Mechanical Engineering

PHYSICS FOR SCIENTISTS AND ENGINEERS
saklviTüal½yPUminÞPMñeBj
mhaviTüal½yviTüasaRsþ
segçbemeron-lMhat; nigdMeNaHRsay
sMrab;fñak;qñaMmUldæan
eroberogeday³ RtYtBinitüeday³
elak hg; sIum saRsþacarürUbviTüa elak hg; can;fun saklviTüaFikarrg

bBa¢IGtßbT
emeron TMB½r
rUbmnþKNitviTüa 1
1-edrIev 1
2-GaMgetRkal 2
3-esrI 4
4-es‘rI Fourier 5
5-es‘rI Taylor 6
6-es‘rI Laureant 6
7-smIkar 6
8-smIkarDIepr:g;Esül 7
9-cMnYnkuMpøic 9
10-viPaKviucT½r 9
11-edrIevvuicT½r 11
12-TMrg;m:aRTIs 12
12-RbmaNviFIkñúgkUGredaenedkat 13
segçbrUbmnþsMxan;²énemkanic 15
RbB½n§xñat 15
EpñkTI1³ sIuenma:Tic 18
1-clnaRtg 18
2-clnaRtg;esμI 18
3-clnaRtg;ERbRbYlesμI 19
4-clnaenAkñúglMh 19
5-clnaekag 22
6-clnavg; 24

7-clnaRtg;suInuysUGIut 25
8-clnaTnøak;esrI 26
9-clnaRKab;)aj; 27
10-clnaeFobeTAnwgtMruyeFIVclnarMkil 28
11-clnaGaRs½y 29
12-kUGredaenb:UElbøg; 29
EpñkTI2³ DINamic 31
1-c,ab ;TaMgbIrbs;jÚtun 31
2-clnarMkilrbs;GgÁFatu 31
3-clnavg;esμI nigclnalMeyal 33
4-famBl nigc,ab;rkSafamBl 34
5-famBlemkanic 38
6-clnaenAkñúgEdn 39
7-m:Um:g;sIuenTicnigm:Um:g;énkMlaMg 39
8-lMnwgénPaKli¥t 40
9-lMnwgénGgÁFaturwg 41
10-clnarNbEpndI 41
11-c,ab;ekEBø 41
12-TgiÁcéncMnucrUbFatuBIr 42
13-sIuenma:Ticbøg;énGgÁFaturwg 44
lMhat; nigdMeNaHRsay EpñksIuenma:Tic 47
lMhat;RtiHriH 162
lMhat; nigdMeNaHRsay 175
lMhat;RtiHriH 333
Éksareyag 372

Faculty of Science Royal University of Phnom Penh
Mr Hang Sim Physic Lecturer , Master of Engineering
1
1-edrIev
1- cxy = ⇒ c
dx
dy
y ==′
2- n
xy = ⇒ 1−
==′ n
nx
dx
dy
y
3- n
uy = , )(xuu = ⇒ 11 −−
′===′ nn
uun
dx
du
nu
dx
dy
y
4- x
ey = ⇒ x
e
dx
dy
y ==′
5- u
ey = , )(xuu = ⇒ uu
eu
dx
du
ey ′==′
6-
nu
ey = , )(xuu = ⇒
dx
du
enu
dx
dy
y
nun 1−
==′
=
nu
euun n 1−
′
7- vuy ⋅= , )(xuu = , )(xvv = ⇒
dx
dv
u
dx
du
v
dx
dy
y +==′
= uvvu ′+′
8-
v
u
y = , )(xuu = , )(xvv = ⇒ 2
v
uvvu
dx
dy
y
′−′
==′
9- xy sin= ⇒ xy cos=′
10- xy cos= ⇒ xy sin−=′
11- uy sin= , )(xuu = ⇒ uuy cos′=′
12- uy cos= , )(xuu = ⇒ uuy sin′−=′
13- uy tg= , )(xuu = ⇒ )tg1(sec
cos
22
2
uuuu
u
u
y +′=′=
′
=′
14- uy cotg= , )(xuu = ⇒ uu
u
u
y 2
2
cosec
sin
′−=
′
−=′
15-
u
uy
cos
1
sec == , )(xuu = ⇒ uuuy tgsec ⋅⋅′=′
16-
u
uy
sin
1
cosec == , )(xuu = ⇒ uuuy cotgcosec ⋅′−=′
17- uuy 1
sinarcsin −
== , )(xuu = ⇒
2
1 u
u
y
−
′
=′
rUbmnþKNitviTüasMxan;²sMrab;GnuvtþkñúgrUbviTüa

2
18- uuy 1
cosarccos −
== , )(xuu = ⇒
2
1 u
u
y
−
′
−=′
19- xy alog= ⇒
ax
y
ln
1
=′
20- uy alog= , )(xuu = ⇒
au
u
y
ln
′
=′
21- uy ln= , )(xuu = ⇒
u
u
y
′
=′
22- u
ay = , )(xuu = ⇒ aauy u
ln′=′
23- vuy ⋅= , )]([ xpuu = , )]([ xpvv = ⇒
dx
dp
p
v
u
dx
dp
p
u
vy ⋅
∂
∂
⋅+⋅
∂
∂
⋅=′
24- cvu =∧ ⇒
dt
vd
uv
dt
ud
dt
cd
∧+∧=
2-GaMgetRkal
1- CxFdxxf +=∫ )()( 2- )()()( aFbFdxxf
b
a
−=∫
3- C
n
u
duu
n
n
+
+
=
+
∫ 1
1
; 1−≠n 4- Cu
u
du
+=∫ ln
5- Cedue uu
+=∫ 6- Cuudu +=∫ sincos
7- Cuudu +−=∫ cossin 8- Cuudu +−=∫ coslntg
9- Cuudu +=∫ sinlncotg 10- Cuuudu ++=∫ tgseclnsec
11- ∫ xdxxn
cossin =
⎪
⎩
⎪
⎨
⎧
−=+
−≠+
+
+
1,sinln
1,
1
sin 1
nCx
nC
n
xn
12- ∫ ⋅ xdxxn 2
sectg =
⎪
⎩
⎪
⎨
⎧
−=+
−≠+
+
+
1,tgln
1,
1
tg 1
nCx
nC
n
xn
13- ∫ xdxxn 2
coseccotg =
⎪
⎩
⎪
⎨
⎧
−=+−
−≠+
+
−
+
1,cotgln
1,
1
cotg 1
nCx
nC
n
xn
14- ∫ xdxxn
sincos =
⎪
⎩
⎪
⎨
⎧
−=+−
−≠+
+
−
+
1,cosln
1,
1
cos 1
nCx
nC
n
xn
15- ∫∫ −= vduuvudv 16- Cudu
u
+=
−
∫ arcsin
1
1
2

3
17- Cudu
u
+=
+∫ arctg
1
1
2
18- ∫∫∫ ±=±
b
a
b
a
b
a
dxxgdxxfdxxgxf )()()]()([
19- ∫∫ =
b
a
b
a
dxxfAdxxAf )()( 20- ∫∫ −=
a
b
b
a
dxxfdxxf )()(
21- ebI ba < nig )()( xgxf ≥ ⇒ ∫∫ ≥
b
a
b
a
dxxgdxxf )()(
22- ∫∫ ≤
b
a
b
a
dxxfdxxf )()( 23- ebI ba < nig 0)( ≥xf ⇒ 0)( ≥∫
b
a
dxxf
24- ⎟
⎠
⎞⎜
⎝
⎛⋅⎟
⎠
⎞⎜
⎝
⎛≥
⎥⎦
⎤
⎢⎣
⎡
∫∫∫
b
a
b
a
b
a
dxxgdxxfdxxgxf 22
2
)]([)]([)()(
25- ebI )(tf nig )(tg Cab;enAelIcenøaH ],[ ba ehIy )()(0 tgtf ≤≤ eK)an³
-ebI ∫
b
a
dttg )( rYm⇒ ∫
b
a
dttf )( rYm
-ebI ∫
b
a
dxxf )( rIk ⇒ ∫
b
a
dxxg )( rIk
-ebI α
)(
)(
tb
A
xf
−
= eBl 0−→ bt , A efr
⇒
⎪⎩
⎪
⎨
⎧
≥
<
∫ 1
1
:)(
α
α
ebIrIk
ebIrYmb
a
dttf
26- RbEvgFñÚénExSekag³ eKeGaysmIkar)a:ra:Em:tkñúgtMruy ),,,0( kjiℜ kMnt;eday³
)(tfx = , )(xgy = , )(thz = . RbEvgFñÚénExSekagKW³
∫ ′+′+′=
t
a
dttztytxtS 222
)]([)]([)]([)(
Edl 2222
dzdydxds ++= .
ebIvaenAkñúgbøg; eK)an³
∫ +′=
1
0
2
)]([)(
θ
θ
θθρθρ dS
27- GaMgetRkalDub ¬BIrCan;¦ sMrab;KNnaRkLaépÞ³
∫∫=
)(
),(
D
dxdyyxfS
28- GaMgetRkalRTIb ¬bICan;¦ sMrab;KNnamaD³
∫∫∫=
)(
),,(
V
dxdydzzyxfV
29- m:Um:g;niclPaBénmaDmYyeFobeTAnwgG½kSmYy³
∫∫∫ +⋅=
)(
22
)(
V
dxdydzyxI ρ , ),,( zyxρ

4
30- TIRbCMuTMgn;énmaDmYy³
∫∫∫ ⋅=
)(
1
V
G dxdydzx
M
x ρ ; M ma:ssrub
∫∫∫ ⋅=
)(
1
V
G dxdydzy
M
y ρ
∫∫∫ ⋅=
)(V
G dxdydzzz ρ
31- karbþÚrGefr³
kUGredaenedkat eTAkUGredaensIuLaMg
∫∫∫)(
),,(
V
dxdydzzyxf = ∫∫∫)(Δ
);sin;cos( dzrdrdzrr θθθρ
32- rUbmnþ Rieman
∫+
+
C
dyyxQdxyxP ),(),( = ∫∫ ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ∂
−
∂
)(D
dxdy
dy
P
dx
Q
33- ∫
∞+ −
⋅=
+0
2
1
2
2
h
hx
dx π
34- ∫
∞+
+
+0 12
)( n
hx
dx
=
hhn
n
n
1
2642
)12(531
2
×
××××
−××××
×
π
35- ∫
∞+
−
0
2
dxe x
=
2
π
¬GaMgetRkal Frenel¦
36- ∫
∞
⋅−
0
22
dxe xω
=
ω
π
2
37- ∫
∞
−
0
2
dxe ix
= ∫
∞
−
0
22
)sin(cos dxxix = )1(
22
i−
π
¬eRBaH ∫
∞
=
0
2
22
1
cos
π
dxx ; ∫
∞
=
0
2
22
1
sin
π
dxx ¦
3-es‘rI
es‘rIBiess²mYycMnYn³
1- ++++++=
!!3!2
1
32
n
xxx
xe
n
x
; ∞<x
2- +
−
⋅−+−+−=
−
−
!)12(
)1(
!5!3
sin
12
1
53
n
xxx
xx
n
n
; ∞<x
3- +
−
⋅−+−+−=
−
−
!)22(
)1(
!4!2
1cos
22
1
42
n
xxx
x
n
n
; ∞<x

5
4- +⋅−+−+−=+ −
n
xxx
xx
n
n 1
32
)1(
32
)1ln( ; 1<x
5- +
−
⋅−−+−=
−
−
)12(
)1(
53
Arctg
12
1
53
n
xxx
xx
n
n
; 1<x
6- +⋅
+−−
++⋅
−
++=+
−
n
n
p
x
n
xnppp
x
pp
pxx
!
)1()1(
!2
)1(
1)1(
12
2
; 1<x
4-es‘rI Fourier
A.f(x) CaGnuKmn_Bit b¤kMupøic EdlmanGefr x ehIymanxYb π2 kMnt;eday³
∫−
π
π
dxxf )(
eyIgbMEbk f(x) Caes‘rI Fourier Kw³
)(xf = ∑
∞
=
++
1
0 )sincos(
n
nn nxbnxaa
Edl ∫−
=
π
ππ
dxxfa )(
2
1
0 ;
∫−
⋅=
π
ππ
nxdxxfan cos)(
1
∫−
⋅=
π
ππ
nxdxxfbn sin)(
1
B. krNIGnuKmn_eBlmanxYb ω
π2
=T
eyIgtag tx ω=
⇒ )(xf = ∑
∞
=
++
1
0 )sincos(
n
nn tnbtnaa ωω
Edl ∫−
= 2
2
0 )(
1 T
T dttf
T
a ;
∫−
⋅= 2
2
cos)(
2
T
Tn tdtntf
T
a ω
∫−
⋅= 2
2
sin)(
2
T
Tn tdtntf
T
b ω
C. ebIvaCaGnuKmn_kMupøic
rUbmnþ A xagelIGacsresrCa³
∑
∞+=
∞−=
=
n
n
inx
neCxf )(

6
Edl ∫−
−
⋅=
π
ππ
dxexfC inx
n )(
2
1
D. RTwsþIbT Parseval
∫
∞+
∞−
dxxf
2
)(
2
1
π
= ∑
∞
=
++
1
222
0 )(
2
1
n
nn baa
cMeBaHGnuKmn_xYb B xagelIeK)an³
∫−
π
ππ
dxxf
2
)(
2
1
= ∑
∞+=
∞−=
n
n
nC
2
5- es‘rI Taylor
ebI f(x) CaGnuKmn_EdlmanedrIevRtg;RKb;cMnucenAkñúgExSekagbiT (C ) eK)an³
+++′′+′+=+ )(
!
)(
!2
)()()( )(
2
af
n
h
af
h
afhafhaf n
n
ebIeKtag hax += ⇒ axh −= enaHeK)an³
+−++−
′′
+−′+= n
n
ax
n
af
ax
af
axafafxf )(
!
)(
)(
!2
)(
))(()()(
)(
2
6-es‘rI Laureant
2 1 2
0 1 2 2
( ) ... ...
a a
f a b a a h a h
h h
− −
+ = + + + + + +
Edl ∫ +
−π
= dx
)ax(
)x(f
i2
1
a 1nn dx)x(f)ax(
i2
1
a; 1n
n ∫
−
− −
π
= ; 1,2,3,4,...n =
ebIeyIgbþÚrGefr 2 1 2
0 1 2 2
( ) ( ) ( ) ... ...
( )
a a
f x a a x a a x a
x a x a
− −
= + − + − + + + +
− −
Edl 1
1 ( )
; 1, 2, 3,...
2 ( )
n n
f
a d n
i a
ξ
ξ
π ξ +
= = ± ± ±
−∫
7-smIkar
k- smIkarbnÞat;³ baxy +=
ebI b = 0, axy = kat;tamKl;0.
x- smIkarbnÞat;kat;tamBIrcMnuc );( 11 yxA nig );( 22 yxB ³
12
1
12
1
yy
yy
xx
xx
−
−
=
−
−
K- smIkarrgVg;³

7
222
Ryx =+ manp©itRtg; 0; kaM R .
222
)()( Rbyax =−+− manp©itRtg;A(a ; b); kaMR .
X- smIkareGlIb³
12
2
2
2
=+
b
y
a
x
, a; b CaG½kSTaMgBIréneGlIb .
g- smIkarGuIEBbUl³
012
2
2
2
=±−
b
y
a
x
c- smIkar)a:ra:bUl³ cbxaxy ++= 2
mancMnuckMBUl ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
−
a
bac
a
b
S
4
4
;
2
2
.
8-smIkarDIepr:g;Esül
A. smIkarDIepr:g;EsüllMdab;mYy³ 0);;( =′yycf
-smIkarmanGefrGacbMEbk)an³
dyygdxxf )()( = ; y
dx
dy
′=
⇒ ∫ ∫ += cdyygdxxf )()(
-smIkar )()( xQxyP
dx
dy
=+
⇒ ∫ +∫=∫ cdxexQye
dxxPdxxP )()(
)(
-smIkar Bernouilli
)()( xQyxPy
dx
dy n
⋅=⋅+
eKGacsresr )()(1
xQxPy
dx
dy
y nn
=+ +−−
tag 1+−
= n
yV ⇒
dx
dy
y
dx
dV
n
n
⋅=⋅
−
−
1
1
⇒ )()1()]()1[( xQnxPnV
dx
dV
−=−+
B. smIkarDIepr:g;EsüllMdab;BIr³ 0);;;( =′′′ yyycf
-smIkarmanrag³
0" =+′+′′ cyybya ,
dy
y
dx
′ = ;
2
2
,
d y
y
dx
′′ =

8
eyIgtag xr
ey = ⇒ xr
rey =′ ⇒ xr
ery 2
=′′
⇒ 02
=⋅+⋅+⋅⋅ rxrxrx
ecebrera
⇒ 02
=++ cbrar ehAfa smIkarlkçN³.
⇒ acb 4Δ 2
−=
-ebI 0Δ > ⇒ bJsénsmIkarKW xrxr
eCeCy 21
21 +=
-ebI 0Δ = ⇒ bJsénsmIkarKW xr
eCxCy )( 21 +=
a
b
r
2
−=
-ebI 0Δ < KμanbJsBit ⇒ bJsénsmIkarKW x
exCxCy α
ββ )sincos( 21 += Edl
a
b
2
−=α ; a
bac
2
4 2
−
=β .
-cMeBaHsmIkarmanrag³ )()()()( xDyxCyxByxA =+′+′′
edaHRsayRsedogxagelIEdr dMbUgeyIgeFVIeGayGgÁTIBIrsUnü ¬sUmemIl]TahrN_¦³
eyIgyk 023 =+′+′′ yyy
manlkçN³smIkar 0232
=++ rr
⇒ 11 −=r ; 22 −=r
eyIg)ancMelIyTUeTAedayKμanGgÁTIBIr
xx
BeAey −−
+= 2
b¤ xx
exBexAy −−
+= )()( 2
⇒ xxxx
exBexBexAexAy −−−−
−′+−′=′ )()()(2)( 22
eyIgeRCIserIslkçx½NÐbEnßm³
02
=′+′ −− xx
eBeA ⇒ 0=′+′ −
BeA x
⇒ xx
BeAey −−
−−=′ 2
2
⇒ xxxx
eBBeAeeAy −−−−
′−++′−=′′ 22
42
⇒ xx
eBeAyyy −−
′−′−=+′+′′ 2
223 = x
e
x
x −−
2
1
⇒ x
e
x
x
A 2
1−
=′ nig 2
1
x
x
B
−
=′
⇒ ∫∫
−
=′= dxe
x
x
dxAA x
2
1
= 1C
x
ex
+−
dx
x
x
B ∫
−
= 2
1
= 2
1
ln C
x
x ++

9
dUecñH xxx
eCeCxey −−−
++= 2
2
1ln .
9-cMnYnkuMpøic
iyxz += ; i ehAfacMnYnnimitEdl 12
−=i .
enHCaTMrg;BICKNit .
-TMrg;gFrNImaRt³ kñúgkUGredaenb:UEl
θcosrx = ; θsinry =
⇒ )sin(cos θθ iriyxz +=+=
EdlkñúgenH 22
yxr += .
-cMnYnkMupøicqøas;³ iyxz −=
-m:UDul³ 22
yxzzz +=⋅=
-rUbmnþ De Moivre :
nn
iyxz )( += = )sin(cos θθ ninrn
+
-rUbmnþ Eulaire:
θθθ
sincos ie i
±=±
10-viPaKviucT½r
k-viucT½rBIr a nig b CaviucT½rkUlIenEG‘ kalNa³
ba λ=
-ebI 0λ > enaH a nig b manTisedAdUcKña.
-ebI 0λ < enaH a nig b manTisedApÞúyKña.
x- plKuNsáaElénBIrviucT½r
);cos( babaabba ⋅=⋅=⋅
ebI
1
1
1
z
y
x
a ;
2
2
2
z
y
x
b eK)an³
212121 zzyyxxabba ++=⋅=⋅
K- plKuNviucT½r

