~I ~lS' l.ill. WJ"'4 ."....s.I1 ~~

.....

A. OB

vIS' !.ill-ISI

IS, L-: oJ ~1 t...,.;~,1:: IS I Io»t-l:.::. oJ ~,

I;.-IS'L-c~~~

..... ~~I y"..rJlvAuJI

-.: l;~ .... ~~ly;oiJ1 JI.u.

,-;X11

-J)'

2 - (Ys-AriB'1)2+

(A,}JX-YS)2+(~Y_YX)2

S.iJI4J.,lAJl 4...cbs.'~. 4S.,.a...JI,,=,,~1 ~V4'JI

tJ WI JL:J I ~ If)l,.l ,-

..... , 2

Ax B

•

2-2

A~

..... -

- (A.B)

2

\A

.:...J ~ lA.J1 r I ~ ~, aJ ~ l~J1 • .lA

,,}J

~.,::J I ) o.4J I .i-.>~,

e.:;..·.... sin e

~L:)I~I.)s. ~~'SI,,:,;.rJ1 JI.J.i. ~ v I

IA xIi 1== AB(1-Cos

-

......

-~

A.C

e )i

--

== AB

~ L..",1jJ1

• B ,A

-.. C == A x B

v~

2

-

(y- \ )

'M;

( \0 J:.:: e ~

,

)

~Ivl~)l; Ir-~ ~~~I":,,r4lI~

-.... ~JS.)s.~~,~~ :B ,A

= AxCx

+ Ay0y + AzC z

= AxOyz-Av!'y)+ Ay(AJ3x-YZ)+A Z (YY-YX) == 0

\oS,:__H .)s. ~~,- v,1: -0

......~.:.J1 v·H B.O ==

.B;A -

~l.; ~I v,s'~} ve ~

JL..~.J1 ve

cO ~,.. LS'

_ wi • '-tl'.~ ...-i,

~I

.J,tJ I ';)l: J.!.t..: C

li.L...~~ "

'·:v JI •

A

.:II~I. tor •• ~ A

t

\

,..:,• 1.i.JI;) • (A_') )

-.J..l W

i":'>4 ~ ~ I

A x B == (AB sin e)~ ~.

1\

, ~

,~I.j

,

\oS I

, ~I .yl;~l;ve ~ n ,~I~,

•

_...

~

~ ~1'4:"¥I~.. ~~ ~t ( II ,.....,.

--

I. L-.J-. t.1.>·11 I':_~ ~ ~ • ,..-..II .. , >:..II • 'Y v-.

,.

v:.

~I

ijl..

~ oJ oJ$ C-;,.S ~".;. ~bJ' ~

Representation of a Given Vector Soalar and a singJ.e Unit Veotor

8S

tW I 1.iA u I ~ J.:=:..: (\Y'_ \) '¥

the Produot of

8

H'

4- ·

-?:- •

OOS()(,

oos;3 ,

~=

.1.::--: 1 ~~j J.:.:: -....

A • A( 1 oos 0( + j oos

-

= A( oos ~

0(

,

13

oos

'0 ,13

+ k oos ~

oos,8 , ooe

)

'0 ) (

A. An

.oos ~, oos,$, OOSDc~ Yl,!.r

.......

B

~

~

~

---

•

-

..b-,.~

,,- ,

--""

n

)

"

'~

B j:. ~l ~~, ...,n.i' --.... ~ B ..b..i-.- v 1 c..-.c ',J I vo

ooe e ;:: B • A • B.n A

-

B ,

--... A

.

~

L

\.1\ v-I••

~'J"

JI'-_-...••

......

,.,.

. .......

...

A = 21+j-k

B • 1-j+2k -

n •

~

Ax B • AxB

1__ :;:::CI AxB ~

= --:;1--.

{l9

A

....

1-'~-'k

•

-

l;~ (n J~IV'

~)W 1,-:,,yA!.1

Triple Produots ~,~ ~)bJ1 is oJ."".Ji,-:,,rAl~

.......

--">

A. (B

-

%

~I ~

0)

~ ~ loS J.JS.'-:'rb ,.;~ ~oJ.JS.~""" •

(, (l f. _,

~u~

-"

--

-....

A.(B x 0):.:

~

a

t

---B , -.....A

JI

oJ ~ I ,-:,,rAI1

Bx

By

13 m

Ox

0y

Oz

"oj ~ ~ ~ J .:; U t .r.* "; oJ.J...>.e

•

-....

) CiJoJWlcl~I ....."...., oJ~

J I.:J I~ I .> J ~ lIS

J

(, C~ \

~)l

&...~ I.:S'

.!4,

oS-J I

('n_')

~ oJ~ , I ~

J.J l;:

..1.4

~'vl~ ~I.J~ L;,~I ~L;JI ........ ~. L.i.l6J1 UWI.~I4IoJWI .."

--:...-....

A. (:8

......

~

CiJ~

~~

x 0) :: (A x B).O

4~_...... ·1I

.....

J

...,.4-::~I,-:,~I"'~' \S.hMJI .....;4JI4..~ JoJ~~1S1 ~)WI~~~I

--"'0

A

-'"0_ x (:8

%

~I~

0)

Triple Veotor Produot

~

~.....:h,

"....2&.....

~

-=--'.

_~

A x (B x 0) = (A.O) 13 - (A.B)O

t:"l;I"b. '11 ,tl;:.; ~. ( ,. _ ,

Change of Ooordinate System

-=

'~H:.II~I ~

fila>.

,--..It...

I.

A,.

til

-...

J.s.

A

'-A

·~7

~... ,~I

II

~#

~~/

V"4)~

- ...

Ay'=

",I

-a... AI

A

AI

Ax

A"j == (1.j )

.-'0,.

... /

AZ / == A.k

,...

(

vJ oJ.:

j

i/

lfIc..

""

k

J:.,

,,~

l,)

11

~ 1;..t->,JI

I"'\,,, I

Ay +

,..,.',/,

(~ .. k )Ay +

), (

" ...

1.i

/

(k.j ) Az .........

J-li ~ , ~ 1 ~ I..i

('{ t,_ ,)

I

(k.k )A z .

) 1S,j~I",:",rAlI...,,-- ~

UJW ~, Coefficients of !ransf'ormation • IJ.-:.i

i

"."J;

1\

.... "1

.... AI'

+ (j.;1 )

~"'/

1.j

1\

(3.1 ) Ay + (k.1 ) Az

Ax +

-..."

(1.k )

==

~,....u.,

I)

Ax +

..... ...1

A.i == (1.1 )

1\

1S.J~I",:"".AJ1

A.. i

~l:S ~ 1.i.U, ~'=

~I

A

";WI~I ~

A'v _ :

A == iAl'+ iL/+ kA'

_~

~

'"

i, j k ~~";~;jJ~ 1\/

III

+ kill

A/ ""'-/ AI'

~ ~ ..... ~.

~I vi ~jt

A

'"

1Ax + 3A,-

A ==

-~

";~I aJ'J~ J:,...u

13k

J.,~I J.~

.., b..J I ~ ~ 1,;.1 1 r G~

_: .; t:J I ,.,..:J I ~ •.r.:> ':II '" b.J 11:.-l.,S",It VS ~ J.: l.:J ~ , ~ .....

4/ ..

~, ,..

",I'"

..... '\.

-VA

'v

"'-I ....

A.

Ax == A.1 == (i.i)Ax'+ (j.i)Ay/+ (k.i)Azl ,#'te./#ft

.. / ...

~~ == A.j == (1.j)Ax / + (j,,;J)Ay/+ (k"j)A .... 1....

z/

(\'0_ \)

"'/"

Az == A.k == (i"k)Ax / + (j.k)Ay'+ (k.k)A z / .."..; 6

I:.-~ .JJ ('{ "_ , ) ~ 'J.J W

I'J-.ti

~

. _II> r--,

.... ,./

1.1

=:

I J J.-.,.,...,.::J I~)t,. L... ~~ u 1

.... / ..... 1 .. i

':J 'i (\'

f._, )

~ 'J

oJ

WI

_ ~,"\> -~ \j

r.-J:.

~

.J.i (, ~ -) ) ~.:~ oJ lAo u"t...

....,;

~ I cll.: .;..!.I ,

• ("...LJ~, (" {_l ) ~~oJ!.A. ~ ~ .J l.-....J \ IJ4 .:J t:s,...~ ..J1 1$#....

...\

u l:: u.

L.-..i...}o ; i;i>.11

J u1;~

J-~' ~~oJlAo.;s ~ul ~

_: J l:J I J!..:.II ~ Ax"

Ay/

=

Az"

J

...

..../

~.~

j .i

......../ i.j

...

..../

['"

.....

....... "

..../

~

... ,,/ k.j ...... k.k

r-J U I

~~oJ~l~ ~ ~Y.. ;)~

k.i

....... " j .k

',-~ - II ....,~

Matrix

(" {_l )

j.j

i.k

J...,,-:J \ v::: I"; u I LA, ' ~~ L.~ 1oP!,.>...t' .(0 ~~~k.i..~

~ ~~ ~ I

(Y'l_l)

Ay

/

Az

~ I u~ ~..; ~..w I 'I"

j

u,.t.-J I ..:!'--:.,

'I"

cill,j,;j~ ~l::;. ~)L~ -...t......srl~IL..i~1 '~I,.;•.;.... ,

• r-> 'J I J

,

1'J ~

~

'.r. I) oJ

y

z " , z .:JIJ'

... A = 3i + 2j + k"

~

,

J..~ JS u;.-i- '7'A·

~

'""I

/

x V"~Iulvjl

>.'1

• lb::..• , I..>--:

z

-

i b

"/

j

...

k

~I~ /

_l

·)tJ1

...".

>JIJ,.........

to

L. P. Smith, Ma"thematical Methods for Scientists and Eng1neers,Prentice-Hall,Eng1ewood Cliffs, N.J., 1953

TY

s,s"

~, I:.IJ~~I l:.~oJ

J"

(

\. _

\

JS:,.:JI

)

•• - ~It---_""'C.-:.J..,I

- •

....... k.1- 0 ..... , 1.k ..

.... " 3.k ..

4)

"

]f • ~' ., .. A,

ci~lr

Jo....:JI J.~

~

0

t.k .. 1

0

,-.

"l-'

At· {} +

}

•

• yll:Jl ~I

uJ-

Ii '

A.' - 1

~I c..IJ~l...J1 -.J~..&e

t . ..J... t-...!.. !' + k' f2 12

s-~I ~. ~~......wl c..IJ ~L..JI

i •. l~. i... 3".

1.i' "!,~1!,

iii

~..u.'

.1n. , .08

8

'"

.,:,!" oJ

'

-,J.A

....

J. ~I..,br~ ~.l.J

..u. ~I ~~ ~ _,.

(-.JWI_.4J .... l.. -.J~ ~WI JI:.JI ) A ..

k.k. 1

~ ~I Io.i"...... ~ J I .

~ "J ~;I ..s oJ ~'-:-~ ~

-'" --::.r.. " " ~.A. A" A "" 2i-j , B=2j+3k , O=i+j+k

vl~ IJI ~

(A x B).a , 1. (a x 0)

~ L,

P1

'"

"

rL.........; ~~ "~j,

N

~

(1)

Fi ""

~l..j r-~,)s

~

"

i

OP1 ",,:r1

)I.M.

c;

e..I

~I

r2 = i + j + k

+

"

--.........--..

.l:l:~"; ~-"'I.·"';

'_l

(,

A,.iJI..:,J.J..-

V_t

'"

21 + j ~lwl5' ~

I!'2

B f

t c:

.

.... j~. k

~I rj-tJ1

1,)1»..u1 ~

4

I.......

-"-..... .i~H:J I .i.-i: L...:J I vA.r.

A_ ,

(Bx C) = (A.C) B - (A.B)C ..sjt,.:. V- ~)i.:o. ~ I,)~ B , A I,)~I vIS' l.:il 11' x 13 I ..s,L..::t)\..p~1 ..sjl,.:.u.L- vl.)s:v-".J'l. ~I

'_1

_

..... --.

-'- _-'>.....

• X

..

...; J.>I ~ 4.~;J I L-", I.iB u I ~ ~ I ~ Li.~ 1JoA;-: \. - \

• ~~ I e

r I ~- ~ ~ I.ili.J I ~ ~":l.::.j1 u"; II v-".J'l \ \- \ --"-

~

-->0.

L...:.J)l:. ~I ~)I: ~ C, B ,A ~~I...-:.,;15I..i1 ..s,; """'" ~)I.lk .1I..sjl,.;. r:- u I .).c. 1JoA.J'l • ~)l.l.~ ~jl,::J '"4

[-;.(13 XC)]

ll_\

t

I· vl..o.u~ ~ ~ ; .=:-:JI}~v'~v-A.r. It_'

,,-';"0;

oos

e

sin

-sin

e

COB

o

o

1

• e laJI.,u. ;..."Ij-! e

i.-.." Ij-!

•

VI

L.olj-!

-11..'-......1--.4./

....

0, b,a

.......:11oo. -....,

a.a

l' -

--:sr. ~

,

--"-../ 1:1I

c.o

1:1I

J,....... Vl..o..u

J..~ I

u,&..

~

,._,

~I J» ~T v 1..0 oJ ~

-......

1:.~'~v~1

b, a

0,

-.a.. -..... /

= b.b

~I

II -

-,.o>-J I J» v l..o..u

Pi

• 11

u,....;....J1

o o

e e

v \S'

1

IJ' W

oJ

11_'

l;:.

~""""":'/

a.b == 0

_

-/

C

-

--4.

(8, x b)

=

Q Q a..

=--.a. ,...

, j ,i

~

a=

-

-a..

(b

~

:It

0)

l{

t

I:i

-

-

(O:lt a)

Q

Q

-'"

(b x 0) ~Ie- -.JoJl;:.

v,$;

I:.~ ~ ~

""

,,~

• i + j + k

,y-,

r""

4frJ I

.I:JI, V I -!II --

~

c I

~ ~I

J.ls::"

~~

- .~'dSr--~;oJ~IdSJ.~.Jl,..l£~)~J.JJI ,..;,.,.

t

yi

c~' ~~I j.l!::.JI, j..-li:Jlrls Jt-.:...~ I~~~l.-J'~ .... , ~I ~

Derivative of a Vector .J....i ~"

-

.;

•e;.J1 •

(\

_"

J,:. ..t>I, ~ JI,oJ "'" A ~Icli". vV"i'

• u

l.,'

~ oJ~I· Parameter • u .i'e!.nJI~, c4J..i '-"I _"..,1 ~,...., ~ ---.. • A ~'~~". ~ u;" ;...,J'~' v'!= .Ai, • t ~,;Jl

= i~(u)

--a.

"

A(u) _~

,.

~ \..!....a.>.;

iA,

l.i.:; ~l..

n

"

+ jAy(u) + kAz(U) ;J~ U;~ ;.,...:J~

A

~I .t-,AtU~,

~ W, •..b...J~ ~ oil .J..AJ I ~ ~,

Lim1 t

..61:-

AI!:iA

1\

J',..u u oJ ~~ I J..;l£:J I !:ilL

III

CUi ='J.im:i:t A~ = limit (i x + ~ ----Z + k !:iu-. o' ,w 6u-o !:i U !:iu vi lSI • ~ ~t::.Ax = ~(u + !:i u) - .Ax(u)

iA "¥X 1, ~ 4J.1S' ;J~ ~t~,. ~~. ~~".

),6.0"1'

~,JI~ ~~II.i.A,. J,6...I1 J..t>1~ Jld~l:~I~I;1 loS I

• J.UJI

J ~I ~)~..ul J,6.JI.j'~

Position Vector

,.. .k'-:Ji:~,JI~ v~ ('_t)

--....

y

= ix

r

"

"-

0"-'n

+ jy + kz

x z ~I~ (\_'f

loSi. ~;.U

x :: x(t) , y

)JU.l1

JI, ... v,!:; ~~ ~~I

= y(t)

, z

=

z(t)

The Velocity Vector ~!Jc \oS'i ;;..,....:J l: ~ IS I J..- \A.:.I .t~,;JI ,...,;.~~ ~~~

• -:;: ~pJ l: .......... V ::

-

A..

~rJI~ (T'_'I')

J,., ~ I ~,...:;J I (\ - '{) ~ I .j ~

r

t4',JI~,.. ~I vI! 1.ilut>;;J~

LrJ ;.,,; u,- ~ I; ~rJI·

dr -a:r

~I.:..~~,

~

t

~;.U ;;..,....:J l: -: J...P li:,

At...

.-.,.

== i.x + ~y + kz

JJ,..-, ~~t>'.J.:r-'ii lolA vi ).

t

~;.U ;;~l:

j.4.li:JI J.:..:: ~\i..:JI ~

UJ- • ... 11 ..".:;J1o/""4 1 ~I~, • (",ri-I JI ,,=,,~I J,I

~

."j;;?J"~ ~ t~.;JI.j~~.jvb'~vl

u- ~

l.i-J

;1

:-; (t + 60t

)~I ~I -; ( t )to",JI ~ ~I 1lJ~·· 6t·' u.}~

tA t ~,;.I'4~IJ)I""'~'j~l~ 4;=Z:-(t+ .4t)-r(t)

~I~ u...;';1 lJ!, .4..>Ij)lJ ~I,. ~ 4? / 4t~'&l>u*&£1J..i.I --'-

~WI ~

/dt

d.r

W' "...,;.11

~

4.l.J..a.e, iSraJI ,~l ~ .w.~1 .,;-'

4t «

~t, t:,Il> ";-;:;I".......li,,.-...I ;~J

4 r I 6 t ~

d.r / dt

0

; II ",-J

~I,. "rJ~ ~ ~I,

_T ) yrb~ b~1J!:J1

yi t';"',. tA

/ "" I

P I

o

I

I

/,

,

.e:::.:'---:;:r~.---:7' 'I

I / /

• p'

,~I

v~

u---I I p .i.b1;J1 ~J~ ..J1 J,.1:»' ~ .~

,,;-A cill-4

,~. v..it "rJ 1

)I"b;~l ~

V=

p ~ p

1.0:41 IlJJ

p" ~I ,,;-f"i=,,;..Jl ~

J Jl-JJV"WI4~' ~

'""W

At i,,;-f"i= I,~

r.l

6, 4 t ~l .~I)~ 1.L. ~I ..

• ..\el.....J1~~",..JI4Jj ...." Sp~ed J)lb;~~ u.,-lI'}.aL ..,...,.

v /_1

=

(.2 •2 e2 x +1' +Z

)*

-. c;l:JlCo-f) J(.:JI, ~

n. ~ .!.J..tJ1

uWI

w.:. 1.;1

~jo-I ~~J)l..b.;~1

vr- ~ 1,)1 ~ L:l

: ~l:Jl ~1 ~clJ';, s ,r.-.)LuWI J,J. ~

v =

~

= limit

6t-o

~

= limit ...w6x)2 +

6t-o

•• (o_'() ;J.)WI ~ ~~,

6'(;

J'..u..u ~,l-

Accelera.tion Vector

a

Cl,)~

l:,

~ dV a = Cit =

........

IJ,...

d2

(6Y12~:. (~z)2 ~

',la,:"'f ..u: .joll,

~, ~

.;.)l:~,~l~~,;JJ~~I.h·.•

( {-'( )

I,)'

r-2

( 1_,( )

dt

:-..

t

" ••

( V_'( )

a = ix + d.Y + kz

~ '-- --q . I ,~, J'. 6••.I1;J~.A..':l.S .. J" ~~ -.-.---.; ~J' .) I -: ~,~ r::r u,.., · .d".,..::J 1~ I~,. ~ ~I "'-> ~

i-,..;~I ~I -

...... ~ ~L-.:J'

~

"£ .

.L4I1 J

•.

(J

....

~~, ~JI,

..... Wl

;J~Wl:u:.J' iSrJI ~

-,

• (A_'()

1-:. I . -

r(t)

• xy

= ibt "

A. + ;Hot -

-

+ )+ t2

....

kO

-s~IJ;Sra-J:.;;J~~li,Ij..o~,l..;,~l: • -~J'.~lS'

U

(

-4

......

• • ~I , t ~)J!~~ r J..,;1.i:..u: v ~rJI ~ ~

1"0 ~

V

drA.

,.

Hc -

= "(!f"'~ib +

gt)

"I • ~ 1 ~ J..-.; ~W I:;"JJ

lrL-li::.o

.......

J'-,. g

~WIJI.u.Jl.J,. 7 _",.......u~WI.~~'"=~ a u'

J'

• (\" _~ ) ~I ~ ~ ~ U' wlf. LJJ u~3S'~1 J)I....J,.;~I )'. u,.." • (~..iiJ' is.,.. ~ I ~ ~ ;...J ..tWI • .:u.

T

=

[b 2 +

(0 _

gt)2

s ;...,H:oJl6.l..tU W,

Ji

~I, _"..J~e....-,JI~~

!b

t.~

(\" -~) J!.:JI

;If.~ J'- ~cd~~

.", ... r =

uI )

.. ~~~I~ul~,,;I_~ sin w t + ;Jb cos wt + ko ~I, J.- ~I·;'w'; v-liWI • iS~I~,

1];"1

= r • (,.,2 :J1n2 w t + b 2 00s2

= (b 2

+

0

2

)

of

c.eJ

t +

0

2 ).

T

: VI

........ v

dr = ,

v=v1"'

;,;-->.;

J

T ~I J~'"~'I

~

..u: ~ II! Ij>~ l..J.4'

(lY,_,,) 4w W I ~ I 4 .J>,J ~

i' ~ I ~,J

~ d-; d (,.,.,) •. dr a = - = - VI =y'l'+T·dt dt dt

-

d~1 LrJ -:; \';~b J'..l.i. W ~I do.b-,J' 0' IJ4 t}~

d T Idt

c

(."

:P \~~

IS;>' ~ JI cl.il- J,Je.

J.t.

6. s ;;sL-,

'!.';--":;

r

P \J~ 4.bt 1~

j

.......

d7"'

=n

d't'

Chain Rule dr

d7

dY'

dt

d'f

dt

-...

d'f

-=--=nde

;o=~ ~

r/ dt

..;a..~ ~S:'

~

-'"

y

dt

;>

'~

d'f

d

ds

-=n-

• -...

P d.hiU I ..) ~.r--=J I r--'~ I J l- ;~ ;-1~ ,j.",.; '-'~ ,~:

y2---,,-

a=v7"+·;0

n

(,,-'{)

, _ - - J llbl

J J.:.;-.:JI~, • ;}I..JI;.b.iu.,..; ~ Ro~ v2/Ro

J ~ .) I"h

t.:~ ~~ .r.f w)lh;~10 IS' l..il 1..1 • ~;I..J1 jJ- ~ ~I.,) ;-",;JbJ'

Y.J ~

~I ~ILl;.J":'~.

(

'\~. 'I' )

.J J.;l.iJ I ~.:.Il~ y

x

6.

a

\V'"

H,.

u.r':'''''',J I ~~".J I ;I.u.

.....

a :: -

dV dt

e

+ r r

e

..... . dlr " ..... . dIe = r1 + r - - - - + (rtt + re)je + re ---r dt dt

_t, _ (y. -" ) ~.JWIV' d3EV'dt, d£:!dt

,

~~-.:.II ~

.:~I.1>~I4.l~~~I_~a.Jl:JI4J.JWI~~

.:,.1

( Yl_ Y)

• ~·I~hill -'0

(n'_y>

&=

8r

a... ~-t1

•• ,,2 == r - rt1

= re

+

2:i:-e = -1.. ..L (r2 S) r dt

(

"0 - Y)

~;";~d}l.J ..,h~clA L.~~I ~. d~WI~~1 ~

.2 be

- f. y-

e

ot

=:

.JI~( ~~_'f) 4.L"WI04

11" ...L.

- ; =:

(41;2) +

dt

~

~(-2b _bt 2 0 2 ) +

= b(2 L-~ "!

(ot)

d1;

(2bt)1r + {bct2 )Se

=:

- : =:

3'e (b1;2) -..L.

t.

~

( H' _

3'e [0

'f )

~ l.:! ~

+ 2(2bt)e]

2 2 - t 0 )1r + 4bct3e

IL·~~~ -, - _-II "! L~II U,,)4-' .11 c..J. I JhJl

A+'

4J .d..."J t 04 j...a:>..:

I~

J.A.:>.)l.; u • t .1.-iJ1 . . ~

IJ

Velocity and Acoeleration in Cylin~ical and Spherioal Ooordinates: Spht'rical coo1"d:lna.te 4...,;1.... .- ., ~ ':it c.:..~I.,\:> "il

.

.. L~~L. ~..~.:.." ; ..1>;:It, xy

..:..~~~ll.:.-lr~l~,!;~ • (\. - T )

i:a

,.:-.J I ..j ~ I ~ I ;..1>, ~

r4>

J!:J I ~ &

a1l:~4..b-, ~A .::...

~ t.f ~ l,!.)!';~



,»>oJ I

z A

.....

:La : -)~

l. .1>

~1".b-'JL.::..I;I..1.>'JIV_ ~

~ ..i 0,f..." lJ~-: "

J

jet> lSI (, r)I~..J.;~.JI-::..~~-I..1.>,J

d..-.,,}J I..::.. t; 1.1> '!! "

i r kg

.....

j;

l~~ I ~ I -J)J~ ) ~l 01 ( 1 )

J

~ll ..1.;-\ I ..;';,~

":"';""-;j •

:L.",Al I,;

- lSI LI. "1 -::.. L....;>.:::., ~..:: . f..."".• - ~jJ ,. • .. '" '-'

.

.." ..II Ij_.,i.;;.),

... ""

ir

C;)l:Jl ~ ~~ l:.11 ~ 1.i)l.••.J1 v lIS;

1> k e

j

Ji:J I ~ I ~} ~ - *' ... ill j

~~ ~~ 45,.. ~.~I !jIll _ t

e•

¢ = wt

r = b

I.,.

"'"'-...1:.: t.S ..u I J W I ....11: ~

..If- (1 + i

1;

OOB iN 1;)

v-j1J. ~J ~.JS ~ t J I~, J;)lh; ~ 1 ~

~

-A. - ciA

dt

= A - ciA

v 1 ~.:.rAJ". 1 Y-

dt

-... 2 ,.. .... ... A =2t i + 3t~ - 2k

...

~

B

I::

i

u IS' I J I 1r _ t

~

OOB W

l

t + j sin w t +

t

2A

~

k

....!.. (1 x B)

-- d (A.B)

dt

dt

• t ~

~-

V--;J.J J ~ .J-.J

-~-

• r. (v x a)

-

-a =dv ~

.j ~---.:JI ~

aa

-...

,

-v-(ti'"' dr

~1: ",,1.1 ~~J~I

d- ~~ Gf>'~ lo -

I. ;.W:.~ I ..;rt I ;..-oj • L•itb....J • I .,; b- ...,..

t

J- ~":""""::·",.s1W tl::-J' u b.

> 11.1: ~J • J

J~I

.. .

....

,.. ("'l_Y)~~.,lWI ~~ d~~dt- - , ; , ~

L-J.,lL-,.

d

~~

_1

•.'""'~ ~L.

4;/4-"3,_

( f 1 _ '{ ), ( f. _ '{ )

• ~jll~I.;I..b-)'1

( \;-' ), • ~ 11",.

~14111- '{

.,l W I ~ ~ I j.!1 l Y_ '{

~I .J~~ "j)llr~~1 '~I lA _"

• ( fA _ " ) l..~

JJ> -

Jl:JI,....:JI Js~~.,lJ j~-:-.j b

•

lJb,FbJ.-i....: ~~ l'\ - '{

.......--. ~w24.:eb~~u.rc "....,.""c..l~ ~..J~. w:J.. ~~~~-J""C

.

,:"~I ~~ ~l~ . &/~I~ 4A.. .:kt~ LN,....

•1

W~uJ.a.a:~~ ~I ..t~~~I~I,:I~~1 t-u.:- I ~ •

1

I,!,"....JI va a..1; ~1 .j ~I wl;..:.i~~

L-J bJ' ;,;:,:......1L.

lJ-J..;..... -:,~

~= --'-

V, V, V,

r

~_.,

W' ".~ I; ..b-,J I SI~;1 '{. _ "

...!... V

....".

a, a

bS'1-

4.I':l~

-"'"

n

~J,....JI ~I d.J>,u;.;~ ~

.r. J,

td~1 ~Ivl ~v,;LiII~,

;.

....;. ',...aJ ~ cr--r. I. k

Inertia

~1.i.ll~I,1

t ~ cJ' I. ~I: ~ ~ ~.J..> ~

~L.: ~~r-J: _.J->.~ ~I~~

--,-"It"

-!,--. cdJ.i ~.JI~I v'oJ 1.;1, ( ..s,...aJI) ~JWI~lkWI~..J.-.i; ~ ~ cdJ.i vti ~ r l

1....-

iil;",11

~

J.

UpJlu..,J r~·- -L...Li.~~...,b..~ ~.LS

'i~~...,b...u-e ~ ~ ~ ~IJoA I. J,~I

Newtonian

~ ~l;

4,;

y,;L;V I

; :;~I ,I i".:..-.JI ~~I..o"-'~

or Iner1rial-

• J, ~l

u:,r

~

• .i.4!.. ~

: ~ l:J i J 1-,

Re.fercnce sys1iem

II ~ v I ~ v~ 1:.Jll

~~Ivl , ~,I~,:,.e; t...,l..... v~ ~...,b..~1S' 1.iII.WJ'A- ~U:tS' ..s,.-iJ I ~ k

t u-e ~ I ~

t~ ~ ~

~)l;. ~ ~ 1-4S J 1-,.- ~ JbJIL·_)~uS-.d Li.... ,· I.L:...J, ..s...,,,.eJlu-e~1;

..,It-=~ iiL- ~I .L..Atc cJ

vW u

6

I. ~I ~ ;-,; Joll ~.i4JI~,. ~ JJ~I

i# '-"", ~I;.:. WoJ uti Lc.,

_~JJ ;";~I...,I....JI~I ~';IV~~J~I~;'_:~J1...,l-.J1

~L..: ~~, ,..;- .... ~~ L.J~',ro1J1 ~p".

lr~I oJI,-J1 ~ u-e c::l::>.; ~ Joll ~I vi>1j~1 J 1.1

'1

J.~ ~~~, u--:Jlj~, l.i'J~ljJ' r~'~

..: Joll ~

~ j;.Q;~I,

l.i-J ~ I i....-Spo ~ I. L: 4....,;~ ~ ..., b.J 1• .iIll J.> uS), • ~J' ~ljJ'JLt:-1 - Jl:JI~ ~ -,.,. ~~I~..;:11 vi • u--:J I' J»

II I

~.

• )-

I

~

~:>

[

to. c.

;;"

-::

-

t

.r ~. f

Ci;

~

t:"

t

~

-

~

c·~ c·-

I

.~

"'"

~ -:

J-I

~

c·

.

f.,

,..

Ci;

-

t !: .r ~

~

r

PJ. ~

'-.

'[

:..

t.-

_

tr. 'r r;'

I~· r"

-

f.,

c·

I

{f \- f f-'-,

T

y'

-a

•

r;'

~

t

I

"I

t.

'll

I

;

'.

I

~ ~1

L.

~. t -.~ ~-.- '.I I

.;- ~~" ~.1 -t· ~ q ~

4

02

~ .=- rr'

•

t:-I

.... s:; ,. ~

a("

.; -'1 ~

,-

• ~ .I'l ~ -

~ \

[

-:;

-

~

c: h ee-.

'C.

t"

t I( F

1-3

f ~ ~. 'k r

0

02

. 1 [ .~ ,"" or

or 1" ~,, "

~

i

c.:.. f L-

t

r£

1:;,.1t:

\r-- f- ., ~~a ~

1= .r" . ~ c.:... tf -~.r ~ 'i.. "' ....

t-c....\·

•

••

t t i= ~ ~ 1~

a

I ~-.-, _,..} .1. .1....., · '1.f-~ ~ _ ~ f !:~. t of I-

_

f1:J'-';;;: ~

\

\ tpQ~

"b

r

'-(.:z.;a 1t1 [F -.

1\

:::

::-

~ ~!~ \ .;-.

\:

~ :1 .r..."-: ( >- ~r.

;1-.

[

t!.

t

c·III'""

~:~ 7'}!;-

..

ttl

~r

~: c.:..

t. \.

\-

b

r (~

L

~ r:. \J~ .1 ...t: ,.- l~ ~ I . .-~ 1.::' \, ; . = . [ .. ~~~:=ff'!:tt.g~. t~·.: or '- E c· l- t· ,.. g .~. E • - I: r >- rr~or..\- ~.r..: ~ E ~ i c, f-'-:

~

t>j

~ ~

f.~ -" 'G..

1~t 1.. \'=t' 1.:.~. ~I ~ .r f .:. t t f ~ c.:.. ,i f11~ e:': 1= • ~ (\::. t: '

~

{

~~ of~d t~~ rs ~

n

~ ~

~ I~

a

~ .r I:.

lor;o'l~' {" ~ r.r ~I ~ ,,'~ ~ .~ :: ~ ~ ' . e. c+

-

r···

~'l.. tt::~~ i.g\.«i-C'~c...~ _.1» ~ ~-

'"

~... 't .

(.0

D

~

.". !"

tost -H

[

cr ~.t: . 1. ~ _ ~

t - ·t·'1='.f" f t- [ i t t-· .f. ~ ,~ . . ):: t ~ (: .: ~. t~

l l~,~

'I. - '1r

'1

~

1t

L-J.aL-J'J.-l!:

~ ~JJI u.L;JI~'JbJl~u'JI~, .:', 'J I

L

J"JJ Lc

: " _,_'6 ,~_.,......:11 -

Constant Faroe

a--Il...J1'~";I· 1.....;b;,....;l1 ~u~~1 ~ ~'JbJl~lul

-if.

;II

-+-

•• III

~b ~Iu~

=a

oonstant

( ll_r )

jX ax = Jt (at + vo)dt x ... at 2 + vat + Xo ~,JJI ~. ~lo.Ut'JI~1 Xo ¥1~'JI~r-l1

J.r. J..--= 2&

ex - Xo )

IA,J\J' a1pJ'

\lloo

J:.::

Vo ~

(ll_r) , ( ll_r ) aJ.a~lI ~

22

III

V

t

( lr_r)

- Vo

~.aLw ~ .~I ;....,s'.i.J1~ 'J .alAJl u Lc~ l1J I jJ:.• r·~ 1;-; " i ~-·'1~I.i

~I ~I

J

,

il':"

J.,.i-aJl"..

"')l....:.; ;~~~L-I~I

.:ll u,s:, '~IL..,li.

n g ,it)LcJ.,i II

JI

.-1 ~J'JI;~I

i;t:b:cdla

d-- Uoe,:-".iILc

~I~I~J:.::,.

(g == 9.8 m/sec

2

L...~

hst

........ liJh,.atMJ1

"\0

( JJ--....Ii:...,JI)

. , u"J ~ I ~.i ~ ."...; v,.,x

Ji-- ~ 1~;

dJ..w U;"

"..t-_"t-~.JIWp"~)..UIU~

WI .... ~!,.:.~.i~I.,.;, mg \oS,L...:,

l"fo.,... ... _ ........_, , . ~I ~ ;''; ..tJJJI, \oSr->I\oS,....;~I~.l;. .. ~,

.nii

,~L.;

v ~I V4

JI...- _•• ~~I ~ e ~I.x J:N u-1.1 x -A

~ ~I .4-:~I1:~1 ..tJ, ( ,

x • ~~ ~.i ~ 1."; ~oJlo:.o ~ ...

Ji-I.;.J~'I..

d--

L..-J 1rtol.>':..

F lila

,. cdJ.i.I ~l: ~ I•

~

ji...JJ1 J V'::1'~ IJ'

v,s::: cUJ .i.I, • V'::1' ~ IJ'. d--l l Ji-I ~

~UpJl.~

a =

l_\" )

?"V l u""il

.u. ~ Lr U,

mg sin

e

\oS, L...:

(l\,,_\,,), (IT-\'') ,( "_\" )

sin &'

~

)Jmg oos 9'

(b)

(a)

u-1. 1 c:L- (1) ~1. c:1-Ji-lcjJ~~ ( l_\") JL.:JI ~

c:h-- loS .s1~~,JI J1.)iJ1 ~fS

L-o ~ ~"

t

je} Ie • oS I.

cU.i-c,4JL. a o,!.; JJ~ .1I4~~Lc.sI. ~UI u~ .i~

a

1:1

f.

~1S"1.i1· cd~~1

• ~,LJIUb. JII~I~I J.-c

d--S I ~I JIU',pJlu,!.;"~'~ (u',J-L.: ;

~;.JLc

) ~I, x - ~

g( sin 9' +./J' oos e-) W.ISlI, &.4pJl WlWl

fhe

JJ~ &.

tan.:?- ~ ts'

&-<E ~ts' l.il LI • ~t ~lc ck--'I

cdtt.-..:...~IJ'; 4~lwlW:..~ t:.J-,

.,..,.....

l.i1 ~I ~I oSloS.i:t-,

't~ L.o Ij! I J:.:, &-;>

.1 ~~ ,.~~.JI~lul ..O

Ji

( , I_T)

&-)

~oroe

L.."

;.,J.ii~ J)JI

WI.aS';,il1

(Y_T

a8 a Funotion ot Position Only.

fhe Conoepts ot Kinetio and Potential Energy

:.

tlLcJ.ii· '-" J& ~ uls.,ill~'= ~ ;",,~w:.1

t&Ir&::"'~ ~ ~ ~J~ I '-:" ..i.~.11 cS'; e.P 4J ~.JI. J.A ~ U'.,...u ~li:6Il

J

cJtW:)l:.;. cS.t>1 t ~ I"'" I

aJ oS L".JI ,. u,ll oi-.AJ I,I v-t1I1 .s'; ~ cU.l1' ~ ,

i V

t

..

~

4JbJl.JA1 pi' ••

.li'(x) • mx

cJJ---Lt ~ -.a...> ~

.w.

( ) o_f)

WJ,1i.:11 ~ ~ ~ l.-J i ~ ~ I 1.iA J-.. ~ \ ~ ~I

t ..,JWI".wJ1 ~~I~~ '" ~ .._ ...JI, _...,iJlu}JI~, •

_#

( ) '_f)

F(x) == mv

~

t

==

2

~

==

-I

~ ~ ~I ~'F' ... ::JJ ~pJ I wUJ ~ ~ !-imv2

(\ Y_f )

;....,JJ I~

: cSl J.~I~ (\Y_f ) W~L-..H

f.(x)

dx ==

J

M!

( lA_f)

~L::~p= ~I~LI •.JIJ.~.J.JlJJ'(X)dx J.~IJ:.c ~~I :

';W'~lja V (x) J:.WI~ u~, F (x) _,ill

d'f - (!i == Ji'(x)

\~~I) ~b~.J-ii

:

J

';':H~I~

F{x)dx == -

!r

J.1" • ...... &gIWUJ~..,......:

S.y

V (x) LJloUl,

V ,"x) .Je~~J.A-:J(J.k: ~dl.i.c, uLr..

dx == - Vex) + oonstant

+ V .: ... mv2 + Vex) • constant == E

....II~

_:_

;;J,JI;,.iJI~~ 1..i1_\S~I.J~. ~'WUJ~

]I ~J

oo. l!J I ,oo~ U - II uUJ I ~ U. 1;, t-- &; • Ju. ~.,.I.. ;S - 11.Wi "'e"'~"'~ ,,;..JI • Conservative (l) ~b... 11 bJ 1• .iA ..j;,.iJ 1 ~,. UpJ 1J~ ~t •

~ ~

v--1.:.)~lv~~l5:UI~

lrl

~I,:; ~ cjJl\S1 ~b·.J1d

IS""''''

• cd I!::> ~'J:. • ~~'tr [("._rHJ.1WOuUJIW~L-.J>v-,... ,=:J1UP" .s~'~ vu)Ll.;)lJ

( l '_f)

: Jl:JI~I~J.t.";

I J~ (E:X~ L...~ x

I.l.-.A, ' l

l;~ f':;

V(x)=E

c::

r--iI..k.i;

=t

x ~ ~'A'

~v~ ~~Iv'

••

4-J 4.oL.-- ,I

-/0,.-,

v,s:. .. ~ ';""

'M,:

E

t..,J-; • .i.A,

\S; (ll_f)

~l UUJI ve

JJ 1~1W I, I;

lI.)WI~

Ji' v: ( x

,;b;.II..j..".....",

) t)~

c), .;..-c

~1

13~~1 ~ dJ..i ~II...U>' • V(x).(.:m J..,....,..I

",ooJJI .kli:JI~..j......::s'P"~' ~vl~p ,,~lIul~ 1.iA,

Turning points ( 'f_r' )

~)I.k~ J.li:JI • .iA...,-.;, • .~L-JIlr;

Ji-tJ1 ~.)~, ~•.:•••H ~liDJl~ aU, • U".JJ

V(x)

uwl

......o--JI-

lw

~~I.bI1J\

x

~ V(x);

;.lnl 1iUJI4JI ...

",~Ju,.

L..JUJ.I ~ ~ ~; 1..J.la: , a1'pJJ ~

JI ~t ~,atl

I;~II";I

'-t-:ti u,J:

c-r-J I ~.iW.J1

:sKI I

E;

•

• l_\" J!.:JI

.j-J14JbJJ..J.,;

"Oonservative" ~b.J1 a1'p.1J

-.

II poll f.r" "'1"11 Up- u I c..-~4Jb...",. .~I ~,~I

......-...;UJI4JI... , • -mgu~~/l'~~'.";uli. ~~I ~I ~ x .~I ~~~I ~t 0 ~la.ol-4,JSJ1

E ~ Vo

i mi: 2

;UJI

to

.....

-Il....

~)lJJ~ IoL.-~

+ mg:x:

~I~I ~)l.J..;lc ~~I ~I ~J.....:

E =

u";I~,L-: L.4b:J1

V = mg:x: + C.

t mv~ = t mi2

u.i.i..u

u.uu + mgx

~ul- J~I ~ ~- vpjl ~I~I~

x =

0

J~I

..a....or.,

;':JI~I~ ;J~~}I4...W:u~ aJb•.II.~ " , . WUJI;J ... W

t'

u J) :i: =

t

0

t"'"= La., ~I ~ JJ I, f.F=,.-;J I ~

2

mvo = ~ax

h

V

= ~ax

2

2~

=

,I

" : ~l:JI~1 ~u~ UUa.lIW.,~ J.~W~~. &.:-SpJl,

JX (v~

- 2gx)"-i dx = t

o V

o

g

~I

t,

X

_..l:..... g

(V2 0

~;"''')l.,J I

vJ.: -g

~

2g:x:)* = t

~ W.)l•.,~J1 • .lA

JlloL- a

u I u- ~ u I

~ UJ I ~

4l ... L.J1 J

~ ~ (' 'f_f )

.Jaii~"...u;JI...s;,.&I1 (A_f only ~,,-.-J Wi., .. ~ ~ ;;tl,JI.,.iJlu~ulul:t>~I;s'1 J~~

The Force as a Function of Velocity ~ ..

J

cd~ ~ ~ ;''; JJI

e-'I,JI 4li. ~b.

~ • ~r-ll e ~ ,:--1:.:: e:---J WI 4" Ii.

J

)l:.

i.iA ,c-"'"

u' ~ 4J.k~Iu-Ji yi

4S""""; cdl...a ~,Jult. v e-J-U-;s" ~1;.; ~A~W,Ier-l1

i.J l:J ~ I ~ U'pJJ ~li.:ll W ... W

t

=

J

mdv

F(T)

= t( v)

I t.....:~

uS-J I u- u It

J

4S"..>" ;',.

( l Lf)

\$1 v

~.r-ll ~L.JI4I",WI

J>

I.:..:~~

u 1l..,.t.Q; \..il

v :: vet) x :: U",WI.j

S

*

vet) dt :: x(t) ~..r-

\$,,>"i I Li-..)JJ I ~~ (H'_f)

F(V) :: my. ~

( l '_f)

• v W"i~ x ~~ J..K:.II-I~~, X --

S F(Y) mvdv

~ J...-: x

= x (v)

e

( lV_f)

.e,JI 411..tS"

v

~r-..ll U '" WI

4.u J> .Mr.,

V :: y(x)

t::

j

yg, :: t(x)

~)l,JI~ (lA_f), (lO_f) ~"i",l.A.JJu~ul~;

i,:b.II..,i . t , x

~_I

ulS",.e,r1.I,:--

ch-- J.s.

~ a ~F(v)= -cv ;

A'

YO

~

__ A,

~1~1~~u.iJ.u ~liul "",,'pI

u l l.5 l •

v t',,:--l::.:: ~11·J,.;JI4...,~ I;'l:.

cli:l14l",WI • ( ZpJI4~~

X

_...,.,.JI) ,,:-1:.:11 ~t

Y'f

dv - ov = m crt'

SV _ ~

t =

vo

ov

= _ l!! ln (....L) C

~ L..:oJ I~A dJ.i,,~,

",s::.

Vo

t. •.;,.jlI;J loSS'

=:-•.:oJli

v

~r-ll;J~ ~ ~

4-J;h!( exponent U"' ~I J..>I, -

i ~~

v = v e- ot/ m o

t

x =

S o

t

... ; _---tiill

v e- ot/ m dt

mv

= ---2 c

0

y

iJ L-

(1 _ e- ot/ m)

IJ..: I u..t..-.; ~ ,,:.Jlill v I • .. )!~I;;.oS..w I ;J ~ ~ IV- • ur mv/c

LA)I.u.

: ~I.:.II~I ~ ~1.i:J1 ;J~~I~l:s' ~..iS'~, -ov

dv = mv di ~,,:-WI

-0

SX ...:, ('f

dv

LrJ.l!:,

vr x

t-

S o

•

= -m In ( To:"'( aim); ) o To

dx V - {o/m)x o

( c.r ~I .l>~

-elm

~~

• ~l.- ~

l.e--A) x e

w.... JJI

~ u.JWI • .iA J... ~,

t , x

~~luoJ; ~

J..-.;

• J..ii ~j.LJ .JI.JS~,.iJ I ('_T' The Foree as a Function of Time Only: : Z'roJl ~.)I.-. J.~~. v-j.H ",.,i.s ~;~.,i1I~~IIJt

F(t) = m

$!5 dt

;

=

c&1I J, • ~~t..

5

F(t) dt m

.....JI.JS x

x

dv dt

=Sv(t)

v,...s,,; • J..ii

t

= v{t)

(f-_f)

~J..--: ~1:.I1.~1 LrJ..~,

dt

=S[}l!'~t'l

-

t

~j.LJ ~I.JS T ; ,h ...

~I 't ~;1J

dt

]

v-;1J ~I.JS; .,1 ... ;,.iI1

dt

~

v,s:: JJt~bJl

L.b)l.~,

It ·:t'~cS.,>~I~ ~bJllJ' lei - ~ ~~ J.~JL..-' ~L.SpJlaJ.JL...J. ~1:.I1~J',JIv-~li:J1 ~~.JWI~;

• t

..,r.:J1

~IJI...·

I~

v-;1J ~I.JS x ~I .J~~

y!-

JII-----.. . . '-i'~_'-"'';'' ~I ~I C6- ~s. v,L-II&Jb. ~ ~1~lvlS'~1J .. F -= at I.SI- ~t; 1~lr &~1;:;"4 ....:i'I.,.;~,·1e1 .. '

.. V4jl!

at

J,

~ d....> Ij'~ I,

~jJl

t- 0

48.r-J I ~

dv

= m at t

1 v = iii

S

2

atdt ::::

....£:L 2m

o

,

3

2 E~_ dt ==

£i..

2m

6m

o (

J;: '" 0

J J>4,,'"iI4-hi~ ~i0i) ~I~~I ~liJl~vl~ 1.Ji'~W I :f.r.)..-., -, f' li;~..6--, ~ A,J,;l.....:J I ;SpJ I

(1 c>_T'

Vertical Motion in a Hesisting Medium;Terminal Velooity Z~~":.1 f.i;" ~ I •__..:.1 ~ Lit! ~ ,I t'~~.J I J l),jiL~~

v

4r:F--lItoJ -0 v" ~~ 1.)1'1') L..ic JJ lit> .;. • ~I •...,s-..u IlI ..l...JI ~ ~.;,JI II ~~I;

~o

TO

.)1..... .;I~I ~L, ••h~1

-.]1 :r.

~a.tIi:JI,,1~ ~~I, .~)~I.~I.,J.i~

J.1.. ,., .:.,. \)~

T2 • To2 +

2. ~I JI

v=O ~clJJ,~tI.i::JI~IISI. ~.i.iJI.s'.p"~JUai: ~J..-..:

.:J,J;.

Xmax

=h =

v2

2g

(1 -

N::1L;

2;.'-

• x ~

1

J>, ( f,o_t' )

~.a ~~ 1....")l..J 1~

v 2

4l.aWI

J..-.; IS.,>I •.,.

o

h=

2g

,,:I:JI ..a.-JIJL.,..I~II";I

~ h ~J-11 v o " ; .a~~ (f,o_t') M:i.a.1,u)L-.H~'~1

~~!,JI ~.acape spee4e)li~1 ",,:)l.Ja.:L,..,.,.....: ~~(·~Y ~

:

~; ~!l_;.J.I~L-c.J"t,.;~'J>I.,~,~ I .ie,;.a~1

ve

= (2gre,i

V

=- 11

e

km/sec.

