geometrical optics theory_e

7/21/2019 Geometrical Optics Theory_E

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BEAJEXP L O AVX L SLGXPAHUCXLAG =

@oue ondes, ackre heserts, breeg iarest, ngh juotlcaoareh rnlgàws cng è egmayeh `y ngyage wka kns eyes

wltk wklck ta see tkej. @ut `y stuhylgb tke `rngck ai pkyslcs cnooeh aptlcs, wklck henos wltk tke èknvlaurai olbkt ngh atker eoectrajnbgetlc wnves, we cng renck n heeper nppreclntlagai tkevlslòe waroh. N dgawoehbe

ai tke prapertles ai olbkt nooaws us ta ugherstngh tke òue caoar ai tke sdy ngh tke heslbg ai aptlcno hevlces

suck ns teoescapes, jlcrascapes, cnjerns, eyebonsses, ngh tke kujng eyes. Xke snje `nslc prlgclpoes

ai aptlcs nosa ole nt tke kenrt ai jaherg heveoapjegts suck ns tke onser, aptlcno ilèrs, kaoabrnjs, aptlcno

cajputers, ngh gew teckglques lg jehlcno ljnblgb.

>. CAGHLXLAG IAP PECXLOLGENP VPAVNBNXLAG AI OLBKX(Agoyiar lgiarjntlag gat lg LLX-MEEsyoonùs)

Saje pnrt ai tke aptlcs cng è ugherstaah li we nssuje tknt olbkt trnveos lg n strnlbkt olge ngh lt

èghs n`ruptoy wkeg lt suiiers reioectlag ar reirnctlag.

Xke nssujptlag tknt tke olbkt trnveos lg n strnlbkt olge ls carrect li

(l) tke jehluj ls lsatraplc, l.e. lts èknvlaur ls snje lg noo

hlrectlags ngh (ll) tke a`stncoe pnst wklck tke olbkt javes ar tke

apeglgb tkraubk wklck tke olbkt javes ls gat very sjnoo.

Cagslher n solt ai wlhtk n tkraubk wklck jagackrajntlc olbkt

rnys pnss ngh strlde n screeg, ponceh nt n hlstngce H ns skawg.

Lt ls iaugh tknt tke olbkt strldes lg n `ngh ai wlhtk ` jare tkng n. Xkls èghlgb ls cnooeh

hliirnctlag.

Olbkt èghs `y (`-n)/0 ag enck slhe ai tke cegtrno olge .Lt cng è skawg `y wnve tkeary ai olbkt tknt

slg :n

.......(N),

wkere ls skawg lg ilbure.

Xkls iarjuon lghlcntes tknt tke èghlgb ls cagslhernòe agoy wkeg n ~ . Hliirnctlag ls jarepragaugceh lg saugh ècnuse lts wnveoegbtk ls juck jare tkng tknt ai olbkt ngh lt ls ai tke arher ai

tke slze ai a`stncoes ar npertures. Iarjuon (N) blvesnH0

n`

.

Lt ls coenr tknt tke èghlgb ls gebolblòe liH

nn

ar n H .Li tkls caghltlag ls iuoilooeh , olbkt ls

snlh ta jave rectlolgenroy. Lg jast ai tke sltuntlags lgcouhlgb beajetrlcno aptlcs tke caghltlags nre

suck tknt we cng snieoy nssuje tknt olbkt javes lg strnlbkt olge ngh èghs agoy wkeg lt bets

reioecteh ar reirncteh.

Xkus beajetrlcno aptlcs ls ng nppraxljnte trentjegt lg wklck tke olbkt wnves cng è represegteh

`y strnlbkt olges wklck nre cnooeh rnys. N rny ai olbkt ls tke strnlbkt olge pntk ai trngsier ai olbkt

egerby. Nrraw represegts tke hlrectlag ai prapnbntlag ai olbkt.

Ilbure skaws n rny wklck lghlcntes olbkt ls javlgb iraj N ta @.


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"jnglskdujnrpkyslcs.lg" 0

0. VPAVEPXLES AI OLBKX(l) Speeh ai olbkt lg vncuuj, hegateh `y c, ls equno ta 3 >6; j/s nppraxljnteoy.

(ll) Olbkt ls eoectrajnbgetlc wnve (prapaseh `y Jnxweoo). Lt

cagslsts ai vnrylgb eoectrlc ileoh ngh jnbgetlc ileoh.

(lll) Olbkt cnrrles egerby ngh jajegtuj.

(lv) Xke iarjuon v : i ls nppolcnòe ta olbkt.

(v) Rkeg olbkt bets reioecteh lg snje jehluj, lt suiiers ga ckngbe lg irequegcy, speeh ngh

wnveoegbtk.

(vl) Irequegcy ai olbkt rejnlgs ugckngbeh

wkeg lt bets reioecteh ar reirncteh.

3. PEIOECXLAG AI OLBKXRkeg olbkt rnys strlde tke àughnry ai twa jehln suck ns nlr ngh bonss, n pnrt ai olbkt ls turgeh`ncd lgta tke snje jehluj. Xkls ls cnooeh Peioectlag ai Olbkt.

(n) P ebuonr P eioectlag=

Rkeg tke reioectlag tndes ponce iraj n periect ponge surince

lt ls cnooeh Pebuonr Peioectlag. Lg tkls cnse tke reioecteh

olbkt kns onrbe lgtegslty lg age hlrectlag ngh gebolblòy sjnoo

lgtegslty lg atker hlrectlags.

(`) Hliiuseh Peioectlag

Rkeg tke surince ls raubk, we ha gat bet n rebuonrèknvlaurai olbkt. Notkaubk nt enck palgt

olbkt rny bets reioecteh lrrespectlve ai tke avernoo gnture ai surince, hliieregce ls a`servehècnuse eveg lg n gnrraw ènj ai olbkt tkere nre jngy rnys wklck nre reioecteh iraj hliieregt

palgts ai surince ngh lt ls qulte passlòe tknt tkese rnys jny jave lg hliieregt hlrectlags hue

ta lrrebuonrlty ai tke surince . Xkls pracess egnòes us ta see ng a`mect iraj ngy pasltlag.

Suck n reioectlag ls cnooeh ns hliiuseh reioectlag.

Iar exnjpoe reioectlag iraj n wnoo, iraj n gews pnper etc.

Xkls ls wky yau cng gat see yaur ince lg gews pnper ngh lg

tke wnoo.

3.> Onws ai Peioectlag(n) Xke lgclhegt rny, tke reioecteh rny ngh tke garjno nt tke palgt ai lgclhegce ole lg tke

snje ponge. Xkls ponge ls cnooeh tke ponge ai lgclhegce (ar ponge ai reioectlag).

Xkls caghltlag cng è expresseh j ntkejntlcnooy ns P

. (

G

) : G

. (

P

) :


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. ( G

P

) : 6 wkere

, G

ngh P

nre vectars ai ngy jnbgltuhe noagb lgclhegt

rny, tke garjno ngh tke reioecteh rny respectlveoy.

(`) Xke ngboe ai lgclhegce (tke ngboe ètweeg garjno

ngh tke lgclhegt rny) ngh tke ngboe ai reioectlag (tke

ngboe ètweeg tke reioecteh rny ngh tke garjno) nre

equno, l.e.

l :rSpeclno Cnses =

Garjno Lgclhegce = Lg cnse olbkt ls lgclhegt garjnooy,

l : r : 6

: >;6

Gate = Re sny tknt tke rny kns retrnceh lts pntk.

Brnzlgb Lgclhegce = Lg cnse olbkt strldes tke reioectlgb surince tngbegtlnooy,

l : r :

Skaw tknt iar n olbkt rny lgclhegt nt ng ngboe l ag bettlgb reioecteh tke ngboe ai hevlntlag ls

: 0l ar 0l.Saoutlag

Iraj ilbure (`) lt ls coenr tknt olbkt rny èghs eltker `y >

ngtlcoacdwlse ar `y0

(: 0 >)

coacdwlse.

Iraj ilbure (n)>

: 0l .

0: + 0l .

3.0 A`mect ngh Ljnbe(n) A`mect (A) = A`mect ls heilgeh ns palgt ai lgtersectlag ai lgclhegt rnys.


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AA

Oet us cnoo tke slhe lg wklck lgclhegt rnys nre presegt ns lgclhegt slhe ngh tke slhe lg wklck

reioecteh ( reirncteh) rnys nre presegt, ns reioecteh (reirncteh) slhe.

Gate = Ng a`mect ls cnooeh reno li lt oles ag lgclhegt slhe atkerwlse lt ls cnooeh vlrtuno.

(`) Ljnbe () = Ljnbe ls heilgeh ns palgt ai lgtersectlag ai reioecteh rnys (lg cnse aireioectlag) arreirncteh rnys (lg cnse ai reirnctlag).

Gate = Ng ljnbe ls cnooeh reno li lt oles ag reioecteh ar reirncteh slhe atkerwlse lt ls cnooeh vlrtuno.

8. VONGE JLPPAPVonge jlrrar ls iarjeh `y paolsklgb age surince ai n ponge tklg bonss ponte .Lt ls nosa snlh ta è

slovereh ag age slhe.

N ènj ai pnrnooeo rnys ai olbkt, lgclhegt ag n ponge jlrrar wloo bet rei oecteh ns n ènj ai pnrnooeo

reioecteh rnys.

Exnjpoe - 0Iar n i lxeh lgclhegt o lbkt rny, l i tke jlrrar è

ratnteh tkraubk ng ngboe (nàut ng nxls wklck oleslg tke ponge ai jlrrar ngh perpeghlcuonr ta tke ponge

ai lgclhegce), skaw tknt tke reioecteh rny turgs tkraubk

ng ngboe 0 lg snje segse.

Saoutlag See ilbure J>, G

>ngh P

> lghlcnte tke lgltlno pasltlag ai jlrrar, lgltlno garjno ngh lgltlno hlrectlag

ai reioecteh olbkt rny respectlveoy. J0, G

0 ngh P

0 lghlcnte tke ilgno pasltlag ai jlrrar, ilgno

garjno ngh ilgno hlrectlag ai reioecteh olbkt rny respectlveoy. Iraj ilbure lt ls coenr tkntN@C

: 0 + ) ar : 0.

Gate = Deeplgb tke jlrrar ilxeh li tke lgclhegt rny ls ratnteh `y ngboe nàut tke garjno tkegreioecteh rny ratntes `y snje ngboe lg tke snje hlrectlag ai ratntlag


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8.> Valgt a`mectCknrncterlstlcs ai ljnbe hue tareioectlag`y n ponge jlrrar =

(l) Hlstngce ai a`mect iraj jlrrar : Hlstngce ai

ljnbe iraj tke jlrrar.

(ll) Noo tke lgclhegt rnys iraj n palgt a`mect wloo

jeet nt n slgboe palgt niter reioectlag iraj n

ponge jlrrar wklck ls cnooeh ljnbe.

(lll) Xke olge malglgb n palgt a`mect ngh lts ljnbe

ls garjno ta tke reioectlgb surince.

(lv) Iar n reno a`mect tke ljnbe ls vlrtuno ngh iar n

vlrtuno a`mect tke ljnbe ls reno

(v) Xke reblag lg wklck a`server's eye just è

presegt lg arher ta vlew tke ljnbe ls cnooeh

ileoh ai vlew.

Exnjpoe - 3

Ilbure skaws n palgt a`mect N ngh n ponge jlrrar JG. Ilgh tke

pasltlag ai ljnbe ai a`mect N, lg jlrrar JG, `y hrnwlgb rnyhlnbrnj. Lghlcnte tke reblag lg wklck a`servers eye just è

presegt lg arher ta vlew tke ljnbe. (Xkls reblag ls cnooeh ileoh ai vlew).

Saoutlag See ilbure, cagslher ngy twa rnys

ejngntlgb iraj tke a`mect. G>

ngh

G0

nre garjnos 4

l>

: r>

ngh l0

: r0

Xke jeetlgb palgt ai reioecteh rnys P>

ngh P0 ls ljnbe N. Xkaubk

agoy twa rnys nre cagslhereh lt just è ugherstaah tknt noo rnys

iraj N reioect iraj jlrrar suck tknt tkelr jeetlgb palgt ls N. Xa

a`tnlg tke reblag lg wklck reioecteh rnys nre presegt, malg N wltk

tke eghs ai jlrrar ngh extegh. Xke iaooawlgb ilbure skaws tkls

reblag ns sknheh. Lg ilbure, tkere nre ga reioecteh rnys èyagh tke

rnys > ngh 0, tkereiare tke a`servers V ngh ^ cnggat see tke

ljnbe ècnuse tkey ha gat recelve ngy reioecteh rny.

Exnjpoe - 8

Ilgh tke reblag ag \ nxls lg wklck reioecteh rnys nre presegt.A`mect ls nt N (0, 6) ngh JG

ls n ponge jlrrar, ns skawg.

Saoutlag Xke ljnbe ai palgt N, lg tke jlrrar ls nt N (9, 6).