10
a b a b c∧ = × =
baab ∧−=∧
m:uDul
αsinabba =∧
ebI a nig b enAkñúgtMruy );;;0( kjiℜ ³
0=∧ ii ; 0=∧ jj ; 0=∧ kk ;
kji =∧ ; ikj =∧ ; jik =∧ ;
kij −=∧ ; jki −=∧ ; ijk −=∧
ebI
1
1
1
z
y
x
a ;
2
2
2
z
y
x
b
⇒ ba ∧ =
222
111
zyx
zyx
kji
= kyxyxjxzzxiyzzy )()()( 122121212121 −+−−−
rebobKNnaedETmINg;¬ma:RTIskaer¦³
=Δ
nnnn
n
n
aaa
aaa
aaa
21
22221
11211
b¤eyIgGacsresr ∑=
+
−=
n
j
ijij
ji
Aa
1
)1(Δ ¬eFobeTAnwgCYredkTI i¦
∑=
+
−=
n
i
ijij
ji
Aa
1
)1(Δ ¬eFobeTAnwgCYredkTI j¦
ijA CaFatuén ija
]TahrN_³
11
32
22322
11
11
)1(Δ
A
nnnn
n
aaa
aaa
a⋅−= +
+
12
31
22321
12
21
)1(
A
nnnn
n
aaa
aaa
a⋅− +
CYredk
CUrQr
k
O j
i
c
b
α
a

11
+
13
21
22221
13
31
)1(
A
nnnn
n
aaa
aaa
a⋅− +
+
X-lkçN³vuicT½r
• wVV =∧ '
• wVwVwVV ∧+∧=∧′+ 21)(
• 2121 )( wVwVWWV ∧+∧=+∧
• VVVV ∧′−=′∧
• )()( wVwV ∧=∧ αα
• )()( WVWV ∧=∧ αα
• 0=∧VV
• ),sin('' VVVVVV ′⋅=∧
11-edrIevvuicT½r
♦
dq
vd
dq
vd
vv
dq
d 21
21 )( +=+ ; ¬ 1v niig 2v CaGnuKmn_én q)
♦
dq
vd
v
dq
vd
vvv
dq
d 1
2
2
121 )( +=⋅
♦
dq
vd
v
dq
vd
2
2
=
♦
dq
vd
vv
dq
vd
vv
dq
d 2
12
1
21 )( ∧+∧=∧
♦ )()())(( 32132
1
321 vv
dq
d
vvv
dq
vd
vvv
dq
d
∧+∧⋅=∧⋅
♦
dq
vd
kvk
dq
d
=)(
· ebI q CaGnuKmn_ p teTAeTot³
♦
dp
dq
dq
vd
dp
vd
×= ⇒ dp
dp
dq
dq
vd
dq
dq
vd
vd ⋅⋅=⋅=
· ebI ( ) jiu sincos +=α
⇒
( )
ji
d
ud
⋅+⋅−= αα
α
α
cossin
ji ⎟
⎠
⎞
⎜
⎝
⎛
++⎟
⎠
⎞
⎜
⎝
⎛
+=
2
sin
2
cos
π
α
π
α

12
⇒
( )
⎟
⎠
⎞
⎜
⎝
⎛
+=
2
π
α
α
α
u
d
ud
12-TMrg;m:aRTIs
CYredk
CYrQr
m
n
aaaa
aaaa
aaaa
aaaa
mnmmm
n
n
n
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎬
⎫
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
321
3333231
2232221
1131211
-m:aRTIsBiess
-ebI n = 1 ⇒
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
1
31
21
11
ma
a
a
a
A ; A manlMdab; m ×1
-ebI 1=m ⇒ [ ]naaaaA 1131211= ; A manlMdab; n×1
-ebI nm = eK)anm:aRTIskaer .
-m:aRTIskaerman jiij aa = ⇒m:aRTIssIuemRTI
-RbmaNviFIelIm:aRTIs
a). m:aRTIsBIresμIKña
][
][
ji
ji
bB
aA
=
=
⇒ BA = ⇔ ][][ jiji ba =
⇒ [ ] [ ]ijij baBA =⇔=
b). plKuNm:aRTIsnwgsáaEl
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
3231
2221
1211
aa
aa
aa
A ⇒ =kA
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
3231
2221
1211
kaka
kaka
kaka
c). plbUkm:aRTIs
⎥
⎦
⎤
⎢
⎣
⎡
=
232221
131211
aaa
aaa
A ; ⎥
⎦
⎤
⎢
⎣
⎡
=
232221
131211
bbb
bbb
B

13
⇒ =+ BA ⎥
⎦
⎤
⎢
⎣
⎡
+++
+++
232322222121
131312121111
bababa
bababa
-plKuNsáaElénBIrm:aRTIs
[ ]naaaaA 321= ; B =
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
nb
b
b
b
3
3
1
⇒ [ ]naaaaBA 321=⋅ · nn
n
bababa
b
b
b
b
+++=
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
22113
2
1
b¤ ⇒ ∑=
=⋅
n
i
iibaBA
1
-plKuNBIrm:aRTIs
[ ]ikiii aaaaA 321= ;
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
kj
j
j
j
b
b
b
b
B 3
2
1
BA⋅ = [ ]ikiii aaaa 321 · kjikjiii
kj
j
j
j
bababa
b
b
b
b
+++=
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
...22113
2
1
b¤ ∑=
=⋅
k
l
jlli baBA
1
13-RbmaNviFIkñúgkUGredaenedkat );;;0( kjiℜ
-Rkadüg;; (Gradient) CaTMhMviucT½r³
ugrad = k
z
u
j
y
u
i
x
u
⋅
∂
∂
+⋅
∂
∂
+⋅
∂
∂

14
-sáaEl Laplace (Laplacien Scalaire)
2
2
2
2
2
2
Δ
z
u
y
u
x
u
u
∂
∂
+
∂
∂
+
∂
∂
=
-DIEvsg; (Divergence) CaTMhMsáaEl
z
z
y
a
x
a
a
yx
∂
∂
+
∂
∂
+
∂
∂
=div
-r:UtasüÚENl (Rotationel)
arot = k
y
a
x
a
j
x
a
z
a
i
z
a
y
a zyzxyz
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
−
∂
∂
+⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∂
∂
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
−
∂
∂
b¤ arot =
zyx aaa
zyx
kji
∂
∂
∂
∂
∂
∂
-viucT½r Laplace (Laplacien Vectoriel)
aΔ = kajaia zyx )(Δ)(Δ)(Δ ++
♦♦ cMNaM³ dudlugrad =×
-Na)øa(Nabla)
∇ = k
z
j
y
i
x ∂
∂
+
∂
∂
+
∂
∂
eK)an³
uugrad ∇= ; uu ⋅∇= 2
)(Δ
aa ∇=div ; aa ∧∇=rot
· 0gradrot = ; Δdivgradrotrot −=
· 0rotdiv = ; Δgraddiv =
· mnnmnm gradgradgrad +=⋅
· AmAmAm ⋅+= )(graddiv)div(
· BAABBA rotrot)div( −=∧
· AmAmAm ∧+= )(gradrot)(rot
· BAABABBABA ⋅⋅−⋅+−=∧ )grad()grad()(div)(div)(rot
sUmEsVgrkKNitviTüasMrab;rUbviTüarbs;elaksaRsþacarü hg; sIum

15
RbB½n§xñat
RbB½n§xñatman³
k-xñatRKwHTak;TgeTAnwgxñatRbEvgma:s nigeBl ¬cMeBaHemkanic¦.
TMhMnigxñatRKwHénRbB½n§GnþrCati
TMhMRKwH vimaRt eQμaHxñat nimμitsBaØaxñat
RbEvg L Em:t m
ma:s M KILÚRkam kg
eBl T vinaTI s
crnþGKiÁsnI I GMEB A
sItuNðPaB θ Eklvin K
brimaNrUFatu N m:Ul mol
GaMgtg;sIuetBnWø J kg;dWLa cd
x- xñatRsLayEdleGayniymn½yedayTMnak;TMngrvagTMhMEdlTak;Tg nigTMhMRKwHTaMgenH.
RbB½n§xñatBIrEdleRbIjwkjab;bMputenaHKW RbB½n§ CGS KitCa skgcm ;; .
nigRbB½n§ MKS KitCa skgm ;; . RbB½n§eRkayenHehAfa SI k_)an.
TMhM xñat CGS xñat KMS TMnak;TMng
RbEvg
ma:s
eBl
sMTuH
cm
g
s
2
/ scm
m
kg
s
2
s/m
cmm 2
101 =
gkg 3
101 =
scmsm /10/1 22
=
dynesN 5
101 =
segçbrUbmnþsMxan;²énemkanic

16
kMlaMg
kmμnþ
sMBaF
DIn )dyne(
erg
)ar )bar(
jUtun )(N
s‘Ul )J(
):asáal; )Pa(
ergsJ 7
101 =
barPa 5
101 =
xñatxøHeTot ]TahrN_ dWeRk ¦ ra:düg;cMeBaHmuM nig atm cMeBaHsMBaFKWxñateRkARbB½n§.
π
0
180
1 =rad Paatm 5
10.013,11 =
smIkarvimaRt
tag ML, nigT CaTMhMRbEvg ma:s nigeBl eKGacsMEdgTMhMTaMgGs;CaGnuKmn_énTMhMTaMgenH.
kenSamEdl)anmkbegI;átsmIkarvimaRténTMhMenH.
]TahrN_ el,Ón 1
. −
== TL
T
L
sMTuH 2−
= LT
kMlaMg 2−
= MLT
kmμnþ 22 −
= TML
efrRKwH
TMhM niimitþsBaØa tMél
el,ÓnBnWø c 299792458 /m s
CMrabsuBaØakas 0μ 7
4 .10 /H mπ −
EBmITIvIetsuBaØakas 0ε 12
8,85481.10 /F m−
efrTMnaj G
11 3 2
6,6725985.10 /m kg s−
efrGavU:kaRdU AN 23
6,022136.10 / mol
bnÞúkdMbUg e
19
1,602177.10 C−
ma:seGLicRtug em 31
9,109389.10 kg−
ma:sRbUtug pm 27
1,672623.10 kg−
ma:sNWRtug nm 27
1,674928.10 kg−

17
xñattaraviTüa
eQμaH ma:s kaM
RBHGaTitü 30
2.10 kg 5
7.10 km
EpndI 24
6.10 kg 3
6,4.10 km
RBHc½nÞ 22
7,35.10 kg 3
1,7.10 km
xñattaraviTüa=cMgayBIEpndIeTARBHGaTitü³ 11
1 . 1,50.10u a m=
cMgayBIEpndIeTARBHc½nÞ³ 5
3,84.10 km.
sUmrg;caMGansñaédepSg²eTotrbs; elak hg; sIum ecjpSay²kñúheBlqab;²

18
EpñkTI1³ sIuenma:Tic (Cinématique-Kinematics)
sInenma:Tic sikSaBIclnarbs;GgÁFatu¬cMnucrUbFatu¦edayminKitBIbuBVehtunaMeGayekItmanclna.
1-clnaRtg;
clnaRtg ; Knøgrbs ;cl½tCabnÞat ;. dUcCaclnatamG½kS oxx′ ³
-viucT½rTItaMg³ ixxOM ⋅==
-smIkarclna³ )(txx =
-smIkarel,Ón³ x
dt
dx
v ==
-smIkarsMTuH³ ⎟
⎠
⎞
⎜
⎝
⎛
==
dt
dx
dt
d
dt
dv
a = 2
2
dt
xd
= x
-edaysÁal;sMTuH³
0 0
x t
x t
dx
a dx a dt
dt =
= ⇒ =∫ ∫
-edaysÁal;el,Ón³
0 0
x t
x t
dx
v dx v dt
dt =
= ⇒ =∫ ∫
-TMnak;TMng³
0 0
v x
v x
dx dv dx dv
a v vdv a dx
dt dx dt dx
= = = ⇒ =∫ ∫
2-clnaRtg;esμI
-Knøgcl½tCabnÞat;
-viucT½rel,Ónnigm:UDulrbs;vaefr
-sMTuHsUnü
smIkar³
00
===
dt
dv
dt
dv
a , 0vv = = efr 0v ³ el,ÓnedIm
dt
dx
v =0 = efr ⇒ dtvdx 0= , 0x ³ Gab;sIusedIm
⇒ ∫∫ =
tx
x
dtvdx
0
0
0
⇒ tvxx ⋅=− 00
⇒ ¬smIkareBl rW smIkarclna¦( ) )(
0
)(
0
)( ms
s
mm
xtvx +⋅=
O i M
x′ x x

19
3-clnaRtg;ERbRbYlesμI
-Knøgcl½tCabnÞat;
-el,ÓnERbRbYl
-sMTuHefr
=a efr ⇒
dt
dv
a = ⇔ dtadv ⋅=
⇒ ∫∫ =
tv
v
adtdv
00
⇒ atvv =− 0 ⇒ ¬smIkarel,Ón¦
xt
dx
v = ⇒ ∫∫ +⋅=
tx
x
vtadx
0
0 )(
0
⇒ tvatxx ⋅+=− 0
2
0
2
1
⇒ ¬smIkareBl rW smIkarclna¦
-TMnak;TMngrvag ,v a nig x ³
-clnaRtg;sÞúH ebI . 0a v > mann½yfa viucT½rel,ÓnnigsMTuHmanTisedAdUcKña.
-clnaRtg;yWt ebI . 0a v < mann½yfa viucT½rel,ÓnnigsMTuHmanTisedApÞúyKña.
4-clnaenAkñúglMh
tMruyedkat )(oxyzℜ rW ),,,0( kjiℜ Edl kji ;; CaviucT½rÉkta.
k-viucT½rTItaMg
tag Mr 0= CakaMvuicT½r rWviucT½rTItaMg³
0vatv +=
00
2
2
1
xtvatx +⋅+=
)(2 0
2
0
2
xxavv −=−
z
M
N
y y
x
x
z
i
j
k
r
0
ℜ

20
kzjyixMr ++==0 rW
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
z
y
x
Mr 0
222
))(( zyxkzjyixkzjyixrrr ++=++++==
x-vuicT½rel,Ón
-el,ÓnmFüm
pleFob mv
t
r
=
Δ
Δ
ehAfa el,ÓnmFüm Edl rrrrrrr Δ−=Δ⇒Δ+= ;'' RtUvnwgry³eBl
ttt −=Δ ' ehIyGaMgtg;suIetrbs;vaKW³ t
r
vm
Δ
Δ
= .
müa:geTot kzjyixr ... Δ+Δ+Δ=Δ rW
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−=Δ
−=Δ
−=Δ
Δ
zzz
yyy
xxx
r
'
'
'
ehIy ( ) ( ) ( )[ ] 2/1222
zyxr Δ+Δ+Δ=Δ
-el,ÓnxN³
r
dt
rd
t
r
v
t
==
Δ
Δ
=
→Δ 0
lim CavuicT½rel,ÓnenARtg;cMnucM RtUvnwgxN³t .
kzjyixk
dt
dz
j
dt
dy
i
dt
dx
kzjyix
dt
d
dt
Md
dt
rd
++=++=++==⇒ )(
0
tag x
dt
dx
vx == el,ÓntamG½kS )'( xx
x
z
M
N
y
y
x
z
i j
k
r
0
M’
rΔmv
'r

21
y
dt
dy
vy == el,ÓntamG½kS )'( yy
z
dt
dz
vz == el,ÓntamG½kS )'( zz
rW
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
==
==
==
=
z
dt
dz
v
y
dt
dy
v
x
dt
dx
v
dt
Md
v
z
y
x
0
GaMgtg;suIetrbs;va³ 222
zyx vvvv ++= rW 222
)()()( zyxv ++=
K-viucT½rsMTuH
-sMTuHmFüm
]bmaenAxN³t eTA 't el,ÓnERbRbYlBIv eTA 'v . dUcenHbMErbMrYlel,Ón vvv −=Δ ' kñúgbMErbMrYl
eBl ttt −=Δ ' . dUcenHsMTuHmFüm³ t
v
am
Δ
Δ
= ehIy GaMgtg;sIuet t
v
am
Δ
Δ
= .
-sMTuHxN³
ebI 0'' →Δ=−⇒→ vvvvv ehIy 0'' →Δ=−⇒→ ttttt eyIg)anlImIt
a
dt
vd
t
v
t
==
Δ
Δ
→Δ 0
lim .
eday →→→→→→→→→→
++=++=⇒++= k
dt
zd
j
dt
yd
i
dt
xd
kzjyix
dt
d
akzjyixv )(
ehIy xax
dt
xd
dt
dx
dt
d
dt
xd
===⎟
⎠
⎞
⎜
⎝
⎛
= 2
2
sMTuHtamG½kS )'( xx
yay
dt
yd
dt
dy
dt
d
dt
yd
===⎟
⎠
⎞
⎜
⎝
⎛
= 2
2
sMTuHtamG½kS )y'y(
M
'M
v
'v
v
'v vΔ
ma
k
j
i

22
zaz
dt
zd
dt
dz
dt
d
dt
zd
===⎟
⎠
⎞
⎜
⎝
⎛
= 2
2
sMTuHtamG½kS )'( zz
rW
⎟⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
===
===
===
==
z
dt
zd
dt
dv
a
y
dt
yd
dt
dv
a
x
dt
xd
dt
dv
a
dt
Md
dt
vd
a
z
z
y
y
x
x
2
2
2
2
2
2
2
2
0
GaMgtg;sIuetKW³ 222222
)()()( zyx aaazyxa ++=++=
CaTUeTA ebIcl½tmYyeeFIVclnaenAkñúglMh rWkñúgbIvimaRt eKsresr ³
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
==→
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=→
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
z
y
x
dt
Md
dt
vd
a
z
y
x
dt
Md
v
z
y
x
M 2
2
00
0
krNIBiess
-ebIcl½teFVIclnaelIG½kSEtmYy ]bmaelIG½kS )'( xx eyIg)an³
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=→
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=→
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
0
0
0
0
0
0
00
x
dt
vd
a
x
dt
Md
v
x
M
-ebIcl½teFVIclnaenAkñugbøg;]bmaenAbøg; ),,0( ji eyIg)an³
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=→
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=→
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
00
0
0
0 y
x
dt
vd
ay
x
dt
Md
vy
x
M
5-clnaekag
sikSaclnarbs;cMnucrUbFatuenAkñúgtMruy ),,,0(
→→→
ℜ kji .
-Gab;sIusekag³ )(tSSAM ==
∩
.
M
i j
k
0ℜ
(C)A
S

23
-viucT½rel,Ón
el,ÓnmFüm³ tt
MM
vm
−
=
'
'
-el,ÓnxN³Rtg;cMnucM ³
u
dt
dS
u
t
MM
MM
MM
t
MM
v
tt
=
Δ
=
Δ
=
∩
∩→Δ→Δ
'
'
'
lim
'
lim
00
-vuicT½rsMTuH
niymn½ysMTuH u
dt
ud
Su
dt
Sd
dt
uSd
dt
vd
a ;+=== manTisedAERbRbYleTAtameBl.
eyIg)an³ dt
dS
dS
d
d
ud
dt
ud α
α
=
eday
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
=⇔=
=
==
ρα
ρ
α
n
dS
ud
n
d
ud
dS
d
vS
dt
dS
1
ehIyρ CakaMkMenagRtg;M nign CaviucT½rÉktaEkgnwgKnøgRtg;M .
elIsBIenHeTAeTot eKman³ ωα
α
==
dt
d
ehAfa el,ÓnmMu ehIymanTMnak;TMng³ ρρ
α v
st
dS
dt
d
==
1
ρ
ωα
v
==⇒ .
dUcenH eyIg)an³ n
v
u
dt
dv
a
ρ
2
+=
tag uSu
dt
dv
at == sMTuHpMÁúb:Hb:HnwgKnøgCanic© .
nn
v
an
2
2
ωρ
ρ
== sMTuHpÁúuMEkgEkgnwgKnøgCanic© .
eyIgsresrCam:UDul 2
2
; ωρ
ρ
===
v
a
dt
dv
a nt
222
22
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
=+=⇒+=⇒
ρ
v
dt
dv
aaanauaa ntnt
M
ta
na
a
M
M’
u
S
mv
pi©t
A
M
M’
u
dS
pi©t
A
n
αd
v
ρ

