PLYTIDT RMWIMS NKVMT: \I D KEPRMLMNTIWM P LYTIDT
RMWIMS NKVMT GKR DAOTT GIRTV YMOR
BY
F R . R O E D LONF P L F( F(P P LYTIDT )
Prindipoa/Lmof kg Plysids Fmportemnt Tlolmmf Bmnozir Bluttk Ckvt Ciras Fmcrmm Dkaamcm, Jlufk, Fistridt Eirpurhlos 04442=?44 04442= ?443=, 3=, roe[r ro e[r2=@ 2=@lkt lkteo eoia. ia.dke dke lttp://www.godmbkkh.dke/roedrocuma
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ROE’T K]VAINM TMRIMT
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Dkpyriclt
d 20?3, Fmportemnt kg Plysids, Tlolmmf Bmnozir Bluttk Ckvmrnemnt Ciras
Fmcrmm Dkaamcm, Jlufk
D KEPKTMF BY F R . R OE D LONF Vlis eonusdript is writtmn in AOVM\. Vlm fiocroes onf ieocms orm drmotmf in kpmn-skurdm
oppaidotikns IPM, AotmxFrow, Grmmpaonm, W]M onf Bamnfmr 4F.
Vlm outlkr is o visitinc sdimntist tk Ospmn Dmntmr gkr Plysidist, ]TO, tlm ]nivmrsity kg Eoaoyo, Huoao Auepur, Eoaoysio, tlm Intmrnotiknoa Dmntmr gkr Vlmkrmtidoa Plysids (IDVP), Itoay onf tlm Dlinmsm Odofmey kg Tdimndms, Bmijinc, Dlino. Vlm outlkr is oask o emebmr kg Oemridon Osskdiotikn kg Plysids Vmodlmrs (OOPT), ]TO. Vlm outlkr’s rmsmordl prkam
don bm gkunf ot lis AinhmfIn pocm & Ckkcam Tdlkaor pocm. roe[
[email protected],
[email protected] lttp://www.godmbkkh.dke/roedrocuma @RoeDRocuma
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Dkntmnts
?
TDKPM KG PLYTIDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . =
2
TDOAORT TDOAOR T ONF WMDVKRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2? 2?
4 EKVIKN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4? 4?
EKVIKN IN VSK FIEMNTIKN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . =4 =4
= TVOVIDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7? 7?
7 CROWIVOVIKN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73 73
3
3= SKRH, PKSMR ONF MNMRCY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3=
<
SOWM SO WM EKVIKN ONF TK]NF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . <4 <4
9
NOV]RM KG AICLV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?03
CMKEMVRIDOA OA KPVIDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??9 ?0 CMKEMVRID
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?. TDKPM KG PLYTIDT
?.? ?. ?
TDIM TD IMND NDM M Tdimndm.. Vlm Vlm fmsirm tk hnkw obkut tlincs, mvmnts onf noturoa plmnkemnkn orkunf us is doaamf Tdimndm wkrf Tdimndm odtuoaay dkems grke tlm Aotin wkrf Tdimntio wkrf Tdimntio emoninc emoninc "tk hnkw". Sm don hnkw obkut tlm univmrsm by kbsmrvinc onf by mxpmriemnt. Lmndm Tdimndm5 tk kbsmrvm + tk fk mxpmriemnt. Tdimnd Tdi mndmm is tlm sue kg lue lueon on hnk hnkwam wamfcm fcm wli wlidl dl eon los coi coinmf nmf tlr tlrkuc kucl l ocm ocms. s. In pos post, t, tli tliss hnk hnkwam wamfcm fcm wos aieitmf but os tlm hnkwamfcm hnkwamfcm mxponfs, its dkepa dkepamtm mtm stufy bmdkems iepks iepkssiba sibam. m. Vlmrmg Vlmrmgkrm krm
sdimndm is fivifmf intk twk eoin brondlms: Plysidoa Tdimndm: It fmoas witl tlm nkn-aivinc kbjmdts onf tlksm gundtikns kg aivinc kbjmdts kbjmdts wlidl
orm sieiaor tk nkn-aivinc kbjmdts. Bikakcidoa Tdimndm: It fmoas witl aivinc kbjmdts knay.
?.2 ?. 2
SLO SL OV IT PL PLYT YTID IDT; T; Vlm wkrf plysids is fmrivmf grke Crmmh wkrf gusis wkrf gusis wlidl wlidl emons "noturm". Vlmrmgkrm, plysids emons "Aows kg noturm". Plysids is fmnmf os tlot brondl kg sdimndm, wlidl stufims noturoa plmnkemno in tmres kg bosid aows onf plysidoa quontitims. Vlm wlkam noturoa univmrsm dknsists kg twk bosid
quontitims : (i) eottmr onf (ii) mnmrcy. Vlmrmgkrm wm fmnm plysids os: Fmnitikn
Plysids is tlm brondl kg plysidoa sdimndm wlidl fmoas witl tlm stufy kg eottmr, mnmrcy,, onf tlm intmrodtikn bmtwmmn tlme. mnmrcy
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Dloptmr ?. TDKPM KG PLYTIDT
7
Vwk prindipams tlrust in tlm stufy kg Plysids orm:‖ ]nidotikn. wlidl emons mxpaoininc figgmrmnt figgmrmnt plysidoa plmnkemno by usinc gmw aows
onf dkndmpts. ‖ Rmfudtiknise. wlidl emons mxpaoininc dkepamx plmnkemno by brmohinc tlme intk
seoaamr dknstitumnts onf stufyinc siepamr ports.
?.4
DAOTTIGID DAOT TIGIDOVI OVIKN KN KG PLYTI PLYTIDT DT Plysids don bm daossimf intk tlrmm eoin dotmckrims: Daossidoa Plysids
Oask hnkwn os Eodrksdkpid Plysids (kr Nmwtknion Plysids) wlidl fmoas witl tlm stufy kg ostrknkeido ostrknkeidoaa onf ktlmr bic mamemnts. Vlm eodrksdkpid wkraf dkntoins tlm tlincs wm don smm witl kur myms. Gkr mxoepam, emdlonids, tlmrekfynoeids, kptids mtd. Gotlmr kg daossidoa plysids is
Nmwtkn. Emsksdkpid Plysids
Slidl fmoas witl tlm stufy stufy kg kg lunfrmfs lunfrmfs kg otkes otkes kr ekamdu ekamduams. ams. Vlm emsksdkpid wkraf is in bmtwmmn tlm eidrksdkpid onf tlm eodrksdkpid wkraf wk raf.. Vli Vliss bro brondl ndl is vm vmry ry nmw onf kri kricin cinotm otmf f ske skemm tmn ymors ymors ock ock.. Oppaidotikn kg tlis maf is aorcmay gkunf in tlm mamdtrknids infustry. Infustry eohms eknmy kut kg tlm einioturizotikn kg tronsistkrs, wlidl switdl tlm mamdtridoa durrmnt kn o dkeputmr dlip. Gotlmr kg Emsksdkpid
Plysids is Prkg. Yksmpl Iery. Ekfmrn plysids is o brondl kg plysids tlot fmoas tlm tkpid wlmrm eottmr
Ekfmrn Plysids
onf mnmrcy orm nkt smporotm, but it is on oatmrnotm gkre kg modl ktlmr. It kgtmn invkavms mxtrmem dknfitikns: mitlmr vmry seoaa tlincs gkr mxoepam otkes onf sub-otkeid portidams KR rmaotivistid mggmdts wlidl invkavm vmakditims dkepormf tk tlm spmmf kg aiclt. Gotlmr kg ekfmrn plysids is
Oabmrt Minstmin.
?.6 BRO BRONDL NDLMT MT KG PLY PLYTID TIDT T Vlmrm orm eony brondlms brondlms kg plys plysids. ids. Vlm kafmst brondl brondl kg plysids is Emdlonids. Fmt Fmtoia oia kg oaa oaa
brondlms is civmn bmakw: Emdlonids ids is tlm brondl kg Plysids wlidl fmoas witl tlm stufy kg porti portidams dams kr EMDLONIDT: Emdlon
bkfims wlmn tlmy orm ot rmst kr in ektikn. Emdlonids is gurtlmr fivifmf intk: ?. Ttotids: Ttotids is tlm stufy kg kbjmdts ot rmst8 tlis rmquirms tlm ifmo kg gkrdms in mquiaibriue. 2. Fynoeids: Fynoeids Fynoeids: Fynoeids is tlm stufy kg ekvinc kbjmdts. It dkems grke tlm Crmmh wkrf pkwmr. Fynoeids is gurtlmr subfivifmf intk hinmeotids onf fynoeis wlidl emons pkwmr.
hinmtids. (o) Hinmeotids is Hinmeotids is tlm stufy kg tlm rmaotiknslip bmtwmmn fispaodmemnt, vmakdity, oddmamrotikn onf tiem kg o civmn ektikn, witlkut dknsifmrinc tlm gkrdms tlot dousm
tlm ektikn. Hinmtids fmoas witl tlm rmaotiknslip bmtwmmn tlm ektikn kg bkfims onf gkrdms (b) Hinmtids fmoas
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?.= PLYTIDOA _]ONVIVIMT ONF ]NIVT
3
?. Mamdtrkstotids: Vlm stufy kg mamdtrid dlorcms ot rmst, 2. Mamdtrkfynoeids: Vlm stufy kg ekvinc mamdtrid dlorcms. 4. Eocnmtk-stotids: Vlm stufy kg eocnmtid pkams ot rmst. emdlonids kg ektik ektikn n in fluif fluifss in bktl aiquif aiquif onf cosmkus stotms orm GA]IF FYNOEIDT: Vlm emdlonids
invmsticotmf in stufims kg fluif fynoeids. OMRKFYNOEIDT: Vlm brondl kg fynoeids tlot fmoas witl tlm ektikn kg oir onf ktlmr cosmkus
fluifs onf witl tlm gkrdms odtinc kn bkfims in ektikn rmaotivm tk sudl fluifs. OVKEID PLYTIDT: Otkeid plysids is tlm brondl kg plysids wlidl wlidl fmoas witl tlm dkepksitikn kg otke oport grke nudamus. It is eoinay dkndmrnmf witl tlm orroncmemnt onf bmlovikur kg
mamdtrkns in tlm slmaas orkunf tlm nudamus. N]DAMOR PLYTIDT: Vlm plysids kg otkeid nudami onf tlmir intmrodtikns, mspmdioaay in tlm
cmnmrotikn kg nudamor mnmrcy. noturm kg plmnkemno ot tlm otkeid _]ONV]E EMDLONIDT: Vlm stufy kg fisdrmtm (quontizmf) noturm
onf subotkeid amvma. PAOTEO PLYTIDT: Vlm stufy kg gkurtl stotm kg eottmr - Paoseo. Tun is eofm kg paoseo. DKNFMNTMF EOVVMR PLYTIDT: Vlm stufy kg prkpmrtims kg dknfmnsmf eotmrioas (skaif, aiquif onf tlksm intmremfiotm bmtwmmn tlme onf fmnsm cos) witl tlm uatieotm ckoa onf
fmvmakpinc nmw eotmrioas witl bmttmr prkpmrtims8 it is mxtmnsikn kg Tkaif Ttotm Plysids. TVOVITVIDOA EMDLONIDT: Vlm brondl kg plysids tlot ottmepts tk rmaotm tlm prkpmrtims kg
eodrksdkpid systmes tk tlmir otkeid onf ekamduaor dknstitumnts. KPVIDT: Kptids is tlm brondl kg plysids, wlidl fmoas witl tlm prkpocotikn, bmlovikur onf
prkpmrtims kg aiclt. ODK]TVIDT: Vlm stufy kg prkfudtikn onf prkpocotikn kg skunf wovms. CMKPLYTIDT: Cmkpl Cmkplysids ysids is tlm brondl kg plysids plysids wlidl fmoas witl witl tlm stufy kg mortl. It is eoinay eoina y dkndmrnmf witl tlm slopm slopm,, strud strudturm turm onf dkepksitikn dkepksitikn kg mortl mortl.. It oask stufims tlm
crovitotiknoa gkrdm, eocnmtid mafs, mortlquohms, eoceos, mruptikn kg vkadonkms mtd. BIKPLYTIDT: Vlm intmrfisdipainory stufy kg bikakcidoa plmnkemno onf prkbames, usinc tlm
prindipams onf tmdlniqums kg plysids. OTVRKNKEY: Vlm brondl kg sdimndm wlidl fmoas witl dmamstioa kbjmdts, spodm, onf tlm plysidoa
univmrsm os o wlkam. OTVRKPLYTIDT: Ostrkplysids is dkndmrnmf witl tlm stufy kg univmrsm i.m., stors, coaoxims onf paonmts usinc tlm aows kg plysids. strudturm, onf DKTEKAKCY: Vlm stufy kg tlm kricin kg tlm univmrsm os o wlkam, kg tlm dkntmnts, strudturm,
mvkautikn kg tlm univmrsm grke tlm bmcinninc kg tiem tk tlm guturm.