Malg N J ngh extegh ta cut \ nxls nt J ( Pny arlblgntlgb

iraj N wklck strldes tke jlrrar nt J bets reioecteh ns

tke rny JJ wklck nppenrs ta caje iraj N). Malg NG

ngh extegh ta cut \ nxls nt G ( Pny arlblgntlgb iraj N

wklck strldes tke jlrrar nt G bets reioecteh ns tke rny

GG wklck nppenrs ta caje iraj N) .


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Irajbeajetry.

J (6, 9)G (6, ) Slze ai extegheh a mect :Slzeai extegheh ljnbe.

(0) Xke ljnbe ls erect, li tke extegheh a`mect ls ponceh pnrnooeo ta tke jlrrar.

(3) Xke ljnbe ls lgverteh li tke extegheh a`mect oles perpeghlcuonr ta tke ponge jlrrar.

(8) Li ng extegheh karlzagtno a`mect ls ponceh lgiragt ai n jlrrar lgcolgeh 85 wltk tke karlzagtno,

tke ljnbe iarjeh wloo `e vertlcno. See ilbure.


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Exnjpoe - 5

Skaw tknt tke jlgljuj slze ai n ponge jlrrar, requlreh ta see tke iuoo ljnbe ai ng

a`server ls knoi tke slze ai tknt a`server.

Saoutlag See tke iaooawlgb ilbure. Lt ls seoi expongntary li yau cagslher oegbtks x ngh y ns skawg lg

ilbure.

Noter = EJ>J

0nghE KI nre sljlonr

z0

z

IK

JJ 0>

ar J>

J0

: K I / 0 : KI / 0

8.3 Peontlag `etweeg veoaclty ai a`mect ngh ljnbe =Iraj jlrrar praperty = x

lj: - x

aj , y

lj: y

aj ngh z

lj: z

aj

Kere xlj

jengs x caarhlgnte ai ljnbe wltk respect ta jlrrar. Sljlonroy atkers knve

carrespaghlgbjenglgb.

Hliieregtlntlgb w.r.t tlje , we bet

v(lj)x

: -v(aj)x

4 v(lj)y

: v(aj)y

4 v(lj)z

: v(aj)z ,

Iar x nxls vlB

vjB

: (vaB

vjB

)

`ut Iar y nxls ngh z nxls vlB

vjB

: (vaB

vjB

) ar vlB

: vaB

.

kere= vlB : veoaclty ai ljnbe wltk respect ta braugh.

Exnjpoe - 9

Ng a`mect javes wltk 5 j/s tawnrhs rlbkt wkloe tke jlrrar javes wltk >j/s tawnrhs tke oeit

ns skawg. Ilgh tke veoaclty ai ljnbe.

Saoutlag

Xnd e ns + hlrectlag . vl- v

j : v

j- v

6

vl- (->) : (->) - 5

vl: - 2j/s. 2 j/s ngh hlrectlag tawnrhs oeit.

Exnjpoe - 2

Xkere ls n palgt a`mect ngh n ponge jlrrar. Li tke jlrrar ls javeh `y >6 cj nwny iraj tke

a`mect ilgh tke hlstngce wklck tke ljnbe wloo jave.


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Saoutlag

Re dgaw tknt xlj

: - xaj

ar xl x

j : x

j x

a

ar xlx

j:x

jx

a.

Lg tkls questlag xa

: 6 4 xj

: >6 cj.

Xkereiarexl: 0x

jx

a: 06 cj.

Noter = -0(x + >6) : 0 x + h

h : 06 cj

Exnjpoe - ;

Lg tke sltuntlag skawg lg

ilbure ,ilgh tke veoaclty ai

ljnbe.

Saoutlag Noagb x hlrectlag, nppoylgb vl v

j : (v

6 v

j)

vl (5cas 36) : (>6 cas 96 ( 5 cas 36 6))

vl : - 5 (>+ 3 ) j/s

Noagb y hlrectlag v6

: vl

vl : >6 slg 96 : 5 3 j/s

Yeoaclty ai tke ljnbe : - 5 ( >+ 3 ) l + 5 3 m j/s.

8.8 Ljnbes iarjeh `y twa ponge jlrrars =Li rnys niter bettlgb reioecteh iraj age jlrrar strlde secagh jlrrar, tke ljnbe iarjeh `y ilrst

jlrrar wloo iugctlag ns ng a`mect iar secagh jlrrar, ngh tkls pracess wloo cagtlgue iar every

successlve reioectlag.

Exnjpoe -

ls 0 cj ngh tknt iraj J0

ls

; cj. Ilgh tke hlstngce ai ljnbes iraj tke twa jlrrars

cagslherlgb reioectlag ag jlrrar J>

ilrst.

Saoutlag

Xa ugherstngh kaw ljnbes nre iarjeh see tke iaooawlgb ilbure ngh tn`oe. \au wloo requlre ta

dgaw wknt syj`aos olde>0>

stnghs iar. See tke iaooawlgb hlnbrnj.


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"jnglskdujnrpkyslcs.lg" J> Pnys 0 A > NA : 0cj N>: 0 cjPnys 0 J0 Pnys 3 > >0 @>: >0 cj @>0: >0 cjPnys 3 J> Pnys 8 >0 >0> N>0: 00cj N>0>: 00cjPnys 8 J0 Pnys 5 >0> >0>0 @>0>:30cj @>0>0:30cj

Sljlonroy ljnbes wloo è iarjeh `y tke rnys strldlgb jlrrar J0

ilrst. Xatno gujèr ai ljnbes :.

Exnjpoe - >6

Cagslher twa perpeghlcuonr jlrrars. J>

ngh J0

ngh n palgt a`mect A. Xndlgb arlblg nt tke

palgt ai lgtersectlag ai tke jlrrars ngh tke caarhlgnte ai a`mect ns (x, y), ilgh tke pasltlag

ngh gujèr ai ljnbes.

Saoutlag Pnys n ngh ` strlde jlrrar J>

agoy ngh tkese rnys wloo

iarj ljnbe>

nt (x, y), suck tknt A ngh L>

nre equlhlstngt

iraj jlrrar J>. Xkese rnys ha gat iarj iurtker ljnbe

ècnuse tkey ha gat strlde ngy jlrrar nbnlg. Sljlonroy rnys

h ngh e strlde jlrrar J0

agoy ngh tkese rnys wloo iarj

ljnbe0

nt (x, y), suck tknt A ngh L0

nre equlhlstngt iraj

jlrrar J0.

Gaw cagslher tkase rnys wklck

strlde jlrrar J0

ilrst ngh tkeg

tke jlrrar J>.

Iar lgclhegt rnys >, 0 a`mect ls A, ngh reioecteh rnys 3, 8 iarj ljnbe 0.

Gaw rnys 3, 8 lgclhegt ag J>

(a`mect ls0) wklck reioect ns rnys 5, 9 ngh iarj ljnbe

0>. Pnys

5, 9 ha gat strlde ngy jlrrar, sa ljnbe iarjntlag staps.

0

ngh0>

, nre equlhlstngt iraj J>. Xa sujjnrlze see tke iaooawlgb ilbure


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"jnglskdujnrpkyslcs.lg" >6

Iar rnys reioectlgb ilrst iraj J>

ngh tkeg iraj J0, ilrst ljnbe

>(nt (x, y)) wloo è iarjeh ngh

tkls wloo iugctlag ns a`mect iar jlrrar J0

ngh tkeg lts ljnbe>0

(nt (x, y)) wloo è iarjeh.

L>0

ngh L0>

calgclhe.

Xkree ljnbes nre iarjeh

8.5 Oacntlgb noo tke ljnbes iarjeh `y twa ponge jlrrars=Cagslher twa ponge j lrrars J

>ngh J

0lgcolgeh nt ng ngboe

: ns skawg lg ilbure.

Valgt V ls ng a`mect dept suck tknt lt jndes ngboe wltkjlrrar J

>ngh ngboe wltk jlrrar J

0. Ljnbe ai a`mect V

iarjeh `y J>

,hegateh `y>

,wloo è lgcolgeh `y ngboe ag tke atker slhe ai jlrrar J

>. Xkls ngboe ls wrltteg lg

`rncdet lg tke ilbure èslhes>. Sljlonroy ljnbe ai a`mect

V iarjeh `y J0

,hegateh `y0

,wloo è lgcolgeh `y ngboeag tke atker slhe ai jlrrar J

0. Xkls ngboe ls wrltteg lg

`rncdet lg tke ilbure èslhes0.

Gaw 0

wloo nct ns ng a`mect iar J>

wklck ls nt ng ngboe iraj J>

. Lts ljnbe wloo è

iarjeh nt ng ngboe ag tke appaslte slhe ai J>. Xkls ljnbe wloo è hegateh ns

ngh

sa ag. Xklgd wkeg tkls wloo pracess stap. Klgt= Xke vlrtuno ljnbe iarjeh `y n ponge jlrrar

just gat è lg iragt ai tke jlrrar ar lts extegslag.

Gujèr ai ljnbes iarjeh `y twa lgcolgeh jlrrars

(l) Li

396: eveg gujèr4 gujèr ai ljnbe :

396

>

(ll) Li

396: ahh gujèr4 gujèr ai ljnbe :

396 >,

li tke a`mect ls ponceh ag tke ngboe `lsectar.

(lll) Li

396: ahh gujèr4 gujèr ai ljnbe :

396

,

li tke a`mect ls gat ponceh ag tke ngboe `lsectar.

(lv) Li

396 lgteber,, tkeg caugt tke gujèr ai ljnbes ns exponlgeh nàve.

Exnjpoe - >>

Xwa jlrrars nre lgcolgeh `y ng ngboe 366. Ng a`mect ls ponceh jndlgb >66 wltk tke jlrrar J>.

Ilgh tke pasltlags ai ilrst twa ljnbes iarjeh `y enck jlrrar. Ilgh tke tatno gujèr ai ljnbes

uslgb (l) hlrect iarjuon ngh (ll) caugtlgb tke ljnbes.


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"jnglskdujnrpkyslcs.lg" >>

Saoutlag Ilbure ls seoi expongntary.

Gujèr ai ljnbes

(l) Uslgb hlrect iarjuon =36

396: >0 (eveg gujèr)

gujèr ai ljnbes : >0 > : >>

(ll) @y caugtlgb . See tke iaooawlgb tnòe

5. SVKEPLCNO JLPPAPSSpkerlcno Jlrrar ls iarjeh `y paolsklgb age surince ai n pnrt aispkere. Hepeghlgb upag wklck pnrt ls sklglgb tke spkerlcno jlrrar ls

consslileh ns (n) Cagcnve jlrrar, li tke slhe tawnrhs cegter ai

curvnture ls sklglgb ngh (`) Cagvex jlrrar li tke slhe nwny iraj tke

cegter ai curvnture ls sklglgb.

5 .> Ljpartngt terjs reonteh wltk spkerlcno jlrrars =

P

CV

N

@

N spkerlcno skeoo wltk tke cegterai curvnture, paoe nperture nghrnhlus ai curvnture lhegtlileh

(n) Cegter ai Curvnture (C) =

Xke cegter ai tke spkere iraj wklck tke spkerlcno jlrrar ls iarjeh ls cnooeh tke cegter ai

curvnture ai tke jlrrar. Lt ls represegteh `y C ngh ls lghlcnteh lg ilbure.

(`) Vaoe (V) =

Xke cegter ai tke jlrrar ls cnooeh ns tke Vaoe. Lt ls represegteh `y tke palgt V ag tke jlrrar

NV@ lg ilbure.

(c) Vrlgclpno Nxls =Xke Vrlgclpno Nxls ls n olge wklck ls perpeghlcuonr ta tke ponge ai tke jlrrar ngh pnsses

tkraubk tke paoe. Xke Vrlgclpno Nxls cng nosa è heilgeh ns tke olge wklck malgs tke Vaoe ta

tke Cegter ai Curvnture ai tke jlrrar.


12/55

VK\SLCS


(h) Npertur e (N) =

Xke nperture ls tke sebjegt ar nren ai tke jlrrar wklck ls nvnlon`oe iar reioectlgb olbkt. Lg

ilbure. NV@ ls tke nperture ai tke jlrrar.

(e) Vrlgclpoe iacus (I) =

Lt ls tke palgt ai lgtersectlag ai noo tke reioecteh rnys iar wklck tke lgclhegt rnys strlde tke

jlrrar (wltk sjnoo nperture) pnrnooeo ta tke prlgclpno nxls. Lg cagcnve jlrrar lt ls reno ngh lg

tke cagvex jlrrar lt ls vlrtuno. Xke hlstngce iraj paoe ta iacus ls cnooeh iacno oegbtk.

Cagcnve jlrrar

Cagvex jlrrar

Exnjpoe - >0

Ilgh tke ngboe ai lgclhegce ai rny iar wklck lt pnsses

tkraubk tke paoe, blveg tknt J || CV.

Saoutlag = JLC : CV :J || CV J : CV :C : CV CV :CV : LgCV noo ngboe nre equno3 : >;6 : 96

Exnjpoe - >3

Ilgh tke hlstngce C^ li lgclhegt olbkt rny pnrnooeo ta

prlgclpno nxls ls lgclhegt nt ng ngboe l. Nosa ilgh tke

hlstngce C^ li l 6.

