NCERT Textbook - समतल में गति Notes | EduRev

सामान्य विज्ञानं (General Science) for UPSC Prelims in Hindi

UPSC : NCERT Textbook - समतल में गति Notes | EduRev

 Page 1


66 HkkSfrdh
vè;k; 4
lery esa xfr
4-1 Hkwfedk
4-2 vfn'k ,oa lfn'k
4-3 lfn'kksa dh okLrfod la[;k ls xq.kk
4-4 lfn'kksa dk ladyu o O;odyu &
xzkiQh fof/
4-5 lfn'kksa dk fo;kstu
4-6 lfn'kksa dk ;ksx & fo'ys"k.kkRed
fof/
4-7 fdlh lery esa xfr
4-8 fdlh lery esa ,dleku Roj.k ls
xfr
4-9 nks foekvksa esa vkisf{kd osx
4-10 iz{ksI; xfr
4-11 ,dleku o`Ùkh; xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
vfrfjDr vH;kl
4-1 Hkwfedk
fiNys vè;k; esa geus fLFkfr] foLFkkiu] osx ,oa Roj.k dh èkkj.kkvksa dks
fodflr fd;k Fkk] ftudh fdlh oLrq dh ljy js[kh; xfr dk o.kZu djus
osQ fy, vko';drk iM+rh gS A D;ksafd ,dfoeh; xfr esa ek=k nks gh fn'kk,¡
laHko gSa] blfy, bu jkf'k;ksa osQ fn'kkRed i{k dks $ vkSj & fpÉksa ls O;Dr
dj ldrs gSa A ijarq tc ge oLrqvksa dh xfr dk f}foeh; (,d lery) ;k
f=kfoeh; (fno~QLFkku) o.kZu djuk pkgrs gSa] rc gesa mi;qZDr HkkSfrd jkf'k;ksa
dk vè;;u djus osQ fy, lfn'kksa dh vko';drk iM+rh gS A vr,o loZizFke
ge lfn'kksa dh Hkk"kk (vFkkZr lfn'kksa osQ xq.kksa ,oa mUgsa mi;ksx esa ykus dh
fofèk;k¡) lh[ksaxs A lfn'k D;k gS \ lfn'kksa dks oSQls tksM+k] ?kVk;k ;k xq.kk fd;k
tkrk gS \ lfn'kksa dks fdlh okLrfod la[;k ls xq.kk djsa rks gesa D;k ifj.kke
feysxk \ ;g lc ge blfy, lh[ksaxs ftlls fdlh lery esa oLrq osQ osx ,oa
Roj.k dks ifjHkkf"kr djus osQ fy, ge lfn'kksa dk mi;ksx dj losaQ A blosQ ckn
ge fdlh lery esa oLrq dh xfr ij ifjppkZ djsaxs A fdlh lery esa xfr
osQ ljy mnkgj.k osQ :i esa ge ,dleku Rofjr xfr dk vè;;u djsaxs rFkk
,d iz{ksI; dh xfr osQ fo"k; esa foLrkj ls i<s+axs A o`Ùkh; xfr ls ge HkyhHkk¡fr
ifjfpr gSa ftldk gekjs nSfud thou esa fo'ks"k egÙo gS A ge ,dleku o`Ùkh;
xfr dh oqQN foLrkj ls ppkZ djsaxs A
ge bl vè;k; esa ftu lehdj.kksa dks izkIr djsaxs mUgsa vklkuh ls
f=kfoeh; xfr osQ fy, foLrkfjr fd;k tk ldrk gS A
4-2 vfn'k ,oa lfn'k
ge HkkSfrd jkf'k;ksa dks vfn'kksa ,oa lfn'kksa esa oxhZÑr djrs gSa A nksuksa esa ewy
varj ;g gS fd lfn'k osQ lkFk fn'kk dks lac¼ djrs gSa ogha vfn'k osQ lkFk
,slk ugha djrs A ,d vfn'k jkf'k og jkf'k gS ftlesa ek=k ifjek.k gksrk gS A
bls osQoy ,d la[;k ,oa mfpr ek=kd }kjk iw.kZ :i ls O;Dr fd;k tk ldrk
gS A blosQ mnkgj.k gSa % nks fcanqvksa osQ chp dh nwjh] fdlh oLrq dh lagfr
(nzO;eku)] fdlh oLrq dk rkiØe] rFkk og le; ftl ij
dksbZ ?kVuk ?kVrh gS A vfn'kksa osQ tksM+ esa ogh fu;e ykxw gksrs gSa tks
lkekU;r;k chtxf.kr esa A vfn'kksa dks ge Bhd oSls gh tksM+ ldrs gSa] ?kVk
ldrs gSa] xq.kk ;k Hkkx dj ldrs gSa tSlk fd ge lkekU; la[;kvksa osQ lkFk
Page 2


66 HkkSfrdh
vè;k; 4
lery esa xfr
4-1 Hkwfedk
4-2 vfn'k ,oa lfn'k
4-3 lfn'kksa dh okLrfod la[;k ls xq.kk
4-4 lfn'kksa dk ladyu o O;odyu &
xzkiQh fof/
4-5 lfn'kksa dk fo;kstu
4-6 lfn'kksa dk ;ksx & fo'ys"k.kkRed
fof/
4-7 fdlh lery esa xfr
4-8 fdlh lery esa ,dleku Roj.k ls
xfr
4-9 nks foekvksa esa vkisf{kd osx
4-10 iz{ksI; xfr
4-11 ,dleku o`Ùkh; xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
vfrfjDr vH;kl
4-1 Hkwfedk
fiNys vè;k; esa geus fLFkfr] foLFkkiu] osx ,oa Roj.k dh èkkj.kkvksa dks
fodflr fd;k Fkk] ftudh fdlh oLrq dh ljy js[kh; xfr dk o.kZu djus
osQ fy, vko';drk iM+rh gS A D;ksafd ,dfoeh; xfr esa ek=k nks gh fn'kk,¡
laHko gSa] blfy, bu jkf'k;ksa osQ fn'kkRed i{k dks $ vkSj & fpÉksa ls O;Dr
dj ldrs gSa A ijarq tc ge oLrqvksa dh xfr dk f}foeh; (,d lery) ;k
f=kfoeh; (fno~QLFkku) o.kZu djuk pkgrs gSa] rc gesa mi;qZDr HkkSfrd jkf'k;ksa
dk vè;;u djus osQ fy, lfn'kksa dh vko';drk iM+rh gS A vr,o loZizFke
ge lfn'kksa dh Hkk"kk (vFkkZr lfn'kksa osQ xq.kksa ,oa mUgsa mi;ksx esa ykus dh
fofèk;k¡) lh[ksaxs A lfn'k D;k gS \ lfn'kksa dks oSQls tksM+k] ?kVk;k ;k xq.kk fd;k
tkrk gS \ lfn'kksa dks fdlh okLrfod la[;k ls xq.kk djsa rks gesa D;k ifj.kke
feysxk \ ;g lc ge blfy, lh[ksaxs ftlls fdlh lery esa oLrq osQ osx ,oa
Roj.k dks ifjHkkf"kr djus osQ fy, ge lfn'kksa dk mi;ksx dj losaQ A blosQ ckn
ge fdlh lery esa oLrq dh xfr ij ifjppkZ djsaxs A fdlh lery esa xfr
osQ ljy mnkgj.k osQ :i esa ge ,dleku Rofjr xfr dk vè;;u djsaxs rFkk
,d iz{ksI; dh xfr osQ fo"k; esa foLrkj ls i<s+axs A o`Ùkh; xfr ls ge HkyhHkk¡fr
ifjfpr gSa ftldk gekjs nSfud thou esa fo'ks"k egÙo gS A ge ,dleku o`Ùkh;
xfr dh oqQN foLrkj ls ppkZ djsaxs A
ge bl vè;k; esa ftu lehdj.kksa dks izkIr djsaxs mUgsa vklkuh ls
f=kfoeh; xfr osQ fy, foLrkfjr fd;k tk ldrk gS A
4-2 vfn'k ,oa lfn'k
ge HkkSfrd jkf'k;ksa dks vfn'kksa ,oa lfn'kksa esa oxhZÑr djrs gSa A nksuksa esa ewy
varj ;g gS fd lfn'k osQ lkFk fn'kk dks lac¼ djrs gSa ogha vfn'k osQ lkFk
,slk ugha djrs A ,d vfn'k jkf'k og jkf'k gS ftlesa ek=k ifjek.k gksrk gS A
bls osQoy ,d la[;k ,oa mfpr ek=kd }kjk iw.kZ :i ls O;Dr fd;k tk ldrk
gS A blosQ mnkgj.k gSa % nks fcanqvksa osQ chp dh nwjh] fdlh oLrq dh lagfr
(nzO;eku)] fdlh oLrq dk rkiØe] rFkk og le; ftl ij
dksbZ ?kVuk ?kVrh gS A vfn'kksa osQ tksM+ esa ogh fu;e ykxw gksrs gSa tks
lkekU;r;k chtxf.kr esa A vfn'kksa dks ge Bhd oSls gh tksM+ ldrs gSa] ?kVk
ldrs gSa] xq.kk ;k Hkkx dj ldrs gSa tSlk fd ge lkekU; la[;kvksa osQ lkFk
67 lery esa xfr
* osQoy leku ek=kd okyh jkf'k;ksa dk tksM+ o ?kVkuk lkFkZd gksrk gS A tcfd vki fHkUu ek=kdksa okys vfn'kksa dk xq.kk ;k Hkkx dj ldrs gSa A
** gekjs vè;;u esa lfn'kksa dh fLFkfr;ka fuèkkZfjr ugha gSa A blfy, tc ,d lfn'k dks Lo;a osQ lekarj foLFkkfir djrs gSa rks lfn'k vifjofrZr
jgrk gS A bl izdkj osQ lfn'kksa dks ge ^eqDr lfn'k* dgrs gSa A gkyakfd oqQN HkkSfrd mi;ksxksa esa lfn'k dh fLFkfr ;k mldh fØ;k js[kk
egÙoiw.kZ gksrh gS A ,sls lfn'kksa dks ge ^LFkkuxr lfn'k* dgrs gSaA
djrs gSa * A mnkgj.k osQ fy,] ;fn fdlh vk;r dh yackbZ vkSj
pkSM+kbZ Øe'k% 1-0 m rFkk 0-5 m gS rks mldh ifjeki pkjksa
Hkqtkvksa osQ ;ksx] 1.0 m + 0.5 m + 1.0 m + 0.5 m =
3.0 m  gksxkA gj Hkqtk dh yackbZ ,d vfn'k gS rFkk ifjeki
Hkh ,d vfn'k gS A ge ,d nwljs mnkgj.k ij fopkj djsaxs %
;fn fdlh ,d fnu dk vfèkdre ,oa U;wure rki Øe'k%
35-6 °C rFkk 24-2 °C gS rks bu nksuksa dk varj 11-4 °C
gksxk A blh izdkj ;fn ,Y;qfefu;e osQ fdlh ,dleku Bksl
?ku dh Hkqtk 10 cm gS vkSj mldk nzO;eku 2.7 kg gS rks
mldk vk;ru 10
–3 
m
3
  (,d vfn'k) gksxk rFkk ?kuRo
2.7×10
3
 kg/m
3
 Hkh ,d vfn'k gS A
,d lfn'k jkf'k og jkf'k gS ftlesa ifjek.k rFkk fn'kk
nksuksa gksrs gSa rFkk og ;ksx lacaèkh f=kHkqt osQ fu;e vFkok
lekukUrj prqHkqZt osQ ;ksx lacaèkh fu;e dk ikyu djrh
gS A bl izdkj] ,d lfn'k dks mlosQ ifjek.k dh la[;k rFkk
fn'kk }kjk O;Dr djrs gSa A oqQN HkkSfrd jkf'k;k¡ ftUgsa lfn'kksa
}kjk O;Dr djrs gSa] os gSa foLFkkiu] osx] Roj.k rFkk cy A
lfn'k dks O;Dr djus osQ fy, bl iqLrd esa ge eksVs
v{kjksa dk iz;ksx djsaxs A tSls fd osx lfn'k dks O;Dr djus
osQ fy, v fpÉ dk iz;ksx djsaxs A ijarq gkFk ls fy[krs le;
D;ksafd eksVs v{kjksa dk fy[kuk FkksM+k eqf'dy gksrk gS] blfy,
,d lfn'k dks v{kj osQ mQij rhj yxkdj O;Dr djrs gSa] tSls
?
