NCERT Textbook - कणों के निकाय तथा घूर्णी गति | NCERT Textbooks in Hindi (Class 6 to Class 12) - UPSC PDF Download

 Page 1


vè;k; 7
d.kksa osQ fudk; rFkk ?kw.khZ xfr
7.1 Hkwfedk
fiNys vè;k;ksa esa geus eq[; :i ls vkn'kZ fcUnq d.k (,d d.k ftls
nzO;eku ;qDr fcUnq osQ :i esa O;Dr fd;k tk,A O;kogkfjd n`f"V ls bldk
dksbZ vkdkj ugha gksrk) dh xfr dk vè;;u fd;k FkkA fiQj] ;g ekurs gq,
fd ifjfer vkdkj osQ fi.Mksa dh xfr dks fcUnq d.k dh xfr osQ inksa esa O;Dr
fd;k tk ldrk gS] geus ml vè;;u osQ ifj.kkeksa dks ifjfer vkdkj osQ
fi.Mksa ij Hkh ykxw dj fn;k FkkA
nSfud thou esa ftrus fi.M gekjs laioZQ esa vkrs gSa os lHkh ifjfer
vkdkj osQ gksrs gSaA ,d foLr`r fi.M (ifjfer vkdkj osQ fi.M) dh xfr dks
iwjs rkSj ij le>us osQ fy, vkerkSj ij mldk fcUnqor~ vkn'kZ vi;kZIr jgrk
gSA bl vè;k; esa ge bl izfrca/ osQ ijs tkus dh ps"Vk djssaxs vkSj foLr`r]
ij ifjfer fi.Mksa dh xfr dks le>us dk iz;kl djsaxsA ,d foLr`r fi.M
izFker;k d.kksa dk ,d fudk; gSA vr% ge viuk foospu ,d fudk; dh
xfr ls gh 'kq: djuk pkgsaxsA ;gk¡ d.kksa osQ fudk; dk nzO;eku osQUnz ,d eq[;
vo/kj.kk gksxhA ge d.kksa osQ fudk; osQ nzO;eku osQUnz dh xfr dk o.kZu djsaxs
vkSj fiQj] ifjfer vkdkj osQ fi.Mksa dh xfr dks le>us esa bl vo/kj.kk dh
mi;ksfxrk crk;saxsA
cM+s fi.Mksa ls tqM+h cgqr lh leL;k,a mudks n`<+ fi.M ekudj gy dh
tk ldrh gSaA vkn'kZ n`<+ fi.M ,d ,slk fi.M gS ftldh ,d lqfuf'pr vkSj
vifjorZuh; vko`Qfr gksrh gSA bl izdkj osQ Bksl osQ fofHkUu d.kksa osQ chp dh
nwfj;ka ifjo£rr ugha gksrhA n`<+ fi.M dh bl ifjHkk"kk ls ;g Li"V gS fd dksbZ
Hkh okLrfod fi.M iwjh rjg n`<+ ugha gksrk] D;ksafd lHkh O;kogkfjd fi.M
cyksa osQ izHkko ls foo`Qr gks tkrs gSaA ijUrq ,slh cgqr lh fLFkfr;k¡ gksrh gSa ftuesa
foo`Qfr;k¡ ux.; gksrh gSaA vr% dbZ izdkj dh fLFkfr;ksa esa ;Fkk ifg;s] yV~Vw]
LVhy osQ 'kgrhj vkSj ;gk¡ rd fd v.kq] xzg tSls fi.Mksa dh xfr dk vè;;u
djrs le;] ge è;ku u nsaxs fd muesa foo`Qfr vkrh gS] os eqM+rs gSa ;k dEiu
djrs gSaA ge mUgsa n`<+ fi.M eku dj mudh xfr dk vè;;u djasxsA
7.1 Hkwfedk
7.2 nzO;eku osQUnz
7.3 nzO;eku osQUnz dh xfr
7.4 d.kksa osQ fudk; dk js[kh; laosx
7.5 nks lfn'kksa dk lfn'k xq.kuiQy
7.6 dks.kh; osx vkSj bldk js[kh;
osx ls laca/
7.7 cy vk?kw.kZ ,oa dks.kh; laosx
7.8 n`<+ ¯iMksa dk larqyu
7.9 tM+Ro vk?kw.kZ
7.10 yEcor~ ,oa lekukUrj v{kksa osQ izes;
7.11 vpy v{k osQ ifjr% 'kq¼ ?kw.khZ
xfrdh
7.12 vpy v{k osQ ifjr% ?kw.khZ xfrdh
7.13 vpy v{k osQ ifjr% ?kw.khZ xfr dk
dks.kh; laosx
7.14 yksVfud xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
Page 2


vè;k; 7
d.kksa osQ fudk; rFkk ?kw.khZ xfr
7.1 Hkwfedk
fiNys vè;k;ksa esa geus eq[; :i ls vkn'kZ fcUnq d.k (,d d.k ftls
nzO;eku ;qDr fcUnq osQ :i esa O;Dr fd;k tk,A O;kogkfjd n`f"V ls bldk
dksbZ vkdkj ugha gksrk) dh xfr dk vè;;u fd;k FkkA fiQj] ;g ekurs gq,
fd ifjfer vkdkj osQ fi.Mksa dh xfr dks fcUnq d.k dh xfr osQ inksa esa O;Dr
fd;k tk ldrk gS] geus ml vè;;u osQ ifj.kkeksa dks ifjfer vkdkj osQ
fi.Mksa ij Hkh ykxw dj fn;k FkkA
nSfud thou esa ftrus fi.M gekjs laioZQ esa vkrs gSa os lHkh ifjfer
vkdkj osQ gksrs gSaA ,d foLr`r fi.M (ifjfer vkdkj osQ fi.M) dh xfr dks
iwjs rkSj ij le>us osQ fy, vkerkSj ij mldk fcUnqor~ vkn'kZ vi;kZIr jgrk
gSA bl vè;k; esa ge bl izfrca/ osQ ijs tkus dh ps"Vk djssaxs vkSj foLr`r]
ij ifjfer fi.Mksa dh xfr dks le>us dk iz;kl djsaxsA ,d foLr`r fi.M
izFker;k d.kksa dk ,d fudk; gSA vr% ge viuk foospu ,d fudk; dh
xfr ls gh 'kq: djuk pkgsaxsA ;gk¡ d.kksa osQ fudk; dk nzO;eku osQUnz ,d eq[;
vo/kj.kk gksxhA ge d.kksa osQ fudk; osQ nzO;eku osQUnz dh xfr dk o.kZu djsaxs
vkSj fiQj] ifjfer vkdkj osQ fi.Mksa dh xfr dks le>us esa bl vo/kj.kk dh
mi;ksfxrk crk;saxsA
cM+s fi.Mksa ls tqM+h cgqr lh leL;k,a mudks n`<+ fi.M ekudj gy dh
tk ldrh gSaA vkn'kZ n`<+ fi.M ,d ,slk fi.M gS ftldh ,d lqfuf'pr vkSj
vifjorZuh; vko`Qfr gksrh gSA bl izdkj osQ Bksl osQ fofHkUu d.kksa osQ chp dh
nwfj;ka ifjo£rr ugha gksrhA n`<+ fi.M dh bl ifjHkk"kk ls ;g Li"V gS fd dksbZ
Hkh okLrfod fi.M iwjh rjg n`<+ ugha gksrk] D;ksafd lHkh O;kogkfjd fi.M
cyksa osQ izHkko ls foo`Qr gks tkrs gSaA ijUrq ,slh cgqr lh fLFkfr;k¡ gksrh gSa ftuesa
foo`Qfr;k¡ ux.; gksrh gSaA vr% dbZ izdkj dh fLFkfr;ksa esa ;Fkk ifg;s] yV~Vw]
LVhy osQ 'kgrhj vkSj ;gk¡ rd fd v.kq] xzg tSls fi.Mksa dh xfr dk vè;;u
djrs le;] ge è;ku u nsaxs fd muesa foo`Qfr vkrh gS] os eqM+rs gSa ;k dEiu
djrs gSaA ge mUgsa n`<+ fi.M eku dj mudh xfr dk vè;;u djasxsA
7.1 Hkwfedk
7.2 nzO;eku osQUnz
7.3 nzO;eku osQUnz dh xfr
7.4 d.kksa osQ fudk; dk js[kh; laosx
7.5 nks lfn'kksa dk lfn'k xq.kuiQy
7.6 dks.kh; osx vkSj bldk js[kh;
osx ls laca/
7.7 cy vk?kw.kZ ,oa dks.kh; laosx
7.8 n`<+ ¯iMksa dk larqyu
7.9 tM+Ro vk?kw.kZ
7.10 yEcor~ ,oa lekukUrj v{kksa osQ izes;