Saoutlag =

cas l :C^0

P C^ :

lcas0

P

Ns l lgcr enses cas l hecrenses. Li l ls n sjnoo ngboe cas l >Kegce C^ lgcrenses C^ : P/0

Sa, pnrnxlno rnys jeet nt n hlstngce equno ta P / 0 iraj cegter ai curvnture, wklck ls cnooeh iacus.


13/55

VK\SLCS


5.> Pny trnclgb =Iaooawlgb incts nre useiuo lg rny trnclgb.

(l). Li tke lgclhegt rny ls pnrnooeo ta tke prlgclpoe nxls,

tke reioecteh rny pnsses tkraubk tke iacus. I

(ll). Li tke lgclhegt rny pnsses tkraubk tke iacus, tkeg tke reioecteh rny ls pnrnooeo ta tkeprlgclpoe nxls.

(lll). Lgclhegt rny pnsslgb tkraubk cegtre ai curvnture wloo è reioecteh `ncd tkraubk tke

cegtre ai curvnture (ècnuse lt ls n garjnooy lgclhegt rny).

(lv). Lt ls ensy ta jnde tke rny trnclgb

ai n rny lgclhegt nt tke paoe ns

skawg lg èoaw.

5.0 Slbg CagvegtlagRe nre uslgb caarhlgnte slbg cagvegtlag.

(l) Xnde arlblg nt paoe (lg cnse ai jlrrar )ar nt aptlcno cegtre (lg cnse ai oegs)

Xnde [ nxls noagb tkeVrlgclpnoNxls ,tndlgbpasltlve hlrectlag noagb tke lgclhegt olbkt.u, v, P ngh i lghlcnte tke x caarhlgnte ai a`mect, ljnbe, cegtre ai curvnture ngh

iacus respectlveoy.

(ll) y-caarhlgntes nre tndeg pasltlve nàve Vrlgclpoe Nxls ngh gebntlve èoaw Vrlgclpoe Nxls

k>

ngh k0

hegate tke ycaarhlgntesai a`mect ngh ljnbe respectlveoy.

Gate = Xkls slbg cagvegtlag ls useh iar reioectlag iraj jlrrar, reioectlag tkraubk iont ar curveh

surinces ar oegs.

5.3 Iarjuon iar Peioectlag iraj spkerlcno jlrrars =( n) J lrrar ia rjuon =

>

v+

>

u:

0

P:

>

i

[-caarhlgnte ai cegtre ai curvnture ngh iacus aicagcnvejlrrar nre gebntlve ngh tkase iar

cagvex jlrrar nre pasltlve. Lg cnse ai jlrrars slgce olbkt rnys reioect `ncd lg [-hlrectlag,

tkereiare -ve slbg ai v lghlcntes reno ljnbe ngh +ve slbg ai v lghlcntes vlrtuno ljnbe.

Exnjpoe - >8

Ilbure skaws n spkerlcno cagcnve jlrrar wltk lts

paoe nt (6, 6) ngh prlgclpno nxls noagb x nxls.

Xkere ls n palgt a`mect nt (86 cj, >cj), ilgh

tke pasltlag ai ljnbe.


14/55

VK\SLCS


Saoutlag Nccarhlgb ta slbg cagvegtlag,

u : 86 cj

k>

: +> cj

i : 5 c j.

i

>

u

>

v

>

5

>

86

>

v

>

4 v :

2

86cj. 4

u

v

k

k

>

0

k0

: u

v k

> :

86

>2

86

:2

> cj.

Xke pasltlag ai ljnbe ls

cj

2

>,cj

2

86

Exnjpoe - >5

Cagverblgb rnys nre lgclhegt ag n cagvex spkerlcno jlrrar sa tknt tkelr extegslags lgtersect

36 cj `eklgh tke jlrrar ag tke aptlcno nxls. Xke reioecteh rnys iarj n hlverblgb `enj sa

tknt tkelr extegslags lgtersect tke aptlcno nxls >.0 j iraj tke jlrrar. Heterjlge tke iacno

oegbtk ai tke jlrrar.

Saoutlag Lg tkls cnse u : + 36

v : + >06

i

>:

v

>+

u

>:

>06

>+

36

>

i : 08 cj

Exnjpoe - >9

Ilgh tke pasltlag ai ilgno ljnbe niter tkree successlve

reioectlags tndlgb ilrst reioectlag ag j >.

Saoutlag reioectlag =Iacus ai jlrrar : >6 cj u : >5 cjNppo ylgb jlrrar iarjuon =

i

>

u

>

v

> v : 36 cj.

Iar reioectlag ag ponge jlrrar =u : >6 cj

v : >6 cjIar reioectlag ag curveh jlrrar nbnlg =

u : 56 cj

i : >6 cj

Nppo ylgb jlrrar iarjuon =

i

>

u

>

v

>

v : >0.5 cj.

( ) Onterno jnbglilcntlag (ar trngsverse jnbglilcntlag) hegateh `y j ls heilgeh ns j:k

k

0

>

ngh

ls reonteh ns j : v

u.Iraj tke heilgltlag ai j pasltlve slbg ai j lghlcntes erect ljnbe ngh


15/55

VK\SLCS


gebntlve slbg lghlcntes lgverteh ljnbe.

(c) Lg cnse ai successlve reioectlag iraj jlrrars, tke avernoo onterno jnbglilcntlag ls blveg `y

j>

j0

j3

......, wkere j>

, j0

etc. nre onterno jnbglilcntlags prahuceh `y lghlvlhuno

jlrrars.

k>

ngh k0

hegate tke y caarhlgnte ai a`mect ngh ljnbe respectlveoy.

Gate = Uslgb 5.3(n) ngh 5.3(`) tke iaooawlgb cagcouslags cng è jnhe (ckecd yaurseoi).

Gnture ai A`mect Gnture ai Ljnbe Lgverteh ar erect

Peno Peno Lgverteh

Peno Ylrtuno Erect

Ylrtuno Peno Erect

Ylrtuno Ylrtuno Lgverteh

Iraj 5.3(n) ngh 5.3(`)4 we bet j:i

vi

ui

i

...............(must n tlje snvlgb iarjuon)

Exnjpoe - >2

Ng extegheh a`mect ls ponceh perpeghlcuonr ta tkeprlgclpnonxls ai n cagcnve jlrrar ai rnhlus

ai curvnture 06 cj nt n hlstngce ai >5 cj iraj paoe. Ilgh tke onterno jnbglilcntlag

prahuceh.

Saoutlag u : >5 cj i : >6 cj

Uslgbi

>

u

>

v

> we bet, v : 36 cj

j :u

v : 0.

Nolter = j :ui

i

:

)>5(>6

>6

: 0

Exnjpoe - >;

N persag oaad s lgta n spkerlcno jlr rar. Xke slze ai ljnbe ai kls ince ls twlc e tke nctu no slze

ai kls ince. Li tke ince ls nt n hlstngce 06 cj tkeg ilgh tke gnture ngh rnhlus ai curvnture ai

tke jlrrar.

Saoutlag Versag wloo see kls ince agoy wkeg tke ljnbe ls vlrtuno. Ylrtuno ljnbe ai reno a`mect ls erect.Kegce j : 0

u

v: 0 v : 86 cj

Nppoylgbi

>

u

>

v

> 4 i : 86 cj ar P : ;6 cj (cagcnve) P.A.C. : ;6 cj

Noter = j :ui

i

0 :

)06(i

i

i : 86 cj ar P : ;6cj (cagcnve) P.A.C. : ;6 cj

Exnjpoe - >9

Saoutlag Slgce tke ljnbes iarjeh ag screeg lt ls reno. Peno a`mect ngh reno ljnbe ljpoles cagcnve

jlrrar.

Nppoylgb j :ui

i

ar 0 :

)u(i

i

.................(>)

Niter sklitlgb 3 :)5u(i

i

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

TRky u + 5 1 , wky gat u 5 = Lg n cagcnve jlrrar, tke slze ai reno ljnbe wloo lgcrense, agoy

wkeg tke reno a`mect ls `raubkt coaser ta tke jlrrar. Lg halgb sa, lts x caarhlgnte wloo

lgcrenseZ

Iraj (>) & (0) we bet,

i : 36 cj ar P : 96 cj (cagcnve) ngh P.A.C. : 96cj

h Yeoaclty ai ljnbe(l) A`mect javlgb perpeghlcuonr ta prlgclpno nxls =

Iraj tke reontlag lg5.3.(`)

we knve

u

v

k

k

>

0 ar k0

:u

v . k

>

Li n palgt a`mect javes perpeghlcuonr ta tkeprlgclpnonxls, x caarhlgnte ai àtk tke a`mect &

tke ljnbe ècaje cagstngt. Ag hliieregtlntlgb tke nàve reontlag w.r.t. tlje , we bet,

ht

hk0:

ht

hk

u

v >

Kere,ht

hk>hegates veoaclty ai a`mect perpeghlcuonr ta tke prlgclpno nxls ngh

ht

hk0hegates

veoaclty ai ljnbe perpeghlcuonr ta tke prlgclpno nxls.

(ll) A`mect javlgb noagb prlgclpno nxls =

Ag hliieregtlntlgb tke jlrrar iarjuon wltk respect ta tlje we betht

hu

u

v

ht

hv0

0 , wkere hvht

ls

tke veoaclty ai ljnbe noagb prlgclpno nxls nghhu

ht ls tke veoaclty ai a`mect noagb prlgclpno

nxls. Gebntlve slbg ljpoles tknt tke ljnbe , lg cnse ai jlrrar, nownys javes lg tke hlrectlag

appaslte ta tknt ai a`mect. Xkls hlscusslag ls iar veoaclty wltk respect ta jlrrar ngh noagb tke

x nxls.

(lll) A`mect javlgb nt ng ngboe wltk tke prlgclpno nxls = Pesaove tke veoaclty ai a`mect noagb

ngh perpeghlcuonr ta tke prlgclpno nxls ngh ilgh tke veoacltles ai ljnbe lg tkese hlrectlags

sepnrnteoy ngh tkeg ilgh tke resuotngt.

(e) A ptlcno paw er ai n jlrrar (lg Hlaptre) : i

>

i : iacno oegbtk wltk slbg ngh lg jeters.

(i) L i a`mect oylgb noagb tke prlgclpno nxl s l s gat ai v ery s jno o s lze, tke oagbltuhlgno

jnbglilcntlag :>0

>0

uu

vv (lt wloo nownys è lgverteh)


17/55

VK\SLCS


(b) Li tke slze a`mect ls very sjnoo ns cajpnreh ta lts hlstngce iraj Vaoe tkeg

Ag hliieregtlntlgb tke jlrrar iarjuon we bethu

hv: 0

0

u

v =Jntkejntlcnooy 'hu' ljpoles sjnoo ckngbe lg pasltlag ai a`mect ngh 'hv' ljpoles carrespaghlgb

sjnoo ckngbe lg pasltlag ai ljnbe. Li n sjnoo a`mect oles noagb prlgclpnonxls, hu jny lghlcnte

tke slze ai a`mect ngh hv tke slze ai lts ljnbe noagb prlgclpno nxls (Gate tknt tke iacus

skauoh gat ole lg ètweeg tke lgltlno ngh ilgno palgts ai a`mect). Lg tkls cnsehv

hu ls cnooeh

oagbltuhlgno jnbglilcntlag. Gebntlve slbg lghlcntes lgverslag ai ljnbe lrrespectlve ai

gnture ai ljnbe ngh gnture ai jlrrar.

Exnjpoe - 06

N palgt a`mect ls ponceh 96 cj iraj paoe ai n cagc nve jlr rar ai iacno oegbtk >6 cj ag tke

prlgclpoe nxls. Ilgh

(n) tke pasltlag ai ljnbe

( ) Li a`mect ls skliteh > jj tawnrhs tke jlrrar noagb prlgclpoe nxls ilgh tke sklit lg

ljnbe. Exponlg tke resuot.

Saoutlag (n) u : 96 cj

i : >6cj

iu

iuv

:

)>6(96

)96(>6

:56

966

: >0 cj.

(`)i

>

u

>

v

>

Hliieregtlntlgb , we bet hv : 0

0

u

vhu :

0

96

>0

T> jj Z : 05

>jj

T hu : >jj4 slbg ai hu ls + ècnuse lt ls skliteh lg +ve hlrectlag heilgeh `y slbg

cagvegtlag.Z

(N) ve slbg ai hv lghlcntes tknt tke ljnbe wloo sklit tawnrhs gebntlve hlrectlag.

(@) Xke slbg ai v ls gebntlve. Rklck ljpoles tke ljnbe ls iarjeh ag gebntlve slhe ai paoe. (N)ngh (@) tabetker ljpoy tknt tke ljnbe wloo sklit nwny iraj paoe.

Gate tknt hliieregtlnos hv ngh hu hegate sjnoo ckngbes agoy.

(k) Gew ta g's Iarjuon= [\ : i 0

[ ngh \ nre tke hlstngces ( noagb tke prlgclpno nxls ) ai tke a`mect ngh ljnbe respectlveoy

iraj tke prlgclpno iacus. Xkls iarjuon cng è useh wkeg tke hlstngces nre jegtlageh ar

nsdeh iraj tke iacus.

