vQm` M�K2
S?vbB+b jkR 6BM�H 1t�K @ hm2b/�v- .2+X RR- kyR3
� T2`72+i b+Q`2 QM i?Bb 2t�K Bb R9y TQBMibX >Qr2p2`- i?2 2t�K Bb ;`�/2/ QM � b+�H2 Q7 Rky bQ i?2H�bi ky TQBMib +�M #2 +QMbB/2`2/ �b 2ti`� +`2/BiX6uA, 6Q` i?2 /Bz2`2MiB�H 2[m�iBQM
x+ 2βx+ ω20x = 0,
i?2 bQHmiBQMb �`2
x = A1e−βt cosω′t+A2e
−βt sinω′t ω′ =!ω20 − β2 (under damped)
x = Ae−βt +Bte−βt, (critically damped)
x = A1e−β1t +A2e
−β2t, βi = β ±!β2 − ω2
0, (over damped).
G�;`�M;2Ƕb 2[m�iBQMb,
d
dt
"∂L∂q
#=
∂L∂q
.
_Qi�iBM; 7`�K2,mdv
dt= Freal −mω × (ω × r)− 2mω × v.
vQm` M�K26BM�H 1t�K @ q2/M2b/�v- J�v k- kyR3
RX *QMbB/2` � T�`iB+H2 Q7 K�bb m BM � QM2@/BK2MbBQM�H ?�`KQMB+ Qb+BHH�iQ` Q7 bT`BM; +QMbi�Mi k�M/ rBi? �M 2ti`2K2Hv bK�HH /�KTBM; 7Q`+2 −bv- BX2X b → 0X �//BiBQM�HHv- i?2`2 Bb �M 2ti2`M�H7Q`+2
F (t) =
$−F0, −τ/2 < t < 0,+F0, 0 < t < τ/2
F (t+ τ) = F (t).
*QMbB/2` i?2 2tT�MbBQM
F (t) = a0 +%
n>0
an cos(nωt) + bn sin(nωt),
U�V U8 TibV q?�i Bb ω\U#V U8 TibV 6Q` r?�i n �`2 i?2 +Q2{+B2Mib an = 0 �M/ 7Q` r?�i n �`2 bn = 0\U+V URy TibV LPh aPGoAL: 6P_ an �M/ bn- 2tT`2bb x(t) BM i2`Kb Q7 an �M/fQ` bn �bbmKBM;
i?2 7Q`+2 ?�b 2tBbi2/ 7Q` � HQM; iBK2 UdzHQM; iBK2Ǵ K2�Mb i?�i i?2 /�KTBM; ?�b 2HBKBM�i2/�HH i`�MbB2Mib 2p2M i?Qm;? i?2 /�KTBM; Bb 2ti`2K2Hv bK�HHVX uQm` �Mbr2` b?QmH/ #2 Q7i?2 7Q`K
x(t) =%
n
· · · ,
Uh?2 /�KTBM; b b?QmH/ MQi �TT2�` BM vQm` �Mbr2`V
a ) w =2.IT/z
b) an = O for all n
b n= O for n = 2,4 ,5
,
6
c) Xlt ) =n€ , ,Basin @art )
I two 'X= Imf bnsincnwt )
⇐NZ w
'
two'
)pn= Lmbn
An = hemf-
f-n' w' two )
Xlt ) = I ←nwysin(nwt )
vQm` M�K2S?vbB+b jkR 6BM�H 1t�K @ hm2b/�v- .2+X RR- kyR3
kX � #�HH Q7 K�bb m Bb /`QTT2/ UBMBiB�H p2HQ+Biv Bb x2`QV 7`QK � ?2B;?i h0X h?2 #�HH 2tT2`B2M+2b� /`�; 7Q`+2 −bvX