7.11 vpy v{k osQ ifjr% 'kq¼ ?kw.khZ
xfrdh
7.12 vpy v{k osQ ifjr% ?kw.khZ xfrdh
7.13 vpy v{k osQ ifjr% ?kw.khZ xfr dk
dks.kh; laosx
7.14 yksVfud xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
7.1.1 ,d n`<+ fi.M esa fdl izdkj dh xfr;k¡ gks
ldrh gSa\
vkb;s] n`<+ fi.Mksa dh xfr osQ oqQN mnkgj.kksa ls bl iz'u dk mÙkj
<wa<+us dh dksf'k'k djsaA izFke ,d vk;rkdkj CykWd ij fopkj djsa
tks ,d ur ry ij lh/k (fcuk b/j&m/j gVs) uhps dh vksj
fiQly jgk gSA CykWd ,d n`<+ fi.M gSA ur ry ij uhps dh vksj
bldh xfr ,slh gS fd blosQ lHkh d.k lkFk&lkFk py jgs gSa]
vFkkZr~ fdlh {k.k lHkh d.k leku osx ls pyrs gSa (fp=k 7.1)A
;gk¡ ;g n`<+ fiaM 'kq¼ LFkkukarj.k xfr esa gSA
'kq¼ LFkkukarj.k xfr esa fdlh {k.k fo'ks"k ij fi.M dk
izR;sd d.k leku osx ls pyrk gSA
fp=k 7.2 ur ry ij uhps dh vksj yq<+drk fl¯yMj (csyu)A   ;g
'kq¼ LFkkukarj.k xfr ugha gSA fdlh {k.k ij fcUnq P
1
, P
2
,
P
3
 ,oa  P
4
 osQ vyx&vyx osx gSa (tSlk fd rhj n'kkZrs
gSa)A okLro esa lEioZQ fcUnq P
3
 dk osx fdlh Hkh {k.k 'kwU;
gS ;fn csyu fcuk fiQlys gq, yq<+drk gSA
vkb;s] vc mlh ur ry ij uhps dh vksj yq<+drs gq,
,d /krq ;k ydM+h osQ csyu dh xfr ij fopkj djrs gSa
(fp=k 7.2)A ;g n`<+ fi.M (csyu) ur ry osQ 'kh"kZ ls
mldh ryh rd LFkkukarfjr gksrk gS] vr% blesa LFkkukarj.k
xfr gSA ysfdu fp=k 7.2 ;g Hkh n'kkZrk gS fd blosQ lHkh d.k
{k.k fo'ks"k ij ,d gh osx ls ugha py jgs gSaA vr% fi.M 'kq¼
LFkkukarj.k xfr esa ugha gSA vr% bldh xfr LFkkukarjh; gksus osQ
lkFk&lkFk ^oqQN vkSj vyx* Hkh gSA
;g ^oqQN vkSj vyx* Hkh D;k gS\ ;g le>us osQ fy,]
vkb;s] ge ,d ,slk n`<+ ¯iM ysa ftldks bl izdkj O;o#¼
dj fn;k x;k gS fd ;g LFkkukarj.k xfr u dj losQA fdlh
n`<+ fi.M dh LFkkukarj.k xfr dks fu#¼ djus dh loZ lkekU;
fof/ ;g gS fd mls ,d ljy js[kk osQ vuqfn'k fLFkj dj
fn;k tk,A rc bl n`<+ fi.M dh ,dek=k laHkkfor xfr ?kw.khZ
xfr gksxhA og ljy js[kk ftlosQ vuqfn'k bl n`<+ fi.M dks
fLFkj cuk;k x;k gS bldh ?kw.kZu&v{k dgykrh gSA ;fn vki
vius pkjksa vksj ns[ksa rks vkidks Nr dk ia[kk] oqQEgkj dk pkd
(fp=k 7.3(a) ,oa (b))] fo'kky pØh&>wyk (tkW;UV Oghy)]
esjh&xks&jkm.M tSls vusd ,sls mnkgj.k fey tk;saxs tgk¡
fdlh v{k osQ ifjr% ?kw.kZu gks jgk gksA
(a)
(b)
fp=k 7.3 ,d fLFkj v{k osQ ifjr% ?kw.kZu
(a) Nr dk ia[kk
(b) oqQEgkj dk pkd
Fig 7.1ur&ry ij ,d CykWd dh v/kseq[kh LFkkukarj.k
(fiQlyu) xfr (CykWd dk izR;sd ¯cnq ;Fkk P
1
, P
2
....
fdlh Hkh {k.k leku xfr esa gSa)
d.kksa osQ fudk; rFkk ?kw.khZ xfr 145
Page 3


vè;k; 7
d.kksa osQ fudk; rFkk ?kw.khZ xfr
7.1 Hkwfedk
fiNys vè;k;ksa esa geus eq[; :i ls vkn'kZ fcUnq d.k (,d d.k ftls
nzO;eku ;qDr fcUnq osQ :i esa O;Dr fd;k tk,A O;kogkfjd n`f"V ls bldk
dksbZ vkdkj ugha gksrk) dh xfr dk vè;;u fd;k FkkA fiQj] ;g ekurs gq,
fd ifjfer vkdkj osQ fi.Mksa dh xfr dks fcUnq d.k dh xfr osQ inksa esa O;Dr
fd;k tk ldrk gS] geus ml vè;;u osQ ifj.kkeksa dks ifjfer vkdkj osQ
fi.Mksa ij Hkh ykxw dj fn;k FkkA
nSfud thou esa ftrus fi.M gekjs laioZQ esa vkrs gSa os lHkh ifjfer
vkdkj osQ gksrs gSaA ,d foLr`r fi.M (ifjfer vkdkj osQ fi.M) dh xfr dks
iwjs rkSj ij le>us osQ fy, vkerkSj ij mldk fcUnqor~ vkn'kZ vi;kZIr jgrk
gSA bl vè;k; esa ge bl izfrca/ osQ ijs tkus dh ps"Vk djssaxs vkSj foLr`r]
ij ifjfer fi.Mksa dh xfr dks le>us dk iz;kl djsaxsA ,d foLr`r fi.M
izFker;k d.kksa dk ,d fudk; gSA vr% ge viuk foospu ,d fudk; dh
xfr ls gh 'kq: djuk pkgsaxsA ;gk¡ d.kksa osQ fudk; dk nzO;eku osQUnz ,d eq[;
vo/kj.kk gksxhA ge d.kksa osQ fudk; osQ nzO;eku osQUnz dh xfr dk o.kZu djsaxs
vkSj fiQj] ifjfer vkdkj osQ fi.Mksa dh xfr dks le>us esa bl vo/kj.kk dh
mi;ksfxrk crk;saxsA
cM+s fi.Mksa ls tqM+h cgqr lh leL;k,a mudks n`<+ fi.M ekudj gy dh
tk ldrh gSaA vkn'kZ n`<+ fi.M ,d ,slk fi.M gS ftldh ,d lqfuf'pr vkSj
vifjorZuh; vko`Qfr gksrh gSA bl izdkj osQ Bksl osQ fofHkUu d.kksa osQ chp dh
nwfj;ka ifjo£rr ugha gksrhA n`<+ fi.M dh bl ifjHkk"kk ls ;g Li"V gS fd dksbZ
Hkh okLrfod fi.M iwjh rjg n`<+ ugha gksrk] D;ksafd lHkh O;kogkfjd fi.M
cyksa osQ izHkko ls foo`Qr gks tkrs gSaA ijUrq ,slh cgqr lh fLFkfr;k¡ gksrh gSa ftuesa
foo`Qfr;k¡ ux.; gksrh gSaA vr% dbZ izdkj dh fLFkfr;ksa esa ;Fkk ifg;s] yV~Vw]
LVhy osQ 'kgrhj vkSj ;gk¡ rd fd v.kq] xzg tSls fi.Mksa dh xfr dk vè;;u
djrs le;] ge è;ku u nsaxs fd muesa foo`Qfr vkrh gS] os eqM+rs gSa ;k dEiu
djrs gSaA ge mUgsa n`<+ fi.M eku dj mudh xfr dk vè;;u djasxsA
7.1 Hkwfedk
7.2 nzO;eku osQUnz
7.3 nzO;eku osQUnz dh xfr
7.4 d.kksa osQ fudk; dk js[kh; laosx
7.5 nks lfn'kksa dk lfn'k xq.kuiQy
7.6 dks.kh; osx vkSj bldk js[kh;
osx ls laca/
7.7 cy vk?kw.kZ ,oa dks.kh; laosx
7.8 n`<+ ¯iMksa dk larqyu
7.9 tM+Ro vk?kw.kZ
7.10 yEcor~ ,oa lekukUrj v{kksa osQ izes;
7.11 vpy v{k osQ ifjr% 'kq¼ ?kw.khZ
xfrdh
7.12 vpy v{k osQ ifjr% ?kw.khZ xfrdh
7.13 vpy v{k osQ ifjr% ?kw.khZ xfr dk
