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A = [Ax, Ay' Az ] ...,6.Jb l ..jJJ4-) L.~""b.. J ..... A vifIJ1)l:.;.
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Equality of' Veotors
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t ~II.S,L-::
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Ay • ~L.::J1
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: Wl:J14.I,jW~ V':.
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Veotor Addition
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_,
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t ~ loS 1 ~un
+ (Bx,By,B z ) ... (Ax+Bx ' Ay+B y ' AIt+B z )
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: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
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=-
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or
0
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-
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=0
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=:-:JI ~u~
The Bull Veotor
--
t
-"
~I
0 == (0,0,0)
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k:..J-
vo
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--'"
• 0 = O,;.} I
u ~ W,
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The Conm1utative Law o:f Addition ~I
J
...,..\.->JI
J.J4: u,.:U -,
v IlSl cl.r:-:JI ~ J v,.:liJll.u. ~ -..
-..=..
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III
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Ax
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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,,) ""
_
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=(Ax+Bx ) + Ox' {Ay+By)+Cy , (A,,+Bs)+Cs)=(A + B) + a The Distributive Law .J,..w.JI(:-'.;,.:v,.:li-A ~ .J ..ts ~
4
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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--=
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Magnitude of: a. Vector
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111
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A
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A= lAI = (~ + ~ + A;)*
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Unit Coordinate Veotors (~I~.b-,JI) ~ ~
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-
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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,
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o J----:r--J----- y j
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Geometrio Meaning of Vector Operations • u L.."L-.
4.-1' 1.::.;:1 ,
~~~
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w,.J.. ~~
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Equality of Veotors
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tl.
y
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11
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t
.1 ...
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r
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u--.,r.S";;-
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A
7: . ~I(J~'" -t- . 6.-;1 vi
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Soalar
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The Soalar Product -..:a...
~
"1\ :B·
r--)~ ~
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,
A. J:.:~ u'J ~ .J.,.,J Ie.:-.,.eJ1
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1~,
_
III
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vi 1&:1 ~..wl ~1 ~ ~,
~
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,J
-=
( .. -H (. _ , )
A 'R_ .. --r""•
:Br-x .1
(J .,
(1-0
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........... A.(B + C) .. ~(Bx + ex> + Ay(B1 + 1 > + A.(B. + 0.>
°
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• 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.:
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A
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u'
.
~
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= 1'A12=-;,,1 A
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k, J , :L
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--=- --. A.,B
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=
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;£~~-7 Some Applications of Vectors Equilibrium of a Partiole
r::---::- vj~
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= (p
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Law of Cosines
~
L>lj~IJI~J
",I 68 .....
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u,.:li
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u,.:li ( Y_') J!:JI
J
~
0
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~
= A.A + 2A.B + B.B
( • - , )~~.)lAJl
e u""""; li,.. I ~ , AB cos
J
ui\,.ill ~v- ~ ~l~H.~I"
40~
0
2
-'--
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= A2
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J I~I,
( 1_ 1 ) ,
+ 2.A:B oos e + »2
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---.
~he Vector Product
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B"
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1S.il1 ~ I.e U;"t." ( -A cros s -"B)
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•
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--
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X
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II
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II
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iJ~ ~
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:1
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x k" = 0
k
to'
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)
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=
k
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IOQmot:r:l.c Intel'J;ll'etat1on. ot t1\..: OroBa Product
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y. ..A.xB-:L
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•
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IA xIi 1== AB(1-Cos
-
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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<:
-4.
~
-;
~)~ I
B , A
or.~
I. (, 0 _
vi
~
~w..-
(ixj=k) ~
-
-
._~..:.J1.~1
C == A x B
-"'"
1,,1""'"",
e.: h •
~I \oS~\oS..iJ1
~
-:0.
J-:.J lO
0
lJ",
~ ;;,
~
13, A
J"..,)
0>.11 L..."I) I J~
~,J r,..J;
B
JI
J'"~._ _>T
"""',r:S
(n_l ) 4J~ '-"J I J~I ~,,, (A - , ) ..\L,..d I
-.
_~
J
• ~L.=t-::~Iyrill
AxB
I
I
I
A
--
....
Alf
-"'.-
A.B
==
=1
-...."
A == 21 + j -k , B
""
"
v I c.::..J- 'JI _,
- j + 2k
(2) (1) +(1) (-1) + (-1)(2)- 2.. 1 - 2 .. -.l "-
..... ..... AxB==
1\
1
j
2
1
1
-1
...
A
= 1'"
-,j... - 3k...
1\
AI
-~I
= 1(2-1) + j(1-4) + k(-2-1)
oos
e
=
B ,A ~.~Il:,1;J1 ~
--
v',L-;
A.B
IS ~.MoI' ":"
"JJ' u",...; ~
-1
AB
-1
1
=------t • -T 6·6
e == oos -1 (-
f) . 99.6°
_t
;,..JII ~ (, ~ - ,
Moment ot a Joroe
-
~ t~,.a ~ ~~ ',;-,,,.rJJ t..-l.;.;~ J ; J.,iJ 140:. ~, ~
--
L.-J'. p ( % , '1 , . ) U.i:J 1J ;'': J ;,tlJ I wi J).t.l wl\S1 ...r j.}1.s OP ~IJ,:..:JJ·(\_')~I~~,.. .
, <'-\ " ) J.f.u I
;,.tJ1 t~·
--_,..
OP = r •
.-
N ==
~
1~
A
,.,
+ jy + kz
~
r x J
• ~~I ,.;1 ...... t.,.J ~~I~ ~L:;-l:.U:
( \
J,:>-
y-, )
;,.tJ1 tr ~I u l
J,:>- ~ I~..J 1~ ~ p ~ I .j :; oJ):o:;,.; Io:-.b.I ••• I..i 1 l......-J,_. vJ." .ill_~ r-=:J 1 ~ ~.J4.' 0 L.b.i:.J I j ....J~ ~ -.... .) ,j I op· f-i~ ".11 J.s., F ;;.ill v-1s. ~ oJ~ u I~ ..J 1 1.ill.;)-""4 ..J~ ~
• ~ I)) ..llIJY- J,l> ~
-N
tr-J I • ~ 1. .::~
t.i--JI J l..u. ~J-.: ~ t:J 111 oJ W I
... 1 ~1 ~I
•
1-r x ~ I • r~
_
sin e .....a.
,•• INL~I~u.j\·
~-e oJ,...J I u WI
J', r
J:.:;.r.:> ~ ',rein e
'-:'J
,.1
oJ,
oJjJ.
,--=: tJt
~ 1 e-,J,;
~ I..., ..J 1LSp-1J
;
~ ...."ljJl~ e ~
;~ I
• ;_:q ,..,
t',;-J 1~
( , A_' )
-...
J ;,.iH J 1.J.i. ~,rio ",,\"" bJ
"t:Ju.. I. J'::'...r-
it-..... J.1.i: J IS,.J; oJ$
u.r li tJt
~
0 ;'';
~I·(;4
L~
~ ~ 1 ~; i~~
U.A,
ui ~ I J.~ tJt' (l \ - \ ,) 11 oJ W' tJt lJl ~ ~ I
IS'· 1).- u,~ t";..JI~ ~~~I ~I u~
~ <-;;' 1 • tJt tJ i J-ill ;
,l
~
,
x -;1) •
J
~ .1 1
• 0
~~;~ ~ ~ ..t.>..
~ 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
(fii= 1
l\
d.A.v
AA ,!i) !:i •
( , _, )
"ciA z
j~+ k
-¥
au
• ~oJ~Ic~ .::~".;.I ~ "'" ~I .to; At v..il ;
i-
A, ~ u,\....; ~ ~ ;i- At vi ll.;WI;J .. WI V-
......
d a:u
--
(A + B) =
~
: ~I ~Jt ~
ciA dB a:u + a:u
(Y _")
4
1 ~ ~~~~,
lo:'' ' J..-tb .. I,; ~~"
Position Vector of a particle ~ ~I ~ (" _" )
..1....->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 =
~,"""F j.,.; li; ~,
' .... == ibwcos w t - ." jbw sinwt + ko
~',. v~ ~rJl ~, I~.s,w z -J~I .. ~~ -;~rJl;~.,...:..,;1S' W,
~
a
-""
dv
... 2 = Cit = -ibw
...
sinwt - jb w
2
coswt
.sIlo
I~
-;.~:::: (bW cos wt)(-b w 2 smwt)+(-bwsin t.Jt)(-b w~cos wt)=O
J!.:J IJ
~ I,,.. \J'
Z
-},.-J I .)r. "'~,.-: v~ ~ I v \; .ill.i \,,11 Ai t.. ~ ~
Vector Integration
fY
~I 4.16.11 ~~ '~I~, .~)Jwl..6l.,J..-.t4',J1 ~ u~
• ;,-;-1' ~ J.~li
~;.U ~--lI..6l.i,~ ~I ~
uA
J~
. -; = =
if Adt ..
~At
-'-1'
+ rJJ3t dt + ...
t2
a.
.
.....,..
.,.,
• ~_t.
~
C, 13, A
itf Ct-ldt
+ jB '2 + kC In t +
-
1'0
.J.~I~~,..
-.... 1'0
~I
~
Relative Velocity
~,.. lS. JI,::JI ~
r 2 ,xl loA ~t~,.~
wi c.JI}'
• (0_")
JUJI J
-
\"A
o v.-. t"....u._lll~ ~
•
Jt
0
...
"~-'-
~
~ n \S .lJl, -;2 _r11J;JI,. J~ ~~l: ~\:JI ~I ~Ijl vi ~
• (r
12
)
: ~ J,)lJ ~~l: ~\:JI ~I_./"'" v.i-I
-~12 v12
~
-a..
v2=v1 +
1 ( "-,, ) = --crt = dt = v 2 - "1 v 12 : J- J..u...; v 1 Jl; ~ • ~I_.".-JI I~.~ •• - - ~\
~
d(i\-r )
-
-
~
~J'-Jl, J~\~' -./"'" 6J~.t..: ~\:J' ~~, -r!1 J:.:: • .iA .~~I
J.a..-JI.):tl.;·\S'~ ~ v....: •• ~II~I_r!I)I~VI~ ~
: "'" ~y.:>~I ~~I, • ~";'-lJ ~,;JI
h I~2 I= h I r2 - r;. 1.
I T;.21
()& . .
~ ~)~u.b;.:;'• .iA,
f'\ • f /~}l.b.U '.s)rJI ~~I ~
~J )lJ~~~;l.b.U .:i)~'''rJI
-
1. • VW
J)U..;~
or
vi \$"'~: l'l"rJl~~u~
\$' ,·I~~~~,.,.,lbJ1.."..., ·I~I~"... ~~ ~
= -'va
~
+ Vw J~' ·~~v'r·~ 3,1 v.·f ~, ~~, vll.:Jt- "" u""uJ v true
.
~
...
~
~."..
J-;
J l:::J 1,).. J \":.)1,
v true = iVa + jvl.l)
"r!1 ..l.l!-.
Vo
1J'l.IJ)U..;~JJ~' ~
• ..;~~~l:
c:r>-'= b
LA):Ju-;~
-"
p J:.~IW6- ,)..~i.::~
v =2v
v
a
0
.-.... ... rap = ib cos & - jb sin 6
e ;;
.;
I,!,
.JI j.,.
