a al av <-l 9)rtu?fro fl 6ruR ltav{'tu?0ufl tnu?10 I

rrui6'ud {5,1"'ukifinurvrqufiua;rruir-ufirfiurrlourac'ldlirrauolun-:doeiotJd

1. uu?narlnv qua4l.l nnflUe,naliU?a (Center of Mass)

l.z nr:urqirurri.rlngudnnru':atolr-erqgrlr:trnfinuriruduut.:dauafinetanf

rur-ruunaqala2i

1.3 lullJustn?,ll.jla-ou (Moment of Inerria)

1.4 vlquAttnuft{aln (orthogonal Axis Theorem)

I.5 nrr n rrir]urlrr'o rlu,o'uonolr-flo: r: I rnfifluururiuu r trlaontisraragr$

luxxrrnaedd

t.o qer: drdrrirunrjr lngudnaxlaa (x"r, Y"" ) unvrirlrunrudnrurfiou

: orlngudnarrurnrorr-nqg r:rrnfieruaiurrr { Ir* )

t.z mqufittnuturutor a'lslfluof (Sreiner's Parallel Axis Theorem)

1.8 en0l!1,lflna (Physical Pendulum)

1 .9 rtu:I#llrosnrrr':orru'ora*ouairutri':oonetrrutruaturu:'rrorn!of,utlnarwaa

l.lo ovnSnn

-- f,;2. .:'lu?nu tflu?10 t

2.1 l]u2tu lull:slYlfl

z.z lruii'uriril:vrrq

u ?nou6v quaI

86lgut flAl.i ?A (Center of Mass)q q

,uo3r#u{srofiturohilr {luon nr: oonu:ln:vrircioe-nnlursuuor{nla [n 1 0]

A a,'. a j " A " ' - qt ' I A d,[ 0 [ ?n0un1:tn0u9t1ufiu{ tnu t lltJu rr:.:mn:vvt'l9to?9tn 0vn0.:H1u n ]ultu {fiu{ cr':nlu

- )- c, , { , terlrrfiu nvlrut uulflu0u 1n t ?!tJ0.ru?a1o':?fiqm{fl0u uac[5 unn lufiu{ulJ cl ?116quunn1{

s i. t d @ d d. ' <l' .i ', "ju?a tnuYr0n[[Hun!nguunal t!?n?nqr[1.:tflr {fl trn4u { nyruulnm0q)Jsc n ]yltt - Aar s 4

o r?nqyl[:Ju 0o! ?ad0 {!o u1n trulmtflu oquuflnu ltnlnnu lofJuun n t{u?a0sogq y i ' o 'y oitnnu?aYllJ'lnn'11 f,]fi:u!?a mr rlas m2 0quuunu x uayoq l t n tnetlt04u.to 1.10{yt!n

df o q o ' 4 u,flltua 0l utsuy rr rrn3 12 nturitgu 0itrrfiu.t10,! !9lfluunat { 1a fi1'[gin]fldunl:

5

tlJ fro

no

no

fr0

fro

O C.M.

ml

m2

It2

lso ]ufiu{nflfluafla1{l'l'l nuo i'l9tt)

ila ' dx?a1 o.:?-9l Q ri?ufi n fi 'r it il rurilu fi I n n irJ

i-,dlu'rnx0{'l-fl odlufi ao r ililriaurilunlnniu

j ,, . ', a .ttSgutYtu'tn mr 0gfi tntlnnlufiu{o'l{o.t 0 x lt'lutuu[!et5

i ,, . ', a dlt3ug11:J'tn m, oufi "l'li'lnfl ]ttfi u':o'1.:on 0 ufl ululuullJglt

----- (l)

----- (2)

----- (3)

q4.!ttt!,^-,jv?'rn?n qlrl {[n:.:1J: rfl0ufi?utJ']nY Uou dl [tu{n?nolu0{nu 10fluun0]{!?410{

q A d, " '-i J, - -A -?n qoo[lJusr]ul.ru.r1r rdu0uflu1lu?dtutJou dl xo.r?nq!1:?rinu l09{na1? n1:fi1u']01'141

/ a 6 lne!frfluu ?afl1{n flvlr todo{?!

d d , n,, zi oenr ol innul ':lfl5.r [[1r lJur.rulJ 1.t uln [t.lfroru0.t linltnlu?ol[[uu snun [un'r:1.r1

r -!o1rrfi u{nofluufl alnll'lan{u

I''tc.M. =

=-I.,i=l

ru'o C.M. 6o rirunrjsloqutlnarwaatorr-nq

4 s 4 - dr -l) m,r no luruuei:1!xo{']ho irrirurilud lani!.urn:l) tti=l

dr as) m no 1,'tat?lJto.:?fto ul,tu'tulUun tantuZrli=l

, id , d / en:ol29rouuur[fl u! 1.1vDJU']aooruot [: 131Mt0 't0r1ultuI!ofl u0na1{u?nn2u

a J v h9'n nfllflnt [[uuunafl nd tfi?1

Irdmc.M. = J- =

Ja'n

rdArpdA

pdA dA

6.14. = mrrr *mzrz

mt +m2

6

[lJo C.M. fro

fio

?v9r'rur4u ilnfJu0na1{ xnIo{?fl q</rd,aluruuri:'lutosisro fi rriruriJu lnnir.rln:

,d,asl, ? 0l ?1110 't't 9to ul4u?u[ un Infl:!

r d, asfl't ]lt14u'l[[u uto iu u ?sto u14u ?u[ u n Idn:tJ/gr r: 1{r!fl :A y - a , I - Jnycrio l2nnrfl'llul.tultruu dlJ ttduo dtlJ l:flnnT.: tn

I ), d d t dtlluYruout ulolafl 1 !fiu'rur uar:1':[lJa:

rdm

dm

IJ

p

crc.i11:0O10nAl?[1JUR1U1tUYr1.:d0A']r lnflUUnnri ?0fr09rrr1ru':raaur0t0r{R1n

J. Z -,, A dAfirixrirxuhdruu?n (flufl{d fimGol. zsaz 'z+6)

nrrfi reiru ri{oofl udnar.rn?axo.tino:rJr:trofi su[! usiuu r{d?unfi oq I osr{

tnjUttnanad?v,jizrel. !nfluun 01{tJ?010{'tn0: dtHa0! r.ruHtultlluHuu I tt 4 ri A a y

lfi uu un a l {x ?a { 0 { n : r 0 r u,.lu ?er 0: 1J dlH n 0 u r.ru r.r'luu u uu u u t,r n J'l i B

01? A !?n V n-lnrn :cnou I dttJ t: oti t'ldd'rd

fio

no

IdA 6o

Y

B

d1

dA:r'dx=Rds

d-A,=rdr'=Adr'

\0

t d - r.i i d ,nlflu:un0u I n r:r11n(uun0 t{!?nr 0 i?et05 rJrilH nu!r.tu,i1[[uuuHuul t

,

7

Y.,

dx

CMli". so*

J'na*

IJ xdx

l,^

x',

(.)

A

A,

L'r'aal,BlAdvJo"

1," lar

(4)

1,"0,

t+)

(s''\[;.]

B

,

B

B

v )(

B

(s)

(a')[;.]

8

. /.q s)fto I -- - |l) ))

& a r a ,j i a e I e,uuno nnfluuflnl{!?a10{2ao: dr aulJflu 'lullullfluul.: n?1{ B ul? A u?a M

la,i2. nafluofral{u?6x0{?nqi drur?rau!ry!alnuuuufluul.:

t'lngutinarllea toln:fi tor:-erqgrJarurr6orrllornuuuuliuurl n1u0'r? A

q{ B !?a u f,rnrnil:snor z arlr:oxrldn"'rd

Y 64=yds=(Bx,'A)d\

B

dA=rdy=[A-(AvB)]dy

\

tti,RlYItJtsn0u 2 fl't:fi'lnnfluufla t.: ?ato.:?9r6r: dtulflau urn tnuuuup.]u1l I I

x. Bx dx

,J"

Bx dx

dr'

0

rX.,

f:.-J'*a*

AT

A2

3

,|

2A.J ----- (6)

GTtt).

Y.u,v'

Bdy

B

-B dy

Ay

t;-r;

9

l"

JJ

d1IJ

dyBJ"

\.'28

( eB' ABr)t_t

[z 3 )AB_4q

2

AB.6

AB2

(7)

ia d v , juun0 1aguundl{ 1010r?nq:udrurYa0}]lJUQlnu!lrlflu1J1.r nlu A q{ B

=93

lJ?d M flo (auvru riu:nnradu. 2553 :41)2AB

33i v , ; ee,

3. 00f,uuno1'uJ'tat0{?9t0t1ld'llJr140uufi utn'llrtlUuHuu'1.:

I n fl u dn a 1.: rt ? a I o { n, fi 1 o{ uri u ?-n qi il fl r u lfi 6 u :r u rir o'auu r ur.i u r r r q r e.

!luu1'r 28 x'Jn M ?1.rgr-?6'.:R1 :3nou 3 fl1r,rtoil rlrin-{fi

l0

d.A: )dri = (Bx A)dx

\

dx

xdx

Y

II

dv

0

t d A gt i'Rryrrl:vno 3 fl'l: tnoduuflnlitxnl0i'r9ro:lJdtulfiauu u10?[luu[[Hu!1{

?E*)0,A)r.I

li

X.,f^r?9,.)ro\A )

J*,'d*

t+)1.

tt)1.