~ • .a

u,~~)~I.}l1 ~'" (N2 ,02) 1; ;·.. ·GJ.iJ,

A

'!,AJIC-t.~ ~r- J~ul

~)l;~1 ~I ~u.iJ1 (0 ':.1;-)

~)l.-J~I u)lJ.;1 u ~ ,~.II ~ IJJl:i.JI • ~I '4£1J.j~

IJ-....AJ,

.~ r-i- ~ tit) ~I eb-.)s.~ ~I

.w .i e-' • ..,.. b

· }'.i.: ~ I ~ l.i ~ Js.

u-

J!,>

~ ~/~I

..,.. b

.s; ~ I I,r.->I p i

~ ~ ~.F.II ,~':'

0

Js.

~I

u~ ~ ~ 1•..0 ~,.. yoio) ~ I ~ u Ii

~~)~~14£I~.!.it - ~u,1l1 ","':;P~f)L.~J~1 .,.~) ~1v-~~I~~lu~ ~~j ~~)~I

J

~L

~~ ~

1.r.J 10) ~ 4£llUb u 1~ (

~J 0) o;tAll ~~ Ll:S'

- ~)~I~)l;1 cj)U.;1 ~ 4....W ~I ~ ~~I ~).Jo;tAll ~t,:JIW'~1

_' .b ...II:i~I.,.iI1 (l'f_T'

Linear Restoring Foroe. Harmonic Motion ~~ ~I ~WI Vt f-i· .... Ju. Js.W'pJl~~l>,.,.' Vt :i..L-.>~

Linear Restoring Foro ~ • ~ .'; t,...~I, )I,,&::-~I~ V- f-

~L..; ,I u.l""::' '.

II

• •

y

~

JJ l.s~ 1 c:U:: ~

~II L..>~Ie- LA)I..u. ~~ .,.iII • .lA

~.";

-

~lj~I.~~ 1~La. wl.J u~ ~u,;liJ u~

F

=-

k(X - a)

=-

kx

j~Iw..~~IJ~u'KineticT.heory~pJl4....rI;.;lllM,

1,$,' m ,

"0

ul..;:J~~l: J:.::

k

~

•

;':.JI.)I~I~).J

T

"~J.J JSl 4£1)' ,,-\. x

(3kT/m)t

u,~ \JT'A

• w..~1 W:S

({)

#

Jan

~ '4~~W' ~,,~ a

J'~\Ht":'t~~~laJl~,,»'I»~1

!)

tJ11O!

:!(;

~I-a

',:j

~J,:,I !,~ ld:

cSti.tneae

',/

"'J

(n 11.

•• ~~, ~i~1 ,,:-J.,.i::J1 ~~~ ( t_T' )

l....-S,pJ1 (y) y;~!1~~1 (1)~!;, m ~h~i~~

L..J.,.iL.

\J I

J • -It (1: - a) + mg Z~~UJbJl~ ~ ~ ~·~H" .. jt.~=-;J

t ,J 1$1" .P""~ Afl~w 1..-SpJJ :,.L.eli:ll dJ oJ W I I ~ , L;~ t ;;;~ F's: x-X-a-mg,!k

~ ;,~,~";;,,. ~k:J:

AY

.-

v,x.

··kx ::: mx

~bJl ~ lSI

J ,I

( u_r) d" .. )kl ;;~.\.)I Zp-lJ ~l&:J1 ;J.,iWI

,>.,._~-,-~~!,.;.r.:S J;,L.

.

k

Com

u;",1..:.:;

v~WI, ~M.J..>~.....: ..s..uIu'"LiJIJ~ ...H"'·.ft,:S"~

kS', • ~jL.........r' xL>I)~I, JL:.;1I~~WI;';,JO' ~I~ u~

v-JJ,

J.~ ~;..~ 14.1b- ~

v ILS·

J,~~I J~,

'~.::. ..::...:~I ;_>~J).iJ

J..n' ". w'"

vSl, (,

;J '" W I ~.iJb

1,:""') ~1~;-.:11

~ ~ 4Si--

~ ~!,:.lI, ~Ij u,....s; ~lj~1

4.J",L,..JI~~

I, I ~ L.Ji..il1 ) 1,:':;'\1 J;...:: x L.;;...S:J1, " 6~1..J I(Parameters) ~IAft'« • ~..I.:: ~ c.:;;.b.: (tA_i ) ;J '" WI J> ~

(i'~~~---'-~-'" I:r"J I .11 V-~, ~ c;L,iJl ),...4&,JI vi

L.

• •,.tll, ~I

V-

( .....; li;: I,I

)~.,..:; ~

J,;-.; ~~

~~

e-

~ \;~ vi ~~ • ~l: ~ c;l.iJl ~ ~1"",'=:.iJI,

V\'

~) ~ ~ La. f ,f-'i I ~ I ~ l: LJ..b-'i 'tt'; 1..iJ I ~ I "'" l,.i.

~IJ~;J,6..4--

....

~ u.JJ

JI',.-JII ..... ~~~ ~l,o:-I ~.~ ..

~)~I V- ~ ~ . . . . )I.:. ~ ~,l.: ~j' ,f-'il ~ LJ..b-I~"~

'.

,••11, .~~l: ~L-.:r.. WI-s vL;,S;:~ • • ~I ~ vi • i}oAJl

. \:.. . .i.>1 , . ~I . .i ~ .1... ~ , - »l..J1 .."

~

.~ U- ~I ~, (J.ii B, A

--

dv, 'dtJ~

~l: 13 t 'yEA

V1,1

.,

\J

•

1

1..)

~?..r

~l: I

.

~

~ J -sl;oJl .w1..JI J~'il

.....

=-

dV B dt

-?

::-li ~l

BA

( \ __ r)

c;l..iJ1 )~I J~' U>JI '" ~7BA~W'

vt-~ ~v~ BA=~ABvl ~ (\_T') ;J-sWIv-

::>1 va .1.; C

•

: WhJI;J-sWl:~L:-'iI,'~ ~, • i)"1

. -:'li·--'Jl-J' l.~ »---,... r' J'

~.,'

I -

~

ttl

,~ . =~ • y BA I!1.A

J....

v~ vi ~

~ l: ~ '" 'i v is' IJI -flBC

-JJ AC

~ / IDA

l.r--i-:

- =YBA

A

..,II

~L.•

~I v'il • cil..iJi

v;:"'- ;J 6.JI • ..... • i ~ I

A't'

t

r~(2E/m). ..

=1

clA

u

.~

- (k/m)x

= ~lID 'It

cos

-1 (

Ax ) + C

1l.aS' x~,..u i.l.~' ;J..tW' J.> ~ • J.~I~l:,.. C

;

.,")l,J' ~ ~ ~ ~ ur- ~ 1 ; ~

•

I.

1 __ ' .~, l:.:1.~

..,-V--V

....,..-

•l>: . L.JI .:......:.JI . .i • ~

W..tL-..;.,. ;~~ M..-JI .. 4 '

v:.

~.'UJ' v l.: ~ t

; "t": iJl • .u. ~, .uJ • ; ~JJ.L.i.;,

.J),..

;~ V~~

~

W- ~ .:JI

;.

"II ~~

~I

(00

_r) WUJI

-J 2E/k,

4-oe1tl1 WUJI wi .. ~..s.il' • E

V-;.U

1 .. L

~I ~,. A

;o;VI~ x 6.Jv l ;. fa~~.i.JJ,

.t

j 2Z/:: ( , - r ) J1-:J' ~

Wl!JI.;.,. ;

;1......

t+'

U~I

Vex)

_-:"_-!..._~-+-~-----x

.Ai, •

eJi \,=

,,:-~.i:oJ

,

.... ltI, iJUJ I ;JI.. JaJ.;.,. ( ,_r)

~I

.~liJUJI.;.,. ~ M...-JI ~".-:; ~1~}'J.\i;~

AA ,1:",,0

~ L~~.~:i: J..,JiuJI~Ivh_;UJI;J~L-.u-~

~ J.-...: ~.l.:, E =

imvma:x2 .,. ikA

vmax

= ~rT iii

w

-,;.11 ,

A = OJA 0

ul..G:J1 ~I,oJ1 W~I ( \ l_f)

WI~ ~I~ • .lA, • JL,:.~I ~ dts::o.>~1

~ WI~ ..s~ ~I; .,1.J't'!1\·;1,.u1.j

~I

..s'; .L.:>.I.j

~~I ;~~I.j

lJ'.

",~Wj>~' Jt:.JI~~, • .....,liJlu-~~

«

l.~ L..L .......-::: 41->,1 ~;.i ". .r - T '1,Al1 ..,li. '.

i • •0;

~b" •

~ ',.i:..-'--'i/ IJ",.--' ,

..sl, (

A- f

• (Y - f )

J:..t • ~ -oi "'" ; :I

mx

Ii

i.

. l:.. ' 1_ ",11 ~ .~

k......

~I.j

lJ' )

",)L..b..;~It'

J!-:J 1 . ..s,.i.J1 • .lA

-, ..JI ;,....lI1 vii • JI;::.- ~I~

;..

]I

•

l...L..,:I.. ""'I l:tJt:. vb" ~I,:;-JI ,,:-.l.:..i:oJJ ~L..JI

~I U;~

~b

' •• 1

2

Damped Harmonic Motion

, I. JI.M.-:"

v max

..J 1;,aH , -la

,->Ij~I ~

"'"

..I~ ~la:;J1 ;J~WI

~ ~,

x ~b" l.,il

~w I ~ ""...,t:: ~ I ~ v.il • c.;--lz.Il

: jWI~I~ JI

ni:i: +

cx

+ lac :;

~J":t' ~ Ae qt

CffA_ r)

II

~_.:~ I 4J L,tJ I J. .':

...... .. Q is"

m

2

l.iI

t.i-.L-liS" IS""'" ~l '

J.-.."., 1.1 ... WI

~ (Ae qt ) + c S- (Ae qt )+ k(Aeqt)

=0 dt q ..:,.;,::-1 l.iI1J6JI v~ ...... ,· t

dt ; ....L.-II4J ...WI

mq2 + cq + k

"i~ ......H:H

I

=0

-c ! (C 2 _ 4mk)~

q

=------2m

J',LA:J I 4.i';)

(over damping

c 2;> 4mk

I...,; vI; ..;JI~ ')lbJl ~

q ~ (critical damping ~~I J',Lr..:oJI ..H...

,=--1

) ~2.

x 1,p,.lj')lI~,~~.J~ ~",...Jlv,s;dJ.iJ.

- ''1'2 ,-"vi~ .

· (A..r ) t(

4mk

,

;..JL-,; wi';''''

JL.:JI ~ ~ tA LS • .;,.,;Jlte,..&..JI ~I q

~

-"n.--=:..-------:::--..-.~-.---~--.-- ..- .,.... .,.".

(A_T')

J',Lw' 101"

J1.U'

~l.JJ .;rt}JI, u..~')ll ~.,,·~I ~

..,..-MI,:: ~~.i:J ~I' J",~"

t

1 ..

;

.I::f ~

~~

..

J-,l.A:6JI~""';;.I6J (o'_T' ) ;.I~W~~'

jW'~I~,WI ~,

: , - '''It x = Ale

. e - ,..;:2 t + A

( 1· _) T'

2

: ~ lJ' ~~ ,W'~" ~,L-::. ;.I~WI.»~ u'!:: ~'J-,L.t.::.l,;.I6. x

=

e- Vt (A .. ~,

.. J

l

+ A t)

(1 l_T')

2

..,..,.,...:;J~~' ~.u, ~l~ ~o r{ = c/2m ~

~bJ' ~~

c

2

< 4mk

u~ ~ InA-

c

~,liJ' ~~ulS'

~bJ' • .iA

J' . (Underdamping

.:rii',:"W,J4

~~"u~; .. L_H;.I~WIIJ4, • ~~

x = Ai-e

e

iu

= cos

u +

'.i'

J-,lA'6Jl v,~;...I6.) ;....:.11:11

: jWI,WI ~~ C-"'+ i(1)t

J

q

u,..s;

W,,-J'';-',

(-,y'-iWl)t + A· e

i sin

("_T' )

u~l J.....

u

I ~

: jWl~' ~....:..:\:i)' ~ ~l Uly~

=e

x

=

-,yt.

(A+e~U\t

.)

+ A e-~uJ It

e-rI't [(iA+ - iA_) sin wit + (A+ + A_)cos wlt]

,I

( If._T')

b

= A+

+ A -

,

a

= i(A+ - A )

~

~1::J1~1 ~ j:.J1 Att!~.l5" ~

,, 9

Ae

-Yt

0

= _ta.n-l(~),

A

=

~

(a2 + b 2 )'t

'""--'I ~I, ~.l.:.1 W'~l ul ~ ; })"I;;

1 Y.l.:.i:UJ",ljJl ~~;dlUI~ t..;l.uJ.1'-"1 u.-,;I- V-;Jl

W

• ~I ~~p~

w 1 ~~p' ~,

eo-- ;"';JUJ~ 1~;.r.:J- t(

Ij'j' ~ .&....-.1; ~I t

WUJI~IJl.t:-t

Ene:t'g;y Oonsideration

~ ~,l

~"_,c

__

ltll WUJI,

rJ

..-~ JJ~ ..}it,:: "r..\.c..i.oJ ~IWUJI ~

mi2

'-J,...o..!' UUJI

~ _

.-J..-J

"I

~ lr:I1-

(

• ~lA:JI.rtf. ":"J.-.~ ~l: ~I I..u. QI

4 1) .»,

J'..i:'-_-A:.ll J......... .J ~ ~ t U";J.l ~ L: .)l.I.""r..uI 1I.J t.-J I J..;u..J ~

~.

"'.

,..;"_H, (

0;" _1" )

....

~~ F o ~'--'

(.IV

~,'j

I.,).,)..... F. xt

;Lit

..JI;,.a.tL,.iAJ vi

•

~jl

l1 t·~~ ~

:: Fo cos (uat +6) ~'J'; ,."Jltl~1 J,.Qi~1 ~,

F

- F

e xt -

0e

i(CI.)t + e)

tt

~)WJI \$,....&JI ~ ~L-. ~I .,AJI v:S= ....,~ •( y) ~l ve ~ ~ • -ox:;

':jjJJ J',WJI~';. -loc .....,~ .~I ~li::JhJoJWI ~dJ..iJ • F ext

-lex -

mX

+

ex + F ext

:0

.,.ul-

~WI

~JLiJI',;J1

mX

ex + loc =:r.xt •

li'oei(wt + Q)

( lA _

r)

~'~'~~ ,,~I •.,,;..wl. ~b;Jl~l'::,)1l1oJW'J.>.vA

=0

\$.i--JI,m,x + ex + kx

&.-.:~llIoJWI~~ J,.~I~WI

W· ;JoJL.JJ "I> J.>.sl ~WI, ~L.J' ~l ~ "l;~ vi, «Jot;.II ""I"....~I~I~ ~~~~.A=A4-:~I;JoJWIV>vl •• ~1) ~l,.. ~

'-vi

.Transient term

~l: ~,iJl ".i.A i.-- t:..Jtf

U, • :; hi

)":W' ~~ ~.s.l.ll,

til .,lJ I:;

-.J- ~

~ IS.lH

u,..s;

J,. oJ4=1 ~l{.lr!':: J,.s..JI.;,.i. ~jJle' ~ ~I.J ~,

ij

I~J...-:J .v.ij· ~I

vejJle'

.~ ~WloJ x ~~~I ~

~l:UI~1 ve)6. ~ ~. )I;:--"'~'

~I~I ~~~l:; ;.lQlI~l::S'~(y)

li'ext • li'o eos( CA;)t + e) + iFo sin (Q;)t + e) :; ,i,i "I ' ~~ I ~,L.: I..I ;";,:0..- u,!; ~ LiJ I ~ la:.11 :;j oJ l••H,

• WoJW'~~ ~~I,

m

d

2

[Ae i ( eu t+e)

dt2

1

+ c

.sL

[Ae (i w t+9)]+

kA.ei( Co,) t+e')

dt • F e i ( at + e)

o

JI---.-_.t,..J')WI,~I:..JI~lJ-J1 'I~I ~ J~4J.A,

-m ufA + i

~cA

, + kA. • F e i (e-6)

o

...

Ja~lI l.J'~}JA-,J~I,

ri ~

"J' .sr-:-J, = F0

+ i sin (9 -

:

.t I

[cos (e - e)

e)]

• ",,-;... I\'~~I lil:L-..L4a,

A(k - m

~

r"

t.~ ...{ lf=... [ t

""t

0

z-.

t:

t~: I rESt _'.L. ~~.' t~· tcr:--\-:' - 1- b t r. ~- .(>\:-.[,

~f~

.t"

'f g

v

I' f .t'

•

~.,~

~

.... - t

~ r.:~

f'

(n

r;"

,,"~r 'f."c..l: --

-.

""t

-

I

-

> fr. (

g;~'t_

"(:'''' ' t. tr,~." i ~ t .- [FtC.~ "I.~ ~tt r • ..... t

~

_

oL

l' t llr f. r ~ ~ J

~_. "i... . .Co',q~'t,

1

.

[

l .. ~ ~ ~ f' t o' I .rk

~

fi: t : ~~ [ t.i~~~ "~~

e \-

II

tlt ~ ~·

tl .r • v v

~~ r-:t~ll

_.

'~ .' c.~' ~~

~t

{

€.. ti

c:'.

r&- 'l-

• It:

"I 'f.

t.

I

II

~~~ ~

fi..-

~rG.

~

_.

't

-

-t

I

.,.,

£

-

,e

Gt

~

o

-

r:;

fc:"1:;

~!'

F~:.;..

r:;

r· ~t

{

'C. '-

.c-

'.t';' " l' ~- crr ~ ~ ~;r: r [..f ; 1" \- \: l \-

-.

l:1i ~Ci'

rt:..

'J

'-

-

C •

f c.:

• I\)

~

~

't.

(i

= i. :-

-I. \-

.t'~:

~,- c:-t

-

~

r-

•

('.

[' oct

Gt

1~

"

t

(), ' -

r-

rt:.. -,.

Gt

I

a

Gt

c·

,t

-

I\).r-

ct-,.

-

II' rt:.. ()

,~

~ ~. 'C.~ a -1'if ?

:

~., ' _ of-

rt:..

n

b;I

~

'G..

,r'

01:r;J

-t

I

,..

-

fi:,~

1:' 'sa

ct

.r.-

-0--,...

~~ ~a

~

oct

-t

-t

I

~

...c

'G..

~'l Co'

- -

[

e

c:'

E '~ 'I..t · - . ,;:, c·

rt:..

-

.~.,

•

~

c;'

,... ~

01:r;J

~

--

r. 1 'f 1·' ~ r ~ J, ~ · \ ; , r ' .. l'_ i ,;.. 1 ~ ~ i- 1 i 1 '_ ,-' ~. 1

't;£~a

~.

.

s..;

-'

'""'::.,, , 't- -,. "

~

rt:..

.,

'J.. ~~:

,. ,. ~ ct

~

'


(A_ r

~.... tJ)l...b.;~le ;,;.11 ~ ~ .;J11I6JJ ('_T' ) v-~I

IS I n ;- :11

n F(v ) = -cv

,-

n J~ 1:~ ~~~ ~ ~,I ~~lf I.H ~~

•

~

L.~I

0,.,·liJ

w,

I··

X

··1

II e

IJL.--'

'- .q ~ I:lI

~..,

11·_:'1 ' •• ~,... ~

....

IS I. Power Law

........ lill WUJI 1110) ~ _ v ~

X-O,

t -0

~;JI~

v · Vo ~lfl.il_y • X;J'--U

J.-I;.bi.; ~

T

wlJS'

• WrJI~J.J.li.: ~-~ b 4.iL- 0,s:-JI ~ b; I~I c:J;: m ~ ~ ( ,._ T'

.. ~I ~.,:J10~liJ lii, j"". ,,~II y~ ~l:;,; ~ ".--,:; F(x) = _kx- 2

' k

r;-

'E:

r

Pi'

,. ·t

'C.

~

~

c.

e-

c.

c·-

[ i:" .f

C

l-

cr )(• i ~ '-t

Y - 'i,. -; ~ .rot. Y C '-

~

~

l.~.

"'1

I

I>

--

i:" ~

·'C

1:

•

_

--

ct:,.

"'1

I

,..

t: ,. C:.~

["c.,c.

'r.: t r. 1:

} - ~:

'-' \- \(."C!. r~ Ci-~ ~

Fz' Fy ' Fx toS,lH~\iJ'~

j~'l/ ~~~.,.b ~'l/ vl.li.>~·,J1 ue vi '~jJl,

~~~~I~I..i&S';J\,.Jjj~.»"""l>tl~I~l..i".~,.~l~'l/Wl

j,

; •'I ."rf.j

I. " ~

LA-.IJ") .L6.w' 4J-... L ~li:oJll. , -l--; . • • ..-.;--.. ......-'l/ jW.L' .... ~I _~Wl

J.A,.:J1; ...li ( 1_0

The Work Principle ~ -'-

...... dP -....

F.v = dt .v

...

~. v.~"~~,.JJ WI;jjL.J\ ..,;~":"~

=

d(mv)

dt

0-0

~

• v

2v.dv dt

'J~

1,.;;JI.wl;jjWI.s, •

~l:

m

~lvl ~j

1.illl.lJ.iJ

->. -'"

F.v ;::

d

J..

""'dt (2

~ ~

rtlV.V)

'--_~; :$',.",,; -vdt:;;dr

= dT dt

u IS" W, •

fF.~=

('f'_O

= tmv2

~~'loiUaH 'T

,La> ~ I ~

~ J.-.=..oJ .;.l!:iJ'

JdT

(l_l)

~I ~1 ~)A, ~J.l{;,.,., ;J.,)W'''~ r"~h;}Jl

;

ur--I-JI~ ,;s'~I)L....tJ,.b~dJ>J~ vi '"

I,

F

;,vI~1::~~I~

't:,I.i1 ~;J4Wli .~~~lloiUJI '" ~U.1.4~ ~~I • ~~lloiLbJI ",;.s~IIS,L...:~ ~~I~1

;,&)11.:- 'j~, Wb.JI.s,AJ1( ~ _l) Conservative Porces and Force Fields

uJa

L..1a;~ ~,

AJWI" ..... ~ ~I,~l v-l!:iJIJI.u. vi

~~~I~I~I ~.s?,l;)~.(l_t)JUJI..la.D.~. v.l!:iJI)t-

'IS?, I~l ;.1.i; u. ~~.; ~ ~ I ~ 1S..u1 ~l:iJ I ~ I ~ :B

( l_l}JUJI Pf -~jiJl'~ .s ;i.t1~t

Si.dX: A

~I Jel!:oJl,.

J! ..H

L-i,-. yJ I •~

ct~

J,A.:J I J.11:; ;.; ,:,L.... L.. ~ ,J IS I

4J ~I vJl-JJ ~oS~~It,,..~I 'IS~'~\,;"", t~I~~JL... ,.. JWl \II • ~I..a.,II ..u. ~~ ~~ J'~l ~ ~'~4:oS v--i ; ;~ I J-UJ I; ...li u I ~. .i3..1.Ot ~L..".. ":aN .;J I '1,.4 ~~ ......1 ~ ~I

v'

v*

t

",'I ;~li vi ~~, .u..1;~ I~;~ v,!;~ I...J (l_l)lIoSWI .s,v~u~ ~I, .s,iJ1 v. IJ'"\> e;; ,r-:l;~ ~l a1"... a..IJ oS J I~; ......

J.i'

.s,..;JI Ue ~

oJ ...

~la JuJI~, Conservative

Foroes

.;JUL. JI

•Uti 6,. .s'; ..;a ~ I ~1,j,&J I

-

~

u~ y~ ~ J~..bii~h::..t;L,•.>~ ~I~ ]' ;,vl u,!;

1...I.Ot

~ .u.Jllo:;.~4-Jltl,..I~ • Statio Foroe Field~l::-I;;';JL:a­

\I"";~~~ vi ·.;.ts:..JI J'-uJa ~~

Si.qr v.·u •HJ.~~t.... ~ l.i; oS )l..:, li; Ii'. dr .,; v~ \I ..iJl ,.. ..lai 6.J I cJ ~ I • ~1cJ ;.Ii.;"- v~ - .itS

-

r~' ~I

~

JelS::oi.laiL.. J~

~ ~~~"..I.Ot tExa,ot di.fferential

~I~I ~~I.,L.JI -;..., .I.~~,...~aj~IUtJ.JI"'ioS~1

"~'iS"...~~ :Potential EnerGY FtUlotion :~ l:J Ip-oiJ I~

J'F.d-; = V (x,y, z)

J.&.:J I J.11:;~ ~, ~ •

..... a....;.UlJ'....JLWU..JI .. (r.... f.) ":;-J ~

.u h:... l;' ."\Il!:~' t , ~,

J(Fxdx + FydY + Fzdz) ~ ~ ~ oS ... .-.JI~ J-1Ac.,u'-..Lijt ~~I

..I.Ot

(-l!!'I\!\>

v!-l' "" vi ~~ i,lbJl~~

Fv

~~

3l' = - --ox ,

. . F Y

7> Y

= --,

-oy

~1;Jl>~ lS, ~l!JluUJI;JI~~~~I.,.i~~~~1v

£JI..JI

L-:l::S' ~ ~.a ,~lS"uu. JI..I\:'.. ~I,; l.il • F(x)= - ~ ~ ..l>1"J1

S.p. .

•

d-r""

= S. b.:l:~ .... I!J\, ~~.i~w~""""I!JIWU:J\

da-

~j~.i~I~\;~~;S~, ~;";JI.::- ~~;SJ>~

,4~~ me; ~~;,;JIJ\.'u. ~.a' ~,.;~

z-J~' L:,;.>II.i\ .~J~'

_~W'':''~,JW' ~'JI,J~~I~~.iI.

oy

Fx

=-

F

=--=0

~

-~v

Y

oY

=0

Z

-J~~W'

• .L'4J;.uh:.. ~..sWI~~bJl;JI..u1

VeX,

\J' ~I,JI V4

y, z)= mgz + C

~.....:o.;, • ~

Ve

(\ \_f.)

~I ~1. ~ ~~I,,-, v..1f.::J1~l: ~

c

~

.. •.... IS:JI~UJ1&.it v-li;vl~u.llI~.,..Hu'::-J~',;J~

.'

~UJ~·~I v~..s~ ~~~l>.."s- ~I ;J1..u~1:J1 J~I ~ ~L;.~I

•.i.......tJ ~ •~L;.I~.,. JI ~4 WI..su~ l..;1, mb. ~~ ;.;.IS:JI ." d .... u;,.; ~ IS:J 1fJUJ' V 1~ I.i;., • j..-I '''''' I ~ L- c ~I:J I ;J 6J' _fJlJJ';J..slA. im(:i: 2 +

i/

~ ~~

+ z2) + L1gz

S'~ ~'~~I~rJ'

• ;

·uj-l1u,:-U

=E

~.,. ,"",)~I

d- v,!:. ( l't'-O

v- • Ja.-:: 4J6.~ ~ E

~'fJUJ';"';';~,

;riJ~I~~Iv,....;li ~

--.-li., uW1t-J-t- ~~4,iJlvl,.J.A'''~J~'~'';~J4-e4.l6.~ 4r---J v,..li ~ • .i;. ~I~~I fJ~ v~~.i! ~ ..u, .~) ~I j-SJ8 ()4 ",_iH .. .lA, .Elementary

particles~I~I~~~L~Ito;:..~~,

.;~I ~~ ~

c.i bJl ~I uJs ...;1:1,.;.,11 ~

]' =-k

1

• ; ....

..,:oJ' t:-'- ~1e.1,..~1 V4

~I~.;JI v,;li ~~~,J

11

.~~

( \ T' _f. )

r

• ":c--U::JI ~l: ~kJr t...;,...J1~.~~ ~I.~I J:.;n~ J------...o~h.W,; ~ ~

,I ~.i~ .."s- ;,&},~,

~ ~WI ;)~~II.I

~ v~\,( J.-~\~ V-'~ y.~I .... jlz;,; ~; .~.,.J1;)~~I,) ..--. ... lSI • r 4J I..u., r \-.,., u··1( ~.~ . ;.......o1L ~. ~ •. ,,; ·_·Il. ~ . . n ~.~~ ...... '"n =rr

_ I It_

• ......=J~"""Ilo:.~I..l.>~I aJ~ ~ ~1~.,:oJi u,-;liJ

r:--r-;:J I u,-;liJ ..l.>l,J I ~ I &.It..

(A _T')

¥ ' ..Il ;,&JI..,J-= V(r)= -

Ii

V(x,y,z)= _k(:x: 2+y2+ z 2) F = _ :x: F

Y

aoXv =

= _ oV ~y

Fz = -

I '-" '~I u ~ '-"'t-

~ &.11"""1 Vi ~1J4-. ;,&1J ~I

~..IU,&JI..,J-=&.II"""I~vl~ oJi,.

-t

4

F(r)

= _ oV = - ~ ~I

or r _ l:.i>I,J~.iJ • :i.:)l:JI ... l-:~Ilo:.l.iaJ6-U

_ lac (:x: 2 +y2 + z2)-3/2

oW.a.:.

= _ ky (:x: 2 + y2 + z2)-3/2

~~

= - kz (:x: 2 + y2 + z2)-3/2 • (\O-l)&.I ... W ;,&JI &.II ... ..,J-= ~ ~j)IJ1 ;,&J Ilo:.L1.,. 1.1.= • ..u.

bJ ... """,,._ ~I&.II ... ~1,:Jt,~ ({_l) Condi~1ons for the Existence of a potential FunctionThe Del Operator. J;,;L...

I ;

I.sI ... vJ= ~~.w. '-" 6S'~1 ul~I:JI J.-iJ1 ,-,,~1J

c:----c IoU. v IS' l.i I u ~ I JJWI ~~

iIl~1S' 1.i1~1 f ~tl "'L...:I~~Uc~llo:.l.iQ~ .. WI~6-U

:;, V

oJi~.JJ l: •.J.;; ~ &.I I... ~;aJllo:WlS' l.iI

u

:. &.I I...

~ 1,= J.t.i .J.ii ~ llo:. ~ I oJ.>~

:iJ I...

~

uJa UJ.-J I

~I • ~'JI',.JII.iA~,,:-~~ ·.)l.I~J1.u1 ('_l)41.sWI WI.s~

Fx F

'1

]I

z

",,;JI~li,,. ~~ ~'~WI~'w; ~IF ~1411.s = Fx (x, '1, z) = F'1 (x, '1, z) = F Z (x, '1, z) • ~l> I'J.L-.

•~~ •e:-: (, _l) ~ ~ .sWI ~I • ",I • ~ ~I,.. ~14J1.s vi ~j.6J L.....:\f x -J,-.!J ~lt ~~ Fy OFX 0'1

__

-

o2V

oy"Ox'

OFy 6X

' Y

=_

-J~ ~~ ~~

L.a..c.li I~I

L'x

.;.~

o2V

QXoy

2

V ---.JloAllvlJj ~) ~~~liJl~y~~ -0 V = 0

•

o'1c>x

2

V

6X 0'1

~,

v----.,,~~I ve J,..-=JI ~ ~l.:oJ~ • ~~l:.II,J,~1 ~ ~.iS',4",,:-

of

oFv

-uL.l. ~(F'1' Fz ), ( F x '

Fz

)

~ ----"-

O"¥ - oX

oFX

oFt

~=.~

of ..:.:.::I. - _oF z ?> z - -0'1

.u 4l1.s

-...-....,~1'"....

~ IF ~ F z' Fy ' Fx

".J ~.a~~I~rJl

~ • .u.

F.d~ Fxdx + F'1d'1 + Fzdz

- .-t:i v~ IS ..iJ lob".:J \ va ~ ~, (l) c.,;1S'.b,~ y~~ vi ~~.iS' Exact Differential c..L.bli:iJI~.s WI.s V- ;la. ~ ~ ~,;JI~li.,. ~li. ( "_l)~~ .sWI~ IJI ",I

•

• ~l: ... l!JI;';u.J~V'~I3Ju.J1 ~ v;"., •

V

(x, '1, z

.;.:. ~ \i:6J I c.I t ..u::. lo;-' l:S J

) ~I

}-.;\ ( l )

A.E. !aylor, Advanced Calculus, Ginn, Boston, 1955.

The Del Operator

" " ~J WJ.s

..J I

...._---.JIl ~; _J;.~.l\ ~ l.i.:w\J l: ~ li).H ~,~ Ui-i b.. .,iJl J ~ u is' I.., ..",..._ _..,JIWI~'~~~'~

F

Jt:.; ~ ~ .......... bJl~UJI

( lA _t)

F=-'\lV ,..,~, u-liJ ".",.11

"0

w.....,1 La

"0

"0

'\l=i~+j-+k-

uX

V

J'l

~

v-- l.l • gra.d Spatial

oy

V

C>z

\lV ~I ~Del Operator

~'.)a v~~, ~~. gradient

~'}JI J-\i:;.\1 j:.:; ~ ~4H""'I J~

L-.J'~ ~WIJ..u..oJ'vii

•

....:..U

~ ~ (~_t)

~ jS", Contour Lines "Ii

•

~1uJ1 ¥-u)l

.;-c .... lSJ1UUJI

v-ilo:: .~l:.s'rJl~~'~I vi ;.JWI.Ju~I~. •).,}

V

• ~~, ~WI ~ •• ~~~ JloAiJl ~ 4Jl.a.U

-:; J~ ~~"~~J" ..,; ~'.,;J'JI""" .~I

"

W., J". .~ ~

JUJI

tS,."=--t..-.:-

·v-SWI.~~I ue ~~~lSJIUUJI

~"-~~";' ~IlIl""::-J~

;>.OJI ~ ~.,,.-: ul.a u,s;..w.: ~I ../ .,&l~ • ~l: .... is'UUJ ~ J:.;

• LA.a~ ~ -,rJ14J.i..iJ1 J)li.. JUI

eU l ,I ~I .s,L.:;J1

+ott: V

I

('f

-0 J!.:JI

~'J:..J~"'; ~ lSJ I.;UJI

"r't-'"

• 'ltI4.lai6..JI.=:-JI;';~lfl.i1 L;';,JAJ~J~W.aY"t~ cSl W.a ,r",J ~~'lI,=""..;JI~II.i;J,.aN..J .......

,.

of

~F

V'xF = i( oyZ

~:rX

--i!-) + j( oZ ... c>F "dl!'x lE( "X - " " 0" A

6F Z aX) +

('(._i)

y

v

II

tSH • (Ourl

"

]I

).

) OJ

F -vl...,~~.,)l.lu,.,.

Lr ~~'lI'="J ....JI~' I:.LiJ' ~ vi (\ --0 ~'tL.JJ la,

"""~ F-vl...,.a ~ k, j, 1

..,.----l W, ~, ~ tpu I.1.~';-:bS' v!ec

'oiU, .;;Ji.; 6.. F

.,&1 I~lf l.i'

• Uaib.- ;,&11 vj; ,JJ

0

\7 x F = 0 ..a..-_PoJoJ 1v~ ~ ~ 1tJ I,

\_t)

cS...,r=J 1.1.~ I 4)l,&PJ/.iJ I iJ .a W' J:.;. ~~

ar

J L.. ~ SF.a.;- J,al1:oJl ~'lcS">';J~,IcH.a~ i. '-:tr-i=cS.ll 1~I vi. ]I - vl...,.a crt)/.;~. ~~I~W( ~ '"

• 4

·cSrIJI·~tl.i;IJt~LA.i ~~IJ'-~ ~'l~~ F a,aJ1 -...

\7. J!' cS"".a...JI,=,,~I..;.--; • W", r",. ~cS,:..w;.:Jb~~ld~ J,..:.; JI.=:-JI;,i;J6..1J'i

• ~ ~ ~ J 4-JI ~~ v-I,i. ~"iSJ1

~ ~ ....l> ~I ~ALI,

;

"i........;A=~~t~

·(divergence of:r)

·~luJ~~~' ~"tl~I ~ v ~

F

=-

\7 V

=

x 2+ yx +

c

-i (2x+y+z) -

f

Wb.- P= ixy +

.....

A

~I4JI..tJ Jb.-JU,; ~-,

%Z

,..

,..' j% "

j%Z

-.J. J...u--

]a 1\

+ kyz

~ J..u..:

bJ

..a

J ~'.'; ~ -,

1 - v '.,t ..a .;.r.. 1,

_\ \A_

,.

j" -0

i

k"

-aX -oy d

x:y

....

...

" = i(z-x)+jO+k(z-x)

"0

~

yz

:x:z

.i.i6...xJ.vJI J4-Jli 1)-..s,U ~I~b' U,

~,vl l.r::i v,!7 ~I

i:i(ax+ by 2) + 3cxy

c, b, a

~I,:JI'; Ie_f ·w6..

A

V'x

F=

i

j"

k"

a

a

~

o -oZ

.:q

o

~ &X+87

2

" = 1(c - 2b) Y a ...- _•..",; • c=2b

v~

vi -.h..,r •

~~

~6..

F-vlJ.'''' J>~

.,vl vi ~14.lA ~

• 4J;~I ~ Forces of theSc~-;a1'able Type

• j~ ~liJI~' ~ ..s,vl (o_f.)

J ~ ... J,..._~~.... I '';~11,;e v,!7~),6.. )1:,;>1 ~ ~~ ~ ~6. '" -lSI.bU ~~I..b-~ ~"

F

= 1Fx (x)

" " + jFy(Y) + kFz(z)

v lU,.....-.: La~v~~

Separable

j"iJ'll:li ~..s,v'~~II.iA

~~'u~ Uai6.. 4J~ v~~1 I~, \~..s,~4~''; V\A'" ·t~·

.!I';I~ _.w.; ...II ...

Lr- JS'~I ... 1e ;,vl~l1~ t-WI JIU~'

;........fJ' JS'~ . . L... v~ • f~ ~~~~li;Jf~~ ... L-.lIJeb; v~ ~.w ~ ~I ~}J~~ ~ ... WI J.> ~ 4J6J 14..iA ",.mX:-F(X)

•t

iii':

...

e,,; Vt v,!7

J-.>~.......s~lv',.....~ c>:L.J'~'~

11,

·;Ja;b.J1 JIh

Y. ;Ja;b.J1 j~1 ~li ",&.II

11:.1 ~".l1&J1 ~I ..j ~

~..,Jil::; J~

..j Me.MJI~~ ('_t)

Motion of a Projectile in a Uniform Gravitational

·r"

::J14,.~.I

c.I ;~I~~I

~iel.

J:L.JI u- Mt.wl-.s;. ~

..,.-i L.L;.lJJI crJl .. WI ..I,&)I ~ l,..;~ ~lc 4JWI ....... ",J.jJ

"'""'"

• ~WI .ar~H

"J,.,J 14li. ul.Al lOl,.cJ I.., li. J..t.; I. ~ Me .MJ I ~ cd~ ~ 14J 6J I ~,I ~j..U • 4J,..-lJ ---~." .~)~I c.;.-~;'; ~ • oW;; ob-~ ;y,.;,; ~ ~l:J14J6..J1 • .u.-.I - ~WI~I ~~~~li:JI4J .. Wlu~ I.J,iU n

d 2;-

~

-.-z-- = -

dt

Z

_"""'JI,::i-I

" mgk

~I,JI ve • ~~ ~)~I ~lul ~j;i.JI:.~4JWI~tdJ .. .;.~~ ;,.-l> 4J b. J:.= l,..;~ ~ 1..Ii.; b.J ~ j~ I ~1&J IerJ I ve u~ ;,&) 1111.. u' .... ...u. d,... V

o

("'_0 4Ic.14.iUJI4J .. L...-

L,.-.Iu~~ .(~~_()

4J ..l...."

~I ~'- ~1~~I~~I)~lcdJ.i,;rI~LJI ~ ~I .....;

UUJI4J .. L.. ...,~ • t . 0 ve,;J1 u~ I.~ J.-~I W,;..j ~I~~I~"

tm(i 2 + y2 + 2: 2 ) + mgz = tmY~ y2

= y2o

'---+0:'_~.L..

_ 2gz

J,.-J I ~ crJ I~ I.,L.J I~ "...... • t~PU 4J 14 ~~ I .;-; • 4.iUJI4J JL.. u- ;.r'''''

~I .~"...u~\a:oJI4J .. WI

•

'-"I .I,-;ul~ .;rI~I~b.;

JJ

"WI ~I~ ~bS'.:J.c ~

d dr dt (""'dt'") ::

" -gk

\

. ;~~

1.r11f:~.

(\ '-'\)

,. .

~'J~ ~1cll:~V'J; ~ .~, ~\. ~".J\ ~~ • ..\>~.J- t.rL~

~~ ~~I:i~ LvJ.~· A.;1~":l'~,,-J\ ~

-1 2" "'2 gt k

r :::

.l....>1 ~.i..i.U

V

+ vot

o j..'~:)\c...:b~ I. ~ I . IS

J I ~ ":l , ~ ' 0 ":l •

1,;..- r

0 \.\.. ~ Ic...: b

IS, ~

- Jl:lI~1 ~c.~"JI4.l":l~,,)~I:i...,s.wI4.l,jL...J1 ~ X

( "'1" _

•

J...

2

J>;

~

V0

A.; I ~ ":l I ~,,-J 1c. ~J- j:..::

•

,x

y

~,jk. ~ t

o

X

YO ::

0 0h' 1..;1 • ul>.~

~,jLw Vt t

1;.,n.

~;ll ;J1..6'~..uII&

v::: bx

•

•

z0' y0' x0

0..\:. •.\:.d~)l.; • ~..uII)L...J ~~ 1.1

YO b = """"X"7" ~..14

0

yot

z ::: Zo t - 1f gt ,":-1- 4.ll0 ..; 1LW'

z,

J ·J-":llll...i: J 4.l bJ 1".u

::: X o t

y :::

0\

&

•

l.:.;~ 1..;1.

0~~1

0' L..::, «.:.-: b

b ~

~,:-..j ~ )L.JI~ cl.I.ll

cl.I..; ~. xz - ~,.:-JI

J

)L...JI~

- Jl:l1 ~I ~ )L...J~I4.l,jk.0~

z :::

0(

X

_

;:;

x2

;3 ::: --fL 2.1:7:o ~ ~~

lJ'. y:::bx~,.:..-lI~~'~~~)l-J10";1 . ('1" _ { ) j!....:J1

1 ,. 1

z

k-----~_+_----y

x iSpo.:.~.iJJL- (\"-0

JSAlI

oJ ~14:)i: .j ~ I 'I,.r,ll .., lie

..s

~..J I :i,.....tJ I l"'; Q,s:. ~;J ~ ~~ ~ I 4J bJ I .j i.i.:.iiI I J> U ~ I~;.:J

;

L..,..;UJ I ~c.iL:.::;. 41~ b.. J'i. ispJ I u,s:. 4J bJ I ,..i.!ll .j · , ~ I .., lie ~ ;;.u l: • dl!:>~l~ul~"'~:i~:i~~1

,.

~ oJ)> .., liJ I :i"; ~~

m ~

ill

1 ~1!.tJ1 ~~l;,:J,,=:.~~ A.:l:S (nlJI v. u~·

rl l .., lie

,~

A

d2; dt

2

V

urJl ~

l..Sl> • ~.iill ~ u I,;',.: u L::,.; u ..; I ~~ • ~ .i.il1ll:S' J:.::

4J.J W I ~ ~ • -nek m'

Jx>- ~ ~ )' I .., lie u,.: L; u I ~).: • 4Jn-!J,

= -m "1

IS, L...: u.:W I .j LS" ~ ~ ~l:ell :i,.aJ ~

~

-m 1 v

. . • i...l.; I.i:J I

--""

'-~ •

v - mgk

~J....-.:

'..l> ~1S v.

m J~~

••

:Y

x·

y

y'

X

= -

0

y

=-

0

z=-

~z-g

~ "'.r- ~ ~h''''': L~

• I.J!-:- - " " " ' ' ' '

JS' ~\.:.. ~ u ,j I • 0" ~ I..:_;}

",..h:_ ~ '" WI .. .iJ1 ~

...s'~:_ _'-.J..JII..-J

.

LJSdSpJl

Xo

YO'

~ "~~I...,1.i. ~~I~I4JbJ~

• L-...tits:, L..J....;(~ ...s,..:-JII..i..Ib .j )W~ • b= • ( fo_ t)

ZO'

~ I .j ~ ~

Yo . X

o

~ Y

= bx

LS' • \,,; ~ I ) WI J.i-I ~ ~ L.: ~ Y,

X

..;J"'~ ~01 ..sl.~

~ ~.J li.JI .k.>-Il

u",. •-JI (L.n ~I

4.fp-:J I ~ ..iill "') l..... ~>~~ .-:) lS.

o

~ -\xJI4..;;I..i.\...:LJ r .1s; JL.~I~.

I

L-,u.~ l..w~.iilI4.fpJ J~l

J;JI (~o_O 4.l.,WI

~ \:) I Li.",,1J l; c ~ I ~ ~~ ~ (~ -'"

r

= ('; 0 +

.&J~

~~

) (1 - e- 'l5' t) _

J.... ldl;4·

oJ.>

k i~t

it

.i;.h.';' 'I,yll

( '( , _ f. )

;.:i>-.~~p-lI ~4-:~I~li:j14..l..lWIJ> 0'1 ru;'; ~ J-~I ~ • 4:.:~ ;b.i; ~

--.

......

411:11

F=-kr

-

(/)--0 jUJl . ..;; 1 ..L .i.:J V:: l .J,.. 'Y.,.... M

......

~)1

&

.J l-.:I, J

\ ,. 0

~~IV6~.b,...J- ~ (, ~I~, ~ (, 4.lbJl,,-u J.;..:~", ( 0_

0 ji:J I J

(; mX

=-

mY = -

IS

.i.-!, ~ rJ:-

~

ch- J4SpJl4..lb•.l

.j .;; ~ I ~ ~ ..i.:J I 4..l..s L,"

~ l.; ~ ..s W I ~ lJ, Vo

~\:}I ~1..,1:-t~'-to J,bJ1 e..:~ ~~..iJ • ( \ "-r)

x

=A

y

=

n

COG

4;ftJl

~\:}I~j,;JI~"WI

lex

lC'J ~

l.f (,

~,.,

4

JS;

1 J .1;..,MJ'"

(w t + ex. )

cos «(.o.J

t + j3 )

w = (m) It: '1r fJI ~ ~:iJb. ~~

ok,;

j3 ,. eX-

(, B

,..A J,.~I~I~~

• ~~ dj..iu,s..", ·~.J~1l0'4 t

~1..t.;~1

~,;Jlu.i::...: 'JL...JI4.l"L~ ,J~'J J\:;)I~I...,h~WI4..l.JWI

y :;: n cos ( uJ t +

0
. ~A _ 1)

Motion of Oharged Particles in Electric and IIagndtic Fields ;;,........;~;''; • loS.,,>1 ~ L,nS..:.- ~ )~ l; L.rc! 0~ ~ u~ L ~ ~

":--_-..:->-:

q ,c.1.:.:: I..u 4

• CGS

F

..:.-"1.1:>-,

V"

li:;

N K S

.j ;:..:.;- ~

(n

J ~ oJ J 4-1 1~ ~J- ~~ ,

~ Ie l; l..bo ~

• z, y, x

4. k ~~

~

E ..l---!,.= ~ Ie l:.:....:J Ie:-: lS'

I.i I •

.,;-

• ~I t

IS I)

v.,;J I~ •~ I c

¥'

~;lI J~oJ'

~ ~ III t...

~,1:c~~t; (

.....,

2;$'"....:.

• (' .~ JL.r..-sJ1 J~I w~A ~Il!ll:' lSI, ~llb. J>~ Ex q

= By = 0 1

.'>-~~ ~

~~·Jl::--JI_4--:~. • ( ( \ -

Z

0 c ~ oJ ~ IS I • .spJ Ic

~ oJ~, • E • EJI

,

~,1: ~ ..i I J l::--J I 1.iA ~ .rJA

x= 0 y=

_~lS',""b.JI.J.>IJ~l;

0

z=~ m = constant "'r_~!;.,-,~ .... I

~/,J1 ~..i~ J~

J ~..i.ilI.s.P'c ~oJ~ ~ CL:: ~,.u., .,

...

• I.ti~ l-W~A )L.JI ~..il -t ~ ~1:

E e:-:tr IJI

IJ ~ -.lloJ ~I,:: lSI, .. 1;.;6.,. ~~ J ~I

~ ~ IS, L...:: I ~ J

•

4-- ~

... mv

2

q ~ ~ t..1!J I ;.;UJ ~

+

~

IS, l-::,

;.; l: u,s: ~ I

~

B

~l- ~1:-"e

J~

.~.~ ~ .it

Q.

J. C. Slater and N. H.

~rank,

Electro-

magnetism, McGraw-Hill, New York, 1947.

:Jt. (r) - .. '

'

~ • ~)l.;~ ~~~I\-,,~I4J'J ~ d~ ~ ~;;-,JI ;,AJII,..\.:..::

"L:jI~1 ~

~

~

(0_,( )

F = q (v x B)

~

•• ~ 4f.r--ll

~li:.Il4J..IL~~

2--"

d r

m dt 2 = .. I ,_I

q

~

• (t)

~I

-""

(v x B)

~I ~ ~.J~ WI.J.u~~I~ul .. )ld;~..wI4J.JWI~

J..u, .I).o...s, L..:: (

to ..,.u;~4.lI..I

V ):LwW I ~~ I ~~ u I -j-"-:. I~, • 4fpJl

D 1.:..:1f,J~dJ..i e--"" ~~~~~10p-o; • v-jJle-~'J 01~r

1,... ..

L..i:: F ~

F

=(

gfo

......

-'"

) ( v x.l:3

) ~01~

--..... r

,rr

J ....l _ _....~• .. em

Ott

z- ~Jl>w.I .~b~~Lu.J~Jv~~.s".~ -; =

ks

~ \.0.:1. ~l. J~l 4~L:

... 1

"j

"Ie

x

y

.

z

.

001 1\

"",

= m(1x + jy + kz)

,..."

= qB(1y

- ji)

qBy

mi =

mY = -qBi

z=0 ill ve~ 4S~ ~l&:;JI~ 'lI.JWI.,;,. ~ • -~ J, 'lI, • L;". U.JL..:

t.' t

ve;.U~L:-rL:e LrJ.~ u.lS'.L:u'll ~~ ~v!l,

ni;: = qBy +

rlY = -qDx

z

+

C

J&~ 1 C

2

= constant = Zo

x= Y=

Wy+C1

- Wx + 02

z = zo

J

•~ l..:-J,:

• j}.U~liJl

o' •

~,.

()J

=

ii,B

~ I:J r.iJ . d.".,J I Vo Y Jat,,.-;..u:,. t )1:.. ~jWI~

{t t_

C2

=

c2 m

I.J4 J,~I ..bWI

J(

to _ t)

l:..~jWI~

~WI x_J~j,;.wI-.JjWI~

..

a +

J..-.;

?

x +

: =

,

°1

C1 - m

·w a

~

coe ( u't +

eo )

( lA _ l)

'J

v-

=b

.t.

t

- A sin (

OJ

t +

eo )

u-b-; • .s"pll}L- ~

j~~,.

b = -

c 1/ 6S"~ I }L.. v I ~ z

a ,b );Jai~H"", ,,~l.:

b.:l: ( ( 0 _ 0

.( Y-O ji.:J1 J~,..lJ" ~WIJ~l,~l.:~'}W'

, T'

0

Ja (tA - t) ;J ... WI ~ J..-..; (o,_t)

(ot _ f.)

:x:

(V_f.)

Z

J!:J1

~Lu.J~ J6d~u~~ ~IJl-J1

U"'",-~1.s., J»c,,;J,;.oJ1 ~I MW. ~1as'rJl

..u ,I 4j,riJ1 Jl:. ..j lJ')l.lS' v~ ..u l)WU,. dJ.. ~ J. ..ill 1.iA ..j, • as'~ ,I ~L: v~ ..u ~ I~ I, • J, ~ I J l:J IS'

__ ...1, ~~ ~ tJ").J.J..-

v~

• .1.iH:...L:.Il ~ 1.wJI •

'U.Jl~I..~.iJH;UJI.J oJl-

W

{ 1 ·_l)

The Energy Equation for Smooth Constraints ~ . ~b.;~l"·.-...'I.~ •• ') biJl.-:1I ,.., " £. ~.."

~ v ..il.~1 ~ ~'J.Ai

I l-=.s''.J-W. -II .we_~..,. L.~",--111JOI.-: 7W. ~I

oJ) "",.~~I.,iJI.

->

R

'wiill.,;, - F

• jWI ~I J...s'rJl6JoJl- ~l::r -"

m-

dv

dt

->-.-'"

= F + R

( --'""

..;. ~ -'"

dv

->0

->. _...

v "'rJ~ ~",-a.6J",WI..j;'

4»

l_O

L...,...

l..il

--'"

m"(i'f"" • ·v = F.v + R.v

L..:"',.. Ii ~I "') v~_cd~~lr::-a. !.II •~

w

(

00

"~I:JJ.iJ. •

~

_

t )

d-JI J:.......,J.~I 'wiill,6J6...j

Ji-l.. -; .....,....II IJr ,_('w l..i- . .~~dl -1..~11 ..,.... Iw y - - ' I c:--

6JoJW~·

w •

~

v

J- ~oJ"'v*

--:..

R

~~

u~ R.v

..s",.J.AJ1

rv

I

( 0-

0 .J.4l1 .,.,..; LS J.lS:::.l1

1;.; IS.

4vA -

*mv .&

2

-->.

,~b...

F c:.: IS !..il , cl\.u

loS I ;a1hl14l ..d Ao

= xes)

y = yes)

z

= s(s)

v-- ~I J,J:- J,s .... liJl uWI J:..:

J

....·~~IJ~'iI~ 1;.i>II..iI-S,,:oJ1

•ims 2

~ v~ ~ 14J1.JJ

lSi"eJltI l ~ ~I~ ~.,~

parametric form

x

'_0

i"eJltI l ) ·aspJI~W ~

s

.,~I~. (~~1~J-1I..i:

v---r-;)'+~pJl ;;UJI ~ • ..b.M

B

t ~1I4J1~~1S)1 ul1.h

",,-$~£J-~~¢-Jl

(OA ... 0

• j.l!:Jlt(z, 7, Z lSI) s JL...:>~

t ~jlJ ~~ .. ~I ;~;1J14JoJWI

.,~lvSec~.,l~H .. ..;..~

J-li:" ~ • lS~l~eiiu.,

''''1

.. me +

Q'S""

dV

me -

JlI s

(/)'\-0

=0

=0 ->.

U lis'

; •.; 1.i.ll>,'

S

"

l;-.:: ~ F

~),:;J 1.,.iJ1 ~)'

J:.:

]Is

~

dV

• Fs=-crs!rhe Simple Pendulum

J, 4 1U-!,-. • :i ~.~

:iJ ~,.,., • ~ 1

~ •

!,.:" 1:j J t. vi

v~ • .yI,I..l.lJ~li~~~,,~u):..:..b,...~~~~

; J~I~l:.:'~I~W'~~' 'J~~I J~ 4~· J';I;~iSpJ' • ( '\ - t) JUJI.j ~ ~ lJ'. J,;I; & J u--le'I.S;I.J dL.. ~ •.ilFJI

o

mg

1 t>

",I~....:~,~:~ ~

o ~

s ;;WI v,li::, .. ~ j":l'~' P

,,~~ mg

e. B

L.J .,\ W ~ .

e

=

sli

~;":III.:-'~';;,.aI Fa ~..w: J,

~~lullS;JUJI~

4' J,.b ~ Jl u b' 1.;(;

~;~I~I ~

_ sin

e

e

4.1~~ ~

J L. l:~ ~

0

( " '/- f.)

... ; _!'l',.t-;,S'-'1.' 1< I :.l~. ;,S'-"w....blA:JI4.loJU . b __ =-< .. ,.., .,.., v •• ...",,\)'#~. ~ ~.. V ."!J""'*' •

Q

,......_ -.....>

........

••

.....

-.J I.,) ~ 1< ~

)b. '-'..:..: 1:.; 1..:.

= A oos wt ~ ~ • ~ ij:J 14.I..l L.-J I ~

-A 41

2

:. ,_

W'.~"

• ,.-

cos

~1 J>JI Q • A 008

wt + B

COS., w

t

( 1A

~()

, {t

.so; • A oos uJt V4 ~I ~...,.i: v~ ~ • ('\ y_t) iJ.)WI J ,)U.I ~.wl J..~.H lw,s Uli

• ;..., .,114.1 bJ I ~ "..i..IIl v I~ .~.... ~ _~,?"I ~.

_ ~t:ll iJ.)WI ~, ~Wlt.:-~~IA.~~ (_A(.&}2 +

+

A -

+ (

~) ai~

2

oos C41 t + (-9B W +....L 13-

t

_Aw 2 +

Ie JS;...JJ

..JL A _ • J.

24~

w t J t.:-lkl.;. ~~ • .r.!11S,.i! ~J' B J .),.»)

w~ 's'tt.li:J1 hi> u.) v,s:::.... vSJ, u

~)oOS

•

t

rJ ~ iJ.) W

Cl

3wt

0

I ~~ IS,>I 4J'

l:, L... r I.:J 1~~ V:J;~ I (x ~ 1J,. ~ ~ JS ~ ~ I I,.it v1ul • .)1).:1

e2. = 0, 8.e

-9Btv 2+ ....Ji. :B -

.t

sA?. = 0

24£ ul"~ 14J .) W 1~

, u. T =

llI. ~ w

2

7T[;g

ry( 1 + -rv 27T Vg ~ To(l

+

-fi

(1 _

i

~~.l.1IV4;J ,i

G2 )1 0

(y "-(,~)

1 Go2 ) -n; + ••••

Q~

+ ••• )

Oo\.--""t'YI Jcl1.j ~ w..~ lJ'j.. ~)I...s.~..-J ~~.iJI~;~ !r o ~ ,.....,;li::lr~)b';I .......... ~~~ ~ulS"v~ .~WI~lul.( )'{-O .;

.6,;.

t-l.:

H