dks.kh; laosx
7.14 yksVfud xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
7.1.1 ,d n`<+ fi.M esa fdl izdkj dh xfr;k¡ gks
ldrh gSa\
vkb;s] n`<+ fi.Mksa dh xfr osQ oqQN mnkgj.kksa ls bl iz'u dk mÙkj
<wa<+us dh dksf'k'k djsaA izFke ,d vk;rkdkj CykWd ij fopkj djsa
tks ,d ur ry ij lh/k (fcuk b/j&m/j gVs) uhps dh vksj
fiQly jgk gSA CykWd ,d n`<+ fi.M gSA ur ry ij uhps dh vksj
bldh xfr ,slh gS fd blosQ lHkh d.k lkFk&lkFk py jgs gSa]
vFkkZr~ fdlh {k.k lHkh d.k leku osx ls pyrs gSa (fp=k 7.1)A
;gk¡ ;g n`<+ fiaM 'kq¼ LFkkukarj.k xfr esa gSA
'kq¼ LFkkukarj.k xfr esa fdlh {k.k fo'ks"k ij fi.M dk
izR;sd d.k leku osx ls pyrk gSA
fp=k 7.2 ur ry ij uhps dh vksj yq<+drk fl¯yMj (csyu)A   ;g
'kq¼ LFkkukarj.k xfr ugha gSA fdlh {k.k ij fcUnq P
1
, P
2
,
P
3
 ,oa  P
4
 osQ vyx&vyx osx gSa (tSlk fd rhj n'kkZrs
gSa)A okLro esa lEioZQ fcUnq P
3
 dk osx fdlh Hkh {k.k 'kwU;
gS ;fn csyu fcuk fiQlys gq, yq<+drk gSA
vkb;s] vc mlh ur ry ij uhps dh vksj yq<+drs gq,
,d /krq ;k ydM+h osQ csyu dh xfr ij fopkj djrs gSa
(fp=k 7.2)A ;g n`<+ fi.M (csyu) ur ry osQ 'kh"kZ ls
mldh ryh rd LFkkukarfjr gksrk gS] vr% blesa LFkkukarj.k
xfr gSA ysfdu fp=k 7.2 ;g Hkh n'kkZrk gS fd blosQ lHkh d.k
{k.k fo'ks"k ij ,d gh osx ls ugha py jgs gSaA vr% fi.M 'kq¼
LFkkukarj.k xfr esa ugha gSA vr% bldh xfr LFkkukarjh; gksus osQ
lkFk&lkFk ^oqQN vkSj vyx* Hkh gSA
;g ^oqQN vkSj vyx* Hkh D;k gS\ ;g le>us osQ fy,]
vkb;s] ge ,d ,slk n`<+ ¯iM ysa ftldks bl izdkj O;o#¼
dj fn;k x;k gS fd ;g LFkkukarj.k xfr u dj losQA fdlh
n`<+ fi.M dh LFkkukarj.k xfr dks fu#¼ djus dh loZ lkekU;
fof/ ;g gS fd mls ,d ljy js[kk osQ vuqfn'k fLFkj dj
fn;k tk,A rc bl n`<+ fi.M dh ,dek=k laHkkfor xfr ?kw.khZ
xfr gksxhA og ljy js[kk ftlosQ vuqfn'k bl n`<+ fi.M dks
fLFkj cuk;k x;k gS bldh ?kw.kZu&v{k dgykrh gSA ;fn vki
vius pkjksa vksj ns[ksa rks vkidks Nr dk ia[kk] oqQEgkj dk pkd
(fp=k 7.3(a) ,oa (b))] fo'kky pØh&>wyk (tkW;UV Oghy)]
esjh&xks&jkm.M tSls vusd ,sls mnkgj.k fey tk;saxs tgk¡
fdlh v{k osQ ifjr% ?kw.kZu gks jgk gksA
(a)
(b)
fp=k 7.3 ,d fLFkj v{k osQ ifjr% ?kw.kZu
(a) Nr dk ia[kk
(b) oqQEgkj dk pkd
Fig 7.1ur&ry ij ,d CykWd dh v/kseq[kh LFkkukarj.k
(fiQlyu) xfr (CykWd dk izR;sd ¯cnq ;Fkk P
1
, P
2
....
fdlh Hkh {k.k leku xfr esa gSa)
d.kksa osQ fudk; rFkk ?kw.khZ xfr 145 146 HkkSfrdh
laca/ esa geus ;g eku fy;k gS fd ;g ,d LFkku ls nwljs
LFkku ij LFkkukarfjr ugha gksrk vkSj blfy, blesa LFkkukarj.k
xfr ugha gSA)  vius vuqHko osQ vk/kj ij ge ;g tkurs gSa fd
bl izdkj ?kwers yV~Vw dh v{k] Hkwfe ij blosQ lEioZQ&fcUnq
ls xqtjrs vfHkyEc osQ ifjr% ,d 'kaoqQ cukrh gS tSlk fd
fp=k 7.5(a) esa n'kkZ;k x;k gSA  (ÅèokZ/j osQ ifjr% yV~Vw dh
v{k dk bl izdkj ?kweuk iqjLlj.k dgykrk gS )A  è;ku nsa fd
yV~Vw dk og fcUnq tgk¡ ;g /jkry dks Nwrk gS] fLFkj gSA
fdlh Hkh {k.k] yV~Vw dh ?kw.kZu&v{k] blosQ lEioZQ fcUnq ls
xqtjrh gSA bl izdkj dh ?kw.kZu xfr dk nwljk ljy mnkgj.k
?kweus okyk est dk ia[kk ;k ihfBdk&ia[kk gSA vkius ns[kk gksxk
fd bl izdkj osQ ia[ks dh v{k] {kSfrt ry esa] nksyu xfr
(bèkj ls m/j ?kweus dh) djrh gS vkSj ;g xfr ÅèokZ/j js[kk
osQ ifjr% gksrh gS tks ml fcUnq ls xqtjrh gS ftl ij v{k dh
/qjh fVdh gksrh gS  (fp=k 7.5(b) esa fcUnq O)A
fp=k 7.5 (a) ?kwerk gqvk yV~Vw
(bldh fVi  O dk /jkry ij lEioZQ fcUnq fLFkj gS)
fp=k 7.5 (b) ?kwerk gqvk est dk ia[kk (ia[ks dh /qjh] fcUnq
O, fLFkj gS )
vkb;s] vc ge ;g le>us dh ps"Vk djsa fd ?kw.kZu D;k
gS] vkSj blosQ D;k vfHky{k.k gSa\ vki ns[k ldrs gSa fd ,d
n`<+ fi.M osQ ,d fLFkj v{k osQ ifjr% ?kw.kZu esa] fi.M dk
gj d.k ,d o`Ùk esa ?kwerk gSA ;g o`Ùk v{k osQ yEcor~ ry
esa gS vkSj budk osQUnz v{k ij vofLFkr gSA fp=k 7.4 esa ,d
fLFkj v{k (funsZ'k izsQe dh z-v{k) osQ ifjr% fdlh n`<+
fi.M dh ?kw.kZu xfr n'kkZ;h gSA ge v{k ls r
1
 nwjh ij fLFkr
n`<+ fi.M dk dksbZ LosPN d.k P
1
 ysaA ;g d.k v{k osQ ifjr%
r
1
 f=kT;k osQ o`Ùk ij ?kwerk gS ftldk osQUnz C
1
 v{k ij fLFkr
gSA ;g o`Ùk v{k osQ yEcor~ ry esa vofLFkr gSA fp=k esa ,d
nwljk d.k P
2
 Hkh n'kkZ;k x;k gS tks fLFkj v{k ls r
2 
 nwjh ij
gSA d.k  P
2 
]  r
2
 f=kT;k osQ o`Ùkkdkj iFk ij pyrk gS ftldk
osQUnz  v{k ij C
2
 gSA ;g o`Ùk Hkh v{k osQ yEcor~ ry esa gSA
è;ku nsa fd  P
1
 ,oa P
2
 }kjk cuk;s x, o`Ùk vyx&vyx ryksa esa
gSa ij ;s nksuksa ry fLFkj v{k osQ yEcor~ gSaA v{k ij fLFkr fdlh
fcUnq] tSls  P
3 
osQ fy,] r = 0 A ;s d.k] fi.M osQ ?kwers le;
Hkh fLFkr jgrs gSaA ;g visf{kr Hkh gS D;ksafd v{k fLFkj gSA
rFkkfi] ?kw.kZu osQ oqQN mnkgj.kksa esa] v{k fLFkj ugha Hkh
jgrhA bl izdkj osQ ?kw.kZu osQ eq[; mnkgj.kksa esa ,d gS] ,d
gh LFkku ij ?kwerk yV~Vw (fp=k 7.5(a))A (yV~Vw dh xfr osQ
fp=k 7.4 z-v{k osQ ifjr% ,d n`<+ fi.M dk ?kw.kZuA fi.M dk
izR;sd fcUnq  P
1
 ;k P
2
 ,d o`Ùk ij ?kwerk gS ftldk