24
BiPakSa
-ebIkaMkMenagρ xiteTArkGnnþ eK)anclnarbs;cl½tCaclnaRtg; BIeRBaH 0lim
2
=
∞→ ρρ
v
¬sMTuHpMÁúEkg
esIμsUnü¦.
-ebIkaMkMenag =ρ efr eK)ancl½teFIVclnavg;.
cMNaM³ ebIcl½teFVIclnaekagEdlmansmIkarKnøg ( )y f x= enaHkaMkMeNagénKnøgRtg;cMnucNamYyKW³
3
2 2
2
2
1
dy
dx
d y
dx
ρ
⎡ ⎤⎛ ⎞
+⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦=
-clnasÞúH ebI . 0ta v > mann½yfa viucT½rel,ÓnnigsMTuHmanTisedAdUcKña.
-clnayWt ebI . 0ta v < mann½yfa viucT½rel,ÓnnigsMTuHmanTisedApÞúyKña.
6-clnavg;
clnavg;CakrNIBiessénclnaekagkalNakaMkMenagefr Rρ = = efr.
k-smIkarclnavg;esμI
v = efr rW ω= efr
-TMhMRbEvg³
dt
dS
v = ⇒ vdtdS = ⇔ ∫∫ =
ts
s
vdtdS
00
⇔ tvSS ⋅=− 0 ⇒
⎪
⎪
⎩
⎪⎪
⎨
⎧
==
==
R
R
v
a
dt
dv
a
a
n
t
2
2
0
ω
⇒ R
R
v
aa n
2
2
ω===
-TMhMmuM (rad)
dt
dα
ω = ⇒ dtd ωα = ⇔ ∫∫ =
t
dtd
00
ωα
α
α
⇒
( ) )(
0
)()( rads
s
radrad
t αωα +⋅= ¬smIkarmuMeBl¦
dt
dω
β = = 02
2
== α
α
dt
d
sMTuHmMu
0StvS +⋅=

25
-eRbkg; nigxYb
2
f
ω
π
= nig 1 2
T
f
π
ω
= =
x-clnavg;ERbRbYlesμI
dt
dv
at = = efr rW dt
dω
β = = efr
dt
dv
at = ⇒ dtadv t= ⇔ ∫∫ =
t
t
v
v
dtadv
00
⇒ 0vtav t +⋅=
dt
dS
v = ⇒ dSdtvtat =+⋅ )( 0
⇔ dtvtadS
t
t
s
s
)(
0
0
0
∫∫ +⋅= ⇒ 00
2
2
1
StvtaS t +⋅+⋅=
dt
dω
β = ⇒ dtd βω = ⇔ ∫∫ =
t
dtd
00
βω
ω
ω
⇒
( ) ( )s
rads
s
rads
rad
t 0
)(
2
ωβω +⋅=
⎟
⎠
⎞
⎜
⎝
⎛
dt
dα
ω = ⇒ dttd )( 0ωβα +⋅= ⇔ ∫∫ +⋅=
t
dttd
0
0 )(
0
ωβα
α
α
⇒ 00
2
2
1
αωβα +⋅+⋅= tt
sMTuHpÁúMEkg ¬sMTuHcUlp©it¦ R
R
v
an
2
2
ω==
TMnak;TMngrvag S nig ω ³
α⋅= RS Rv ⋅= ω, βRaT =,
ehIy³ )(2 0
2
0
2
ααβωω −=−
)(2 0
2
0
2
SSavv t −=−
7-clnaRtg;suInuysUGIut
smIkarclna³ )sin( ϕω +⋅= txx m
x ³ eGLuúgkasüúg
mx ³ GMBøITut
ω ³ Bulsasüúg
ϕ ³ pasedIm
x
+ mx
O
mx−
x′
S
R
α

26
el,Ón )]sin([ ϕω +== tx
dt
d
dt
dx
v m
⇒ )cos( ϕωω += txv m
sMTuH )]cos([ ϕωω +⋅== tx
dt
d
dt
dv
a m = )sin(2
ϕωω +− txm
⇒ xa ⋅−= 2
ω b¤ xx ⋅−= 2
ω
⇔ 02
=+ xx ω ¬smIkarDIepr:g;EsüllMdab;2¦
xYb³ ω
π2
)(
=
s
T
eRbkg;³ π
ω
2
1
z)(H
==
T
f
8-clnaTnøak;esrI
clnaTnøak;esrIrgEtkMlaMgEdnTMnajdI¬sikSaenAkñúgEdnTMnajdI¦. eRCIserIsG½kSQrsMrab;sikSa.
-Tnøak;esrIKμanel,ÓnedIm³ )0,0( 00 == zv
sMTuH a g=
smIkarclna³ 2
2
1
gtz =
smIkarel,Ón³ dt
dz
v = = gtgt
dt
d
=⎟
⎠
⎞
⎜
⎝
⎛ 2
2
1
TMnak;TMng³ gzv 2=
-Tnøak;esrImanel,ÓnedIm³eRCIserIs 00 =z
- smIkarclna³ -smIkarclna³
tvgtz ⋅+= 0
2
2
1
tvgtz ⋅−= 0
2
2
1
gzvv 2
2
0
2
=− gzvv 2
2
0
2
=−
z′
O
g
z
O
0v
g
z
0v
O
g
z

27
-smIkarclna³ -smIkarclna³
tvgtz ⋅+−= 0
2
2
1
tvgtz ⋅−−= 0
2
2
1
gzvv 2
2
0
2
−=− gzvv 2
2
0
2
−=−
9-clnaRKab;)aj;
k-)aj;tamTisedk³ tMruy (Oxy)
clnatamG½kS l½kçx½NÐedIm sMTuH el,ÓnxN³ smIkareBl
(0 )x 0
0 0
0
x
x
v v
=
=
0xa = 0xv v= 0.x v t=
(0 )y
0
0
0
o
y
y
v
=
=
ya g= − yv gt= − 21
2
y gt= −
smIkarKnøg 2
0
2
2
1
v
x
gy =
x-)aj;tamExSeTrbegáIt)anmuM α mYy³
clnatamG½kS l½kçx½NÐedIm sMTuH el,ÓnxN³ smIkareBl
(0 )x 0
0 0
0
cosx
x
v v α
=
=
0xa = 0 cosxv v α= 0 cos .x v tα=
(0 )y
0 0
0
sin
o
y
y
v v α
=
=
ya g= − 0 sinyv gt v α= − + 2
0
1
sin .
2
y gt v tα= − +
-smIkarKnøg³ x
v
x
gy ⋅+−= α
α
tg
cos2
1
22
0
2
z
g
0v
O
z
O
0’v
O
0v x
M xv
yv v
y

28
-cMgayFøak;³ g
v
d
α2sin
2
0
=
-kMBs;eLIgdl;³ g
v
Ym
2
sin22
0 α
=
10-clnaeFobeTAnwgtMruyeFIVclnarMkil
PaKli¥tBIrA nigB eFIVclnaenAkñúglMh.
viucT½rTItaMg Ar nig Br eFobnwgtMruy ),,,0( kjiℜ .
tMruy ''' zyAx eFIVclna rMkileFobnwg xyz0 .
eK)an³
ABAB += 00 rW ABAB rrr /+=
eFVIedrIeveK)an³
dt
ABd
dt
Ad
dt
Bd
+=
00
rW dt
rd
dt
rd
dt
rd ABAB /
+= rW ABAB rrr /+=
rW ABAB vvv /+= ehIy eKehA Av Cael,ÓnnaM.
-sMTuH
dt
vd
dt
vd
dt
vd ABAB /
+= rW ABAB vvv /+=
rW ABAB aaa /+= eKehA Aa CasMTuHnaM.
cMNaM tMruy 'ℜ eFIVclnarMkilpg nigclnargVilpgCamYyel,ÓnmMuω . eK)an³ ABvv Ae ∧+=ℜ ω/ .
y
g
oyv 0v S
·
α p xv
O xov x
yv v
z
x
y0
k
ji
z’
x’
y’A
'k
'j'i
B
Br
Ar
ABr /
ℜ
'ℜ

29
kenSamel,ÓnsresreRkamTMrg;³
ABAB vABvv /+∧+= ω
)andUcKñaEdrcMeBaHkenSamsMTuH
ABABAB vABAB
dt
d
aaa // 2)( ∧+∧∧+∧++= ωωω
ω
kñúgenH )(// ABAB
dt
d
aa ABe ∧∧+∧+=ℜ ωω
ω
ehAfa sMTuHnaM
ABC va /2 ∧= ω ehAfa sMTuHCoriolis
11-clnaGaRs½y
TItaMgrbs;cl½tmYyGaRs½yeTAnwgTItaMgrbs;cl½tmYyeToteBlmanclnaeFobtMruyEtmYy.
eyIg)an³
A Bx x+ =efr
-eFIVedrIeveFobnwgeBl³
0A Bdx dx
dt dt
+ = rW 0A Bv v+ = A Bv v⇒ = −
-eFVIedrIevel,ÓneFobnwgeBl³
0A Bdv dv
dt dt
+ = rW 0A B A Ba a a a+ = ⇒ = −
sBaØa( )− mann½yfa GgÁFatuA pøas ;TIeLIgelIehIyGgÁFatuB pøas;TIcuHeRkam.
12-kUGredaenb:UElbøg;
M enAkñúgbøg; )0( yx . bnÞat; )(D CabnÞat;tam 0 nigM . bnÞat;enHedAedayviucT½redayvuicT½rÉkta ru
NamYy.
k-viucT½rTItaMg³ rurM =0 eday ] [∞+∞−∈ ,r .
y
x
)(D
θ
j
θu
ru
i0
M
Ax
A
Bx
B

30
TMnak;TMngCamYykUGredaenedkat³
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
=
θ
θ
sin
cos
0
ry
rx
M
x-viucT½rel,Ón
rurOM =
dt
ud
ru
dt
dr
dt
urd
dt
Md
v r
r
r
+===
)(0
θθ ururv r += rW ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
θr
r
v
rr urv = ehAfa el,Ónra:düal; manm:UDul rvr =
θθ θ urv = ehAfa el,ÓnGrtUra:düal;manm:UDul θθ rv =
m:UDulel,Ón³ ( )22
θrrv +=
K-viucT½rsMTuH
( )
dt
ud
ru
dt
d
ru
dt
dr
dt
ud
ru
dt
rd
dt
ururd
a r
r
r θ
θθ
θ
θ
θ
θ
θ
++++=
+
=
( ) ( ) θθθ urrurra r ++−= 22
rW ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
=
θ
θ
rr
rr
a
2
2
( ) rr urra 2
θ−= ehAfa sMTuHra:düal; manm:UDul ( )2
θrrar −=
( ) θθ θθ urra += 2 ehAfa sMTuHG½rtUra:düal;
manm:UDul ( ) ( )
⎭
⎬
⎫
⎩
⎨
⎧
=+= θθθθ
21
2 r
dt
d
r
rra
m:UDulsMTuH³ ( ) ( )222
2 θθθ rrrra ++−=

31
EpñkTI2³ DINamic (Dynamique-Dynamics)
1-c,ab ;TaMgbIrbs;jÚtun
k-c,ab ;TI1 rW c,ab ;niclPaB³ cl½tpøas;TIedayclnaRtg;esIμ rWenAnwgfál ;¬rkSadUcPaBedIm¦.
0F =∑
x-c,ab;TI2 rW c,ab ;RKwHDINamic³ plbUkviucT½rkMlaMgTaMgGs;EdlmanGMeBIelIGgÁFatuesIμnwgma:sGgÁFatu
enaHKuNnwgviucT½rsMTuHrbs ;va.
.
dp
F m a
dt
= =∑
K-c,ab ;TI3 rW GMeBIeTAvijeTAmk³ GMeBIesIμnwgRbtikmμ³ 1 2 2 1 1 2 2 1F F F F→ → → →= − ⇒ =
2-clnarMkilrbs;GgÁFatu
k-brimaNclna³ Gvmp = , p brimaNclnaKit /sm.kg
x-p©itniclPaBénGgÁFatu¬RTwsIþbT)arIsg ;¦³
nn
M
n OMmOMmOMmOGmmm +++=⋅+++ 221121
∑=
=⇒
n
i
ii AmGM
1
00.
M
Am
GO
n
i
ii∑=
=⇒ 1
0
G CaTIRbCMuTMgn;.
enAkñúgkUGredaenedkat
-G mankUGredaen ),,( GGG zyx
-cMnucma:snImYy²mankUGredaen ),,( iii zyx
dUcenHeyIg)anTMnak;TMng³
∑∑∑ ===
===
n
i
iiG
n
i
iiG
n
i
iiG zm
M
zym
M
yxm
M
x
111
1
,
1
,
1
-krNIcMnucma:senACab;²KñarwRbB½n§Cab;
eK)an³ ∫= dmM ehIy ∫∫∫ === dmz
M
zdmy
M
ydmx
M
x GGG .
1
,.
1
,.
1
K-RTwsþIbTénp©itniclPaB³

32
G
ext
vmp
F
dt
pd
=
= ∑ ⇒ G
G
am
dt
vd
m = = ∑ extF ¬kMlaMgeRkA¦
X-kMlaMgkkit f ³ manTisedApÞúyBITisedAclna³
Nf .μ= , μ ³ emKuNkkit N kMlaMgRbtikmμEkg
emKuNkkitmanBIrKWemKuNkkitsþaTic sμ nig emKuNkkitsIuenTic cμ ehIy s cμ μ≥ .
g-kMlaMgRbtikmμénTMr³
amRP =+
tn PPP +=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
⋅
α
α
cos
sin
P
P
P
0=+ RPn
⇒ RPn =
c-tMnwgExS³ ExSKμanma:s
q-rebobedaHRsaylMhat;DINamic³
-kMnt;kMlaMgEdlmanGMeBIelIGgÁFatu
-sresrRTwsþIbTénp©itniclPaB ∑ = Gamf
-eRCIserIsTisedAclna
-eFVIcMeNalTMnak;TMngxagelIG½kSclna
-edaHRsaysmIkar
-sMTuH = ssrub;ma;
kMlaMgTb;kMlaMgTaj−
y
R
nP xP
P α x
AT BTA B
BA TT =
G
R
P
0=+ RP

33
3-clnavg;esμI nigclnalMeyal
k-clnavg;esμI³
kMlaMgcUlp©it namF = b¤ nmaF =
eday R
R
v
an
2
2
ω== na
⇒ Rm
R
v
mmaF n
2
2
ω===
x-clnasIunuysUGuItb¤clnalMeyal
-kMlaMgyWtrbs;rWusr³
xkF .−= , k ³ efrkMrajrWusr(N.m)
x ³ sac;lUt (m)
T ³ kMlaMgrMlwk(N)
xmamF .. ==
⇒ xkxm .. −= ⇔ 0=+ m
x
k
x
smIkarenHmanbJs³
)sin( 0 ϕ+= txx m ω
BulsasüúgpÞal;³ m
k
=0ω ⇒xYbpÞal;³ m
k
T π20 =
-lMeyale)a:leTalnigrgiVle)a:lekaN
FPT =+ 2
cos
ω
α
g
=⇒
xYbpÞal;énlMeyal³ g
T π20 =
e)a:lgakecjBITItaMglMnwg)ankalNa³ g
=≥ 0ωω ¬CaBulsasüúgpÞal;¦
F O
x
m
P
x
m
0 F

34
smIkarDIepr:g;EsüllMeyaléne)a:l ¬minEmnlMeyalGam:Unic¦³
sin 0
g
α α+ =
krNIlMeyaltUc sin ( )radα α= ¬CalMeyalGam:Unic¦
( )2
0 00 sinm tα ω α α α ω ϕ+ = ⇒ = +
4-famBl nigc,ab;rkSafamBl
4-1- famBlsIuenTic
k-rUbFatu b¤GgÁFaturwgeFVIclnarMkil³ 2
2
1
mvEC =
x-GgÁFatueFVIclnargVilCuMvijG½kS(Δ) mYy³ 2
2
1
ω⋅= JEC , J³ m:Um:g;niclPaB( 2
g.mk )
K-GgÁFatueFVIclnarMkilpg rgVilpg³ 22
2
1
2
1
ωJmvEC +=
cMnucrUbFatu 2
2
1
mrJ = sIuLaMgRbehag 2
2
1
mrJ = suILaMgesμIsac; 2
2
1
mrJ =
O′
l T
F m
O P
¬e)a:leTal¦
O′
α
l
T
F m
O P
¬e)a:lekaN¦

35
Es‘VesμIsac; 2
5
2
mrJ = r)ar 2
12
1
mrJ = kgmUl 2
mrJ =
fasesμIsac; RTwsþIbT h‘uyEhÁn
2
2
1
mrJ = 2
)()Δ( mdJJ += Δ′
rebobKNnam:Um:g;niclPaB³ 2
J r dm= ∫
K-bMErbMrYlfamBlsIuenTic³ 12
Δ CCC EEE −= = 12W kmμnþ
4-2-kmμnþ nigGanuPaB
k-kmμnþ³ ebIcl½tpøas;TIBIA eTAB eRkamkMlaM F eKsresr³
∫∫∫ ===→
B
A
t
B
A
B
A
BA dsFrdFdWFW ..)(
ebIkmμnþénkMlaMgbMlas;TIenAkñúglMh
éntMruy ),,,0( kji eK)an³
rdFdW .=
Edl
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
z
y
x
F
F
F
F ehIy
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
dz
dy
dx
rd
r
r
r
r
)(Δ
)'(Δ
G
d
tF
tF
A ds B
dsFdW t .=
W

36
dzFdyFdxF
dz
dy
dx
F
F
F
rdFdW zyx
z
y
x
++=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
==⇒ .
ebIkmμnþbMlas;TIBIA eTA B ³
∫∫∫∫ ++==→
B
A
z
B
A
y
B
A
x
B
A
BA dzFdyFdxFdWFW )(
cMNaM³ebIkmμnþelIExSbiTesIμsUnü BIeRBaH³
0....)( =−=+=== ∫∫∫∫∫∫→
B
A
B
A
A
B
B
A
A
A
BA rdFrdFrdFrdFdWdWFW
-kmμnþénkMlaMgefrkñúgbMlas;TIRtg;
GgÁFatumYypøas;TIBI BA → eRkamGMeBIénkMlaMgefrF enaHeK)an³
αcos...)( ABFABFFW BA ==→ Edl );( ABF=α
ebIFatukmμnþénkMlaMgF kñúgbMlas;TId¾tUcd enaH eKsresr³
dFdW .=
ebIvapøas;TIBI BA → enaHeK)an ∫∫ ==
B
A
B
A
AB dFdFW αcos...
-ebI ABFFWAB .)(1cos0 =⇒=⇒= αα
-ebI 0)(0cos900
=⇒=⇒= FWABαα
-ebI ABFFWAB .)(1cos1800
−=⇒−=⇒= αα
eday 00
1800 ≤≤ α
dUcenHeyIgsniñdæan)anfa³
-ebI 0900 00
>⇒<≤ ABWα kmμnþclkr
-ebI 018090 00
>⇒≤< ABWα kmμnþTb;
-ebI 0=ABW kMlaMgmin)ancUlrYmbegáItkmμnþ
-kmμnþénkMlaMgefrkñúgbMlas;TINamYy
kmμnþénkMlaMgkñúgbMlas;TIΔ ³ αcos...)( Δ=Δ=Δ FFFW ehIykmμnþsrubkñúgcMgay ∩
AB KW
∑∑∑ Δ=Δ=Δ=
B
A
B
A
B
A
AB FFFWFW αcos...)()( .
F
α
TisedAbMlas;TI
A B

37
ebIbMlas;TId¾xøI ∫=⇒→Δ
B
A
AB dFFWd αcos..)( .
vtßúpøas;TIBI A eTA B
ABFWAB ⋅= = αcos⋅⋅ ABF
o
900 << α ⇒ 0cos >α ⇒ 0>W kmμnþclkr
o
90=α ⇒ 0cos =a ⇒ W = 0
oo
18090 << α ⇒ 0cos <α ⇒ 0<W kmμnþTb;
kmμnþénTMgn;³
mghhPPW =⋅=)(
x-GanuPaB
GanuPaBsMEdgedayniymn½y dt
dW
P = EdldW kmμnþKitCas‘Ul )(J ehIydt ry³eBlKitCa
vinaTI )(s nigP GanuPaBKitCava:t; )(W .
eday rdFdW .= vF
dt
rd
F
dt
rdF
P ..
.
===⇒
rW αcos..vFP = Edl ),( vF=α
-ebI 0>P kMlaMgCakMlaMgclkr
-ebI 0<P kMlaMgCakMlaMgTb;
-ebI 0=P kMlaMgmanGanuPaBsUnü )( vF ⊥
-kñúgclnarMkil
eyIgBinitüFatukmμnþénkMlaMg xdFdW .= xd; CavuicT½rbMlas;TI F; CavuicT½rkMlaMg
vF
dt
xd
F
dt
xdF
dt
dW
P .
.
====⇒
),(;cos... vFvFvFP === αα
4-3-famBlb:Utg;Esül
k-famBlénEdnkMlaMgrkSa
F
α
α
A B
F Fα α
d d
F
vα

