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7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" >
BEAJEXP L O AVX L SLGXPAHUCXLAG =
@oue ondes, ackre heserts, breeg iarest, ngh juotlcaoareh rnlg`aws cng `e egmayeh `y ngyage wka kns eyes
wltk wklck ta see tkej. @ut `y stuhylgb tke `rngck ai pkyslcs cnooeh aptlcs, wklck henos wltk tke `eknvlaurai olbkt ngh atker eoectrajnbgetlc wnves, we cng renck n heeper nppreclntlagai tkevlsl`oe waroh. N dgawoehbe
ai tke prapertles ai olbkt nooaws us ta ugherstngh tke `oue 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 il`ers, kaoabrnjs, aptlcno
cajputers, ngh gew teckglques lg jehlcno ljnblgb.
>. CAGHLXLAG IAP PECXLOLGENP VPAVNBNXLAG AI OLBKX(Agoyiar lgiarjntlag gat lg LLX-MEEsyoon`us)
Saje pnrt ai tke aptlcs cng `e ugherstaah li we nssuje tknt olbkt trnveos lg n strnlbkt olge ngh lt
`eghs 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 `eknvlaur 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 `eghlgb ls cnooeh
hliirnctlag.
Olbkt `eghs `y (`-n)/0 ag enck slhe ai tke cegtrno olge .Lt cng `e skawg `y wnve tkeary ai olbkt tknt
slg :n
.......(N),
wkere ls skawg lg ilbure.
Xkls iarjuon lghlcntes tknt tke `eghlgb ls cagslhern`oe agoy wkeg n ~ . Hliirnctlag ls jarepragaugceh lg saugh `ecnuse 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 `eghlgb ls gebolbl`oe 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 `eghs agoy wkeg lt bets
reioecteh ar reirncteh.
Xkus beajetrlcno aptlcs ls ng nppraxljnte trentjegt lg wklck tke olbkt wnves cng `e 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 nppolcn`oe 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 `aughnry 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 gebolbl`oy sjnoo
lgtegslty lg atker hlrectlags.
(`) Hliiuseh Peioectlag
Rkeg tke surince ls raubk, we ha gat bet n rebuonr`eknvlaurai olbkt. Notkaubk nt enck palgt
olbkt rny bets reioecteh lrrespectlve ai tke avernoo gnture ai surince, hliieregce ls a`serveh`ecnuse eveg lg n gnrraw `enj ai olbkt tkere nre jngy rnys wklck nre reioecteh iraj hliieregt
palgts ai surince ngh lt ls qulte passl`oe tknt tkese rnys jny jave lg hliieregt hlrectlags hue
ta lrrebuonrlty ai tke surince . Xkls pracess egn`oes 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 `e expresseh j ntkejntlcnooy ns P
. (
G
) : G
. (
P
) :
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"jnglskdujnrpkyslcs.lg" 3
. ( 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 `etweeg garjno
ngh tke lgclhegt rny) ngh tke ngboe ai reioectlag (tke
ngboe `etweeg 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 `eghs 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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"jnglskdujnrpkyslcs.lg" 8
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 `e
slovereh ag age slhe.
N `enj ai pnrnooeo rnys ai olbkt, lgclhegt ag n ponge jlrrar wloo bet rei oecteh ns n `enj ai pnrnooeo
reioecteh rnys.
Exnjpoe - 0Iar n i lxeh lgclhegt o lbkt rny, l i tke jlrrar `e
ratnteh tkraubk ng ngboe (n`aut 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 n`aut 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 `e
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 `e
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 `e 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 `eyagh tke
rnys > ngh 0, tkereiare tke a`servers V ngh ^ cnggat see tke
ljnbe `ecnuse 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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"jnglskdujnrpkyslcs.lg" 9
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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"jnglskdujnrpkyslcs.lg" 2
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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"jnglskdujnrpkyslcs.lg" ;
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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VK\SLCS
"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 `e iarjeh `y tke rnys strldlgb jlrrar J0
ilrst. Xatno guj`er 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 guj`er 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
`ecnuse 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 `e iarjeh ngh
tkls wloo iugctlag ns a`mect iar jlrrar J0
ngh tkeg lts ljnbe>0
(nt (x, y)) wloo `e 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 `e lgcolgeh `y ngboe ag tke atker slhe ai jlrrar J
>. Xkls ngboe ls wrltteg lg
`rncdet lg tke ilbure `eslhes>. Sljlonroy ljnbe ai a`mect
V iarjeh `y J0
,hegateh `y0
,wloo `e lgcolgeh `y ngboeag tke atker slhe ai jlrrar J
0. Xkls ngboe ls wrltteg lg
`rncdet lg tke ilbure `eslhes0.
Gaw 0
wloo nct ns ng a`mect iar J>
wklck ls nt ng ngboe iraj J>
. Lts ljnbe wloo `e
iarjeh nt ng ngboe ag tke appaslte slhe ai J>. Xkls ljnbe wloo `e hegateh ns
ngh
sa ag. Xklgd wkeg tkls wloo pracess stap. Klgt= Xke vlrtuno ljnbe iarjeh `y n ponge jlrrar
just gat `e lg iragt ai tke jlrrar ar lts extegslag.