v  A bl izdkj v rFkk 
?
v  nksuksa gh osx lfn'k dks O;Dr djrs
gSa A fdlh lfn'k osQ ifjek.k dks izk;% ge mldk ^ije eku*
dgrs gSa vkSj mls |v| = v  }kjk O;Dr djrs gSa A bl izdkj
,d lfn'k dks ge eksVs v{kj ;Fkk A ;k a, p, q, r, ..... x,
y ls O;Dr djrs gSa tcfd buosQ ifjek.kksa dks Øe'k% ge A
;k a, p, q, r, .... x, y }kjk O;Dr djrs gSa A
4-2-1 fLFkfr ,oa foLFkkiu lfn'k
fdlh lery esa xfreku oLrq dh fLFkfr O;Dr djus osQ fy,
ge lqfoèkkuqlkj fdlh fcanq O dks ewy fcanq osQ :i esa pqurs
gSa A dYiuk dhft, fd nks fHkUu&fHkUu le;ksa t  vkSj t'  ij
oLrq dh fLFkfr Øe'k% P vkSj P' gS (fp=k 4.1a) A ge P
dks O ls ,d ljy js[kk ls tksM+ nsrs gSa A bl izdkj OP le;
t ij oLrq dh fLFkfr lfn'k gksxh A bl js[kk osQ fljs ij ,d
rhj dk fu'kku yxk nsrs gSa A bls fdlh fpÉ (eku yhft,)
r ls fu:fir djrs gSa] vFkkZr~ OP = r A blh izdkj fcanq P'
dks ,d nwljs fLFkfr lfn'k OP' ;kuh r' ls fu:fir djrs gSaA
lfn'k r dh yackbZ mlosQ ifjek.k dks fu:fir djrh gS rFkk
lfn'k dh fn'kk og gksxh ftlosQ vuqfn'k P (fcanq O ls ns[kus
ij) fLFkr gksxk A ;fn oLrq P ls pydj P' ij igqap tkrh
gS rks lfn'k PP' (ftldh iqPN P ij rFkk 'kh"kZ P' ij gS)
¯cnq P (le; t) ls P' (le; t') rd xfr osQ laxr
foLFkkiu lfn'k dgykrk gS A
fp=k 4.1 (a) fLFkfr rFkk foLFkkiu lfn'k] (b) foLFkkiu lfn'k
PQ rFkk xfr osQ fHkUu&fHkUu ekxZ A
;gk¡ ;g ckr egRoiw.kZ gS fd ^foLFkkiu lfn'k* dks ,d
ljy js[kk ls O;Dr djrs gSa tks oLrq dh vafre fLFkfr dks
mldh izkjfEHkd fLFkfr ls tksM+rh gS rFkk ;g ml okLrfod
iFk ij fuHkZj ugha djrk tks oLrq }kjk fcanqvksa osQ eè; pyk
tkrk gS A mnkgj.kLo:i] tSlk fd fp=k 4.1b esa fn[kk;k x;k
gS] izkjfEHkd fLFkfr P rFkk vafre fLFkfr Q osQ eè; foLFkkiu
lfn'k PQ ;|fi ogh gS ijarq nksuksa fLFkfr;ksa osQ chp pyh xbZ
nwfj;ka tSls PABCQ, PDQ rFkk PBEFQ vyx&vyx gSa A
blh izdkj] fdUgha nks fcanqvksa osQ eè; foLFkkiu lfn'k dk
ifjek.k ;k rks xfreku oLrq dh iFk&yackbZ ls de gksrk
gS ;k mlosQ cjkcj gksrk gSA fiNys vè;k; esa Hkh ,d ljy
js[kk osQ vuqfn'k xfreku oLrq osQ fy, bl rF; dks HkyhHkkafr
le>k;k x;k Fkk A
4-2-2 lfn'kksa dh lerk
nks lfn'kksa A rFkk B dks osQoy rHkh cjkcj dgk tk ldrk
gS tc muosQ ifjek.k cjkcj gksa rFkk mudh fn'kk leku gks** A
fp=k 4.2(a) esa nks leku lfn'kksa A rFkk B dks n'kkZ;k
x;k gS A ge budh lekurk dh ij[k vklkuh ls dj ldrs
gSa A B dks Lo;a osQ lekarj f[kldkb;s rkfd mldh iqPN Q
lfn'k A dh iqPN O osQ laikrh gks tk, A fiQj D;ksafd muosQ
'kh"kZ S ,oa P Hkh laikrh gSa vr% nksukssa lfn'k cjkcj dgyk,axs A
lkekU;r;k bl lekurk dks A = B osQ :i esa fy[krs gSa A bl
Page 3


66 HkkSfrdh
vè;k; 4
lery esa xfr
4-1 Hkwfedk
4-2 vfn'k ,oa lfn'k
4-3 lfn'kksa dh okLrfod la[;k ls xq.kk
4-4 lfn'kksa dk ladyu o O;odyu &
xzkiQh fof/
4-5 lfn'kksa dk fo;kstu
4-6 lfn'kksa dk ;ksx & fo'ys"k.kkRed
fof/
4-7 fdlh lery esa xfr
4-8 fdlh lery esa ,dleku Roj.k ls
xfr
4-9 nks foekvksa esa vkisf{kd osx
4-10 iz{ksI; xfr
4-11 ,dleku o`Ùkh; xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
vfrfjDr vH;kl
4-1 Hkwfedk
fiNys vè;k; esa geus fLFkfr] foLFkkiu] osx ,oa Roj.k dh èkkj.kkvksa dks
fodflr fd;k Fkk] ftudh fdlh oLrq dh ljy js[kh; xfr dk o.kZu djus
osQ fy, vko';drk iM+rh gS A D;ksafd ,dfoeh; xfr esa ek=k nks gh fn'kk,¡
laHko gSa] blfy, bu jkf'k;ksa osQ fn'kkRed i{k dks $ vkSj & fpÉksa ls O;Dr
dj ldrs gSa A ijarq tc ge oLrqvksa dh xfr dk f}foeh; (,d lery) ;k
f=kfoeh; (fno~QLFkku) o.kZu djuk pkgrs gSa] rc gesa mi;qZDr HkkSfrd jkf'k;ksa
dk vè;;u djus osQ fy, lfn'kksa dh vko';drk iM+rh gS A vr,o loZizFke
ge lfn'kksa dh Hkk"kk (vFkkZr lfn'kksa osQ xq.kksa ,oa mUgsa mi;ksx esa ykus dh
fofèk;k¡) lh[ksaxs A lfn'k D;k gS \ lfn'kksa dks oSQls tksM+k] ?kVk;k ;k xq.kk fd;k
tkrk gS \ lfn'kksa dks fdlh okLrfod la[;k ls xq.kk djsa rks gesa D;k ifj.kke
feysxk \ ;g lc ge blfy, lh[ksaxs ftlls fdlh lery esa oLrq osQ osx ,oa
Roj.k dks ifjHkkf"kr djus osQ fy, ge lfn'kksa dk mi;ksx dj losaQ A blosQ ckn
ge fdlh lery esa oLrq dh xfr ij ifjppkZ djsaxs A fdlh lery esa xfr
osQ ljy mnkgj.k osQ :i esa ge ,dleku Rofjr xfr dk vè;;u djsaxs rFkk
,d iz{ksI; dh xfr osQ fo"k; esa foLrkj ls i<s+axs A o`Ùkh; xfr ls ge HkyhHkk¡fr
ifjfpr gSa ftldk gekjs nSfud thou esa fo'ks"k egÙo gS A ge ,dleku o`Ùkh;
xfr dh oqQN foLrkj ls ppkZ djsaxs A
ge bl vè;k; esa ftu lehdj.kksa dks izkIr djsaxs mUgsa vklkuh ls
f=kfoeh; xfr osQ fy, foLrkfjr fd;k tk ldrk gS A
4-2 vfn'k ,oa lfn'k
ge HkkSfrd jkf'k;ksa dks vfn'kksa ,oa lfn'kksa esa oxhZÑr djrs gSa A nksuksa esa ewy
varj ;g gS fd lfn'k osQ lkFk fn'kk dks lac¼ djrs gSa ogha vfn'k osQ lkFk
,slk ugha djrs A ,d vfn'k jkf'k og jkf'k gS ftlesa ek=k ifjek.k gksrk gS A
bls osQoy ,d la[;k ,oa mfpr ek=kd }kjk iw.kZ :i ls O;Dr fd;k tk ldrk
gS A blosQ mnkgj.k gSa % nks fcanqvksa osQ chp dh nwjh] fdlh oLrq dh lagfr
(nzO;eku)] fdlh oLrq dk rkiØe] rFkk og le; ftl ij
dksbZ ?kVuk ?kVrh gS A vfn'kksa osQ tksM+ esa ogh fu;e ykxw gksrs gSa tks
lkekU;r;k chtxf.kr esa A vfn'kksa dks ge Bhd oSls gh tksM+ ldrs gSa] ?kVk
