Work, Power & Energy_E

8/12/2019 Work, Power & Energy_E

1/32

RF_XOHX

"anlosfguanrpfysohs.ol" >

UKPG RKUCP CLCPE_OLTPKDSHTOKL =

Tfc tcra wkrg ns uldcrstkkd ol cvcrydny joic fns n doiicrclt acnlole ol shocltoioh sclsc. Oi n hkkjoc os

hnrryole n jknd kl fos fcnd nld wnotole ikr tfc nrrovnj ki tfc trnol, fc os lkt pcrikraole nly wkrg ol tfc

shocltoioh sclsc. Ol tfc prcsclt study, wc sfnjj fnvc n jkkg oltk tfc shocltoioh nspcht ki tfos akst hkaakljy

uscd tcra o.c., wkrg.

UKPG DKLC @_ HKLXTNLT IKPHC =Tfc pfysohnj acnlole ki tfc tcra wkrg os cltorcjydoiicrclt irka tfcacnlolenttnhfcd tk ot ol cvcrydny joic. Ol

cvcrydny joic, tfc tcra wkrg os hklsodcrcd tk `c sylklya ki jn`kur, tkoj, ciikrt cth. Ol pfysohs, tfcrc os n

spchoioh wny ki dciolole wkrg.

Ukrg os snod tk `c dklc `y n ikrhc wfcl tfc ikrhc prkduhcs n dospjnhcaclt ol tfc `kdy kl wfohf

ot nhts ol nly dorchtokl cxhcpt pcrpcldohujnr tk tfc dorchtokl ki tfc ikrhc.

Ikr wkrg tk `c dklc, ikjjkwole twk hkldotokls aust `c iujiojjcd.

(o) N ikrhc aust `c nppjocd.

(oo) Tfc nppjocd ikrhc aust prkduhc n dospjnhcaclt ol nlydorchtokl cxhcpt pcrpcldohujnr tk tfc dorch-

tokl ki tfc ikrhc.

Xuppksc n ikrhc I

os nppjocd kl n `kdy ol suhf n wny tfnt tfc `kdy suiicrs n dospjnhcaclt X

ol tfc

dorchtokl ki tfc ikrhc. Tfcl tfc wkrg dklc os eovcl `y

U 1 IX

Fkwcvcr, tfc dospjnhcaclt dkcs lkt njwnys tngc pjnhc ol tfc dorchtokl ki tfc ikrhc. Xuppksc n hklstnlt

ikrhcI

, nppjocd kl n `kdy, prkduhcs n dospjnhcacltX

ol tfc `kdy ol suhf n wny tfntX

os olhjolcd tkI

ntnl nlejc. Lkw tfc wkrg dklc wojj `c eovcl `y tfc dkt prkduht ki ikrhc nld dospjnhcaclt.

U 1I

. X

Xolhc wkrg os tfc dkt prkduht ki twk vchtkrs tfcrcikrc ot os n shnjnr qunltoty.

U 1 IX hkskr U 1 (I hks)X

U 1 hkapklclt ki ikrhc ol tfc dorchtokl ki dospjnhcaclt anelotudcs ki dospjnhcaclt.

Xk wkrg os tfc prkduht ki tfc hkapklclt ki ikrhc ol tfc dorchtokl ki dospjnhcaclt nld tfc anelotudc ki tfc

dospjnhcaclt.Njsk, U 1 I (X hks)kr wkrg os prkduht ki tfc hkapklclt ki dospjnhcaclt ol tfc dorchtokl ki tfc ikrhc nld tfc anelotudc ki tfc

dospjnhcaclt.


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RF_XOHX

"anlosfguanrpfysohs.ol" 0

Xpchonj Hnscs =Hnsc oUfcl 1 ?2, tfcl U 1 IX hks ?2 1 2Xk, wkrg dklc `y n ikrhc os zcrk oi tfc `kdy os dospjnhcd ol n dorchtokl pcrpcldohujnr tk tfc

dorchtokl ki tfc ikrhc.

Cxnapjcs =

>. Hklsodcr n `kdy sjodole kvcr n fkrozkltnj surinhc. Tfc wkrg dklc

`y tfc ikrhc ki ernvoty nld tfc rcnhtokl ki tfc surinhc wojj `c zcrk.

Tfos os chnusc `ktf tfc ikrhc ki ernvoty nld tfc rcnhtokl nht lkr-

anjjy tk tfc dospjnhcaclt.

Tfc snac nreuaclt hnl c nppjocd tk n anl hnrryole n jknd kl fos fcnd nld wnjgole kl n rnojwny

pjntikra.

0. Hklsodcr n `kdy akvole ol n horhjc wotf hklstnlt spccd. Nt cvcry pkolt ki tfc horhujnr pntf, tfc

hcltropctnj ikrhc nld tfc dospjnhcaclt nrc autunjjy pcrpcldohujnr (Ioeurc).Xk, tfcwkrg dklc `ytfc

hcltropctnj ikrhc os zcrk. Tfcsnac nreuaclt hnl `c nppjocd tk n sntcjjotc akvole ol n horhujnr kr ot.

Ol tfos hnsc, tfc ernvotntoklnj ikrhc os njwnys pcrpcldohujnr tk dospjnhcaclt. Xk, wkrg dklc `y

ernvotntoklnj ikrhc os zcrk.

8. Tfc tclsokl ol tfc strole ki n soapjc pcldujua os njwnys pcrpcldohujnr tk dospjnhcaclt. (Ioeurc).Xk, wkrg dklc `y tfc tclsokl os zcrk.

Hnsc oo =Ufcl X 1 2, tfcl U 1 2.

Xk, wkrg dklc `y n ikrhc os zcrk oi tfc `kdy suiicrs lk dospjnhcaclt kl tfc nppjohntokl ki n ikrhc.

Cxnapjc =

N pcrskl hnrryole n jknd kl fos fcnd nld stnldole nt n eovcl pjnhc dkcs lk wkrg.

Hnsc oo o =Ufcl 2 : ?2 WIoeurcQ, tfcl hks os pksotovc. Tfcrcikrc.U (1 IX hks ) os pksotovc.Ukrg dklc `y n ikrhc os snod tk `c pksotovc oi tfc nppjocd ikrhc fns n

hkapklclt ol tfc dorchtokl ki tfc dospjnhcaclt.

Cxnapjcs =

>. Ufcl n fkrsc pujjs n hnrt, tfc nppjocd ikrhc nld tfc dospjnhcaclt nrc ol tfc snac dorchtokl. Xk,

wkrg dklc `y tfc fkrsc os pksotovc.

0. Ufcln jknd os joitcd, tfc joitole ikrhc nld tfc dospjnhcaclt nht ol tfc snac dorchtokl. Xk, wkrg dklc

`y tfc joitole ikrhc os pksotovc.

8. Ufcl n sprole os strcthfcd, ktf tfc strcthfole ikrhc nld tfc dospjnhcaclt nht ol tfc snac dorch-

tokl. Xk, wkrg dklc `y tfc strcthfole ikrhc os pksotovc.


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RF_XOHX


Hnsc ov =Ufcl ?2 : >62 (Ioeurc), tfcl hks os lcentovc. Tfcrc-ikrc U (1 IX hks ) os lcentovc.Ukrg dklc `y n ikrhc os snod tk `c lcentovc oi tfc nppjocd

ikrhc fns hkapklclt ol n dorchtokl kppksotc tk tfnt ki tfc

dospjnhcaclt.

Cxnapjcs =

>. Ufcl `rngcs nrc nppjocd tkn akvole vcfohjc, tfc wkrg dklc `y tfc `rngole ikrhc oslcentovc. Tfos os

`chnusc tfc rngole ikrhc nld tfc dospjnhcaclt nht ol kppksotc dorchtokls.

0. Ufcl n `kdyos drneecd njkle n rkuef surinhc, tfc wkrg dklc`y tfc irohtoklnj ikrhc os lcentovc. Tfos

os `chnusc tfc irohtoklnj ikrhc nhts ol n dorchtokl kppksotc tk tfnt ki tfc dospjnhcaclt.

8. Ufcl n `kdy os joitcd, tfc wkrg dklc `y tfc ernvotntoklnj ikrhc os lcentovc. Tfos os `chnusc tfc

ernvotntoklnj ikrhc nhts vcrtohnjjy dkwlwnrds wfojc tfc dospjnhcaclt os ol tfc vcrtohnjjy upwnrds

dorchtokl.

Cxnapjc >. Ioeurc sfkws ikur sotuntokls olwfohf n ikrhc nhts kl n `kx wfojc tfc `kx sjodcs roeftwnrd n dostnlhc

d nhrkss n irohtokljcss ijkkr. Tfc anelotudcs ki tfc ikrhcs nrc odcltohnj, tfcor krocltntokls nrc ns

sfkwl. Pnlg tfc sotuntokls nhhkrdole tk tfc wkrg dklc kl tfc `kx durole tfc dospjnhcaclt, irka

akst pksotovc tk akst lcentovc.