A3

3

A2

2

2A.J ----- (8)

ll

!2a M flO

Iuruuslnarudou (Moment of Inertia)+ / zi d, i -J zi - d"a .i[1l[!u0rR?rurd0u I uduuao01{fi u.rlfl o1,utlJ0?fl q14qu rijulJ:lJ 1tuYlu0n

Rrrlrdoolunr: rryu (Rotationat Inerria) 10.:{nq 1u nr: d ovnutu-t!fns ran rfl16rt,uo!nr:

nryurorl{ lnue-nqihruueinrlr,door',n fiwrir1#r-nqriur:Jdrrarlnruun',rnrlurGrldurne v &." t A , d .dro '- ,A - he'lrasnllnnxrulr tlrr:.r1rqn11lflDouufllJfln3r1t11']9ruuuntnuuflntlI10.:n1: lJutftlj tfi.:1u

l. d i - -,1q{ t! r:.r u9rfr?lurqoull0 i ?9t0ufl'luun unufllJu :11:'ll10i't9tflu0tnnu gn1t[:u{et?10t

?-nq:ouunu Iu flr:fi1u?rufi1rirIurrudnarrrdourronnt:r.t'ot:tut'[rlr{Ju z n:dnidai I - A t e e d s a A . h,n: vl?jU{ :'lglQyluSgnoUn?U!'lnflOU[nn "l lutUuslfl']1![noun lU?tU tn0lndUnl:

It,{'N

i=l

(e)

rn'o tari-d,aslttrljunn?t rflouto.:iao !Hu']ur ufl tnfl5u.tltgl5-

!;, ,"", -!?aruYr i yrn:3n1u0q tuluo1qq fiu']ur un tnnru

: c u r na{ n r n 0 r n u ? n'hJ dr r r n u r u u fi r ri r u rtl u ur n :

-) - ;a' a - .i- ' :i d'ntOlYldO.t: ?9lQYllUUlUYll {[:lJ'ln n r:0?fiqT!U?An: g i lusl0[U0lnUlflu011[1JU

I ^ r A - uc)-!rlrorou'tnu n1 ?01fllnt tuluunt o.tfi?1![qout 0.1?flq tao.ru

(10)

[:J no

9ro

s, A - dr aso 2

t!luunn2 tlJlaou10{ la0 !flu1ur1]un t n n: !.[1r 9]: -

v ,,-:gugFtiq lnn IflLr Id llIItrrfl1rfitlu rl..tu tti11Iur q:

i- d' ardm n0 !?afl? uou.l flflsltillt !Yu'tu[!un tnn:!, -.{; d, a<

n?1u1{u]t[1-lu[1{11Uylu0{[[r]u'rfiq lJfi u ?uluun tan:!a0q]t'l,tr!n:I A, d d , d,lluYltJoUl'Utfi tnfl 1 Ut4U?U[lJUgI1;1{rtJ91:

mi

r

p

ft0

fto

fio

fto

n0

Jr'pdA,= J. dm

I

r

o

dA

riu6o lngu tinar{!?axo{inqgildrrrrfi AurLfi #r,i'?rruuuriuu r{ g{ A n ru 28

2A

-,03

I

t2

s./ziar?A.vllur!u nn'l1u re o0rllu :rl xudrnnr: lnu [!lrlufln?1 lqtoutstJn'Mfl1150 0u1u

firi!an uov:vu;nrnrrxa1 u-':unuHqu dlunufirlunJiuu thcfir{ndrl#n-nuturnr:rrlu's, ? a

uayfl r lljluugrfi?rxlaoul0.:?9ro[ 6uuhjrirtr

fl qgfi llnusr'{qrn (Orthogonal Axis Theorem)rA

fi1nn1fiuo [14 lx , Iy tIny Iz ft0 tlJluunn?]1lrao01o.:'t910:0urlnu x. Y u0, z

nlrl.irnix rirnru :vnou q

Y

ornrhvflou 4 vrquA[lnu01'{arni a aa a J aYl!1 :nlfl?Yly{flnfl lmn{n: !14l?YlUlAU.2543 : 135

t A A!nvlIU?1 [ltuJ1.l9tfl?'tlJ[a0u:0utlnu n{a1nflu:vu1u10{?0t0lJflltmlnu nu?n10{

s ? A - o f,, .[lJluunnl'lulaoul0{']9to Iu rY0tuu'][[nuYln{a'lnnu[[nv1u1 n1:9u11lti0l[lHu?n0

riuur{lualnr:'[d'ir

Z

I I* + I" ------- (l l)z

A - a A i ' ? A'- -luo lz no t[tJ1-l9rfl?1utQ0u:0utlnu11a]lulnguunal{u?aylet.:Q1nfl!