= wt
...1'" • ~~I;'w'; J»~WI ~P .. ~~;;~;I.J:iS» J:.; .. ..iA
I-:-JI j.,.J ;;~~ p ~I ~J'W ~ ~~ ~;JI ;;~
-,.
vrel= -ibw sin
e-
;;.IWI • ..iA .j : loS I
...
jb
A.i •
IJJ
cos
e
.~k"
.
\"... S"'" ,lJ~;
)[1·(~ oiL ,I
•
...
........
- ib w sin e - j'b w cos a (1 - sin e) - jvo cos e
v == iv o #0-
;"0
ivo
:::
. e t...j I;.1J ~il..:;,",;"
~
u..J I<;~ 4~ ~ ( 1 -
" ) J{.~l I;
Deriva.tives of Proclucts of Vectors ..:.~I,-;-'yp J..;lc y ~ . . " ....":,,.
_-.:t.
~
cA.x13, A.:B, nA
. L-; ~
J..o\cll rW "-i.,~ I ~.
.....
( l _
-
f ) .,.:.J, I ,..I '
-'" d(nA~ == limit n(u + .6u) A(u + .6u) -n(u) A(u)
.6u-O
du
d(A.B)
=:
--'-0.
A (u + .6 u) .B ( u + .6 u) - A( u) .B ( u~ ~
.6u-O
....... : .
d (.Ax.B 1 du
-
== lind t
du
.6u ~
~u
--.
....,:...
-....-a...
-=::,A,},,;(u=-.,:.+-=.6:..:u=-,).::x-:B:.l(~u;-:..+ -:.6;;;:;".;;;u.)-_ _A~(_u,,",)..,;;x...'=B.... ( u-.r.) 6u
li.mit 6u-O
~
lli~l..:.'Y .JWll~ ~ ( n(u + 6,u) A(u)
d.---;:J l:;j , ~~ l,il' -
denA) == du
~
A+ n
~......:llr.......
d.(A.B) = ~ du du
dew) du
=
aT du
x
t
-0.
(\._'() -..
;B
+,A.~ du
i
+
J& J..a::-: }'..iJ'
A x eli du
'!'rolgential and Normal Components of Aocelera.tion:
- LSI ~I 4$;> ,,4-=1 --a.",
<..>..h....::.lr ~ 4.1>,
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
d
v~'
_
•
I
"'"'", __
I 'III 0_<.. r ~14 ~
I __ .\ ~
I ~J4 •
OJ..
0
I •
•
"",..s.;....,
d4~~~~~~I"-~0t.;~bf..:::.~I;S.r-~I!I
.
0
. jf 1..1. (A _ Y)
p .---- . ---.
('It)
lS~l., .It.- ~~.r--; p--;J ~...:.II\::. ~ (A _T ) J.!Al1 ~~ JJU.:I_~
..
II)..u. ~ ~ )
V
~1: J)lb;I-
T
."
·c.J.:
...
~ di l' ...... dr .... v=-=rir+r-
dt
dt
.~,:J I .} ':)'~ I..:;..l~!~vo;:l: "-
JL~;
..:->
r
~4.:1~
k..w
6
dt;/dt
d~' 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 :
k
·l~ •
I,)
k
(\. -" )
~I
~~I..1»U ~l ~1.1>,J1 ~l,J:.-~1
....
l,,.,."':"_41;.1 ~ u 'J 1~ i r ~~"""
---AJ~I~~t;. (11 _ i' ) ~l J~y.l$
{n
Ji-.-All
~~~
lotS .3r
4-=t;
k&t
;Jep
:J I .y' ~)l~ 0~ ~ j ~ k e
j15
z
d
i
r
kg
~".-.,;.j- ---+-----.------ Y
ftA"
'"
-"
-"" dr a .... V = ---- = ri dt r
+ r-
dt
-- -----------------0'
!.:'" ~~)l.
J I to.,,}] 1..::..1; I.~>-)~
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
.
..... j>- j¢
ka
(f'{_'()
ill
=
k
cos 9 -
sin
e
;JY4~ ~.,,; ...) I ~ ( ~,
~. = i
"-
=
kg
..L_ .... -~
I
•
"
i
~
S.in
e
COS
cps
e
eo,s~
~
l' +
j sin
dt
A
.
.
".I-
Q\,
dJ:" dt
8
e
e
sin
¢ i r
.
=
e
si~c
i'" + r
.
e
sin
sin ep +,¢sin
e
cos ¢' )
.
....
e - 0 ke ,."
0j¢
cos
_.
lee sin
r
.....
e ke
+
ep)
cos > -
e
cos
'-
.
e
cos
jee
= 1 Jp
j
,..,
k sin
,. "
+
dj
e
ijk
= i"' (8
_2.._ =
A
ein¢+ k cos
+ j cos G s:Ln
dt'
lr
e
e
CO2
e
e
.
(
-
v
\
1'\ --
'(
~
__ ':
1"0
I
·\L-S,rlI;W ~ dlJS r-;~ 41l>
(a) -;
= rot
-
~....
j..JL ... 2
JS J
vo;.u J~~ ~~b
t2
(b) r = iot + jA sin w t
r
( 0) = !A sin w t + 3:8 OOB w t (d)r = iot + jb oos(Ut + kb sin ~t
..... r
=i
W
t + 2j sin
dyio is;l,,) Jt....
r2 ~
...
:: i1' sin w t + jb .oos
=
Ib
oos w t - Jb sin
.:: 4J WJ ~,;J' J;l-:-;.:ll
J..l.Ao,
Ct
tV
a = tit + jBt
>-
i
2" ~ + kt'Jt
f - '\'
t
wt ~ I ~yJ 1;I.J.1 ,,~ 1~.r-ll .J.:;-
• t ~,.
A.:J l:J I ~';)!....J' ....t.:...: '\'
t
J.s. ~ I5A~;-~ wl;H::.l1 0 1,:;;,WI ",L:...;
-->-..
1
uJ
vo,;J I J ~.rJ I ~, \".~...:.ll ~ ..:.~ L." I,;J I ~
t = 11 /4 w
r
r~ ;;$.r
.. oos
vo;lJ
JI;I..6~1 " v-~~I
41..1 ~ ti;, v-) I t" ~ ~...:: r.:-~ ~ - '\'
Df,
• ~IS;u.;--U d!'.J~I, ;.....WI~~~I~t.;~tl.J.:i..:-~
a, v
_·;~.J!-::..l;;I~ ~l~
• 4
a "l'
~ll..il
-v:a ..
vl~1
0
Ivl
LJw· -~
8n = (a2
_~
2'" - a-r )
'\' ) IJ"'~I J~j.ll
t
• ~'r""'I\~ .• ',.,.
0 {o
:L.l·' ~ --II ~ ..., ) ~ •
~~;J ~1.:.iJ1 ~l;i~~I
(a)
r
= be
kt
(b)
r
=A
oos c.;t
v I ~ lJl '\ - '\'
E}=(.)t
e=
owt
• t
=0
• x-;.:; ~ ',1-_...:~ I .,; L; I ~ Ie.:..: If 1..; (; , ~,J J 1- ~ ~ ~po.;
(---,:1 ~ ',..:.l;)~ R
T
= a.v
• ( 1-
Co
_.
=A
, ~ = Bt
2
,
z
= at 2
\.i;,
\ • - '\'
V4,;) I
00
_ J~l!i...",;J1 ~a.:I..\> ~~ ~~ 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<1r,
.aJ~J1i} ,~- Jl:J1 ~
}UIuS),UbJl~.r"=,1 ..s~li~ r~~ L1l1
V"'"
~ Li.• , • ~ lAJ L- v I.; • ::r. u,.......; y 1:.:. _!,.;.r .., b.. ti!,J I J ~ l.i'J~ I ~ ~_;:..J1..., b.J 1l£1J JJ ,6l.i'J~ 1
v I..,.a '.. • • L
• ..sr->~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 ~
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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
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J....
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l.r--i-:
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A
..,II
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~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
;
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•
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,
.... 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 =
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vmax
= ~rT iii
w
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A = OJA 0
ul..G:J1 ~I,oJ1 W~I ( \ l_f)
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~ WI~ ..s~ ~I; .,1.J't'!1\·;1,.u1.j
~I
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~~I ;~~I.j
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«
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i • •0;
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• (Y - f )
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mx
Ii
i.
. l:.. ' 1_ ",11 ~ .~
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•
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' •• 1
2
Damped Harmonic Motion
, I. JI.M.-:"
v max
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,->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 :;
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CffA_ r)
II
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...... .. Q is"
m
2
l.iI
t.i-.L-liS" IS""'" ~l '
J.-.."., 1.1 ... WI
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=0 dt q ..:,.;,::-1 l.iI1J6JI v~ ...... ,· t
dt ; ....L.-II4J ...WI
mq2 + cq + k
"i~ ......H:H
I
=0
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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';''''
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~
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(A_T')
J',Lw' 101"
J1.U'
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t
1 ..
;
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~~
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jW'~I~,WI ~,
: , - '''It x = Ale
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( 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
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J' . (Underdamping
.:rii',:"W,J4
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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 )
~
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,, 9
Ae
-Yt
0
= _ta.n-l(~),
A
=
~
(a2 + b 2 )'t
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W
• ~I ~~p~
w 1 ~~p' ~,
eo-- ;"';JUJ~ 1~;.r.:J- t(
UJ
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ee '==-'
~, y.l.:..i:oJi Vt
Li)l.J'
o
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2 t( 2W
_
~~
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i.
000
( ll_\" )
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- • J.ii ~,~ I v==.uJ" \i.;o- I, uc.uJ h::. I .1
;; "I
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i- ;, - .s'~1~""-J (\_\" )
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_r )
t ~~, ...\ rlw~'J.~u~' W'~,~
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J.l..-.i6J I • .iA ~"
= Ae-'ft
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~
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1_,\ +
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... ~I ~tJJJcsJ~.,ai,.
x 1...>~",,,,wJ,,~'~l
x --Ae
-it
1.1.:
~ ~
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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 ~
~.
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,..;"_H, (
0;" _1" )
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<
UlX
dE
"I
• EJ
.;.;JI
+ k:xx "" (mx + kx) x
+ kx .. -2
-ex
-ex
,;J .Jl...JI..s1
•
;s',.Jh~~I6J.Jla.J1
va ~ J
~~ ('V_f)
Ae
- '/T,~
A
:: e
_ --/rr'l .1-
e -1.56 :: 0.21.
Forced Harmonic Hotion. Resonanoe .~ ~ -.\J' ~l.G!oJ 1
...,u I,:.)'
':- ~..i.:J 1W
j>
4
1 1.lA
J..., J .JW..,.""
. o-;Jl~ sinusoidally ~ ;JI~ .;:.~~,; ~I· ~'" ~Jl> ~~ 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
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':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
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~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 <.&)2) = F 0 cos ¢
( '" _
r )
( y. _ r ) 1..-J ... WI~, ¢ ,i-)l: ~ sin ¢/cos ¢ = tan ¢
•
e - e"o-bJl a-",lj ,I...,JoJ1 i~lb_J1tl.»w-l, ~,~I
,j) ~
Ja
~WI
~J..u..: ( Yl _ r )
tan ¢ = --,c,-t.P __ k - ml.o\2
'':~, (y. _ r ) ,( '" _ r) ~ ... L..II . j.,J. ~..;..:,
"I ~ 8in2 ¢ + cos 2 ¢
=1
~~l
\" .A,
L1,l;
"w;,,'"
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]' ".,,"'''''-~''"'~-..