(N) (@) (H) (D)

Nlswcr = D, H, @, N

Cxpjnlntokl = Ol (D)1 2, hks1 > (anxoaua vnjuc). Xk, wkrg dklc os anxoaua.Ol (H): ?2, hksos pksotovc. Tfcrcikrc, U os pksotovc.Ol (@)1 ?2, hksos zcrk. U os zcrk.Ol (N)os k`tusc, hksos lcentovc. U os lcentovc.

UKPG DKLC @_ ASJTORJC IKPHCX=Oi scvcrnj ikrhcs nht kl n pnrtohjc, tfcl wc hnl rcpjnhc I

ol cquntokl U 1I

.X

`y tfc lct ikrhc I

wfcrc

I

1I

>+ I

0+I

8+ .....

U 1 WI

Q . X

...(o)

Tfos eovcs tfc wkrg dklc `y tfc lct ikrhc durole n dospjnhcacltX

ki tfc pnrtohjc.

Uc hnl rcwrotc cquntokl (o) ns =

U 1I

>. X

+I

0. X

+I

8. X

+ .....

kr U 1 U>+ U

0+ U

8+ ..........

Xk, tfc wkrg dklc kl tfc pnrtohjc os tfc sua ki tfc oldovodunj wkrgs dklc `y njj tfc ikrhcs nhtole kl tfc

pnrtohjc.


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RF_XOHX


Oapkrtnlt pkolts n`kut wkrg =>. Ukrg os dciolcd ikr nl oltcrvnj kr dospjnhcaclt. Tfcrc os lk tcra jogc olstnltnlckus wkrg soaojnr tk

olstnltnlckus vcjkhoty.

0. Ikr n pnrtohujnr dospjnhcaclt, wkrg dklc `y n ikrhc os oldcpcldclt ki typc ki aktokl o.c. wfctfcr ot

akvcs wotf hklstnlt vcjkhoty, hklstnlt nhhcjcrntokl kr rctnrdntokl cth.

8. Ikr n pnrtohujnr dospjnhcaclt wkrg os oldcpcldclt ki toac. Ukrg wojj c snac ikr snac dospjnhc-

aclt wfctfcr tfc toac tngcl os sanjj kr jnrec.

4. Ufclscvcrnj ikrhcs nht, wkrg dklc `yn ikrhc ikrn pnrtohujnr dospjnhcaclt os oldcpcldclt ki ktfcr

ikrhcs.

3. N ikrhc os oldcpcldclt irka rcicrclhc irnac. Ots dospjnhcaclt dcpclds kl irnac skwkrg dklc `yn

ikrhc os irnac dcpcldclt tfcrcikrc wkrg dklc `yn ikrhc hnl `cdoiicrclt ol doiicrclt rcicrclhc irnac.

9. Ciicht ki wkrg os hfnlec ol golctoh clcrey ki tfc pnrtohjc kr systca.

5. Ukrg os dklc y tfc skurhc kr neclt tfnt nppjocs tfc ikrhc.

Slots ki wkrg =>. Slot ki wkrg =

O. Ol hes systca, tfc ulot ki wkrg os cre.

Klc cre ki wkrg ossnod tk`cdklcwfcl n ikrhc ki klc dylc dospjnhcs n `kdytfrkuef klc hcltoactrc

ol ots kwl dorchtokl.

> cre 1 > dylc > ha 1 >e ha s0

> ha 1 > e ha0

s0

Lktc,Cre os njsk hnjjcd dylc hcltoactrc.

OO. Ol XO o.c., Oltcrlntoklnj Xystca ki ulots, tfc ulot ki wkrg os mkujc (n``rcvontcd ns M). Ot os lnacd nitcr

tfc inakus @rotosf pfysohost Mnacs Rcrsklnj Mkujc (>6>6 >69?).

Klc mkujc ki wkrg ossnod tk `cdklcwfcln ikrhc kiklc Lcwtkl dospjnhcs n `kdytfrkuef klc actrc

ol ots kwl dorchtokl.

> mkujc 1 > Lcwtkl > actrc 1 > ge > a/s 0 > a 1 >ge a0 s0

Lktc Nlktfcr lnac ikr mkujc os Lcwtkl actrc.

Pcjntokl `ctwccl mkujc nld cre

> mkujc 1 > Lcwtkl > actrc

> mkujc 1 >23 dylc >20 ha 1 >25 dylc ha> mkujc 1 >25 cre

> cre 1 >25mkujc

DOACLXOKLX KI UKPG =WUkrgQ 1 WIkrhcQ WDostnlhcQ 1 WAJT0Q WJQ 1 WAJ0T0Q

Ukrg fns klc doaclsokl ol anss, twk doaclsokls ol jcletf nld 0 doaclsokls ol toac,

Kl tfc `nsos ki doaclsoklnj ikraujn, tfc ulot ki wkrg os ge a0 s0.

Lktc tfnt > ge a0 s0 1 (> ge a s0) a 1 > L a 1 > M.

Cxnapjc 0. Tfcrc os nl cjnstoh njj nld n roeod wnjj. @njj os tfrkwl tkwnrds tfc wnjj. Tfc wkrg dklc `y tfclkranj rcnhtokl cxcrtcd `y tfc wnjj kl tfc `njj os -

(N) +vc (@) vc (H*) zcrk (D) Lklc ki tfcsc

Xkjutokl = Ns tfc pkolt ki nppjohntokl ki ikrhc dkcs lkt akvc, tfc w.d `y lkranj rcnhtokl os zcrk.

Cxnapjc 8. Ukrg dklc `y tfc lkranj rcnhtokl wfcl n pcrskl hjoa`s up tfc stnors os -

(N) +vc (@) vc (H*) zcrk (D) Lklc ki tfcsc

Xkjutokl = Ns tfc pkolt ki nppjohntokl ki ikrhc dkcs lkt akvc, tfc w.d `y lkranj rcnhtokl os zcrk.

Cxnapjc 4. Ukrg dklc `ystntoh irohtokl ikrhc wfcl n pcrskl stnrts rullole os [[[[[[[[ .

Xkjutokl = Ns tfc pkolt ki nppjohntokl ki ikrhc dkcs lkt akvc, tfc w.d `y stntoh irohtokl os zcrk.


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RF_XOHX


UKPG DKLC @_ YNPOKSX PCNJ IKPHCX

Ukrg dklc `y ernvoty Ikrhc.

Cxnapjc 3.>2 a/s

>2 a/s

(>)

(0)

(8)(4)

>22 a

Tfc anss ki tfc pnrtohjc os 0 ge. Ot os prkmchtcd ns sfkwl ol ikur doiicrclt wnys wotf snac spccd ki

>2 a/s. Iold kut tfc wkrg dklc `y ernvoty `y tfc toac tfc stklc injjs kl erkuld.

Xkjutokl = U 1 I X hks1 0222 M ol cnhf hnsc.

Ukrg dklc `y lkranj rcnhtokl.

Cxnapjc 9.02ge N @>2ge

>22a

I1>02L

(o) Iold wkrg dklc y ikrhc I klN durole >22 a dospjnhcaclt.

(oo) Iold wkrg dklc y ikrhc I kl @ durole>22 a dospjnhcaclt.

(ooo) Iold wkrg dklc y lkranj rcnhtokl kl @ nldN durole tfc eovcl dospjnhcaclt.

(ov) Iold kut tfc golctoh clcrey ki `jkhg N & @ iolnjjy.

Xkjutokl = (o) (UI)kl N

1 IX hks1 >02 >22 hks 2

1 >0222 M

(oo) (UI)kl @

1 2

I dkcs lkt nht kl @

(ooo) >2 ge@

L

4 a/s0

L 1 >2 4 1 42 L

02 geN

>02 42

(UL)kl @ 1 42 >22 hks 2

1 4222 M

(UL)

kl N1 42 >22 hks >62 1 4222 M

(ov) v0 1 u0 + 0ns u 1 2


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RF_XOHX


Oi a os acnsurcd ol erna nld v ol ha s>, tfcl tfc golctoh clcrey os acnsurcd ol cre. Oi a os

acnsurcd ol gojkerna nld v ol a s>, tfcl tfc golctoh clcrey os acnsurcd ol mkujc. Ot any `c lktcd

tfnt tfc ulots ki golctoh clcrey nrc tfc snac ns tfksc ki wkrg. Olicht, tfos os truc ki njj ikras ki

clcrey solhc tfcynrc oltcr-hklvcrto jc.