ldrs gSa] xq.kk ;k Hkkx dj ldrs gSa tSlk fd ge lkekU; la[;kvksa osQ lkFk
67 lery esa xfr
* osQoy leku ek=kd okyh jkf'k;ksa dk tksM+ o ?kVkuk lkFkZd gksrk gS A tcfd vki fHkUu ek=kdksa okys vfn'kksa dk xq.kk ;k Hkkx dj ldrs gSa A
** gekjs vè;;u esa lfn'kksa dh fLFkfr;ka fuèkkZfjr ugha gSa A blfy, tc ,d lfn'k dks Lo;a osQ lekarj foLFkkfir djrs gSa rks lfn'k vifjofrZr
jgrk gS A bl izdkj osQ lfn'kksa dks ge ^eqDr lfn'k* dgrs gSa A gkyakfd oqQN HkkSfrd mi;ksxksa esa lfn'k dh fLFkfr ;k mldh fØ;k js[kk
egÙoiw.kZ gksrh gS A ,sls lfn'kksa dks ge ^LFkkuxr lfn'k* dgrs gSaA
djrs gSa * A mnkgj.k osQ fy,] ;fn fdlh vk;r dh yackbZ vkSj
pkSM+kbZ Øe'k% 1-0 m rFkk 0-5 m gS rks mldh ifjeki pkjksa
Hkqtkvksa osQ ;ksx] 1.0 m + 0.5 m + 1.0 m + 0.5 m =
3.0 m  gksxkA gj Hkqtk dh yackbZ ,d vfn'k gS rFkk ifjeki
Hkh ,d vfn'k gS A ge ,d nwljs mnkgj.k ij fopkj djsaxs %
;fn fdlh ,d fnu dk vfèkdre ,oa U;wure rki Øe'k%
35-6 °C rFkk 24-2 °C gS rks bu nksuksa dk varj 11-4 °C
gksxk A blh izdkj ;fn ,Y;qfefu;e osQ fdlh ,dleku Bksl
?ku dh Hkqtk 10 cm gS vkSj mldk nzO;eku 2.7 kg gS rks
mldk vk;ru 10
–3 
m
3
  (,d vfn'k) gksxk rFkk ?kuRo
2.7×10
3
 kg/m
3
 Hkh ,d vfn'k gS A
,d lfn'k jkf'k og jkf'k gS ftlesa ifjek.k rFkk fn'kk
nksuksa gksrs gSa rFkk og ;ksx lacaèkh f=kHkqt osQ fu;e vFkok
lekukUrj prqHkqZt osQ ;ksx lacaèkh fu;e dk ikyu djrh
gS A bl izdkj] ,d lfn'k dks mlosQ ifjek.k dh la[;k rFkk
fn'kk }kjk O;Dr djrs gSa A oqQN HkkSfrd jkf'k;k¡ ftUgsa lfn'kksa
}kjk O;Dr djrs gSa] os gSa foLFkkiu] osx] Roj.k rFkk cy A
lfn'k dks O;Dr djus osQ fy, bl iqLrd esa ge eksVs
v{kjksa dk iz;ksx djsaxs A tSls fd osx lfn'k dks O;Dr djus
osQ fy, v fpÉ dk iz;ksx djsaxs A ijarq gkFk ls fy[krs le;
D;ksafd eksVs v{kjksa dk fy[kuk FkksM+k eqf'dy gksrk gS] blfy,
,d lfn'k dks v{kj osQ mQij rhj yxkdj O;Dr djrs gSa] tSls
?
v  A bl izdkj v rFkk 
?
v  nksuksa gh osx lfn'k dks O;Dr djrs
gSa A fdlh lfn'k osQ ifjek.k dks izk;% ge mldk ^ije eku*
dgrs gSa vkSj mls |v| = v  }kjk O;Dr djrs gSa A bl izdkj
,d lfn'k dks ge eksVs v{kj ;Fkk A ;k a, p, q, r, ..... x,
y ls O;Dr djrs gSa tcfd buosQ ifjek.kksa dks Øe'k% ge A
;k a, p, q, r, .... x, y }kjk O;Dr djrs gSa A
4-2-1 fLFkfr ,oa foLFkkiu lfn'k
fdlh lery esa xfreku oLrq dh fLFkfr O;Dr djus osQ fy,
ge lqfoèkkuqlkj fdlh fcanq O dks ewy fcanq osQ :i esa pqurs
gSa A dYiuk dhft, fd nks fHkUu&fHkUu le;ksa t  vkSj t'  ij
oLrq dh fLFkfr Øe'k% P vkSj P' gS (fp=k 4.1a) A ge P
dks O ls ,d ljy js[kk ls tksM+ nsrs gSa A bl izdkj OP le;
t ij oLrq dh fLFkfr lfn'k gksxh A bl js[kk osQ fljs ij ,d
rhj dk fu'kku yxk nsrs gSa A bls fdlh fpÉ (eku yhft,)
r ls fu:fir djrs gSa] vFkkZr~ OP = r A blh izdkj fcanq P'
dks ,d nwljs fLFkfr lfn'k OP' ;kuh r' ls fu:fir djrs gSaA
lfn'k r dh yackbZ mlosQ ifjek.k dks fu:fir djrh gS rFkk
lfn'k dh fn'kk og gksxh ftlosQ vuqfn'k P (fcanq O ls ns[kus
ij) fLFkr gksxk A ;fn oLrq P ls pydj P' ij igqap tkrh
gS rks lfn'k PP' (ftldh iqPN P ij rFkk 'kh"kZ P' ij gS)
¯cnq P (le; t) ls P' (le; t') rd xfr osQ laxr
foLFkkiu lfn'k dgykrk gS A
fp=k 4.1 (a) fLFkfr rFkk foLFkkiu lfn'k] (b) foLFkkiu lfn'k
PQ rFkk xfr osQ fHkUu&fHkUu ekxZ A
;gk¡ ;g ckr egRoiw.kZ gS fd ^foLFkkiu lfn'k* dks ,d
ljy js[kk ls O;Dr djrs gSa tks oLrq dh vafre fLFkfr dks
mldh izkjfEHkd fLFkfr ls tksM+rh gS rFkk ;g ml okLrfod
iFk ij fuHkZj ugha djrk tks oLrq }kjk fcanqvksa osQ eè; pyk
tkrk gS A mnkgj.kLo:i] tSlk fd fp=k 4.1b esa fn[kk;k x;k
gS] izkjfEHkd fLFkfr P rFkk vafre fLFkfr Q osQ eè; foLFkkiu
lfn'k PQ ;|fi ogh gS ijarq nksuksa fLFkfr;ksa osQ chp pyh xbZ
nwfj;ka tSls PABCQ, PDQ rFkk PBEFQ vyx&vyx gSa A
blh izdkj] fdUgha nks fcanqvksa osQ eè; foLFkkiu lfn'k dk
ifjek.k ;k rks xfreku oLrq dh iFk&yackbZ ls de gksrk
gS ;k mlosQ cjkcj gksrk gSA fiNys vè;k; esa Hkh ,d ljy
js[kk osQ vuqfn'k xfreku oLrq osQ fy, bl rF; dks HkyhHkkafr
le>k;k x;k Fkk A
4-2-2 lfn'kksa dh lerk
nks lfn'kksa A rFkk B dks osQoy rHkh cjkcj dgk tk ldrk
gS tc muosQ ifjek.k cjkcj gksa rFkk mudh fn'kk leku gks** A
fp=k 4.2(a) esa nks leku lfn'kksa A rFkk B dks n'kkZ;k
x;k gS A ge budh lekurk dh ij[k vklkuh ls dj ldrs
gSa A B dks Lo;a osQ lekarj f[kldkb;s rkfd mldh iqPN Q
lfn'k A dh iqPN O osQ laikrh gks tk, A fiQj D;ksafd muosQ
'kh"kZ S ,oa P Hkh laikrh gSa vr% nksukssa lfn'k cjkcj dgyk,axs A
lkekU;r;k bl lekurk dks A = B osQ :i esa fy[krs gSa A bl
68 HkkSfrdh
ckr dh vksj è;ku nhft, fd fp=k 4.2(b) esa ;|fi lfn'kksa
A' rFkk B' osQ ifjek.k leku gSa fiQj Hkh nksuksa lfn'k leku ugha
gSa D;ksafd mudh fn'kk;sa vyx&vyx gSa A ;fn ge B' dks
mlosQ gh lekarj f[kldk,a ftlls mldh iqPN Q', A' dh
iqPN O' ls laikrh gks tk, rks Hkh B' dk 'kh"kZ S', A' osQ 'kh"kZ
P' dk laikrh ugha gksxk A
4-3 lfn'kksa dh okLrfod la[;k ls xq.kk
;fn ,d lfn'k A dks fdlh èkukRed la[;k ? ls xq.kk djsa
rks gesa ,d lfn'k gh feyrk gS ftldk ifjek.k lfn'k A osQ
ifjek.k dk ? xquk gks tkrk gS rFkk ftldh fn'kk ogh gS tks
A dh gS A bl xq.kuiQy dks ge ?A ls fy[krs gSa A
| | | | A A ? = ? ;fn 0 > ?