~~~,; ";l;~,PJI'-;"'~..i.:U 0-'~~Vo'~ ~.....,;I ~I

u~'-;"'~.l.::.lI

r' jl?')I;.......J

~I~ ~ ;l,i.::..'!I~1 i..c~..i ~j vl~,

c~J,.:lIV4~U4)l:;..I..I.:,)~I~ l.;~ ~JI,J

" harmonics

.;i: ~;;""';I,J;~et..w· ~.r.fj~ j~~lvl uk~~I~

;

.'HJ crJ 1~ )1.:.,; •

c l:.U I,.: ~ I.S~ U,- 4.-oi 1loS 1• l4: 1 ur,.:.. v~ ~ ~ 1~ 1dl:: ~ ~ (c. l::ii 1,.:) L~ ~..t>..: .J.i I.S L; loS lrJil j'r:- "1~ I) 1 flo:

..,I

• ~,.,,:!l~1 ~I ;~I v. ;~I..:;..)J..~I4J~~~1J,,~I4S'pJ Jki..J1 ~I (, f._0

Exact Solution of the Motion of the Simple Pe~dulum by 11eons of Elliptic Integrals:

_

~"JI!~I~~

4.-..l!Il ;';lbJl~...:;~ ;.;lh]1 AJ~lA..~~

tm(j.Q)2 + mgj, (1 - oos G) = E

.

J...-J~ ,( Go =

e( Y\_f.)

0 ) ~~ (;;.......JI) Go

I..\-i: ~ .~I ~~~I AJ~WI ~.

E

~~ ~~ J,41~ l.il

= l1J8 j,

( 1 -

- Jl:JI~1 ~~. mJ

92 = ~ .~b'

~

(oos G-008 Go>

t ~. ~ ~ .< •

~

oos 9-1-2sin 2 ( Q/2 'I

2

oos

Go)

~4.-il~ ~,..1:>Jt

\t

0

~~,

. (oos ~) ~ ~~

= T1

t ~;1J.~~ ~li.::~

9 9 oos (2) 2

(YO_O

(Y\"'-O i..!"WI ~ ~(vo_t) ,( yt-O v:-=J"W1v-. lSI, ~ 4J~~4J!;h.iJ"l-.~1

~2 =

-t

(1 - k

2

sin

2

~)

J.~~~I~lj~ t

t -

J4 J

~cW

----y .g . (1";';k2 sin 2,fJ)'I' o

~) = ~~(l.-

7(k.

k

2

Bin2

, ~ ~4.i)W1 ,,~I~~~

·Ut

= --g

( vv_O

F(k,~)

~)-i ~ J, 'J 1erJ I ~ ~W I ~.I J. t!:J ~

Incomplete elliptio integral-

of the first kind • • 4..L>~ 4"""t'~

9 0 ~I 0

~ 9 4"~~)l.-. J,.,L~JI~~'; ~j~

'"---;:~~".u';l. ~;JI;,,:iJ1 ~ T = 4

J:

K(k)

=

f

o

ul l

~~ ~ u1e,S;edJ.lJ

0

'wf~l!

J717 o

TT/2

..1[-

(1_k2

2

(1_k2

dtJ.

8m2 ~)i

8m2 ~)-t ~

=4

J;

= l!'(k, TT/2)

T ~~..iJ1 cJ4j ~I:S K(k)

(YA_t)

, f. ,

r ,:; ~~J (0) J,I~~~, 'J,~ltrJlv-- tL:JI~I~l.~~ J--- ~ ~ cll';' 4.~ 4...Jfl' 4J~ .) ~I ~ J b.. ~ I ~. ~;=:,J I~)l. ts::J I • 1

,...i

I

1..1.> .j.1!:J ~ •

loX ••bo.lIl.:-lJ ~

r' ~ ~ (Y.A- f.) 4J.., W 1

2 IT f1Tl k2 2 T= 4 ~7 . (1 + 2"" sin ~ + ••• )

r-e 7

0

=27T~

'

(1 +

Q2

Q

o

k2

4

"-oJ

.~~~

dJ5

+ •••••• ) ~J-:>.;'

v

.•

( V,_t) Qo

t-1J~~~v~~

0

2-4

... ... -

~~I~l:S~~.i1

)

(A' _ t )

JL-:. c:.. )l.ts:ll J,I~ t ~, • ~J.s ( \ . ) ~.M-::~JJ~ 6.:~.i v--i ~

• 4~1'..,.s'.wII.:,.~~~ ..~ •..,.JI~I~ ~I L..-:JL;' ~~I

L.

M. Uilne-Thomaon, Jacobian Elliptic

Function Tables, Dover, New York, 1950, or B. O. Peiroe, A. Shor~

Table of Integrals, Ginn, Boston, 1929.

\ {y

k • sin 10° • 0.17365

,

Qal2 • 0.17453 radians

~ I.! ~ tf!,.oJ I 4.I",W~, J,I~Iv.

(YA_O T = 4

f Jig K(100) = ~ Jig (6.3312) ( Y'\_f.)

T

= 27ri Pig

(1 +

t

sin 2 10°)

=I j/g

4.I",WI ~

(6.,3306)

(,,·-0 4.1 ",WI v. T

= 27TllIg

(1 + .e~16) =-

To = ,27TJ)/g =

l.J/g

I fig

(6.3310)

~1.J.:.:~I~)l,J1

(6.2832)

The ISOChronous Problem --.,; ,,:-,.i;J.;~. ~ r 1 ~ ~ d1:.A v ~ UI ~ Jl',...Jl~~e;:.Jl ~

."'l!~~ ~ ~ ~ ~~ ~ I • ~ L...::. v lej ~ ~/~ I 4::,j '=:J I .w t:~.,:o.; ~ I

-----,--

"1

l

.--.11 ~

x ~jJ I ~, L...:: 4.1 b.

J.P',J I ~,.&II ( \ •-

f.)

J!.tJ I

ltA

~ is ~I~ IJ"U~ ~~'..b.;J'~~I;JI~ Q vl.)o}1

•

.$'.rJ 1, ~~ ...".cJ ~ I'":- ~ I ~JO v~

~ ~. (I,. -

0

~ 1~ ~

(u--l.I-..a}1 ) ~I JL..JI J,J- ~4S'r-J.l ~1.;.:.l14.lA.~~ -mg sin Q

v,x: ~~ IllS = -mg sin

......w I J,Je.

,. "='

Q

~~

4.:ii ~ :is.P''

~~" ~I

;.."s oW I 4.l .j l".•J1..::..: IS' I..i , ~, - .)s. \.;1...:.:..; v , '-:-~

ms

(A ~ _f. )

= -ks

:J..lWI J~~.j)' ~1 ~I,. v..;l (A\"

s == c sin Q

• ~~~ ,,~, ;~..wl ;J.jL~1 ~

~ x

==

5 ==

dQ

y:: ~..i.I

~ ~ ds

S

c

d

y

••

,x

"'» ~ .. d

.,l~1 ~ v~'

== (cos Q) (c cos Q)

c 00s2

==

~

~ ~

Q :iJ~~

Ua...........~I·;S

_n

Q dQ

=

+

(2Q + sin 2Q)

== (sin Q) (c cos Q)

dQ

sin Q cos QdQ = -

• cycloid

6~) I~ ~ t,.:

~ ..

'~, cos 2Q

l$X'rJ';:'J4I'~'.I \L... (Ao_.f.) ,(A f._t)

S lf~,t:'";";;;S,j>-

(Ao _ f.)

\.;l:.:

1o$.JX. oJ ~ ~ u.j) 1 ~ I ~ ,~

.~.ill~lvllS;

.,ill JI

J

• ~~. ~1~~~..iJlvej ~"I~

\".L.~ voyl ~ ~ vJ.IISJO.J ~ c.Pu,1-Jlv- t~ • ~ 1.t.tJ I tlt.i: ~;..u 1~

Ohrietiaan H~gen~v~}IS~~1~~~ J~Iu.:.:sJ.J.iI • clt.WI J,~~ u.l".::-I ~Ie ~,~~ ~")LJ .. ~I-;~.ul J'1bJ1 vl~, (evolute )'~'Jljl.JJ~~,JI~~I"'J::'h.U:SI.dJ~

J,~I~ ~v,jl· ~11S.xJ.J~ISJO..J1'b.:>.;lj~ "....s.:;JI~1

••» ~.J l;l-~vl ~. J,4 Iij6S'p-ul;. L"",.J (.J,~)

v

J

pO

''":,

rl.......: 1 ')II 'tlr>') I 4&1.,H vo t)lo · "J..-JI ~ ~ ')I u..il ~~..iJ1

· ~c~J '"",I The Spherical Pendulum

1S,,,iJ1 J,~I

( 11_{)

'I lS"y '!I.J -;~ 1l:S

IS '"' L-~ ~Jb, ~ I,1.k..U ~L;.J ~,I ,,:; ~Y J,4~~

I.u, (, '-0

~.,.~ 'J}

J1:JIJV:tte~ IJ'. ~tLb..i:J~ulS'.. ~1

\1,)31

Z

mg

x

J

~J~.iJI~t.:lob- 'i~ ~~J.JI

J-i- -Sl J>'; ~~.:J t ~J ~ I,,:-~ I~"; LA • ~ I ~u 1;",.: u ~~Ia urU ~li:)t~.JWI u'!: ~~.;:,JI,I ~I 4.;.;)1 ~ S .u,J1.,Jj L.d

~

mr = mg -'>

--'"

(

+ S

J,.;l:J 1 b.--:'i It

L-J'i ..lot 4S"".,J 14.I.J l.-. ~ ~J- u,!:,

4

A1_ t

)

z _ ~ I 1:.1.6-1 IJ I

Jl:JI~1 ~ • ~J~.iJt~trl ....'i1

mi

= SX

mY

= S

(I\y_t)

y

mz = Sz

- mg

;~JI~'JI~VS'A>lj'JIU'!:

Ix I «J,

~IS'

l.:-L;)W~ ~

mg ~I

W,

d

. ' .;

J

y_

li:) I

i +

Y

X

+ x=

X

~t ..s.:J1 JI..u. U~~ .I~

~~J- ~~

• ::::. 0

S ~ _mg..:L.

~ X S --mg-

x

1;, L.., 4.-."i:

--'-

s..L:JJ y,

L~4.I~ ~~ .J~I~

l.:- 'i .J~,

1

• j1.\.l1 4....i:...CIo ~ 4.1~

•

11'1«1, iJ WI a...,iJ I

li i.....·

uSee c;J ~

~~~CSpJJ

0

-1-

y+ Y == 0 (AA ~ t) ~ .. 1:::....: u ~ ~ \S ..u11.X~1~..i ~.!,.:tI,,:-, ..l:j:JI~ 'i .. W a.L:L. 4;'" ~ J,bJ~ ( y_t ) X

= A oos (

W

y

= Boos

w t + /3

(

t + tX )

)

(A , ... t)

\

/)

\

• ~IIS,.:-JI IS,.;..-J I ~ d$pJ I k.i-. 0~ Lb;..b.i-II

J,4 1~ l.S"

J,..-,sJ I 4..,,J l- L.:; 4.-...;: ~ 0,5.; \.S ..iJ I ..t.>J I ~, ~ 0~ ~L>. ~ ~ b. • ¥ • ~ ~ . Wi; t.....bJ x y _ Ii

• ~I~ 'J1.b,rJl ~clJ.i ~." ~;loJ,1 ~ L-.,,;J1~L;loJ>~I rl~~ J~I oJ

J, ..L..4I 1~ W ( \ \_ t) Ji:J I ~ ~,; t.n.. ,,;J1 ~ L; I oJ> ~ I r.b ".•:~

;;;~~~~". S

b

.-\

4.b~,

J..iJ -mg sin g

-

~~O .,,~I~j..;~lvePl~;~IS,jll

mg oos

Q~

0l:4S'". mg

J.iill

~

- ~ WI ~ I ~ 4..,."jll ~ l; I oJ> ~ ~ ~.r--il ~ lA:J I 4.loJ W 1J;.b-..: ~ rna

r mag malJ

= Fr = rng

= F g = -mg

oos g - S

(, \ - 0

sin g

= FIJ = 0

~~1 ~I J a,0,ag ,Sr

Ii

r = j, :: constant

- vt

~ .....: u~'J10l:4S'~~ • ~~I~~IJ~I ~

a g = ) Q - ) ~2 sin g cos g a,0 :: ; ~ sin g + 2 j~Q cos g JJ

;

••

Q

•2

-,0 1

,:.lA:J I ~ 'J oJ W I ~ 4....»~ I..=-'J l..:..::> 'J I 'I~ ~ ,,\,.J.:>J I J.i: ~

sin g oos g + d'

---- (,0 sin sin g dt • h

2

-+

- Jl:.lI~1 ~

sin Q = 0

Q) = 0

IJ ,

g ~

( , l _ t) ( H'-

0

1'-_~fi'4"_";loW'li. d.;b ~,.aI1 ~~~I ~101 ~~~I4..I.JWI

\ 0'"

•

15

h = sm2 Q

~_ _~lsJ--a:...:

,(

n-O 4J..lWI ~,,~I;~..wl i5 _~L:JI

4..;~~

Q~~~I~..lWI

('\0_0

I ..l W I ~ \oJ..i I • h

..

e+ -t- sin

0 d1.iJ,

15 =0

15

~ .J4 , ~ b

- ~~ ~ ( '\ 0 -.0

=0

; ~~4Sr-J~ .~IJ,~~I.i:)I4J..lWI¥~~,,~,

1./

¢

Q

II:

~ = Po

= constant

~_.

( ~b)

1\

~

conical Dendulum

-

Q

II:

0

;jbJI,,~~,.

Q

II:

Q

o

II:

constant

(~b) ~~..ill

..

Q ~

f,

.

sin Qo - h

~

cos Q o -sm -...---3 Q o

II:

0

,

=0 ,I ('\1_0

~;;;',JIl.S,.;J1 ;l.~'J1 F

~1: ~I. (1

1.;..i..:;..1 !..il " ;.s.-L: 14.J ""WI J,,~I cll..i.f ~

"-0 ~l ~~~~ lJ'" 4.;;1..1.11

~.f""

~1..JI)Wlf)4~~Y'~~ =~ ve~II.S';

(\·'-0

0

.s

;J~ ~ ;SpJli· b=3 cos

cos

(rt;

{

= 2TrJ

g

( \ • ~- f.)

JJ>

~I,.:

9 ~ ~J.:: 0 J I

,f,

( \. \,,':"0

g(3 cos Qo + sec go)

.

iJ

~~I;'.:il\~ I~ ~~ • ('\ f._0 :Ll.JL~1 e,;... (,

J»

/J

9 ~.i:..\.: J)L;.. )I~~

/J

oJ 1oJ';: cll..il •

i10

;"';0 ';/~

;.;,rJ I ~,A1I;SpJ I

~1~~~.JloJB·(\\'·_f.)~I~~I)L...~~.

) I~

/J1

== 2 7T (3 cos

~~

lJ'.

9

:0

T , 1

2 Q0 + 1) -t

~l::S'" ~ cl.I J ~.~

cos

== 27T (4- 3;02/$2)-

t

2

Q 0=1

~o

~V4

( \ · t_ 0

o,s,= L.J4d;I.,J1 ~u..: ~t.:.:

90

90

a...." 1;1J :t1.1! d ..1> ~ 4..:..i..:"; ~P!';;"

Q

o I;J~ ~ (, JW

x., -

J!:JI

~,:-ll ~.hi..J1

~,~IJ,41

v~

)C

1- \s,:-JI ~ J, 4 1; j

~1~v~1 ~

;

J

• ;~

L.... .hi- vi, ~ 11.illI ~ 1~J JaJ'C Q

~ljJl ~IS'

1..;1, Wl: w.;~

• ~L.Preoess(f3t~ \.;~ ,~~~I~WI~I~vl~1L.WI

;

'.IS'~~"; J! J)I.> 6 ~ ~~ ~WI~I ~ A~~

.

• ( H' - t )

.' j)

.J~.Jjl

J!.'.II JV:Ht'" IJ'. e J

~";l:J1 ZpJl~~l_ ~lhIl~IJ~1

Energy

Oonsiderations. Limits of the Vertioal Motion

~~.lJI;'-

;

V\K;"")l,J1 .J~~ • ~UJI~.Jt... tl~I~ •.JIVe

• :. tQ I WUJ I

v = -mg)

v,s:;. l:rJ 41 ~ ~ • 41 t..J I~ ~~ ~,jJ I J, ~ ~,.;Lt.n

OOB

Q

, II "(

~~II.:;,..L;I.1> ~1~~..t..:u~II.:;,..~Jf r~

..

c=.l.JJJ r

E ""

1:1

m.i 2

).

•

~t ~IS' W• u.sl· (r,rQ,

Q ...

riS

sin Q)

ul!JI~~L..

2 '2 "2 2 (Q, + ~ sin·~) - mg)oos ~

L-.....t")l,JI rl~I ~, ctlJ.s j~~ • cos

• ~,PJI i;l!J1 ~~~

u vi ~I ~;.u,

•

92

--' .. ~I •..o!.wl ~~WI

· i1 • h/s1n2 Q

2E mt

Q u~

2 Q

+...,..

h2

#

~

U -

l-u

J:...U

~~, ~L... •• ~:':"I~'

..

'2

~~

( \ ., _ t)

~I

Q =

--=

~.

2 • feu)

. - ~ • ( \ J \)

rei': •••11.1>". (G) l) ~I...,:>

lb.iU I

~

x _~I

y, • (. J') J.-~ I ti:.i:

.1>,,>~IJL.J.I,.

x • '7

.( 'J \ ) a.Jojiul i

x. ,

• wb..J1 \oS"w .... lSJI ;;UJI aJloJ ..s.=:-' ( , - t) V'~I" ' - { • cj-..J~Itt

Q

J.,.."

ToJI~I~L.'J.-~Iti:.i:V4~.li

~I y_ t

~~J~ly~ ~.i.ill ul~)'C. ~ ~J"1I..:.~~ ,'1~TJla.,li.~IIJI

T:

8U 29

I

Wt...

• 'J.i;"~ 1\oS .aJ i.e

vt

."

u L.J I , ~ • J.-"11 U:r.i; V4

uL-.JUlvl ~~~. ~ '1~U.,li.~1S'1.11.( y _

{)~I"

A_ t .

J.

~ u, L.c J.-"11 U:r.i; V4 i-"11 u ~ 1

r.

'-

[

-

ot

•

i:'";

0

i

f

,~

~I .

-

~,

E:



1;.

V == %.2 +2y'2 + 411I

0 V4jJl

J ( , J'

2

l\

-

\1_ {

_

E

A

..»,.

J' ) .~lo "»~.JI..u.

_

Vi ~;I

'0 - {

c.JO~1

JL.n-! J~V4vA""J~Jv",~J1td~

.....

.i->, - B=kJ ,E=jE -

~ J clpo-; U::SJ I IS, 0.\:>, ~

~ ".J,.r. B

.vO=1vO~I~Iu.~j..,'1IlW:

J

~ ~~

JI.J,~'1lv,~'1I~

1S~.JJI~4SpJIJL-vI4~ .~4SpJl~ ~ X

= a sin w t + bt.

Y=

~(l

z = 0

t· 6

.;...

- cos

~t)

".l .J.i, magnetron

v,?!J I J v,,~)lJ ;",Jo::' .JJ I4SpJ I ve .J ~

• ",,,,?Ii.:tJ~~,,,IJ~~ ~J~ ~,.,:!1'1ltlwJl

iJt- ~ b ltl};Ju....: 'L.J.'j~~ ~~~. \ Y - { • L j.5,,; u,- U:..i; cS~ • .~I~~ Ji,wl i...~ 1 \.3'1,;:1 .~. ISF".,H lAt,.:.V-- b/2

r

,~

u....: u I! Ul; • j';L.t.~ vJ.1 uj.b.. c£lL., ~;»>- ~F - ''A - {

. v-;.U w I~ 4j,riJ 1~

~.

J,J:J 1•

..», J -.iJ

n cl~, b

v,jbJ 1

• 0,w1~... t~ 4j,riJ1 u I J>;I

,

";,

•• u IS' l.iIi·

\1 \

"rbJu....: u.1. 1,,;;I~ ~ ~;j,,> ~r:: ,,_ f.

b

it ~,J' j.". ,,;~ Lb.i: ~ uP I ~ ~j~ I ,.::.l.".::-:~

.;...

..

• ~1".iA J;j~1 ~dWl

J....i

, ~,;u ~ I

~~ J.i-~I J;j;:iJ1 ~I ~

• i...i.6J1 J.i.-I utI JJ~ ~ ;j~ ri~I~;J1 ~ ~WI~I J ~._ f.

· r "--~IS'

uJ I

"L-

IJI. g ~ .,~ ~ ~ ...~ ~ ~ J,J.~ ,,~I

- ;J 1:11 ...1"" ~ I .Go

1·

J

~I n _ f. b

".i.:

I£1 ".;,:..1 ....... ".ill Il..J I ~ • • f · J., ..:.,II.,.. !l! • 271 )/g y,~1 ~I;JI J,:a- ;~'-'=-~.i.t.i~ ~~ r L.J,J:. ,,;,J J,~ '0_ 1.

G ~.i.t.i~;"

~,;:i.J I .spJ I ~.i.t.i V4i ~ ~

0 If·

";, l-:

Q0 c.; IS'

I.it

• 6 ~ r..u:ll ~'.;, Go J~

;

'.,WI,,;I • f(u)=O

4.I.,l..-U Uo...:i,:i>-JI u-.,J~I vi ~VJbJ1 H'_ • ~~I

•. \. 4oIj~~JI.t.;I~ J t

vU; • ~I ~ ,~~ T~

...

J.",.-:J I ~ - ~

To

f.

J,4 14.1b. Jv~~ ( \. y-

f.)

.....J,J:» IS'} J,~d1-~ ~ 1. -

f.

yluJ"""'«;~1 ~t~11.i1; • J";l:JI~

g'£ ~j>' J)l..> J, ~ I;~ ...J.....:..s,.:- U, I ~

J·

· [feU) ..

0

4.I~L••1I ~ ..i>.1Sel.>1 ~

u· 0

•

~1~'It

\H

.!>JI

U"

J

.LlI

Moving ReferenceSy.tems ·':"U'J>u..,JZ'p-:;.""b...tl~lvt.>~I~1 J 'J,.;.;~lc>

• t;

c.IJ~ t.,.,;~""b...tl~l~ ~ ~1""""~J~I~J>~f)l,;

• 4.i.i11 U'J> 4.:tJli::~1 ""b.J1

!eranslation of the Ooordinate System

UJ> ( 1 ••

J..:.:: ( \_0 ) JU.lI ~ •~ IS'p J It~ I ~I ~ ~ li:: ~ I"",b..J I ~J> v I A..-J b.. ~ • Z')~..:.J1 "" b..J I Q.%yz ,( ~ t ~;);.,.... L.. ~ I"" b..J I J1.XYZ

v LS l.il; .~!,.:. • I)":" ~ Ix ,QX ~L..:JI ""b..J1 ~~li::~1 Z'pJl

nb.Jl~ '.-; ,..;;;hJl,I~L.~I""b..JI~:p ~1~~,J:.;t' . v.il. go Z',po.:.J1 J.- ~I ~ ~Ijl ~ Ro c.:.;~. ~po:Jl ~

~

......

~

·R=r+R 0,

Cl_.)

y

p

-=.,F-------X ( 1 _0)

Z

-., =-' ~..:.....ll - 'I. --

'J

I.

J1:J1 , ........

II

~ W~~-..a.AAI

;

<J> ~ JSecJ~ .""b...

J

~~""~Vli::1

~I (~ ~

~

( r _ °)

+~ A o

A = a

_ 0)

a, v ., c;b;Jl~ 4Spo-:J 1J.., '11 -.u ~ ~r v~ Ae , V ~ · J I::J 1~ 4Spo-:J I"" b.J1 JP rlt'...~·Jl ~ ~r -->0

---'"

0

~

~

~~

=a ·;1-.16. 'rf'J:.M ~ . ..u., .""b.J1 ~ 'rf"t...:. ~1'vl~1 A

r I,

~ 1~ ~ I""..J l4SpJ 1~

lJ"J,J;

u,- • 4Spo-:J I..., b.J 1~ ~ I""..J l-.s'pJ f' ·r.slill(r_ o )

Inertial :Forces -....

~

:F = mA

..,,_ _~; -.s'pJl-.1.sL... • (r _0) 4.I.sL..J1 Ve

• ~..w .~L. '11"" b.J1 ~

c-=

v,s; .4S~1 ""b.JI ~

~

~

F - mAo = ma

"

·1;) I o.bJ ~ 1.l....A

I

~

u,.-,.

-.... F

-

;,...iJ1 ~ I ";mA Q

• inertial term 4.~~. • ..

-.... n

:F "

--" :I

ma

~ • 'J 1'1; .s (

• _

0

)

'H e.:.- .~ ~ 1;1 1 oi>J I w..>..t I I..i I •

;, 'oJ

ill

.

.spo.:J I J' b.J I '" 6S'p J 1;J ..t W

LS .~~ ~ r~ ~I r..t~ e.:.- ~,J ..t.:aJ11.u, •

---" P

;.l"R·':"~ I A 6-J ~ .~". JJ b.. )~I e.:.- ~~, '4.:..t ~'j 11S,.aJ I '"" 4J bJ I

IS,

~>" 'oJ

,;11 "" 6..11 cfll: ~•.I.A,:H~

~ , cJ W1J...e..i.II ~h:..>

•a .iJ I, I CJ !jJ 11S"al ~ .s,.J I ;j ..t '""'- ~ a

loS,

r

l.S":-':~ I

b ..t,.bo ~ l;-;.spJ I;J ..t l-. 1;1 I ..t,..t.:aJ Iv l;p. 'oJ I~ ..,-:;

i.;i>.f1 ~ I¥\S,; ~vl~xV"'cllav l5 UI'Jb. ~I ~ .;J~I

;; • '=

-,J~""b..rl~1~ .,j,~14.i;llJT~'~1......,

•..;;

dJ:-.

.~

up-

JL...:. ,~I,4--:~4.i,LhJI~IJ(j.

~;

.;;I;J,lb ~t~ ~~\i

~ ~lill ~~oJ~ ~1.b,rJl ~ ~~ ·;J,llJl ~~LiII v-:-:ld~'jl

•~ liJ I U:S ~

III

~.

,;U JIg

~~ ~ • .d~'Jli";~ -~

J.lA. J:.:: p

~ ~~

-...J,d;lli,.i)1

~

W

vi ~~ .

F ~ l!::>. 'oJ 1;,.aJ v~

~vl~d'oJ.r'jI...br,

,.. ~'J;'JI.br • v..il • ;J,UJI ~

l-mAo l >pmg A o >pg

\1

(I

..ob-U .. WI iSpJI (r_. General Motion of the Ooordinate System ~1li::1 W.f"'~~IJ'6.J1L~~~~I~bJlv~l~

II

--

& ~ ~W\5 J.W· ~~I",,6.J1 u11~~.

4S'r=J 1""b.J1 ~, ~

r

= ix A

~

"

~

0

+

(, ~

~

= Ro +

ix +

~

A

/It.

II-

kz

~

A.

d1

'"

jj + kZ + x -

dt

,.

-!..>.

r

~

+ Y

d -\ .;;.M.

dt

~

dk

(A _ 0)

+ z-

dt

A

~rllll..i..lll ~ ~.,I

~I •

4!,...:..II""U ~~r~···.:~I"r J:.: p

z

)

....

ix + jy +

ix + jy + kz

•

~;";I.., ...,

j.)~~,.,JI",,6.J1 J~I

R

A

~

--JIo.

dt

..bo

+ jy + kz

R = Ro + r

_d.lt =~ V

'1,-

~

~.

A.

r = ti: + jy

+

,..

kz

y (T - 0)

J,b.J 1v I..., oJ,

x

J!:..I'

Jli::~ ... WI ~bJl~

\ '\ '\

..

~I ~ ~l ti:.itJ10\1S\'J-&1\\~

v- ~ Wj,.}I~ t~u,...

WI:.JI4:)W\""..t::Jl.~;I..Jl~b.J1 J~;U~~~'c&U.ia.A. ~ #I.

"

x di +

at

'U'

~ + z

., at

1\

dk

Cit .Oxyz

A -.~I 4"\>,J~ oxyz •

w

~b.J1

J 01~..JI ~ .~l J:.;J

.r-)~~II~ J~ 01~..uJ IS,IjJl ~~~ • (

4..--.l:;s" ~ ~ I • ~; I..J I ~ b...U ~ IjJ I ~rJ l:

u:J ).

r -

0

JUJU

~"..aJ 1j..c, b. ~ u,...,

z

x

(r_o)JL.:J1 ~ t-..J.l ~ IjJ , ".rJ I ~ ~)

·I..JI

.\".1.UJ1J~~ lJ' .~I ~I;~li.: ~ ~\jJ\".rJl~.~I~

JS'

~, Jo}\'

~

W

~',L,,":

"

A

~

d.k/dt, dj/dt, di/CIt

"'~~

, 1Y ~

IS'~

~

~

Ai )1..u.vllS.r jiJJl~.( ~ k,j

~

d

IAil ~ (sin~) A Q .:.j;;--:i J~ ~ 1S..iJ1 Oxyz

"" WI v I"" oJ)..u. ~

~

v,il •

w,

i

~~I;H A

1*1 = Lim

w

~t--O

~Q~

1\

I~~

j1,

6t

1 =(sin~)

(t_

O )

~

~~.w sin ~ ~I

A.

i~l;;o.\>,J1 ,,.,; ~I .D L-...~-.J

I (; 1)5 oJ ~~

o

, w A o

d --l.=

dt

~

wx

A

i

~

-ti... dt 1\

=

GJX j -..:0

...

di/dt

It

-elk =wXk dt

(,

0_

..:r 0)

'1A

'"

x

'"

...

-

di ij, dt + Y dt + z ~ = x(Z;-x

~

i)

+ y(

...

'"

. .... :::: £...IX (ix + jy + l:z) ~...

cTI{

r::

(,,~_c)

r

ji.."J I d:. (A _

4J '" W I ~ , dl.il ~"

c:»

.:._... + c;.~ X --""r .L·V~ 0

crt=r ... ~ &

"

. ~ ~...) I..:-li:..:... ~ d';~ I vs- ,,:)jd d:;,5 jJ I 4J .J W

'-.:~, ' ~}I..JI-J,b.J1 ~

J~ ~

(.4)

'"

J

~

q

..

1"4.U ~)I ~I \.-l..u-... ~

~

A.

Xq ~L:--='.II~r&l ~.

uI ~ (di/'(,t) +

.J

....

c

-'>.

:::: Wj{

~ l:J I

~

A

Xj) + Z(WX k)

VJ

1i)J

cay

a ...'JI~~

b.J I I,) I." '" ...

i~ +

V4

"".

"".

jcay

t=W 1 q

+ kqZ

-q

~

~ I~, L..::

~ ~,;J 1~ 1

h

(dj/dt) + qz (dk/dt) • ~d";)WI "'~~ ( 1(,) _

I ~s.:;

c) tl",WI ~

d~ = ~ + 2~X~ + ~Xr" + ~X(;:;X;) + ~ l,!~ .J ,

L..J ..I L-.J I 0'" t:'+:: '~I ~ I~ I ,i lJ, ':J I ..1>J I • (r~ I:;.I~.;J 1J,1 ~ '-:?'~ 1$,p.)l1 ~:"::J I

J _~ Y" ~ 'lI1

..I,

~ ~ • ~ WI

"b,..11 1.1 • AS'~ I

J' b.J I .j ~ I \..\.:;.-:........;

J' L.,.,J I ,)h:,.)\.:; lof ~::J I IJ.~.~~ ~ I." ..J ! • AS'~ I", WI

.)~

~.:' 'i I

LL,~ 14o!"";;

o.1:>J ~ • Transverse

." CelltripetaJ.

J.
.:.:J ,

!II

lJ' ~ '.:':" .·~L.> oJ': .,,:II ~I Ve ~ x - ~~ • ~I.?J- ~ .(,_o)~I~~

u·I.J,::>o;~~ . --"-

r

" = ib

r= 0

~

r= 0 ~ ~ 1;1 I~"...J I ~

......." "v w=kw=kb

~~I4IoJWI ~ ~I.J~~~ ~)l:;u..il.

-J.: lJ'~.

yl;;Jl:.=..II...,b-U

~}~I~~I..t>JII~

L

\ Y\

'? .;

I~. I ~ L:~

~;-.:.

y

b

Ifc..I.JJ1 , ~t. .,..~ • 'J '-'")LWI

\•• v v. I 1.Pr·.~d v--

I x ~ I 0~ ~ I j.J- .j L.-J-l LW; ,;;,s;~ JJ b..JL....::..;..U

\..l!..--AlL ve C'.?~ ~ 1.5' ~..l~';;~ ~

~Ivel.S~.;.W: ~I~ Ie

,..

Z -

~~Ij~~

v

w=k-r z p

v

"x--+---«

( Y_o)

Ji:J1

~ J-..)!:> ~ r::..J>.c.:; ~

~~

•

oxyz

Ab.JIJJ~,;;..iI.(Y_o}

c

~.)all '~I

j.J-'-:

1 y"

d...--!,..i; .) I W

~ b

l1l>},:"h~;;; \ oJ.: d~ ;;~>,J I

L--L:-J I ...,.....k ;;.1.i; 4.; ~

); WI..} ...k~ ~ r. dSp.:.J I ~ 1I1 2 cil.il· V / b ,j:I~ L'~ 0 y.; """'::",..j I ~I L;

o~

r. r.

~~ ~

I

I;.... 0A

i

:u.rJ I ~ ts' W (.s}~ I ,y.w I J.::>.-...: ciJ,j) :i....,j I) I

=0

y ~~ ;;..bi; .).sl

Dynamics of a Particle in a Rotating

F = m

0 j

V'·· ...

27

z:5

-'" -->0 V2 " A A (~X r) .. /,,2 k X (k X bk)

-\.

If

?

2(7 k) X (-jv) =

L-J ..I W 10,5; ~ ~ ,

,_,k:i.b.£; J$ c..; ts' W

J

,}ill \..~ 10 j I

Coo~dinQte

System

..:..J1 '" b..J I ~ ~pJ 14.l '" L... 1' ••

4

F

J t.s'

== mr == m(ix ... jy + kz)

P

Ii -... It

c. rJ1

~

--'-

-

;l1!1I. _,....ill

--"

~I,.:

tJ4

oJ

(L\_o)

-->..

== F + F eor + F trana + F cent -

J' b..J Ie.": ~ ~ ~

• ~ c.:.il..i1

mAO

W I ~ ~ ~ I":'"' 4J 1J

a

~~_\,,)l.U ~..Jlr=-I f/~'>tJ4 b~,

I) 1~,) ':1'1

·

ZpJl

oJ,

~ 1~

~~~ ~I - l,$,P-1 .. 4 1

,» 1

J

>.e

~ I ~~

L..w..,b.i; n.h..:; • .¥ l.> ~ I '-:tl~ I_,.....w

-4..-Sp-.:JI,»b.J1 ~ ~I ~r ~ ~ ~oJ~ wi,,) 0~ ~~I, - _;1", 4

1

eo I)

-)

~l~ ~")~,, ~~ ~~ ~ U,...:: Y ts; 4;l,.......,sJ I_,....s.ll ,~~..i1 ;I':'=.II ~ -ll:J.ilI)L- yl-> SJI

2- ~

II

J

.)~

~ I,J ~ ~ W I .b...i...cJ 1..:.-1.> L-

• i...-(" _,.ill,,~,

• ......:$,r--

Jj> 'I,.-.J I u 1J3 oJ 4.:tl,..~ I

Jf J...,J.J.:JI '~f J,~ JWI.Ja..A..;JI..:.-I..i..:.-1.> WI ~6.

J

~..i1 • ~) ':1'1

\ Yto "~~01J'..Jlv~~ (,JI..:JI~),),ldpJl~ ",,";~I d~ -II J_l •

)J ,

.~ t LwtJ ~ MtII Vltt t:J ~ '.n

~'I';-;'¥I'~

'.)-I mg

J.lA. V k .••1d J I '13'1,;: 'I ~ 1...,:

~~c..,; -; cl~"'14,....iJ~I!U

~

~u

1ll'I=pmg

(2m wv)2

... ~ ~ dJ.il • ( l • _1» ~

("

·_/)~.JWI~. E

mre~2 cos A

+ 90 0

• '.J

j.!.:J I ~ ~(.; ~ lJ' doJ ~, I

4~~I~u~ I.:~, I.I.:~J~14jll

~ J.-..: t ":-~ I u": L; ~ • (\ \ - /) ~ I) = sin ').

___s,;;;in;("-E

t(A:

IllI'e oos ).. , ) c...v 2

Vo''';~ I v': .. 1,;.11..:...\ b ~ 'r-) ~ m8 ~ I j;..:

riG

- mAo'

~

)~I ~ (

i.: ;...11 •

mg

A =0

) ,~.•:~I..h>J

t

~)l:;.;dJ..iJ

.J,..,J I u& ' l:..,ll J"; ~..b.:iJ ~~ Iu lpo.: ~ I u~ • 1:....;:: IJ' ~A = 45 u~ I.u 0

r tA:J2

C max

= --.;e~_ .:::= 1. 7 x 10-3 radian 2g

.:::::::....l.. 10

degree

, y '\

bi.,1 ~ • ti.i: ~ I ~

eo-

l.r-J.- ~ ~ ~ 'l;.,) I J";U .la.> ~ ~/~ I ~,

,~I ~.w I crJ I". ( , ~'.:.. 0 JUlI) Wi:

l.-.b,;

~ v~ c.fW I ~rJ I

( H _ 0)

J(._~'I

t-

..b L--~ I c..J,t..,; Jl•• t""'.)

L..J!" oJ ~.~WI.Jt.) ~I

--"

~~ "",,~II..iA vi. "",,/~Ij~~~ ~h mG

J 4:J If .~I~Ii)t.;>~I

~~14";VI L:..,;}

.)t).i ;''':; dJ..iS' ·4)..i> .j~ ..P; ~I v ~ .L..tl..:

. '~I J,.;Lt\.d.~' ~ l~rl1!>, ~~'~~;;J~ Dynamic Effects. Motion of a Projectile ull:JI~' uk(' y- o)~pJl~ ..d.",,~b'~ --a..

~

fir

=F

1

--"-

-=...a.

~

~

-----=:-

~

+ (mG - mAo) - 2m.c.V X r - m w X «(.A.) X r)

'bJ I ~ ,

I ~,

-~

• IJ'"") ~ 1~ ~ I.,.; .~ ~ :J:J...... 0.$"; ISIJ,:.:: F ~ -'-

~·II • L;., '•• ' J " " ' .

mg

LA 1:.>r...;-'"':t":::'" • -~11 ~ < .l::.-J I

Jl:JI~I..,k4SpJl4.l.Jh.~~~ ~

--10.

--"

_~

-..:..,..

~

~

mr = F + mg - 2m uJ X r - m'~ X ( ....,.ncdJJ\

ul l ;s",-JI~ .a L-. ~ &:I J I •

..J..,..-

(l~_.)

.i.,J~I;,JII~~1

ll5-...,.......JI~,!:~

~.

.-.)1 ;~~

",,1.••Jlc.:..LA~I)~~UlaJoJl,JIJ-J,

y-~~. ~rJl~ ~ .-~~ '(·b.J'J,iLtJu..~"')

Ie..,.-

J£:J I) J I..:J I,...:

""""

• (U· - .)

•

~J..-..: ' ' '1-..11 J~~II.i.H ~

"

g . -kg

"

i

-.... -:..... WX r =

w x

.

x

= 1(~. cosA - wj sin + ~k( -

. cos

()J Z

A)

A

,.

3

k

"-'.,.

W

.,.•

•z

~JI

z

A. ) + 3(wi.

sin-,'). ) •••••••••

(t , _ •

)

(lS~)

(~)

"'--!>o

JS v-

m } ..",:>~

(

~ If - 0)

-'*"

~

4JoJL-J1 J wX r

V&~I ~

4..-JpJJ ~.:I ,.;.li.::J Ic~ ~ oJ L-J 1c.- ~". U 1 ~ 4J oJ L-J I J,,1 V:K c.- ~"J I A;j t,. ,: ~

i: ::

-2e.;1(Z cos A -

:; = -2 w (x

'1

sin A )

ai"A ). )

i ... _g + 2 tv:i: cos A. ; .bo ~ ;".-

('f 1....