Guj`er ai ljnbes iarjeh `y twa lgcolgeh jlrrars
(l) Li
396: eveg guj`er4 guj`er ai ljnbe :
396
>
(ll) Li
396: ahh guj`er4 guj`er ai ljnbe :
396 >,
li tke a`mect ls ponceh ag tke ngboe `lsectar.
(lll) Li
396: ahh guj`er4 guj`er ai ljnbe :
396
,
li tke a`mect ls gat ponceh ag tke ngboe `lsectar.
(lv) Li
396 lgteber,, tkeg caugt tke guj`er ai ljnbes ns exponlgeh n`ave.
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 guj`er ai ljnbes
uslgb (l) hlrect iarjuon ngh (ll) caugtlgb tke ljnbes.
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Saoutlag Ilbure ls seoi expongntary.
Guj`er ai ljnbes
(l) Uslgb hlrect iarjuon =36
396: >0 (eveg guj`er)
guj`er ai ljnbes : >0 > : >>
(ll) @y caugtlgb . See tke iaooawlgb tn`oe
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 `e heilgeh ns tke olge wklck malgs tke Vaoe ta
tke Cegter ai Curvnture ai tke jlrrar.
7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" >0
(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.
7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" >3
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 `e reioecteh `ncd tkraubk tke
cegtre ai curvnture (`ecnuse 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 `eoaw.
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 n`ave Vrlgclpoe Nxls ngh gebntlve `eoaw 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.
7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" >8
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
7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" >5
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 `e 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 `atk tke a`mect &
tke ljnbe `ecaje cagstngt. Ag hliieregtlntlgb tke n`ave 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 `e lgverteh)
7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" >2
(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 `etweeg 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 + `ecnuse 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 `e 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 ?
7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" >;
Li tke olbkt ls lgclhegt garjnooy tkeg lt baes ta tke secagh jehluj wltkaut `eghlgb, `ut 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 pra`oejs rejej`er
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 `eghs nwny iraj garjno.
Rkeg olbkt javes iraj rnrer ta hegser jehluj lt `eghs tawnrhs tke garjno.
7/21/2019 Geometrical Optics Theory_E
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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.
7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 06
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 reversl`oe.
2. PEIPNCXLAG XKPAUBK N VNPNOOEO SON@Rkeg olbkt pnsses tkraubk n pnrnooeo son`, knvlgb snje jehluj ag `atk slhes, tkeg
(n) Ejerbegt rny ls pnrnooeo ta tke lgclhegt rny.
Gate = Ejerbegt rny wloo gat `e pnrnooeo ta tke lgclhegt rny li tke jehluj ag `atk tke slhes ai
son` nre hliieregt.
(`) Olbkt ls skliteh onternooy, blveg `y (stuhegtsskauoh `e n`oe 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.
7/21/2019 Geometrical Optics Theory_E
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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.
7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 00
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.
7/21/2019 Geometrical Optics Theory_E
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 03
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.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 08
; . 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 `e 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 `e seeg iraj tke rnrer jehluj. Pnhlus ai C.A.L
cng `e ensloy iaugh.
Exnjpoe - 36 Ilgh tke jnxljuj ngboe tknt cng `e 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
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 05
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 :
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 09
(`) 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 `ecnuse ai reversl`loltyprlgclpoe ai olbkt
Gate = (l) Iar nppolcntlag ai n`ave resuot jehluj ag `atk slhes ai prlsj just `e snje.
(ll) @nseh ag n`ave brnpk we cng nosa herlve iaooawlgb resuot, wklck snys tknt l ngh e cng `e
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
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 02
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 `e 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 passl`oe 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 `e 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
+
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 0;
>6. HLSVEPSLAG AI OLBKXXke ngbuonr spolttlgb ai n rny ai wklte olbkt lgta n guj`er ai cajpagegts ngh sprenhlgb lg hliieregt
hlrectlags ls cnooeh Hlsperslag ai Olbkt. TLt ls iar wkaoe Eoectra Jnbgetlc Rnve lg tatnoltyZ. Xkls
pkegajegag ls `ecnuse wnves ai hliieregt wnveoegbtk jave wltk snje speeh lg vncuuj `ut 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 `etweeg 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 `enj (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
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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.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 36
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.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 3>
>>.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 hlstrl`utlag ai olbkt ejltteh `y tke saurce. Xke ejlsslag spectrn nre blveg
`y lgcnghescegt saolhs, olqulhs ngh bnses wklck nre eltker `urgt hlrectoy ns n ionje (ar n
spnrd) ar `urgt ugher oaw pressure lg n hlscknrbe tu`e.
>>.3 Lg N`sarptlag spectrujHnrd olges lghlcntes irequegcles n`sar`eh.
Rkeg n `enj ai olbkt iraj n kat saurce ls pnsseh tkraubk n su`stngce (nt n oawer
tejpernture), n pnrt ai tke olbkt ls trngsjltteh `ut rest ai lt ls n`sar`eh. Rltk tke keop ai n
spectrajeter, we cng dgaw tke irnctlag ai olbkt n`sar`eh carrespaghlgb ta enck wnveoegbtk.