mnkgj.kLo:i] ;fn A dks 2 ls xq.kk fd;k tk,] rks ifj.kkeh
lfn'k 2A gksxk (fp=k 4.3a) ftldh fn'kk A dh fn'kk gksxh
rFkk ifjek.k | | A dk nksxquk gksxk A lfn'k A dks ;fn ,d
Í.kkRed la[;k ? ls xq.kk djsa rks lfn'k ?? ? ? ? ? izkIr gksrk gS
ftldh fn'kk A dh fn'kk osQ foijhr gS vkSj ftldk ifjek.k
| | A dk &? xquk gksrk gS A
;fn fdlh lfn'k A dks Í.kkRed la[;kvksa &1
o&1-5 ls xq.kk djsa rks ifj.kkeh lfn'k fp=k 4-3(b) tSls
gksaxs A
HkkSfrdh esa ftl ?kVd ? }kjk lfn'k A dks xq.kk fd;k
tkrk gS og dksbZ vfn'k gks ldrk gS ftldh Lo;a dh foek,¡
gksrh gSa A vr,o ?? ? ? ? ? dh foek,¡ ? o A dh foekvksa osQ
xq.kuiQy osQ cjkcj gksaxh A mnkgj.kLo:i] ;fn ge fdlh
vpj osx lfn'k dks fdlh (le;) varjky ls xq.kk djsa rks gesa
,d foLFkkiu lfn'k izkIr gksxk A
4-4 lfn'kksa dk ladyu o O;odyu % xzkIkQh fofèk
tSlk fd [k.M 4-2 esa cryk;k tk pqdk gS fd lfn'k ;ksx osQ
f=kHkqt fu;e ;k lekUrj prqHkqZt osQ ;ksx osQ fu;e dk ikyu
djrs gSa A vc ge xzkiQh fofèk }kjk ;ksx osQ bl fu;e dks
le>k,axs A ge fp=k 4.4 (a) esa n'kkZ, vuqlkj fdlh lery
esa fLFkr nks lfn'kksa A rFkk B ij fopkj djrs gSa A bu lfn'kksa
dks O;Dr djus okyh js[kk&[k.Mksa dh yackb;k¡ lfn'kksa osQ
ifjek.k osQ lekuqikrh gSa A ;ksx A + B izkIr djus osQ fy, fp=k
4-4(b) osQ vuqlkj ge lfn'k B bl izdkj j[krs gSa fd
mldh iqPN lfn'k A osQ 'kh"kZ ij gks A fiQj ge A dh iqPN
fp=k 4.2 (a) nks leku lfn'k A rFkk B, (b) nks lfn'k A' o
B' vleku gSa ;|fi mudh yackb;k¡ ogh gSa A
fp=k 4-3 (a) lfn'k A rFkk mls èkukRed la[;k nks ls xq.kk
djus ij izkIr ifj.kkeh lfn'k] (b) lfn'k A rFkk
mls Í.kkRed la[;kvksa &1 rFkk &1-5 ls xq.kk djus
ij izkIr ifj.kkeh lfn'k A
fp=k 4-4 (a) lfn'k A rFkk B, (b) lfn'kksa A o B dk xzkiQh fofèk }kjk tksM+uk] (c) lfn'kksa B o A dk xzkiQh fofèk }kjk tksM+uk]
(d) lfn'kksa osQ tksM+ ls lacafèkr lkgp;Z fu;e dk izn'kZu A
Page 4


66 HkkSfrdh
vè;k; 4
lery esa xfr
4-1 Hkwfedk
4-2 vfn'k ,oa lfn'k
4-3 lfn'kksa dh okLrfod la[;k ls xq.kk
4-4 lfn'kksa dk ladyu o O;odyu &
xzkiQh fof/
4-5 lfn'kksa dk fo;kstu
4-6 lfn'kksa dk ;ksx & fo'ys"k.kkRed
fof/
4-7 fdlh lery esa xfr
4-8 fdlh lery esa ,dleku Roj.k ls
xfr
4-9 nks foekvksa esa vkisf{kd osx
4-10 iz{ksI; xfr
4-11 ,dleku o`Ùkh; xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
vfrfjDr vH;kl
4-1 Hkwfedk
fiNys vè;k; esa geus fLFkfr] foLFkkiu] osx ,oa Roj.k dh èkkj.kkvksa dks
fodflr fd;k Fkk] ftudh fdlh oLrq dh ljy js[kh; xfr dk o.kZu djus
osQ fy, vko';drk iM+rh gS A D;ksafd ,dfoeh; xfr esa ek=k nks gh fn'kk,¡
laHko gSa] blfy, bu jkf'k;ksa osQ fn'kkRed i{k dks $ vkSj & fpÉksa ls O;Dr
dj ldrs gSa A ijarq tc ge oLrqvksa dh xfr dk f}foeh; (,d lery) ;k
f=kfoeh; (fno~QLFkku) o.kZu djuk pkgrs gSa] rc gesa mi;qZDr HkkSfrd jkf'k;ksa
dk vè;;u djus osQ fy, lfn'kksa dh vko';drk iM+rh gS A vr,o loZizFke
ge lfn'kksa dh Hkk"kk (vFkkZr lfn'kksa osQ xq.kksa ,oa mUgsa mi;ksx esa ykus dh
fofèk;k¡) lh[ksaxs A lfn'k D;k gS \ lfn'kksa dks oSQls tksM+k] ?kVk;k ;k xq.kk fd;k
tkrk gS \ lfn'kksa dks fdlh okLrfod la[;k ls xq.kk djsa rks gesa D;k ifj.kke
feysxk \ ;g lc ge blfy, lh[ksaxs ftlls fdlh lery esa oLrq osQ osx ,oa
Roj.k dks ifjHkkf"kr djus osQ fy, ge lfn'kksa dk mi;ksx dj losaQ A blosQ ckn
ge fdlh lery esa oLrq dh xfr ij ifjppkZ djsaxs A fdlh lery esa xfr
osQ ljy mnkgj.k osQ :i esa ge ,dleku Rofjr xfr dk vè;;u djsaxs rFkk
,d iz{ksI; dh xfr osQ fo"k; esa foLrkj ls i<s+axs A o`Ùkh; xfr ls ge HkyhHkk¡fr
ifjfpr gSa ftldk gekjs nSfud thou esa fo'ks"k egÙo gS A ge ,dleku o`Ùkh;
xfr dh oqQN foLrkj ls ppkZ djsaxs A
ge bl vè;k; esa ftu lehdj.kksa dks izkIr djsaxs mUgsa vklkuh ls
f=kfoeh; xfr osQ fy, foLrkfjr fd;k tk ldrk gS A
4-2 vfn'k ,oa lfn'k
ge HkkSfrd jkf'k;ksa dks vfn'kksa ,oa lfn'kksa esa oxhZÑr djrs gSa A nksuksa esa ewy
varj ;g gS fd lfn'k osQ lkFk fn'kk dks lac¼ djrs gSa ogha vfn'k osQ lkFk
,slk ugha djrs A ,d vfn'k jkf'k og jkf'k gS ftlesa ek=k ifjek.k gksrk gS A
bls osQoy ,d la[;k ,oa mfpr ek=kd }kjk iw.kZ :i ls O;Dr fd;k tk ldrk
gS A blosQ mnkgj.k gSa % nks fcanqvksa osQ chp dh nwjh] fdlh oLrq dh lagfr
(nzO;eku)] fdlh oLrq dk rkiØe] rFkk og le; ftl ij
dksbZ ?kVuk ?kVrh gS A vfn'kksa osQ tksM+ esa ogh fu;e ykxw gksrs gSa tks
lkekU;r;k chtxf.kr esa A vfn'kksa dks ge Bhd oSls gh tksM+ ldrs gSa] ?kVk
ldrs gSa] xq.kk ;k Hkkx dj ldrs gSa tSlk fd ge lkekU; la[;kvksa osQ lkFk
67 lery esa xfr
* osQoy leku ek=kd okyh jkf'k;ksa dk tksM+ o ?kVkuk lkFkZd gksrk gS A tcfd vki fHkUu ek=kdksa okys vfn'kksa dk xq.kk ;k Hkkx dj ldrs gSa A
** gekjs vè;;u esa lfn'kksa dh fLFkfr;ka fuèkkZfjr ugha gSa A blfy, tc ,d lfn'k dks Lo;a osQ lekarj foLFkkfir djrs gSa rks lfn'k vifjofrZr
jgrk gS A bl izdkj osQ lfn'kksa dks ge ^eqDr lfn'k* dgrs gSa A gkyakfd oqQN HkkSfrd mi;ksxksa esa lfn'k dh fLFkfr ;k mldh fØ;k js[kk
egÙoiw.kZ gksrh gS A ,sls lfn'kksa dks ge ^LFkkuxr lfn'k* dgrs gSaA
djrs gSa * A mnkgj.k osQ fy,] ;fn fdlh vk;r dh yackbZ vkSj
pkSM+kbZ Øe'k% 1-0 m rFkk 0-5 m gS rks mldh ifjeki pkjksa
Hkqtkvksa osQ ;ksx] 1.0 m + 0.5 m + 1.0 m + 0.5 m =
3.0 m  gksxkA gj Hkqtk dh yackbZ ,d vfn'k gS rFkk ifjeki
Hkh ,d vfn'k gS A ge ,d nwljs mnkgj.k ij fopkj djsaxs %
;fn fdlh ,d fnu dk vfèkdre ,oa U;wure rki Øe'k%
35-6 °C rFkk 24-2 °C gS rks bu nksuksa dk varj 11-4 °C
gksxk A blh izdkj ;fn ,Y;qfefu;e osQ fdlh ,dleku Bksl
?ku dh Hkqtk 10 cm gS vkSj mldk nzO;eku 2.7 kg gS rks
mldk vk;ru 10
–3 
m
3
  (,d vfn'k) gksxk rFkk ?kuRo
2.7×10
3
 kg/m
3
 Hkh ,d vfn'k gS A
,d lfn'k jkf'k og jkf'k gS ftlesa ifjek.k rFkk fn'kk
nksuksa gksrs gSa rFkk og ;ksx lacaèkh f=kHkqt osQ fu;e vFkok
lekukUrj prqHkqZt osQ ;ksx lacaèkh fu;e dk ikyu djrh
gS A bl izdkj] ,d lfn'k dks mlosQ ifjek.k dh la[;k rFkk
fn'kk }kjk O;Dr djrs gSa A oqQN HkkSfrd jkf'k;k¡ ftUgsa lfn'kksa
}kjk O;Dr djrs gSa] os gSa foLFkkiu] osx] Roj.k rFkk cy A
lfn'k dks O;Dr djus osQ fy, bl iqLrd esa ge eksVs
v{kjksa dk iz;ksx djsaxs A tSls fd osx lfn'k dks O;Dr djus
osQ fy, v fpÉ dk iz;ksx djsaxs A ijarq gkFk ls fy[krs le;
D;ksafd eksVs v{kjksa dk fy[kuk FkksM+k eqf'dy gksrk gS] blfy,
,d lfn'k dks v{kj osQ mQij rhj yxkdj O;Dr djrs gSa] tSls
?
v  A bl izdkj v rFkk 
?