38
pE
F
r
∂
= −
∂
rW
2 2
1 1
( )
( )
( )
P
p
E r r
p
E r r
dE F r dr= −∫ ∫
x-famBlbU:tg;EsülTMnajdI³
mghEP = , h: kMBs;
K-famBlbU:tg;EsülyWt³
2
.
2
1
xkEP = , k ³ efrkMrajruWsr ( 1
mN −
⋅ ) , x ³ sac;lUt (m)
X-famBlbU:tg;EsüleGLicRtUsþaTic³
ABFWAB ⋅= , BA VVABE −=⋅
⇒ )( BAAB VVqW −= , AV , BV b:Utg;Esül
famBlbU:tg;Esül qVEP =
⇒ )()( BPAPAB EEW −=
g-famBlb:Utg;EsülrmUl³ 21
2
PE Cθ=
c-tMhyfamBlb:Utg;EsülesIμnwgkmμnþénkMlaMg³
PW E= −Δ rW ( )1 2 (2) (1)P PW E E→ = − −
5-famBlemkanic
CPM EEE +=
krNIRbB½n§Rtemac b¤RbB½n§biT b¤RbB½n§rgEtGMeBIbU:tg;Esül famBlemkanicCaTMhM)anrkSa
CPM EEE += = efr
k-famBlemkanicEdnTMnajdI³ mghmvEM += 2
2
1
= efr
x-famBlemkanicénkMlaMgyWtrbs;rWus½r³
22
.
2
1
.
2
1
xkvmEM += = efr
K-famBlemkanicénkMlaMgGKÁisnI³
qVmvEM += 2
2
1
= efr
X-RTwsIþbTfamBlemkanic³ ( )ME W fΔ = kmμnþénkMlaMgminrkSa.
q F
AVA,
BVB,
E

39
6-clnaenAkñúgEdn
k-clnaenAkñúgEdnTMnajdI
clnaenAkñúglMhesrI GgÁFaturgkMlaMgEtmYyKt;KW kMlaMgTMnajdI edayminKitkMlaMgkkitnana.
f P=∑
tamTMnakTMngRKwHDINamic³
P ma a g= ⇒ =
clnaTaMgenH manclnaTnøak;esrI clnaRKab;)aj; clnarNbCMuvijEpndI.
smIkarclna nigsmIkarKnøgGaRs½yeTAnwgkar)aj; rWpøas;TIrbs;GgÁFatu.
x-clnaenAkñúgEdnGKiÁsnI
edayminKitkMlaMgkkitnana pg;pÞúkGKiÁsnIrgEtkMlaMgEdnGKiÁsnI³ f F qE= =∑
tamTMnak;TMngRKwHDINamic³ q
qE ma a E
m
= ⇒ =
smIkarclna nigsmIkarKnøgGaRs½yeTAnwgkar)aj;pg; rWpøas;TI.
K-clnarbs;pg;enAkñúgEdnma:ejTicÉksNæan
edayminKitkMlaMgkkitnana pg;pÞúkGKiÁsnIrgEtkMlaMgma:ejTic³ f F qv B= = ∧∑
tamTMnak;TMngRKwHDINamic³ q
qv B ma a v B
m
∧ = ⇒ = ∧
smIkarclna nigsmIkarKnøgGaRs½yeTAnwgkar)aj;pg; rW pøas;TI.
7-m:Um:g;sIuenTicnigm:Um:g;énkMlaMg
k-m:Um:g;énkMlaMg
m:Um:g;énkMlaMgCaTMhMviucT½rkMnt;eday³
FrFAFM ∧=∧= 0)(0
-TisedArbs; )(0 FM tamviFanxñÜgqñúkedaybgVilBI r eTA F .
-m:UDulrbs;va );(;sin..)(0 rFrFFM == αα .
tag αsin.rd = ehAfaédXñas;. xñatm:Um:g;KitCa ).( mN
ebIm:Um:g;énkMlaMgenAkñúglMh eK)ankMubU:sg; TItaMg nigkMlaMgdUcxageRkam³
F
α
A0
r
d

40
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
z
y
x
F
F
F
F
z
y
x
rA ;0
dUcenH m:Um:g;³
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
∧
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
z
y
x
z
y
x
F
F
F
z
y
x
FM
FM
FM
FM
)(
)(
)(
)(
0
0
0
0
eyIg)anm:Um:g;tamG½kSnImYy²KW³
⎪
⎪
⎩
⎪⎪
⎨
⎧
−=
−=
−=
xyz
zxy
yzx
FyFxFM
FxFzFM
FzFyFM
..)(
..)(
..)(
0
0
0
m:UDul³ ( ) ( ) ( )[ ] 2/1222
0 ......)( xyzxyz FyFxFxFzFzFyFM −+−+−=
x-m:Um:g ;sIuenTic³
-niymn½y³ , , .A r p r M p m vσ = ∧ = =
-RTwsIþbTm:Um:g;sIuenTic³ ( )A
A
d
M F
dt
σ
=
8-lMnwgénPaKli¥t
-lMnwgenAkñúgbøg;
0x yf F i F j= + =∑ ∑ ∑
rW 0 , 0x yF F= =∑ ∑
-lMnwgenAkñúglMh
0x y zf F i F j F k= + + =∑ ∑ ∑ ∑
rW 0 , 0 , 0x y zF F F= = =∑ ∑ ∑
z
x
y0
A
r
xF
zF
yF
x
y
z
z
x
y
0
A
r
F
)(0 FM
¬G½kSm:Um:g;¦

41
9-lMnwgénGgÁFaturwg
0
( ) 0i i i
F
M F r F
=
= ∧ =
∑
∑
10-clnarNbEpndI
m ³ ma:srNb TM, ³ ma:sEpndI
-tamc,ab;TMnajsakl³
221
)( zR
Mm
GFF T
+
⋅
⋅==
USI1067,6 11−
⋅=G , R ³ kaMEpndI nig h ³ kMBs;
eday mgPF ==1
⇒ mg = 2
)( zR
Mm
G T
+
⋅
⋅ ⇒ 2
)( zR
M
Gg T
+
⋅=
-krNIvtßúenAelIEpndI³ z = 0, 0gg = ⇒
R
M
Gg T
⋅=0
⇒
G
Rg
MT
0
= ⇒
2
0 ⎟
⎠
⎞
⎜
⎝
⎛
+
=
zR
R
gg
-krNIrNbeFVIclnavg;CMuvijEpndI
-el,ÓnrNb³ ,0
zR
g
Rv
+
=
zR
GM
v T
+
=
-xYbrgVil³ 2
3
0
)(
2
zR
gR
T +=
π
11-c,ab;ekEBø
c,ab;enHRtUv)anEcgedayelakekEBømanbI³
k-RKb;PBTaMgGs;eFVIclnaCMuvijRBHGaTitümanKnøgCaeGlIb ehIymankMnMumYysißtenAelIRBHGaTitü.
x-kñúgry³eBlesμIKñakaMviucT½rekos)anépÞesμIKña. ebIPBcr)an ∩
AB rW ∩
CD kñúgry³eBlesμIKñaeK)anépÞ
SAB esμInwgépÞ SCD .
RkLaépÞeGlIb abaS ;..π= knøHG½kSFM b; knøHG½kStUc
tamkenSamel,OnépÞ C
C
dt
dS
;
2
= ehAfa efrépÞ
K-kaerénxYbbrivtþrgVilsmamaRteTAnwgKUbénknøHG½kSFMrbs;eGlIb.
h
1F
2F
m
TM
R
A
B
C
D
S

42
efrsmamaRtmantMéldUcKñaRKb;PB.
ebIvaeFVIclna)anmYyCMumYyxYb Tdt =⇒ xYb ehIy baSdS ..π==
2
.. C
T
ba
T
S
==⇒
π
4
.. 2
2
222
C
T
ba
=⇒
π
tag GMGkkpC S ;.;.2
== efrTMnajskl
SM; ma:sRBHGaTitü ehIy a
b
p
2
=
==⇒
ka
T 2
3
2
4π
efr
SMGa
T
.
4 2
3
2
π
=
X-rebob)aj;begðaHrNbEdlmanKnøgepSg²
-el,ÓnecjBIdI
0
2 . E
e
G M
v
r
=
-el,ÓnelIKnøgvg;
0
. E
C
G M
v
r
=
-Føak;mkEpndIvij ebI 0 Cv v<
12-TgiÁcéncMnucrUbFatuBIr
k-GaMBulsüúg
2
1
t
t
I p F dt= Δ = ∫
x-karrkSabrimaNclna
1 2 2 1F F→ →= − ¬GMeBInigRbtikmμ¦
1 2
1 2 2 1,
dp dp
F F
dt dt
→ →= = ¬c,ab;TI2jÚtun¦
1 2dp dp⇒ =
1 1 2 2' 'p p p p⇔ − = −
1 2 1 2' 'p p p p⇒ + = + ¬brimaNclnamunTgiÁcnigeRkaTgiÁcesIμKña¦.
a
b
G½kStUc
G½kSFM
pi©tkMnMu kMnMu
EpndI
Knøgvg;
KnøgeGlIb
Knøg)a:ra:bUlKnøgGIuEBbUl
0v
rNb
0r
Faøk;mkdI
1 2F→ 2 1F →
1 2

43
K-TgiÁcxÞatl¥tex©aH
-rkSabrimaNclna³ 22112211 '' vmvmvmvm +=+
-rkSafamBlsIuenTic³ 2
22
2
11
2
22
2
11 '
2
1
'
2
1
2
1
2
1
vmvmvmvm +=+
-el,ÓneRkayTgiÁc³
21
12211
2
21
21122
1
)(2
';
)(2
'
mm
mmvvm
v
mm
mmvvm
v
+
−+
=
+
−+
=
X-TgiÁcxÞatl¥minex©aH
-brimaNclnarkSa³ 22112211 '' vmvmvmvm +=+
-famBlsIuenTicminrkSa
-el,ÓneRkayTgiÁc³ )()''( 1212 vvevv −−=−
eday 10 << e ¬emKuNbdiTan¦ ehIy ebI 1=e )anTgÁicxÞat .
21
21211
2
21
12122
1
)()1(
';
)()1(
'
mm
vemmvem
v
mm
vemmvem
v
+
−++
=
+
−++
=
g-TgiÁcsÞk;
eBlTgÁicnigeRkayTgÁic eyIgeXIjGgÁFatuTaMgBIrenACab;Kña.
-rkSabrimaNclna³ ')( 212211 vmmvmvm +=+
-famBlsIuenTicminrkSa³
)
2
1
2
1
(')(
2
1 2
22
2
11
2
21 vmvmvmmEC +−+=Δ
)(
)(
2 2
22
2
11
21
2
2211
vmvm
mm
vmvm
EC +−
+
+
=Δ
12
21
2122
21
21
21
';;'.)(2 vvv
mm
mm
vvv
mm
mm
EC −=
+
=−=−
+
−=Δ μμ
c-TgiÁcxÞatenAkñúgbøg;
-karrkSabrimaNclna³ 221111 '' vmvmvm +=
-karrkSafamBlsIuenTic³ 2
22
2
11
2
11 '
2
1
'
2
1
2
1
vmvmvm +=
1M
1v 2M
y
1'v
2θ
1θ
2'v
x

44
2
21
11
2 cos
2
' θ
mm
vm
v
+
=⇒
13-sIuenma:Ticbøg;énGgÁFaturwg
clnaGgÁFaturwgEckCa³
k-clnarMkil³ RKb;cMNucTaMgGs;énGgÁFatumanel,ÓndUcKña ehIyKUs)anKnøgRsb²Kña. clnarMkilrYm
man clnarMkilRtg; rMkilekagnigrMkilvg;.
x-clnargiVlCMuvijG½kSnwgmYy³ RKb;cMnucénGgÁFaturwgKUs)anKnøgCargVg;manpi©tsißtenAelIG½kSrgVilman
el,ÓnmMudUcKña Etel,ÓnRbEvgxusKña¬GaRs½ykaMKnøg¦.
K-ebIGgÁFaturwgeFVIclnarMkilpgrgiVlpg clnarbs;vaCaclnasmasrvagclnaRtg;nigclnargiVl.
1-clnarMkil
-TItaMg³ /B A B Ar r r= +
-el,Ón³ B Av v=
-sMTuH³ B Aa a=
2-clnargiVlCMvijG½kSnwgmYy
-TItaMgmMu³ ( )tθ θ=
-el,ÓnmMu³ d
dt
θ
ω θ= =
-sMTuHmMu³ d
dt
ω
β θ= =
-TMnak;TMng³ d dβ θ ω ω=
-cMeBaHel,ÓnmMuefr³
0tω β ω= +
B•
'y
A
'x
x
y
/B Ar
Br
Ar
0
P
θ
ω
v
r0

45
2
0 0
1
2
t tθ β ω θ= + +
( )2 2
0 02ω ω β θ θ− = −
-cMeBaHcMnucP eFobnwgcMnuc0minenARtg;pi©t
k-el,ÓnRbEvg³ Pv rω= ∧
x-sMTuH³ ( )P Pa r rβ ω ω= ∧ + ∧ ∧
-ebIcMnucP nig0Capi©ténrgVg;mankaMr ³
k-el,ÓnRbEvg³ v rω= ∧
x-sMTuH³ ( ) 2
a r r r rβ ω ω β ω= ∧ + ∧ ∧ = ∧ −
enAkñúgeKaleRbeN³ t na a a= +
eyIg)an³
( ) 2
,t na r a r rβ ω ω ω= ∧ = ∧ ∧ = −
3-clnaeFob
-TItaMg³ /B A B Ar r r= +
-el,Ón³ /
/
B A
B A A B A
dr
v v v v
dt
= + = +
edaycMnucB eFVIclnavg;eFobnwgcnucA ³
/ /B A B Av rω= ∧
-sMTuH³ 2
/ /B A B A B Aa a r rβ ω= + ∧ −
4-pi©txN³énel,ÓnsUnü
el,ÓnéncMnucB NamYysißtenAelIGgÁFaturwgGacRtUvTTYl)anedayviFIpÞal; ebIeyIgeRCIserIscMnuceKalA
Edlmanel,ÓnsUnüenAxN³Binitü KW /0A B B Av v rω= ⇒ = ∧
P
φ
ω
v
Pr
0
0'
0
na
ta
a
Pr
B•
'y
A
'x
x
y
/B Ar
Br
Ar
0

46
cMeBaHclnarbs;GgÁFatuenAkñúgbøg; cMnucA RtUv)anehAfa pi©txN³énel,ÓnsUnü kMnt; IC .
dUcenH /B B ICv rω= ∧
-TItaMgIC
edIm,Idak;TItaMgIC eyIgGaceRbIel,ÓnéncMnucBinitü
enAelIGgÁFatuCanic©kalEkgeTAnwgviucT½rTItaMgeFob
Edlsnw§gBIIC eTAcMnuc.
/A ICr
B
AvA
/B ICr
Bv
IC
ω
sUmEsVgrkGansñaédelak hg; sIum Edl)anpSayrYcehIy

47
EpñksIuenma:Tic
lMhat; nigdMeNaHRsay
1-cUreRbIkarviPaKvimaRtedIm,IkMNt;vimaRtsmIkarxøHxus³
2
. , , ,
2
m mv v
v t F F h
a t g
λ = = = =
Edl ,hλ CaRbEvg nig [ ] 2
F MLT −
⎡ ⎤= ⎣ ⎦ .
cMelIy
[ ] [ ] [ ]1
vt LT T L−
⎡ ⎤= =⎣ ⎦ b:uEnþ [ ] [ ]Lλ = dUcenHsmIkar vtλ = RtwmRtUv.
[ ] 2 1 1 2m
M T L ML T
a
− −⎡ ⎤
⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦⎢ ⎥
⎣ ⎦
b:uEnþ[ ] 2
F MLT −
⎡ ⎤= ⎣ ⎦ dUcenHsmIkar m
F
a
=
minRtwmRtUv.
1 1 2mv
MLT T MLT
t
− − −⎡ ⎤
⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥
⎣ ⎦
eday [ ] 2
F MLT −
⎡ ⎤= ⎣ ⎦ dUcenHsmIkar mv
F
t
=
RtwmRtUv.
[ ]
2 2 2
2
2
v L T
L
g LT
−
⎡ ⎤ ⎡ ⎤
= =⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
eday [ ] [ ]h L= dUcenHsmIkar
2
2
v
h
g
= RtwmRtUv.
2-ebIs CacMgay ehIyt CaeBl cUrrkvimaRt 1 2 3, ,C C C nig 4C enAkñúgsmIkarnImYy²dUcteTA³
( )2
1 2 3 4
1
, , sin
2
s C t s C t s C C t= = =
cMelIy
vimaRténs KW [ ]L
BIsmIkar eyIg)an³
[ ] 1
1 1
s
C C LT
t
−
⎡ ⎤= ⇒ = ⎣ ⎦ CavimaRtel,Ón.
[ ] 2
2 32 2
2 2s s
C C LT
t t
−⎡ ⎤
⎡ ⎤= ⇒ = = ⎣ ⎦⎢ ⎥
⎣ ⎦
CavimaRtsMTuH
eday ( )4sin C t KμanvimaRt dUcenH 3C manvimaRtdUcs KW [ ]L .
edaysarmMuénGnuKmn_RtIekaNmaRtKμanxñat dUcenH [ ] 1
4C T −
⎡ ⎤= ⎣ ⎦

48
3-eRbkg; f énrMj½rénma:sm enAcugrWusrEdlmanefrkMrajk Tak;TgeTAnwgm nigk edayTMnak;TMngman
TMrg;³ ( tan ) a b
f cons t m k= . cUreRbIkarviPaKvimaRtedIm,Irka nigb . edaydwgfa
[ ] [ ]1 2
,f T k M T− −
⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦ .
cMelIy
0 1 2 2a b a b b a b b
f m k M T M M T M T− − + −
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤∝ ⇒ = =⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
dUenH 0a b+ = nig 1
2 1
2
b b a− = − ⇒ = − =
4-el,Ónv énrlkelIExSGaRs½yeTAelItMnwgF enAkñúgExSnigma:senAkñúgmYyxñatRbEvg /m énExS.
ebIvaRtUv)andwgfa [ ] [ ][ ]
2
F ML T
−
= . cUrbgðajefra nigb enAkñúgsmIkarcMeBaHel,ÓnrlkelIExS³
( )( tan ) /
ba
v cons t F m= .
cMelIy
vaRtUv)aneGaydwgfa [ ] [ ] [ ]/
a b
v F m=
eyIgsresr³ [ ] [ ]0 1 1 2 1 2a b aa b a b
M LT MLT ML M L T
+ −− − − −
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
0, 1, 2 1a b a b a⇒ + = − = − = −
dUcenH eyIg)an³ 1 1
,
2 2
a b= = −
5-Bak;kNþaldMbUgénry³eBlrbs;va rfynþmYyeFVIclnaedayel,Ón 1 80 /v km h= nigBak;kNþaeTot
edayel,Ón 2 40 /v km h= . cUrrkel,ÓnmFümrbs;rfynþ.
cMelIy
el,ÓnmFümrbs;rfynþ
tagt Cary³eBlsrub
1t Cary³eBlBak;kNþaldMbUgént
2t Cary³eBlBak;kNþalQb;ént
x
1x 2x
1 1,v t 2 2,v t

49
eyIg)ansmIkar³
1 1 1
2 2 2
x v t
x v t
=⎧
⎨
=⎩
eday 1 2
2
t
t t= = rW 1 2t t t+ =
1 1 1 1
2 2 2 2
2
2
t
x v t v
t
x v t v
⎧
= = ×⎪⎪
⇒ ⎨
⎪ = = ×
⎪⎩
( )1 2 1 2
1
2
x x v v t⇒ + = +
el,ÓmmFüm ( )1 2
1 2
1
2
m
x x x
v v v
t t
+
= = = +
( )
1
80 40 60 /
2
mv km h= + =
6-Bak;kNþaldMbUgéncMgaycrrbs;va rfynþmYyeFVIclnaedayel,Ón 1 80 /v km h= nigBak;kNþaleTot
edayel,Ón 2 40 /v km h= . cUrrkel,ÓnmFümrbs;rfynþ.
cMelIy
el,ÓnmFümrbs;rfynþ
1 2
2
x
x x= =
eday 1 1 1 2 2 2,x v t x v t= =
ry³eBlsrub 1 2
1 2
1 2 1 22 2
x x x x
t t t
v v v v
= + = + = +
el,ÓnmFüm 1 2
1 2
1 2
2
2 2
m
x x v v
v
x xt v v
v v
= = =
++
2 80 40
53,33 /
80 40
mv km h
× ×
= =
+
x
1x 2x
1 1,v t 2 2,v t