osQUnz  (C
1
 ;k C
2
) v{k ij fLFkr gSA o`Ùk dh f=kT;k (r
1
;k r
2
) v{k ls fcUnq (P
1
 ;k P
2
) dh yEcor~ nwjh gSA
v{k ij fLFkr  P
3 
tSlk fcUnq fLFkj jgrk gSA
Page 4


vè;k; 7
d.kksa osQ fudk; rFkk ?kw.khZ xfr
7.1 Hkwfedk
fiNys vè;k;ksa esa geus eq[; :i ls vkn'kZ fcUnq d.k (,d d.k ftls
nzO;eku ;qDr fcUnq osQ :i esa O;Dr fd;k tk,A O;kogkfjd n`f"V ls bldk
dksbZ vkdkj ugha gksrk) dh xfr dk vè;;u fd;k FkkA fiQj] ;g ekurs gq,
fd ifjfer vkdkj osQ fi.Mksa dh xfr dks fcUnq d.k dh xfr osQ inksa esa O;Dr
fd;k tk ldrk gS] geus ml vè;;u osQ ifj.kkeksa dks ifjfer vkdkj osQ
fi.Mksa ij Hkh ykxw dj fn;k FkkA
nSfud thou esa ftrus fi.M gekjs laioZQ esa vkrs gSa os lHkh ifjfer
vkdkj osQ gksrs gSaA ,d foLr`r fi.M (ifjfer vkdkj osQ fi.M) dh xfr dks
iwjs rkSj ij le>us osQ fy, vkerkSj ij mldk fcUnqor~ vkn'kZ vi;kZIr jgrk
gSA bl vè;k; esa ge bl izfrca/ osQ ijs tkus dh ps"Vk djssaxs vkSj foLr`r]
ij ifjfer fi.Mksa dh xfr dks le>us dk iz;kl djsaxsA ,d foLr`r fi.M
izFker;k d.kksa dk ,d fudk; gSA vr% ge viuk foospu ,d fudk; dh
xfr ls gh 'kq: djuk pkgsaxsA ;gk¡ d.kksa osQ fudk; dk nzO;eku osQUnz ,d eq[;
vo/kj.kk gksxhA ge d.kksa osQ fudk; osQ nzO;eku osQUnz dh xfr dk o.kZu djsaxs
vkSj fiQj] ifjfer vkdkj osQ fi.Mksa dh xfr dks le>us esa bl vo/kj.kk dh
mi;ksfxrk crk;saxsA
cM+s fi.Mksa ls tqM+h cgqr lh leL;k,a mudks n`<+ fi.M ekudj gy dh
tk ldrh gSaA vkn'kZ n`<+ fi.M ,d ,slk fi.M gS ftldh ,d lqfuf'pr vkSj
vifjorZuh; vko`Qfr gksrh gSA bl izdkj osQ Bksl osQ fofHkUu d.kksa osQ chp dh
nwfj;ka ifjo£rr ugha gksrhA n`<+ fi.M dh bl ifjHkk"kk ls ;g Li"V gS fd dksbZ
Hkh okLrfod fi.M iwjh rjg n`<+ ugha gksrk] D;ksafd lHkh O;kogkfjd fi.M
cyksa osQ izHkko ls foo`Qr gks tkrs gSaA ijUrq ,slh cgqr lh fLFkfr;k¡ gksrh gSa ftuesa
foo`Qfr;k¡ ux.; gksrh gSaA vr% dbZ izdkj dh fLFkfr;ksa esa ;Fkk ifg;s] yV~Vw]
LVhy osQ 'kgrhj vkSj ;gk¡ rd fd v.kq] xzg tSls fi.Mksa dh xfr dk vè;;u
djrs le;] ge è;ku u nsaxs fd muesa foo`Qfr vkrh gS] os eqM+rs gSa ;k dEiu
djrs gSaA ge mUgsa n`<+ fi.M eku dj mudh xfr dk vè;;u djasxsA
7.1 Hkwfedk
7.2 nzO;eku osQUnz
7.3 nzO;eku osQUnz dh xfr
7.4 d.kksa osQ fudk; dk js[kh; laosx
7.5 nks lfn'kksa dk lfn'k xq.kuiQy
7.6 dks.kh; osx vkSj bldk js[kh;
osx ls laca/
7.7 cy vk?kw.kZ ,oa dks.kh; laosx
7.8 n`<+ ¯iMksa dk larqyu
7.9 tM+Ro vk?kw.kZ
7.10 yEcor~ ,oa lekukUrj v{kksa osQ izes;
7.11 vpy v{k osQ ifjr% 'kq¼ ?kw.khZ
xfrdh
7.12 vpy v{k osQ ifjr% ?kw.khZ xfrdh
7.13 vpy v{k osQ ifjr% ?kw.khZ xfr dk
dks.kh; laosx
7.14 yksVfud xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
7.1.1 ,d n`<+ fi.M esa fdl izdkj dh xfr;k¡ gks
ldrh gSa\
vkb;s] n`<+ fi.Mksa dh xfr osQ oqQN mnkgj.kksa ls bl iz'u dk mÙkj
<wa<+us dh dksf'k'k djsaA izFke ,d vk;rkdkj CykWd ij fopkj djsa
tks ,d ur ry ij lh/k (fcuk b/j&m/j gVs) uhps dh vksj
fiQly jgk gSA CykWd ,d n`<+ fi.M gSA ur ry ij uhps dh vksj
bldh xfr ,slh gS fd blosQ lHkh d.k lkFk&lkFk py jgs gSa]
vFkkZr~ fdlh {k.k lHkh d.k leku osx ls pyrs gSa (fp=k 7.1)A
;gk¡ ;g n`<+ fiaM 'kq¼ LFkkukarj.k xfr esa gSA
'kq¼ LFkkukarj.k xfr esa fdlh {k.k fo'ks"k ij fi.M dk
izR;sd d.k leku osx ls pyrk gSA
fp=k 7.2 ur ry ij uhps dh vksj yq<+drk fl¯yMj (csyu)A   ;g
'kq¼ LFkkukarj.k xfr ugha gSA fdlh {k.k ij fcUnq P
1
, P
2
,
P
3
 ,oa  P
4
 osQ vyx&vyx osx gSa (tSlk fd rhj n'kkZrs
gSa)A okLro esa lEioZQ fcUnq P
3
 dk osx fdlh Hkh {k.k 'kwU;
gS ;fn csyu fcuk fiQlys gq, yq<+drk gSA
vkb;s] vc mlh ur ry ij uhps dh vksj yq<+drs gq,
,d /krq ;k ydM+h osQ csyu dh xfr ij fopkj djrs gSa
(fp=k 7.2)A ;g n`<+ fi.M (csyu) ur ry osQ 'kh"kZ ls
mldh ryh rd LFkkukarfjr gksrk gS] vr% blesa LFkkukarj.k
xfr gSA ysfdu fp=k 7.2 ;g Hkh n'kkZrk gS fd blosQ lHkh d.k
{k.k fo'ks"k ij ,d gh osx ls ugha py jgs gSaA vr% fi.M 'kq¼
LFkkukarj.k xfr esa ugha gSA vr% bldh xfr LFkkukarjh; gksus osQ
lkFk&lkFk ^oqQN vkSj vyx* Hkh gSA
;g ^oqQN vkSj vyx* Hkh D;k gS\ ;g le>us osQ fy,]
vkb;s] ge ,d ,slk n`<+ ¯iM ysa ftldks bl izdkj O;o#¼
dj fn;k x;k gS fd ;g LFkkukarj.k xfr u dj losQA fdlh
n`<+ fi.M dh LFkkukarj.k xfr dks fu#¼ djus dh loZ lkekU;
fof/ ;g gS fd mls ,d ljy js[kk osQ vuqfn'k fLFkj dj
fn;k tk,A rc bl n`<+ fi.M dh ,dek=k laHkkfor xfr ?kw.khZ
xfr gksxhA og ljy js[kk ftlosQ vuqfn'k bl n`<+ fi.M dks
fLFkj cuk;k x;k gS bldh ?kw.kZu&v{k dgykrh gSA ;fn vki
vius pkjksa vksj ns[ksa rks vkidks Nr dk ia[kk] oqQEgkj dk pkd
(fp=k 7.3(a) ,oa (b))] fo'kky pØh&>wyk (tkW;UV Oghy)]
esjh&xks&jkm.M tSls vusd ,sls mnkgj.k fey tk;saxs tgk¡
fdlh v{k osQ ifjr% ?kw.kZu gks jgk gksA
(a)
(b)
fp=k 7.3 ,d fLFkj v{k osQ ifjr% ?kw.kZu
(a) Nr dk ia[kk
(b) oqQEgkj dk pkd
Fig 7.1ur&ry ij ,d CykWd dh v/kseq[kh LFkkukarj.k
(fiQlyu) xfr (CykWd dk izR;sd ¯cnq ;Fkk P
1
, P
2
....