v  nksuksa gh osx lfn'k dks O;Dr djrs
gSa A fdlh lfn'k osQ ifjek.k dks izk;% ge mldk ^ije eku*
dgrs gSa vkSj mls |v| = v  }kjk O;Dr djrs gSa A bl izdkj
,d lfn'k dks ge eksVs v{kj ;Fkk A ;k a, p, q, r, ..... x,
y ls O;Dr djrs gSa tcfd buosQ ifjek.kksa dks Øe'k% ge A
;k a, p, q, r, .... x, y }kjk O;Dr djrs gSa A
4-2-1 fLFkfr ,oa foLFkkiu lfn'k
fdlh lery esa xfreku oLrq dh fLFkfr O;Dr djus osQ fy,
ge lqfoèkkuqlkj fdlh fcanq O dks ewy fcanq osQ :i esa pqurs
gSa A dYiuk dhft, fd nks fHkUu&fHkUu le;ksa t  vkSj t'  ij
oLrq dh fLFkfr Øe'k% P vkSj P' gS (fp=k 4.1a) A ge P
dks O ls ,d ljy js[kk ls tksM+ nsrs gSa A bl izdkj OP le;
t ij oLrq dh fLFkfr lfn'k gksxh A bl js[kk osQ fljs ij ,d
rhj dk fu'kku yxk nsrs gSa A bls fdlh fpÉ (eku yhft,)
r ls fu:fir djrs gSa] vFkkZr~ OP = r A blh izdkj fcanq P'
dks ,d nwljs fLFkfr lfn'k OP' ;kuh r' ls fu:fir djrs gSaA
lfn'k r dh yackbZ mlosQ ifjek.k dks fu:fir djrh gS rFkk
lfn'k dh fn'kk og gksxh ftlosQ vuqfn'k P (fcanq O ls ns[kus
ij) fLFkr gksxk A ;fn oLrq P ls pydj P' ij igqap tkrh
gS rks lfn'k PP' (ftldh iqPN P ij rFkk 'kh"kZ P' ij gS)
¯cnq P (le; t) ls P' (le; t') rd xfr osQ laxr
foLFkkiu lfn'k dgykrk gS A
fp=k 4.1 (a) fLFkfr rFkk foLFkkiu lfn'k] (b) foLFkkiu lfn'k
PQ rFkk xfr osQ fHkUu&fHkUu ekxZ A
;gk¡ ;g ckr egRoiw.kZ gS fd ^foLFkkiu lfn'k* dks ,d
ljy js[kk ls O;Dr djrs gSa tks oLrq dh vafre fLFkfr dks
mldh izkjfEHkd fLFkfr ls tksM+rh gS rFkk ;g ml okLrfod
iFk ij fuHkZj ugha djrk tks oLrq }kjk fcanqvksa osQ eè; pyk
tkrk gS A mnkgj.kLo:i] tSlk fd fp=k 4.1b esa fn[kk;k x;k
gS] izkjfEHkd fLFkfr P rFkk vafre fLFkfr Q osQ eè; foLFkkiu
lfn'k PQ ;|fi ogh gS ijarq nksuksa fLFkfr;ksa osQ chp pyh xbZ
nwfj;ka tSls PABCQ, PDQ rFkk PBEFQ vyx&vyx gSa A
blh izdkj] fdUgha nks fcanqvksa osQ eè; foLFkkiu lfn'k dk
ifjek.k ;k rks xfreku oLrq dh iFk&yackbZ ls de gksrk
gS ;k mlosQ cjkcj gksrk gSA fiNys vè;k; esa Hkh ,d ljy
js[kk osQ vuqfn'k xfreku oLrq osQ fy, bl rF; dks HkyhHkkafr
le>k;k x;k Fkk A
4-2-2 lfn'kksa dh lerk
nks lfn'kksa A rFkk B dks osQoy rHkh cjkcj dgk tk ldrk
gS tc muosQ ifjek.k cjkcj gksa rFkk mudh fn'kk leku gks** A
fp=k 4.2(a) esa nks leku lfn'kksa A rFkk B dks n'kkZ;k
x;k gS A ge budh lekurk dh ij[k vklkuh ls dj ldrs
gSa A B dks Lo;a osQ lekarj f[kldkb;s rkfd mldh iqPN Q
lfn'k A dh iqPN O osQ laikrh gks tk, A fiQj D;ksafd muosQ
'kh"kZ S ,oa P Hkh laikrh gSa vr% nksukssa lfn'k cjkcj dgyk,axs A
lkekU;r;k bl lekurk dks A = B osQ :i esa fy[krs gSa A bl
68 HkkSfrdh
ckr dh vksj è;ku nhft, fd fp=k 4.2(b) esa ;|fi lfn'kksa
A' rFkk B' osQ ifjek.k leku gSa fiQj Hkh nksuksa lfn'k leku ugha
gSa D;ksafd mudh fn'kk;sa vyx&vyx gSa A ;fn ge B' dks
mlosQ gh lekarj f[kldk,a ftlls mldh iqPN Q', A' dh
iqPN O' ls laikrh gks tk, rks Hkh B' dk 'kh"kZ S', A' osQ 'kh"kZ
P' dk laikrh ugha gksxk A
4-3 lfn'kksa dh okLrfod la[;k ls xq.kk
;fn ,d lfn'k A dks fdlh èkukRed la[;k ? ls xq.kk djsa
rks gesa ,d lfn'k gh feyrk gS ftldk ifjek.k lfn'k A osQ
ifjek.k dk ? xquk gks tkrk gS rFkk ftldh fn'kk ogh gS tks
A dh gS A bl xq.kuiQy dks ge ?A ls fy[krs gSa A
| | | | A A ? = ? ;fn 0 > ?
mnkgj.kLo:i] ;fn A dks 2 ls xq.kk fd;k tk,] rks ifj.kkeh
lfn'k 2A gksxk (fp=k 4.3a) ftldh fn'kk A dh fn'kk gksxh
rFkk ifjek.k | | A dk nksxquk gksxk A lfn'k A dks ;fn ,d
Í.kkRed la[;k ? ls xq.kk djsa rks lfn'k ?? ? ? ? ? izkIr gksrk gS
ftldh fn'kk A dh fn'kk osQ foijhr gS vkSj ftldk ifjek.k
| | A dk &? xquk gksrk gS A
;fn fdlh lfn'k A dks Í.kkRed la[;kvksa &1
o&1-5 ls xq.kk djsa rks ifj.kkeh lfn'k fp=k 4-3(b) tSls
gksaxs A
HkkSfrdh esa ftl ?kVd ? }kjk lfn'k A dks xq.kk fd;k
tkrk gS og dksbZ vfn'k gks ldrk gS ftldh Lo;a dh foek,¡
gksrh gSa A vr,o ?? ? ? ? ? dh foek,¡ ? o A dh foekvksa osQ
xq.kuiQy osQ cjkcj gksaxh A mnkgj.kLo:i] ;fn ge fdlh
vpj osx lfn'k dks fdlh (le;) varjky ls xq.kk djsa rks gesa
,d foLFkkiu lfn'k izkIr gksxk A
4-4 lfn'kksa dk ladyu o O;odyu % xzkIkQh fofèk
tSlk fd [k.M 4-2 esa cryk;k tk pqdk gS fd lfn'k ;ksx osQ
f=kHkqt fu;e ;k lekUrj prqHkqZt osQ ;ksx osQ fu;e dk ikyu
djrs gSa A vc ge xzkiQh fofèk }kjk ;ksx osQ bl fu;e dks
le>k,axs A ge fp=k 4.4 (a) esa n'kkZ, vuqlkj fdlh lery
esa fLFkr nks lfn'kksa A rFkk B ij fopkj djrs gSa A bu lfn'kksa
dks O;Dr djus okyh js[kk&[k.Mksa dh yackb;k¡ lfn'kksa osQ
ifjek.k osQ lekuqikrh gSa A ;ksx A + B izkIr djus osQ fy, fp=k
4-4(b) osQ vuqlkj ge lfn'k B bl izdkj j[krs gSa fd
mldh iqPN lfn'k A osQ 'kh"kZ ij gks A fiQj ge A dh iqPN
fp=k 4.2 (a) nks leku lfn'k A rFkk B, (b) nks lfn'k A' o
B' vleku gSa ;|fi mudh yackb;k¡ ogh gSa A
fp=k 4-3 (a) lfn'k A rFkk mls èkukRed la[;k nks ls xq.kk
djus ij izkIr ifj.kkeh lfn'k] (b) lfn'k A rFkk
mls Í.kkRed la[;kvksa &1 rFkk &1-5 ls xq.kk djus
ij izkIr ifj.kkeh lfn'k A
fp=k 4-4 (a) lfn'k A rFkk B, (b) lfn'kksa A o B dk xzkiQh fofèk }kjk tksM+uk] (c) lfn'kksa B o A dk xzkiQh fofèk }kjk tksM+uk]
(d) lfn'kksa osQ tksM+ ls lacafèkr lkgp;Z fu;e dk izn'kZu A
69 lery esa xfr
dks B osQ fljs ls tksM+ nsrs gSa A ;g js[kk OQ ifj.kkeh lfn'k
R dks O;Dr djrh gS tks lfn'kksa A rFkk B dk ;ksx gSA D;ksafd
lfn'kksa osQ tksM+us dh bl fofèk esa lfn'kksa esa ls fdlh ,d osQ
'kh"kZ dks nwljs dh iqPN ls tksM+rs gSa] blfy, bl xzkiQh fofèk
dks 'kh"kZ o iqPN fofèk osQ uke ls tkuk tkrk gS A nksuksa lfn'k
rFkk mudk ifj.kkeh lfn'k fdlh f=kHkqt dh rhu Hkqtk,a cukrs
gSa A blfy, bl fofèk dks lfn'k ;ksx osQ f=kHkqt fu;e Hkh
dgrs gSa A ;fn ge B+A dk ifj.kkeh lfn'k izkIr djsa rks Hkh
gesa ogh lfn'k R izkIr gksrk gS (fp=k 4.4c)A bl izdkj
lfn'kksa dk ;ksx ^Øe fofues;* (lfn'kksa osQ tksM+us esa ;fn
mudk Øe cny nsa rks Hkh ifj.kkeh lfn'k ugha cnyrk) gS A
A + B = B + A (4.1)
lfn'kksa dk ;ksx lkgp;Z fu;e dk Hkh ikyu djrk gS tSlk fd
fp=k 4.4 (d) esa n'kkZ;k x;k gS A lfn'kksa A o B dks igys
tksM+dj vkSj fiQj lfn'k C dks tksM+us ij tks ifj.kke izkIr
gksrk gS og ogh gS tks lfn'kksa B vkSj C dks igys tksM+dj fiQj
A dks tksM+us ij feyrk gS] vFkkZr~
(A + B) + C = A  + (B + C) (4.2)
nks leku vkSj foijhr lfn'kksa dks tksM+us ij D;k ifj.kke
feyrk gS \ ge  nks lfn'kksa A vkSj –A ftUgsa fp=k 4.3(b)
esa fn[kyk;k gS] ij fopkj djrs gSa A budk ;ksx A + (–A) gSA
D;ksafd nks lfn'kksa dk ifjek.k ogh gS fdUrq fn'kk foijhr gS]
blfy, ifj.kkeh lfn'k dk ifjek.k 'kwU; gksxk vkSj bls 0 ls
O;Dr djrs gSaA
A – A = 0 |0| = 0 (4.3)
0 dks ge 'kwU; lfn'k dgrs gSa A D;ksafd 'kwU; lfn'k dk