50
7-cl½tmYyeFVIclnaRtg;esIμ. enAxN³edImeBl vaenARtg;cMnucEdlmanGab;sIus 0 4x m= . enAxN³
1 4t s= vaenARtg; 1 8x m= .
k-cUrsresrsmIkareBlénclna
x-cUrtagRkabGnuKmn_ ( )x x t=
cMelIy
k-smIkarclnaRtg;esIμmanTMrg;³
0 0x v t x= +
dUcenHvakøayCa³
0 08 4 4 1 /v v m s= + ⇒ =
4x t⇒ = +
x-Rkab ( ) 4t x t t= + CabnÞat;
8-cl½tmYyKUsKnøgCabnÞat; tamsmIkareBl³ 2
3 2x t t= − xñatKitCa SI .
k-cUrKNnael,ÓnmFümenAcenøaHxN³ 0 0t = nig 1t s= bnÞab;el,ÓnenAxN³ 0 0t =
x-cUrKNnasMTuHrbs;cl½t
K-cUrtagRkabrvagxN³ 0 0t = nig 1t s=
cMelIy
k-smIkareBlCadWeRkTI2éneBl. dUenHclnaCaclnaERbRbYlesIμ.
1 , 1 1 /m
x
t s x m v m s
t
Δ
Δ = Δ = ⇒ = =
Δ
nig ( )0
0
0 0
0
6 2 2 /t
t
dx
v t m s
dt =
=
⎛ ⎞
= = − = −⎜ ⎟
⎝ ⎠
x-sMTuHrbs;cl½t
2
2
2
6 /
d x
a m s
dt
= =
K-smIkar 2
3 2x t t= − CasmIkar)a:ra:bUlkat;tamKl;0.
bnÞat; 1
3
t = CaG½kSsIuemRTI.
x
0 4x =
4x t= +
0
t
0
x
1
1
t
1
3
−
1
3

51
9-cl½tmYycrelIbnÞat;edayclnaERbRbYlesIμ. sMTuHrbs;vaKW 2
5 /m s . enAxN³ 0t = vaenA20m xag
eqVgcMnucEdleRCIserIsCaKl;tMruyehIyel,Ónrbs;va10 /m s . cUrsresrsmIkareBlénclna
cMelIy
smIkareBl ( )x x t=
tamniymn½ysMTuH
2
5 /
dv
a m s
dt
= =
5dv dt⇒ =
0 10 0
5
v t
v t
dv dt
= =
⇒ =∫ ∫
5 10v t⇒ = +
müa:geTot tamniymn½yel,Ón³
5 10
dx
v t
dt
= = +
( )
0 20 0
5 10
t t
x t
dx t dt
=− =
⇒ = +∫ ∫
2
2,7 10 20x t t⇒ = + −
10-cl½tmYyeFVIclnaRtg;ERbRbYlesIμ. eKniyayfa cMeBaH 0 00, 10t x m= = cMeBaH 1 11 , 5t s x m= =
cMeBaH 2 22 , 10t s x m= = . cUrsresrsmIkareBlénclna
cMelIy
smIkareBl
smIkareBlTUeTAénclnaRtg;ERbRbYlesIμ
2
0 0
1
2
x at v t x= + +
-cMeBaH 1 1t t s= = smIkarkøayCa³
2
0 0
1
5 1 1 10 2 10 (1)
2
a v a v= × + × + ⇔ + = −
-cMeBaH 2 2t t s= = smIkarkøayCa³
2
0 0 0
1
10 2 2 10 0 (2)
2
a v a v a v= × + × + ⇔ + = ⇒ = −
0 20x m= −
0t =
0 10 /v m s=
0 i x

52
yk(2) CMnYskñúg(1) eyIg)an³
0 10 /v m s= − nig 2
10 /a m s=
dUcenH smIkareBlénclnaKW³
2
5 10 10x t t= − +
11-ekμgRsImñak;edIrtamTisBIekIt-lic ehIyRkabénbMlas;TIBIpÞHRtUv)anbgðajdUcrUb. cUrrkel,ÓnmFüm
rbs;nagEdlcMnucelIRkabbgðajBIel,ÓnxN³Rtg;cMnucnImYy².
cMelIy
el,ÓnmFümsUnü edaysarbMlas;TIsUnü.
el,ÓnxN³Rtg;cMnucnImYy²CaemKuNbnÞat;b:HExSekagRtg;cMnucenaH.
-cMeBaHcMnucA el,ÓnKW 40
6,7 /
6
m mn= sMedAeTATisxagekIt
-cMeBaHcMnucB el,Ón 40
13,33 /
3
m mn= sMedAeTATisxagekIt
-cMeBaHcMnucC el,Ón 65
13 /
5
m mn− = − sMedAeTATisxaglic
12-rfePIøgmYypøas;TIedayel,Ón ( )20 1 /t
v e m s−
= − Edl t KitCavinaTI. cUrkMNt;cMgaycr nigsMTuH
kñúgry³eBlbIvinaTI.
cMelIy
tamniymn½yel,Ón
( )t mn20
1412
11
C
96 1,5
B
A
0
40
20
10
21−
cMgayeTAekIt( )m

53
dx
v
dt
=
( )20 1 t
dx e dt−
⇒ = − edayeRCIserIs 00, 0t x= =
( )
3
0 0
20 1
x
t
dx e dt−
⇒ = −∫ ∫
( ) ( )3
20 3 0 1 79x e m−
⎡ ⎤= − − − =⎣ ⎦
nigsMTuH
3 2
3
20 0,995 /
t s
dv
a e m s
dt
−
=
= = =
13-Rkabel,Ón ( )v f x= rbs;kUnrfynþkMsanþelIpøÚvRtg;mYybgðajdUcrUb.cUrkMNt;sMTuHenARtg;³
50x m= nig 150x m= . cUrKUsRkabsMTuH ( )a f x=
cMelIy
eyIgEckclnarbs;kUnrfynþCaBIrvKÁ³
-vKÁTI1 enAcenøaHeBl 0 100t s< <
sMTuHenAcenøaHeBlenHCaemKuNR)ab;TisénbnÞat;
24
0,08 /
50
a m s= =
-vKÁTI1 enAcenøaHeBl 100 200s t s< <
sMTuHenAcenøaHeBlenHCaemKuNR)ab;TisénbnÞat;
24
0,08 /
50
a m s= − = −
-Rkab ( )a f x=
0
100 200 ( )x m
8
( / )v m s

54
14-smIkareBléncMnuccl½tmYy pøas;TIedayclnaRtg;ERbRbYlesμI tambeNþayG½kS )( xx′ KW³
342
+−= ttx , t > 3 .
k-rkkenSamel,Ón nigsMTuH.
x-KUsdüaRkamrbs;el,Ón.
K-etIcenøaeBlNa eTIbcl½tmanclnayWtesμI-sÞúHesμI?
xñatRtUvyktamRbB½n§SI.
cMelIy
k-kenSamel,Ón nigsMTuH
-kenSamel,Ón³ tamTMnak;TMng³
dt
dx
vx x == ¬cl½teFVIclnaEttamG½kS ¬ xx′ ¦
⇒ )34( 2
+−= tt
dt
d
vx = 2t – 4
-kenSamsMTuH
dt
dv
ax x
x == = )42( −t
dt
d
= 2 2
s/m
x-KUsdüaRkamel,Ón
eyIgman³ 42 −= tvx
ebI t = 0 ⇒ 4−=xv m/s
t = 1s ⇒ 2−=xv m/s
ebI 0=xv ⇒ 2=t s
xv (m/s)
42 −= tvx
O 1 2 t (s)
– 2
– 4
0
100
200
( )x m
0,08
2
( / )a m s
0,08−

55
1=xv → 5,2
2
5
==t s
K-etIcenøaHclnaeBlNaeTIbcl½tmanclnayWtesμI-sÞúHesμI?
- clnayWtesμI
Binitü³ xx va ⋅
iaa xx ⋅= , ivv xx ⋅=
⇒ iviava xxxx ⋅⋅⋅=⋅ = xx va ⋅ , 1=⋅ii
cMeBaHclnayWteyIg)an³
0<⋅ xx va ⇔ 0)42(2 <−t ⇒ 2<t s b¤ st 20 <≤
- clnasÞúH
eyIg)an³ 0>⋅ xx va
⇒ 0)42(2 >−t ⇒ st 2>
15-BinitüclnaRtg;ERbRbYlesμImYymansmIkar 2
2
1
atx = . bgðajfa kñúgcenøaHeBlCabnþbnÞab; ehIy
esμInwg θ cMgaycrbegáIt)ansVIútnBVnþmYyEdlmanersug 2
θar = .
cMelIy
bgðajfa 2
θar =
sikSaclnarbs;GgÁFatuenAelIG½kS )( xx′
eyIg)an³
2
00
2
1
atx =
2
11
2
1
atx = = 2
0 )(
2
1
θ+ta
2
22
2
1
atx = = 2
0 )2(
2
1
θ+ta
2
11
2
1
−− = nn atx = 2
])1([
2
1
0 θ−+ nta
0t i 1t 2t nt
x′ 1x 2x nx x

56
2
2
1
nn atx = = 2
0 )(
2
1
θnta +
eyIg)an³
122Δ xxx −= , 1Δ −−= nnn xxx , nnn xxx −= ++ 11Δ
KNna nxΔ
nxΔ = 2
0
2
0 ])1([
2
1
)(
2
1
θθ −+−+ ntanta
edayBnøateyIg)an³
)22(
2
1
Δ 0 θθθ −+= ntaxn
eyIg)an³
])1(22[
2
1
Δ 01 θθθ −++=+ ntaxn
⇒ nn xxr ΔΔ 1 −= +
⇔ )22(
2
1
])1(22[
2
1
00 θθθθθθ −+−−++= ntantar
= 2
θa
dUecñH kMenInnBVnþKW 2
θar = .
16-sikSaclnarbs;XøImYyEdleKecaleLIgelItambeNþayTrRtg; ehIyeRTt sßitelIbøg;eT. enAkñúg
tMruy );0( i clnaenHkMnt;eday ,2ia = ,60 iv −= m,50 =x 0≥t G½kS )( xx′ RsbnwgTr
ehIytMrg;cuHeRkam .
cMelIy
sikSaclnarbs; M
x′
B
clnacl½tM
O
i
A
x

57
cl½trbs; M mansMTuHefr 2
s/m2=a
cl½tpøas;TItambeNþay )( ixx′
dt
dv
a = ⇒ dtadv ⋅= ⇔ ∫∫ =
tv
v
dtdv
0
2
0
⇒ 62 −= tv
smIkarclna³ dt
dx
v = ⇒ vdtdx =
⇔ ∫∫ −=
tx
x
dttdx
0
)62(
0
⇒ ttxx 62
0 −=−
⇒ 0
2
6 xttx +−= , m50 =x
⇒ 562
+−= ttx
ebIcl½tqøgkat;Kl; O
⇒ x = 0 ⇔ 0562
=+− tt
⇒ 11 =t s, 52 =t s
Rtg; B el,Ónrbs;cl½tmantMélsUnü
0=v ⇔ 062 =−t ⇒ 33 =t s
eyIg)antMél t = 1s; 3s; 5 s
eyIg)antarag³
t 1 3 5
a + +
v – 0 +
x – 4
av – +
tamtaragsBaØa va ⋅ xagelIeyIg)an³
-ebI st 3< clnayWtrhUtdl; x = – 4 m .
-ebI t = 3s cl½tsßitenARtg;kMBUl B Rtg; 0=Bv .
-ebI st 3> clnasÞúH.

58
17-rfePøIgBIrmanRbEvgesμIKña 150m= rt;elIpøÚvRsbBIr mYyedayel,Ón km/h60 mYyeToteday
el,Ón 90 /km h .
k-rfePøIgrttamTisedApÞúyKña . etIGs;ry³eBlb:unμaneTIbvaTaMgBIreCosKñaput?
rkcMgayEdlrfePøIgnImYy²eFVI)an.
x-sMnYrdEdl kalNarfePñIgTaMgBIrrt;tamTisedAdUcKña.
cMelIy
eyIgykknÞúyrfePøIgTImYyCaKl;Gab;sIus
m150=
A: hkm60=Av
B: hkm90=Bv
k-TisedApÞúyKña
smIkartagcMgaycrCaGnuKmn_éneBlrbs;rfePøIgTImYy
-yk O CaKl;Gab;suIsRtg;knÞúyénrfePñIg A
+ cMeBaHrfepøIg A
hkm60=Av = sm
3
50
= efr CaclnaesμI
eyIg)ansmIkarclna³ OAAA xtvx +⋅= , 0=OAx
eyIg)an³ tvx AA ⋅= b¤ txA
3
50
=
+ cMeBaHrfePøIg B
OBBB xtvx +⋅= , enAeBlt = 0, m3002 ==BOx
edayrfePñIg B rt;tamTisedApÞúy
hkm90−=Bv = sm25−
⇒ 30025 +−= txB
A B
O O′

59
edIm,IeCosKñaputkalNa Ax = Bx
⇒ 30025
3
50
+−= tt ⇒ st 2,7=
-cMgayrbs;rfePøIgnImYy²
+ cMeBaHrfePøIg A: 2,7
3
50
×=Ax = 120 m
+ cMeBaHrfePøIg B: 1203002 −=− Ax = 180 m
x-krNITisedAdUcKña
ry³eBleCosKña
ykO CaKl;Gab;suIs smIkarclna³
-cMeBaH A
AOAA xtvx ′+⋅= , 0=′AOx ⇒ txA
3
50
=
-cMeBaH B
OBBB xtvx +⋅= , m300−=OBx
⇒ 30025 −= txB
eCosputKñakalNa³ BA xx =
30025
3
50
−=⋅ tt ⇒ st 36=
cMgaycr³
-cMeBaH A: 36
3
50
×=Ax = 600 m = Ad
x(m)
300
200
·
100
O 1 2 3 4 t (s)

60
-cMeBaH B: tvx BB ⋅= = 3625× = 900 m
18-rfynþGñkdMeNIrmYyRtUvQb;es¶ómeBlmanePøIgRkhm. enAeBlmanePøIgexov GñkebIkbrrfynþenH
begáInel,Ónkñúgry³eBl 8s EdlmansMTuH 2
m2 s . bnÞab;mkrfynþenHpøas;TIedayel,Ónefr. enAxN³
ecjdMeNIrrbs;va manrfynþdwkTMnijpøas;TIedayel,Ónefr 12m/s. etIGs;ry³eBlb:unμan nigcMgay
b:unμanBIePøIgsþúb eTIbrfynþGñkdMeNIreTATan;rfynþdwkTMnij?
cMelIy
ry³eBltamTan;nigcMgaycr
-cMgaycrrbs;rfynþGñkdMeNIr
rfynþenHmanclnaBIrKW sÞúHesμInigclnaesμI.
-cMgaycrcMeBaHclnasÞúHesμI
2
1
2
1
atx = , ( 00 =v , 00 =x )
-cMgaycrcMeBaHclnaesμI
tav
txd
M
M
⋅=
′⋅=2
⇒ ttad ′⋅⋅=2
cMgaysrub³ 21 ddx += , 11 xd =
⇒ ttaatx ′⋅⋅+= 2
2
1
tag θ Cary³eBlEdlrfynþdwkGñkdMeNIr tamTan;rfynþdwkTMnij³
tt ′+=θ , st 8=
⇒ 8−=′ θt
⇒ )8(8282
2
1 2
−⋅+⋅⋅= tx
= )8(1664 −+ t
smIkarclnarbs;rfynþdwkTMnij³
θ⋅′=′ vx = 12.θ
i 1x M
x′
1d
MO
2d
x

61
tamTan;³x = x′
⇔ θθ 12)8(1664 =−+ ⇒ s16=θ
cMgaycrEdleFVI)an³
m1921612 =×=′x
19-el,Ónrbs;rfynþmYyman 90 km/h eKeFVIeGayclnarbs;vayWtesμI ehIyQb;kñúgry³eBl 5s .
rkcMgaycrenAeBlEdleKcab;RhVaMgenH.
cMelIy
cMgaycrenAeBlEdlrfynþcr)ankñúgry³eBl5s
eyIgman³ smhkmv /25/900 ==
smIkarel,Ón³ 0vtav +⋅=
eBlrfynþQb;eK)an³ 00 =v
⇔ 00 =+⋅ vta ⇒ 20
sm/5
5
25
−=−=−=
t
v
a
tamTMnak;TMng³ axvv ⋅=− 20
2
⇒
a
vv
x
⋅
−
=
2
2
0
2
⇔
)5(2
)25(0 2
−×
−
=x = 62,5 m
20-rfynþmYyecjdMeNIredayKμanel,ÓnedImedayclnasÞúHesμI. enAeBlcr)an 500m rt;edayel,Ón
72 /km h . rkry³eBledIm,IeGayvaeTAdl;el,ÓnenH .
cMelIy
KNnary³eBl
eyIgman³ 00 =v , 500=x m, v = 72 km/h = 20 m/s
smIkarclna³ 0
2
2
1
vtax +⋅= = 2
2
1
ta ⋅
⇒
a
x
t
2
=
eday axvv 2
2
0
2
=− ⇒
x
v
x
vv
a
22
22
0
2
=
−
=

62
⇒ 50
20
50022
2
2
2
=
×
===
v
x
x
v
x
t s
21-smIkar)a:ra:Em:ténclnarbs;rUbFatuEdleKecaleTAkñúglMhKW³ x = 2t, y = 0, ttz 45 2
+−= .
cMgaycrKitCa (m), ry³eBl (s) ehIyG½kS )( kzz′ CaG½kSQr. eKyk 0≥t .
a). rksmIkarKnøg
b). kMNt;viucT½rel,Ónrbs;rUbFatu
k-kalNacMnucenHkat;tamkMBUlénKnøg
x-kalNacMnucenHkat;bøg;Edlman z = 0
K-enAxN³ t = 5s .
cMelIy
a). rksmIkarKnøg
eyIgrkGnuKmn_ )(xfz = .
⎪
⎪
⎩
⎪
⎪
⎨
⎧
+−=
=
=⇒=
)2(45
0
2
)1(2
2
ttz
y
x
ttx
(2) ⇒ ⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
−=
2
4
2
5
2
xx
z = xx 2
4
5 2
+−
dUecñHsmIkarKnøgKW z = xx 2
4
5 2
+− .
b). kMNt;viucT½rel,Ón
k-Rtg;kMBUlénKnøg C
eyIg)an³
0==
dt
dz
vz
viucT½rel,Ón³ kzixv +=
dt
dx
vx x == = st
dt
d
m2)2( =
z
0
0v
C
Cv
A
x
Av

63
dUecñH )sm(2iv =
x-Rtg; z = 0
⇒ 02
4
5 2
=+− xx ⇒ 00 =x ,
5
3
1 =x
viucT½rel,Ón³
kzixv +=
s
dt
dx
x m2== ,
dt
dz
z = = )45( 2
tt
dt
d
+− = 410 +− t
⇒ ktiv )410(2 +−+= (3)
ry³eBlRtg;Kl; O tx 2= ⇒ 0
2
0
0 ==
x
t
sm2=x , 410 +−= tz
t = 0 ⇒ sz m4=
eyIg)anRtg;Kl; O KW³ iiv 420 +=
m:UDul³ 22
0 42 +=v = sm52
ry³eBlenARtg; A EdlmanGab;suIs 5
8
1 =x
x = 2t ⇒
2
1
1
x
t = =
25
8
⋅
= 0,8s
⇒ 410 +−= tz = sm448,010 −=+×−
(3) ⇒ kivA 42 −=
m:UDul³ 22
)4(2 −+=Av = sm52
K-enAxN³ t = 0
410 +− tz ⇒ sm4=z
⇒ kiv 42 +=
m:UDul³ 22
42 +=v = sm52 .
segát³ eyIgeXIjfa cab;BIcMnuc AO → el,Ónrbs;cl½tfycuHbnþicmþg²
rhUtdl;el,ÓnKittamG½kS )( zkz′ sUnü.
cab;BIcMnuc AC → el,Ónrbs;cl½tekInbnþicmþg² rhUtdl;Rtg; A nig O ¬enA
elIG½kSEtmYy (ox) mantMélesμIKña.