Typohnj g olctoh clcreocs G =X.Lk. K`mcht Anss (ge) Xpccd (a s>) G(M)

> Nor akjchujc >209 322 >20>

0 Pnol drkp nt tcraolnj spccd 8.3 >23 ? >.4 >28

8 Xtklc drkppcd irka >2 a > >4 >20

4 @ujjct 3 >23 022 >28

3 Pullole ntfjctc 52 >2 8.3 >28

9 Hnr 0222 03 9.8 >23

PCJNTOKL @CTUCCL AKACLTSA NLD GOLCTOH CLCPE_ =Hklsodcr n `kdy ki anss a akvole wotf vcjkhoty v. Jolcnr akacltua ki tfc kdy, p 1 av

Golctoh clcrey ki tfc `kdy, Cg1

0

>av0

Cg1

a0

>(a0v0) kr C

g1

a0

p0kr p 1 gaC0

Cxnapjc 6. Tfc golctoh clcrey ki n `kdy os olhrcnscd y 0>%. Ufnt os tfc pcrhcltnec olhrcnsc ol tfc anelo-

tudc ki jolcnr akacltua ki tfc `kdy;

Xkjutokl = Cg0

1>22

>0>C

g> kr

0

>a v

0

0 1>22

>0>

0

>av

>

0 kr v01

>2

>>v

>

kr av01

>2

>>av

> kr p

01

>2

>>p

>

kr>

0

p

p > 1

>2

>> > 1

>2

>

kr>

>0

p

pp >22 1

>2

> >22 1 >2

Xk, tfc pcrhcltnec olhrcnsc ol tfc anelotudc ki jolcnr akacltua os >2%.

Cxnapjc ?.>2ge

I 1 >2Lsakktf

Ikrhc sfkwl nhts ikr 0 schklds. Iold kut w.d. `y ikrhc I kl >2 ge ol 8 schklds.

Xkjutokl = w 1 I

X

w 1 I

X

hks 2 w 1 >2 X

Lkw >2 1 >2 n n 1 > a/s0 X 10

>nt0 1

0

> > 00 1 0 a

w 1 >2 0 1 02 M

Cxnapjc >2. Iold Golctoh clcrey nitcr 0 schklds.

Xkjutokl = Y 1 2 + nt Y 1 > 0 1 0a/s

GC0

> >2 00 1 02 M.


8/32


9/32

RF_XOHX

"anlosfguanrpfysohs.ol" ?

Ukrg dklc `y Ynron`jc Ikrhc U 1 dsIUd

Cxnapjc >8. Nl k`mcht os dospjnhcd irka pkolt N(>, 0) tk @(2, >) `y nppjyole ikrhc I

1 xo+ 0ym .

Iold kut wkrg dklc `y I tk akvc tfc k mcht irka pkoltN tk @.

Xkjutokl = dU 1 I

. ds

,0)

@

(2,>)

dU 1 (xo+ 0ym ) (dxo+ dy o )

dU 1 2

>

dxx +>

0

dyy0

U 1 8.3 M

Cxnapjc >4. Tfc jolcnr akacltua ki n kdy os olhrcnscd `y>2%. Ufnt os tfc pcrhcltnec hfnlec ol ots golctoh

clcrey;

Xkjutokl = Rcrhcltnec olhrcnsc ol golctoh clcrey 1 0>%Q

>22

>0>

>2

>>

C

C,v

>2

>>v,av

>22

>>2av.oltF

0

>

0>0>0

>22

C

CCclcreyol golctoholhrcnscRcrhcltnec

>

>01 0>%

Cxnapjc >3. N toac dcpcldclt ikrhc I 1 >2 t os nppjocd kl >2 ge `jkhg ns sfkwl ol ioeurc.

>2ge I 1 >2 t

t 1 2

Iold kut tfc wkrg dklc `y I ol 0 schklds.

Xkjutokl = dU 1 I

. ds

dU 1 >2 t. dx

dU 1 >2 t v dt .............. (>) dx 1 vdt

njsk >2 dt

dv1 >2 t

v

2dv 1

t

2dtt v 1

0

t0................. (0)

irka (>) & (0)

dU 1 >2 t .

0

t0

dt

dU 1 3t8dt

U 1

4

3 024t 1 02 M

Njotcr = G.C. 10

> >2 (00 2) 1 2


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RF_XOHX

"anlosfguanrpfysohs.ol" >2

UKPG DKLC @_ XRPOLE IKPHC

Cxnapjc >9.

Olotonjjy sprole os rcjnxcd.N pcrskl stnrts pujjole tfc sprole `y nppjyole n vnron`jc ikrhc I. Iold kut

tfc wkrg dklc `y I tk strcthf ot sjkwjy tk n dostnlhc `y x.

Xkjutokl = dU 1 dsI 1 x

2dxxG U 1

x

2

0

0

Gx

1

0

Gx0

Cxnapjc >5. Ol tfcn`kvccxnapjc

(o) Ufcrcfns tfc wkrg eklc ;

(oo) Ukrg dklc y sprole kl wnjj os zcrk. Ufy;

(ooo) Ukrg dklc y sprole ikrhc kl anl os [[[[[[[ .

Xkjutokl = (o) Ot os stkrcd ol tfc ikra ki pktcltonj clcrey ol sprole.

(oo) ]crk, ns dospjnhcaclt os zcrk.

(ooo) 0Gx

0

>

Cxnapjc >6. Iold kut wkrg dklc y nppjocd ikrhc tk sjkwjy cxtcld tfc sprole irka x tk 0x.

Xkjutokl = Olotonjjy tfc sprole os cxtcldcd `y x

I

JL

x1xx12

0x

x 1 0x

U 1 I

. ds

U 1 dx.xGx0

x

U 1

x0

x

0

0

Gx

1

08 Gx0

Ot hnl njsk ikuld y doiicrclhc ki RC.

o.c. Si1

0

>G (0x)0 1 0gx0 S

o1

0

>gx0 S

i S

o1

0

8gx0

Cxnapjc >?. N `jkhg ki anss a os rcjcnscd irka tkp ki n sakktf ioxcd olhjolcd pjnlc ki olhjolntokl.

Iold kut wkrg dklc `y lkranj rcnhtokl & ernvoty durole tfc toac `jkhg hkacs tk `kttka.

Xkjutokl = UL1 2 ns I X

Ue1 I

. X

1 ae .X . hks (?2 ) 1 aeX sol1 aef

Cxnapjc 02. Iold kut tfc spccd ki tfc jkhg nt tfc kttka nld ots golctoh clcrey.

Xkjutokl = Y0 1 u0 + 0ns

Y0 1 2 + 0(e sol)sol

f Y0 1 0ef Y 1 ef0

GC 10

>av0 1 aef


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RF_XOHX

"anlosfguanrpfysohs.ol" >>

UKPG DKLC @_ TCLXOKL

Cxnapjc 0>. N `k` ki pcldujua os rcjcnscd nt rcst irka cxtrcac pksotokl ns

sfkwl ol ioeurc. Iold wkrg dklc `y tclsokl irka N tk @ , @ tk H

nld H tk N.

Xkjutokl = ]crk `chnusc IT

dX nt njj toac.

Cxnapjc 00. Ol tfc n`kvc qucstokl iold kut wkrg dklc `yernvoty irka N tk @ nld @ tk H.

Xkjutokl = Ue 1 I

. X

1 aeX hksU

e1 ae( hks) ikrN tk @

Ue1 ae ( hks) ikr @ tk H

Cxnapjc 08.

Tfc systca os rcjcnscd irka rcst. Ufcl >2 ge `jkhg rcnhfcs nt erkuld tfcl iold =

(o) Ukrg dklc y ernvoty kl >2 ge (oo) Ukrg dklc y ernvoty kl 3 ge

(ooo) Ukrg dklc ytclsokl kl >2 ge (ov) Ukrg dklc ytclsokl kl 3 ge.

Xkjutokl = (o) (Ue)

>2 ge1 >2 e x 0

1 022 M

(oo) (Ue)

3ge 1 3 e 0 hks >62

1 >22 M

(ooo) (UT)

>2 ge1

8

022 0 hks >62

18

422M


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RF_XOHX


(ov) (UT)

3 ge1

8

022 0 hks 2 1

8

422M

Lct w.d. y tclsokl os zcrk. Ukrg dklc `y oltcrlnj tclsokl o.c. (tclsokl nhtole wotfol systca) kl tfc

systca os njwnys zcrk oi tfc jcletf rcanols hklstnlt.

Cxnapjc 04. Tfc vcjkhoty jkhg N ki tfc systca sfkwl ol ioeurc os YN

nt nly

olstnlt. Hnjhujntc vcjkhoty ki `khg @ nt tfnt olstnlt.

Xkjutokl = Ukrg dklc y oltcrlnj tclsokl os zcrk.

N

TT

0T0T

4T4T

6T

6T

>9T

@ VN

V@: : : :

>3 T V@ T VN1 2

VN

1>3V@

YN1 >3 Y

@

Cxnapjc 03. N `jkhg ki anss a os rcjcnscd irka tkp ki nl olhjolc pjnlc ki olhjolntokl . Tfchkciiohoclt ki irohtokl`ctwccl tfc `jkhg nld olhjolc surinhc os ( : tnl ). Iold wkrg dklc `ylkranj rcnhtokl, ernvoty &irohtokl, wfcl jkhg akvcs irka tkp tk tfc kttka.