U�V U8 TibV q?�i Bb i?2 i2`KBM�H p2HQ+Biv\U#V URy TibV q?�i Bb i?2 p2HQ+Biv �b � 7mM+iBQM Q7 iBK2\ Uv b?QmH/ #2 M2;�iBp2 7Q` /QrMr�`/
KQiBQMVU+V U8 TibV q?�i Bb i?2 ?2B;?i �b � 7mM+iBQM Q7 iBK2\U/V U8 TibV � b2+QM/ #�HH Bb `2H2�b2/ 7`QK i?2 b�K2 TQBMi- #mi BMBiB�HHv KQpBM; /QrMr�`/
rBi? i?2 i2`KBM�H p2HQ+BivX �7i2` � HQM; iBK2 ?�b T�bb2/- �M/ #Qi? #�HHb �`2 KQpBM; rBi?i?2 i?2`K�H p2HQ+Biv- r?�i ?2B;?i b2T�`�i2b i?2 irQ #�HHb\
terminal
a) def =-
g- bar
7- erm
=
-
m 8/1,
b ) dir -
1mA*MI)
= dt
*
In =
- hatV term
- blunt
✓ t Veeru= V
term
E
- by t
y=
- Vtermt Veteran
e
c ) X =- rennet
- Farm. !E ( e- Kt
-I )
=
- rennet t mix ( I - e-It )
d) Xz =- term t
a
D X = niff = Veering
vQm` M�K2S?vbB+b jkR 6BM�H 1t�K @ hm2b/�v- .2+X RR- kyR3
jX � T�`iB+H2 Q7 K�bb m Bb �BK2/ �i � bT?2`B+�H i�`;2i Q7 `�/Bmb RX h?2 T�`iB+H2 722Hb � `2TmHbBp2TQi2MiB�H 7`QK i?2 i�`;2i
V (r) =α
r,
r?2`2 r Bb i?2 /Bbi�M+2 iQ i?2 T�`iB+H2 7`QK i?2 +2Mi2` Q7 i?2 i�`;2i bT?2`2X
U�V U8 TibV q?�i Bb i?2 KBMBKmK 2M2`;v i?2 T�`iB+H2 Kmbi ?�p2 BM Q`/2` iQ BKT�+i i?2bm`7�+2 Q7 i?2 bT?2`2\
U#V UR8 TibV �bbmKBM; i?2 2M2`;v 2t+22/b i?Bb i?`2b?QH/- r?�i Bb i?2 +`Qbb b2+iBQM 7Q` +QHHBbBQMrBi? i?2 bT?2`2\:Bp2 vQm` �Mbr2` BM i2`Kb Q7 R,α,m �M/ i?2 2M2`;v EX
a) Emin = Mr
b ) dbL = mr ,
b = on VfR
n' =
- Fe
if = to FEI ,
b = R TEE6 =
ITR' ( I
- In )
vQm` M�K2S?vbB+b jkR 6BM�H 1t�K @ hm2b/�v- .2+X RR- kyR3
9X *QMbB/2` irQ 2[m�H K�bb2b m +QM}M2/ iQ KQpBM; BM i?2 z = 0 TH�M2X h?2 K�bb2b �`2 +QMM2+i2/#v � K�bbH2bb bT`BM; Q7 bT`BM;@+QMbi�Mi k r?Qb2 `2H�t2/ H2M;i? Bb ℓX AM i2`Kb Q7 i?2 +QQ`/BM�i2bQ7 i?2 irQ T�`iB+H2b- i?2 TQi2MiB�H 2M2`;v Q7 i?2 bT`BM; Bb
U =k
2
&!(x1 − x2)2 + (y1 − y2)2 − ℓ
'2.