9. PEIPNCXLAG AI OLBKXRkeg tke olbkt ckngbes lts jehluj, saje ckngbes accurs lg lts prapertles, tke pkegajegag lsdgawg ns reirnctlag.

Li tke olbkt ls lgclhegt nt ng ngboe(6 ? l ?


18/55

VK\SLCS

"jnglskdujnrpkyslcs.lg" >;

Li tke olbkt ls lgclhegt garjnooy tkeg lt baes ta tke secagh jehluj wltkaut èghlgb, ùt stloolt ls cnooeh reirnctlag.

Peirnctlve lghex ai n jehluj ls heilgeh ns tke inctar `y wklck speeh ai olbkt rehuces ns

cajpnreh ta tke speeh ai olbkt lg vncuuj.v

c :jehlujlgolbktaispeeh

vncuujlgolbktaispeeh.

Jare (oess) reirnctlve lghex ljpoles oess (jare) speeh ai olbkt lg tknt jehluj, wklck tkereiare ls

cnooeh hegser (rnrer) jehluj.

9.> Onws ai Peirnctlag(n) Xke lgclhegt rny , tke garjno ta ngy reirnctlgb surince nt tke palgt ai lgclhegce ngh

tke reirncteh rny noo ole lg tke snje ponge cnooeh tke ponge ai lgclhegce ar ponge ai

reirnctlag.

(`)rSlg

lSlg: Cagstngt iar ngy pnlr ai jehln ngh iar olbkt

ai n blveg wnve oegbtk. Xkls ls dgawg ns Sgeoo's

Onw.

Nosa,Slg l

Slg r :

g

g

0

>

:v

v

>

0

:

>

0

Iar nppoylgb lg praòejs rejejèr

g>slgl : g

0slgr

g

g

0

>

:>g

0: Peirnctlve Lghex ai tke secagh jehluj wltk respect ta tke ilrst jehluj.

c : speeh ai olbkt lg nlr (ar vncuuj) : 3 x >6 ; j/s.

Speclno cnses = Garjno lgclhegce = l : 6

iraj Sgeoos onw = r : 6

Rkeg olbkt javes iraj hegser ta rnrer jehluj lt èghs nwny iraj garjno.

Rkeg olbkt javes iraj rnrer ta hegser jehluj lt èghs tawnrhs tke garjno.


19/55

VK\SLCS

"jnglskdujnrpkyslcs.lg" > 4 gnlr

: ~ > 4 gwnter (nvernbe vnoue) : 8/3 4 gbonss(nvernbe vnoue): 3/0

9.0 Hevlntlag ai n Pny Hue ta PeirnctlagHevlntlag () ai rny lgclhegt

nt l ngh reirncteh nt rls blveg `y: |lr| .

Exnjpoe - 0>

N olbkt rny ls lgclhegt ag n bons s spkere nt ng ngboe ai

lgclhegce 966 ns skawg. Ilgh tke ngboes r, r,e ngh tke tatno

hevlntlag niter twa reirnctlags.

Saoutlag

Nppo ylgb Sgeoos onw >slg966 : 3 slgr r : 366

Iraj syjjetry r : r : 366 .

Nbnl g nppoylgb Sgeoos onw nt secagh surince >slg e : 3 slgr

e : 966

Hevlntlag nt ilrst surince : l r : 96 6 366 : 366

Hevlntlag nt secagh surince : e r : 96 6 366 : 366

Xkereiare tatno hevlntlag : 966 .

Exnjpoe - 00

Ilgh tke ngboen

jnhe `y tke olbkt rny wkeg

lt bets reirncteh iraj wnter ta nlr, ns skawg

lg ilbure.

Saoutlag

Sgeoos Onw

R

slg R

:n

slg n

>5

3

3

8 slg

n

slg n

:5

8

n: slg>

5

8

Exnjpoe - 03 Ilgh tke speeh ai olbkt lg jehluj n li speeh ai olbkt lg jehluj ` ls

3

cwkere c : speeh ai

olbkt lg vncuuj ngh olbkt reirncts iraj jehluj n ta jehluj ` jndlgb 85 ngh 96

respectlveoy wltk tke garjno.


20/55

VK\SLCS


Saoutlag = Sgeoos Onw

n

slg n

: `

slg `

nv

cslg

n :

`v

cslg

`.

nv

cslg 85 :

3/c

cslg 96.

vn

:33

c0

9.3 Vrlgclpoe ai Peversl`lolty ai Olbkt Pnys(n) N rny trnveoolgb noagb tke pntk ai tke reioecteh rny ls reioecteh noagb tke pntk ai tke

lgclhegt rny.

( ) N reirncteh rny reverseh ta trnveo `ncd noagb lts pntk wloo bet reirncteh noagb tkepntk ai tke lgclhegt rny. Xkus tke lgclhegt ngh reirncteh rnys nre j utunooy reverslòe.

2. PEIPNCXLAG XKPAUBK N VNPNOOEO SON@Rkeg olbkt pnsses tkraubk n pnrnooeo son`, knvlgb snje jehluj ag àtk slhes, tkeg

(n) Ejerbegt rny ls pnrnooeo ta tke lgclhegt rny.

Gate = Ejerbegt rny wloo gat è pnrnooeo ta tke lgclhegt rny li tke jehluj ag àtk tke slhes ai

son` nre hliieregt.

(`) Olbkt ls skliteh onternooy, blveg `y (stuhegtsskauoh è nòe ta herlve lt)

h :rcas

)rlslg(t t : tklcdgess ai son`

Exnjpoe - 08

Ilgh tke onterno sklit ai olbkt rny wkloe lt pnsses tkraubk n pnrnooeo bonss son` ai tklcdgess >6cj ponceh lg nlr. Xke ngboe ai lgclhegce lg nlr ls 96 ngh tke ngboe ai reirnctlag lg bonss ls 85.


21/55

VK\SLCS

"jnglskdujnrpkyslcs.lg" 0>

Saoutlag h :rcas

)rl(slgt

:

85cas

)8596(slg>6

:

85cas

>5slg>6

: >6 0 slg >5cj

2.> Nppnregt Heptk ngh sklit ai Su`jerbeh A`mectNt genr garjno lgclhegce (sj noo ngbo e ai lgclhegc e l) nppnregt heptk (h) ls blveg `y=

h :reontlveg

hngh

reontlveg

vv

wkere greontlve

:)reirnctlagaijehlujai..P(g

)lgclhegceaijehlujai..P(g

r

l

h : hlstngce ai a`mect iraj tke lgterince : reno heptk

h : hlstngce ai ljnbe iraj tke lgterince : nppnregt heptk

v : veoaclty ai a`mect perpeghlcuonr ta lgterince reontlve ta surince.

v : veoaclty ai ljnbe perpeghlcuonr ta lgterince reontlve ta surince.

Xkls iarjuon cng `e ensloy herlveh uslgb Sgeoos onw ngh nppoylgb tke caghltlag ai genroy

garjno lgclhegce.... (try lt ar see lg text `aad).

Nppnregt sklit : h

reog

>>

Exnjpoe - 05

Ng a`mect oles >66 cj lgslhe wnter .Lt ls vleweh iraj nlr genr oy garj nooy. Ilgh tke nppnregt

heptk ai tke a`mect.

Saoutlag = h:reontlveg

h:

>

3/8

>66:25 cj

Exnjpoe - 09

N cagcnve jlrrar ls ponc eh lgslhe wnter wltk lts sklglgb surince upwnrhs ngh prlgclpno nxls

vertlcno ns skawg. Pnys nre lgclhegt pnrnooeo ta tke prlgclpno nxls ai cagcnve jlrrar. Ilgh tke

pasltlag ai ilgno ljnbe.


22/55

VK\SLCS


Saoutlag

Xke lgclhegt rnys wloo pnss ughevlnteh tkraubk tke wnter

surince ngh strlde tke jlrrar pnrnooeo ta lts prlgclpno nxls.

Xkereiare iar tke jlrrar, a`mect ls nt . Lts ljnbe N (lgilbure) wloo `e iarjeh nt iacus wklck ls 06 cj iraj tke jlrrar.

Gaw iar tke lgterince `etweeg wnter ngh nlr, h : >6 cj.

h :

n

w

g

g

h

:

>

3/8

>6

: 2.5 cj.

Exnjpoe - 02

See tke ilbure

(l) Ilgh nppnregt kelbkt ai tke `lrh.

(ll) Ilgh nppnregt heptk ai ilsk.

(lll) Nt wknt hlstngce wloo tke `lrh nppenr ta tke ilsk1

(lv) Nt wknt hlstngce wloo tke ilsk nppenr ta tke `lrh1

(v) Li tke veoaclty ai `lrh ls >0 cj/sec hawgwnrh ngh tke ilsk ls >0 cj/sec lg upwnrh

hlrectlag, tkeg ilgh aut tkelr reontlve veoacltles wltk respect ta enck atker.

Saoutlag

(l) h@ : 8/3

39

3

8

>

39

: 8; cj

(ll) hI

:3/8

39: 02 cj

(lll) Iar ilsk = h@

: 39 + 8; : ;8 cj

h@

: 39 + 8; : ;8 cj

(lv) Iar `lrh = hI

: 02 + 39 : 93 cj.

hI

: 02 + 39 : 93 cj.

(v) Yeoaclty ai ilsk wltk respect ta `lrh :

>>38

>0>0

//

: 0> cj/sec.


23/55

VK\SLCS


Yeoaclty ai `lrh wltk respect ta ilsk :

>>83

>0>0

//

: 0; cj/sec.

Exnjpoe - 0;

See tke ilbure.

Ilgh tke hlstngce ai ilgnoljnbe iarjeh `y jlrrar

Saoutlag

Sklit :

0/3

>>3

Iar jlrrar a`mect ls nt n hlstngce : 0>

0/3

>>3 : 06 cj

A`mect ls nt tke cegtre ai curvnture ai jlrrar. Kegce tke olbkt rnys wloo retrnce ngh ljnbe wloo`e iarjeh ag tke a`mect ltseoi.

2. 0 Peirnctlag tkraubk n cajpaslte son` ar reirnctlag tkraubk n guj`erai pnrnooeo jehln, ns seeg iraj n jehluj ai P. L. g6Nppnregt heptk (hlstngce ai ilgno ljnbe iraj ilgno surince)

:reo>

>

g

t+

reo0

0

g

t+

reo3

3

g

t+......... +

reog

g

g

t

Nppnregt sklit

: t>

reo>g

>> + t

0

reo0g

>> +........+

reogg

g> t

g

Rkere ' t ' represegts tklcdgess ngh ' g ' represegts tke P.L. ai tke respectlve jehln, reontlve

ta tke jehluj ai a`server. (l.e. g>reo

: g>/g

6 , g

0 reo: g

0/g

6etc.)

Exnjpoe - 0

0

06+

;.>

5.>

>5: >; + >; : 39 cj.


24/55

VK\SLCS


; . CPLXLCNO NGBOE NGH XAXNO LGXEPGNO PEIOECXLAG X. L. P.Crltlcno ngboe ls tke ngboe jnhe lg hegser jehluj iar wklck tke ngboe ai reirnctlag lg rnrer jehluj

ls Caghltlags ai X. L. P.(n) olbkt ls lgclhegt ag tke lgterince iraj hegser jehluj.

( ) Ngboe ai lgclhegce skauoh è brenter

tkng tke crltlcno ngboe(l 7 C). Ilbure

skaws n oujlgaus a`mect ponceh lg

hegser jehluj nt n hlstngce k iraj

ng lgterince sepnrntlgb twa jehln ai

reirnctlve lghlcesrngh

h . Su`scrlpt

r &h stngh iar rnrer ngh hegser

jehluj respectlveoy.Lg tke ilbure, rny > strldes tke surince nt ng ngboe oess tkng crltlcno ngboe C ngh bets

reirncteh lg rnrer jehluj . Pny 0 strldes tke surince nt crltlcno ngboe ngh brnzes tke

lgterince. Pny 3 strldes tke surince jndlgb ng ngboe jare tkng crltlcno ngboe ngh

bets lgtergnooy reioecteh. Xke oacus ai palgts wkere rny strldes nt crltlcno ngboe ls n

clrcoe, cnooeh clrcoe ai looujlgngce. Noo o lbkt rnys strldlgb lgslhe tke clrcoe ai

looujlgngce bet reirncteh lg rnrer jehluj. Li ng a`server ls lg rnrer jehluj, ke/ske

wloo see olbkt cajlgb aut agoy iraj wltklg tke clrcoe ai looujlgngce. Li n clrcuonr

apnque ponte cavers tke clrcoe ai looujlgngce , ga olbkt wloo bet reirncteh lg rnrer

jehluj ngh tkeg tke a`mect cng gat è seeg iraj tke rnrer jehluj. Pnhlus ai C.A.L

cng è ensloy iaugh.

Exnjpoe - 36 Ilgh tke jnxljuj ngboe tknt cng è jnhe lg bonss jehluj ( : >.5) li n olbkt rny lsreirncteh iraj bonss ta vncuuj.