64
22-cl½tmYypøas;TIrgnUvsMTuH .a k v= − .
k-cUrsMEdg v CaGnuKmn_eBl t .
x-cUrrk x CaGnuKmn_eBl t .
K-cUrsMEdg v CaGnuKmn_ x .
cMelIy
k- sMEdg v CaGnuKmn_eBl t
tamniymn½ysMTuH
.
dv dv
a k v
dt dt
= ⇒ = −
0 00
. ln .
v t
v t
dv v
k dt k t
v v=
⇒ = − ⇒ = −∫ ∫
dUcenH .
0
k t
v v e−
=
x-sMEdgx CaGnuKmn_eBl t
tamniymn½yel,Ón³
.
0
k tdx dx
v v e
dt dt
−
= ⇒ =
0
.
0
0 0
x t
k t
x t
dx v e dt−
= =
⇒ =∫ ∫
dUcenH ( ).0
1 k tv
x e
k
−
= −
K- sMEdg v CaGnuKmn_
tamTMnak;TMng³ dv dv dx dv
a v
dt dx dt dx
= = =
. .
dv
v k v dv k dx
dx
⇒ = − ⇔ = −
0 0
.
v x
v
dv k dx⇒ = −∫ ∫
dUcenH 0 .v v k x= −
23-cl½t M mYyecjdMeNIredayKμanel,ÓnedImBIcMnuc0 enAxN³ 0t = . cl½tenHpøas;TIenAelIG½kS
);( ixx′ edayclnasÞúHesμIEdlmanvíucT½rsMTuH 1a Edl 2
1 m/s8,1a = enAxN³ s11 =t .
v
0v
0
t
x
0v
k
0
t
v
0v
0
x0v
k

65
sMTuHbþÚrTisedAy:agrh½s ehIym:UDulkøayeTACa 2
2 m/s4,3=a .
rkel,Ón nigTisedArbs;cl½t enAxN³ st 22 = .
cMelIy
rkel,ÓnnigTItaMgrbs;M enAxN³ st 22 =
eyIgeRCIserIsenAxN³ t = 0, 00 =x , 00 =v
t1 t2
x′ O A B x
smIkarel,ÓnRtg; A
,011 vtavA += 00 =v
edayyk 0 CaKl;Gab;suIs
8,118,111 =×=×= tavA m/s
smIkarel,ÓnRtg; B
edayyk A CaKl;Gab;sIusRtUvnwgxN³ 0t =
AB vtav +⋅= eday 11212 =−=−= ttt s
⇒ 2,58,114,3 =+×=Bv m/s
-kMnt;TItaMgrbs;cl½tM
dMNak;kalTImYy )1( 1 st =
smIkarclnaenAxN³ t = 0 ⇒
⎩
⎨
⎧
=
=
0
0
0
0
v
x
⇒
2
11
2
1
taxA =
dMNak;kalTIBIr enAxN³ 0t = cl½tenARtg;cMnuc A edayel,Ón Av ³
⇒ AAB xtvtax ++= 2
2
2
1
edayyk A CaKl;Gab;suIs ⇒ 0=Ax
⇒ tvtax AB += 2
2
2
1
cMgaycrEdlcl½t)anBI 0 dl; A
tvtataxxx ABA ++=+= 2
2
2
1
2
1
2
1

66
⇔ 18,114,3
2
1
18,1
2
1 22
×+××+××=x = 4,3 m
24-rfynþmYyecjdMeNIredayclnaRtg;sÞúHesμI ehIyeTAdl;el,Ón 90km kñúgry³eBl 25s .
KNnasMTuH nigcMgaycrkñúgry³eBl25s enH.
cMelIy
k- KNnasMTuHrbs;rfynþ
tamTMnak;TMng³ 0vatv += enAxN³ 0,t = ,00 =v 00 =x
⇒ atv = ⇒
t
v
a = , 90=v km/h = 25 km/s, t = 25 s
⇒ 2
sm/1
25
25
==a
x-KNnacMgaycr
2
2
1
tax ⋅= = 2
251
2
1
×× = 312,5 m
b¤mü:ageTot axvv 20
2
=− , 00 =v ⇒ 5,312
2
25
2
22
==
×
=
a
v
x m
25-sMTuHéncMnucA RtUv)ankMnt;edayTMnak;TMng ( )2
200 1 .a x k x= + Edl a KitCa 2
/m s nigx KitCa
( )m ehIyk CacMnYnefr. edaydwgfael,ÓnénA KW 2,5 /m s enAeBl 0x = nig 5 /m s enAeBl
0,15x m= . cUrkMNt;tMélk .
cMelIy
kMNt;tMélk
tamniymn½ysMTuH
dv
a
dt
=
rW dv dv dx dv
a v
dt dx dt dx
= = =
eyIg)an³
( )
( )
2
0,155
2
2,5 0
200 1 .
200 1 .
dv
v x k x
dx
vdv x k x dx
= +
⇒ = +∫ ∫
D
x
A
E
CB

67
5 0,152 2 4
2,5 0
200
2 2 4
v x x
k
⎡ ⎤ ⎡ ⎤
⇒ = +⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
( )
4
2 2 2 0,15
2 5 2,5 100 0,15 0
2
k
⎛ ⎞
− = + −⎜ ⎟
⎝ ⎠
2 2
37,5 2,25 0,02531
1392,73
k
k m s− −
⇔ = + ×
⇒ =
26-ebIsMTuHcMnucA eGayeday 3
200 3200a x x= + Edl a KitCa 2
/m s nigx KitCa( )m . edaydwgfa
el,ÓnénA KW 2,5 /m s nig 0x = enAeBl 0t = cUrkMNt;el,ÓnnigTItaMgéncMnucA enAeBl 0,05t s= .
cMelIy
tamniymn½ysMTuH
3
200 3200
dv
a x x
dt
= = +
rW dv
a v
dx
=
( )3
200 3200v dv x x dx= +
( )3
2,5 0
200 3200
v x
vdv x x dx⇒ = +∫ ∫
2
2 4
0
2,5
100 800
2
v
xv
x x
⎡ ⎤
⎡ ⎤= +⎢ ⎥ ⎣ ⎦
⎣ ⎦
( )2 2 41
6,25 100 800
2
v x x− = +
2 4
200 800 6,25v x x⇒ = + +
eday dx
v
dt
=
0,05
2 4
0 0200 800 6,25
x
dx
dt
x x
⇒ =
+ +
∫ ∫
27-PaKli¥tmYyeFVIclnatampøÚvkMnt;eday)a:ra:bUl 2
0,5y x= . ebIkMub:Ysg;énvuicT½rel,ÓntamTisx KW
5 ( / )xv t m s= Edlt KitCavinaTI. cUrKNna cMgayBIPaKli¥teTAKl;tMruy0 nigtMélsMTuH enAeBl 1t s= .
enA 0, 0, 0t x y= = = .

68
cMelIy
-KNnacMgay 2 2
0M x y= +
eyIgBinitü
0 0
5
x t
x
dx
v dx t dt
dt
= ⇒ =∫ ∫
25
2
x t⇒ =
nig dy dx
x
dt dt
=
25
5
2
dy
t t
dt
⇒ = ×
3
0 0
25
2
y t
dy t dt⇒ =∫ ∫
4
25
2 4
t
y⇒ = ×
cMeBaH 1 , 2,5 , 3,125t s x m y m= = =
( ) ( )
2 2
0 2,5 3,125 4M m= + =
-sMTuH 2 2
x ya a a= +
eday 275
0 ,
2
x ya a t= =
eBaH 1 , 0, 37,5 /x yt s a a m s= = =
( )
22 2
0 37,5 37,5 /a m s= + =
28-cMnucrUbFatumYyeFVIclnaenAkñúgbøg;(0 )xy edayel,Ón , ,v i x jα β α β= + CacMnYnefr. enAxN³
edImeBlcl½tsißtenARtg;cMnuc 0 00, 0x y= = .
k-cUrsresrsmIkarKnøgrbs;cl½t ( )y f x=
x-cUrkMNt;kaMkMeNagénKnøgCaGnuKmn_énx .
cMelIy
k-smIkarKnøg
eyIgman³ v i x j xi y jα β= + = +
0 x x
My
y
2
0,5y x=

69
dx
x
dt
α⇒ = = nig dy
y x
dt
β= =
müa:geTot dy dy dx dy
x
dt dx dt dx
β α= ⇔ =
0 0
y x
dy xdx
β
α
⇒ =∫ ∫
dUcenH 2
2
y x
β
α
= ¬Knøgrbs;cl½tmanrgCa)a:ra:bUl¦.
x-kaMkMeNag ( )xρ ρ=
tamrUbmnþkaMkMeNagénExSekag
3
2 2
2
2
1
dy
dx
d y
dx
ρ
⎡ ⎤⎛ ⎞
+⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦=
eday
2
2
2
2
dy d y
y x x
dx dx
β β β
α α α
= ⇒ = ⇒ =
3
2 2
1 x
β
α
ρ
β
α
⎡ ⎤⎛ ⎞
+⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦⇒ =
⎛ ⎞
⎜ ⎟
⎝ ⎠
rW
3
2 2
1 x
α β
ρ
β α
⎡ ⎤⎛ ⎞
= +⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦
29-cMnucrUbFatumYyeFVIclnaelIFñÚrgVg;mankaMR . el,ÓnvaGaRs½yeTAnwgcMgaycr sMEdgedayc,ab;³
,v k S k= CacMnYnefr nig S CaGab;sIusekag. cUrkMNt;mMuϕ pMÁúeLIgrvagviucT½rel,Ón nigsMTuHCaGnuKmn_
énS .
cMelIy
eRCIserIseKaleRbeNmksikSa( , , )M u n
-kenSamel,Ón³ v vu k S u= =
-kenSamsMTuH³ t na a u a n= +
Edl
2
22
t
dv d S S k
a k k
dt dt S
= = = =

70
nig
2 2
n
v k S
a
R R
= =
2 2
2
k k S
a u n
R
⇒ = +
mMu ( ),a vϕ = edayeRbIplKuNsáaElrvagvuicT½rTaMgBIr.
cos cos
a v
a v av
av
ϕ ϕ= ⇒ =
( )
2 2
2 2 2 22 2
2
cos
4
2
k k S
k S u u n
R R
R Sk k S
k S
R
ϕ
⎛ ⎞
+⎜ ⎟
⎝ ⎠= =
+⎛ ⎞ ⎛ ⎞
+ ×⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
1
2 2
cos
4
R
R S
ϕ − ⎛ ⎞
⇒ = ⎜ ⎟
+⎝ ⎠
30-enAxN³mYy TItaMgedkén)aLúgGakasFatumYy dUcrUb kMNt;eday 9x t= KitCaEm:t. ebIsmIkar
cMNr ¬pøÚv¦
2
30
x
y = . cUrkMNt;³
k-cMgayén)aLúgBIsßanIy_ A enAeBl 2t s= .
x-GaMgtg;sIuet nigTisrbs;el,Ón enAeBl 2t s= .
K- GaMgtg;sIuet nigTisrbs;sMTuH enAeBl 2t s= .
cMeelIy
k- cMgayén)aLúgBIsßanIy_ A enAeBl 2t s=
enAeBl 2 18 10,8t s x m y m= ⇒ = ⇒ =
bnÞat;Rtg;BI A B→ KW³
( )
22
18 10,8 21r m= + =
x-GaMgtg;sIuet nigTisrbs;el,Ón enAeBl 2t s=
kMub:Usg;el,Ón³
9 /
2 .
10,8 /
30
x
y
dx
v x m s
dt
dy x x
v y m s
dt
⎧
= = =⎪⎪
⎨
⎪ = = = =
⎪⎩
y
A
B
x
2
30
x
y =

71
2 2
14,1 /x yv v v m s⇒ = + =
Tisrbs;el,ÓneFobnwgG½kSedk³
0
tan 50,2y
x
v
v
θ θ= ⇒ =
K- GaMgtg;sIuet nigTisrbs;sMTuH enAeBl 2t s=
kMub:Usg;sMTuH
2
2
0
2. 2. .
5,4 /
30 30
dx
x
dt
dy x x x
y m s
dt
⎧
= =⎪⎪
⎨
⎪ = = + =
⎪⎩
2 2 2
5,4 /a x y m s⇒ = + =
Tisrbs;sMTuHeFobnwgG½kSedk³
0
tan 90
y
x
α α= ⇒ =
31-enAeBlGñkelgsIÁmañk;mkdl;cMnucA tampøÚv)a:ra:bUldUcrUb Kat;manel,Ón6 /m s EdlekIn 2
2 /m s .
cUrkMNt;TisedAénel,Ón ehIyTisedA nigTMhMénsMTuHenAxN³enaH. minKitTMhMénGñkelgsIÁkñúgkar
KNna.
A
5m
x
Av
y
10m
21
20
y x=
θ

72
cMelIy
-viucT½rel,Ón
viucT½rel,ÓnCanic©kalb:HeTAnwgKnøgRKb;xN³.
eyIgman 21
0,1
20
dy
y x x
dx
= ⇒ = dUcenH
10
1
x
dy
dx =
=
enHCaemKuNR)ab;TisénbnÞat;b:HRtg;cMnucA .
dUcenHel,ÓnmanTissißtenAelIbnÞat;enH Edl 0
tan 1 45θ θ= ⇒ =
dUcenHviucT½rel,ÓnsMEdgenAkñúgedkat³
cos sin 3 2 3 2A A Bv v i v j i jθ θ= − − = − −
-vuicT½rsMTuHnigTMhMva
edayeRCIserIseKaleRbeN ( ), ,A u n
viucT½rsMTuH t na a u a n= +
eday t
dv
a
dt
=
2 2
,
dx dy
v
dt dt
⎛ ⎞ ⎛ ⎞
= +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
ehIy 0,1
dy dx
x
dt dt
=
2
2 /ta m s⇒ =
nig
2
n
v
a
ρ
= Edl
3
2 2
2
2
1
28,28
dy
dx
m
d y
dx
ρ
⎡ ⎤⎛ ⎞
+⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦= =
2
26
1,732 /
28,28
na m s= =
{ } 2
2 28,28 /a u n m s⇒ = +
nigGaMgtg;sIuet³
2 2 2
2,37 /t na a a m s= + =
32-enAxN³ t sMTuHmuMrbs; Rotor énm:UT½rmYymantMél 2
rad40 s . enAeBlenaHel,ÓnmuMmantMél
srad30 . kMNt;el,Ón v nigsMTuH a rbs;cMnucM én Rotor EdlsßitenAcMgay 10cm BIG½kS.

73
cMelIy
kMNt; v nig a rbs;cMnucM
muMcr )(tαα =
el,ÓnmuM α
α
ω ==
dt
d
sMTuHmuM α
αω
β =⎟
⎠
⎞
⎜
⎝
⎛
==
dt
d
dt
d
dt
d
ehIy αRS = ⇒
R
S
=α
dt
R
S
d
dt
d
⎟
⎠
⎞
⎜
⎝
⎛
==
α
ω =
R
v
R
S
=
⇒ Rv ω= , R = 100 cm = 0,1 m, srad30=ω
⇒ 1,030×=v = sm3
sMTuHRbEvg
αβ = =
R
a
R
s
dt
sd
R
==⋅
1
⇒ Ra ⋅= β , 2
srad40=β
⇒ 2
m41,040 sa =×=
33-PaKl¥itmYypøas;TIenAelIrgVg;tamc,ab;mYyEdlman tt 34 2
+=θ , θ KitCa ( )rad nig t KitCa( )s .
k-KNnael,ÓnmuMnigsMTuHmuMrbs;PaKl¥itkñúgry³eBl 4s.
x-ebIkaMénKnøgenHmanRbEvg 1,6m KNnael,Ón v nigsMTuH a enAxN³dUcKñaenH.
cMelIy
k-KNnael,ÓnmuMnigsMTuHmuMrbs;PaKl¥itkñúgry³eBl 4s
eKeGay tt 34 2
+=θ .
-el,ÓnmMu³ dt
dθ
ω = = )34( 2
tt
dt
d
+ = 38 +t
eday 4=t s ⇒ srad35348 =+×=ω
-sMTuHmMu³ )38( +== t
dt
d
dt
dω
β = 8
⇒ 2
srad8=β
M
R S
α
O A

74
x-el,ÓnRbEvg v nigsMTuHRbEvg a
-el,ÓnRbEvg³ smRv /566,135 =×== ω
-sMTuHRbEvg³ 2
/8,1286,1 smRa =×== β
34-enAkñúgtMruyedkat );;,(O kji cMnuc M mYypøas;TIedayclnavg;Edlmanp©it0 nigkaMR eday
el,ÓnmMu kω enAkñúgtMruy ¬bøg;¦ ( 0 )x y .
k-bgðajfakUGredaenrbs; M Gacsresr³
⎩
⎨
⎧
=
=
tRy
tRx
ω
ω
sin
cos
EdleKnWgbBa¢ak;edImeBl.
x-rksmIkarnKnøg nigkMub:Usg;énvíucT½rrbs;cMnucenH enAelIG½kS ),( ixx′ nig ),( jyy′ .
K-Tajrkm:UDulénel,ÓnmMuenHCaGnuKmn_én R nig ω .
X-KNnakMub:Usg;pÁúMb:HnigpÁMúEkgrbs;sMTuHenAxN³nImYy².
cMelIy
k-bgðajfacMnuc M
tRy
tRx
ω
ω
sin
cos
=
=
eyIgeRCIserIs A CaKl;Gab;suIsRtUvnwgxN³ 0t = .
tag 0 , 0x H y P= = eyIg)an³
OM
OH
cos =θ ⇒ θcosOM=OH
⇒ θcosRx =
OM
OP
OM
MH
sin ==θ ⇒ θsinOMOP =
⇒ θsinRy =
θCamMuekosrbs; M ³
enAxN³t = 0 ⇒ θ = 0
enAxN³ t mMuenHmantMél tωθ = .
dUecñHeyIg)an³ M
tRy
tRx
ω
ω
sin
cos
=
=
.
P M
R
j θ A
O i H x

75
x-rksmIkarKnøg - rkkMub:Usg;énvíucT½rTItaMg OM
viucT½r OM ³
jyixOPOHOM ⋅+⋅=+= = jtRitR ωω sincos +
eday tRx ωcos= ⇔ tRx ω222
cos=
tRy ωsin= ⇔ tRy ω222
sin=
⇒ 22222
sincos RtRtRyx =+=+ ωω
⇔ 222
Ryx =+ CasmIkarrgVg;Edlmanp©it0 kaM R .
K-m:UDulénel,ÓnmMu
M
tRy
tRx
ω
ω
sin
cos
=
=
⇒
dt
OMd
v = = )sincos( jtRitR
dt
d
ωω +
= jtRitR ωωω cossin +−
⇔ 2
v = 22
)cos()sin( jtRitR ωωω +−
= tRtR ωωωω 222222
cossin +
= )cos(sin 2222
ttR ωωω + = ωR
⇒ ωRv =
X-KNnakuMb:Usg;pÁúMb:HnigpÁMúEkgrbs;sMTuHenAxN³nImYy²
-kMub:Usg;pÁúMb:H³
0
)(
===
dt
Rd
dt
vd
at
ω
eRBaH ωR = efr
-kMub:Usg;pÁúMEkg³
2
2
22
)(
ω
ω
R
R
R
R
v
an ===
⇒ nRan ⋅= ω
35-viucT½rel,Ónénpg;mYymancMeBaHkuMb:Usg;edkat tx sin2= ; ty cos= ; yx, KitCa m/s ehIy t
KitCa s .
k-cUrsMEdgkUG½redaenCaGnuKm n_eBledaydwgfa enAxN³ 0=t enAelI Ox Rtg; m2=x .
x-edaybM)at;eBl t rvagkenSamkUGredaen cUrsresrsmIkaredkaténKnøg.