6Q` ;2M2`�HBx2/ +QQ`/BM�i2b- mb2 i?2 +2Mi2`@Q7@K�bb +QQ`/BM�i2b- X = (x1 + x2)/2- Y =(y1 + y2)/2- i?2 /Bbi�M+2 #2ir22M i?2 K�bb2b r =
((x1 − x2)2 + (y1 − y2)2 �M/ i?2 �M;H2
Q7 i?2 bT`BM; `2H�iBp2 iQ i?2 x �tBb- φX
U�V URy TibV q`Bi2 i?2 G�;`�M;B�M BM i2`Kb Q7 X,Y, r,φ, X, Y , r �M/ φXU#V URy TibV q`Bi2 i?2 7Qm` 2[m�iBQMb Q7 KQiBQMXU+V URy TibV q`Bi2 �b K�Mv +QMb2`p2/ [m�MiBiB2b �b vQm +�M i?BMF Q7 BM i2`Kb Q7 m, k, ℓ, X, Y, r,φ, X, Y , r
�M/ φX
a) T= Tom t Tree = zzm.ci- tin
-1+-7i' * Emrick
= mC i
' tri ) t mr t m g'
P 2
L = mix'
'- i ) tryin'
t m e
b )I = o
- Ecr- e )'
if = O
dad' g) = o
my= her
if'
-
k Cr - e )
c) E,
r' ie
,I,Vi
vQm` M�K2S?vbB+b jkR 6BM�H 1t�K @ hm2b/�v- .2+X RR- kyR3
x1x2
k kmm
k8X *QMbB/2` i?2 irQ K�bb2b m +QMM2+i2/ iQ i?2 i?`22 B/2MiB+�H K�bbH2bb bT`BM;b- 2�+? rBi? bT`BM;
+QMbi�Mi k- i?�i �`2 }t2/ �i i?2 7�` 2M/b U/�b?2/ HBM2bVX �i 2[mBHB#`BmK- 2�+? bT`BM; Bb �i Bib`2H�t2/ H2M;i? ℓX
U�V URy TibV q`Bi2 i?2 G�;`�M;B�M mbBM; i?2 +QQ`/BM�i2b x1 �M/ x2XU#V U8 TibV q`Bi2 i?2 2[m�iBQMb Q7 KQiBQM 7Q` x1 �M/ x2XU+V URy TibV .2b+`B#2 i?2 irQ MQ`K�H KQ/2b �M/ i?2B` 7`2[m2M+B2bX
a) T = Emi .
't Im
u = Ek ( x,
- e )'
t Inc x.
- x,
- e)'
+ Ik ( xa
- 2e )'
= thx ?kcx .
- x.) 't thx
:- kex ,tRK,
- xd l - 2k X. l
Ikx
,
't
k Xi -k x. x
.
- 3k Xue t 3 he-
I = Em t In- taxi - kxitkxix .
- 3k€ - e) e
b ) m Ii =-2kx
,
t k X,
m Ii =-
2KX
,
t k X,
-3Kl
a ) x,
= l t.
A e'- WE
,x a
=2e it Bei
-t
.
- m w? A =- 2k A t k B
- mu'
B =- 2 KB t KA
A
B-= .
2 -
mR
vQm` M�K2S?vbB+b jkR 6BM�H 1t�K @ hm2b/�v- .2+X RR- kyR3
1ti`� rQ`F bT�+2 7Q` O8
.
- m w? A =- 2k A t kI
- mu'
B =- 2 KB t KA
A
B-= .
2 - in
- m w' ( 2
- MRI ) =- 2k ( 2 - IF ) t k
m wit - 4 inw
'
t 3 k = O
k
w?
-
4 wo
'
w' t 3 wit =O
,
wtf Klm
4miw
"=
-
2 -
=2 wi I Wo
'
=
3 wi ,no
ft =2
- ww÷.=
- I,
d
#,X,=e-Bws#wot)X
,
= 2et Bus ( two t )
Made # 2,
X,
=l t B cos Carrt
Xu-
-
zet Boos @o
IIe )
vQm` M�K2S?vbB+b jkR 6BM�H 1t�K @ hm2b/�v- .2+X RR- kyR3
eX � K�bb m Bb /`QTT2/ UBMBiB�H p2HQ+Biv Bb x2`QV 7`QK � H�`;2 ?2B;?i U#mi MQi bQ H�`;2 vQm +�MǶi+QMbB/2` g +QMbi�MiV �#Qp2 � TQBMi QM i?2 2[m�iQ`X L2;H2+i �B` `2bBbi�M+2X
U�V UR8 TibV _2H�iBp2 iQ /`QTTBM; 2t�+iHv #2HQr r?2`2 Bi r�b /`QTT2/- }M/ i?2 /Bbi�M+2 iQr?2`2 i?2 K�bb ?Bib i?2 ;`QmM/X
U#V U8 TibV AM r?B+? /B`2+iBQM UMQ`i?- bQmi?- 2�bi Q` r2biV Bb i?Bb TQbBiBQM `2H�iBp2 iQ /B`2+iHv#2HQr i?2 /`QTTBM; TQBMiX
Let Earth Ians Iet Hadvelvety =w
I = EAST
m drag=
- 2 m ur #yn x C - I )
= 2 mw ( g
t )
d rx z w gt
z=
2
I=
u gt
X = Igu g
t3
2
h = Iz gt
' le
← = ( 2h19 )3h
( 2h )
x = I w GTb ) EAST
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