vè;k; 1 hkwfedk - delan5sxrj8jj.cloudfront.net · cgqfodyih; iz'u (mcq) fo|kfkhz osq...

vè;k; 1

Hkwfedk

jk"Vªh; ikB~;p;kZ :ijs[kk (NCF) – 2005 us ikB~;p;kZ iqujhdj.k osQ ,d u;s nkSj dh 'kq:vkrdhA loZizFke LowQyh f'k{kk osQ lHkh pj.kksa osQ fy, foKku ,oa xf.kr osQ ubZ ikB~;Øe fodflrfd, x,A bu ikB~;Øeksa osQ vkèkkj ij ubZ ikB~;iqLrosaQ fodflr dh xb±A blh Øe esa d{kk 11,oa 12 osQ fy, HkkSfrdh dh ikB~;iqLrosaQ Hkh Øe'k% 2006 ,oa 2007 esa izdkf'kr gqbZ gSaA

jk"Vªh; ikB~;p;kZ :ijs[kk (NCF)–2005 esa O;Dr ,d izeq[k ljksdkj ijh{kk iz.kkyh esa lqèkkjls lacafèkr gSA

NCF–2005 osQ vuqlkj] ¶,d vPNh ewY;kadu ,oa ijh{kk iz.kkyh vfèkxe izØe dk vfHkUuvax cu ldrh gS vkSj blls Lo;a fo|k£Fk;ksa vkSj fo'oluh; iqu£uos'ku }kjk f'k{kk iz.kkyh] nksuksadks gh ykHk gks ldrk gSA¸ blosQ vfrfjDr bl :ijs[kk dk ;g Hkh dguk gS fd ¶f'k{kk dkljksdkj ukxfjdksa dks lkFkZd ,oa mRiknd thou osQ fy, rS;kj djuk gSA ewY;kadu ,d ,slk lkèkugksuk pkfg, ftlls gesa mu lhekvksa esa ,d fo'oluh; iqu£uos'ku izkIr gks losQ ftuesa ge ,slhf'k{kk iznku djus esa liQy gks ik, gSaaA bl ifjizs{; esa ns[ksa rks ewY;kadu dh orZeku izfØ;k,¡ tks;ksX;rkvksa osQ cgqr gh lhfer ifjlj dk vkdyu ,oa ewY;kadu djrh gSa vR;ar vi;kZIr gSa rFkkf'k{kk osQ mís';ksa dh iw£r dh fn'kk esa fdlh O;fDr dh ;ksX;rk ;k izxfr dk iw.kZ fp=k.k izLrqrugha djrhA¸

18-04-2018

iz'u izn£'kdkµHkkSfrdh

2

,d vksj rks ewY;kadu dk mís'; ;g Kkr djuk gS fd fdl lhek rd vfèkxe esa liQyrkizkIr gqbZ gS vkSj nwljh vksj bldk mís'; f'k{k.k&vfèkxe izØe ,oa f'k{k.k lkexzh esa lqèkkj ykukgSA vU; ckrksa osQ lkFk ;g ekirs gq, fd LowQyh f'k{kk osQ fofHkUu pj.kksa esa f'k{kkFkhZ esa fdllhek rd ;ksX;rk,¡ fodflr gqbZ gSa blosQ }kjk iwoZ fuèkkZfjr mís';ksa dk iqujh{k.k Hkh dj ldus;ksX; gksuk pkfg,A ijh{k.kksa dh vfHkdYiuk bl izdkj dh tkuh pkfg, fd ge ;g eki losaQfd cPpksa us D;k lh[kk gS vkSj bl Kku dk mi;ksx leL;kvksa dks gy djus rFkk okLrfod txresa blosQ vuqiz;ksxksa dks ysdj mudh ;ksX;rk fdruh gS\ blosQ vfrfjDr ¯pru izfØ;kvksa dkijh{k.k Hkh fd;k tkuk pkfg, rkfd ;g tkuk tk losQ fd D;k f'k{kkFkhZ us ;g Hkh lh[kk gS fdtkudkjh dgk¡ ls izkIr gksxh] ubZ tkudkfj;ksa dk mi;ksx] fo'ys"k.k ,oa ewY;kadu oSQls fd;ktk,xkA ewY;kadu esa bl izdkj osQ iz'u fn, tkus pkfg, ftuosQ gy osQ fy, mls iqLrd ls vkxsvè;;u dh vko';drk iM+sA izk;% cPpksa dk vfèkxe blfy, izfrcafèkr gks tkrk gS D;ksafdf'k{kd ekxZn'kZd iqLrdksa esa fn, x, Kku ls vyx muosQ mÙkjksa dks Lohdkj ugha djrsA vPNsiz'uksa dh vfHkdYiuk ,d dyk gS vkSj f'k{kdksa dks ,sls iz'u lkspus ,oa mudh jpuk djus esale; yxkuk pkfg,A

ns'k osQ fofHkUu fo|ky;h f'k{kk cksMks± dh dk;Z&iz.kkfy;ksa dk izs{k.k djrs gq, jk"Vªh; -iQksdllewg osQ vkys[k esa ijh{kk lqèkkjksa osQ lacaèk esa dgk x;k gSµ

¶---D;ksafd iz'u&i=k U;wu xq.koÙkk osQ gksrs gSaA muosQ fy, izk;% jVus dh gh vko';drk gksrhgS rFkk roZQ ,oa fo'ys"k.k tSlh mPpdksfV osQ dkS'kyksa dk ijh{k.k os ugha dj ikrs] fiQj ik£'odfparu] l`tu'khyrk rFkk fu.kZ;ijdrk osQ ijh{k.k dk rks iz'u gh iSnk ugha gksrkA¸

blesa cgqfodYih; iz'uksa (MCQ) dks iz'u&i=kksa esa lfEefyr djus dh odkyr dh xbZ gS vkSjbUgsa bl izdkj osQ iz'u dgk x;k gS ftuesa mPp viz;qDr {kerk,¡ gSa ijarq lkFk gh blesa osQoycgqfodYih; iz'uksa (MCQ's) osQ vkèkkj ij fd, tkus okys ijh{k.k dh lhekvksa dk Hkh mYys[k gSA¸cgqfodYih; iz'u (MCQ) fo|kFkhZ osQ ladYiukRed vocksèku osQ Lrj dh vfèkd xgjkbZ ls tk¡pdj ldrs gSa rFkk fo"k; dh tfVyrkvksa osQ ikafMR; dks eki ldrs gSaA ijarq] fdlh Hkh ijh{kk esaosQoy blh izdkj osQ iz'u ugha gks ldrsA cgqfodYih; iz'u (MCQ's) rc lcls vfèkd izHkkodkjhifj.kke nsrs gSa tc iz'u&i=k osQ nwljs [kaM esa eqDr mÙkjksa okys fucaèkkRed iz'u Hkh iwNs tkrs gSa tksvfHkO;fDr vkSj laxr rF;ksa osQ vkèkkj ij roZQ djus dh ;ksX;rk dk ijh{k.k djrs gSaA¸

bl leL;k osQ lekèkku osQ fy, foKku ,oa xf.kr f'k{kk foHkkx us o"kZ 2007-08 osQ nkSjkud{kk 11 osQ fy, ¶iz'u&izn£'kdk HkkSfrdh esa vuqdj.kh; vH;kl iz'uksa dk fodkl¸ dk;ZØeij dk;Z djuk izkjaHk fd;kA bl dk;ZØe esa ,u-lh-bZ-vkj-Vh- }kjk izdkf'kr d{kk 11 dhikB~;iqLrd osQ fofHkUu vè;k;ksa ij vkèkkfjr vH;kl&iz'uksa dk fodkl fd;k x;kA ;svH;kl&iz'u LFkwy :i ls fuEufyf[kr ik¡p Jsf.k;ksa esa oxhZo`Qr fd, x, gSaµ1. cgq fodYih; iz'u I (MCQ I): ftuesa osQoy ,d fodYi lgh gSA

2. cgq fodYih; iz'u II (MCQII): ftuesa ,d ;k vfèkd fodYi lgh gks ldrs gSaA3. vfr y?kqmÙkjh; iz'u (VSA): ftuosQ mÙkj ,d@nks okD;ksa esa fn, tk ldrs gSaA4. y?kq mÙkjh; iz'u (SA): ftuesa oqQN fo'ys"k.kkRed@vk¡fdd dk;Z djus dh vko';drk gks

ldrh gSA5. nh?kZ mÙkjh; iz'u (LA): ftuesa foLr`r fo'ys"k.k vk¡fdd ifjdyuksa dh vko';drk gks

ldrh gSA

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Hkwfedk

3

;|fi fdlh vè;k; fo'ks"k esa fn, x, vfèkdka'k iz'u ml vè;k; esa o£.kr ladYiukvksaij gh vkèkkfjr gSa fiQj Hkh oqQN ,sls iz'u Hkh cuk, x, gSa tks ,d ls vfèkd vè;k;ksa dhladYiukvksa ij vkèkkfjr gSaA

f'k{k.k&vfèkxe izfØ;k esa f'k{kk£Fk;ksa dks leL;kvksa dk gy djus esa lfEefyr djus dk ,dizeq[k mís'; ;g gS fd vfèkd lfØ; vfèkxe okrkoj.k dks izksRlkfgr fd;k tk,] f'k{kk£Fk;ksaosQ vfèkxe esa lqèkkj yk;k tk, vkSj lkFk gh ;qok f'k{kdksa dks muosQ izkjafHkd fuekZ.kkRed f'k{k.kvuqHkoksa osQ nkSjku] O;kolkf;d fodkl osQ iz;Ruksa esa mudh lgk;rk dh tk,A bl mís'; dhizkfIr osQ fy, mÙke iz'uksa ij vkèkkfjr leL;kvksa dks gy djuk lh[kuk f'k{k.k vfèkxe izfØ;kdk vfHkUu vax gksuk pkfg,A mÙke iz'u fo|kFkhZ dks mÙkjksÙkj fparu vkSj fo'ys"k.k osQ xgureLrj ij ys tkrs gSaA ;g vk'kk dh tkrh gS fd bl iqLrd osQ iz'u f'k{kdksa dks mÙke iz'uksa dhvfHkdYiuk osQ fy, izsfjr djsaxsA fdlh mÙke iz'u osQ D;k y{k.k gSa\ jksfcu ,y- feyj osQvuqlkj1 fdlh mÙke iz'u osQ oqQN vfHky{k.k bl izdkj gSa %• f'k{kkFkhZ dh vfHk#fp vkSj ftKklk dks izsfjr djrk gSA• f'k{kkFkhZ dh vocksèku osQ izcksèku esa lgk;rk djrk gSA• f'k{kkFkhZ dks vuqeku yxkus vkSj mudh oSèkrk osQ fo"k; esa roZQ djus osQ fujarj volj iznku

djrk gSA• f'k{kkFkhZ osQ iwoZ&Kku] vocksèk vkSj@vFkok nqcksZèk dks ckgj fudkyrk gSA• f'k{kd dks fujarj vkSipkfjd ewY;kadu }kjk ;g tkuus osQ fy, fd fo|kFkhZ D;k lh[k jgs

gSa] lkèku iznku djrk gSA• lfØ; vfèkxe okrkoj.k iksf"kr djus osQ f'k{kd osQ iz;klksa esa lgk;rk djrk gSA

fo|k£Fk;ksa osQ fy, ekxZn'kZubl iqLrd esa dkiQh cM+h la[;k esa iz'u fn, x, gSaA buesa ls oqQN ljy] oqQN vkSlr dfBukbZLrj osQ] oqQN dfBu gSa vkSj oqQN iz'u vkiesa loZJs"B osQ fy, Hkh pqukSrhiw.kZ gks ldrh gSaaA vkidks;g ijke'kZ fn;k tkrk gS fd igys vki viuh ikB~;iqLrd esa nh xbZ ladYiukvksa dks Hkyh&Hkk¡frle> ysaA viuh ikB~;iqLrd esa fn, x, mnkgj.kksa ,oa vH;klksa dks gy djsaA mlosQ i'pkr~ ghbl iqLrd esa fn, x, iz'uksa dks gy djus dk iz;kl djsaA ,slh dksbZ fuf'pr fofèk ugha gS ftldkmi;ksx djosQ vki HkkSfrdh dk izR;sd iz'u gy dj ldrs gksa rFkkfi HkkSfrdh f'k{k.k ls lac¼'kksèk dk;ks± ls ;g fu"d"kZ fudyrk gS fd oqQN fuf'pr pj.kksa dks Øeokj viuk dj HkkSfrdhdh vfèkdka'k leL;kvksa dks gy fd;k tk ldrk gSA Mu LVs;j2 }kjk izLrqr ,slh gh ,d dk;ZokghosQ fofHkUu pj.k uhps izLrqr gSa µ

1. j.kuhfr vfHkdYiukµ

(a) gy djus dh fofèk osQ vuqlkj iz'u dk oxhZdj.k dhft,A(b) fdlh vkjs[k lfgr fLFkfr dks la{ksi esa izLrqr dhft,A(c) y{; dks vk¡[kksa ls vks>y er gksus nhft, (dnkfpr bls fy[kdj jf[k,)A

1 http://www. math.cornell.edu/~ maria/mathfest_educationpreprint.pdf

2 http://www.oberlin.edu/physics/dstyer/SolvingProblems.html

18-04-2018


4

2. vfHk;kstu dk;Zuhfr µ(a) izrhdksa dk mi;ksx dhft,A(b) lacafèkr pjkadksa dks ,d lkFk lewgksa esa jf[k,A(c) lqO;ofLFkr ,oa laxfBr jfg,A(d) bls ljy cuk, jf[k,A

3. mÙkj dh tk¡p&iM+rky djuk µ

(a) foeh; lqlaxfr\(b) la[;kRed :i ls roZQ laxr (fpÉ lfgr)(c) chtxf.krh; :i ls laHko\ (mnkgj.kr% dksbZ dkYifud vFkok vuar mÙkj ugha)(d) -iQyukuqlkj roZQlEer\ (mnkgj.kr% ftruh vfèkd izkFkfed pky mruk gh vfèkd

iz{ksI; dk ijkl)(e) fof'k"V izdj.kksa ,oa lefefr dh tk¡pA(f ) la[;kvksa dk mYys[k lnSo fof'k"V ek=kdksa osQ lkFk rFkk mi;qDr lkFkZd vadksa esaA

ge bl ckr ij tksj nsuk pkgsaxs fd bl iqLrd esa fn, x, iz'uksa dk mi;ksx HkkSfrdh dhf'k{k.k&vfèkxe izfØ;k dh xq.koÙkk esa lqèkkj ykus osQ fy, fd;k tkuk pkfg,A buesa ls oqQN iz'uksadks lhèks ewY;kadu osQ fy, viuk;k tk ldrk gS ijarq vfèkdka'k dks fuèkkZfjr le;@vadksa osQvuqlkj mi;qDr :i ls vuqowQfyr fd;k tk ldrk gSA y?kqmÙkjh; iz'u ,oa nh?kZmÙkjh; iz'u osQvarxZr lfEefyr fd, x, vfèkdka'k iz'u Øe'k% vfry?kqmÙkjh; iz'u ,oa y?kqmÙkjh; iz'u Js.khosQ vkSj cgqr ls iz'uksa dh jpuk djus osQ fy, mi;ksx esa yk, tk ldrs gSaA

18-04-2018

cgq fodYih; iz'u I (MCQ I)

2.1 0-06900 esa lkFkZd vadksa dh la[;k gSµ

(a) 5 (b) 4 (c) 2 (d) 3

2.2 436.32, 227.2 ,oa 0.301 la[;kvksa dk ;ksx mi;qDr lkFkZd vadksa esa gSµ

(a) 663.821(b) 664(c) 663.8(d) 663.82

2.3 ,d fiaM dk nzO;eku vkSj vk;ru Øe'k% 4.237 g ,oa 2.5 cm3 gSA bl fiaM osQinkFkZ osQ ?kuRo dk lgh lkFkZd vadksa esa eku gSµ

(a) 1.6048 g cm–3

(b) 1.69 g cm–3

(c) 1.7 g cm–3

(d) 1.695 g cm–3

vè;k; 2

ek=kd vkSj ekiu

18-04-2018


6

2.4 ;fn 2.745 ,oa 2.735 la[;kvksa dks 3 lkFkZd vadksa rd iw.kk±fdr dj O;Dr fd;ktk, rks izkIr la[;k,¡ gksaxhµ

(a) 2.75 vkSj 2.74

(b) 2.74 vkSj 2.73

(c) 2.75 vkSj 2.73

(d) 2.74 vkSj 2.74

2.5 ,d vk;rkdkj 'khV dh yackbZ ,oa pkSM+kbZ Øe'k% 16.2 cm vkSj 10.1 cm gSAmi;qDr lkFkZd vadksa esa vkSj mi;qDr =kqfV osQ mYys[k osQ lkFk 'khV dk{ks=kiQy gksxkµ

(a) 164 ± 3 cm2

(b) 163.62 ± 2.6 cm2

(c) 163.6 ± 2.6 cm2

(d) 163.62 ± 3 cm2

2.6 HkkSfrd jkf'k;ksa osQ fuEufyf[kr tksM+ksa esa ls fdl tksM+s dk foeh; lw=k leku ugha gS\

(a) dk;Z vkSj cy&vk?kw.kZ(b) dks.kh; laosx vkSj Iyk¡d fu;rkad(c) ruko vkSj i`"B ruko(d) vkosx vkSj js[kh; laosx

2.7 nks jkf'k;ksa dh eki] mudks ekius esa iz;qDr gq, eki ;a=kksa dh ifj'kq¼rk osQ lkFk O;Drdjrs gq, gSaµ

A = 2.5 m s–1 ± 0.5 m s–1

B = 0.10 s ± 0.01 s

A B dk eku gksxk

(a) (0.25 ± 0.08) m(b) (0.25 ± 0.5) m(c) (0.25 ± 0.05) m(d) (0.25 ± 0.135) m

2.8 nks jkf'k;ksa dks eki dj vki mudk eku A = 1.0 m ± 0.2 m, B = 2.0 m ± 0.2

m izkIr djrs gSaA AB dk lgh eku gksxkµ

(a) 1.4 m ± 0.4 m(b) 1.41m ± 0.15 m(c) 1.4m ± 0.3 m(d) 1.4m ± 0.2 m

18-04-2018

ek=kd vkSj ekiu

7

2.9 fuEufyf[kr esa dkSu&lk eku lokZfèkd ifj'kq¼ gS\

(a) 5.00 mm(b) 5.00 cm(c) 5.00 m(d) 5.00 km

2.10 fdlh fiaM dh vkSlr yackbZ 5 cm gSA fuEufyf[kr esa dkSu&lk eki lokZfèkd ;FkkFkZgS\

(a) 4.9 cm(b) 4.805 cm(c) 5.25 cm(d) 5.4 cm

2.11 LVhy dk ;ax izR;kLFkrk xq.kkad 1.9 × 1011 N/m2 gSA ;fn bls CGS ek=kdksa] Mkbuizfr lsaVhehVj esa O;Dr fd;k tk, rks bldk eku gksxkµ

(1N = 105 dyne 1m2 = 104 cm2)

(a) 1.9 × 1010

(b) 1.9 × 1011

(c) 1.9 × 1012

(d) 1.9 × 1013

2.12 ;fn laosx (P), {ks=kiQy (A) ,oa le; (T) dks ewy jkf'k;k¡ eku ysa rks ÅtkZ dk foeh;lw=k gksxkµ

(a) (P1 A–1 T1)(b) (P2 A1 T1)(c) (P1 A–1/2 T1)(d) (P1 A1/2 T–1)

cgq fodYih; iz'u II (MCQ II)

2.13 foekvksa osQ vkèkkj ij fu.kZ; dhft, fd ljy vkorZ xfr djrs gq,] fdlh d.k osQfoLFkkiu osQ fy, fuEufyf[kr lacaèkksa esa dkSu&ls lacaèk lgh ugha gSa\

(a) y = a sin 2 /t Tπ

(b) y = a sin vt.

(c) y = sina t

T a

(d) y = 2 2

2 sin cost t

aT T

π π −

18-04-2018


8

2.14 ;fn P, Q,R ,slh HkkSfrd jkf'k;k¡ gSa ftuosQ foeh; lw=k fHkUu gSa rks buosQ fuEufyf[krla;kstuksa esa fduls dksbZ lkFkZd jkf'k O;Dr ugha gksrh\(a) (P – Q)/R

(b) PQ – R

(c) PQ/R

(d) (PR – Q2)/R

(e) (R + Q)/P

2.15 iQksVksu] fofdj.k dk ,d DokaVe gS ftldh ÅtkZ E = hν gksrh gS] tgk¡ ν fofdj.kdh vko`fÙk gS vkSj h Iyk¡d fu;rkad gSA h dh foek,¡ ogh gSa tksµ

(a) js[kh; vkosx dh

(b) dks.kh; vkosx dh

(c) js[kh; laosx dh

(d) dks.kh; laosx dh

2.16 ;fn ge Iyk¡d fu;rkad (h) rFkk fuokZr esa izdk'k osQ osx (c) dks nks ewy jkf'k;k¡ ysysa rks fuEufyf[kr esa ls dkSu&lh jkf'k rhljh ewy jkf'k yh tk,xh rkfd yackbZ]nzO;eku vkSj le; dks bu rhu ewy jkf'k;ksa osQ inksa esa O;Dr fd;k tk losQ\

(a) bysDVªkWu dk nzO;eku (me )

(b) lkoZf=kd xq#Rokd"kZ.k fu;rkad (G )

(c) bysDVªkWu dk vkos'k (e )

(d) izksVksu dk nzO;eku (mp )

2.17 fuEufyf[kr vuqikrksa esa fdu&ls nkc O;Dr gksrk gS\

(a) cy @ {ks=kiQy

(b) ÅtkZ @ vk;ru

(c) ÅtkZ @ {ks=kiQy

(d) cy @ vk;ru

2.18 fuEufyf[kr esa dkSu le; osQ ek=kd ugha gSa\

(a) lsoaQM

(b) ikjlsd

(c) o"kZ

(d) izdk'k o"kZ

vfr y?kq mÙkjh; iz'u (VSA)

2.19 ge ,d gh HkkSfrd jkf'k osQ fy, fHkUu&fHkUu ek=kdksa dk mi;ksx D;ksa djrs gSa\

2.20 ijek.kq dh f=kT;k 1 Å dh dksfV dh gS vkSj ukfHkd dh f=kT;k iQehZ dh dksfV dh gSA ijek.kqdk vk;ru ukfHkd osQ vk;ru dh rqyuk esa fdrus ifjek.k dksfV vfèkd gS\

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ek=kd vkSj ekiu

9

2.21 ijek.kqvksa rFkk v.kqvksa dk nzO;eku ekius osQ fy, iz;qDr gksus okyh ;qfDr dk ukecrkb,A

2.22 ,dhÑr ijek.kq nzO;eku bdkbZ (a.m.u) dks kg esa O;Dr dhft,A

2.23 iQyu f (θ ) uhps fn, vuqlkj ifjHkkf"kr fd;k tkrk gSµ

( )2 3 4

1– + – + 2! 3! 4!

fθ θ θ

θθ =

θ dk ,d foekghu jkf'k gksuk vko';d D;ksa gS\

2.24 ;kaf=kdh esa] yackbZ] nzO;eku ,oa le; dk p;u vkèkkj jkf'k;ksa osQ :i esa D;ksa fd;ktkrk gS\

y?kq mÙkjh; iz'u (SA)

2.25 (a) i`Foh panzek osQ chp dh nwjh yxHkx i`Foh dh 60 f=kT;kvksa osQ cjkcj gSA panzekls ns[kus ij i`Foh dk O;kl (yxHkx fMxzh eki esa) fdruk gksxk\

(b) i`Foh ls panzek dk O;kl (½)º fn[kkbZ iM+rk gSA i`Foh dh rqyuk esa panzek dkvkisf{kd lkbt fdruk gksxk\

(c) yacu ekkiu }kjk lw;Z dh nwjh i`Foh&panzek osQ chp dh nwjh dh 400 xquk ikbZxbZA lw;Z&i`Foh O;klksa osQ vuqikr dk vkdyu dhft,A

2.26 fuEufyf[kr le; ekid ;a=kksa esa dkSu lokZfèkd ifj'kq¼ gS\

(a) nhokj ?kM+h(b) fojke ?kM+h(c) fMftVy ?kM+h(d) ijek.kq ?kM+h

vius mÙkj osQ leFkZu esa roZQ nhft,A

2.27 fdlh eankfduh dh nwjh 1025 m dh dksfV dh gSA bl eankfduh ls gekjs ikl rdigq¡pus esa yxus okys le; dh dksfV dh x.kuk dhft,A

2.28 fdlh py lw{en'khZ osQ ofuZ;j iSekus esa 50 Hkkx gSa tks eq[; iSekus osQ 49 Hkkxksa lslaikrh gksrs gSaA ;fn eq[; iSekus osQ 1 Hkkx dk eku 0.5 mm gS rks nwjh ekiu esa vkusokyh U;wure =kqfV dh x.kuk dhft,A

2.29 iw.kZ lw;Z xzg.k dh fLFkfr esa panzek lw;Z osQ xksys dks iw.kZr% <d ysrk gSA lw;Z ,oa panzekdh nwfj;ksa rFkk lkbtksa esa lacaèk fyf[k,A

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10

2.30 ;fn cy dk ek=kd 100 N, yackbZ dk ek=kd 10 m rFkk le; dk ek=kd 100 s

gS rks ek=kdksa dh bl iz.kkyh esa nzO;eku dk ek=kd D;k gS\

2.31 ,d mnkgj.k nhft,µ

(a) fdlh HkkSfrd jkf'k dk] ftudk ek=kd rks gksrk gS ij foek,¡ ugha gksrhaA(b) ,d HkkSfrd jkf'k dk] ftudk u rks dksbZ ek=kd gksrk gS vkSj u gh foek,¡A(c) fdlh vpjkad dk] ftldk ek=kd gksrk gSA(d) fdlh vpjkad dk] ftldk dksbZ ek=kd ugha gksrkA

2.32 31.0 cm f=kT;k osQ o`Ùk osQ ml pki dh yackbZ Kkr dhft, tks osaQnz ij 6π

dks.k

cukrh gSA

2.33 1cm2 {ks=kiQy dh ifjfèk (ifjjs[kk) }kjk blls leferr% 5 cm nwj fLFkr ¯cnq ijcuus okys ?ku dks.k dh x.kuk dhft,A

2.34 ,d izxkeh rjax dks foLFkkiu y = A sin(ωt – k x ) }kjk fu#fir fd;k tkrk gS tgk¡x nwjh vkSj t le; gSA (i) ω ,oa (ii) k osQ foeh; lw=k fyf[k,A

2.35 fdlh yksyd osQ 20 nksyuksa dk dky t1= 39.6 s; t2= 39.9 s; t3= 39.5 s ekikx;kA ekiu dh ifj'kq¼rk fdruh gS\ ekiu dh ;FkkZFkrk fdruh gS\

nh?kZ mÙkjh; iz'u (LA)

2.36 ek=kdksa dh ,d ubZ iz.kkyh dk izLrko fd;k x;k gS ftlosQ nzO;eku dk ek=kd α kg,

yackbZ dk ek=kd β m rFkk le; dk ek=kd γ s gSA bl ubZ iz.kkyh esa 5 J dk eki

D;k gksxk\

2.37 l yackbZ vkSj r f=kT;k osQ fdlh ikbi ls izfr lsoaQM fuxZr gksus okys nzo dk vk;rufdlh fo|kFkhZ }kjk fuEufyf[kr lehdj.k }kjk O;Dr fd;k x;k gSµ

4

8Pr

vl

π

η=

tgk¡ P ikbi osQ nksuksa fljksa osQ chp nkckarj gS rFkk η nzo dk ';kurk xq.kkad gS ftldkfoeh; lw=k ML–1 T–1gSAtk¡fp, fd lehdj.k foeh; n`f"V ls lgh gS ;k ughaA