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Dloptmr ?. TDKPM KG PLYTIDT
<
BOTID _]ONVIVIMT: Vlksm plysidoa quontitims wlidl don nkt bm fmrivmf onf orm bosid in
noturm. Tmvmn bosid quontitims orm dlksmn gkr tlmir dknvmnimndm. quontitims orm fmrivmf grke knm kr ekrm kg tlm bosid quonFMRIWMF _]ONVIVIMT: Oaa ktlmr quontitims titims. titi ms. Vlmsm don bm mxpr mxprmssmf mssmf in tmres kg gunf gunfoemnto oemntoaa plysidoa quontitims quontitims,, m.c., spmmf 5
fistondm/tiem.
?.7
DOVM OVMCKRY CKRY KG PLYTI PLYTID DOA _] _]ONVIV ONVIVIMT IMT Fiemnsiknoa Dkstonts Vlmsm orm tlm quontitims wlidl pkssmss fiemnsikns fiemnsikns onf lovm o xmf voaum. Gkr mxoepam, Crovitotiknoa dknstonts C, eoss kg mortl E M , spmmf kg aiclt d mtd. Fiemnsiknoa Woriobams Vlmsm orm tlm quontitims wlidl pkssmss fiemnsikns onf fk nkt lovm o xmf voaum8 gkr mxoepam, vmakdity v, oddmamrotikn o, gkrdm G mtd. Fiemnsiknamsss Dknstonts Vlmsm orm tlm quontitims wlidl fk nkt pkssmss fiemnsikns onf lovm o Fiemnsiknams
xmf voaum8 gkr mxoepam, π . Fiemnsiknamss Woriobams Vlmsm orm tlm quontitims wlidl orm fiemnsiknamss onf fk nkt lovm o
xmf voaum8 gkr mxoepam, Ttroin, Ttmrofion, Tpmdid Crovity mtd. ROE’T EINF EOP
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?.< TI Ttonforfs
9
Vobam ?.?: TI units
?.< ?. <
_uontity
]nit Noem
Tyebka
Amnctl
emtmr
e
Eoss Viem
hiakcroe smdknf
hc s
Vlmrekfynoeid Vmepmroturm
Hmavin
H
Mamdtrid Durrmnt Oekunt kg substondm Aueinkus Intmnsity
Oepmrm ekam donfmao
O eka df
TI Tt Tton onfo forf rfss In ?970, on intmrnotiknoa dkeeittmm ocrmmf kn o smt kg fmnitikns onf stonforfs tk fmsdribm tlm
plysidoa quontitims. Vlmsm stonforfizmf units orm smvmn in nuebmr onf orm civmn os unfmr: Amnctl Amnctl is fmnmf os tlm fistondm bmtwmmn twk pkints. Vlm TI unit kg amnctl is emtrm. Knm stonforf emtrm is mquoa tk ?7=0374.34 wovmamnctls kg tlm kroncm-rmf aiclt meittmf by tlm infivifuoa infivifuoa otkes kg Hry ptkn − <7 in o hryptkn fisdlorcm aoep. quontity ity kg eottmr dkntoinmf dkntoinmf in o bkfy. bkfy. It is infmpmnfmnt infmpmnfmnt kg tmepm tmepmrotur roturmm onf Eoss Eoss is tlm quont
prmssurm. It fkms nkt vory grke paodm tk paodm. Vlm TI unit kg eoss is hiakcroe. Vlm hiakcroe is mquoa tk tlm eoss kg tlm intmrnotiknoa prktktypm kg tlm hiakcroe (o paontinueirifiue oaaky dyainfmr) hmpt ot tlm Intmrnotiknoa Burmou kg Smiclts onf Emosurms ot Tmvrms, nmor Poris, Grondm.
Viem In ?973, on otkeid stonforf wos ofkptmf gkr smdknf, tlm TI unit kg tiem. Knm stonforf smdknf is fmnmf os tlm tiem tohmn gkr 9?9274?330 pmrikfs kg tlm rofiotikn dkrrmspknfinc tk unpmrturbmf tronsitikn bmtwmmn lypmrnm amvmas kg tlm crkunf stotm kg Otkeidd dakdhs orm bosmf kn tlis. In otkeid dakdhs, dakdhs, on mrrkr kg knm Dmsiue ?44 otke. Otkei
−
smdknf kddurs knay in =000 ymors. Oepmrm Vlm oepmrm is tlm dknstont durrmnt wlidl, flkwinc tlrkucl twk stroiclt poroaama innitmay aknc dknfudtkrs kg nmcaicibam drkss-smdtikn, onf paodmf in voduue ?e oport, wkuaf prkfudm
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Dloptmr ?. TDKPM KG PLYTIDT
?0
Gkr mxoepam, tlm fistondm bmtwmmn twk pkints in spodm don bm emosurmf in gmmt, emtmrs, kr eiams, wlidl wli dl orm fig figgmr gmrmnt mnt woy woyss kg mxp mxprms rmssin sinc c tlm fie fiemns mnsikn ikn kg amnctl systme tme kg emo emosur surmem memnt, nt, sud sudl l amnctl. In ony sys os tlm emtrid systme, dmrtoin quontitims orm dknsifmrmf gunfoemntoa, onf oaa ktlmrs orm dknsifmrmf tk bm fmrivmf grke tlme. Vlm syebkas tlot wm usm tk spmdigy tlm fiemnsikns kg amnctl, eoss, onf tiem orm A , E , onf V , rmspmdtivmay. Vlm mxprmssikn kg ony portiduaor quontity in tmres kg gunfoemntoa onoaysis onf kgtmn prkvifms plysidoa insiclt intk tlm rmsuats kg quontitims is hnkwn os fiemnsiknoa os fiemnsiknoa onoaysis onf o eotlmeotidoa doaduaotikn. Furinc fiemnsiknoa onoaysis kn civmn mquotikn, tlm gkaakwinc ruams
eoy oppay: ?. Fiemnsikns kn amgt lonf sifm (ALT) kg tlm mquotikn eust bm mquoa tk tlm fiemnsikns kn tlm riclt lonf sifm (RLT):
ALT 5 RLT 2. Fiemnsikns don bm bm trmotmf os os oacmbroid quontitims. quontitims. dkmgdimnts kg ony hinf in mquotikns slkuaf bm icnkrmf. Gkr 4. Furinc fiemnsikn onoaysis, dkmgdimnts mxoepam: v g 5 vi + ot ∽ v g
5
⇒
A V
5 e/s 5 A/V 8 5
A V
+
A
vi 5 e/s 5 A/V 8 A
A
A
V 5 + 5 2 2 V V V V
But 2 in obkvm mquotikn is dkmgdimnt wlidl eust bm frkppmf: A V
A
5 2
V
Vlmrmgkrm, A V
5
A V
o 5 e/t 2 5 A/V 2
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?.< TI Ttonforfs
?? ROE’T EINF EOP
PLYTIDOA _]ONVIVIMT ONF VLMIR FIEMNTIKNT Gkaakwinc plysidoa quontitims witl tlmir fiemnsikns orm civmn. Nktm tlot pkwmr tk fiemnsikn slkws lkw eony tiems tlm plysidoa quontity is usmf in tlm mquotikn kr gkreuao. Ig nk plysidoa quontity
(nk unit) is usmf tlmn pkwmr tk tlot fiemnsikn is civmn os zmrk. PLYTIDOA _]ONVIVIMT LOWINC TOEM FIEMNTIKNOA GKRE]AO
• Fistondm, Fispaodmemnt, rofius, aiclt ymor, wovmamnctl, wovmamnctl, rofius kg cyrotikn (A) . • Tpmmf, Wmakdity, Wmakdity kg aiclt . • oddmamrotikn, oddmamrotikn fum tk crovity (c), intmnsity kg crovitotiknoa maf, dmntripmtoa oddmamrotikn . • Iepuasm, Dloncm in ekemntue Gkrdm, Smiclt, Smiclt, Vmnsikn, Vlrust
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Dloptmr ?. TDKPM KG PLYTIDT
?2
_uontity
Vobam ?.2: TI units onf Fiemnsikns ]nit Fiemnsikn
Ormo ( O)
e2
E 0 A2 V 0
Wkauem (W )
e4
E 0 A4 V 0
Fmnsity (Ϗ )
hc/e4
E ? A−4 V 0
Wmakdity (v)
e/s
E 0 A? V −?
Oddmamrotikn (o)
e/s2
E 0 AV −2
Ekemntue ( p)
hc.e/s
E ? A? V −?
Gkrdm (G )
(hc.e/s2 ) N (
E ? A? V −2
Prmssurm ( p)
Po (hc/e.s2 )
E ? A−? V −2
Mnmrcy ( M )
(hc.e2 /s2 ) J (
E ? A2 V −2
Pkwmr (P)
S (hc.e2 /s4 )
E ? A2 V −4
Grmqumndy (ε )
Lz (?/s)
E 0 A0 V −?
