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

1ti`� rQ`F bT�+2 7Q` OR

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

1ti`� rQ`F bT�+2 7Q` Ok

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

1ti`� rQ`F bT�+2 7Q` Oj

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

1ti`� rQ`F bT�+2 7Q` O9

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

vQm` M�K2S?vbB+b jkR 6BM�H 1t�K @ hm2b/�v- .2+X RR- kyR3

1ti`� rQ`F bT�+2 7Q` Oe