1; ,""I~~

:i: :: -2 w (z oos A - ., sin A ) + Xo

~

)

J.l!: u I ~ vS), • jPJ ~ Lill (,.oJ I v-..:..-.J ~ c.- ~ oJ L-J I.. .i...ft, ~J,.~

Z ::

0

-gt + 2 WX

COB

'X + 2: 0

• ~".....u ~I~ ~Ic.:.. ~;JI ~ i ot

( 'f y_ 0)

( 'f

~

_0 )

Yot Xo J..b::.Il~~~

• ( t {.... 0) 4J..lWI ~v.:':i.:LJI ~.x>~1 ~..lWI V-

Zt Y

~d>"'-;

'AT

~~~

i =.2wgt cos A - 2"'-'(zo COS')" -1 0 sin)..) (r. _ 0 ) 2 ~ ~ ~"p-I .JO Jelf; v 'S~ , w ~~,::.>c: ~I ~ ~I ~I ~ x = (,t.) p2 cos )", -2 (>.}i; (z cos A - Y ~1n )..) + X · 0 0 o

,

•..........

(f'_o)

01.~. ("'_0)" " \A _ 0) ~ ...d~ I ~ ")~';.,,5' oW I x ~

lit,.....:

<J1:. J...--: ~W I ~.J W' •

Y = Yot -

= - l-

z:

c..r-i

x =

W

..

x0

g+'? +

Ii

2

sin

zoJ~

.u,

J.l!:

'\ A

+ wx t 2 coe·-X

(ir_o )

0

0.:.: In.i-: .~~J: u i litj ~ w 2 ~,It V- -?.bJ' , ~ b .,.,. • c.::.J, ./Ill. ~

t w

gt 3 cos

~ J.-:

A

(:i:0 = 0 if =0 z =

0

)

y = •

z = -

t

gt2

o.>--_.--.;,~~ v~ h

~ L...

!:,J';l;.hi.. U l; • JrJ I ~ U ~ ~ I u I ~ I

3~I~ul,.-.;'Slu~fL!j~. i

W cos

A(

8h' g

f 4

2

~ 2h/g

r~

• ",:-J"".ll ~ u~ ul~ r:--~I u~~ ..u 'JrJ' ~ .~'Sbt':-'A

t

",oJ.::

4J;J 'iIc..;1!

W,

dJ.iJ I~ ~u"ISJlill~J...t.i

~l: 4k~r-:c.il.61 ..u~.j.;v 11it~1 .~b ~l>-.Jb.S

1 • - w

TO t

2

sin).

i;'~' \of.,1 H o.:..:l! U~ .~1~~,-..;1 oJJ4.S:..i..iIlul ~~;,

'..,.

t1

~, 8

,.-S',; J, ~ ( 1 .... • *

The Pouoault Pendulum

• \of ,,r---S J,~;SP- ~ u";'SI u l""oJ;;1:: 4 1 l.u cj V,,),J;u,u,. • •( n' _I» 4 1"" u, l.:j..i ..;J I -s,,~ I J, 41 ~ I ~ W t;

;;'1,J I ; ,....ill u,s; •( \ {_I> ) jUJl J

•hol l ""

-'"

S

-"

~~ mg

V:tt"

Y' ~ • ~ ok t...:J I "" b.J I t..u.:-;

J,iL:Jl ~ ~lb~~I~1 ~ J,41~j~ ~pJJ ~k:.Il-.J,JWlv,s; ~~

~

-..lIlI.~

-a-'

mr=mg+ S - 2m ~ Ie L.,s'.1" ~I

~ I "" l.S...;A

W

u I c.>:-

X r ~

•

......Jrrrr.~

-!Xl "-']; (

--31-

s

~

W

--JII"

X

r

)..bJ I j..A I ~ ---"-'

.s..:JJ

1-X

91

I:.

41'.,., •( n"_0) -.t oJ WI"" "'-'x

-+ s

....:.-.

r

z,z"

1-. 'Y sin ').. )

mx..

= 7-x

rrii

= :::L S - 2m (.II:i: sin A

S - 2m ""' (z cos

.,l

mz = S Z

- mg + 2m

.,;,~ ~ ~~ JWI~

.....,

(.AJ

i:

COB

A

h.r: ~ 141 bJ I.,;, I lc,L-, l;l:, ~ s

J,Jl:J I.;;sholj ~ I.,;,,!; L.i4..,r-

,41bJI4.iA ~ ,cdJ.iS'· mg )

~

UJ>~~.w,.(to_O)41oJlAJl~ y

_

e~JliJlt

.A.4J1

z

~ l:J I ~lA;) II.:. ~ oJ lAJ ~

x = - ..JL x + 2 CAl'i .t (t'\_o)

•

'G..

[

b

N'

0 ).1,

~'l

No

·r.·

.y

~ _ ~\ II

-t'

~

fr'

~

IT

-

_

c· I

0

..!

<j' ,.

(;.

v(;.

i

~

~

-

Hl

t. 'b

u

~.

..;,

-,

•

-

'

-

.~ ~~ \ ~ .iJ:, c:~

,:"" "J"

.uJ • ~ L

'I ~

=-: ~i..: . iJ I > ,J

i.JS- L~

...v.;J I \:..1.r.= ~ I j.•..,d

-.j l(..o::>;.h; I (; b' I.> lao i...P) 'oJ I

a o c'":~ ~ v0

c.,": b

c:--: 1: J-:-~ fr"'; ~ )

d 1~J L, ~~ /"..d Y

v)!.h; 1.:

l.:- ( '"

_.1>

x l:J I ~ 4.'.,.i; IS I ,.\..~ • • ") 1...,.;., I."l.:-:"",-.,.;;J I

~;hi;

i.b, I J:-~p l.. (, L_"

~(

n J t.. :i.;; !,-\-' I ~ I)..J I 4S.f"" ~ ( r ~I> '?~.JI

u....-: P". ..j '" l-J I . . .L.c,l ;J.:>oi: .,;;,i: l....w (, '"_'" ,1

'(,J~

z -

~~ 41:;....J1 P".

~ -...:JI

.J.':'

..L;-- 4-l.;. ; h'~ • ; r~ I

•cD

~

o

-1

l

-,-

&-

't"..

i:s

~

},

(;..

,e.f· .f·

(i

f

-
-

~

- ,....·r

B ~1:"'-"J~

-a.

mr :; qE + qv X B

J q/2m) B

......-..,Ij C$~;.r-JI..s ",,6.. JIU,JI~ l.il. ; .!,=,..ilI""b.J1

~4J.>Wlvl~I.( ....::..

->

mr=qE "';J

_,.]32

; -t:Jv-e

..s,~IJl.AI~~;;~

• Larmore Theorem

~

B ~;~

(J'

oS L.J 1J---il'

~,l-JI ~~~ ;""p~1 ~,....aJ1

Oentral

~oroes

and Oelestial Mechanics

&'~ 1~,.al ~ ~;,.; FJ- ,I ~ b ;1~

•

J

LA,,; 1:.h.> Je:: ~, ;,i1l

~~,.J~' L,.;~ '~I J~l-'~I"""p~I~,w, Oentral Jlorce o

~he

L~J:f' ~ I::...,~ ~ II$~ u"J 'J 1~ J.:::. ~,.;

Law of Gran V

___...:_l; l.j 1;..;J ~ ~ ~ ~"="~ 1.~:i.,U J!:J,4J;,.:W1 ~."sJI.,r.IJ1 .".....sv-;!JM~,j·~J.ilI~

~

.lA;!,It J~.u Yrs,J:i.,1.

~

;

vIS'" ~JL;. ~vJI~~~lo~,~I~li •• .u.., ~

\;S

~1e'~tloJl.~IJi.;;:U ~II~ ~. j~IJ~~I;~1

••

... ; ---.od~j;".:J J.>loJ ei~

~ ~~ ~,

~

)1=1..:.

;,s u I ~J1;: v I ~ lhU 1 ~ ~.H.l!" J4-t J

J

5

~'I J~~ lJ' .l:JJb.. j...,1LW:" • ~u..u, ~,~u"'"

.(1",_,) 2

~

JI--(GJIII/r

~

~ J~'JI ~ ~

)n

--a.

li' :i,JlIc.;..;15lJ

~J L;.;",,; ~ ~ ;".i! I .. .iA.>,.J..;.:lJ ~ ~

-

~ 4r -"' ..... dW • -Jt.dr ..

-

(~I~.,JI)

n

r

dr

1f • 111m

~I

.....

2 n.ar

....

JI#....J~ ndr : ~J-JI

~!tJ1 ~ • ~~)

...n.dr. v JI

Gb-

uL...J~'JI~~~ iW

f - 1 •

dr ~ ~u'J~ JS..t..II)J Xi ~~~"''''> 'J~

I 1t

Z'

-----'1

•

(r_1) J!..-.JJI x

(~)

( i)

Lhi: ~ ) \.;.>1 ~ d..~ rj)lj 1J....uJ 1 oJ ~~ ~

• ..

.:~.i~ 1J ~

J 1oS,..1 ul l

~~y,)WI~I-.\.:-,!~vAj~.-..ilI~WI V~ r 2t rl~ .k........;; ~ 1.;~ .~I~IoS.ilI)WI~J-;...:J1

~~v.iI.~l:.:J1

u I c..r"J 1.+o;J""W Jt- 4~ c.....c ~·X I~, ') L-lJ ~ ~ ~ ~ 1~ I ~ ~ • ~ b... 4,.ilJ ~ 1C:.,:) I v,; L; J~ JW~~H~ J4.l.:S'IoSJ~""'ISJI;';I1JIu..~ ~

.l.-:>.; (loS) 1.::>1 ) ~ ~ ) ~ '11 ~ ed...,p-:! ~ 1J-;...:J ~ ,..1 ~ ~..i l=:.~~)UI J~~I& ..il>.::I~)WI~ oJI',....!1 J~I4..b.£l1 ull~~

I 1

v(r)=-~ U>)l..

.... ; _ .... :.tol~;UJI4l1,J

,.....-J I v-)

1:1

01,~Ul (\_~) ~Jl.j~ij~J!

-->. k • ., • - ( ~ r

.....

-"'"

~

)n

;,....;.u ~ 1~..rJ I 0~ t;

J.l{:.l l.: ......... is' Lilb wL..,......: uJ--.w I V"v~ ~

U I ~ ~~ ~ I.. :P. dr

---""

• ,J ~..i4J1 .:.y.;-;;\

•

c'-

1v 1, ~b., J

--.

r

(

:1:,

y,

Z

•. ,j~I J~;~.......:J.:S' JI r,h. ~ ~~-4J1 ;";v=-:WI...,.......:

.S

upJ ~ L~ J-,/fJ

'l 't

l

a-

c·

:-:,

i- _

I( 'Go ~ ~ '[ ' -'£ , ~ I -'

t:

t .• at - -

r\-:E~~ f~· .:\

:.~~.~

'~t..

.F -

t·

!:- ~ "~

• oS ~

It

'1

-.

,,-"

~

~

-

,

r.

'-""

-

It

<j'

t i.v .l'

I.....

•.

t

~

'

~

'G-o

.

~. .t '

~~

;.~

-

\

_L

([. I \- <J <J at. ~ •

aCt-

.r" ~ .. ~

..

~f:

tr

. t... _ _ c. c't.. E~~ir}!'" _ • ,.

~

;f

If '"

t.

'r, J l;

R

JI

r~I.SF".I" Jl~ ~~tol~UJI (f._l)

Potential Energy 1n

v 'j 1J>P,

8.

General Uen:1;:ral J!lield.

. IE.; b.. v~ ~ 1t:="'? I tf V- loS?rJ I J ~ 1vi li.: L.. ~ t)

: J \:J I J 1-,.-\ 1 v-:~ll.Sp"J1

J 4-J1 ~~. ~ y

; ... f(r) -; .. 1'(E2 r

(tx

&.;~

....:0.

\7xl ...

-?>/ox ~x

- ~ lJ'r Wl (\ L.l)

+ iT + 'its)

~.jJ, '~1~1;'1>,J1 ~

"i

u:A.;,.;.I ..sPr J ~..sl jJb ..

i. (

"

1'7

~

jy

1\

~

o/?J

~

1.x +

k '7

-1)/0

f

s

is

+ k~ )/r ~

ix.f(r) ~

,fy.f(r)~, f ••f(r) ; Jl;}I~1

o

fx lJy

• x

0%

ox

)

d'1

..s.. ( 1"( rr L) dr

"0 r

0

= ~(x2+y2+.2)*. ox

~_...z.. 0'1 - r

lJ r

oy

r

J L.,- ':

". ~ •

ar

f y .•

x

X(X

+.2 )-i.

2+72

;

.-!..

'os

o f x = ...!.y... 4

r

r

"0

r

ar

"01"y = '1

~

4 ( f( r)

uh ..;y;J1 J.;l;;JI~

r

(

fer) ) • of'l r 0 x ;t. , f x 6 f'. ~jlJ ~ I.. I.::.;~ oJ ~ ~J L:Jll,

u~dJ.il, 1';-\S,~ul.»..u1014J~ ..L~ ul~

i

~WI~I

Lo. tsJ 1ullJ I LJ 1..1 u....r--= ~ u .; 1• lliJ b.. ,,~I ;,f..w 1~"p.,J 1;~ 1J ~

1

r

VCr)

=-

-;.d-;- ... -

J.j 1, ~ tsJ IUUJ I U

v.i' "0 VCr)

fer) dr

( '/)_1)

Cl()

• ---,iA. .,....iJ1 LJ loJ c.,; IS'

'b r

(t _1 ) J.i.1H ~ l,S'

r

()O

fer) ... _

1

J l;} I ,-:J I ~,

loJ ~ L...:.. W ~ l.i.lll,

L..:.tsJIUI.WILJI.Ju~~

1.;1

~~

".. _ ~ ~I ~;'Jl..

.Angular Ko.entum

-,

~4S'pJ ~WliJ.JWI it} I

• j-)'I ~~

J? ;,.....ill r;-..s J:.:: • 4.I.,WI .. .itJ ~ ~lu):J1

• ~It>

L-:..w. dJ..i~~, or x mY 4.~••Sllv-j~~4,4svec~lu~~

-h (r x my)

• -.; x mT +

r

x ~ ...

r

x

mt

......----.

--.

( ,y ...:,)

rX1lIV=L

~ ~ ~ c:.lU.I -

IS, IjJl ~ I~}c ""'.: ( lA

r~-.:>o';') Iv-; ~ ..s, l-:: r:-~ ~ ,p,; 4,sJ

4.,1.-. J....11b.i: J» t rJ I u I "I - 4.hi:J I .. .iA

L.,.p"J I c

~ l~ I

-on

J» ",1,;.11

J 1S,ljJ I ,.>,;J I

Angular Ko.eJltWI in Oentral :I1elcla

J~ ~ ~45p'~' ..... 1>4.16. ~.)I~U~J,JI "WI#~IiII~ '-:-J

..s'-

,;11 0 1

~

r

,,1JJ1u...;~ .. ~lc.,....,:

-..

;,.iI16t:...

..s ~ c:J.I.i.I • I~ IS, ~ t ~ 1vi. ~ IJ.A, ; I"i- u, L..c -; z

i

ulJ&

~~11

-~ul".J~ dL

-rr- . . L

a

0 eonstant

u JI

T• ,

.~~ ~I,) ~';PJA , ..1.-.,..

~

v J I,

,.:.--

.j u~ .

• ~t '----'I'l)loJ.i.~j;,ro:;~~LSJ>u~ r + ~ r ~~I~w~1

~A.*I;x ~rl

~ A Jl~I~I~L...

T• ,.

o )~Iw......: ....~ ~~ ~I~WI (O--1}Ji:..l1

*=.

- .); J.,-.: ~WI

,

.i>~ 6 t ~4J.,lL-.JI • ..i.A ""';. ~ ~

l~r%;1

(T·_1)

Jl:JI~1 ~4J.JWI"..i.Ai...:~ ~

iA J. ar""-s-

I -....

-"

i rXDlT

I

L

-2m --'-

dA/dt

L.p.WI",rJluldJ..i ve ~ 'lSjJ-J~ISIJ l;b L ISj"J1 J ~I

J

~I ~b uY-:

Kepler'a La".

u I L-.~ I u.~ , 1 • '\

4.:.... Johannes Kepler

1..i.A ~ ~ I ..uI • ~b ~ W t ylr.r'

......

).IF L.,H ~I

u,s:: v-:J' J,=-- ...,..L: L ~~!,.1J I

I~rliul ~. v--~i~~ u,.:1.iJ1 •• ' 161 1 I < •t :.. 11 . ~t ,J I.S, IjJl r>jJl ~ h u..;1

~1.: ~ ~~ L ~ l; I: u~ ~ 1.S,ljJ4 r>jJl u I ...r" • ~ I: ;1" ~,tAJ I · --.ft)ll.,J WI..u~" t_l),( n_l)ul:!..sWI] ~li:)I~..sW1J>~ ~

'I • t

~I ,""," ..

., ~ ...

~;lJ J~4 Q J r ~J,.-JJ ~ ( r ) ~~

~;l' c)6~1 ~ ( )I..-JI) -1...iU...; )'-'I.J..ii~c. ~I.JI

- Jt. lJ'u,;-J1

..

.1'•••11 roAA'lw!. )1.)1 4J.,1..-.

(T1 _ 1 )

1 :r - . .

I,) JI

• ('(,\_,>, (T t..l> ~ J.--: ..

r .. -11

..

d~

•

~ dQ

dO

J,.= ('U_l)

2 4 ; +

U •

-

4J.,l".JlI,)I~ ~

21 2

~u

Ui,

• ~ t)l 4.,JJ J-1i:J 1 ~

d~

2 2

4i 11 .. -llQ ::! • - h-u

~, ~

d.u

~.,L...U

( t , - ,)

r 0, t

~(u-1)

r

,,;t>t (f-_')

~~1::~ ...~~ u,..li:J1 JI.a.JI4J ... t.-. ~.~U..s'JJU.J.aL.J1 ~If

IJI- ~lo· 0 ~ 4J14 ( r I,)JI) u

611...... ~I ~ JJ~' r-r(.)_u-J.

- ;;

.1 ..li::llaJ ... t.".JlJ', .~ ;;

J-- J-J~ -'-11".

"I t-a.,l-. )1.aJJ ~14J.,L.J1

d2.a!,.2 J-J,..JJ lrU~.,...tI1 ':.1

/

T•

/I

• .,......iJ1 W I~ J.1:.' 1 oQ

~

L.;~

u=~

,

1 6 ou2 +U'" - 22 mhu

f(r)

= _mh2

( 60 + 1)

,

-;r~

t.1I ~I)I .,.iI~ ~I~I .p;liv-".il ",sz.,.iIl;

• '"

• v-;Jle- 9 ~I;JI...,..a::uJ~U.WI UL.tl J ".il.

+= 5

l:cl:

2 ho- t

C =oonstant • l5hc

-2 1/5

)

h

lit

r

2

9

",s'~ h.a J..-.:-:

_'f

t •,

Energy Equation of the Orbit

(T'l_1J

..:;-.-, • u. = -~ ~1;'.I'~"",=U:Wlll-slAJI.L.l:S.uJ~ ~ ~J-; ('fA _1) ,(l Y _1) vrI-s""Ji

-linh 2 [t~)2 + .,'

•

U

2 ]

+ v(u-~) ... E

~. Q ,u

(r\'_1)

J.,..J..ij v I~ ldJ.:.. ;.L,Wlll-sL-J1 '" • JI.u iJU..H ~-s~v.il ~oJL.J1

JL..:.. u-,_--..1. r =09 2 ~IJI..LJJ w....~ .~WI ~I ~. Jl:Jl ve du <m

1:

imh 2

~2 --3 -0 e

i 3/2

-20 U

r-

7

(401l.,.Ij-

U

2

)

+ 'f •

E

,I

;fer)

a

-dV/dr

~I~I J~ ..;c:.IJI.,.J1 (\_1

Orbits in an Inverse-square

~ield

LiWl~ee ~',....;Jl-';~\J"i.li~ ~"J;~1~lt;"I,..I-:>­ ~I

.

.

fer) '"' v~

2

(T''r _

-k;/l"

.j 4l Q... ,! J:).:;:1: W,r..,.,f;; lIS" W

k ,-;_",!';:!l . .,~l:;j i\,!-lI:oJ ",I-.li

k=GJ4m

( Y_1)

..t.:.J I ,j ~ t,

.[n' · - '\ ) L1 ••d.All.J1J d~

........",0;

dQ'"

+

J 1.JJ

«

,~L;L~."J ~ ~ ,\ ~ 1~,Jl ~,.

I 4Ld...,~ c~' .i~ ,~f • O-.t1.i ~ i J ~

k ,,"" mho..

=

U

L.S)

"n

(r t

-\1

,... r WI JJoJlvl ~~I~ k

m.n 2 )

r

= -_._._....... .l

A GO

~c.,;lf

lt~1

JI

COB (q - Qo)+

1..1,

~

('i'l_'li.l ,

~

k/mh

.~I~ ~1J.,,,.:Ji ~

;J.,WI ~~;,) 1.J.ii ~,~

Qol' A

J.IS:JI~~~

~ cdI.il ~.*~

Ji.-.!I..;,:)l,. ~

_~t'JI.-JI~ ~ ~

1

l' _

- A OOB Q +

9,,"

k/mh2

'''''~'v-iL:eW) ~~eW~,JL....

""" • ~IJI..tI;J,Jl- ~ • .iA,

~~I;Ji:. 1"

~. • • •

I.~

1 a .u~e--I-J

r

a.,; ~ r.

< 1 : u-ibe-W

... 1:

.~I{.r:-W

(fA _1)· lJ .. ""J1 ~

~ (;.7f"~ 1 + •

1- •

«(\ _1)

ro

UWI.."..; 104~1 J~~ ~~Ic.:;..~I.a..It..,J14,-UL,

r 1 wW~ .(~I..,JI~'-y;l):r.:riUU_ ..,...:JI~~ e: WI..••JI

1.1 • ( ~ I ~ ~t... J.A.c1)aph.11on

• • ~,~ ~ ~

, ~. yh..aJ1 ti'; ~Iec~ ~~l~, .~)~, J,.....,..t1, JI~ lrl4.bWI • cr--..J!=JI ~ .peg•• 0 'j~ Per1getf

ts"..ill ~I~lit...

J ,~~I) ,~.~ v-tJ1 J,.,..~Ii.¢t.!...., • e

II:

0.017 ~J 'j1;1''''';;'' ~b.

V.,..!I.c.:;..I;"'''ilul

J • JtJl J.,- ~. {.1:~I·

\_1 •

a,.b ~ .rl.95'OOO~OOO atl.s, r o • 91,0('0,000 miles iJ

1 c.:;..~I .... )."'!VJl'c.:;..Ii~Ilr' .. l. ii....-c..:.t.J.iJ1 ull,,,,,..1

..

;-11-,-,\' y "'~ .~c.:;..lt:.u, ·u~ JI

Ju. ~.u "jS';Jlu~~' ~ • UJlJ-M 'jl ~~I ~

• .;...

.....

• ""~ ,1.Jl{.e,J-J.;:.~'. (JI.t1 ~ ttl') yl~~1J.,;.UI.,.lql;,iJt~~"';1 ~l

- U=--"ol'4lJ'~~'~1;J'u~1 u·l ( t·_')U.jL.4Jl~ ~1i

u,s. .»~ 1"J1~~1i

••

-1

( f.J ....,)

'"

\

.

cj jpJl u)C.i..1 U~~bS" ~

~.w.

VloJ )1.tJ1,,~

J-t,lJ't (U._1) . 'V 0

( '. v)

2

( 0_1)

- 1

c

JL:. w

) ~I'":-t- ~.

1'0

(!).k,h...... I.$J'~loJ )i.ae.: vi-) ~I

J.P' ~oJ.:. ~)L.~I;

)..t.JI ~ ..d.-. ~ • ~W ~ ;J'~~ ......;)U.;I d oJ~ -~l&J1 QJLJI~.P'"­ • ~,~IL..J".i: uL..~~ oJ.:.~1 d ~I---.J...:~I v 0" 'u)I..u! )1.aJI.j ~~I J.:...:: va u I~;.u

"Iult oJ.:.~1 JI.a.::.,~1

• ( Y _1)

J1.tJ i J 10:.1) 1..t...11 ~ .a.;

~I~I J~ J~I..t.JIIo:.L;UJI (\. _1

Orbital Energies in the Inverse-square Field

TI I

(Y,_'>JUJI 41)1 .......

."'~ ~1~U,;Jb:-J'

r

fbe-W..,JIJJI.a...,.,

1 + (v2 _

o

•"J-Ju..a, v.

\lj,.,Jlu~~1~c),s::u oil

2 2 r2 v 8in ~ .D!!) ....... o~0i-· .....

r.

.2;2

_ ] i-

~J 11 ~ 1111 Joe III "r';JJ I J.-L.. u- J'f"'C ~ ,,1.("JI.iJI.aJlc)l~J';')

2& • \.r__ + 1'1

to:I' •

(e ( - , ) J {

r. \lJI.a..f1

.,....

.r -''0

~.aW·I~ , .• ~ JI-w

r

I (

• ~, aJUJ 1f.)t l,JS ~ a 6.,i'

&:I ..iI

J;~ a ."wu...:~..uI~;IJJIJI.aJUJ~ "" V

lI:S

-

~~~ a

'"

'i3\l!i.stant

~ ~ - ~: • Gonetant ~ • 'imv2 a E - V ~ ~2a J L-;-. .j ~~ "I14SpJ,J • ~!,J' uUJi ~~; J ~ v I tP .....,H)I ~ y. 4.o..b11:.;u.JJ ~;Jl

-11 a

J .,j,#,J~ 'Lac' l"/2a..,. ~'~I

·~l)u..J'tHd,;...., .~WI~ !~)I...,..JIJ:..

a ~

·Perj.odic Time o~ Orbital llotion -.,.,1..tJIU~ ... ..tJI ..... (\ \ _,

J

?.r J t.::.. ~ 1~ d ;:-~!, ("Ifc~ At.,-I-J U.5~ Iu I ( 1 -

IS I

;,.,..J I

PI

J:. Uai: .,:,.. ~ td~ t 12 tj)lll v..iJl , v JI •

{ 'I'i ._, } ~

t 12 =

A 12

T

'\ ) .a.;.J I

.J

L..J I Vt ~ J.....J( A ~

It:

A12T

•

2 ~2T

'\

"JUJ I)

.J I:.,.c

~b v'!:

:P2 ,,~I1J.JJ

,..., (\'. _1) J

:P2

o (A -, ) JL..\II

,..1.\ .;JI.L-II ~1u..J~

,P2 'PI ~t ~).iJI~~ 'r-laic .;I U... L.J I Js.: ~2 ~ • I.J

..!~ I~.,:J I J ~ ~ ~ (JIIlil:

J-= v~ b , L-.... I.a. ~l- ~ ~ Jl

~)I~I

r.

c:J.i JI.a. aJ I.J 1icW I..~••~JI t~

a~ 7Tab .... ....-JW1c:Jail1~l- ~b' U

7' ,pul ~;JI.~~. yll::J1 e.P '\$,;tJ~

,..I...... ~

(., -,)

211" ab 11

f =

i.cl:$ w:r.ev..iI·\$/.,Jlu~~I .....

2 7T a 2

/1 _.2

h

~)I..o=--Jlvl'

(t \ _1) ,( t,_,)

D1h2

2a= r o+ r 1 = ~ / '" 2 7T

~

r----

1

1

u-rJ.JL.JI~ ~

,.....

2Jlh2

(1+. + 1-.'· k(1-.2)

(+>'"

.tdJ..i e.P4,)l.

Je I.f;.. eLI I ~j v- ~ I w:r.e GJ J.o a'/2

(. Y_1)

~.1J1~IJ,J.~.J.i~~~~;';J~;.eJJI~J'v..il

• ~leJ-iIl JI.a.l ,,--_·~~~.~I~..i~J~JcJA.

~t~lD= "'vb'u,

- Je 1.f~1.&.1I t ~1;.1'"... i.e.J ~j L...cl:$ T = ca3/ 2

( .A _ 1 )

W.JL.J~ 'tl~~I~ ~ .... \ III

0

vi c:J1,J1~.C=27T"(GM)-i~

.(. 1_1

~1).~tJl)Jv,;li1"""YU

a .artil

= J.:. H'

••• • •• ) a

(,J"

" ••)1

Js.:(eA_l)

~ ~i.ll~ 1.,\>,.11.· 1••,:,•• 1 I.i 1

........-11I; • l.b~ ~oJ.a..J1 ~I

~v,s: ~~. ~~

0

T' ~

(~u.....,

.,lle W)Iu..; "'" 1..J~.:..t.,..\.I1 V lejl ( , _1) J,oJ,~.J1 J.:.."J-.-

•

• ~I ~I J tl~ ~1.:..IJI.aJ ~."JI.:..Ii~~I~.i.SJ ~I ( , _1 ) ...;~~~

:i"",..ul v-j

c.SJ--.f",.J1

.:..~.

·'le

• J'f {,

•J'f. 1 • J. • Y • J. , Y

• .1 "

I>

, J. • •

,

..

J,~I

..; "",b.J1

;

..

S)i)

ttJl

I

•.I T'A Y

Mercury

oJ)L-J.

• .I Y'fT'

Venus

.".....a;J1

Earth

~J~I

',Jo'f{ .I. ··

·.I., T'

}JAo

e J. tA

" .lAo 1

• J. I> 1 e J. ( Y

'H .I f.1

• J'feT' 'J0T''\

A{.Ie 'f

• J. • ,

• J'f H

.

tJ

.:.. l.-.,1le ~)I

Mare

~I

Jupiter

IS p.J1

Saturn

J..-....j

1'\.11'\

Uranus

!J"

11 tJA

T' • J. ,

Neptlme

v~

'f ( YJ Y

T''\) t 1

Pluto

A'

,

,;1...1

;11

tJ.iJI.:..~I~- c.S,,;t.:Il ~1eF-.,:J1 J~ J11'",.JI.( ,"_1

Motion in an Inverse-square Repulsive J1.ld. Scattering of Atomic Particles t~ v-e y ~,.ill v,; J.; • ISP;- J ~ J ,.,.-:t' 11',. ~ t1"" ~ lq; ~ clau. ~L...:,:,~I) ~~I~WI4cl.ll1~~lulpoJlS 'ISJb.:Il.,..t-J'~1

;"'-L.~II!..b...e~lvl.~I~I~IJ.lII.:..'rw~( • ,- -

~~ lASt.:.......... 6-:.11 ~J~ ~ c.S,,:J~ c.SJ.aJI~.,:1J VI.JI h:1.,L-. J""" ~I L,J vd l

• ~IJ

v

r

vle:-~. r-O(, Q.O .JrIJo:.c~I~I~v~

~I • cos (9 - 9 0 )-1

v,s: l.oW~".,wJ1

..,trJl ~ .l6-b )1.aJI.;JoJl...

v ..il ·.Jb:JI~ V=t'.JlioJl~Iv.tc40~l;· .... 2'0 I..... a.c~ ~1. ~J.JWI~1 ~u~ ~I ~ ~~~. 2'0 ""L.:

=

~

• ..."

(1,. _1)

IT-H o

1

.... oW'

(1 1 - 1 )

11 oJ W I u.c r li. """~ .eU..i e.P v..il ..... 2.0

.,.)l. ,

vl~~~

tan

Q

o = (2Em)* hQ-lq-l • cot

-4-

(1"" _1)

• (1' _1) lIoJW'v..~~I;,.wJ1 rF1 h ~t.J1 ~ ~I,,:-L.JI~. ~I ~L.. ~i..L.WllloJWI ~ ~ jJ/&

roJ t-:.::JI JMJ'~ • Impae"

Paraaeter r.J L..:oJ I.,:..x~ p ",,,...UIJ.S'lI~ Joe

",:"11 ~pJ ..; 1Jo:.c ~ 1~ ~

( .._

(~,.:.. t.:l1

j ~ ) J- ~ I ~ Vltf ~..,...J I u L..JI

vl",l. ('_')JUjI"'Qfte)A1J' h

= IrxTI = pVo

(,(_,)

~,L..:; ~1: E UUJI vll..icI,L-;, • ~ JIJo:.c~1 ~~I

I"...... "" L..: ~ IJo:.c'i I.... ~ IulJ.J Iv ~. imv~ (

H' _ 1)

cot

J:..:

y IJo:.c':II ~~ IullJ I

11..1 W 1Vt ~ I .,;~ ~ I::S' I.wS..e • GJ..u lii", • (r

I>

""2 =

To~

... t;;I:J

)

211:J1~1~ 2pB V'

plDV0

Qq

...

Qq

( 1. _1)

(\_,)

~l

J~J4d~u~~~I~"'~JL..

1u,....:.. ~ ~;~JI ~1 ~n

J

\S,l

....

";~u,~IV:tc)l.~\S,L.: E ) r"'oJ')'~ liJ'~~_'

u- ":".rJI"= •...,J-

~

\.

~'.ku~

H

( cd)1

-,. x 1

J' x ,

,.~

x•

• \' r oJ I.-:.t' ;;'.Ii ~ lei • ":-" JJ U ~ ~,.~IL:,..,.\1' ~e ":,,,,'Q

.e=4.ax10-10 esu· l:.:1~ J.

P =

-*

= 2.1

= 798 ":-,,.llJ,'

2

...')4loJL.J'~ . -20

2 x 79 x { 4.8 ) + 10. 2 x 5 x 1.6 x 10-0

x 10-12 om

•

= 2eliJl~~

-)I,1J ".,w'" I

)I

(l • _1)

" ;

II ,i'l

J{.AJI

L...obli

rrJl.>- ..,.~I ul'JbJl J.t-..,). '"JrJ :t(r) • c)t

~

-

4

(A' _1)

r r" e:-~'" 1.1 • ...... c:l.- ",.,all aJ 101 tt,.. loiA ) • I...,. ....... E ~ " 1 ) L...J0It....J1c)-6.; .... IL...obl'· .a.Lc' .. JIIJI

( y, -

0"-'

2ka-'

a- ...._-2~+ -4 EE. -4 -_ a

Cf/.7f (, + a • 7T (

r

-1 -2

5 )1

......1.

1 - or::; k a ),. ~ .".( 1+ 2 1 1 + E k- a-

c. Co

2 )

(A l _1)

ka

\

1

c)-eJl ......1~la.,I"J1 f/ka 2 ~I ~ ......:~I.,wJI c.i b.1.A f .....; \ ·aJL. ~ ~IS loil"...6 ~ • ~ E ~IS' l.il r otein 'l;~1 uJ "'; • .~ ,,!;.Sall J",,,...~I~I.,........~~~.i~I"r'~I"r'J'" ~I '.,...au. Jt.:-I u£o • (A 1 _l) aJaaWI J f!-/r4 .-J~a..-:JI

3

.,....Jw 1) • J.L ~ ~.:.".a.=1,11.; f.r-i: ..,. .;,1

~

......~I~I. aaju....Jc:.I.e~~I':'-L. ~ (IT' _, aJaaWI

.~J~~I

a.......r t

~.Jt..-J 1~ 1.. ~

Ja,J1s eT'l .JI... "~~.a...c~IIJ.,J..,...aJ1~1

(T' •J 1.t1. ",.ill c.hiJ 11.i.A •u"il f .(l)

c.i a,J b • Y t ,.. Col" 1':11 t ~ 1~

.:N~~LWI~I~I.e""JJ

";J ~I·u ,. ~I~ 1.:nJ Ii ~ -)lJJ";J ~I ~ "r'}J I.e 1,c.i~'

J~

~

.sWI.rb...Jlecl:lJ ",,,.iII~1roJJ: 1.1A~ • "1.:1...oJ~ I~ aJI..)I. u I •ec bJlU,. • t,.:~ Jt.,.:.lc '1,1- ~ 1..,,- ~ "r'~ I.e •JI. c:i&

•

c. u....~.II .i.A c:.J,c .aM• ..;J~ I J1a 'r W

~

..... 11 Jt.M:~ .. ~I~w~· ';""..JI JUlI~YNJ~IJ!.'ul

Jl.wl~ pr•••••

~ Urcl..,.......,.J~~"}t.dt8'=ll(ljiltf

."'-J'I J~'I Uz

u,;.tJ I ~ I",..I~ IS' t~ I

c:J.-J 1",,:- J JI.aJ1 uSc ~ l.il

<J, -=-cis ~ t~ I 1.i.A· ..,...&J I ~ 'r-.-c .... cJtle t.a...i: .a."c ( l • .;,)llI ,"n..... ~ a...,: b ~ J-il u~ JJ

)

~

; .. 1$,1-:.: .il~Lo};~~loa ~ ~~J~I;"";vI41 (\ -,

• ~t ~J~I ~ v I ~Wa.,..ll (y) ,;""L,. ;,.,'1 oa~1 ( 1) ~ ~1l,.tJ.J~1 jL:Jl4-tWoe~~~J~I;)Jlvl ~j IJI (l_'

~~~~v~....:S~vi J.v.a~ .,.,a:-JI~Ic./~J.i.lr

• UpJI • .iA ;...,oa oei ~ .........--sl.-JY c./;~~U.J.~I~IS' IJI~ ~;,atlv,.:l: ~ (t'_,

• ,0 ~ts'~ v I~I· c.i'J ~I J~ml. ;.....-.cJeC ~I ,.,a:-~I J.-.I..a JI", ~ ( L'

~ ~b J.,-J.I ( l

- \) ~ I •..., oa .>j ..,..i:

~

u..-c...~ u,s:.. ispJ I • u!",..ul

.:

\2;J~"""" ~~~I..soa~I~~;,aJ~~J~1

•UbJ I P~ V. r • ;,Al1u,.:Li

u L.. ~ J Le:6-~ I ~ u IS' IJ I

~. r=a.kt

.J4

~ ~(o_,

l.,.:J.::S; a

L."Jaiu....

",;L-JIJI.aJL.~p~J~ c./~cd.•,~4d'-' •

t

ue;Jle- 0 ~u.j ~ ~

u,.:Li,...... t.; . ;}lo.iJlJ.,...~;,aJlp~c:io~;loa~JI~u~lJl (Y_,

, ;,...ill .;

:S.JI~~I.aJ1 ~. ;,au~i~1J~ c./,,-:-tlJ~ IJI (A_' ~;bJIJI.aJ1c./~cd~ IJI

o=

ot

,

(,-,

t

t'

\ ..1 :1" ", Y \-

.

Ii

t"I

k'.

[-

-

\it

•~

-

r ...- \- _

••

f, ' i\.t . :. t

~'-. ·r ~

•

~ ~. ~

. ..,

II

13

to

~

[

E \.

t:

-

'C

t'i

-.:.

'o - .

IF

~, t.r:t'·

~ --

•

~• ~

Jt ' . . .., t

~i~ ~ · i

_0

..

t.

~-

n "'.

r

"'~ -

--'

I

.foF

~" -'·r ~...... ~ ). 'f -~ '-~ i

't

'G~..

". I:.~ ""'r.~\ ~'

~ .. \ -

(0

\:ltt~t t;- ~, c· ( ". ~- ... r l' fi'"

'-

+

&

\ gFr

--'

.

I

.r.~

.'

t'

•. ~-"= "r. -~ '( -

1•I'

i }

1=~~. tt ~

I

"1

.. 't .

{ t .. ~ t {' of. ~ .'11" ;; ~ 'E \.-,' '!: 't"'.- ~ { · i - '" t--~.f.~ ~ ~ .t -: f IFi· ~~f

1: '~I

.

'-

• ~t t:'-tt~ ~ t, .f~ f!. 1=- '\ • 1 r l:f' I '- I 1 . ·

rr~tt~

IF.

~ ~

,-. C '-,

·r

i:

11,'- . . •

ee' ~I..

.11 J

".JI;

...;11

.,l;,,, 4beloa

Dynamics of a System of , , - - -... h I.a I

Part1c~es

p; u,.. , ;pJ I ~ ~ I u- .",.i;

~~, 1.;.....,1;.;. i-IJ .a.1

• ; ,>••~..JI.d1: U,J t W I ~ I ~ u1"~ I ~J..u I.t

';-••11 ,...,;1 ~ ._J.:~]1 j-SJI

( 1_Y

Center of Mass and Linear J(oaentum

• ~I::JI .;..

rn

, •••• ,

( 1 _YJ1:J1) ~

r cm

=

~r;.

--a...

--3rI,.

r,

'

.........

~

~!,.~~

r 2 , r1

~ ~ ~11..J.iU1.t; .,b.;.II';W:S'j-S.... u~,

+ m2r 2 + ••• +

~+m2+

•••

~rn

+~

=

LJl1~

( 1 _V)

m

~Iul e:--~IVIJ· L..,LW'~I~IJ:.' JI~=L~ ~

x cm =

L

Jl1%1

m

Yom •

L

a,JWI~~I~ 1 oaL..J I II~~ .:>l.I..,s.ul

-171 _.

15011'1

=

~I t'joAJ ~~~I t~1.t L.,I:.oJI

L

~1I1

111.

-......

...

P ..,JuJ1,...,;Ilu.,-;,

"I , ; oa".U p

= L

i1 = L

~T1

~J..--.;~}1J~I.t (1 _V)

(, -y) UoaL..J1 J.-li:~ ( " •• Y)

TtT

• •

• •• • •

• "1

0-

y)

JUJI

~~IV't~~ljJ-

~

)In

~

-

~

1'1 •••••• )12 • ]!II ~~J~~ ~ u ~vJI

•••••

~..J. ~''; 1'2 , ~~;~]!II ul..s l ) .~h.,JI~~1 ~;''::

.u ,d,Li '"" I &i"-~ 1.0 • < i:' I • •• -2 "C~I.~H .s"il' •.lrl,;.;-, • • ..,\;,;,'1 ~

~ ..s 1VIK .......1... r'" L.: ..s" ~ ~

;.,....~I~I ~ ..s..iJI1'13

u~ J.t~ 1

-

...:10

]it +

J

J-

.,.,..

~~'

~ -S,,-JI &l ... lA.i. ~ ~I J.ti~ i _

~.; :l1~· 1

...::...:-..

Iltl"i

= ~1

~I ~ (

~

)

t_y

t' .~I.;.;".",JI~I ~JLaJU,....atI";'-; )11 4-

.;-,JIy..IJJId)ilI~'J"'~"jI~1UeUllJ ...l.J1 ~ .ttl 1.-JI

t""

• ~I.c

,,!,IIL-:JI.~~I~);""JJ..s,,..~I~~I~V't

.,.11 V't n

;-l< f.- y) aJ ... l,.JI~

~ •( 3

1

,...-.JI

= 1 .-Jl'l&.l

Jl

II.

L'j"1~ -

L

3-1

:,...; J.iJ c) I •

-

c)

I"I 1

c)

1-1

~

i-_v)

.)l.I..oS.u I '1'1=.1 I ~, .,... --""

b.. u.:a., c) be L.:a. c) &:;.ill c) bLa,

~

J11~.-J131

JI ~ 1

(l_Y)

£ I J J~ ~I o&J I tI,aJ I ~ tIlJ.iJ • J,.AJ I "11

-

L.~. ,.. • 3

J.-iU ~ bJ I.;,=,; c)~ iii Iii" ';"IlS (-- y) lJ .. laJ I L.cbf W£c c) JI • JJ;:JI~I ~~

L

Pi-P· ....

(y-y)

O II

'If ~ ~ ,.. c.l.,..-.JI ~ ~ U:Q II.... ~ c) II I"rJa.J I ",.;I I ~ ".,b~; IJ Ir..;JU•

c.lJ:ill..

.,w U..J:4i I '" I-: ~ ..~

-

·r"'·h l.ltJ~ J~ .... td"..::c.~l~ I...... 1 JI:JI J.,-';' 1';';1 ~

- .it.a.:...; ~ J.iJ J'1- ~I

c)

JI ·~b

~

c

c) "1

c) IS'

U,

•...,."11 ;,JuJ1 ~

--.

lA _V)

a• • g

• We .UII ,I .....A.JI ~ UJ -' L....,..;: .",. • .iA

.,.-J

.~ r;-J.... ~ ~ ;"., lJ.:;J I

.,

; L..ac .all • jJ

•

1clL...&J i L.J.:$ }-oS.... JL- ~~ (. J'

tloW I '''''';''-11e-J-iIl J t.- ~ JoA

'~ I

• laJ~lf"""U~

• ~"a..,JP;. II ..,JuJ1,...-...;J 1~ tdJ JJ • ~"

L~

(, _v)

,: IS;'.JI .; """"1 ,...,;J 1~ &) 1• """"1 ,..,;J I..... :; ...li ..". .. JA

'" ;;

•P • .y.. • oout.t

~ ,.,. u,ilJl

.j, • ~ ItJ 1~ ~ lie :;.,.at.. J...:JlI u,.;....,.., .p:wJ1

•.,.......~~1.:.......,.1~""1W~ .;J1c:.11.J1"; ~ • ....J 1."1 •

tJIi

1f..:J 1c:.1.I....A.I1

'* a,....J.1uJ 1~ J,:. • J..&J 1 ~JI ~ 1~ ..

J L-.-J~ c:.~ .J91 ..,JuJi,-;J1 ~ .. ..a. ~ ..,JuJ1 ~jJ I.........11 • ( l)

~h.i. ,;:011

.....,1;;." '" IjJ 1~,;J I (, _ Y Angular • •a.na of

&

STat.

~~11,:-".JJ1e"';'~~ ~ ",1jJ1 ,...,;JI • (._,) .JJ.J1";~.o IS _

"...-;J ..". ~11 ~ Ie c:.~""" •

"i

a:::

)1.1

-->0.

1. "" IjJ I,...;J h.;J""C . . JJ.r :r; aT

..". • '" IjJ1c:.~ I

(~:r; -iT1)

1...:-" ·'" IjJ 1~}1Il,)e,;J 1uu. ~

....--- ~11 ,:-"..u ~I ~ li J • •

-i-

a:::

~

L1-1 (~:r; ~T,,)+ L: (~:i: ~a;.) 1-1

.... .,)1

~

(l. - Y)

JbJl J..- ~"Ji.:1 ( l) W. !. Beott, !AO l'D1'a1ea of noev1c1V and Jlaga.e1i1aa,

2Jl.d

eel.,

10. 91107 .. 8aa,

aOe • •ew

York, 1966,

v'-- _ ! U, V1 .x ~1:&0 v'J. '.1i-.s, t...c ~ tot l.J I ~ J

J, 'J I .a.J I &) I

~~ ~CJJJ. 1 ~I ul-jj;-,JI~ljj""'I.."L.c .1~

.....

-* = =

n

S

[-;1

t

x

(~

n

+

~

L: -...

r=/ ;13)J j=l

~

C' ---'"

1=1 r 1 % ~1 + 1=1 3=1 r i ..."..__....~.J I ul- ~ I ~J l:iJ Ijj,.ill J:..: '"i1 ' ( l_ Y) ~T ~.sl ve

J!

~

7 ~13

.u.J I J

1 ~I ul-:;;-,JH;.,Lo.I.aJ1 )jj~1

,:,11 ul-~tot';' ~"\>ve~totWI~

J

J:..:

(, , _ y

)

lJ' • .:...,.. ---'"

J11j

, 1

JLWI~lv~· ~ ~

_

~I:JI

( l,( _ Y)

~3 ,;-)~ 1 ~a..-J~ j ~~~'JI~~~ vi ( '( - Y ) J!.'JI

J U;tJ1~1 ve .s; ( U' _ Y)

-j

o

('( _ ~

-

-ri~

%

~

_V) ;

'1j

;')I,JI.J....e:CJJJ

i;,3

JS'

.11

~I~

• :r'1j

~I ( H

Y)

=

-~jviS' U, ( 1 f. - y)

J,...J. ~ ;-,: ~ is' loiI"I • 1..cJS'J - ~1.aJ I.s,.ill v,s:; I. ~ I;.-.s, L.c .s .iJ'

.( \ \ - v) 1I",t.J1 'ri' ';~I~I; .~l.,-~.JI e,)e ~,j JS.J.u= ~1J1l,JuJI ,( ' l - , )

~I 'ri ' U~ LS r1

~ltrJlu.il~ i,l

n

Li ;1~

Kli cr-JlaJltrJl

:J:"i1

~~~I",:,~~. I;...

\$,'-' u.il

~~. ~ ~Jt..JI.,...JJlt}-$,.,.

w=.. I.Hi· ~I ~.;~I~JIaJI~,.aII~ - .;.,. LS ( \ \ _v) 1I",l,.J1

et-=

~.,...,

-:..

Jr "(

( \.. - y)

~JL;JllS.~lt'.i-~~'~~~,ljJl,.,;)I~ ~j11 J.w.JI. ul~1

• ; .,JeiJI ~;J"",.JI

rJ -=:J ( 'A -,) 11 '" L..U ".-; ~ .~, I,;) 1~;J I...... ~lil -t...- •.i.A It!

~,Ij

1.iS'. ~WI ~I 'ri~".lI1 ~••JI~jJl~ • ..sj~ J~ II r"jJ 1J ~I ....... l:ct u~ 4),,;.-.

;!.,..:. ~ W

J

-"}4

~ ~,ljJ' ,...;1 I

.. {O ~h.A.,,n111J~ ~~~V',pJI4iUJI{r

_ y

Kinetic Energy of a STste. ot Particles

'"

; ~~I~liU.~~,L.:~~~ !

~1V'pJl4iUJI ~I

• ; .."1;.-..,, ( \ y - y)

~ r:l ~ JS'~v-~1~.