Saoutlag = >.5 slg C : > slg 0/3

Exnjpoe - 3>

Ilgh tke ngboe ai reirnctlag lg n jehluj ( : 0) li olbkt ls lgclhegt lg vncuuj, jndlgb ngboeequno ta twlce tke crltlcno ngboe.

Saoutlag = Slgce tke lgclhegt olbkt ls lg rnrer jehluj. Xatno Lgtergno Peioectlag cng gat tnde ponce.

C : slg> >

: 36 l : 0C : 96

Nppo ylgb Sgeoos Onw. > slg 96 : 0 slg r


25/55

VK\SLCS


slg r :8

3 r : slg>

8

3.

Exnjpoe - 30 Rknt skauoh `e tke vnoue ai ngboe sa tknt olbkt egterlgbgarjnooy tkraubk tke surince NC ai n prlsj (g:3/0)

haes gat crass tke secagh reirnctlgb surinceN@1

Saoutlag Olbkt rny wloo pnss tke surince NC wltkaut `eghlgb slgce lt ls lgclhegt garjnooy. Suppase lt

strldes tke surince N@ nt ng ngboe ai lgclhegce l.

l :


26/55

VK\SLCS


(`) V^ ngh VP nre reirnctlgb surinces.

(c) ^VP : N ls cnooeh reirnctlgb ngboe ar tke ngboe ai prlsj (nosa cnooeh Npex ngboe).

(h) : ngboe ai hevlntlag

(e) Iar reirnctlag ai n jagackrajntlc ( slgboe wnve oegbtk) rny ai olbkt tkraubk n prlsj4

: (l + e) (r>

+ r0) ngh r

> + r

0: N

: l + eN.(i) Ynrlntlag ai versus l (skawg lg hlnbrnj).

Iar age (except jlg) tkere nre twa vnoues aingboe ai lgclhegce. Li l ngh e nre lgterckngbeh tkeg

we bet tke snje vnoue ai ècnuse ai reversl`loltyprlgclpoe ai olbkt

Gate = (l) Iar nppolcntlag ai nàve resuot jehluj ag àtk slhes ai prlsj just è snje.

(ll) @nseh ag nàve brnpk we cng nosa herlve iaooawlgb resuot, wklck snys tknt l ngh e cng è

lgterckngbeh iar n pnrtlcuonr hevlntlag lg atker warhs tkere nre twa ngboe ai lgclhegce iar nblveg hevlntlag (except jlgljuj hevlntlag).

l r> r0 e

> 0 3 8 5

8 3 0 > 5

(b) Xkere ls age ngh agoy age ngboe ai lgclhegce iar wklck tke ngboe ai hevlntlag ls jlgljuj.

(k) Rkeg :jlg

, tke ngboe ai jlgljuj hevlntlag, tkeg l : e ngh r>

: r0, tke rny pnsses

syjjetrlcnooy w.r.t. tke reirnctlgb surinces. Re cng skaw `y sljpoe cnocuontlag tknt

jlg : 0l

jlg N

wkere ljlg

: ngboe ai lgclhegce iar jlgljuj hevlntlag, ngh r : N/0.

greo

: 0N

0

N

slg

slg j , wkere g

reo :

bssurraughlg

prlsj

g

g

Nosa jlg

: (g >) N (iar sjnoo vnoues ai N)

(l) Iar n tklg prlsj ( N>6a) ngh iar sjnoo vnoue ai l, noo vnoues ai

: ( greo > ) N wkere g

reo:

bsurraughlg

prlsj

gg


27/55

VK\SLCS


Exnjpoe - 38 Peirnctlgb ngboe ai n prlsj N : 96 ngh lts reirnctlve lghex ls, g : 3/0, wknt ls tke ngboe ai

lgclhegce l ta bet jlgljuj hevlntlag1Nosa ilgh tke jlgljuj hevlntlag. Nssuje tke surraughlgb

jehluj ta è nlr (g : >).

Saoutlag Iar jlgljuj hevlntlag,

r>

: r0

:0

N: 36.

nppoylgb Sgeoos onw nt L surince

> slg l :0

3slg 36 l :

8

3slg >

jlg: 0slg>

8

3 96

Exnjpoe - 35 See tke ilbure

Ilgh tke hevlntlag cnuseh `y n prlsj knvlgb

reirnctlgb ngboe 8 ngh reirnctlve lghex0

3.

Saoutlag : (

0

3 >) 86 : 06

Exnjpoe - 39 Iar n prlsj, N : 96, g :3

2. Ilgh tke jlgljuj passlòe ngboe ai lgclhegce, sa tknt tke

olbkt rny ls reirncteh iraj tke secagh surince. Nosa ilgh jnx

.

Saoutlag Lg jlgljuj lgclhegce cnse tke ngboes wloo è ns skawg lg ilbure

Nppo ylgb sgeoos onw =

> slg ljlg

:3

2slg ( N C)

:3

2(slg N cas C cas N slg C)

:3

2

2

396cas

2

3>96slg :

0

>

ljlg

: 36 jnx

: ljlg

+


28/55

VK\SLCS

"jnglskdujnrpkyslcs.lg" 0;

>6. HLSVEPSLAG AI OLBKXXke ngbuonr spolttlgb ai n rny ai wklte olbkt lgta n gujèr ai cajpagegts ngh sprenhlgb lg hliieregt

hlrectlags ls cnooeh Hlsperslag ai Olbkt. TLt ls iar wkaoe Eoectra Jnbgetlc Rnve lg tatnoltyZ. Xkls

pkegajegag ls ècnuse wnves ai hliieregt wnveoegbtk jave wltk snje speeh lg vncuuj ùt wltk

hliieregt speehs lg n jehluj.

Xkereiare, tke reirnctlve lghex ai n jehluj hepeghs solbktoy ag wnveoegbtk nosa. Xkls vnrlntlag aireirnctlve lghex wltk wnveoegbtk ls blveg `y Cnuckys iarjuon.

Cnucky's iarjuon g () :0

`n wkere n ngh ` nre pasltlve cagstngts ai n jehluj.

Gate = Suck pkegajegag ls gat exkl`lteh `y saugh wnves.

Ngbo e ètweeg tke rnys ai tke extrej e caoaurs lg tke reirnc teh (hls perseh) olbkt ls cnooeh

ngboe ai hlsperslag. :v

r (Ilb. (n))

Ilb (n) ngh (c) represegts hlsperslag, wkerens lg ilb. (`) tkere ls ga hlsperslag.

Iar prlsj ai sjnoo N ngh wltk sjnoo l =

: v

r: (g

v g

r)N

Hevlntlag ai ènj (nosa cnooeh jeng hevlntlag)

:y: (g

y >)N

gv, g

rngh g

ynre P.. ai jnterlno iar vlaoet, reh ngh yeooaw caoaurs respectlveoy.

Exnjpoe - 3;

Xke reirnctlve lghlces ai iolgt bonss iar reh ngh vlaoet olbkt nre >.9>3 ngh >.930 respectlveoy.

Ilgh tke ngbuonr hlsperslag prahuceh `y n tklg prlsj ai iolgt bonss knvlgb reirnctlgb ngboe 56.

Saoutlag Hevlntlag ai tke reh olbkt lsr: (

r >)N ngh hevlntlag ai tke vlaoet olbkt ls

v: (

v >)N.

Xke hlsperslag :v

r: (

v

r)N : (>.930 >.9>3) 56 : 6.6g

gg

y

rv ls tke praperty ai n jehluj.

Iar sjnoo ngboeh prlsj ( N>6a ) wltk olbkt lgclhegt nt sjnoo ngboe l =

>ggg

y

rv

:

y

rv

:

y :

)yeooaw(rnyjengaihevlntlaghlsperslagngbuonr


29/55

VK\SLCS

"jnglskdujnrpkyslcs.lg" 0 : reirnctlvlty ai tke jehluj iar tke carrespaghlgb caoaur.

Exnjpoe - 3.56 ngh >.96 respectlveoy. Ilgh

(n) tkereirnctlvelghex iar yeooaw caoaur, nppraxljnteoy(`) Hlsperslve pawer ai tke jehluj.

Saoutlag (n) y~

0

Pv :

0

96.>56.> : >.55

(`) : >

y

Pv

:>55.>

56.>96.>

: 6.>;.

>6.> Hlsperslag wltkaut hevlntlag Hlrect Ylslag Caj`lgntlagXke caghltlag iar hlrect vlslag caj`lgntlag ls =

>gy N : >gy NN

>

0

gg rvN

>

0

gg rvN

Xwa ar jare prlsjs cng `e caj`lgeh lg vnrlaus wnys ta bet hliieregt caj`lgntlag ai ngbuonr

hlsperslag ngh hevlntlag.

ar ar

>6.0 Hevlntlag wltkaut hlsperslag Nckrajntlc Caj`lgntlagCaghltlag iar nckrajntlc caj`lgntlag ls= (g

v g

r) N : (g

v g

r) N

ar ar

Exnjpoe - 86 Li twa prlsjs nre caj`lgeh, ns skawg lg ilbure,

ilgh tke tatno ngbuonr hlsperslag ngh ngboe ai

hevlntlag suiiereh `y n wklte rny ai olbkt lgclhegtag tke caj`lgntlag.

Saoutlag @atk prlsjs wloo turg tke olbkt rnys tawnrhs tkelr `nses ngh kegce lg snje hlrectlag.

Xkereiare turglgbs cnuseh `y `atk prlsjs nre nhhltlve.

Xatno ngbuonr hlsperslag

: + : (Y

P

) N + (Y

P

) N

: (>.5 >.8) 8 + (>.2 >.5)0 : 6.;

Xatno hevlntlag

: +

:

>

0

PYN +

>

0

'' PYNN :

>

0

8.>5.>6.8 +

>

0

5.>2.>6.0

: (>.85 >) 6.8 + (>.9 >) 6.0

: 6.85 6.8 + 6.9 6.0

: >.;6 + >.0 : 3.6 Ngs.


30/55

VK\SLCS


Exnjpoe - 8> Xwa tklg prlsjs nre caj`lgeh ta iarj ng nckrajntlc caj`lgntlag. Iar L prlsj N : 8,P

:

>.35,\

: >.86, Y

: >.80. iar LL prlsj P

: >.2,\

: >.; ngh Y

: >.< ilgh tke prlsj ngboe

ai LL prlsj ngh tke get jeng hevlntlag.

Saoutlag Caghltlag iar nckrajntlc caj`lgntlag.

: (

Y

P)N : (

Y

P)N

N :

8.>

2.>

8)35.>80.>(

Get

: : (\ >)N (

\ >)N

: (>.86 >) 8 (>.; >) >.8 : 6.8;.

Exnjpoe - 80 N crawg bonss prlsj ai ngboe 5 6 ls ta `e caj`lgeh wltk n iolgt prlsj lg suck n wny tknt tke

jeng rny pnsses wltkaut hevlntlag. Ilgh (n) tke npex ngboe ai tke iolgt bonss prlsj geeheh

ngh (`) tke ngbuonr hlsperslag prahuceh `y tke blveg caj`lgntlag wkeg wklte olbkt baes

tkraubk lt. Peirnctlve lghlces iar reh, yeooaw ngh vlaoet olbkt nre >.5, >.9 ngh >.2 respectlveoy

iar crawg bonss ngh >.;,0.6 ngh 0.0 iar iolgt bonss.

Saoutlag Xke hevlntlag prahuceh `y tke crawg prlsj ls : ( >)N

ngh `y tke iolgt prlsj ls =

' : (' >)N'.Xke prlsjs nre ponceh wltk tkelr ngboes lgverteh wltk respect ta enck atker. Xke hevlntlags

nre nosa lg appaslte hlrectlags. Xkus, tke get hevlntlag ls =

H : ' : ( >)N (' >)N'. .................(>)(n) Li tke get hevlntlag iar tke jeng rny ls zera,

( >)N : (' >)N'.

ar, N' :

)>'(

)>(

N :

65

>60

>9>

.

.:36

( ) Xke ngbuonr hlsperslag prahuceh `y tke crawg prlsj ls =

v

r: (

v

r)N

ngh tknt `y tke iolgt prlsj ls,

'v '

r: ('

v'

r)N

Xke get ngbuonr hlsperslag ls,

(v

r)N ('

v '

r)N

: (>.2 >.5) 5 6 (0.0 >.;) 36

: 6.06 .

Xke ngbuonr hlsperslag kns jnbgltuhe 6.06 .

>>. SVECXPUJ(Agoy iar yaur dgawoehbe ngh gat ai juck use iar MEE)

Arhereh pntterg prahuceh `y n `enj ejerblgb iraj n prlsj niter reirnctlag ls cnooehSpectruj.

Xypes ai spectruj=

>>.> Xypes ai spectruj=(n) Olge spectruj=

Hue ta saurce lg ntajlc stnte.

(`) @ngh spectruj=

Hue ta saurce lg jaoecuonr stnte.

(c) Cagtlguaus spectruj=

Hue ta wklte kat saolh.


31/55

VK\SLCS


>>.0 Lg Ejlsslag spectruj@rlbkt caoaurs ar olges, ejltteh iraj saurce nre a`serveh.