fdlh Hkh {k.k leku xfr esa gSa)
d.kksa osQ fudk; rFkk ?kw.khZ xfr 145 146 HkkSfrdh
laca/ esa geus ;g eku fy;k gS fd ;g ,d LFkku ls nwljs
LFkku ij LFkkukarfjr ugha gksrk vkSj blfy, blesa LFkkukarj.k
xfr ugha gSA)  vius vuqHko osQ vk/kj ij ge ;g tkurs gSa fd
bl izdkj ?kwers yV~Vw dh v{k] Hkwfe ij blosQ lEioZQ&fcUnq
ls xqtjrs vfHkyEc osQ ifjr% ,d 'kaoqQ cukrh gS tSlk fd
fp=k 7.5(a) esa n'kkZ;k x;k gSA  (ÅèokZ/j osQ ifjr% yV~Vw dh
v{k dk bl izdkj ?kweuk iqjLlj.k dgykrk gS )A  è;ku nsa fd
yV~Vw dk og fcUnq tgk¡ ;g /jkry dks Nwrk gS] fLFkj gSA
fdlh Hkh {k.k] yV~Vw dh ?kw.kZu&v{k] blosQ lEioZQ fcUnq ls
xqtjrh gSA bl izdkj dh ?kw.kZu xfr dk nwljk ljy mnkgj.k
?kweus okyk est dk ia[kk ;k ihfBdk&ia[kk gSA vkius ns[kk gksxk
fd bl izdkj osQ ia[ks dh v{k] {kSfrt ry esa] nksyu xfr
(bèkj ls m/j ?kweus dh) djrh gS vkSj ;g xfr ÅèokZ/j js[kk
osQ ifjr% gksrh gS tks ml fcUnq ls xqtjrh gS ftl ij v{k dh
/qjh fVdh gksrh gS  (fp=k 7.5(b) esa fcUnq O)A
fp=k 7.5 (a) ?kwerk gqvk yV~Vw
(bldh fVi  O dk /jkry ij lEioZQ fcUnq fLFkj gS)
fp=k 7.5 (b) ?kwerk gqvk est dk ia[kk (ia[ks dh /qjh] fcUnq
O, fLFkj gS )
vkb;s] vc ge ;g le>us dh ps"Vk djsa fd ?kw.kZu D;k
gS] vkSj blosQ D;k vfHky{k.k gSa\ vki ns[k ldrs gSa fd ,d
n`<+ fi.M osQ ,d fLFkj v{k osQ ifjr% ?kw.kZu esa] fi.M dk
gj d.k ,d o`Ùk esa ?kwerk gSA ;g o`Ùk v{k osQ yEcor~ ry
esa gS vkSj budk osQUnz v{k ij vofLFkr gSA fp=k 7.4 esa ,d
fLFkj v{k (funsZ'k izsQe dh z-v{k) osQ ifjr% fdlh n`<+
fi.M dh ?kw.kZu xfr n'kkZ;h gSA ge v{k ls r
1
 nwjh ij fLFkr
n`<+ fi.M dk dksbZ LosPN d.k P
1
 ysaA ;g d.k v{k osQ ifjr%
r
1
 f=kT;k osQ o`Ùk ij ?kwerk gS ftldk osQUnz C
1
 v{k ij fLFkr
gSA ;g o`Ùk v{k osQ yEcor~ ry esa vofLFkr gSA fp=k esa ,d
nwljk d.k P
2
 Hkh n'kkZ;k x;k gS tks fLFkj v{k ls r
2 
 nwjh ij
gSA d.k  P
2 
]  r
2
 f=kT;k osQ o`Ùkkdkj iFk ij pyrk gS ftldk
osQUnz  v{k ij C
2
 gSA ;g o`Ùk Hkh v{k osQ yEcor~ ry esa gSA
è;ku nsa fd  P
1
 ,oa P
2
 }kjk cuk;s x, o`Ùk vyx&vyx ryksa esa
gSa ij ;s nksuksa ry fLFkj v{k osQ yEcor~ gSaA v{k ij fLFkr fdlh
fcUnq] tSls  P
3 
osQ fy,] r = 0 A ;s d.k] fi.M osQ ?kwers le;
Hkh fLFkr jgrs gSaA ;g visf{kr Hkh gS D;ksafd v{k fLFkj gSA
rFkkfi] ?kw.kZu osQ oqQN mnkgj.kksa esa] v{k fLFkj ugha Hkh
jgrhA bl izdkj osQ ?kw.kZu osQ eq[; mnkgj.kksa esa ,d gS] ,d
gh LFkku ij ?kwerk yV~Vw (fp=k 7.5(a))A (yV~Vw dh xfr osQ
fp=k 7.4 z-v{k osQ ifjr% ,d n`<+ fi.M dk ?kw.kZuA fi.M dk
izR;sd fcUnq  P
1
 ;k P
2
 ,d o`Ùk ij ?kwerk gS ftldk
osQUnz  (C
1
 ;k C
2
) v{k ij fLFkr gSA o`Ùk dh f=kT;k (r
1
;k r
2
) v{k ls fcUnq (P
1
 ;k P
2
) dh yEcor~ nwjh gSA
v{k ij fLFkr  P
3 
tSlk fcUnq fLFkj jgrk gSA
d.kksa osQ fudk; rFkk ?kw.khZ xfr 147
tc ia[kk ?kwerk gS vkSj bldh v{k b/j ls m/j nksyu
djrh gS rc Hkh ;g fcUnq fLFkj jgrk gSA ?kw.kZu xfr osQ
vf/d lk£od ekeyksa esa] tSls fd yV~Vw ;k ihfBdk&ia[ks osQ
?kweus esa] n`<+ fi.M dk ,d fcUnq fLFkj jgrk gS] u fd ,d
js[kkA bl ekeys esa v{k rks fLFkj ugha gS ij ;g ges'kk ,d
fLFkj fcUnq ls xqtjrh gSA rFkkfi] vius vè;;u esa]
vf/dka'kr%] ge ,slh ljy ,oa fof'k"V ?kw.kZu xfr;ksa rd
lhfer jgsaxs ftuesa ,d js[kk (;kfu v{k) fLFkj jgrh gSA vr%
tc rd vU;Fkk u dgk tk;] gekjs fy, ?kw.khZ xfr ,d fLFkj
v{k osQ ifjr% gh gksxhA
,d ur ry ij uhps dh vksj csyu dk yq<+duk nks rjg
dh xfr;ksa dk la;kstu gS& LFkkukarj.k xfr vkSj ,d fLFkj v{k
osQ ifjr% ?kw.khZ xfrA vr%] yq<+du xfr osQ lanHkZ esa ftl
^oqQN vkSj vyx* dk ftØ igys geus fd;k Fkk og ?kw.khZ
xfr gSA bl n`f"Vdks.k ls fp=k 7.6(a) ,oa (b) dks vki i;kZIr
f'k{kkizn ik;saxsA bu nksuksa fp=kksa esa ,d gh fi.M dh xfr]
leku LFkkukarjh; xeu&iFk osQ vuqfn'k n'kkZbZ xbZ gSA fp=k
7.6(a) esa n'kkZbZ xbZ xfr 'kq¼ LFkkukarjh; gS] tcfd fp=k
7.6(b) esa n'kkZbZ xbZ xfr LFkkukarj.k ,oa ?kw.khZ nksuksa izdkj dh
xfr;ksa dk la;kstu gSA (vki Lo;a Hkkjh iqLrd tSlk ,d n`<+
fi.M isaQd dj n'kkZbZ xbZ nksuksa izdkj dh xfr;k¡ mRiUu djus
dh dksf'k'k dj ldrs gSaA)
vkb;s vc ge izLrqr [k.M esa o£.kr egRoiw.kZ rF;ksa dk
lkj fiQj ls vkidks crk;saA ,d ,slk n`<+ fi.M tks u rks
fdlh pwy ij fVdk gks vkSj u gh fdlh :i esa fLFkj gks]
nks izdkj dh xfr dj ldrk gS & ;k rks 'kq¼ LFkkukarj.k
;k LFkkukarj.k ,oa ?kw.kZu xfr dk la;kstuA ,d ,sls n`<+
fi.M dh xfr tks ;k rks pwy ij fVdk gks ;k fdlh u fdlh
:i esa fLFkj gks] ?kw.khZ xfr gksrh gSA ?kw.kZu fdlh ,slh v{k osQ
ifjr% gks ldrk gS tks fLFkj gks (tSls Nr osQ ia[ks esa) ;k fiQj
,d ,slh v{k osQ ifjr% tks Lo;a ?kwerh gks (tSls b/j ls mèkj
?kwers est osQ ia[ks esa)A bl vè;k; esa ge ,d fLFkj v{k osQ
ifjr% gksus okyh ?kw.khZ xfr dk gh vè;;u djsaxsA
7.2 nzO;eku osQUnz
igys ge ;g ns[ksaxs fd nzO;eku osQUnz D;k gS vkSj fiQj blosQ
egRo ij izdk'k MkysaxsA ljyrk dh n`f"V ls ge nks d.kksa osQ
fudk; ls 'kq#vkr djsaxsA nksuksa d.kksa dh fLFkfr;ksa dks feykus
okyh js[kk dks ge  x- v{k ekusaxsA (fp=k 7-7)
fp=k 7.7  nks d.kksa vkSj muosQ nzO;eku osQUnz dh fLFkfr
fp=k 7.6(a) ,d n`<+ fi.M dh xfr tks 'kq¼ LFkkukarjh; gS
fp=k 7.6(b) n`<+ fi.M dh ,slh xfr tks LFkkukarjh; vkSj ?kw.khZ
xfr;ksa dk la;kstu gS
fp=k 7.6 (a) ,oa 7.6 (b) ,d gh fi.M dh fofHkUu xfr;k¡ n'kkZrs
gSaA è;ku nsa] fd  P  fi.M dk dksbZ LosPN fcUnq gS_  O  fi.M dk
nzO;eku osQUnz gS] ftlosQ fo"k; esa vxys [k.M esa crk;k x;k gSA
;gk¡ ;g dguk i;kZIr gksxk fd fcUnq  O  osQ xeu iFk gh fi.M