A\{(k~m <J:)2i +
0
2
(yt __ ,.. l
~2
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~ vi o-bo; ,..; r{=O/2m,
(Yt_r)
, ( Yf. .. r ) -~~2_ J)2 ~, 4-{2 j . o e,,,_..J~..u1 ,J.J.rJ', A M-JI ~u")l,JI ~..;JI, ~I .W.i.ii lI.,)L.JI A..
• ~lhJI ~l... ~161J)L.J1 VUI
'J."
tr~
e"A JJnpressed
.,).,)'!I ,I:...~ 1.1 A ,•• 1 '.r- ~ ~"1""'....
I
driving frequencYw;~1
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__,,
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CAl. I
"
t
J Il.\o
~1...J1.,).,).rJIc:.-a..-JI ~ ~ (,. -
= 1/8
0 .:
m~o
1/16 m ~o
, tv r ) J!-:JI
t-~~, (yt_r) ;J.,)WIVtdA/d~. . . ~. ~)I "l.J~vl ~W
""J' .,)~~
J ~I.iJI1J.,)W~J..~,· ~ ~",L- a ~.JI
s v~~)1
I~,~.......t:-'~
0
J',~'~1: u~ i.~"I. ~I J',tA:JI1Jl....j
oioJ~ u'lS,;-l ~~ ~ «WoulS' IJIlf.LJ~
lI""tre 1.,1 c«
2
r.
iJ
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...
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'v
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a J-J...u..:..W; .;~" J,~'~....JLc ~l,
2
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( Y1 _,..
)
<.j)o
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a 1...tl::S' ~ .iS~ • if=/w o~' J.:.:: -t ~1 ~
CI.) 0
-
)i
E
l,;,'
~jJJ • J',lAoJi
(
YY - f )
l£lJ~ ~'~1 ......~1~ u)
r'C= c:,) 0 -
E
(
YA
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4:0'1 ... 1 I
..".~I,
Amax
-sJJI, ( Y. - f ) , \ Yf. _ f )
'lit
( Y,\ _ f )
y2 JI.AI~, A
max
~o
(A._f)
"C::! - . . - : ; . . , _
.'
2~me..t
o
c..eW~lS' IJII~ •.M' ~)IJ.~..,. -s;\;,11 y~J:H4.- ~uJI J,..JI_~,s:, ~ ,I.....,;~"JI ~
ui-cL..;,1 ~J••. ..,;1
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lC~· ~IS;I.JI.~~I .. ..,.~~a.JI.
• II • .u. a....;",. J ~ ,
I:. ~
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J L.b.;1 J:,li;J
c;L~ I &d
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(Yf._t') W,jL.JI~"I.JIJ:t-~' IU.. a...(.A2 _
o
(.b2. ...
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l ....... bJl ~Lc
A...
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j<
+ (,)1)( ~ -~) ~ 2 ~
a........J I;Jolt..
(40-
cA)EJ-
2 A ...
J:.:;
(
.,.,2 q)
;
00
Amax.~
a.,,/wo-w 1• '"
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0
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Q;;
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•
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a.....,.;I ..,U .,aU ~ S' .."JJI \oij ( YT' _T' ) 6J ~L.J1
•• Il
• LI
~;I L..iJ ",~I
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) J!.tJI .........WI • .iA..:--J
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t»
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_0,. -e ...,JJ'UllP-.~~~I~j;
I~;
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cs1JJJ'~.IJl ....~ .... lr/2JI_
1. ,.,iJ7T ~ _ 4i,:-;fb I~I,
i1 '
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· ~;I"
a.....,.;IJoII.,a.u
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"""'-'~i:...-_..:.----,
e.il ••JI~~.i6U"
2 ~o
~1~lj ~a."j",J1 ~
-~tu
("
_ T' ) J£.tJI
II
\I
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1... '
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r
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•
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II
u\
o
J+ · J¥. 8 m
J:L" 8
: ..}t. IS 4,1-,.:)\ J.1a ~, • -" Q.
.-;,
) ............
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wr
<.Po (7/8)'1l
2...(
2( ~0/4)
r:-::.
- . -----.
*I.r-..u ''r.l.:.i::JJ
tan ,. 2(
~ '" I,iJ I""; ~~} I .J ..;;J
(,))0
~074)( QO/2)
I 2. -', L.c ..I.1-J I ....;;J 1.,;, IS' 1J I _
•
~ 2 _ (Q /2)2
o
2 "'7/8 • 1.87 -~
.-J.JI~~ ~
! .~L;..t.». (vr _ r ) lI ..l".JI c,)t J4
0
, • tan-1(~) .. 18.50 i
"I~ Ji. ~'" ;....".;1.. i,; h\:;~ ;.f~' (1' __r
Motion under a lTonsinusoidal Driving Force ~I.J .:...~I.;JI~
L,Ii I,: ..,.;1.. :i,...J
I.a,&.-: ~ u.,). , ,......-:
'Wi .; lr (t)
ib'
c,)t
;.;'= ~ ..,u'" Y.l.:.a:..s'".. ~ J-..~ ~L_JI J.'-i.&::J I ;; ..li
_:
.1\f~1
,
I
".... . I ,J)l.JI c,)t ,
3 ••
~"';'.,;,fJ,
Principle of superposition
Jli,lJlJ,-..,;,SeI.,;,1
:i .. liJl.~ &JI"*';
.WI
~I ~I 'r.l.:.a:. ~
••1
a.,J bJ I ~ li:AJl ..:. ~ oJ W, .;". .)AS" u I~.~ ~+ein +
kr:o. •
l!'n(t)
~ • x.n.(t)
mi +
ex + lex: ... l!'(t) ...
L
.~:r".(t)
x
:P (t) n
(A f._f)
L11~~~I'~')
Jl
1,,"'-11 (J,..s.:J4
1
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~ 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 .. .iJ<, u I .6)L
~I a.i; .j ~I~~I ~.i.ilI~ .i>1 ~
...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'<.5 1
L....wu l ,1.1:;-;;~ '~'4., li. ~ JI;---'II I..L.-!, •
-? ~
J.:.:
~ 04
A
0;: ~I dJbJl ).~I ~l V--
.,.;~ I 4.;- ~II ;:j I..J I
,i
~ I.:J 14.::-'.II
•. u
=-
rt
~Il ) I.~~
0A
;;\. I •.:J I r~u,.",
~ t ~ vD~ l. J4 ~ I,
J: l.S ..I,..l.,J I <.'f
y _ f. }
• L..J....>. < ~ (, y:w - ~ 1\ \j)1.> u..ill! t.L.il1 45:r- 1\ '-:!'• • ..., l;W,.vY•\,; v. r:. • M
M
I",L.- ~ I ~.i...J.L.~ ~..1 ~8'~ ) W I ~ l.-~o ~r..ll I.J.::- ~ ..u.- tJ 1..1 ....sJ,. o
o
~~I~k..::..~6J1 ;..1!:-~
.J l.A.: 1~)l:J ~ ~ ~ I .j
1.S.J...vJ1
.;; I,.:J 1":-' ..1.:.i.:J 1 (
'f_
0
The Harmonic Oscillator in two and three Dimensions ~1.J~~;~;';~;.';~';$f'"411~ .jJ>~
L.i)W~ ~~ ~.i~;'; 01;;[;1.1>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<
+ 6
) '~
y :;: B [OOPC wt + 0< ) cos
~
- sin (wt +0() Si116J ~ ( 't' • _ t) ~
+
I;
x A
cos 6
x2 1- (1 - ~) A
sin 6
oJ
I.-
~,1 ~ ~,
,.2
. 2
+ ~.J4r = 8m ~
("'L_O
B""'
I_'~M'" ~ LJ I ;jJ L..J I to 0 ~ ~. Y j
x
J ~ WI
..J I V" ..L~ t~!I • j;. ~:hJ! ~JJJf
ax 2. + bxy + cy 2 + ax +
~y
.., f
')
b"" - 4ac r i
.:-lJ ~ J ~ .)--"'; ~ I LI bJ I ~ • I·
I
I '.('
~......" l':'''' ) ~j ! 0:tL:.
.•II..i.J
',:U
•
II
6 ~ Lw
J I.::J I ~ 4-,.
,,1 , I~,
41 l-
• <' -I ..s I - ( 2 • ""=--'...... . sin.6 Al3 '. )2 is,~.
( 1_ 0
ji.:J I J
V:1- ~
LS WI:
B r-----~-~
I
'
I
-A A _~~--~~.l..-:--._p.-+ x
I I
==--....::r-:.:; - - -
I .J
-B
(, - t) J,tAJ1 ~ ~..i.:. .s'p J ~1: ~ .It-
u
_~~..,-.~
\ ,. y
.w~
(,
6 ~I 0} uA l..w , ~WI 4.l\;.J1 -.}
~,~
71/2
~l::J1
2
4JJWI.)I JUI 4.l J6..~;
2
-2L+....L-l A2
B2 -
J} 0 I! I.i I 1.1 • c I; I.b- ~ I "" b.. ~;~ "» b... .......; l; ~ ;J ..I lAo
~I (, ~ B
Y=±T .-
J
1l' ,I I;.... ~l
,
6=0 u,.·-<-L~~--";b'11 t,. • ..1.>-~ J J ~~I . ~ ,...,
':.~ yaiW I ~ 1) t.- ~ b.. 0 1 (, do. W 14.l b.lI ~~ 1 ~. 6 ~
x -
~ 1t;-
1-~ ~ .s'.r--1I ~l;.;j 14J oJ W 1(,
mi
= -lac
mY
= -ky
mz = -kz )
Y = B COS (G<.Jt"l'" ;3
)
( w t + 't
)
Z
=C
cos
'l)
L....: 1~Je J 45'.r-J 14J b. .j ~ l::J I ~~ I ;j,.,sJ I
(iT_tJ
cos (w t + 0<-
x=A
..I
= it
'f,~J I;~!
Cn _
c ~ ., W
..i...,)
X
L <-L.w4...lW .,
u~
1h>. ~ JLJI il.JW
A
(n._t)
, ,. A
x = A1 sin W t + B1 oos Y = A 2 sin w t + B oos 2 z = A3 sin oJ t + B OOS 3
<-&'
t
GV
t
W
t
(to _ t)
•..;x:.; 1.J.:4'¥ 1~ 1~",..., ~ V- v-:".::-J 1V- JS" J
4.:...J I J,. b::J Ic-. ~ ~
~ ~. (fO_O \:"'¥,,)WI~v-~~~~,':lI~..sWIu";'v~1
Pl'
B2 , A2 ,
)l
B's
"I'
~~~ Y' , x ~~~
~~,
OOS
"'t ,sin ""'t ~ vi
A's ~1,:~Voe~ b, a
c-.~JI~
.j )UI • .j..c.':lILb.i; v-~,.:- J u ';I~ is,PJ1 I£lj.; ~',) ~ • C:x ..u.,J.1 ~ I.; ;Spol I J ~ 6J I ...r- l.S'). ~ I WI;
~ uA 4.lbJl';, ~
I
V--
z;
A1
y~x"
j:.; "\:,,)..1" J' b.. '"'" I ~pol 1\:,. ~..s L.. W>=,," I.; I • ~ l;Il :i ~hJ I
;;;.'y/ _~~·I ~,~ 1:...1-',,)' ~~I""b.JI~I~ ~ L...J~~;S,roJJ ~l£:JIt.:- ~.)WI ~ ~':lu,.... ~.w.