2.38 ,d HkkSfrd jkf'k X, pkj es; jkf'k;ksa a, b, c ,oa d ls uhps n'kkZ, lw=k }kjk lacafèkr gaSµ

X = a2 b3 c5/2 d–2

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ek=kd vkSj ekiu

11

a, b, c ,oa d osQ ekiu esa izfr'kr =kqfV Øe'k% 1%, 2%, 3% vkSj 4%, gSA X osQekiu esa fdrus izfr'kr =kqfV laHkkfor gS\ ;fn mijksDr lw=k osQ vkèkkj ij ifjdfyrX dk eku 2-763 gS rks bl izkIr ifj.kke dk iw.kk±fdr eku D;k gksxk\

2.39 O;atd P = E l 2 m–5 G–2 esa E, m, l ,oa G Øe'k% ÅtkZ] nzO;eku] dks.kh; laosx ,oaxq#Roh; fu;rkad gSaA n'kkZb, fd P ,d foekfoghu jkf'k gSA

2.40 ;fn fuokZr esa izdk'k dk osx c, Iyk¡d fu;rkad h ,oa xq#Roh; fu;rkad G dks ewyjkf'k;k¡ eku ysa rks nzO;eku] yackbZ ,oa le; dks bu jkf'k;ksa osQ inksa esa O;Dr dhft,A

2.41 ,d Ñf=ke mixzg] M nzO;eku vkSj R f=kT;k osQ xzg osQ pkjksa vksj r f=kT;k dh d{kk esaifjØek dj jgk gSA fdlh osaQnzh; fiaM osQ pkjkssa vksj ifjØek djrs fdlh mixzg osQfy,] osQIyj osQ r`rh; fu;ekuqlkj ifjØe.k dky T dk oxZ d{kk dh f=kT;k r

osQ ?ku osQ lekuqikrh gksrk gSA foeh; fo'ys"k.k fofèk dk mi;ksx djosQ n'kkZb,]

3k rT

R g= , tgk¡ k ,d foekfoghu fu;rkad gS vkSj g osaQnzh; fiaM osQ xq#Ro osQ

dkj.k Roj.k gSA2.42 vksfyd vEy osQ ,d v.kq dk lkbt Kkr djus osQ fy, fd, x, ,d iz;ksx esa

vksfyd vEy osQ 1 mL dks vYdksgy osQ 19 mL esa ?kksyk x;k gSA vc bl foy;uosQ 1 mL esa vYdksgy feykdj bldk vk;ru 20 mL dj fy;k x;kA bl ruqÑrfoy;u dh ,d cw¡n] fNNyh ukan esa fy, x, ty esa Mky nh xbZA foy;u dh cw¡nus ty osQ i`"B ij leku :i ls iSQydj ,d v.kq tSlh eksVh ijr cuk yhA vc blfiQYe osQ Åij leku :i ls ykbdksiksfM;e ikmMj fNM+d dj fiQYe dk O;kl ekify;kA cw¡n dk vk;ru vkSj fiQYe dk {ks=kiQy Kkr gksus ls ge fiQYe dh eksVkbZ dhx.kuk dj ldrs gSa tks vksfyd vEy osQ v.kq osQ lkbt osQ cjkcj gSA

mijksDr m¼j.k dks è;kuiwoZd if<+, vkSj fuEufyf[kr iz'uksa osQ mÙkj nhft,A

(a) ge vksfyd vEy dks vYdksgy esa D;ksa ?kksyrs gSa\(b) ykbdksiksfM;e ikmMj dh D;k Hkwfedk gS\(c) rS;kj fd, x, foy;u osQ izR;sd mL esa vksfyd vEy dk fdruk vk;ru gS\(d) vksfyd vEy osQ bl foy;u dh n cw¡nksa dk vk;ru vki oSQls ifjdfyr djsaxs\(e) bl foy;u dh ,d cw¡n esa vksfyd vEy dk vk;ru fdruk gksxk\

2.43 (a) 1 ikjlsd esa fdrus [kxksyh; ,dd (A.U.) gksrs gSa\(b) eku yhft, fd lw;Z tSlk dksbZ rkjk] 2 ikjlsd nwjh ij fLFkr gSA ;fn bl rkjs

dks 100 vkoèkZu osQ nwjn'kZd }kjk ns[kk tk, rks bldk dks.kh; lkbt D;k gksxk\i`Foh ls ns[kus ij lw;Z dk dks.kh; lkbt (1/2)º fn[kkbZ iM+rk gSA ok;qeaMyh;mrkj&p<+ko osQ dkj.k vk¡[kksa }kjk 1 pki feuV ls NksVs fiaMksa dk foHksnu ughafd;k tk ldrk gSA

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12

(c) eaxy xzg dk O;kl iFoh osQ O;kl dk yxHkx vkèkk gSA tc ;g iFoh ls fudVrenwjh ij gksrk gS rks bldh i`Foh ls nwjh yxHkx 1/2 A.U. gksrh gSA x.kuk dhft,fd mijksDr nwjn'kZd }kjk ns[ks tkus ij ;g fdl lkbt dk fn[kkbZ nsxk\(fVIi.khµ bl iz'u }kjk ;g Li"V gksrk gS fd nwjn'kZd ;a=kksa }kjk xzgksa dk vkdkjD;ksa c<+ tkrk gS tcfd rkjksa osQ izdj.k esa ,slk ugha gksrk)A

2.44 vkbalVkbu osQ izfl¼ lkisf{kdrk osQ fl¼kar ls nzO;eku (m) ,oa ÅtkZ (E) esa lacaèk E= mc2 O;qRiUu gksrk gS] tgk¡ c fuokZr esa izdk'k dh pky gSA ukfHkdh; Lrj ij ÅtkZosQ ifjek.k cgqr de gksrs gSaA ukfHkdh; Lrj ij ÅtkZ dks izk;% MeV esa ekik tkrkgS] tgk¡ 1 MeV = 1.6 × 10–13 J; rFkk nzO;ekuksa dk ekiu ,dhÑr ijek.kq nzO;ekubdkbZ (u) esa fd;k tkrk gS] tgk¡ 1u = 1.67 × 10–27 kg.

(a) n'kkZb, fd 1 u osQ lerqY; ÅtkZ 931.5 MeV gSA(b) dksbZ fo|kFkhZ1 u = 931.5 MeV fy[krk gSA f'k{kd laosQr djrk gS fd ;g

lacaèk foeh; :i ls lgh ugha gSA lgh lacaèk fyf[k,A

18-04-2018

cgq fodYih; iz'u (MCQ I)

3.1 fn, x, xzkiQksa (fp=k 3-1) essa osQoy ,d xzkiQ ,slk gS ftlesa le; varjky (0, T) osQfy, vkSlr osx] ,d mi;qDr :i ls pqus x, le; T osQ fy, 'kwU; gks ldrk gSA ;gdkSu&lk xzkiQ gS\

vè;k; 3

ljy js[kk esa xfr

fp=k 3.1

x

t

x

t

x

t

x

t

(a) (b)

(c) (d)

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14

3.2 ,d fyÝV vkBoha eafty ls uhps vk jgh gS vkSj pkSFkh eafty ij igq¡pus okyh gSA ;fnlHkh jkf'k;ksa osQ fy, Hkwry dks ewy ¯cnq rFkk Åij dh vksj èkukRed fn'kk ysa rksfuEufyf[kr esa dkSu lgh gS\

(a) x < 0, v < 0, a > 0

(b) x > 0, v < 0, a < 0

(c) x > 0, v < 0, a > 0

(d) x > 0, v > 0, a < 0

3.3 ,dfoeh; xfr esa] rkR{kf.kd pkyv osQ fy, 'krZ 0 ≤ v < v0 iwjh gksrh gS rks

(a) T le; esa foLFkkiu dk eku dHkh ½.kkRed ugha gksrkA

(b) T le; esa foLFkkiu x osQ fy, – vo T < x < v

o T gksrkA

(c) Roj.k dHkh ½.kkRed ugha gksrkA

(d) xfr dh fn'kk esa dHkh ifjorZu ugha gksrkA

3.4 ,d okgu vkèkh L dks pky V1 ls rFkk 'ks"k vkèkh nwjh dks pky V

2 ls r; djrk gSA bldh

vkSlr pky gSµ

(a)1 2

2

V V+

(b)1 2

1 2

2V V

V V

+

+

(c) 1 2

1 2

2V V

V V+

(d)1 2

1 2

( )L V V

V V

+

3.5 fdlh d.k dk foLFkkiu x = (t – 2)2 fu#fir fd;k tkrk gSA tgk¡ x ehVj esa rFkk tlsoaQM esa ekik x;k gSµ

(a) 4 m

(b) 8 m

(c) 12 m

(d) 16 m

3.6 fdlh esVªks LVs'ku ij dksbZ yM+dh ,d #osQ gq, ,LosQysVj ij t1 lsoaQM esa Åij p<+rh

gSA ;fn og ,LosQysVj ij [kM+h jgs rks ,LosQysVj mls t2 lsoaQM esa Åij ys tkrk gSA ;fn

og pyrs gq, ,LosQysVj ij viuh iwoZ xfr ls gh Åij p<+s rks mldks Åij rd igq¡pusesa yxus okyk le; gksxk&

18-04-2018

ljy js[kk esa xfr

15

(a) (t1 + t

2)/2

(b) t1t2/(t

2–t

1)

(c) t1t2/(t

2+t

1)

(d) t1–t

2

cgq fodYih; iz'u (MCQ II)

3.7 fdlh d.k dh ljy js[kk esa xfr dk fooj.k nsus osQ fy, jkf'k B osQ lkFkjkf'k A esa gksus okys ifjorZu dk xzkiQ fp=k 3-2 esa n'kkZ;k x;k gS%

(a) jkf'k B le; fu#fir dj ldrh gSA

(b) ;fn xfr ,dleku gks rks jkf'k A osx gSA

(c) ;fn xfr ,dleku gks rks jkf'k A foLFkkiu gSA

(d) ;fn xfr ,dleku Rofjr gS rks A osx gSA

3.8 x vkSj t osQ chp ,d xzkiQ fp=k 3-3 esa n'kkZ;k x;k gSA uhps fn, x, dFkuksaesa ls lgh fodYi pqfu,µ

(a) t = 0 ij d.k fojkekoLFkk ls NksM+k x;k Fkk

(b) B ij Roj.k a > 0

(c) C ij osx ,oa Roj.k 'kwU; gksrs gSa

(d) A ,oa D osQ chp xfr dk vkSlr osx èkukRed gS

(e) D ij pky E ls vfèkd gS

3.9 x = t–sint }kjk fu#fir ,d foeh; xfr osQ fy,µ

(a) lHkh t > 0 ekuksa osQ fy, x (t) > 0

(b) lHkh t > 0 ekuksa osQ fy, v (t) > 0

(c) lHkh t > 0 ekuksa osQ fy, a (t) > 0

(d) v (t) dk eku 0 ,oa 2 osQ chp gksrk gSA

3.10 ,d fLiazx dks ftldk ,d fljk ,d nzO;eku ls vkSj nwljk ,d n`<+ vkèkkj ls tqM+k gS][khapdj NksM+ fn;k tkrk gSµ

(a) Roj.k dk ifj.kke vfèkdre rc gksrk gS tc fLizax dks NksM+k tkrk gSA

(b) Roj.k dk ifj.kke lkE;koLFkk esa vfèkdre gksrk gSA

(c) pky vfèkdre rc gksrh gS tc nzO;eku lkE;koLFkk esa gksrk gSA(d) foLFkkiu dk ifj.kke vfèkdre osQoy rHkh gksrk gS tc pky U;wure gksrh gSA

3.11 ,d xsan 10 m yacs ,d jsy osQ MCcs dh foijhr nhokjksa osQ chp xfr dj jgh gSA munhokjksa ij ;g 1ms–1 dh pky ls muosQ yacor~ izR;kLFk la?kV~V djrh gSA jsy xsan dh

fp=k 3.2

x

A

B

C

t

E

D

fp=k 3.3

18-04-2018


16

xfr dh fn'kk osQ lekarj 10m/s–1 osQ fu;r osx ls xfreku gSA Hkw&i`"B ls ns[kus ijµ

(a) xsan dh xfr dh fn'kk izR;sd 10 lsoaQM esa cny tkrh gSA

(b) izR;sd 10 lsoaQM esa xsan dh pky cny tkrh gSA

(c) 20 lsoaQM osQ fdlh Hkh le;&varjky esa xsan dh vkSlr pky vpj jgrh gSA

(d) xsan dk Roj.k Hkw&i`"B ls ns[kus ij Hkh ogh gksxk tks jsy ls ns[kus ijA

vfr y?kqmÙkjh; iz'u (VSA)

3.12 fp=k 3-1 eas n'kkZ, xzkiQksa dk lanHkZ yhft,A fuEufyf[kr dk feyku dhft,µ

xzkiQ vfHky{k.k

(a) (i) iwjh le;kofèk esa v > 0 rFkk a < 0

(b) (ii) iwjh le;kofèk esa x > 0 rFkk blosQ ,d fcanq ij v = 0 rFkk,d fcanq ij a = 0.

(c) (iii) blesa t > 0 ij ,d ,slk fcanq gS tgk¡ foLFkkiu 'kwU; gSA

(d) (iv) v < 0 rFkk a > 0 gSA

3.13 fdlh ,dleku xfr ls vkrh gqbZ fØosQV xsan dks cYyk ekjdj okil ykSVk fn;k x;kAxsan vfr vYidky osQ fy, gh cYys osQ laioZQ esa jghA le; osQ lkFk xsaan osQ Roj.kesa gksus okys ifjorZu dks n'kkZb,A ¹foijhr fn'kk esa Roj.k dh fn'kk dks èkukRed ekuyhft,º

3.14 ,dfoeh; xfr osQ ,sls mnkgj.k crkb, tgk¡]

(a) èkukRed x-fn'kk esa pyrk gqvk d.k vkorhZ :i ls fojke esa vkrk gS vkSj fiQj vkxsc<+ tkrk gSA

(b) èkukRed x-fn'kk esa pyrk gqvk d.k vkorhZ :i ls fojke esa vkrk gS vkSj fiQj ihNsykSV tkrk gSA

3.15 ,d ,slh xfr dk mnkgj.k nhft, ftlesa fdlh fof'k"V {k.k ij x > 0, v < 0, a > 0A

3.16 fdlh rjy esa fxjrs gq, fiaM dk Roj.k ge a = g – bv }kjk O;Dr djrs gSa tgk¡g = xq#Roh; Roj.k rFkk b ,d fu;rkad gSA fiaM dks rjy esa fxjus osQ fy, NksM+us osQdkiQh le; osQ ckn ;g fu;r pky ls fxjrk gqvk ik;k tkrk gSA bl fu;r pky dkeku D;k gksuk pkfg,\

fp=k 3.4


3.17 ,d xsan dks oqQN Å¡pkbZ ls fxjk;k x;k gS vkSj mldk foLFkkiu≤xzkiQ fp=k 3-4 esa n'kkZ, vuqlkj izkIr gksrk gS (foLFkkiu x Hkwryls ekik x;k gS vkSj lHkh

(a) xq.kkRed i{k è;ku esa j[krs gq, osx≤ xzkiQ cukb,A

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ljy js[kk esa xfr

17

(b) xq.kkRed i{k è;ku esa j[krs gq, Roj.k≤ xzkiQ cukb,A

3.18 ,d d.k dh xfr dks ( )( ) ;1 −= −t

ox t x e γ 0≥t , x0 > 0 }kjk fu#fir fd;k tk

ldrk gSA blesa

(a) d.k dgk¡ ls vkSj fdrus osx ls xfr izkjaHk djrk gS\

(b) x (t), v (t), a (t) osQ vfèkdre ,oa U;wure eku Kkr dhft,A n'kkZb, fd x (t) ,oaa (t) osQ eku le; osQ lkFk c<+rs gSa rFkk v (t) dk eku le; osQ lkFk de gksrk gSA

3.19 ,d i{kh ,d lhèkh lM+d ij ,d nwljs dh vksj pyrh gqbZ nks dkjksa osQ chp] ,d dkjls nwljh dkj rd ckj&ckj mM+dj tkrk gSA ,d dkj dh pky 18m/h tcfd nwljhdkj dh pky 27km/h gSA ftl le; igyh dkj vkSj nwljh dkj esa 36 km dh nwjhgSA i{kh 36 km/h dh pky ls ,d dkj ls nwljh dkj rd mM+uk izkjaHk djrk gSA i{khoqQy fdruh nwjh r; djrk gS\ i{kh dk oqQy foLFkkiu fdruk gS\

3.20 ,d O;fDr ,d Å¡ps Hkou dh Nr ij nkSM+rk gS vkSj bl vk'kk ls {kSfrt fn'kk esa Nykaxyxkrk gS fd og ikl osQ ,d vU; vis{kkÑr uhps Hkou dh Nr ij igq¡p tk,xkA ;fnmldh pky 9 m/s gS] nksuksa Hkouksa osQ chp dh {kSfrt nwjh 10 m gS vkSj Hkouksa dh Å¡pkbZesa varj 9 m gS rks D;k og nwljs Hkou rd igq¡p ik,xk\ (g = 10 m/s2 ys ldrs gSa)

3.21 45 m Å¡ph bekjr ls fxjkbZ tkrh gSA Bhd mlh le; ,d nwljh xsan 40

m/s dh pky ls Åij dh vksj isaQdh xbZ gSA nksuksa xsanksa osQ lkis{k osx dhle; osQ iQyu osQ :i eas x.kuk dhft,A

3.22 fdlh d.k dh xfr dk osx&foLFkkiu xzkiQ fp=k 3-5 esa n'kkZ;k x;k gS

(a) v ,oa x osQ chp lacaèk fyf[k,A

(b) Roj.k ,oa foLFkkiu esa lacaèk izkIr dhft, vkSj bldk xzkiQ cukb,A


3.23 ;g ,d lkekU; izs{k.k gS fd o"kkZèkkjh es?k Hkwry ls yxHkx 1km dh Å¡pkbZij gks ldrs gSaA

(a) o"kkZ dh ,d cw¡n ;fn bruh Å¡pkbZ ls osQoy xq#Ro osQ vèkhu fxjs rks Hkwry ijigq¡pus ij bldh pky D;k gksxh\ bl eku dks km/h esa Hkh ifjdfyr dhft,A(g = 10m/s2)

(b) ,d izk:fid o"kkZ cw¡n dk O;kl yxHkx 4mm gSA laosx] nzO;eku ,oa pky osQxq.kuiQy osQ cjkcj gksrk gSA cw¡n osQ Hkwry ls Vdjkrs le; mlosQ laosx dk vkdyudhft,A

(c) cw¡n dks pkSjlkus esa yxs le; dk vkdyu dhft,A

(d) laosx ifjorZu dh nj cy gksrh gSA ml cy dk vkdyu dhft, tks ;g cw¡n vkiij vkjksfir djsxhA

xo

x

vo

v

o

fp=k 3.5

18-04-2018


18

(e) Nkrs ij yxus okys cy dh ifjek.k dh dksfV vkdyu dhft,A o"kkZ dh nks cw¡nksaosQ chp izk:fid ikf'Zod i`Fkdu 5 cm gSA

eku yhft, fd Nkrk o`Ùkkdkj gS vkSj mldk O;kl 1 m gS rFkk o"kkZ dh cw¡nsa NkrsosQ diM+s dk osèku ugha dj ldrhA

3.24 72km/h dh pky ls xfreku dkj 3-0 lsoaQM ls de le; esa fojke esa ugha vkldrh tcfd Vªd osQ fy, ;g le; vofèk 5.0 s gSA fdlh jktekxZ ij dkj Vªd osQihNs gS vkSj nksuksa 72 km/h dh pky ls xfreku gSaA Vªd ;g flXuy nsrk gS fd mlsvkikr fLFkfr esa #duk iM+ jgk gSA dkj Vªd ls fdrus ihNs gksuh pkfg, fd ;g mllsVdjkus osQ cp losQA ekuoh; izfrfØ;k dky 0.5s gSA ;fn Vªd dkj osQ ihNs gksrk rksmÙkj D;k gksrk\

(fVIi.khµ ;g iz'u ;g Li"V djrk gS fd xkfM+;ksa osQ ihNs ^^lqjf{kr nwjh cuk,jf[k,** dk lans'k D;ksa fy[kk jgrk gSA)

3.25 25 ,d canj ,d fpdus LraHk ij 3s rd Åij p<+rk gS vkSj fiQj 3s rd uhps fiQlytkrk gSA fdlh {k.k t ij ms–1 esa bldk osx α t < 3s osQ fy, v(t) = 2t (3 –t) lsrFkk 3< t < 6s osQ fy, v(t) = –(t –3) (6 – t) ls fu#fir fd;k tkrk gSA canj dh xfrdk ;g Øe rc rd pyrk gS tc rd fd ;g 20m Å¡pkbZ ij ugha igq¡p tkrk

(a) fdl {k.k ij bldk osx vfèkdre gksxk\

(b) fdl {k.k ij bldk vkSlr osx vfèkdre gksxk\

(c) fdl {k.k ij blosQ Roj.k dk ifj.kke vfèkdre gksxk\

(d) canj dks 'kh"kZ rd igq¡pus esa fdrus Øe uhps dh xfr osQ gksaxs (fHkUukRed va'kksadh x.kuk Hkh djsa)

3.26 ,d O;fDr 100 m Å¡ps Hkou osQ Åij [kM+k gqvk gSA og nks xsanksa dks ÅèokZèkjr% isaQdrkgSA ,d t = 0 ij vkSj nwljh 2 lsoaQM ls de le;&varjky osQ cknA nwljh xasn igyhxsan osQ vkèks osx ls isaQdh tkrh gSA t = 2 s ij nksuksa xsanksa osQ chp ÅèokZèkj nwjh + 15m

gSA ;g ik;k tkrk gS fd ;g nwjh vifjofrZr jgrh gSA mu osxksa dh x.kuk dhft,] ftulsxsansa isaQdh tkrh gSa vkSj mudks isaQosQ tkus osQ {k.kksa osQ le;&varjky dh Hkh x.kuk dhft,A

18-04-2018

cgq fodYih; iz'u-I (MCQ I)

4.1 ˆ ˆ= +A i j rFkk ˆ ˆ= −B i j osQ chp dks.k gSµ

(a) 45° (b) 90° (c) –45° (d) 180°

4.2 fuEufyf[kr esa dkSu&lk dFku lR; gS\

(a) vfn'k jkf'k og gksrh gS tks fdlh izfØ;k esa lajf{kr jgrh gSA

(b) vfn'k jkf'k og gksrh gS ftldk eku dnkfi ½.kkRed ugha gks ldrkA

(c) vfn'k jkf'k og gksrh gS ftldk eku vkdk'k esa ,d ¯cnq ls nwljs ¯cnq ij ughacnyrkA

(d) vfn'k jkf'k dk eku v{kksa osQ fofHkUu foU;klksa esa fLFkr izs{kdksa osQ fy, lekugksrk gSA

4.3 fp=k 4.1 esa XY ry esa nks lfn'kksa u ,oa v osQ foU;kl n'kkZ, x, gSaA ;fn

ˆ ˆa b= +u i j vkSj

ˆ ˆp q= +v i j

vè;k; 4

lery esa xfr

u

Y

X

v

fp=k 4.1

18-04-2018

20


rks fuEufyf[kr esa dkSu&lk dFku lgh gS\

(a) a ,oa p èkukRed gSa tcfd b vkSj q ½.kkRed gaSA(b) a, p vkSj b èkukRed gaS tcfd q ½.kkRed gSA(c) a, q vkSj b èkukRed gSa tcfd p ½.kkRed gSA(d) a, b, p vkSj q lHkh èkukRed gSaA

4.4 fdlh lfn'k r osQ X-v{k osQ vuqfn'k ?kVd dk eku vfèkdre gksxk ;fn

(a) r èkukRed Y-v{k osQ vuqfn'k gSA(b) r èkukRed X-v{k osQ vuqfn'k gSA(c) r X-v{k ls 45° dk dks.k cukrk gSA(d) r ½.kkRed Y-v{k osQ vuqfn'k gSA

4.5 15° osQ dks.k ij iz{ksfir fdlh iz{ksI; dk {kSfrt ijkl 50 m gSA ;fn bls 45° osQ dks.kij iz{ksfir fd;k tk, rks bldk ijkl gksxk&

(a) 60 m

(b) 71 m

(c) 100 m

(d) 141 m

4.6 jkf'k;ksa nkc] 'kfDr] ÅtkZ] vkosx] xq#Roh; foHko] fo|qr vkos'k] rki vkSj {ks=kiQy ijfopkj dhft,A buesa osQoy lfn'k jkf'k;k¡ gSa&

(a) vkosx] nkc vkSj {ks=kiQy(b) vkosx vkSj {ks=kiQy(c) {ks=kiQy vkSj xq#Roh; foHko(d) vkosx vkSj nkc

4.7 fdlh f}foeh; xfr esa rkR{kf.kd pky v0 ,d èkukRed fu;rkad gSA rc fuEufyf[kr

esa dkSu&lk dFku vfuok;Zr% lR; gS\

(a) vkSlr osx fdlh Hkh le; 'kwU; ugha gksrkA(b) vkSlr Roj.k lnSo 'kwU; gksuk pkfg,A(c) leku le; varjky esa gq, foLFkkiu leku gksrs gSaA(d) leku le; varjkyksa esa leku iFk nwfj;k¡ r; dh tkrh gSaA

4.8 fdlh f}foeh; xfr esa rkR{kf.kd pky v0 dksbZ èkukRed fu;rkad gSA fuEufyf[kr esa

dkSu&lk dFku vfuok;Zr% lR; gS\(a) d.k dk Roj.k 'kwU; gSA(b) d.k dk Roj.k ifjc¼ gSA(c) d.k dk Roj.k vfuok;Zr% xfr osQ ry esa gSA(d) d.k dks ,d leku o`Ùkh; xfr djuh pkfg,A

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lery esa xfr

21

4.9 rhu lfn'kksa A,B ,oa C dk ;ksx 'kwU; gSA fuEufyf[kr esa dkSu&lk dFku vlR; gS\

(a) (A×B) × C 'kwU; ugha gksrk tc rd B,C lekarj u gksaA

(b) (A×B).C 'kwU; ugha gksrk tc rd B,C lekarj u gksaA

(c) ;fn A,B,C fdlh ry dks ifjHkkf"kr djsa rks (A×B)×C ml ry esa gksxkA(d) (A×B).C=|A||B||C|→→→→→ C2=A2+B2

4.10 ;g ik;k x;k gS fd |A+B|=|A| rc blls vfuok;Zr% ;g èofu gksrh gS fd

(a) B = 0

(b) A,B izfr lekarj gSa

(c) A,B yacor~ gSaA

(d) A.B ≤ 0

cgq fodYih; iz'u-II (MCQ II)

4.11 nks d.k ok;q esa V0 pky ls iz{ksfir fd, x, gSaA nks d.k {kSfrt ls Øe'k% θ

1 rFkk θ

2

(nksuksa U;wu dks.k) osQ iz{ksi dks.kksa ij ok;q esa v0 pky ls iz{ksfir fd;s tkrs gSaA ;fn igys

d.k }kjk izkI; Å¡pkbZ nwljs d.k dh rqyuk esa vfèkd gS] rks lgh fodYiksa dk p;udhft,µ

(a) iz{ksi dks.k : θ1 > θ

2

(b) mîó;u dky : T1 > T

2

(c) {kSfrt ijkl : R1 > R

2

(d) oqQy ÅtkZ : U1 > U

2

4.12 dksbZ d.k fdlh ijoyf;d (y = x2) ?k"kZ.kjfgr iFk(A – B – C) ij fcanq A ls fojkekoLFkk ls uhps dh vksjfiQlyrk gS (fp=k 4.2)A fcanq B ijoy; osQ 'kh"kZ ij gS rFkkfcanq C dh Å¡pkbZ fcanq A ls de gSA C osQ i'pkr~ d.k eqDr :ils ok;q esa iz{ksI; dh Hkk¡fr xfr djrk gSA ;fn ;g d.k mPprefcanq P rd igq¡prk gS] rks&