ifjek.k 'kwU; gksrk gS] blfy, bldh fn'kk dk fuèkkZj.k ugha
fd;k tk ldrk gS A njvly tc ge ,d lfn'k A dks
la[;k 'kwU; ls xq.kk djrs gSa rks Hkh ifj.kkeLo:i gesa ,d
lfn'k gh feysxk fdUrq mldk ifjek.k 'kwU; gksxk A O lfn'k
osQ eq[; xq.k fuEu gSa%
A  + 0 = A
? 0 = 0
0 A = 0 (4-4)
'kwU; lfn'k dk HkkSfrd vFkZ D;k gS \ tSlkfd fp=k 4.1(a)
eas fn[kk;k x;k gS ge fdlh lery esa fLFkfr ,oa foLFkkiu
lfn'kksa ij fopkj djrs gSa A eku yhft, fd fdlh {k.k t ij
dksbZ oLrq P ij gS A og P' rd tkdj iqu% P ij okil vk
tkrh gS A bl fLFkfr esa oLrq dk foLFkkiu D;k gksxk \ pwafd
izkjafHkd ,oa vafre fLFkfr;ka laikrh gks tkrh gSa] blfy,
foLFkkiu ¶'kwU; lfn'k¸ gksxk A
lfn'kksa dk O;odyu lfn'kksa osQ ;ksx osQ :i esa Hkh
ifjHkkf"kr fd;k tk ldrk gS A nks lfn'kksa A o B osQ varj
dks ge nks lfn'kksa A o –B osQ ;ksx osQ :i esa fuEu izdkj
ls O;Dr djrs gSa %
A – B = A + (–B) (4.5)
bls fp=k 4-5 esa n'kkZ;k x;k gS A lfn'k –B dks lfn'k A esa
tksM+dj R
2 
= (A – B) izkIr gksrk gS A rqyuk osQ fy, blh fp=k
esa lfn'k R
1
 = A + B dks Hkh fn[kk;k x;k gS A lekUrj
prqHkqZt fofèk dks iz;qDr djosQ Hkh ge nks lfn'kksa dk ;ksx Kkr
dj ldrs gSa A eku yhft, gekjs ikl nks lfn'k A o B gSaA
bu lfn'kksa dks tksM+us osQ fy, mudh iqPN dks ,d mHk;fu"B
ewy fcanq O ij ykrs gSa tSlk fp=k 4.6(a) esa fn[kk;k x;k gSA
fiQj ge A osQ 'kh"kZ ls B osQ lekarj ,d js[kk [khaprs gSa vkSj
B osQ 'kh"kZ ls A osQ lekarj ,d nwljh js[kk [khapdj lekarj
prqHkqZt OQSP iwjk djrs gSa A ftl fcanq ij ;g nksuksa js[kk,a
,d nwljs dks dkVrh gSa] mls ewy fcanq O ls tksM+ nsrs gSaA
ifj.kkeh lfn'k R dh fn'kk lekUrj prqHkqZt osQ ewy fcanq O
ls dVku fcanq S dh vksj [khaps x, fod.kZ OS osQ vuqfn'k
gksxh [fp=k 4.6 (b)]A fp=k 4.6 (c) esa lfn'kksa A o B dk
ifj.kkeh fudkyus osQ fy, f=kHkqt fu;e dk mi;ksx fn[kk;k
x;k gS A nksuksa fp=kksa ls Li"V gS fd nksuksa fofèk;ksa ls ,d gh
ifj.kke fudyrk gS A bl izdkj nksuksa fofèk;k¡ lerqY; gSaA
fp=k 4.5 (a) nks lfn'k A o B, –B dks Hkh fn[kk;k x;k gS A (b) lfn'k A ls lfn'k B dk ?kVkuk&ifj.kke R
2
 gS A rqyuk osQ fy,
lfn'kksa A o B dk ;ksx R
1
 Hkh fn[kyk;k x;k gS A
Page 5


66 HkkSfrdh
vè;k; 4
lery esa xfr
4-1 Hkwfedk
4-2 vfn'k ,oa lfn'k
4-3 lfn'kksa dh okLrfod la[;k ls xq.kk
4-4 lfn'kksa dk ladyu o O;odyu &
xzkiQh fof/
4-5 lfn'kksa dk fo;kstu
4-6 lfn'kksa dk ;ksx & fo'ys"k.kkRed
fof/
4-7 fdlh lery esa xfr
4-8 fdlh lery esa ,dleku Roj.k ls
xfr
4-9 nks foekvksa esa vkisf{kd osx
4-10 iz{ksI; xfr
4-11 ,dleku o`Ùkh; xfr
lkjka'k
fopkj.kh; fo"k;
vH;kl
vfrfjDr vH;kl
4-1 Hkwfedk
fiNys vè;k; esa geus fLFkfr] foLFkkiu] osx ,oa Roj.k dh èkkj.kkvksa dks
fodflr fd;k Fkk] ftudh fdlh oLrq dh ljy js[kh; xfr dk o.kZu djus
osQ fy, vko';drk iM+rh gS A D;ksafd ,dfoeh; xfr esa ek=k nks gh fn'kk,¡
laHko gSa] blfy, bu jkf'k;ksa osQ fn'kkRed i{k dks $ vkSj & fpÉksa ls O;Dr
dj ldrs gSa A ijarq tc ge oLrqvksa dh xfr dk f}foeh; (,d lery) ;k
f=kfoeh; (fno~QLFkku) o.kZu djuk pkgrs gSa] rc gesa mi;qZDr HkkSfrd jkf'k;ksa
dk vè;;u djus osQ fy, lfn'kksa dh vko';drk iM+rh gS A vr,o loZizFke
ge lfn'kksa dh Hkk"kk (vFkkZr lfn'kksa osQ xq.kksa ,oa mUgsa mi;ksx esa ykus dh
fofèk;k¡) lh[ksaxs A lfn'k D;k gS \ lfn'kksa dks oSQls tksM+k] ?kVk;k ;k xq.kk fd;k
tkrk gS \ lfn'kksa dks fdlh okLrfod la[;k ls xq.kk djsa rks gesa D;k ifj.kke
feysxk \ ;g lc ge blfy, lh[ksaxs ftlls fdlh lery esa oLrq osQ osx ,oa
Roj.k dks ifjHkkf"kr djus osQ fy, ge lfn'kksa dk mi;ksx dj losaQ A blosQ ckn
ge fdlh lery esa oLrq dh xfr ij ifjppkZ djsaxs A fdlh lery esa xfr
osQ ljy mnkgj.k osQ :i esa ge ,dleku Rofjr xfr dk vè;;u djsaxs rFkk
,d iz{ksI; dh xfr osQ fo"k; esa foLrkj ls i<s+axs A o`Ùkh; xfr ls ge HkyhHkk¡fr
ifjfpr gSa ftldk gekjs nSfud thou esa fo'ks"k egÙo gS A ge ,dleku o`Ùkh;
xfr dh oqQN foLrkj ls ppkZ djsaxs A
ge bl vè;k; esa ftu lehdj.kksa dks izkIr djsaxs mUgsa vklkuh ls
f=kfoeh; xfr osQ fy, foLrkfjr fd;k tk ldrk gS A
4-2 vfn'k ,oa lfn'k
ge HkkSfrd jkf'k;ksa dks vfn'kksa ,oa lfn'kksa esa oxhZÑr djrs gSa A nksuksa esa ewy
varj ;g gS fd lfn'k osQ lkFk fn'kk dks lac¼ djrs gSa ogha vfn'k osQ lkFk
,slk ugha djrs A ,d vfn'k jkf'k og jkf'k gS ftlesa ek=k ifjek.k gksrk gS A
bls osQoy ,d la[;k ,oa mfpr ek=kd }kjk iw.kZ :i ls O;Dr fd;k tk ldrk
gS A blosQ mnkgj.k gSa % nks fcanqvksa osQ chp dh nwjh] fdlh oLrq dh lagfr
(nzO;eku)] fdlh oLrq dk rkiØe] rFkk og le; ftl ij
dksbZ ?kVuk ?kVrh gS A vfn'kksa osQ tksM+ esa ogh fu;e ykxw gksrs gSa tks
lkekU;r;k chtxf.kr esa A vfn'kksa dks ge Bhd oSls gh tksM+ ldrs gSa] ?kVk
ldrs gSa] xq.kk ;k Hkkx dj ldrs gSa tSlk fd ge lkekU; la[;kvksa osQ lkFk
67 lery esa xfr
* osQoy leku ek=kd okyh jkf'k;ksa dk tksM+ o ?kVkuk lkFkZd gksrk gS A tcfd vki fHkUu ek=kdksa okys vfn'kksa dk xq.kk ;k Hkkx dj ldrs gSa A
** gekjs vè;;u esa lfn'kksa dh fLFkfr;ka fuèkkZfjr ugha gSa A blfy, tc ,d lfn'k dks Lo;a osQ lekarj foLFkkfir djrs gSa rks lfn'k vifjofrZr
jgrk gS A bl izdkj osQ lfn'kksa dks ge ^eqDr lfn'k* dgrs gSa A gkyakfd oqQN HkkSfrd mi;ksxksa esa lfn'k dh fLFkfr ;k mldh fØ;k js[kk
egÙoiw.kZ gksrh gS A ,sls lfn'kksa dks ge ^LFkkuxr lfn'k* dgrs gSaA
djrs gSa * A mnkgj.k osQ fy,] ;fn fdlh vk;r dh yackbZ vkSj
pkSM+kbZ Øe'k% 1-0 m rFkk 0-5 m gS rks mldh ifjeki pkjksa
Hkqtkvksa osQ ;ksx] 1.0 m + 0.5 m + 1.0 m + 0.5 m =
3.0 m  gksxkA gj Hkqtk dh yackbZ ,d vfn'k gS rFkk ifjeki
Hkh ,d vfn'k gS A ge ,d nwljs mnkgj.k ij fopkj djsaxs %
;fn fdlh ,d fnu dk vfèkdre ,oa U;wure rki Øe'k%
35-6 °C rFkk 24-2 °C gS rks bu nksuksa dk varj 11-4 °C
gksxk A blh izdkj ;fn ,Y;qfefu;e osQ fdlh ,dleku Bksl
?ku dh Hkqtk 10 cm gS vkSj mldk nzO;eku 2.7 kg gS rks
mldk vk;ru 10
–3 
m
3
  (,d vfn'k) gksxk rFkk ?kuRo
2.7×10
3
 kg/m
3
 Hkh ,d vfn'k gS A
,d lfn'k jkf'k og jkf'k gS ftlesa ifjek.k rFkk fn'kk
nksuksa gksrs gSa rFkk og ;ksx lacaèkh f=kHkqt osQ fu;e vFkok
lekukUrj prqHkqZt osQ ;ksx lacaèkh fu;e dk ikyu djrh
gS A bl izdkj] ,d lfn'k dks mlosQ ifjek.k dh la[;k rFkk
fn'kk }kjk O;Dr djrs gSa A oqQN HkkSfrd jkf'k;k¡ ftUgsa lfn'kksa
}kjk O;Dr djrs gSa] os gSa foLFkkiu] osx] Roj.k rFkk cy A
lfn'k dks O;Dr djus osQ fy, bl iqLrd esa ge eksVs
v{kjksa dk iz;ksx djsaxs A tSls fd osx lfn'k dks O;Dr djus
osQ fy, v fpÉ dk iz;ksx djsaxs A ijarq gkFk ls fy[krs le;
D;ksafd eksVs v{kjksa dk fy[kuk FkksM+k eqf'dy gksrk gS] blfy,
,d lfn'k dks v{kj osQ mQij rhj yxkdj O;Dr djrs gSa] tSls
?