Ttroin (ν )
Nk unit
E 0 A0 V 0
Ttrmss (τ )
Pos ( N /e2 )
E ? A−? V −2
Turgodm Vmnsikn (δ )
N /e
E ? A0 V −2
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?.< TI Ttonforfs
?4
nuebmrs, aorcm onf seoaa. Tdimntid nktotikn is nktotikn is fmnmf os o stonforfizmf woy tk rmprmsmnt ony nuebmr os tlm prkfudt kg
o rmoa nuebmr onf o pkwmr kg ?0. o
Ó ?0
b
In tlis gkre, o is doaamf tlm dkmgdimnt onf b is tlm mxpknmnt. Vlm dkmgdimnt is tlm voaum kg ony nuemridoa mxprmssikn in rmoa nuebmr. ?.<.? E]A E]AVIPAIM VIPAIMRT RT ONF PRMGI PRMGI\ \ In Plysids, euatipaimrs orm fmnmf in pkwmrs kg ?0 ?0 grke ?0−26 tk ?026 , prkdmmfinc in indrmemnts kg tlrmm krfmrs kg eocnitufm ( ?04 kr ?,000). Vlmsm euatipaimrs orm fmnktmf in Vobam Nk. ?.4 onf in
Vobam Nk. ?.6.
Prmx
Vobam ?.4: Biccmr euatipaimrs Tyebka
Euatipaimr
Yktto
Y
?026
Zmtto
Z
?02?
Mxo
M
?0?<
Pmto
P
?0?=
Vmro
V
?0?2
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Dloptmr ?. TDKPM KG PLYTIDT
?6
Prmx
• •
Vobam ?.6: Teoaamr euatipaimrs Tyebka
Euatipaimr
Ykdtk
y
?0−26
Zmptk
z
?0−2?
Ottk
o
?0−?<
Gmetk
g
?0−?=
Pidk
p
?0−?2
Nonk
n
?0−9
Eidrk
½
?0−7
eiaai
e
?0−4
Dmnti
d
?0−2
Fmdi
f
?0−?
Vlm undmrtointy in tlm emosurmemnt kg o plysidoa quontity is doaamf on mrrkr. oddurody kg Vlm oddurody Vlm kg o emosurmemnt is o emosurm kg lkw daksm tlm emosurmf voaum is tk tlm trum
voaum kg tlm quontity
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?.< TI Ttonforfs
?=
OBTKA]VM MRRKR Vlm eocnitufm kg tlm figgmrmndm bmtwmmn tlm infivifuoa emosurmemnt onf tlm trum voaum kg tlm
quontity is doaamf tlm obskautm mrrkr kg tlm emosurmemnt. ∇o 5 o
| −o | emon
RMAOVIWM MRRKR Vlm rmaotivm mrrkr is tlm rotik kg tlm emon obskautm mrrkr tk tlm emon voaum kg tlm quontity emosurmf. emon 5 ∇ooemon Rmaotivm Mrrkr 5
PMRDMNVOCM PMRDMNVO CM MRRKR
Slmn tlm rmaotivm mrrkr is mxprmssmf in pmr dmnt it is doaamf tlm pmrdmntocm mrrkr: emon Ó ?00 5 ∇ooemon Pmrdmntocm Mrrkr 5 ?.<.2 DOA DOAD]AO D]AOVIKN VIKN KG MRRKRT MRRKR KG T]E KR FIGGMRMNDM
• Mrrkrs orm oawoys offmf. • Slmn twk quontitims orm offmf kr subtrodtmf, tlm obskautm mrrkr in tlm noa rmsuat is tlm sues kg tlm obskautm mrrkrs in tlm infivifuoa quontitims: 5 O + B, tlmn eoxieue pkssibam mrrkr in D is ∇D 5 5 ∇ O + ∇ B. — Ig D D 5 5 O − B, tlmn eoxieue pkssibam mrrkr in D is ∇D 5 5 ∇ O + ∇ B. — Ig D D 5
MRRKR KG PRKF]DV KR FIWITIKN
•
Mvmn tlkucl quontitims orm euatipaimf kr fivifmf, tlm mrrkrs orm oawoys offmf.
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Dloptmr ?. TDKPM KG PLYTIDT
?7
• •
Zmrkz bmtwmmn ony sicnidont curms orm sicnidont. sicnidont. Gkr mxoepam: in 607, tlm nuebmr 6 onf
7 orm sicnidont sk is zmrk. Vlis is skemtiems doaamf "dopturmf zmrk". Vroiainc zmrks in o wlkam nuebmr: 200 is dknsifmrmf tk lovm knay KNM sicnidont curm ig tlis is bosmf kn tlm woy modl nuebmr is writtmn. Slmn wlkam nuebmr orm writtmn os obkvm, tlm zmrks, BY FMGINIVIKN, fif nkt rmquirm o emosurmemnt fmdisikn, tlus tlmy orm nkt
sicnidont.
•
Ig 200 rmoaay los twk kr tlrmm sicnidont curms tlmn it eust bm writtmn in sdimntid nktotikn. nktotikn. 2 . . Ig 200 los twk sicnidont curms, tlmn 2 0 ?0 is usmf. Ig it los tlrmm, tlmn 2 00 ?02 is
usmf. Ig it los gkur, tlmn 200.0 is sugdimnt.
Ó
Ó
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?.< TI Ttonforfs
?3
_]MT _] MTVIK VIKNT NT ON ONF F ON ONTSM TSMRT RT _:? Fmnm gkaakwinc; (i) Tuppamemntory ]nits (ii) Rofion (iii) Ttmrofion ?. Tuppamemntory ]nits: Vlm Cmnmroa Dkngmrmndm kn Smiclts onf Emosurms los nkt ymt Ons daossimf dmrtoin unit kg TI unfmr mitlmr bosm kr fmrivmf units. Vlmsm TI units orm doaamf fmrivmf suppamemntory units. Rofion onf orm suppamemntory suppamemntory units. units. Tmm Rofion onf Ttmrofion Ttmrofion orm
Vobam.
_uontity
Vobam ?.=: Tuppamemntory units ]nit Noem
Tyebka
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Dloptmr ?. TDKPM KG PLYTIDT
?<
Vobam ?.3: TPMDIOA KR PRODVIDOA ]NIVT: AMNCVL: Noem Tyebka & voaum Fmsdriptikn ? Gmrei
? g e 5 ?0−?= e
? \-roy unit
axu 5 ?0−?4 e
? Oncstrke
? Í5 ?0−?0 e
? Eidrkn
?½ e 5 ?0−7 e
? Ostrknkeidoa
? Ou 5 ?.69 Ó ?0??e
]nit
]smf in Otkeid Plysids
Ovmrocm fistondm bmtwmmn sun onf mortl, i.m., rofius kg mortl’s
krbit
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?.< TI Ttonforfs
?9
Noem
Vobam ?.9: TPMDIOA KR PRODVIDOA ]NIVT: ORMO: Tyebka & voaum Fmsdriptikn
? Born
?0−2< e2
? Odr drmm
6063 60 63e2 5 6<60 yorf 2 5 64=70
tk emosurm drkss smdtikn kg nudami g t 2
Vobam ?.?0: TPMDIOA KR PRODVIDOA ]NIVT: PRMTT]RM: Noem Tyebka & voaum Fmsdriptikn ? Ote Otekspl ksplmri mridd Prm Prmssur ssurmm ?.0?4 Ó ?0= N /e2 5 370ee kg Lc
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Dloptmr ?. TDKPM KG PLYTIDT
20
Pkints tk Nktm:
• Vlmrm orm tlrmm systmes kg units: TI (kafmr noem EHT systme), DCT systme(oask hnkw os Coussikn Tystme) onf GPT Tystme (Gkkt, Pkunf, Tmdknf
Tystme).
• •
Vwk kr ekrm plysidoa quontitims orm offmf kr subtrodtmf wlmn tlmir units onf
fiemnsikns orm soem. Vwk kr ekrm plysidoa quontitims orm euatipaimf kr fivifmf mvmn tlkucl tlmir unitss onf fiemn unit fiemnsikn siknss orm nkt soem. Ogtmr euatipaido euatipaidotikn tikn kr fivisikn, fivisikn, tlm
rmsuatont quontity eoy lovm figgmrmnt unit. • Vlm unit is oawoys writtmn in sincuaor gkre, m.c., emtmr nkt emtmrs. • Nk pundtuotikn eorhs orm usmf ogtmr unit, m.c., smd nkt smd.
• Ig o unit is noemf ogtmr o pmrskn, tlm unit is nkt writtmn witl dopitoa initioa amttmr, m.c., nmwtkn nkt Nmwtkn.
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2. TDOAORT ONF WMDVKRT
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Dloptmr 2. TDOAORT ONF WMDVKRT
22 Fmnitikn
O vmdtkr is o plysidoa quontity tlot los bktl o eocnitufm onf o firmdtikn.
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24 Croplidoa rmprmsmntotikn kg vmdtkrs Wmdtkrs orm frown os orrkws. On orrkw los bktl o eocnitufm (lkw aknc it is) onf o firmdtikn (tlm firmdtikn in wlidl it pkints). Vlm stortinc pkint kg o vmdtkr is hnkwn os tlm toia onf tlm mnf pkint is
hnkwn os tlm lmof.
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Dloptmr =. TVOV TVOVIDT IDT
72
eoss orm \ , Y onf Z : 5 \ 5
?
?
∟ ei xi , Y 5 E ∟ ei yi , E i
5 Z 5
i
? E
∟ ei zi i
5 ∟ iei , is tlm tktoa eoss kg tlm systme. Akdotm tlm pkint witl dkkrfinotms ( \ , Y , Z ). Vlis wlmrm E 5 eoss kg tlm civmn dkaamdtikn kg tlm portidams. Ig tlm pksitikn vmdtkr kg pkint is doaamf tlm dmntrm tlm dmntrm kg eoss kg tlm i tl portidam is r i , tlm dmntrm kg eoss is fmnmf tk lovm tlm pksitikn vmdtkr: DE 5 R
? E
∟ iei r i
=.?.? =.? .? M_ M_]IA ]IAIBR IBRI]E I]E O bkfy wiaa bm in mquiaibriue ig tlm gkrdms odtinc kn it eust bm dondma tlm mggmdt kg modl ktlmr. In
tlm ktlmr wkrf wm don oask writm tlot: O bkfy is soif tk bm in mquiaibriue dknfitikn ig tlmrm is nk unboaondm kr nmt gkrdm odtinc kn it.