osQ LFkkukarjh; xeu iFk  Tr
1
  ,oa  Tr
2 
 gSaA rhu vyx&vyx {k.kksa
ij] fcUnqvksa  O  ,oa  P dh fLFkfr;k¡ fp=k  7.6(a) ,oa  7.6 (b) nksuksa
gh Øe'k% O
1
, O
2
, O
3
, ,oa  P
1
, P
2
 , P
3
 }kjk izn£'kr dh xbZ gSaA
fp=k  7.6(a) ls ;g Li"V gS fd 'kq¼ LFkkukarj.k dh fLFkfr esa]
fi.M osQ fdUgha Hkh nks fcUnqvksa  O  ,oa P  osQ osx] cjkcj gksrs gSaA
;g Hkh KkrO; gS] fd bl fLFkfr esa  OP, dk fnfXoU;kl] ;kfu fd
og dks.k tks  OP ,d fu;r fn'kk (ekuk fd {kSfrt) ls cukrk
gS] leku jgrk gS vFkkZr~  a
1 
= a
2
 = a
3 
A fp=k 7.6 (b) LFkkukarj.k
,oa ?kw.kZu osQ la;kstu ls fu£er xfr n'kkZrk gSA bl xfr esa fcUnqvksa
O  ,oa  P osQ {kf.kd osxksa osQ eku vyx&vyx gks ldrs gSa vkSj
dks.kksa  a
1
, a
2
, a
3
 osQ eku Hkh fHkUu gks ldrs gSaA
Page 5


vè;k; 7
d.kksa osQ fudk; rFkk ?kw.khZ xfr
7.1 Hkwfedk
fiNys vè;k;ksa esa geus eq[; :i ls vkn'kZ fcUnq d.k (,d d.k ftls
nzO;eku ;qDr fcUnq osQ :i esa O;Dr fd;k tk,A O;kogkfjd n`f"V ls bldk
dksbZ vkdkj ugha gksrk) dh xfr dk vè;;u fd;k FkkA fiQj] ;g ekurs gq,
fd ifjfer vkdkj osQ fi.Mksa dh xfr dks fcUnq d.k dh xfr osQ inksa esa O;Dr
fd;k tk ldrk gS] geus ml vè;;u osQ ifj.kkeksa dks ifjfer vkdkj osQ
fi.Mksa ij Hkh ykxw dj fn;k FkkA
nSfud thou esa ftrus fi.M gekjs laioZQ esa vkrs gSa os lHkh ifjfer
vkdkj osQ gksrs gSaA ,d foLr`r fi.M (ifjfer vkdkj osQ fi.M) dh xfr dks
iwjs rkSj ij le>us osQ fy, vkerkSj ij mldk fcUnqor~ vkn'kZ vi;kZIr jgrk
gSA bl vè;k; esa ge bl izfrca/ osQ ijs tkus dh ps"Vk djssaxs vkSj foLr`r]
ij ifjfer fi.Mksa dh xfr dks le>us dk iz;kl djsaxsA ,d foLr`r fi.M
izFker;k d.kksa dk ,d fudk; gSA vr% ge viuk foospu ,d fudk; dh
xfr ls gh 'kq: djuk pkgsaxsA ;gk¡ d.kksa osQ fudk; dk nzO;eku osQUnz ,d eq[;
vo/kj.kk gksxhA ge d.kksa osQ fudk; osQ nzO;eku osQUnz dh xfr dk o.kZu djsaxs
vkSj fiQj] ifjfer vkdkj osQ fi.Mksa dh xfr dks le>us esa bl vo/kj.kk dh
mi;ksfxrk crk;saxsA
cM+s fi.Mksa ls tqM+h cgqr lh leL;k,a mudks n`<+ fi.M ekudj gy dh
tk ldrh gSaA vkn'kZ n`<+ fi.M ,d ,slk fi.M gS ftldh ,d lqfuf'pr vkSj
vifjorZuh; vko`Qfr gksrh gSA bl izdkj osQ Bksl osQ fofHkUu d.kksa osQ chp dh
nwfj;ka ifjo£rr ugha gksrhA n`<+ fi.M dh bl ifjHkk"kk ls ;g Li"V gS fd dksbZ
Hkh okLrfod fi.M iwjh rjg n`<+ ugha gksrk] D;ksafd lHkh O;kogkfjd fi.M
cyksa osQ izHkko ls foo`Qr gks tkrs gSaA ijUrq ,slh cgqr lh fLFkfr;k¡ gksrh gSa ftuesa
foo`Qfr;k¡ ux.; gksrh gSaA vr% dbZ izdkj dh fLFkfr;ksa esa ;Fkk ifg;s] yV~Vw]
LVhy osQ 'kgrhj vkSj ;gk¡ rd fd v.kq] xzg tSls fi.Mksa dh xfr dk vè;;u
djrs le;] ge è;ku u nsaxs fd muesa foo`Qfr vkrh gS] os eqM+rs gSa ;k dEiu
djrs gSaA ge mUgsa n`<+ fi.M eku dj mudh xfr dk vè;;u djasxsA
7.1 Hkwfedk
7.2 nzO;eku osQUnz
7.3 nzO;eku osQUnz dh xfr
7.4 d.kksa osQ fudk; dk js[kh; laosx
7.5 nks lfn'kksa dk lfn'k xq.kuiQy
7.6 dks.kh; osx vkSj bldk js[kh;
osx ls laca/
7.7 cy vk?kw.kZ ,oa dks.kh; laosx
7.8 n`<+ ¯iMksa dk larqyu
7.9 tM+Ro vk?kw.kZ
7.10 yEcor~ ,oa lekukUrj v{kksa osQ izes;
7.11 vpy v{k osQ ifjr% 'kq¼ ?kw.khZ
xfrdh
7.12 vpy v{k osQ ifjr% ?kw.khZ xfrdh
7.13 vpy v{k osQ ifjr% ?kw.khZ xfr dk
dks.kh; laosx
7.14 yksVfud xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
7.1.1 ,d n`<+ fi.M esa fdl izdkj dh xfr;k¡ gks
ldrh gSa\
vkb;s] n`<+ fi.Mksa dh xfr osQ oqQN mnkgj.kksa ls bl iz'u dk mÙkj
<wa<+us dh dksf'k'k djsaA izFke ,d vk;rkdkj CykWd ij fopkj djsa
tks ,d ur ry ij lh/k (fcuk b/j&m/j gVs) uhps dh vksj
fiQly jgk gSA CykWd ,d n`<+ fi.M gSA ur ry ij uhps dh vksj
bldh xfr ,slh gS fd blosQ lHkh d.k lkFk&lkFk py jgs gSa]
vFkkZr~ fdlh {k.k lHkh d.k leku osx ls pyrs gSa (fp=k 7.1)A
;gk¡ ;g n`<+ fiaM 'kq¼ LFkkukarj.k xfr esa gSA
'kq¼ LFkkukarj.k xfr esa fdlh {k.k fo'ks"k ij fi.M dk
izR;sd d.k leku osx ls pyrk gSA
fp=k 7.2 ur ry ij uhps dh vksj yq<+drk fl¯yMj (csyu)A   ;g
'kq¼ LFkkukarj.k xfr ugha gSA fdlh {k.k ij fcUnq P
1
, P
2
,
P
3
 ,oa  P
4
 osQ vyx&vyx osx gSa (tSlk fd rhj n'kkZrs
gSa)A okLro esa lEioZQ fcUnq P
3
 dk osx fdlh Hkh {k.k 'kwU;
gS ;fn csyu fcuk fiQlys gq, yq<+drk gSA
vkb;s] vc mlh ur ry ij uhps dh vksj yq<+drs gq,
,d /krq ;k ydM+h osQ csyu dh xfr ij fopkj djrs gSa
(fp=k 7.2)A ;g n`<+ fi.M (csyu) ur ry osQ 'kh"kZ ls
mldh ryh rd LFkkukarfjr gksrk gS] vr% blesa LFkkukarj.k
xfr gSA ysfdu fp=k 7.2 ;g Hkh n'kkZrk gS fd blosQ lHkh d.k
{k.k fo'ks"k ij ,d gh osx ls ugha py jgs gSaA vr% fi.M 'kq¼
LFkkukarj.k xfr esa ugha gSA vr% bldh xfr LFkkukarjh; gksus osQ
lkFk&lkFk ^oqQN vkSj vyx* Hkh gSA
;g ^oqQN vkSj vyx* Hkh D;k gS\ ;g le>us osQ fy,]
vkb;s] ge ,d ,slk n`<+ ¯iM ysa ftldks bl izdkj O;o#¼
dj fn;k x;k gS fd ;g LFkkukarj.k xfr u dj losQA fdlh
n`<+ fi.M dh LFkkukarj.k xfr dks fu#¼ djus dh loZ lkekU;
fof/ ;g gS fd mls ,d ljy js[kk osQ vuqfn'k fLFkj dj
fn;k tk,A rc bl n`<+ fi.M dh ,dek=k laHkkfor xfr ?kw.khZ
xfr gksxhA og ljy js[kk ftlosQ vuqfn'k bl n`<+ fi.M dks
fLFkj cuk;k x;k gS bldh ?kw.kZu&v{k dgykrh gSA ;fn vki
vius pkjksa vksj ns[ksa rks vkidks Nr dk ia[kk] oqQEgkj dk pkd
(fp=k 7.3(a) ,oa (b))] fo'kky pØh&>wyk (tkW;UV Oghy)]
esjh&xks&jkm.M tSls vusd ,sls mnkgj.k fey tk;saxs tgk¡
fdlh v{k osQ ifjr% ?kw.kZu gks jgk gksA
(a)
(b)
fp=k 7.3 ,d fLFkj v{k osQ ifjr% ?kw.kZu
(a) Nr dk ia[kk
(b) oqQEgkj dk pkd
Fig 7.1ur&ry ij ,d CykWd dh v/kseq[kh LFkkukarj.k
(fiQlyu) xfr (CykWd dk izR;sd ¯cnq ;Fkk P
1
, P
2
....