v  A bl izdkj v rFkk 
?
v  nksuksa gh osx lfn'k dks O;Dr djrs
gSa A fdlh lfn'k osQ ifjek.k dks izk;% ge mldk ^ije eku*
dgrs gSa vkSj mls |v| = v  }kjk O;Dr djrs gSa A bl izdkj
,d lfn'k dks ge eksVs v{kj ;Fkk A ;k a, p, q, r, ..... x,
y ls O;Dr djrs gSa tcfd buosQ ifjek.kksa dks Øe'k% ge A
;k a, p, q, r, .... x, y }kjk O;Dr djrs gSa A
4-2-1 fLFkfr ,oa foLFkkiu lfn'k
fdlh lery esa xfreku oLrq dh fLFkfr O;Dr djus osQ fy,
ge lqfoèkkuqlkj fdlh fcanq O dks ewy fcanq osQ :i esa pqurs
gSa A dYiuk dhft, fd nks fHkUu&fHkUu le;ksa t  vkSj t'  ij
oLrq dh fLFkfr Øe'k% P vkSj P' gS (fp=k 4.1a) A ge P
dks O ls ,d ljy js[kk ls tksM+ nsrs gSa A bl izdkj OP le;
t ij oLrq dh fLFkfr lfn'k gksxh A bl js[kk osQ fljs ij ,d
rhj dk fu'kku yxk nsrs gSa A bls fdlh fpÉ (eku yhft,)
r ls fu:fir djrs gSa] vFkkZr~ OP = r A blh izdkj fcanq P'
dks ,d nwljs fLFkfr lfn'k OP' ;kuh r' ls fu:fir djrs gSaA
lfn'k r dh yackbZ mlosQ ifjek.k dks fu:fir djrh gS rFkk
lfn'k dh fn'kk og gksxh ftlosQ vuqfn'k P (fcanq O ls ns[kus
ij) fLFkr gksxk A ;fn oLrq P ls pydj P' ij igqap tkrh
gS rks lfn'k PP' (ftldh iqPN P ij rFkk 'kh"kZ P' ij gS)
¯cnq P (le; t) ls P' (le; t') rd xfr osQ laxr
foLFkkiu lfn'k dgykrk gS A
fp=k 4.1 (a) fLFkfr rFkk foLFkkiu lfn'k] (b) foLFkkiu lfn'k
PQ rFkk xfr osQ fHkUu&fHkUu ekxZ A
;gk¡ ;g ckr egRoiw.kZ gS fd ^foLFkkiu lfn'k* dks ,d
ljy js[kk ls O;Dr djrs gSa tks oLrq dh vafre fLFkfr dks
mldh izkjfEHkd fLFkfr ls tksM+rh gS rFkk ;g ml okLrfod
iFk ij fuHkZj ugha djrk tks oLrq }kjk fcanqvksa osQ eè; pyk
tkrk gS A mnkgj.kLo:i] tSlk fd fp=k 4.1b esa fn[kk;k x;k
gS] izkjfEHkd fLFkfr P rFkk vafre fLFkfr Q osQ eè; foLFkkiu
lfn'k PQ ;|fi ogh gS ijarq nksuksa fLFkfr;ksa osQ chp pyh xbZ
nwfj;ka tSls PABCQ, PDQ rFkk PBEFQ vyx&vyx gSa A
blh izdkj] fdUgha nks fcanqvksa osQ eè; foLFkkiu lfn'k dk
ifjek.k ;k rks xfreku oLrq dh iFk&yackbZ ls de gksrk
gS ;k mlosQ cjkcj gksrk gSA fiNys vè;k; esa Hkh ,d ljy
js[kk osQ vuqfn'k xfreku oLrq osQ fy, bl rF; dks HkyhHkkafr
le>k;k x;k Fkk A
4-2-2 lfn'kksa dh lerk
nks lfn'kksa A rFkk B dks osQoy rHkh cjkcj dgk tk ldrk
gS tc muosQ ifjek.k cjkcj gksa rFkk mudh fn'kk leku gks** A
fp=k 4.2(a) esa nks leku lfn'kksa A rFkk B dks n'kkZ;k
x;k gS A ge budh lekurk dh ij[k vklkuh ls dj ldrs
gSa A B dks Lo;a osQ lekarj f[kldkb;s rkfd mldh iqPN Q
lfn'k A dh iqPN O osQ laikrh gks tk, A fiQj D;ksafd muosQ
'kh"kZ S ,oa P Hkh laikrh gSa vr% nksukssa lfn'k cjkcj dgyk,axs A
lkekU;r;k bl lekurk dks A = B osQ :i esa fy[krs gSa A bl
68 HkkSfrdh
ckr dh vksj è;ku nhft, fd fp=k 4.2(b) esa ;|fi lfn'kksa
A' rFkk B' osQ ifjek.k leku gSa fiQj Hkh nksuksa lfn'k leku ugha
gSa D;ksafd mudh fn'kk;sa vyx&vyx gSa A ;fn ge B' dks
mlosQ gh lekarj f[kldk,a ftlls mldh iqPN Q', A' dh
iqPN O' ls laikrh gks tk, rks Hkh B' dk 'kh"kZ S', A' osQ 'kh"kZ
P' dk laikrh ugha gksxk A
4-3 lfn'kksa dh okLrfod la[;k ls xq.kk
;fn ,d lfn'k A dks fdlh èkukRed la[;k ? ls xq.kk djsa
rks gesa ,d lfn'k gh feyrk gS ftldk ifjek.k lfn'k A osQ
ifjek.k dk ? xquk gks tkrk gS rFkk ftldh fn'kk ogh gS tks
A dh gS A bl xq.kuiQy dks ge ?A ls fy[krs gSa A
| | | | A A ? = ? ;fn 0 > ?