(a) P ij xfrt ÅtkZ = B ij xfrt ÅtkZ

(b) P dh Å¡pkbZ = A dh Å¡pkbZ

(c) P ij oqQy ÅtkZ = A ij oqQy ÅtkZ

(d) A ls B rd pyus esa yxk le; = B ls P rd pyus esa yxk le;

4.13 fdlh d.k dh O;kid xfr osQ fy, uhps foLFkkiu] osx ,oa Roj.k ls lacafèkr pkjfofHkUu O;atd fn, x, gSaaA mu O;atdksa dk p;u dhft, tks lgh ugha gSaµ

Ay

-x2 -x1 B -xo

( 0)x =

P

Vo

x

C

fp=k 4.2

18-04-2018

22


(a) [ ]1 2

1( ) ( )

2av t t= +v v v

(b)2 1

2 1

( ) ( )av

t t

t t

−=

−

r rv

(c) ( )2 1 2 1

1( ) ( ) ( )

2t t t t= − −r v v

(d)2 1

2 1

( ) ( )av

t t

t t

−=

−

v va

4.14 ,d leku orZqy xfr djrs fdlh d.k osQ fy, lgh dFku@dFkuksa dk p;u dhft,µ

(a) d.k osQ osx dk ifjek.k (pky) vpj jgrk gSA

(b) d.k dk osx èkzqokarj js[kk osQ yacor~ fn"V gksrk gSA

(c) xfr djrs le; d.k osQ Roj.k dh fn'kk ifjo£rr gksrh jgrh gSA

(d) dks.kh; laosx dk ifjek.k fu;r jgrk gS] ijarq fn'kk ifjo£rr gksrh jgrh gSA

4.15 nks lfn'kksa A ,oa B osQ fy, + = −A B A B rHkh lnSo lR; gksxk tcµ

(a) 0= ≠A B

(b) ⊥A B

(c) 0= ≠A B rFkk A ,oa B ;k rks lekarj gS ;k izfr lekarj

(d) ;k rks A vFkok B 'kwU; gSA


4.16 dksbZ lkbfdy lokj 1 km f=kT;k osQ o`Ùkkdkj ikoZQ osQ osaQnz O ls pyuk vkjaHk djrkgS vkSj vkoQfr 4.3 esa n'kkZ, x, iFk OPRQO osQ vuqfn'k xeu djrk gSA ;fn 10ms–1

dh fu;r pky cuk, j[ks rks R fcanq ij mlosQ Roj.k dk ifjek.k vkSj fn'kk D;k gS\

4.17 dksbZ d.k ok;q esa {kSfrt ls dksbZ dks.k cukrs gq, iz{ksfir fd;k tkrk gS vkSj ;gfp=k 4.4 esa n'kkZ, vuqlkj fdlh ijoyf;d iFk ij xfr djrk gSA ;gk¡ x ,oa y Øe'k%{kSfrt ,oa ÅèokZèkj fn'kk,¡ lwfpr djrs gSaA fp=k esa fcanq A, B ,oa C ij osx ,oa Roj.k dhfn'kk,¡ n'kkZb,A

Q

R

PO

fp=k 4.3

fp=k 4.4

y

x

A

H

B

C

18-04-2018

lery esa xfr

23

4.18 fdlh Hkou dh Nr ls dksbZ xasn {kSfrt ls 45° osQ dks.k ij Åij isaQdh tkrh gSA oqQNlsoaQM osQ ckn ;g èkjrh ls Vdjkrh gSA viuh xfr osQ nkSjku fdl fcanq ij xsan

(a) dh pky vfèkdre gksxh](b) dh pky U;wure gksxh](c) dk Roj.k vfèkdre gksxk\

vius mÙkj dh O;k[;k dhft,A

4.19 fdlh iqQVcky dks fdd ekjdj ÅèokZèkjr% Åij isaQdk x;k gSA mPpre fcanq ij bldk(a) Roj.k] vkSj (b) osx D;k gS\

4.20 A, B ,oa C rhu vlajs[kh] vleryh lfn'k gSaA A × (B × C) dh fn'kk osQ fo"k; esafVIi.kh dhft,A


4.21 lery lM+d ij] fu;r osx ls] [kqyh dkj esa ;k=kk djrs gq, dksbZ yM+dk fdlh xsandks ok;q esa ÅèokZèkjr% Åij mNkyrk gS vkSj fiQj mls yid ysrk gSA iqQVikFk ij [kM+sfdlh vU; yM+osQ }kjk izsf{kr xsan dh xfr dk vkjs[k [khafp,A vius vkjs[k dkLi"Vhdj.k dhft,A

4.22 dksbZ yM+dk fdlh xsan dks lM+d osQ vuqfn'k {kSfrt ls 60° dk dks.k cukrs gq,10 m/s osx ls isaQdrk gSA ogha ls xqtjrh fdlh dkj esa cSBk dksbZ yM+dk bl xsan dhxfr dk izs{k.k djrk gSA ;fn dkj dh xfr (5m/s) gks] rks dkj esa cSBs yM+osQ }kjk iszf{krxsan dh xfr dk vkjs[k [khafp,A vius vkjs[k dk Li"Vhdj.k dhft,A

4.23 ok;q esa iz{ksI; dh xfr dk vè;;u djrs le; ge xfr ij ok;q izfrjksèk osQ izHkko dhmis{kk dj nsrs gSaA blls tSlk fd vkius vè;;u fd;k gS] gesa izrhr ijoyf;d izkIrgksrk gSA ;fn ge ok;q izfrjksèk dks lfEefyr djssaa rks iz{ksI; iFk oSQlk izrhr gksxk\ bliz{ksI; iFk dk vkjs[k [khafp, vkSj le>kb, fd vkius bls ,slk D;ksa cuk;k gSA

4.24 dksbZ yM+kowQ foeku] 1.5 km Å¡pkbZ ij] 720 km/h pky ls {kSfrtr% mM+ jgk gSA({kSfrt osQ lkis{k) fdl n'kZ dks.k ij y{; fn[kkbZ iM+us ij ik;yV dks ce fxjkukpkfg, rkfd og y{; ij Vdjk,\

4.25 (a) i`Foh dks 6400 km f=kT;k dk ,d xksyk ekuk tk ldrk gSA dksbZ Hkh fiaM (;kO;fDr) i`Foh dh ?kw.kZu xfr osQ dkj.k i`Foh osQ v{k osQ ifjr% orqZy xfr dj jgkgS (ifjØe.k dky ,d fnu)A i`Foh osQ i`"B ij (fo"kqor o`Ùk) ij fLFkr fdlhfiaM dk ;g Roj.k v{kka'k ij fdruk gksxk\ bu Roj.k ekuksa dh g = 9.8 m/s2

osQ lkFk rqyuk dhft,A

(b) i`Foh Hkh lw;Z osQ pkjksa vksj × 111.5 10 m f=kT;k dh o`Ùkkdkj d{kk esa pDdjyxkrh gS tks o"kZ esa ,d ckj iwjk gksrk gSA i`Foh (;k mlosQ i`"B ij fLFkr fdlh

18-04-2018

24


fiaM) dk lw;Z osQ osaQnz dh vksj Roj.k fdruk gS\ ;g Roj.k g = 9.8 m/s2 dhrqyuk esa fdruk gS\

4.26 uhps dkWye I esa lfn'kksa a, b vkSj c osQ chp lacaèk fn, x, gSa rFkk dkWye II esa a, b

vkSj c osQ XY ry esa foU;kl fd, x, gSA dkWye I osQ lacaèkksa dk dkWye II esa fn, x,muosQ lgh foU;klkas osQ lkFk feyku dhft,A

dkWye I dkWye II

(a) + =a b c

(b) a – c = b

(c) b – a = c

(d) a + b + c = 0

Y

Xa

c b

Y

X

a

cb

Y

X

b a

c

(ii)

(iii)

(iv)

Y

X

ba

c

(i)

π =

2 2

2

4V R

R TlaosQr µ Roj.k

18-04-2018

lery esa xfr

25

4.27 ;fn = =2 4A B,oa , rks dkWye I esa fn;s x;s lacaèkksa dk dkWye II esa fn;s x;s

A ,oa B osQ chp dks.k θ ls feyku dhft,A

dkWye I dkWye II

(a) A.B = 0 (i) θ = 0

(b) A.B = +8 (ii) θ = 90°

(c) A.B = 4 (iii) θ = 180°

(d) A.B = –8 (iv) θ = 60°

4.28 ;fn = =2 4A B,oa , rks dkWye I esa fn, x, lacaèk dk dkWye II esa fn, x,

A vkSj B osQ chp osQ dks.k θ ls feyku dhft,A

dkWye I dkWye II

(a) 0× =A B (i) θ = 30°

(b) 8× =A B (ii) θ = 45°

(c) 4× =A B (iii) θ = 90°

(d) 4 2× =A B (iv) θ = 0°


4.29 dksbZ igkM+h 500 m Å¡ph gSA fdlh rksi }kjk] tks iSdVksa dks 125 m/s dh pky lsiz{ksfir dj ldrh gSA bl igkM+h osQ ikj dksbZ vkiwfrZ dh tkuh gSA rksi igkM+h osQ vkèkkjls 800m dh nwjh ij fLFkr gS vkSj igkM+h ls bldh nwjh lek;ksftr djus osQ fy, blsi`Foh ij 2 m/s pky ls pyk;k tk ldrk gSA og vYire le; ifjdfyr dhft,ftlesa dksbZ iSosQV igkM+h osQ ikj Hkwry rd igq¡p ldrk gSA (g =10 m/s2 yhft,)

4.30 dksbZ canwd vfèkdre pky ov ls xksyh nkx ldrh gS vkSj blls izkIr gks ldus okyk

vfèkdre {kSfrt ijkl 2

ovR

g= gSA ;fn dksbZ y{; (R osQ ijs) x∆ nwjh

ij gks] rks n'kkbZ, fd ml ij mlh canwd ls izgkj djus osQ fy, bl canwd

dks de ls de 1x

h xR

∆ = ∆ +

Å¡pkbZ rd mBkuk iM+sxkA

(laosQr% bl leL;k dks nks fHkUu&fHkUu fofèk;ksa ls gy fd;k tk ldrk gS)A

(i) fp=k 4.5 dk lanHkZ yhft, % y{; T dh {kSfrt nwjh x = R + ∆x gS vkSj;g iz{ksi.k fcanq ls y = – h nwjh uhps gSA

vo

vo

P

Tx

R

q

q

h

fp=k 4.5

18-04-2018

26


q

L

P

fp=k 4.7

(ii) fp=k esa fcanq P ls % ov pky ls {kSfrt ls uhps dks.k θ ij iz{ksi.k Å¡pkbZ h

rFkk {kSfrt ijkl ∆x.)

4.31 dksbZ d.k ok;q esa fdlh ,sls lery i`"B ls β dks.k cukrs gq, iz{ksfir fd;k x;k

gS tks Lo;a {kSfrt ls α dks.k cukrk gS (fp=k 4.6)A

(a) lery i`"B ij ijkl osQ fy, O;atd O;qRiUu dhft, (lery i`"B ijiz{ksi.k fcanq ls ml fcanq rd dh nwjh tgk¡ iz{ksI; tkdj Vdjk,xk)A

(b) mîó;u dky Kkr dhft,A

(c) β dk og eku Kkr dhft, ftl ij vfèkdre ijkl izkIr gksxkA

(laosQr % ;g leL;k nks fHkUu fofèk;ksa }kjk gy dh tk ldrh gS)A

(i) og fcanq P ftl ij izf{kIr d.k lery ls tkdj Vdjkrk gS mls iz{ksI; iFk(ijoy;) rFkk lery osQ vuqfn'k js[kk osQ dVku fcanq osQ :i esa ns[kk tkldrk gSA Lej.k jgs] d.k {kSfrt ls dks.k ( )α β+ ij izf{kIr fd;k x;k gSA

(ii) ge x-fn'kk dks lery osQ vuqfn'k vkSj y-fn'kk dks blosQ yacor~ ys ldrsgSaA rc bl izdj.k esa g (xq#Roh; Roj.k) dks nks fofHkUu ?kVdksa g

x lery

osQ vuqfn'k vkSj gy blosQ yacor~ esa fu;ksftr dhft,A vc bl leL;k dks

Øe'k% x rFkk y fn'kkvksa esa le; dks mHk;fu"B izkpy osQ :i esa ysdj nksLora=k xfr;ksa osQ :i esa gy fd;k tk ldrk gSA

4.32 fdlh Å¡pkbZ ls ÅèokZèkj uhps fxjrk gqvk dksbZ d.k v0 pky ls fdlh ,sls

lery i"B ls Vdjkrk gS tks {kSfrt ls θ dks.k cukrk gS rFkk ry ls izR;kLFkla?kêð djosQ izfrf{kIr gksrk gS (vkÑfr 4.7)A lery osQ vuqfn'k og nwjhKkr dhft, ftl ij ;g lery ls nwljh ckj Vdjk,xkAlaosQr: (i) Vdjkus osQ ckn Hkh d.k dk izkjafHkd osx oV gh gksxk\

(ii) ml dks.k dk ifjdyu dhft, tks izfrf{kIr gksus osQ i'pkr~ d.k dkosx {kSfrt ls cukrh gSA

(iii) 'ks"k foospu mlh izdkj gS tSls fd d.k dks vkur ry ij Åij dhfn'kk esa izf{kIr fd;k tk,A

4.33 dksbZ yM+dh tks lkbfdy ij mÙkj fn'kk esa 5 m/s osx ls tk jgh gSA ;g izsf{krdjrh gS fd o"kkZ ÅèokZèkjr% fxj jgh gSA ;fn og viuh pky c<+k dj 10 m/

s dj nsrh gS] rks o"kkZ ÅèokZèkj fn'kk ls 45° dk dks.k cukrs gq, fxjrh izrhr gksrhgSA o"kkZ dh pky Kkr dhft,A i`Foh ij [kM+s fdlh izs{kd dks o"kkZ dh fn'kk D;kizrhr gksxh\

(laosQr µ mÙkj fn'kk dks i vkSj ÅèokZèkjr% uhps dh fn'kk ˆ− j ekudj o"kkZ dkosx rv = ˆ ˆa b+i j yhft,A)

vo

fp=k 4.6

18-04-2018

lery esa xfr

27

yM+dh }kjk izsf{kr o"kkZ dk osx lnSo r girl−v v gksxkA iznÙk lwpuk osQ vkèkkj ij lfn'k

vkjs[k cukb, rFkk a vkSj b osQ eku Kkr dhft,A vki lHkh lfn'k i`Foh ij [kM+s izs{kdosQ lanHkZ izsQe esa fu#fir dj ldrs gSaA

4.34 dksbZ unh iwoZ fn'kk esa 3m/s pky ls cg jgh gSA dksbZ rSjkd fLFkj ty esa4 m/s–1 pky ls rSj ldrk gSa (fp=k 4.8)A

(a) ;fn rSjkd mÙkj fn'kk esa rSjuk izkjaHk djs] rks mldk ifj.kkeh osx(ifjek.k vkSj fn'kk) D;k gksxk\

(b) ;fn og nf{k.kh rV osQ fcanq A ls rSjuk izkjaHk djosQ mÙkjh rV ij cnqA osQ Bhd lkeus osQ fcanq B ij igq¡puk pkgs] rks

(a) mls fdl fn'kk eas rSjuk pkfg,\

(b) mldh ifj.kkeh pky D;k gksxh\

(c) Åij o£.kr nks fHkUu izdj.kksa (a) vkSj (b) esa ls fdlesa og de le;esa foijhr rV ij igq¡psxk\

4.35 fØosQV dk dksbZ {ks=k j{kd fØosQV xasn dks vo pky ls isaQd ldrk gSA ;fn og u osx

ls nkSM+rs gq, xsan dks {kSfrt ls θ dks.k ij isaQdrk gS] rks Kkr dhft, µ

(a) fdlh n'kZd }kjk izsf{kr {kSfrt ls cuk ok;q esa izf{kIr xsan dk izHkkoh dks.k

(b) mîó;u dkyA

(c) iz{ksi.k fcanq ls ml fcanq rd dh nwjh ({kSfrt ijkl) tgk¡ tkdj xsan fxjrh gSA

(d) og dks.k θ ftl ij xsan isaQdus ls (ii) esa ifjdfyr xsan dk {kSfrt ijkl vfèkdre gksxk\

(e) ;fn u >vo, u = v

o, u < v

o θ gS] rks vfèkdre ijkl osQ laxr θ dk eku fdl

izdkj ifjo£rr gksrk gSA

(f) u = 0 osQ fy, θ osQ eku (vFkkZr~ 45o) dh rqyuk (V) esa izkIr θ osQ lkFk

dhft,A

4.36 fdlh lery esa f}foeh; xfr dk vè;;u] fLFkfr] osx vkSj Roj.k dks

dkrhZ; funsZ'kkadksa esa lfn'kksa dh Hkk¡fr ˆ ˆx yA A= +A i j osQ :i esa O;Dr

djosQ fd;k tk ldrk gS] ;gk¡ ˆ ˆi j,oa Øe'k% x ,oa y fn'kk esa ,dkad

lfn'k gS rFkk Ax ,oa A

y lfn'k A osQ laxr ?kVd gSaA (fp=k 4.9)A xfr dk

vè;;u lfn'kksa dks o`Ùkkdkj èkzqoh funZs'kkadksa osQ :i esa ˆˆrA Aθ= +A r θθθθ dh

Hkk¡fr O;Dr djosQ Hkh fd;k tk ldrk gS] ;gk¡ ˆ ˆˆ cos sinr

θ θ= = +r

r i j rFkk

ˆ ˆˆ sin cos= − θ + θi jθθθθ mu fn'kkvksa esa ,dkad lfn'k gSa ftuesa ‘r’ ,oa ‘θ ’

osQ ekuksa esa o`f¼ gks jgh gSA

N

E

B

A

3m/s

fp=k 4.8

fp=k 4.9

P (x, y) = (r, )q

v

Y

y

r

j

i

Xx

q

q

18-04-2018

28


(a) ˆ ˆi j,oa dks ˆr vkjS θθθθ osQ inksa esa O;Dr dhft,A

(b) n'kkZb, fd ˆr vkjS θθθθ nksuksa ,dkad lfn'k gSa vkSj ,d nwljs osQ yacor~ gSaA

(c) n'kkZb, fd ( ) ˆˆd

dtω=r θθθθ tgk¡

d

dt

θω = rFkk ˆ( )

d

dtθθθθ = ˆω− r

(d) ,d d.k Likbjy (lfiZy oqaQMy) ˆaθ=r r , osQ vuqfn'k xfr djrk gS] ;gk¡

a = 1 gSA blosQ fy, ‘a’ dh foek,¡ Kkr dhft,A

(e) Åij (d) esa o£.kr Likbjy osQ vuqfn'k xfr djrs gq, d.k osQ osx ,oa Roj.kèkzqoh&lfn'k fu:i.k esa Kkr dhft,A

4.37 dksbZ O;fDr fdlh oxZ osQ ,d dksus A ls mlosQ foijhr osQ dksus C (fp=k 4.10) ijigq¡puk pkgrk gSA oxZ dh izR;sd Hkqtk dh yackbZ 100 m gSA bl oxZ osQ osaQnz ij ,dvU; jsr ls Hkjk 50m × 50m vkeki dk oxZ gSA ;g O;fDr bl oxZ osQ ckgj1 m/s dh pky ls py ldrk gSA osaQnzh; oxZ esa og osQoy v m/s (v < 1) dh pkyls py ldrk gSA v dk ,slk U;wure eku D;k gksxk ftlosQ fy, og ljy js[kh; iFkij jsr ls gksdj xqtjrs gq, jsr osQ ckgj oxZ esa gksdj tkus dh rqyuk esa] rhozrk ls igq¡plosQxk\

fp=k 4.10

RP

A

50m

100m

Q

C

18-04-2018


5.1 dksbZ xsan ,d leku LFkkukarjh; xfr dj jgh gSA bldk vFkZ gS fd&

(a) ;g fojke voLFkk esa gSA

(b) bldk iFk ljy js[kh; vFkok o`Ùkkdkj gks ldrk gS vkSj xsan ,d leku pky lspy jgh gSA

(c) xsan osQ lHkh Hkkxksa dk osx (ifjek.k ,oa fn'kk) leku gS rFkk ;g osx fu;r gSA

(d) xsan dk osaQnz vpj osx ls xfr djrk gS rFkk xsan vius osaQnz osQ ifjr% ,d leku?kw.kZu djrh gSA

5.2 dksbZ ehVj LosQy ,d leku osx ls xfreku gSA bldk vFkZ gS fd

(a) LosQy ij yxus okys cy dk ifjek.k 'kwU; gSA ijarq LosQy ij nzO;eku osaQnz osQifjr% dksbZ cy&vk?kw.kZ dk;Z dj ldrk gSA

(b) LosQy ij yxus okys cy dk ifjek.k 'kwU; gS vkSj LosQy osQ nzO;eku osaQnz osQifjr% dk;Z djus okyk cy vk?kw.kZ Hkh 'kwU; gSA

vè;k; 5

xfr osQ fu;e

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30


(c) bl ij yxus okyk oqQy cy 'kwU; gksuk vko';d ugha gS ijarq bl ij dk;Z djusokyk cy&vk?kw.kZ 'kwU; gSA

(d) LosQy ij dk;Z djus okys u rks cy vkSj u gh cy vk?kw.kZ dk 'kwU; gksukvko';d gSA

5.3 150 g nzO;eku dh fdlh fØosQV dh xsan dk izkjafHkd osx 1ˆ ˆ(3 4 )m s−= +u i j

vkSj fgV gksus osQ ckn vafre osx 1ˆ ˆ(3 4 )m s−= − +v i j gSA xsan dk laosx ifjorZu

kg m s1 gS µ

(a) 'kwU;

(b) – ˆ ˆ(0.45 0.6 )+i j

(c) – ˆ ˆ(0.9 1.2 )+i j

(d) – ˆ ˆ5( )+i j

5.4 iz'u (5.3) esa fgV gksus dh izfØ;k esa gLrkarfjr laosx dk ifjek.k gS µ

(a) 'kwU; (b) 0.75 kg ms–1 (c) 1.5 kg ms–1 (d) 14 kg ms–1

5.5 d.kksa osQ chp la?kêð esa laosx laj{k.k dk vocks/u fdl vk/kj ij fd;k tk ldrkgS \

(a) ÅtkZ laj{k.k(b) osQoy U;wVu dk izFke fu;e(c) osQoy U;wVu dk f}rh; fu;e(d) U;wVu osQ f}rh; ,oa r`rh; fu;e

5.6 gkWdh dk dksbZ f[kykM+h foi{kh ls cpus osQ fy, mÙkj fn'kk esa tkrs&tkrs iwoZorhZ pkyls gh vpkud if'pe dh vksj eqM+ tkrk gSA f[kykM+h ij yxus okyk cy gS %

(a) if'pe fn'kk esa ?k"kZ.k cy(b) nf{k.k fn'kk esa is'kh; cy(c) nf{k.k&if'pe fn'kk esa ?k"kZ.k cy(d) nf{k.k&if'pe fn'kk esa is'kh; cy

5.7 2 kg nzO;eku dk dksbZ fiaM lehdj.k 2 3( )x t pt qt rt= + + osQ vuqlkj xfr djrkgS] ;gk¡ 13m sp −= , 24 m sq −= vkSj 35m sr −= gSA

t = 2 s ij fiaM ij yxus okyk cy gS µ

(a) 136 N

(b) 134 N

(c) 158 N

(d) 68 N

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xfr osQ fu;e

31

5.8 5 kg nzO;eku osQ fdlh fiaM ij dksbZ cy ( )ˆ ˆ= N–3i + 4jF dk;Z dj jgk gSA ;fnt = 0 ij ¯iM dk izkjafHkd osx ( ) –1ˆ ˆ m s6i -12j=v gks] rks og le; tc bldkosx osQoy y-v{k osQ vuqfn'k gksxk] gS µ

(a) dHkh ugha(b) 10 s

(c) 2 s

(d) 15 s

5.9 fojke voLFkk ls xfr vkjaHk djus okyh m nzO;eku dh fdlh dkj dk 2s esa iwoZ fn'kk

esa osx ( )ˆv v > 0= iv gks tkrk gSA ;g ekurs gq, fd dkj ,d leku Roj.k ls xfr

djrh gS] dkj ij yxus okyk cy dk ifjek.k µ

(a)2

mv iwoZ fn'kk osQ vuqfn'k gS vkSj dkj osQ batu }kjk yxk;k tkrk gSA

(b)2

mv iwoZ fn'kk osQ vuqfn'k gS vkSj lM+d rFkk Vk;jksa osQ chp ?k"kZ.k osQ dkj.k gSA

(c)2

mvls vf/d iwoZ osQ vuqfn'k gS rFkk ;g batu }kjk lM+d osQ ?k"kZ.k ls ikj ikus

osQ fy, yxrk gSA

(d)2

mvgS tks batu osQ dkj.k yxrk gSA


5.10 m nzO;eku osQ fdlh d.k dh xfr bl izdkj O;Dr dh xbZ gS µx = 0 tc t < 0 s,

x(t) = A sin4π t tc 0 < t <(1/4) s (A > 0), rFkkx = 0 tc t >(1/4) s

bl xfr osQ lanHkZ esa fuEufyf[kr esa dkSu ls dFku lR; gSa\

(a) t = (1/8) s ij d.k ij yxus okyk cy –16π2 A m gSA(b) t = 0 s ,oa t = (1/4) s ij d.k ij yxus okys vkosx dk ifjek.k

4π A m gSA(c) d.k ij dksbZ cy ugha yxrkA(d) d.k ij dksbZ vpj cy ugha yxrkA(e) d.k ij dksbZ vkosx ugha yxrkA

5.11 fp=k 5.1 esa] iQ'kZ vkSj fiaM B osQ chp ?k"kZ.k xq.kkad 0.1 gSA fiaM B ,oa fiaMA osQ chp ?k"kZ.k xq.kkad 0.2 gSA dksbZ cy F fiaM B ij fp=k esa fn[kk, vuqlkj yxk;kx;k gSA A dk nzO;eku m/2 rFkk B dk nzO;eku m gSA fuEufyf[kr esa dkSu ls dFkulgh gSa\

fp=k 5.1

18-04-2018

32


(a) ;fn F = 0.25 mg, rks fiaM ,d lkFk xfr djsaxsA(b) ;fn F = 0.5 mg, rks ¯iM A ¯iM B osQ lkis{k fiQlysxkA(c) ;fn F = 0.5 mg, rks ¯iM ,d lkFk xfr djsaxsA(d) ;fn F = 0.1 mg, rks ¯iM fojke esa jgsaxsA(e) F dk vf/dre eku ftlosQ fy, ¯iM ,d lkFk xfr djsaxs] 0.45 mg gSA

5.12 nzO;eku m1 fdlh vkur lery ij j[kk gS tks {kSfrt ls θdks.k ij >qdk gSA nzO;eku

m1 dks fp=k 5.2 esa n'kkZ, vuqlkj nzO;eku m

2 ls /kxs }kjk] mls ?k"kZ.kghu f?kjuh ls

xqtkjrs gq,] tksM+k x;k gSA m1 ,oa vkur lery osQ chp ?k"kZ.k xq.kkad µ gSA fuEufyf[kr

esa ls dkSu ls dFku lR; gSa µ

fp=k 5.3

m1m2

B

�

fp=k 5.2

(a) ;fn 2 1 sinm m θ> , rks iM ry ij Åij dh vksj xfr djsxkA

(b) ;fn ( )2 1 sin cosm m θ µ θ> + , rks iM ry ij Åij dh vksj

xfr djsxkA

(c) ;fn ( )2 1 sin cosm m θ µ θ< + , rks iM ry ij Åij dh vksj

xfr djsxkA

(d) ;fn ( )2 1 sin cosm m θ µ θ< − , rks ¯iM ry ij

uhps dh vksj xfr djsxkA

5.13 fp=k 5.3 esa] m nzO;eku dk dksbZ iM A {kSfrt ls 1θ dks.k ij >qosQ lery ij fiQlyldrk gSA ¯iM A vkSj lery osQ chp ?k"kZ.k xq.kkad 1µ gSA A dks gYdh Mksjh ls ck¡èkdj Mksjh dks ?k"kZ.kghu f?kjuh ls xqtkjk x;k gS vkSj m nzO;eku osQ gh fdlh vU; iMB ls tksM+ fn;k x;k gSA B {kSfrt ls 2θ dks.k ij >qosQ ?k"kZ.kghu lery ij fiQlyldrk gSA fuEufyf[kr esa dkSu ls dFku lR; gSaµ