Xke hlstrl`utlag 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 `enj), 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.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 30
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 n`ave iarjuon ns
v : u
greo,
lt blves tke reontlag `etweeg tke nppnregt heptk ngh reno heptk, ns we knve seeg `eiare.
Exnjpoe - 88
Uslgb iarjuon ai spkerlcno surince ar atkerwlse, ilgh tke nppnregt heptk ai ng a`mect ponceh
>6 cj `eoaw 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.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 33
Iar n spkerlcno, tklg oegs knvlgb tke snje
jehluj ag `atk 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 `ecaje 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. (Pejej`er tkls)
Iraj tke reontlag
>
i :(g reo >)
0> P>
P
>
lt cng `e seeg tknt tke secagh iacno oegbtk hepeghs ag
twa inctars.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 38
(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 `atk 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 n`errntlag.
Exnjpoe - 85
Ilgh tke`eknvlaurai n cagcnve oegs ponceh lg n rnrer j ehluj.
Saoutlag
Inctar (N) ls gebntlve , `ecnuse tke oegs ls cagcnve.
Inctar (@) ls pasltlve , `ecnuse 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 `eknve 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.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 35
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 `aughnrles
l.e., knvlgb snje rnhll ai curvnture. Hlscuss n`aut 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 npprecln`oe `eghlgb.
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 `e P>
ngh reirnctlve lghex ai oegs `e . Oet pnrnooeo rnys `e lgclhegtag tke oegs. Nppoylgb reirnctlag iarjuon nt ilrst surince
>
v>
:>P
>...(>)
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 39
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 `e tke
vnoue ai h sa tknt ilgno rnys nre nosa pnrnooeo.
Saoutlag Ilgno rnys skauoh `e pnrnooeo. Iar tkls tke iacus ai O>
just calgclhe wltk iacus ai O0.
h : >6 + 06
: 36 cj
Kere tke hlnjeter ai rny `enj `ecajes wlher.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 32
Exnjpoe - 55
See tke ilbure
Ilgh tke pasltlag ai ilgno ljnbe iarjeh.
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 `e 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
`e nt paoe ngh lts ljnbe wloo `e
iarjeh tkere ltseoi.
Cnse LL =
Li tke rnys strlde tke jlrrar garjnooy, tkey wloo retrncengh tke ljnbe wloo `e 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 `atk slhes ls hliieregt, tkeg we knve ta nppoy tkeiarjuon blveg iar reirnctlag nt spkerlcno surinces step `y step.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 3;
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
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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 `etweeg
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 passl`oe. 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 `etweeg 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, `ecaje 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 j0`e 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
>
>
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 8>
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 `eknves 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
Ilgh tke pasltlag ai ilgno ljnbe iarjeh.
(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 `e iarjeh ag tke a`mect ltseoi
Saje lgterestlgb incts n`aut 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 `egh
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
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 80
sa tke vertlcno hlnjeter ai tke sug nppenrs ta `e 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 `eghlgb 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 lggujern`oe 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
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 83
Vra`oej >.
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.
Vra`oej 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@.
Vra`oej 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
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 88
Vra`oej 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 `eextegheh 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.
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////
////
////
/////////
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.
Vra`oej 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>
Vra`oej 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
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 85
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>
Vra`oej 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
Vra`oej ;.
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.
Vra`oej j n`ave tke surince ai n olqulh ai reirnctlve lghex : 0 ngh
kelbkt 0j. Ilgh x.
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 89
>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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VK\SLCS
"jnglskdujnrpkyslcs.lg" 82
Vra`oej >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 :.............TPejej`er 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.
Vra`oej >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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VK\SLCS
"jnglskdujnrpkyslcs.lg" 8;
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.
Vra`oej >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.
Vra`oej >9.
Ilgh tke nppnregt heptk ai a`mect A `eoaw surince N@, seeg
`y ng a`server lg jehluj ai reirnctlve lghex
Saoutlag =
hnpp.
:0>
>
/
t
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 82.
Lg n`ave 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
Vra`oej >;.
Lg n`ave questlag li a`server ls lg jehluj , wknt ls tke nppnregt heptk ai a`mect seeg `eoawsurince CH.
Saoutlag =
Li tke a`server ls lg jehluj 3. nppnregt heptk `eoaw surince CH : ^
0.
:lreo
l
)g(
t:
3>
>
30
0
/
t
/
t
Vra`oej >
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VK\SLCS
"jnglskdujnrpkyslcs.lg" 56
@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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VK\SLCS
"jnglskdujnrpkyslcs.lg" 5>
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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VK\SLCS
"jnglskdujnrpkyslcs.lg" 50
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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VK\SLCS
"jnglskdujnrpkyslcs.lg" 53
Vra`oej 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 `e iarjeh >6 cj rlbkt ai A.
Vra`oej 02.
Ilgh tke iacno oegbtk ai n hau`oe-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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VK\SLCS
"jnglskdujnrpkyslcs.lg" 58
Vra`oej 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 .
Vra`oej 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
Vra`oej 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
Vra`oej 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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VK\SLCS
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.