:su1u1o{[lr.iuinq--as/rl;I*,I" 6o lluurinrrrrfrou:oullnufri, luril:vuru1ufia x uar y

slruiirnix lnounu x uns v #lq rneionluluuu'tssururo{i'nnA4^L

nrty{-qifrquAufl ucr-ie rn lnoordtrvrqufl uvrroiilyt'tnoffl

o,.x

v

l3

rlnilquAumlo{fl lno;drrasR1fl :sao! I 1.{u'jt

[!O r

r *'*y' ------ (12)

x

v

fi0

Ft0

fi0

,dt:vuuntflrrnu z hjo.:ounrn ln 1 !flu'turljul!9t:

:s us i'lflrrnu v'hJdroqnrnln 1 firri:orilun:n:

:vusnrnrrnu x 1il6ror1nrn1n 1 fiurj:orflluruor:

aA0]fluullfuo { ttJrtJucrfl'l tutflou n{dltJ1:urtluu [o ]1

I r'dm

J(x'+ y')4,n

fx'd,n + Jv'am

tAvato'ttfitn1l t usto?-t taou,t o{igto:rJllxln fluuultsluu1{o?unflflfl1dglI

uluunanfros/A-,jizrel. tullJufln?1![n0010'1?nqg d[?lauu uu'rulruuHuuN< / i d - ,; A a qt

lrJtrrunn?1rlliiouroul9fluunal{u?010 1?nq! dn4 nu!fi uH rrruuuHuu1.ry f,t i

u?o M nir{ B ul'r A:oullnufiilruloqudnorr!']orururiuut.r?rrnu x n-in1 rl:vnou 5

u Yw idlu"t5i]ltl too{u

Y

dA:rdr-:Adv

dy

\

rs"A?q,jizr,Rl?$Jrsnou 5 [lJrlunfl?lxlnou: ou !nflu0n n 1rtJ ?010 r let0:lJ drr nu Nu,iluljlj[[Huu 1.t

.i, ? stouunum ru!nquunal{!'t0lulunuuu?[[nu x

Z

Ix + ly

B

0

l'

(Nc*r. Yc.r)

T

l4

I Jv'oae

[s)a1],,'a,

Xc"

j,'[#) xdy

)^( 3

))t^14-r-q,/ [za \ 24

y'

)1",

IMB2

t2 ----- (13)

s /A t -.1 ;4y.,lu0Junn?"tlJlQ0u: 0uloquuna l.rrJx nfl 0 r'Jfi qdl au fl u H'luuuufl uul{ n11.: B

i, r eoll a:ouunufinirulfifludndl.:rJlaru.turi'ttuurunu y n-rnrn :vnorfi o arur:orr'ld

fi {u

dA=r'dr:Bdr

Y/

0 Llx

fs?A4vriiaeR1?ltJ:vnou 6 t!l!u9lft?lutnou: ounnf,uunn'l {! ?a10 {'t9to:lltYl au!flu H'lrnruuflulJ 1.:

.i, t o:ox[[num,.l1u1ft4uufl a1{!?alu1ufl lluu']unu Y

MAB

MAB

,--,!-t--./B

N

(\.r'lYatr)

I I

l5

2 pdAI,.I-

uay0lndxnl:fi (11)

/vr\t*]BIJ-at2

dx

yd*I,.'(#)

x

(#) IlBx

,

(#),[# ( #))

MA,I

12

I I*.* * I".'

= lNas'*aMA't2 t2

zcu

----- (14)

----- (1s)I

)M A.2+822I

s/iz. Iuruurinru:rfi ooto.::-oro: artlrr6u!ur:qrnrrurur.iurrr

s/Aa;'tur!unn?llj[q0u:0u !nfluuna]{1l ?a10 i 1nq;u dlu[110u:.r1l1la ]nuuuur.ruu'r {

;, . @

!iu A qr e:orrrnufiairuloqudnarrrraturutirrrulunu x rirnrn :vnor-r 7 dulT:ou1

too iu

(\gr,:\_.")

l6

Y

2B3

-B3

dy

\(t'):(2A'3+AY'B)dA = xd1' = (1A'j + AY,B)dy

cA

Jv'oaa

tAto,iRlVturvn0u 7 tlfllrusrfr']]![a ourou!stJuunnl{u']a10{?nqi dl!lltaulwalnrluuufluulr

i, ? stolJunufl 1u!ogu0na1{u'tn1 ll-ln rru?unu x

I Xc^,

2B/l

-B/lp

p

I,,(zd Av\-l_-___1 d\.: B) v

( 2au' auo )l'1., -;il/

( zu\( ztv3 ero )l't_ ll __! _ ____! [(ee.l[ e 4B )l_

B/l

B/l

B/3

B/l

IM82

18 ------- (16)

tAdqrf,tuvlru0 ilfiu'lnu t lrJufl n?'rlJ[aout0lJlofluunali ?a10{?aqgudlulfi nu rytJ, -it / . nr,.i

a'rflllulrrlHuu'r.1 s1u A q{ B :ouunuTfllulofluunal{!']aljulunuuu?uflu Y tonlu

II

t8MA ----- (17)

y(x):CBxA+2B,3)

t7

[[ovolndrnl:fi ( 11)

I I I+XcvZcu

lun'*!ua'l8 t8

1M ,{2+82

l8( )