.. ,
"I
1
21 ::::
~~
;SpJIIS':'- ~
"
-loe .
nIX ::::
my
L.,J
/
= -ley
c:-
0
IJ,:;...-ll
J ~l;ehi~I,v=.L~lIt.:-I.i;Spoll)t-J!.: vJ;)W. v.i'
~1..t.:i.c,¥'~.,..J1J.r.;Sf~.·)II,$':'-ul.."..s ~. JbJl~. X/7'
. ~",....¥Jt'" <,,;-.-,;~I ,,:,,~.l:J~~I.;S'p"'L~,.b-, (ff_OI:..~,,)WIJ:.=
4~1;,aJ1
J
...:..:~.....,.,
•
~)l:J1 oS~~11S.i
isotropic os-cillator
,.;; •• ~~I,~I ~;~I ;,.iIlc~1 l.il
LI·
;';"lj~I,L;--:1 ~
":' ~.i.:J IaJ bJ a.,l..,:, li:Il e ~ oS L..J I ~ l:S' ~ • ~ ~ I* ~ ~ .i.:J I4J b. ~ .)I:JI~I ~~)l. JJb.. )~~ci.I..i, (j..;~I*
mz
-k~z
III
w 2= " kim,
~llll Jk1/m ~ ":' ~.i.:U llJ.:.:;.. c. loS oJ.r 4:~ 4.l b. 1; ~ l:~ ~I:JI J,.w~~pJl ~
=A
x
OOB (
U..)
w 3 =Vlc3/m
1 t + ct. )
( r V' _ f.)
y = B ooe ( w 2 t +;.3 )
=0
z
COB (
vv t + ~ )
3
k,,,.-:JI ve ,)lsI •...,s..w Ie ~ oS L..J I J 4.::-11 J.h:;l I~~ ~ ilb- ... lIlSjl,.:.u,~
e)l
~l;.:; ~I aJbJl
J
~ ~ efWI,,:,~..i.:IIt':J
J · ~~I tl:J; '" .j"., (20
6
IS;>I .".
• ~I.s.:.:~1 ~)l..ol)
, 2D ,2A
I
~\ (W~, W
1
III
J11 U~ I 211" ~
_ _ _-
w 1
w 1 )
w 2 = Wi
n2 jI " ) •
w2,
n~
L.J I v*-
2 TT n 2 w 2
III
• ~ loS.. I~II",L:.:
w,
2
TT n'2J
(fA -
L~
0
n" n 2 , n 1 ~ •J I",....· ..u...• r=---_........ 11 04) JJ".
-;_-J1oSWI ~ cJ;?I} • IS;>I.". ~pJl oSL..::, JI~ ~I ~ .)1 oS~ ___...~,.. ..' )WIU~ 6(j.!..JL,.
(~I .s.id~ J"~lSlvl
0'''' _
i)
\
;;
,. .
.:...... 4JJ ~tlJ.....J14~1 d,.ill ~ ~~10,s;.~..u:...:.- ':lb. ~
:i.iW. J ~.J ~II.;-' .i.., .i.:ll..:.-I.) .J.r U..,.:.... ~ • ~ 4-~;;..iJ..... ~ ;; .) L J • ;,.;t.l1 J~~J \ f. \. ~I \ '( \. :·'I~!..:.->.:¥ ~W~~L~I..:.-':lb.-JI~~,.-.:.JI~~I~,t>. ~A _ 1)
Motion of Oharged Particles in Electric and IIagndtic Fields ;;,........;~;''; • loS.,,>1 ~ L,nS..:.- ~ )~ l; L.rc! 0~ ~ u~ L ~ ~
":--_-..:<~; 4"" I.::-J I c. ~ E ~
F
J L.r~ I J h:.J 1va
t
~
u.;,; F
;;,.i1l. ~
-'0.
= qE
. . . I. I J~I ~~I~L.n..1lI~I\".~
·<'<)JI.,,...
q
~
~l:lI~1 ~,-.-~I4S'p" 4.l.J~ 0,s;~ <~._f.)
.. = .. Ex .. JIQ"= .. :By
c.~.,JI4.l':l~,1
mx
({\_f. )
...
mE = .. :BE
c.I ,
...,
J"....:. ~ c:J,sJ~
E, t,J~~ _q , ~4 F V"li:
c:J,cl:..J 4 E , ~~~ ~ Ic.1.1:>->-:
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
'.n-'<11 ... '.,-.I\~.
..
4-...b1:..Ao
~
..w1~1 '.Jll • l.J'v 'J'l
'J
~xE=O
I..i.....--r:.
~
J .sr~1 ~j ~
. E = - ~ IJ •
Hh.! ~
0,5;~ gfl
I~, • ~L...c1:.>-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/ <-J
~
~J..-wi(l'\ - t ) ,(lY_l) ~jWI
')
(x - u)~ + (y - b)
2
= A2
l...Aj."., A LA#u.-.:~.r=I.J'" ry-
IS,:-JI ~ c~1 }l-J:-&.- vi",
l:..'JjW'~;.:J\:.lI-.JjWIV4~)u";~lvIS"U· (
)~
v,...!-.." • Ji.:..11 ~,;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"'",-~<: A u,;6J I .,w u.-: u I tS.~' v 1 = (x2 + y2)'t
~ ~"Jts
A =
.A
v
1 c;;-
tV
.s,I..;JI
tPl~, UlAWI
;;
iJ.. .)a.
m
V1-qB'"
."wu.-: ;;Jl <~
;;
=
u* Jl-Jl;.
(or - f.) Z
.~L:"rU:i.1". ~"J ~ts Lil
.,,:J~. v 1 ~)lJ.;"J't' ~ ...).":-~ A u l e-~' .;:c (A ..,..;, ·~'¥IJa ~'¥ .sJI.iJIJl-JI..,.
II' .~-tS~L, ~AWlErnest_Lawre.ce~J:I"\~JI
,,.., • v~ 1~ 1",,)U.;l u1-.a.:;..::~
4 ~l ~l '6~ ('\_l)
Conotrai.ned Motion of a :Particle ~ ~
"""= v I~ ..s .iJ I t~ ~ 1st-
oi.OA
I~ 4d.)~.H ,.,_~.H v~ l. ..
'",I ~ .; ~ Y~.s'rJI ~ ~ .. Jli:ti • oJ,ob.e ~
•
vI
oJ
d:- ,I ~
1.iJrJI4j,riJ1 ,I..s,j u.-: 'l;1 J>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 <,..,..j; ~
Ll,.-J ~
+ V (x,y,z) = constant = E
.:~) ~
•
..••.II, I e;t--J I ~ ~,3, ~_,:.1I v 11.$1 • uJ. 'J I ¥:J 141 b. ¢ L""tb ~,. ~~ ~ 1~U:J 1v,x. l.tl,....; ~
,~.>
4
~I ~, ~~;JI C-,jIIJI. a
~)-.Ju.,.; ~L.J..d}W~~~ ~ d~ I d;; Ur dj."i;
~J--"; .J) d"; ~u..... 'J I ~ I ~) 'J I ":-- ..4J 1d";
--....
v~v JI ~r-JI4I.JW· R 1.$,;11 cUI
~
dv
I.:f' ~ I ~;;~,J II.$,.;J i
-lo"'::'
m --ci-t = mg + R
v,..-..-x ~ ~ 4:-.lSJ 1;..;lhIli· (A -
0
Jf..~ 1 ~
()o:HO
~ ~..:.-l; 1..1> 'J 1~
~d";UJI4I.J~'
.. L-_ _ ""~
"""""'~
Y
mgz
, ,.. A
•
E
= mga
So: ( z =
V,
v~
v
2
.,.-J
v = 0
) ~1~~I..b,r'JI~
Ji- 'i I J I I-i'iy I ~ r=-=:J I ~ I •
oJ lA. A.: l:S'
=-
~
• 4S,,:oJ 14.1 oJW ~ 1~ ~~U b J.p.1 l.iI .~ ~ ~ ~I:JI~I~
.
e + R = - mg -!... a + R 2 R = mg --!.. a - mv a = mg.I---!... a - ...!!.. a 2g( a - z) - E!!.. a
\f.I)JJ
= 2g(a - z)
;r-iJ I W 2
a
= -4-
mg
OOS
C3z - 2a)
u JI
-¥
L.--~ ~A ~ I ~ I "" ~ I. 7- = v'!:' L ~ l"i-lS' L..: R u JI • I v--- • l...:.Jll R ~J ~ 1~ v,s •.i:»- ~ • .. .iJl; U~ ~ • 4"....!11 r=-=:J I ·~WIJ'~' ~ J,s4SpJl ( ,
Motion on a OlU'Te
c:-- WlW 14.1 oJ W , v-:- ~ ~; ~ r=-=:J I.s'j >
= xes)
y = yes)
z
= s(s)
v-- ~I J,J:- J,s .... liJl uWI J:..:
J
....·~~IJ~'iI~ 1;.i>II..iI-S,,:oJ1
•ims 2
~ v~ ~ 14J1.JJ
lSi"eJltI l ~ ~I~ ~.,~
parametric form
x
'_0
i"eJltI l ) ·aspJI~W ~
s
.,~I~. (~~1~J-1I..i:
v---r-;)'+~pJl ;;UJI ~ • ..b.M
B
t ~1I4J1~~1S)1 ul1.h
",,-$~£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~~ ~
<"
=0
~.J,...JI;';WI ~
.. -mg sin
~ZpJJ ~li::J'
ms + mg sin( ~ .. ) = 0
e+ +
,'"';I....U
~ mgz
z
.u,1
_1.)
V ..... Kl'uUJlul&~~
t.T'
-lSI. 0 ~Ive~ v = mgz = mg; (1 - oas e) oos ( ~ )
= mg J - mg J" dv -
sin
e
~
-mg sin
( S )
7
= -rng
sin
e = Fa
•
e ~~edJ.1o (, l _t) ~
e
= 9 0 cos ( w ot +
, u .i I....,J:J I J. ~ I.a"" IJ0
,
( A _ f)
.J.4l1 J ~1; LS' • 4J oJ L~.Jl4 .u. J-., (, r
fJ o )
I.:-' ~.i:.l1 4- ~
L......S'pJl u~ J,....,sJIIS;l.. 9
ul l ain jJb
9
90
,
Wo =
{j
_t)
~
":""~...; v~ lS..iJ1 ~I ~I
~o ~~";veJJ' ~~~
e
1{1
• 4~1~1 1f")LJ1 ""
~1~~~..i.:J~~I J,4 1 4JW u.s 'p~1 J:.JI (H'_f) More Aoourate Solution of the Simple Pendulum Problem and the Nonlinear osoillator:
J, 41.s'~ ~I&:J 14J.s W
~1
Q+
I
- } sin Q ;;: 0
r:-i- 4~;"; ~l:~.s'~ 4.WI~l&:Jlc "l.sWIVt ..... u.W~ 'rI' ~~~. ~lj~lt'r~l~l:.:oJlJi. 1.~~4"; ,,,,I, ~ Jl:JI~1 ~ J;~v,..L.:~.h>cj~WI4J.slAJl
~
+ t(
~
(, 0_0
) = 0 \
~,JI;:....~I&V'~'j"~I~
1'(0) ;;: 0
t
~,~
• L~~~16):J1~, ~oJ~1 ,~Ir.f.~li.:Jlc"t.,lAJl~
, t 1'(
4:l-L...I.L.I",;S~.s.i:f.'(~) 4J1.1l'vl~j'
cjpower series
~
) :::
~t
+
82
t2 + ~ t '
+ ••••
l~.s'pJJ ~l.i:ll ~.,WI ~~
2 d
1
dt 2
8t
+ 1
+
a2~2
+
8,1"
+ ... = 0
("_t)
,l,
..t-:>J 1J W-: v,..L.: ~ ~ ~ ~.i.:J 4. WI 4S'p.J14J oJ L... ~
J1:.