(T' _v)

J1:.:Ji ve e:-~"" tJ; Jl:JI~1 ( \A _ Y )

;,

;Jlc J,.t1i:J1 ....., •

~I

j". ~I ~lc

1 ~I~ ~ ~

c.P J.....,;.

t

~

vejll

( \ \ - y)

.~Ij". .)I~lc 1 ~I~"....

vi

,U:~lIjJ-~""" j:.:-;em ~

Jl:JI~I~ T

T =

~l:S~I.J..i1

L ~ (~em + -;1)· (TCII+T;,)

= L ifniv em

2

+ L m1

(vcm ·~1 )+L-ivi

1;~. (\A -

L~~=O L m1r"1-~m • 1:1

0

V) 4J.)L.Jl ve

JI4JpJ 14.iUJ 141 ~ lAo. ~ v .H ( '( • - Y) ~li;~I4jpJl";UJI ~4S'W' v.iI~~~·~I4JpJ';';UJI

"~ ~pJl~l;UJI~ r~lj (~I ~J,~I obJl )W:~:lIj~

c..I'l!

yo-Jl~~.1

4.iUJI '-"lJilll..u,. (~~lobJl )~lj"J~l:4!;;.!1

'-t! IJ'. ~#c..~6. '-t! ~ y~1 .. c~ oil I ;;UJ I I~ Ij J..!S a.~ 4tIli:; ~ I ;.;LJ..l1 V4 v,s::.: ~,;.::.11 ~ I

IpJl;';UJI·v~ .4-~1 '~I

; ;

•

~I J.>I~ 4-.:'1..,..tJ~

• ;;".-....H u.:sJ I • ~~ I ~ 1..Ill.,b.1 ;'~ v.l' "':" U'po ( t_ Y

Motion of Two Interaoting Bodies. The Reduced Mass ~

1..A..bo1 ;.~ (~ J..lA:;) ~ V4 W,s.. ~ Zpo c,;~

L;:QI j$'". cd~ v .i I • U,,;.-.; ..,ia ;J I v' c,;~ •. ~".;;,....ic";~'

;

IsJ..-..':

U'

~~.• J.-~I~ ~~Ij". .....L4I~,. ~b~J-C

r;, + .;:~ = 0 e---~J.,J.: "i'2 ,r;, ~

--'"

II c.:1S' '.iii • ~'j$'".

~

R

= r;. -

( '( , - y)

~~Iv"( LY)J!:J'V4 e-~"" IJ'~

JI;.•...tJ~ • ~l::Il ~.

~~

-2-

1 + - -r; =r;.(

~IJ'~~

-1

-2)

2

, ~ ~~I

"l"

~I~J:.:

1Il

( '( '( -

y)

• (n _Y)-.J~~IV';.r-o>~';,..JuJ'~

- ~U:{Jlj". c.J1~~ 2:::;.

4 r 1 JIll ~

dt

. =~F 1 = feR)

~

R

R

"I"

~IZ,pJ~Ii:JI4J.sWlvl (

. U' - Y}

( ( - v)

.,,1141'--

Vii-

;

'..aWl Jl..a.:.-L, ·V•• _.~I

= :f(R)

p.

=

~4J..a4:JI.,....il1 )I~ ~ f'(R)

+

-~

( T /) _ V )

2

W."...u;; ~~I';J..aL.JI. reducec. mass 6-J..al.,.•.lI • .ia

2

~I ~I~~

;;~I ~~ )J. ~I ~

1 ~I.s'J'l" ~.(

J~ Jd~)A .....:L::S'..a~~.s'pJ 4c..a~~I4J..aWI v-&: ~~m2 I';~

,-

• m1

~.P')~~I~to:...i>ldJ..iJ· ~

f'

~

_~l:S ~ ('fl _V)

m~2 m + m 1

J1..-:.l I

t (R)

'( (_V)

1.1.:..,-

V4~4S:I.J-'~

;;,wI~1J)b.~cd!';' lJ:e=1.,:,1~I:I.J-JI

:IS' l.il • ~~.)' =m/2

'":'r.::A: ~ ~. I ~ • ;;~ t:t-=

..lS~. m

-11m2

~I ~ V. . . ~

~. ~ V4 Jtif!c JJI ~

• DI:t• 1;~v~

•

vIS'

V4)A 1.....,i

1J.il:;~~Il..a..boI'":'~vil' • .,,~

\I.

r

'(

t{

•

~

~

·f

-

m

~

-

(i '--

i

la, ~

1 t-

t:- -

~Co

Co.

.

~_ fl,

r,t.

t:- ~.... t \-

Co

~

}

•

IT

_

-

I-
.r ~.

-
;1 ~ erJ I '"'" I. ~ WIS'. 1a ~ b..:iI , ~ 4-

~L...~I~~I.;;..~ Q vl·r.,l.:lI~~I,':i.s,.iiJIWUJIu....

~ L".~••II ,I :i)r.:.J1 Wu.JI ~ Lr:~

• ~,;J~ ;"',J.ilI·~1 ,-,,4.,.J~

~~,I u~1 ~~ • .~~ ~b.

J

.~~ ;"',J.ilt~L..)l...:jt

~I~~I ~r..tl."..:Jll:J? ~I~~t~ • • -4S~ ~b. '-" • r.)L.:.!1

d 'J

.:~I\ ..,~ , -~,; . •

ih;,.

10;1IJ:...b. v,;li ~ :i.»

e.

~ ~ I ~ ~ 14....J:S" J"- it

• .iA.....................J.>.t.: .---J

6

J b. ~I ~. m4

U

•

, m:;

~ I;,...):..ill lii,~,

vi J,......aIll.. ..

Center-of-ma.8 OOOrd.1nates

J'b-. ~I~~~~~~ ltll ~l '~I ~#"'~I~~L...:.JIIS~ ~ L.Ji..)lJ IIS~ .c=lJ.i ~ • hSl.. .. ~ L.:JI ~ ~ I -....J.:S' FJ" ~ vA "c' v.iI~lvei· ~I..ob.JI~~~~~I.·· .. • "p~i

.•

• ';~I~IV:tt~1 ~i~J~I~t..}...::,>~

,;--.SJ' rlJi..:, IS~I

rr

r~1 ,,~rll~~ J.J.:;". ('\ - y) J1.:Ji~

'Nr~vl ~.J..;WI~I~lpo.:I4Ij ~1 J:.:~. ~I

~IZP".J..>e-~~I~Z~.J..>~...,::JI~!;1IJ:.: ¢2 ,u~1

1::SJ1 j

(i ...

J"

~ I) I l:lS ·.wW I

v IS' W, • IS~ 1r lIa.i.I ~ 4-li. I> 2 ,1>1

......._jf*',.:l; 'VJ41.:·.~l\ ~ ~J.... J,J 1J..:iJ I ~

WI.) ~ v I~ , 4l:SJ I jJ" rlli..; ~

• ~ ~ l..i.::. ~ ~~......:. v I ~ ,: v Ie ~ ~ ~ I j

"'J.

t

J" ~

v LA

vi

..s~lr~1

.. _._ _l:{)""", I ,;---S'J' t I

F

.!

..,-i J.JL-l i

J~-~l; I}.- 4l:SJ I J" r U=.; ~

~I

u lpo.: 1~ Ij • J,:..;

JUJ I r>jJl vA .. U:Sl1 j J"....-..~ ~ ~l:S ~v.il •• ~ r~L,.:)1 (r'\_Y)

•LJ.::SJ I'p~ r~

u-l l ~ J I',...J I J

~ I u I ~ lu. ~I\·j• .:...1 ..t.; ~ ;;.i l.IJ ,

-2 :PI

--;::----2m1 +

-2 :P2

2m2

=

-..2 Pl 2m

1

-~

+

P2

2m2

-2

----= 2/J. PI

( l 1 _ y)

+ Q

• r>;Jlo.::..U~ .. 1~4dJ.i, -.iUJI iJ ..do"'o V4

uj t,.: 4l.J L...,

'P 2'

,

U.,b. ;~I ~

P2

-,2

PI

2fl

( l" _

+ Q

'~....::l>-...... ;II ~erJl iJ~ ~ ( l·_ y) , (r,

-0

0.::.. ~

.JWI

to

y)

r>jHo.::..li~~

~l:l1

( lr _ y)

~~Ijyt4.s~ V

em

( l- _ Y)

=

trHo.::.- ~ ~;;)l,J'~ ..s.; -uleS; J!JJI~ vi. sin ~I = v{ sin

Q

Tty

( lA _V)

..s~ I

r&.;,II J erJ I~ ~ V:H ~.;)l,J I : ( Y• \:Sll ~~ rU:.:'

Y) Ji:..l1

~l:JI~1 ~v~~.:j1 ~~jJ....f ~1;,i)l.J1 vi ~ ~oJWl~

sin Q

tan ~l =

't +

( t1 _ Y)

oos Q

~ I.::J I;,i)l,J ~ ~-; ~ ..s oJ..uJ I ;:.",J I J:.:

=

To'

1.

JllVl (Jl1 + Un)

b

~

( ...- y)

...

I'l!rJ~_.~~ .... ~I~"jI;,iUJI~"j~~~ V{ ~",="L.:,.. ~;"j~ ~t-JI ~".ia,. (H _V)

~ b, PI

.:

iJoJW1..s1.

;"';lJ,JliJoJL..~.J.JWI

J • )l:..; • ~ I ~ ~j V:H ;,i)LJ I ~ I.w., = p{

v I;...;UJI iJoJL... V- ~. Q

t;;11 vl:J-.: 1 ov=_m m2

,(

=0

~ oJ ~ "j L.r.#~ I

• W,.,.J1 rl.::Jl roJL...:JI

f.1 - Y ) ~oJlA.J1 t'': ~I,,;"". VI

= V{ ,t

(.'_Y)

·vr-J I (oJ L.:.U

o ;;1 '"' -2' ~ u iJ""" ~ I ~ I; 0.-: l.l.; IS, I-; \$.~;"....lI !" Uiw 1,\ fUr.,;

.J u I,p.: ~ \ Lo ~ • v I IS I

,j 7T~ '.$,t-::u~l~ui~\L.",~~ts' U, .~ij-S.I"tl!2J

rL-_Jit.Ji J~I;L vi.:, ~.a.:.c.. • \ 1 - Y) JUJI-JVftfe>Jt U'. al::11ijJ"

IT,_

Q

2 i !•• " " ...,,)""'' . ".' '-;'Ji "., r~"·' 1 ·,l

q '.. , • :"-.1\ I , t~, 1< "~'?~' ~ ~ ~,.I ~ ~~ -r- lJ" •

Impull:Je

( ,'

L...........>·~i I

~

.r,..

.~ ~ ,:i!~ 'C

,'6

'J-'

.I"b._........;~.:f . J. or-

#

I -\ .. ' 1;' ~...."

u-.-_..,..--'

.

.. .:.1.11

4--

•

~

'

tf .~" 'r'~i~~~

l.:J':.1 W"' :,1 _:~ !$..J.,.JI ~J

J~ ..i1',,"\::i-!I~b4."*.:"JI_~i r4'li~ "~,,,'f:,'l ~~, ~JI>.""';J.Y-: ~ ~,l oJloJ,;':

• ~ I~ I ,

~..:..~.~

..u Jl~ C.t-fX;: L~...:..~::,

.,

)

...u. .j.:.. .r"' bJ ! ~ I •.;? J"'l'" ..;,,"" . . J b. t.~ I ,,).: '

Center of Mass of a Rigid Body

~~J:.JI o..'\:,~4~ (\_y

~ !..S X om

'Lximi m i

= L

' Y em

4

"'*' (

LYimi m i

= L

1) ~I}J- J.:j;ul~-

L

'

zO-lll'-

I.SI'r.~JI~ .)sj..I1:J~~I~J1~1iwS..::

X

om =

-c.;ox dv

Sv I' dv

_~ . ~I;-SJ-":"~oJW

~I'x

ds

X om= -~---

Sf) ;0 ds

Yem

'

X cm

h

'

zcm)

Zimi

--1.= mi

.

~ .J.:..~,

~J-~I ~ ·~WI;~~

;0 ,-.L.JI.,...~ de ~ .~I~'­

< =-

1r

L.:~v~ • ~J~

x

dl ~y

5..£ I'd!

om

= om

1ty

J1.' ~t...~.HvlS'

d.f ,Zell=~f'1i -:--JJl'd£

dt

-&.t dJ

L.1I ~ d) ,J,.JJI;..uo." U:S J:.:

• J,..

~.iJ,c, 'i6Ji~ A;b jY

6i~1

l.il, J't.:H...

J.""v,s;.

(f_A)

;0 • 1I&.J14~ cJ

4.:~I~' t~

• ~Wlc,'i"L..JIJS~ l,JJ.,.~ ,I.i--=l" 'i I J.:S plJ- ~ l5, pi, I ~ V- vA IS I ~J- ~ 4&1 .... u IS' I J~

- ~ l:S L.l:SJ I PJ" ~ V- ~ .,..; J .iJ ..... e:-~ I x·om

=

+ xzm2 + •••••

Xl~

c>J..,.,.... (LA)

m1 + m2 + •••••••

• I~,u.,. ~ '~I1..,J:$' j JI!

rJi. t:1I ~ t.-J u

_ • ':J ".....ID 1:.:11 IJ.A v. ; oJ li::... 'i 1~ I v-i ,;'1;.:. ~ -dla ~ IS' IJI

r·IS'

=:-

J.-Jl u IS' l.i 1• •• 11 1dJ.i

IS I ,

)it lclJ , - j

v IS' l.il •

.J ~ 1PJ" eic .iJ ~ • I. ,.:..J ~ lc

_1;~v~~~·

~ JI • ....J::s:U JfJ'

~

J:....... ~

)it1:.:l 11S":- ~ rr 1S,.:-J1~1~}6J .1.i.AUoA~ JJ,

"

~ ,/ ) L.- l zim1 + zi-i

ZCIll.

~

= -----:;;--=--~-=---

L. (mi

+

-1)

ry - ~,:._1I '" ei: ~_J.::~1.1 ;S''''u'' ~ '.ia, • ~ ... U::~)\ j J- V I~

Zcm

=0

v' J.r.J I v-J }a 1:.:1.1 J.u. ~ v ~ I.il • J: l.:llo ·~vl.AJJI4d~ ~ .~'ld.I.i ~

" ~ 1;.:) 1 ~

}

.;.. . "0

z _~ .(a 2_ z2

6

a

.~,.:-11

t..1l..h.A,..; Ul.:.. ;...; ~ ~rS ~ ~ j".

Lr-J- y

If' ~~l).ill ~ ~LJ.:{lljJ-~

~H ~ WI, t.J.:D t ~". v~

f

III

• ( \-

.s'-..o.J.e t~

."wu..-;, dz dv

..s ~~

11 ( a 2

....2

A ) JUJI ~ ~,..

lJ'..o b..

~~

yt.-J,

Zom

v I ~I • J!.UI ~ VI:fa"" lJ

) dz

fa jOrr z(a2 _ z2)dS =

E

om

0

J~rr(a2 ...

~

...

v.il ( __ A )

a

182 . .

z2)dz

o

•

z2

•

1--..:...-..;,."....... •

•

"-~--x

o

(' _A )

J!.-4J1

';,J u...= a...J.::S' ;,J;t y t.-J ..0 I...J I

z-~I

2 dQ ; (a 2 - z)

~i

dz

2 71 a dz

ds -

~

l.S -......l=sJ I j

J- ~ u~ ~,jj ~,

{l_A }

'. ;.." .:.

Z ::

a

JU '>~J~ Ll.:S F,p .,\~~ 1; oW u,.i...t • ( l_ A) j!.:J I J Y IS'.» b...

lJo).Ju...,.; ~;I"\u...:

,

a. sin Q

./

u"\'

,if

j I .t..

.,m =

o

~

i

..J

(a ~in w)e. dQ

o

a

d~

" ,2- a

Tf

(Y _ A )

n\

('f - A )

J.{-tJ I

lSy I", l.kN c:.ll.. 4...L:$ isJIt y W

;:~

- Jr-:-J I ~..Ji&

~....J..:~~)1

j

J- v~

,

~~I

~ 1.,) I", u..: ~..s1...

•

.J

-vi ~ ~. H1HJWdJj~;.a.; • (l _A J1.:J1) = ..• L

:.:; eLl

Sts:;;i~

a

37!

(A _A)

.u..~ Jr\:Wluj~1 (l_A EC.i.·

N = I ""'CJt

.....

.

.1li::~1

•

;I..,.JI

L ::: I w u, !;J I ,.=i-,;ll

P = my

J»JI~;JI

.11

.... aT

~,

= I~ ,. j

Ii

T= tIw2~pJlulJ,J1

!I!. jmv2

~I..,JJI ~1.iJ1~ U"~~ ~ t..iJ 1.o-iU

6

-ill

a,f~ I UlJ.) I

~lu.iI'Lo.c ..;1.il1~ltJo"i

l- ~ ~ I u,s: lJ' l.1.;.6

U

I.., .uJ ~I: .,... uI 14... ~ ~ ~li:;~1

.... ..;I.ill ,r-iI1 t .i--":- L... ( l - A Calculation of the Moment of Inertia

LmR 2 ..;1.ilI,r-iIlti-J~I~I.sL.JI" • L.1.::SJ IF". ~ l - ~ w....; lJ' t I.; 6 ~ I ~ J.1{:l I.s ~ I J I.s.c:- I ws:., -'l(4)U

• 4,L

I =

r

J

R

_~~~"I

2

dm

~I) ~h. ~ ~u~I~".. J.-b. u,L.:, 41:01~J:.; dJI .:..,.

u--~.)~luWI'w't'

R

ulj.i.:i ul,..,.JI~. (J,JJI,I,

.'-1'

• u~.ul ~ ~._l:SU."... ~l.~I.ilI~I~v-~ul ~~~I~U.lbJ • e-~I~ I

= II

+ I 2 + •••

(

'f' .... A)

" 1y

~I J~

)j

'Ifi. ~I~ ~1.Al1 ~I t';'-' cr•

e~ 1 2 • 11 ~ .J~I

r= IS JJ I ,,~,

• y l ~~11o:,.~bJl ~ ~1.Al1",....»1t'ja~~I~

~~

J'-' J~ I

=

f

m

....:1:S;

a .....J,J.~tJJ~ ~1.iJl""""lt~~'

- ~(

a ;x:2/' dx = ..

I'

(n

t- A JUlI) .......; . .1 ""'-'. .

U'·_"'~

2

a 3 . . . ma

o

•• -/'&

~~ ......~I~I~~

-..,JI 1+---

a

-a/2

(i_A)

J1..--U1

•

--.,;;' .bol J~ (1) ~ ~ I.AI I ..,.-ill t ';--y L..J ........1

j-S.,.JI ~ (y)

"1A

Boop or Cylindrical Shell e".J; ~ ~~I~~a.,;"J ••"~14rJ1l,1 ~..\II \,$;1..\11 u,..I.J14J1> ..j

. - v~ v oil ~I.ill..o--ill t}Ai

'..0-1 1v.

~:i

( f ~ _ A)

I • Dla2

Circular Diso or Cylinder

.j

:r'

..;~

~J'oaV'~~I.il'..o--iIlt;S~l-J~I...,b.Jlt~";

·)aiu..:~oaUb.~;J'+la~I"..,. m ~ a

.~i

"·fJ2 1f rdr

,;-sJ- v.e J-C ...... J,-.. ~'.ill ..o--iI 1t}Ai • ~ L.J I;

.

U.:S ~ j:J 4 ,

.,\Do ,

- ~ .is u ( • - A J1.t.l1 ) ~,::-J I ~ I

•

~

a

f•

I'

. (1'2)( 271" r dr) •

]I

= 2 7TjJ

""I' 7T ~ a 2

+ = ... 4

ma 2

LJ)l.J I v.

vI.s \,$ ~ ",,till ( f f _ A)

~;~~ I ;,JuJ ~I

·'IJ~I.aa.:~~cU~t~(ff_A)4JoaWlvl. e:-~Iv.e

.L.-...:~~ ,,;S'~'...,...JI,.....,....J~'

JI

~

a

u.)tJu...; ~I ;.1-.~J1

J>..-,... •

~ a

u."J-Ju...; ~ ; ~ ;)l ~I.ill ..o--iI' t.i-- ~

• L..4iJ1Jloa ,,~I ~I;)l''''' u,..

• u.j~ v. ~ ( z - ~I)

..0>"1

(._A) JS--:JI -I "J-,,'" .. :11

!II

-_:'1'

-1- ~

~ t~ tI"':'C

I

'I

~...,

.;I.&ll..,.-ill ,~~ (rr _A) ·&JtlWI~ ('\ - A) JS.:.11 ~ ~>A lJ' U1 • J2iZ~~~.·

I •

[/"

t lf f 4

V2

d. •

dDl JiA 7 ..~u-: 1S.iJ1 ~~I UlI"ilII.i..tJ

L:

v .il tTf/"{a2 _.2 )2 d •

= htr,oast n

_A)

•

.....,.JI ('\ _A )

JL-:JI

;Z _~l J,.. ;;I.;'J1t..,...i11,,;... tI~~ ~L...JI ~ •

,

u:g I ~ If U, .;~~ U,.wJ I c:.1::.:.. 1~ u.J I ult-

m • ....L .". a'l'

~

= ...1...

I

5

ma.

y.

2 Spherical Shell

<J

;.lH-f; .1:.:'. 4 } ; VJ;;".:.il ~l.,i.ll ...,....all t,j-Jo oJ ~ I oS-c

.1.1. ~ ~

~J..-.:

~ 7Tf? a 4

a ~I~~ I..w.-I; I.W· (ft-_A) ;.loJl••JI

da

•~u.-:, de.

• a I

I ""

f

ma

4S-;;r-'I "I.i.ll ...,....all t

j-$ ~; t'::+ II • .i.a,

~l:S ~ uoil 4-71 a~dacS,L..:;r-'l1 W:S~IS U,

2

IlJJ I ~. m ~ a 1...,J..;u.-: L.i,iJ ;;r-'I ~I.i.ll ..~...iU t j--AJ

..

• ~ltJl J.1S:l~; ~n •.iA - - ~ c,ji....:ll Perpendicular-axis Theorem

.~e-:J 'JUJI~l;.r.I~; iJ~~JU ~ f.u.- ~ului'jl

J,

.... ~I.i.ll ~I t j--ai • ( Y-

A

J1:J1)

'q

-

cS,::-J1

u~uoil

I z=

L

u ~.

i

mi

(x~

+ Yi)...

Y -...,...J I J,..

Iy

l:.i

mixi +

1:= 1

J ~I

z _ ..~•••H

yi

m1

2 ,,'.i.ll ..o-iI 1. t,j-Jo,.. Lm 1 i x1

~ . I c:A

rr~~ ~Y12 j:l.;1lo 'c:..~I~ l"a-cS,'-c s1 _~l:H,....oJ1 ~~WI~oJWI~~ uSecu.H· x - ...,...JI J,.. Ix ~1..iJ1 .JJ

..ill

I z = Ix + I y

(fY·_A)

tY\

.. ""~ •• ~~~ ~1.i.ll~lr~vl~l. ~~L....:JI",b.Jly.; ~ • .:u. u.---,.".,...

~ I J,.. ~l.ill ~ I '-I',j$o ~ ~,~ ~ ~ ~.s,.. ~

J,..

• ~~l~~vl~~l~,.-. .jv~~~L.:.

-~~-~f---X

z ( V_A) J1:J1 ~ ....... L....:J 1J1 b.J I.....).:

xy -

IS,- ... 11 .j l¥J V I~ l.."; ~~ • ~ I• .:u. r l..u..:.. ~ J I:.S' _ L:.w (ff _A) 4J-sWl ve (A _A -

Iz

=t

ma

2

= Ix

+ Iy

- ~J..-.; vl~v.il. }»~Ive Ix I _4

x

= Iy =i

_...J'-S W ~ • . j

J-

JS.AJI)

= Iy

ma 2

(fA _A )

ve ~

•

v 1u".... 4JbJl ...... ~

V""aJ I~,.:- .j .1J>ot" ~ 1J~ ~l.ill ~ I r.;-l

"..t~l

Parallel-axis Theorem

,j..l{:oIlve

~1lrl-l,o.:o..l~ (fA _ \")

I •

~ Jl.tl~

7~)

+

b81:J.l-+~IrI ... IW'j~ 71 ,xi

;

~~I:'jl ~

7

• a '''''I ;.SJ-·J'~lc ( xi' '1' .1 ) ~lrl ... 'j~ ( %011,7 am' Scm

- Jc I.S' (, _A JUJI)

_. ~ - xCJIl + %1

)

(t'_A)

71 • 1 em + 11

~J,.-.J ~.-JI~ ~I ~

I • L1

II.t (i~ + if)+

+ 2:1:..

L1 -1 (X~_ + 7:'>

~~i" + 2T.. ~

( i-_A)

IIj.11

r.-. 55 J~'" J~ -;1.il1"".-iI1 t;-'''' ~I u-J, 'jl~1 ,.-.J11.-.bf",L..c .;bJl~lul e-~IUr"· I cm ......... ~Ij". ~

-i

a.L..JI. JA r4

- • -.,..J ~ f..-l:n I p". ~ 4i t-J I eve ~ 4...0~ .J, 2 • '$~ +".2 -em "om

u lIS I

;~~I;~~ v~~

( (o_A)ij..aL.J1v.-c c.lvl"...~lv . . .,.

- ~I:II,•••JI ~~ ( .. , _ A)

2

= 1 0m

I

r

Mt"e •

~JI

+ m.t

..tl-~ISI J.. ~ ~ ~~ ° au!,:JI...,I..JIy.; ~ • .u.

J~ ~I~ ~1.il1~lt.;--vlJ..c:i~lc.I~lcA

°

a.+-,I

r~lj ~1jS'.,.~~ 4-l j!,.~ J~ ~1.il1~lt";'-'-IS,L.c...... "I • ~ I VlK iJL.J I eu-

.J ~ I t...J.:r '-:'~ J.tl..

t-'..aWI ~ J..--; • 1S;'oiJl ue~1 ul-UcWIy..J1 ~ .a.IIj

V

~«(,

_A) J (rr_A)

v- ~ ue~ I d- J.. IS ~ ~ J,.,.. ,.w.. IS; l..aue~ ~1.il1"""'" t.;.-...J v 1 ~ ( ( \ - A) , ( r, - A) v.:J..a L.J I v. • 4C1J.i .) 1IoU1° ~.. I

= ima 2

+ ma 2 =

-2... ma 2 4

( U' _A)

•....:;bJ U" I.e, ue~ I IS'::- c.I c:i~ ...,..,. J,. .;I.itI....' t .......

•

RadiuB of Gyration ..a..-.......I~ ..I\

~ ~I.ill ..,-ill t ~ 0- ~ 1•

4-J..a W lc U J""C I

= mk2

k

=

k

c,; Ij ~ 1~ • t')l.h.:.-

~ ',..0 .a.:l1.,J.J~ ..,.: .;J ~ .t UI-JUJ1...

·-LJWI ~I(ll-AJ

I -;

~ I ~ ~,..

~.a.:I'.,w~

J,.. ~..a ~ f""'O ~1"J.Ju...: c) I • J ItJI J.- ~ -)A~ ro_A) 6J.. L..lJt~"]~

k

=

a

-1f

TYl

eLl.i, J,~.Jt-- .).~~ ,4 ~ ~1.iJ I ~ I, J}-So~r ~ ( '_A) JJ~I J~ I.S ~ t"""'O~IJlJ.Jh..iL.;Ieu- c:.J~ (, _A) ";JJJ~

.,:"~I~

,1 (

k

2

x • .L:~J1.

k

2

~

~ 1.iJ1,,o_ill , }-So ) ....JI

t

•j--JJI' V- H 1·:·t.-iU .). U ~ ~",J. 0»0 I V-~ ~ I~ u ~ L~t • .w -v,... .:J ~ •

L.w.... t:~~I b J

u-"ill u':;- J ei~ ,is'~ I V- ~ u-,,__ill

~

1iJ.""J 1u,:;-

u""" ,is'

u...; c.hiJ u'; l.su-";

~, V- .J-C

a

J ei~J ,is'~ 1V- .J-C

I •

, • .J.. ';'u""" j~1 V-~

.;,.u 1 LA~

uJ~'_~5"J 1

a

W

'J

.

..' " " " "J

a

..al..

LA"J..;

u...; ~J ~ ~ I.,r..b b

lrJ,J.,

aL.)..i

t

u'p~1 J,JJI L.~

~livl"" ..:J,J.-.I

LA~.. ~ \oS ..l".." LA~.J- yo.. ~

L-.."wu.,.; • ~

o: 4..~ ~J~ u..iJ I ..J..,J I ~ I

• Compound Pendulum

V,

b

J ~ IJ,.a.411 (-_ A

The Physical Pendulum. LJ In....u ~l:

a

J?' (

QJ(

V:t< ~L.J~ °

0

~ p".

J..aJ I VlH ~ I;) I ~

IJ' j....;)

~J:.:J I ~,all r .J-- J IoU.

~L.~lo..-JI41""L-

~Yi

..

mgt

~

I

+

sin

Q =

0

A

( \._A}J1:J1 ~~IJ,......,;.JI

~ U bJ I 'a"" • 4~ ,-=-~~.i.I

t.s; • J1.1J ~ ~ I J, .1.••••11 iJ ""LA. J: L:: iJ "" W I • .i.a Q

~ljJ~ sin Q V6'~L..:.. "il

t:..

~ • .J.,....II

J,.a.lJ' \II

mg) Q

+

Q

I

(n _A)

=0

( lV_A) u~ f

f

=

1 211"

~ .i.: j,:) I

/~O

.J..I;:' •

...,J.J I ~ Ij E , " - ' I 'a"" e0

~

~yy

T

= - f1 = 27f,(;f; mg j,

.s.s.,.....:J 1~ ~ I.;.) J....:-; ~ u,..

•

~ ~ \S .u' .d l.... ~, .J)l:J)

".......J.J U-:4J~..l.1v-...,.-JI".,J ~. ~~.illv-j~ ''p~I~ \S, 1,;.11

r-->Y ~ \,..UA;; 4. ~ ...,;..: ( ,

- A

A General Theorem Concerning Angular Momentum Jj-.I>wt

~uA .~I\SI. ~I ~I-S,.J ~?'JI;JbJl;""'I)~ ~~

oi.-4J1 LI

~ ~1) •\S, I;J I r>,;.U ~ L:-.I ....)a.; .J. ~ I ~ 1 c;.ba..; • ~ b ~ u IJ' ..J 1 •• 1\ rrJ1 ~,~ ~ ~~ ~,lyl r>,;.U ~yl ~I J ~ u 1(~_ y) \II

.~

e-~'-'; ~,I

-£:e

L(rix~vi)" i

L{r 1 x-;1) 1

(--_A.)

• ; .;,:;.; "" ~ ~ I ,,~I •..oS'.u I ~ ~ L.J I~ I,J' ~~ ~

r1

,.

':"

~ ~ I ~ V- ~ It cdJ .;, ~ I'j". ~~ I ~.A.tJ

..... ; ---..liil.l=sJ I j". ~ I ~ It 1 ~ I ~ ~ rom

;..J=sJ I j". ~

~ .(t'_ Y

41J

~~ ~ .:::;.:

IJ') r 1

,

...&om .. Yom ....:....

~~I ve a-,1:.;,a.: I ;...L:S j"J ~li::;~Ias'pJI~ I( ,- y) 4.1J

u.-t)

~~L.JJe-:

i: F1

;:1

1

•1

f

L 1

W

m~

..

~

(-A._A)

ve J, ~I .»..lIe- (. Y- A );j~L.J1 JL..cv. J, ~I .»..II ~ cdJ.iJ ~~~t.,..oJl~~

'G..

f

co..

~

-t. ~

tt.

r- bt·

.

f'

.,

,-~

no

8. rv

a= .... §

J ~ ~ 1

C' F

L'"

~."

\-

L

"L'


'b (; [

~ v" t ~,f ~

-tl I ~ '- -~

_

_

••

:-

{,.,£

L

Ilk

C

•

(0

_

't..

t·~

.:

".L ••

•

-" ~ 'Iii 'i

t( Ci.

- -.

-

)-o.L

-

tl.

l.

>

~-

t· 't

-

..,.. ._.c.

It

.

>-

l.- __- 'f ...

~.'-

[.t. \

~ q

't:::

E" - ~

j

t:

. ! : },

t.

\-

L

'J,

t ' 1. ' ~.c: r. '( : i' r -; i-. r ~ ~ i' f { f' = '[t '[t ~- = ::I"r:t:-h ~.~ i . · 'i t. t. r t - ~ i_ .....

•

L •_ ~ u~~pJl

jJ.

ori t =

tan

Q

2

( YA - A )

1 + (a/kcm ) • c.;'~;: I u' ~ ~ I c::~ ~ ~ ~ • .. ~I..oS'.u IJ1.uJ Ive.,.ll

u""'!:u l ~e,;~j;lu,~ c::~,J:-0 to ~ 2/7

,t

d-- ~

tan 45°/(1+

P.

c.::.; I!

4~~ IJI.)I.:..;

+)

v-.,.II jJ

wIJ;,.~t::~ k~~~ Motion of a Rigid Body Under an Iapaleive ~I

I.i I

('_A

Force

Impulsive ForoeiAil.J1 ;,.iI1 rnJ. 1o.Li.JL~WI J..-iJ1 ~

J .~li. ~~I..bol,... • ~.JI ,I. i,.ilI .. .iA ~t:: u 11:~ • ~.).r. ;.'; ~l&.JI;SpJUjbJ~.J'rnJ.J~ u>""

4

11 .iA".,. ~14s".. .~k~ ;~I4-SpJl

Free Motion ~

J:. ei J~.J..L. .s.;,.:- J

• .L.....J ~ • P

4S',.J I ~ ~ u I vi' j I

..t..-4I1 J~ ~I"WI~ ~ &r1"".J~ ;.,Jli::~I4S'pJl

Ve

JS' .i>1 ~

c..H:)J .. WI"'jL,JIc~;.,Jli::~I~pJl.~,I.( V_A) ~

Jr

a

.:..a mv

~

.~

.).r. J..-.;,; • ;"".; IJ ~ ~ F dt ... P ... m A V em

S,,-a.

~

P

om =m

~v'

"JloIJ.~I;S'J-u"..

F

i,AJ1 W ~ I! I.i I

J ~ei.Jl~

~u·H

( Y~ _A)

4.Jl:.Il4I..It-!1 ~(-..~ ~--.Il ., J ' I-I: . .r--r-:-v~ 'It"

H.Y

~l:JImL,J1 ~J~ t

~;1J~l~JeI!:JI ~

S

N dt = I om 6w

.11 ~J u)lJ

-•

..

6. w PJ-t:.1"

L

(A-_A)

je} 1t..u..:..MJ, • ~ I..., .u 1e:i.u ~

J N dtJeI!:J 1~,

JI.s.iJ~ ~1jJ1 ~I~.I"" ~ ~ • .;.,~ ~I...,.ul~.ul ~'=

=t=

(A\_A)

em

b.AAc~ •.,.:t:.Ju,.v~~~I~ lill..iJ

iii"

:t ~"~le:i.ul~ l.i~

N=Fb \S,~ ';"~r;-'.ll;

• ~l

(A \' _A)

Jt. U'e:i.u 1~ ~W I ~ 1,;J I~,.,..l' ~ ~ I v$ ~ I .;., ~ ~

r A

dw=

(Ar _A)

om

~1..L.->1

( \) ~ ~li...-.s'p" ~ ~pJl'p" .al- ~ ~e:i.ul..wl:- -"-)LU~

~ '~li. ~~I.»-l ('f) ,.}\i:=~I.,.:l:JI_U::~)\,PJ-~.I"" ~ '~li. ~

• ~ "'" .u I ~ l:J t_ ~ I,;) t ~ 16s.l"" ~~I~pJl

Constrained Motion

.---.1 c-. t ~ J,......,~ ~~, ~pJl'p" ~~ ~~~J-1... l.il voil.N=Iw J,.i;~I...,.uI..J.rJIJ~~I~~1;vl ~b..;

SN dt ... i ,-

.WI J~I

= I A"" J,... 'wf1..iJI...,-iI1 r';-'- J:.c

I • .~I ~...,s-.wl ~~WI ~

~I...,.ul~.ul. iJ6JI • ..a.- J, . ~Ilill.i J,... t.r-JI~ ...,

.... vo b

N,. v l...,.L1J uL-u1s.~l:~~ P ~;.:...},I~~v$ ~\S..iJ1 L

fAA

...

...

L = Pb

(Al_A)

• ~e.l'vl...,.~JI~ JJ> ~1)J1"rJl...,.~, ~ ~lt';.) ;.a.,n:l::

Effeot

Several Simultaneous Impulses

o~

~~I41.-.i.~~..::...;~ ~~~~~.);.a.~I..il

\;l::JI~~ I.S r'~~tt.sJI~.> rr~ ~ljJl"rJ~ ~'j.}'t~J'"" ,-,.),rJJ' ~~~I~'.>t';JJIv. .).J.AJ JU:;~lwl;J.1 ~J..-..:' veil :..... ~ ~ ( y,\ - A) 11 .) t,••J1 v' '-' I • t..J -" 6.v

om

Pl + P 2 + •••

=

~J..-; ...

A

~W=

J-.-i t

(Ao _A)

Dl.

(Ar _A)lI.)~I~~ ...

.~I...,JJlwl:1J .J:I.:lL,

Plb l + P 2b 2 + ••• I

(A1

om

-

A)

,JJ~IlU""1;~t~.) ~~t~J~;~.......:S.P"~lI""...,.

,,?,~.>~pJl~~.t.i.s. ~Jl>t'.).J.J... I. u=- ~I ~...,...JI

• .~I_.o!.ul~ ~.)L..U lii, t';..&J1

.. tapalss Applied to a PTe_Rod

.,.--Jil d-~~JJPc ~vl~;I •• )laI;.s.u,~ ~

jJ".~>

b

1il.o. ~~I J~ ul-,-,~.~ls

P e;i.)JJ..

• ( "_A) JuJl JVttte~

~ • c"J.~ LS • ....:J:S ,

.

.,

1t

,---~

( H _A)

Ji-tJ1

.~ .....:S,. ~ ~J.J.-

r;-iol

... ; _---r.ni::'J
~"J1 w~u~ ~~ .lo$L.~I~I.J~lulS' l.,il

.. . .. P

~

T

- .} l::J I ~ &,: 1.0 JJ ~

~

.-

.

•." Fe ....... I\

1_--.2;3

J.I.a.a.. 2& ",L.c...J,J.,.~ ~lulS' l.,il~l.>;~ - c=UJJ

w. P~

ti"

(A'_A)

11&

e elotJ1 ...b ~ .,. ~ ~ ~ ..,. ~ I;S'.,J ~ ~ I_.,..J 1u,s; dlJJ, 'L.icI"neJ.l-J1ei.tJ1 ~~I ~ ~1"1jI1a..,..J1 ~ ~ '2

"2

."~~YJI.,s'~IUlJ.J1·1 v· "

, • pv:' + tI_",,2 • Ii + if ( : )2

e e.tJ1 WJ.J...,~liWl~ ~ I~ul e-~

n· ~t ...... J~",,~

¥

~~.J..L..e-s_'f

Iapulse Applied to a Rod Constrained to Rotate about & J1ixed Axis ~ .... ~"".a,J r¥...a:~1 ~u~~I4JbJl~.i ~c,;;.J

JS

.aJlytu-", U'~I.J). ...1.... 0 J~•••JI~u'c,;jl.~t

JI:J~I ~~ar""'oaJla!..,.J1 aJ-slA.

...

:r. • p( &

+ 'It) •

Ie w

u' .a-.-: aJl.Jl.J.a yt • ( U'

_A)

( \ •_ A )

1'"

..J-J...-w. 1 0 •

i. v..'.,,,....,:

ma2 oJ ~.l

u Lf W• .i:.w

(" - Al

j

a+'It ...

2&

r" .......--~ J1-tJ1 • .,;)....,J,... "".a,J ¥ ~ ..J-.J..L.. n ••n .... t-i oaJl ~ -s),.. (\"'_A)

• 0 aJiJl J~ ""oic~luLf U T

-

.. &"""

-t e-s 1;.

.;~~ • ~I ~ ~I~I;II ..".JJ .)I.a.i. ~ L. lidA U::Sl 11".; ,I

,."

~~. ~I..,.,.( Y'\

a

cd'

_AJ41.Jl.".H.

~li:;~I~pJJ

.. WI

I.i...-A, • uJ,I~.JS4..i:~I.j~1 ~~,,>i~.J ~,..;~ .~l:.: r

.·~·~I~ 0

.j~I. L..,,,,,u.iJIJ-iJI.JJ~.J,..~CJI~..J1

r!..~••H,.. ~ I ~ JJ_H '1111\ e;i..J' v~ ~ ~. ~ - 'a"" •..J.:~)IFJ- ~ ~lur1lv~dJ.iJ

~ V

cm

=

f+~

v.iI.('\l_A)

Po - P

V,

5. J...iH

~

~

P + Po

'a""~~1 ('\\"_A)

Dl

1J.JWI~ ~ voa ~Jr....:..~ ~ .t'o ~

~ 3(a + b) _ P ~ma ~... _~

u.".

,;... J-iJ I .JJe;i.J

-

~c.;.-t-.v~1 ~

~

P + Po

m

3b-a

4a

.J..., J....Hv": li ve • -'::oiJ I ~ I ve ~ I ~ IS .iH •

0

~ ~le.uI,n:L: 4.J..i:

e..\ll

~

.j ~I ~~I. L..'-, 1S.iJ1e;i.uI- Po

c.#J l.il ~)l; J-iJ1 tJJetJ v I ;tr.l. ~

, ~11JL.. ~ . . .1ier of percUB810nt.wlIFJ-':~I.. ;... ~. ~ b =

aI.,

IJ'

~ w..i:J I.. ;... v,f:; .. ~I ~.u I

OeJ.J.181eas of Rigid Bodies

.. ..u...J1 t 4 ~I ttJL.= ( , ._ A

t4~1 ~;~ ~I u,all • ;;~ ·.al.-r4IttJL.=~ c;J1 JH...JI '"

v---cJ ~ c-: v .i I • ...S'l..:., ~ L..:o. L....,Jl v loASl-r- juJ ~

~ ItJ v~

UII4l)l:lI Ill:: I ~ ~ ~

~ I..., ..J Ie;i.u 1..,.,.,.. vi. IS, I;J ~ ~ I....;J 1.Ji.i.,.

• JSI....J1 .. .u. ~.J

J ... ! ---...., ~;;-St"W

.m.. ..

+~ 2~....J,J. r&.:..~

I.i: U ~

m~.} rolL.: 8)1:.8

c,;JI

d-.)s L:.Sl.. vl!~IJ.JI.J~1 vi c,;A 1I .JV:He'" lJ'~IjS'J-ve b ;';L..~rolL.:lI

J.:Wl!. v-1e 1~I

• ( \ (- A )

J-5

,t.

W~~rolWI ~~I~P'v~ (AA _A), (AY_A)

vrJolL.Jlvl

.~I...J-:;I.S.ilI~..Jlul~I,.L-; ••~lve~I~",jJlt

r,:ioiJl

"-

-4.

~

lJ'i..:J li:; ~ I ~pJ II.:- ~ ol l.-. ~ l:S' ~ dJ jJ. .p

~

= mv-~cll

P

,.. ~.alI ~ (

1--1.

' ._A

)

CJ!2a

v i m'" ~-.o_~ _ _

"':---""',,;.~; ~ rol L..:: ~

;;"..!U ~ lot.; ~ I U.,..J I v 0

• rol W

I ~ ~ U.1,'$ jS'J- ~J'""

J:.:

~

T_

.:..,.

~

mvo

,.......

= mV1

~

+ mVem

,-->;!IJ:.». ;;.1Sl;rl~1 ~ 'r.,)L.:JI~~'v'''''.,) ,,:,L.... ~~ IS ,I p' r>;!~ bmv0

,.,.

~I jS'". J,=o-;;~ ~I~~' IS";!' ,>;!' vi. IS";! 1

1S,1jJ1 ,.->-;!~ I~ IS'~~ ~I~~I\S,I;!I ,>;J~ •

bmV1

v .i I. I em

,.,.

~YJI

\S, ~ ~

l.r'J,

('\A _A)

..

J;15~L,.l.-

W I.:J-.:~ ,~, ~...,s-.iJ1 ~I...,..J~ ~Li::;~1

~.,)L..

vi

":' L.... ~)I; • vJ • Y em • Y 1 \S I • ~ l.r'J I ;s'p JJ erJ 1~ I;:../:f .,) ~~ .. l.,)lA. .. .iA v,s:: ..u •\S".>I4J.,)lA. ~, c;.~' iJ.1!;;~~LrJI~pJl • WUJlvj~ ('\'\_A)

J-lA. 4J .,) L.. t I ~ I Lw.S..e • \S".>,;,a..)a.o ·;;,;UJ I ~ ;;J WJ I J:..: •

Q

~

.,)I~J~I

€. • ":'I".....::J~I ~I .,)~~I~I

• ,,:,I?~I ~1\S,l-:

,.a....,•• u-e)l:JI..w: ~~I ~,;.;~ ;

Yo

1:..t4. ~1~4Jt..J1 ~

c;.~' .~~I ~I .,)~~

w·lldl:l ~I...,..JI ~~~ a:sJ1jS'~ JLi:;~1 ~~I~ve~

•v em +b

W

,..

t'.,) L.:J I ~;;".£ L..

CJ"")l:J I a..a: ~ Iv .i I • jS'~ ~ 4

:; t

~_

lo.

~.

rt'

lID

t[:.

rf

t· [0 ..

.. It

I

~

.!,

• -

'r..

r

1;-

~

~ ~

,

c.

1_ --

~,

t.

\T"

t

[~

- ,.. . ,-:"...'J ~t ,. ~ ~ . : f l. 1=£

~

•

_ ~

'"

+

_

,-."

~:

~

~

1

[ u-

kl'..

a

~"

;

_

f'"

lID

_

+

"

'tlo.

lD

'i:' (

'G..

. . t ~,r.- c·-;

I=.. (' ... tl.." ~ lo. E

..

t.

,.

t..:

--e..

A r-.

~.

~; {' .£, ..

~.~

1

l'-

'$: {l: 1 l" -

~ l: t-

I

- ...

I

$

I

0

U t4

~

I

t

"t.

~. L

r

.f· .,£ -

c.•

0

..

+

t I ~

I

~-e

t1\

• o~ ,,.,

$

"'l + ~ -~ t

....

'" ~

+

·,1- ,

+ ...

(t\

_

It

If

i + IID~.J_;:::~'" '1gO --1"1

-

H

I~ .~

I

t ~~ o~

n

a-e eM•

1..

.-

\S

~

1" f·

~

f.

'"

I-

.y

I-

.y

_

c.

I-

c..y

,.).

c.

'-.

r.-c..

lr.,£

~

L.

... ~. r -r.- 'J r ; (. f' -f. .. t- ~ - CiA ~ t· ~. f [ f ~ i ~ ~ 1 t .1-' ~ ~'. ~ f;; C;,. - ~. f ~ ~ ~ ~ ~:: r f c;... \. ~ ... c-. t [ ~'. c:. c:. !: &i. • • -' I t • • • • > I r ~ r i 1. ~ =j > > > > >

=• .c.~

~ ~

!~

1f l'S 't r. - '" 1: c- -

-L ...

\T"

~. ~ t .~ ~~. ~ { r , ~ t \ '~"'" ~ ~ JI: - H HI\) t:" It \;": ~.. 'b \- o'b lID f z.1\). I=.. t: ~,~ u'I...... '-" 1\)" ~·

r. ~ "'\~~ :'\" CO'

I

~

(oS

~ ~ l' 'I' ~ 'C: ~

t

-"l=: ]...~

,...

-4 -I>

~

l

l\:)

'i-:- t.

't-

I»

-.

t

t·

b

t -

Y I:.:

c;:' "'.

>-

I

-e

..-

.

r

~ '-"

_~

L

I

~-:

't!.

>-

I

..",.

•

l c· c.... -t. ,__f,'

--

•

'-, 0'

+

I\)

1-

c:

r

t=-

r%

'F' -oft.e -~") (,

rf.

0

-t

F ,,"'

~1 1:.."'-

~

,.

'G.. -

.,~.

6'

€>

t~(~

C.

[

~)

r\.

tt

i-:- tc.

\-

l- ~- ~, r. 'l~)~fJ~r

r~'

'l..

et

r£

......,

>-

• I

--

N.

IlII\t

-

I:.:

t.:..

r

cr-

t:.

~ ~

'-.

Co

'i

l

l-'

"'d

I\)"'d

..

~

• c.

jto"

't".

i

~ >I

•

(f .... ---

•

>-

I

>-

t::

~

t

b

-

-

•

I

,.

~

•

f

~

c.

- ~. ~ ~-f'} ~ '-- t t. -; t- : crr of' ~ "- c· t·;;' (0 ~ f! • - t 'i.:. c. ~ )" _,-~ ~~al= ..+ f__ .-t.'r:_ t'· '-I : i'!:.- it!"'~{~'-~[" .rrl,\r" ~ l __ r

~., ~

'fA.

~

\- t:

-I\)

0

~rC\.- ~~Jtt ~~~J-:: 1- ~., • t- f." r ,t'.. .;; '-. '.:. : ., .r. t ·t · C\.- r {_ 'f." to . ... ~. ~ t ~ t r.c· c· .. l -~ t~, 1• eft.c· c· cr ., 'G.. 'C t t i: r\:" tg1t-E;:\ ·t~~t ~ ~ .I: l i' ~ 1 \c· 'G.. '"

~-:''i-:-~

I:..

'i

_Ill

I\)

~

e.-

.... \Jr" 1.

c: '

C

,.

""

1-

r

0

1 · l'

~l\:) It'':'

I»N

-l'-IS ..-..

~... " •. ~ ~Iif • Y •

-\

~

•

'-~

~-...-.. ~1 I» 1 c·

1

~

c....

to._

t

,",'

'lot~·

~

't~·~t

-

~

~

t'

". I:.:

rI':

L

F tt ~ ~ ..r 'b t. i 't.