__-_-___-_ (18)

< / A - , i ,'&3. ttJltJu9tft?'l![a0ux0.:?nqgufl r!lfi nuu u rn?ulJlJIrr.ruu'N

n:fi .uo.rinq;rJrrrLrHAu!fi dr#?rruxrrr.iuur{ nriH rl!ruu gin'nilrdou:ou!o4 t9 t 4 t A d @

fluunar.:!2n d'uJ1:ol|l tnrxulnu?nu tur!unfl?'lrJra0u:0u!oguunal{!?n10{'tnq,j: dlulfl aulJUUq'lnrruu[lflulJ'l {

tdeAg9l td-ltt09l-lttfiu{lnoBuunar{ ?a (xcM, YcM) uavn r lulrJl{fl n ?rrrsourou !o

augudnor{il?ato'tiorqgrJrttrnfioruuurriuut.r ( Iz." )

" --.J - r )tu'tlu?ouu s3fl n! ]fl 't5t4'ln 1 [![1lunfl i11l[a out 0ulfigu un n 1.:!?d ( Iz.. ) 10t

:-nn:rir:rrnfieruuurlriuur'!iirl.r?u 3 r rr! ld'uri :rjfrrriulfiudr eruru6urlrain rrau; e i, A - ? ^ / o ! -"dllJr nu!ru1n? cr{0rnn1:flf,0uylr':niuerfl'rd9r: ?nq;t]r:rln nridllJlJe|r[r u{

t t< d A tl6fludnnr{u'la unsrirlrrruudo?r!r6ouro!lnfludnar'! ?a o-'!udorlunrrir I

nrr1.r I giilrfi ri{ !nflu dnar.irt?a lrayrir]ilrr uein'lrrJrd o u: o! !n fludn n rruln ro.:?-aQ!il

[5 I',r fi ru st[ [lJ lJ [ lfl u u'r {

;ilto.ruriuurs lnsuftnal':lJ?a (c.M.)llruuein'rrlrdoo

:oulqqurinal{u?0 ( Iz", )

.i i aedrfiautHllHl

n?1{ B u]? A !2n M

dlU[ Aurr]Unln

!1uu1? A q{ B ll?n Mi r,&

6t1lJrfiAU!lJU 1t',l

(tIl.\2(rol.;(+

;)

1

-Mt2

Ml8

I

-Ml8

l' * n')

B

3

0

22A +B

2

,luu]? 28 flr A !?a M.u')

l8

.tvalnqUQltnUXUlUllO{d tflUtUOt (Steiner's Parallel Axis Theorem)

dr r-n q l r o t,t fi Il nr u ein a r u rfi o u : o u l n gu tin a r.: r r a r{J u I., o v'ld'j r fu rr u o{

i - ! "

i - i, t tt qfr?1![aou I Io.r?nqu:oxunu to dl ry{Iu ]u nulrnu mfl ]u !ag unn1{!']ouasoqH1{nudr -d' d,

[111-rtsus d unlrIu

I=lcu+Md2 ------- (le)

rdo I

I.,

M

d

?;'dra.no trruu9tfr?lurnou10.r?nn ! u'lurl-lufi tan5r. rlr9l:-t a ],

", ? ano I u u u ei n r rl rfi o u t o.r'lin q fi d r u 1 n qu dn a r.r l aa i n il rurfl u

filan:-l.rrn:'dr ^+fl0 u'1n10{'tgto 1lfiu2u[1]un lnn:u

6u rrorrirulu,ru?,r,u'ru?-n0rnnnautinnxrurafi vrirurfl urlsr:

flrlnr: (19) QnBun'jt tqufrunututu (Parallel Axis Theorem) durrrllntrtrnoil alnfruof

(Jakob Steiner) linnfi ngranfvrraidlryotuoue{

nr:fi qorivrqufi unilrururo{dlndruot

.ln

\

nry{rhsnou 8 rquArrnurururo.: alsrrfruofi d v 4 qaYlll'l : tr&Y{Ut dusllfl?qd. 2547 :342

ornnu :snou a 1# o rilulnqudnnl{1,?nro.ri'nqln 1 Inuns{unu x, Y, z

l#oqjfilnrirru^n o rirgil lntlrnu z ri':oonnrnrrirn:vnru

(\

t-bI'

J,I,

i - i:i-

19

4 - . i i' dluo IcM flo Iuruugrn?llJtli0tJ: o![[nuYtHlu!n8uunn1.:lJxn ([nu Z)4ai-I fro turuu9lfr2lurDot t oullfluYtllulunutlnu z Hlunn P

Iou:vosmtruullnu x uav y ro{ln p fi rir a uav u errltirnlr

s / d - ', ! i,tuluuflnl txJlqou10I tflnflouu: ol rrrfluq ifl'tu00 P 14'l tottn

!t

J[(" -")'*(v -b)']

A dr ly q JIoont: o-uvt-rn:n rilunr:o-un-rn:nr*rtiou:hg n:lnrunoriunvoh;rJrrlri'ir