'L..;lb ~I I.u.. vIS' I..il • ~I .~.J1~ .)"~I;~.wI4J.sWI ~ L--~.~ a.i'po.H v'!:: ~..w
".$; ~I c)l. W~ ~ ~I
~
).lit I~
&:1
v IS' I.;1•
t IS I
, f. Y
L-.---....' L II J,.".w . -\ '" ~ I I.il; • ~ . I-.J L. ....,.,'I ~ -r- ~..l • clJ.;.~.~ sin 9 = Q - -
g'
+-
'I
g5
-
51
.
J.s. J....' .j • 9 + ...L 1"
Q -
g' =
...s.... 6;"
..b.U ~ ~ ~
J, ~ 1(x..l.>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
<.0
t +
+
A cos cut -
6~
•
".u. i,.i~"':
oJJoAi
=0
A' cos" wt
~1~U~lrl~1 ~,I
oos' (-AW
v
U •
2
+
i~ooe u + t
-+
OOB
"u
.,
A-
~
.,
)C08
wt - ~.t oos 3wt =
~~ .~I;,.tS'..wI4J.,)L.JI~I\S;.
.. )L:..,,~v I ~~ A 001
'I. ~'..l~I~ U-u , \",,,. J
0
A = 0 ~oJ~'oII-.JbJl 'L:.:.:...l:,
wt l:Jlt:>1 ;.II.,) \£l).lJ. t
v I e;-;:: ..tJ~, • ,)is.I;.,s-.w 1;J..l L..J I J oos .,
w t
r:i~ C~v I ~ I "",",t"~ '-:--t-:
AISJ4:.>~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$.JX. oJ ~ ~ u.j) 1 ~ I ~ ,~
.~.ill~lvllS;
.,ill JI
J
• ~~. ~1~~~..iJlvej ~"I~
\".L.~ voyl ~ ~ vJ.IISJO.J ~ c.Pu,1-Jlv- t~ • ~ 1.t.tJ I tlt.i: ~;..u 1~
Ohrietiaan H~gen~v~}IS~~1~~~ J~Iu.:.:sJ.J.iI • clt.WI J,~~ u.l".::-I ~Ie ~,~~ ~")LJ .. ~I-;~.ul J'1bJ1 vl~, (evolute )'~'Jljl.JJ~~,JI~~I"'J::'h.U:SI.dJ~
J,~I~ ~v,jl· ~11S.xJ.J~ISJO..J1'b.:>.;lj~ "....s.:;JI~1
••» ~.J l;l-~vl ~. J,4 Iij6S'p-ul;. L"",.J (.J,~)
v
J
pO
''":,
rl.......: 1 ')II 'tlr>') I 4&1.,H vo t)lo · "J..-JI ~ ~ ')I u..il ~~..iJ1
· ~c~J '"",I The Spherical Pendulum
1S,,,iJ1 J,~I
( 11_{)
'I lS"y<_J..~'~ '-',,~,r-'IS'. ~ C-- ...,- ~ ;S'J> '!<)I1lI-;·!!'I6S'-'1-'6I, ~ ~ '.J""W ~ vi I,
.
~~)I. i ~ 4.l bJ I j:.; ~ • uJ.1 1S,;'1: I
J», J..>I.J -;~ 1l:S
IS '"' L-~ ~Jb, ~ I,1.k..U ~L;.J ~,I ,,:; ~Y J,4~~
I.u, (, '-0
~.,.~ 'J}
J1:JIJV:tte~ IJ'. ~tLb..i:J~ulS'.. ~1
\1,)31
Z
mg
x
J
~J~.iJI~t.:lob- 'i~ ~~J.JI
J-i- -Sl J>'; ~~.:J t ~J ~ I,,:-~ I~"; LA • ~ I ~u 1;",.: u ~~Ia urU ~li:)t~.JWI u'!: ~~.;:,JI,I ~I 4.;.;)1 ~ S .u,J1.,Jj L.d
~
mr = mg -'>
--'"
(
+ S
J,.;l:J 1 b.--:'i It
L-J'i ..lot 4S"".,J 14.I.J l.-. ~ ~J- u,!:,
4
A1_ t
)
z _ ~ I 1:.1.6-1 IJ I
Jl:JI~1 ~ • ~J~.iJt~trl ....'i1
mi
= SX
mY
= S
(I\y_t)
y
mz = Sz
- mg
;~JI~'JI~VS'A>lj'JIU'!:
Ix I «J,
~IS'
l.:-L;)W~ ~
mg ~I
W,
d
. ' .;
J
y_
li:) I
i +
Y
X
+ x=
X
~t ..s.:J1 JI..u. U~~ .I~
~~J- ~~
• ::::. 0
S ~ _mg..:L.
~ X S --mg-
x
1;, L.., 4.-."i:
--'-
s..L:JJ y,
L~4.I~ ~~ .J~I~
l.:- 'i .J~,
1
• j1.\.l1 4....i:...CIo ~ 4.1~
•
11'1«1, iJ WI a...,iJ I
li i.....·
uSee c;J ~
~~~CSpJJ
0
-1-
y+ Y == 0 (AA ~ t) ~ .. 1:::....: u ~ ~ \S ..u11.X~1~..i ~.!,.:tI,,:-, ..l:j:JI~ 'i .. W a.L:L. 4;'" ~ J,bJ~ ( y_t ) X
= A oos (
W
y
= Boos
w t + /3
(
t + tX )
)
(A , ... t)
\
/)
\
• ~IIS,.:-JI IS,.;..-J I ~ d$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'~~ =<-£sin ~O)~~
• l.S)
-t ~I
l.SI').J.iJ1
~J.a:...;. ~..l,...J~~)llc.~"JI.i>I~.. u..il
'1 0
n
•2
( v\ - 0 =(m,;:.sin Qo ) /J o S cos Qo = mg ( 'I y_ 0 4.J..l W I u1- ~ ~ "" W I ~ S U..b. ~ S sin '1
0
(1\-0 ~I
t) ;;J""WI V- h
,( '\ 1 -
2
)..,.;. US
J.w,s. l..it.;· ~o ~v.~"; ~ ~
'~:4.J1 u t.; { '\ 0 - t );;J "" W 1l.S I •
:;, ..
Q +
....A.. II
""
;;,;,;J I ~ ~ I 4J "" W I ~
u,....s.;
sin4~ 0
cos '1 ,:5) = 0 0 sin Q ~ lJ' ~~ ~ ~ ,r.-A:e
t ., :
J J.i,.. oJ I • ~ ~)lJ I Vo (
= Q - Q
i.:t- ~ I mLJ.J lii,
t)
~ ~
(sin ~ - cos Q'
t
t(
J, ..s.W.I ~ L .. ':- II;;J bJ I • ~.r-
t
o
~ ~ L 1:;, L J
= teO) + t"(O)
.:,$ ( , '1- 0 ;;J "" W
t + t~(O)
k:
1 •• -
0
I ~ V",.;J I d£.:,
L...:J l:;j I +
.....
" =3 cos .1'(0) ~14i~1
Qo+
t
sec9 0 ,f( 0) = 0
~.;~
_~WI J!:JI ~ ('\
f +-} s:-: 0 oil
0'
f
= Q - 90 =
= 2Tf J
r:il~o,s;~I;JbJly~I!W,
f
;JoJWI4..:l::S'"~~~~ '..1>~ 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/$2)-
t
2
Q 0=1
~o
~V4
( \ · t_ 0
o,s,= L.J4d;I.,J1 ~u..: ~t.:.:
90
90
a...." 1;1J :t1.1! d ..1> ~ 4..:..i..:"; ~P!';;"
Q
o I;J~ ~ (,
/J1
~
Q + sec Q o o
t + E )
T 1
t
I • 4.,.».rA' I ~I:J,....J I 'I~I ~ ~
bf=O
,I
Q
'-0
0
- fJ2 /.t
2
~.J4.
~l:JI~1
;0
0 1J>~
(\ ~ - 0
uk ( \.
f._
~1,!,:J I '-Ji
0 ;J.,sWI
, 00
-+ ••••••••••••••••••••
x
( H'-.. t)
as'j> JW
x., -
J!:JI
~,:-ll ~.hi..J1
~,~IJ,41
v~
)C
1- \s,:-JI ~ J, 4 1; j
~1~v~1 ~
;
J
• ;~
L.... .hi- vi, ~ 11.illI ~ 1~J JaJ'C Q
~ljJl ~IS'
1..;1, Wl: w.;~
• ~L.Preoess(f3t~ \.;~ ,~~~I~WI~I~vl~1L.WI
;
'.IS'~~"; J! J)I.> 6 ~ ~~ ~WI~I ~ A~~
.
• ( H' - t )
.' j)
.J~.Jjl
J!.'.II JV:Ht'" IJ'. e J
~";l:J1 ZpJl~~l_ ~lhIl~IJ~1
Energy
Oonsiderations. Limits of the Vertioal Motion
~~.lJI;'-
;
V\K;"")l,J1 .J~~ • ~UJI~.Jt... tl~I~ •.JIVe
• :. tQ I WUJ I
v = -mg)
v,s:;. l:rJ 41 ~ ~ • 41 t..J I~ ~~ ~,jJ I J, ~ ~,.;Lt.n
OOB
Q
, II "(
~~II.:;,..L;I.1> ~1~~..t..:u~II.:;,..~Jf r~
..
c=.l.JJJ r
E ""
1:1
m.i 2
).
•
~t ~IS' W• u.sl· (r,rQ,
Q ...
riS
sin Q)
ul!JI~~L..
2 '2 "2 2 (Q, + ~ sin·~) - mg)oos ~
L-.....t")l,JI rl~I ~, ctlJ.s j~~ • cos
• ~,PJI i;l!J1 ~~~
u vi ~I ~;.u,
•
92
--' .. ~I •..o!.wl ~~WI
· i1 • h/s1n2 Q
2E mt
Q u~
2 Q
+...,..
h2
#
~
U -
l-u
J:...U
~~, ~L... •• ~:':"I~'
..
'2
~~
( \ ., _ t)
~I
Q =
--=
~.
2 • feu)
. -<.. clJ.L v,-.,
( \ • Y - {)
JIo;o-~.i.:Jlt~J.hli;,II.:;,..~I$~ feu) == 0 ~~WI",,~
• ~ L.. )# f( u) ~ v,s; ~,
Q
~ ~ ;;S,PJ ~ •"" .i.~.II .. '"'-tJ ~)l:.:
~ • ( \ {- {) J1:J1).;1 • ~I ~;I~ Vr: ~";L:JIIo;o-~.i.:JIei:clJ.iJ .~I..t..:;;S,PJI~ ~~ v~lv~~1 ~cS'\....;: ~1""'l=JI4IbJl
( , t -
;
t) JUt I
,1";WI4!pJl
~
IS,~IJ,~
(a) (b)
( 0) (d) (e) (f) (g)
(h)
a = b
) l:! I44 b11 uUJ I J ~ ~ V'
;
(a)
V
2
=.k(x +
2xy
*
y2)
(b) V = 8:%2 + bxy + oy2
= k::q/z2 V = kea(x+y+z)
(0) V (d)
2 2 2) (e) V • ke a ( x +y +z
(t)
V
= k(x + Y +
z)n
#
~ Ji I,; ~ I_,.ill
4J I~
~
,,_ f.