Ttotid Mquiaibriue Slmn o bkfy is ot rmst onf oaa gkrdms oppaimf kn tlm bkfy dondma modl ktlmr tlmn it is soif tk bm in
stotid mquiaibriue. Fynoeid Mquiaibriue Slmn o bkfy is ekvinc witl unigkre vmakdity onf gkrdms oppaimf kn tlm bkfy dondma modl ktlmr
tlmn it is soif tk bm in tlm fynoeid mquiaibriue. =.?.2 DKNFIVIKNT DKNFIVIKNT KG M_]IA M_]IAIBRI] IBRI]E E GIRTV DKNFIVIKN KG M_]IAIBRI]E O bkfy wiaa bm in rst dknfitikn kg mquiaibriue ig sue kg oaa gkrdms oaknc \-oxis onf sue kg oaa gkrdms oaknc Y-oxis orm orm mquoa tk zmrk, tlmn tlm bkfy is soif tk bm in rst dknfitikn kg mquiaibriue. ∟ G x 5 0 onf ∟ G y 5 0
TMDKNF DKNFIVIKNT KG M_]IAIBRI]E O bkfy wiaa bm in smdknf dknfitikn kg mquiaibriue ig sue kg dakdhwism(Ekemnt) tkrqum eust bm mquoa tk tlm sue kg ontidakdhwism tkrqum(Ekemnt), tlmn tlm bkfy is soif tk bm in smdknf dknfitikn
kg mquiaibriue. 5 0 ∟ φ 5
=.2 =. 2
VKR KR_] _]M M Vlm turninc mggmdt kg o gkrdm witl rmspmdt tk skem oxis, is doaamf ekemnt kg gkrdm kr tkrqum fum tk tlm gkrdm. Vkrqum is emosurmf os tlm prkfudt kg tlm eocnitufm kg tlm gkrdm onf tlm pmrpmnfiduaor fistondm kg tlm ainm kg odtikn kg tlm gkrdm grke tlm oxis kg rktotikn. It is fmnktmf by Crmmh amttmr φ .
Eotlmeotidoaay, 5 φ 5 r G
Ó
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=.4 ONC]AOR EKEMNV]E
74
=.2.? =.2 .? DK] DK]PAM PAM KG GKR GKRDM DM Vwk gkrdms wlidl orm mquoa in eocnitufm but kppksitm in firmdtikn onf nkt odtinc oaknc tlm soem
ainm dknstitutm o dkupam. Dknsifmr twk mquoa onf kppksitm gkrdms odtinc kppksitmay oaknc poroaama ainms kn twk G onf G odtinc pkints O onf B. Amt r ? onf r 2 orm tlmir pksitikn vmdtkrs witl rmspmdt tk kricin.
−
Vkrqum fum tk 5 r ? Ó G G 5 Ó G Vkrqum fum tk - 5 −r ? G 5 5 φ ? + φ 2 5 Vktoa tkrqum 5 r ? Ó G − r 2 Ó G ( Vktoa tkrqum 5 ( r ? − r 2 ) Ó G 5 But r 5 r ? − r 2 is tlm fispaodmemnt vmdtkr grke B tk O, tlmrmgkrm: 5 Vktoa tkrqum 5 r Ó G Eocnitufm kg tkrqum is civmn by: φ 5 rG sin μ , wlmrm μ is tlm oncam bmtwmmn r onf onf G . is tlm pmrpmnfiduaor fistondm bmtwmmn tlm ainm kg odtikn kg tlm twk gkrdms. Amt it is fmnktmf r sin μ is by f . Vlus tlm eocnitufm kg tlm tkrqum kg dkupam wiaa bm: φ 5 5 G Gf f
Slmrm f is is doaamf tlm ekemnt ore kg tlm dkupam. Nkw tlm eocnitufm kg tlm dkupam 5 Eocnitufm kg ony kg tlm gkrdms gkreinc dkupam
kg dkupam.
Ó ekemnt ore
Mxoepams
Mxoepams kg dkupam orm ?. Gkrdms oppaimf tk tlm lonfam kg o sdrmw prmss, 2. Kpmninc kr daksinc o wotmr top. 4. Vurninc tlm dop kg o pmn. 6. Ttmmrinc o dor.
=.4 ONC ONC]AO ]AOR R EKE EKEMNV MNV]E ]E Vlmm em Vl emos osur urmm kg tl tlmm qu quon onti tity ty kg ek ekti tikn kn pk pkss ssms mssm smf f by o bk bkfy fy in rk rkto toti tikn knoa oa ek ekti tikn kn is do doaa aamf mf ON ONC] C]AO AOR R
EKEMNV]E. "Vlm oncuaor ekemntue ekemntue kg o bkfy is mquoa tk drk drkss ss prkfudt kg its ainmor ekemntue onf tlm vmdtkr fistondm grke tlm oxis kg rktotikn." Ig o bkfy kg eoss e is ekvinc in o dirdam kr rofius witl vmakdity r witl v, tlm ainmor ekemntue kg bkfy is P. Vlm oncuaor ekemntue kg tlm bkfy is civmn by:
5 5 A r P
Ó
Puttinc tlm voaum kg P, wm cmt 5 5 A r e v 5
Ó
⇒
5 e( A r v)
Ó
Eocnitufm kg oncuaor ekemntue is civmn by:
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Dloptmr =. TVOV TVOVIDT IDT
76 =.4.? AO AOS S KG DKNTMRW DKNTMRWOVIK OVIKN N KG ONC]AOR ONC]AOR EKEMNV]E EKEMNV]E
"Slmn tlm nmt mxtmrnoa tkrqum odtinc kn o systme obkut o civmn oxis is zmrk , tlm tktoa oncuaor ekemntue kg tlm systme obkut tlot oxis rmeoins dknstont."
Eotlmeotidoaay, 5 dknstont 5 0 tlmn A φ 5 Ig ∟ Prkkg Oddkrfinc tk tlm smdknf aow kg ektikn nmt gkrdm odtinc kn o bkfy is mquoa tk its rotm kg dloncm kg
ainmor ekemntue, i.m., f P 5 G 5 ft
Vohinc vmdtkr prkfudt kg r kn kn bktl sifm kg obkvm mxprmssikn:
Ó f ft P
5 r G 5 r
Ó
φ odtinc But r Ó is tlm tkrqum odtinc kn tlm bkfy: G is
Ó f ft P −−−− > (?)
5 φ 5 r
Oncuaor ekemntue is fmnmf os: 5 5 A r P
Ó
Figgmrmntiotinc bktl sifms witl rmspmdt tk t : f A f A ft
5
P) f (r
Ó
ft fP
ft
Ó ft + P Ó fr ft
5 r
f A ft
5 + φ + P
But f A ft
Ó fr ft
fr ft
5 v
φ + 5 + P v
Ó
Tindm P 5 e v f A ft
f A ft
5 + e φ +
5 + e( φ + v v)
Ó
Ó 0(∽ v Ó v 5 0) f A ft
5 φ
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=.4 ONC]AOR EKEMNV]E
7=
Intmcrotinc bktl sifms:
0 5 f A
A 5 dknstont
Vlus tlm oncuaor ekemntue kg o portidam is dknsmrvmf ig onf knay ig tlm nmt mxtmrnoa tkrqum odtinc
kn o portidam is zmrk. Pkints tk Nktm:
?. Vlm dmntrm kg eoss eoss nmmf nkt tk aim in tlm bkfy. bkfy. 2. Intmrnoa gkrdms fk nkt dloncm tlm dmntrm kg eoss. 4. Slmn o drodhmr mxpakfms in oir, oir, tlm dmntrm kg eoss kg grocemnts trovma trovma oaknc porobkaid potl. 6. Vlm sue kg ekemnt kg eossms obkut its dmntrm dmntrm kg eoss is oawoys oawoys zmrk. =. Vlm pksitikn kg dmntrm kg eoss fkms nkt fmpmnf upkn tlm dk-krfinotm dk-krfinotm systme dlksmn. rktotms obkut its oxis in onti dakdh dakdhwism wism firmdtikn, firmdtikn, 7. Pksitivm tkrqum: Ig o bkfy rktotms tlmn tlm tkrqum is tohmn pksitivm . 3. Nmcotivm tkrqum: Ig tkrqum: Ig tlm bkfy rktotms in tlm dakdhwism firmdtikn, tlmn tlm tkrqum is tohmn os nmcotivm . oncuaor vmakdity vmakdity kg oaa pkints kg o rici ricif f bkfy orm soem. But in tlm dosm <. Vlm oncuaor kg nkn-ricif bkfy, crmotmr tlm fistondm kg tlm pkint grke tlm oxis kg rktotikn,
crmotmr wiaa bm its oncuaor fispaodmemnt. fmpmnfs kn tlm pkint obkut wlidl rktotikn is dknsifmrmf. 9. Vlm oncuaor vmakdity fmpmnfs ?0. Vlm sue kg ekemnt kg eossms obkut its dmntrm dmntrm kg eoss is oawoys oawoys zmrk. ??. Ekemnt kg inmrtio fmpmnfs fmpmnfs upkn tlm pksitikn kg tlm oxis kg rktotikn. rktotikn. ?2. Ig o nuebmr kg tkrqums odtmf kn o systme onf tlm systme is in rktotiknoa mquiaibriue, tlmn dakdhwism tkrqum 5 ontidakdhwism tkrqum. ?4. Ig o bkfy kr systme is in Dkepamtm mquiaibriue, tlmn nmt gkrdm onf onf nmt tkrqum kn tlm bkfy kr systme orm zmrk. ?6. In tlm dosm kg dkupam, tlm sue kg ekemnt kg oaa gkrdms obkut ony pkint is tlm soem.
(Daoss Rmvimw Nktms gkr \I Plysids)
By Fr. Roe Dlonf, Ckvmrnemnt Ciras Fmcrmm Dkaamcm, Jlufk Gkr vifmk amdturms pamosm visit dkaamcm gb pocm: www.godmbkkh.dke/ccfdjlufk
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7. CROWIVOVIKN
7.?
HMPAMR’T HMPA MR’T AO AOST ST KG PAONM PAONMVOR VORY Y EKVI EKVIKN KN Hmpamr’s rst aow (aow kg maaiptidoa krbit):-
O paonmt ekvms rkunf tlm sun in on maaiptidoa krbit witl sun situotmf ot knm kg its gkdi. Hmpamr’s smdknf aow (aow kg ormoa vmakditims):-
O paonmt ekvms rkunf tlm sun in sudl o woy tlot its ormoa vmakdity is dknstont. Hmpamr’s tlirf aow (aow kg tiem pmrikf):O paonmt ekvms rkunf tlm sun in sudl o woy tlot tlm squorm kg its pmrikf is prkpkrtiknoa tk tlm dubm
kg smei eojkr oxis kg its maaiptidoa krbit. V 2 ∕ R4
Lmrm R is tlm rofius kg krbit. 2
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Dloptmr 7. CRO CROWIVOVIKN WIVOVIKN
7<
Dknsifmr twk bkfims kg eossms e ? onf e 2 witl tlmir dmntrms smporotmf by o fistondm r . Vlm crovito-
tiknoa gkrdm bmtwmmn tlme is
∕ e? e2 G ∕ e ? e2 ∕ ?/r 2 5 G ∕ ∕ G ∕ r 2 e? e2 5 C 2 G 5 r
⇒
Slmrm C 5 7.73
stotmf os: 5 G 5
Ó ?0−?? Ne2 /hc2 is univmrsoa crovitotiknoa dknstont. In vmdtkr gkre, it don bm
−C e r ?2e2 r rˇ ?2?2 ?2
Einus slkws tlot gkrdm is ottrodtivm. ]nit vmdtkr ˇr r ?2 ?2 slkws tlot gkrdm odts oaknc tlm ainm jkininc tlm e? onf e 2 .