J(x' + v')om - 2aJxdm - 2bJrdm + J(a' + t')am

, j )- ay[[srruo{0'rfln{unu rrlnnrrunrmrnir!fi fludnnrw']a ufln{'jr xcM = o rirbfld'jr

J(x'+y')am+J(a'+b')

I dm

dn.r

I

0rauu1 fl0{19flu0fla1{:J'ra

Jxdm = 0

!1 ,tuvl'luo.:[nu?nu 0s to

t - - -LJ vdm = o rirriu r: rriuuaunr:lnrjLi' 6o

dm

I., + M(a'?+b')

X.n l]r-M

I

I., + Md2

20

!!nelxlY{dnA (Physical Pendulum), qau t I e d a Aa i

0nfl !?ldna u:snoue?u?aolnj{rfl :':1ltJ1u10 uayllfl ?n:ouufru:1rou14u.1

nrnnr :snou 9lnlJurtflnrru?oo{r4rrornlnrrlurflu:yuv dsinO uovrirrlr 0 ri'ruua6r? I - i . o v.TO:nTfl:vYt]fl 'ton tUmf,mml l 'toflflal rdolrrt4xl idtJnn flo

-mgdsin0 ----- (20)

lu0

"c

1

mg

dsin0

n0

fio

n0

tAanrdr-sTO:nYrO.:'lnonouqgi'l[lfl u.: duq 0rJ1.ru1u[rJuu ?sru. mfl :| - , d,asu1 unlj0.:1nnu u?unluu19lu

tgugn'lflnnfluuna rw?a6':lnn{u1!,^uuul9t{ilnu u?utilurun:

ol)ir o t

0

( ]I\'Ir/ siu 0

ntg

I 9 4aoRll',ttl:sno 9 nnfiufldna

TU'l : Serway and Jewett. 2004 : 469

a?[[ognln ul 10{r0:fl

'c Icr

d':eI

dt2 ------ (21)

2t

ttJ oaJ;Au,d'Aoflo T o: nmo{1clnnatl flo Itfiu i d:.ro n:Jfiu ?fln luu?olu.mot

4A'd,6ano t!tlJugrft't"tulQotjtjo i'tgtoufi u'tut ufl l nn: u. r:.tct: "

6o nrtlri.rriwltor{noir:16auliurfi 'aid,dno ryumluu tuolnllu? uqnufiu']ur ur:louu

0

1

I

0

e

0rndunl: (20) lray (21) nyld

rg'edt' -mgd sin 0 (22)

,i'n; rJaru n r: 1n ri ki'jr

d20 . mgd sin 0

dCt I ------ (23)

0lnrona'n!01ilr'rsrilnoriG rfio e riluqlrdn 1 rrld'ir sin0 ^,0 ry'ldcunr:nr:jita

lRa0uTltull El:lJounou'1.:.1'lufr0

d'e med ^

-+-----udt' I

0

rur'ouGtrrrfi trrnir:rlrirtltorarnr: nr:rn6oufi urr orfuofi notir.riru

------- (24)

(2s)

(26)

d2e-""':- * (D

dt',e

0

031iu?1

mgd

Io

t 1nu 01lJ 10.1fi'lT lJ filryi !

22

o 2n

T

I.cd

T

----- (27)

uuno

rravld''i1

2tr----- (28)

,i "i !r, qau d ' d, an'lun t:rnnouvrl0{nnaxr,{dfl n u u'lurllu?u11r

s/,i-drasu)tuluustfi ?'ltJ[ 0u10 {']9rfr u u'turllun tafl 5u'rusr:-

vddatdtu?aroi thorri{tni r ilrri torilufi lnnfir

, I se * dln?1ult ilu 0 i0']flr[: i tutJ0?i10{ tafl fi u?urlll] rtJgr5/'ru r11-

i- r d ' dr:vuu1'r?o01n1fl fluufl n r{u'tn tJ u?u[Llurugr:

drlunr:nnao.rfiryrrn'i{;o0 lnufilorrtauoqjrir':orncirrrurjrlafludnar{!?a

r{Ju:gt,v d rnjonrrirlrurruuo{orrrrdoo:or-,unufi#rqrn6'u:yurxro.irr.iurr{ uavdr'Iil

rirurardounair d:ovrqufiunuturuuoid'lafruof rvr'onreirlun:uo{naru,dou:oron

fludna1i rnro.:rrriulrI ( Ir., t loorJitgilalnr:b?rflu

2nIr." + md2

T ----- (2e)mgd

s o/ ^

Y - it dttJr u9fl 1rrJ[q 0u:0]Juflu9tr D tnflu:vu llJ1']fl1uEofluufla1{ ?aIltJ o Zcu fi0

dtosultJ?tu')utltufl t n nt!.tlJst:

ruo l' fio

lnom fto

gn0d n'o

2n

T

rE"d!t

rE"dii

23

rfi oo-ogr.lalnr: o;rrrir Ir.,'ld6o

msdT2--:- - md'4n'

mpdT2--= . - mcl'4tr'