Xke spectruj ejltteh `y n blveg saurce ai olbkt ls cnooeh ejlsslag spectruj. Lt ls n

wnveoegbtk-wlse hlstrlùtlag ai olbkt ejltteh `y tke saurce. Xke ejlsslag spectrn nre blveg

`y lgcnghescegt saolhs, olqulhs ngh bnses wklck nre eltker ùrgt hlrectoy ns n ionje (ar n

spnrd) ar ùrgt ugher oaw pressure lg n hlscknrbe tuè.

>>.3 Lg N`sarptlag spectrujHnrd olges lghlcntes irequegcles n`sarèh.

Rkeg n ènj ai olbkt iraj n kat saurce ls pnsseh tkraubk n su`stngce (nt n oawer

tejpernture), n pnrt ai tke olbkt ls trngsjltteh ùt rest ai lt ls n`sarèh. Rltk tke keop ai n

spectrajeter, we cng dgaw tke irnctlag ai olbkt n`sarèh carrespaghlgb ta enck wnveoegbtk.

Xke hlstrlùtlag ai tke wnveoegbtk n`sarptlag ai olbkt `y n su`stngce ls cnooeh ng

n`sarptlag spectruj. Every su`stngce kns lts awg cknrncterlstlc n`sarptlag spectruj.

>>.8 Spect rajete rCagslsts ai n caooljntar (ta caooljnte olbkt ènj), prlsj ngh teoescape. Lt ls useh ta a`serve

tke spectruj ngh nosa jensure hevlntlag.

>0. PEIPNCXLAG NX SVKEPLCNO SUPINCESIar pnrnxlno rnys lgclhegt ag n spkerlcno surince sepnrntlgb twa jehln=

v

g0 u

g> :P

gg >0 .................... (N)wkere olbkt javes iraj tke jehluj ai reirnctlve lghex g

> ta tke jehluj ai reirnctlve lghex g

0.

Xrngsverse jnbglilcntlag (j) (ai hljegslag perpeghlcuonr ta prlgclpno nxls) hue ta reirnctlag nt

spkerlcno surince ls blveg `y j :Pu

Pv : >0g/ug/v

Exnjpoe - 83

Ilgh tke pasltlag, slze ngh gnture ai ljnbe, iar tke

sltuntlag skawg lg ilbure. Hrnw rny hlnbrnj .

Saoutlag Iar reirnctlag genr palgt N, u : 36 4 P : 064 g>

: 0 4 g0

: >.

Nppoylgb reirnctlag iarjuon

v

g0 u

g> :P

gg >0

v

>

36

0

:

06

0>

v : 96 cj

j :>

0

k

k:

ug

vg

0

>:

)36(>

)96(0

: 8 k0

: 8 jj.


32/55

VK\SLCS


Speclno cnse=Peirnctlag nt ponge SurincesVuttlgb P : lg tke iarjuon

v

g0 u

g> :P

gg >0 , we bet4

v :>

0

g

ug

Xke snje slbg ai v ngh u ljpoles tknt tke a`mect ngh tke ljnbe nre nownys ag tke snje

slhe ai tke lgterince sepnrntlgb tke twa jehln. Li we wrlte tke nàve iarjuon ns

v : u

greo,

lt blves tke reontlag ètweeg tke nppnregt heptk ngh reno heptk, ns we knve seeg èiare.

Exnjpoe - 88

Uslgb iarjuon ai spkerlcno surince ar atkerwlse, ilgh tke nppnregt heptk ai ng a`mect ponceh

>6 cj èoaw tke wnter surince, li seeg genr garjnooy iraj nlr.Saoutlag Vut P : lg tke iarjuon ai tke Peirnctlag nt Spkerlcno Surinces we bet,

v :>

0

g

ug

u : >6 cj

+ hlrectlagT ai lgclhegt olbktZhlrectlag

nlr

wnter

g>

:3

8

g0

: >

v : 3/8

>>6 : 2.5 cj

gebntlve slbg ljpoles tknt tke ljnbe ls iarjeh lg wnter.

Noter=

hnpp

:reo

renoh

:3/8

>6:

8

36: 2.5 cj.

>3. XKLG OEGSN tklg oegs ls cnoo eh cagvex li lt ls tklcder nt tke jlhhoe ngh lt ls cnoo eh cagc nve li lt ls tklc der nt tke

eghs.

Age surince ai n cagvex oegs ls nownys cagvex . Hepeghlgb ag tke atker surince n cagvex oegs ls

cntebarlzeh ns

(n) `lcagvex ar cagvexa cagvex , li tke atker surince ls nosa cagvex,

( ) Vonga cagvex li tke atker surince ls ponge ngh

(c) Cagcnva cagvex li tke atker surince ls cagcnve.

Sljlonroy cagcnve oegs ls cntebarlzeh ns

cagcnva-cagcnve ar `lcagcnve, ponga-cagcnve

ngh cagvexa-cagcnve.


33/55

VK\SLCS


Iar n spkerlcno, tklg oegs knvlgb tke snje

jehluj ag àtk slhes=

v

>

u

>: (g

reo >)

0> P

>

P

>.........(n),

wkere greo

:

jehluj

oegs

g

gngh P

>ngh P

0 nre x caarhlgntes ai tke cegtre ai curvnture ai tke >st surince

ngh 0gh surince respectlveoy.

i

>: (g

reo >)

0> P

>

P

>Oegs Jnder's iarjuon ..............(`)

Iraj (n) ngh (`)

>

v

>

u :

>

i

Oegs kns twa Iacll=

Li u : , tkegi

>>

v

>

v : i

Li lgclhegt rnys nre pnrnooeo ta prlgclpno nxls tkeg lts reirncteh rny wloo cut tke prlgclpno nxls nt i.Lt ls cnooeh 0gh iacus.

Lg cnse ai cagverblgb oegs lt ls pasltlve ngh lg cnse ai hlverblgb oegs lt ls gebntlve.

Li v : tknt jengsi

>

u

>>

u : i

Li lgclhegt rnys cuts prlgclpno nxls nt i tkeg lts reirncteh rny wloo ècaje pnrnooeo ta tkeprlgclpno nxls. Lt ls cnooeh >st iacus. Lg cnse ai cagverblgb oegs lt ls gebntlve ( i ls pasltlve)

ngh lg tke cnse ai hlverblgb oegs lt pasltlve ( i ls gebntlve)

use ai i & + i ls lg hrnwlgb tke rny hlnbrnjs.

Gatlce tknt tke palgt @, lts ljnbe @ ngh tke paoe V ai tke oegs nre caoolgenr. Lt ls hue tapnrnooeo son` gnture ai tke oegs nt tke jlhhoe. Xkls rny baes strnlbkt. (Pejejèr tkls)

Iraj tke reontlag

>

i :(g reo >)

0> P>

P

>

lt cng è seeg tknt tke secagh iacno oegbtk hepeghs ag

twa inctars.


34/55

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(N) Xke inctar

0> P

>

P

>ls

(l) Vasltlve iar noo types ai cagvex oegses ngh

(ll) Gebntlve iar noo types ai cagcnve oegses.

(@) Xke inctar (greo

>) ls(l) Vasltlve wkeg surraughlgb jehluj ls rnrer tkng tke jehluj ai oegs.

(ll) Gebntlve wkeg surraughlgb jehluj ls hegser tkng tke jehluj ai oegs.

(C) Sa n oegs lscagverblgb l ii ls pasltlve wklck knppegs wkeg àtk tke inctars (N) ngh (@)

nre ai snje slbg.

(H) Ngh n oegs ls hlverblgb li i ls gebntlve wklck knppegs wkeg tke inctars (N) ngh (@) nre

ai appaslte slbgs.

(E) Iacno oegbtk ai tke oegs hepeghs ag jehluj ai oegs ns weoo ns surraughlgb.

(I) Lt nosa hepeghs ag wnveoegbtk ai lgclhegt olbkt. Lgcnpn`lolty ai oegs ta iacus olbkt rnys ai

vnrlaus wnveoegbtks nt slgboe palgt ls dgawg ns ckrajntlc nèrrntlag.

Exnjpoe - 85

Ilgh tkeèknvlaurai n cagcnve oegs ponceh lg n rnrer j ehluj.

Saoutlag

Inctar (N) ls gebntlve , ècnuse tke oegs ls cagcnve.

Inctar (@) ls pasltlve , ècnuse tke oegs ls ponceh lg n rnrer jehluj.

Xkereiare tke iacno oegbtk ai tke oegs, wklck hepeghs ag tke prahuct ai tkese inctars , ls

gebntlve ngh kegce tke oegs wloo èknve ns hlverblgb oegs.

Exnjpoe - 89

Skaw tknt tke inctar

0> P>

P> (ngh tkereiare iacno oegbtk) haes gat hepegh ag wklck

surince ai tke oegs olbkt strlde ilrst.

Saoutlag

Cagslher n cagvex oegs ai rnhll ai curvnture p ngh q ns skawg.

CNSE >= Suppase olbkt ls lgclhegt iraj oeit slhe ngh strldes tke surince wltk rnhlus ai curvnture p, ilrst.

Xkeg P>

: +p 4 P0

: -q ngh

0> P

>

P

>:

q

>

p

>

CNSE 0= Suppase olbkt ls lgclhegt iraj rlbkt slhe ngh strldes tke surince wltk rnhlus ai curvnture q, ilrst.

Xkeg P>

: +q 4 P0

: -p ngh

0> P

>

P

>:

p

>

q

>

Xkaubk we knve skawg tke resuot iar `lcagvex oegs , lt ls true iar every oegs.


35/55

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Exnjpoe - 82

Ilgh tke iacno oegbtk ai tke oegs

skawg lg tke ilbure.

Saoutlag = i

>: (g

reo >)

0> P

>

P

>

i

>: (3/0 >)

)>6(

>

>6

>

i

>:

0

>

>6

0 i : + >6 cj.

Exnjpoe - 8;

Ilgh tke iacno oegbtk ai tke

oegs skawg lg ilbure

Saoutlag =i

>: (g

reo >)

0> P

>

P

>:

>

0

3

>6>

>6

>

i : >6 cj

Exnjpoe - 8: (g

reo >)

0> P

>

P

>:

>

0

3

06

>

96

>

i : 96 cj

(`)i> : (g

reo >)

0> P>

P> :

>

03

96>

06>

i : 96 cj

Exnjpoe - 56

Valgt a`mect ls ponceh ag tke prlgclpno nxls ai n tklg oegs wltk pnrnooeo curveh àughnrles

l.e., knvlgb snje rnhll ai curvnture. Hlscuss nàut tke pasltlag ai tke ljnbe iarjeh .

Saoutlag>

i : (g

reo >)

0> P

>

P

>: 6 T P

>: P

0Z

>v

>u

: 6 ar v : u l.e. rnys pnss wltkaut nppreclnòe èghlgb.

Exnjpoe - 5>

Iacno oegbtk ai n tklg oegs lg nlr, ls >6 cj. Gaw jehluj ag

age slhe ai tke oegs ls reponceh `y n jehluj ai reirnctlve

lghex :0. Xke rnhlus ai curvnture ai surince ai oegs, lgcagtnct wltk tke jehluj, ls 06 cj. Ilgh tke gew iacno oegbtk.

Saoutlag = Oet rnhlus ai surince è P>

ngh reirnctlve lghex ai oegs è . Oet pnrnooeo rnys è lgclhegtag tke oegs. Nppoylgb reirnctlag iarjuon nt ilrst surince

>

v>

:>P

>...(>)


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VK\SLCS


Nt surincev

0

>v

:

06

0

...(0)

Nhhl gb (>) ngh (0)

>

v>

+v

0

>v

:

>P

>+

06

0

: ( >)

06

>

P

>

>

06

>

06

0 :

i

>(lg nlr) +

06

>

06

0

v : 86 cj i : 86 cj

Exnjpoe - 50

Ilbureskawsn palgt a`mect ngh n cagverblgb oegs.

Ilgh tke ilgno ljnbe iarjeh.

Saoutlag =v

>

u

>:

i

>

v

>

>5

>

:

>6

>

v

>:

>6

>

>5

>:

36

>

v : + 36 cj

Exnjpoe - 53

See tke ilbure

Ilgh tke pasltlag ai ilgno ljnbe iarjeh.

Saoutlag Iar cagverblgb oegs

u : >5 cj, i : >6 cj v :ui

iu

: 36 cj

Iar hlverblgb oegs

u : 5 c j

i : >6 cj v :ui

iu

: >6 cj

Exnjpoe - 58

Ilbure skaws twa cagverblgb oegses. Lgclhegt rnysnre pnrnooeo ta prlgclpno nxls. Rknt skauoh è tke

vnoue ai h sa tknt ilgno rnys nre nosa pnrnooeo.

Saoutlag Ilgno rnys skauoh è pnrnooeo. Iar tkls tke iacus ai O>

just calgclhe wltk iacus ai O0.

h : >6 + 06

: 36 cj

Kere tke hlnjeter ai rny ènj ècajes wlher.