1; ti 'E·' ~ ~ f:'\

•

f;l

f r _ i:f

s

~ k t:,~

t

\J1~N I»

· 1;-

\-

-

tr

S' {

so

,.

j-:

~. l

.£

Jo'



-.....-..",

~. ~ l.

.....f.tT

,.. .,...

>.

~• 'to

b. ~.

~!=~ ~,y k'

b

'"

,r \'L ~ fi-

.~,

'r.

t'

. . \- r b

•

1- c·

,:,...

·c.c

'L

.s [n~ .t

r ~

I-'

I:l

~

rva

- ~. -

~

C\

c,

'L

t

~~,

-

:t

' '\..

~."

-

C.

l'U

F~~~:r f. l:l ~. f

c·

ti

t.

r.--' '"-~".

~

'Ge.

f ~; ~'f." f ~ t-

~', 1.

:r f t ! t 'f.f~~~ f \!}{ {

·r

Co·

C\

~l

-~.~

~ .r,

s,

k

t...

~ t:-

-

t t~ ~

III

,",.~

'r..~'

·L.

t·

0

i >

..

-

'.

e..-

'&~...,

>

I~

..",

Co..,

.~.

~tt

Ed

~ .~ -,., I:.: .r' ~ . 'G-,I~" a. ;:

t-

I:.:

o

1

I

--"

-.

.~

-

L

&'( ~-

"...: ""', " . . '-"\ lCo --. . 'L

C"

1'"

l.

,.

Co

'too

~.

~

e..-

~

.t

;'u ~4w~I~I

",I .....).i. ,j>..,; 1..u:1.:.-1.. ~

-';)l,.J I .J:. ~

W

-"

(ri x mvi

)

,...->jJJ ~ ~ ~ I "I,....:>J I J. • J--A.l I 1..i....I. J • L..o. \.:;.A I r-Ji' u,..- , • L;,..:'J,oJoll W~ 4.,:-L- ~I W ... Wl:.......:J~, IS,I;-JI -a.

dL .N = -

~

dt

,,--...,~) ~ ~ I ...;,,r--W I ~~ ..t-i, • J..I ...11 r,;--J I J..-:.:: • ~liJI~I .~l:;.l:.i.; J,.-~~ ~r

J

N

~

~L.JIW,JWI

"=:elIS,~Ir->jJl ~,I..

....'

r· .

a

_J:J.: v,J;~'

J

~

J' L-.. ~

• ( ,_'\ Jf-.:J') ,"r'tli~U~ ~

~.r-_J1v'~ (

t

L...IbJ' • .L.A

~~I~I j

)

J

J

~

0 ~,

~'J'~)l.:,

~\:.ll "",4-;~I ~raJl.: ~~, ~ ~I ~1.4

.»"'----'

~~

~

J

vi I.j':'..t"':~"'r-~

: (\ - , ) ~,

W ~,';JI"'r-Il~ u~ ~~v,:>' ~[~ x(~x ~)] ~)WI ~~~I,;-,tAU x~._ ~J-' v'¥l

f~

x (wx-rt>]

"",~~~.r'Ji

=e.vx(7~ + S~)-Wtt17i-W!ixiZi

x

oIoL...Ae

~ 4Je iJ~ I

f

(T _'\ )

.~4Je ~I ~,

v'

• ( u-.J.::n.iA ~ ~Ll:J' ~, ) v..tlv,....s;\S,!;lI,.>,;lJ x 1...1~

J-~ ~I cJ~6 ~..w..o~ IS,I}!I r->}!I~L-J

: WbJI..:..I;l..a:::>. ~I J>~, • ~L.Jls'

6

~I ~

x- ~ I cJ,.:'" .; I ..u I ~ I t)--o$

I:x:x =

I

Y7

=

I ZZ =

L

(yi +

2" (Zi + xi)mi L (Xi + yf>mi

Iyx = I:q

=- L

I EX

= I xz

= -

I zy

= I yz

=

; b.

Zi>~

f

=

=

=

(y2 + Z2)dIn

S

(Z2

r

(:1:

xiYiJli = -

2' xi l5i Jli

-L ziYiJli

J'-~I J,.:'" .;1..iJi J,.-LJI

= -

= -

+ X2 )dIn Z- ~ I

2

tr

J,.:'" .; I..~.lI J,..-aJ I t)--o$

... y2)dIn

.;UJI ~ xy- ~"..-..P

J

xy dIn .;UJIJ~

S

xz- ~yP

xz dIn

f

zy dIll

~I..:..~L>JI~ ..l~~I..ul..:..IJ,..-i.Ilt~ '-4--vl~

~l-....>;

i;"h.:

~UJ",: . IJ,..-i.I'~yP

J.r..

~,

• u..:L.J1 J.-iII J

: .;~ls' IS,I}!I ,.>}!I v&- ~I ~ll.:L.J1 j,.)' t Ia>w.-~, • L..J.:l..

'" ZX

J~ .~I v-Li J WI.s v~ "I L IS,IrJlr->.;Jl ~ vl~, . W ~,I.;JI4sr-l1 ~,l vl~..L...l1 >.;JI ~.

t

z- ~I Jr->.»~ ~I ~~I r--'t- 1

.L Js. J..=...:

~~ w

~.

s

=W

, W • W

x

~

L = 1Ixz w

=0 LJ WI • ..i.A

Y

IS,\;-JI

.,j

"Ii"

v Is" W

~

+ ;)1115 W

+ kIzzw

IS'~ " Iyz,I I xz .;1..iJ1),.-iJ1 ~r--PV' v! vIS' l.il l.::-- ", Ii I, w JI. ~.J~ YJt L IS,I.;JI,...->;JJ.J...-:::-;- ~~. \)....>

• v IJ, ..L...l1 ),..-e • 4 1v-Li

W

~I.:~",I; u.~J'~ ~

II

J

IS, I.;J \ ,...-» I v~ "

~,J.-

-J,1 ~J~

->0

~. ~I ~

0(

~~. ( y - \ ) jS:....:J1

J

•

L

~",I; ~ , ;-S~\ V4 ~;,.-. J~ ~ ~ lJ' ~I

J.~

2 I

xx

=I

yy

=

mJ.

12

IS,L....= IS;-> ~I ~1..iJI~IJ,...-iJi

U; ~L...1""

J ~.»~ Jl::;...;

~~, ~1.iJ1

),.-iJ1 rjj-t-~,

YZ-IS,::.-JIJ~ VlJ,..JI~ vb'W,· ~

J.:. lJ' v,..x wx

=0

w

=w

sin oc

=

cos ex.

U)

Y

z

W

_l

u- .r-:. ~ L ~ Jp

u 1.».1..J I ~ ..t..-.L- ~ • ( y - 'l )

J!-:J I • ,;-5...,1 I

e-- .» ~, (, JL.:J I J V - ';-.;:- oS 'j

~.w ex ..

....

9UO

~

rx nv

L::--: ~ L ~ u .i 1 ----.. u I V- cJ.....i>.:l1 J;--J I v-) • W J,.:--

cJ::ofe

~ 1J',

J

y

~I

J,.....1 ~U~ '~I ---....

oSl, 4~'j1~ JUI~'>-:: w

• ~

4

..:-:lS' l..il (, 4.-l>.:;~, • (

• y- ~14~1

..

y

.~ u:.

>}.\ ~1..i,J1~I),....-.ilII.;-'r?,~1.iJ1 ~1~lr'}-S- ~

LrL-1

L.J,..i; ~",6..J b,a

• ( l" _, )

~w...., m

J.L..:..l I J

-

(, L ;\-:;". V.......--.. ·~6.;L ,.,1;-< dU ..i, . . . "'" J.

cJ::ofe ~

~

_l"

Jt-...:JI ~Io· ..

IS ~ I

~~)

.L-> I

,... £

y

b

x jS:.:.J1.il.:b-:-- • ....~ ( \" - '\-.) ~.

L:..w.

z =

u I~

0

~L..II J...i.II ~l;; V4

ulS' L..J, · ~~L..::JI ""L...JI L..")'; V4~ ~.r.:--->~I ~.)WI z ~ u,.....::>.::

cjJ I ~ 1..i.Ilc I~ I

~"..a;..

",, ;.. I\,J.li: ~ . UI~1 ~)l;;

V4 ~ 1..i.I1...,..-iU Xl '-:'~ ~ ~.

I:xy

=-

f

:xy dm

=-

Jb Ja 4)

0

:xy J"d x dy

= _j>

I,r.:->I,

2 2

a :

£-

u. • Cc4H Cof

H +

~~

H

.:.

~

~

'

I:tI

'-

.t

i\)

•• t,

0

It~

.

t

C.

.r

b

c,

•

~

-.\t

C'\

b c-

ft

o~

l'"

rio

~

~

&-~

r' • ~ ~ ~

t-I

r t t .:'

~ i:.. [.

· . b i €of" .t ,. ~ 1; [ f ~ r :1 \; !::.. f.' 'L rF fr 1-r i ~'~'~. t:. t: ~' r: ~ f f, t' ~ r ~ ~ 1f r -c £ : 1:' · ,r J" t.'

;.. ~~ I

'

ulJ:..-)l;, I..LA t: -.:.;,.:.l, •

~Z

Jxy

dm J.l!:.Il

~~~, ~loJ

~u~'O,. ~~ Oxy ",L-l1 ulJ)oJ ~.::)wl~ J-~I ~ ~ ~l.S.il1 ~I~.J IJ4 ~ L .... ).II

y

y'

~ :[ [ i' . '~ ,--" '~-C"

_r,

['

t

?i9

L

t-i ~

c·

,t"_ b- \- ". f:..

t

-

0

.... m

'[

c

-

c·

u ..

J

~

C. ":-

,t: 'C t 1.

-

[''}

t.'E:

-

~. t [

c;

Otl

":-

u..t

'f:..

'C.

~

b

•

t-

..t

b'''

I

,

u.

t.

t.

1"""1 .::

'C

h1·1. f' -c. 'r' ,£'f ~

~

~r' -~1" -f-.: ,r 1: 'f:"~.:[1-~,

I.l~ 1.:~

J ~~ r J1 t. 1..: r k" ~ ! fr '~- ~ r )~ i t bE, t... I c.:..

'J ~ ~.. -:\.,

'C

s· 10

•

rt I

,;;

Il...[

I

c


N

t [

t t~..

1 " w>~ "t

1:~ ,t.. =- -l

!:: c::.. .. ..

~ ~ ~' t.~ t-i

c-

1 t' u~

t. ~:i.:

,";) I."

b.J' ." ~,

.,,~I~~ »ldIV4~ t.~u~""; .#I.:"~L-.,,

WbJ.I 4.u~1S" \.jjj

0

~W~ J..,-;~' ~...J .L->~ ~

: ~ 1J'U--..,,>~IV'.. • ~)l.~ 1.>..

L

1,yi- L l.S, I) 1r>,;J 1~ r--x o

I zx

~ ~.,l L.; j

0

Ji:.l' t. oW u I." ..J , J>->'4 J}>

;;;1." L>,~

0

'\

-

'\

)

~~,~

v-.r> ~ I 0"•.:

":0") I ~~ I

ds.r--l l

~. ~Iv-A:

~,jj j--:, L", I) I

.J> I

...;) I

v-~ 1u 1l.S'

Jr-> .".1:: ~ I u I! \.j ~

.j l.S, Ij-J 1r>,;J I ~. ~I

~

Iw=L p

oJI',

(

= I zy = 0

...5..:. (, ;> ~ I ,I v-..:-

->0>

z- ..,.-I' u 'vi'''; ,

I,:;. # L>,,,J 1u~

• xy- l.S,.::..-.--.J I ~ ~ u I ~ U'

""')'V-~' "'41~ ~~

J-' L.b.i; ~ ~J rr:-S u,.J-"4

-

.:"J ~

.. \1..1 ~I.ill~ ~)I~rJl.L->1

-l::S"

~.

1.. ~

~

: ~-JL:.II~~ J:> L.b.i;J I d~·

.).~

I y,li.

0 L.;~ LJ.,.i;

~ oJ ~

I.S, L...: OP e

OJ:>:::

: uh J.-;

4$ L-II

u,s:.

cos

0

:1

::::

-

z _..

==

IL

:::

Y

fL

...

z

fL

Op

J

z, Yt

vl~;Du~1

X

'I .;3. ex

uI

rL.:lII.;-'~~l.A~~ ~I".LA ~ IJli

" ( \ f.

- '\ )

£.l.sW I ~ ~I.i.ll.",..-.ill

yy + z2! zz + 2yzI yz + 2zxI zx + 2ryIry == 1 •• • ••• ( ,'\ -- , )

~Iu.,.....: ~ • ( '\ - '\ ) J1.:J1" ~.J..JIV4~l! W, .Op

W,·

J&

,»>.1 I J,.:o- ~ 1.l.II ~ I rrJ ~t4J I

"" :x:

~ J..-;

v IS'

~ U I.#..L..J I J!t->4

r.-..i::.....-J I ~~j ~lo. ~I ~

1. . OX

:=

.J L..WIW)WI

%2I :xx + y 21

~}I

rI

OJ:' cos~

u I.» .) ~

1

~..a.:.s.. OP

cos 0(

I

~ ~I

, i ; l , (, p L..hi;jL.::.,l;I..b-I~

,J-o)-

rj

J,::--

c:L-

£J.s~

J:..::

~I ,,~I~ L~ P

.s ~~ I tf'::b

~L...Ual.aWI

J.liUJ -r-.a.aJ1

d--J 4e W i 41.,) W I J:.:: ~

~ oil ~ 1:.11

fly

i.J1~vJI ~ .~,~ ~I;" .J.~~IS~ I~ I.S,L-::vl~ ~

~ r.r-ll ~W I ~ ~ ~,.

ellipsoid

~ .. ~ u~ ,

~ 1v IA oJ

~t J:I->wt J,=o-

4-".)a;

J ~ l-I

v I""...J 1 ~ ~

!ad,. Oonstrained to Rotate about a .1%&4 .An. d

J; .u..-~~WIWbJl~~,~i ~~.>W~

~~. ~t ~'j~.J--{ ~t~J,=o-"";'..r,.>

11];

=: W

By ==

74)z lI zs - 1 ) 11

W Zw];

(In: - Iss>

.z = w];w., (I ... In:> 11 • ~I ~..t-:iJ' J.O--JI ~

v~lv~";'.a.:...J ~J

;'>-t.

~I

'.0:- v'""...J'~v~ ' ..

tr-J 1 ~lI~~ .. ..t...

; I. ~l>;,~,

T't ,

J':,e~..iJbj!,:l~""l:iJ1

-Jo.

10 ~ ~::L;:.H w ~lsS',.,. yo. ~L..JI ~I~ ~ I..,... ,. I' t~~~1

~ l-t!~ 1~ ..,-')1.:: dJ.lI

Li".

1~ u

~

.(,. -,)

~I"

JL

A"'-'JJ r:;-;J

~~

L-S'~Ie-

0
;J \; 6 t.r-a- t,;-J I ~ I. ~

~ ~~ ~ ~.i~

& WJ , Vi 4S,)l:

l.S 6

\S,I;--Jl,.->;JI..JiJ> U,...liI ~~.u ?-l

~ ~T J l....:., • uJ..1:Jt-->- ~

6

Lii,

JI.,L..J.JI, .. ~~I

J

~I: ~Ul~

~IJ~ .~I.)...,L:-J ~~~, • rUI

..:II~,. ~I: ~UI~ ..}..u.UI,J, .. ~~IJ 4S,I;JI,....>,;l1

1..;1 t:l t W.) W _

L • L

;; i,in..II. ~ ~

c

L4,)*J1 w ~.i.e,

:

~~l5 ~L-II

= const.

~.) W I ~

I

~

--0,.

r

=:J.l ~) I..., t-.J I '" I ;;'0

;.11

2 w 2 + I 2w 2 + I 2 w 2 -L 2::;. constant xx·x T1 '7 ss z

~I~ ~ uI~~"

(H'_ , )

zt> c~~~.s,i6 ~lul.c.J~, • .)l........sl;."....s.w I LJ oJ WJ

~.,.

_

nr • J~~I~vIA.illYpJIWUJI..L...>1 ~ ~b4-J~ ~~, V IA '

II

.... .L ~

IJ}

WpJIWUJIi. l;-i.e \1'~r;--J1 vb' W , U~14>­ : ~ \J' ~ ~.;J • .i.A. ~l; ~vl~ Will

-= 2! ... oenstant u I .. .;;:..1~....,J1 W ~ ~ c!.ll..i ..J ~ I., I

I

XX

w X 2 + IT1w~2 + IZSwZ 2= 2f",. constant

( yo _ '\ )

~JLA. ""->~ v 1~ J,.:--; v I~ 0J c.~". v I v ~! \.Sf

V':

v l:l .1 L..J I, \1,1.;1 I ,->,;J I.) IJ.i., ~pJ I WUJ I ~ ~ v J.r.-.:: ~. "ihioe cX..uJ~WI ~•• ~I \:.IJ~ l-.A v l:lJao • (l 0 _ 'I ) ,( H' _ 'I )

, J, ~I t I

-:-11. ~ ~~)I Al..>.JI~ iJ-:lk.:::: . "

..••) I•J LJ..i1..

....; .. (l 0

-

'\ )

J:l..,.sa ,P01n80t ",1)1 4".....J1~

;

~-" b...

4.1 .1 L.J I '~l~.l1 r

I-I-:1-1-, I-I xx T1 •• ~

w:..)1

.".~)I\.)l1.iL.._w .. (n'_'l ) 4"J~WI

-l·I -lsI -1 xx • " BS

...

i..;

,.".

ellipsoid lJ"l';l~"J1c.J-W~r

...-.y

.-;-.3 1

r-r-

~ ur-t. lolA,

~I vl""J ~ • rjA.Ue,p.iWI

-:-1 1 ~U::,........ ~ v l:).) WI .. ~W I ~ I ~ lA: J-J oJ LA. ve

.( \ \_'\) J!.:JI~ ~ l.u,. ~1:JI~ ,

( H' _ '\)

vl~.,t.

1.) ~ ~ ~u.;

e---J-U:JI ~

, .." l

tA

~.w.

ii....ar,J1 ~ .. ('{o _ 'l ) ",->I~ C;I~~I

~I ~ \1~I'J~' • L.J.J; ~I

• ..-.J I 1,iJo. ~ .~ •...,.....-c ~ I A ~ , 1S.u l~ I JJ> .r l ~ 'i I v I;,.L.J J v~ L. ~ J-ij

v~ 1.lIa vSJ,

~I.l-J I •...,....; r~

J

r . ~ ~~1I...,l.>.J1

wh;WJvL.iWlvl. >.p",

.jJ14.lbJl

rn. .... JIS',.J...-,.:JI~I

IJ ,

hi: Q~ "i..:r.te

d

~I~li::~..a.:. Ixx:

~,. (lY_, ) ~I

J

~~

lJ',

> In > I zs

l-tJ»

I,.1S'

~

T-

...,~~~,

_ ... ~I J»~ ulJ.)..JI~u "i UJ.; vl...,..Jlv~ Wb,J1 • .i.a

• r ..

J~vl!I.,i1,· ~~I,I ~I~~J

J;l..i

_L.A.b.i u~ 1£lI..i,~ ~lb.J1 • .iA ~~,

• "~I J":'l:S' ,I

( H _,) ~I

T,L lr;~ J» vl...,..JI.f'" ..J...~ v~ J...

W

-,:.. (

0'

~) ~I (":') .rA-1

(1)

J ..;,.u, r;.-oJ' ~.

:Body and Spaoe Cones W~ .LI;;~.~.H~I;JI~r-J1

~~

__.. . . ' 1,,1a..--,I-,,.'IIIL~.:-~ ,---;~LS"... u...,~

J..,~II..u.~, • ~I J~I';I..JI.ob.J1

J

u.,~ r-~

~ ~1.J'l>;..U~~~..lS:1J.dy-J11xed oone ~hJl ~IJ.,~

,..,.

j)

~,Ij----ll r>jJl~ J~ .Precesses

rW

b,~ r--

I

~ ~1;JI~r-J16 'LJJU

v I W,.i.: u I,J.J I L.;~!b u...,

.~ cill..iS',.L

Spaoe-Fixed cone ~l:.Il "L.dII..J.,~ ~J.,~II.iJt. .......

e:..J:J I

--->-

L ~•

~

;.i..II U,.Jli . w

.41~LJ:..:,J~1 J~·vl"".iJI~ ~":;J.

I.

;..~, • .. L..dJIJ.,.rA ~ ~IJ.,~ ~~ ~ .:J ~

.( ""- '\ ) JUlI J4JbJl".u.~ oJ-i, ~ w .41 ,,'il tJ"'tI)I,::JI ~;l:

L.-.._..

L-W ~ I, ..L....;.ill J-,~ L.....J..4 v~ 4.- W 1W bJ 1~ ,

• ~ \; v:--~ 103 ~p...

V I;~ ~..&,;...J ,

;:.1;; ~ r-4

v IS' lwi I ~ I

L

.. L....dJI, ;;

~1l1,.r;;."

(H· - '\ )

.'.'>.:lI~WI. ).1:;~J ..w-~r-Jlvl""..J1

~I

(A - '\

he. Rotation of a Rigid :Body with an Axis of

Symmetry.

Analytical Treatment.

~L.-II ~IJ~I\S.i.lI.u..JI~lw~~4k.t.••,.J1~!,Ji

U.,J.JI,.iA ~'r,,;--JI,r.;L:.I ~Wlp ~lvl..,..JI~ ~ ~~

II 1.iJl> rriJ u ~I

1.-1...,

Je-

1:"-'· 4~~;~

• ~~I ;L."I..::.. ~~~

r

l'=:-U

J-li

~ u~

~~.a.»,.J

J.b::: ~ ~ JJI;

~LiJILJbJJ ~I..::..~~L..

J--; ~

),,1>0:11 J"...JJ~

J:..;u,.-

~I ~) W)WI ~1.il1~lr,~~ul.:.:llS'l.-..;~');h;~ .(~ ~~,).1:.::~r,;.oJJ~Wlr -:-lJu~ul~ d1J u l

Wl:l1 j,.-)I Is

J.>..a.D, )-);l.:.::.U ~

= I zz

( ).I;,:JI~

= In = I yy

I

z-

~I ~

J» ~1.ilIJ'_""J1t~)

().1:.:.I1 ~ ~~~~I ""b..J1

J» rj-"H)

: ~ lJ' )-.J I e. ~ oJ ~ ~"i-U ~ L- r,;.-J 1 L.r; u~ JJ 1LJ b..U I ~x +

w

yw

d> y +

w

z eu x (I - I s )

I

Is

W

W

z

z (I s - I) = 0

=0

('"

-,

)

z = 0

=

constant

, TY_' )

~~.o.. ~I:JI u~lu~,

.1'L=w z

,~

x

Is - I I

+..Qw

y

==0

W y -Jl.(...) x =0

.'I~~,.~I~l&; ••)l...s.1 ~...,!.wI~~WI~ ~1~lj~

;

~. J.-..:.I

;;;

z

+.rLw •

0

'1

~I~(\'·. _\)1I ... lAJl

J

Wz +1t 2wz

~jlJ

t

4 ..::;.11

~,

. '1

C/.)

.IF.;.;JJ ~ ~,

• 0

(T'l _\)

,.. L,J>, .'Ie, -.II ...... t,.:J 1upJ 1W.a L-. ..... ..i..A w x .w 1 ••• (..!l. t)

( T'l -., )

Lie L-J 111 ... lAJ I J-li; w 'I ......,..: JJ • J,. ~ 1 .:..r b ~ w 1 .:..."... L ..~, • (l \_, ) 1J ... lAJl J ~=J~ I' " ~jlJ a.,-:.JLc ;J,. -JJ (AJ.,. ~ ~ .iS~ GU'I • u> 1 ria

(f''' _ ~ )

(-0. t)

..,.-i~l..Jla.",...n. "'~ ....~~.;Ile ~~~I~

(.V'I'W Z c)JI

",::--I I J. G:; .J..1- r-JC UJ • .."./2 )1........ ...,...u 1 • ..0... ",'.;JI oJ ..jtJl oJ W 1 t....;.;~ 2JI.a

rr-

.JI ~'..o.a..J1 J a Jt. is 3.icL...J1 ~1:.iJ1 tr .. oJ" ~,

J

foP" 4'.;J I ..."......J I ~ r-Jf,

J

",I.;JI ......;.H,

Y~IIJ.AJ

J.,.-:.:.

...

0(

, ~ h:: ~.....,;

Je is 11.

(

.. ~

."....Ji»WJI~J,..... (u~) ~~

~I ~,,&tI. (lA_ \ ) W.alAJl Ju.... .,.LII

(tv ;-~i) ~ 1..0 .&..JI ......... ,

......J.-,.

II-

11) .,...Ji>lcll ~

•.,.......

J.:.: ~ Jt~ ( 1 ( - ,

.n..

)

JL...tJ1 J

~hJl,..

~ 4'.;J 1

~ J--A W (T'(_')

'" ~.4.

•

(

Jo.

T

-

1) woo. ex

y

.,iiJ ~

c.i ~ 10 I;J 1a.J-Hc.l&

• .,.,Jol L..c~). 1...:ct .. ~ *-t-

'.

(, t.-\)

J1.tJ I

;I~ ~ ~ v.l.o oS u...,

'l"'I,JJILS~c.:lS',........L.~vl,.".LJ~L.JI J.,b..:)IJ

.;lic U..,.,J Iu.., ~~, • ..-. .. .a-:, ~ I" ...... .0 t.-.J U._\)JUJI ..... ~t .. t-... J..e; lvl ~~I G/W- oiAL:.J a.

,.. elf.' I ~

~

.. '=--••.11, •

'L.i.iJI" c:.et~

IJ 0Xf.

ut .. b... u~,·

....-...o.a-:,~I.j

••11,. ~I):.4=~,1 ....- ...,......JI~

• ~/ - .s,:--J I e' Xl'-.sf •II e--l-li: _ ;-.)ls x-.o >ell, X - .. ,..JI~

"Jl> 1.:.:.11 ~ JP' ~ I v 1.0 oS '.

""" ,

e .;.) ls

.. L...U

OXIZ

z -..o---JI~

J..J.,.....,.

:z:-

~ljJl~ ~,

s, z ~

c.j--J I,

y

..., L=..J I ~ )..,., I ~ ~j v::-: ";)l,J I Ji:J I ~ (

~J.l.ll,~l:J1

• .)

. f/

i--}~~JJ~ x"_~I, x - ~I~

• .rJ.::, I ~~}l 5V , i, ~

.->-) I ~. v~ , • OXYZ ,I

'0 -'l ) Ji:J 1

4.::..:~1

;-.II,r.:---S.k.:iJ~....;~1..iJb,

..., b.J I J

~ 0 xyz

~

=0

~

=L

L

JJ ILJ bJ 1.j ,

)1..t-..iJ1.j l;b

.~l:

Z

L ~ W J' vi uf jL.:J I V-

.J J

-...,.-..11 ~

invariable

sin Q

t", L...&;I

'--......,..,. , ' ~ I),..w I rr-U~W 1r~ 1v~ ~ ( • OxYZ"

L \$,ljJl

--'"

);,1.:.; ~...J u..uI~IL.lb.

)

,

~

.~~I,

.

1;J1v-

~)l....:J I ~~;J 1 ..."

~I ~,.,}-$ ~ y.,,: ~

""L>...U ~~

4

L-.....J1s' ~) "" b..

OxyZ

~ \$r->I.~ B -

..,....-J 1)

..., L-.J 1 v~ dJ ..iJ

rr. ~

w

=0

~ V.j I • ;J I ~

J,

(

'>-

.....

\'l - '\ ) JL-:.l1 ..,...; vl........:;jbJl ~ .J--£I

z

>oJ I ) ,r-l:> 1:.:J 1.»

-.b

(u I,.---b )

:j

b,~ :;

.:

...... r--'"r!;> r .~ I (0

~

;"

U;"

V4~I,,1 ~II ~lrlJl L ~ cill..i

=:J I d.I b. .j v 1J.'..u I ~, )-\:,1:.:J 1~ ~ c.s, ';J1 r>;J I

~ (I

fA - '\ )

..... ~ •

...i'~I.j, • L ~

• ..t..t,,:)1 ~

L

J,.-. (

~bJ' ~,I;-

~

(.L)

~::J I)

v ';"

L

( . )

~ ";,. ~ (I.:--) , uo,t-iJ (1) "...-11 u I»~I (\'\ - '\ ) ~I . •

-

d

i -•• 1 "'1 __ ... ~\I !e:>.144·" L..---..aJ , ~

Lb.,r-.

IS,::-J I u I),..u IS, I}! I t3)L....:~ I u lIS; , ( \:) - 'l ) J!,.-JJ I J I~) ~

z-

,;,.1 ~}!'J~IIS'~

}i;;

~I J,.....:>

yz-

• ,r-l:> 1::.1 I ~ u I;-iJ ~}! I J .L-AJ' ~ 15 u .j I • ~ 4.-." I}! I V't IS,}-t. IJ' ( L ~') ~I #~, J,.....:> ( ~I,) ~

-4.

CA.)

~

w CLJ

--"-

W

I:- ~J4 u , J.L:J I L- ').J U4

c-"'> ~,.

G' WJ ,

x =0.

Y = ~ sin

w z

=~

Q

0013 Q

V"--4 ~L:.l'W.JW"

+ If/

4LI ~~..wl; ~..=;-U W.J~ til:~,

sin ex

u'

~

(

f f - '\ ) I:-~.JW'

(

(.

-

'\

)

try

(;) v---s ~ ~ cll..i , ~ I..

:~ ~

~ =w .

•

\.

pi • ~;;.. •_"'\; ll: L..II iJ .J l".)\ t-" v!'·o

J..--;, • (

L1 + (

1

fA - '\ ) 4.I.JWI~~

2

;~

_1)008

~ I# ~ I

2

J,:>' r-t 1:.:J I

wl~

J

Js.L~

J*

( l\ _ '\ )

~

~;ll

u!,-.JJ

J ..s..-ll

I" II

..

Free Preoession

0 ~,L.-: '>-

lJ'

lJ'

J

~>JlllJ'

·r,\, Y JI,> ~,L-::

~u';l~

J u,)-".! 1v-.

1;.11, • ,,-J; 1.:.::.11 ~ ~ ~.ill "" I~ I

oc. y

u-dlJS'- ( ~eJ~1 1 Y- \ ~I '-.

_'\'

) ~I v-.~b

l / l ~Iul u,~1

( rf._ \ ) U",W1v-..

uO>;~1

c-hli.:

v-.

.n. • 0.00327 W JI-_-.lS' J.H u; ~ Iu!,JJ I ~ ~ J

v-.i u lJ ,(

£P

== 2TT/ :L .."...

2-d'

". ..u.

1 0.00327

~) ~b'W,

lJ'.·. ....., .il

.

L..-

.

• ,. •

days == 305 days

~,~ ~IJ". uO>/~lul~",~ u~ ~)WIL-e~JJlv-.j, ~I;,..

......J1,~)WI~I~~J,.:llt..t...S~~,·

~ U.· ~."i::

• ~ -s)l...J1 L.l:: ~ uO>; ~ Iu,....s

Ll",L...\lt • '~I v-. ~y. lJ' uO>; ~I)o1:.:: ~ u~ ~ l.

J'

fn. ~ = 1.00327 uJ

~. ( t\ _1'\ ) u~ ~u~I~1 ~~.iJ1

=

21T

w

1 1.00327

~

~).;!.~.!1u!,.1J 1v. J-.>I..t.; '~I

~

;''':; JJ I r'j-AJ I V4

0.997

v.j,

d~

J ~/~ 1...,..-..1 ~ I u ~J I I ~

~.r.:->'j ~ , -.:.- ",

~ 1,>.'

A JrJ.. 'j I

;vo ~I~~I,~1 "....;oJ I u!,lJ I v..~ J,..b I ~~ I u I,...b

~~Jv..ju,s'~ul. (~~

~) WI

r'j-AJ 1 J l.A I)~

• "....;oJ I u 1,_.~1I ~ ~ J v.j ~ L.-I

1#~tl~1 ex ~ljJl.cJ»'jlul""..l,):.ldl""b..

(\ y_,\) JUJI

\;')~IW'p" - ~,~lu~1 ( '\ - '\ Gyrosoopic Precession. Motion of a Top. 4.....b.i; J~L~"",",-,~.ill oJ....1.Jl~IW.P"

4

11

.u. J(j'J~

rro • t~..,h ;'", 'J \oS ..iJ 1~ 1 u ~ I U 6. v- 'J ~ • ri-J- ~ ;'"" L:.: ~ • ( u,J~1 ,I) I.. JI ",:",L-;::4J 1 J~~bJl~, w

15,

..,b..JI~)J w""b...j>-J ~

1, Z

d-- ~

, ..

\oS.J~

(i \A_" ) jUJl

~I~ ("':" \A_" ) J1'..-tl1 ~ ~ CO .:.:,:ill

X -

• ~I LrJ,>..,~ JJI~~I~I ~

0 J-",~IL1..ii,. W~I

z y

o

b

i

rr

~ mg£ Bin

Q

y

.~I~,!-~~I

( lA _ 't ) Ji.:..l1

yo J~I V- ~WI 0 J» r,;.",JI JI.1.....i. u I

~ I 1.i..Jb ~,.

0

Ll.:~11 j ~

JI

0 V- ~ L-J I ~ V1u1

By

X -

~I J»

(0 _ 't )

=0

liz = 0 ~~~ul ~~,. ~

;.) ~

0 x y z

.» L-l.I ~ I;J I ".J--H ~ , ~~I ~~j Lj'J~

--'"'

w

tV

.

=Q

x

.

= pj

(n' _ "; )

sin Q

W

y

~

z = pj cos Q

.

T. =1 u.J =IQ -.ox xx x

~_

-y

= I yy (,)JY =

L = I z

zz

(W

z

.

( tt_'\)

I pj sin 9

+ cj ) = I (~ oos s

•

Q

+ ...., -,II:" ._, ••

I ":11 . ........." ~ . '-. , . '... WI -"-.J .. • 't!

Wei!,JI

J' · ~WI

J L..,J .... ~ ~ J vl:•.:;~14J

;,

·L...;::aJI~

J~1:J14J..J1

• ~.J~I k~l.S.lII~"LI~.w1 ~~I ~;:1I, UUJI,,:,:~.JL....

The Energy Equation and Nutat1.n

4JUJI vi; LJLJI ~~ ":,,,L-x4J 1~ ;.'; ~l!:o>ll.S"; ~1:..A~,J 1.;1

• ~1: ~ +(I w 2 + I x

(..lJ

2 + I S 2 ) + mg j s y

oos

Q

111

!r + V

~I

E

.r-lt" I ~~j fJ~ ~ Y;~ I.,! '2 '2 2 2 i(IQ + I~ sin Q + IsS )+ mg £ oos Q

,,)~I;;~.wI;j.JWIJ~"""'::'

J5 J

III

E

(t'l_'l) fJoJ~IJa..~

u~4:::J1,

,

(B-I S cos Q)2

... IQ 2 + s 2 I s1n 2 Q '9

~.: I , ~ I ~

.J I

Vt

+ *I s S 2+mg .R. oos .

oos Q •

vi ~J. u

III

W j:.~ 4.I.J WI"..i.A •

:- Jl:l1 ~I J..a.J, U =

Q

= -(sin

• Q)Q

E

Q

• J.~I Li.~

2 i . = -(l-u) Q

LJ ~ ~ 1,1!

JJ I,

t ~jlJ WI~

T'T"I

'2

U

J.l1::J~

t

,I

= feu) t

=J

~jll WloJoS'

( 9 ~,) u

..lL~1 ~ III

du

[t:(u)

v..il ,~t.l14)'.t.l1 ~ J?olynOJiial ..l,~1 ..l~~ f(.) v"Jl elliptic functions ; ~.:~) III 'j I J ~..J I W "J ~

c.rJ.

u,,~1

-I

J.~I·I~~I ~14 i;i...JI

J

J. ~ I ~

~u...: "J ,

t v~.jJ 4~v~vl~ f(U!v l

);t",

..l L~ I ~

J6. ~I ~

loS;, - LSpll4.-WI

,,----+,.I,.f(u)=O 4.I..lWI",,4~v..ill.•• ..>.: 9 JUpJl ~~W I

u .l-...>L-vl~~~'4)..l'·,~~~vl~ 9 ~IS' f(u)4JI..J1 ( " - '\ ) JL.~I

..coMt.'

1 +,~ ~ u 2 ,u1

y";~I~pJl~~~ ~

J

~I

J.....:.:: -

1

+, ~ ~ ~I

L..r~vl~vl)41~ ~ ~1;JbJJ

9 2 , 91

uI1rl4)-J~J.WI

G

~v,...s.;

9 ...." Ijll ~ I ~ \;h ~ Ul.::J I, r'- 'j I ~ I u,)..i.::J l;r->.A ~.i: .i.:..~, I i.

,.,~,

- (~- _ '\ ) ~I

J,.;UI J» u,)..i.::J1 -J~ ~

!.il Nutation f(U)

c:-'?~ ~~.i.:J I

u

-1

• feu)

4.l1..JI ~~ ( ,'\_'\) J.U.l1

"'f. • .w~

=

ul

U

2 ~lS" IJII.SI

\" -_ ; ;_1> II ~, • .,,;.:.--;~

I.J

Qo)>-J~cdh.AulS'

Double root

u,J ~ "-~,

~,,: ~ ~

• (0. _ '\ ) #J.)!.AJI ve Q.),;-)I J~IJ.~

,.-: WI u,J-..i.:iJ 1

Sleeping !fop

~~ 4s~ ~";L:JI~~ 1~ l.il u~ u,J~~ ~veJS'

•

I.J

til ,:-,i-)::~I ~~ ~ (\'. - \ ) J!:.11

• W J.,r-:S' ~";L:JI. 4-:~ 1J ~ l: ~

u. ~li;

u' ~ r,:J I u I IS";

, 9

.

=Q =0

~lS'

u,J ~ I ...,-.; L.:ti lS'

, ~L.J I J.:.b..;Ilw~ ~,.Sleeping

W, #J6JI • ..i.A

W.)WIi :B

II:

J

I.S ,

v.. +1

J

• • mgh

+ i 1.8 2

1'(u) = 0

(1 - u) 2

[2m!~

(1 + ...)_

r W~

]

• 0

Q.),;-;4

~~)~~ J-~~ It,,L-4~L...JIW.JWIJ ~I ~

vl~,.~)-A 2

___6 I

8

2

J.

2 I mgh

"3

c.:..:1S'1.i1~1.

J~L.;~~~~~ u 3

Q

4.J

I, •• ...ill>, •

~K·

... ~I~

4,ti:-- ~ WI LSpJ I v'!':

00

)~lvlS'\.i1

~~

(,

..1.->

~ Vot ,r.$'1

u,).i:iJ1 tb~ \.it; • rWI u,).i.:iJ1 )IA-~ ~

l..-.:>.:,,:u,).i:iJ1 ~~ ~~(, L""J>", .. )!~I~..wI.k".-.:JI~ ~~ ~ 1.r.,.>I,

.w...J1~1 ~~.J

J

u~ltl.l:i..·

.. I (\. - ,

~1~1~4~1~1

Use of Matrices in Rigid Body Dynamics. The Inertia Tensor tl.~ J...llll.ill>

J

.·J.~I J-JI Vot

4,#

~~.J~~l::S'~

, ~, Ij--ll r>;.lJ r LJ I ~ I c,;.j I , ~ • u,....&-ll. ; ..... ~)l. 4»-:, ~u,.....s.-JIW~~cJ.JWI.. .iA (f.

•x

Lx

rIxx

lxy

I xz

W

Ly =

lIyx

l yy

I

yz

w

I

l zz

W

Lz

lzx

zy

_ \ ) W.J~1

(00 _'\ )

y

z

*

~I

,

=:-:J I ~

I.;'~ ,

( 10 - 1 ) ~ 1 J

,.,>--J 1 ~ IS~ IS .lJ I

--!,.:> '"'" 1~ lc ..u.J 1r

""

"" ~ I~~

~,.s

lJ'

....

1..c.." .."JI ~ 4;

\" x \" u,..J..:aJ 1•

Il~

,~ L.l. ~ I ,,!,-:;J 1~; .:-1.lJl ~ I~ 1

,~u d .s.-:.J 1~ 1 J:t:..:J 4...-1>. i~,.a U, ..; '-jJ I V& ~ 1~ -->..

--A.

L=Ic.V

(0' _ '\ )

---..

-"--~.~~Li,i-. ~

w ,

~

~, .~I~I~

J,

KiDetic Energ ~I..o~I~pJIWu..urWI~I~1 ~41,. ~l:H,.~,.;JI~u,.w1 i..J~~ ~.

T=t [wx

wy

~zJ

~pJlUl.lJl

c t....:...~I~

( \'f_ '\ ) U.sW1, .u..J1 r

I xx

1%1

1 XB

w

Iyx

I yy

I yz

w

I zx

I zy

I

w

zz

x (oy_,\)

"7

s 4~;

!f =

U, •

i it; T

I~

i ... \\!ranspose

;,..... ,I

( oA _

, )

matrix ~''';~J.AwT ~'Slu~IJ.;Jb

I

~L-u;t ~.ilI",,-au:;~I~I~ Principal Axes

J,>W4.JW 1~I J.-;.:u li

:W

~

6

r . -;J.l ~

l:J I diagonal

';

'~IS~I '':~) I ..0 L>.J I

""t..>.. ~ ""L-.Jlc.,;b' \.il L...",wt I ~ I ..i->~ .;1.lJ I .o--il' h

~I ~ ~.IJ1""t.-.J1 .a~~ 4..-..WI4JL-Jl v l. e:---I,JIV--J

u,r--a' • ~.rlJu,.....s- JI \"'x\'" ..;, ;,.11 ~ ~luJllIL_lI 'Jb:: '-., ; ,~ J I" \~ ~".J:.1;.;;.e-.... u, ;... ~ I v I. c:.. L.,;....11 a.-~ v-.~".>~I ~~j)U j:l.::Jlo

L.,J~~

1xy = 1yx

~~ I"" b.J I Vt u..r. ~ I,.; u 1~

ejUl4JbJi J. ~).J

ed.I..L..l, .. ~ 1:.:. ~ v.i I u,....&_H; L..i;~1

.:i

I.) lA. "" ~ .d.~~ ~)-.i,~

... :

J I u,...4J1 ~ "'" :

I. - All

J

~1.:l1 .)J.

..J 1

= 0

l.-.J ..d .-II .. ..i.A ~ , unit matrix

4 ~ I t.J, ; ""

~

J.-:.c .L

~

: ~1.:l1~1~~~4~

I xx -). Iyx

I=Q" I Y7

Izx

I zy

J

-A

I XZ

-"A I,...

¥;;.;

,.

,

(1·_\)

I zz -).

2 + A'}., +B A+C=O

_:_ _·)wI..c~1 • If. -lJ ;\'"

:J~.) ~

a, :at

• W)l:jl~~t: .-1.ll1..::.-1...,..-iJIt'H ~

A.

~

A,. ~2' ~l

.....,..;1 ~,~~I: i.:i,Jlr,w;......" • ~)I ...,b..JI v~ ~J',

rtf.

v I J>)D, • 41) I ~)-..J1 ~ • ~I =' Ii ~.r-:» ~ ~ I v IJ»;..J, 't ( 13 'oc ..,.. ~)I v~ ~~ IS,I)I ,...->;J1i ~11.iA J)P'

..., ,'---> ....... .1 I 0.1-> ~ 4..-." I) Ic..1.lll ~~ I ~

W..

L=A~=I~ ..__ ...,.. . _ _ ...x"""". ,. ' \ ,

,I

(1'_')

).. 2 '

:

I xx

Ixy

AW oos/J

= Iyx

In I zy

Izx

w)l:Jl ",~I oL..>l J:.: ~

Ix. I 1's

W COB

O(]

woos

IB

I sz

w coS ~

: ;.,J 1.:11 ~)l:j I ~ oJ .M.l1 c.. ~ oJ L.J I ' J

(Ixx -

).) OOSIl(+

Ixy

.:..,.

';I.:II~I ~ ~--"~;~ ~L-.-JIU ..WI

AW oos IX AW oos r

A1

cos ;3 +

18+

~

• ed.I..u

I xz

oos

1

oos ~

~

0J,.b~~L ... ) ... 1 ~1.iJ1...,--iU;~I~1 ~

• '>}.JI ~~j lSobol"'; 0 ~ Oxyz

....:.;,1

""b.J1

J •

~,

_,

• ~v:. .1. J,....b J.s. y , x vl~l, 1: ~ ( 1 _ '\ )..t...N T' _ J L:J I ,....:; I . r 1.J.:i..:...- L. •

I xx = I yy = m j,

0,;.

,..

2 2 2 /, , ISS = 2m 1, I"~ Ix:y= -ml/4 t IxZ=Iyz=O ~ ~1.i11 ~;.J.::.JI~10..i1

mJ- 2/3

J

-m ),2/4

2

-mJ

=

,

ml

0

=-

2

0

0

0

- t

1

0

0

0

2

2

m£ I;

/4

- !

1

0

2

2m)

I;

Jp .» ~ L... ~ 4)~14;-$..w I 4 -> •..i.... U loS, I) I r» I ~ u,.;..:..J ~ ;--." I) 1 ~~ 1~ VS ~ 1~ ~ 6J 14.u. ~. l;l.) lb.;1

_-.::-,.,1

- "

: J l:J 1loS .3,.-11 w ~

(IJ

=

w

[~]

X

Y

=

=

o

Y' d.l.il

1 ...,).

L =

~

I w =

m£

U;, " I.S, I) I r--) I,

- !

0

1

1

0

1

0

2

0

2 w

3r;

- i0

t

2

=

• .1 w 3 {2"

..1,

2

1 tV

1

.--.....~

r2

12

o

0

~L.J';Jl-J1 ~ v'''''~ i,J'pJIWUJI ~

_f

. [lJ .J ",,2 OJ 1 . - - ,j

L.:~.

T=t w

T~1u)=*~L= ~!.-& ml ul 24

"--"'~~j lSobol J~ ........".

a.L.J1 .tol' J~L .~,....

[1 1

2

- t m1-

-ill.£.

2

•

12

- ... }.J ~)I ~1.il1,."...al1t'j-$ ~ ~

t .. .1-A

;l

t .. t

_f.

(,. _ , ) ;J.slAJlla

2 0

2

-A

0

• 0

2

[.. • ± i .. .£ 2

"

J

,

'\

1 2 A=-m).. 12

• W~I~)lr,rJl ~.J.p. c)W1 .)~I~~..wI~L-U~)1

",,6.J1c:.. 4~1

~

(i

m..e 2

- i m.t 2

(f m.£ 2

A)

-

cosO(

.08 0
~I

•• J,~I ~ ~~ ~~I,I r-WJI • 1...>;ti..IlIS":'- ~

J,J. ~~.)H""I-.JI

"..s,.s u~ ~ ~ I ~,,1 I ~ I,.

~ ,./ 1~1I

I \jJ..)!'-- - - - ,

"" ~

b , a \.Jl, L.J.# "

m

J,....- :s, 1)1 r.>;J I • ~ I , ) 1.J.i.

yL::S' 4. i;.::. ~ •

JL..:..l1 ;;1.: b-

loS, I; J)L1.:'~

W

... ~

Lr-';";

_1

..l> I

J9"

• u I"" oJ....J I .,b--"':>-'" L.,,,, I;

J j~1 J» loS,I;.lI ,--.»1 .~I, JI.J.i. ~ ~~IS,I;J)l1.;~ul.»..J1 ~ a .;-Wu....:, m ..±)~IS;I..lur.; ~L.JI JI-,.-JI

.~I ~. <J'"~I~ ~

(0

~1;~,F'~lve ~J}>..t J»

-~ _f

W

IS, I) I ,.>;.ll ) 1.1.-i. , J~

\..1 ~c..)I.;h- '.. IS;!,.:J ~1..iJ1c..1~I~rb, ,..J I, ~~;J lIS .a> 1 J ~ 1 Lb.i;.» b..J

.» I

J,.-. .» ~ c..)l.;h a .11 IS; J,..:. u l! 1..iI, •

0, b,

r"r ~ -(

a

c..~1-,a "J lIS; I,.:. c..W b..

~)L..- u,~ ~~..i v-j

~'.nJ ~I J» }tli:J1 ~ul~ • &/4 It!.

•"J..,ll,;. ~..slA. IS, I

~ ;;~L.e

". r,jAJ' ~

(~~':,"~..sW~',

W

»- _\,

('( 0_ \

) , ('(

;;

r _\ )

u,...s: ..rJ' WbJJ "J...,l

I,;.

»

... (y)

'~J

~..sWI~'

~..sl-.~ a ...... )

z "Ca.:Jw~Wy"C~l:Jl,w%"C J'~',,:,~l~'~

_\ V

,. 0 •

ul'

• (t, _, ) ~-,l.".H J.l,M-'

_'A J»LzJrN ~..L...1.,~ _1'

~ -""':' ~I~~I '~I

v,~~l:~J»4-.~J'~'

y

J.

;~ I ",,_~•••J1 , ;-it L:U' ..,...... ~ 4 1jJ h::...Jlf 1..1 • r>--.r.r-'b 8JI, vlJJ"u1 ~ ~4'.;J' u l J- ~x • tX "" vl",,""-olJ I

tan-1 [ (Is - I) timex Is + I tu 2 DC

--"

~

~) ~''''

(L

J

;\ "'" ( "JitL:U1 ~ J,.. &jI.&.lI.tt-_Jl,.H) Is ~ ( ."..JL:UI ~ J- .s -',..,J I ~ I J,.. ..;1.&.1 I ~ I ,Jt) I ~ • tan-1 (8- i).s"'-=ul~~1-tJ~jJl • .i...Avl~1

(" ) ~ J

,»-- t&lUa

~l.JJ

~ ~ • ~ J-S~

- ,oJ---.

~6..01;J1

. ..;~

4-..JI;J~

11

J,......;.....,Jh:

......-.. _t· ......-.. _t ,

JJ"'-c .a..J... ~ _t t

i!·.. w)l:JI ~)I J...,:I.&.II~I..."..,.-iJI~~lf1..1• ;~,.

;

• u~ u,...- ,.---.J I v I..., -' v I A, I ~ ~ -' L.. u..- ~.::...,:I

I};

&-J .s.L-II...,.,JI 11 .;1 ........ ,,;-.,-Ji-'.J .s"wI...,......J1 J,.. J!)

....

HJ~ ul...,~1 u lf 1..1 Lei • .;1 .. ~ ,r.,...,wl

I~ v~ ~u,

r-.~·n ulJJoU .;1.&.11 ~I,,;-J.L-,:;JI

• ( vJ'¥w......&1 m • y~

;!O;(,

I.Hi •

....,......J 1tJt,.

a

~·t,..JI

J

.~u..-; ~J'

."..Ji.L;; ~ J,..

-t •

y~u.t....1: ~ .. ~ ~)

.~,

JS'

."~;'-,';Li.i..,~ _H"

e..v ~,Ij ~~ ·L.iaiJ1

,,If, P t-i ~ .u." A

J JJ"'-c

..,,Jil' W~ cl;,r-.;

• ,,}JIU'~; I..... ~ • ~",ul

f

-

'

1

~

'-

"l

a

1

c.

".

I"

a

0

L.

...

)

;

"

m

..

L.

't:.. ~

tC. ."1'

I»

Co-

_.

..

...

'-

~ .~. 1t · r: 1· t ~

c·_

NC>

t.

•

a

Y

lU

....

~ ..

1M

_

!=

':

t. ~~ .".

~ 'to ~

_~'.._

~.

-L

I

.... 1 r~: ! ~ ~ f' 11 ~ r[ 1 ~ - ~ ~o ;0 f J~ ~~. ~ '-cl::t~ } ~.

Ir'

I

J

(y)

.;IJJI

~J J-S",

J

~

.J__

II;'_••~JI .......... t,

(1) LtJ.-I w.;.;

...,t.....J

.. tSi I,-:J .;1 JJI ~ ••~_:;••JI._••~JI ........ -,. L...rJ.-I~

• ~)ll1k .. JI lSi'':'

""l..J 4a, 2a, a

~I

ut,.-... ..JI~I Ja""L..JI ~ ....... ,

,,--~.s-~IWUJI,tS,'.;JI

,.->;.1'

..J~~ u,_J.JI~,~I

~'IS:.JI "" ~ I. ...... ('( '\ - '\ ) ~I .... ~ (r. - '\ ) v-~I J ~I ~ J....-' • ;-S.,JI J JUI, ~l V'~, J ~ ........,)1 ""L..JI ~LA~I "'"""'"

.r\

;,... _ _.....JoN.lili J,.,..

("':'), ('( '\ -'\ )

.('._')~IJ~I

_,t

T'oT'

,,-'WI J..--Lli .;I~

f!:

10:.0 ~ oJL...

Lagrange'. Iqua'U.Jl8 ~.J>

;\

J- ,,-' '-tJ 1 .:,,-;~

I~ .;

..

~t,i ~J I v~' u L..i.: c;,......,

.:T .)U

~)~.".H ...,~I

droJl,

~qk

JI r.:-

J 1,.1 4,:-1

r> 1'''' ~

IS

,;-)1

i

~~ II:" \i:......:.J \;

t~ 1oS..i...J1

c;

b ..~1

L-c,~)I.J 1 b--)' ~ I ~, • ; to..,J I I:" L.::: I ..l>:~ I

J 1,,~

;;'-,4l ~..il • ~J~ ..JI...,L:-JI lSI JI.1'.

,;-)1 ,job .;.H,

J'

.3N

~ Xi

,

~ ~1.i>Lo:- 1

1

1:,,1..

F

4""; frb ~

N

ve~,....J:.:;

~I IS,....LII

Generalized Forces

~ w J.A-.Al' ~I~

~I V4

S-; L>.ljl

(l

"..1 k

('( -, •

~..,.'- 1.;1

~~ ~~;,....LII~.HWJI ~W

~

a

->.

s'r

P.

CI

P.:x:~X +

Hy 5;,.

+ 'z bz

J.A-.All ~~ J,-; Ls.~ ~I L;j,.)w~~,

~

W:::

L i

F 1 [; Xi

' -, •

~'i: Jt[t 14M

()f'\

I

I

~. &; r '- t 'Lb

j-

(i

: 'f : '"}

~

Li -.... -.... r

~

:. J'. 1

~

. t"

\-

L

['

1-

-

E

'''-.

_

w

• ~.r

01'

#

t.

tf>

'-.

to

r[

01

t?

~M

••

~

Ii l.

-:-

(i'

., 1 f t I.... - [ t ~ J ~£: #

...

~ ", \r ......cil l: tf ." .14 ~tt -

•

~

0"\

#

01\

-

#.14

cYltI

..ttIfI

.. M

~M

••

t>'"

#~

01101

..ttIfI

\M.

~

t

1-

t i

\a-

I

•

01

>t

e>I'

. . . . .M

:1

. .ttIfI

..M ..M wM "}1

•

CY\

...

-

(i

t

~

-

to

r'f t;:

'I

rr

'-

~ l:l~ {t ~ ~~ r f_b~. It ;:tC.:..1 -; ~ 1. ·r.

{_ltt

~

~ ~ L

r t ~

to

t• ~·1 ~

'-

..

,l'

"

.t

E ~

[ ~ {I

[

f• i·~· rl~~

•r[

-
:~ l ~ ui-i) ,!"II...,~ l..t..i.a~Jl:>J'~,....;J1 ~;-~•.:.,H ~Ii

f

,.-,,...J-. ~

J'olJ.,.-

L 9 ~ 1.9 il e ...,......--ALeu~W6JI .. .u.

ui,· ~IJr-> •e

I, •)-'. .,.H;,....uJ

~~I~ ~lrj-tJ1

~loL...>)lI

f....U1.,J1; .. ..JIIS~'

Qk

=-

(1

~,.u-)lI ~ ~ T

..JIIS~I

~'Jb' ~ ..li.iL.>.. ',...J J~ ..,,---

W w..~ ~~.u Lii"

)

It

; "'- L-::J 1 lo:. Li~ I "I e:-:I) I ~ I " 41J

bV P1 = - -0 Xi

;,...

I~ ~

Jr-> be 4.-."b J)l.J.

W=J6.Jllo:.L..~ i

u

6

aT

oX1

• ~IWlb ....1101

J~

~ V ~

~

lJ'i

It

..JI

C Xi

Oqk)

W l...u ; ..~ 1 .........-""~...,II ,.- ..:n--,iJ 1~ ~ I, u.il .qk ( t _, • )

.d· .,a,~ Q2=e,

t r

J uiol ..".

; ••b;1I J,L-JI L:.~I l.il)l.:.i

Qlar

T ~b' l.il Qe=

-'OvAe,

• Qe = 0

Lagranges Equations

~=-dvAr;

.,a, ~

.. -.11 ~,....;JI

( 4.-.I.".;,...J ) Ja;

~1~lo:.'JoIL....

f_\.

~

-

(i

\.

~I

H'

~

t.: t

-."'-

-

tK I-.t ·

1:.

or-

•

'ct..

t..

_•

[ '- " e P r .. [ 'L- ,..

-t

\

.., tt.. t

L

~

~

W

f·

·r

~

..

1,

N

_

~

{1

't:t:

.. r j ~

01101H

+

w

.a •

l e t ....

I\)

.t:.. .

'r

~ "L t; r Je.

III

t' ;..

•

r : s·~ ..~

1::.

~ ,~ '~~ ell ()/ [~ t ~ #.14

1 wM

'[t'[l " ,.

t..

F

-

C' t...

'vi

....

8

et

_

t'-

~

L

:l

•

f

~

c....

_ ' t.. .. " -;G,

-: .-

I

I\)

21

f

E ~

~

''i:.'

~

. , - 'Je \- '- l

N' .... 1\)

It.

+

~~

+

.... 1\)

...... H'

t-P

II

L:::'.-J

" ~lt-;~ (i

~1.~t t.. ~ l~ t ~f{

l

...

I!f

...

,r(......H·

't i" ~,1' \-'-. rf

l

_

(i

,

.L.~

~ ~

r. t' ..... ~

. [

.

..

't. ,f'

~

,t-' 6..

. t r. r:'

i

.. • ••

I\)

~

Ft

II

12I

fo

t..,

~.

[-

1-"

II

-

.~ E t - K ... r--

,ro

-to

rl:: ~'l~ t[ ~

'tr:'

fj c..

,~

fo

t..

~

r: ~

E:::

f t ".

.1",

K

t

J. y

[t..

~'tE~t. ~[ ~'[t.- .... ~f. t '-f t1 l. t ~1 {~{f~.f. ~~

~f\-l &:"'cEt-r

r.-

f:..

lD

~....

'r [Kt:.'~

•

,ft

;:[b[ r: [ "~' El-c· ro-

(

[

H:

}- f "t.

II

~

~

.. H

r

H

,r!,

II

....

.a

......

,.a.

~ ; F 't. "-

ft c~:- l' [~!'if [ M . ·fo. /!.. - . .M ~ .: 'i::~. 'er (.~. ~ t·~·~......

r'

'u::

H

o

....

"'1

t .~ ,::

l .

ul

xi

It' -.IlI:"L.:,:\~

W)I.. V-

WI~

JL...;.Il....;.;IS.~

!

~...ol" V-;.U Wi.,) v~

.J..,j, •

v'.. ".;

~jJJ ~~ ( y- ,. )

=

qk

v-;.u ;.".....J~J..olA:oJl ~y u-S-~IS.I ~~~I

t

'

.I.::S' ~ )11%1 .. 1'i

• d

0

dt 'bqk(

·2

~ xi

D xi 2·)" 1'i -aqk +

.~J\ ~

~'-1 ~ ~ l.il,:. ~

u·1

,I

2

~~-

¢

ocik

':It

(-y--)

........... •

1

J-

t1 .~I ....... lcv.i'

(J._,. )

r"l \

• ...;,.~, 1o:..L..::'-b-~1 J~li;J1 W~IIo:..~"'L...

4.i'

...,-,:u.

• W~ ctl~ Io:..~"'~ ~

~ ~6..W,.J1 ~v,...s:;t.iJlw6JIJ

Qt.

:';WI~'~ctl~I.:-'l/... L...i..:l::S'~ ~~. (f._,e) W""W1!J'"

('e_le) d

4J"'14-.~. ( 1 - ,. Jt.:JI);~~ ~, j,

;

x ~, r W I ~,h.oJ I J!..-; ~ :x ~, j)'~;;

J

IIl

5'

~, ~ ~.,..1J'

4.-.l,b

U,.- • • ~l,

m1 ~I JI .~I V4~.J,-JI~L-.lI~

-r al

i-x

oi""';'

~,~ J....:.::

a

~

i/a

4....Sl. l _ ,. JL;J I

:r-A;~ lS,1.;J1 ~~'JI vi t:-"I" ~~~~~, WUJlv.i1

L "" i(m1 +-

D1

2 ....

-'~) i

2

+ g(mI - m2 ) x +'m 2 g)? ~1;~W",l- ~,

m1 m2

> m2 ~if l.il • t-,.l1wJ1 ~ ~ I""", ;;y ~~ m < m 1 2 ..:..,;l! UI ~'..:;-.l: ~~.~

"I ~

t

M1

• ;j.JJ u-i 1.lJI ~ I .r.r b

~, LJJ I J IIa 2 ""-"'011" • ~l: J

'-=:, "";-J I "',...;1 LSi.

The Double .Atwood Machine tJtoboIW~Il...:A • ( ,. - ,. ) Ji-tJ1

"....>1 ~

,1

J

~It-,.wl ~jl

~,...,. ~ ~ ..sr->I;~ ;~"",,,,,ll

LrJt-.:

=-,....:.

"',,;1;..$1.

~,,.....w

u,.-' '-::rJ1":,,b,.J'" V- "~) ... ,, ~I ~ ~}JI Jl.1:;S' J-t-6J • J!-:J I J ~"...,. IJ' :x;/, X .,:,WI ~ L.; .11

~

l; ....... •

T

•2 = .1lP'llX

+ .1TLU2(-x• + x
;c1J"1J'l41:.;..U ! T

= imv2 +

jMi2

= is (i 2 +

~,..U'~~I~,~

Q

/ %'2 + 2iX coe

v.;l. ~'Cb

Q) +

iMX2

..;. ~~ ",:;-JI v ~

V = -mgx sin Q + constant

= im(i2 +

i"2 + 2ii/cos 9) + jMX2 + mgx'sin 9

.,. constan1i

• .:.,....

.~'1...l:$~ . ,

I

L

-..-II

1..J~'UU..II~.i1

,.~UI~'1...l:$~

x ~,,~ ~ .....~ ....... b:lIWUJI .". ~l:S' ~

J-

,

m(:'

+ il' 0013

i

m~x

0013

+ Mi =

G)

G) • mg

0

8in

Q -. I

%

.. -g sin Q Q :x: ::: - - - - - - -

__ /

COS

% = - - - -...2 --

m + J4 _ oos 2 Q m

J--.i

gsinQ

1 _ •

~J~n ~,iJ1 ~ve ~I.~..wl

4..t1.uI~IPIo:.~~L..;

0013

a+1

; •.~...J1

•

~ J,........JI~J

;~ ve;s1 ;~ ~1,.iA ~ J' .....,iwJ1

• QI"".u1 r-'" ,

,~ W ~

,~I

..w..., , ~I ~I 10:. ~..aL....;~1

).'" 110:. ~ ~ l- ..; ~ ~ tt"""1"N ; i~ t 1~';".

rJ;..s.;,;b QJ~ ""~ .a...J.,.~W6.~;;- ~11.iA

1~

J•

.a....ls

ve ~ .L..J...~4~IWUJlul ~IJ

~. t(IXX W%2 +

2 (J.J1 + Is"w/) T1 uol l WJ· ~)I~IJJL...JI~,..-.... w's~!;JItr---l1~ I

J!-:.IIL..-IJ~ve.Qi lfl ¢ ,rJ..",1 ~1,j~1S.iJ1 (It _\ ) JL-..Ul

......_..... ..;I.j 1 1o:.1.i.:.-:..,,rJ..,,, 1 ~ I,j J

W • 8 ... , !;J1er-ll ~ L.nI.. II _&

W

x

= Q cos If

+

. sin 9

/J

sin 'fI

QI IS""

~IJ'~

wy

""..Q

W

=

d

dt

-!L dt

-!L dt

J

I .

z

sin 'fI + ~ sin 9 co s If

~ + ~ oos

°9 oT

oT

----- =--'o~ b~ oT --,.... = dc."rT c.l.f1

~

·U v,.:\i ~

'u ~l • l"...a.-

a~ -

'(,w

Z

il T

~

9

GT oT -=---09

'oT '6 T ~wz - - - --I

d"i' -

(H_' .. )

= 1xx

W

:z:

= I xxW :z:

ZZ

~,L-:: Q'e

a..--Jl ~,...;Jl~ u ~

w II

'ow:z: ~ 'd'f + InvJy 'fJ

a

(..Q ein

--

01 01 tfl:'l 1

•

~o

W

I

'i..

f.= ~

•

I

. .- --r -

::

~

1;-

-.

o

I

J~

•

'd'

Y

1:1

-

...

-
-- i.-l>

W'po l£Jl.o..-i t 2 -t1 u---a ~ 1 ~,. ",,.-'..all,' "1:...,,,lI ~ Y ~ ,)l.1

w~

~ 'i-;:-iej

...,..soW 1J.. ~ 1

: i-JWI ~L. -'1\ 1-iA • • • L....:.L • 'fJ '-""""

I

I

I I I

~----:+-----.-l~-- 1;

t1

Q.k ;1 qk

1-.J~W've;r-'L, ~ W'pU

UJ.I""- W, ~ vI>:

t2

~ ~. (•• - ,. )

e-""N

J!-:JI