mnkgj.kLo:i] ;fn A dks 2 ls xq.kk fd;k tk,] rks ifj.kkeh
lfn'k 2A gksxk (fp=k 4.3a) ftldh fn'kk A dh fn'kk gksxh
rFkk ifjek.k | | A dk nksxquk gksxk A lfn'k A dks ;fn ,d
Í.kkRed la[;k ? ls xq.kk djsa rks lfn'k ?? ? ? ? ? izkIr gksrk gS
ftldh fn'kk A dh fn'kk osQ foijhr gS vkSj ftldk ifjek.k
| | A dk &? xquk gksrk gS A
;fn fdlh lfn'k A dks Í.kkRed la[;kvksa &1
o&1-5 ls xq.kk djsa rks ifj.kkeh lfn'k fp=k 4-3(b) tSls
gksaxs A
HkkSfrdh esa ftl ?kVd ? }kjk lfn'k A dks xq.kk fd;k
tkrk gS og dksbZ vfn'k gks ldrk gS ftldh Lo;a dh foek,¡
gksrh gSa A vr,o ?? ? ? ? ? dh foek,¡ ? o A dh foekvksa osQ
xq.kuiQy osQ cjkcj gksaxh A mnkgj.kLo:i] ;fn ge fdlh
vpj osx lfn'k dks fdlh (le;) varjky ls xq.kk djsa rks gesa
,d foLFkkiu lfn'k izkIr gksxk A
4-4 lfn'kksa dk ladyu o O;odyu % xzkIkQh fofèk
tSlk fd [k.M 4-2 esa cryk;k tk pqdk gS fd lfn'k ;ksx osQ
f=kHkqt fu;e ;k lekUrj prqHkqZt osQ ;ksx osQ fu;e dk ikyu
djrs gSa A vc ge xzkiQh fofèk }kjk ;ksx osQ bl fu;e dks
le>k,axs A ge fp=k 4.4 (a) esa n'kkZ, vuqlkj fdlh lery
esa fLFkr nks lfn'kksa A rFkk B ij fopkj djrs gSa A bu lfn'kksa
dks O;Dr djus okyh js[kk&[k.Mksa dh yackb;k¡ lfn'kksa osQ
ifjek.k osQ lekuqikrh gSa A ;ksx A + B izkIr djus osQ fy, fp=k
4-4(b) osQ vuqlkj ge lfn'k B bl izdkj j[krs gSa fd
mldh iqPN lfn'k A osQ 'kh"kZ ij gks A fiQj ge A dh iqPN
fp=k 4.2 (a) nks leku lfn'k A rFkk B, (b) nks lfn'k A' o
B' vleku gSa ;|fi mudh yackb;k¡ ogh gSa A
fp=k 4-3 (a) lfn'k A rFkk mls èkukRed la[;k nks ls xq.kk
djus ij izkIr ifj.kkeh lfn'k] (b) lfn'k A rFkk
mls Í.kkRed la[;kvksa &1 rFkk &1-5 ls xq.kk djus
ij izkIr ifj.kkeh lfn'k A
fp=k 4-4 (a) lfn'k A rFkk B, (b) lfn'kksa A o B dk xzkiQh fofèk }kjk tksM+uk] (c) lfn'kksa B o A dk xzkiQh fofèk }kjk tksM+uk]
(d) lfn'kksa osQ tksM+ ls lacafèkr lkgp;Z fu;e dk izn'kZu A
69 lery esa xfr
dks B osQ fljs ls tksM+ nsrs gSa A ;g js[kk OQ ifj.kkeh lfn'k
R dks O;Dr djrh gS tks lfn'kksa A rFkk B dk ;ksx gSA D;ksafd
lfn'kksa osQ tksM+us dh bl fofèk esa lfn'kksa esa ls fdlh ,d osQ
'kh"kZ dks nwljs dh iqPN ls tksM+rs gSa] blfy, bl xzkiQh fofèk
dks 'kh"kZ o iqPN fofèk osQ uke ls tkuk tkrk gS A nksuksa lfn'k
rFkk mudk ifj.kkeh lfn'k fdlh f=kHkqt dh rhu Hkqtk,a cukrs
gSa A blfy, bl fofèk dks lfn'k ;ksx osQ f=kHkqt fu;e Hkh
dgrs gSa A ;fn ge B+A dk ifj.kkeh lfn'k izkIr djsa rks Hkh
gesa ogh lfn'k R izkIr gksrk gS (fp=k 4.4c)A bl izdkj
lfn'kksa dk ;ksx ^Øe fofues;* (lfn'kksa osQ tksM+us esa ;fn
mudk Øe cny nsa rks Hkh ifj.kkeh lfn'k ugha cnyrk) gS A
A + B = B + A (4.1)
lfn'kksa dk ;ksx lkgp;Z fu;e dk Hkh ikyu djrk gS tSlk fd
fp=k 4.4 (d) esa n'kkZ;k x;k gS A lfn'kksa A o B dks igys
tksM+dj vkSj fiQj lfn'k C dks tksM+us ij tks ifj.kke izkIr
gksrk gS og ogh gS tks lfn'kksa B vkSj C dks igys tksM+dj fiQj
A dks tksM+us ij feyrk gS] vFkkZr~
(A + B) + C = A  + (B + C) (4.2)
nks leku vkSj foijhr lfn'kksa dks tksM+us ij D;k ifj.kke
feyrk gS \ ge  nks lfn'kksa A vkSj –A ftUgsa fp=k 4.3(b)
esa fn[kyk;k gS] ij fopkj djrs gSa A budk ;ksx A + (–A) gSA
D;ksafd nks lfn'kksa dk ifjek.k ogh gS fdUrq fn'kk foijhr gS]
blfy, ifj.kkeh lfn'k dk ifjek.k 'kwU; gksxk vkSj bls 0 ls
O;Dr djrs gSaA
A – A = 0 |0| = 0 (4.3)
0 dks ge 'kwU; lfn'k dgrs gSa A D;ksafd 'kwU; lfn'k dk
ifjek.k 'kwU; gksrk gS] blfy, bldh fn'kk dk fuèkkZj.k ugha
fd;k tk ldrk gS A njvly tc ge ,d lfn'k A dks
la[;k 'kwU; ls xq.kk djrs gSa rks Hkh ifj.kkeLo:i gesa ,d
lfn'k gh feysxk fdUrq mldk ifjek.k 'kwU; gksxk A O lfn'k
osQ eq[; xq.k fuEu gSa%
A  + 0 = A
? 0 = 0
0 A = 0 (4-4)
'kwU; lfn'k dk HkkSfrd vFkZ D;k gS \ tSlkfd fp=k 4.1(a)
eas fn[kk;k x;k gS ge fdlh lery esa fLFkfr ,oa foLFkkiu
lfn'kksa ij fopkj djrs gSa A eku yhft, fd fdlh {k.k t ij
dksbZ oLrq P ij gS A og P' rd tkdj iqu% P ij okil vk
tkrh gS A bl fLFkfr esa oLrq dk foLFkkiu D;k gksxk \ pwafd
izkjafHkd ,oa vafre fLFkfr;ka laikrh gks tkrh gSa] blfy,
foLFkkiu ¶'kwU; lfn'k¸ gksxk A
lfn'kksa dk O;odyu lfn'kksa osQ ;ksx osQ :i esa Hkh
ifjHkkf"kr fd;k tk ldrk gS A nks lfn'kksa A o B osQ varj
dks ge nks lfn'kksa A o –B osQ ;ksx osQ :i esa fuEu izdkj
ls O;Dr djrs gSa %
A – B = A + (–B) (4.5)
bls fp=k 4-5 esa n'kkZ;k x;k gS A lfn'k –B dks lfn'k A esa
tksM+dj R
2 
= (A – B) izkIr gksrk gS A rqyuk osQ fy, blh fp=k
esa lfn'k R
1
 = A + B dks Hkh fn[kk;k x;k gS A lekUrj
prqHkqZt fofèk dks iz;qDr djosQ Hkh ge nks lfn'kksa dk ;ksx Kkr
dj ldrs gSa A eku yhft, gekjs ikl nks lfn'k A o B gSaA
bu lfn'kksa dks tksM+us osQ fy, mudh iqPN dks ,d mHk;fu"B
ewy fcanq O ij ykrs gSa tSlk fp=k 4.6(a) esa fn[kk;k x;k gSA
fiQj ge A osQ 'kh"kZ ls B osQ lekarj ,d js[kk [khaprs gSa vkSj
B osQ 'kh"kZ ls A osQ lekarj ,d nwljh js[kk [khapdj lekarj
prqHkqZt OQSP iwjk djrs gSa A ftl fcanq ij ;g nksuksa js[kk,a
,d nwljs dks dkVrh gSa] mls ewy fcanq O ls tksM+ nsrs gSaA
ifj.kkeh lfn'k R dh fn'kk lekUrj prqHkqZt osQ ewy fcanq O
ls dVku fcanq S dh vksj [khaps x, fod.kZ OS osQ vuqfn'k
gksxh [fp=k 4.6 (b)]A fp=k 4.6 (c) esa lfn'kksa A o B dk
ifj.kkeh fudkyus osQ fy, f=kHkqt fu;e dk mi;ksx fn[kk;k
x;k gS A nksuksa fp=kksa ls Li"V gS fd nksuksa fofèk;ksa ls ,d gh
ifj.kke fudyrk gS A bl izdkj nksuksa fofèk;k¡ lerqY; gSaA
fp=k 4.5 (a) nks lfn'k A o B, –B dks Hkh fn[kk;k x;k gS A (b) lfn'k A ls lfn'k B dk ?kVkuk&ifj.kke R
2
 gS A rqyuk osQ fy,
lfn'kksa A o B dk ;ksx R
1
 Hkh fn[kyk;k x;k gS A
70 HkkSfrdh
gy %  o"kkZ ,oa gok osQ osxksa dks lfn'kksa v
r
 rFkk v
w
 ls fp=k
4-7 esa n'kkZ;k x;k gSA budh fn'kk,a iz'u osQ vuqlkj iznf'Zkr
dh xbZ gSa A lfn'kksa osQ ;ksx osQ fu;e osQ vuqlkj v
r
 rFkk v
w
dk ifj.kkeh R fp=k esa [khapk x;k gS A R dk ifjek.k
gksxk&
Rv v
r
2
w
2
=+ = + =
--
35 12 m s 37 m s
22 1 1
mQèokZèkj ls R dh fn'kk ? gksxh&
12
tan 0.343
35
w
r
v
 
v
?== =
;k ? = tan
–1
 (0.343) = 19°
vr,o yM+osQ dks viuk Nkrk mQèokZèkj ry esa mQèokZèkj
ls 19°
 
dk dks.k cukrs gq, iwoZ fn'kk dh vksj j[kuk
pkfg, A ¶ ¶ ¶ ¶ ¶
4-5 lfn'kksa dk fo;kstu
eku yhft, fd a o b fdlh lery esa fHkUu fn'kkvksa okys
nks 'kwU;srj ('kwU; ugha) lfn'k gSa rFkk A blh lery esa dksbZ
vU; lfn'k gS A (fp=k 4-8) rc A dks nks lfn'kksa osQ ;ksx osQ
:i esa fo;ksftr fd;k tk ldrk gS A ,d lfn'k a osQ fdlh
okLrfod la[;k osQ xq.kuiQy osQ :i esa vkSj blh izdkj nwljk
lfn'k b osQ xq.kuiQy osQ :i esa gS A ,slk djus osQ fy, igys
A [khafp, ftldk iqPN O rFkk 'kh"kZ P gS A fiQj O ls a osQ
lekarj ,d ljy js[kk [khafp, rFkk P ls ,d ljy js[kk b osQ
lekarj [khafp, A eku yhft, os ,d nwljs dks Q ij dkVrh
gSa A rc]
A = OP = OQ + QP (4.6)
ijarq D;ksafd OQ, a osQ lekarj gS rFkk Q P, b osQ lekarj gS
blfy,
 ? µ == OQ a QP b rFkk (4.7)
tgka  ? rFkk µ dksbZ okLrfod la[;k,¡ gSa A
fp=k 4.6 (a) ,d gh mHk;fu"B fcanq okys nks lfn'k A o  B  ij, (b) lekUrj  prqHkqZt fofèk }kjk A+B ;ksx izkIr djuk] (c) nks lfn'kksa
dks tksM+us dh lekUrj prqHkqZt fofèk f=kHkqt fofèk osQ lerqY; gS A
• • • • • mnkgj.k 4-1 fdlh fnu o"kkZ 35 m s
–1
 dh pky ls
mQèokZèkj uhps dh vksj gks jgh gS A oqQN nsj ckn gok
12 m s
–1
 dh pky ls iwoZ ls if'pe fn'kk dh vksj
pyus yxrh gS A cl LVki ij [kM+s fdlh yM+osQ dks
viuk Nkrk fdl fn'kk esa djuk pkfg, \
fp=k 4.7
fp=k 4-8 (a) nks vjSf[kd lfn'k a o  b, (b) lfn'k A dk a
o b osQ inksa esa fo;kstu A
vr% ? µ =+ Aa b (4.8)
ge dg ldrs gSa fd A dks a o b osQ vuqfn'k nks
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