76
cUrR)ab;RbePTclna.
K-KNnakuMb:Usg;énvuicT½rsuMTuH. cUrbgðajfa eBlvaqøgkat;edaykUGredaenedIm ehIybgðajfa
m:UDulrbs;vasmmaRteTAnwgcMgayenAKl;.
cMelIy
k- kMnt;kUGredaen
eyIgman³ ty
tx
v
cos
sin2
=
=
⇒
y
x
OM
eday dt
dx
x =
⇒ tdtdx sin2=
cMeBaH m2;0 == xt
⇒ ∫∫ =
tx
dttdx
02
sin2
⇒ 2cos2 0
+−=
t
tx
⇒ 4cos2 +−= tx
ehIy dt
dy
y = ⇒ tdtdy cos=
cMeBaH 0=t ; 0=y
⇒ ∫∫ =
ty
dttdy
00
cos ⇒ ty sin=
⇒
ty
tx
OM
sin
4cos2
=
+−=
x-smIkarKnøg
eyIgman ³
4cos2 +−= tx ⇒
2
2cos
x
t −= ⇒
2
2
2
2cos ⎟
⎠
⎞
⎜
⎝
⎛
−=
x
t
ehIy ty sin= ⇒ ty 22
sin=
⇒
2
222
2
2sincos ⎟
⎠
⎞
⎜
⎝
⎛
−+=+
x
ytt
⇔ 1
2
2
2
2
=⎟
⎠
⎞
⎜
⎝
⎛
−+
x
y ⇔ 4)4(4 22
=−+ xy

77
⇔ 4)4()2( 22
=−+ xy
tag³ yY 2= ⇒ ( ) 44
22
=−+ xY
CasmIkarrgVg;mankaM m2=R p©it ( )4;0A .
clna Caclnavg; .
x- KNnakMub:Usg;sMTuH
-enAkñúgtMruyedkat³
yx aaa +=
Et dt
xd
ax =
⇒ tax cos2=
ehIy t
dt
yd
ay sin−==
⇒
t
t
a
sin
cos2
−
-enAkñúgtMruyeRbeNénclnavg;³
tn aaa +=
eday³ R
v
an
2
= ; 222
yxv +=
= tt 22
cossin4 +
= 1sin3 2
+t
ehIy dt
dv
at =
( ) ( )2
1
22
1sin31sin3 +=+= t
dt
d
t
dt
d
at
2
1
2
2cos5
⎟
⎠
⎞
⎜
⎝
⎛ −
=
t
dt
d
2
1
2
2cos5
2
2cos5
−
⎟
⎠
⎞
⎜
⎝
⎛ −
⋅
′
⎟
⎠
⎞
⎜
⎝
⎛ −
=
tt
2
1
2
2cos5
2sin
−
⎟
⎠
⎞
⎜
⎝
⎛ −
⋅=
t
t
⇒ u
t
tn
t
a ⋅⎟
⎠
⎞
⎜
⎝
⎛ −
+⋅
+
=
−
2
1
2
2
2cos5
2sin
2
1sin3
M
v
x
R
A
2 4
y
x
y
0

78
36-)øaTInéneGLicRtUpUneFVIclna)anknøHCMumunnwgdl;el,ÓnmMu 0
45 CMukñúg 1mn. eKcat;Tukfa sMTuHmMu θ
efrkñúgry³eBlknøHCuMenH.
k-KNnary³eBlkñúgdMnak;kalvakMBugsÞúH rYcKNnatMél θ .
x-cUrkMnt;kuMb:Usg;b:H nigEkgénviucT½rsMTuHéncMnucmYysßitenAcMgay10cm BIG½kSrgVileBlfaseFVI
)an 4
1
Cu¿ .
cMelIy
k- KNnary³eBlvakñúgeFVI)anknøHCu¿
eyIgmansMTuHmuM θ efr ehIy dt
θd
=θ
⇒ dtθθd =
cMeBaH 0;0 == θt
45;0 == θt CuM /mm
rd/sπ
30
45
/
60
45
⋅== strθ
⇒ ∫∫ =
t
dtθθd
00
θ
⇒ .tθθ =
ehIy 0;0; === θt
dt
dθ
θ
⇒ ∫∫ ⋅=
tθ
dttθdθ
00
⇒ 2
2
1
tθθ ⋅=
dUcenHeyIg)an³ rd/s
30
45
,rd ππ == θθ ⇒
⎪
⎪
⎩
⎪⎪
⎨
⎧
⋅=
⋅=
)2(t
2
1
π
)1(tπ
30
45
2
θ
θ
eyIgEck )2(
)1(
⇒
2
2
1
30
45
tθ
tθ
⋅
⋅
=
π
π
⇒
t
2
5,1 = ⇒ st
5,1
2
= ⇔ st
4
3
=
-KNna θ ³ (1) ⇒
3
4
30
45
×=θπ
M
A
O

79
⇒ 45,4 ⋅= θπ ⇒ 2
/
4
5,4
srdθ π=
x-sMTuH
nt aaa += eday dt
dv
at = ehIy Rθv ⋅=
⇒ θR
dt
Rθd
at =
⋅
=
2
cm/s
4
45
4
5,4
10 ππ =×=ta
Rθ
R
v
an ⋅== 2
2
37-cMnuccl½t M mYyeFIVclnaekag EdlmansmIkarkUGredaenb:UElbøg; θRr cos2= edayel,Ón mMu θ
efrKW 0ω .
k-kñúgkUGredaenb:UElbøg; KNnakMub:Usg;el,Ón nigsMTuH rYcKNna
x-C Cap©íténKnøgvg;rbs;clna bgðajfa sMTuHkUlIenEG‘nwg CM rYcbMNkRsaytamrUb.
cMelIy
kalNaGgÁFatucrCaclnakñúgkUGredaensIuLaMgedayGvtþmanclnatamG½kS
);( kO eBlenaH clnaenHkøayCaclnakñúgkUGredaenb:UElbøg;.
k-Knøgrbs; M CargVg;EdlmankaM R p©ít C kat;tam O .
2θ
θ
θ
M
uθ
ru
r
C
xθ
y
0 i
j

80
-kuMb:Usg;énel,Ón v ³
θθθ sin2)cos2( Rr
dt
d
dt
dr
vr −===
⇒ θRvr sin2 0ω−=
θRvrθrv cos2 0θ0θ ωω =⇒==
m:UDul³ 2
θ
2
vvv r +=
⇒ 2
0
2
)cos2()cos2( θωθ RθR +− 02 ωR=
-kuMb:Usg;énsMTuH a ³
θRθRrrar cos2cos2 2
0
2
0
2
ωωθ −−=−=
⇒ θRar cos4 2
0ω−=
θθ rra += 2θ eday 0=θ
⇒ θωωθω sin4)cos2(2 2
000θ RRa −=⋅−=
m:UDul³ 2
θ
2
aar +
dUecñH 2
04 ωRa = .
x-CM enAkñúgeKal );( θuur
θθθ uRuRCM r sincos +=
edÍm,Ibgðajfa CM nig a kUlIenEG‘Kña eyIgBinitüemIl³
plKuN³ aCM ∧
θθθ )()θsincos( uauauRuR rrr +∧+θ
= 0)]cos4)((sinsin4)(cos[(
2
0
2
0 =−−− kRθθRR θωωθ
dUcenH a kUlIenEG‘nwg CM .
-bMNkRsaytamrUb³
( θ2), =CMi , M eFVIclnaedayel,ÓnmuM kka 02ωθ =
el,Ónrbs;M ³ CMkv ∧= 02ω
el,ÓnvamantMélefr dUcenHvaeFVIclnavg;esμIEdlmansMTuHcUlp©ít³
2
0
2
0
22
4
4
ω
ω
R
R
R
R
v
a ===

81
39-cMnucrUbFatuA pøas;TIelIrgVg;mankaMR pÞúyTisedAclnaRTnicnaLika. Gab;sIuskMenagéncMnucrUbFatu
ERbRbYltamc,ab;³ . ,S k t k= cMnYnefr. cUrsresrsmIkar)a:ra:Em:teFobeTAnwgtMruy0xy EdlKl;tMruy
enARtg;pi©tvgVg;. ebIG½kS0x kat;TItaMgedIméncMnucA .
cMelIy
smIkar)a:ra:Em:t ( ) , ( )x t y t
kUGredaencMnuc ( )( ), ( )M x t y t
Edl .cos , .sinx R y Rθ θ= =
ehIy .
( )
S k t
rad
R R
θ = =
dUcenH eyIg)an³
( ) .cos
( ) .sin
k
x t R t
R
k
y t R t
R
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
40-kñúgry³eBl 20sτ = el,Ónrbs;cMnucrUbFatumYyEdlpøas;TItamFñÚrgVg;mankaM 200R m= ERbRbYlBI
15 /m s eTA12 /m s . edaysnμtfa m:UDulénsMTuHpMÁúb:HkñúgcenøaHeBlenHsmamaRteTAnwgkaerénel,Ón.
KNnacMgaycrrbs;cMnucrUbFatukñúgry³eBl10s .
cMelIy
KNnacMgaycrS kñúgry³eBl10s
eyIgman³ 2
,ta k v k= CaefrsmamaRt
tamniymn½ysMTuHpMÁúb:H
2
t
dv dv
a k v
dt dt
= ⇒ =
2
15 0
v t
v
dv
k dt
v=
⇒ =∫ ∫
1 1
15
k t
v
⇒ − =
1 1 1 15
115 1 15
15
k t v
v k tk t
⇒ = − ⇒ = =
−−
, 0A t =
x
θ
R
M
0
y
v
S
x
y

82
cMeBaH 12 /v m s= nig 20t sτ= =
4
8,33 10k −
⇒ = − ×
dUcenH 15
1 0,0125
v
t
=
+
müa:geTot 15
1 0,0125
dS dS
v
dt t dt
= ⇔ =
+
10
0 0
15
1 0,0125
S t
dt
dS
t
=
⇒ =
+∫ ∫
( )
10
0
15
ln 1 0,0125
0,0125
S⇒ = +⎡ ⎤⎣ ⎦
15
ln1,125 141,34
0,0125
S m⇒ = =
41-cl½tmYyeFVIclnaRtg;sIunuysUGIut Gab;sIusrbs;vaRtUv)ankMnt;CaGnuKmn_eBl t ³ tAx ωsin=
xYbénclnaKW 6s . cMeBaH s0,5=t el,Ónrbs;cl½t cm/sπ+=v .
k-KNna ω nigA .
x-KNnasMTuHéncl½tkalNavasßitenARtg;0,5cm BITItaMglMnwg.
K-KNnael,ÓnvaRtg;cMnucenH.
cMelIy
]bmacl½tpøas;TItambeNþay (x'x)
k-KNna ω nig A
eyIgmansmIkarclna³ ωtsinAx =
eday T
π
ω = ; s6T = ⇒ sradω /
3
π
6
π2
==
-KNna A
⇒ tAω
dt
dx
v ωcos==
enAxN³ s0,5=t ; cm/sπ=v
⇒ 0,5
3
cos
3
×××=
ππ
Aπ

83
⇒ cm46,0
3
0,5
cos
3
==
π
A
x-KNnasMTuHRtg; cm0,5=x
ωtAx sin= ⇒ ωtAωx cos=
⇒ xωωtAωx 22
sin −=−=
⇒ 2
2
cm/s0,550,5
3
π
−=×⎟
⎠
⎞
⎜
⎝
⎛
−=x
K-KNnael,Ón
eday ωtAωx sin=
⇒ 2
2
2
in
A
x
ωts = (1)
ehIy tAωxv ωcos==
⇒ 22
2
2
cos
ωA
v
ωt = (2)
bUk (1) nig (2)
⇒ 122
2
2
2
=+
ωA
v
A
x
⇒ 2
2
22
2
1
A
x
ωA
v
−= ⇒ 2
2
1
A
x
Av −±= ω
⇒ ( ) ( )222222
105,01046,3
3
−−
⋅−⋅±=−±=
π
ω xAv
dUcenH cm/s3,59±=v
42-PaKl¥itmYydMbUgenAnwgRtg;cMnucmanGab;sIus 0x pøas;TItambeNþaybnÞat;edaysMTuH³
)3( xka −=
cUrkMnt;el,Óncl½tCaGnuKmn_énGab;sIus .
cMelIy
clnaeFVIclnatambnÞat;(x'x) edaysMTuH³ )3( xka −=
cm/s59,3±=v
x′ 0 x

84
eday dt
dv
a = ⇒ ( )xk
dt
dv
−= 3
KuNGgÁTaMgBIrnwg dx
⇒ ( )dxxk
dt
dv
dx −=⋅ 3
⇔ ( )dxxkdvv −= 3
⇒ ( )∫∫ −=
x
x
v
o
dxxkvdv 3
0
⇒
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−=
2
2
1
2
1
32
xxk
x
xo
v
⇔ ⎟
⎠
⎞
⎜
⎝
⎛
+−−= 2
00
22
2
1
3
2
1
3
2
1
xxxxkv
⇒ ( )2
00
2
66 xxxxkv +−−=
43-enAkñúgtMruyGrtUNrem (Ox, Oy) kUGredaenéncl½tKW³ ttx cossin −= ; tty cossin +=
a). cUreGaysmIkaredkaténclna nigRbePTKnøg.
b). ]bmafa KnøgTisedAsßitenAkñúgTisedARtIekaNmaRt Kl;énFñÚRtYtsIuKñanwgedImeBl.
k- KNnael,Ón v nigel,ÓnmMu.
x-KNnasmIkareBl.
K-KNnasMTuH.
cMelIy
a). smIkaredkat nigRbePTKnøg
eyIgman³
ttx cossin −=
tty cossin +=
elIkCakaer eyIg)an³
· ( )2
cossin ttx2
−=
ttttttx cossin21coscossin2sin 222
−=+−=
· ( )22
cossin tty +=
tttttty cossin21coscossin2sin 222
+=++=

85
⇒ 222
=+ yx CasmIkarrgVg;mankaM 2=R dUecñHKnøgrbs;vaCargVg; .
b). k-KNnael,Ón v nigel,ÓnmMu
eyIg)an³
tt
dt
dx
x sincos +==
tt
dt
dy
y sincos −==
⇒ m/s222
=+= yxv
el,ÓnmMu³ rd/s1
2
2
===
R
v
ω
x-smIkareBl
edayel,Ónefr³
dt
ds
v = ⇒ vdtds =
⇒ dtds
ts
∫∫ =
00
2 ⇒ tS ⋅= 2
K- KNnasMTuH
xtt
dt
xd
x −=+−== cossin
ytt
dt
yd
y −=−−== cossin
( ) ( ) 22222
ms2 −
=−+−=+= yxyxa
b¤Gacrktam³
( ) 2
22
m/s2
2
2
====
R
v
aa n
44-cl½t M mYyeFVIclnaRtg;sIunuysUGIutelIG½kS (x'ox). TItaMgcugeFobcMnuc0manGab;sIuserog 4cm nig
+ 4cm . xYbénclnaKW s4=T .
a). edaydwgfa enAxN³ 0=t cl½t M enHpøas;TIedayel,ÓnmYymanTisedAviC¢manenARtg;
cMnucM0 manGab;sIus cm20 =x . cUrsresrsmIkarclna.
b). etIry³eBlb:unμancl½tM qøgkat;cMnuc0elIkTImYy.

86
c). cUreGayTMnak;TMngrvagv nig x; a nig x .
d). KNnael,Ón nigsMTuHéncl½teBlvasßitenAcMnucM0 manGab;sIus+ 2 cm .
cMelIy
a). cl½teFVIclnaRtg;sIunuysUGIuutdUcenHsmIkarclnamanrag³
( )ϕω += txx m sin
eday cm4+=mx ¬GMBøITut¦
ehIy rd/s
24
2
T
π2 ππ
ω ===
enAxN: 0=t ; cm20 +=x
⇒ ⎟
⎠
⎞
⎜
⎝
⎛
+×=+ ϕ
π
0
2
sin42
⇒
2
1
sin =ϕ ⇒
6
π
ϕ =
dUcenH ⎟
⎠
⎞
⎜
⎝
⎛
+=
62
sin4
ππ
tx
b). ry³eBlcl½tqøgkat;0elIkTImYy luHRtaEt³ 0=x
⇒ 0
62
sin4 =⎟
⎠
⎞
⎜
⎝
⎛
+⋅
ππ
t
⇔ 0
62
sin =⎟
⎠
⎞
⎜
⎝
⎛
+
ππ
t
⇔ π
ππ
=+⋅
62
t
⇔ s66,1
3
5
==t
c). eyIgmansmIkarclna³
⎟
⎠
⎞
⎜
⎝
⎛
+=
62
sin4
ππ
tx
⇒ ⎟
⎠
⎞
⎜
⎝
⎛
+==
62
cos2
ππ
π t
dt
dx
v
0v
A′ O
x′ – 4 + 2 + 4 x

87
⇒
⎪
⎪
⎩
⎪
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
+=
⎟
⎠
⎞
⎜
⎝
⎛
+=
62
cos
2
62
sin
4
ππ
π
ππ
t
v
t
x
elIkCakaer eyIg)an³
⎟
⎠
⎞
⎜
⎝
⎛
+=
62
sin
16
2
2
ππ
t
x
⎟
⎠
⎞
⎜
⎝
⎛
+=
62
cos
4
2
2
22
ππ
π
v
dUcenH 1
416 2
22
=+
π
vx
ehIy ⎟
⎠
⎞
⎜
⎝
⎛
+−==
62
sin2 ππ
π t
dt
dv
a
⇒ ⎟
⎠
⎞
⎜
⎝
⎛
+×−=
62
sin4
4
2
πππ
ta
dUcenH xa
4
2
π
−=
d). KNnael,Ón nigsMTuH
cMeBaH cm2=x
eday 1
416 2
22
=+
π
vx
⇒ ( ) 22
2
316
4
π
π
=−= xv
⇒ cm/s44,5±=v
cMeBaHsMTuH xa
4
2
π
−=
⇒ 2
2
cm/s93,42
4
−=×−=
π
a
45-cl½tmYyeFVIclnaRtg;sIunuysUGIutEdlGab;sIusvakMnt;edayGnuKmn_éneBl³ tAx ωsin= xUbén
clnaKW 6s . cMeBaH s5,0=t el,Óncl½t sv cm/π+= .
k-KNna ω nig A .
x-KNnasMTuHéncl½tkalNacl½tsßitenAcMgay 0,5cm BITItaMglMnwg.

88
K-KNnael,ÓnRtg;cMnucenH.
cMelIy
k-KNnaω nig A
eyIgman³ rd/s
36
22 πππ
ω ===
T
eyIgmansmIkarclna tAx ωsin= .
⇒ tA
dt
dx
v ωω cos==
cMeBaH s5,0=t ; cm/sπ+=v
⇒ ⎟
⎠
⎞
⎜
⎝
⎛
⋅⋅=
2
1
cos
3
ω
π
π A
⇒
2
3
3
1 ×=
A
⇒ cm32=A
x-KNnasMTuH
xAtA
dt
dv
a ⋅−=−== 22
sin ωωω
cMeBaH cm5,0=x
⇒ 2
2
cm/s548,05,0
3
−=×⎟
⎠
⎞
⎜
⎝
⎛
−=
π
a
K-el,Ón
eday 122
2
2
2
=+
ωA
v
A
x
cMeBaH cm5,0=x
⇒ 22
xav −= ω∓
⇒ cm/s59,3
4
1
12
3
±=−±=
π
v
46-RKab;)aj;mYyRtUv)aneK)aj;BIcMnuc0enAkñúgbøg; (xOy) edayel,Ón m/s100 =v .
k-kMnt; αtg ¬α ekItBIviucT½rel,Ón 0v nig G½kSedk (ox)) eBlRKab;)aj;mkdl;cMnuc A eKeXIj kUrGr
edaen m20=x ; m60=y .
x-KNnael,ÓnRtg;cMnuc A nigry³eBl. ]bmafa kMlaMgTb;énxül;minKit. yk 2
m/s10=g .

