Mecanica_B5-1

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CUPRINS CUPRINS............................................................................................................... 5 PREFA ............................................................................................................... 9 1. INTRODUCERE................................................................................................ 11

1.1. Generaliti.......................................................................................................11 1.2. Scurt istoric al mecanicii ....................................................................................11 1.3. Obiectul mecanicii .............................................................................................14 1.4. Sisteme i uniti de msur..............................................................................15

CAPITOLUL 2 ...................................................................................................... 17 2.1. Noiuni de calcul vectorial..................................................................................17 2.2. Operaii cu vectori.............................................................................................18

2.2.1. Adunarea a doi vectori a i b . ...................................................................18 2.2.2. Produsul scalar a doi vectori a i b . ...........................................................20 2.2.3. Produsul vectorial a doi vectori liberi a i b . ...............................................21 2.2.4. Produsul mixt a trei vectori a , b i c .........................................................22 2.2.5. Dublu produs vectorial a trei vectori liberi a , b i c ....................................23 2.2.6. Descompunerea unui vector dup trei direcii. ..............................................24

CAPITOLUL 3 ...................................................................................................... 25 3.1. Statica punctului ...............................................................................................25

3.1.1. Punctului material liber. Punct material supus la legturi ...............................25 3.1.2. Echilibrul punctului material liber .................................................................25 3.1.3. Probleme rezolvate .....................................................................................26

3.2. Punctul material supus la legturi ......................................................................29 3.2.1. Axioma legturilor. Legturile punctului material...........................................29 3.2.2. Echilibrul punctului material supus la legturi fr frecare .............................30 3.2.3. Echilibrul punctului material supus la legturi cu frecare................................33

3.3. Probleme rezolvate ...........................................................................................37 3.4. Probleme propuse.............................................................................................46

CAPITOLUL 4 ...................................................................................................... 49 4.1. Statica rigidului.................................................................................................49

4.1.1. Caracterul de vector alunector al forei ce acioneaz un rigid......................49 4.1.2. Momentul unei fore n raport cu un punct. ..................................................50 4.1.3. Momentul unei fore n raport cu o ax. .......................................................52 4.1.4. Cupluri de fore ..........................................................................................53 4.1.5. Caracterizarea unui vector alunector. .........................................................54 4.1.6. Teorema momentelor (Teorema lui Varignon). .............................................55 4.1.7. Sisteme de fore echivalente. Operaii elementare de echivalen. .................56 4.1.8. Reducerea unei fore aplicat ntr-un punct al unui rigid................................56 4.1.9. Reducerea unui sistem de fore aplicate rigidului. Torsorul de reducere. Variaia torsorului cu punctul de reducere. Invariani .........................................................57 4.1.10. Torsorul minimal. Axa central...................................................................59 4.1.11. Cazurile de reducere ale unui sistem de fore oarecare................................60

4.2. Reducerea sistemelor particulare de fore...........................................................61 4.2.1. Reducerea sistemelor de fore concurente....................................................61

+

4.2.2. Reducerea sistemelor de fore coplanare ..................................................... 61 4.2.3. Reducerea sistemelor de fore paralele ........................................................ 62 4.2.4. Reducerea forelor paralele, distribuite ........................................................ 65

4.3. Probleme rezolvate .......................................................................................... 67 4.4. Probleme propuse ............................................................................................ 74 4.5. Centre de greutate (centre de mas) ................................................................. 76

4.5.1. Centrul de greutate al unui sistem de puncte materiale................................. 76 4.5.2. Centrul de greutate al corpurilor.................................................................. 77 4.5.3. Teoremele Pappus - Guldin ......................................................................... 80

4.6. Centre de mas pentru corpuri uzuale ............................................................... 81 4.7. Probleme rezolvate ........................................................................................... 83 4.8. Probleme propuse ............................................................................................ 96

CAPITOLUL 5 Echilibrul rigidului ...................................................................... 99 5.1. Echilibrul rigidului liber ....................................................................................99 5.2. Echilibrul rigidului supus la legturi fr frecare ...............................................101

5.2.1. Generaliti .............................................................................................101 5.2.2. Legtura rigidului ....................................................................................102 5.2.3. Cazurile particulare de echilibru ................................................................109

5.3. Echilibrul rigidului spus la legturi cu frecare ...................................................113 5.3.1. Generaliti asupra fenomenului de frecare ...............................................113 5.3.2. Frecare de alunecare ...............................................................................114 5.3.3. Frecarea de rostogolire ............................................................................116 5.3.4. Frecarea de pivotare ............................................................................... 120 5.3.5. Frecarea n lagrul radial (articulaia cilindric) ..........................................122

5.4. Probleme rezolvate ........................................................................................125 5.5. Probleme propuse .........................................................................................132

CAPITOLUL 6 Statistica sistemelor materiale ..................................................135 6.1. Echilibrul sistemelor materiale ........................................................................135

6.1.1. Sistemul material .....................................................................................135 6.1.2. Torsorul forelor intercalare ......................................................................135 6.1.3. Teoreme i metode pentru studiul echilibrului sistemelor materiale .............136 6.1.4. Sisteme static determinate i static nedeterminate .....................................138

6.2. Probleme rezolvate ........................................................................................139 6.3. Probleme propuse .........................................................................................155 6.4. Grinzi ci zbrele .............................................................................................158

6.4.1. Ipoteze simplificatoare .............................................................................158 6.4.2. Eforturi de bare .......................................................................................159 6.4.3. Grinzi cu zbrele static determinate ...........................................................160 6.4.4. Metode pentru determinarea eforturilor din bare ....................................... 161

6.5. Probleme rezolvate ........................................................................................162 6.6. Probleme propuse .........................................................................................175 6.7. Statica firelor ................................................................................................176

CAPITOLUL 7 Cinematica punctului .................................................................185 7.1. Noiuni fundamentale ....................................................................................185

7.1.1. Legea de micare ....................................................................................185 7.1.2. Traiectoria ..............................................................................................185 7.1.3. Viteza .....................................................................................................186

)

7.1.4. Acceleraia ..............................................................................................187 7.1.5. Viteza i acceleraia unghiular .................................................................188

7.2. Studiul micrii punctului n sistemele de coordonate cartezian i natural............189 7.2.1. Sistemul de coordonate cartezian .............................................................189 7.2.2. Cinematica punctului material n coordonate polare ...................................190 7.2.3. Sistemul de coordonate intrinseci .............................................................192

7.3. Micarea circular ..........................................................................................194 7.3.1. Studiul micrii circulante n coordonate carteziene ...................................194 7.3.2. Studiul micrii circulante n coordonate naturale ......................................195


CAPITOLUL 8. Cinematica rigidului .................................................................213 8.1. Micarea general a rigidului ..........................................................................213

8.1.1. Mobilitatea rigidului .................................................................................213 8.1.2. Distribuia de viteze .................................................................................214 8.1.3. Distribuia de acceleraii ...........................................................................216

8.2. Micarea de rotaie ........................................................................................217 8.2.1. Distribuia de viteze .................................................................................218 8.2.2. Distribuia de acceleraii ...........................................................................219

8.3. Micarea plan paralel ...................................................................................220 8.3.1. Distribuia de viteze .................................................................................222 8.3.2. Centrul instantaneu de rotaie ..................................................................222 8.3.3. Distribuia de acceleraii ..........................................................................224

8.4. Micarea rigidului cu un punct fix ....................................................................225 8.4.1. Generaliti .............................................................................................225 8.4.2. Studiul vitezelor ......................................................................................226 8.4.3. Studiul acceleraiilor ................................................................................231

8.5. Micarea general a rigidului ..........................................................................232 8.5.1. Generaliti .............................................................................................232 8.5.2. Studiul vitezelor ......................................................................................232 8.5.3. Studiul acceleraiilor ................................................................................234

8.6. Probleme rezolvate........................................................................................235 8.7. Probleme propuse..........................................................................................254

CAPITOLUL 9 Micarea relativ........................................................................257 9.1. Micarea relativ a punctului material .............................................................257

9.1.1. Derivata absolut i relativ (local) a unui vector .....................................257 9.1.2. Studiul vitezelor ......................................................................................258 9.1.3. Studiul acceleraiilor ................................................................................259

9.2. Micarea relativ a rigidului ............................................................................261 9.2.1. Generaliti .............................................................................................261 9.2.2. Studiul vitezelor ......................................................................................261


CAPITOLUL 10 Dinamica punctului material ..................................................271 10.1. Noiuni fundamentale ..................................................................................271

10.1.1. Lucrul mecanic .....................................................................................271 10.1.2. Funcia de for ....................................................................................272

*

10.1.3. Puterea ................................................................................................273 10.1.4. Randamentul mecanic ...........................................................................274 10.1.5. Impulsul .............................................................................................. 274 10.1.6. Momentul cinetic ..................................................................................275 10.1.7. Energia mecanic ................................................................................. 275

10.2. Teoreme generale n dinamica punctului material ..........................................276 10.2.1. Teorema impulsului ...............................................................................276 10.2.2. Teorema momentului cinetic ................................................................. 277 10.2.3. Teorema energiei cinetice ..................................................................... 277

10.3. Ecuaiile difereniale ale micrii punctului material ........................................278 10.3.1. Generaliti ..........................................................................................278 10.3.2. Ecuaiile difereniale ale micrii punctului material .................................279

10.4. Probleme rezolvate ......................................................................................281 CAPITOLUL 11 Dinamica sistemelor de puncte materiale i a rigidului ..........289

11.1. Noiuni fundamentale ..................................................................................290 11.1.1. Momente de inerie mecanice ...............................................................290

11.2. Probleme rezolvate .....................................................................................298 11.3. Probleme propuse ......................................................................................302 11.4. Lucrul mecanic elementar al unui sistem de fore care acioneaz asupra

unui rigid ............................................................... .................................. 303 11.4.1. Cazul general ..................................................................................... 303 11.4.2. Cazuri particulare ............................................................................... 304 11.4.3. Impulsul ............................................................................................ 304 11.4.4. Momentul cinetic ................................................................................ 305 11.4.5. Energia cinetic ...................................................................................309

11.5. Teoreme generale n dinamica sistemelor de puncte materiale i a rigidului.... 314 11.5.1. Teorema impulsului ............................................................................. 314 11.5.2. Teorema momentului cinetic ................................................................ 317 11.5.3. Teorema energiei cinetice .................................................................... 320

11.6. Probleme rezolvate .................................................................................... 323 11.7. Probleme propuse ..................................................................................... 335

BIBLIOGRAFIE................................................................................................. 337

(

PREFA

35 &56178

51149:161;

$$

1. INTRODUCERE

1.1. GENERALITI

! 1!