fdlh Hkh {k.k leku xfr esa gSa)
d.kksa osQ fudk; rFkk ?kw.khZ xfr 145 146 HkkSfrdh
laca/ esa geus ;g eku fy;k gS fd ;g ,d LFkku ls nwljs
LFkku ij LFkkukarfjr ugha gksrk vkSj blfy, blesa LFkkukarj.k
xfr ugha gSA)  vius vuqHko osQ vk/kj ij ge ;g tkurs gSa fd
bl izdkj ?kwers yV~Vw dh v{k] Hkwfe ij blosQ lEioZQ&fcUnq
ls xqtjrs vfHkyEc osQ ifjr% ,d 'kaoqQ cukrh gS tSlk fd
fp=k 7.5(a) esa n'kkZ;k x;k gSA  (ÅèokZ/j osQ ifjr% yV~Vw dh
v{k dk bl izdkj ?kweuk iqjLlj.k dgykrk gS )A  è;ku nsa fd
yV~Vw dk og fcUnq tgk¡ ;g /jkry dks Nwrk gS] fLFkj gSA
fdlh Hkh {k.k] yV~Vw dh ?kw.kZu&v{k] blosQ lEioZQ fcUnq ls
xqtjrh gSA bl izdkj dh ?kw.kZu xfr dk nwljk ljy mnkgj.k
?kweus okyk est dk ia[kk ;k ihfBdk&ia[kk gSA vkius ns[kk gksxk
fd bl izdkj osQ ia[ks dh v{k] {kSfrt ry esa] nksyu xfr
(bèkj ls m/j ?kweus dh) djrh gS vkSj ;g xfr ÅèokZ/j js[kk
osQ ifjr% gksrh gS tks ml fcUnq ls xqtjrh gS ftl ij v{k dh
/qjh fVdh gksrh gS  (fp=k 7.5(b) esa fcUnq O)A
fp=k 7.5 (a) ?kwerk gqvk yV~Vw
(bldh fVi  O dk /jkry ij lEioZQ fcUnq fLFkj gS)
fp=k 7.5 (b) ?kwerk gqvk est dk ia[kk (ia[ks dh /qjh] fcUnq
O, fLFkj gS )
vkb;s] vc ge ;g le>us dh ps"Vk djsa fd ?kw.kZu D;k
gS] vkSj blosQ D;k vfHky{k.k gSa\ vki ns[k ldrs gSa fd ,d
n`<+ fi.M osQ ,d fLFkj v{k osQ ifjr% ?kw.kZu esa] fi.M dk
gj d.k ,d o`Ùk esa ?kwerk gSA ;g o`Ùk v{k osQ yEcor~ ry
esa gS vkSj budk osQUnz v{k ij vofLFkr gSA fp=k 7.4 esa ,d
fLFkj v{k (funsZ'k izsQe dh z-v{k) osQ ifjr% fdlh n`<+
fi.M dh ?kw.kZu xfr n'kkZ;h gSA ge v{k ls r
1
 nwjh ij fLFkr
n`<+ fi.M dk dksbZ LosPN d.k P
1
 ysaA ;g d.k v{k osQ ifjr%
r
1
 f=kT;k osQ o`Ùk ij ?kwerk gS ftldk osQUnz C
1
 v{k ij fLFkr
gSA ;g o`Ùk v{k osQ yEcor~ ry esa vofLFkr gSA fp=k esa ,d
nwljk d.k P
2
 Hkh n'kkZ;k x;k gS tks fLFkj v{k ls r
2 
 nwjh ij
gSA d.k  P
2 
]  r
2
 f=kT;k osQ o`Ùkkdkj iFk ij pyrk gS ftldk
osQUnz  v{k ij C
2
 gSA ;g o`Ùk Hkh v{k osQ yEcor~ ry esa gSA
è;ku nsa fd  P
1
 ,oa P
2
 }kjk cuk;s x, o`Ùk vyx&vyx ryksa esa
gSa ij ;s nksuksa ry fLFkj v{k osQ yEcor~ gSaA v{k ij fLFkr fdlh
fcUnq] tSls  P
3 
osQ fy,] r = 0 A ;s d.k] fi.M osQ ?kwers le;
Hkh fLFkr jgrs gSaA ;g visf{kr Hkh gS D;ksafd v{k fLFkj gSA
rFkkfi] ?kw.kZu osQ oqQN mnkgj.kksa esa] v{k fLFkj ugha Hkh
jgrhA bl izdkj osQ ?kw.kZu osQ eq[; mnkgj.kksa esa ,d gS] ,d
gh LFkku ij ?kwerk yV~Vw (fp=k 7.5(a))A (yV~Vw dh xfr osQ
fp=k 7.4 z-v{k osQ ifjr% ,d n`<+ fi.M dk ?kw.kZuA fi.M dk
izR;sd fcUnq  P
1
 ;k P
2
 ,d o`Ùk ij ?kwerk gS ftldk
osQUnz  (C
1
 ;k C
2
) v{k ij fLFkr gSA o`Ùk dh f=kT;k (r
1
;k r
2
) v{k ls fcUnq (P
1
 ;k P
2
) dh yEcor~ nwjh gSA
v{k ij fLFkr  P
3 
tSlk fcUnq fLFkj jgrk gSA
d.kksa osQ fudk; rFkk ?kw.khZ xfr 147
tc ia[kk ?kwerk gS vkSj bldh v{k b/j ls m/j nksyu
djrh gS rc Hkh ;g fcUnq fLFkj jgrk gSA ?kw.kZu xfr osQ
vf/d lk£od ekeyksa esa] tSls fd yV~Vw ;k ihfBdk&ia[ks osQ
?kweus esa] n`<+ fi.M dk ,d fcUnq fLFkj jgrk gS] u fd ,d
js[kkA bl ekeys esa v{k rks fLFkj ugha gS ij ;g ges'kk ,d
fLFkj fcUnq ls xqtjrh gSA rFkkfi] vius vè;;u esa]
vf/dka'kr%] ge ,slh ljy ,oa fof'k"V ?kw.kZu xfr;ksa rd
lhfer jgsaxs ftuesa ,d js[kk (;kfu v{k) fLFkj jgrh gSA vr%
tc rd vU;Fkk u dgk tk;] gekjs fy, ?kw.khZ xfr ,d fLFkj
v{k osQ ifjr% gh gksxhA
,d ur ry ij uhps dh vksj csyu dk yq<+duk nks rjg
dh xfr;ksa dk la;kstu gS& LFkkukarj.k xfr vkSj ,d fLFkj v{k
osQ ifjr% ?kw.khZ xfrA vr%] yq<+du xfr osQ lanHkZ esa ftl
^oqQN vkSj vyx* dk ftØ igys geus fd;k Fkk og ?kw.khZ
xfr gSA bl n`f"Vdks.k ls fp=k 7.6(a) ,oa (b) dks vki i;kZIr
f'k{kkizn ik;saxsA bu nksuksa fp=kksa esa ,d gh fi.M dh xfr]
leku LFkkukarjh; xeu&iFk osQ vuqfn'k n'kkZbZ xbZ gSA fp=k
7.6(a) esa n'kkZbZ xbZ xfr 'kq¼ LFkkukarjh; gS] tcfd fp=k
7.6(b) esa n'kkZbZ xbZ xfr LFkkukarj.k ,oa ?kw.khZ nksuksa izdkj dh
xfr;ksa dk la;kstu gSA (vki Lo;a Hkkjh iqLrd tSlk ,d n`<+
fi.M isaQd dj n'kkZbZ xbZ nksuksa izdkj dh xfr;k¡ mRiUu djus
dh dksf'k'k dj ldrs gSaA)
vkb;s vc ge izLrqr [k.M esa o£.kr egRoiw.kZ rF;ksa dk
lkj fiQj ls vkidks crk;saA ,d ,slk n`<+ fi.M tks u rks
fdlh pwy ij fVdk gks vkSj u gh fdlh :i esa fLFkj gks]
nks izdkj dh xfr dj ldrk gS & ;k rks 'kq¼ LFkkukarj.k
;k LFkkukarj.k ,oa ?kw.kZu xfr dk la;kstuA ,d ,sls n`<+
fi.M dh xfr tks ;k rks pwy ij fVdk gks ;k fdlh u fdlh
:i esa fLFkj gks] ?kw.khZ xfr gksrh gSA ?kw.kZu fdlh ,slh v{k osQ
ifjr% gks ldrk gS tks fLFkj gks (tSls Nr osQ ia[ks esa) ;k fiQj
,d ,slh v{k osQ ifjr% tks Lo;a ?kwerh gks (tSls b/j ls mèkj
?kwers est osQ ia[ks esa)A bl vè;k; esa ge ,d fLFkj v{k osQ
ifjr% gksus okyh ?kw.khZ xfr dk gh vè;;u djsaxsA
7.2 nzO;eku osQUnz
igys ge ;g ns[ksaxs fd nzO;eku osQUnz D;k gS vkSj fiQj blosQ