~~~L..v' ~

..jJ,

Lvi ,,;.j ~ ~ ~ 1 \_ .hJ , ,)l.I;;,-S oW'

l:........

qk

'i-;:-ie;J 1 l.t-=li::....:.., qk

~ 1 ~ Lrl ~

ryy

1;2' 1;1 ....i&qk=O ~I v~

1;2' 1;1 ~LUJ ~l: ~ ~,

vi cUoi ~ r!f'-" J.~I ~I ~~ v oil

~I~

v'!: .i!~ il;· .. qk ~I 1:..L;lobo~l~lo:..ilS'l..il. v ~I

J--~I

J'

~,; ~ oboJS' ~~ vl~ v..il • ;.1;;; -

v oil • ~ d ~ oqk - "Ctt "lis: = 0

(k

• 1 , 2 ,•••••• n ) I:.. ~

.I Lw I. L.:: t.;'" • .:u.

v'\S' ~;...JI.I ~I,; .)lsI...,..s.w'J~~1J L.o..i.j

~~~,~ ~ ~ • L.hi6..

;

J.~I ~ ~ ~

.......i.;

• J.=L.J I J L.l; ~ ~ I W'~ ~ 1j'J a-~I

~I b qk

J. LAJL;;r.1

~

n v-J.ii

'~,;JI ~ ~,~I ..t....>1 ~~~

l...ii" ' ; ~~ ~

I.j~

u JI 'O:L

_

; . ' :..

11 n-l

(' ~ qk) dt

=0

n - 1

•

">--;...-JI ~ v ~I J~

~..uJ1 JL;;r.1 ~ ~~.

J,.~I.a.-l1

• ~ I V1:-,;J I ~ ~ I ~,.a.-l I ~)l;;.;

oqk -

d o :L crt -

'b ~ ~I 4-.Sla ~ u,J:;;

i-J6.U; .i,;.J1

~,;$'~, ~j ... L.. ~I.,::'

~

~.ii 1.S~.a.....,. ( ,_,

~I ~Ia....sl. ~ ...jt _~ ...jJ W'~' ~j ...

~~.,. (

( t -, • J...JJ J.Ji ~

W,UJIWb.J-r-'-jl~WI

I

.

''.HJ'4li. u'~' ~

b..I:$ u LJIi

J-:;

]14=

I

'm, .,

ct'~ ~

u'

.;I"')jl.-a J':;'-

J-J-4(.

~'~I~

= 211

~I. (

J

~

"_'t

.d~,.,.

~\

(i

.r

~

-:

..

l •

r.

(

-u

~

1..

-

... .

~

.t

co.

•

-L.

~

-

"'.

C·

""~' &- L..

-

ts

.

MN

.+

N

+N

~

e .

Co

~

rCo

.r

t.

·r

-

=.

~.

• ...-

-

r-

'G..

c·

'f.:. '"'

~ 'f-

I

_. ~.

:

Ci

-:

[

~~

Ii>

+'

f-

Ccl

H' I

,,-..

•

I

..

--

1-

~_

'-' H

;

-!::~., Cel'

Co

[~

~

(i

-

r:= ~

0/

H~ ~ ~

:-c

01

~

01

1 ~

~

'.

1;:- i'

.t'

L..

t:

~.t ~l -" r ·[r .. · .. t -_.:.t f c." ' i .-

~

~ «> ~.r·

-

c. _s

-

&- ' ;

'i

f:: . '--

l-. • . . ',,-

.[ ._ c·

~ f-~

. L.,i'"'

~

r

~' •_

.

'ro

I

f ' ·ro -co.

".1

N

..:-

(i

-

~'.•

Co

I

r tE 1 "r "4~-

&--.

- 1;:-[ 1-r• "f'· c-[~ r.:.~.~ it'~. 1

f'

1.r

_

C['

L..

[

ttL J I

..'a. -

.t 'l f ~ -;,~ fi--i.r '~I ... r::-

«>..

'lSI.. •

..... Co

. I

(i

-

I\)

I:S

-

NC>[

~

~ ""-e...,..

lD

r'~

N

-

~):.~

N

+

l'I

L..~.

.,~ l. [

u

~.fe.L ~N'~. F

Co

N

..

r~

C-;~...,

t ;:.t

f

~

''114..i,p.-.J'I.J.: ) I,ri:- '11_ u)!,.:I ~ .... ~ I;,,;l1.l1 ( ,-, \ Potential EnerBY and Equilibrium. Stability • .......&: uj!,.:ll ) t-::.>~ ~ (, u) t,.: ~ CIn,··Q2' ql

J» ~ isP' 4e, I).J.: t.~ u I J.:i

~1.:..l;I..\.>'1lu~ (, i...f"'~)..s n ~4.~ulJ:'}I

V ~~I~;UJlu~

a..ib..4.,.J;..W1 u l J:'};u,...· )1. If ~~I~ IS I • .h.i;

Qk

=-

OV ()'qk

(k

= 1,2, ••• •

n)

q' s .:..l; I..\.»U -.J l..s

.

_

t·

r r " "l\7 ~. . 'G-. e;

c·~·

• or:

0;; ('I

.r. ~ ~)... t: ~ r - ~ ~ Y';.JI

4

v,......s.: v I ~ v J I }:-ll vj ~ I J ~

•

I.,i::- v~ ~ bJ I.;..

J ~i ~ I u I

• u~1 ~Lr "';4-i.WIUUJI

v = V(q)

( L1l)

vi 1,.:1' '" ~-o dq -

(O_ll)

~ l.S'J~~I(J$~ ~~

... )

( ; J ;-

(1_11)

( Y_ 11)

v (X)

o

X

(1 -

~ C J B

v l:..bi.U1 • ~ uj~

\l)

J.L:J'

A ~I J:,.; • T(Z) ~I ~l.b llloJ

~

&:

f:.

1...

c·

o

n

C>

c·

c.

'&

.! C'i

t. k

C> ~

g

o

gO

-

cr 't:

c·

rr

f

t-

~. c'

_

1

:l~ c.

~

•

~

~

~.

-

......

-

~

L.

-


III

0'

PI

'f."

\..

c·

u

o

C>f


II

~

C> 1=lo41=loI\)

~I~ ~

t

'::...

'

t

• l" ':"t; 't 6'

_

f

keto

't~~E

=i

k=

~:

fl

l.-

W

~

i-

'1

'f:'

~.

\:,

_

~.

i

j l'

.r \~ t

'f."

<j' ~. c· o'£::. \..

~ .~

c·

~PI~

~.'

~. t: •

_

!..

\r"t :.

t.

fi

o -.f o -
, \r" tC.. r. .• -u. 'to , ~. ., ;::

1 ~ ~ 'l. .r CI1;I

-
~ 4..,p- ~).) '-tJ ~,4 ~,I vi'P L...w. • q V( q)

~I

a

+

;:: (

.-:' d,w'

V( q) ;:: k

V (q

o

J

1 + 2T

.J.:>J I ~ ..,r'Y.::...r.f.

.J>

r k

e::-...e,..

~.A

q=a

~'..:,.;I5I..iI, v ~I

l ~I u~ I~ • k1=(dV/dq)q=a 2

(q - a)

J, I ~

q

2 +. • • • • •

=a

a.£;J I

~ l- ;~ I.,::'; b' l..i ~ • 4,.

dnV/dqn

=0

(A - \ \ )

J vi J,:j I },i.~... 1 ~

u~ ~...w.'~i n .J.:>J11~v-'ul5l..iI.~Wld,lWIJ'

n

) L.:.. ~,

dnv )q=a dqn

~----,'~. , v.i~

"

JP- ~ l-14..J .. bS

1 2 + k 1 (q - a) + 2'T k 2 (q - a) +. • ••• ~! ~ (q _ a)n +

= ko

~

c

k o J,~'

;~ 10.:..:15 l..i' , I;i::- ui J,:j I

v1vi'P' clJ..iu,i: I..iW..s; ~ ·;.a.J:fu,i:viJ,:jli4::.)"; n ~b',1

L: ~ uj !,.:J I tl.i: ~ ~ '" L..::.: ~ ~ ~ F

= _ .k oq

~

_k II

• J, ~ I ~)t:J I Ji.

.1>J I ~J J,:.::

(q _ a)n-1

..::.--'IS' 1.;14;lL... yl \$1, a po: F .~I u~ u I~ ~I ujt,.:i! u ~~ .::-; IS' I.; I.hi; iJ bJ I • ~

u,s: u I ~

• q

.:~ul ~ 'L..P'c-~J"';;~ ~L..,I;;.iJbJ '~L:l~

•

uj!,-::~ ~!,.;:ulS' \.il .1;j!,.::

L...i,

......~H,;ll~~

~2V k 11 ... (0 k

_ (

12 -

q1

~~ ~~L;..~ILilhlIUI~

2) q1... q2=u"'~"0

~

.

()2V

()

q1 = q2= •• ~~ = 0 u~~

-,.

Q1 oQ 2

.J,~I~I

)

q1=q2= •• =~=0

.J.il. V(O, 0, ..... 0) ... 0

~~Il:...-...:, .1~rU,

• uj !,.:: e'" J» u IS' dill u ~ c¥iJ 1~ ~ I

T' '\ /)

• ~l:.lI~)"JI~ ~ (\ ._\ l) ;J..sWI ~~I V:H~I

~ ~,I ~ v'!:: v 1 1. 1..s I ~ •

Y ~ 4~ I"..u. ( \)

~v~ ql..q2= .... ~= 0 v;J,.:JI~ ~~ ~'s

Osoillations of a System with one Degree o£

•

c. oJ.1> IJ l;

~

~eedom.

f.rJ I Ulhll L-: l:S' ~ 4 .1> ~ 4...P" 4)..s LrJ L.., t:w ..::..: If IJ I :~l.S" f i l i i )J q2

~I"p' q

:;

II

( 1 1 _ \ I)

.,Jlc.~I.l>~ wl..s ,I ~1:

4 )U .. )U (0) +

d,u

(~)

aq

q=

q

P JoWl v~ ..u Ja

J ~'-I ;1 ..,1 ·s

)A c:J.; ~

0 q + ••••••

J');u,...;

'v;~~ ~

q=O

• ~

,JJ. .. )J

(0) ...

'.J b.

constant

: ~ l.! ~l:S'~ L

I.:-lIfJ.H .,wl..s.)-

(1T'_") t!1~;JI..svl..sY (,\_1 \) ;J..sWI~

( \ ._\ \ );J..sWI J~l:.lI~)..ulc.l.i~ ",bJ~ ..s""r'J 1vlbrJl ( I)

~~v,s;~

• k ll

k12

>0

kn

k 12

k 13

k 2l

&22

k 23

k:51

k 32

k:53

>0

L

=T

- V

= 1- p

q2

*

-

kq

2

( \ L\\)

~p-1I il~ 4.l.JL... • k=k2 = (d2V/dq2) q=O d

---at

'bL ~ L q=1)q ~

).lQ. + ..J----....J~ ~ k

kq • 0

> 0 c:..: IS' l.i lIS I ."a::.- vj ~ ~..",. IS'~ .J.J~vj!,.::Jl~

q

J»

= 0 c:..: IS' I;I v ; I ~~ q

~~.i:;;

(\1_\\)

, q :: qo oos (w t +

€ )

( l Y_l l)

J..ts:::ll~!,: ~ ~ • ~1~lj yrA € , . ~.i..:.i::JI;-- j:.:: qo ~

• ~1~~IJ.,rJlo-

JL.:. jL:J1) ~WI 41 JI:. ~~1S.il1;~1iI1_..., ..".1I~1~,.. ~;I

v,

S'" • .hi; ~ I..., oJ ~,..

L-j,..J I ;.;l\J I c41.il Lu". T

=t

~

Q ;;~ ~~;J

m (bQ)2 + t1 em

v--s: ~ I ~ •

J..-: 4,;,.,uJ I r~

w

= mg :: mg

l.:) I v IS'

c41...i!. 4.l::SJ I j J-

a -(a - b)(l-

J-

~I

: ~ t.S v'!: !

J» ~ l.ill ...,.-iJ I r j-So ~ t.S V

Q2 G4 ~ + ~ -

.

.

I oa ~

Lo..~IUUJU....II",

[a -(a - b) cos GJ [

I.i I • ( \' _ H }

,.oa ~;iJ I 4.l::SJ I j

92 ~

V(Q)

(J"

•••• )

]

nv

L

= *(mb2 +

l om )G2 - * mg(a _ b) 92

~..lWIe;.W)l.iJ1

.w,'

~I ..l,~~~~1 Jl.A1 ~

u I uy ( , ~_l 1) , ( 1 t _1 1)

,u= mb 2 + 1 0m k

:ih,:

= mg(a

- b}

... :d ,:u\,.:4S'p-

~u,s; Q

= 0

ujJ,.:11

~J»4SpJlul,;~

c.s,ljll IA '" "'or ( lA -)

vb.,,J;,ul:iil,;v~~J,;,

Two Coupled Harmonic

n

t _1)

O~oillators

• "',.pJI~~)..l u.o ..l~c.s~~~J.:JII:-l.,J::.WJ ;"WI"'~I~ vi JJ

• L-..~..l;.~\,.:~~J.:.~w,s:.4~.~ UP~~~J.f~J ~ ~;;-.

,

JIl ~J$~~

~

~l&.

...

'¥b.

(\"_) )}J1AJIQf1;:

k ~)l..,~l: ~IN, L... ~l~l a.,..l;~. k

•( vw,.s- lJ'

:It

V'J~+

~)l..,

.b:-=~l) r:t-~ J-;$pJI l~~lvl

k

~ oJ;' ~ I,.: VlH ~..i:J t:.';": •

r -, , JS.:.l1

~~~, ~llI",'-"JJ~WI",,~.~H~I..Jb,~

,,-J ~ loil; • " Ii,:,.

B 2 • B 1 • A 2t .11

~ I. ~ J.-...: WlIIlW

a

ktlt' k k" (--m. ... ---).1 m. 1 + --m .1 2 == 0

~l-J~.

( ~ 1_11)

C08

w

..::. ~ '" lAJ I

1; •

J

~

I.~(l)

t.

k+k" k+2k.... k" (_. m - ---)B m l - - m B2

Xl

= A oos

w at

1,""-; u.w I..::..l: ~.l.:l ~ • ot•._ - - :

X

2

J

=0

=A

COB

wt a

(ll _l l)

~ Iu,pJ I rI~ I "" J I~ ':J 11S"",r<J IV4e.-....,.I,

.r.- ~. 1S.iJ l..brJ ~ . normal

~).?-~ ~b.. "" • .,l.,l,rll

u..i:.:

modes

4..,) ~ I~ l: v-:: ,,~I J,.bJ 1

",:-~.l:::..::..t;I.1> ':JI~ u 1~ 4..,)~I~1 ul~ ~ a .,l.,l?I....,.;":'~.l.:l1

Xl ~,r--s

= x2

":' ~.l.:ll u~

Xl

• Symmetric mode

= -x2

.·~I..::..l.b.W..~ t _11

~I. i»l.;,:JI~i~l:".lAu--:;

~..JI.:-aJI~ v.j- ~Ijl..:..~ :

( f. _\ \) JUJI

~I,;~~..i.:. ~~jJ

The Complete Solution 4

r-Jl y.... ~ I ~ 1s:l1..:.. ~ ~ W I c,;.}:, ·I..,J I ~ I ~,....D

I ~ WI, ~

~ x' B (,)

j:

Y

~ IS ..i.l11.J'""~ I J.:oJ I u I ~"J 4.l~ ~ • (

~~ ~

COB

W

UIISI4.....J~I~~..:..I.J~?1

t V4 ~ ~

Bin

c.V t

'f._\\)

J. L-.l1 J)L;.. V4 v.;J I

u..i: u41S! ol.i.:L..lrJs. w..-

~I ~l::.ll

, (\"1_\\)

(. r

.u J,kJ I v I ,J--.: ~ ~li.:ll.::.:~ oS W

~,.b- ~ I

V'!:

~l:1I~1 ~~~I ~I~I.:S ~dJ.i1 ou~I~,b. ~

l:,-

t-. ~

,

=A

%1

0

,

cos wat + A sin

W

at + B cost.,\t + B sinwltt (ry_,,)

,

/

=A

%2

I~ IS' W,

cos tV a t + A sin.' fA) a t - B cos Wbt - B. sin u)b t l:~ t = 0 . ''1.; . .i1.UI •.I.~,:~1 Loll· c.:..l-.ll V4J" '-Ii' u . -,,r- V4

=A + B

%1(0)

c. L."')LJ I ~ t=O ve,;J I ""

.

%2 ( 0)

= A wa = A. cJ a

J.-:.

- B

vejolJ ~ 1.: ~L;.::

t

oW •

dJ..iS'

WlJl

,

~

%1(0)

=A

%2(0)

•.••

~

+ B ""b

-

Ii "'" b ~.::..L..-lJ J=.JI ~~~~

i

A =

[%1( 0) + %2 ( 0)]

'

:B

= ..

[Xl( 0 )- %2 ( 0 )] (rA _II)

i. = --L.. 2W

J..,J

[Xl(0)

a

+ :i:2 ( 0) ] ' B' = --L. [ xl ( 0 ) -i 2 ( 0 )] 2W b

.tJlve~..J~I~Ic.:..~

·.a,.ve ·~I ~

'J

"l~1.:W

~ .u~lul JhJl ~ ~,"",}I ·41~~1

t b.:~I ....i; ...i. 4..-J1..s.:..~1 L ."~IS'~ .~,r_lil.b . ,... v - ":T.7

%1(0) o

= x 2 (0)

c-: c.~"lWI".u.

,.:i:~(0)

= %2(0) =

0

1.J.i...• t -"" L...:.. V"..J

·.t. -r),..

I,,'--JlJl~l~

A 1..a.,_-"rl...::...:!,:JI~ ~)l:;u ~ • .b.i;~P1;.:JI~~1 ~ ~ ~U

~~~I ~ ~L...:. v--..JI.l.i..:~I~dSpJlc.:..

t..\.: '..il ~~

~ (0)=-x2 (0) ,xl (O)=x (O)=O ~ ~I~ ~1.bJ.r:.l1 ~~, lilbl 2

r

{

S• .hi; ).1;.:J I r-i

v,

.~,

.

{

v I loS I B I,"",L I~ loS, L...: ~ I,:J I ~ 4.J bJ 14.iA l:i tV J4:i ....,!:.;.II'-;--.l.:.i::v~L.~;;~• ~

Normal Coordinates

;; ~ ~ ~ I; I.J.> , L.,J;.;. r~

~a; I,.: ~.l.:.i.:.

t' o.l';' W j>' u...,J

~~ts"4,;,,-J~

1 q =-

a

,

[2

qb '

qa

(l'\_\:)

1

q b = - (Xl -

{2

x2 ) l;~. ~1;'.J.>~'4~i.I~~ l!1~i.llo.lVS~

k~

1

xl

= {2

(qa + qb)

x2

I =----12

(q

=k

+ 2k'

a

- q ) b

mg,a •

-kq

a

: ~ lJ'~W~ J,L-JI ~~ ~ • t:.jj .u~ Jlt:. ~ -slAJli

~

= (1oos

qb

=113

eN

a

•

(

~at + ~a)

oos (""b t +

(....L)i m

"'b • ( :. )i. (17.

~b)

(

2k)*

k :

"'01~ ~I ... qa

';lo.bo~I~~~~~I~,PJ'~~IU'y

qb ,qa • .i.:~I~l;lo.bo 'lI1 ~. "" b

.J.J~ qb

~l.». ~l~~.l;,

• Normal Coordinates .&

J... :~l;I.bo)U

\:=.h>

Y'';U'J~I ';I..bo~I~~. a.WI4.lbJl ~

ci1j'J ~~.Jl-j;: ~..w. L...~~~,"",I ~ltJ~~roJlv::al1JI~ ~Iy.l'" ..bo.~ .J.J~ ~~.lI,

• (n _\ \) ~'ll.JWb' l.~t.,.::I~poll

~~ .J~ ~,!~~~.i.:.~4..,J~I~ljJo··, .U'J~~I..bo1 JS'

;

~4.....,J:.WI~~.l::

L..s.:..:,

'U'J~I .. .J.J';~ ~1~U'J~~I.J.:>I4..,J~

..b....i; .J.:>~ U'J~ ';1.J.:>ld~, .J.:>~ "'.JA~~I~~~.i:.:::i

.:,. L,J~

• ~IS,L...:: 'lI

't

.t

c'.

c..

tT

o£

c~

~

}- {: 'r-

,F -[ c-:- ;:

-

-

J..

~

'f:"

.r

r0'f-.

o

..

ft'

r; ~.

.0

..

0'

to

of

.

II)

to

0'

to

$l'

0

r

,....

fo '~

..J ''Slv.il. qa

=~

qb

D

~

+ -

~

Q

.f2

9

)~_~ ••t-"_...oI:11 ~ Mo...,. ..:,~

1"2

±.

J I ~ li I~ 4J loJ V I ~~ v--~ ~.r ..u, ")lsI •.,,s..i.J I ~,J~I..:,L;I..1> 'S14J'S ~ 4.:~.i.:Jl..:,l.,4.U

.. WI4.-pl

~

(1 _, ,

.

General Theory of Vibratins Systems ~ L:.::.;I , ~,.pJI..:,~)oJ <J4 ~)oJ ~ h.ll ~J"J I <J4

4J loJ

v~ '1'

n

~

.. ~~ .)I v 'SI oJ""";

YpJ I..,;UJ Iv I ( 1_' . 41)

0-: WI J....iII

u,....-i' L..+vjl,.:Jl~ J,:-- -SpJl~1S' W, • -S~..:,I~ ~'S v l .1A o.J'II ~ lvi,

( I I _II )4.JoJWI

dl.i 04 pi J;>;A:

u~

1_114 1

• vj I,.:J I~

J

IS' J;>j.;

J W IS, L..:,

04~vjl,.:Jl~~r~I"";

«. 1 • T=tk 11

q~

+ k 12 q1 q2 +

ik22q~+···-

L L ikjkq3qk ~

c}Wl J1:..Il

~

lJUjkqj + kjkqj) • 0

(k

( tA -, , )

k l!l~

111., .l>l:

~~

= 1,2, ••• n)

j

(k.l,2, •••• n)

(oY_ll)

u l A' 8

J.!,.oJI..loboe

-).( 11"'" 2 + k 11

-.fJ 12 W2 +

-P

-

21W 2 + k 21

• • •

P

~)l:~u.,WI ~ J=.J~

k 12 • • •

22W2 + k 22 • • •

• ••

=0 (0 r

_, 1)

•••

J,~l. c.o)2 'ri n ~J" v-. 1I .,l- ~ • ~1.,r,S..wI~, ~11I..lWI • 4-,1;.. 11 ~~I~I..l.J;;JI~L....,.

~

n

LA..l~ ~I

Ul

~..J~Ic:..I;I.1>~I ~~

Existence of Normal Coordinates

clhJ.,· ;..l~;~ ~v~vl~ (t Y_l \ ;J..lWI )~Ic:..I;I.1>~1

~J~I ~J.~~IS' I..il: ~

('()

~1c:..~I;"'J;.;J~l..l~ ~1:.l1

~IJ..o>.:lI ~I,.; ~.w., ;..l~;~~ ~,~I~lS, ~=

Lj

CkjY~ ..sl,

(k

~.,.c:..~ ~1~I:.lI~J~1 ~ ~ ~

L Lajkxjxk = ~lyi + j

2 2 0(21 2 + ••• + CXn1n

.It

L Lbj~jXJt ... j

= 1,2, •••• n)

k

~ lyi

+

(J2Y~

2 + ••• + (J n1 n

See, for example, L. P. Smith, Mathematical Methods for Soientists and Engineers, Prentice-HaLl, If.

J.,

195'.

Eng~e~ood

Cliffs,

..l-'='-'I;j..lt-",,~vl ~ ~I"'..;...JI~

- K811 -

+ b 11

-

~ 8 12 + b 12

•••

=0

~ 8 21 + b 21

...

• • •

• • •

• • •

...

...

·..

·..

.~J.:JI~I\SI ~..l.Mot.:V-: y:J1;j6JI~ ...pl~ ~

•,

'rl~l:~~~

Lj

~jqj

~···,q2

'~

..::-l;I.1>I~d1:...ll>vl\Si

(k .. 1,2, •••• n)

4.:tll:lI;;......~I~~1 ~1~ !

=0

= t(}1i~i

+

P2q~

+ ••• +

v

vl~

,T

)in~ )

- "':"2 - -2 - -2 V .. t(k1 q1 + k 2 q 2 + ••• + ~~ )

Jl:lI,-.,J1 ~tl,L.....t.:;j~1 ctlj~ ;jl..l ~ ~~ L ..

L:t

t()i k1i

- kkqi)

(

0

V_I I)

{

Wk

\

,.

= {kii }lk)*

L-.rJ ~ I"J I ~ ~ 1..:- J., .,,,:J ~ i....J ~ l..:-l; J...\> 'S J..rL-....J ~ 1..:..1., .,,,:J I ~ ;.,;J""'t

qk ..:- ~ J u .i J

".

'J;.; ld:.,JJ I ~ I 4. W I i...J:.ill LU" • W k ../'

u, ~ .,.J.:>.J I IoU ~ l:S' ~ • ( 0 r - \ \) 4J ., WI, •

:i ..,1;,.;,\1, (0 ( _, \) :i.__~.H.....

J

J, 'S I .,.J.:>.J I V4

~~I..:..l;l...\> ~I

4.1., W I i....J~ b I.;'...\> 'J I J.::,.>.: ., ~I 4J L. u J

I,i;:. 'J~l. ~J .J.il ·u~ ~I~J

J::,:..; ~:i..WI

. u.:L.J14IJ~I~~ ~WI

l>.r: ~- t~}.>:i""";J.,u~

.,~I..:......;~

d

Ubi •

~~I

J',u:: u"'; "'-=-:-1,= ~4.~~ AS'p-

U"';uJ

.,~

1.;.l...tbl •

L..~L..,l::...o.. ~.i..:.i:.I

~..:..~ u-','JI~J..JIV4trt'~l:.::::4jJ J',U::u"';~I. Jl:.lI~1 ~ ~1~..:..~.,t.....~l:S'

, V4 ~ Qk ,

Qk

·li

=-

o· -

•

-~b:.Il..r..::I~I.:..A. ~jJle-I.:-I;"'..Jl:-....J1~I..:..t.-J'L..:.::.:u.i1

• , ""

>t,.J1

~I

~WI ~,I

CpJl

~l.-

.....11..:..1; 1.1>- ~ I ~ ~W l-SpJ I..:.. ~ .) t- .i.>b· (1· _ \ \ ) ~.) L-\I ~l:.Il;;~1

• .::.--: I$' l..i 1•

:i.i:bJ 1..j, . ;;;W 1- ~ bJ 1.1r

~

-..- ~I .i.>l:: '-t-l: l.i::

4.S,~ ~.).)~ ~I ;"'..J~I ;;~I ~~..-j';A-l1 ~ yc.:.

• ~I ;;~,JI

J ',1..A:l1

~ u,s::~I..J1

.::...:~

.).);:.11

tto

J..->..

Vibration of a Loaded String V--

J

(y

_l 1

u,s:.:. ~ ~ ~ L., 1;;.. U.r- J ~':J 1~ 41 IoU> 1.:1 .i:;..L

~u~ l.: ~ L..:...

II

yr:J I ~ I ispJ I4..'):":

Jh

~ ~).JI ..l,..u..U#u.JOh

~,~ ~ JH_~ -l~ u1:~L...:. ~ t;.~A~.~b •~ ~ I \.;..l,.s:, ~ ..i.: ..i.:.U

• r..ll4l1 41 ~ .~ ••• , Q2' ql u1:

L......:tJ"J.

u,!: u I 1.I..spJ Iv";:"" 4J~, • ( 1

_I 1) ~ I ~ ~ ~

~IVt{!"'""~~~ ..u".i.:i>II~~I4J6J1 ~~"

Vt

IJ'

~""'~JI~

~UpJILilh.lI~.w

'W,lJl4SrJ l 4.lb. '"' ~oW. (, ~ISI ~ljor.J V urJl w......:...11..i1 ,.rt>

J)

+ 1 , JJ ~ I t.J-:-:

oJ, ~ I hl l .~

tK(qJJ+1 - q.JI )2 V----..", ~ I ~ I.h..r.:

..r' [h IS

2

+

(

J/ + 1, J/

J......--li ,,::,J I '~.hh.::

-k

2

K

~

~I~;';WI (, A...:."....:.-lI:i5j ...Jl4.lbJ

Jt =

h +

(q>J +1 - q.v )2

..i..->IIS ..iJ I .~ ~ ~ I L;1lJ t; (,

s ~;= ~

,,::,J I thi. L;,J4 j. L... ~

1 2 2li (q;.? +1 - q.JI ) + ••• ~ oW. • V-..J-' 4.:. ~ ~ ..:> j t,.::J I 4.S L... ..r' h ~

q.v +1 - q,P )

A.t=

IS .i-J I

;:,J I l:! .J..:J I ~ s

015' l.il (, 0.il ~ )~,),I)..:..:

(q.v +1 - q.v )2

~,lJ1 er'1000":>~011.1L.~~1~~1L;l!~101d1.i000~

v= ~2

• •• +

('In-~-1 ) 2

+

~]

("

_ \ \

J

L~,

k=+

k=K

( A...:."....:.-ll 4.....SrJ I )

f. \ Y

~L

d ') L . dt 1l i ) q,V

p

V.l,2, ••••• n

JI_,.

.,.

l,,:-loJl ~ • e,;,..;lle- I.:»~ ~ q' s"""; u";': ~.ill ~I 4:Jl:}I~~I~~1

~1~1~~·.vth~~~.l:H~-~a.v ~

.:..L-l.l ~l:}IL.i)l,J1

ri? (1A

_\ \) ~\.;.:II.:..~.,)WI~ ~WI

(Y·_\l)

L:-..i, loi I ;:,J I ~).

~ ~~,.::>-:.-;;Y..A.\ I 4 ..i.Jb

(Yl_ll)

UA

_mtJJ

2 + 2k

o

-k

-k

-mw

2

+ 2k

o

-k

o

o

o

0

o

0

o

-k

o

o 0 0

.0

(V,\, _, \ )

................................................

u.,,) ~ \ l;,; ~ ~ 0 \ ~ ,

o

4..,)'P." .;.a..~

••• -mw 2 +2k

"" ~ \ V4 • .u.

t

n

,,) ~ I V4 ~ ~ ,

v!I,

• rL,J\ ( y._\ \) ;j,,)lAJ\ rl..l.:>.::.w~

~l:.Il;j,,)W~ a.v~L-U4-W1 11 ~u;"';, l.:.A a)l

=A

sin ( JJ

11)

~J.-: ~~

( V T' _ ( Yo_\ \)

\ \ )

;-i)W1 ~ rL,J1 ~ ~

-m ",,2A sine IJ ~)= kA [ sin(.v 11~)- 2 sin(V~)

+ sin (JI ~ +

11)] ('(0_\\)

-7 1

w

= 2wo

Sin-4-

( n _\ \)

( YV _\ \)

r--' ~,I

~

~14..I~~~~1

..::..I,,"rl l (Vl_l 1 )4J.,WI

4Js. w..- ~IU)l,J1 u.J: ~~~I J,. 'ap

~ A.-1JJ~

,

A.1JJ~,

A

J-.:.

v~l..:?>- ~

cOSJJ~~l:Il..::..L..2." ..=HvelS~

~ aJ,l-.A s1n(.v~~U,... 'Jb.~I~. ~~W''':ISI,I

4£1.-IJ,'

~

,,:.~ i.;i.i....lli..}I1":-l...> ~~,. a o

J

~~ I ~ YJ I J.r

=0

~YJI.br

J-::-: ' ":- ~.i.:;.J1 ,,:,1J 4..,J ~ 1..::..1., .,rl I ., ~~

l:.....i, 1.i1J..rJ11.l.A~·'.""·8u+1· 0 (n + 1)~ ... 1l71"

(VA _l l)

~ ~~J....-; ..iI, ~ r.,..u~ 8u+1 ... A sin (1l7T)

ve

lSI

J-.: ~I ,

1l

~

0

III

4..,J~I..::..I""rll,,:-L- V ~I ~ ~

.J~to

Iff

wlJ'"

2c.)0

sin ( 2n + 2 )

(Y'l_ll)

..::..L,,__JI v I ( VA - 1 l) , ( VT' _I 1 ) ..::.. ~., Wive IS; , cd.l..i cJ I uL.; ~ to ve

l), =

.il~"

r ·i

ll71"u A sin ( . n + 1

.,L;;.11

J..;..

)

(A •- 1 1 )

~,,: "'L..l> ~

J:.:

~,,:-~J.:;~I4..,J~I~lcJlr

V-- ~I

lJ"..::..l-.J I r-J'l cd.l J,

ul-: ~~ Ie:-..u

P =1,2, ... n ~I 1:..A

N ... 1,2, ...... n

I:t: ~ ~ I ~,J ~ I ~ I ~ .J.iJ

( Y _l l) JU.!I", ~)J' lJ"~~ ~ea:'.l.A, • (A· _l 1)4..I.,WI

(A\_\\)

~,'JI~~I

N S

a

,

/ /

I

\ \

,

\

I

~

I

a

- a1 ,

sin

, 3 rr

J:I{) •

~WI""~:-..JI

\

I (

I

A dn(

, a2= 0

A '\

I

v=

=1

I

I

,

I

\

-A

\

\

,

J I

(

I

-A

I

N

=2

sp = A

Sin(~)

/ /

~

... '\

~WI.~I

N

A.s1.l1 1f"

\

= :3

a.v:: A sine ':7")

U\

I...... i --"~.r--t.••.:;H~· ~~I~I~~~r~~po1JrWlerJl

J:l.S • ~1~'SIJ.,rJlv.~

L.A.:-li J,.....:o.

oJ.JStt 4;)liJ~ I~

Jl

JI~I ~,~~ ~_;: 1; ~ u~

,~

~I~

c. ~I oJ.JS ~ u~ ~I iJbJl

If .,,-/( 2Jl + 2

u .i I •

~N

)

J

~Iu~~ (,

( Y'I - \ \) iJ oJ L..J 1J

~ I) I ~ Ij 'S ~

N

~I

(A\_\\)

oJoJ~ LJ., 'S ~c.tu.L..:.. ,; •

!

C-."i: u~ ~~ 1e-loJoJ? lui ~~ \,ip"

;\·.;.JIa.....J~lc.loJ,)?I;l;.1~ (, IS,,>-I~)~. 1!'(..c)c!(n + 1) ..

;....;oJ ~ • '.Po-~' •

(, ~WI ~J,:JI (, ~WI ~J,:JI (, ~L.I

c.~1

oJ.JS ~

4,.)1 iJoJL.. •

lJS"4lo~1 ~J,:JI

~".:- ~,,:-,.i.:.i..:

;....;)WI A _\ 1

Vibration of a Continuous System. The Wave Equation. c.J.. )

~I oJ.JSub;~;~~~~4.k,...."Jlc.~Iv.u..J~.r-J1J>p

; ..)\...,--' ~ 1.:.:., Vo',.A 4-..:. ~

, t,......:J I 1.i.A v. ~

...,1;.:.

JS' ~ ;.; W

J.:.6..:l • ~,I J::.l r-;:'

L-..IoJ L..J I , ; oJ, ~ __,.W ;Spo1J

~ li..~H

~ ~I

J

oJ, ~ ;+

I.: ~ (, IS;>I ;)~

c. 'S.,) L..J I ~ l;j' ; ..dd ~)W I ~

Jl:JI~I...,h

(, ( 1A _\ \)

!..il • v~~ '.V--""'~~~~ vjl,.:ll ~~uWI j:.:: h ~ 1.:-~I..ljSv~ 'J~I..t~~1 ~6.~;~c.L;WI ~ ~~~~~ 'uU1IJ~lte4....;;liJl.:;~

~

qp +1 - q.l1 h qJl - q;.r -1 . h

• h

2..9(1)X

~

x ~lvlS'

hu~~I~~n

)x:Vh+ b/2

..ll.. ("bX ):x:aJl h - h/2

,,~~ ~ WI ;.i:..:.J I ~ W I u--~ I ~ '"'~ I ~,~ (, ~..iJ

fD" ~I

q

.p

- q

h

JI

2

-1 ~ h (.:L-s) '"Zlx2 x =.vh

(Ao_\\)

Jl:JI~1 ~;SpJl4.1..l~~l:S~u..il (;\1_\\)

•• ...

~1-~I";Jn-:.JI~k:JIc.~..lWI~(AY_\\) 4.l..lWI ;;;l· .... ;~t""'!,..~U..lL..::~

• .b-~I ~1c.1..i;~I4.1,,)L.-.~

J> v I ~ J.i>.:J1 J.t-J I ~ • J.i:.:J I,-;-I"h.:> ~ I ~ Ir,.; ~ I 4.l ,,) ~ j,.l> J:..:: ~

1.5',... ~14.1..lW

rWlt~1

Or

q = r(x + Tt)

(A'\ _II)

,I q

= f(x

~1:.l1~jWI

- Tt)

J,:..:; •

( '\ ._ \ I)

•

~~~WI x

T

4~~r-; ~1"h..Q1 J;~I

• ~I x 4~~

• Yj!,::~ vo ~ ~ '.B- ~Ijl,-D' ~

4-b

~

l:.l.rc:=

t=t1

~

4';..A

,

1::JL-.