(a) dHkh Hkh A ry ij Åij dh vksj ugha pysxkA(b) A ry ij Åij dh vksj rHkh xfr djuk vkjaHk djsxk tc

2 1

1

sin sin

cos

θ θµ

θ

−=

(c) A dks ry ij Åij dh vksj xfr djus osQ fy,] 2θ dks 1θ ls

vf/d gksuk pkfg,A(d) B lnSo vpj osx ls uhps dh vksj fiQlysxkA

5.14 5 m s–1 pky ls 50 g nzO;eku dh nks fofy;MZ xsan foijhr fn'kkvksa esa xeu djrsgq, ,d nwljs ls la?kêð djrh gSa vkSj la?kêðð osQ i'pkr~ mlh pky ls okil ykSV tkrhgSaA ;fn la?kêð dky 10–3 s gks] rks fuEufyf[kr esa dkSu ls dFku lgh gSa\

(a) izR;sd xsan dks fn;k x;k vkosx 0.25 kg m s–1 gS vkSj izR;sd xsan ij dk;Zjr cy250 N gSA

18-04-2018

xfr osQ fu;e

33

(b) izR;sd xsan dks fn;k x;k vkosx 0.25 kg m s–1 gS vkSj izR;sd xsan ij dk;Zjr cy25 × 10–5 N gSA

(c) izR;sd xsan dks fn;k x;k vkosx 0.5 Ns gSA

(d) izR;sd xsan ij vkosx vkSj cy ifjek.k esa cjkcj rFkk fn'kk esa foijhr gSaA

5.15 10 kg nzO;eku osQ fdlh ¯iM ij 6N ,oa 8N osQ nks ijLij yacor~ cy ,d lkFkyxs gSaA ¯iM dk ifj.kkeh Roj.k gS µ

(a) 1 m s–2, tks 6N cy ls 1 4tan

3

−

dks.k cukrk gSA

(b) 0.2 m s–2, tks 6N cy ls 1 4tan

3

−

dks.k cukrk gSA

(c) 1 m s–2, tks 8N cy ls 1 3tan

4

−

dks.k cukrk gSA

(d) 0.2 m s–2, tks 8N cy ls 1 3tan

4

−

dks.k cukrk gSA


5.16 dksbZ lh/h lM+d ij 5 m s–1 dh pky ls ckbfldy ij xfreku dksbZ yM+dh HkwryosQ lkis{k 15 m s–1 dh pky ls] 0.5 kg nzO;eku dk ,d iRFkj] viuh xfr dh fn'kkesa isaQdrh gSA ckbfldy ,oa yM+dh dk oqQy nzO;eku 50 kg gSA iRFkj isaQdus ij D;kckbfldy dh pky esa dksbZ varj vkrk gS\ ;fn gk¡] rks pky esa varj Kkr dhft,A

5.17 50 kg nzO;eku dk dksbZ O;fDr fyÝV esa Hkkj ekius dh e'khu ij [kM+k gSA ;fnfyÝV uhps dh vksj 9 m s–2 osQ v/kseq[kh Roj.k ls tkrh gS rks Hkkj ekius dhe'khu osQ LosQy dk ikB~;kad D;k gksxk\ (g = 10 m s–2)

5.18 2 kg nzO;eku osQ fdlh iM dk fLFkfr≤ xzkiQ fp=k 5.4 esa n'kkZ;k x;k gSAt = 0 s vkSj t = 4 s ij ¯iM dk vkosx fdruk gS\

5.19 dksbZ dkj pkyd lkeus lM+d ij fdlh cPps dks ns[kdj vpkud czsd yxkrkgSA ;fn mlus lhV csYV ugha ck¡/h gS] rks og vkxs dh vksj >Vdk [kkrk gS vkSjmldk flj fLV;¯jx Oghy ls tk Vdjkrk gSA ,slk D;ksa gS\

5.20 2 kg nzO;eku osQ fdlh iM osQ osx dks le; osQ iQyu osQ :i esa 2ˆ ˆ( ) 2t t t= +v i j

ls fu#fir djrs gSaA t = 2s ij] bl ij yxus okys laosx ,oa cy dk ifjdyu dhft,A

5.21 [kqjnjs {kSfrt lery i`"B ij j[kk dksbZ xqVdk fdlh {kSfrt cy F }kjk [khapk tkrkgSA ekuk fd f [kqjnjs i`"B }kjk xqVosQ ij yxk;k x;k cy gSA f vkSj F esa xzkiQ [khafp,A

fp=k 5.4

18-04-2018

34


5.22 ifjogu osQ fy, iS dx ls iwoZ ikslhZfyu dh oLrqvksa dks dkx”k ;k Hkwls esa D;ksa yisVktkrk gS\

5.23 ckx dh uje feêðh ij fxjus ls yxus okyh pksV dh rqyuk esa lhesaV osQ dBksj iQ'kZij fxjus ls yxh pksV ls fdlh cPph dks vf/d nnZ D;ksa gksrk gS\

5.24 dksbZ efgyk 500 g nzO;eku osQ fdlh ¯iM dks 25 m s1 dh pky ls isaQdrh gSA

(a) ¯iM dks iznku fd;k x;k vkosx fdruk gS\

(b) ;fn ¯iM fdlh nhokj ls Vdjk, vkSj ewy pky dh vk/h pky ls okil ykSVs rksblosQ laosx esa fdruk ifjorZu gksrk gS\

5.25 igkM+ ij lM+osaQ lh/s [kM+h p<+kbZ dh u cukdj Åij dh vksj p<+rh gqbZ l£iykdkjcukbZ tkrh gSa D;ksa\


5.26 2kg dk dksbZ nzO;eku fdlh /kxs AB }kjk yVdk;k x;k gS (fp=k 5.5)A blh izdkjdk ,d /kxk CD 2kg nzO;eku osQ nwljh vksj tksM+k x;k gSA /kxs CD dks uhps dhvksj /hjs&/hjs cy c<+krs gq, [khapk tkrk gSA dkSu&lk /kxk VwVsxk\ D;ksa\

5.27 Åij fn, x, iz'u (5-26) esa ;fn /kxs CD dks >Vdk ekjdj [khapk tk,] rks D;kgksxk\

5.28 5 kg vkSj 3 kg osQ nks nzO;eku] nzO;eku jfgr vforkU; /kxs osQ }kjk fp=k 5.6 esan'kkZ, vuqlkj yVdk, x, gSaA laiw.kZ fudk; 2ms–2 osQ Roj.k ls Åij dh vksj xfrekugSA T

1 ,oa T

2 ifjdfyr dhft,A (g = 9.8 m s–2 dk mi;ksx dhft, )

5.29 {kSfrt ls 30° dks.k ij >qdk dj j[ks x, fdlh ?k"kZ.kghu lery ij 100N Hkkj dkxqVdk A j[kk gS (fp=k 5.7)A A ls ,d yphyk /kxk tksM+ dj bls ,d ?k"kZ.kfoghuf?kjuh osQ Åij ls xqtkjk x;k gS vkSj blosQ nwljs fljs ij W Hkkj dk dksbZ nwljk xqVdkfp=k 5.6

fp=k 5.5

f

N A BF

w

mg

mg sin 30°

mg cos 30°

30°

fp=k 5.7

18-04-2018

xfr osQ fu;e

35

B tksM+ fn;k x;k gSA Hkkj W dk og eku Kkr dhft, ftlosQ fy, ;g fudk; larqyuesa jgrk gSA

5.30 M nzO;eku osQ fdlh xqVosQ dks vaxqyh ls fdlh [kqjnjh ÅèokZ/j nhokj ij nckdjfxjus ls jksdk x;k gSA ;fn nhokj vkSj xqVosQ osQ chp ?k"kZ.k xq.kkad µ rFkk xq#Ro osQdkj.k Roj.k g gks] rks xqVosQ dks fxjus ls jksdus osQ fy, vaxqyh }kjk bl ij yxk;k tkusokyk U;wure cy ifjdfyr dhft,A

5.31 100 kg dh dksbZ rksi 500m Å¡ph pêðku ls 1kg dk dksbZ xksyk {kSfrtr% nkxrh gStks pêðku osQ vk/kj ls 400m nwjh ij tkdj fxjrk gSA rksi dk izfrf{kIr osx Kkrdhft, (xq#Roh Roj.k = 10 m s–2)

5.32 fp=k 5.8 esa nks foekvksa esa xfr'khy d.k osQ (x, t), (y, t ) xzkiQ n'kkZ, x, gSaA ;fn d.kdk nzO;eku 500g gks] rks d.k ij yxus okyk cy (ifjek.k ,oa fn'kk) Kkr dhft,A

5.33 2 m s–2 osQ Roj.k ls Åij dh vksj tkrs gq, fdlh ,fyosVj ls dksbZ O;fDr ,d flDdk20 m s1 dh pky ls ÅèokZ/j Åij dh vksj mNkyrk gSA fdrus le; osQ i'pkr~flDdk okil mlosQ gkFk esa vk fxjsxk\ ( g = 10 m s–2)


5.34 fdlh iM osQ cnq P ij F1, F

2 ,oa F

3 rhu cy yxs gSaA bu cyksa osQ izHkko esa iM

,d leku pky ls xfr djrk gS %

(a) n'kkZb, fd cy leryh; gaSA

(b) n'kkZb, fd iM osQ fdlh cnq osQ ifjr% bu rhu cyksa osQ dkj.k oqQy cy&vk?kw.kZ'kwU; gksxkA

5.35 tc dksbZ ¯iM fdlh ,sls fpdus vkur lery ij tks {kSfrt ls 45° dk dks.k cukrkgS] fojkekoLFkk ls fiQlyrk gS rks] bldks uhps igq¡pus esa T le; yxrk gSA ogh iM

fp=k 5.8(a) (b)

18-04-2018

36


tc mrus gh dks.k ij >qosQ gq, [kqjnjs vkur lery ij fojkekoLFkk ls mruh gh Å¡pkbZls fiQlyrk gS rks ;g ik;k tkrk gS fd bldks uhps igq¡pus esa pT le; yxrk gS] ;gk¡p dksbZ la[;k gS ftldk eku 1 ls vf/d gSA iM vkSj [kqjnjs ry osQ chp ?k"kZ.k xq.kkadifjdfyr dhft,A

5.36 fp=k 5.9 esa ,dkad nzO;eku osQ fdlh ¯iM osQ ( , ), ( , )x yv t v tvkSj vkjs[k n'kkZ, x,

gSaA le; osQ iQyu osQ :i esa cy Kkr dhft,A

fp=k 5.10

AF

E

D

C

R

R

90°

B

2RO

5.37 dksbZ js lx dkj fdlh /kou iFk ABCDEFA (cSa dx jfgr) ij py jgh gS(fp=k 5.10)A ABC dksbZ o`Ùkkdkj pki gS ftldh f=kT;k 2 R gSA CD ,oa FA ljyjs[kh; iFk gSa ftuesa izR;sd dh yackbZ R gS] rFkk DEF o`Ùkkdkj pki gS ftldh f=kT;kR = 100 m gSA lM+d dk ?k"kZ.k xq.kkad µ = 0.1 gSA dkj dh vf/dre pky50 m s–1 gSA ,d iwjk pDdj yxkus esa yxus okyk U;wure le; ifjdfyr dhft,A

5.38 m nzO;eku osQ fdlh d.k osQ foLFkkiu lfn'k dks bl izdkj O;Dr fd;k x;k

gS µ = ω + ωˆ ˆ( ) cos sin .t A t B tr i j

(a) n'kkZb, fd d.k dk xeu iFk dksbZ nh?kZ o`Ùk gSA

(b) n'kkZb, fd ω= − 2mF r

fp=k 5.9

vx

(m s )–1

2

2s1s t

1

vy

(m s )–1

2

2s 3s1s t

1

O

(a) (b)

18-04-2018

xfr osQ fu;e

37

5.39 dksbZ xasnckt fØosQV dh xsan dks nks fHkUu <axksa ls isaQdrk gS µ

(a) bldks osQoy {kSfrt osx nsdj vkSj

(b) {kSfrt osx osQ lkFk&lkFk uhps dh vksj vYi osx nsdjA

xsan tc mldk gkFk NksM+rh gS rks nksuksa fLFkfr;ksa esa mldh pky vs leku gksrh gSA

nksuksa ckj xsan Hkwry ls leku Å¡pkbZ H ls isaQdh tkrh gSA Hkwry ls Vdjkrs le; fdlxsan dh pky vf/d gksxh\ ok;q izfrjks/ dh mis{kk dhft,A

5.40 fdlh ¯cnq P ij fp=k 5.11 esa n'kkZ, vuqlkj Mksfj;ksa dh lgk;rk ls pkj cy yxk,x, gSaA ¯cnq P fojkekoLFkk esa gSA F

1 ,oa F

2 cyksa osQ eku Kkr dhft,A

5.41 dksbZ vk;rkdkj fdlh [kqjnjs vkur lery ij j[kk gSA vkur lery vkSj ckWDl osQchp ?k"kZ.k xq.kkad µ gSA eku yhft, ckWDl dk nzO;eku m gS µ

(a) ry osQ {kSfrt ls fdrus dks.k θ ij >qdk gksus ij ckWDl ry ij uhps dh vksjfiQlyuk vkjaHk dj nsxk\

(b) ;fn ry dk vkufr dks.k α > θ rks ckWDl ij uhps dh vksj fdruk cy yxsxk\

(c) ckWDl dks fLFkj cuk, j[kus osQ fy, ;k ,d leku pky ls Åij dh vksj xfr izkjaHkdjus osQ fy, bl ij Åij dh vksj ry osQ vuqfn'k fdruk cy yxkus dhvko';drk gksxh\

(d) ckWDl vkur lery ij Åij dh vksj a Roj.k ls xfr nsus osQ fy, bl ij Åijdh vksj ry osQ vuqfn'k fdruk cy yxkus dh vko';drk gksxh\

5.42 2000kg nzO;eku dk dksbZ gsfydkWIVj 15 m s–2 osQ ÅèokZ/j Roj.k ls Åij mBrkgSA dehZny ,oa ;kf=k;ksa dk oqqQy nzO;eku 500 kg gSA fuEufyf[kr dk ifjek.k ,oafn'kk Kkr dhft, % (g = 10 m s–2)

(a) dehZny ,oa ;kf=k;ksa }kjk gsfydkWIVj osQ iQ'kZ ij yxus okyk cyA

(b) gsfydkWIVj osQ jksVj }kjk pkjksa vksj dh ok;q ij fØ;kA

(c) pkjksa vksj dh ok;q osQ dkj.k gsfydkWIVj ij yxus okyk cyA

fp=k 5.11

2N

45° 45°

1N

45°

F190°

F2

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38


6.1 ,d bysDVªkWu ,oa ,d izksVkWu ikjLifjd cyksa osQ izHkko ls xfreku gSaA xfr osQ nkSjkubl ra=k dh xfrt ÅtkZ osQ ifjorZu dh x.kuk djrs le; ge ,d osQ }kjk nwljs ijyxus okys pqacdh; cyksa dh mis{kk dj nsrs gSaA ,slk blfy, gS D;ksafd]

(a) nksuksa pqacdh; cy ifjek.k esa cjkcj vkSj fn'kkvksa esa foijhr gksrs gSa blfy, os dksbZusV (ifj.kkeh) izHkko mRiUu ugha djrsA

(b) pqacdh; cy bu nksuksa esa ls fdlh Hkh d.k ij dksbZ dk;Z ugha djrsA

(c) pqacdh; cy izR;sd d.k ij cjkcj (ijarq 'kwU;srj) vkSj foijhr dk;Z djrs gSaA

(d) pqacdh; cy vfuok;Zr% ux.; gksrs gSaA

6.2 ,d izksVkWu fojkekoLFkk esa j[kk x;k gSA blosQ {ks=k esa ,d vU; èku vkos'k;qDr d.k]blls d nwjh ij fojke voLFkk esa gh foeqDr fd;k tkrk gSA nks iz;ksxksa ij fopkjdhft,µ igyk og ftlesa nwljk vkosf'kr d.k Hkh izksVkWu gh gS vkSj nwljk ogftlesa nwljk èku vkosf'kr d.k ikWftVªkWu gSA leku le; t esa nksuksa xfreku d.kksa ijfd;k x;k dk;Zµ

vè;k; 6

dk;Z] ÅtkZ vkSj 'kfDr

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(a) leku gS D;ksafd bu nks iz;ksxksa esa ,d gh cy fu;e ykxw gksrk gSA

(b) ikWftVªkWu osQ izdj.k esa de gksrk gS D;ksafd ikWftVªkWu vfèkd rhoz xfr ls izfrdf"kZrgksrk gS vkSj ml ij cy de gks tkrk gSA

(c) ikWftVªkWu osQ izdj.k esa vfèkd gksrk gS D;ksafd ikWftVªkWu vfèkd nwjh rd izfrdf"kZrgksrk gSA

(d) mruk gh gksrk gS ftruk vosf'kr d.k }kjk fLFkj izksVªkWu ij fd;k x;k dk;ZA

6.3 tehu ij mdM+w cSBk gqvk ,d O;fDr mBdj lhèkk [kM+k gksrk gSA bl izfØ;k esa O;fDrij yxus okyk i`Foh dk izfrfØ;k cyµ

(a) vifjofrZr jgrk gS vkSj ifjek.k esa mg osQ cjkcj gksrk gSA

(b) vifjofrZr jgrk gS vkSj ifjek.k esa mg ls vfèkd gksrk gSA

(c) izkjaHk ifjorhZ ijarq ifjek.k esa lnSo mg ls vfèkd

(d) izkjaHk esa mg ls vfèkd gksrk gS ijarq ckn esa mg osQ cjkcj gks tkrk gSA

6.4 ,d ckbfldy lokj czsd yxkus osQ ckn 10 m dh nwjh fiQlyrs gq, tk ldrk gSAbl izfØ;k esa lM+d }kjk ckbfldy ij yxk;k x;k cy 200N gS vkSj xfr osQ Bhdfoijhr fn'kk esa yxrk gSA lkbfdy }kjk lM+d ij fd;k x;k dk;Z gSµ

(a) + 2000J

(b) – 200J

(c) 'kwU;(d) – 20,000J

6.5 ,d fiaM fuokZr esa osQoy xq#Ro osQ vèkhu Lora=krkiwoZd fxj jgk gSA blosQ fxjus osQnkSjku fuEufyf[kr esa ls dkSu&lh jkf'k vpj jgrh gS\

(a) xfrt ÅtkZ

(b) fLFkSfrt ÅtkZ

(c) oqQy ;kaf=kd ÅtkZ

(d) oqQy js[kh; laosx

6.6 nks fiaMksa osQ chp gksus okys vizR;kLFk la?kV~V osQ nkSjku fuEufyf[kr esa ls dkSu&lh jkf'klnSo lajf{kr jgrh gSµ

(a) oqQy xfrt ÅtkZ

(b) oqQy ;kaf=kd ÅtkZ

(c) oqQy js[kh; laosx

(d) izR;sd fiaM dh pky

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40

6.7 nks ?k"kZ.k foghu ur iFkksa esa ls ,d nwljs dh vis{kk] {kSfrt ls de dks.k ij >qdk gSA;s nksuksa iFk fcanq A ij feyrs gSa] tgk¡ ls nks iRFkj fojkekoLFkk ls NksM+s tkrs gSa] ftuesals izR;sd iRFkj vyx iFk ij fiQlyrk gS tSlk fp=k 6.1. esa n'kkZ;k x;k gSA

fuEufyf[kr esa dkSu&lk dFku lR; gS\

(a) nksuksa iRFkj ,d gh {k.k ij ryksa osQ vkèkkj ij igq¡prs gSa ijarq ogk¡ mudh pkyleku ugha gksrhA

(b) nksuksa iRFkj] ryksa osQ vkèkkj ij ,d gh pky ls igq¡prs gSa vkSj iRFkj I, iRFkj IIls igys igq¡prk gSA

(c) nksuksa iRFkj ryksa osQ vkèkkj ij ,d gh pky ls igq¡prs gSa vkSj iRFkj II iRFkj Ils igys igq¡prk gSA

(d) nksuksa iRFkj ryksa osQ vkèkkj ij vyx&vyx le; ij rFkk vyx&vyx osx lsigq¡prs gSaA

6.8 ljy vkorZ xfr (SHM) djrs fdlh d.k dk fLFkSfrt ÅtkZ iQyu gSµ 21

( )2

V x kx=

tgk¡ k nksfy=k dk cy fu;rkad gSA k = 0.5N/m osQ fy, V(x) vkSj x dk xzkiQfp=k 6.2 esa n'kkZ;k x;k gSA E ÅtkZ dk dksbZ d.k mx x= ± ij igq¡p dj okil ykSVrk

gSA ;fn x = +xm ij V ,oa K Øe'k% d.k dh fLFkSfrt ÅtkZ (P.E.) ,oa xfrt ÅtkZ

(K.E.) fu#fir djrs gksa rks fuEufyf[kr esa dkSu&lk dFku lgh gS?(a) V = O, K = E(b) V = E, K = O(c) V < E, K = O

(d) V = O, K < E.

6.9 nks loZle ckWy fc;fjax ,d ?k"kZ.k foghu est ij ,d nwljs osQ laioZQ esa fojkekoLFkkesa j[ks gSa] vkSj leku nzO;eku dk ,d rhljk ckWy fc;fjax V pky ls pyrk gqvkvkdj buls lEeq[k la?kV~V djrk gS tSlk fp=k 6.3 esa n'kkZ;k x;k gSA

;fn la?kV~V izR;kLFk gks rks fp=k 6.4 esa n'kkZ;h xbZ dkSu&lh fLFkfr la?kV~V osQ i'pkrlaHko gS\

fp=k 6.2

fp=k 6.1

fp=k 6.3

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41

6.10 0.5 kg nzO;eku dk ,d fiaM ,d ljy js[kk esa v = a x3/2 osx ls xfreku gS tgk¡a = 5 m–1/2s–1gSA blosQ x = 0 ls x = 2 m rd foLFkkiu esa oqQy cy }kjk fd;kx;k dk;Z gSµ(a) 1.5 J

(b) 50 J(c) 10 J(d) 100 J

6.11 ,d fiaM fdlh fu;r 'kfDr iznk;d ÅtkZ lzksr osQ izHkko esa ,d gh fn'kk esa py jgkgSA fp=k 6.5 esa dkSu&lk vkjs[k bldh xfr dk lgh foLFkkiu≤ xzkiQ gS\

fp=k 6.5

t

d

t

d

d

t

d

t

(a) (b)

(c) (d)

(a) (b)

(c) (d)

fp=k 6.4

18-04-2018


42

6.12 fp=k 6.6 esa n'kkZ;s x;s vkjs[kksa esa ls dkSu&lk vkjs[k lw;Z osQ pkjksa vksj nh?kZ o`Ùkkdkjd{kk esa ?kwerh gqbZ i`Foh dh ,d ifjØek esa xfrt ÅtkZ esa ifjorZu dk fudVrefu:i.k djrk gS\

6.13 fp=k 6.7 esa n'kkZ, x, vkjs[kksa esa ls dkSu&lk vkjs[k ok;q esa nksyu djrs gq, fdlhyksyd dh oqQy ;kaf=kd ÅtkZ esa le; osQ lkFk gksus okys ifjorZu dk lgh fu:i.kdjrk gS\

fp=k 6.6

K.E

t

K.E

t

K.E

t

K.E

(d)

(a)

t t

(b)

(c)

fp=k 6.7(d)

t

E

t

E

(b)

t

E

t

E

(a)

(c)

18-04-2018


43

6.14 5 kg nzO;eku dk ,d fiaM 1 m f=kT;k osQ o`Ùkkdkj iFk ij xfreku gSA ;fn ;g fiaMizfr feuV 300 pDdj yxkrk gks rks bldh xfrt ÅtkZ gksxhµ

(a) 250π2

(b) 100π2

(c) 5π2

(d) 0

6.15 ,d o"kkZ dh cw¡n tks i`Foh osQ Åij h Å¡pkbZ ls fxjuk izkjaHk djrh gSSA (3/4)h Å¡pkbZls fxjus osQ ckn vafre osx (Terminal Velocity) izkIr dj ysrh gSA fp=k 6.8 esan'kkZ, x, vkjs[kksa esa ls dkSu&lk vkjs[k bl cw¡n osQ Hkw&i`"B rd fxjus esa bldh xfrtrFkk fLFkSfrt ÅtkZ esa ifjorZuksa dks lgh izdkj n'kkZrk gS\

(a)t

KE

PE

h

t

h

h/4

KE

PE

(b)

fp=k 6.8

t

h KE

PE

t

h

KE

PE

(c) (d)

6.16 xksyk isaQdus dh izfr;ksfxrk esa ,d f[kykM+h 10 kg nzO;eku osQ ,d xksys dks1m s –1 osQ vkjafHkd osx ls i`Foh ls 1.5 ehVj dh Å¡pkbZ ls 45° ij isaQdrk gSA ;gekurs gq, fd ok;q izfrjksèk ux.; gS ,oa xq#Ro osQ dkj.k Roj.k 10 m s–2 gS] tc xksyki`Foh ij fxjrk gS rks bldh xfrt ÅtkZ gksrh gSµ(a) 2.5 J

(b) 5.0 J

(c) 52.5 J

(d) 155.0 J

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44

6.17 fp=k 6-9 esa dkSu&lk vkjs[k fdlh >hy esa Lora=krkiwoZd fxjrs gq, yksgs osQ xksys dhxfrt ÅtkZ osQ ifjorZu dk lgh fu:i.k djrk gS tcfd >hy dh xgjkbZ bruh gS fdxksyk vafre osx (Terminal Velocity) izkIr dj ldrk gSµ

6.18 126 km h–1 dh pky ls pyrh gqbZ 150 g nzO;eku dh ,d fØosQV xsan] cYysckt}kjk n`<+rkiwoZd idM+s x, cYys osQ chpksa&chp Vdjkrh gSA cYys ls Vdjkdj xsan lhèkhxsanckt dh vksj ykSV tkrh gSA ;g ekurs gq, fd xsan vkSj cYys osQ chp la?kV~V iw.kZr%izR;kLFk gS vkSj ;s nksuksa 0.001s osQ fy, ikjLifjd laioZQ esa jgrs gSa] og cy tkscYysckt dks viuk cYyk n<+rkiwoZd idM+s jgus osQ fy, yxkuk vko';d gS] og gSµ(a) 10.5 N

(b) 21 N

(c) 1.05 ×104 N

(d) 2.1 × 104 N


6.19 m nzO;eku dk ,d O;fDr L Å¡pkbZ dh lh<+h osQ vkèkkj ij [kM+k gksrk gSA fiQj og

lh<+h ij p<+dj blosQ 'kh"kZ ij [kM+k gks tkrk gSA(a) O;fDr ij yxs lHkh cyksa }kjk fd;k x;k dk;Z mldh fLFkSfrt ÅtkZ esa o`f¼ mgL

osQ cjkcj gksrk gSA(b) O;fDr ij yxs oqQy cyksa }kjk fd;k x;k dk;Z 'kwU; gksrk gSA

(d)

fp=k 6.9

K.E

xgjkbZ

K.E

xgjkbZ

(c)

(a) (b)

K.E

xgjkbZ

K.E

xgjkbZ

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45

(c) O;fDr ij yxs xq#Rokd"kZ.k cy }kjk fd;k x;k dk;Z mgL gSA(d) lh<+h dh fdlh ikS<+h }kjk yxk, x, izfrfØ;k cy }kjk dksbZ dk;Z ugha gksrk