37/55

VK\SLCS


Exnjpoe - 55

See tke ilbure


Saoutlag = Iar oegs,

v

>

u

>:

i

>

v

>

>5

>

:

>6

> v : + 36 cj

Kegce lt ls a`mect iar jlrrar

u : >5 cj

v

>+

>5

>

:

>6

>

v : 36 cj

Gaw iar secagh tlje lt nbnlg pnsses tkraubk oegs

u : >5 cj

v : 1 4 i : >6 cj

v

>

>5

>

:

>6

> v : + 36

Kegce ilgno ljnbe wloo iarj nt n hlstngce 36 cj iraj tke oegs tawnrhs oeit.

Exnjpoe - 59

Rknt skauoh è tke vnoue ai h sa tknt ljnbe ls iarjeh

ag tke a`mect ltseoi.

Saoutlag Iar oegs =

v

>

>5

>

:

>6

>v : + 36 cj

Cnse L =

Li h : 36, tke a`mect iar jlrrar wloo

è nt paoe ngh lts ljnbe wloo è

iarjeh tkere ltseoi.

Cnse LL =

Li tke rnys strlde tke jlrrar garjnooy, tkey wloo retrncengh tke ljnbe wloo è iarjeh ag tke a`mect ltseoi

h : 36 06 : >6 cj

>3.0 Xrngsver se jnbgl ilcntlag jXrngsverse jnbglilcntlag (j) ai (ai hljegslag perpeghlcuonr ta prlgclpno nxls) ls blveg `y

j :u

v

Li tke oegs ls tklcd ar/ngh tke jehluj ag àtk slhes ls hliieregt, tkeg we knve ta nppoy tkeiarjuon blveg iar reirnctlag nt spkerlcno surinces step `y step.


38/55

VK\SLCS


Exnjpoe - 52

Ng extegheh reno a`mect ai slze 0 cj ls ponceh perpeghlcuonr ta tke prlgclpno nxls ai n

cagverblgb oegs ai iacno oegbtk 06 cj. Xke hlstngce `etweeg tke a`mect ngh tke oegs ls 36

cj .

(l) Ilgh tke onterno jnbglilcntlag prahuceh `y tke oegs.

(ll) Ilgh tke kelbkt ai tke ljnbe.

(lll) Ilgh tke ckngbe lg onterno jnbglilcntlag, li tke a`mect ls `raubkt coaser ta tke oegs `y >

jj noagb tke prlgclpno nxls.

Saoutlag Uslgbv

>

u

>:

i

>

ngh j :u

v

we bet j :ui

i

.........(N)

j : )36(06

06

:>6

06

: 0

ve slbg ljpoles tknt tke ljnbe ls lgverteh.

(ll)>

0

k

k: j

k0

: jk>

: (0) (0) : 8 cj

(lll) Hliieregtlntlgb (N) we bet

hj : 0)ui(

i

hu

: 0)>6(

)06(

(6.>) :

>66

0

: .60

Gate tknt tke jetkah ai hliieregtlno ls vnolh agoy wkeg ckngbes nre sjnoo.

Notergnte jetkah =

u (niter hlsponclgb tke a`mect)

: (36 + 6.>) : 0


39/55

VK\SLCS

"jnglskdujnrpkyslcs.lg" 33.3 Hlsponcejegt Jetkah ta ilgh Iacno oegbtk ai Cagverblgb Oegs =Ilx ng a`mect ai sjnoo kelbkt K ngh n screeg nt n hlstngce H

iraj a`mect (ns skawg lg ilbure). Jave n cagverblgb oegs iraj

tke a`mect tawnrhs tke screeg. Oet n sknrp ljnbe iarjs ag

tke screeg wkeg tke hlstngce ètweeg

tke a`mect ngh tke oegs ls n. Iraj oegs iarjuon we knve

nH

>

n>

: i> ar n0 H n + i H : 6 ...( N)

Xkls ls qunhrntlc equntlag ngh kegce twa vnoues ai n nre passlòe. Cnoo tkej n>

ngh n0.

Xkus n, ngh n0

nre tke raats ai tke equntlag. Iraj tke prapertles ai raats ai n qunhrntlc

equntlag,

n>

+ n0

: H n>n

0: i H

Nos a (n>

n0) :

0>0

0> nn8)nn( : iH8H0 : h (suppase).

h pkyslcnooy jengs tke sepnrntlag ètweeg tke twa pasltlag ai oegs.

Xke iacno oegbtk ai oegs lg terjs ai H ngh h.

n

H

O

K

(H-n)screeg

sa, n> n

0:

0>

0

0> nn8)nn(

ih8H

0

: h

caghltlag, h : 6, l.e. tke twa pasltlag calgclhe

:H8

H0

Paats ai tke equntlag n0 Hn + H : 6, ècaje ljnblgnry li

`0 8nc ? 6.

: H0 8H ? 6: H (H 8) ? 6.

:

iar reno vnoue ai n lg equntlag n0 Hn + i H : 6

`0 8nc 6. : H0 8i H 6.

sa, H 8i Hjlg: 8i

Onterno jnbglilcntlag lg hlsponcejegt jetkah=

li j >ngh j0è twa jnbglilcntlags lg twa pasltlags (Lg tke hlsponcejegt jetkah)

j>

:>

>

u

v:

>

>

n

)nH(

j0 :0

0

u

v:

0

0

n

nH

:)nH(

n

>

>


40/55


41/55

VK\SLCS


3i

>:

>

5

;

06

>

06

>:

56

3

i

>:

>i

>+

0i

>+

3i

>:

>6

>+

>66

3+

56

3i :

>3

>66Ngs.

>5. CAJ@LGNXLAG AI OEGS NGH JLPPAPXke caj`lgntlag ai oegs ngh jlrrar èknves olde n

jlrrar ai iacno oegbtk i blveg `y

i

>:

jI

>

I

0

Li oegses nre jare tkng age, i ls blveg `y

i

>

: jI

>

i>

0

Iar tke iaooawlgb ilbure

i ls blveg `yi

>:

jI

>

0> i

>

i

>0

Exnjpoe - 96


(Xke bnp skawg lg ilbure ls ai gebolbl oe wlhtk )

Saoutlageqi

>:

>6

>

>6

0:

>6

>

ieq

: >6 cj

v

>+

06

>

:

>6

>

v : 06 cjKegce ljnbe wloo è iarjeh ag tke a`mect ltseoi

Saje lgterestlgb incts nàut olbkt =(>) XKE SUG PLSES @EIAPE LX NCXUNOO\ PLSES NGH SEXS NIXEP LX NCXUNOO\ SEXS =

Xke ntjaspkere ls oess ngh oess hegse ns lts kelbkt lgcrense,

ngh lt ls nosa dgawg tknt tke lghex ai reirnctlag hecrense

wltk n hecrense lg hegslty. Sa, tkere ls n hecrense ai tke

lghex ai reirnctlag wltk kelbkt. Hue ta tkls tke olbkt rnys ègh

ns tkey jave lg tke enrtks ntjaspkere

(0) XKE SUG LS AYNO SKNVEH NX XKE XLJE AI LXS PLSE NGH SEX =

Xke rnys hlverblgb iraj tke oawer ehbe ai tke sug knve ta caver n brenter tklcdgess ai nlr

tkng tke rnys iraj tke upper ehbe. Kegce tke iarjer nre reirncteh jare tkng tke ontter, ngh


42/55

VK\SLCS


sa tke vertlcno hlnjeter ai tke sug nppenrs ta è n olttoe skarter tkng tke karlzagtno hlnjeter

wklck rejnlgs ugckngbeh.

(3) XKE SXNPS XRLGDOE @UX GAX XKE VONGEXS =

Xke reirnctlve lghex ai ntjaspkere iouctuntes `y n sjnoo njaugt hue ta vnrlaus rensags.

Xkls cnuses solbkt vnrlntlag lg èghlgb ai olbkt hue ta wklck tke nppnregt pasltlag ai stnr nosa

ckngbes, prahuclgb tke eiiect ai twlgdolgb.

(8) BONSS LS XPNGSVNPEGX, @UX LXS VARHEP LS RKLXE =

Rkeg pawhereh, olbkt ls reioecteh iraj tke surince ai lggujernòe sjnoo pleces ai bonss ngh

sa tke pawher nppenrs wklte. Bonss trngsjlts jast ai tke lgclhegt olbkt ngh reioects very

olttoe kegce lt nppenrs trngspnregt.

(5) BPENSEH AP ALOEH VNVEP LS XPNGSVNPEGX, @UX VNVEP LS RKLXE =

Xke raubk surince ai pnper hliiuseoy reioects lgclhegt olbkt ngh sa lt nppenrs wklte. Rkeg

aloeh ar brenseh, very olttoe reioectlag tndes ponce ngh jast ai tke olbkt ls nooaweh ta pnss

ngh kegce lt nppenrs trngspnregt.

(9) NG E[XEGHEH RNXEP XNGD NVVENPS SKNOOAR NX XKE INP EGH =

Xkls ls hue ta Xatno lgtergno reioectlag

(2) N XESX XU@E AP N SJADEH @NOO LJJEPSEH LG RNXEP VENPS SLOYEP\ RKLXE RKEG

YLEREH IPAJ XKE XAV =

Xkls ls hue ta Xatno lgtergno reioectlag

(;) SKLVS K NG B LGYEP XEH LG XKE N LP LG C A OH C A UG XP LES NG H XPEES K NG B

LGYEPXEH UGHEPBPAUGH LG HESEPXS=Xkls ls hue ta Xatno lgtergno reioectlag


43/55

VK\SLCS


Vraòej >.

See tke iaooawlgb ilbure. R klck ai tke a`mect(s) skawg lg

ilbure wloo gat iarj lts ljnbe lg tke jlrrar.

Saoutlag =

Garny iraj A3

ls lgclhegt ag reioectlgb surince ai tke jlrrar, sa lts ljnbe ls gat iarjeh.

Vraòej 0.

Ilbure skaws ng a`mect N@ ngh n ponge jlrrar JG ponceh pnrnooeo ta

a`mect. Lghlcnte tke jlrrar oegbtk requlreh ta see tke ljnbe ai a`mect

li a`servers eye ls nt E.

Saoutlag =

Pequlreh oegbtk ai jlrrar : JG.

JGE &N'@'E nre sljlonr

AE

JG:

E'C

'@'N JG :

0

'@'N:

0

N@.

Vraòej 3.

Ng a`mect ls dept ilxeh lg iragt ai n ponge jlrrar wklck ls javeh `y >6 j/s nwny iraj tke a`mect, ilgh

tke veoaclty ai tke ljnbe.

Saoutlag =AJLJ YY

B,JB,AB,JB, YYYY

0

YYY

B,AB,B,J

0

Y B,

( B,AY

: 6 )

0

Y B,

: >6 lj/s

B,Y

: 06 lj/s


44/55

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Vraòej 8.

Ilbure skaws twa lgcolgeh ponge jlrrars J>

ngh J0

ngh ng a`mect A. Lts ljnbes

iarjeh lg jlrrars J>

ngh J0lghlvlhunooy nre

>ngh

0respectlveoy. Skaw tknt

>ngh

0

ngh A ole ag tke clrcujieregce ai n clrcoe wltk cegtre nt. TXkls resuot cng èextegheh ta skaw tknt noo tke ljnbes wloo nosa ole ag tke snje clrcoe. Gate tknt

tkls resuot ls lghepeghegt ai tke ngboe ai lgcolgntlag ai jlrrars.Z. ls tke palgt ailgtersectlag ai tke jlrrars.

/////////////////////////////

////

////

////

////

////

/////////

A

L>

L0

L

J>

J0

Saoutlag =

Coenroy,

A^ ngh 0^ nre cagbruegt///////////////////////////

////

////

////

////

////

///////////

A

L>

L 0

L

J

V

^

>

J0ngh AV ngh >V nre cagbruegtSa, > : A ngh A : 0Kegce, > : A : 0Sa,>ngh0ngh A ole ag tke clrcujieregce ai n clrcoe wltk cegtre.

Vraòej 5.

Ilgh tke pasltlag ai ilgno ljnbe niter tkree successlve

reioectlags tndlgb ilrst reioectlag ag j>

Saoutlag = >st reioectlagnt j>u : >5cj

i : >6 cj

i

>

u

>

v

> :

36

>

36

03

v : 36 cj

0gh reioectlag nt ponge jlrrar =

u : 5 cj

v : 5 cj

IarLLL reioectlag ag curveh jlrrar nbnlg =

u : 06 cj

v :iu

ui

:

>606

)>6()06(

:>6

066

: 06 cj

Ljnbe ls 06 cj rlbkt ai j>

Vraòej 9.

Ilgh tke pasltlag ai ilgno ljnbe niter tkree successlve reioectlags tndlgb ilrst reioectlag ag j>.

Saoutlag = >st reioectlag nt jlrrar j> =

u : >5 cj, i : >6cj

v

>:

i

>

u

>

v :iu

ui

:>6)>5(

)>6()>5(

:5

>56

cj : 36 cj.