----- (30)Zcu

s'/ d d v .ull? [LlrJlro{n rurla orao 9rlrrlfl{{oonn lu u?l,tJ tl.l1flo tnlflfluun0] { ?a

01ndrn1: (30)

I

6'o::Jornr:1#od1u:rlarunr:rirn-.:fl0{ Ax2 + Bx +C = o ovld'

.ndt - tBdJ' *t

4n'

Zcu

T 2x

0zcM ----- (31)

----- (32)

nvr,{u'irdunr:ou1u: aln r:nru r!ri1n-{do { fi1#: r.Jn: rylrtlu tstluoiu.:tuA y o ' v r j

"ay a, y -icr{o o ilJfrl: yu:rlu'llu l lnnlnlofluunn1{ ?a d Tytl tl1n lu[?a1 [un1:rrn?{ T UflluouTdn

^ g I sv i s ,l ' rrrs v drnnxu (1n n rqn 10 { tn rYlu0n l:1tun1) cr rrn'r:yus d tn [0un15 'r0u11u n 1o.l T

rfi or niu d u a y r r 1 a in q o I o': : r u r a Io o n r : 1 #o q n-u f oiu oiu fi n d r fi ri r ril u qu d

d

dd

nn: dlJnl: tl1': lulu

d

aI + md2

)(Zcy =0

mgd

uufro

,,(;)[I + mdz

mgd

zcv =0

I

24

1lmga(zma) - (r,., + md' )(me)

(-gd)'-n

IZcu + md2

mgd

n:v orun oririrr 1 ov'ld

mgTr md2 - I r.,

(rgd)' zcu +md2=0

I

mgd

031IU 11

md2 - Ir., = 0

uuflo

arldr rnmqu0ufl unululuu0{d tnulu0t

I = Icu +mdl

d, =+r-!;

----- (33)

--------- (34)

umurir d = - rF nlndunt: (34) alludtnlt (le) icy{u'jl

IZI Zcu +mI-

; 2lzc* ----- (3s)

& a o a i"s,uuflo nrlrl.rl.r{ d tn tnm lfiFn lz

dr o ' i.cr,- r, .izt, 0sluugt'l[[fi u{ltvt'l lfi [nflnruuoumda

I12

A2+82

25

(36)

(a' + e')

; .i ure'tufl t0llJo.:dlfi au!fl 'l I 2.,

.i.oy y A a dt:gu3 d l]vn tfi T u00mfl910{r u

( )

Im

t2A2+82( )

dr

1

,E

n1

A2+82( )

.ltu fl : 0l u o {d lulH n ull u lJ Q 1nu0

i"oqr u .i a d,:cuc d vrm'l tl.t T uoumqon{r u

gdITJI AUUfiIl8

u 1n? l,

1

-ml8 A2+82( )dr

nt

----- (37)

0vn?anad'/sjoun:afl 41d9rflll:0 tltat!Taluml tfl:laa fi50 PPMA njulyro: [r.Jy{a'ldglfl1uft14u.:

<zittAy.tdte1JS01l l.:nl:fl1fi n'lUAOn?UnU tYlJ Plexiglas. Lucite. Perspex [UUtlU u0yqfltlRlJl0{Y{AldAn

lfini16o cjH8o, ardn-nyundgnriuril:rqndldlurruraluorir': rliu n:u!nx#ld1{,l - r s et e t*t e) A -uuln:0{uu lJ'lu tgtu0lt n5!nnaun'l 2dqm1ifl 1: [r1r11u [uusru [u 0.t0 rfl? dou duun [nn[nu

q .l - A the, & -.' , ; AtU[:0{fl']1lJlllUU'] (Toughness) dM:0tU;U [6'l lU UOflnlnUUUilJn]n?'llJ14U'l[[UUgi] ry.1

Cr6r.oa,6<drqaaaa[lJU fllJ!nU:v 0]n?10.1?do15slnflfi nldnnrrn ? osfl: nnun rflafl! {ruu ?dnlu Fll.rlJ.rvtuuu

90u'ru1 txul1uufl2 [u{lufl nlooti'I.1 finetluutrriluil:vrJliu l.l5 - l.l9 a;!/anljlflrirryuerlrror:

lnuilnraorruoaiiqonr4fi r30 - 140 o{flrrsnr6ufl rrnsln160odqorr4l' 200 olnrrrardoaJ ', - v '1, rlJU IU Un?lXvtUYtlU0t0flt:n:v[[vtn (lmpact strength) qtn? uln? [ra9ilfl?'l t an]tu0[u91

= #JG1.f

26

- .J o.{ , a ^ d hy,unsllnl dnn?fl?n::!i n0u utu000ui {lfla5 0u1nl,n tn{lu !n']llJytulllunodn'l1lrr10a 0!