$"

? 1 4! 6 :; =! >

$%

EL1&.$)$)B$)*%/&6DMH1N8

$,

1.3. OBIECTUL MECANICII

$-

1.4. SISTEME I UNITI DE MSUR ?>:161;5@

$+

&$$

"( (-$ %#$

( #!# 6#

# ("!"

6#A'1 A 1

$)

782401

CAPITOLUL 2

2.1. NOIUNI DE CALCUL VECTORIAL ?

$*

B

a

b

c

O

C

A

78224''1

a

upri Ox= uprj Oy= uprk Oz= .""/Q

$(

c

P'"%

615

"#

' =1 ! 56 1 @ 5

=

=

n

iixx VV

1!

=

=

n

iiyy VV

1!

=

=

n

iizz VV

1 ."$,/

615

"$

apr u

a

7829/3=@

a

b

c

O

782:45

""

?1;=

"%

H

v

h

a

b

c

O

P'")'16

",

X

Z

Y

O

V

782;/

"-

CAPITOLUL 3

3.1. STATICA PUNCTULUI

3.1.1. Punctului material liber. Punct material supus la legturi Q 1 5= :

"+

3.1.3. Probleme rezolvate - (54

3 1

")

- (542

"*

( ) ( ) ( )kPjPiP

aaa

kajaiaPMEMEPuFF

++=

=

++

++===

432

4324322929

222333 !

( ) jPajaP

MBMBPuFF ==== 2

3322

2444!

( ) ( ) jPiPaajaiaP

MDMDPuFF +=

+

+=== 64

3232132132

22555

45

"(

'R

NR

R

TR

N

TM

( )P

( )S P'%-31;:&

( )[ ]jyixakMBkFB == 22

== jyaixakMCkFC 2

3233

;:&

%#

A'6 ; '6 ; . : 1

'! ;= R R : 5 &

%$

:5151

%"

= &

%%

3.2.3. Echilibrul punctului material supus la legturi cu frecare. Legile frecrii uscate ?5&

%,

@=;6

%-

31

%+

56=&

21

1

11

+

nRnR

%)

3.3. PROBLEME REZOLVATE - (554Q1' G ;66; F 0=tg 0=

> 0=F =tg 2pi

=

> GF = 1=tg 4pi

=

- (5526564' G !

%*

- (555Q ' G !6=:

:=;!56 ( ) cossin + QP A> :

%(

&56'6.;'%$,/!

,#

&56 '6 .;'%$- ;'%$-&/

,$

- (55=

,"

- (55>3:'W;=656!

,%

- (55;Q ' G! 6 ;66

; 56 ! ;

,,

- (55?

'G!6;

&6

,-

- (5543

3

,+

3.4. PROBLEME PROPUSE - (594

,)

7852:

- (5993 ' '&6!

,(

CAPITOLUL 4

4.1. STATICA RIGIDULUI

4.1.1. Caracterul de vector alunector al forei ce acioneaz un rigid.

Q

-#

4.1.2. Momentul unei fore n raport cu un punct. 1 ;= : @16 ;=

-$

FrFABrFrFM

FrFM

BO

AO

=+==

=

)()()(

.,,/

:> 0= FAB ! AB F ; 1 ;= : 56 '

-"

=

=

=

xyz

zxy

yzx

yFxFMxFzFMzFyFM

.,*/

A 5

$@6.;',,!/

-,

A F

F

)(P P',-;=

B

O

O

b

Br Ar

'

Ar '

Br

M

M

- 3=@6

--

!1

-+

4.1.7. Sisteme de fore echivalente. Operaii elementare de echivalen. ?> : ;

-)

4.1.9. Reducerea unui sistem de fore aplicate rigidului. Torsorul de reducere. Variaia torsorului cu punctul de reducere. Invariani

-(

4.1.10. Torsorul minimal. Axa central.

P6>

+#

[ ] [ ] [ ]kyRxRMjxRzRMizRyRMRRRzyxkji

kMjMiMROPMM

xyzzxyyzx

zyx

zyxOP

)()()( ++=

=++==.,,%/

= PM R

+$

4.2. REDUCEREA SISTEMELOR PARTICULARE DE FORE

4.2.1. Reducerea sistemelor de fore concurente Q

+"

5" 0=R 0OM

+%

45

+,

95=3!@6% ;=

+-

4.2.4. Reducerea forelor paralele, distribuite P= !

++

22 0

2

0

pll

pxdxlxpR

ll

=== .,)#/

32

2

3

0

20

3

0

0 lx

x

dxlxp

xdxlxp

x l

l

l

l

C ===

.,)$/

+)

4.3. PROBLEME REZOLVATE - (954

+*

( ) jaPPPa

kjiFOEFrFM =

===

00022220

( ) jPaiaPPaaa

kjiFOGFrFM ====

0033330

( ) kPaiPaPP

a

kjiFOCFrFM ====

00044440

++=

++=

kMjMiMMkFjFiFR

iziyix

iziyix

00

=

=

kPajPaiPaM

kPR22

4

00

0

0

00

M

R

?

+(

- (952

)#

( ) jPaiPaPPPa

kjiFOFFrFM =

=== 0022220

kPaiPaaa

kaiaPaAFAFMM =

+

==2211

2

+=

+=

kPajPaiPaMkPjPiPR

3200

r

)$

- (9556&'1 !=

)"

iPOAOAFF 1011 ==

kPjPaa

kajaPCBCBFF 86

1694310

2222 =

+

==

( ) kPjPiP

aaa

kajaiaPDCDCFF 8610

16925435210

22233 ++=

++

+==

( ) 00010005110 ===

Pa

kjiFOAFM

( ) iPaPP

a

kjiFOBFM 24

860030220 =

==

( ) jPaiPaPPPa

kjiFOCFM 4024

8610400330 +=

==

=

=

=

kPajPaiPaMR

4020

00

===

=

=

000435402

0

0 aaa

kjikPajPaiPaROEMM

R

EE

kPajPaiPa = 402 5

= 00MR

)%

- (95:

),

+=

=

=

jPaiPaMjPiPR1212

63

00 @

aPMRTkPajPaiPaROAMM

jPiPRA

A2

0

0

6612122

63==

+==

=

=

aPMRTkPaPP

a

kjiROA A 26612

063002 ===

=

5

)-

- (99231*1245!&56

6 :6=1 5! 161 ;=

)+

4.5. CENTRE DE GREUTATE (CENTRE DE MAS) A!

))

3=@ Cr

)*

0@

)(

61'

( )( )

=

s

sc dA

dArr ! .,**/

'

*#

4.5.3. Teoremele Pappus - Guldin # !;1111

*$

4.6. CENTRE DE MAS PENTRU CORPURI UZUALE 4!.;',"(/

3cos2

coscos

0

0

2

R

drdr

ddrr

drrddrrdr

dAxdA

R

R

==

==

+

+

!

drrddA = 16

1

*"

x y

z

O

O

O

A

A

dz

z

h r

R

78953

5!

*%

4.7. PROBLEME REZOLVATE - (9>46

x

y

O

B C

E

D 2l

4l

l

78952

x

y

O

78952

C l C1

B

*,

- (9>26

*-

y

O x

A

B C

D

E

a

a

a

P',%,

- (9>56

*+

- (9>96

*)

561:6

**

561=6

*(

561>3 1'6 ;' ,%*

(#

- (9>;31'6;',%(

($

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("

- (9>4331';',,"

(%

- (9>4431';',,%

(,

- (9>4261'6;',,,1

(-

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(+

4.8. PROBLEME PROPUSE - (9;46

()

- (9;531'6;',,*

(*

456( )

.

32

,

32

,

32

,02-3

21

2

22

RlRl

Rl

Rl

==

=

=

CAPITOLUL 5 Echilibrul rigidului

5.1 ECHILIBRUL RIGIDULUI LIBER

!

=

=

00

OO M

R "#$%

&'

==

=

iiiO

i

MFrMFR

"#(%

"#$%'

=

=

00

i

i

MF

"#)%

*"%

'

=

=

=

000

iz

iy

ix

F

FF

=

=

=

000

iz

iy

ix

M

MM

"#+%

*"% '

=

=

=

000

iz

iy

ix

M

FF

"##%

$,,

- '%

.%

/

*

/

-

'

)z,y,x(A),z,y,x(A),z,y,x(A 333322221111

/

-'

=++=

=++=

=++=

32

312

312

3113

22

232

232

2332

12

122

122

1221

)()()()()()(

)()()(

dzzyyxxAA

dzzyyxxAA

dzzyyxxAA

"#0%

*

! "#0% " ! *

# $ ! "%

- " %1

"#$%'

.

&'()&" *+2**%.)&!" +%.)&!" * %

3#$1

,%+!"!

$,$

* -.-.4'

dyyxxAA =+= 2122

1221 )()( "#5%

!#$%

-

5.2 ECHILIBRUL RIGIDULUI SUPUS LA LEGTURI FR FRECARE

5.2.1 Generaliti

/ & $ $ " $ 0 0! "

- '

.

"% "% "#(% 6

OT R OM O R OM

OO M

RT

OO M

R "#7%

"#% .