) 1..J.J:J1
, ( 1 , 1 , 1
t...t.lSJl LiUJltlloJ ~ L".~L.. ~tli:/JI~~J~I~~tV:H {_ { v
= mgz
~
WI:H
B
(1 - T )
• .. ~I:i.",s-.w 1 ~ 141loJ V4 ;,.iJ I ~ • ...i'/~ I ~u...: ~ R ~ • , .ltoH,J
.w1.: ~ ~.lill &S'p J ~ k:J I..:. ~ oJ l,.J I..:.l,f". ~ -
_
"
•
.SF.dr . . . -.. 3Y
(a)F = ix + ~~
~l:JI~I..r'IoJ
.-."
A
,.
~1cJi>' ~b..J:f
(
t
0 _
3x
JP = 1y -
(b)
...i'}1
b ) u~Wb..
(
a)
u1v;.
I t :,.
..J I J.> V'..J L..J I
0.\> ~ • ( \ J \)
rei': •••11.1>". (G) l) ~I...,:>
lb.iU I
~
x _~I
y, • (. J') J.-~ I ti:.i:
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F = m
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27
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L ~ WI.;, 1;.11 r.;.;J 1,,-;""':' ..:> \
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Magnitude of the Angular Momentum -->.
~I.i1I~r ,)\ v
.;,1;.11~)101.;1
1) 1...I.i.
ds..rJI~ ~~, \S,!;lIr>;.11)1~,;-kJ
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.~
.~
v = rn + rQn'"
->.
~
, n x n L
= mr 2
III
0 u IS'
W,
.
9 = oonstant
• L.jf~;";J~
J.
c:J~~
;) ~ I ~~ I Z'p J)J ~ l,.; • ~ b. WI u": l; ( '\ - '\ The Law of Areas. Kepler's Laws of Planetary Motion.
•
_-->.
• ~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
,
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l~r%;1
(T·_1)
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iA J. ar""-s-
I -....
-"
i rXDlT
I
L
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dA/dt
L.p.WI",rJluldJ..i ve ~ 'lSjJ-J~ISIJ l;b L ISj"J1 J ~I
J
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u I L-.~ I u.~ , 1 • '\
4.:.... Johannes Kepler
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r# =
J 4-e ~ ~ is.)-,,,, ~_')..iJ
n
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(T'l_1J
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Pluto
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,
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tJ.iJI.:..~I~- c.S,,;t.:Il ~1eF-.,:J1 J~ J11'",.JI.( ,"_1
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v,s: l.oW~".,wJ1
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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~
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..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...
•
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-vi ~ ~. H1HJWdJj~;.a.; • (l _A J1.:J1) = ..• L
:.:; eLl
Sts:;;i~
a
37!
(A _A)
.u..~ Jr\:Wluj~1 (l_A EC.i.·<..t11ibrium of a Rigid Body
:.t.r·...:.) \...... ~L.:,:~l ~jlu,"""'.;. ~La;. 4...L:$ pJlt ~vl ( I,.;.._l.:il>- ~:i-;.~i~~
til, .... L;.;...,..-o
,-y..) Loc'
J
.;..J.:\JI~..,...~JWI
\$1- 1~\$,I..c~JlPJl\S,aJI~~u~ ~
'';'1 + :F 2 + 11 , '>,
",,~
0.' ;: 0
'"
( \ - A)
A ) ~ oJ l.:...J IJ • 1·5';,., i....; i..y I ,i v lS' IjI, 1oS'l- ~ I PJIt ~ ~ "'':.to
•
• .J.-li ~
J la::; ~ I vi ~,:H J.r u- v .i I .,...,..:
~ll ~ tlJ.-J I\S,aJ I~ t " ..r~ v I , 1'2 x -" F 1
i'
'r";:,
J:
~
-F
2
+
.00
= 0
J: l.:Ilo ( ,
0_)..
)
,., ,. r ,.. I..LA·
~I"".ul viJ,.:JlJ.
0- y~) ~'X ~ ~,I,;JI r>,;J1 vi ~
•v In .,.,.j ~ 1~ 'X u,...
J
r-~-J1 v lS' I..;1~ 1 It
.u..lJ ~
•
~I v~ v 'X ~".-ll v-::J.rJl vL:~ 1- ( 1 . - A), ( 't_ A) ~ l:1oJL-J~
• viJ,.:J1 rL: .J...JI
,.-1.:.:. ~..; ~ J ~ J vi 1,.:.11 Equilibrium in a Uniform Gravitational Field
.~J'X14~1 do- .)s.~~~~.i4- J~.j 1..L1... ~vl
u'p
~l:S~1t~1~~ m ~ mg ~,~~.i~IIS";~vlS'W
r
- ~ l.! J li::; 'X I vi J,.:J I J.
( 1 l_ A)
• ..
.: J ~ I -1:.:.::... ~ ~;l,~J I ~,.iI1 ~...". ,I~ ~ F 2
- ~I.S .j I"".u Ivi 1,.:)1 J.r ~l::S' ~
r 1 x -F;.
+
r; :x: F2+
••• +
L
-;1 :x:
i
F1
It
mJ = 0
It
.::-.,.
J:I..:I Lo ( IT_A)
- 4..:l:S' ~ c:U.iJ It ~b ~ . g' v1I,
Li ;, C;
r
r;. :x: ~g =( Li
r
m )xg i 1
= m:rem
:x:
g =r em
x
mg ( H'_A)
; r ,j--& IS, ~ • LI.J: ~ I J,:a- ~ . iL~J 14"; r r v I ~ ~ W 1~ oJ W I 1--:
~~ .;1""..t..I1 vil,.:ll ~oJW • ( l) ~I j~ .)s.4;-,.
~;. ~ .H~
mg
4oJ~
J 4-. ~ • jJ:J I j J"'( IS"A &..II ":-' ..i l~...::.JI."'; j ~ ~ ( 1 ) • LJ:SJI j~
~I
jJ- ~~ ~1:..;.; 1S.iJ1S'
Equilibrium under Coplanar Forces
""'-_~"~ ,.:0- ~ ~ .t- ~ ~ tiJ.- \oS~ io-,J;.,;.. ~ t::.J.,.Ju. ~ is' IJ t ~
'v)!,.-::114o:".~.aL-.4o:".l.t1'J-~ .I~,.a, F1
,.
=Ul
1\
+jYl ~vl~ ~~
,(~,~l '~Iv-~UJI la}J.:::= ~I) ,( \ '_A), ('_A) 4o:".~.aWl\oS1
Xl + X2 + •••
=0
=0
y 1 + Y2 + , ••
( \ 0_ A)
~ t.., "'" Iv)!,.-::11 X 1Yl
- y l X1 +
X 2Y2
- yiZ2 + ."
(\1_1..)
=0
Rotation of a Rigid Body about a Fixed Axis, Moment of Inertia. ~ I"" "'" I -.::S'p o , ~~ I ~ li:: ~ I ~.,:oJ I I.as. ,""'.....11 r...~.l1 ~ is';'' ~ I V-
JL....i • v I"".ill ~ ~h. "" b.. ~
• v~ ~ ~ ( Xi ' Yi
J:.o.
z
-..0-1 1 ~~
,zi L-;S".,
~
-..o>-J I ~ •..:...:t ~ J,.. 4..s.:iJ I
) tl.i:J I ~ \oS.il1 mi ~ J,..:J I ~ 1
B~t = (x~
xy - \oS,.:.-Jf
+
y~ r~
la)..iu.,.; 4}1 ••
I:v ~ ~pJ.l l.t--JS LJ..i. (T')--16
i
A)
J1:J I
~ I J)lJ.; I
(T'_A)
2. -,.J I J"... ..o.Ac
J!-:JI
.u... ~ ..,.-~ e-J.i.
(lA_A)
( "
~ ~ .. W I ~ J,...J I dJ..iS' ~ • J1.~,)1
'" ~,.,. lJ' 11
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~ IjJ I~~~
_~WI4l .. WI~lt1'J- .. ~~~WI
"'0
T =
L
-Im1V~
... i<
1
L m1R~) W 2 = iIW 2 1
L
L
~R~. .1(X~ + yf> 1 1 <~"'IJ"*J""l>~llrI' -.acWI4J.aL",.J1J~~..,:J1
I ..
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~~ ~J..-;,' (1' _A) ,( lA _A) ~.aWI w....:..1~
z _
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a 1Rf w
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• ~JL;JI ",.nJ..,w1 rjoAJl (Y' _A)
v 1.0 .u I...,.... J,. UJ.-II ",.aJ I~ ~ I r JAIl J,:.;
- ~vl ~~I: I
Ii ~
v,!; ...,~. j"..t.L..~lvlfl.il. (~IJ,J.
T1 1
dt.e.>
N = I ""'CJt
.....
.
.1li::~1
•
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L ::: I w u, !;J I ,.=i-,;ll
P = my
J»JI~;JI
.11
.... aT
~,
= I~ ,. j
Ii
T= tIw2~pJlulJ,J1
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~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
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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
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I •
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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.,.; • ~
<~I ~,J.JI
\oS j
b
~I~ u~'p~IVe~ c
L..-J,J.,
~~~Iuj~
--.;bJJ j~"'" ab ' .
~I. ~
c.
o
Q
dL=IW
-dt
Q
= I9
l "'"
CK
,
OA J";W IJuJ ~ OOlf
~ 1::11 ~ I ~ v I"" .1.11 ~
mg,£ sin
O~ .J~J,~~ (lo_A) j1.tJ~
~,.. l.S"
Q';') ~
-mgj sin
a
<."JI J,.a.41 I,! J~I J,.a.41l: ~
CM "vl"".uI~~J:.::
S,
•
~ I ~ J?' -.l.i: ~ b::.o>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.<~~.ulvejV5~'~jS'~·(
2"" f
\S,~I
j gJ 2
T • 27f
k
.k 2/
...J,J.~ J,.a.:..c~~Jvej~"'" vJ'~~.iJlve;;
J
J,......... "'""' &,.;.; J, 4 ~)t:e
&l,J. ~ ~J ~. J tJ, J..- ~ ••*--f..Lc.s •• v--~ (k;2 • a 2/,%) ~ L 1 .) 1.". ~ ~,J- obo a
Center of Osoillation k
".s.:JI"wu-: V5~1 ~. 4d~I...,&...J'~,I~~
r
~
= I om
I
+ mj
lJ' k om ~ 1jS'J' J~ r-".s.:J 1J '-Ju-: W ~..l.<
2
~J..-; k
2
= k 2em
+
IJ ok
2
obo
JSve I m's )~I ~ (.'_A)
,. VA
"
). i.iL.-~ 0" ~i&~1 oV l ~J~J,~ulJ'~ ~ul~}1 1,.i......A
J,=a-
T" ~~oiJl ~ji. (\
0-
A) J.U.lI
J ~,..
IJ' • ~ I j". c.?'