Crovitotiknoa dknstont (C) It is mquoa tk tlm gkrdm kg ottrodtikn ottrodtikn odtinc bmtwmmn twk bkfims modl kg unit eoss, wlksm dmntrms orm paodmf unit fistondm oport. Woaum kg C C is dknstont tlrkuclkut tlm univmrsm. It is o sdoaor quontity. − Vlm fiemnsiknoa gkreuao is C 5 X E ? A4 V −2 U. Crovitotiknoa gkrdm is dmntroa gkrdm onf dknsmrvotivm in noturm. Vlm voaum kg C is fmtmreinmf by
Dovmnfisl emtlkf in ?39<. Crovitotiknoa gkrdm is oawoys ottrodtivm in noturm. 7.2.? 7.2 .? Cro Crovit vity y It is tlm gkrdm kg ottrodtikn mxmrtmf by mortl tkworfs its dmntrm kn o bkfy ayinc kn kr nmor tlm surgodm kg mortl. Crovity is tlm emosurm kg wmiclt kg tlm bkfy. Vlm wmiclt kg o bkfy kg eoss e is mquoa tk eoss oddmamrotikn fum tk crovity. Vlm unit kg wmiclt kg o bkfy wiaa bm tlm soem os tlksm kg gkrdm.
Ó
Oddmamrotikn fum tk crovity (c) It is fmnmf os tlm oddmamrotikn smt up in o bkfy wliam goaainc grmmay unfmr tlm mggmdt kg crovity oaknm. It is vmdtkr quontity. Vlm voaum kg dloncms witl lmiclt, fmptl onf rktotikn kg mortl. Vlm
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7.2 NMSVKN’T AOS KG CROWIVOVIKN
79
7.2.2 7.2 .2 Vlm vo voriot riotikn ikn kg c Woriotikn kg c witl oatitufm (lmiclt) Amt P bm o pkin pkintt kn tlm surg surgodm odm kg tlm Mortl onf _ bm o pkint ot on oatitufm l . Amt tlm eoss kg tlm Mortl bm E onf onf rofius kg tlm Mortl bm R . Dknsifmr tlm Mortl os o splmridoa slopmf bkfy.
Vlm oddmamrotikn fum tk crovity ot P kn tlm surgodm is: c 5
CE R2
−−−− > (?)
Amt tlm bkfy bm paodmf ot _ ot o lmiclt l grke tlm surgodm kg tlm Mortl Mortl.. Vlm oddmamrotikn oddmamrotikn fum tk crovity ot _ is cl 5
CE
( R + l)2
−−−− > (2)
fivifinc (2) by (?): cl c
5
R2
( R + l)2
⇒ c 5 c( R +R l )2 5 c(? + l R )−2
5
l
By siepaigyinc onf mxponfinc usinc binkeioa tlmkrme (suppksinc l 11 R): cl 5 c(?
− 2 Rl )
Vlm voaum kg oddmamrotikn fum tk crovity fmdrmosms witl indrmosm in lmiclt obkvm tlm surgodm kg tlm
Mortl.
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Dloptmr 7. CRO CROWIVOVIKN WIVOVIKN
30 Ϗ bm tlm fmnsity, tlmn, tlm eoss kg tlm Mortl is: Ig Ϗ
6 4 π R Ϗ 4 6 c 5 π CR CRϏ 4
5 E 5
Vlm oddmamrotikn fum tk crovity ot _ ot o fmptl f grke grke tlm surgodm kg tlm Mortl is: cf 5
CE f f
( R
− f )2
wlmrm E f f is tlm eoss kg tlm innmr splmrm kg tlm Mortl kg rofius ( R − f ). E f f 5 c 5
6 π ( R − f )4 Ϗ 4
6 π C( R − f )Ϗ −−−− > (2) 4
fivifinc (2) by (?), cf c
5
cf 5 c(?
R
− f
R
− Rf )
Vlm voaum kg oddmamrotikn fum tk crovity fmdrmosms witl indrmosm kg fmptl.
•
Fum tk rktotikn kg mortl, tlm voaum voaum kg c fmdrmosms c fmdrmosms os tlm spmmf kg rktotikn kg mortl indrmosms.
Vlm voaum kg oddmamrotikn fum tk crovity ot o aotitufm χ is cχ 5 c Rω 2 dks2 χ
−
Ot mquotkr, χ 5 0k onf ot tlm pkam, χ 5 90k ?. Ot tlm mquot mquotkr, kr, c M 5 c − Rω 2 2. Ot tlm pkam, c pkam 5 c
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7.4 Crovitotiknoa maf
3?
7.4.? Crovi Crovitotik totiknoa noa maf intmn intmnsity sity Crovitotiknoa maf intmnsity kr strmnctl ot o pkint is fmnmf os tlm gkrdm mxpmrimndmf by o unit eoss paodmf ot tlot pkint. It is fmnktmf by M . It is o vmdtkr quontity. Its unit is N hc˜?. Dknsifmr o bkfy kg eoss E paodmf ot o pkint _ onf onktlmr bkfy kg eoss e paodmf ot P ot o fistondm r grke grke _. Vlm eoss E fmvmakps o maf M ot P onf tlis maf mxmrts o gkrdm G 5 eM . Vlm crovitotiknoa gkrdm kg ottrodtikn bmtwmmn tlm eossms e onf E is is Ee 5 C 2 G 5 R
Vlm crovitotiknoa maf intmnsity ot P is M 5 5 G : e 5 M 5
CE r 2
Crovitotiknoa maf intmnsity is tlm emosurm kg crovitotiknoa maf. Crovitotiknoa pktmntioa figgmrmndm Crovitotiknoa pktmntioa figgmrmndm bmtwmmn twk pkints is fmnmf os tlm oekunt kg wkrh fknm in ekvinc unit eoss grke knm pkint tk onktlmr pkint ocoinst tlm crovitotiknoa gkrdm kg ottrodtikn. Dknsifmr twk pkints O onf B smporotmf by o fistondm ∇r in in tlm crovitotiknoa maf. Vlm wkrh 5 S O→ B . Crovitotiknoa pktmntioa figgmrmndm is: fknm in ekvinc unit eoss grke O tk B is ∇] 5
5 M ∇r ∇] 5
−
Lmrm nmcotivm sicn infidotms tlot wkrh is fknm ocoinst tlm crovitotiknoa maf. 7.4.2 Crovi Crovitotik totiknoa noa Pkt Pktmntio mntioaa Mnmrcy Mnmrcy Vlm crovitotiknoa pktmntioa mnmrcy kg o pkint eoss e eoss e paodmf paodmf in tlm crovitotiknoa maf kg o pkint eoss don bm gkunf kut by tlm wkrh fknm in ekvinc tlot pkint eoss e grke innity tk tlm pkint ot E don
wlidl crovitotiknoa pktmntioa mnmrcy is tk bm fmtmreinmf i.m., Fmnitikn Crovitotiknoa pktmntioa ot o pkint is fmnmf os tlm oekunt kg wkrh fknm in ekvinc unit eoss grke
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Dloptmr 7. CRO CROWIVOVIKN WIVOVIKN
32 Krbitoa vmakdity
Ortidioa sotmaaitms orm eofm tk rmvkavm in on krbit ot o lmiclt kg gmw lunfrmf hiakemtrms. Ot tlis oatitufm, tlm gridtikn fum tk oir is nmcaicibam. Vlm sotmaaitm is dorrimf by o rkdhmt tk tlm fmsirmf lmiclt onf rmamosmf lkrizkntoaay witl o licl vmakdity, sk tlot it rmeoins ekvinc in o nmoray dirduaor krbit. Vlm lkrizkntoa vmakdity tlot los tk bm ieportmf tk o sotmaaitm ot tlm fmtmreinmf lmiclt sk tlot it eohms o dirduaor krbit orkunf tlm paonmt is doaamf krbitoa vmakdity. Amt us ossuem tlot o sotmaaitm kg eoss e ekvms orkunf tlm Mortl in o dirduaor krbit kg rofius r witl witl 5 R + l, unigkre spmmf v k . Amt tlm sotmaaitm bm ot o lmiclt l grke tlm surgodm kg tlm Mortl. Lmndm, r 5 wlmrm R is tlm rofius kg tlm Mortl. Vlm dmntripmtoa gkrdm rmquirmf tk hmmp tlm sotmaaitm in dirduaor
krbit is: 5 G 5
ev2k r
5
ev2k R + l
Vlm crovitotiknoa gkrdm bmtwmmn tlm Mortl onf tlm sotmaaitm is: Ee eE 5 C 2 5 C G 5 ( R + l)2 r
Gkr tlm stobam krbitoa ektikn, ev2k R + l
5 C
eE
( R + l)2
Tindm tlm oddmamrotikn fum tk crovity kn Mortl’s surgodm is c 5
cR2
, tlmrmgkrm:
CE R2
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7.4 Crovitotiknoa maf
34
Ig tlm sotmaaitm krbits vmry daksm tk tlm Mortl, tlmn l 11 R:
5 2π V 5
R c
Msdopm Wmakdity It is fmnmf os einieue spmmf kg prkjmdtikn witl wlidl ig o bkfy is prkjmdtmf upworfs, tlmn it
fkms nkt rmturn bodh tk mortl. 2CE 5 Eotlmeotidoaay, vmsd 5
R
2cR
Slmrm E is is tlm eoss onf R is tlm rofius kg tlm paonmt. Cmkstotiknory sotmaaitm O sotmaaitm wlidl rmvkavms orkunf tlm mortl witl tlm soem oncuaor spmmf in tlm soem firmdtikn os is fknm by tlm mortl orkunf its oxis is doaamf cmkstotiknory kr cmksyndlrknkus sotmaaitm. Vlm lmiclt kg cmkstotiknory sotmaaitm is 47000he onf its krbitoa vmakdity is 4.?hes−? .
7.4.6 7.4 .6 Rmo Rmoaa Sm Smicl icltt
Vlm rmoa wmiclt kg tlm kbjmdt is tlm crovitotiknoa puaa kg tlm mortl kn tlm kbjmdt. 7.4.= Oppor Oppormnt mnt Sm Smiclt iclt Vlm rmofinc kg wmiclt kn tlm sdoam kg o sprin sprinc c boaondm is doaam doaamf f oppormnt wmiclt. wmiclt. Cmnmro Cmnmroaay aay tlm wmiclt kg tlm kbjmdt is emosurmf by sprinc boaondm. Vlm gkrdm mxmrtmf by tlm kbjmdt kn tlm sdoam is mquoa tk wmiclt kg tlm kbjmdt. Vlis is nkt oawoys trum, sk wm doaa tlm rmofinc kg tlm sdoam os oppormnt
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Dloptmr 7. CRO CROWIVOVIKN WIVOVIKN
36 5 2V π wlmrm V is tlm pmrikf kg tlm rmvkautikn kg spodmslip: But ω 5 od 5 R(
π 2 )2 5 6 R 2 V V
2π
Os tlm grmqumndy g 5 V ? , tlmrmgkrm 2 2
od 5 6 Rπ g 5
⇒
2
g 5
od
6 Rπ 2
5
⇒
? g 5 2π
od
R
Vlm grmqumndy g is indrmosmf tk sudl on mxtmnt tlot od mquoas tk c. Vlmrmgkrm, ? g 5 2π
c
R
Vlis is tlm mxprmssikn kg grmqumndy gkr prkfudinc tlm ortidioa crovity in sotmaaitm mquoa tk tlot kg
mortl. Pkints tk Nktm:
• Ig tlm mortl stkps spinninc, tlmn tlm voaum kg c c wiaa indrmosm saicltay ( c). • Vlm mortl los o buacm ot tlm mquotkr bmdousm kg tlm spinninc ektikn. • Msdopm vmakdity is infmpmnfmnt kg tlm eoss kg prkjmdtiam, but it fmpmnfs kn tlm eoss kg paonmt. • Msdopm vmakdity fkms nkt fmpmnf kn oncam kg prkjmdtikn. • Ig o portidam portidam kg eoss e is frkppmf grke tlm mnf kg tunnma oaknc fioemtmr kg
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3. SKRH, PKSMR ONF MNMRCY
3.?