=+

=+

0MM0RR

OO

"#%

3#(

$,(

5.2.2 Legturile rigidului

8'*'

*'

.

9

:*:

!

;': "#)%!

9"%

)M,R(T OO '

.

,%+!"!

$,)

tM "% "% :

$,+

/

!0!

; "%

>"%

- "#+%

' "% 9>> > >

!

""%%

-

)M,R(T OO R "%"%? OM "% "%!

"% R "#+% 4 R '###

6 )( zyxO RRRR ++= '

% %

3#+/

,%+!"!

$,#

=

=+

0OMR 0R

"#$(%

'

===

=+

=+

=+

0000

zyx

z

y

x

MMMRRR

z

y

x

R

RR

"#$)%

*!!"%

"% "%"%"##%

@ "%"%

!

/

R OM > 4"##%'

+=

+=

yxO

yxO MMM

RRR "#$+%

% %

3##/

$,0

"% '

++=

++=

zyxO

zyxO MMMM

RRRRT "#$#%

:"#%- "#$#%

'

=

=+

=+

0R0RR0RR

z

yy

xx

=

=+

=+

0M0MM0MM

z

yy

xx

"#$0%

*

OM *

)( yxO RRR += "#$0%'

====

=+

=+

0MMMR0RR0RR

zyxz

yy

xx

"#$5%

*

>"#0%9

! > '

R

3#0/

,%+!"!

$,5

4 R "% H V "#0%

*'

==

+=

kMMM

RRRT

OzO

yxO . )( VHRO += "#$7%

'

=

=+

0M0RR

O

"#$%

-

"#$%'

=

=+

=+

0

0V0H

O

y

x

M

RR

"#(,%

>

H V "#0%

*"%

!"%

-> iR "%"%"#5%

6" % OT

'

=

=

iiO

iO FrM

FRT

=

=

iiO

iO R'rM

RR "#($%

"#%'

A R OM

!

$,7

!

$ !

* R OM ' # # # , , , "#5%

9

'

++=

++=

zyxO

zyxO MMMM

RRRRT

++=

++=

zyxO

zyxO MMMM

RRRR "#((%

9 '

=+

=+

=+

000

zz

yy

xx

RR

RRRR

=+

=+

=+

000

z

y

x

M

MM

z

y

x

MMM

"#()%

!

$

*

R OM '23,"#7%

9'

==

+=

kMMM

RRRT

OzO

yxO

"#(+%

==

+=

kMMMVHR

OzOO

9 '

=+

=+

=+

000

OO

y

x

MM

VRHR

"#(#%

3#5*

3#7*

,%+!"!

$,

8

-

" "#%

1!'$4

"

(!>

)*"% '

5.2.3 Cazurile particulare de echilibru

4 nFFF ...,, 21 4"#$,%3'-5.-5.B-5.

6 F OM

++=

++=

kMjMiMMkFjFiFR

zyxO

zyxO "#(0%

9"%

nNNN ...,, 21 @9 '

3#-

$$,

=+=

==

==

0:00:00:0

izz

yy

xx

NFFFFFF

==

=+=

=+=

0:00:00:0

zz

iiyy

iixx

MM

xNMMyNMM

"#(5%

*"#(5%

0! 0!6%C"#(5%

'

==

i

ii

i

ii

NyNy

NxN

x ; (5.28)

nNNN ...,, 21 4- iN

"#(5%* : 7

- '

=++

=++

=++

x

y

z

MyNyNyNMxNxNxN

FNNN

332211

332211

321

"#(%

-"#(%

'

0111

321

321 =yyyxxx "#),%

'! "#(5%

"#(%+757575'8 95,95,75# 0!!

6"%

3#$,

,%+!"!

$$$

: !4

!"#"#

4 :&

nFFF ...,, 21 "#$$%

1R 2R

++=

++=

kRjRiRRkRjRiRRzyx

zyx

2222

1111 "#)$%

4 1OO F OM

++=

++=

kMjMiMMkFjFiFF

zyxO

zyxO "#)(%

9 '

=++=

=++=

=++=

0:00:00:0

21

21

21

zzzz

yyyy

xxxx

RRFF

RRFFRRFF

==

=+=

==

0:00:00:0

2

2

zz

xyy

yxx

MM

hRMMhRMM

"#))%

; "#))% ' 0!

00-"#))%'

zzzx

yy

xyx

yxy

x FRRhMR

hM

RFh

MRFh

MR =+==== 212211 ;;;; "#)+%

!# #

1

'

02 =zR "#)#%

3#$$9

$$(

"

4

nFFF ...,, 21 "#$(%9

R

kRjRiRR zyx ++= "#)0%

6 F OM

++=

++=

kMjMiMMkFjFiFF

zyxO

zyxO "#)5%

3

'

==

==

==

=+=

=+=

=+=

0:00:00:0

0:00:00:0

zz

yy

xx

zzz

yyy

xxx

MM

MMMM

RFF

RFFRFF

"#)7%

-"#)7%

' 0! F 0

3#$(9

,%+!"!

$$)

5.3 ECHILIBRUL RIGIDULUI SUPUS LA LEGTURI CU FRECARE

5.3.1 Generaliti asupra fenomenului de frecare

* > * > tR

/

9 >@"#$7%

4

6 nFFF ...,, 21

'

=

=

iiO

iO

FOAM

FRT "#)%

6 ip '

=

=

iiO

iO pOBM

pR "#+,%

"#%-

' :

$$+

3 nR "%> N

3 tR "% : /

fF 0!0

nM "%: @ 1

pM 0 1

tM "%:@

rM 0

9 "%'

=+

=+

00

ft

n

FRNR

=+

=+

0MM

0MM

rt

pn "#+(%

-

5.3.2 Frecarea de alunecare

4 "%

)RRR(T tnO += )( fO FNR += "#+)%

* "#(,%

R N fF tR

3#$

3#(,

,%+!"!

$$#

/ fF 0! 0

3

!

>0

, 0! 0 & ! ! 0 ! 1

$$0

5.3.3 Frecarea de rostogolire

4"%

"#($%'

=

+=

tO

tnO MM

RRRT "#+5%

=

+=

rO

fO

MM

FNR

- '

=+

rM

R

tMR 0

"#+7%

? tM "%"% rM /

-

# F G "#((%

* #(( > 1

N fF '

3#((4

3#($

,%+!"!

$$5

==

==

==

0:00:00:0

FrM

GNFFFF

O

y

fx

"#+%

!"#+%8:5

F !

n t "#((%

4 fF t

4 N n

@

"#((% *

N "%

-

N : 0 0

/ mm15,0s mm01,0005,0s

#((' N fF rM

=

=

NM

MNsM

r

r

r maxmin 0 "##,%

?

-

"%

NF f 3 fF

F

$$7

#%4 G F "#()%4

F

# 1%C N fF rM

9 '

=+=

==

==

sNMNF

GrFrMMGNF

FGFF

r

f

rO

y

fx

0sin:00cos:0

0sin:0

!'

=

=

=

rGFMGFF

GN

r

f

)sin(sin

cos

'

sGrGFGGF

)sin(cossin

8'

+

+

)cos(sin)cos(sin

r

sGF

GF

1

'

r

s> )cos(sin

r

sGF +

8

r

s

,%+!"!

$$

#%4 G F M "#(+%4

F M

# 1%4 fF rM F M

- '

=++=

==

==

sNMNF

GrFrMMMGNF

FGFF

r

f

rO

y

fx

0sin:00cos:0

0sin:0

!'

+=

+=

=

rGFMMGFF

GN

r

f

)sin(sin

cos

'

+

sGrGFMGGF

)sin(cossin

8,'

++

rr

sGFM

GF

)cos(sin

)sincos(

-

cos

sinG

GF +

!

cos

sinG

GF +<

,

3#(+

$(,

5.3.4 Frecarea de pivotare

4"%

"#(0%'

=

=

nO

n

O MMRR

T

=

=

pOO MM

NR "##$%

- '

=+

pM

N

n

n

MR 0

"##(%

9

"%"% : pM

3"%14 nM :

pM 4 #"#(5%

- "+:8%

)( 22 rRNp

=

pi "##)%

9

dddA = "##+%

)( 22 rRddNdApdN

==

pi

"###%

3#(0

3#(53

,%+!"!

$($

'

)( 22 rRddNdNdF f

==

pi

"##0%

?

'

)( 222

maxrR

ddNdFdM fp

==

pi

"##5%

?

'

NrRrRdd

rRN

rRddNdMM

R

r

R

rA pp

=

=

=

==

pi

pi

pi

pi

22

332

0

222

2

022

2

)( maxmax

32

)(

)( "##7%

3'

2233

32

rRrR

= "##%

'

NM maxp = "#0,%

'

=

=

NM

MNM

p

pp

max

min 0 "#0$%

?

* # " : 5%

'

NRM p 32 "#0(%

3 :

$((

5.3.5 Frecarea n lagrul radial (articulaia cilindric)

4

> :"%*#(

> -"> -

M

?: "% ? : 6

F "

% M?

,06"%

' N > fF > rM >

'

=+=

==

==

sNMNFFrMMM

FNFFFF

r

f

rI

y

fx

0sin:00cos:00sin:0

"#0)%

!"#0)%

=

=

=

sinsin

cos

FrMM

FFFN

r

f "#0+%

3#(8

,%+!"!

$()

"#0)%

'

+

)cos(sin

r

sFrM

tg "#0#%

- *

"#0#%'

tgtg == "#00%; '

tgsinsin1coscos

"#05%

C"#05%"#0#%'

)(r

sFrM + "#07%

1

r

s+= ' "#0%

'

+=

=

22 VHF

MM f "#5,%

"#0% "#5,% "#07%0

22' VHrM f + "#5$%

9 "#5,%

M fM F

" % VHR += H V " 22 VHR += %

3>

*

*"#),% *>

"#),% iN fiF riM

$(+

9 '

=+= 0:0 MMrFM rifiC "#5(%-

80,

#),:

4'

== 022:0 1 rifiiO MrFM "#5)%

!"#5(%"#5)%'

+=+= riri Mrrrr

rMM )11()1(11

"#5+%

& ,

,0 iN #,+'

+==

=

iri

i

f

sNMVHRN

MM22 "#5#%

C"#5#%"#5+%'

22

1)11( VHr

rrsM f ++ "#50%

a) b) c)

3#),8

,%+!"!