':JA "" ~ 1...,.-J I
• 0
..uu ....•.i...L:.I1 .< -u.WI ;J~WL• • J".J'>"l ~ •
0
~L.J...; - ~II 0 • .~ ' "
/
~l;
~)~a -J,J.~u.il. O"4J..iill~~.l:j".~I~ Oule-~' • (or _A) ;J~WI v- loiJ .j = a/2 , k~m = a 2j12 uA ..bo~u). J,=a'" a/6 W L.- ~ ~ J,=a- c::) t; \S .i.l1 ~ u~ dJ.iI, .J. = a/6 o~). ..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'
<;'.
tI
~
~
~ f··
~
C'
r
~~ lli
ri'
'I
~
f :- J:
't::: -
r
0<;..
t':"
L
-
&- r- ,t' ~ "E:
•
f'1 i..r' 1:., B~ ".\,
>;.,
>
L
'
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1~ ~ _ '1
•
t'
~ k-·t ~".,
~
t
i Ii:
v·
10.
B
-
{
r· ~~' 1.~ ~.I \-~:
II
~
,J
!
~
'C..
'b (; [
~ v" t ~,f ~
-tl I ~ '- -~
_
_
••
:-
{,.,£
L
Ilk
C
•
(0
_
't..
t·~
.:
".L ••
•
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t( Ci.
- -.
-
)-o.L
-
tl.
l.
>
~-
t· 't
-
..,.. ._.c.
It
.
>-
l.- __- 'f ...
~.'-
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~ 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 •_ <\. _ ..,.. C\.....c."
.b
1 t~ c;.. ~
-
.t'
.- ..::-
e'
0
PI
II'
r:t:"
-~.,,: ~
•
r.I-t- ~kr~ ~I~tl : t~
t!::! 1: -'Go.C'tfr~~ f t ~ 1 1 ~ f ~ t ~ t.:-
r
f:..
~ fr ~ f 'l ~ 1 :v- t ",-·r ' l t, f~,., t·-- 11..:• 1'::. 'tt.L.l"~ of.[.. "-e:.
~~ ~ ~ i'
~
" r po.f' - r ~ ,r E;; ~ t j' " no ~ 'a.1 ~ l:: 'f" 'L ~ 1- ~ ~ :f· txt ,. L - . t.: ~ 'L f ·t ~ ~' ~ ~
~
}
f
r: .. ~
j '.'
r
fl
~
~'
'-!:
e' e:.
L '[ F
..
'}-
t> F
.-
-
[
'a
-
,,{ f ~l ~ f [f f f ~ b 'Go, .r
-... ~ -:..... .. • !DrOll = mVcm
,;
\:!II
~
= ma om
(l'_A)
m, ~.I ~ ;-,.: ~I ~Jl:;"11 u,.ill~~ J:.::
--a..
JI.:..,..
.~IF.J-~
&.
-...
vl~li..J1 ~1~1~'p"41bJ ('(0 _A)t1jWI~ve ~
L=I J~
om
w
(l'(_A)
k~I ~ w ~
, ~I F.J- ve.ll" 0 ~ J,» u,I;11 ,.->;11 JI.uJ
,~Ivl..,j ~~ ~1~l.~ltljW\;.~ldJ..i J,» yl"".uJ 4.1,1;.11 ~ .i:.w(l·_A)
4.!jL".J1 u l (IT'_A)
• 0
~I J,» tiJ-J1 u,.iU JSJI r,;.-..JI J:.c Ii ~ ~1. u':- Ji-I ~~..t:.e ~ (A_A
:Soq Rolling Down an Inolined Plane
L..,. i}' 4:~I»)lC~~~p'IJ"J~' y.s1&..J1~"..JJ ~ ~;.":u,i~~~ ( \ \ _A) ~I • ~1. d- Ji-I ~~..t:.e (GJ.i uJl J .&11 jJ (n Ji- ~I ul l l,J,iu ;-,.: ~I ~..i~1 ~ ( \) .."., ~l ..... nL...J1 J~lo
• d-ll ~~I
miOIl
Q - ]I
= -mg cos Q +
.. \d~ ~I.,i (t),
:r. d-U ,,~I
F.J- ~~~L.S'.J-~ ~jl- v,s:. ~I 'ti~~
~U:;~I~I
ai om = mg sin
-
:r.
(
IJ'
1 1. _ .\ )
{ll_A)JL..U1
J--J L. ~ Ji-I '---_ ••)l.
Yom
=
~ ~ I v ~ U, • ...;;~ I ~
~P' ~ ,-=:-
e:U I J.,. io Ij J,:.:
Q
~
constant
"I
yem = 0 ~J..-;.(
FN
= mg
008
Q
("_A)
~
• F ~I
..
lIoJWlvev~H
1._A)
i-.:>- ~ I ;;,; ~ ~ I j".
loJ WI· ~ I)..iu...:
J» L.j5 ..LJ..: ~J1 - ~ I -,.aJ I v I
J,:.::
a ~ J.i'a
'" ~
t,;..J I 1.iA JloJoi.,
v..il ~. {lr _A)lIoJWI"I~I..o.AJ1 I em W CO I" .. II
= Fa
( 1 Y - A)
~ voe)l:)1 ~I.:.-L:-,,&J~~
uH
~l:;,..; • ;s"llIWI c:. rJ,
v::JbJ .s'pJll.:.- ~oJL....
~ • ~~
••tion with ~ L-.,,")l.,J I 1; oW o~
•
.0
Slipping
J'jy I ~ ~'j ~ 1. L; h..u. ~ I 0 If I J I ~1:I1
(1A _A)
.·.I~ .... • i am
..
g sin 0
mg sin Q
( yo_ A)
x am = - - - - - = - - - - - 2 m + (Iomfa2 ) I + (k~. /a )
JL.I ~".~ 0 JI ~l; ° 41::n1 jS'J- J~ r-o~I).Ju.-: ~ k cm ~ .~ (1A _A)~~U lii,~l: lS,lj ~ ~l: J-.>~ cUI
. ., . o. .I'
g sin 0 I +
*
=
f
~
)~;;lL 1 __ .,.- I ~
g sin Q
2 lS,l..c ( k om
= 2a2/5
)~I;~I ~
= -2..
g sin 9 1 + • Energy Considerations g sin Q
2
L-. (k 2 =...!2
7
L....iUJI~~;
v = L. (~gai) •
mgs om
• (~~l) ~".\$"'" ulIU:~lIj".v-~..,..,JI~L.JI..,... Iloa ~ • )l.,u ~~ ~l ~;.,:. ~I
• ~.i~U"; .h:::.-~. \$,.iJI.::..,;lf l.il;~l ~ l:S
+ V = ! + mgll cm
!
=B •
h.lS.o aJiJ I-.. as'".J I ",s: .is.a.1a
constant a....J~IWUJIJ:.: !
~
• "_A
J1.tJ1.~UI cUI Ji-I (t.P'~cS.,iJI~UJ~ '" • \$, L....: ~ I.A' .AJ I is'pJJ, imi~. \$, L..: ~ li:; ~ 1;S~ ~,J IwUJ I
~UUJlilol .... "I\$I"I~w2
iai::'
+ iIOII w 2 + agsoa
" JI
_
.~
.1-.:. 2 lJWoA
..1._lr 2
em _ ap. -a-
cm + FlAcm
k.2
mXcmicm(l +
II
x
-
OJI
=E
em = -x em siD aiD Q =: E
/a ~
~
.
~ ~ ~~I~toVU~~ 4U~
-:;> - mgicm siD 9 = 0 a
~ (Xcm " 0
Q. w.:i:
"I~;~~) :i:cm
• tJ}Al ~ \$,all t I~ ~ ~ L. ~
cd~IJ.WIJL.:oD-~
w..- ..,:I I ~ Iv-&: u1- J-...;
Ocourrence
o~
%0.
J
Slipping
,,1\$11;~1 rl;"'; cUle- v-)l:J1 ~,,~ ~I~'-JI. ;~I~;.u JI.u.v~u~yl~ob
l.ili·
,.,ll
'JlrJJ. ~ ~~y~lcdl$',;,.)U )l.~cdla
miom = mg
Xcm
= g( sin
sin Q -,)i JIg cos Q
Q -
( Vl _ A)
(Vi_A)
)J. cos Q)
: I.;~ \oS,ljJl ~I v~~~1 ~ UJ = "umga cos 1 0m
Q
=
•
V- I~ r-~.II v I c,;.j
",J.Jga cos Q
u1-.
t VI;1J ~lc~~L.J1 ~LJ.
u1- J..-.; • if-o, XOJIl
= I( sin
w =
Q -
j • g( JI a
( vo.. A)
2 k Oll
.p cos
XC.= 0,
t
=0
J,.b:.J
1..aJ.a\oSI. v,1-J1
C08 . ) .
(
Y1 ~ A )
Q/lt:' ) t
~l:S~ ~.aJ.a. ~~v~\oS,ljJ~ ~I~~I~~I. 4£1J.iJ
XOIl = 1
~_
aw
sin Q -,Ilcos Q = Itcm - )J a 2 cos Q/lr:~
2
7
Jil v,s:. v I ~~ ~
r cd.!.iJ. x
OlI
(tj}Q _ 1) VI
HI v,s:. vi ~~ a
(.,V
~,
... ~ob.JUlbJl ~ \oS I (~ )J..it ctp'.J.:cdla v~ 1..aJ.a. v I \oS I • Xom = a
't =
1
Lii"
t.J
~~ VI I)e
,. A' ~ -'_ _",~;ul~. )' • '~p
J.L..U
(YY_A)
~JWI J> ~ u~~pJl
jJ.
ori t =
tan
Q
2
( YA - A )
1 + (a/kcm ) • c.;'~;: I u' ~ ~ I c::~ ~ ~ ~ • .. ~I..oS'.u IJ1.uJ Ive.,.ll
u""'!:u l ~e,;~j;lu,~ c::~,J:-0 to ~ 2/7
,t
d-- ~
tan 45°/(1+
P.
c.::.; I!
4~~ IJI.)I.:..;
+)
v-.,.II jJ
wIJ;,.~t::~ k~~~ Motion of a Rigid Body Under an Iapaleive ~I
I.i I
('_A
Force
Impulsive ForoeiAil.J1 ;,.iI1 rnJ. 1o.Li.JL~WI J..-iJ1 ~
J .~li. ~~I..bol,... • ~.JI ,I. i,.ilI .. .iA ~t:: u 11:~ • ~.).r. ;.'; ~l&.JI;SpJUjbJ~.J'rnJ.J~ u>""
4
11 .iA".,. ~14s".. .~k~ ;~I4-SpJl
Free Motion ~
J:. ei J~.J..L. .s.;,.:- J
• .L.....J ~ • P
4S',.J I ~ ~ u I vi' j I
..t..-4I1 J~ ~I"WI~ ~ &r1"".J~ ;.,Jli::~I4S'pJl
Ve
JS' .i>1 ~
c..H:)J .. WI"'jL,JIc~;.,Jli::~I~pJl.~,I.( V_A) ~
Jr
a
.:..a mv
~
.~
.).r. J..-.;,; • ;"".; IJ ~ ~ F dt ... P ... m A V em
S,,-a.
~
P
om =m
~v'
"JloIJ.~I;S'J-u"..