SKRH Vlm tmres wkrh onf mnmrcy orm quitm goeiaior tk us onf wm usm tlme in vorikus dkntmxts. In mvmryfoy aigm, tlm tmre wkrh is usmf tk rmgmr tk ony gkre kg odtivity tlot rmquirms tlm mxmrtikn kg emntoa kr
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Dloptmr 3. SKRH, PKSMR ONF MNMRCY
37
Dknvmrsikns bmtwmmn Figgmrmnt Tystmes kg ]nits
? Jkuam 5 ? N .e 5 ?0= fynm 5 ?03 mrc 5 ? Jkuam /s 5 ?03 mrc/s ?wott 5 ?hwl 5 ?04 wott .lr 5 5 4.7 Ó ?07 Jkuam ? LP 5 367wott 5 ?07 wott ? ES 5 ?doa 5 6.2 Jkuam ?mW 5 ?.7 Ó ?0−?9 Jkuam (m 5 eocnitufm kg dlorcm kn tlm mamdtrkn in dkuakebs ) TICN DKNWMNVIKN (i) Pksitivm wkrh:
5 0 kr μ 1 1 90k μ 5 Ig gkrdm onf fispaodmemnt orm in tlm soem firmdtikn, wkrh wiaa bm pksitivm kr ig μ (ii) Zmrk wkrh:
Ig gkrdm onf fispaodmemnt orm pmrpmnfiduaor tk modl ktlmr, wkrh wiaa bm zmrk. i.m., dks90 k 5 0. (iii) Nmcotivm wkrh:
In gkrdm onf fispaodmemnt orm in tlm kppksitm firmdtikn, wkrh wiaa bm nmcotivm: S 5 G s dks?<00 5 G Gss
Ó (−?) 5 −G s.
NKVM:- O pksitivm wkrh don bm fmnmf os tlm wkrh fknm by o gkrdm onf o nmcotivm wkrh os NKVM:- O tlm wkrh fknm ocoinst o gkrdm
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3.4 MNMRCY
3.4 3. 4
33
MNMR MN MRC CY Fmnitikn
Mnmrcy don bm fmnmf os tlm dopodity tk fk wkrh.
Mnmrcy don eonigmst itsmag in eony gkres aihm emdlonidoa mnmrcy, tlmreoa mnmrcy, mamdtrid mnmrcy,
dlmeidoa mnmrcy, aiclt mnmrcy, nudamor mnmrcy, mtd. Emdlonidoa Mnmrcy Fmnitikn
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Dloptmr 3. SKRH, PKSMR ONF MNMRCY
3< Mxprmssikn gkr Hinmtid mnmrcy
Dknsifmr o bkfy kg eoss e storts ekvinc grke rmst. Ogtmr o tiem intmrvoa t its vmakdity bmdkems v. Ig initioa vmakdity kg tlm bkfy is v i 5 0, noa vmakdity v g 5 v onf tlm fispaodmemnt kg bkfy is f .
Vlmn usinc mquotikn kg ektikn: 5 W g 2 − W i2 2oT 5 Puttinc tlm obkvm emntiknmf voaums
2of 5 5 v2 − 0 5⇒
o 5
v2
2f Nkw gkrdm is civmn by 5 eo G 5
Puttinc tlm voaum kg oddmamrotikn 5 e(v2 /2f ) G 5
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3.4 MNMRCY
39
Slmrm tlm amgt lonf sifm kg tlm obkvm mquotikn civms tlm wkrh fknm kn tlm bkfy onf tlm riclt lonf sifm civms tlm dloncm in hinmtid mnmrcy kg tlm bkfy. Vlis is tlm eotlmeotidoa gkre kg wkrh
mnmrcy prindipam. It don oask bm writtmn os: wkrh-fknm 5 ∇HM 3.4.6 Dknsmrvotivm gkrdms onf nkn-dknsmrvotiv nkn-dknsmrvotivm m gkrdms Dknsmrvotivm gkrdms Ig tlm wkrh fknm by o gkrdm in ekvinc o bkfy bmtwmmn twk pksitikns is infmpmnfmnt kg tlm potl
gkaakwmf by tlm bkfy, tlmn sudl o gkrdm is doaamf os o dknsmrvotivm gkrdm. Mxoepams : gkrdm fum tk crovity, sprinc gkrdm onf maostid gkrdm. Vlm wkrh fknm by tlm dknsmrvotivm
gkrdms fmpmnfs knay upkn tlm initioa onf noa pksitikn kg tlm bkfy. Vlm wkrh fknm by o dknsmrvotivm gkrdm orkunf o daksmf potl is zmrk.
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Dloptmr 3. SKRH, PKSMR ONF MNMRCY
<0 3.4.7 3.4 .7 Pk Pkwm wmrr
It is fmnmf os tlm rotm ot wlidl wkrh is fknm: 5 Pkwmr 5
Skrh fknm tiem
5
⇒
P 5
S t
Pkwmr is sdoaor quontity onf its TI unit is wott (S) onf fiemnsiknoa gkreuao is EA 2 V ˜4 . Pkwmr is soif tk bm knm wott, wlmn knm jkuam kg wkrh is soif tk bm fknm in knm smdknf. Ig ∇S is is tlm wkrh fknm furinc on intmrvoa kg tiem t tlmn: tlmn: P 5
∇S ∇t
5 (G dks μ )∇s, wlmrm μ is tlm oncam bmtwmmn tlm firmdtikn kg tlm gkrdm onf fispaodmemnt. But S 5 is dkepknmnt kg tlm gkrdm in tlm firmdtikn kg tlm seoaa fispaodmemnt ∇s. Vlmrmgkrm: G dks μ is
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3.4 MNMRCY
Pkints tk Nktm:
• Skrh fmpmnfs upkn tlm groem kg rmgmrmndm. • Skrh is usmf tk dknvmrt mnmrcy grke knm gkre tk onktlmr gkre. • Skrh fknm by dknsmrvoti dknsmrvotivm vm gkrdm fkmsn’t fmpmnf upkn potl gkaakwmf by tlm kbjmdt. • Skrh fknm by dknstont gkrdm fkmsn’t fmpmnf upkn potl. • Vwk bkfims kg eoss e? (lmovy) onf eoss e2 (aiclt) orm ekvinc witl soem
hinmtid mnmrcy. Ig tlmy orm stkppmf by tlm soem rmtorfinc gkrdm, tlmn ?. Vlm bkfims dkvmr dkvmr tlm soem fistondm bmgkrm dkeinc tk rmst. rmst. gkr e2 onf it los amss ekemntue 2. Vlm tiem tohmn tk dkem tk rmst is amssmr gkr e i.m., t 5 5 P/G 4. Vlm tiem tohmn tk dkem tk rmst is ekrm gkr e? os it los crmotmr ekemntue.
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<6
Dloptmr <. SO SOWM WM EKVIKN ONF TK]NF
wlmrm ω is o dknstont hnkwn os oncuaor grmqumndy kg tlm siepam loreknid ektikn. Vlm nmcotivm sicn infidotms tlot tlm oddmamrotikn is kppksitm tk tlm firmdtikn kg fispaodmemnt. Ig e is tlm eoss kg tlm portidam, rmstkrinc gkrdm tlot tmnfs tk brinc bodh tlm portidam tk tlm emon pksitikn is civmn by
Lkkhm’s Aow
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<.2 Ainmor siepam lor eknid ksdiaaotkr
<=
Viem Pmrikf (V) Vlm tiem rmquirmf tk dkepamtm dkepamtm vibrotikn vibrotikn is hnk hnkwn wn os tiem pmrikf. pmrikf. Vlm TI unit kg tiem pmrikf is
smdknf (s).
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<7
Dloptmr <. SO SOWM WM EKVIKN ONF TK]NF
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Dloptmr ?0. CMKEMVRIDOA KPVIDT
?22 Wirtuoa Ieocm
On ieocm tlot is kn tlm soem sifm kg tlm amns os tlm kbjmdt onf donnkt bm prkjmdtmf kn o sdrmmn is
doaamf o virtuoa ieocm. ?0.?.4 ?0. ?.4 Vli Vlin n amn amnss Gkre Gkreuao uao Amt OB Amt rmprmsmnts on kbjmdt paodmf ot riclt oncams tk tlm prindipoa oxis ot o fistondm crmotmr tlon tlm OB rmprmsmnts gkdoa amnctl g kg kg tlm dknvmx amns. Vlm ieocm O B is gkremf bmyknf 2G 2 onf is rmoa onf invmrtmf.
Gicurm ?0.6: Vlin amns gkreuao KO 5 kbjmdt fistondm 5 p KO 5 ieocm fistondm 5 q KG 2 5 gkdoa amnctl 5 g ∇KOB onf ∇KO B orm sieiaor, tlmrmgkrm:
O B OB
5
KO KO
−−−− > (?)
Tieiaoray ∇KDG 2 onf ∇G 2 O B orm sieiaor: O B KD
5
G 2 O KG 2
5 OB, tlmrmgkrm, obkvm mquotikn don bm writtmn os: But wm hnkw tlot KD 5 O B KD
5
O B OB
5
G 2 O O B KG 2 OB
5
Grke mquotikn (?) onf (2), wm cmt: KO KO
5
G 2 O KG 2
G 2 O KG 2
−−−− > (2)
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?0.? AMNT
?24
Fivifinc wlkam mquotikn by pq g , wm cmt: ? p
? g
5 5
?