$(#

1>'

)11("1rr

s += "#55%

'

22

" VHrM f + "#57%

"#0%"#55%'

'"

$(0

! 0 =%

?

4 > 2 3 ?4 B5 ? 10 Flq = 4 #)+

( ) 0sin: =+ QHOx ( ) 0cos: 1 =++ QFNPVOy B ( ) 0cos3

252: 1 =+ QlMlFlNlPM BA

? '

+++= cos23

245

2 0Q

lMlqPNB . = sinQH . +++= cos2422

3 0 Qlql

MPV

!- 0 =%= @

? 4 aAB 12=

aAD 9= aAE 4= aAH 5= 4

-

0=%=

$

$

$

,%+!"!

$(5

/'

aAxA

xi

iic 16,4=

. aAyA

yi

iic 25,3=

4 "0 =%C%6 ' ? * '

( ) 0:0 321 =++= GSSSZ i ( ) 0:0 1 == cix xGABSM

( ) 0:0 2 == ciy yGADSM '

E,)+@. E,)[email protected],(@!D 0=%D lAB = @ A

.- E ( )lAC 2= F>4

/

#+F +F )F ($0F $0(F

>0F

38

>$0F >0F

+7F - - >(,,F >$#0F $ %

$ &

$ '

$(7

4"#)7%>2

3* '

( ) 0cos: = HSOx ( ) 0sin: =+ SGVOy ( ) 0cos2sin2cos

2: = lSlSlGM A

23 '

( )

cossin4cos

+=

lGS . ( )

cossin4coscos

+

=

lGH . ( )

cossin4sincos

+

=

lGGV

!0=%G

?

4 >23 ?"#+,%

4 B5

10 Flq = B5 ?' 20 Flq = :?'

4 '

( ) 0: =HOx ( ) 02: 21 =+ BNFPFVOy ( ) 04

274

32

: 21 =+ lNlFPlMlFM BA

23?'

0=H . Pl

MlqNB ++= 42425

0 . PlMlqV 3424

230 ++=

$ (

$ )

,%+!"!

$(

!% 0 =%H

4 >

2 3 , 4 B5 1Flqo = : 4 #+(

4 '

( ) 0sin: =+ QHOx ( ) 0cos: 1 =+ QPFVOy ( ) 0cos64

23

: 1 =+ lQlPlFMM A

23,'

= sinQH . += cos6423 2

0 lQPllqM . += cos0 QPlqV

!& 0 =%H

?

4 > 2 3 ?4 ?>

1Flqo = 202Flq =

4#++ '

( ) 0cos: =+ PHOx ( ) 0sin: 21 =+ BNFFPVOy ( ) 03cos

27

25

: 21 =+ lNlPlFlFMM BA

23?

= cosPH . += cos339

110

Pl

MlqNB . lqlMpPV 018

53

cos3

sin ++=

$

$

$

$

$),

!'D ?

?0=%H=

?

4 >

2 3 ?4 B5 1Flqo =

20

2Flq =

4

#+0 '

( ) 0cos2: = PHOx ( ) 0sin2: 21 =+ PNFFVOy B ( ) 0sin423

34

21

: 21 =+ MlPlNlFFM BA

23

'

= cos2PH . ++= sin38

3187

0 PlMlqNB . lql

MPV 0910

3sin

32

+=

!(@ ? :

/ #+5 B5 B5/,4

$

$ %

$ &

,%+!"!

$)$

4

> 2 3 4 B5 1Flqo =

? 20

2Flq =

4#+54 '

( ) 0: =+ PHOx ( ) 0: 21 =+ CNFPFVOy ( ) 04

310

25

31

: 21 =+ lNPllFMlPFM CO

23'

PH = .l

MPlqNC 487

127

0 ++= .

!)D lAB 2= @ 0

=%HI A ?* E

F

/,4?

4 > 2 3 ? F 4

#+

( ) ( ) 0cossin: =+ QNHOx B ( ) ( ) 0sin: =++ QNGVOy B ( ) 02

23

cos: =+ lNlQGlMM BA

$ '

$ (

$ &

$)(

23?'

+= sin43

cos22

QGl

MNB

( ) += cossinsin43

sincos2

sin2

QQGl

MH

( )+++= sincossin43

cos2

cos2

2 QQGl

MGV

5.5. PROBLEME PROPUSE

!G ##, J

: ' / ///

4

'

4 4

>23'4 > !'

PND 2= . PH = . PV =

!

G ? J

: 0 =%= ' / / H/ / 4

'

PND 23

= .

PH 3= .

2PV =

$ )

$

,%+!"!

$))

!@ ?

: 3 0 =%=% 4

PH 3= . lqPV 03 += .

PlMlqMO 1237 2

0 +=

!@ ?

: ##) /B5 B5 / A4

= cos2PH .

+= cos34

1819

0 PlqNC .

+= cos34

sin21817

0 PPlqV

!D ?: @ 0

=%=H A D ?

?:4

=

sin4GS . = ctgGH

4.

43GV =

$

$

$

$)#

CAPITOLUL 6 Statica sistemelor materiale

6.1. ECHILIBRUL SISTEMELOR MATERIALE

6.1.1. Sistemul material ;

:3'$ 3

( 3

6.1.2 Torsorul forelor interioare 4 :

?$?(???:"0$%

/?

iF ijF ":E$(B%

'

jiij FF = "0$%

??: ir jr >@'

===

=+=+=

=+=

0)()(

0

ijijijji

ijjijijijijiO

jiij

O

FMMFrr

FrFrFrFrM

FFR

"0(%

30$

$)0

"0(% > "0$%

ij MM ijF " ijij MMF = %4

6@?'

=

=

jijiOi

jiji

Oi FrM

FR

"0)%

G'

====

===

0FrFrMM

0FRR

i jiji

i i jijiOiintO

i jij

iiint

intO "0+%

"0+% > "0(%

'

6.1.3. Teoreme i metode pentru studiul echilibrului sistemelor materiale

%*!!#!

-

? iF

j

ijF ?'

0FFj

iji =+ "E$(B% "0#%

-

,%+!"!

$)5

%+!!#

*

*"0#%'

0FFi j

iji

i =+ "00%

* "00% ir ?

'

0FrFri i j

ijiii =+ "05%

C"00%"05%"0+%

'

=

=

iii

ii

0Fr

0F "07%

@'

=

=

iiiO

ii

O FrM

FR "0%

C"07%"0%

=

=

0M0R

OO "0$,%

"07%"0$,% .

/

0 $&! 0! 0 - 0 . $ & ! 0! " %

$)7

*

"07% "0$,% "

%

9"0$,% "0#%

%+!#!

!

6

6.1.4. Sisteme static determinate i static nedeterminate 9

9

1 0)

!

!

1

*

,%+!"!

$)

$ %

30(

6.2. PROBLEME REZOLVATE !%6 aAB 2= aCDBC ==

@@?"0(%4 A> ? 4

?

4

*KL-$"0(%'-("0(%'

( )( )( )

==

==

==

0:00:0

0:0

aHM

GVVYHHX

CB

BCi

BCi

-)"0(%'

( )( )( )

=+=

==

==

0sincos2sin2:0

0:0

0:0

DCCD

CDi

CDi

MGaaHaVM

GVVY

HHX

4'+2?3?

2323,%'

GVN BA == .0=== DCB HHH .

GVC 2= .GVD 3= .

= sin5aGM D

30(

30(

$+,

30)

Fig. 6.3.b

Fig. 6.3. c

!%- 0) A E

4

/@?D

4

*KL-$"0)%

( )( )( )

=++=

==

==

0sinsin2cos2:00:0

0:0

GllVlHMGVVY

HHX

OOA

OAi

OAi

-("0)%

( )( )( )

=+=

==

=+=

044:002:0

0:0

GllVMGVVY

HHX

MO

MOi

OMi

-)"0)%'

( )( )( ) ( ) ( )

( )

=

+=

==

==

0sin2coscos2:0

0:00:0

pipipi

lHGllVM

GVVYHHX

B

BB

BMi

BMi

30)

,%+!"!

$+$

30+

Fig. 6.4.c.

4 ' 2 3 2 3 2, 3, 2? 3? %

' GgHHHH MOBA 23

====

GVV bA 2== . GVV MO = !%40+

/H: / 5 / : A - @ M 4

@

aCAOC 2==

4

*KL-$"0+%'

( )( )( )

=+=

=+=

==

0cos432:0

0cos2:0

0sin:0

2 aNaPaPM

NPaPVY

NHX

COO

COOi

COi

-("0+%'

( )( )( )

==

==

=+=

042:00:0

0:0

1

1

1

aNaNMMNNVY

THX

ACOO

CAOi

Oi

-)"0+%'

( )( )

==

==

0cos:00sin:0

GNYGTX

Ai

i

Fig. 6.4.a.

Fig. 6.4.b.

$+(

8'

ANT '

( ) 0cossin G 47'/+23,

+% 0cossin G

'

== sinGTH O . = cosGNA . ( )PapN OC 32cos41

+

= .

( )PaptgH OO 324 +

= .4

6 PapV OO+

= . ( )PapGV OO 32cos41

cos1 ++= .

( ) ++

= cos464cos41

aGPapaM OO

!%D 0#

?

! lAB 4= @

lCDBC 2== @ 8

A E ?