- - j I s a / 4 a a, enn?'lna]dnnsuooultu tlIafl r:uotufl n{uulJ lioyfl : nn 01dfl nnu{]unn1tu,[i

Av.r4 7){'tu20umlng?10rt

.t llt?oul t.l:rlllfl

n:-d q ur::u!r (ooulnf.2545) 'ldlilnrr yln n o.: rlu[rru g{n?lrrrfiior,r 0'!! ?nj<. rn,, zlfl!anu y tun0[u0'! tfrufl'r5l.l ryu !?auu 01u]r { u unsfi1n1: 6no{ 'lR?1lJ[:tr,0.trru1l

ru'o'ldrirnrrlrirudr 6lrirhlnrllruurinarurfi oo'lrirrnan:

I., _ m(e-a)R'a

t A ia-rln&nal?0{n?1![[nnn]ltyfi?1.1flt: tnI[!tlJunnlluttoul0{ 2ar!anu,5,, zi a A .i- -hjsiorfio.: uocn rr r rril luluucinrrurqoulorr?nm anu yflorfio.:"1{'j'l

* t = I., t' rfl u arnr:fi u rluruudnrrurdoulun:fi fi rrar{Ju!o ugirirulat-xtlJrrrru1n[?r uaulu o']non: s n ruodr idrirrd!o oydo rldiG nr:6ufinforrru lnuuril

. r " i' d, < r i! ?aIo r?nq oonruurulan 1 nl dm oqu I rt rnlrnufl ryuluu: EUs r ll]rlJugrfi'r'r![a 0u10.1

oqnrnr6n f docrilu I = Jr'dmrravtil prilunerrurururjuro{?-ero uay dV riiuil3rt10t:

rf;n 1 dm = pdv rfiouvrudrluaunt: nslri'ir I = [r'pdv "

.nui,t'usirrrJrsrnq

Peter E. Banks (l9s+ : :as) lrili rnr: n r llruurio rrurdooornyio pvc lnunr:rirvio

' o d, ,a s !," , j q 4 e.Ulgtoflutllu: gtlt4 ttnlflqn1]50fl0tfluu tryr1an'litltofi"lo9l:'lt:1 ua?ultJ1[1ul]fl:lfl: . he, !.

:v11?r.:oet:r[:?nu[-]nt lfifitFl-]lrJsunoogr:1r:.110{1an ornuudl'1ilr4 rrit]rLnrudnatrLrdou

[1]: UltlYtUUfl Ufnfl IU?tUyt'1.:1161!I

Chartes J. Burstone (l9g8 : a26 - a3l ) '[r i]n1:Tnno{fi 1]tullupinarlrd ourtfi utrfiutniu! - i r, , aet o, v i cv

fl'ltfrlu10lYr'ri11qua tnu t5 trJ2r{9r.ifl1rYluun1ufl?r tl.r t0nrluu4nnra qn cr{ tnulwfl'r

fu ra n rlu n r : l o n fl 5 u : o u ua u d r lil nJ3 o u rfi uu riu n r : ti r u ? iu lu t u dn e ril rfi o ufi r 1dd A t Ad ya

11f)U0Uf1U1U]U 11',1t14 tfrnl [ [:JUAn1',] [A0U:0U!FlflUUnA] tm [nA[nU inUn',lrl iYlq!A

Russeva, Tsutsumanova and Russev (2010 : 59 - 62)'lflliln1:mnno{91:lndouylquAn I d tit a d u / /s q9

ltflu1u]u10.1 d'lfluru05 'l9f,nu1ron0{ 9]:?ndoullquaunululul0id tgluluot touts

'r-nqrr.iuul riililarrrwriluqn{rfl firin mrailnnirlogudna xlrodruiidlmrttartrrr-nq o tlu

27

, d g " . .i-

nrortu r rnriuvirnr: rrrv: : ou igr6lrrr.iur rr: nrarw druru z : fr fr :vuvri r.:rrn/ n' ' s

nnqudnnxrrahjrvirdu uarrJriool#gndlflfidngrJr.lorerwfiHnnr:if,uurir rirrir

orrnr:un'ir lrriou n: rv,l:vo;r.ir.::vn'jrlloqudnar.:l:o drl#n: rlfuiJuxr: rluarr rrur la I t

lr a 3 rl u u n : 111 o nt U fl u { n o n'lfr l1J n 1: llfl ? { fl 1 o.r d 0.: I ol : v uv 11 1{ n lR ! n qu un o 1'r x'] a

rirl:yrJv?i-lr:sfi'jrrlergutinarrl:aunrira-raor (r'.a riu a') dr'[dn:rr,{rilurduo:r

finaufuriluu'ln lnu'ldhrn:rvrin:rr',lrdovrrirnrrruriudorornu:rlrllsir'rtorlnn 6.rnuir

irir s.s*o.snrn:.iurfi'' uasdn:rdru:vil'ir{ I- rm 6o 0.0t+*0.0005rusr:'Lcu

i-dJ.i,unulotvvllt?d 26 ntu nu l6n1 l![ltunn?]![a0ur (0.27 t 0.01) x l0 - n tnnlu.[un: -