D;ksafd cy rks fo|eku gksrk gS] ¯drq ftl ¯cnq ij cy yxrk gS] og ¯cnqLFkkukarfjr ugha gksrkA

6.20 m nzO;eku dh ,d xksyh canwd dh uyh ls {kSfrt ls 30° dk dks.k cukrs gq, v osxls ckgj vkrh gSA xksyh èkjkry ls h Å¡pkbZ ij j[ks ,d dksey y{; ls uhps dh vksjvkrs gq, Vdjkrh gS vkSj ftl xfrt ÅtkZ ls y{; ls Vdjkrh gS mldh vkèkh xfrtÅtkZ ls blosQ ckgj fudyrh gSA y{; ls ckgj fudyus ij xksyh osQ fo"k; esafuEufyf[kr esa dkSu&ls dFku lR; gSa\(a) xksyh dk osx blosQ izkjafHkd osx dk vkèkk gksxk\(b) xksyh dk osx blosQ izkjafHkd osx osQ vkèks ls vfèkd gksxk\(c) xksyh ml gh ifjofy;d iFk ij pyrh jgsxhA(d) xksyh ,d fHkUu ifjofy;d iFk ij pysxhA(e) y{; ls Vdjkus osQ ckn xksyh ÅèokZèkjr% uhps dh vksj fxjsxhA(f) y{; osQ d.kksa dh vkarfjd ÅtkZ esa o`f¼ gks tk,xhA

6.21 nks CykWd M1 vkSj M

2 ftuosQ nzO;eku cjkcj gSa ,d {kSfrt] ?k"kZ.kfoghu lrg ij pyus

osQ fy, Lora=k gSaA M2 osQ lkFk ,d nzO;ekufoghu fLizax tqM+k gS tSlk fp=k 6.10 esa

n'kkZ;k x;k gSA izkjaHk esa M2 fojke voLFkk esa gS vkSj M

1, M

2 dh vksj v

1 osx ls py

jgk gS vkSj blls lEeq[k la?kV~V djrk gSµ

(a) tc fLizax iw.kZr% laihfM+r gS rks M1dh laiw.kZ xfrt ÅtkZ

fLizax dh fLFkSfrt ÅtkZ PE osQ :i esa laxzfgr gks tkrhgSA

(b) tc fLizax iw.kZr% laihfM+r gksrk gS rks bl fudk; dk laosxlajf{kr ugha gksrk ;|fi vafre laosx izkjafHkd laosx osQcjkcj gksrk gSA

(c) ;fn fLizax nzO;eku jfgr gS rks M1 dh vafre voLFkk fojke dh voLFkk gSA

(d) ;fn og i`"B ftl ij CykWd py jgs gSa ?k"kZ.k;qDr gSa rks la?kV~V izR;kLFk ugha gksldrk gSA


6.22 ,d [kqjnjk vkur lery ,d xkM+h ij j[kk gS tks {kSfrt Hkwfe ij fu;r osx u lsxfreku gSA M nzO;eku dk ,d CykWd vkur lery ij j[kk gSA D;k CykWd vkSj vkurlery osQ chp yxus okys ?k"kZ.k cy }kjk dksbZ dk;Z gksrk gS\ D;k rc ÅtkZ dk dksbZ{k; gksrk gS\

fp=k 6.10

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46

6.23 ,fyosVj tgk¡ uhps dh vksj vkrk gS rks bls fo|qr 'kfDr dh vko';drk D;ksa gksrh gS\bl fLFkfr esa ;kf=k;ksa dh la[;k lhfer D;ksa gksuh pkfg,\

6.24 ,d fiaM dks i`Foh dh lrg ls h Å¡pkbZ rd Åij mBk;k tk jgk gS] rks

(a) yxk, x, cy rFkk

(b) xq#Rokd"kZ.k cy }kjk fd, x, dk;Z dk fpÉ D;k gS\

6.25 ,d lhèkh {kSfrt lM+d ij xfreku dkj }kjk xq#Ro osQ fo#¼ fd;s x;s dk;Z dh x.kukdhft,A dkj dk nzO;eku 400 kg rFkk pyh xbZ nwjh 2 m gSA

6.26 ,d fiaM ok;q esa i`Foh dh vksj fxjrk gSA D;k fxjus osQ nkSjku bldh oqQy ;kaf=kd ÅtkZlajf{kr jgrh gS\ vius mÙkj osQ leFkZu esa roZQ nhft,A

6.27 ,d fiaM ,d can ywi ij pyrk gSA D;k fiaM osQ pyus esa fd;k x;k dk;Z vfuok;Zr%'kwU; gksrk gS\ ;fn ugha rks og 'krs± crkb, ftuosQ varxZr can iFk ij pyus esa fd;kx;k dk;Z lnSo 'kwU; gksrk gSA

6.28 fcfy;MZ dh nks xsanksa osQ izR;kLFk la?kV~V esa fuEufyf[kr esa ls dkSu&lh jkf'k xsanksa osQla?kV~V osQ vYidky esa (vFkkZr tc os nksuksa laioZQ esa jgrs gSa) lajf{kr jgrh gS\

(a) xfrt ÅtkZ

(b) oqQy js[kh; laosx

izR;sd fLFkfr esa vius mÙkj osQ fy, roZQ nhft,A

6.29 ml ozsQu dh 'kfDr okV esa ifjdfyr dhft, tks 100 kg nzO;eku osQ ,d fiaM dks20s esa 10 m dh Å¡pkbZ rd Åij mBkrh gSA

6.30 ,d ckj èkM+dus esa ekuo ân; vkSlru 0.5 J dk;Z djrk gSA ;fn ân; 1 feuV esa72 ckj èkM+drk gS rks bldh 'kfDr dh x.kuk dhft,A

6.31 ,d ,slh fLFkfr dk mnkgj.k crkb, ftlesa yxk, x, cy osQ dkj.k xfrt ÅtkZ esaifjorZu ugha gksrkA

6.32 vleku nzO;eku osQ nks fiaM ,d gh fn'kk esa leku xfrt ÅtkZ ls xfreku gSaA nksuksafiaMksa ij cjkcj ifjek.k dk cy yxkdj mUgsa fojkekoLFkk esa yk;k x;k gSA fojkekoLFkkesa vkus rd muosQ }kjk pyh xbZ nwfj;ksa dh rqyuk dhft,A

6.33 L yackbZ dh gYdh Mksjh ls yVdk gqvk m nzO;eku dk ,d yksyd fp=k 6.11 esan'kkZ, vuqlkj ,d ÅèokZèkj o`Ùk esa ?kqek;k tkrk gSA d.k dk xeu iFk D;k gksxk tcMksjh dks ml le; dkV fn;k tkrk gS tc ;g]

(a) ¯cnq B ij gSA

(b) ¯cnq C ij gSA

(c) ¯cnq X ij gSAfp=k 6.11

x

B

A

L

C

m

18-04-2018


47


6.34 fLFkSfrt ÅtkZ V ( x ) vkSj nwjh x osQ chp xzkiQ fp=k 6.12 esa n'kkZ;k x;k gSA ;gk¡ E0

ÅtkZ dk ,d d.k xfr dj jgk gSA ,d iw.kZ pØ AFA osQ laxr osx ,oa nwjh rFkk xfrtÅtkZ ,oa nwjh osQ chp xzkiQ [khafp,A

6.35 m nzO;eku dh 2v0 osx ls pyrh gqbZ ,d xsan fojkekoLFkk esa j[kh gqbZ ,d vU; loZle

xsan ls vizR;kLFk (e > 0) :i ls Vdjkrh gSA n'kkZb, fd](a) lEeq[k la?kV~V gksus ij nksuksa xsansa vkxs pysaxhA(b) lkekU; la?kV~V esa] la?kV~V osQ i'pkr~ izdhf.kZr xsanksa osQ osxksa osQ chp dks.k 90°

ls de gksxkA

6.36 ekuk fd ,d d.k ftldh oqQy ÅtkZ E gS ,d foeh; xfr dj jgk gSA pkj A, B,

C ,oa D {ks=kksa esa fLFkSfrt ÅtkZ V, xfrt ÅtkZ (K) rFkk oqQy ÅtkZ E osQ chp lacaèk uhpsfn, x, gSaµ

{ks=k A : V > E

{ks=k B : V < E

{ks=k C : K > E

{ks=k D : V > K

izR;sd {ks=k osQ fy, dkj.k lfgr le>kb, fd d.k dk fn, x, {ks=k esa ik;k tkuklaHko gS ;k ughaA

6.37 fdlh yksyd dk xksyd A {kSfrt fLFkfr ls foeqDr fd, tkus ij yVdu cnq osQÅèokZèkjr% uhps est ij fojkekoLFkk esa fLFkr leku nzO;eku osQ ,d vU; xksydB ls Vdjkrk gS tSlk fp=k 6.13 esa n'kkZ;k x;k gSAyksyd dh yackbZ 1 m gS] rks ifjdyu dhft,µ(a) ml Å¡pkbZ dh ftl rd la?kV~V osQ i'pkr~ xksyd A tk,xkA(b) ml pky dh ftlls xksyd B pyuk izkjaHk djsxkA

xksydksa osQ lkbt dh mis{kk dj ldrs gSa vkSj la?kV~V dks iw.kZr% izR;kLFk ekuk tkldrk gSA

6.38 1.00 g nzO;eku dh ,d o"kkZ dh cw¡n 1 km Å¡pkbZ ls fxjrh gS vkSj Hkwry ij50 m s–1 dh pky ls Vdjkrh gSA x.kuk dhft,&(a) cw¡n dh fLFkSfrt ÅtkZ esa gqbZ {kfr dhA(b) cw¡n dh xfrt ÅtkZ esa o`f¼ dhA(c) D;k fLFkSfrt ÅtkZ dh {kfr xfrt ÅtkZ dh o`f¼ osQ cjkcj gS\ ;fn ugha rks D;ksa\

( -2= 10 m sg ys ldrs gSa)

6.39 leku yackbZ vkSj loZle xksydksa ls ;qDr nks yksyd ,d mHk;fu"B vkèkkj ls bl izdkjyVdk, x, gS a fd fojkekoLFkk es a nk suk s a xk syd laioZQ es a jgrs gS aA(fp=k 6-14)A ,d yksyd dks 10° ij foLFkkfir djosQ NksM+ fn;k tkrk gS vkSj ;gnwljs xksyd ls izR;kLFk lEeq[k la?kV~V djrk gSA

fp=k 6.12

V

A

Eo

xDC

B

F

x

fp=k 6.13

fp=k 6.14

18-04-2018


48

(a) nksuksa xksydksa dh xfr dk o.kZu dhft,A(b) 0 2≤ ≤t T osQ fy,] le; osQ lkFk izR;sd yksyd dh ÅtkZ dk ifjorZu n'kkZus

osQ fy, xzkiQ cukb,A tgk¡ T izR;sd yksyd dk nksyu dky gSA6.40 ekuk fd o"kkZ dh cw¡nksa dk vkSlr nzO;eku 3.0 × 10-5kg rFkk mudk vkSlr vafre

9 m s-1 gSA fdlh LFkku ij tgk¡ o"kZ esa 100 cm o"kkZ iM+rh gS izfr oxZ ehVj {ks=kesa o"kkZ }kjk gLrkarfjr ÅtkZ dk ifjdyu dhft,A

6.41 ,d batu vkSj ,d oSxu dks 1.5m yackbZ dk iz?kkr vo'kks"kd yxkdj tksM+k x;k gSA50,000 kg oqQy nzO;eku dk ;g fudk; ml le; 36 km h-1 osQ osx ls py jgkFkk tc bls jksdus osQ fy, bl ij czsd yxk, x,A fojke esa ykus osQ izØe esa iz?kkrvo'kks"kd dk fLizax 1.0 m laihfM+r gksrk gSA ;fn oSxu dh 90% ÅtkZ ?k"kZ.k osQ dkj.k{kf;r gks tk, rks fLizax fu;rkad dk ifjdyu dhft,A

6.42 600 N Hkkj dk ,d izkS<+ O;fDr tkWfxax djrs le; 1 m yackbZ dk izR;sd dne j[kusesa vius 'kjhj osQ xq#Ro osaQnz dks 0.25 m Åij mBkrk gSA ;fn og 6 km dh nwjhrd tkWfxax djrk gS rks ;g ekurs gq, fd gok vkSj i`Foh osQ ?k"kZ.k osQ dkj.k dksbZ ÅtkZ{kfr ugha gksrhA ml O;fDr osQ }kjk tkWfxax esa O;; dh sxbZ ÅtkZ dk ifjdyu dhft,A;fn ;g eku ysa fd O;fDr osQ 'kjhj esa Hkkstu osQ :i esa xzg.k dh xbZ ÅtkZ osQ 10%

dks ifjofrZr djus dh {kerk gS rks tkWfxax esa iz;qDr ÅtkZ dh iwfrZ osQ fy, vko';drqY; [kk| dh x.kuk dhft,A

6.43 ,d yhVj isVªksy osQ iw.kZ ngu ij 3×107 J osQ rqY; Å"ek izkIr gksrh gSA Mªkboj lfgr1200 kg nzO;eku dh dkj dks pykus osQ ijh{k.k esa ;g ik;k x;k fd ,d lhèkhlM+d ij leku pky ls pyus ij ;g 1 yhVj isVªksy esa 15 km pyrh gSA ;g ekursgq, fd bl ijh{k.k osQ nkSjku lM+d dh lrg vkSj gok osQ dkj.k ?k"kZ.k ,d lekujgrk gS vkSj dkj osQ batu dh n{krk 0.5 gSA dkj osQ Åij yxus okys ?k"kZ.k cy dhx.kuk dhft,A


6.44 {kSfrt ls 30° osQ dks.k ij >qosQ ,d lery i`"B ij 1 kg nzO;eku dk ,d CykWdbl vkur lery osQ i`"B osQ lekarj 10 N dk cy yxkdj Åij dh vksj èkosQyktkrk gSA (fp=k 6-15)A CykWd ,oa vkur lery osQ chp ?k"kZ.k xq.kkad 0.1 gSA ;fnCykWd vkur lery osQ vuqfn'k 10 m dh nwjh rd èkosQyk tk, rks] ifjdyudhft,&

(a) xq#Ro osQ fo#¼ fd, x, dk;Z dhA(b) ?k"kZ.k cy osQ fo#¼ fd, x, dk;Z dhA(c) fLFkSfrt ÅtkZ esa gqbZ o`f¼ dhA(d) xfrt ÅtkZ esa gqbZ o`f¼ dhA(e) yxk;s x;s cy }kjk fd;s x;s dk;Z dhAfp=k 6.15

30o

m

F

18-04-2018


49

6.45 fp=k 6.16 esa ,d oØ i`"B n'kkZ;k x;k gSA bldk BCD Hkkx ?k"kZ.kjfgr gSA loZlef=kT;kvksa vkSj nzO;ekuksa dh rhu xksykdkj xsansa nh xbZ gSaA xsanksa dks ,d&,d djosQ cnqA ls fojkekoLFkk esa NksM+k tkrk gSA ¯cnq A ¯cnq C ls oqQN vfèkd Å¡pkbZ ij gSA

i`"B AB osQ lkFk xsan&1 dk ?k"kZ.k bruk vfèkd gS fd ;g bl ij fiQlys fcukyq<+drh gS_ xsan&2 dk ?k"kZ.k de gS vkSj xsan&3 dk ?k"kZ.k ux.; gSµ(a) fdu xsanksa osQ fy, oqQy ;kaf=kd ÅtkZ lajf{kr jgrh gS\(b) dkSu&lh xsansa ¯cnq D rd igq¡p ikrh gSa\(c) tks xsansa D rd ugha igq¡p ikrh muesa ls dkSu&lh xsansa okil A ij ykSV ikrh gSa\

6.46 ,d jkWosQV uhps dh vksj xSl fu"dkf"kr djrk gqvk lhèkk Åij dks Rofjr gksrk gSA ;g,d vYi le; ∆t esa] ∆m nzO;eku xSl] lkis{k osx u ls fu"dkf"kr djrk gSA laiw.kZfudk; dh K.E., t + ∆t ,oa t {k.k ij ifjdfyr dhft, vkSj n'kkZb, fd bl le;

varjky esa xSl fu"dkflr djus okyh ;qfDr = ( ) 212

m u∆ dk;Z djrh gSA (xq#Ro dh

mis{kk dj ldrs gSa)

6.47 nks loZle LVhy osQ cus ?kukdkj Bksl ( nzO;eku 50g, Hkqtk 1cm) izR;sd10cm/s–1 dh pky ls vkdj i`"B ls i`"B feykrs gq, lEeq[k la?kV~V djrs gSaAizR;sd ?ku esa gksus okys vfèkdre laihMu dh x.kuk dhft,A LVhy osQ fy, ;axizR;kLFkrk xq.kkad, Y= 2 × 1011 N/m2 gSA

6.48 ghfy;e ls Hkjk xqCckjk xq#Ro osQ fo#¼ Åij mBrk gSA ftlls bldh fLFkSfrt ÅtkZesa o`f¼ gks tkrh gSA tSls&tSls xqCckjk Åij tkrk gS bldh pky Hkh c<+ tkrh gSA blizs{k.k dk rkyesy vki ;kaf=kd ÅtkZ laj{k.k osQ fu;e ls oSQls cSBk,axs\ vki ok;q osQ';ku d"kZ.k dh mis{kk dj ldrs gSa vkSj ;g eku ldrs gSa fd bldk ?kuRo ughacnyrkA

fp=k 6.16

A

B

C

D

18-04-2018

50


7.1 fuEufyf[kr esa ls fdl fiaM dk nzO;eku osaQnz mlosQ ckgj fLFkr gksrk gSA

(a) isafly

(b) 'kkWViqV (xksyk)

(c) (iklk)

(d) (pwM+h)

7.2 fp=k 7-1 esa n'kkZ, x, fudk; esa vafdr dkSu&lk cnq blosQ nzO;eku osaQnz dh laHkkforfLFkfr gS?

(a) A

(b) B

(c) C

(d) D

vè;k; 7

d.kksa osQ fudk; rFkk ?kw.khZ xfr

R/2

R/2

A

B

C

D

ok;q

[kks[kykxksyk

jsr

fp=k 7.1

18-04-2018

51


7.3 m nzO;eku dk dksbZ d.k ,d leku osx v ls YZ ry esa bl izdkj xfreku gS fd bldkiFk + y-v{k osQ lekarj jgrk gS vkSj z-v{k dks z = a ij izfrPNsfnr dj jgk gS(fp=k 7-2)A ;fn ;g y = vpjkad osQ laxr nhokj ij ewy cnq osQ ifjrµ blosQ dks.kh;laosx esa ifjorZu dk eku gSµ

(a) mva êx

(b) 2mva êx

(c) ymv êx

(d) 2ymv êx

7.4 tc dksbZ fMLd ,d leku dks.kh; osx ls ?kw.kZu djrh gS] rks fuEufyf[kr esa dkSu&lkdFku lR; ugha gksrk\

(a) ?kw.kZu dh fn'kk leku jgrh gSA

(b) ?kw.kZu v{k dk fno~Q&foU;kl leku jgrk gSA

(c) ?kw.kZu dh pky 'kwU;srj gksrh gS rFkk leku jgrh gSA

(d) dks.kh; Roj.k 'kwU;srj gksrk gS rFkk leku jgrk gSA

7.5 fdlh ,d leku oxZdkj IysV ls dksbZ vfu;fer vko`Qfr dk NksVk VqdM+k Q dkVdjbls IysV osQ osaQnz ls fpidk fn;k x;k gS vkSj IysV esa iwoZ LFkku ij fNnz NksM+ fn;k x;kgS (fp=k 7-3)A rc z-v{k osQ ifjr% bl IysV dk tM+Ro vk?kw.kZ

(a) c<+ tkrk gSA

(b) ?kV tkrk gSA

(c) leku jgrk gSA

(d) vuuqesf;r :i ls cny tkrk gSA

7.6 iz'u 7-5 esa] vc IysV dk nzO;eku osaQnz x-y ry osQ uhps fn, x, fdl prqFkk±'k esa gS\

(a) I

(b) II

(c) III

(d) IV

fp=k 7.2

fp=k 7.3

fNnz

18-04-2018


7.7 1 m yach fdlh vleku NM+ dk ?kuRo bl izdkj O;Dr fd;k x;k gSρ (x) = a(1+bx2)

;gk¡ a rFkk b fLFkjkad gSa rFkk 1o x≤ ≤

bl NM+ dk nzO;eku osaQnz gksxk

(a)3(2 )

4(3 )

b

b

+

+

(b)4(2 )

3(3 )

b

b

+

+

(c)3(3 )

4(2 )

b

b

+

+

(d)4(3 )

3(2 )

b

b

+

+

7.8 f=kT;k R rFkk nzO;eku M osQ NYys tSls IysViQkeZ dk cuk dksbZ esjh&xks&jkmaM >wykdks.kh; pky ω ls ifjØe.k dj jgk gSA M nzO;eku dk dksbZ O;fDr bl >wys ij [kM+kgSA fdlh {k.k fo'ks"k ij ;g O;fDr bl >wys ls] bl >wys osQ osaQnz ls ijs f=kT;r%(>wys ls ns[kus ij) owQnrk gSA blosQ i'pkr~ >wys dh pky gSµ

(a) 2ω (b) ω (c) 2

ω(d) 0


7.9 lgh fodYi pqfu,µ

(a) fdlh O;kid ?kw.khZ xfr osQ fy, dks.kh; laosx L rFkk dks.kh; osx ωωωω dk lekarj gksukvko';d ugha gSA

(b) fdlh fLFkj v{k osQ ifjr% ?kw.khZ xfr osQ fy, dks.kh; laosx L rFkk dks.kh; osxωωωω lnSo lekarj gksrs gSaA

(c) fdlh O;kid LFkkukarjh; xfr osQ fy, laosx p rFkk osx v lnSo lekarj gksrs gSaA(d) fdlh O;kid LFkkukarjh; xfr osQ fy, Roj.k a rFkk osx v lnSo lekarj gksrs gSaA

7.10 fp=k 7-4 esa nks loZle d.k 1 ,oa 2] ftuesa izR;sd dk nzO;eku m gS lekarj js[kkvksaosQ vuqfn'k foijhr fn'kkvksa esa leku pky v ls xfr djrs n'kkZ, x, gSaA fdlh fo'ks"k{k.k ij lekarj js[kkvksa osQ ry esa fdlh cnq A ls [khaps x, bu d.kksa dh fLFkfr lfn'kØe'k% r

1 ,oa r

2 gSA lgh fodYi pqfu,

(a) d.k 1 dk A osQ ifjr% dks.kh; laosx =1 1m vdl

(b) d.k 2 dk A osQ ifjr% dks.kh; laosx =2 2m v rl fp=k 7.4

18-04-2018

53


(c) A osQ ifjr% fudk; dk oqQy dks.kh; laosx = +1 2( )mvl r r

(d) A osQ ifjr% fudk; dk oqQy dks.kh; laosx = −2 1( )mv d dl ⊗

⊗

xi"B oQs cfgZ keh ,dkda lfn'k dk s fu:fir djrk gAS

i"B osQ vra xkeZ h ,dkda lfn'k dk s fu:fir djrk gAS

7.11 d.kksa osQ fdlh fudk; dk fdlh v{k osQ ifjr% usV cká cy vk?kw.kZ 'kwU; gSAfuEufyf[kr esa dkSu&lk dFku blosQ lkFk lqlaxr gS\

(a) bl v{k ij fdlh ¯cnq ls cy f=kT;r% dk;Z dj jgs gks ldrs gSaA(b) cy ?kw.kZu v{k ij dk;Zjr gks ldrs gSaA(c) cy ?kw.kZu v{k osQ lekarj dk;Zjr gks ldrs gSaA(d) oqQN cyksa osQ dkj.k cy vk?kw.kZ] oqQN vU; cyksa osQ dkj.k cy vk?kw.kksZa osQ cjkcj

,oa foijhr gks ldrs gSaA

7.12 fp=k 7.5 esa x-y ry esa fLFkr ,d iVy n'kkZ;k x;k gSA nks v{k z ;k z ′ blosQ ry osQyacor~ gSaA dksbZ cy F iVy osQ ry esa ¯cnq P ij n'kkZ, vuqlkj dk;Z djrk gSAfuEufyf[kr dFkuksa esa dkSu&lk dFku lR; gS\ (fcanq P, z-v{k dh rqyuk esa z′-v{kosQ vfèkd fudV gS)A

(a) z-v{k osQ ifjr% F osQ dkj.k cy vk?kw.kZ τ, ˆ-k osQ vuqfn'k gSA

(b) z′-v{k osQ ifjr% F osQ dkj.k cy v?kw.kZ τ′ , ˆ-kosQ vuqfn'k gSA(c) z-v{k osQ ifjr% F osQ dkj.k cy v?kw.kZ τ τ τ τ τ ifjek.k esa z′&v{k osQ ifjr% vk?kw.kZ

τ′τ′τ′τ′τ′ ls vfèkd gSA(d) oqQy cy vk?kw.kZ = τ τ τ τ τ + τ′ τ′ τ′ τ′ τ′.