? − g q
? p
+
? q
Vlis is hnkw os tlin amns mquotikn. It don bm oask usmf gkr dkndovm amns by oppayinc opprkpriotm
sicn dknv dknvmntikn. mntikn. ?0.?.6 ?0.?. 6
Gkreotikn Gkreoti kn kg kg Ieocm by Dknv Dknvmx mx Amns Vlm noturm kg ieocms gkremf by o dknvmx amns fmpmnfs upkn tlm fistondm kg tlm kbjmdt grke tlm Kptidoa Dmntmr kg tlm amns. Amt us nkw smm lkw tlm ieocm is gkremf by o dknvmx amns gkr vorikus
pksitikns kg tlm kbjmdt. ?. Slmn tlm Kbjmdt is Paodmf bmtwmmn G ? onf K :
Gicurm ?0.=: Vlm kbjmdt is paodmf bmtwmmn G ? onf K Lmrm wm dknsifmr twk roys stortinc grke tlm tkp kg tlm kbjmdt paodmf ot G ? onf kptidoa dmntmr. Vlm roy poroaama tk tlm prindipoa oxis ogtmr rmgrodtikn possms tlrkucl tlm gkdus ( G 2 ). Vlm roy possinc tlrkucl tlm kptidoa dmntmr ckms tlrkucl tlm amns unfmviotmf. Vlmsm rmgrodtmf roys oppmor tk emmt knay wlmn prkfudmf prkfudmf bodhworfs. bodhworfs. Vlus, wlmn on kbjm kbjmdt dt is paodm paodmf f bmtwmmn G ? onf K kg o dknvmx amns, o virtuoa, virtuoa, mrmdt onf eocnimf ieocm kg tlm kbjmdt is gkremf kn tlm soem sifm kg tlm amns os tlm
kbjmdt.Vlot is:• Gkremf kn tlm soem sifm kg tlm amns • Wirtuoa • Mrmdtmf • Eocnimf 2. Slmn tlm Kbjmdt Kbjmdt is Paodm Paodmf f ot G ?
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Dloptmr ?0. CMKEMVRIDOA KPVIDT
?26
Gicurm ?0.7: Vlm kbjmdt is paodmf ot G ?
• Gkremf ot innity • Rmoa • Invmrtmf • Eocnimf 4. Slmn tlm Kbjmdt is Paodmf bmtwmmn G ? onf G 2
Gicurm ?0.3: Vlm kbjmdt is paodmf bmtwmmn G ? onf G 2 Amt us dknsifmr twk roys dkeinc grke tlm kbjmdt. Vlm roy wlidl is poroaama tk tlm prindipoa oxis ogtmr rmgrodtikn possms tlrkucl tlm amns onf possms tlrkucl G 2 kn tlm ktlmr sifm kg tlm amns. Vlm roy possinc tlrkucl tlm kptid dmntmr dkems kut kg tlm amns witlkut ony fmviotikn. Vlm twk rmgrodtmf roys intmrsmdt modl ktlmr ot o pkint bmyknf 2G 2 . Tk, wlmn on kbjmdt is paodmf bmtwmmn bmtwmmn G ? onf 2G ? kg o dknvmx amns tlm ieocm is gkremf bmyknf 2 G 2 . Vlot is: Gkremf bmyknf 2 G 2
• • Rmoa • Invmrtmf
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?0.? AMNT
?2=
Gicurm ?0.<: Vlm kbjmdt is paodmf ot 2 G ? tlrkucl tlm gkdus. Vlmsm twk rmgrodtmf tlrkucl rmgrodtmf roys emmt ot 2G 2G 2 . Vlus, wlmn on kbjmdt is paodmf ot 2G ? kg o dknvmx amns, invmrtmf onf rmoa ieocm kg tlm soem sizm os tlm kbjmdt is gkremf ot 2G 2 kn tlm ktlmr
sifm kg tlm amns. • Gkremf ot 2G 2 • Rmoa • Invmrtmf • Toem sizm os tlm kbjmdt
Gicurm ?0.9: Vlm kbjmdt is paodmf bmyknf 2G ? =. Slmn tlm Kbjmdt is Paodmf bmyknf 2G ? Vlm roy poroaama tk tlm prindipoa oxis ogtmr rmgrodtikn possms tlrkucl G 2 onf tlm roy wlidl possms tlrkucl tlm kptidoa dmntmr dkems kut witlkut ony fmviotikn. Vlm rmgrodtmf roys intmrsmdt ot o pkint bmtwmmn G 2 onf 2G 2 . Vlm ieocm is invmrtmf, rmoa onf fieinislmf. Vlot is: Gkremf bmtwmmn G 2 onf 2G 2
• • Rmoa • Invmrtmf
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Dloptmr ?0. CMKEMVRIDOA KPVIDT
?27
wlidl possms tlrkucl G ? bmdkems poroaama tk tlm prindipoa oxis ogtmr rmgrodtikn onf tlm roy wlidl
possms tlrkucl tlm kptidoa dmntmr fkms nkt suggmr ony fmviotikn. Vlot is: • Gkremf ot G 2. • Rmoa • Invmrtmf • Liclay fieinislmf
Gicurm ?0.?0: Vlm kbjmdt is paodmf ot innity Vlm tobam ?0.= civms ot o caondm tlm pksitikn, sizm onf noturm kg tlm ieocm gkremf by o dknvmx amns
dkrrmspknfinc tk tlm figgmrmnt pksitikns kg tlm kbjmdt onf oask its oppaidotikn. ?0.?.= ?0.?. =
Gkreotikn Gkreoti kn kg kg Ieocm by Dkndov Dkndovm m Amns Amns Bmdousm tlm roys oawoys fivmrcmf by o dkndovm amns, tlm memrcinc roys fk nkt odtuoaay intmrsmdt. But tlmy fmme tk intmrsmdt kn tlm indifmndm sifm by trodinc bodhworfs tlm memrcinc roys. Lmndm dkndovm amns ieocms orm oawoys virtuoa ieocms. Amt us nkw frow roy fiocroes tk slkw tlm pksitikn
kg tlm ieocms wlmn tlm kbjmdt is paodmf ot figgmrmnt pksitikns. ?. Slmn tlm Kbjmdt Kbjmdt is ot Innity: Innity:
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?0.? AMNT
?23
• Gkremf ot G ?. • Mrmdtmf • Wirtuoa • Fieinislmf 2. Slmn tlm Kbjmdt Kbjmdt is Paodm Paodmf f bmtwmmn bmtwmmn K onf G:
Gicurm ?0.?2: Kbjmdt is Paodmf bmtwmmn K onf G In tlis dosm tlm noturm kg ieocm is civmn bmakw: • Gkremf bmtwmmn K onf G ? • Mrmdtmf • Wirtuoa • Fieinislmf 4. Slmn tlm Kbjmdt is Paodmf ot ony Pksitikn bmtwmmn K onf Innity:
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Dloptmr ?0. CMKEMVRIDOA KPVIDT
?2<
• Wirtuoa • Fieinislmf Dkndovm Amns Mxoepams Vlm “fkkr mym’ is tlm ekst prodtidoa prodtidoa usm kg o dkndovm amns. O fkkr mym is o seoaa dkndovm dkndovm amns ttmf in tlm mntrondm fkkr kg o lkusm. Tindm o dkndovm amns prkfudms tlm ieocm kg o rmoa kbjmdt eudl daksmr, yku don ifmntigy tlm pmrskn wlk is hnkdhinc tlm fkkr mvmn ig lm/slm stonfs gor grke tlm fkkr.
Fkkr myms orm mxtmnsivmay usmf os sogmty emosurm. ?0.?.7 ?0. ?.7 Eoc Eocni nidot dotikn ikn Amt us dknsifmr on kbjmdt KK paodmf kn tlm prindipoa oxis witl its lmiclt pmrpmnfiduaor tk tlm prindipoa oxis. Vlm roy KP roy KP possinc possinc tlrkucl tlm kptid dmntrm wiaa ck unfmviotmf. Vlm roy K roy K O poroaama O poroaama tk tlm prindipoa oxis eust poss tlrkucl tlm gkdus G 2 . Vlm ieocm is gkremf wlmrm K PI onf OG 2 I intmrsmdt. Frow o pmrpmnfiduaor grke I tk tlm prindipoa oxis. Vlis pmrpmnfiduaor I II I is tlm ieocm kg KK .
Gicurm ?0.?6: Eocnidotikn Vlm ainmor kr tronsvmrsm eocnidotikn is fmnmf os tlm rotik kg tlm sizm kg tlm ieocm tk tlot kg tlm kbjmdt.
Eocnidotikn e 5
Tizm kg tlm ieocm II l 2 5 5 Tizm kg tlm kbjmdt KK l?
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?0.? AMNT
?29
Vlm eocnidotikn is nmcotivm gkr rmoa ieocm onf pksitivm gkr virtuoa ieocm. In tlm dosm kg o dkndovm
amns, it is oawoys pksitivm. ]sinc amns gkreuao tlm mquotikn gkr eocnidotikn don oask bm kbtoinmf os e 5
l2 l?
5
q p
5
g
−q 5 g
g
+ p g +
Vlis mquotikn is voaif gkr bktl dknvmx onf dkndovm amnsms onf gkr rmoa onf virtuoa ieocms. ROE’T EINF EOP
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Dloptmr ?0. CMKEMVRIDOA KPVIDT
?40
Vobam ?0.?: Ticn dknvmntikns gkr tlin amns Tyebka In grknt
_uontity Kbjmdt akdotikn
p
+
Ieocm akdotikn Amns rofii
q
− −
R? , R 2
In bodh
−
+ +
Gicurm ?0.?=: Dkebinotikn kg amnsms prkfudms on ieocm ot I ot I ? . Vlis ieocm ieocm I amns B.. Vlm noa ieocm ieocm is I ? odts os tlm kbjmdt gkr tlm smdknf amns B prkfudmf ot I . Tindm tlm amnsms orm tlin, o dkeekn kptidoa dmntrm P is dlksmn. 5 q, noa ieocm fistondm Amt PK 5 p , kbjmd kbjmdtt fistondm gkr tlm rst amns (O), PI 5 fistondm onf PI ? 5 q? ,
ieocm fistondm gkr tlm rst amns (O) onf oask kbjmdt fistondm gkr smdknf amns (B). Gkr tlm ieocm I ? prkfudmf by tlm rst amns O, ? q?
+
? p
5
? g ?
−−−− > (?)
Gkr tlm noa ieocm I , prkfudmf by tlm smdknf amns B,
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?0.2 Obmrrotikns
?4?
Lmrm G is tlm gkdoa amnctl kg tlm mquivoamnt amns gkr tlm dkebinotikn. Vlm fmrivotikn don bm mxtmnfmf gkr smvmroa tlin amnsms kg gkdoa amnctls g ? , g 2 , g 4 ... in dkntodt.
Vlm mggmdtivm gkdoa amnctl kg tlm dkebinotikn is civmn by ? G
5
? g ?
+
? g 2
+
? g 4
+ ... 5
⇒
? G
n
?
i5?
g i
5∟
−−−− > (7)
In tmres kg pkwmr, mquotikn (7) don bm writtmn os P 5 P? + P2 + P4 + ....
Vlm pkwmr kg o dkebinotikn kg amnsms in dkntodt is tlm oacmbroid sue kg tlm pkwmrs kg infivifuoa
amnsms. Vlm dkebinotikn kg amnsms is cmnmroaay usmf in tlm fmsicn kg kbjmdtivms kg eidrksdkpms, doemros,
tmamsdkpms onf ktlmr kptidoa instruemnts.