89
cMelIy
k-kMnt; αtg
eyIgsikSaclnaenAkñúgtMruyEkg (oxy) .
eyIgBinitüenAlkçx½NÐedIm 0;0;0 00 === yxt
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
α
α
sin
cos
0
0
0
vv
vv
v
Oy
Ox
tamTMnak;TMngRKwHDINamic³
amf =Σ edayRKab;)aj;rgEtTMgn;va
⇒ gmPam ==
O x
α
0v
y
0
α
0xv x
x
M
0voyv
y
y
v
.P m g=

90
⇒ ga = (1)
-eFVIcMenal(1) elI(Ox)
⇒ 0=xa ⇒ 0==
dt
dv
a x
x
⇒ === αcos0vvv Oxx efr
smIkareBl³
dt
dx
vx = ⇒ ∫∫ ⋅=
tx
dtvdx
0
0
0
cosα
⇒ tvx ⋅= αcos0 (2)
-eFVIcMenal(1) elI(Oy)
gay +=
g
dt
dv
a y
y +== ⇒ ∫∫ +=
tv
v
y gdtdv
y
0sin0 α
⇒ αsin0vgtvy ++=
smIkareBl³ αsin0vgt
dt
dy
vy ++==
⇒ ( )∫∫ +=
ty
dtvtgtdy
0
0
0
sinα
⇒ tvgty ⋅++= αsin
2
1
0
2
(3)
(2) ⇒
αcos0v
x
t = CMnYskñúg (3)
⇒ ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+=
α
α
α cos
sin
cos2
1
0
0
2
0 v
x
v
v
x
gy
⇒ α
α
tg
cos2
1
2
0
2
2
x
v
x
gy +=
tamTMnak;TMng α
α
2
2
tg1
cos
1
+=
⇒ ( ) αα tgtg1
2
1 2
0
2
2
x
v
x
gy ++=
enAcMnuc A: m20=x ; m60=y
⇒
( )
( )
( ) αα tg20tg1
10
20
10
2
1
60 2
2
2
++×=
⇔ ( ) αα tg20tg12060 2
++=
⇔ 02tgtg2
=−+ αα

91
tag αtgU = ⇒ 02UU2
=−+
9241Δ =×+=
⇒
2
31
U
±−
= ⇒ 2U1 −= ; 1U2 =
bJsénsmIkarykEt³ 1U =
⇒ 1tg =α ⇒ o
45
4
==
π
a
x-KNnael,ÓnRtg; A
yx vvv += ⇒ yx vvv 22
+=
( )
2
0
2
0 sin
2
1
cos ⎟
⎠
⎞
⎜
⎝
⎛
++= αα vgtvv
gyvv 2
2
0 +=
m/s3660102102
=××+=v
ry³eBlmkdl;cMnuc A
(2) ⇒ tvx ⋅= αcos0
⇒ s82,2
45cos10
20
cos o
0
===
αv
x
t
47-smIkarclnaéncMnuccl½tmYy³ 1+= tx ; 2
2
2
+=
t
y
k-cUreGaysmIkaredkaténclnanigRbePTKnøgclna.
x-viucT½rel,Ón nigsMTuH.
K-KNnakMuub:UUsg;sMTuHpÁúMEkg nigsMTuHpÁúMEkg.
X-cUreGaykenSamkaMkMeNagénKnøgCaGnuKmn_éneBl. KNnael,ÓncMeBaH 1=x ; 2=y .
cMelIy
k-eyIgmansmIkareBl³
1+= tx ⇒ 1−= xt
2
2
2
+=
t
y ⇒
( ) 2
2
1
2
+
−
=
x
y
⇒
2
5
2
2
+−= x
x
y CasmIkaredkat
eyIgBinitüemIlRkahVik³ )(xfy =

92
t 0 1 2 4
x 1 2 3 5
y 2 2
5
4 10
smIkarenHmanrag cbxaxy ++= 2
CasmIkar)a:ra:bUl .
x-kenSamviucT½rel,Ón
jvivvvv yxyx ⋅+⋅=+=
ehIy ( ) 1
1
=
+
==
dt
td
dt
dx
vx
t
t
dt
d
dt
dy
vy =⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+== 2
2
2
⇒ 22222
11 ttvvv yx +=+=+=
kenSamviucT½rsMTuH
jaiaaaa yxyx ⋅+⋅=+= 22
0
1
===
dt
d
dt
dv
a x
x
1===
dt
dt
dt
dv
a y
y
y
10
2
j

93
⇒ 22222
m/s110 =+=+= yx aaa
K-kMuub:UUsg;b:HnigEkgsMTuH
eyIgman³
tn aaa +=
( ) 2
2
1
1
1
t
t
dt
d
dt
dv
at
+
=+==
R
t
R
v
an
22
1+
==
R ³ CakaMkMenagCaGnuKmn_éneBl
tn aaa 22
+=
( )
2
2
2
22
1
1
t
t
R
t
a
+
+
+
=
X-kenSamkaMkMenag R CaGnuKmn_éneBl
tamsMnYrx- 2
m/s1=a
tamsMnYrK- ( )
2
2
2
22
1
1
t
t
R
t
a
+
+
+
=
⇒
( )
2
2
2
22
1
1
1
t
t
R
t
+
+
+
=
⇔
( )
2
2
2
22
1
1
1
t
t
R
t
+
+
+
=
⇒ ( )2
3
2
1 tR +=
cMeBaH 1=x ; 2=y enAxN³ 0=t
eyIg)an³ m1=R
48-cl½tM mYyeFVIclnaenAelIrgVg;mankaM R p©it0edayel,ÓnmMu dt
dθ
ω = .
k-KNnakUGredaenkaetesüógéncMnuc M CaGnuKmn_ R nig θ . KNnakMb:Usg;énel,Ón
nigsMTuHéncMnuc M elIG½kS (Ox) nig (Oy) .
x-etIsMTuHeTACay:agNa ebIm:UDulénel,ÓnmantMélefr? cUrR)ab;RbePTclnaénM.

94
K-LÚveyIg]bma 0α=
dt
dω
0, α cMnYnefrxusBIsUnü. cUreGaykenSam ω nig θ CaGnuKmn_én
eBl edaydwgfa enAxN³edImeBl 0=t ; 0=θ nig 0ωω = . rYceGayTMnak;TMngrvag ωnig θ .
cMelIy
k-eyIgeFVIcMenalM elIG½kSTaMgBIr³
θcosRx = ; tωθ = ; θsinRy =
-el,ÓntamG½kSnImYy²³
θω sinR
dt
dx
vx −==
θω cosR
dt
dy
vy ==
-kuMb:Usg;sMTuH
⎥⎦
⎤
⎢⎣
⎡
+−== θω
ω
θ cossin 2
dt
d
R
dt
dv
a x
x
⎥⎦
⎤
⎢⎣
⎡
−== θω
ω
θ sincos 2
dt
d
R
dt
dy
vy
sikSaenAkñúgtMruyeRbeN ( )nuM ,,
eyIg)an³
nROM ⋅= ; ( )nox,θ =
⇒
( ) ( )
dt
nd
R
dt
nRd
dt
OMd
v =
⋅
==
eday θ CaGnuKmn_éneBl ehIyCaGnuKmn_ θ
⇒
dt
d
d
nd
R
dt
nd
Rv
θ
θ
⋅==
eday u
d
nd
=
θ
b:HnwgKnøg
⇒ u
dt
d
Rv
θ
=
ehIysMTuH³ dt
vd
a =
⇒
dt
ud
dt
d
Ru
dt
d
Ra
θθ
+= 2
2
b¤ dt
d
n
dt
d
d
ud
dt
ud θθ
θ
−==
0 θ
y
u n
x
M

95
⇒
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+⎟
⎠
⎞
⎜
⎝
⎛
−= 2
22
dt
d
u
dt
d
nRa
θθ
eyIg)an³
dt
dθ
³ el,ÓnmMu nig 2
2
dt
d θ
³ sMTuHmMu
2
⎟
⎠
⎞
⎜
⎝
⎛
dt
d
R
θ
³ sMTuHpMÁúb:H nig
2
⎟
⎠
⎞
⎜
⎝
⎛
dt
d
R
θ
³ sMTuHpMÁúEkg
x- ebI =v efr
⇒ 02
2
==
dt
d
dt
d θω
⇒ sMTuH a eFob (Oxy)
θω cos2
Rax −= , θω sin2
Ray −=
⇒ Raaa yx
222
ω−=+−=
cMeBaHtMruyeRbeN³
0=ta nig R
dt
d
Ran
2
2
ω
θ
−=⎟
⎠
⎞
⎜
⎝
⎛
−=
⇒ Ra 2
ω−= dUcenHcl½teFVIclnavg;esμI .
K-eGaykenSam ω nig θ
eyIgman³ == 0α
ω
dt
d
efrxusBIsUnü
⇒ dtd ⋅= 0αω
⇒ ∫∫ = dtd 0αω ⇒ A0 += tαω
ehIy dt
dθ
ω =
⇒ ( )∫∫ += dtAtd 0αθ
⇒ BtAt ++= 2
0
2
1
αθ
cMnYnefr A, B kMnt;enAl½kçx½NÐedIm³
0=t ; 0=θ ⇒ 0=B
0=t ; 0=ω ⇒ 0ω=A
dUcenH eyIg)an³
00 ωα += tω
ttθ 0
2
0
2
1
ωα +=

96
TMnak;TMngrvag ω nigθ
00 ωαω += t ⇒
0
0
α
ωω −
=t
⇒ ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
=
0
0
0
2
0
0
2
1
α
ωω
ω
α
ωω
αθ
dUcenH θαωω 0
2
0
2
2=−
49-enAelIrUb x tagGab;suIséncl½tenAelIKnøgehIyenAxN³ t . ExSekagCaFñÚsuIunuysUGuIt.
k-cUrkMnt;smIkareBlénclna )(tfx = .
x- kMnt;el,ÓnedIm.
K-kMnt;sMTuHGtibrma.
cMelIy
k- kMnt;smIkareBlénclna )(tfx =
smIkarénclnaKW x = f(t)
clnaenHCaclnaRtg;suInuysUGuIténeBlEdlmanTMrg;³
( ) )cos( ϕω += txx mt
=mx GMBøITutGtibrma
=ω Bulsasüúg
=ϕ pasedIm
eyIgBinitüeTAelIRkab
-cMeBaH 0=t ; 0=x
⇒ ( )ϕω +×= 0cos0 mx
x (cm)
2
1

97
⇒ 0cos =ϕ ⇒
2
π
ϕ =
⇒ ⎟
⎠
⎞
⎜
⎝
⎛
+=
2
cos
π
ω txx m
ehIy T
π
ω
2
= kñúgRkabvaeFVIknøHxYb
⇒ s1
2
=
T
⇒ s2=T
⇒ rd/s
2
2
π
π
ω ==
-cMeBaH cm2=x ; s5,0=t
⇒ ⎟
⎠
⎞
⎜
⎝
⎛
+×=
2
5,0cos2
π
πmx
⇒ cm2=mx
⇒ smIkarclna cm
2
.cos2 ⎟
⎠
⎞
⎜
⎝
⎛
+=
π
π tx
x- el,ÓnedIm
⎟
⎠
⎞
⎜
⎝
⎛
+−==
2
sin2
π
ππ t
dt
dx
v
-cMeBaH 0=t
⇒ ⎟
⎠
⎞
⎜
⎝
⎛
+×−=
2
0sin20
π
ππv
⇒ cm/s20 π−=v
K-sMTuHGtibrma
eyIgman³
⎟
⎠
⎞
⎜
⎝
⎛
+−==
2
cos2 2 π
ππ t
dt
dv
a
edIm,IeGay aGtibrmaluHRtaEt 1
2
cos −=⎟
⎠
⎞
⎜
⎝
⎛
+
π
π t
dUecñH 22
cm/s2πα =maa
50-k-c,ab;eBlTUeTAénclnasuInuysUGuItsMEdgeRkamTMrg;³ CtBtAx ++= ωω sincos Edl
A, B, C nig ω CacMnYnefr. cUrsMEdgxñatrbs;vaTaMgBIrenH.

98
x- eyIgeRCIserIsKl;Gab;suIscMeBaH C = 0 mü:ageToteyIgsÁal; 0x nig 0v énGab;suIs x esμInwg
el,Ón dt
dx
enAxN³ 0=t . cUrkMnt; A nig B .
K- cUrbgðajfa eKGacsresr x eRkamTMrg;³ )cos( ϕω += txx m .
cUrkMnt; xm nig ϕ edaysÁal; 0x nig 0v .
cMelIy
k-eyIgmansmIkareBl³
CtBtAx ++= ωω sincos
ebI x KitCa m ehIy tωcos nig tωsin CatMélemKuNKμanxñat.
dUcenHeyIg)anxñatrbs; A; B nig C KitCa m ehIy ω
CaBulsasüúgKitCa /rad s .
x-kMnt; A nig B
kñúgsmIkareBl CtBtAx ++= ωω sincos edayeRCIserIsedImeBl³
0=t , 0=C
dUcenHeyIg)an³
00sin0cos.0 +×+×= ωω BAx ⇒ 0xA =
ehIy tBtA
dt
dx
v ωωωω cossin +−==
cMeBaH 0=t ⇒ 0cos0sin0 ×+×−= ωωωω BAv
⇒ ω.0 Bv = ⇒
ω
0v
B =
dUcenHeyIg)an ;0xA =
ω
0v
B = .
K- smIkarxagelIkøayeTACa³
t
v
txx ω
ω
ω sincos 0
0 +=
smIkarenHeyIgGacsresreRkamTMrg;³
)cos( ϕω += txx m

99
ebIeyIgKuNGgÁTIBIrénsmIkarnwg
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
+
ω
ω
02
2
02
v
x
v
x
⇒
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+
+
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
+= t
v
x
v
t
v
x
xv
xx ω
ω
ωω
ω
ω
sincos
2
02
0
2
02
0
2
02
eyIg)an³ ϕ
ω
cos
0
0
2
0
=
⎟
⎠
⎞
⎜
⎝
⎛
+
v
x
x
ϕ
ω
ω sin
2
0
0
2
0
=
⎟
⎠
⎞
⎜
⎝
⎛
+
v
x
v
tag
2
0
0
2
⎟
⎠
⎞
⎜
⎝
⎛
+=
ω
v
xxm .
⇒ [ ]ttxx m ωϕωϕ sinsincoscos +=
⇒ )cos( ϕω += txx m
kñúgenH
2
02
0 ⎟
⎠
⎞
⎜
⎝
⎛
+=
ω
v
xxm
nig ω
ωϕ
0
0
0
0
tg
x
v
x
v
== ⇒
ω
ϕ
0
0
Arctg
x
v
=
ω
2
02 v
x +
0v
ω
ϕ
0x

101
51-rkcMgayrbs;GgÁFatumYykñúgry³eBlTI n s rbs;Tnøak;esrI.
cMelIy
KNnacMgaycr
eyIgeRCIserIsenAxN³ t = 0, 00 =x , 00 =v
-smIkarenAxN³TI n
2
2
1
ngxn =
-smIkarenAxN³TI n – 1
2
1 )1(
2
1
−=− ngxn
cMgaycrKW³ 1−− nn xx
⇒ 22
1 )1(
22
1
2
1
−−=− − ngngxx nn = ⎟
⎠
⎞
⎜
⎝
⎛
−
2
1
ng
52-enAcMnuc O EtmYy eKTMlak;GgÁFatuTImYy A . 0,1s eRkaymk eKTMlak;GgÁFatuTIBIr B . A nig
B manclnaTnøak;esrI. etIry³eBlb:unμanbnÞab;BIkarecjdMeNIrrbs;A cMgay 1AB = m ?
rkcMgaycr nigel,Ónrbs;GgÁFatunImYy². g = 9,8 USI .
cMelIy
k-ry³eBl
eyIgeRCIserIsenAxN³ t = 0, 00 =x CaxN³eBlEdl GgÁFatuA ecjdMeNIr.
-smIkarrbs;GgÁFatu A
2
2
1
tgxA =
-smIkarrbs;GgÁFatu B
22
)1,0(
2
1
2
1
−== tgtgxB
cMgay³ 22
)1,0(
2
1
2
1
−−=−= tggtxxAB BA
= ggt 005,01,0 −
O
1x 1n
2x 2n
1−nx 1−n
nx n
0
B
1 m
A

102
⇔ 18,9005,08,91,0 =×−× t , 1AB = m
⇒ t = 1,07 s
x-KNnacMgaycr
4,5)07,1(8,9
2
1 2
=⋅=Ax m
4,414,5 =−=Bx m
c). KNnael,Ón
48,1007,18,9 =×=⋅= tgvA m/s
)1,0)07,1(8,9 −×=Bv = 9,5 m/s
53-XøImYyRtUveKTMlak;BImat;GNþÚgedayclnaTnøak;esrI. 4s eRkaymkGñksegátEdlenAmat;GNþÚg
eTIblWkarTgÁicrvagXøInigTWk. el,ÓndMeNalrbs;sMelgman 340 /m s .
KNnaCMerAGNþÚg ¬sUmbBa¢ak;fa épÞTwkenACab;)atGNþÚg¦.
cMelIy
KNnaCMerAGNþÚg
-enAxN³XøImanclnaTnøak;esrI eRCÍserIs
t = 0, 00 =x , 00 =v
smIkar³ 2
1
2
1
tgx ⋅=
tag 2t Cary³eBlEdlel,ÓnsMelgedalBIépÞTwkdl;mat; GNþÚg³ 2tvh ⋅=
Et xh =
⇒ 2
2
1
2
1
vttg = eday s421 =+ tt ⇒ 12 4 tt −=
⇔ )4(
2
1
1
2
tvtg −= , 340=v m/s
CMnYseyIg)an³
013603409,4
2
1 =−+ tt
)1360(9,4)170(Δ 2
−×−=′ = 3564 ⇔ 58,188Δ =′
⇒
9,4
58,188170
1
±−
=t ykEttMélvíC¢man eRBaH 01 >t
⇒ =
−
=
9,4
17058,188
1t 3,8 s

103
C¿erAGNþÚgKW³
22
1 )8,3(8,9
2
1
2
1
××=⋅== tgxh = 70,7 m
54-eKTMlak;GgÁFatumYyBIkMBs; 1000 m . etIGs;ry³eBlb:unμan ehIymanel,Ónb:unμan enAeBlva
Føak;mkdl;dIebIeKKitkMlaMgTb;énxül;? g = 9,8 2
sm/ .
cMelIy
ry³eBlGgÁFatuFøak;mkdl;dÍ
-edATisvíC¢mancuHeRkam ga +=
-enAxN³t = 0, 00 =x , 00 =v
-smIkarclna³
2
2
1
tgx =
⇒
g
x
t
2
=
⇔
8,9
10002×
=t = 14,28 s
55-eKecalGgÁFatumYyeLIgelItambeNþayExSQredayel,ÓnedIm 3m/s BIkMBs; 300 m .
-etIvaeLIgeTAelI)ankMBs;b:unμan?
-etIGs;ry³eBlb:unμaneTIbvaqøgkat;TItaMgedImeLIgvíj?
-etIel,ÓnvaesμIb:unμan eBlqøgkat;TItaMgedImrbs;va?
-etIGs;ry³eBlb:unμaneTIbvaeTAdl;dI?
-rkel,ÓnvaenAeBlvamkdl;dI. g = 9,8 2
s/m
cMelIy
-kMBs;eLIg)an ¬eFob 0¦
-edATisvíC¢mancuHeRkam
-eyIgyk0CaKl;Gab;suIsenAxN³t = 0, 00 =x , 30 −=v m/s
O
g
x (+)

104
smIkarclna³
tvtgx 0
2
2
1
+=
ttx 38,9
2
1
−×=
tamTMnak;TMng³ )(2 00
2
xxgvv −=− , 00 =x
Rtg;cMnuc A³ 0=Av ¬Gs;el,ÓnRtUvFøak;mkvij¦
⇒
g
vv
xA
2
2
0
2
−
=
= 45,0
8,92
)3(0 2
−=
×
−−
m
-ry³eBlEdlvaqøgkat;TItaMgedIm
0=x ⇔ 038,9
2
1 2
=−× tt
⇔ 0)39,4( =−tt ⇒ 0=t ; 61,0=t s
t = 0 RtUvnwgeBlecjdMeNIr
t = 0,61s Cary³eBlRtLb;mkKnøgedImvíj.
-el,ÓneBlqøgkat;TItaMgedIm 0³
38,938,9
2
1 2
−=⎟
⎠
⎞
⎜
⎝
⎛
−== ttt
dt
d
dt
dx
v
eday t = 0,61s ⇒ 3361,08,9 =−⋅=v m/s
-ry³eBlEdlecjBI 0dl; A
Rtg; A: 0=Av
⇒ 038,9 =−⋅t ⇒ ==
8,9
3
t 0,306s
⇒ ry³eBlBI OA → esμI AO → .
-ry³eBlFøak;dl;dÍ
smIkarclna³
tvatx 0
2
2
1
+= , ga += , 30 −=v m/s
⇒ ttttx 39,438,9
2
1 22
−=−⋅=
·A
0v
O
g
x
(+)

105
eBlFøak;mkdl;dÍeyIg)an³ x = 300m
⇒ tt 39,4300 2
−=
⇒ 030039,4 2
=−− tt
5889)300(9,443Δ 2
=×−= ⇒ 73,76Δ = s
⇒
9,42
73,763
×
±
=t ¬bJsGviC¢manminyk 0<t ¦
⇒
9,42
73,763
×
+
=t = 8,13s
-el,ÓneBlFøak;dl;dÍ
313,88,938,9 −×=−= tv = 76,67 m/s
56-fμmYydMu)ancMnayeBl edIm,IFøak;dl;)atGNþÚg.
k-rkCMerAGNþÚg.
x-rkry³eBledÍm,IeGayvaFøak;dl;)atGNþÚg EdlmanCMerA4 dg, 9 dg , 16 dg eRCACagmun.
yk g = 9,8 m/s2
cMelIy
k-CMerAGNþÚg
eRCIserIs
-0CaKl;Gab;suIsCakEnøgecjdMeNIr 00 =x
-TisedA (+) cuHeRkam ga +=
smIkarclna³
2
2
1
tgx = ⇔ 2
9,4 tx =
ry³eBlFøak;dl;)atGNþÚg 2=t s
CMerAGNþÚgKW³ 2
39,4 ×=x = 44,1m
x-ry³eBl
2
2
1
gtx = ⇒
g
x
t
2
=
-eRCACagmun 4 dg
O
g
x
(+)

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