Xkjutokl = UL1 2 I

L X

Ue1 ae sol

Ui1 ae hks.

Cxnapjc 09. Ufnt os golctoh clcrey ki `jkhg ki anss a nt kttka ol n`kvc prk`jca.

Xkjutokl = Y0 1 u0 + 0ns

Y0 1 0(esol e hks) ()

GC 1 0

>

a 0 (e sol e hks)

1 ae (sol hks)


13/32


14/32

RF_XOHX


wkrg dklc `y GI kl systca ki N & @

1 853 + >03 1 032 M

Ukrg dklc `y GI kl n systca os njwnys lcentovc.

Fcnt eclcrntcd 1 (UGI

) kl systca

(UGI

) kl systca 1 (iG X

rcjntovc)

1 >2 03 1 032 M

Truc / Injsc =Cxnapjc 82. Ukrg dklc y golctoh irohtokl kl n `kdy os lcvcr zcrk.

Nlswcr = Injsc

Cxnapjc 8>. Ukrg dklc `y golctoh irohtokl kl n systca os njwnys lcentovc.

Nlswcr = Truc

UKPG DKLC @_ RXCSDK IKPHCGolctoh Clcrey ki n `kdy irnac dcpcldclt ns vcjkhoty os n irnac dcpcldclt qunltoty. Tfcrcikrc

pscudk ikrhc wkrg fns tk `c hklsodcrcd.

Cxnapjc 80. N `jkhg ki anss >2 ge os pujjcd `yikrhc I 1 >22 L. Ot hkvcrs n dostnlhc 322 a ol>2 sch. Irka olotonj

pkolt. Tfos aktokl os k`scrvcd y tfrcc k`scrvcrsN, @ nld H ns sfkwl ol ioeurc.

N

n 1 2v12 @

n12>2a/s

H

u12n1>2 a/s

0

Iold kut wkrg dklc `y tfc ikrhc I ol >2 schklds ns k`scrvcd `y N, @ & H.

Xkjutokl = (UI)

kl `jkhg w.r.t N 1 >22 322 M 1 32,222 M

(UI)kl `jkhgw.r.t@

1 >22 W322 >2 >2Q 1 42,222 M

(UI)

kl`jkhgw.r.tH 1 >22 W322 322Q 1 2

UKPG DKLC @_ OLTCPLNJ IKPHC

IN@

1 I@N

o.c. sua ki oltcrlnj ikrhcs os zcrk.

@ut ot os lkt lchcssnry tfnt wkrg dklc `y oltcrlnj ikrhc os zcrk. Tfcrc aust `c skac dcikrantokl kr

rcikrantokl `ctwccl tfc systca tk dk oltcrlnj wkrg. Ol hnsc ki n roeod `kdy wkrg dklc `y oltcrlnj ikrhc os

zcrk.

Ukrg-Clcrey Tfckrca =Nhhkrdole tk wkrg-clcrey tfckrca, tfc wkrg dklc `y njj tfc ikrhcs kl n pnrtohjc os cqunj tk tfc hfnlec ol ots

golctoh clcrey.

UH+ U

LH+ U

RX 1G

Ufcrc, UHos tfc wkrg dklc `y njj tfc hklscrvntovc ikrhcs.

ULH os tfc wkrg dklc `y njj lkl-hklscrvntovc ikrhcs.U

RXos tfc wkrg dklc `y njj psucdk ikrhcs.


15/32

RF_XOHX


Akdoiocd Ikra ki Ukrg-Clcrey Tfckrca =Uc glkw tfnt hklscrvntovc ikrhcs nrc nsskhontcd wotf tfc hklhcpt ki pktcltonj clcrey, tfnt os

UH1 S

Xk, Ukrg-Clcrey tfckrca any `c akdoiocd ns

ULH

+ URX

1G +S

ULH

+ URX

1C

Cxnapjc 88. N `kdy ki anss a wfcl rcjcnscd irka rcst irka n fcoeft f, fots tfc erkuld wotf spccd ef.

Iold wkrg dklc `y rcsostovc ikrhc.

Xkjutokl = Odcltoiy olotonj nld iolnj stntc nld odcltoiy njj ikrhcs.

Ue+ U

nor rcs.+ U

olt ikrhc1G

aef + Unor rcs

+ 2 10

>a 0ef 2

Unor rcs.

10

aef

Cxnapjc 84. Tfc `k` ki n soapjc pcldujua ki jcletf j os rcjcnscd wfcl tfc strole os fkrozkltnj. Iold ots spccd nt

tfc `kttka.

Xkjutokl = Ue+ U

T1G

@u

NK

ae+ 2 10

>au0 2

u 1 e0

Cxnapjc 83. N `jkhg os eovcl n spccd u up tfc olhjolcd pjnlc ns sfkwl.

x

u

Ssole wkrg clcrey tfckrca iold kut x wfcl tfc jkhg stkps akvole.

Xkjutokl = Ue+ U

i+ U

L1G

ae x sol aex hks+ 2 1 2 0

>au0 x 1

)hks(sole0

u0

Cxnapjc 89. Tfc ansscs A>nld A

0(A

0< A

>) nrc rcjcnscd irka rcst.

A> A0

Ssole wkrg clcrey tfckrca iold kut vcjkhoty ki tfc jkhgs wfcl tfcy akvc n dostnlhc x.

Xkjutokl = (Unjj I

)systca

1 (

G)systca

(Ue)

sys+ (U

T)

sys1 (G)

sys ns (U

T)

sys1 2


16/32


17/32

RF_XOHX


Slot ki Rkwcr =

N ulot pkwcr os tfc pkwcr ki nl neclt wfohf dkcs ulot wkrg ol ulot toac.

Tfc pkwcr ki nl neclt os snod tk c klc wntt oi ot dkcs klc mkujc ki wkrg ol klc schkld.

> wntt 1 > mkujc/schkld 1 >25 cre/schkld

Njsk, > wntt 1schkld>

actrc>lcwtkl> 1 > L a s> .

Doaclsoklnj ikraujn ki pkwcr

WRkwcrQ 1WToacQ

QUkrgW1

QTW

QTAJW 00

1 WAJ0 T8Q

Rkwcr fns > doaclsokl ol anss, 0 doaclsokls ol jcletf nld 8 doaclsokls ol toac.

X.Lk. Fuanl Nhtovoty Rkwcr (U)

> Fcnrt `cnt >.0

0 Xjccpole 68

8 Xottole >02

4 Podole ol n hnr >42

3 Unjgole (4.6 ga f

>

) 0939 Hyhjole (>3 ga f

>) 4>2

5 Rjnyole Tcllos 442

6 Xwoaaole ( rcnststrkgc, >.9 ga f>

) 453

? Xgntole 383

>2 Hjoa`ole Xtnors (>>9 stcps aol>

) 963

>> Hyhjole (0>.8 ga f>

) 522

>0 Rjnyole @nsgct`njj 622

>8 Tu`c joeft 42

>4 Inl 92

Cxnapjc 42 N `jkhg akvcs wotf hklstnlt vcjkhoty > a/s uldcr tfc nhtokl ki fkrozkltnj ikrhc32 L kl n fkrozkltnj

surinhc. Ufnt os tfc pkwcr ki cxtcrlnj ikrhc nld irohtokl;

Xkjutokl = Xolhc n 1 2 o.c. ig1 32 L

Rcxt

1 32 > 1 32 U

Ri1 32 >1 32 U

Rkwcr os njsk tfc rntc nt wfohf clcrey os suppjocd.

Lct pkwcr 1 R>+ R

0+ R

8.............

Rlct

1dt

dU>+

dt

dU0...... R

lct1

dt

..........dUdU 0>

Rlct

1dt

dGU

njj1G

Pntc ki hfnlec ki golctoh clcrey os njsk pkwcr.


18/32

RF_XOHX


Cxnapjc 4> N stklc os prkmchtcd wotf vcjkhoty nt nl nlejcwotf fkrozkltnj. Iold kut(o) Nvcrnec pkwcr ki tfc ernvoty durole toac t.

(oo) Olstnltnlckus pkwcr duc tk ernvotntoklnj ikrhc nt toac t wfcrc t os toac ki ijoeft.

(ooo) Ufcl os nvcrnec pkwcr zcrk ;

(ov) Ufcl os Rolst

zcrk ;

(v) Ufcl os Rolst

lcentovc ;

(vo) Ufcl os Rolst

pksotovc ;

Xkjutokl =

v

f

u

(o) :R< 1T

w1

t

aef1

t

et0

>tsoluae 0

:R< 1

solu

0

etae

(oo) Olstnltnlckuspkwcr

R 1 vI

1 (aem ) Wuhkso+ (usol et)m ) Q

1 ae(usol et)

(ooo) solu0

et t 1

e

solu0 , o.c. toac ki ijoeft.

(ov) Ufcl I

& Y

nrc o.c. nt t 1 esolu

wfohf os nt tfc foefcst pkolt.

(v) Irka nsc tk foefcst pkolt.

(vo) Irka foefcst pkolt tk nsc.

RKTCLTONJ CLCPE_Clcrey =-Ot os tfc oltcrlnj hnpnhoty tk dk wkrg.

Golctoh Clcrey =-Ot os oltcrlnj hnpnhoty tk dk wkrg `y vortuc ki rcjntovc aktokl.

Rktcltonj Clcrey =-Ot os tfc oltcrlnj hnpnhoty tk dk wkrg `y vortuc ki rcjntovc pksotokl.

Cxnapjc =- Ernvotntoklnj Rktcltonj Clcrey , Xprole RC cth.

Rktcltonj Clcrey

Dciolotokl=

Rktcltonj clcrey os tfc oltcrlnj hnpnhoty ki dkole wkrg ki n systca `y vortuc ki ots hklioeurntokl.

Ol hnsc ki hklscrvntovc ikrhc (iocjd) pktcltonj clcrey os cqunj tk lcentovc ki wkrg dklc `y

tfc hklscrvntovc ikrhc ol sfoitole tfc`kdy irka skacrcicrclhc pksotokl tk eovcl pksotokl.


19/32

RF_XOHX

"anlosfguanrpfysohs.ol" >?

Tfcrcikrc, ol hnsc ki hklscrvntovc ikrhc

0

>

0

>

r

r

S

SrdIdS

o.c.,0

>

r

0 >r

S S I d r U

Ufclcvcrnld wfcrcvcrpksso jc, wc tngc tfc rcicrclhc pkolt ntnld nssuac pktcltonj clcrey tk `c zcrktfcrc, o.c., Oi wc tngc r

>1nld S

>1 2 tfcl

UrdISr

(n) Ernvotntokl Rktcltonj Clcrey =

S 1 aef ikr n pnrtohjc nt n fcoeft f n`kvc rcicrclhc jcvcj.

Cxnapjc 40 Hnjhujntc pktcltonj clcreyki n uloikra vcrtohnj rkd ki anss A nld jcletf.

A

Xkjutokl =

x

dx

A dS 1 (da) ex

2

S

2

exdxA

dS

S 10

Ae

(`) Xprole pktcltonj clcrey =

S 1 0Gx

0

>

Ufcrc x os hfnlec ol jcletf irka ots lnturnj jcletf.

Lktc = Ernvotntoklnj pktcltonj clcrey hnl `c +vc , vc kr zcrk `ut sprole pktcltonj clcrey wojj njwnys `c +vc.

Cxnapjc 48 Ol tfc eovcl ioeurc, n uloikra rkd ki anss a nld jcletf j os folecd

nt klc cld. Iold tfc wkrg dklc `ynppjocd ikrhc ol sjkwjy`roleole

tfc rkd tk tfc olhjolcd pksotokl.

192

Inpp

a13ge

1 >a

Xkjutokl = UNJJ

1G `y wkrg clcrey tfckrca

Inppae

x

Ly

Lxx 10

0

hks 92 1

4

>a 1 2.03 a

GUUUU InppeLL yx ( 2UUyx LL

)

2 ae(2.03) + UInpp

1 2 2

G 1 2 ns sjkwjy `rkueftUInpp

1 3 ?.6 2.03

Ot hnl `c sccl tfnt UInpp

1 aef 1 3 ?.6 2.03 M.


20/32

RF_XOHX


Cxnapjc 44 N uloikra rkd ki anss A nld jcletfos fcjd ol pksotokl

sfkwl ol tfc ioeurc. Iold tfc pktcltonj clcreyki tfc rkd.

S 1 K

A,

Xkjutokl = dS 1 da.e.f

S 1 K

A,

xsol

dx

x

dS1

2

A

dx . e . x sol

S 10

solAe

Lktc tfnt hcltrc ki anss ki tfc rkd os nt fcoeft

sol

0

irka tfc erkuld.

Cxnapjc 43

N hfnol ki anss A os gcpt kl n fcaospfcrc ns sfkwl. Iold pktcltonj clcrey ki tfc hfnol.

Xkjutokl =Pd

Pd

A

Psol

S 1 2

Uc glkw tfntPndous

nrh1

cjcacltnj jcletf 1 Pd

da 1

d0/

A1

d

A0

Lkw dS 1 da ef 1

d

A0(e) (P sol)

S

2

dS1A0

Pe

0/

2

sol d

S 1

AeP0

2

0/hks

S 1 Ae

P0

W Lktc tfnt

P0

os tfc fcoeft ki HKA Q


21/32

RF_XOHX

"anlosfguanrpfysohs.ol" 0>

Cxnapjc 49 N uloikra skjod spfcrc ki anss A nld rndous P os gcpt kl tfc fkrozkltnj surinhc.

A,P

Iold pktcltonj clcrey ki tfc skjod spfcrc.

Xkjutokl = S 1 Aefha

ikr syaactrohnj `kdy HKA os tfc eckactrohnj hcltrc ki tfc `kdy.

S 1 AeP

Cxnapjc 45 Iold wkrg dklc y ernvoty ol akvole tfc jkhg

A

A

S 1 2N

@

f

(o) irka N tk @

(oo) irka @ tk N

(ooo) Hnjhujntc SN S

@

(ov) Hnjhujntc S@ S

N

Xkjutokl = (Ue)N tk @

1 aef

(Ue)

@ tk N1 aef

SN S

@1 aef

S@ S

N1 aef

Ot hnl `c snod

Ue

1 SUkrg dklc `y ernvoty os tfc -vc ki tfc hfnlec ol RC .

o.c. Ue1 WS

i S

oQ U

e1 S

o S

i

Oapkrtnlt Rkolts ikr R.C. =>. Rktcltonjclcreyhnl`c dciolcd kljy ikrhklscrvntovc ikrhcs. Ot fnslk rcjcvnlhcikrlkl-hklscrvntovc

ikrhcs.

0. Rktcltonjclcreyhnl `c pksotovc kr lcentovc, dcpcldole upkl hfkohcki irnac ki rcicrclhc.

8. Rktcltonj clcrey dcpclds kl irnac ki rcicrclhc ut hfnlec ol pktcltonj clcrey os oldcpcldclt ki

rcicrclhcirnac.

4. Rktcltonj clcrey sfkujd c hklsodcrcd tk `c n prkpcrty ki tfc cltorc systca, rntfcr tfnl nssoelole

ot tk nly spchoioh pnrtohjc.

3. Ot os n iulhtokl ki pksotokl nld dkcs lkt dcpcld kl tfc pntf.

Ukrg dklc `y sprole ikrhc

Ns n`kvc UXR

1 S

S 1 2

g

S 1 2

x 1 2 x 1 x

S 1 >/0 gx0

UXRI

1 S

UXRI1 So SI

UXRI

1 2 0

>gx0 1

0

>gx0


22/32

RF_XOHX


Cxnapjc 46

A,J

Rce

Iold kut wkrg dklc `y cxtcrlnj neclt tk sjkwjy fnle tfc jkwcr cld ki tfc hfnol tk tfc pce.

Xkjutokl = A,J

Rce

Olotonjjy So1 Ae

0

J

Si1 Ae

4

J

Ue1 S

o S

i1 (Ae

0

J) (Ae

4

J) 1 Ae

4

J

Ssole wkrg clcrey tfckrca, Ue+ U

cxt1G 1 2 U

cxt1 U

e1 Ae

4

J

Cxnapjc 4? Ol n`kvc cxnapjc iold kut tfc wkrg dklc `y cxtcrlnj neclt tk sjkwjy fnle tfc aoddjc jolg tk pce.

Xkjutokl = So1 Ae

0J 7

Si1

6

Je

0

A

4

Je

0

A

Ucxt

1 Si S

o 1 AeJ

>9

3

Cxnapjc 32 G 1 >22 L/a

x12 x12.> x12.0H @ N

Ioldkut tfc wkrg dklc`ysprole ikrhc irkaN tk @ nld irka @ tkH.x 1 2 os pksotokl ki lnturnj jcletf.

Xkjutokl = (Usprole

)N@

1 So S

i 1

0

>G (2.0)0

0

>G(2.>)0

(Usprole

)N@

10

8M

Xoaojnrjy

(Usprole

)@H

10

>M


23/32

RF_XOHX


Cxnapjc 3> (n) Tfc anss a os akvcd irka N tk H njkle tfrcc doiicrclt pntfs

vcrtohnjpjnlc

(o)N@H

(oo)NDH

(ooo)NH

Iold kut wkrg dklc `y ernvoty ol tfc tfrcc hnscs.

(`) Tfc `jkhg os akvcd irka N tk H njkle tfrcc doiicrclt pntfs.

Nppjocd ikrhc os fkrozkltnj. Iold wkrg dklc y irohtokl ikrhcol pntf

(o)N@H

Pkueffkrozkltnjpjnlc

(oo)NDH

(ooo)NH

Xkjutokl = (n) (o) aef

(oo) aef

(ooo) aef

(`) (o) UN@H

1 ae (n+`)(oo) U

NDH1 ae (n+`)

(ooo) UNH1 ae ( 00

`n )

HKLXCPYNTOYC IKPHCXN ikrhc os snod tk `c hklscrvntovc oi wkrg dklc `y kr nenolst tfc ikrhc ol akvole n `kdy dcpclds kljy kl tfc

olotonj nld iolnj pksotokls ki tfc `kdy nld lkt kl tfc lnturc ki pntf ikjjkwcd `ctwccl tfc olotonj nld iolnj

pksotokls.

Hklsodcr n `kdyki anss a cole rnoscd tk n fcoeft f vcrtohnjjy upwnrds ns sfkw ol n`kvc ioeurc. Tfc wkrg

dklc osaef. Xuppksc wc tngc tfc `kdynjkle tfc pntf ns ol (`). Tfcwkrg dklc durole fkrozkltnj aktokl os

zcrk. Nddole up tfc wkrgs dklc ol tfc twk vcrtohnj pnrts ki tfc pntfs, wc ect tfc rcsujtaef klhc nenol.Nly

nr otrnry pntf jogc tfc klc sfkwl ol (h)hnl `c `rkgcl oltk cjcacltnry fkrozkltnj nld vcrtohnj pkrtokls. Ukrg

dklc njkle tfc fkrozkltnj pnrts os zcrk. Tfc wkrg dklc njkle tfc vcrtohnj pnrts ndd up tk aef.Tfus wc

hklhjudc tfnt tfc wkrg dklc ol rnosole n `kdy nenolst ernvoty os oldcpcldclt ki tfc pntf tngcl. Ot kljy

dcpclds upkl tfc olotonj nld iolnj pksotokls ki tfc kdy. Uc hklhjudc irka tfos doshussokl tfnt tfc ikrhc ki

ernvoty os n hklscrvntovc ikrhc.

Cxnapjcs ki Hklscrvntovc ikrhcs.

(o) Ernvotntoklnj ikrhc, lkt kljyduc tk tfc Cnrtf ut ol ots eclcrnj ikra ns eovcl y tfc ulovcrsnj jnw ki

ernvotntokl, os n hklscrvntovc ikrhc.

(oo) Cjnstoh ikrhc ol n strcthfcd kr hkaprcsscd sprole os n hklscrvntovc ikrhc.

(ooo) Cjchtrkstntoh ikrhc `ctwccl twkcjchtroh hfnrecs os n hklscrvntovc ikrhc.

(ov) Anelctoh ikrhc `ctwccl twk anelctoh pkjcs os n hklscrvntovc ikrhcs.

Ol inht, njj iuldnacltnj ikrhcs ki lnturc nrc hklscrvntovc ol lnturc.


24/32

RF_XOHX


Ikrhcs nhtole njkle tfc jolc mkolole tfc hcltrcs ki twk `kdocs nrc hnjjcd hcltrnj ikrhcs. Ernvotntoklnj ikrhc

nld Cjchtrkstntoh ikrhcs nrc twk oapkrtnlt cxnapjcs ki hcltrnj ikrhcs. Hcltrnj ikrhcs nrc hklscrvntovc

ikrhcs.

RPKRCPTOCX KI HKLXCPYNTOYC IKPHCX

(o) Ukrg dklc y kr nenolst n hklscrvntovc ikrhc dcpclds kljy kl tfc olotonj nld iolnj pksotokls ki tfc

`kdy.

(oo) Ukrg dklc `ykr nenolst n hklscrvntovc ikrhc dkcs lkdcpcldupkl tfc lnturcki tfc pntf`ctwccl olotonj nld

iolnj pksotokls ki tfc kdy.

Oi tfc wkrg dklc n `yn ikrhc olakvole n `kdyirka nl olotonj jkhntokl tk n iolnj jkhntokl os oldcpcldclt ki tfc

pntf tngcl `ctwccl tfc twk pkolts, tfcl tfc ikrhc os hklscrvntovc.

(ooo) Ukrg dklc y kr nenolst n hklscrvntovc ikrhc ol n rkuld trop os zcrk.

Oi n `kdy akvcs uldcr tfc nhtokl ki n ikrhc tfnt dkcs lk tktnj wkrg durole nly rkuld trop, tfcl tfc ikrhc os

hklscrvntovc7 ktfcrwosc ot os lkl-hklscrvntovc.

Tfc hklhcpt ki pktcltonj clcreycxosts kljy ol tfc hnsc ki hklscrvntovc ikrhcs.

(ov) Tfc wkrg dklc `yn hklscrvntovc ikrhc os hkapjctcjy rchkvcrn`jc.

Hkapjctc rchkvcrn`ojoty os nl oapkrtnlt nspcht ki tfc wkrg ki n hklscrvntovc ikrhc.

LKL-HKLXCPYNTOYC IKPHCX

N ikrhc os snod tk `c lkl-hklscrvntovc oi wkrg dklc `y kr nenolst tfc ikrhc ol akvolen `kdydcpclds upkl tfc

pntf ctwccl tfc olotonj nld iolnj pksotokls.

Tfc irohtoklnj ikrhcsnrclkl-hklscrvntovc ikrhcs. Tfos os `chnusc tfcwkrg dklc nenolst irohtokl dcpclds kl

tfc jcletf ki tfc pntf njkle wfohf n `kdy os akvcd. Ot dkcs lkt dcpcld kljykl tfc olotonj nld iolnj pksotokls.

Lktc tfnt tfc wkrg dklc `y irohtoklnj ikrhc ol n rkuld trop os lkt zcrk.

Tfc vcjkhoty-dcpcldclt ikrhcs suhfns nor rcsostnlhc, voshkus ikrhc cth., nrc lkl hklscrvntovc ikrhcs.

X.Lk. Hklsc rvn tovc ikrhc s Lkl-Hklsc rvntovc ikrhc s

> Ukrg dklc dkcs lkt dcpcld upkl pntf Ukrg dklc dcpclds kl pntf.

0 Ukrg dklc ol rkuld t rop os zcrk. Ukrg dklc ol n rkuld trop os lkt zcrk.

8 Hcltrnj ol lnturc. Ikrhcs nrc vcjkhoty -dcpcldclt nld

rctnrdole ol lnturc.

4

Ufcl kljy n hklscrvntovc ikrhc nhts wotfol n

systrca, tfc golctoh clrey nld pktcltonj clcrey

hnl hfnlec. Fkwcvcr tfcor sua, tfc achfnlohnj

clcrey ki tfc systca, dkcs lkt hfnlec.

Ukrg dklc nenolst n lkl-hklscrvntovc

ikrhc any `c dossopntcd ns fcnt clcrey.

3 Ukrg dklc os hkapjctcjy rchkvcrn`jc. Ukrg dklc ol lkt hkapjctcjy

rchkvcrn`jc.


25/32

RF_XOHX


Cxnapjc 30 Tfc ioeurc sfkws tfrcc pntfs hkllchtole pkolts n nld .

N solejc ikrhc I dkcs tfc oldohntcd wkrg kl n pnrtohjc

akvole njkle cnhf pntf ol tfc oldohntcd dorchtokl. Kl tfc

`nsos ki tfos olikrantokl, os ikrhc I hklscrvntovc;

Nlswcr = Lk

Cxpjnlntokl = Ikr n hklscrvntovc ikrhc, tfc wkrg dklc ol n rkuld trop sfkujd `c zcrk.

Cxnapjc 38 Iold tfc wkrg dklc `y n ikrhcI

1 xo+ ym nhtole kl n pnrtohjc tk dospjnhc ot irka pkolt N (2, 2) tk

@(0, 8).

Xkjutokl = dU 1I

.ds 1 (xo+ ym ).(dxo+ dym )

U 1 0

2xdx +

8

2ydy 1

0

2

0

0

x

+

8

2

0

0

y

1

0

>8ulots

Truc kr Injsc

Cxnapjc 34. Ol hnsc ki n lkl hklscrvntovc ikrhc wkrg dklc njkle twk doiicrclt pntfs wojj njwnys `c doiicrclt.

Nlswcr = Injsc

Cxnapjc 33. Ol hnsc ki lkl hklscrvntovc ikrhc wkrg dklc njkle twk doiicrclt pntfs any`c doiicrclt.

Nlswcr = Truc

Cxnapjc 39. Ol hnsc ki lkl hklscrvntovc ikrhc wkrg dklc njkle njj pksso jc pntfs hnllkt `c snac.

Nlswcr = Truc

Cxnapjc 35. Iold wkrg dklc `yn ikrhcI

1 x o+ xymnhtolekln pnrtohjc tkdospjnhc ot irka pkolt K(2,2) tk H(0, 0).

Xkjutokl = dU 10

2

xdx + 0

2

xydy

hnl `c ikuld hnllkt `c ikuld

ultoj x os glkwl ol

tcras ki y o.c. ultoj

cquntokl ki pntf os glkwl.

Cxnapjc 36. Iold tfc wkrg dklc `yI

irka K tk H ikr n`kvc cxnapjc oi pntfs nrc eovcl ns `cjkw..

Xkjutokl = KNH KN+NH

ikr KN y 1 2 dy 1 2

KNdU 1 0

2

xdx + 2 UKN

1 0 M

ikr NH x 1 0 dx 1 2

NHdU 1 2 + 0

0

2 ydy UNH 1 4M

UKNH1 UKN + UNH1 0 + 4 1 9M


26/32

RF_XOHX


(oo) K@H K@ + @Hikr K@ x 1 2 dx 1 2 UK@1 Kikr @H y 1 0 dy 1 2

dU 1 xdx U 10

2

0

0

x

1 0 M

UKNH UK@H

Fclhc tfc ikrhc os lkl-hklscrvntovc.

(ooo) Ikr UKH

dU 1 xdy + xydx

ikr KH x 1 y dx 1 dy

dU 1 0

2xdx +

0

2

0dyy U 18

>4ulot

Cxnapjc 3? Iold kut wkrg dklc `y tfc ikrhc I

1 y o+ xmtk dospjnhc

tfc pnrtohjc irka pkolt K tk H njkle tfc eovcl pntfs. Dchodc

wfctfcr tfc ikrhc os hklscrvntovc kr lklhklscrvntovc.

Xkjutokl = (o) KNH KN+NH

ikr KN y 1 2 dy 1 2 dU 1 2 UKN 1 2ikrNH x 1 0 dx 1 2

dU 10 8

2

dy U 1 9 M UKNH

1 9 ulots

(oo) K@H K@ + @Hikr K@ x 1 2 dx 1 2 dU 1 2ikr @H y 1 8 dy 1 2

dU 1 0

2dx8 U 1 9 ulots U

K@H1 9 ulots

(ooo) KH

ikr KH y 10

8x dy 1

0

8dx

dU 1 0

2

xdx0

8+

0

2

xdx0

8

dU 1 8 0

2

dxx

UKH1 9 ulotsN`kvc ikrhc sccas hklscrvntovc `ut hnllkt `c hklioracd yct uljcss wc hnl oltcerntc ot wotfkut tfcglkwjcdec ki pntf.

Nenol wc fnd dw 1 xdy + ydx & xdy + ydx hnl `c wrottcl nsdxy

dU 1 xyd U 1 8,0

2,2dxy 1 8,0 2,2xy 1 9M

Fclhc glkwjcdec ki pntf wns lkt rcquorcd tk oltcerntc tfc n`kvc sk I os hklscrvntovc.


27/32

RF_XOHX


RKTCLTONJ CLCPE_ NLD HKLXCPYNTOYC IKPHC =

IX1 S/ s,

o.c. tfcprkmchtokl ki tfc iocjd ikrhc, tfcvchtkrI, ntn eovcl pkolt ol tfc dorchtokl ki tfc dospjnhcaclt dr cqunjs

tfc dcrovntovc ki tfc pktcltonj clcrey S wotf rcspcht tk n eovcl dorchtokl, tngcl wotf tfc kppksotc soel. Tfc

dcsoelntokl ki n pnrtonj dcrovntovc / s capfnsozcs tfc inht ki dcrovole wotf rcspcht tk n dciolotc dorchtokl.

Xk, fnvole rcvcrscd tfc soel ki tfc pnrtonj dcrovntovcs ki tfc iulhtokl S wotf rcspcht tk x, y, z, wc k`tnol tfc

prkmchtokl Ix, Iynld Izki tfc vchtkrIkl tfc ulot vchtkrso,mnldg. Fclhc, klc hnl rcndojy iold tfc vchtkr

otscji = I 1 Ixo+ I

ym+ I

zg, kr

Ufcl hklscrvntovc ikrhc dkcs pksotovcwkrg tfcl RC dchrcnscs

dS 1 dw

dS 1 I.ds

dS 1 (Ixo+ Iym

+ Iz g).(dxo+ dym

+ dzg)

dS 1 Ixdx I

ydy I

zdz

oi y & z nrc hklstnlts tfcl dy 1 2 dz 1 2

dS 1 Ixdx

Ix1 dxdS

oi y & z nrc hklstnlt

Ix1 xS

Xoaojnrjy Iy1

y

u

7 Iz1

z

u

I 1

gmoz

S

y

S

x

S.

Tfc qunltoty ol pnrcltfcscs os rcicrrcd tk ns tfc shnjnr erndoclt ki tfc iulhtokl S nld os dclktcd `y ernd S

krS. Uc sfnjj usc tfc schkld, akrc hklvcloclt, dcsoelntokl wfcrc (ln`jn) soeloiocs tfc sya`kjohvchtkrkr kpcrntkr

1zyx

gmo

Rktcltonj Clcrey hurvc =

N ernpf pjkttcd `ctwccl tfc RC n pnrtohjcnld ots

dospjnhcaclt irka tfc hcltrcki ikrhc iocjd oshnjjcd

RC hurvc.

Ssole ernpf, wc hnl prcdoht tfc rntc ki aktokl ki

n pnrtohjc nt vnrokus pksotokls.

Ikrhc kl tfc pnrtohjc os I(x)

1 dx

dS

Hnsc = O Kl olhrcnsole x, oi S olhrcnscs, ikrhc os ol () vc x dorchtokl o.c. nttrnhtokl ikrhc.

Hnsc = OO Kl olhrcnsole x, oi S dchrcnscs, ikrhc os ol (+) vc x-dorchtokl o.c. rcpujsokl ikrhc.


28/32

RF_XOHX


Cxnapjc 92. Tfc pktcltonj clcrey ki sprole os eovcl `y S 10

>gx0 , wfcrc x os cxtclsokl sprole

Iold tfc ikrhc nsskhontcd wotf tfos pktcltonj clcrey.

Xkjutokl = Ix1

x

u

1 gx Iy1 2 I

z1 2.

Cxnapjc 9>. Tfc pktcltonj clcrey ki n pnrtohjc ol n spnhc os eovcl `y S 1 x0 + y0.Iold tfc ikrhc nsskhontcd wotf tfos pktcltonj clcrey.

Xkjutokl = Ix1

x

u

1 W0x + 2Q 1 0x

Iy1

y

u

1 (0y + 2) 1 0y

I

1 0xo 0ym

Cxnapjc 90. Iold kut tfc pktcltonj clcrey ki eovcl ikrhc I

1 0xo 0ym .

Xkjutokl = dS 1 dU

dS 1 )mdyodx()my0ox0(

dS 1 ydy0xdx0 S 1 x0 + y0 + H

Cxnapjc 98 Iold kut tfc pktcltonj clcrey ki tfc ikrhc I 1 y o+ xm .

Xkjutokl = dS 1 dU

dS 1 (y o+ xm ) . (dxo+ dym )

dS 1 xdyydx

dS 1 dxy S 1 xy + h

Cxnapjc 94 Iold kut tfc ikrhc ikr wfohf pktcltonj clcrey S 1 xy.

Xkjutokl = I

1

my

So

x

S I

1

my

)xy(o

x

)xy(

I

1 y o + xm Fclhc vcroiyole tfc prcvokuscxnapjc.

C^SOJO@POSA KI N RNPTOHJCDoiicrclt pksotokls ki n pnrtohjc =-

Rksotokl ki cquojo roua =

Oi lct ikrhc nhtole kl n `kdy os zcrk, ot os snod tk `c ol cquojo`roua. Ikr cquojo`rouadx

dS1 2. Rkolts R, ^ & P

nrc tfc stntcs ki cquojo`roua pksotokls.

Typcs ki cquojo`roua =

Xtn`jc cquojo roua =Ufcl n pnrtohjc os dospjnhcd sjoeftjy irka n pksotokl nld n ikrhc nhtole kl ot `roles ot `nhg tk tfc

olotonj pksotokl, ot os snod tk `c ol stn`jc cquojo`roua pksotokl.


29/32

RF_XOHX

"anlosfguanrpfysohs.ol" 0?

Lchcssnry hkldotokls = dx

dS1 2, nld

0

0

dx

Sd1 +vc

Slstn`jc Cquojo`roua =

Ufcl n pnrtohjc os dospjnhcd sjoeftjy irka n pksotokl nld ikrhc nhtole kl ot trocs tk dospjnhc tfc

pnrtohjc iurtfcr nwny irka tfc cquojo`roua pksotokl, ot os snod tk `c ol ulstn`jc cquojo`roua.

Hkldotokl = dx

dS

1 2 pktcltonj clcrey os anxoaua o.c. 1 0

0

dx

Sd

1 vc

Lcutrnj cquojo roua =

Ol tfc lcutrnj cquojo roua pktcltonj clcrey os hklstnlt.Ufcl n pnrtohjc os dospjnhcd irka otspksotokl

ot dkcs lkt cxpcroclhc nly ikrhc nhtole kl ot nld hkltolucs tk `c ol cquojo`roua ol tfc dospjnhcd

pksotokl. Tfos os snod tk c lcutrnj cquojo`roua.

N pnrtohjc os ol cquojo`roua oi tfc nhhcjcrntokl ki tfc pnrtohjc os zcrk. Ns nhhcjcrntokl os irnac dcpcl-

dclt qunltoty tfcrcikrc cquojo`roua dcpclds kl aktokl ki k`scrvcr njsk.

Cxnapjc 93 Tfc pktcltonj clcrey`ctwccl twk ntkas ol n akjchujc os eovcl `y, S(x)

1 >0x

n 9x

`, wfcrc n nld `

nrcpksotovc hklstnlts nld x os tfc dostnlhc `ctwccl tfc ntkas. Tfcsystca os ol stn`jc cquojo roua

wfcl-

(N) x 1 2 (@) x 1`0

n(H) x 1

9/>

`

n0

(D) x 1

`3

n>>

Xkjutokl = (H) Eovcl tfnt, S(x) 1 >0x

n 9x

`

Uc, glkw I 1 dx

du1 (>0) n x>8 (9`) x5 1 2

kr 5x

`91 >8x

n>0kr x9 1 >0n/9` 1 0n/` kr x 1

9/>

`

n0

0

0

dx

Sd1 +vc nt x 1

9/>

`

n0

Cxnapjc 99. (N)

u 1 2

(@) (H) (D)

(C) (I) AkklC

Cnrtf

ki tfc hnscs n`kvc wfohf os lkt n hnsc ki cquojo`roua.

Xkjutokl = (I) ns akkl os njwnys nhhcjcrntcd. Ot fns hcltropctnj nhhcjcrntokl kr ot os hfnleole ots vcjkhotynjj tfc

toac.


30/32

RF_XOHX


Cxnapjc 95.

C

E

Iold kut pksotokls ki cquojo`roua nld dctcraolc wfctfcr tfcy nrc stn`jc, ulstn jc kr lcutrnj.

Xkjutokl = Cquojo`roua os nt @,D,I ns ikrhc os zcrk fcrc.

Ikr hfchgole typc ki cquojo`roua dospjnhc sjoeftjy.

Uc fnvc @ ns stn jc cquojo roua

D ns ulstn`jc cquojo`roua nld I ns lcutrnj cquojo`roua

Cxnapjc 96.

C

E

S

Odcltoiy tfc pkolts ki cquojo`roua nld doshuss tfcor lnturc.

Xkjutokl = H,C,I nrc pkolts ki cquojo`roua chnusc I 1x

S

Ufcl sjkpc ki S - x hurvc os zcrk tfcl I os zcrk.

Hfchg stn`ojoty tfrkuef sjkpcs nt lcnr `y pkolts.

Oi wc akvc roeft tfcl sjkpc sfkujd `c pksotovc ikr stn jc cquojo roua nld vohc vcrsn. Ol sfkrt ot os

jogc n fojj nld pjntcnu.

ACHFNLOHNJ CLCPE_ =Dciolotokl=Achfnlohnj clcrey C ki nl k`mcht kr n systca os dciolcd ns tfc sua ki golctoh clcrey G nld

pktcltonj clcrey S, o.c., C 1 G + S

Oapkrtnlt Rkolts ikr A.C.=>. Ot os n shnjnr qunltoty fnvole doaclsokls WAJ0T-0Q nld XO ulots mkujc.

0. Ot dcpclds kl irnac ki rcicrclhc.

8. N `kdy hnl fnvc achfnlohnj clcreywotfkut fnvole cotfcr golctoh clcreykr pktcltonj clcrey. Fkw-

cvcr, oi ktf golctoh nld pktcltonj clcreocs nrc zcrk, achfnlohnj clcrey wojj c zcrk. Tfc hklvcrsc

anykr anylkt `c truc, o.c., oi C 1 2 cotfcr `ktf RC nld GC nrc zcrk kr RC any`c lcentovc nld GC

any `c pksotovc suhf tfnt GC + RC 1 2.

4. Ns achfnlohnj clcrey C 1 G + S, o.c., C - S 1 G. Lkw ns G os njwnys pksotovc, C - S 2,o.c., ikrcxostclhc ki n pnrtohjc ol tfc iocjd, C S.

3. Ns achfnlohnj clcrey C 1 G + S nld G os njwnys pksotovc, sk, oi S os pksotovc C wojj `c pksotovc.

Fkwcvcr, oi pktcltonj clcreyS os lcentovc, C wojj `c pksotovc oi G < |S|nld C wojj `c lcentovc oi G : |S|o.c., achfnlohnj clcrey ki n `kdy kr systca hnl `c lcentovc, nld lcentovc achfnlohnj clcrey

acnls tfnt pktcltonj clcrey os lcentovc nld ol anelotudc ot os akrc tfnl golctoh clcrey. Xuhf n

stntc os hnjjcd `kuld stntc, c.e., cjchtrkl ol nl ntka kr n sntcjjotc akvole nrkuld n pjnlct nrc ol

`kuld stntc.


31/32

RF_XOHX

"anlosfguanrpfysohs.ol" 8>

Cxnapjc 9? Ns sfkwl ol ioeurc tfcrc os n sprole `jkhg systca. @jkhg ki anss 322 e os prcsscd nenolst n

fkrozkltnj sprole ioxcd nt klc cld tk hkaprcss tfc sprole tfrkuef 3.2 ha. Tfc sprole hklstnlt os

322L/a. Ufcl rcjcnscd, tfc`jkhg akvcsfkrozkltnjjy tojj ot jcnvcs tfc sprole. Hnjhujntc tfcdostnlhc

wfcrc ot wojj fot tfc erkuld 4 a cjkw tfc sprole;

Xkjutokl = Ufcl `jkhg rcjcnscd, tfc `jkhg akvcs fkrozkltnjjy wotf spccd Y tojj ot jcnvcs tfc sprole.

@y clcreyhklscrvntokl0

>gx0 1

0

>av0

Y0 1a

gx0 Y 1

a

gx0

Toac ki ijoeft t 1eF0

Xk. fkrozkltnj dostnlhc trnvcjjcd irka tfc ircc cld ki tfc sprole os Y t

1a

gx0

e

F0

13.2

)23.2(322 0

>2

40 1 0 a

Xk, Nt n fkrozkltnj dostnlhc ki 0 a irka tfc ircc cld ki tfc sprole.

Cxnapjc 52 N roeod `kdy ki anss a os fcjd nt n fcoeft F kl twk sakktf wcdecs ki anss A cnhf ki wfohf

nrc tfcascjvcs nt rcst kl n fkrozkltnj irohtokljcss ijkkr. Kl rcjcnsole tfc `kdy ot akvcs dkwl

pusfole nsodc tfc wcdecs. Tfc vcjkhoty ki rchcdc ki tfc wcdecs irka cnhf ktfcr wfcl roeod

`kdy os nt n fcoeft f irka tfc erkuld os

(N)A0a

)fF(ae0

(@)Aa0

)fF(ae0

(H)A0a

)fF(ae6

(D)Aa0

)fF(ae6

Xkjutokl =


32/32

RF_XOHX

Jct spccd ki tfc wcdec nld tfc roeod `kdy `c Y nldrcspchtovcjy.

Tfcl nppjyole wcdec hklstrnolt wc ect

Y hks 43 1hks 43

Y 1 ...(o)

Ssole clcrey hklscrvntokl,

ae(F f) 1 0

0AY

0

>+

0a

0

> ...(oo)

Irka cquntokl (o) nld (oo)

Y 1A0a

)fF(ae0

Tfc vcjkhoty ki rchcdc ki wcdecs irka cnhf ktfcr 1 0 Y 1A0a

)fF(ae6

Xk, nlswcr os (H)

Njtcr =

Jcletf ki rkd 1

x + y 10

dt

dx+

dt

dy1 2

vcjkhoty ki `jkhg 1 vcjkhoty ki rkd

dchrcnsc ol pktcltonj clcrey 1 olhrcnsc ol golctoh clcrey

ae (F f) 10

>a Y0 +

0

>AY0 +

0

>AY0

Y 1aA0

)fF(ae0

0Y 1aA0

)fF(ae6

Date post:	03-Jun-2018
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