?0.2 ?0 .2 Ob Obmr mrro roti tikn knss Amnsms usuoaay fk nkt civm o pmrgmdt ieocm. Tkem dousms orm: obmrrotikn is dousmf by tlm godt tlot n 5 n(θ ). Vlis don bm portioaay dkrrmdtmf ?. Dlrkeotid obmrrotikn is witl o amns wlidl is dkepksmf kg ekrm amnsms witl figgmrmnt gundtikns n gundtikns n i (θ ). ]sinc N ]sinc N amnsms amnsms eohms it pkssibam tk kbtoin tlm soem g gkr N wovmamnctls. wovmamnctls. 2. Tplmridoa obmrrotikn is dousmf by smdknf-krfmr mggmdts wlidl orm usuoaay icnkrmf8 o splmridoa surgodm fkms nkt eohm o pmrgmdt amns. In dkeinc roys gor grke tlm kptidoa oxis wiaa ekrm
bmnt. 4. Dkeo is Dkeo is dousmf by tlm godt tlot tlm prindipoa paonms kg o amns orm knay flot nmor tlm prindipoa oxis. Gurtlmr owoy kg tlm kptidoa oxis tlmy orm durvmf. Vlis durvoturm don bm bktl pksitivm kr
nmcotivm. kptidoa oxis tlm ieocm is on maaipsm 6. Osticeotise: Osticeotise: grke modl pkint kg on kbjmdt nkt kn tlm kptidoa bmdousm tlm tlidhnmss kg tlm amns is nkt tlm soem mvmrywlmrm. =. Gimaf durvoturm don bm dkrrmdtmf by tlm lueon mym. ms obmrrotikns obmrrotikns nmor tlm mfcms kg tlm ieocm. Vlis don bm dkrrm dkrrmdtmf dtmf witl o 7. Fistkrtikn civ Fistkrtikn civms dkebinotikn kg pksitivm onf nmcotivm amnsms. ROE’T EINF EOP
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Dloptmr ?0. CMKEMVRIDOA KPVIDT
?42
?0.4 ?0. 4
Kptido Kpt idoaa Ins Instrue truemnt mntss
?0.4.? ?0.4 .?
Tiepam Tiepa m Eidrk Eidrksdkpm sdkpm Dknvmx amns is doaamf siepam eidrksdkpm os it is kgtmn usmf os o eocnimr wlmn on kbjmdt is brkuclt witlin tlm gkdoa amnctl kg dknvmx amns. Vlm eocnimf onf virtuoa ieocm is gkremf ot amost fistondm kg fistindt visikn f . O amns is paodmf in grknt kg tlm mym in sudl o woy tlot o virtuoa ieocm ieocm kg tlm kbjmdt is gkremf ot o fistondm f fistondm f grke grke tlm mym. Vlm sizm kg tlm ieocm is nkw eudl aorcmr tlon witlkut tlm amns. Ig β β onf ο orm rmspmdtivm oncams subtmnfmf by tlm kbjmdt wlmn smmn tlrkucl amns (siepam eidrksdkpm) onf wlmn vimwmf firmdtay, tlmn oncuaor eocnidotikn E is is fmnmf os:
5 E 5
β ο
−−−− > (?)
Slmn oncams orm seoaa, tlmn tlmy orm nmoray mquoa tk tlmir toncmnts: 5 ton ο 5 5 ο 5
Tizm kg kbjmdt K 5 −−−− > (2) fistondm kg kbjmdt f
Tieiaoray: β 5 5 ton β 5 5
Tizm kg ieocm I 5 −−−−− > (4) fistondm kg ieocm q
Tindm tlm ieocm is ot tlm amost fistondm kg fistindt visikn, sk q 5 f . Vlmrmg Vlmrmgkrm, krm, tlm mquotikn mquotikn (4)
bmdkems: 5 β 5
I q
5
I f
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?0.4 Kptidoa Instruemnts ?0.4.2 ?0.4 .2
?44
Dkepkunf Dkepk unf Eidrk Eidrksdkpm sdkpm Dkepkunf Dkep kunf eidrksdkpm eidrksdkpm is on kptidoa instruemnt instruemnt wlidl is usmf tk kbtoi kbtoin n licl eocni eocnidoti dotikn. kn. It
dknsists kg twk dknvmrcinc dknvmrcinc amnsms: Kbjmdtivm: Vlm amns in grknt kg kbjmdt is doaamf kbjmdtivm. Its gkdoa amnctl g ? 5 g k is tohmn tk bm vmry seoaa .Vlm kbjmdtivm gkres o rmoa, invmrtmf, onf eocnimf ieocm kg tlm kbjmdt paodmf just bmyknf
tlm gkdus kg kbjmdtivm. Mym pimdm Vlm amns tkworfs tlm kbsmrvmr’s mym is doaamf pimdm. Gkdoa amnctl kg mym pimdm is crmotmr
tlon tlm gkdoa amnctl kg kbjmdtivm. Mym pimdm wkrhs os o eocnigyinc caoss. Vlm kbjmdtivm is sk ofjustmf tlot tlm kbjmdt is vmry daksmf tk its gkdus. Vlm kbjmdtivm gkres o rmoa, invmrtmf onf eocnimf ieocm kg tlm objmdt bmyknf 2 g k kn tlm riclt lonf sifm. Vlm mym pimdm is sk ofjustmf tlot it gkres o virtuoa ieocm ot tlm amost fistondm kg fistindt visikn f . Vlm noa ieocm is
liclay eocnimf.
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Dloptmr ?0. CMKEMVRIDOA KPVIDT
?46
]sinc tlin amns gkreuao gkr mym pimdm : ? g
5
? ? p
+
⇒ g ?2 5 q?2 + p?2
5
q
Lmrm g 2 5 g m , q2 5 −f onf onf p 5 p 2 : ? g m
5
?
−f
+
? p2
− f ? + p?2
5
Euatipayinc bktl sifms by f , f g m f
5 ?+
f
5 ? +
−−−− > (4)
p2
−
p2 f
g m
Dkeporinc mquotikn (2) onf (4): E m 5 ? +
f g m
−−−−−−−−(6)
Vktoa eocnidotikn is mquoa tk tlm prkfudt kg tlm eocnidotikn prkfudmf by tlm kbjmdtivm onf tlm
mym pimdm: 5 E k E 5 q?
5 ( E 5
p
Ó E
)(? +
m
f
g
)
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Gicurm ?0.?7: Ostrknkeidoa Vmamsdkpm Eocnigyinc Pkw Pkwmr mr
Vlm eocnigyinc pkwmr (E) kg ostrknkeidoa tmamsdkpm is civmn by: 5 E 5
Oncam suspmnfmf by noa ieocm @ mym Oncam suspmnfmf by noa ieocm @ mym 5 Oncam suspmnfmf by kbjmdt @ mym Oncam suspmnfmf by kbjmdt @ kbjmdt:
It is bmdousm tlm kbjmdt is ot innitm fistondm onf lmndm tlm oncam subtmnfmf by tlm kbjmdt ot mym β eoy bm tohmn os tlm oncam subtmnfmf by tlm kbjmdt ot kbjmdtivm: E 5 ο . Tindm ο onf β orm seoaa
oncams, tlmrmgkrm wm don tohm: ο 5 ton ο onf onf β 5 ton β . Lmndm 5 E 5
ton β ton ο
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?47
Dloptmr ?0. CMKEMVRIDOA KPVIDT
Dkaaieotkr Vlm dkaaieotkr is on orroncmemnt tk prkfudm o poroaama bmoe kg aiclt. It dknsists kg o aknc dyainfridoa tubm witl o dknvmx amns ot tlm innmr mnf onf o vmrtidoa sait ot tlm kutmr mnf kg tlm tubm. Vlm fistondm bmtwmmn tlm sait onf tlm amns don bm ofjustmf sudl tlot tlm sait is ot tlm gkdus kg tlm amns. Vlm sait is hmpt godinc godinc tlm skurdm kg aiclt aiclt.. Vlm wiftl kg tlm sait don bm ofjustmf. Vlm dkaai dkaaieotkr eotkr is ricifay xmf xmf
tk tlm bosm kg tlm instruemnt. Vurn tobam Vlm turn tobam is usmf gkr ekuntinc tlm prise, crotinc mtd. It dknsists kg twk dirduaor emtoa fisds prkvifmf prk vifmf witl tlrmm amvmaainc amvmaainc sdrmws. It don bm rktotmf obkut o vmrt vmrtidoa idoa oxis possinc tlrkucl its dmntrm onf its pksitikn don bm rmof witl vmrnimrs W ? onf W 2 . Vlm prise tobam don bm roismf kr akwmrmf
onf don bm xmf ot ony fmsirmf lmiclt. Vmamsdkpm
Vlm tmamsdkpm is on ostrknkeidoa typm. It dknsists kg on mympimdm prkvifmf witl drkss wirms ot knm mnf kg tlm tubm onf on kbjmdtivm amns ot its ktlmr mnf dk-oxioaay. Vlm fistondm bmtwmmn tlm kbjmdtivm amns onf tlm mympimdm don bm ofjustmf sk tlot tlm tmamsdkpm gkres o damor ieocm ot tlm drkss wirms,
wlmn o poroaama bmoe grke tlm dkaaieotkr is indifmnt kn it. Vlm tmamsdkpm is ottodlmf tk on ore wlidl is dopobam kg rktotikn obkut tlm soem vmrtidoa oxis os
tlm prise tobam. O dirduaor sdoam crofuotmf in loag fmcrmm is ottodlmf tk it.
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?0.= Lueon mym
Vobam ?0.2: Ticn dknvmntikns gkr tlin amns Pksitivm wlmn ... Nmcotivm wlmn...
_uontity Kbjmdt akdotikn ( p) Ieocm akdo Ieocm akdotikn tikn (q) Ieocmm Ieoc (l )
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(Rmoa ieocm)
tuoa ieocm)
Ieocm is upriclt
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kg amns Dknvmrcinc amns
kg amns Fivmrcinc amns
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Dloptmr ?0. CMKEMVRIDOA KPVIDT
Vobam ?0.4: Gkreotikn kg Ieocm by Dknvmx Amns
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Dloptmr ?0. CMKEMVRIDOA KPVIDT
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Noem
Tyebka
Nuebmr π Nuebmr m Mamemntory dlorcm Crovitotiknoa dknstont
π
m m C
Woaum
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]nit
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?0.7 ]smgua Gkr euao
?6?
Vricknkemtrid Ifmntitims + dks2 μ 5 5 ? sin2 μ + sin( O + B) 5 sin O dks B + dks O sin B dks( O + B) 5 dks O dks B − sin O sin B + ? 5 smd2 μ ton2 μ + ? + dkt2 μ 5 5 dsd2 μ dks(−μ ) 5 dks μ sin(−μ ) 5 − sin μ dks( O − B) 5 dks O dks B + sin O sin B ? ? sin O dks B 5 sin( O − B) + sin( O + B) 2 2 ? ?
(?0.?0) (?0.??) (?0.?2) (?0.?4) (?0.?6) (?0.?=) (?0.?7) (?0.?3)
(?0.?<)