4

*KL-$"0#%'

( )( )( )

=+=

==

==

04:0

0:0

0:0

1

lHMM

VGVY

HHX

BAA

BAi

BAi

-("0#%'

( )( )( )

=++=

==

==

0sin2cos2sin:00:0

0:0

2

2

lVlHlGMVGVY

HHX

CCB

CBi

CBi

30#

30#

30#

,%+!"!

$+)

-)"0#%'

( )( )( )

====

=+=

0cos2sin:0

0:0

0:0

12

2

1

lFlGM

GVY

FHX

C

Ci

Ci

4'23,2?3?23AE%'

1FHHH CBA === .

21 2GGVA += .

14lFM A = .

22GVB = .

2GVC = .

2

1

32GF

tg = .

2

12GF

tg =

!%4*?

?* aBCABBBAA ==== '' > ??*??@*: A

?*?* 00

4

*KL

$ %

300

$++

-$"00%'

( ) 0:0 == AAi HHX ( ) 02:0 ' =+= GVVY AAi ( ) 0:0 == aHM AA -("00%'( ) 0:0 == CBAi HHHX ( ) 04:0 =++= CBAi VGVVY ( ) 024:0 == aVaVGaM BBA ( ) 042:0 =+= aVGaaVM AAB

-)"00%'( ) 0:0 == BBi HHX ( ) 02:0 == BBi VGVY ( ) 0sincos2cos2:0 == aHaVaGM BBB

4> $,

'232*3*2?3?2*?3*?23%4'

0' ==== CAAA VVHH .GVA 2

'

= .

GVB 4= .GVB 6' = .

( )=== ctgGHHH CBB 6cos2'

300

300

300

,%+!"!

$+#

!%%* 05

(F@/,/ A 4

/ , , 4 '

11 2lMO = . 22 2lMO = "$%/' .-"05%'( ) 0:0 21 == HHX Mi ( ) 0:0 12 == QVVY Mi ( ) 02:0 212 == lHM MO -"05%'( ) 0cos:0 11 == Mi HPHX ( ) 0sin:0 11 =+= PVGVY Mi ( ) 0sin22:0 11111 =+= lPlVGlM MO .("(%/"05%'

( ) 0cos:0 22 == PHHX Mi ( ) 0sin:0 22 =+= PVQVY Mi ( ) 022cos:0 2222 =+= lHlPM MO -05'

( ) 0:0 21 =+= Mi HHX ( ) 0:0 21 =+= Mi VGVY ( ) 02:0 1211 == lVGlM MO

305

305 305

305c

305

$+0

.(/"05%'

( ) 0:0 232 == HHX Mi ( ) 0:0 322 == QVVY Mi ( ) 02:0 2322 == lHM MO

-"05%'

( ) 0:0 311 == Mi HHX ( ) 0:0 311 =+= Mi VGVY ( ) 02:0 13111 == lVGlM MO

-?!"05%'

( ) 0cos:0 3231 == PHHX MMi ( ) 0sin:0 3132 == PVVY MMi

*)

'

++=

=

=

=

2sin

02

cos

2

2

1

1

GPQV

H

GV

PH

* ,

%2

sin; 11GPVOH MM +==

%2

;cos 22GVPH MM ==

%2

sin;cos2;2

;cos 32323131GPVPHGVPH MMMM +====

305

305

305

,%+!"!

$+5

!%&* C%I

( : = 30 / M - " # @%@4

@

min11 GG = ? 3= lBCAB ==

4

*KL

-$"0C%I%%% ( )

( )

==

==

0cos:0

0sin:0 2PNY

PTSX

Bi

i

BNT -("07%

( )( )( )

==

=+=

=+=

02:0

0:0

0:0

lNlNM

VNNY

THX

CBA

ABCi

Ai

- ) " 07%

'

( )( )( )

==

==

==

0:0

0sin:0

0cos:0

21

21

2

rSRSM

SSGVY

SHX

O

Oi

Oi

307.

307

307

307

$+7

30.

-+-0C%I%.(

( ) 0:0 11 == GSYi 4

'@32+?+23

/'

( ) ( )

+ cossin;cossin1 PR

rPRrG

-'

( )== sincosmin11 PRrGG

'

( )= cossin1 PRrS

( )= cossin2 PS . TH A = . === cos22PNNV CBA .

( )= cossincosPHO .( )

++= sin1cossin

r

RPGVO

!%'*>/

#@D F > 0

4

N O

'4

'

( ) ( )++= sinaRR 4'

= 0PM .( ) 0sinsin =+ aQaRG

'( ) sinsin Qa

aRG +=

( )( ) RaR =++ cossinsinsin

307.

,%+!"!

$+

( ) ( )aR

RQa

aRGaQ

aRG+

=+++ cossinsin1sin 22222

*' 22322

2321121 2sin2sin

22cos

4212

cos4

2 =++

' ( )

22

22

1aQ

aRG += .

aRR+

=2 .( )Qa

aRG +=3

( )

+= sin1arcsinaQ

RG

-!%(4 0$,

6 # # @ M D @ RAO 81 = RAB 2= ! /

4'%8 .%

8

% 4 " 0$,% >

'

( ) 0:0 01 =++= PHNX i %( ) 02:0 1 =+= Oi VGTY %( ) 0810:01 =+= lNlPMO %

NT %

30$,

30$,

$#,

4 " 0$,% '

( ) 0:0 2 =+= NHX Oi ( ) 0:0 2 == SGTVY Oi ( ) 02:02 == RSRTMO 9

"0$,%'( ) 0:0 3 == Oi HX ( ) 0:0 3 =+= PVSY Oi ( ) 042:03 == lSlPMO * -.

+

8

! -. ' PN45

= -6. 2PS =

-.4PT =

' PP45

4 '

,

5PF

%

/

25-.'

41FFNHO ==

--.'

4221

PGTGV O ==

! -. FNHO 45

2 ==

-0.'

GPPGPSGTVO +=++=++= 43

241

!-&.-.'03 =OH

23PSPVO ==

30$,

30$,

,%+!"!

$#$

!%)40$$>8

@#/ AMD?

* : / B!

4'%/ .%?

% - -

4 C

0$$9'( ) 0cos:0 == PNX Ci %

( ) 0sin:0 2 == PTSFY %( ) 0sin2:0 2 =+= RPRSMM rC %

* '

CNT %Cr sNM %

30$$

30$$

$#(

* " 0$$%

'

( ) 0cos:0 21 =+= SHX Oi %( ) 0sin2:0 211 == SSGVY Oi %( ) 02:0 211 == RSRSMO %

*

?"0$$% '( ) 0cos:0 2 =++= BOCi NHNX %( ) 0sin:0 2 =++= BOi NTVY :%( ) 036cos:02 == RNRNM CBO H%- " 0$$%

' ( ) 0:0 1 =+= OOi HQHX %

( ) 0:0 1 == OOi VVY % ( ) 0989

32

:0 0101 =+= RHRVQRMM OO %

'2

92

qRqOAQ == / -. '

09827 12

=+ RHRVqRM OqO 3

@"0$$%

( ) 0:0 1 == Oi HX %( ) 0:0 13 =+= GSVY Oi %( ) 023:0 13 == RGRSMO %

!-.' = cosPNC

6

-.' GS32

1 =

*-&.

'32

12

GSS ==

30$$

30$$

30$$

30$$

,%+!"!

$#)

3-.' == sin3

sin2 PGPST

!-.' cossin3

PPG

' ( )+ cossin3GP

8 "%'

( ) cossin3GP

/!-.,'

RPRGRPRSM r == sin32

sin2 2 -.'

cossin3

2sPRPRG ( )+ cossin3

2sR

RGP

8 /

'

( ) cossin32

sRRGP

/'

( ) ( )+ cossin3cossin3GPG

( ) ( )

+

cossin3,

cossin3GGP

/'

( ) ( )+ cossin3cossin32

sRGP

sRRG

( ) ( )

+

cossin32

,

cossin32

sRRG

sRRGP

( )

( )

+

+

cossin

2,

cossin1

min3

.

cossin

2,

cossin1

max3

Rs

G

Rs

G

P

$#+

%

/?-P.'

2cos2cos

cos2PPNN CB =

=

=

'23!-6.'

3sin

2sin

23sinsin2

GPPGPNTV BO ===

2

-.'

=== cos2

cos2

coscos2PPPNNH BCO

--0.-.

== cos3

cos21GSHO

+=++=++= sin33

8sin

3322sin2 211

GGGGGSSGVO

*'32,3-.(

+== sin33

81

GGVV OO

!-.

2'

2

9cos

32

1qRGQHH OO ==

?-.'

( ) +++= cos3sin83

827 21 GRGqRM

,%+!"!

$##

6.3. PROBLEME PROPUSE !%* 0$(

> # : M? ? @ J ?

@ ! : 4

/ LABOA ==

( )+ tgGP

211

PHO 2= . GPVO 3= . ( ) GLtgPLMO 521 += PTHH CB === . ( )GPtgVB = 2 .

GPtgVC = 2 . ( ) GtgPNA += 12 !%! F

??/ABOA = : I

0$)

4

lDBAD ==

QTHH AO === .

2GVV AO == .

= 0sincos2

2 QGlMO .

0=rM . GND =

30$(

30$)

$#0

!%;#/@

H0$+9 FM

lOB 3= F ?'

P

RSPQ

2,min .

= P

RSPQ

2,minmax

QTHO == . GPVO += . ( )PGlRQMO 322 += .

PNB = . RQM r 2= !%@@ A

"%9?

M4

/

?

'

== ctgGRrPTHO 2

. PVO = .

RrPS = . GNA =

= ctgGNB 2.

+

2,0 ctg

r

RGP

30$+

30$#

,%+!"!

$#5

!%30$0D:

"M%6($

)H5

) -M % + D # B

4'%/ .%/%

30$0

$#7

6.4. GRINZI CU ZBRELE

6.4.1. Ipoteze simplificatoare G

G

"0$5%

*

*'$ 8( 1

! :

"0$7% "0$7%

) 3 >

+ G : !

30$7

30$5

,%+!"!

$#

6.4.2. Eforturi n bare 4 Q6 6

iR jR "0$%

=

=+

0Rij0RR

j

ji "0$$%

0R,0ij j "0$$%ijR j = "0$(%

6Q6!"0$(%

Q6'

ji RR = "0$)%

1 ijS jiS 6 Q6"0$)%'

jiij SS = "0$+%

/

! 09'

!R>"0$%

!R>"0$%

30$

a

)

b)

c

)

$0,

6.4.3. Grinzi cu zbrele static determinate !

4

8 .

1 1' 2

.

:*'

% S:

G

:

% S7

G

*:'

%:).7

8

%7):

1)

%7)7

G

% S9

G!

,%+!"!

$0$

6.4.4. METODE PENTRU DETERMINAREA EFORTURILOR DIN BARE

%*!!#!!

9

?

'

0SFj

iji =+ "0$#%

iF ijS 6 )6

)k1j(,Sj

ij = P

#'$ .( *

.) 9R>9

R> * RT>>.

+ ! ) .

# -

?

: >>>

%*!!

?

*

>

3'

#'$ 4 1 P

P7P)

$0(

( 4

) * " %

> !

6.5. PROBLEME REZOLVATE !%4 0 C%5

kNP 11= = 30 lAB = ;

?'

( ) 00 == PHF Ax ( ) 00 =+= BAy NVF ( ) 030 == lNlPM BA

/

+?(

kNPl

lPABACNB 05,19

3===

' kNNV BA 05,19== kNH A 11= 4 -0

C%5%).4

4 '

( ) 0cos0 2 =+= TPFx ( ) 0sin0 21 == TTFy

'

===

==

=

kNT

kNPT

35,63

1121

322

7,123

22cos

1

2

30(,

30(,

,%+!"!

$0)

4 U

'

( ) 060sin0 32 =+= TTFx ( ) 060cos0 62 == TTFy

4'

==

==

kNT

kNT35,630sin7,12

1130cos7,12

6

3

4' '

( ) 030cos0 53 == TTFx ( ) 030sin0 541 == TTTFy

'

==

=

=

kNTTT

kNTT

7,1230sin

7,1230cos

1

514

35

*J'

( ) 030cos0 57 =+= TTFx ( ) 030sin0 5106 =+= TTTFy

4'

=+=

==

kNTTT

kNT

7,12217,1235,630sin

11237,12

5610

7

*

( ) 030cos0 97 =+= TTFx ( ) 030sin0 984 == TTTFy

'

==

=

=

kNTTT

kNTT

5,1930sin

7,1230cos

1

948

79

9 ?

30(,

30(,

30(,

30(,

$0+

!%4

0 C%

4

? "0 C%%% '

( ) 020 =+= PHF Ax ( ) 00 =+= BAy NVF

( ) 02220 == aPaNM BA '

=

=

=

pVPNPH

A

B

A

222

4

4

( ) 0245sin0 1 == PTFx ( ) 045sin0 12 =+= TTFy

'

==

=

PPT

PT

22

22

42

4

2

1

4 '

( ) 00 3 == TFx ( ) 00 24 == TTFy

'

=

=

PT

T2

0

4

3

4 '4 '

( ) 045sin0 35 =+= TTFx ( ) 045cos0 361 =+= TTTFy

30($

30($

30($

30($

30($

,%+!"!

$0#

'

==

=

24

0

61

5

PTT

T

*?'

( ) 045sin0 57 =+= TTFx ( ) 0245cos0 54 =++= PTTFy

' 07 =T 9 '

PTT 245cos54 = PP 22 =

! !%4

? 0C% kNP 4=

-

"0C%%.

.

( ) 00 == PHF Ax ( ) 00 =+= TVF Ay ( ) 0430 == aTaPM A

'

==

==

==

kNPV

kNPT

kNPH

A

A

34

3

34

34

4

/

'

( ) 060cos0 12 =+= TTFx ( ) 060sin

430 1 =+= TPFy

30($

30((

30((

30((

$00

30((

'

===

==

kNTT

kNT

121260cos

23

23

12

1

44

' ( ) 060cos60cos0 143 =+= TTTFx ( ) 030cos60sin0 14 =+= TTFy 4'

==

==

060cos60cos2

413

14

TTTkNTT

4' '

( ) 060cos60cos0 2456 =+= TTTTFx ( ) 060sin60sin0 54 =+= TTFy

'

=++=

==

kNTTTTkNTT

160cos60cos2

5426

45

*8'

( ) 060cos60cos0 573 =++= TTTPFx ( ) 060sin60sin0 75 == TTFy

! 85' kNTT 257 ==

30((

30((

,%+!"!

$05

!%30C%

/ 4

-

? -0 C%%%.

'

( ) 00 =+= ABx HNF ( ) 020 == PPPVF Ay ( ) 0

23

23

2320 =++= lPlPlNlPM BA

4'

=

=

=

235

42

35

PN

PV

PH

B

A

A

4''

( ) 030cos0 21 =+= TTFx ( ) 030sin0 2 =+= TPFy '

=

==

PT

PPT

2

3232

2

1

4

'

( ) 060sin0 24 == TTFx ( ) 060cos0 23 =+= PTTFy

'

==

==

323

221

24

23

PTT

PTPT

30()

30()

30()

30()

$07

*

'

( ) 060sin30cos0 156 =+= TTTFx ( ) 030sin260cos0 635 =++= TPTTFy

'

=

=

=

=+

PT

PTPTT

PTT53

82

6

5

65

56

-

5 '

( ) 060sin2350 6 == TPFx

( ) 060cos40 67 == TTPFy '

2360cos4 767

PTTPT ==

9 ?

!%D mAB 3=

?' ? ! ?/-0C%H. * ' -0C%H%.% 4

4

* 4 '

( ) 060cos20cos0 21 =+= TTFx ( ) 060sin20sin0 21 =++= PTTFy

'(,JE,)+(

(,JE,)#$('

:,557/:>$+0$/

$(

30()

30()

30(+

30(+

,%+!"!

$0

*?. '

( ) 045cos60cos45cos60cos0 45 == TTFx ( ) 020cos45sin60cos45sin60cos0 145 =+= TTTFy ( ) 020sin60sin60sin0 1453 =++= TTTTFz

* + # )

PTTT 032.145cos60cos2

20cos154 =

==

PTTT 524.160sin220sin 413 == !%%60C%=

lCECDDEDOADEOBECBAC =========

4#

4

'%

:.%?

$5,

9 '

( ) 00 0 == HFx "%( ) 00 0 == GFVFy "%( ) 000 == rFRGM "%

!"%'r

RGF =

%-?"0C%=%%'9 '( ) 00 0 == Ax HHF "%

( ) 00 0 =+= VNVF BAy "%( ) 0300 =+= lNlVlHM BAA "%

! "% 00 =H . "% 0=AH *"%8'

+=+=+= 10

r

RGGr

RGGFV

9 "% "%'

+=== 1

222 00

r

RGVNlVlN BB

!"%

3'

+== 1

2 rRGNV BA

%9 / "0 C%=%%

'

04

cos4

cos 890 =+pipi TTH "%

04

sin4

sin 890 =pipi TTV " %

-

0C%=%'

04

cos21 =pi

+ THT A "%

04

sin2 =pi

+ TVA ":%

30(#

30(#

30(#

30(#

,%+!"!

$5$

C-0C%=%. '

04

cos4

cos 3514 =pi

pi+ TTTT "H%

04

sin4

sin 35 =pi

+pi TT "%

9 ? -0 C%=%0.

'

04

cos64 =pi

+ TT "%

04

sin6 =pi

+ TNB "%

- 0 C%=% '

'

04

cos4

cos4

cos 6957 =++pipipi TTTT "%

04

sin4

sin4

sin 659 =pi

pi

pi TTT "%

C-0C%=%&.

'

04

cos4

cos 3728 =pi

pi+ TTTT "%

04

sin4

sin 37 =pi

+pi TT "%

4 00 =H "%

98 TT = !" %'

222 80 = TV

20

8VT = '

+== 1

298 rRGTT

A":%'

+= 1

22 rRGT

/"%'

+= 11

r

RGT

90

"%'

30(#

30(#

30(#

30(#

$5(

+= 1

26 rRGT

H"%'

+== 1

22 6

4r

RGTT

*="H%'

0222 145 =+ TTT =E,

"%E,95"%DE,!%&4 0 C%C / >

>F # - ? * 2 @ !

4

'%8 .%+.%$()

30(0

,%+!"!

$5)

% 4 -0C%C%.9 '

== 00 0HFx "% == 00 0 TQFVFy "% == 000 RTrFM "%4

"0 C%C%% 4 B

=+= 0cos20 0 PqlHHF Ax "% == 0sin0 0VPVF Ay "%

== 0452sin20 0020 lHlVlPqlMM A "%4 "0 C%C%%

'

== 00 ox HF "% =+= 00 GVTF oy " % == 030 lTlGM B "%!"%'

r

RTF =

!

"%'

3GT =

8'r

RGF =3

% ' 3 2 , /

/"%' += sin0 PVVA

35

"%'330GQ

r

RGTQFV ++=++= '

++

+= sin1

3PQ

r

RGVA

30(0

$ 0(0

30(0

$5+

!"%2( += cos20 PqlHH A '

= cos2 PqlH A ?"%'

Qlr

RGlPlqlVlPlqlM OA +

+++=++= 51

35sin225sin22 22

% - $ ( )

"0 C%C%.

>

9 '

== 04cos0 231

piTTTHF Bx ":%

=pi

+= 04

sin0 2TGVF By "H%

=pi

pi= 02

4cos

4sin20 223 lTlTlTlGM B "%

-(

3?" %'

3

23

GGGTGVB ===

*"H%'

3

22

23

22

32

2GG

GGT ==

=

A)

"%'

03 =T !":%'

34cos231

GTTHT B ==pi

30(0

,%+!"!

$5#

6.6. PROBLEME PROPUSE !%%4

0 C%D kNP 12= = 60 aAE 2=

!%%G 0 C%I%

? 4 ' lCABCAB === 4

?

!%%4 0 C%G

? 4

lHBFHAF === 3lAD = 332lEGDE ==

!%%4 0 C%5% 4

$ ( +

"%

30(5

30(7

30(

30),

$50

!%%4

*

0C%%4 lAB 2= /

!%%%4

0 C%

?//4:

"

%

6.7. STATICA FIRELOR !%&;

?&4J A ? "0 C%. 4 ?

6?

a

xchapypT BBa ==

4'

tgdxdy

a

xsh

B

B=

=

30))

30)$

30)(

,%+!"!

$55

cos12 ap

a

xshapT BB

=+=

' 21 LLL += 12 hhh = 8

+=+=

a

lsh

a

lshaLLL 2121

'

==

a

lch

a

lchahhh 1212

;'

+++=

a

lch

a

lch

a

lch

a

lch

a

lsh

a

lsh

a

lsh

a

lshahL 121222212212222 22

a

lshahL

24 2222 =

22

22 hL

a

lsha =

-

!%&4

? "0 C%H.% 6 4

8 I

9

a

xchay =

30)+

$57

6?'

BBA ypTT ==

4?

Lpa

lchapTT BA ===

!

Lpa

lchap =

a

lshaL = 2

a

lshap

a

lchap = 2

;

2

22

a

la

la

la

l

eeapeeap

=+

3ln2 =a

l

-

3ln

2la =

*

3ln334

313

3ln2

222 lleea

a

lshaL

a

la

l

=

=

==

4

( )

3ln333221

23

13

3ln21

2

=

+=

+===

lleeaa

a

lchaayf

a

la

l

B

; A'

33

23

13

2=

=+

===

a

la

l

ee

a

lshytg

6pi

=

,%+!"!

$5

!%&4

0 C%= A ?4

2

?J0

9

Hxp

a

xy22

22

=

=

4'

H

xptg

dxdyy ==='

4 ?

2l

x = "0C%=%.

62

12 =

tgplH

40

1

2

2

max 4822

tglH

plH

lpfyB ==

==

6?'

11 sin2cos

=

==

plHTT BA

*'

( )2'222 11 ydxdxdydxdydxds +=

+=+=

4 6L

dxdy

( )[ ]2'1 ydxds + C'

2

322

22

222

2 2421

Hlpldx

HxpdsL

l

l

l

l

+=

+==

+

+

30)#

30)#

$7,

!%&4 J

"0C%C%4

= tg 4

4?-0%C%C%. ( ) 0cos 0 == TFF fix

( ) 0sinN 0 == TFiy NF f

!'

= cosTF f = sinTN

-' NF f = 480+' = sincos TT

=

tg1 = tgctg

4

== tgxHp

shdxdy

4

= ctgxHp

sh

30)0

30)0

,%+!"!

$7$

ctgeeHpx

Hpx

=

2

012 = ctgee Hpx

Hpx

4

2

= ctge Hpx

4'

2

ln = ctgpH

x

!J

Hpx

shpHL 2=

:

= ctgpHL 2

'

= LtgpH2

!?'

2

ln2 === ctgpH

xlAB

2

ln = ctgLtgl

-

2

ln ctgLtgl

$7(

!%&;

0 C%D *,4

4 ' G$ G( 4

$"0%C%D%.' ( ) = 0iyF 011 = GT (-0%C%D%. ( ) = 0iyF 022 = GT 3"0%C%D%"%4@ '

23

1

pi eTT "%

23

2

pi eTT "%

!

11 GT = 22 GT =

"%"%'

23

1

pi eTG

23

2

pi eGT

* pi 321 eGG pi

< 3212 eGGG !

pi 32

1 eGGl

2

1ln31

GG

pi

30)5

30)5

,%+!"!

$7)

!%&%

4

4 , "0%C%I%.

-

r

MF f = "%

-F9 '

( ) 0 0 21 =+= QTTFix ( ) 0 0 21 == MrFrTrTM fiO

9 9'

eTT 21

4

r

MQT +=1

r

MQT =22

4'

+ e

r

MQr

MQ

9F'

r

Me

eQ

+

11

* '

eTT 21

=sin

'

E :

sin21 eTT

91

sin

1sin

+

e

e

r

MQ

30)7

CAPITOLUL 7 Cinematica punctului

!"

# $

!

7.1 NOIUNI FUNDAMENTALE

7.1.1 Legea de micare " "

$! r %&'

)t(rr = %&'(

! %&' !)

7.1.2 Traiectoria

! * +,

!! )t(r %&-'

# ! )t(r ,

ktzjtyitxtr )()()()( ++= %&-'

k,j,i !$

.&

.&-

/

($! )t(r

)t(zz);t(yy);t(xx === %&0'(%&0'1

%',

0)z,y,x(f;0)z,y,x(f 21 == %&2'

%'

#

+ %&0'

(%' (

(

! ,

)t(ss = %&'

7.1.3 Viteza " !

!

%' ! # ! )t(r ! ( )ttr + %&2' 3!

! r)t(r)tt(rMM 21 =+= *

t

r

$ %

%' ! 4 + ! 0t 12 MM

5+$&&&1,

.&0

.&2

&'&%

&

rdtrd

t

rlimt

MMlimv0t

21MM 12

&====

%&/'

* %&/' ! ! !

%!

!!'

6!,

vdtds

dsrd

rdrd

dtrd

v === %&&'

,

sdtds

;1dsrd

;rdrd

&=== %&'

!

7.1.4 Acceleraia 4!

! !

!

%' ! # !+ !

v)t(v = vv)tt(v +=+ %&' 6!!,

*

t

v

!!

%(+1+! 0t 12 MM &&&,

rvdt

rddtvd

t

vlima 22

0t&&& =====

%&7'

# !

! r

!%%&4!

! !dtad

.&

7.1.5 Viteza i acceleraia unghiular +

89 + ,

)t( = %&:' !

! # !+ 9 =)t(

+=+ )tt( %&/' 6 9 !,

=+=+ )()t()tt( *

t

$&!(%(

+1 + ! 0t 12 MM $&!(&&&,

&===

dtd

tlim

0t %&'

+ !

!#!+!9 =)t( +=+ )tt( !!9!,

=+=+ )()t()tt( *

t

! ! 9

&!( % ( +1 + ! 0t 12 MM &!(&&&,

&&& ===== 2

2

0t dtd

dtd

tlim %&-'

(!!9!

! ! ! 9 !9

.&/

&'&%

7

7.2 STUDIUL MICRII PUNCTULUI N SISTEMELE DE COORDONATE CARTEZIAN I NATURAL

7.2.1. Sistemul de coordonate cartezian 4

! r ! v

a %&&'

6$,

kzjyixr ++= %&0'),

)t(zz);t(yy);t(xx === %&2'5 %'

6!!

,

kzjyixdtrd

rv &&&& ++=== %&'

!,

zvyvxv zyx &&& === ;; %&/'

"!,

2222z

2y

2x zyxvvvv &&& ++=++= %&&'

# ! ! $ ,

v

z)k,vcos(;v

y)j,vcos(;v

x)i,vcos( &&& === %&'4!

!!

!,

kzjyixdt

rdrakvjviv

dtvd

va zyx &&&&&&&&&&&& ++===++=== 2

2

; %&7'

,

zva;yva;xva zzyyxx &&&&&&&&& ====== %&-:'

",

2222z

2y

2x

2z

2y

2x zyxvvvaaaa &&&&&&&&& ++=++=++= %&-'

.&&

7:

# ! $ ,

a

z

a

vkaa

ya

vjaa

x

a

via zyx &&&&&&&&&

====== ),cos(;),cos(;),cos( %&--'

# 0z = %&'6$,

yv,xv;jyixv yx &&&& ==+= %&-0'"!,

222y

2x yxvvv && +=+= %&-2'

!91!!$,

x

yv

vtg

x

y

&

&== %&-'

4,

yvaxvajyixjviva yyxxyx &&&&&&&&&& ====+=+= ,; %&-/' ",

222y

2x

2y

2x yxvvaaa &&&&&& +=+=+= %&-&'

91!$,

x

yv

v

a

atg

x

y

x

y

&&

&&

&

&=== %&-'

7.2.2. Cinematica punctului material n coordonate polare ( 4

8%'

)(trr = ; )(t = %&-7') %&-7'

)+

)(rr = %&0:'6 u

u

.&

y

O

A

x

r

u

u a

a

a

C

v

v

v

.&7

&'&%

7

6 u u +14!!

! u u +

.,

jiu sincos += ; jiu cossin += %&0'#!+

,

==

=+=

ujiuujiu&&&&

&&&&

sincos

cossin %&0-'

6$! u

urr = %&00'

6

+=+== ururururrv &&&&& %&02'

#

uvuvv += %&0'

!

rv &= ; &rv = %&0/' v v !

22222 && rrvvv +=+= %&0&'4 ( ) ( ) urrurrurururururva &&&&&&&&&&&&&&&&&&& ++=++++== 22 %&0'# . uauaa += %&07'

#+

2 &&& rra = ;( )dtrd

rrra

&

&&&&212 =+= %&2:'

",

( ) ( )22222 2 &&&&&&& rrrraaa ++=+= %&2'

7-

7.2.3. Sistemul de coordonate intriseci % %& &

&& % )& %&:'

!+

$,

! ! ;

! !;

! + , = ,,

(!,*&&( ! . !

,

dsrd

= %&2-'

.,

dsd1

= %&20'

+ ,

)t(ss = %&'6 r $,

[ ])t(sr)s(rr == %&22'6 !+ !

+

%&-&',

sdtds

dsrd

rv && === %&2'

!$.,

0v;0v;sv === & %&2/'*!8,

svv &== %&2&'

.&:

&'&%

70

4!+!

+%&20',

2s

sdtds

dsd

ssdtd

sdtsd)s(

dtd

dtvd

a&

&&&&&&&

& +=+=+=== %&2'

$.,

0a;vsa;vsa22

===== &

&&& %&27'

",

2

42

2

4222 vv

ssaaa

+=+=+= &

&&& %&:'

4

! %&'

+$, # 0a.,ctv ==

; 4

,

===

===

rectiliniemiscare;01;0v;01v;0a

uniformamiscare.ctv;0v;0a2

&

a $ ! !

a !!

# 0av > 0av

72

7.3. MICAREA CIRCULAR

7.3.1. Studiul micrii circulare n coordonate carteziene (,!+!

9$,

==== &&&& ;);(t %&'

%&-' ) ,

)(sin);(cos tRytRx == %&-'(

-. ! , %

',

222 Ryx =+ %&0'

! $ ,

====

====

xcosRcosRyv

ysinRsinRxv

y

x

&&

&& %&2'

6!$,

jxiyv += %&' OM ,

0xyxy)jyix()jxiy(OMv =+=++= "!,

Ryxvvv 222y2x =+=+= %&/'

!!,

====

====

yxRRRRva

xyRRRRva

yy

xx

22

22

sincossincoscossincossin

&&&

&&& %&&'

6$,

jyxixya )()( 22 += %&'

.&-

&'&%

7

,

4222242222

22222y

2x

R)yx()yx(

)yx()xy(aaa

+=+++=

=+=+= %&7'

7.3.2 Studiul micrii circulare n coordonate naturale ( , !+ !

9$,

==== &&&& ;);t( %&/:')%&0',

( ) )t(Rtss == %&/'

6!$,

RRsv === && %&/-'!,

0v;0v;Rv === %&/0',

Rvv == %&/2'

6,

+=

+=

+= RRR

RRssa 222 )(&

&& %&/'

,

0a;Ra;Ra 2 === %&//' ",

4222 Raaa +=+= %&/&'

.&0

7/

&* ! 9 .ct0 == 0== &

9,

00 ;;0 +=== t %&/' &*$ 9 .ct0 ==

9,

00

2

00 t2t

;t.;ct ++=+=== %&/7'

7.4. PROBLEME REZOLVATE (&2

,

+=

=

+=

136

12

tz

tytx

,(

,4,

21

=x

t ;6y

t = ;3

1=

zt

)+1,3

162

1 ==

zyx

=

=

2233266zx

yx %'

9!

=

=

12333

zx

yx

&'&%

7&

!0,

==

==

==

362

zv

yvxv

z

y

x

&

&

&

.7499364222 ==++=++= zyxv &&&

0$,

==

==

==

000

za

yaxa

z

y

x

&&

&&

&&

0222 =++= zyxa &&&&&&

* !

4t

r = t8pi = !

+$,(

pi

8=t 3+

4t

r =

),pi

2=r

!,

==

==

32

41

trv

rv

pi

&

&

"!,64

141

641

161

3241 22222222 tttvvv pi+=

pi+=

pi+

=+=

&'&%

77

!

, tbear = ; ct= ; !

,(

ct

= 3+

cb

aer

=

),c

b

aer

= !,

==

==

bt

bt

acerv

aberv

&

&

"!,

( ) ( ) 222222 cbaeaceabevvv btbtbt +=+=+=

-::

1!

&2 lABOA == "! !9 .ct= !

dAC =

$

$ ! $ 98 / (

! $ !-.,

t)t( = = $ !

jyixjtsin)dl(itcos)dl(jsin)dl(icos)dl()t(rOC

+=++=

=++==

$ 8 -.,

=====

+=+=+=+==

tsin)dl(sin)dl(sindsinlAAAACOytcos)dl(cos)dl(cosdcoslCAAOCOx

,

tdl

yt

dlx

=

=+

sin;cos

(-.!

%',

1)dl(y

)dl(x

2

2

2

2=

++

$ )dl( + )dl( 6,

tcos)dl(yv;tsin)dl(xv yx ==+== && 6!,

[ ]jtdlitdljvivv yx ++=+= cos)(sin)(

.&2

&'&%

-:

4,

ytsin)dl(yva;xtcos)dl(xva 22yy22xx =====+=== &&&&&& 6,

rjyixjtdlitdla 2222 )(sin)(cos)( =+=+= ! r (&2/"! ,

! 9 8

,

98 & !

,$ $$! $

98 + $ )t(xxC = $!, t =

IAAOOIxC +==

,

=

=+=

==

==

===

=

==

)tsin1(tsinR2)1tsin4tsin4(1R

)1tsin2(1RIA)1tsin2(R)1sin2(R

RsinR2ACAACACAACIA

tcosR2cosR2AO

2

2

22

))tsin1(tsint(cosR2)t(xxC +==

))tsin1(tsin2)tsin21(tcos

t(sinR2xv CC

++== &

.&

-:-

(&2&1

/ , ! 9 !

%&/'

> $ $

$ 98 6 $ )t(xx = + .ctl = + 9 -. ! 9 % '

)x( = ,

BMBAAA llll 00 ++=

,

=

=

==

=

22BM

BA

AA

Rxl

)x

Rsin:OBM(

x

RarcsinRRl

tRl

0

0

,22 Rx

x

RarcsinRtRl ++=

#!+$

,

0)Rx(xRxxR;Rx2

xx2

x

R1

x

xR

RR0 222222

2

2

2=+

+

+= &&

&

22 Rx

xRvx

==

&

.&/

&'&%

-:0

(&2 !

, dr = ; ( )tcos12

=pi

!,5 dr = [ ]pi ,0 !,

==

==

td

rv

rv

sin2

0pi

&

&

"!,

td

td

vvv sin2

sin2

222 pipi =

=+=

4,

+=

=

&&&&

&&&

rra

rra

2

2

=

=

=

pi

pipi

cos2

sin4

sin2

222

da

td

tda

!,

pipipipi 24222

22

22 cossin42

cos2

sin4

+=

+

=+= t

ddt

daaa

"82pi =

-:2

(&27

&'&%

-:

,

===

===

=

0)0()0(Rv)0()0(

0t021

0021

",

+=

+=

/

=++

=+

pi

2t2t

t

0t

B0

2B

2B0

B20

B

02 t

= ,

000B

B0

v3R4

Rv3

434

t;22

t3 pipi

pipi

====

(/9 )t( BB1 = ,

34

v3R4

Rv

t)t(0

0B0BB1

pipi ====

.&

-:/

7.5. PROBLEME PROPUSE

2t

r = t4pi = !

+$

22tr = t2pi = ! + $

98

+ ! $

&'&%

-:&

-

= MPOP *@: tt pi 3)( = @B7

0 4A @ 2 >4 @ 0: 4" @ -&tt pi =)( @B-

2

= MPOP @0: tt pi 6)( = @B-

>>-@->C@#@-A@2:>@4A@4"@& tt pi =)( @B/

/4A@/:"A@-:@B-

tts pi2sin260)( =

A

B

x O

y

M

3

M O

r

y

4

r

P

x

5 A

B

x O1

y M

O2

C D

K

A

B

x O

450

y

M

6

s

M C

P

R

y

x

2

O

-:

&

= MPOP *@-:@: tt pi 5)( = @B

>@4@A@4"@: tt pi 3)( = @B-

74A@/:4"@-:>4@0tt pi =)( @B/

:4A@2:A"@: tts pisin40)( = @B2

>>-@->C@#@-A@/:>@4A@-24"@- tt pi 2)( = @B/

7

M O

y

x

P

r

R

C

A

B

x

O

y

M 9

10 A

B

x

O y

M

s

11 A

B

x

O1

y

M

O2

C D

K

M

y

B

C

A

x

O

8

&'&%

-:7

-4A@24"@@B0

tts picos330)( =

0

= APOP 4"@*@: tt pi 5)( = @B

2>@4@A@-:A"@:tt pi 2)( = @B-

4A@/:A"@-:>4@2:tt pi 3)( = @B-

/4A@2:4"@-: tts pi3sin40)( = @B7

12 y

M

B

A

x

O

600

s

13

M y

x

O

A

C R

P

M

y

B

C

A

x O 14

A

B

x

O

y

M

15

16

A

B

x

O

y

M s

-:

&>>-@->C@#@-A@2->@4A@0:A"@: tt pi 3)( = @B-

4A@2:4"@-:@B/tts pi2sin80)( =

7

= APOP *@:4"@-tt pi 4)( = @B/

-:>>-@->C@#@/:4"@>@4A@-- tt pi 2)( = @B/

-

= MPOP *@-@-tt pi =)( @B/

17

A

B

x

M

C

D

O1

y

O2

K

18 A

B

x O

300

y

M s

19

M C

P

R

y

x

O A

20 A

B

x O1

y M

O2 C D

K

300 300

21 x O M

R

y

P r

&'&%

-

--4A@24"@ tts pisin45)( = @B0

-04A@2-4"@2 tts pi2cos42)( = @B/

-24A@24"@

tts pisin245)( = @B/

->>-@->C@#@/:>@4A@- tt pi =)( @B0

-/4A@0:A"@:

tts pi3sin320)( = @B

22 y

M

B

A x

O s

23 A

B x

O

y

M

s

25

A

B

x

O1 y

M

O2 C D

K

300 300

24

y

M

B

A

x

O

s 450

26 y

M B

A

x O s 60

0

--

-&

= MPOP *@-0@:tt pi 2)( = @B

-

= MPOP *@2@0:tt pi 3)( = @B2

-74A@:4"@-tts pi2cos100)( = @B

0:>>-@->C@#@/:>@4A@-2 tt pi 3)( = @B-

27 x

O M

R

y P r

28

y

x

P R r

O M

29

y M

B

A

x O s 300

30 A x

O1 y

M

O2 C D

K

300 300

B

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