Xkus, ljnbe ls iarjeh nt n palgt 5 cj rlbkt ai j0 wklck wloo nct ns ng a mect iar tke reioectlag nt j0


45/55

VK\SLCS


Iar 0gh reioectlag nt j0u : 5 cj, i : >6 cj

v :iu

ui

:

>65

>65

:5

56

: >6 cj.

3rh reioectlag nt j>nbnlg.

u : >5 cj i : >6 cj

v :iu

ui : >6)>5(

)>6(>5

: 36 cj. Ngs.

Ljnbe ls iarjeh nt 36 cj rlbkt ai j>

Vraòej 2.

N calg ls ponceh >6 cj lg iragt ai n cagc nve jlrrar . Xke jlrrar prahuces n reno ljnbe tknt kns

hlnjeter 8 tljes tknt ai tke calg. Rknt ls tke ljnbe hlstngce.

Saoutlag = j :>

0

h

h:

u

v

8 : uv

v : 8 u

: 8 (>6)

: 86 cj

Vraòej ;.

N sjnoo stntue kns n kelbkt ai > cj ngh ls ponc eh lg iragt ai n spkerlcno jlrrar . Xke ljnbe ai tke

stntue ls lgverteh ngh ls 6.5cj tnoo ngh oacnteh >6 cj lg iragt ai tke jlrrar. Ilgh tke iacno oegbtk ngh

gnture ai tke jlrrar.

Saoutlag = Re knve j :>

0

k

k:

>

5.6: 6.5

v : >6 cj (reno ljnbe)

@ut j :i

vi 6.5 :

i

>6i i :

3

06cj

sa, cagcnve jlrrar. Ngs.

Vraòej j nàve tke surince ai n olqulh ai reirnctlve lghex : 0 ngh

kelbkt 0j. Ilgh x.


46/55

VK\SLCS


>j

0j

856

x

calg

eye

Saoutlag = 0 :rslg

85slg

slg r :0

> r : 36

x : P^ + ^V

: >j + 0tng36 j

:

3

0> j Ngs.

Vra`oej >>.

N rny ai olbkt inoo s nt ng ngboe ai 36 agta n ponge-pnrnooeo bons s ponte ngh oenves lt pnrnooeo ta tke

lgltlno rny. Xke reirnctlve lghex ai tke bonss ls >.5. Rknt ls tke tklcdgess h ai tke ponte li tke hlstngce

`etweeg tke rnys ls 3.;0 cj1 TBlveg = slg>

3

>: >;Z

Saoutlag =

: >.5Uslgb s :

rcas

)rl(slgh

h :)r36slg(

rcas;0.3

..................... (>)

Nosa, >.5 :rslg

36slg

slg r :3

>

sa, r : >6slg

;.6


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Vraòej >0. N olbkt pnsses tkraubk jngy pnrnooeo son`s age `y age ns skawg lg ilbure.

Vrave tknt g>slgl

>: g

0slgl

0: g

3slgl

3: g

8slgl

8 :.............TPejejèr tklsZ. Nosa prave tknt li g

>: g

8 tkeg

olbkt rnys lg jehluj g>

ngh lg jehluj g8

nre pnrnooeo.

Saoutlag = Re knve,

0

>

lslg

lslg:

>

0

g

g

g> slg l> : g0 slg l0 ....(l)

Sljlonroy g0slg l0 : g3 slg l3

sa ag

sa, g> slg l> : g0 slg l0 : g3 slg l3 : ..............

g>slg l> : g8 slg l8 slg l> : slg l8 (g>: g0)

sa, l> : l8

Kegce, olbkt rnys lg jehluj g>ngh lg jehluj g8nre pnrnooeo.

Vraòej >3.

Ng a`me ct oles

cj8.

N cagcnve jlrrar ls ponc eh lgslhe wnter wltk lts sklglgb surince

upwnrhs ngh prlgclpno nxls vertlcno ns skawg. Pnys nre lgclhegt

pnrnooeo ta tke prlgclpno nxls ai cagcnve jlrrar. Ilgh tke pasltlag ai

ilgno ljnbe.


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Saoutlag =

Re knve,

>

0

h'

: >

u : , i : 06 cj

Sa, v : 06 cj

Sa, h :>6 cj

h :reo

h

:

3/8

cj>6

:8

36cj : 2.5 cj Ngs.

Vraòej >5. Vrave tknt tke sklit lg pasltlag ai a mect

hue ta pnrnooeo son` ls blveg `y sklit

:

reog

>>h wkere g

reo :

'g

g.

Saoutlag =

@ecnuse ai tke rny reirnctlag nt tke ilrst surince, tke ljnbe ai A ls iarjeh nt>. Iar tkls reirnc-tlag, tke reno heptk ls NA : x ngh nppnregt heptk ls N>.

Xkus = N>:rl g/g

NA:

g/g

NA:

g

)NA(g.

Xke palgt > ncts ns tke a`mect iar tke reirnctlag ai secagh surince. Hue ta tkls reirnctlag, tke ljnbeai> ls iarjeh nt 0. Xkus,

g

g

)g/g(

)@(@ >0

(@>)

: g'/g (N@ + N>)

:g

g

)NA(

g

gh :

g

gh + NA.NA.

Get sklit : A0: @A @0

: h + (NA) g

gh NA

:

g

g>h :

reog

>>h

wkere greo: g

g. Ngs.

Vraòej >9.

Ilgh tke nppnregt heptk ai a`mect A èoaw surince N@, seeg

`y ng a`server lg jehluj ai reirnctlve lghex

Saoutlag =

hnpp.

:0>

>

/

t


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"jnglskdujnrpkyslcs.lg" 82.

Lg nàve questlag wknt ls tke heptk ai a`mect carrespaghlgb ta lgclhegt rnys strldlgb ag surince CH

lg jehluj.

Saoutlag =

Heptk ai tke a`mect carrespaghlgb ta lgclhegt rny strldlgb ag tke surince CH lg jehluj 0: t

0+V

>

: t0+

0>

>

/

t

Vraòej >;.

Lg nàve questlag li a`server ls lg jehluj , wknt ls tke nppnregt heptk ai a`mect seeg èoawsurince CH.

Saoutlag =

Li tke a`server ls lg jehluj 3. nppnregt heptk èoaw surince CH : ^

0.

:lreo

l

)g(

t:

3>

>

30

0

/

t

/

t

Vraòej >


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@ut C : slg> rh/>

sa, r : k tng

rh

>

/

>slg

: 0r

0h

r.k

Vra`oej 06.

N sklp ls snlo lgb lg rlver. Ng a`s erver ls sltunteh nt n heptk k lg wnter (w

). Li x 77 k, ilgh tke ngboe

jnhe iraj vertlcno, ai tke olge ai slbkt ai sklp.

Saoutlag = C : slg>

w

n

c

wnter

Vra`oej 0>. Ilgh r, r, e, iar tke cnse skawg lg ilbure.

Saoutlag =Kere : >;6 25a : >65

slg 85a : 0 slg r


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r : slg>0

>: 36.

r'

r' : >;6 (r +): >;6 36 >65 : 85a

slg e : 0 slg r'

slg e : 0 slg 85a : >

Sa, : l + e N : 85a + ;6

: N + r +


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Saoutlag =

Iraj tke brnpk 4: l + e N.36 : 36 + 96 N

(use tke resuot = Li l ngh e nre lgterckngbeh tkeg we bet snje vnoue ai

Vra`oej 08.

Li twa prlsjs nre caj`lgeh, ns skawg lg ilbure, ilgh tke get

ngbuonr hlsperslag ngh ngboe ai hevlntlag suiiereh `y n wklte

rny ai olbkt lgclhegt ag tke caj`lgntlag.

Saoutlag = Get ngbuonr hlsperslag

: (v r) ('v 'r): (

v

r) N ('

v '

r) N'

: (>.5 >.8) 8 (>.2 >.5) 0a : 6

Ngboe ai hevlntlag :

>

0

rvN

>

0

'' rvN'N'

:

>

0

8.>5.> 8

>

0

5.>2.> 0 : 6.9

Vra`oej 05.

Xke hlsperslve pawers ai crawg ngh iolgt bonsses nre 6.63 ngh 6.65 respectlveoy. Xke reirnctlve

lghlces iar yeooaw olbkt iar tkese bonsses nre >.5>2 ngh >.90> respectlveoy. Lt ls heslreh ta iarj ng

nckrajntlc caj`lgntlag ai prlsjs ai crawg ngh iolgt bonsses wklck cng prahuce n hevlntlag ai > 6 lg

tke yeooaw rny. Ilgh tke reirnctlgb ngboes ai tke twa prlsjs geeheh.

Saoutlag = c: 6.63 : >g

gg

y

rv

(gv g

r) : 6.63 (>.5 >2>) : 6.6>55

ngh, i: 6.65 :

>'g

'g'g

y

rv

g'v g'

r: 6.65 (>.9 0> >) : 6.63>

: (gv gr) N (g'v g'r) N': 6.6>55 N 6.63> N' .........(>)

@ut get

: >

Sa, (gy >) N (g'

y >) N' : >

: 6.5>2 N 6.90> N' : > ........(0)

N : 8.; ngh N' : 0.8


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Vraòej 09. See tke sltuntlag skawg lg ilbure

(>) Ilgh tke pasltlag ai ljnbe ns seeg `y a`server N.

(0) Ilgh tke pasltlag ai ljnbe ns seeg `y a`server @.

Saoutlag = (l) Ns seeg y a`server N .

P : >6 cj.

u : >6 cj.

a`server

Sa,v

g0

u

g>:

P

gg >0,

v

>

)>6(

5.>

: )>6(5.>>

v : >6 cj(nt A)

(ll) Ns seeg `y a`server @

P :

u : 06 cj

v

>

06

6.0:

6.0>

v : >6cj

ljnbe wloo è iarjeh >6 cj rlbkt ai A.

Vraòej 02.

Ilgh tke iacno oegbtk ai n hauòe-cagvex oegs wltk P>: >5 cj ngh P

0: 05 cj. Xke reirnctlve lghex

ai tke oegs jnterlno g : >.5.

Saoutlag =

i

>: (g >)

0> P

>

P

>

: (>.5 >)

05>

>5> .

: 6.5

>56

9>6

:>56

;.

i :;

>56: >;.25 cj


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Vraòej 0;.

Ilgh tke iacno oegbtk ai n ponga-cagvex oegs wltk P>

: >5 cj ngh P0

: . Xke reirnctlve lghex ai tkeoegs jnterlno g : >.5.

Saoutlag =

i

>: (g >)

0> P

>

P

>: (>.5 >)

>

>5

>. : 6.5

>5

>

: 36 cj .

Vraòej 0

: >5 cj ngh P0

: 05 cj.

Xke reirnctlve lghex ai tke oegs jnterlno g : >.5.

Saoutlag =i

>: (>.5 >)

05

>

>5

>: 6.5

>56

9>6. :

8

366: 25 cj

Vraòej 36. Ilbure skaws n palgt a mect ngh n hlverblgb oegs.

Ilgh tke ilgno ljnbe iarjeh.

Saoutlag = v

> u

>: i

>

v

>:

>6

>

+

)>6(

>

:

>6

0. v : 5 cj

Vraòej 3>.

Ng extegheh reno a`mect ls ponceh perpeghlcuonr ta tke prlgclpno nxls ai n cagcnve oegs ai iacno

oegbtk >6 cj, suck tknt tke ljnbe iaugh ls knoi tke slze ai a`mect.

(n) Ilgh tke a`mect hlstngce iraj tke oegs

(`) Ilgh tke ljnbe hlstngce iraj tke oegs ngh hrnw tke rny hlnbrnj

(c) Ilgh tke onterno jnbglilcntlag li a`mect ls javeh `y > jj noagb tke prlgclpno nxls tawnrhs

tke oegs.Saoutlag = (n) Re knve, i: >6 cj.

j :>

0

k

k: 6.5 :

u

v

Sa,v

>

u

>:

i

>blves.

u5.6

>

u

>:

>6

>

u: >6cj

( )u

v: 6.5.

v : 6.5 ( >6) cj.: 5 cj

Pny hlnbrnj =-

(c) j :ui

i

hj : 0uii

hu : 0)>6>6(

)>6(

(6.>) : 6.6605 cj

sa, ilgno onterno jnbglilcntlag (j+hj) : 6.5605 cj Ngs.


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Vra`oej 30.

Ilgh tke equlvnoegt iacno oegbtk ai tke systej

iar pnrnxlno rnys pnrnooeo ta nxls.

Saoutlag =

0>0>eq ii

h

i

>

i

>

i

> :

5

>

)>6(>6

06

>6

>

>6

>

ieq

: 5 cj

Vra`oej 33. See tke ilbure

Ilgh tke equlvnoegt iacno oegbtk ai tke

caj`lgntlag skawg lg tke ilbure ngh

pasltlag ai ljnbe.

Saoutlag = Iar tke cagcnve oegs

i

>: (g >)

0> P

>

P

>:

>

0

3

>6>

>6

>:

0

>

>6

0:

>6

>

Ngh, ij

: 0

P:

0

>6: 5 cj

eqi

>:

ji

> 0

i

>

:5

>+ 0

>6

>:

5

0Ngs.

geometrical optics theory_e

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