~

u,!; .J.i • (A _I \) ji:J1

t=t ".- 2

--+

"-

~~d~ ~I;"':'I

q ,-:,1"h.jl

1.;1 v!J, .,;..JI ~ h

4' ~ ~ t;

S ~ S/h ~I~,~

":"?=

1.~4.::Y ~1."..:-J1;:,lJ

1: ~ 0~

" ~ J~ 1~

.,\>, 4.I..::S' , I

. ----s ~

k=+ ,...-...,II ~ (

AA - II ) LJ"\WIL..:~~

(0

d..J.i.I

LU"

'-:--_~;,pJrt»'- J~,.hJ I..::.. ~ I ).:.::: I ~ I v ,j I • ~ h

v lIS; 1Sr'-1 4)0 ~v)o

v

= Sinusoida.l Wa.ves ~1~

..d•...J ;...,WI J,.bJ1

6

~,JI4SpJl4-l)oJ

J

1>2

~

2:

q

= A sin cos

[

q

= A::

[ 2{

A

(X

J

+ vt)

(x - vt

~

( 'IA _I 1)

.."...--; L> I; ':J I ~ ~ 4.r.::.:.....::.. ~ I~ I J,.bJ \ • .u j:.; • ~ L...I ~ 1..::..\ ,j

t - - ~~ 4>1;~1 ~

6

t =0

J:..

6

t v-o;1J ~ ~

~1S.~-,•••J1,1 ~...JI~~I J~L:::.~I;I~~WI

6

dI..i ~4,)~ •

%

4JWI

( 'l y _ \ \ ) 4J oJ W

~ 6X..J I ~~ ~ I ~ • ~ I J""~'

)..

~1:.l1

, ~I 3: "l;-..:~ ('lA _II )4J,",W~ ~ ~1clJ.:; !~.:,~WI X "~~

J.-W I r W I t.,.;J I ~ j:....U 4..,."l:;...:.. ~l> ~'

• ( '\ _\ \) JUlI ~ V:He ~ l.S

• ('l. _ \ I ) , (A~'\ _ \ \) .:.. ~ "" W ~

- - - - - A -----~f

" ,\

, "

" ... ~4,. ('\ _\ \ ) JL:..l1 Standing Waves ""~ISI~~, ~(A Y_I \ )4J""WI, ~14.l",,~..:a:lS'W d

b,:;.) I ~I;J lIS .J.:> I •

•

~

""""""'~~~~ ~j.a:...; ~I yrA~L:JI~~I.:..I.i~1

uq =

L,;,r--J I J"w I ~ ~~I~ J-.: J"w I Q4

t

~ I ~ J:.. J.;;"'J ~ • ~L..::. ~ 6.-=1 ~

Asin [ 2.[ (x+vt) J+ iA

I(

sin

u Ir.:, u l;, L....:.

[2{ (x-vt)]

···4J""WI·1 ~ ""..b.!1 ~ . • d.......~I4....±Jlall. 'I 1..1.>.:-1":'. I.J. ." • • u::.....J r

....r--' ~

q

IJ'

=A

sin ( ~ x) cos (wt) _.L ;~WI4J""WI J:..;,

• ('\ '\

(-\ •• _\\) _I \) 4J""WI ~

w

~~

• x ="'-';e-~ L;~4:.:t~~~I;~I;--ullS; L;,;.. ;~I.:..~~

=

0

H

H



H

tr £i c·

f ,r 0{"

~

~.

-

{

1= {

c.

f."

l'f

fl

f ••

(..

to

I-

.f-~

-

(i

~ ~ ~ -

- t., t

~

c·

{• ~, --

l'~,

' ,-'C

__

~

t":"

,,L.

F - t: ~ ~ 'l

t

- li•

•

(i

II

....

-

r-

.......

.-

-,-•

W

t- ~ t t i: t r- ~- { fj-----

j-

l

rr7 t-.. {~ e- f ~ ~ t r 1,.

glD ~

&II

I\)

,. 'f: ~

"

t' c -.

-

}

.

t'.

f

,

•

.~

(

·

:.

[

'5~c.. .- \- b

• "" e

r~ 'i!:

l'

~

-.

\T"

r

t

"

..

'(0:::::. 0-

{~.f '. --

...

t.' 7 ~ ~ ~

•

-~ f;

C.

1,'1 ~ Ej- = ,,-.,·t· . rL......J·e k ~~ ,--.

,

f

~ \-

c..

a'

PI +

-~

-~,

CD

•1"£"

f. ~';':).

c,

-

Ci'

.'b

~ ~

-:.

_.Il

r t: i:: f: ~ ~ t·..., "

[1.. ~ {. t -· '-· ~\ ' > 1. .. . .~ [ a c·- t . r 1i . 1 - 't. J- 1 ".- B:: -~~f

. t' c, ... "'. 't: ..

'tT

fa'"

•

c,

1. • . :. f; e ~ .C"l ~. · · E'f·r. -r[;-:

..:.

t'. '(

b'

(. .r to· 'c.. ~.

l.

,

1~,~

r, 1: r~ I\)

0

•

+ r:rI\)

g

N

I

,r

g'

+

~VI

I\)

I

~.j::o.

c.

-

,..

.[E ~~ l:~. f.'5 t · f' c: f: ::I ~r ·tE-

c·

~

"'.

.1-

~

r-

~

f

~

t

"-

"'4

1

C(;

· t 1i : c..

~.

t'G..

[

f!t.

~ Ci'

'".~.

1

t-

-0

.... ....

c..~~.i~j ~ .• l_l) u--~I ~

J

m.......:l.:S ~dA· '\ _\ 1 • uj!,.:1l) 1;::-1 eo"!,.

v---~I

J ~ uj1,:}1 ~ J.» 4:tJ,.;~ c.~.i:..u

J.» ~~

oJ oJ".:.! 11.:->1. •

•

n

Y-

~I

'----f""f;j !,.: ~

ol> _ \

t_\\

1 V'~I J~I~.i:.i 1.:->1. \ I _I I

J i..:.i:.i.:J 14.r-SJ 1u.,.; ~~.iu-jt..:-I

J~ I.AJ~ ~

1·_ I \

r I. ~I ~ I

t:,p".so:: 4~ ~ ~,b. 4j

.\Y_\\

• H' -

I\

• ~~..ul~) ~.~ l.S,j~J.>.loJ .~ A. JoWl':"';),,::,;· m ~

.J -.l,J. AB

~ ';~lu).H~ • 11'2 ~14...I,J. U" ~

• 2.t~1~L-

AC

;';L.Jlu,s:~~1 u,;4~4e

~.~ • It _I I :B

l.S,LJlu}Jl~

ei= ~I

C

i-W:

J

u.i~l ~1~~J";L:JI~lu~ ~;:,..ll~1 k ~)l.J1 ~ • ~I J.»~ ;~c.~~.j"l.jldl.i!.a.::- .~I •

(T' y _ \ I) 11 oJ WI. t:,J oJj4 ~ \,; ":' ~.i.:J

J.w I ~ I ~I

• I I) _ I \

4:tJ l:J I ~ I ~~ 1J.,;.uJ t ... 0

• t ...

O,G=A,j.o,e.;,=o • ~1r-J1c.1;1,b.~I-.I~~4S;aJ14.i.a ~~

0'\

.~L.:.Jlf~I~~v~~U.lb...ll

f_"

UbJ~I.s.s;oJ1 ~ 4..-l>..~

.... ' .... ..,..k

w

•

.(...5..).

•

• ..)lsl..o! oi•.11 ~••J-:1H..,J t,...J I ~ l; lob- ~ I ~ •

•c:.'

.tlob- ~1 .. .i.tJ M..J-~~

( • - \, ) JL:JI 'tf

~,.sjJl

u1 1~

'A -, ,

cil~ UI.s v I v-.~

J,oL4U i...ut,...JI~I.s.s;oJ1

~

., '\ -, ,

• J 2 JWI J,JJ~ .a 1 ~,LJI nJl J,J. ~ v,1: ~11Ib.JJ •

J

...::.l:S,.,.-:e- ~ ~,LJ I.....;). c..; k ~)l.. u.,;;... vJ- ~1; • y• -, 1 1~ .....,,,,,=,...,~I'tf k~~~~1;~,J'JWlu)JIJ

~Lc.i.c.i.::1.l j!

.,1;.1." ~t,...JI~I.s.s;oJ1

~ • JWI""';)' J

2m ......:L::S' ~

• i...u t,...J I~ I; lob-I ~ I ~ •vj!,:oll ~ J~ 4.t.!,.;L:J I

•

•

\-

(.

~

~

~ c·

:

4'

~

''I.

e-

~ ~.•

00(

t:

~ r

'f.-.

1,fc:.·t

~

'to

.t-. ~

~: .f. _.F '- .( r \T"

v

to i.

-

c·

.C'

$-

b· •

-:

-

-'

~ ';f!..

'"

;:..c

lao'

..

• YO. _ 1 1 ~v~vlve ~~ I'

I-'

go •

t r:

-'•-\;;;.

t.

r

~

E 'G..

'G.

,.

£

-.

t'

'I. r

-

- c

>-.t

i

~,t::

~

r

Ch

Gt_rr

..' ~ ~ ~ c· C (;.

C':'. r

do.

I

(i

...

-.

t

L.

r ~

I

oF

(1-.

-

I.

e!.

.1.

~."

t::

~

~ ~. . ; :

-

CD

lLt

,ti

tA.

j- , .

{

•

1.~

r-

-

~

0

-

fo

oCC:.

CD

....

t!~

f

~

_ - -t

r..

,-,

Ct.. c· [ E.-

'r' 1:

~.f..:.~ ;-

~

f: ( . : ~ft.t-c+ E~ ~ li"",

F ,

f ~

J-.,.: I "et' ::;~:

Co.

•

J. .. E 1 of- '., :-

l:.

L ' r.o

t f ! ~. ~. .:. t ':

t

:

.,

~ ).. 'I.... [. I' ., l!e

-

[t·· . r~~. ~• .of:-~ ~t 1#

t:.-' "ttft

~ ~ -; ~.~ t~ ~r . ,. t· -t ; {.1:E: ~. ~ {td - I := g ~ ~ : r:'~ ~ . ~".:,' ~. r; ~ ~ ,:r. 01-' _ E: 'r~ ~. \- f 'l.~. t r. l:" ~ i c\. !!' t c : .~ ~ ". ~ t [ f., ~ 1. ~ ~ ~ ~ f. ~ ': .r f f t '~. ~ ~ 1: t ~ r ~""l.; f ~ t or; r r.k ~ {- ~I fc:: . . bf t-",~ J.'t r -c r ort- },'.~~ {'_ . t, - · - ~ ~ ·r. ct 1. tr. -;~c. L~~~'i'~,! ' i t ~- ~r' t~ I~ {-. ~t, [ ffS'"t ~ 1.~ S 'to ~ .'rr1: "~ f r., t· ..:. ...: • -; t '\. ":" ~. ,. ~ ~ et "' ~ t ~ I r~ _ ~." { 'G... t~... c E. -& ~. c· -

t~ t·~ ~ ~1 u I"",j .w.~,JJ!, ~ ,j~1 "--.J~

• J J.~~ulS'

U)L-~ ~l;,:;,-,,,,)'JI;Sp"~ cfWI (~~I c.~) ~t::~I~, •

',.b) L...

J I.. ~... ~ \.\.\ \

~~, I :i..,p,--: ;;

~,;.JI ~~I~I~~luL;

t

.w I ~ W I 1S,p.-I ..uJ •~ 'J-dll

(, cl.I..;~ • ;"~H

t---u,L,..~ll~c.t,.:.....::- ~ • ~u~ ,j~ (,

•r--::.. \\

~I~\s';.;..

l)l

)~I v")~ICS'p"~

u,~ ~I J,..<JI.JL...v-- Jl>..I.. ~ E.W. Morley

J->1.J.:l1.b.>,li (, ~;JI

0,.

J}}4

v. LS~ Julb~I~1 ~~L;~jLr:JI~ ~

•

-.r~ I J IJ', ~ I: ; ~ •

"

'S'!I~lulS'· :i~lfl..i.Jll,J, _~~~.JJ/J ~1S1.b~~·~WI

0

\'"

1·

~ Ij..: ~ lill u I"",j .w. ;I.~:.... ~ 'J"l ~ I

.

..

;r~ : ...:

}. 'ii.

r ·

~ t• t .. t

c.-- 'G-.

C't..,3>

.

I

~

f

.,r

~.

- s· ..!i ~ • t··

'k

'. l

~I

-

o£

!::

_ """

•

~:.t f -

J;

b

\;.

•

i"

-f t rt -t.. [c. i= f' 'f.""

.:.{. -:'-t ~:~. ~ ~ .r.

;, "

( t. \ . ~-;-

[

~ •

-f• •

'-:

~•

•('

o£.

(:; . v

,

.

c_ ~ 'i

_

I:"l

0",

•

~

-0

~tt! l · t:. t· of ,t'! t 1 r··

l

~ ~~'t f.· ~ f'

r :r tt ~"~~ ~'t [b ~ 8 ~ { ~ r t-[- 1" .

-("

-• • ". 0'~ ~IS"

r:rI;



... .

f 'rfa lD':- ~•

[

" f'H~

•

,. ,..

,

x=x+vt Y ... 7

, ,

t = t

v l. ~).

loX..U> L:.JJ

1,)4) I .::.Il ~ v ~ 'J ~ (~U~

:x:

--,f~-----

~L...

:x:

,

,

__ II ~ .... • v;;--; ......:..i...lJ..:. 0 ,

~

~

0 vI -~Il:b.i: v,... I;J~ • .»b..J 1 ~ v,s:vl~· 0

C -

J

c u~~I~

u)IJ..;~~lJIl~~1 ~~;~~ ~I

I J,JIlL:..J1

Jl:lI~1 u4L...:iJI~ ~1.»b..J1 ~ •• ~I~4J-sL...

(A_H) (Y_H)aJ-sWl ve

~~

·(l_H) 4J-sl-Jl ve {A_lT)aJ-sWI ~~-s~I;~I~

{tT'

8.:J..lX

+ ~2t

= a 21x

+ a 22 t

x"= t

,

~14!pJlc=..:l5 W • 1.:-)l.l.-.ll,,:-Lo.)~1 ~;

L.tWI ~~lo

..

'-., {r::y

It'b.\J;;;

~l ~

~;:~I ~-' .. ~~

~I ~ .;~l:.:Jl ~~I J.l-.;~~~ ~ ~.il

2

~12 - c a 21 ~2

2

2

- c &22

2

2

• 1.

all ~2 -

2 0

&21&22 • 0

.-02

( , , _H)

x'=0 i.b.i:J1 ~l ~ v!J, .J..i;c':i..tl-.~~~, J.,.a~ ~Jl 1;~ , ;':il .x .. ~~.

T ~~~p.::: ~l.

,

0

J-':iIUU; 'a""

),L:..Jl\S..bo1 ~~I4!pJl~ ~~. ~,.,; ~l~l5l.il «() ~

1

JS~':i;;; .i_.~ .. ~ .~h11...,~IJ .u.~;;;

'=:- ...

• \S,,>':il '-"'I 4.t--U~;;; ,~ ... ~ ...,~l y -,,!;.;.

1.1'II~·I~ ~-v ..•

~ Lt..cIJ T •

_

&:1.2

a1 ~ La.. .)s. J..-.: dJ ~ (H'_H)

all c:..~l..:>JI~)lJJtl~1~ ~ (H' _il), (H _l~) ~~WIVe

"'" c:..)l. W Iv I ~ I

~.~I~I~a1~L...vl,.,.,

,

x ..

,

'" b..J I ..;. ,..JAi... J

J

4-...i:

(x - n) ("_H)

....

7-7

,

(.

l:.::1J.b,:. ~ til:J1 Jo=..:Jlv.i1

~ (t -

-'!f-) o

{{"

..Wid•...:11 ....):..ill ~~Ir'~Ic.:J$~\S.ll'F~~""~

•:

~ ~W I ~ I x ...

'I

10...

' ~ (10 "+ ~>

]

(10 2 - ;1>]

li'"

U,1

L.J..lWlr~+~~it t 2 = t 1

it VlloS llt (...J ~l:) ..:...;Jlv.i;

J

J I.. ,;"~I ~,."s.w I

( L\ _) \ )

~"tJ.

B

o.Ul

LW I....J~ I.. .r~1 ~~ ~ I J,J;, v I A .J.h L..W I J~ v ..; I

~..l ..IJ;. loS, ~ loSrdl.:J I ./":_.zJ I \.i..D. l/~

- ,;::...,J

vJ.i; J'~::d

4-~ )d'!: ..s.S ~.r~ I":";t':.ill Ll,';

..s.iI • v I.i.l.:.i.~ II:t"-rio~, • ..sJ I):p."? - .P...,J tJJ.i::

'V"L.-.:i:; -;;b:.I~ ~,;WI ~ it L~~~~ ~r!:>L::' -:1,.1, ~ ..s.ll~;-? • loS.r~

11-,,; L: -: 1,.1,

'--__~ it 4~~L..~~~ ~ l.5 • L>.::i;ll ~

it

it

~ I v~ I

~j~'?VJ~~

J-...-': ~ I "b.J I J v,w I ~ b. ~ L.W I v I .:l li;,.

&).l~lcJ04,;.llv~ed.l..i~'~.

J

J.W,,=~I-.;I ~I

• 1 ( 2!0

T

.JL.t,,=

2! 0

l

,..

JuWJ

~~lc?;1

J1111b.) Ic)ej

c)e,;.lI.Ac~J.I5;1:c'J

),.. ~J~I4.:-~,-loJ I~,..u-, • ~ I • ~ I~ ~ ~ • c)ejJ lc .,....: b;

•(

~,~I~;JIe:- ~

~

lolA).

U',...:JI~WIS' ~jJIJ~1 ~~

-.; Ii r''': t I ...J v IS','; ) L;.; ~y. r ~ \oS .iJ Iv Ii .,df.iS' oJ- ~l v IS' I..I

v.,;.ll ~~ v I ~ v I ~~, • ~I '" ~ \oS.il1 r',:iI~1 c)e J'"".-I,.-,

;. r,

~

0 td

:r

-

,..

Ii

-

\-

I\)

txt

I-'

· fa

-

-

'S

~

I

r-

...

c.:..

'i-

Ii

~

t

c+

~

I!.

..:.

t

-

[

l

t r1

•

" l::. ,

at

~ ,,~ .. l::. ~ -

't'

v

~

)

,

t'·" .

u

~.

:-.

N

I'

}

o·

E ~ -

.....

•

.

).

,

"

\-

~.

-

~~

['G....'

[_

't....

~

II--

j'

II--

['t... _ .r

C,

[..

E ' , : ; ;• _ tT

....,

[...

.r

:

'1-

';.

_,

•

:

'I'

Ii'

"',£

..:.

c.:..

J" , _ IT .r

I v 1..:-...~1I ":'Al . ~ 1 ~)U.; I ~ ~ v, Ool v ~ 1 ~ fl J l::..:;..

dJ:)

J:)

b... ~ ~L..:.,~L..:. ~~ ~J I,.:. V'.)L.... J~ ~

I.S,.:-J 1..j1.:lS' W pJ I ~ ~ • (i) y - \" Ji:.ll '" !J'. OXT

v

.·.}1~~11..~)"'" ~~

, ~;JI \Joe

2, 1 ~)l:v~lu.t". ~.

JS ~~ - J~I ~

("=, 7 )

v,Ool ~ x fl~l: Y~J- 1.1, ~"...J '1 ~,..o> IS (-x c..r"IL,

.1.".,::; ••

1;J.r --

YJ-

~

(

i: ,

~ l-h:..:1

X7 -

-7 ) ~

tOol t...:l I ~

, -i), ( :i,;'). l.A v 1:; LrJ Iv ~~ Iv .i I • ~

,,,

OYV ...,

-JUlIJ \•• ...... '\.. v-"

I J:)~

y1 f"~.11 v"""" , '

I

- A->.:i11 j.>.;'J ~ I J:) ~ V-- ~

j:. 4..t-lo.

..iJ,'); 1:.:11 rl.: rOol L...:: • ;'>.::iJl J.>:: ~ I

J:)

_.1\'

~Ool~

, .1 ..........

,a.' l: X ,,~l:

T"~

b..J I J

~,.J,

• dt:..

.~I~r.)I~~b-JJ~II.SJ'rJJl

(xi, -yi )...r- L>.:i.II J.>:: ~I J:)b..J1 ~ 41~ 'JI ~"...JI~~J- v I L4J- ~L..:: r"sWI ~

• Jl::l1 ~ 2,1 V,l•• ==:Jj ( -i:~ ,

7; ),

y

(x,-j)

('x.l.)

,....):x( ~/

~)

to

o

.

~

x

~ '-F -l.'I I·WI.Jt..:Jlo.::..~ u".. V' r •

u ~ ~ L~ l.--il L; • ~ u,.J )(' l;5,Je ~ ~. 4--::~ I '" ~ • ~Y_l'\JU.lI ~~,.,. t.r ( -:i:~

,-1;

),( %J., ji

)vJ1

~ ( T) ~ J..-; Y- \ 'i' 4 1 , ~"..J Io.::..~ oiSl,illii" • 7 -'1'(1-

x•

.~

•

-;x •

~X2

°1 +

- v

i; v/o 2

• -7 -

. '" ii.v/ o2 ) 71

."

-72 "'(1 +

X;T/O 2 )

~J.-.;~Ive

7, i

U.b. ~

- v 22/ c 2

2

v1 2

V2

"2 = xl

./2 + Yl

• /2

./2 + Y2

= x2

(, v 1)::::,.\ 1;.. L.I v ..; I

.j:,~

~r--"""" II ~ v I J"s.: v I ""'::-->.

v""-; I.s ~

cJ;

v.

)! ~ '1,)-:A4

""L.;..

l.:.:h

r-j

I.; ~ v,l:;I

.J,..... ($1 (, 4. jJ -" ... ):: oJ l.i...:

~,I (, .)~ \ i';'~lIt~ vY I.s ~

I,)

I tol

..J..d.::.J ~ ~ ~ I d-Sp- ...,ld.. d-L.)~

~

r»

Lt-b.

~ I r::--:J 14.....l.:$ vi

J

'~I J~' ~'v~ • J'b.JI .. .u. ~

. (c.:--) A _lY)

Ji.:J1

~ I.S' r.sL..::.lI ~~~I ~'~.1'""v~ ~ .v,s...JI

f. '1

• ~ I ~ ~ I ..., 6.J l..t

0

~ r..l k.:!1 ~ ~~ I r~-~.H u,)"'" u I

-V

~ 1J',.so ~ I J.:>.: ~ I ..., 6.J I ,

-my

~

~ (,f)

= -mo

= (1

V

0

~

(v)v

..t ~ I r>Y I u .j I

~

_ ..: 2;0 2 )- ...

-mY = -Iiio

~

(v)v

• r..l k.:!1 ~ u,s::J I ~~ I ~ I u,i... L1::lJ in o Ja,;.".:

JI~~I ~I

r>yl u l

~

mo =

2mo LbL...: Ji:

1;.......,

1.j10~1

L..H.,~I JI~x c41.j~ u l ·L_::.i.ll ~ ~1...,6.JI..t JYJI l.S,~~

.i->I ~~, 'r..lk.:!1 ·~I.)Ip' JI J~l.S.i.ll T ypJl L..iUJI

,«

j

..1\ l.iJo

~r>.)"

l.S1

= 2m o

+ 2mo ( '1 - 1)

• 2m o

l

.

., -Co

~

. ~}

II

-

v

i= r : t .

r

~:

~

<j'

,

~

'(

t v

Co.

~

.

_

{

•

\

_.r Co

Co

G.

~ ~

I

•

t=.

Ko

c·

•

CD

ett1

'"

~

'd

Co

t

-

t

'G..

.-

.[

,.

f

&&

~c.:..

'£ -

-.

Ko

c·

1

~. 1

et ....

I

~ c ~.

.... f f

[

'[ c~

b

~ -

•

..:

t t

°1 c:· ~ 1

p:" Of ~.

l:I"

1 (. ~ . ~~ ~~ r· ~1· ().

-Co

\,..

r - - ~'.'b . tr e · . fr· ~.'[ t: ~ :c.~.£ - r f~ F'" .f'f1.· t:.:. v [:,. ~ 1

~!

't

+

-

~'r' . E:

Co

v

- '

-

!..

, v ,

r"r

t

...

Co

•

r";

'

p.

~ 'f·.t~I ~ ~ ~J.JC

r~ L.:: Ve ~ Li.. v":;)'l- u":;)'l

..s~I";u.JI ~

r' ~ I l.y,

cJ,; .... u,,,:sJI u~~I..i

t:.'; t:.l:; ~ L.;)lI1 L$UJ I ~

• v,wl~b.."";'

...;..t.rI I ,

~,.::,.,..JI .J>lu~~

~v,wI~1> ~ ~I t:.';J~ u~~lu~1 ~ u~ ~J~6JJ~ L.Jt,~1 ¢1;,.:;~I~A' v..il. ojH t:.l:;I~ ~-r4e~IPJ-""b..

;

':;S'~JSJ~4_J.:~J1PJ-""'-.j

_Vi,

V'

~,,-J~~':JIVe

~ J..-;

• • 211

o

=.

0

12 (1 _ ~)-

1.....~IJJL-.JI~

.J.. 11"

mO

~,s-

,I

0'"

T

J..il.-ll ~~IJ,)LJ..;lv~~II~, r-A

..I'

Y •

-y-+-y--. 1 + y" 2; 0 2

..c L..:>.J I ~ J:,.;LJ I

• 2(i}

0

(t;,.

•

0

1 + (f)

~ I .,;lb u I ~

•

~~ • ~"..J I ~ v.:: I,.il

r.,

ve~L~1

5.5 BeV ~1~L......,I~I_~~~~~lrl~1 ~he

Use of Matrices and

~our-Yectors

·ILllo .•. _I r- r'.»"

U

= x2

1 __ ·\

o)IWh:JI·1 U

~ ~.

L:..1.'}

invariant 4.....::.:1:~ ~~l.S~ul, ~

+

i

*

in Relativity

I~. ~"JI",,6.JI~,:!~I~lrJ,1

x 2 + y2 + z2 _ 02 t 2

.\\_\'t

2 + ; 2 _ o2 t " 2

L ~

2

1 }J

II:

L .P

';~'6I..\Jl. ~

12 x/l

J: 1.S't..,i,i-J I ~~ oW; .P~ ..::..~ ~ ~ I ~

r;

, xl , x2

,

x3 ,

x4

=

h:'l

0

0

1 ;9'1

1

0

0

0

1

0

x3

0

0

~

x

xl x2

4

(f.._\

n

IJ'_H_~ li:: ~ I ~pJ I ~

4...l:iJ I4Jb.lI ( i· - \'f) 4J,J W I c.J u,;..J I ~

~li::~IU'plJ4.l~ ~I~~,J~ ,J~I~.&.1.

_~,.~,

• IS,,>~I~LA~~I

c.J

u,.........i- --: ~ .P...,.u ~ Iu,A- J..-.=:. ~ I ~ ....,; I ~)l. e:=-J I ve Jl:.Il ~I J L:;..,J!.: ~J, IS,J~~I 'l;d.Il ~J.:t.-.H v 1..0""'" ~

II:

OOS

'f/

~ J.-; ~ ~. ~ I::>

;),M

,

5Co.

~):

~.

L-I

... - ..-.

,,-

\....-.

1[.

I\)

~

;{

'1;,

Mt f~' f

...

f. "' _ l. b t f- .: '. "s 1 IJ I (00_\'0

dx

..

m

dt

tYV

P4

III

10m

~, ~ oJ ~~ 1....->,;.1 I ~ 1IJ-

I. W ~ ~L...J 1- ,-.>,;U #.; LQiH ~ 1I"J 1v .iI

• u,.-. . -< 1\ L.I:S.v e ~~ ' _.. . t..-U •. I t ....l:~lI .'lu. v---

,,~ •",s."I...... LS. r.J.:.o~ ....LJI- ....jJ,:f"l LP

p~ r

=:

.2u 2 ....2 0 2 . . . 2(u2 ...

0

' 2 ) l2(u)_ jI1l .." 2 0 2

~~IS'W.,.,..JI • ~I UlbJ~ ~I. E

II - - 1cP4 ..

\;.

1.-.' .....

0 , . 0

hJl v'; li v I (0 Y _

110

\

' I J.>.,\,O,\

2

n ~ ~ ..sL••J1 v-~; • \S..s~~I.....->,;.IIJ:.i> v,;liI

'lie vi. ed.l..; ~1u..;.1· v I!. JS ,,~I t ...1.~:lIC" 1. --:... 11..;1 ~ 1S..s~~1 U:SJ I, I. L.. ~ I ~ I ~ Iv I ~ ..,fL.; 1- .....->jlJ .~I;J 14"J I

• ~~J-",,~~IJWb..v,s=1:.II WUJ~

£. VA

~L.~~ ~Ipair oreation ~,;.l'Jl>~Jl:J'~~~;'

\.SJ

-....JI

lr;y ~W~ L",l.:

.u~J.nIoJ'~;.;.I· \ ._\l

4' ~

1oJ,LJ I ~ 1oJ,s; ~ ~. ~ 1- j Jt rU=a.: yi.a l6."".1JU ~J.x ~ ~j)U' - j---S'". r U=a.: ~ L:SL. 1oJ,s; ~ LrJ I ~J ~ I '-=- ~ , v ~ 4ao ~ l.,..oJ' L-iUJ~ Ll:ill E + m 0 2\.S~I rlJ;..;J1 ~ 1: ~, r",l..:11 J.ti ·;'....l.:m o

~L) 1_)I ~ '-=-,.; ~ ~ 1~ v .i I •

u.y ~ J...;W I v-c:,.::,,>tU ~, ~l.f

2 2 -m0 c -0 2Em

:B

= -15

m0

2 2 0

= Li..WI 1o.:LL...~.-A.a • • ~--~-.1 !he

-

d

I.J.

~ J.>~

lour-~oree

4L) 1- .,...ill

. ~I:.II ~.,.....ilILl",L....UI:t-J~ ~;~I~

'\I~I ~~. 4L)I_.,....il~~J1".u

~~~~

f. y,\

--'t ~1:S~ll:WI4l..l~lv\~;'( 01_H) 41..lW\ ~\~)Lo

u....

1!'p

=~

(u)

:t

..,ll:J1 ~I

it)

dx

(m

(1\ _\'f)

,~L)\_.,.....au J, ~1~)w\~4S""J\ v.il

.~I ~I "'"

u

~

~

~~IS':t

F1 = ~ (u);!x

J1 2 •

~ (U);!7

1!',.

~..l~~\.,.....il\ JI~

r (u);!. L-.:~IY~ 1;~

;

~

F4

c:

1!'4

c:...!.. ~ o

10

(u)

(u)

:

~ dot

l~\,\~\~.....;~~ ..i.ll~;1\J..LAJI~1

1'4 ~u.il

• ~L)\

• 0

r->;1\ ;I.J.i.~

WUJ~

~~I;J' vi'jl

tA··

~ ~

Jp " 0

~~~;j"L-,JI~l:S

ow,. J p

,p~

"'- L...:t ~ IJJrt""""'~,

~,J.-..; p

I'"

4:r... .r.J1~~li.;J1 (1

4-\j~

v'!:'

(H' - H), • ( H _H) • ( I» Y- H) c:.~ .. WI.>e'

- - .. -(1011) mv.!

_ ..... v.t }

f

a

"-n-

~1S.l\1 J~I,...

:I ..

80

~ ~jJl~IJoMe'~~~f

2

'::'......'.'.

r ee ~ IJ.A, • ~ I ~)l,u • 1· _'"

v ,( _ll'

v--

U"L-,JI.

~ ~

( \ Y-

!

;".. Ic:t~u,;'t

41 • UUJ~~I.;

Cll.::

j-.o::~ ~

J.L.. dli:.£1 Jc:.~1 •~I_' _, t

H) 4.J ... L".J1

'.i--=...,J ~ ~ v I ~ -y, -l).JY

• ( " - \ 'f) aJ .. WI.

rt.J1 ~.

aJWIc:.~U_.... ~I_"

(a)

1~

(b)

!:;::;.,

""'.

Jl,

1 a 0dr'

e.,p_1-oN;J IJ.cW I ~

•

Y_'U:,

2 1 +~

20

j2€ 0

it where

T

.c:

i· ~

-

{~~t ·:-~.t ~~ tf -' 'f[~.t ~-t. c.

L. ~ .

=

t't

..

t

Q

l.

E;.,.

I'

•

f. ';.

~~.

iC-~

't ~

-

I,

{l.r "'"

~

.

r

N

-

~ t ~ [

It r.'.

=~ ·

'\r"

r.

t

.

'..r t ~.· .,:¥

-t

f.

s'

I

-

~

'yG .- ) .~.. £~ -

'-

~!f.p; · \- ~Co ~~~ 7 :..... .~ ~ ~..j. . t ~. . " 'f c [_'i:.. , · F \ S 1 ., I '4. • . l.-. 1- [~t:~,.l: t: -G:. .f,. 1: t· t:.: '(Cl- r.- f ~ t. £: .~ ,. ;- ~ -[ ."._.·r ca. ~ 1;-.' ~ c;. : ~ ' ( ~ ~ .t· ~ - t 'G.. '\C f t£: ~.C\. 'it' ~.£ , f j 'i.t 'G.;.

f:

~

., I I

{;tIl "=

-II(

;: f·

'.

Co.

.,0(

_

-


tAl ~)O ",,6...

J

...i.l·~ A 6...

(, B

J

(j"li:

L [( c + V)/( c - v} ]

l..::-S',,r.-

l.r: 1j1. ..i ti» J v-)I~ ~

J

l: - ,~'JI

0
vts;· ~~".""6..~ 0/2 ~~lvb"I.j\_\o_\'f ",,6..,J v-li.: IJ'~IU?,,,~~ ~I ~. 1-~1~ 0

(0

~~ ~

• ~!,.:.~U:...;JI",,6..vlc I.k • x .. ~~I,JO/4 ~lc~~~".

• ~~ *~I ~1.1,)e )lS'~1

lot..., ,

A ~".JI""b.J1 ~

• l.i • A ~I ~lc •

LA~IIoL-.~~~J~~_\1_\'f u)U.;lc~~

B 1S,ri-1 ",,6.. ~ ~ ~I

"b.~ .. ~I J~IS'L5pJl~vlc

I.k \' ~ ~,wl J,.kJ1

T

~ I ~ ~ l:ll U)W lJ"ot:.:•••;\

i ::

_\ Y_ \'f

_ .. " 6'(1 + vir/' 0 2)3 X

• • , 1 ~~".~4l:Wlcli)L,J1 ~

..,.... ~T ~I ~ ",,6.. ~ ~ IJ'

• x .. ~lc v

~lc~~

a .:.-.:1:~.....J ~

\A _\ '(

~~I ~lvI41.~ ~)tJlrU:...;J1 ~'vllSl 'v,wI~6.

r.

.y

• "'


...

y. ~ (.. ,. I\) [ l· ~ 't(.' r-

~

+

'r-

;; •

.....

:t.. ~

· ,.

~ t t- !..-~ c:F~. L t Cj' C. ~.

"C..

u. ~ '1:

~

•

~ ~. t

-

l. r.'

t

•

-

~

0

r

[

""r

Iifa t~.

fl c·

t

t .• ' •

~ c:

r-: J_N r

r

('

""

e;t- r'" L..:J I ~

v I ~ I '4 1~ 'J I J.;W I 2 cot Q cot

~

= (1

r .....--:;tpo.... XJl' X ~

_

'f • 1 +

~

T2

/c 2 )- •

"" b.J I ~ ~ F..oJ J~.>.:h.;,.i.-.ll ~ _l'\ - '" ~

~l:c£Jp-:.::~1J-.:: ~I ""b..JI~ 'v~!"::' Xl ' Xl vl~1 .. , • Xl _~I J~ Q ."I~ I"".J X, , X2 vl~~ Xl .~l: T

•

..J~'JI ~l;I..\> 'JI ':!~ ~.J t-.e--) 4,J'J ~ '1..oJ1 ~ I '4l.rU 1~I

, .. , • x, 7, -, t, x, 7, _, t

..

f':'';

(AY c;lJl_~I~~lvs~~d.l..i,

Y_l'f v--~IJ>_t·_\'{

· 4L) 1- .~ 1J I.,..:..., ~ \' t _l'f

J> _"" _ \'{

v--~ 1

v---r~ ~ c;L.J-~ J~~ ylvJl-4,iJ1 v,s~ve_tl _\

~

, •

.-_...-._#/0) ~.

)'(1 -

i

' - " 1... "..a;·c~vl~.~1~vl~ ~WI~1

- • -,+ ( ~

~

"I

L..r!:;

\'

J-:.-I_tt _\ T ~l.!

~/.-.......~"

1 - 1)~

...

x (...L x ~ ) p +)'...!.. 0 0

-" T •

iT

,

~ I • ..u.). ~_=!.H~.)-..,..

-/

.. " + kt ... ,

~p. j~;r

~

Z

~

';'

"J""

, ~ 1J.>.: ~ I "" 6.J 1

-'

U

;

IF

~L.nS."";..,.. ~" ~IS' l.il v.il • ~l:.A.,~ ..;~~

J1 '-'" ~I JJA J~~.iJ l;)I>J Q 000-1(2/') ~ L. .~I ' • "lo 'J I '-"I

;.1


J6.. •• • . 1.1'.

u. Y

t.:.---i !,:: IS","" I£J I ~ l:::J I

{ 0 ,

; !,::

~

",L... ~

{11 \ \ \

,

'\'

¥I~ t.:.~

,Y

t.:.~I~

\""

~,j4J1 ~

\ '\ 0

~~.~

'\"\'A

y

.L..;

c. AA

~rJj~ 4J6.

A'\

Jj Uo::JI U' '" JjUo::JI ~

AA

~~

\1'\'

~~

'\T'

'\' A'\' '\ 'I

J~;;I YJ~ ~r

~ • d.J.i1.. wl",.J :1: t.:."-: •••• ~ • ,., ; r Jt"

A'

~!,::

'\'\'

........ ~I ~!,.:

AY

4..b~ ~!,:

'\' A 1:1

.a-_

wi", ."; ~ 1: t.:.~

.u...:. ~

,t

T'n.

~J.J.i..;..

fT'

"Ii...b-J ~ ~L:

to.

i:}.JL-.

t V'\

.L..J.., ~ ~(w

• 0'

~~uJ. ~~uJ.

\TV ~/

t \0

u';~1 ~I~..;JI JL,. ~

t t·

.....;I.JJI ue J.:>.J= I::,..LI-'e ~

to,

J-

~

-

-

. "'I

\VA

~ ~..J,N

,t

.&:,..~";;

\\A

~

ylil:: J~ yi ~..;;

\1\

:

11 •

0:-",. ""

L....

~~I~L- ..... W~

\1'

t A' \T'O

~.~ -IJJI - -:11·

Y'

...-- fI'

t,V

&,:!,J-l

t,v

~

IF. ............

t,y t , '\

t.

n· t \, \ \ 0

~l: ~J.J.i-..;.. .....J l4.J I

.... .,- U.

.b....>. LL,

~

rot 'fty 'f

0 •

lty

U.r---tJ,,)

~ Id;.~..s \ Y1

.,;I..s"",L... yi ~

.J

t

0

1

.:Jt-----rr!J '\0

..s..s~

.J

'f

r· • ,.

~r uJ~

'f • • • 1 n

1 '\ '\ 'f

rr

'f'li.

tV, lVV

"i4.. "j uJ~ L-"w uJ~

&...-JJO ~

~

r-:J

..sJ~ ifL.J -

J-.r-Jl Jb. ~~

",:,,-L:.. yt

r-J v-j

y ( • yr 't'y

(( (y

~1~1~l:,:I~~1~ ~~ ~;:J~ ~!,..b- ~I

lJ"•

vi' \ 0

~

l (

c.r.~ cr'~1

\l liA \l

•Y

cS..t~

L-,;IJ

..,

10 \'\l

• \\\ T'A, l • ,

A

~~

.

·..tob ~ ~

10 T' ••

...... j - - I P.

.. 6.=1 .

$~

1

.&..oJ., ~

L......ilb

-.= ~ .AJ14S'~ I i.. If

&..f_ _

---..".; ~ .J~

~ 4i.lf

u.i~~ ~1br"jJl J~ yi~l",- ~l;lb ~,

.~

trb

rrr rrY rry

,J-...JI J-J ~I ..;~ 1w1'~

,;;-

t T\

;;~

«., \

A A

A

;;..trJ ~ Ai U:J I Ai)l.

~I ..; "'J 0I..J1 J0I 1 oIJ oI..J I

f

~l;

,-:,";;r

~0I..J1~

\AA

~..i~

\0

t~I,-:,~

o,

J, ~ I U-=.".:

o It.

~l:.J1 .:;~

oA

-wl:.J1

T· •

J

J

ryo

,;-s

u-::J.+J

~h'

11 ..;.1 ~ ;;..\..-sl; ~

J;-: 1

oy \A'\

61*ul ~!1;J~ ~

.;I..i

~

~~;;-j

annihilation

.... ~I

antipartiele

",L...;.. ~

~

apsides

~l.t~

asidal angles aSJJIptote

~Jli. ~

B

ballistie

c ~l;

canonical canical pendulum cartesian ooordinates center of mass coordinates celestial aeohanics central :torce centripetal conSerTatiTe

~J,~ ~Jl.t", U;s:J I ;!",.

\$J'- ~k:,.

~",.

configuration compound pendulum. oompatible oouple

-.,...;

;!",..Jl ~ ,,:,.4-

.wL....

constraint contour lines

L.... L....

~

"'" .T:- l;.

LI...

~

ta.

~ .,..!",. J, ~

e4'

~ Q"';"-

creation curve ~.J:--"IJ 01

cycloid

yA~1 '":'~

oross produot Charaateri8tio time

~~ ~L ....

coordinates

coe~r1cient o~

~

'I

oor10118. 001li8ion r8etitut1an

D

del operator divergenoe

loJoI

~,.

&-i~

td,.~01

dynamic

J,.-Ju...

dampe driving

~requency

driving

~orce

e:-i I 01

01 01';

~

.. .,.

degree of freedom

L-....~..:,.b.0I ..r :;J

exact

c;.i---,.i 01 J,#li:

d1f~erent1al

elemntar,r particles esoa~e

speed

effective potential exoergic extreme

~l.ac ..:,.~

..:,.)\;')1

J~I

c.r--W ~ ;.J~~ ~IJI~

exact solution elliptic

Jl

field

j-SjJ J,.ue

foucaul t pendulum

L-"ii!,: .r~

forced harmonic mot1on four-vector form

'!It

cA,)1- ~l~

frequency G

gradient (grad.)

y---Ut::

gravitat10nal

~.roJ1 ~ ~!Jj

geocentric lat1tude generalized coordinates

~ ~l.,:l~

H

harmonic hoop I

isotrop1c incomplete elliptio isochronous inertialre:terence system in phase J.!I1pact

~)bJi

01

L,., ~ I .; fJ"J ~ ~l;~

~l "JW ;~ ~Je ~L.., ~lW~ toiL.-....::

lIIpalse

t;---J.J

1am1a

..;IJ.~

J..--...

19aorab1e

JL.....>ol.t

1Jlter~erODleter

inertial :toro.e. inertial teaa

il!;~,J

•

~~ .JJoJ..>

L

11Jlear 11J11t 18J11»&I"

• ...ent o:t 1Jlert1a ...enta1 e111p.o14 .000entD

a04e

..; l..iJ I ",...u I ,;-

,;--JJ ~W' r-JI

. 1

.....j r";t-

nonlinear nul

nutation neutral normal restoring foroe resonance response reduoed :mass rigid boc17 radius of Qration reotilinear

o osoillator

~r-:

rt' ~ I ~

"';..-a:

~~~

p

potential pro3eotile par8Jll.eter pendulWl poyer series

~J~ J,~ ~L..I

i' , 'i.:

prooess pllDl) line polar ooordinates prinoiple power la... pqsioal padula principle ues proper tille

r

..L.o..-i::

'~I~G ~I::,.~I~

• ",---"Ii ~ L.. 'JI

y,.;1i

'l· .

L..u.

Y"~~.

~ • • 'J _.L.;..

,,:-le.

~j

Q

quality factor R

radian refr8210e re1ati",0 resisting

..

I/P

~jO

'I '

spatial ••parabl. .phenoal padulWl statio soalar

y---I'~

.J.r-.i.I1 ~ Ii .s J.I J, oUt

c1r1:L.-., oJ ~

;.,s

J~I

sp••d

.

sti1'D.•••

1

!1Je.

soatt.ring .i117 pu.t1;J • •11 J~;;I

alippiq spm

r-o ~

staole sinusoidal wav•• st_ding wav.s

~1~~4:f. .~..::.~

811mltan.i1;J

81I.p.rpositiOJl seoular equation

J

~ .A:

~I.at 4JJL...

translation transv.rse terminal velocity

u#J".! ' ..

~l~.r ~~

transiat thrust top

oJ L-...;..

e:--i'"

~..\:

\

'fO transp ose mat"i x tllle dilata tion twin

parade .~

transf orJlla ' ~ion

u

J

~-.Jl~,&­

v--;Jl .a.,..L...,.o:: ""-- UJI ~ lo;J'

J

;!,;

,,-e: