4.4. ΤΡΙΠΛΑ ΟΛΟΚΛΗΡΩΜΑΤΑ (ΛΥΜΕΝΕΣ)

Studies & Publishing

TPIIIAO OAOKAHPOMA

M eOo 6o l,oyfa ozrol"oyropof

14 Aopfva 06paru

E6vtcriq - Earp6iercr

IOANNHI KPOKO'

A_P.xeEsrudies & Publishing EOAQMoY 29 (fIoA'l'TEXNEIo) TILA..: 210.38.22-157 ww$'.arnos.grOPONTETHPIAKA MAOHMATA OOITHTfIN A.E.I. - T.E.I. _ E.A.II. - LNIJI.

fysikoblog.blogspot.com

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MAOHMATA KATATAKTHPIfTN E=ETAEEQN AtrO T.tt XE AJrt


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?.l f'{rioodot, l{laouuoiI

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J{ Jnqur&r,otn opr,puuu, vtnsi,, T= t(t,u,il - \ of xt1 ^u l=p%;nt1?(p,qft) -

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+(ys,,nq,o{o p,,nl1 l:-;,[ ":;ri'^

ry dp d q de'flPo

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[tn tpu uofupwfld ra]v q,0 om, 6tdqopo, ltiw, cr]oLr qurp ilu, rn,

^o'm0r,wXtitu*0 UrhjWu[n, ro'rdpo uoornl,b vd tllt, I 0

bna l3 r Jlurpuor i,o q* qrrixuodltr w toq wtiNrro ,gulowudc rtt rttdriwd -.trlpaioJo0th'df ulou ifpra flrlv qfr, u,nuoqsoilmLbprFior,lc m 0 tuoui,rg, oniird ., '

lil X'.0,ij i0

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0r, 6qufu/i nrrl

d ftpeloo', .T-t-

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MAOEMATAEOA(IMOY 29 (IIOAYTEXNEIO) TIIA: 38.22.1 57_nryw.enro$.gri


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fuyurupltrrho:n)

"yn'tJc (tulirUi,bo q0:)

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fu$CI,gi r,rclyr, I'urpilo r$u; qpno 6rrfr,1 ml}t; uu}ltilido ,6ipo{po ui,uii)?io6oo

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uo.L ,iq,*tl=[ 'I:p:I I

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APN6iESrudies & Publishrng XOAAMOY 29 GIOAyTEXNEIO) THA: 38J2.157_www.arnos.gr

MAOHl\,fATA


- 1l-

3= - rrl

2= au

1=e{'Iq

0rryt j =!tl.rq ,ouJ0r,r Y= p

1/: p lin0aefr

1i ttiwpinur buulrrullirur: ,rr

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fiu tn 0'pt;t r,rt}ofrotrlEc upurfli ofic'rr?rpur'il0, bl'd01, o,6v.lulo un urlolo Mur,

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i,ur,oulriil0lr0'd,05 tfgli'dLu0toi, u,iltj vrrir mt^*tnrfn inr $duo"rns'il rfnn-

f,xO/ttr:

1=_affip

Utitt V: p-00:0<lfft eP,

{iuuun g =fo rraptowir bL uuJ,n

drd,l dl = lfl'\[r{qr,u), qiquiuri, r{44u}) tll duolv d,u' 'i* ln' .- tw

iu,obui,rt!r0t0: tfloti'd',u0rei, u,,dd vrrir lt11r*l;rron ,h udyom '+t u,qpo-

tiutlttt, ilu: uj ilrtuuofio un'eruio rq; d006frnJsr,05

0ulwt, WI u,[ ur,l*ru,i drdtlde= r]l,du,ivalut

\=U(qr;,ruJt = l ttt y,'iI

.l

?lq u'ttt1

npo JJ({

(x 11, Iv

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I rio tlr t.}ur

J au. Iv tul

l+oI

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e^v_H ffin - fflH|€tlTPtiln0y 0n0,{nH P LNAT0l

I

lur

trr0"

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0

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0.

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xtc , yro ,?>,0 uo, =t

2=C

IO.,\QMO\ 29 (flOAYTEXNEIO) THA.: 010 38.22.157 - 010 38.22.{ei

Fas.: 33.06.{63 \\ !\ \\.ilIl'lOS. gr e-mail: info(oarnos.gr

x+tl+ ?

1

I

x-td=0, ?=1.

X: t= A,\=LA (c,o,r)

, 6 (o,4ol

, r( l,(]r0Jfn T= 7=A, X=L

0a truoi ,

'll,0,t

APN&I

A :7 / l-x-|/.A tl:L0.t1 t!-X

:lrii;te. c\ l'ttllltslttnL

MAOHMATA KATATAKTHPIfIN E=ETAIEC}NOOITHTON A.E.I. _ T.E.I.


-,f3-

u)qdi{ =

,-x-u Lt=rJ,'il,

I

Apo.

tl+g +?+L]3

+ -" l'-uu dlltx=Xll+tl+t+tlt ,0.

@+W)^f't*^. ,_^+ 4 ldudx=ft -!-u- 1 I'dx=ett+r1+4)e/o Jo g o

etr+q+dls

4

=fi-jjl;"t

JjJ," (

f( rj__ LlbB

i- + L)tr= {-t- L*L,l,nlx+tL l'=1't(r+fl $-g {.t' Io'

t4

q

l_

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tc(o,c,cl^'rU

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fr,g,i)eA3 l\t1itl:ot, x>,c, U>C, rro lt t'= u.i iru Uol B=0.J{ 6qupo

:OAOMO\ 29 (nOAYTEXNEIO) THA.: 010 38.22.1s? - 010 38.22.'{e5

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MAOHMATA KATATAKTHPIfiN EEETAEEON<DOlTHTfln* A.E.l. - T'E.1.

APNGIit l)lrblrritutt,',1\'


-{q-

0 tu,u x= l.rmq

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t;=psinq,rin0I=prcSq

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; ; [;I tu,i'; 7,,a)

,to.r 0 l0 tl (6ririr, n,yr.a)

Apo r= f* f i.tp,*grind) p*r,ngdp,rlg,is=

, I Jo Js Jg,.[

= It tn'yo.i'

i* sindot

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= J:" TSo.1 t-*'eilJ'^ glo=

= f ts-W)/j =f,'(fi J:,*oJ .Y(r\ght)dxdydriJ

'= Illf , llq\ n

f

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APN*T\'. !l:e

}lAOHMATA KATATAKTHPIAN E=grN'EOXOOITHTAN A.E.I. - T.E.I.


-f5-

0wu x=pimti,r,or0,ti=Pfrflq,5iI0

=pe

2 = 0'tm0n,l -l

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s\\'l\t^,

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= l* sinxq,uoici

0uyori

ti'.fU

APo

= ?tf

0rpr{0 r qlI0 r 0:lff

X,=U-5

,.lff ii{ rtJ= i .l, 'J^-

P*'Parirrri'i

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lJ = f

5*,

, ,ef iil rL

pdtid0 = Jo

ot Jo

*n,gA$ .J,

p'Op =

{{lttlral )

I nq t I otl du,lr

,@1 6

n 0\ ^ ^. At' U(L\l{I

0t uit

0 r,ulprr r= b J^Jl

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lrffrtj

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ttr''p li?lti lim0

1=f 'ffi(.f9t ' o':P: b

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C r c: *cx'

1 = p\ing11"&t

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OOITHTfiN A.E.I. - T.E.I.

{


dp{d0 =

-16'

px.trt?t1 rnitp srng i,os g, p\rntg

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'{ l'^,rlr -t,tt5e0li-l, il,

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0

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J^t

p'*p J

g'l'5 16

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rql uofl{o6pou gld+ [.?Jga ;

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&LU0,U:

= l(,

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t tt e dr dq drJoJ I

il\r1t , xlqt< or

0piutiu,rgb wu r,ouri' 0,1'= Xe+1d 1

, x**d*=*t'JflPe n !ryn ToUs

1=

0?

E

0.

vo rflopa0o? artlilt7=U

IOAOMO\'29 (nOAYTEXNEIO) THA.:010 38.22'ls7 - 010 38'22''les

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IVTEOHIT'IATA KATATAKTHPION E=ETAIEON

APNGIOOITHTON A.E.I. _ T.E.I.

\iLrrltc: & l)ublt<htng


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0iflori o:t < xbfs.

{a

0,

eil

gtoe

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6

*:ltJ

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Fas.: 33.06.463 \\ \.\'\\.al'l'loS.gf e-mail: info(n,arnos'8r

MAOHIV1ATA KATATAKTHPICIN EEETAIEON

d,p dO =

7t

nz-l^at

=

,II

III

Jg

ff

0ftfl x=P ffi0^ uo,r J= p

U = 0 51110

Apooo'rasf 145

ot

D le+{x= o,e J 0 t

Apo r= Jl- tl

( f?d?),

-oll

['n f

n f;ifi'p dpdo= fn= J, J, eln 'l I

r-J.,

= eil {^ ln= an-u'-:{i o.? lo {?oa

6l t= JJ/

rldtt

V i

tlq,rl .'h3 tx-t) ** (t1-e)'. t , o

c i,ru?rtilpor h-al\(11+)e=l L^tL utrJ

wr, 0 r?<ffum 0, [n-ll1(ti-l)Lt

OOITHTON A.E.I. - T.E.I.

APNfuI


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Fas.: 33.06..163 \\'\\'\\ -al'lloS.gr e-mail: info(oarnos.gr

f -- 1.+ptcl8

)+plrn0cosg

dfd0=

/ar I

.l, f rrnio do]

=

+p m5e)(e*pl'n 0)= 4 +ep,(ln0 +n50

{p + ? pa(:rnO t h J O) +p3r,nO,,lt g] nr

a(r,n 0 + mj0l * p,ln ocosg hpdg =

f 'ntr,ro,tnroloB I e p'dp .*

['f 'ofJ6 Js t Ju

= 4irff

t1 = ?"+p iin00,rYcli

"40r11i= lq =

( Ifulni C:ptl

0g'0stf, rtrf f,. 1tu

ffPo t=Jo I J, L

= J;'.f, *LqP *'o

=-n | ,'.Jl{pdp +

= fi | rn * a - oJ

APN&EMAOHMATA KATATAKTHPION E=ETA:EAN

@OITHTCIN A.E.I. _ T.E.I.


^^1,1^+U^+7":AJ

! !'(b)=;b, I tr,r1,e)rft3 ttlrtrt)?

-{q -

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NAEOHTUN.U KATATAKTHPION EEETA'EAN

tl

I

0tfu 1= pi,os0

t=blrh0rldori, \= finu og'1. @t fnfrnfid ogr.,f[T

hrttns D , h--Ut)*r ?, (ltue)fnn ufn 0 :'p :

-ils tX _r rr f{Itffio 1' 1Ja-Pa-, \ j ;Apov=rl' | \l'drJtotdo=

)-s Jo Jo

,d ko r r€l t\= o lo, ftaur)cv

(f;--P'iF .iP do= 4'{ 'J-[ Js jr

'latnslg+ (a'-P*)

nt l de' =J fo

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-etr

,tItr l1++ I '. e{ad0=c J-a

1

B,la I-

.- 33

4{-er1 _3

33

(roeo -La''lf/ n!=

It-tcor?o )ult*o

( {-cosl o) '/tio *

(,

if

14Iu

\ft

{4JX

tlit I1-rI

J_[1

_u3

=_u3

(llrag]"tdo *

Itirr3olao + 4{afl

-=3

d

= -3! (+3 )-n

3t ( 'l=-t\ J

I r,nro oo = [ (r-nJJ0t)r'* tos0 =t

, -onrBdB=dL,

Apo Y= -0X.f, 3\= _S (t_r__G

J\ j A"

= 3'3 6

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5rra0

Ia

onodd.o

(-rrnaB) aingdo + ftjoI

lz

-**Lln W--J'3

q{inj-

= j ttlr)s e0 )sin 0ci0

l5'f t,cle-(-S) |' 3 ',l_tn/7 ^ -!, efi e{-a'-i Tr :-

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OOITHTAN A.E.I. _ T.E.I.


-l {-

s) t=\mF)''lr1r+?r"=16

V =,

,toptnU $p[r,U,oi''li tgUJ ttrdn ftrn OOodp s5 ,;lt, ,w,i n.t,tyta

z*=t{rlue) l' o tel I tae-aal J 7r--4e4tla4?2=48 ,zxt+gt+11) rc I

. 4=. 11=4? a ?=r[a = e$ftPn n U',rn CI,5 v,lnv \ l^+yr74| | t r=?"13

a

0r0gln

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-lr -

{3)fudo=l- (_1@_Egly

f) u-"an( ,*

u*)=

ptm0

o sin0

!p uo,i {lut}J=,ffi= o

i.t < 'tr--f 6n.? o'. r L p

tior fiS t/ t= fi.x 6nl c'. p

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tp'Ii*pu- p'

u._3

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0rm x=nl=

0ii:or,t 'Jiiuui o

dr{dt

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64rtr-{-3J3

r--n ffill 0

v

e)

9 l-0, ^it1*=lx

,?=W

r lr,ol0{ r{l

ftpo I=

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-ls-

1 ?tn50 ,los

lo

d \t+nttd*d0 =

-

u

/

(t 1rofi 1 r i

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t-

J-ra

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IA /a n{v: I f-

J-r 1|\d

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fll4

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It*ttotlo+J-[

6

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{,

I=Pi t{:p,5rn0 , flpa0turt

x

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tL

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+p5ino/p dr ldpdg =| | ll

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{rl { (p,q,o ) =

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P

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I

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t 4r . , I

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.(tb

flF

131 ,lr=x\ur' l*= xt*t4t . t >0

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' l?--lbq^ J L ?x=4t a v(r-q)=c .+ { ?=c

tt=l'*d* I L*4nro tyirt'

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r\Lt,'o [r-ll'rrl'=!a xLt"t+I( ={.

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tXLt fit0rfrJ

P

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le

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l\,/*=tfr fnl

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/il01

h rV [?r*llcu l/= | ?. { -

lllJ-T J,

nl*

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I

Ii { iA- I v-J-r \-.l,\

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0t,urt 6h0 =itpt0 d0, dt

, stn?0 ,T--1q

+ 51n40.+c

3? n'5in40) | [nll-tr=

,

qg

+

+{do=

J t,or'ro d,o

= Lo* I_\1

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u

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6:

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\=tw f,f-utl

1 = 0tl'uJ

T*IIII

J

1-v

v (l-u)

yrf

=urU-wl

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I 7u lv gri)

{ = u(4-v)

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J rr,gd0 =

J txinro)m50d0 =Jt,r-l

r)d,+, = L

tr = sL,rt, _ yr,

flpo t/

(,{+ trrst o + arzerj,ig =

0 , 5ifl?01' -4- +

3I_

I

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-e9-

, U=l , Uy({-CI)=0 ilor ,ttyuj =0 tni'

=L,IJ)=0,Iil=1.

5*, u(,1-vl=0

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-il= i t-u*'t'MlJ -J,

= t,(-t g -e(,'., L L

fipo

npo

trfunr

0 ttt'- L|_tvtL0:u :INh=

t4 tt 1t=lllJo Jo Je

#v r,-rdudy,do = jJn,

J, ,nr ,l w^rudu

=

L_w,-T+tlr*l =

fipo I

q,,l ,t

?.u rul * eJ r-uou) = It- t =

: *( t t

llt'u

1)!,

0

L*xJ =ttt,L t a

APbL@TStudies & Publishing EoAoMoY 29 (IIOAYTEXNEIo) THA.: 210.38.22.157-ww*'.arnos.grOPONTTETHPIAKA MAOHMATA OOITHTON A.E.I. _T.E.I. _ E.A.[I. _ E.M.I].


4.4. ΤΡΙΠΛΑ ΟΛΟΚΛΗΡΩΜΑΤΑ (ΛΥΜΕΝΕΣ)

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Transcript of 4.4. ΤΡΙΠΛΑ ΟΛΟΚΛΗΡΩΜΑΤΑ (ΛΥΜΕΝΕΣ)