7.13 fp=k 7-6 esa nh xbZ Hkqtk a rFkk nzO;eku m osQ ?ku osQ lanHkZ esa vafdr dhft,fd fuEufyf[kr dFku lR; gS vFkok vlR; (0 ?ku dk osaQnz gSSS)A

(a) z-v{k osQ ifjr% ?ku dk tM+Ro vk?kw.kZ Iz = I

x + I

y

(b) z′ -v{k osQ ifjr% ?ku dk tM+Ro vk?kw.kZ 2

'2

= +z z

m aI I

(c) z″-v{k osQ ifjr% ?ku dk tM+Ro vk?kw.kZ 2

2z

m aI= +

(d) Ix = I

y


7.14 i`Foh ij fdlh oLrq dk xq#Ro osaQnz y?kq ¯iM osQ fy, mlosQ nzO;eku osaQnz osQ laikrh

fp=k 7.5

fp=k 7.6

18-04-2018


gksrk gS tcfd foLr`r iMksa esa laHkor% ,slk ugha gksrkA bl lanHkZ esa y?kq ,oa foLr`r dkxq.kkRed vFkZ D;k gS\ fuEufyf[kr esa fdlosQ fy, ;s nksuksa laikrh gksrs gSa\

dksbZ Hkou] rkykc] >hy] ioZrA

7.15 leku nzO;eku ,oa leku f=kT;k osQ nks xksyksa osQ vius lefer v{kksa osQ ifjr% tM+Rovk?kw.kksZa esa Bksl csyu dk tM+Ro vk?kw.kZ [kks[kys csyu dh rqyuk esa de D;ksa gksrk gS\

7.16 fdlh ?kw.khZ n`<+ ¯iM osQ fdlh ¯cnq dh dks.kh; fLFkfr θ esa le; t osQ lkFk ifjorZudks fp=k 7-7 esa n'kkZ;k x;k gSA ;g ¯iM okekorZ ?kw.kZu dj jgk gS vFkok nf{k.kkorZ\

7.17 Hkqtk a rFkk nzO;eku m dk dksbZ ,d leku ?ku fdlh ?k"kZ.k jfgr {kSfrt i"Bij j[kk gSA vko`Qfr 7-8 esa n'kkZ, vuqlkj blosQ fdukjs ij dksbZ ÅèokZèkj cy F vkjksfirfd;k tkrk gSA fuEufyf[kr (lcls mi;qDr fodYi) dk feyku dhft,µ

(a) mg/4 < /2F mg< (i) ?ku Åij mBsxkA

(b) F > mg/2 (ii) ?ku xfr iznf'kZr ugha djsxkA(c) F > mg (iii) ?ku A ij ?kw.kZu djus yxsxk rFkk fiQlysxkA(d) F = mg/4 (iv) vfHkyac izfrfØ;k A ls a/3 ij izHkkoh] dksbZ

xfr ughaA

7.18 f=kT;k R rFkk nzO;eku m dk ,d leku xksyk fdlh :{k {kSfrt i`"B ij fLFkr gS(fp=k 7-9)A iQ'kZ ls h Å¡pkbZ ij xksys ij {kSfrtr% vk?kkr fd;k tkrk gSA fuEufyf[krdk feyku dhft,µ

(a) h = R/2 (i) xksyk fcuk fiQlys fu;e osx ls ?kw.kZu djrk gS rFkkÅtkZ dk ßkl ugha gksrkA

(b) h = R (ii) xksyk nf{k.kkorZ pØ.k djrk gS] ?k"kZ.k osQ dkj.k ÅtkZdk ßkl gksrk gSA

(c) h = 3R/2 (iii) xksyk okekorZ pØ.k djrk gS] ?k"kZ.k osQ dkj.k ÅtkZdk ßkl gksrk gSA

(d) h = 7R/5 (iv) xksys esa osQoy LFkkukarjh; xfr gksrh gS] ?k"kZ.k osQdkj.k ÅtkZ dk ßkl gksrk gSA


7.19 fdlh ¯iM ij dk;Zjr vlajs[kh cyksa osQ fudk; dk lfn'k ;ksx 'kwU;srj fn;k gqvk gSA;fn fudk; osQ lHkh cyksa osQ fdlh fuf'pr cnq osQ ifjr% cy vk?kw.kksZa dk lfn'k ;ksx'kwU; gS] rks D;k bldk ;g vFkZ gS fd fdlh ;kn`fPNd ¯cnq osQ ifjr% ;g vko';d:i ls 'kwU; gSA

fp=k 7.7

fp=k 7.9

fp=k 7.8

18-04-2018

55


7.20 vius ry osQ yacor~ rFkk vius osaQnz ls xqtjus okys v{k osQ ifjr% ,d leku xfr djrsfdlh ifg, dks ;kaf=kdh; (LFkukarjh; rFkk ?kw.khZ) lkE; esa ekuk tkrk gS D;ksafd bldhxfr dks cuk;s j[kus osQ fy, fdlh usV cká cy vFkok cy vk?kw.kZ dh vko';drk ughagSA rFkkfi ftu d.kksa ls feydj ;g ifg;k cuk gS os osaQnz dh vksj fufnZ"V vfHkosaQnz cydk vuqHko djrs gSaA ifg;s dh lkE;koLFkk osQ lkFk vki bl rF; ls oSQls lkeatL; cSBk,¡xsA

vki fdlh vkèks ifg;s dks ifg;s osQ ry osQ yacor~ rFkk mlosQ nzO;eku osaQnz ls xqtjusokys v{k osQ ifjr% ,d leku xfr esa oSQls LFkkfir djsaxsA D;k vkidks bldh xfr cuk,j[kus osQ fy, fdlh cká cy dh vko';drk gksxh\

7.21 fdlh njokts osQ ,d fljs ij pwy gS rFkk ;g ÅèokZèkj v{k osQ ifjr% ?kw.kZu osQ fy,Lora=k gS (fp=k 7-10)A D;k bldk Hkkj bl v{k osQ ifjr% dksbZ cy vk?kw.kZ yxkrk gS\vius Lrj osQ fy, dkj.k fyf[k,A

7.22 fdlh fu;fer n- cgqHkqt osQ 'kh"kksZa ij m nzO;eku osQ (n-1) leku cnq nzO;eku fLFkrgSaA blosQ [kkyh 'kh"kZ dk cgqHkqt osQ osaQnz osQ lkis{k fLFkfr lfn'k a gSA nzO;eku osaQnz dhfLFkfr lfn'k Kkr dhft,A


7.23 fdlh ,d leku (a) vèkZ&pfØdk (b) prqFkk±'k pfØdk dk nzO;eku osaQnz Kkr dhft,A

7.24 nks pfØdk,¡] ftuosQ viuh laxr v{kksa (pfØdk osQ vfHkyacor~ rFkk muosQ osaQnz lsxqtjus okyh) osQ ifjr% tM+Ro vk?kw.kZ I

1 rFkk I

2 gSaA dks.kh; pkyksa w

1 rFkk w

2 ls ?kw.kZu

djrs gq, vius&vius iQydksa osQ lkFk ?kw.kZu v{kksa dks laikrh j[krs gq, laioZQ esa ykbZtkrh gSaA

(a) D;k bl fLFkfr ij dks.kh; laosx laj{k.k fu;e ykxw gksrk gS\ D;ksa\(b) nks pfØdkvksa osQ fudk; dh dks.kh; pky Kkr dhft,A(c) bl izfØ;k esa fudk; dh ÅtkZ esa gksus okys ßkl dh x.kuk dhft,A(d) ßflr ÅtkZ dk D;k gqvk\ crkb,A

7.25 f=kT;k R dh dksbZ pfØdk {kSfrt v{k osQ ifjr% dks.kh; pky oω ls ?kw.kZu dj jgh gSA

bls fdlh {kSfrt es”k ij j[kk tkrk gSA xfrt ?k"kZ.k xq.kkad µk gSµ

(a) es”k osQ laioZQ esa ykus ls iwoZ blosQ nzO;eku osaQnz dk osx D;k Fkk\

(b) es”k osQ laioZQ esa j[kus ij bldh usfe (fdukjs) osQ fdlh ¯cnq osQ jSf[kd osx dkD;k gksrk gS\

(c) tc pfØdk dks es”k osQ laioZQ esa j[kk tkrk gS rks blosQ nzO;eku osaQnz osQ js[kh;osx dk D;k gksrk gS\

fp=k 7.10

18-04-2018


(d) dkSu&lk cy (b) rFkk (c) esa izHkkoksa osQ fy, mÙkjnk;h gS\

(e) yq<+duk (yksVu) vkjaHk gksus osQ fy, fdl 'krZ dk iw.kZ gksuk vko';d gS\

(f) yq<+duk (yksVu) vkjaHk gksus esa yxus okyk le; ifjdfyr dhft,A

7.26 leku Å¡pkbZ h osQ R ,oa 2R f=kT;kvksa osQ nks csyukdkj [kkyh Mªe Øe'k% ω (okekoÙkZ)rFkk ω (nf{k.kkorZ) dks.kh; osxksa ls ?kw.kZu dj jgs gSaA buosQ v{k fu;e ,oa lekUkkarj rFkk{kSfrt ry esa gSa\ buosQ chp (3 )R δ+ i`Fkdu gSA bUgsa vc laioZQ esa yk;k tkrk gS

( 0)δ → A(a) laioZQ osQ Bhd i'pkr~ ?k"kZ.k cyksa dks n'kkZb,A(b) laioZQ osQ Bhd i'pkr~ fudk; osQ ckgj osQ cyksa rFkk cy vk?kw.kks± dh igpku

dhft,A

(c) ?k"kZ.k lekIr gksus ij vafre dks.kh; osxksa dk vuqikr D;k gksxk\

7.27 fdlh ,d leku oxkZdkj IysV S (Hkqtk c) rFkk fdlh ,d leku vk;rkdkj IysV R(Hkqtk,¡ b, a) osQ loZle {ks=kiQy ,oa nzO;eku gSa (fp=k 7-11)An'kkZb, fdµ

7.28 f=kT;k R dh dksbZ pfØdk fdlh est ij viuh usfe (fdukjs) ij fVdh gSA est rFkkpfØdk osQ chp ?k"kZ.k xq.kkad µ gSA (fp=k 7-12) vc vko`Qfr esa n'kkZ, vuqlkj cy F}kjk pfØdk dks [khapk tkrk gSA og vfèkdre cy Kkr dhft, ftlesa vuqiz;ksx lspfØdk fcuk fiQlys yksVu djrh gSA

fp=k 7.12

fp=k 7.11

< > >( ) / 1; ( ) / 1; ( ) / 1xR xS y yS zR zSRi I I ii I I iii I I

18-04-2018


8.1 i`Foh ,d xksys dk lfUudV :i gSA ;fn blosQ vH;arj esa gj LFkku ij ,d leku?kuRo dk nzO; ugha gS] rks i`Foh osQ i`"B ij xq#Roh; Roj.k

(a) osaQnz dh vksj fufnZ"V gksxk] ijarq gj LFkku ij leku ugha gksxkA(b) dk gj LFkku ij leku eku gksxk ijarq osaQnz dh vksj fufnZ"V ugha gksxkA(c) ifjek.k esa gj LFkku ij leku rFkk osaQnz dh vksj fufnZ"V gksxkA(d) fdlh Hkh ¯cnq ij 'kwU; ugha gks ldrkA

8.2 i`Foh ls izs{k.k djus ij lw;Z yxHkx o`Ùkkdkj d{kk esa xfr djrk izrhr gksrk gSA cqèktSls fdlh vU; xzg dh xfr osQ fy, i`Foh ls izs{k.k djus ij Hkh ;g ckr(a) blh izdkj lR; gksxhA(b) lR; ugha gksxh D;ksafd i`Foh ,oa cqèk osQ chp cy O;qRØe oxZ fu;e osQ vuqlkj

ugha gksrkA(c) lR; ugha gksxh D;ksafd cqèk ij izeq[k xq#Rokd"kZ.k cy lw;Z osQ dkj.k gSA(d) lR; ugha gksxh D;ksafd cqèk xq#Rokd"kZ.k cyksa osQ vfrfjDr vU; cyksa ls Hkh

izHkkfor gksrk gSA

vè;k; 8

xq#Rokd"kZ.k

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8.3 i`Foh osQ fofHkUu cnq lw;Z ls oqQN fHkUu nwfj;ksa ij gksrs gSaA vr% xq#Rokd"kZ.k osQ dkj.kfHkUu cyksa dk vuqHko djrs gSaA ,d n`<+&¯iMksa osQ fy, ge tkurs gSa fd ;fn blosQfHkUu ¯cnqvksa ij fHkUu&fHkUu cy dk;Z djsa] rks bldh ifj.kkeh xfr bl izdkj gksxhAtSls fd ,d usV cy blosQ nzO;eku osaQnz ij vkjksfir gksdj blesa LFkkuarjh; xfr mRiÂdj jgk gks rFkk usV cy&vk?kw.kZ nzO;eku osaQnz ls xqtjus okys v{k osQ ifjr% ?kw.khZ xfrmRiÂ dj jgk gksA i`Foh&lw;Z fudk; osQ fy, i`Foh esa ,d leku ?kuRo osQ xksys osQln`'; ekudj

(a) cy vk?kw.kZ 'kwU; gSA(b) cy vk?kw.kZ i`Foh dks pØ.k djkrk gSA(c) n<+&¯iM ifj.kke ;gk¡ ykxw ugha gksrk D;ksafd iFoh n<+&¯iM osQ ln'; Hkh ugha gSA(d) cy vk?kw.kZ i`Foh dks lw;Z osQ pkjksa vksj xfr djkrk gSA

8.4 i`Foh dh ifjØek dj jgs mixzgksa dh vk;q ifjfer gksrh gS rFkk dHkh&dHkh mixzgksadk dpjk i`Foh ij fxjrk gSA bldk dkj.k ;g gS fdµ

(a) lkSj lsy rFkk cSVfj;k¡ lekIr gks tkrh gSaA(b) xq#Rokd"kZ.k fu;e Hkhrj dh vksj lfiZy iz{ksi dk laosQr nsrk gSA(c) ';ku cy mixzg dh pky dks de djrs gSa vkSj bl izdkj mixzg dh Å¡pkbZ

èkhjs&èkhjs ?kVrh gSA(d) vU; mixzgksa ls la?kV~V gksrk gSA

8.5 i`Foh rFkk panzek nksuksa ij lw;Z dk xq#Rokd"kZ.k cy dk;Z djrk gS] lw;Z ls izs{k.k djusij panzek dh d{kkµ

(a) nh?kZo`Ùkh; gksxhA(b) iw.kZ:i ls nh?kZo`Ùkh; ugha gksxh D;ksafd ml ij yxk oqQy xq#Rokd"kZ.k cy osaQnzh;

ugha gSA(c) nh?kZo`Ùkh; ugha gksxh] ijarq vko';d :i ls ,d can oØ gksxhA(d) i`Foh osQ vfrfjDr vU; xzgksa osQ izHkko osQ dkj.k nh?kZo`Ùkh; ls dkiQh fHkUu gksxhA

8.6 gekjs lkSj ifjokj osQ varjkxzkfgd {ks=k esa nzO; osQ VqdM+s (xzgksa dh rqyuk esa] vkekiesa cgqr NksVs) fo|eku gSa ftUgsa {kqnzxzg dgrs gSaA(a) lw;Z dh rqyuk esa cgqr de nzO;eku osQ gksus osQ dkj.k lw;Z osQ pkjksa vksj xfr ugha

djsaxsA(b) vius y?kq nzO;ekuksa osQ dkj.k vfu;fer <ax ls xfr djsaxs rFkk cká varfj{k esa

pys tk,¡xsA(c) can d{kkvksa esa lw;Z osQ pkjksa vksj xfr djsaxs] ijarq osQIyj osQ fu;eksa dk ikyu ugha

djasxsA(d) xzgksa dh Hkk¡fr d{kkvksa esa xfr djsaxs rFkk osQIyj osQ fu;eksa dk ikyu djsaxsA

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59

8.7 vlR; (xyr) fodYi dk p;u dhft,µ

(a) tM+Roh; nzO;eku fdlh cká cy }kjk fdlh ¯iM dks Rofjr djus esa dfBukbZdh eki gS tcfd xq#Roh; nzO;eku ml ij fdlh cká nzO;eku }kjk xq#Rokd"kZ.kcy osQ fuèkkZj.k esa izklafxd gksrk gSA

(b) xq#Roh; nzO;eku rFkk tM+Roh; nzO;eku leku gksrs gSa ;g ,d iz;kSfxd ifj.kke gSA(c) xq#Roh; nzO;eku rFkk tM+Roh; nzO;eku leku gksus osQ dkj.k i`Foh ij lHkh

oLrqvksa osQ fy, xq#Roh; Roj.k leku gksrk gSA(d) izksVkWu tSls d.kksa dk xq#Roh; nzO;eku vkl&ikl osQ Hkkjh iMksa dh mifLFkfr ij

fuHkZj dj ldrk gS tcfd tM+Roh; nzO;eku ,slk ugha dj ldrkA

8.8 2M, m rFkk M nzO;eku osQ d.k Øe'k% A, B rFkk C ¯cnqvksa ij bl izdkj fLFkr gSafd AB = ½ (BC) gS rFkk M dh rqyuk esa m cgqr NksVk gS vkSj le; t = 0 ij ;s lHkhfojke esa gSa (fp=k 8-1)A rnuarj ] fdlh la?kV~V ls iwoZµ

(a) m fojke esa jgsxkA(b) m, M dh vksj xfr djsxkA(c) m, 2M dh vksj xfr djsxkA(d) m nksyuh xfr djsxkA


8.9 uhps fn, x, dkSu ls fodYi lgh gSa\

(a) xq#Roh; Roj.k Å¡pkbZ c<+us ij ?kVrk gSA(b) xgjkbZ c<+us ij xq#Roh; Roj.k c<+rk gS (i`Foh dks ,d leku ?kuRo dk xksyk

ekfu,)A(c) v{kka'k c<+us ij xq#Roh; Roj.k c<+rk gSA(d) xq#Roh; Roj.k i`Foh osQ nzO;eku ij fuHkZj ugha djrkA

8.10 ;fn xq#Rokd"kZ.k fu;e O;qRØe oxZ fu;e ls O;qRØe ?ku fu;e gks tk,] rks

(a) xzgksa dh d{kk nh?kZo`Ùkh; ugha gksaxhA(b) xzgksa dh o`Ùkkdkj d{kk,¡ laHko ugha gksaxhA(c) gkFk ls i`Foh osQ i`"B ij isaQosQ x, iRFkj dh iz{ksi xfr yxHkx ijoyh; gksxhA(d) ,d leku ?kuRo osQ xksyh; [kksy osQ Hkhrj dksbZ xq#Rokd"kZ.k cy ugha gksxkA

fp=k 8.1

2M

A B C

Mm

18-04-2018

60


8.11 ;fn lw;Z dk nzO;eku 10 xquk NksVk rFkk xq#Roh; fu;rkad G ifj.kke esa 10 xquk cM+kgks] rks

(a) i`Foh ij pyuk vfèkd dfBu gks tk,xkA(b) i`Foh ij xq#Roh; Roj.k esa ifjorZu ugha gksxkA(c) o"kkZ dh cw¡n vR;fèkd rsth ls fxjsaxhA(d) ok;q;ku dks vfèkd rhozrk ls pyuk iM+sxkA

8.12 ;fn lw;Z rFkk i`Foh ij fo'kky ek=kk osQ fotkrh; vkos'k gksa] rks

(a) osQIyj osQ lHkh rhuksa fu;e fiQj Hkh oSèk jgsaxsA(b) osQoy rhljk fu;e oSèk gksxkA(c) nwljs fu;e esa dksbZ ifjorZu ugha gksxkA(d) igyk fu;e fiQj Hkh oSèk gksxkA

8.13 ,sls laosQr gSa fd xq#Roh; fu;rkad G dk eku Hkfo"; esa vR;fèkd cM+s le; osQ i'pkr~(yk[kksa@djksM+ksa o"kZ esa) de gksrk tk,xkA ;fn gekjh iFoh osQ fy, ,slk gks] rks

(a) dksbZ ifjorZu ugha gksxkA(b) yk[kksa@djksM+ksa o"kZ osQ i'pkr~ ge (i`Foh) vfèkd rIr gks tk,axsA(c) i`Foh ifjØe.k (lw;Z dh) djsxhA iw.kZr% can d{kkvksa esa ughaA(d) dkiQh vfèkd le; osQ i'pkr~ ge (i`Foh) lkSj ifjokj dks NksM+ nsaxsA

8.14 eku yhft, r1 vkSj r

2 ij fLFkr nks nzO;ekuks a m

1 rFkk m

2 osQ chp

xq#Rokd"kZ.k cyks a F1 rFkk F

2 osQ fy, U;wVu osQ xq#Rokd"kZ.k fu;e dks

1220

– = –

n

1 22

03

12

m mGM

Mr

1 2

rF F� }kjk O;Dr fd;k tkrk gSA ;gk¡ M

0 ,d fLFkjkad

gS ftldh foek nzO;eku dh gSA r12

= r1 –

r

2 rFkk n dksbZ la[;k gSA ,sls izdj.k esa

(a) i`Foh ij xq#Roh; Roj.k fofHkUu ¯iMksa osQ fy, fHkUu&fHkUu gksxkA

(b) osQIyj osQ rhuksa fu;eksa esa ls dksbZ Hkh oSèk ugha gksxkA

(c) osQoy rhljk fu;e gh voSèk gks tk,xkA

(d) n osQ ½.kkRed gksus ij ty ls gYosQ ¯iM ty esa Mwc tk,¡xsA

8.15 fuEufyf[kr esa dkSu lR; gS\

(a) dksbZ èkqzoh; mixzg i`Foh osQ èkqzoksa osQ ifjr% mÙkj&nf{k.k fn'kk esa xfr djrk gSA

(b) dksbZ rqY;dkyh mixzg i`Foh osQ ifjr% iwoZ&if'pe fn'kk esa xfr djrk gSA

(c) dksbZ rqY;dkyh&mixzg i`Foh osQ ifjr% if'pe&iwoZ fn'kk esa xfr djrk gSA

(d) dksbZ èkqzoh; mixzg i`Foh osQ ifjr% iwoZ&if'pe fn'kk esa xfr djrk gSA

8.16 i`Foh osQ i`"B ij fdlh foLrkfjr ¯iM dk nzO;eku osaQnz rFkk bldk xq#Ro osaQnz

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61

(a) lnSo ,d gh ¯cnq ij gksrs gSa pkgs ¯iM dk lkbt+ oqQN Hkh gksA(b) osQoy xksyh; ¯iMksa osQ fy, lnSo ,d gh ¯cnq ij gksrs gSaA(c) dHkh Hkh ,d ¯cnq ij ugha gks ldrsA(d) vkeki esa 100 m ls NksVs ¯iMksa osQ fy, ,d nwljs osQ fudV gksrs gSaA(e) ;fn ¯iM dks i`Foh osQ Hkhrj cgqr xgjkbZ rd ys tk,¡ rks ;s nksuksa ifjofrZr gks

ldrs gSaA


8.17 ok;qeaMy esa ok;q osQ v.kq i`Foh osQ xq#Ro cy }kjk vkdf"kZr fd, tkrs gSaA Li"Vdhft,] ;s lHkh o`{k ls lsc dh Hkk¡fr i`Foh ij D;ksa ugha fxjrs\

8.18 osaQnzh; cy rFkk vosaQnzh; cy dk ,d&,d mnkgj.k nhft,A

8.19 eaxy osQ fy, {ks=kh; osx rFkk le; osQ chp xzkiQ vkjksfir dhft,A

8.20 lw;Z osQ ifjr% i`Foh osQ {ks=kh; osx dh fn'kk D;k gS\

8.21 nks cnq nzO;ekuksa osQ chp xq#Rokd"kZ.k cy fdl izdkj izHkkfor gks tk,xk] ;fn buosQchp osQ i`Fkdu dks leku j[krs gq, mUgsa ty esa Mqck fn;k tkrk gS\

8.22 D;k ;g laHko gS fd fdlh ¯iM esa tM+Ro gks ijarq Hkkj gks\

8.23 ge fdlh vkos'k dk fo|qr {ks=kksa ls ifjj{k.k mls [kks[kys pkyd osQ Hkhrj j[kdj djldrs gSaA D;k ge fdlh ¯iM dk ikl osQ nzO; osQ xq#Roh; izHkko ls ifjj{k.k mlsfdlh [kks[kys xksys osQ Hkhrj j[kdj vFkok fdlh vU; fofèk }kjk dj ldrs gSa\

8.24 fdlh NksVs varfj{k;ku esa] tks i`Foh dh d{kk esa ifjØek dj jgk gS] cSBk dksbZ varfj{k;k=kh xq#Ro cy dk lalwpu ugha dj ldrkA ;fn iFoh dh d{kk esa ifjØek djus okysvarfj{k LVs'ku dk lkb”k cM+k gS] rks og xq#Ro cy osQ lalwpu dh vk'kk dj ldrk gS\

8.25 fdlh [kks[kys xksyh; [kksy (f=kT;k R vkSj ,d leku ?kuRo dk) rFkk ¯cnq nzO;ekuosQ chp xq#Rod"kZ.k cy F gSA F rFkk r osQ chp xzkiQ dh izo`Qfr n'kkZb, tcfd r cnq,d leku ?kuRo osQ [kks[kys xksyh; [kksy osQ osaQnz ls nwjh gSA

8.26 vilkSj vkSj milkSj fLFkfr;ksa esa ls fdl ij i`Foh dk osx vfèkd gksxk\ D;ksa\

8.27 fo"kqor js[kh; lery rFkk fuEufyf[kr osQ d{kh; lery osQ chp fdruk dks.k gksrk gS\(a) èkqzoh; mixzg(b) rqY; dkyh mixzg

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62



8.28 ekè; lkSj fnu mu nks Øekxr nksigjksa osQ chp dk dky&varjky gS tc lw;Z f'kjks cnq(èkzqo o`Ùk) ls xqtjrk gSAu{k=k fnu fdlh nwjLFk lkSj osQ eè; ¯cnq (f'kjks ¯cnq) (èkzqo o`Ùk) ls nks ØekxrlaØe.kksa osQ chp dky&varjky gSAmi;qDr vkjs[k [khapdj] tks i`Foh dk pØ.k rFkk d{kh; xfr n'kkZrk gks] ;g n'kkZb,fd ekè; lkSj fnu u{k=k fnu ls 4 feuV vfèkd vofèk dk gksrk gSA nwljs 'kCnksa esa]nwjLFk rkjk izR;sd Øekxr fnuksa esa 4 feuV igys mn~; gksxkA(laosQr% vki i`Foh dh d{kk o`Ùkkdkj eku ldrs gSa)A

8.29 nks loZle Hkkjh xksyksa osQ chp dh nwjh mudh f=kT;kvksa dk 10 xquk gSA bu nksuksa dksfeykus okyh js[kk osQ eè; ¯cnq ij dksbZ ¯iM LFkk;h lkE; esa gksxk vFkok vLFkk;hlkE; esa\ vius mÙkj osQ fy, dkj.k nhft,A

8.30 i`Foh dh d{kk esa ifjØek djrs mixzg osQ fy, fuEufyf[kr xzkiQksa dh izo`Qfr n'kkZb,A(a) xfrt ÅtkZ vkSj d{kh; f=kT;k R osQ chp(b) fLFkSfrt ÅtkZ vkSj d{kh; f=kT;k R osQ chp(c) oqQy ÅtkZ vkSj d{kh; f=kT;k R osQ chp

8.31 fp=k 8-2 esa dbZ oØ n'kkZ, x, gSaA (ok;q osQ ?k"kZ.k dh mis{kk djosQ) roZQ lfgr;g Li"V dhft, fd buesa ls dkSu&ls oØ fdlh iz{ksI; osQ laHkkfor iz{ksi&iFk gksldrs gSaA

(a) (b) (c)

��

i`Fohi`Foh

(d) (f )

fp=k 8.2

i`Foh

i`Foh

(e)

i`Foh

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63

8.32 m nzO;eku osQ fdlh iM dks i`Foh osQ i`"B ls i`Foh dh f=kT;k osQ cjkcj Å¡pkbZ rdÅij mBk;k tkrk gS] vFkkZr~ mls i`Foh osQ osaQnz ls R ls 2R nwjh rd ys tk;k tkrkgSA bldh fLFkSfrt ÅtkZ esa yfCèk fdruh gSA

8.33 r f=kT;k rFkk M nzO;eku osQ fdlh irys o`Ùkkdkj NYys osQ osaQnz ls xqtjus okysvfHkyac osQ vuqfn'k h nwjh ij fLFkr ¯cnq P ij dksbZ nzO;eku m j[kk gS(fp=k 8-3)A

;fn bl nzO;eku dks vkSj vfèkd nwjh ij bl izdkj ys tk,¡ fd OP = 2h gks tk;s]rks ;fn h = r gS] rks xq#Rokd"kZ.k cy fdruk xquk de gks tk;sxk\


8.34 lw;Z osQ leku fdlh rkjs osQ pkjksa vksj dbZ iM fofHkUu nwfj;ksa ij xfr dj jgs gSaA ;gekfu, fd lHkh o`Ùkkdkj d{kkvksa esa xeu djrs gSaA eku yhft, fd rkjs osQ osaQnz ls nwjhr gS rFkk bldk jSf[kd osx v, dks.kh; osx w, xfrt ÅtkZ K, xq#Roh; fLFkSfrt ÅtkZU, oqQy ÅtkZ E rFkk dks.kh; laosx L gSA tSls&tSls d{kk dh f=kT;k r esa o`f¼ gksrhgS] rks mijksDr jkf'k;ksa esa ls fduesa o`f¼ gksrh gS vkSj fduesa deh gksrh gS\

8.35 Hkqtk L osQ fdlh fu;fer "kVHkqt osQ 'kh"kks± ij m nzO;eku osQ N% cnq&nzO;eku fLFkrgSaA buesa ls fdlh Hkh nzO;eku ij oqQy xq#Rokd"kZ.k cy ifjdfyr dhft,A

8.36 lapkj osQ fy, i`Foh dh fo"kqorh; rqY;dkyh d{kk esa fdlh mixzg dks LFkkfir fd;ktkrk gSA

(a) bl izdkj osQ mixzg dh Å¡pkbZ ifjdfyr dhft,A(b) mixzgksa dh og U;wure la[;k Kkr dhft, tks lapkj osQ fy, leLr i`Foh dh

O;kfIr osQ fy, vko';d gks rFkk fo"kqor~ o`Ùk osQ fdlh cnq ls de ls de ,dmixzg vo'; fn[kkbZ nsA

fp=k 8.3

r

h

P

mo

M

18-04-2018

64


[M = 6 × 1024 kg, R = 6400 km,

T = 24hr, G = 6.67 × 10–11 SI bdkbZ]

8.37 i`Foh dh d{kk 0.0167 mRosaQnzrk dk nh?kZo`Ùk gSA bl izdkj] i`Foh dh lw;Z ls nwjhrFkk pky ftlls ;g lw;Z dh ifjØe.k djrh gS izfrfnu ifjofrZr gksrh gSA bldk ;gvFkZ gS fd iwjs o"kZ esa lkSj fnu fu;r ugha gSA eku yhft, fd i`Foh dk ?kw.kZu v{kblosQ d{kh; ry osQ vfHkyacor~ gS] rc lcls NksVs rFkk lcls cM+s fnu dh vofèk(yackbZ) Kkr dhft,A nksigj ls nksigj rd osQ le; dks ,d fnu ekfu,A D;k bllso"kZ dh vofèk esa fnu dh yackbZ esa ifjorZu Li"V gksrk gS\

8.38 dksbZ mixzg i`Foh osQ pkjksa vksj fdlh nh?kZo`Ùkh; d{kk esa 6 R osQ vilkSj rFkk 2 R

osQ vilkSj lfgr ifjØe.k dj jgk gS tcfd R = 6400 km i`Foh dh f=kT;k gSAd{kk dh mRosaQnzrk Kkr dhft,A HkwfemPp rFkk Hkwfeuhp ij mixzg osQ osx Kkr dhft,A;fn bl mixzg dks 6 R f=kT;k dh o`Ùkh; d{kk esa LFkkukarfjr djuk gks rks D;k fd;ktkuk pkfg,\

[G = 6.67 × 10–11 SI units and M = 6 × 1024 kg]

18-04-2018

cgq fodYih; iz'u I (MCQ I)9.1 fdlh vkn'kZ nzo dk vo:i.k xq.kkad gksrk gSµ

(a) vuar(b) 'kwU;(c) ,dkad(d) dksbZ ifjfer] NksVk] 'kwU;srj fu;reku

9.2 ;fn fdlh rkj dh viuh ewy yackbZ ?kVdj vkèkh jg tkrh gS] rks og vfèkdre yksM]tks ;g rkj fcuk VwVs lgu dj ldrk gSA(a) nksxquk(b) vkèkk(c) pkj xquk(d) mruk gh (leku)

9.3 fdlh rkj dk rki nksxquk dj fn;k tkrk gS rks bldk ;ax izR;kLFkrk xq.kkad(a) Hkh nksxquk gks tk,xk(b) pkj xquk gks tk,xk

vè;k; 9

Bkslksa osQ ;kaf=kd xq.k

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(c ) ogh jgsxk(d) ?kV tk,xkA

9.4 fdlh dekuh osQ ,d fljs ij yksM vuqiz;qDr djosQ bls [khapk tkrk gSA dekuh esamRiUu foo`Qfr gSµ

(a) vk;ruh(b) vo:i.k(c) vuqnS?;Z ,oa vo:i.k(d) vuqnS?;Z

9.5 M nzO;eku dh dksbZ n`<+ NM+ rhu rkjksa] ftuesa izR;sd dh yackbZ l gS] ij lefer :ils fVdh gSA buesa nksuksa fljksa okys rkj dkWij osQ rFkk eè; okyk rkj vk;ju dk gSA ;fnizR;sd esa ruko leku jgrk gS] rks bu rkjksa osQ O;klksa dk vuqikr cjkcj gS

(a) Ycopper

/Yiron

(b)iron

copper

Y

Y

(c)

2

2

iron

copper

Y

Y

(d) iron

copper

Y

Y

9.6 yackbZ 2L, vuqizLFk dkV {ks=kiQy A osQ fdlh e`nw bLikr osQ rkj dks bldh izR;kLFkrklhek osQ Hkhrj nks LraHkksa osQ chp {kSfrtr% rkfur fd;k tkrk gSA dksbZ nzO;eku(fp=k 9-1) m blosQ eè; fcanq ls fuyafcr fd;k tkrk gSA rkj esa foo`Qfr gS&

(a)2

22

x

L

(b)x

L

(c)2x

L

(d)2

2

x

L.