egRo ij izdk'k MkysaxsA ljyrk dh n`f"V ls ge nks d.kksa osQ
fudk; ls 'kq#vkr djsaxsA nksuksa d.kksa dh fLFkfr;ksa dks feykus
okyh js[kk dks ge  x- v{k ekusaxsA (fp=k 7-7)
fp=k 7.7  nks d.kksa vkSj muosQ nzO;eku osQUnz dh fLFkfr
fp=k 7.6(a) ,d n`<+ fi.M dh xfr tks 'kq¼ LFkkukarjh; gS
fp=k 7.6(b) n`<+ fi.M dh ,slh xfr tks LFkkukarjh; vkSj ?kw.khZ
xfr;ksa dk la;kstu gS
fp=k 7.6 (a) ,oa 7.6 (b) ,d gh fi.M dh fofHkUu xfr;k¡ n'kkZrs
gSaA è;ku nsa] fd  P  fi.M dk dksbZ LosPN fcUnq gS_  O  fi.M dk
nzO;eku osQUnz gS] ftlosQ fo"k; esa vxys [k.M esa crk;k x;k gSA
;gk¡ ;g dguk i;kZIr gksxk fd fcUnq  O  osQ xeu iFk gh fi.M
osQ LFkkukarjh; xeu iFk  Tr
1
  ,oa  Tr
2 
 gSaA rhu vyx&vyx {k.kksa
ij] fcUnqvksa  O  ,oa  P dh fLFkfr;k¡ fp=k  7.6(a) ,oa  7.6 (b) nksuksa
gh Øe'k% O
1
, O
2
, O
3
, ,oa  P
1
, P
2
 , P
3
 }kjk izn£'kr dh xbZ gSaA
fp=k  7.6(a) ls ;g Li"V gS fd 'kq¼ LFkkukarj.k dh fLFkfr esa]
fi.M osQ fdUgha Hkh nks fcUnqvksa  O  ,oa P  osQ osx] cjkcj gksrs gSaA
;g Hkh KkrO; gS] fd bl fLFkfr esa  OP, dk fnfXoU;kl] ;kfu fd
og dks.k tks  OP ,d fu;r fn'kk (ekuk fd {kSfrt) ls cukrk
gS] leku jgrk gS vFkkZr~  a
1 
= a
2
 = a
3 
A fp=k 7.6 (b) LFkkukarj.k
,oa ?kw.kZu osQ la;kstu ls fu£er xfr n'kkZrk gSA bl xfr esa fcUnqvksa
O  ,oa  P osQ {kf.kd osxksa osQ eku vyx&vyx gks ldrs gSa vkSj
dks.kksa  a
1
, a
2
, a
3
 osQ eku Hkh fHkUu gks ldrs gSaA
148 HkkSfrdh
ekuk fd nks d.kksa dh] fdlh ewy fcUnq O  ls nwfj;k¡
Øe'k% x
1
 ,oa  x
2
  gSaA bu d.kksa osQ nzO;eku Øe'k%  m
1
 ,oa
m
2
 gSaA bu nks d.kksa osQ fudk; dk nzO;eku osQUnz  C ,d ,slk
fcUnq gksxk ftldh  O  ls nwjh]  X  dk eku gks
11 2 2
12
mx m x
X
mm
+
=
+
(7.1)
lehdj.k  (7.1) esa  X  dks ge  x
1 
 ,oa  x
2 
 dk nzO;eku
Hkkfjr ekè; eku ldrs gSaA ;fn nksuksa d.kksa dk nzO;eku cjkcj
gks rks m
1
 = m
2
 = m,
 
 rc
12 1 2
22
mx mx x x
X
m
++
==
bl izdkj leku nzO;eku osQ nks d.kksa dk nzO;eku osQUnz
Bhd muosQ chpksachp gSA
vxj gekjs ikl  n d.k gksa] ftuosQ nzO;eku Øe'k%  m
1
,
m
2
, ...m
n
  gksa vkSj lcdks x- v{k osQ vuqfn'k j[kk x;k gks]
rks ifjHkk"kk osQ vuqlkj bu lc d.kksa dk nzO;eku osQUnz gksxk
11 2 2
12
....
....
ii
nn
ni
mx
mx m x m x
X
mm m m
++ +
==
++ +
?
?
    (7.2)
tgk¡  x
1
, x
2
,...x
n
 d.kksa dh Øe'k% ewyfcUnq ls nwfj;k¡ gSa_  X
Hkh mlh ewyfcUnq ls ekik x;k gSA laosQr 
?
(;wukuh Hkk"kk
dk v{kj flXek) ladyu dks O;Dr djrk gS tks bl ekeys
esa n d.kksa osQ fy, fd;k x;k gSA ladyu iQy
i
mM =
?
fudk; dk oqQy nzO;eku gSA
ekuk gekjs ikl rhu d.k gSa tks ,d ljy js[kk esa rks ugha]
ij ,d lery esa j[ks x, gSaA rc ge ml ry esa ftlesa ;s
rhu d.k j[ks x, gSa x  ,oa  y-v{k fu/kZfjr dj ldrs gSa] vkSj
bu rhu d.kksa dh fLFkfr;ksa dks Øe'k% funsZ'kkadksa (x
1
,y
1
),
(x
2
,y
2
) ,oa (x
3
,y
3
) }kjk O;Dr dj ldrs gSaA eku yhft, fd
bu rhu d.kksa osQ nzO;eku Øe'k% m
1
, m
2
 ,oa m
3 
gSaA bu rhu
d.kksa osQ fudk; dk nzO;eku osQUnz  C funsZ'kkadksa  (X, Y)  }kjk
O;Dr fd;k tk;sxk ftuosQ eku gSa
1 1 22 33
12 3
m x mx mx
X
mm m
++
=
++
(7.3a)
1 1 22 33
12 3
m y m y m y
Y
mm m
++
=
++
(7.3b)
leku nzO;eku okys d.kksa osQ fy, m = m
1
 = m
2
 = m
3
,
12 3 1 2 3
()
33
mx x x x x x
X
m
++ + +
==
12 3 1 2 3
()
33
myy y y y y
Y
m
++ ++
==
vFkkZr~ leku nzO;eku okys d.kksa osQ fy, rhu d.kksa dk
nzO;eku osQUnz mudh fLFkfr fcUnqvksa dks feykus ls cus f=kHkqt
osQ osQUnzd ij gksxkA
lehdj.k (7.3a,b) osQ ifj.kkeksa dks] ljyrkiwoZd] ,sls
n  d.kksa osQ ,d fudk; osQ fy, lk£od fd;k tk ldrk gS
tks ,d lery esa u gksdj] varfj{k esa iSQys gksaA bl rjg osQ
fudk; dk nzO;eku osQUnz  (X, Y, Z) gS] tgk¡
ii
mx
X
M
=
?
(7.4a)
ii
my
Y
M
=
?
(7.4b)
vkSj  
ii
mz
Z
M
=
?
(7.4c)
;gk¡ M = 
i
m
?
fudk; dk oqQy nzO;eku gSA lwpd i
dk eku 1 ls  n  rd cnyrk gS] m
i
  i osa d.k dk nzO;eku
gS] vkSj i osa d.k dh fLFkfr  (x
i
, y
i
, z
i
) ls O;Dr dh xbZ gSA
;fn ge fLFkfr&lfn'k dh vo/kj.kk dk mi;ksx djsa rks
lehdj.k (7.4a, b, c) dks la;ksftr djosQ ,dy lehdj.k
osQ :i esa fy[kk tk ldrk gSA ;fn 
i
r , iosa d.k dk
fLFkfr&osDVj gS vkSj R nzO;eku osQUnz dk fLFkfr&lfn'k gS%
 
µ
ii i i
xy z   ri j k
$$
,oa   
µ
X YZ   Ri j k
$$
rc   
ii
m
M
=
?
r
R (7.4d)
lehdj.k osQ nkfguh vksj fy[kk x;k ;ksx lfn'k&;ksx gSA
lfn'kksa osQ bLrseky ls lehdj.kksa dh laf{kIrrk ij è;ku
nhft,A ;fn lanHkZ&izsQe (funsZ'kkad fudk;) osQ ewy fcUnq
dks] fn, x, d.k&fudk; osQ nzO;eku osQUnz esa fy;k tk, rks
0
ii
m =
?
r A
,d n`<+ fi.M] tSls fd ehVj&NM+ ;k Ýykb Oghy] cgqr
ikl&ikl j[ks x, d.kksa dk fudk; gS_ vr% lehdj.k
(7.4a, b, c, d) n`<+ fi.M osQ fy, Hkh ykxw gksrs gSaA bl izdkj
osQ fi.Mksa esa d.kksa (ijek.kqvksa ;k v.kqvksaa) dh la[;k bruh