F
i,AJ1 W ~ I! I.i I
J ~ei.Jl~
~u·H
( Y~ _A)
4.Jl:.Il4I..It-!1 ~(-..~ ~--.Il ., J ' I-I: . .r--r-:-v~ 'It"
H.Y
~l:JImL,J1 ~J~ t
~;1J~l~JeI!:JI ~
S
N dt = I om 6w
.11 ~J u)lJ
-•
..
6. w PJ-t:.1"
L
(A-_A)
je} 1t..u..:..MJ, • ~ I..., .u 1e:i.u ~
J N dtJeI!:J 1~,
JI.s.iJ~ ~1jJ1 ~I~.I"" ~ ~ • .;.,~ ~I...,.ul~.ul ~'=
=t=
(A\_A)
em
b.AAc~ •.,.:t:.Ju,.v~~~I~ lill..iJ
iii"
:t ~"~le:i.ul~ l.i~
N=Fb \S,~ ';"~r;-'.ll;
• ~l
(A \' _A)
Jt. U'e:i.u 1~ ~W I ~ 1,;J I~,.,..l' ~ ~ I v$ ~ 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
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Analytical Treatment.
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*
Equations of Motion with Constraints
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See, for example E.T. Whittaker, Anal7ticalf ~amica,
or C.
Cambridge UniTersity press, Oam)riig., 1"7.
LanOSG8,
The Variational principles of ••Uaai•• ,
University of Toronto press, Tormte, 1949.
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Theory of Vibrations
ujl,--:
t-"'J J~
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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.
-<
:- ·f
w!
~
or:
(i
,
~
\..
c·
'::;
t"- ""'
E1
C'
6
f·'
•
. , t r lit
. t
-PI
PI
II
of ~.
~s
l~
~.
~
t·
t:-
•
C>
III
0'
PI
'f."
\..
c·
u
o
C>f<
1...
III
Jj:ao
~
C>
-g
• 0'
II'
-
a ~
I
-• &: .., r tt .
II
o
C>
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 -<
'\
~
~
>-
~.
'f."
f'.
~ 1. .~ f: ~ If. 1: 1" ~._CS~i' k ,r r.
lL
S.
_
5
•
F'C: !:: - --\t ~'. 'Ga. ~ ~~ 1: ~.=i C > , \r" tC.. r. .• -u. 'to , ~. ., ;::
1 ~ ~ 'l. .r CI1;I
-<
"1
...
nr
, =0
g
•
-:-II-<~,...i"
.(J""'-.YJ4
",-
Expansion of the Potential-Energy Function in a Power Series. 4..-...9UJ I U 1.,1 cli; L;.; 1vi'''; ~ • ;.J> ~ 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
u I \$ I
n
=2
~ IS' \.i I ~,
~j
n,~
u~ L oW ~ ~ L.,r.; ~ I ~ ~ I c- ~ bJ I
• ~WI", .,.....iI~ 4..>lj)U L::b.l 14 ~~ V(O)=O
l:...-...:, L:1~~ q=a
J .J1 V"
q
>a ~
.-h- ~
Wi", u~L;..1S)1 LiUJI
L.hi;J1 ~I ~~ILhi:
w.a:
I..il;
L:~~
t
V( q) ...
k2P2
( '\ - \ \ )
• q J ~~I \$,.;.!I ~I \.il ~J..:,>.:~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
<,,-, \)
V
' = tUl2 + tk <x 2
- xl)
2
2 + t kx 2
j'J t5 ...rL
2 = .J-.... WUAI
L ~J.tJ;JI.Jv.i1
. 2 - t lex 2 - ..L..' ~ 2· + 1mX2 .k (x 2 - xl ) 2 - .k:x <. ,.,_, , ) l 2
= -kxl + mX 2 = -kx2 -
mXl
k/(x2 - xl) k/(x 2 - xl)
t 11
,I "
k
o.
k
xl + D1 x2 +
mk" + m-
xl -
""'iii"" X2
k"
LS~ 4.~ ~ JS ~~,
(X 2 - Xl) = 0 (x 2 - Xl) = 0
v-:::J oJ W
I j } v!. ~, k' c:.~ oJj ~ I ~1; ~
"'--_"~·.IS.i.l1 j:.J1;"~u..iI~LWIv-i. ~
OJ
~
COB
wt
('n'_\l)
~oJoJ;:~~t,.;
Jklm
J.WI J)l.>.~ ~jJl ~ eel!>
X
2 ,Xl
••• ~,
;r-~
Xl %2
= Al = A2
C08
wt
cos w t
.J ,.I,
1
~
JS ~
w.:... ..... 1 ~I.J
t .•
lS,±,!;' ... w 2
k' -m
m
.......... m k'
k.t.Js.' ...
== 0
w2
m
eeo~lar eq~ation
~ J....,.; ,~I I
,
... ~ 2)2 ... (....L) (~ In a
2 111
';.oS.w I ~ I~ 111
0)
~ I cd;
.w,
(~A _ll)
0
~)~ ~ ~1.il.lt~IJ~~ • w 2 J~l:.lU'=:-J.1llvello)L-...,.,
(1)
(
~'_l
~oa( ... (,I,)ll)
,)
~ ~ ~~,1> ~~~, ~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-:- ;:
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r0'f-.
o
..
ft'
r; ~.
.0
..
0'
to
of
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II)
to
0'
to
$l'
0
r
,....
fo '~
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-
........
Cl\
'"
I
......
II
I\)
•
~
'l:
fl.
_
.. r. n
c..
I:
~
F
_ V
o
I\)
H
0
I-'
'f:"
(i
.~
,r
....
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f
r
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}-
;f
c.
[.•
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c-..
r.C\
~
{o
0
o£
ro'
C.
f
Co
oC.~
; ~
o£
~ ~~. ~. c' [ 'tr ,[ (i 'h _ 'f:"
(i
;: £: c..
[t·
o£
Co
'f.'-[-[
-
I
-' c· ~ ...." \.f', ",; "=
\,;
:; Ci' Ci'
,r:
.,..• of".y c. c.
[: r.- r.-
~l..l.
~ ~.
..
~
r
·tt' 1"~.
r
t-
<j' G.-
C.
o£
v ' .f [0£ u:: -
~
o£
fr.-c'of." f
c.
~
Cl\
t
H
......
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II
0
H
;: I\)
[. -
,[
e'
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t f.
0
't, ~
c.
'.~c.
~
'G..
E: F'
;c[
..
1':,
......
l, : r"1,r
0
I\)
H
I\)
I
H
H
I
II
II
... . . .
c'
~
-
J. Y
t
. '";:: r
~
e' f ,[
,~
e.
k--
c·
.r
~
F ~..... r. b
'i""-
~.
(
e -
~ c..
r·.
'f"
r.
-
. •
_
.r e~l.
;- 1.
e
t:
l f
[
;:- .r:
(i
·r
E -t.r
.;::::.
. E
H I\)
I
II
.J4
• ..o
lit
rQ
t
r..
,#
0'
..t
II
...
........
f ·r
~.
[
'&:-
I\)
• H
......H
• .o
#
\0
(
II)
..t .r
t
n
t-
'G..
~
y
~.
=
"[ :::.
[ 3
F'
... .r· '&:-
~.
t·.
l":'"
. ".,
fA
{. y
JL...:.
J---: li Ji. ~ ,;, v- vA loS .lJ I ( rlo);J I J, -4~)
•• -:a:s r:=-==- ~~ I u)J I J ~ ~,,1 vl~;I.jI.(o_\\)J!.:JI~I.S.
~,
,..--.. I.S'
.l>
~'I.;s.P' ~;.tJ
I~ ~
2.J,
w~ .bJa.,::lJ
l.A:1~~;-:,.llj".~~
G ~ I,;) ~ ~ I ~ ~ •
.l> ~,.:-
~ ~ 4..-, J:,. .11
~~"ot·=:-Ill;)lJ.;1
-rag.1
vA 'uj~l~ J» .~c.~~.i.:.I • JUJI ~ ~ ub.~Il..al:.iu.,). ( i + if ),..£ 9 ~~I ~
oos G
·IJJl:II~1 ~ C!!I~ #Jlo) ~~..t4.
L
c1t2i2
+
,J
2
-mg.i ( cos 9 + cos ~ )
+ ~)2 + 2mg.J, cos G + mg
(9
J, cos ~
LSp-lJ C!! I~ ~~ 0) L...
m)2 0 + •
.t2 (8 -: j) •
-2ag'/ 8in G
m.t 2 (i + j) • -
ag.i
~.uJI~~~. 8in ~ ~ ~
2e +
~ 9
+
i .
8in; ,
sin 9:;" •
0
i+j++~.O ~W~I )s:v~~ ~,~lo)..boJ1 ~..t4
_",2
= 0
{. A
w 4 _ 4W
2
<-f-> + 2(+>2 = 0 t... u ~ I.:....H ~ I", '"J':1l u .i I
m
( -2 w 2 + 2
+)
, ,,)1....'-' d~..iJI4.1.JWI ~
Q
(
=- W 2~ f.1 _\ \)
~=+f2Q,(N.Wa ~=-{2g,W.Wb
..:.'J.JWI vo w
~~ ~
{
.,
..",.._--l-'-, , v l:.;...J v~ ( u -, , )•.::.:~ oJ W I .j'
c -=-: l:.J1 v I ~
~4.-,J~I..:,L;I..l>'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 <:..-....:~ ~
".u's Jlvl
..",.._I.II\,;~:>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· -
•
•
c lkq l - c 2k Q 2 - ••• - 0nll:'ln
n)..:.. 'J.,wg
:i.o......JI J',~I:i"';~ ( 1 , -, \ )
~l.;..;JI~ 4.lbJl ~I.:~WI ~L;.;;,\I ~~I..:.. 'J.,t.....
( ~I., ~~,) ~WI
J' q'8
~u~ ~I ."..bJ1 ~I,= I~l..
L.tJ...;1i.:J1 ..:.. ~ .)L-\I u,s::~ 4.S)~ ~1.1>-1 ~ .)~I ~bJl"..i..A ~ ~ Jl:JI~I~~WI
(H'_\\)
LJb.
~,.1>-~b:.Il..r..::I~I.:..A. ~jJle-I.:-I;"'..Jl:-....J1~I..:..t.-J'L..:.::.:u.i1
• , ""
>t,.J1
~I
~WI ~,I
CpJl
~l.
;~;;~4.S";~I~L.:.~~~~I.......,\;,.;.lIi.....SpJ I~I
Jw..)~
W:>-::
~~I ~VejJle- ~~~~)L;.;;..-jl.) 4.S,..il~.
,........sj>~.)~ JS ~ "
:LJbJ
(Qkext
e1Wt
,I ) Qkext
QOS
wtt~l~ .),.1>-
.....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~ <:..~IVt
•
~J '-:' ~.l;;
11
~ ..l~-::i
,
.WI4...):..:JI &4JWI. L..:>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 <:..l;I..bo ':Jl.:
Y;J,;e-<.1,. Vt<:..~I~ <:..b.I;':J jt;.!
• ,r-:,) 1J,1 ~ ~..l~ ~ I l~ d ~ yr:J I .....r=:- ~ I; ~ ,,:,J I J,J:> 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.<.~1 • ~WI
;..~ ~.,..u 4--~~1
-_---..../'
T
"-- - -
_/
--
4,J I ~ 1,-:, L-. • •
,;"......=-JI J..:-li :HJ I ~;=oJ"
k -=-: h.ll 0 1J,:L..J I
~14~~~~.:HJ1~~1~ ~I ~~I W'>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
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