9.7 fdlh vk;rkdkj izsQe dks nks leku yackbZ dh Mksfj;ksa }kjk nks voyacksa ls lefer :ils fuyafcr fd;k tkuk gS (fp=k 9-2)A bls uhps fn, rhu <a+xksa ls fd;k tk ldrkgSµ Mksjh esa ruko&

fp=k 9.1

2L

x

m

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67

(a) lc izdj.kksa esa leku gksxkA(b) (a) esa lcls de gksxkA(c) (b) esa lcls de gksxkA(d) (c) esa lcls de gksxkA

fp=k 9.2

(a) (b) (c)

9.8 loZle foekvksa dh nks csyukdkj NM+sa ftuesa & ,d jcM+ dh vkSj nwljh LVhy dhgS] ij fopkj dhft,A nksuksa NM+ksa dk ,d fljk Nr ls n`<+rkiwoZd tM+ fn;k x;k gSAizR;sd NM+ osQ eqDr fljs osQ oasQnz ij dksbZ nzO;eku M layXu fd;k x;k gSA(a) nksuksa NMksa esa o`f¼ gksxh vkSj bldh vko`Qfr ifjofrZr gksxhA(b) LVhy dh NM+ esa o`f¼ gksxh o mldh vko`Qfr ifjofrZr gksxh ijarq jcM+ dh NM+

esa osQoy o`f¼ gksxhA(c) LVhy dh NM+ esa] vko`Qfr esa cksèkxE; ifjorZu gq, fcuk o`f¼ gksxh] ijarq jcM+

dh NM+ esa o`f¼ gksxh rFkk blosQ fupys fljs dh vko`Qfr nh?kZo`Ùk esa ifjofrZr gkstk,xhA

(d) LVhy dh NM+ esa] vko`Qfr esa cksèkxE; ifjorZu gq, fcuk o`f¼ gksxh] ijarq jcM+dh NM+ esa o`f¼ gksxh rFkk bldk fupyk fdukjk osaQnz ij iryk gksdj uksad cutk,xkA

cgq fodYih; iz'u -II (MCQ II)

9.9 fp=k 9-3 esa nks inkFkks± osQ izfrcy&foo`Qfr oØ n'kkZ, x, gSa] (leku LosQy ekfu,)A

pje ruulkeF;Z foHkax fcanq

izfrcy jSf[kdlhek

foÑfrinkFkZ (i)

fp=k 9.3

jSf[kdlhek

pje ruulkeF;ZfoHkax fcanq

foÑfrinkFkZ (ii)

izfrcy

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68


(a) inkFkZ (i) dh rqyuk esa inkFkZ (ii) vfèkd izR;kLFk gS vkSj bl izdkj inkFkZ (ii) vfèkdHkaxqj gSA

(b) inkFkZ (i) ,oa inkFkZ (ii) nksuksa cjkcj izR;kLFk rFkk cjkcj Hkaxqj gSaA(c) inkFkZ (i) dh rqyuk esa inkFkZ (ii) foo`Qfr osQ vfèkd {ks=k esa izR;kLFk jgrk gSA(d) inkFkZ (ii) dh rqyuk esa inkFkZ (i) vfèkd Hkaxqj gSA

9.10 dksbZ rkj Nr ls yVdk gS rFkk nwljs fljs ij yVosQ Hkkj F osQ }kjk rkfur gSA Nr }kjkrkj ij yxk;k x;k cy yVdk, x, Hkkj osQ leku ,oa foijhr gSA

(a) rkj dh fdlh Hkh vuqizLFk dkV A ij ruu izfrcy F/A gSA(b) rkj dh fdlh Hkh vuqizLFk dkV ij ruu izfrcy 'kwU; gSA(c) rkj dh fdlh Hkh vuqizLFk dkV A ij ruu izfrcy 2F/A gSA(d) rkj dh fdlh Hkh vuqizLFk dkV A ij ruu izfrcy F gSA

9.11 l yackbZ rFkk mis{k.kh; nzO;eku dh dksbZ NM+ vius nks fljksa ij leku yackbZ nks rkjksals yVdkbZ xbZ gS ftuesa ,d rkj LVhy (rkj A) dk rFkk nwljk ,Y;wfefu;e (rkj B)

dk gS (fp=k 9-4)A rkj A rFkk B dh vuqizLFk dkV osQ {ks=kiQy Øe'k% 1.0 mm2 rFkk2.0 mm2 gSaA

( )9 2 9 –270 10 Nm 200 10 NmAl steelY Y−= × = ×vkjS

(a) nksuksa rkjksa esa leku izfrcy osQ fy, fdlh nzO;eku m dks rkj A osQ fudVfuyafcr djuk pkfg,A

(b) nksuksa rkjksa esa leku izfrcy osQ fy, nzO;eku m dks rkj B osQ fudV fuyafcr djukpkfg,A

(c) nksuksa rkjksa esa leku izfrcy osQ fy, nzO;eku m dks rkj osQ eè; ij fuyafcr djukpkfg,A

(d) nksuksa rkjksa esa leku foo`Qfr osQ fy, nzO;eku m dks rkj A osQ fudV fuyafcrdjuk pkfg,A

9.12 fdlh vkn'kZ nzo osQ fy,(a) vk;ru xq.kkad vuar gksrk gSA(b) vk;ru xq.kkad 'kwU; gksrk gSA

fp=k 9.4

LVhy ,Y;wfefu;e

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69

(c) vo:i.k xq.kkad vuar gksrk gSA

(d) vo:i.k xq.kkad 'kwU; gksrk gSA

9.13 leku O;kl osQ dkWij ,oa LVhy osQ rkjksa dks fljs ls fljk feykdj tksM+k x;k gSA blla;qDr rkj ij dksbZ fo:id cy F vkjksfir fd;k tkrk gS tks blesa 1 cm dh oqQyo`f¼ dj nsrk gSA bu nksuksa rkjksa esa&(a) leku izfrcy gksrk gSA(b) fofHkUu izfrcy gksrk gSA(c) leku foo`Qfr gksrh gSA(d) fofHkUu foo`Qfr;k¡ gksrh gSaA


9.14 jcM+ dh rqyuk esa LVhy dk ;ax xq.kkad dkiQh vfèkd gSA leku vuqnS?;Z foo`Qfr osQfy, fdl esa ruu izfrcy vfèkd gksxk\

9.15 D;k izfrcy lfn'k jfk'k gS\

9.16 LVhy rFkk dkWij dh loZle dekfu;ksa dks cjkcj leku :i esa [khapk tkrk gSA fdlij vfèkd dk;Z djuk gksxk\

9.17 iw.kZr% n`<+ fiaM osQ fy, ;ax xq.kkad D;k gksrk gS\

9.18 iw.kZr% n`<+ fiaM osQ fy, vk;ru xq.kkad D;k gksrk gS\


9.19 cy F rFkk f=kT;k r osQ rkj osQ ,d fljs dks n`<+rkiwoZd tdM+k x;k gSA tc bl rkjosQ nwljs fljs dks cy F }kjk [khapk tkrk gS] rks bldh yackbZ esa l o`f¼ gks tkrh gSAmlh inkFkZ osQ 2L yackbZ rFkk 2r f=kT;k osQ rkj dks 2F cy ls [khapk tkrk gSA blrkj esa o`f¼ ifjdfyr dhft,A

9.20 nksuksa fljksa ij n`<+rkiwoZd tdM+h xbZ 1 m yackbZ rFkk 1 cm2 vuqizLFk dkV {ks=kiQydh LVhy (Y = 2.0 × 1011 Nm–2; rFkk α = 10–50 C–1) dh NM+ dks 0°C ls200°C rd bl izdkj xeZ fd;k x;k gS fd u rks bldh yackbZ esa o`f¼ gks u gh ;geqM+sA NM+ esa mRiUu ruko fdruk gS\

9.21 fdlh xgjs leqanz esa ,d jcM+ dh xsan dks fdruh xgjkbZ rd ys tk,¡ fd bldk vk;ru0.1% ?kV tk, [jcM+ dk vk;ru izR;kLFkrk xq.kkad 9.8×108 N m–2; rFkk leqanz osQty dk ?kuRo 103 kg m–3]

9.22 dksbZ Vªd 9.1 m yach 5 mm f=kT;k dh LVhy dh rkj }kjk [kkbZ esa i¡Qlh fdlh dkjdks ckgj [khap jgk gSA tc dkj xfr djuk vkjaHk djrh gS rc rkj esa ruko 800N

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70


gSA rkj dh yackbZ esa fdruh of¼ gqbZ\ (LVhy dk ;ax xq.kkad 2 × 1011 Nm–2)

9.23 nks loZle xsansa ftuesa ls ,d gkFkh nk¡r dh gS rFkk nwljh xhyh feêðh dh gSA lekuÅ¡pkbZ ls i`Foh ij fxjkbZ tkrh gSaA buesa ls dkSu&lh i`Foh ls Vdjkdj vfèkd Å¡pkbZrd Åij mBsxh vkSj D;ksa\


9.24 LVhy dh yach NM+ ij fopkj dhft, ftlosQ fljksa ij yackbZ osQ vuqfn'k yxs cyF osQ dkj.k ruu izfrcy gS (fp=k 9-5)A ,d ,sls ry ij fopkj dhft, tks yackbZls θ dks.k cukrk gSA bl ry ij ruu&izfrcy rFkk vo:i.k&izfrcy D;k gS\

(a) fdl dks.k osQ fy, ruu izfrcy vfèkdre gS\(b) fdl dks.k osQ fy, vo:i.k izfrcy vfèkdre gS\

9.25 (a) fdlh LVhy osQ rkj dk izfr ,dkad yackbZ nzO;eku µ rFkk bldh o`Ùkh; vuqizLFkdkV dh f=kT;k 0.1 cm gSA {kSfrt j[kdj ekius ij bldh yackbZ 10 m gSA blrkj dks nhokj esa yxs gqd ls ÅèokZèkjr% yVdk;k x;k gS rFkk fupys eqDr fljsls 25 kg dk dksbZ nzO;eku yVdk;k tkrk gSA ;g ekurs gq, fd rkj ,d lekugS rFkk vuqizLFk foo`Qfr;k¡ << vuqnS?;Z foo`Qfr;k¡] rkj dh yackbZ esa o`f¼ Kkrdhft,A LVhy dk ?kuRo 7860 kg m–3 gSA (;ax xq.kkad Y=2×1011 Nm–2)

(b) ;fn LVhy dh ijkHko lkeF;Z 2.5×108 Nm–2 gS] rks rkj osQ fupys fljs lsvfèkdre fdruk yVdk;k tk ldrk gS\

9.26 yackbZ 2l, vuqizLFk dkV {ks=kiQy A rFkk nzO;eku M dh dksbZ LVhy dh NM+ vius osaQnzls xqtjus okys v{k osQ ifjr% {kSfrt ry esa ?kw.kZu djkbZ tkrh gSA ;fn LVhy dk ;axxq.kkad Y gS] rks NM+ dh yackbZ esa o`f¼ Kkr dhft,A (NM+ dks ,d leku ekfu,)A

9.27 dksbZ leckgq f=kHkqt ABC nks rkacs dh NM+ksa AB rFkk BC ,oa ,d ,Y;wfefu;e dhNM+ AC ls feydj cuk gSA bls bl izdkj rIr fd;k tkrk gS fd bldh izR;sd HkqtkosQ rki esa ∆T of¼ gksrh gSA dks.k ABC esa ifjorZu Kkr dhft,A rkacs dk jSf[kd izlkjxq.kkad α

1 rFkk ,Y;wfefu;e dk jSf[kd izlkj xq.kkad α

2 gSA

fp=k 9.5

FF

a'

a

�

18-04-2018


71

9.28 izo`Qfr esa] izk;% lajpukRed vo;oksa osQ {k; dk dkj.k ruu vFkok laihMu foo`Qfr;ksaosQ ctk; ,saBu vFkok cadu osQ dkj.k mRiUu fo'kky cy vk?kw.kZ gksrs gSaA lajpukvksaosQ bl izdkj Hkax gksus dh izfØ;k dks vkoqaQpu dgrs gSa rFkk o`{kksa tSlh fo'kkycsyukdkj lajpukvksa osQ izdj.kksa esa ;g cy&vk?kw.kZ vius Lo;a osQ Hkkj osQ dkj.k mRiUugksdj lajpuk dks cafdr dj nsrk gSA vr% xq#Ro osaQnz ls xqtjus okyh ÅèokZèkj js[kklajpuk osQ vkèkkj ls ugha xqtjrhA bl cadu osQ dkj.k o`{k dh osaQnzh; v{k osQ ifjr%

izR;kLFk cy vk?kw.kZ 4

4

Y r

R

π}kjk O;Dr fd;k tkrk gS] ;gk¡ Y ;ax xq.kkad] r rus dh

f=kT;k rFkk R o`{k dh yackbZ osQ vuqfn'k xq#Ro osaQnz ;qDr cafdr i`"B (mnklhu i`"Bdh oØrk f=kT;k gS)A o`{k osQ rus dh fdlh nh x;h f=kT;k osQ fy, o`{k dh ØkafrdÅ¡pkbZ dk vkdyu dhft,A

9.29 dekuh fLFkjkad k rFkk mis{k.kh; nzO;eku dh fdlh izR;kLFk Mksjh ls m nzO;eku dkdksbZ iRFkj caèkk gSA vrkfur Mksjh dh yackbZ L rFkk nzO;eku mis{k.kh; gSA Mksjh dk nwljkfljk fcanq P ij fdlh dhy ls tqM+k gSA vkjaHk esa iRFkj fcanq P osQ ry esa gSA iRFkjdks fcanq P ls ÅèokZèkj fxjk;k tkrk gSA

(a) 'kh"kZ ls og nwjh y Kkr dhft, tc iRFkj igyh ckj fdlh {k.k osQ fy, fojkeesa vk tkrk gSA

(b) bl ikr esa iRFkj }kjk izkIr vfèkdre osx D;k gS\(c) fuEure fcanq ij igq¡pus ij iRFkj dh xfr dh izo`Qfr D;k gksxh\

18-04-2018

Exemplar Problems–Physics


10.1 dksbZ Å¡pk flfyaMj ';ku rsy ls Hkjk gSA blesa dksbZ xksy iRFkj blosQ 'kh"kZ ls 'kwU;vkjafHkd osx ls fxjk;k tkrk gSA fp=k 10-1 esa n'kkZ, xzkiQksa esa og xzkiQ pqfu, tks le;(t) osQ iQyu osQ :i esa iRFkj osQ osx (v) dk fu#i.k djrk gSA

vè;k; 10

nzoksa osQ ;kaf=kdh xq.k

fp=k 10.1

v

t

v

t

v

t

v

t

(a) (b) (c) (d)

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73

10.3 fdlh èkkjk js[kk osQ vuqfn'k(a) fdlh rjy d.k dk osx fu;r jgrk gSA(b) fdlh nh xbZ fLFkfr ls xqtjus okys lHkh rjy d.kksa dk osx fu;r gksrk gSA(c) fdlh fn, x, {k.k ij lHkh rjy d.kksa dk osx fu;r gksrk gSA(d) fdlh rjy d.k dh pky fu;r jgrh gSA

10.4 dksbZ vkn'kZ rjy] o`Ùkh; vuqizLFk dkV osQ vleku ikbi ls izokfgr gksrk gS ftlosQ nksvuqHkkx osQ O;kl 2.5 cm rFkk 3.75 cm gSaA bu nks ikbiksa ls izokfgr rjy osQ osxksadk vuqikr gSµ

(a) 9:4

(b) 3:2

(c) 3 : 2

(d) 2 : 3

10.5 Li'kZ dks.k dk eku ty&dk¡p] varjki"B ij 0º, ,sfFky vYdksgy&dk¡p varjki"B ij0º, ejdjh&dk¡p varjki`"B ij 140º vkSj feFkkby vk;ksMkbM&dk¡p varjki`"B ij30º gSA fdlh nzks.kh esa Hkjs bu pkjksa esa ls fdlh ,d nzo esa dk¡p dh dksf'kdk dksj[kk x;kA ;g ik;k tkrk gS fd esfuLdl mÙky gSA nzks.kh esa Hkjk nzo gSµ

(a) ty(b) ,sfFky vYdksgy(c) ejdjh(d) esfFky vk;ksMkbM


10.6 fdlh i`"Bh; v.kq osQ fy,

(a) bl ij yxus okyk usV cy 'kwU; gksrk gSA

10.2 fuEufyf[kr esa dkSu&lk vkjs[k (fp=k (10-2) èkkjkjs[kh] izokg dks fu#fir ugha djrkgS\

fp=k 10.2

(a) (b) (c) (d)

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74


(b) bl ij usV vèkkseq[kh cy yxrk gSA(c) Hkhrj osQ xq.k dh rqyuk esa de fLFkSfrt ÅtkZ gksrh gSA

(d) Hkhrj osQ v.kq dh rqyuk esa vfèkd ÅtkZ gksrh gSA

10.7 nkc vfn'k jkf'k gS D;ksafd

(a) ;g cy ,oa {ks=kiQy dk vuqikr gS rFkk cy ,oa {ks=kiQy nksuksa lfn'k gSaA

(b) ;g cy ,oa {ks=kiQy osQ ifjek.kksa dk vuqikr gSA

(c) ;g {ks=kiQy osQ vfHkyacor~ cy osQ vo;o dk vuqikr gSA

(d) ;g p;u fd, x, {ks=k osQ lkb”k ij fuHkZj ugha djrkA

10.8 dksbZ ydM+h dk xqVdk] ftlosQ Åij fp=k 10-3 esa n'kkZ, vuqlkj dksbZ flDdk j[kkgS] ty ij rSj jgk gSA

nwjh l rFkk h vko`Qfr esa n'kkZ, vuqlkj gSA oqQN le; i'pkr~ flDdk ty esa fxj tkrkgS] rc

(a) l ?kV tkrk gS

(b) h ?kV tkrk gS

(c) l c<+ tkrk gS

(d) h c<+ tkrk gS

10.9 rki c<+us ij

(a) xSlksa dh ';kurk ?kVrh gSA

(b) nzoksa dh ';kurk c<+rh gSA

(c) xSlksa dh ';kurk c<+rh gSA

(d) nzoksa dh ';kurk ?kVrh gSA

10.10 èkkjkjs[kh izokg laHkor% mu nzoksa eas vfèkd gksrk gSµ

(a) ftudk mPp ?kuRo gksA

(b) ftudh mPp ';kurk gksA

(c) ftudk fuEu ?kuRo gksA

(d) ftudh fuEu ';kurk gksA


10.11 D;k ';kurk ,d lfn'k jkf'k gS\

10.12 D;k i`"B ruko ,d lfn'k jkf'k gS\

10.13 dksbZ fge'kSy vius oqQN Hkkx dks tyeXu djrs gq, rSjrk gSA ;fn fge dk ?kuRoρ

i = 0.917g cm–3 gS] rks fge'kSy osQ vk;ru dk fdruk Hkkx tyeXu jgrk gS\

10.14 ty ls Hkjs fdlh crZu dks Hkkj.k e'khu ij j[kdj blosQ LosQy dks 'kwU; ijlek;ksftr fd;k x;k gSA dekuh fu;rkad k dh fdlh Hkkjghu dekuh ls dksbZ M

fp=k 10.3

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75

nzO;eku ,oa ρ ?kuRo dk dksbZ xqVdk fuyafcr gSA bl xqVosQ dks crZu osQ ty esatyeXu fd;k tkrk gSA LosQy dk ikB~;kad D;k gS\

10.15 ?kuRo ρ dk dksbZ ?kukdkj xqVdk ty osQ i`"B ij rSj jgk gSA bldh Å¡pkbZ L dk xHkkx ty esa Mwck gSA ;g ty ls Hkjk crZu ,sfyosVj ij j[kk gS tks Roj.k a ls mifjeq[khRofjr gks jgk gSA xqVosQ dk fdruk Hkkx tyeXu gS\


10.16 o`{kksa esa jl (tks xfeZ;kssa esa eq[;r% ty gksrk gS) f=kT;k r = 2.5×10–5 m

dh dksf'kdkvksa osQ fdlh fudk; esa Åij mBrk gSA jl dk i`"B rukoT = 7.28×10–2 Nm–1 rFkk Li'kZ dks.k 0º gSA D;k lHkh o`{kksa osQ 'kh"kZ rd ty dhvkiwfrZ osQ fy, osQoy i`"B ruko mÙkjnk;h gks ldrk gS\

10.17 fojke esa fdlh VaSdj esa Hkjs rsy dk eqDr i`"B {kSfrt gSA ;fn VSadj Rofjr gksuk vkjaHkdjs rks eqDr i`"B θ dks.k ij >qd tk,xkA ;fn Roj.k a m s–2 gS] rks eqDr i`"B dk<yku D;k gksxk\

10.18 ejdjh dh 0.1 cm rFkk 0.2 cm f=kT;k dh nks c¡wn ijLij feydj ,d cM+hcw ¡n cu tkrh gSA bl izfØ;k esa fdruh ÅtkZ eqDr gksrh gS\ ejdjh dk i`"Bruko T= 435.5 × 10–3 N m–1 gSA

10.19 ;fn nzo dh dksbZ cw¡n NksVh&NksVh cw¡nksa esa VwVrh gS] rks cw¡nksa dk rki ?kV tkrk gSA ekuyhft, R f=kT;k dh dksbZ cw¡n N NksVh&NksVh cw¡nksa esa VwVrh gS ftuesa izR;sd dh f=kT;kr gSA rki esa fxjkoV dk vkdyu dhft,A

10.20 20º C ij ty dk i`"B ruko rFkk ok"i nkc Øe'k% 7. 28×10–2 Nm–1 rFkk2.33×103 Pa, gSA ml y?kqÙke xksyh; ty cw¡n dh f=kT;k Kkr dhft, tks 20ºC

ij fcuk okf"ir gq, cu losQA


10.21 (a) ok;qeaMy esa Åij tkus ij nkc ?kVrk gSA ;fn ok;q dk ?kuRo ρ gS rks Å¡pkbZ esadh varj gksus ij nkc esa varj dp D;k gS\

(b) ;g ekurs gq, fd nkc p ?kuRo osQ vuqØekuqikrh gS] ;fn i`Foh osQ i`"B ij nkcp

0 gS rks Å¡pkbZ h ij nkc p Kkr dhft,A

(c) ;fn p0 = 1.03×105 N m–2, ρ

0 = 1.29 kg m–3 rFkk g = 9.8 m s–2 gS] tks

i`Foh ls fdl Å¡pkbZ ij nkc ?kVdj i`Foh osQ i`"B ij nkc dk (1/10) jgtk,xk\

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(d) ok;qeaMy dk ;g ekWMy lkis{kr% de nwfj;ksa ij dk;Z djrk gSA mu iwokZuqekuksadh igpku dhft, tks bl ekWMy dks lhfer dj nsrs gSaA

10.22 v.kqvksa osQ chp vkd"kZ.k cy gksus osQ dkj.k nzo i`"B ruko n'kkZrs gSaaA rki esa o`f¼gksus ij i`"B ruko ?kVrk gS rFkk DoFkukad ij ;g 'kwU; gks tkrk gSA fn;k x;k gSµty osQ ok"i.k dh xqIr Å"ek L

v= 540 K cal kg–1] Å"ek dk ;kaf=kd

rqY;kad J = 4.2 J cal–1, ty dk ?kuRo ρw = 103 kg m–3, vkoksxknzks la[;k

NA = 6.0 × 1026 k mole –1 rFkk ty dk vkf.od nzO;eku M

A = 18 kg 1 k mole

osQ fy,A

(a) ty osQ ,d v.kq osQ ok"i.k osQ fy, vko';d ÅtkZ vkdfyr dhft,A

(b) ;g n'kkZb, fd ty osQ fy, varjkv.kqd nwjh 1/3

A

A

1

w

Md

N ρ

= ×

gS rFkk

bldk eku Kkr dhft,A(c) 1 atm nkc ij ok"i voLFkk esa 1 g ty 160 1 cm3 LFkku ?ksjrk gSA ok"i

voLFkk esa DoFkukad ij varjkv.kqd nwjh vkdfyr dhft,A(d) ok"iu osQ le; varjkv.kqd nwjh d ls d ′ rd c<+kus esa v.kq cy F dk lkeuk djrs

gSa ftls vki eku ldrs gSa fd ;g fu;r gSA F dk eku vkdfyr dhft,A(e) F/d ifjdfyr dhft,] tks i`"B ruko dh eki gSA

10.23 dksbZ xje ok;q ls Hkjk xqCckjk 8 m f=kT;k dk xksyk gSA blosQ vanj Hkjh ok;q dk rki60°C gSA tc ckgj dk rki 20°C gS] rc ;g xqCckjk fdrus cM+s nzO;eku dks mBkldrk gS\ (eku yhft, ok;q ,d vkn'kZ xSl gS] R = 8.314 J mole–1K-1,

1 atm = 1.013×105 Pa; f>Yyh ruko 5 N m–1 gS)A

18-04-2018


11.1 ,d f}èkkrqd i=kh vY;wfefu;e ,oa LVhy ( )Al steelα α> dh cuh gSA xje djus

ij ;g i=kh&

(a) lhèkh jgsxhA(b) O;kofrZr gks tk,xhA(c) vY;qfefu;e dks vory ik'oZ cukdj eqM+sxhA(d) LVhy dks vory ik'oZ cukdj eqM+sxhA

11.2 dksbZ ,d leku /krq dh NM+ vius yacor~ f}Hkktd osQ ifjr% ,d leku dks.kh;pky ls ?kw.kZu djrh gSA ;fn bldk FkksM+k rki c<+kus osQ fy, bls ,d leku rIrdjsa rks bldh

(a) ?kw.kZu&pky c<+ tkrh gSA(b) ?kw.kZu pky ?kV tkrh gSA(c) ?kw.kZu pky vifjofrZr jgrh gSA(d) ?kw.kZu&pky blosQ tM+Ro vk?kw.kZ esa o`f¼ osQ dkj.k c<+ tkrh gSA

vè;k; 11

nzO; osQ rkih; xq.k

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11.3 fp=k 11-1 esa nks rkiØeksa A rFkk B osQ chp xzkiQ n'kkZ;k x;k gSA LosQy A rFkk LosQyB ij fuEu fu;r rkikad rFkk mPp fu;r rkikad osQ chp Øe'k% 150 rFkk 100

leku Hkkx gSaA

nksuksa LosQyksa osQ chp :ikarj.k osQ fy, fn;k x;k lacaèk gSµ

(a) 180

100 150A Bt t−

=

(b)30

150 100A Bt t−

=

(c)180

150 100B At t−

=

(d)40

100 180B At t−

=

11.4 fdlh vY;qfefu;e osQ xksys dks ty esa Mqcks;k x;k gSA fuEufyf[kr esa dkSu&lkdFku lR; gS\

(a) 4°C osQ ty dh rqyuk esa 0º C osQ ty esa mRIykourk de gksxhA(b) 4°C osQ ty dh rqyuk esa 0º osQ ty esa mRIykourk vfèkd gksxhA(c) 0°C osQ ty esa mRIykourk 4°C osQ ty dh mRIykourk vfèkd gksxhA(d) 4°C osQ ty esa mRIykourk de ;k vfèkd gksuk xksys dh f=kT;k ij fuHkZj gSA

11.5 rki esa o`f¼ gksus ij yksyd dk vkorZ dky

(a) c<+ tkrk gS D;ksafd yksyd dh izHkkoh yackbZ xksyd dk nzO;eku osaQnz mlosQosaQnz ij jgus ij Hkh c<+ tkrk gSA

(b) ?kV tkrk gS D;ksafd yksyd dh izHkkoh yackbZ xksyd dk nzO;eku osaQnz mlosQosaQnz ij gh jgus ij Hkh c<+ tkrh gSA

(c) c<+ tkrk gS D;ksafd yksyd dh izHkkoh yackbZ] xksyd dk nzO;eku osaQnz xksydosQ xq#Ro osaQnz osQ uhps LFkkukarfjr gksus osQ dkj.k c<+ tkrh gSA

(d) ?kV tkrk gS D;ksafd yksyd dh izHkkoh yackbZ leku jgrh gSA ijarq xksyd dknzO;eku osaQnz xksyd osQ xq#Ro osaQnz ij LFkkukarfjr gks tkrk gSA

11.6 Å"ek lac¼ gksrh gSµ

(a) v.kqvksa dh ;kn`fPNd xfr dh xfrt ÅtkZ lsA(b) v.kqvksa dh O;ofLFkr xfr dh xfrt ÅtkZ lsA(c) v.kqvksa dks ;kn`fPNd ,oa O;ofLFkr xfr;ksa dh oqqQy xfrt ÅtkZ lsA(d) oqQN izdj.kksa esa ;kn`fPNd xfr dh xfrt ÅtkZ ls rFkk oqQN vU; izdj.kksa esa

O;ofLFkr xfr dh xfrt ÅtkZ lsA

fp=k 11.1

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79

11.7 rki T ij fdlh èkkrq osQ xksys dh f=kT;k R gS rFkk èkkrq dk jSf[kd izlkj xq.kkadα gSA xksys dks FkksM+k rIr djosQ blosQ rki esa ∆T o`f¼ dh tkrh gS ftlls bldku;k rki T T+ ∆ gks tkrk gSA xksys osQ vk;ru esa gqbZ yxHkx o`f¼ gSµ

(a) 2 R Tπ α ∆

(b) 2R Tπ α ∆

(c) 34 /3R Tπ α ∆

(d) 34 R Tπ α ∆

11.8 leku nzO;eku rFkk leku inkFkZ osQ cus ,d xksys] ,d ?ku ,oa ,d o`Ùkkdkj IysVdks leku mPp rki rd vkjaHk esa rIr fd;k x;k gS

(a) IysV lcls vfèkd rhozrk ls vkSj ?ku lcls èkhjs BaMk gksxkA(b) xksyk lcls vfèkd rhozrk ls vkSj ?ku lcls èkhjs BaMk gksxkA(c) IysV lcls vfèkd rhozrk ls vkSj xksyk lcls èkhjs BaMk gksxkA(d) ?ku lcls vfèkd rhozrk ls vkSj IysV lcls èkhjs BaMh gksxhA


11.9 lgh fodYiksa dks vafdr dhft,µ

(a) dksbZ fudk; X fudk; Y osQ lkFk rkih; lkE; esa gS] ij Z osQ lkFk ugha gSA fudk;Y rFkk Z ,d nwljs osQ lkFk rkih; lkE; esa gks ldrs gSaA

(b) dksbZ fudk; X fudk; Y osQ lkFk rkih; lkE; esa gS] ij Z osQ lkFk ugha gSA fudk;Y rFkk Z ,d nwljs osQ lkFk rkih; lkE; esa ugha gSA

(c) dksbZ fudk; X u rks Y osQ lkFk rkih; lkE; esa vkSj u gh Z osQ lkFkA fudk; Y

rFkk Z ,d nwljs osQ lkFk rkih; lkE; esa gksus pkfg,A(d) dksbZ fudk; X u rks Y osQ lkFk rkih; lkE; esa gS vkSj u gh Z osQ lkFkA fudk;

Y rFkk Z ,d nwljs osQ lkFk rkih; lkE; esa gks ldrs gSaA

11.10 xqykc tkequ (xksy ekudj) fdlh Hkêðh ij rIr fd, tkrs gSaA ;s nks lkb”kksa esamiyCèk gSaA ,d nwljs ls nksxquk (f=kT;k esa) cM+k gSA fiT”kk (fMLd ekudj) dks HkhHkêðh ij rIr fd;k tkrk gSA ;s Hkh nks lkb”kksa esaa gSaA ,d nwljs ls nksxquk (f=kT;k esa)cM+k gSA pkjksa dks ,d lkFk HkV~Vh osQ rki ij rIr fd;k tkrk gSA fuEufyf[kr esa lslgh fodYi dk p;u dhft,µ

(a) nksuksa lkb”kksa dh xqykc tkequ leku le; esa rIr gksaxhA(b) NksVh xqykc tkequ cM+h ls igys rIr gks tkrh gSA(c) NksVk fiT”kk cM+ksa ls igys rIr gks tkrk gSA(d) cM+k fiT”kk NksVksa ls igys rIr gks tkrk gSA

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11.11 fp=k 11-2 esa n'kkZ, x, xzkiQ osQ lanHkZ esa] ftlesa rIr djus ij ciZQ dh voLFkk n'kkZbZ(LosQy osQ vuqlkj ugha) xbZ gS] dkSu lk fodYi lgh gS\

fp=k 11.2

Øe la[;k l (m) ( C)T∆ (m)l∆

1. 2 10 44 10−×

2. 1 10 44 10−×

3. 2 20 42 10−×

4. 3 10 46 10−×

;fn igyk izs{k.k lgh gS] rks vki 2] 3 rFkk 4 izs{k.kksa osQ ckjs esa D;k dg ldrs gSa\

11.15 gekjs 'kjhj osQ rki ls mPpre leku rki dh èkkrq dh NM+ ydM+h dh NM+ dhvis{kk vfèkd rIr izrhr gksrh gS\ blh izdkj ;fn ;s nksuksa gekjs 'kjhj osQ rki ls derki ij gSa rks èkkrq dh NM+ ydM+h dh NM+ dh vis{kk 'khry izrhr gksrh gSA

(C

)o

100

O

A B

tm

C D

E

rki

le;

(a) {ks=k AB rkih; lkE; esa cjiQ ,oa ty dks fu:fir djrk gSA

(b) B ij ty mcyuk vkjaHk djrk gSA(c) C ij ty Hkki esa :ikarfjr gks tkrk gSA(d) C ls D rd DoFkukad ij ty rFkk Hkki osQ rkih; lkE; dk

fu:i.k gSA

11.12 rIr nwèk ls iw.kZr% Hkjs fxykl dks est ij mM+syk x;k gSA ;g èkhjs&èkhjs 'khry gksukvkjaHk dj nsrk gSA fuEufyf[kr esa ls dkSu&lk fodYi lgh gS\

(a) 'khryu dh nj nwèk dk rki ifjos'k osQ rki rd igq¡pus ij fu;r jgrh gSA(b) nwèk dk rki le; osQ lkFk pj?kkrkadh :i ls ?kVrk gSA(c) 'khryu osQ le;] Å"ek&izokg nwèk ls ifjos'k esa gksrk gS vkSj lkFk gh ifjos'k

ls nwèk esa Hkh gksrk gS] ijarq usV Å"ek&izokg nwèk ls ifjos'k esa gh gksrk gS blhfy,nwèk BaMk gks tkrk gSA

(d) nwèk ls ifjos'k esa Å"ek gkfu osQ fy, pkyu] laokgu ,oa fofdj.k rhuksa ghifj?kVuk,¡ mÙkjnk;h gksrh gSaA


11.13 D;k FkekZehVj dk cYc Å"ek&ik;Z vFkok #¼ks"e nhokj dk cuk gksrk gS\

11.14 dksbZ Nk=k fdlh NM+ dh vkjafHkd yackbZ l, rki&ifjorZu T∆ rFkk yackbZ esa ifjorZu

l∆ dks bl izdkj fjdkMZ djrk gSµ

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11.16 og rki ifjdfyr dhft, ftldk vkafdd eku lsfYl;l rFkk iQkjsugkbV LosQynksuksa ij leku gksrk gSA

11.17 vktdy yksx dkWij dh ryh okys LVhy osQ crZu mi;ksx djrs gSaA bls ,d lekurkiu osQ fy, Js"B ekuk tkrk gSA bl rF; dk] fd dkWij vPNk pkyd gS] mi;ksxdjosQ bl izHkko dks Li"V dhft,A


11.18 fdlh ,d leku NM+ (jSf[kd izlkj xq.kkad α) osQ vius yacor~ lefoHkktd osQifjr% tM+Ro vk?kw.kZ I esa o`f¼ Kkr dhft, tcfd blosQ rki esa y?kq o`f¼ ∆T dhtkrh gSA

11.19 Hkkjr esa xzh"e dky esa ,d lkekU; pyu osQ vuqlkj vius dks 'khry j[kus osQ fy,ciZQ dks owQVdj mlosQ xksys dks 'kcZr esa Mqcksdj pwlrs gSaA blosQ fy, ,d rhyhdks oqQVh cjiQ esa èkalkdj gFksfy;ksa ls nckdj xksyk cukrs gSaA rqY;r% lfnZ;ksa esa]fgeikr okys {ks=kksa esa yksx fge osQ xksys cukdj bèkj&mèkj isaQdrs gSaA ty osQP–T vkjs[k osQ lanHkZ esa oqQVh gqbZ cjiQ vFkok fge ls xksys cukus dh izfØ;kLi"V dhft,A

11.20 100 g ty dks –10°C rd vfr'khfrr fd;k x;k gSA bl fcanq ij oqQN ;kaf=kdfo{kksHk osQ vFkok vU; dkj.ko'k ;g ty ;dk;d te tkrk gSA ifj.kkeh feJ.k dkrki D;k gksxk rFkk fdruk nzO;eku tesxk\

o w

FusionwS = 1cal/g/ C = 80cal/gLrFkk

11.21 ,d fnu izkr% jes'k us xhtj osQ xeZ ikuh ls 1@3 ckYVh Hkjh 2@3 ckYVh dks BaMsty (d{k rki ij) }kjk feJ.k dks vkjkenk;d rki ij ykus osQ fy, Hkjk tkukFkkA vpkud jes'k dks dksbZ dk;Z djuk gS ftlesa ugkus ls iwoZ 5&10 feuV dk le;yxsxkA vc mlosQ ikl nks fodYi gSa% (i) ckYVh osQ 'ks"k Hkkx dks iwjk Hkjus osQ i'pkr~dke ij tk, (ii) igys dke djs vkSj fiQj ugkus ls rqjar igys 'ks"k ckYVh dks HkjsAvkiosQ fopkj ls fdl fodYi ls ty vfèkd Å".k jgsxk\ Li"V dhft,A


11.22 ge ,slk LosQy cukuk pkgrs gSa ftldh yackbZ rki ifjorZu osQ lkFk ifjofrZr u gksA;g lq>ko fn;k x;k gS fd bl izdkj dk tks LosQy cus mldh yackbZ 10 cm jgsAge ihry ,oa yksgs dh cuh f}èkkrqd i=kh dk mi;ksx dj ldrs gSa ftlesa izR;sddh yackbZ fHkUu gks ftudh yackbZ (nksuksa vo;oksa) esa bl izdkj ifjorZu gks fd

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mudh yackb;ksa eas varj lnSo fu;r jgsA ;fn 51.2 10 /Kiron−= ×α rFkk

51.8 10 /Kbrass−= ×α , gS] rc gesa izR;sd i=kh dh yackbZ D;k ysuh pkfg,\

11.23 ge ,d ,slk crZu cukuk pkgrs gSa ftldk vk;ru rki esa ifjorZu osQ lkFk ifjofrZru gks (mijksDr iz'u ls laosQr yhft,)A bls 100 cc vk;ru dk cukus osQ fy, geihry ,oa yksgs dk mi;ksx dj ldrs gSa ( β

brass = 6 × 10–5/K rFkk β

iron= 3.55

×10–5/K)A vki bls oSQls cuk ldrs gSa\

11.24 rkack Hkjh nk¡r dh dksVj esa 57oC rki dh pk; ihus ij mRiÂ izfrcy ifjdfyrdhft,A vki 'kjhj dk rki 37o C,α = 1.7× 10-5/oC rFkk rkacs dk vk;ruxq.kkad = 140 × 10 9 N/m2 ys ldrs gSaA

11.25 10 m yach LVhy dh cuh jsy dh iVjh fdlh jsyos ykbu ij vius nksuksa fljksa ijtdM+h gS (fp=k 11-3)A fdlh xehZ osQ fnu rki esa 20° C dh o`f¼ gksus ij ;gvko`Qfr esa n'kkZ, vuqlkj fo:fir gks tkrh gSA ;fn steelα = 1.2× 10-5 /°C gS rks

x dk eku (osaQnz dk foLFkkiu) Kkr dhft,A

11.26 0ºC ij L0 yackbZ dh dksbZ iryh NM+] ftldk jSf[kd izlkj xq.kkad α gS osQ nksuksa

fljksa dks 1θ rFkk 2θ rki ij j[kk x;k gSA bldh ubZ yackbZ Kkr dhft,A

11.27 LVsiQkWu osQ fofdj.k fu;e osQ vuqlkj dksbZ Ñf".kdk vius ,dkad i`"Bh; {ks=k ls gjlsoaQM σT4 ÅtkZ fodfjr djrh gS] ;gk¡ T Ñf".kdk osQ i`"B dk rki rFkkσ = 5.67 × 10–8 W/m2K4 dks LVsiQkWu fu;rkad dgrs gSaA fdlh 0.5 m f=kT;kdh ckWy dks ukfHkdh; vL=k osQ :i esa eku ldrs gSaA foLiQksV djus ij bldk rki106K igq¡prk gS vkSj bls Ñf".kdk eku ldrs gSaA

(a) blosQ }kjk fodfjr 'kfDr dk vkdyu dhft,A(b) ;fn ifjos'k esa 30º C dk ty gS] rks mRiÂ ÅtkZ dk 10% fdrus ty dks 1

s esa okf"ir dj ldrk gSA

= = × 54186.0 J/kg K 22.6 10 J/kgw vS LrFkk

(c) ;fn ;g leLr ÅtkZ U fofdj.kksa osQ :i esa gks] rks rnuq:ih laosx p = U/c gS];g 1 km nwjh ij izfr ,dkad {ks=kiQy dks izfr ,dkad le; esa fdruk laosxiznku djrk gS\

fp=k 11.3

18-04-2018


12.1 dksbZ vkn'kZ xSl ,d gh vkjafHkd voLFkk ls izkjaHk djosQ fofHkUu izØeksa lsxqtjrh gS (fp=k 12-1)A ;s pkj izØe gSa & #¼ks"e] lerkih;] lenkch; ,oalevk;rfurA 1] 2] 3 vkSj 4 esa ls dkSu&lk #¼ks"e gS\

(a) 4(b) 3(c) 2

(d) 1

12.2 ;fn dksbZ lkekU; O;fDr eaFkj xfr ls pyrk gS rks og14.5 × 103 cal/min

Å"ek mRiÂ djrk gSA ;g Å"ek ilhus osQ ok"iu ls 'kjhj ls fudy tkrh gSA (;gekurs gq, fd1 kg ilhus osQ ok"iu osQ fy, 580 × 103 cal pkfg,) rc izfr feuVokf"ir ilhus dk ifjek.k gS(a) 0.25 kg(b) 2.25 kg

(c) 0.05 kg(d) 0.20 kg

vè;k; 12

Å"ekxfrdh

fp=k 12.1

P

1

2

3

4

V

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12.3 fp=k 12.2 esa n'kk, x, fdlh vkn'kZ xSl osQ P-V vkjs[k ij fopkj dhft,A

fp=k 12-3 esa fn, x, vkjs[kksa esa ls dkSu&lk xzkiQ blosQ laxr T-P vkjs[k dksfu#fir djrk gS\

(a) (iv)

(b) (ii)

(c) (iii)

(d) (i)

12.4 dksbZ vkn'kZ xSl fp=k 12-4 osQ P-V vkjs[k esa n'kkZ, vuqlkj pØh; izfØ;kABCDA djrh gSA

fp=k 12.4

fp=k 12.3

fp=k 12.2

P1

2

V

��P�=

V

T

1

2

T

P(a)

T

1

2

P

T

(b)

T

12

P

T

(c)

T

1 2

P

T

(d)

xSl }kjk fd, x, dk;Z dh ek=kk gS

(a) 6PoV

o

(b) –2 PoV

o

(c) + 2 PoV

o

(d) + 4 PoV

o

18-04-2018

Å"ekxfrdh

85

12.5 nks A rFkk B ik=kksa ij fopkj dhft, ftuesa leku nkc] vk;ru rFkk rki ij vkn'kZxSl Hkjh gSA ik=k A dh xSl dks lerkih; izØe }kjk mlosQ ewy vk;ru osQ vkèksvk;ru rd laihfMr fd;k tkrk gS tcfd ik=k B dh xSl dks #¼ks"e izØe }kjkmlosQ ewy vk;ru osQ vkèks vk;ru rd laihfMr fd;k tkrk gSA B esa xSl rFkk Aesa xSl osQ vafre nkcksa dk vuqikr gS

(a) 12γ −

(b)

11

2

γ −

(c)

21

1 γ

−

(d)

21

1γ

−

12.6 dkWij osQ rhu xqVosQ ftuosQ nzO;eku Øe'k% M1, M

2 ,oa M

3 kg gSaA lkE; voLFkk

esa vkus rd rkih; laioZQ esa j[ks x, gSaA laioZQ ls iwoZ buosQ rki T1, T

2 ,oa T

3 (T

1

> T2 > T

3 ) FksA ;g ekurs gq, fd ifjos'k esa dksbZ Å"ek ßkl ugha gksrh] larqyu rki

T dk eku gksxkµ

(a) 1 2 3

3

T T TT

+ +=

(b) 1 1 2 2 3 3

1 2 3

M T M T M TT

M M M

+ +=

+ +

(c)( )

1 1 2 2 3 3

1 2 33

+ +=

+ +

M T M T M TT

M M M

(d) 1 1 2 2 3 3

1 2 3

+ +=

+ +

M T s M T s M T sT

M M M


12.7 uhps o.kZu fd, x, izØeksa esa dkSu&ls vuqRØe.kh; gSa\(a) gFkkSM+s ls ihVrs le; yksgs dh NM+ osQ rki esa o`f¼ gksukA(b) T

1 rki ij y?kq ik=k esa Hkjh fdlh xSl dks mPp rki T

2 osQ cM+s ik=kksa osQ laioZQ

esa ykrs gSa ftlls xSl osQ rki esa o`f¼ gks tkrh gSA(c) ?k"kZ.kghu fiLVu yxs flfyaMj esa Hkjh fdlh vkn'kZ xSl dh LFkSfrddYi

lerkih; vk;ru o`f¼A(d) #¼ks"e nhokj dh fiLVu&flfyaMj O;oLFkk esa dksbZ vkn'kZ xSl Hkjh gSA fiLVu

ij dksbZ Hkkj W j[kus osQ ifj.kkeLo:i xSl laihfMr gksrh gSA

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P

V

B

I

IV

AII

III

12.8 dksbZ vkn'kZ xSl viuh fdlh vkjafHkd voLFkk i ls vafre voLFkk f rd lerkih;izØe djrh gSA lgh fodYi dk p;u dhft,µ(a) dU = 0

(b) dQ = 0

(c) dQ = dU

(d) dQ = dW

12.9 fp=k 12-5 esa A ls B rd fdlh vkn'kZ xSl dh voLFkk ifjorZu dk P-V vkjs[kn'kkZ;k x;k gSA blosQ pkj fofHkUu Hkkx I, II, III rFkk IV vkjs[k esa fn, vuqlkj lekuvoLFkk ifjorZu dh vksj laosQr djrs gSaA

(a) IV rFkk III izdj.kksa esa vkarfjd ÅtkZ esa ifjorZu leku gSa ijarq I ,oa II esa ,slkugha gSA

(b) vkarfjd ÅtkZ esa ifjorZu lHkh pkjksa izdj.kksa esa leku gSaA(c) izdj.k I esa fd;k x;k dk;Z vfèkdre gSA(d) izdj.k II esa fd;k x;k dk;Z fuEure gSA

12.10 fdlh batu }kjk viuk, x, pØ ij fopkj dhft, (fp=k 12-6)

1 ls 2 lerkih;2 ls 3 #¼ks"e gS3 ls 1 #¼ks"e gS

,slk izØe O;ogkjr% vfLrRoeku ugha gksrk D;ksafd

(a) ,sls izØe esa Å"ek iw.kZr% ;kaf=kd ÅtkZ esa :ikarfjr gksrh gS] tks laHko ugha gSA(b) bl izØe esa ;kaf=kd ÅtkZ iw.kZr% Å"ek esa :ikafjr gksrh gS] tks laHko ugha gSA(c) nks #¼ks"e izØeksa dks fu:fir djus okys pØ izfrPNsnu ugha djrsA(d) fdlh #¼ks"e izØe rFkk fdlh lerkih; izØe dks fu:fir djus okys pØ

izfrPNsnu ugha djrsA

12.11 fp=k 12.7 esa n'kkZ, vuqlkj fdlh Å"ek batu ij fopkj dhft,A Q1 rFkk Q

2 Øe'k%

batu osQ ,d pØ esa Å"eu T1 dks nh xbZ Å"ek rFkk T

2 ls yh xbZ Å"ek,¡a gSaA bl

batu ij fd;k x;k dk;Z W gSA

;fn W > 0 rks laHkkouk,¡ gSa fd

(a) Q1 > Q

2 > 0

(b) Q2 > Q

1 > 0

(c) Q2 < Q

1 < 0

(d) Q1 < 0, Q

2 > 0

fp=k 12.5

fp=k 12.7

fp=k 12.6

18-04-2018

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87


12.12 D;k ;g laHko gS fd fdlh fudk; dks Å"ek nh tk, fiQj Hkh mldk rki fu;r jgs\

12.13 fp=k 12-8 esa n'kkZ, P-V vkjs[k esa dksbZ fudk; P ls Q rd nks fofHkUu iFkksa }kjktkrk gSA iFk 1 ij fudk; dks nh xbZ Å"ek 1000 J gSA fudk; }kjk iFk 1 osQvuqfn'k fd;k x;k dk;Z iFk 2 dh rqyuk esa 100 J vfèkd gSA iFk 2 esa fudk;}kjk Å"ek fofue; D;k gS\

12.14 ;fn fdlh jsfÚtjsVj dk njoktk [kqyk j[ksa rks dejk xje gksxk vFkok BaMk\ Li"Vdhft,A

12.15 D;k fdlh xSl dks fcuk Å"ek fn, mlosQ rki esa o`f¼ dh tk ldrh gS\ Li"Vdhft,A

12.16 dkj pykrs le; blosQ Vk;jksa esa ok;q nkc c<+ tkrk gSA Li"V dhft,A


12.17 1 500KT = rFkk T2= 300K osQ chp izpkfyr fdlh dkuksZ&pØ ij fopkj

dhft, ftlesa izfrpØ 1 k J ;kaf=kd ÅtkZ mRiÂs gks jgh gSA Å"ek HkaMkj }kjk batudks LFkkukarfjr Å"ek dh ek=kk ifjdfyr dhft,A

12.18 dksbZ O;fDr ftldk nzO;eku 60 kg gSA 10 m Å¡ph lh<+h p<+&mrj dj viuk 5 kg

nzO;eku ?kVkuk pkgrk gSA eku yhft, uhps mrjus dh vis{kk Åij p<+us esa nks xquh olktyrh gSA ;fn 1 kg olk dks tykus osQ fy, 7000 fdyks oSQyksjh [kpZ djuh iM+rhgSa] rks 5 kg nzO;eku ?kVkus osQ fy, mls fdruh ckj Åij&uhps tkuk gksxk\

12.19 eku yhft, fdlh lkbdy&Vk;j esa iai }kjk ok;q Hkjh tk jgh gSA eku yhft, Vk;jdk vk;ru (fu;r) V gS rFkk ,d pj.k esa iai #¼ks"e izØe }kjk ( )V V∆ ok;qdks V~;wc esa LFkkukarfjr djrk gSA V~;wc esa nkc dks P

1 ls P

2 djus esa oqQy fdruk dk;Z

fd;k tkrk gS\

12.20 fdlh jsfÚtjsVj esa de rki osQ izdks"B ls Å"ek dks gVkdj mPp rki osQ ifjos'kesa fuf{kIr fd;k tkrk gSA bl izØe esa] ;kaf=kd dk;Z djuk gksrk gS] ftls fo|qr eksVjiznku djrk gSA ;fn eksVj dh 'kfDr 1 kW gS rFkk Å"ek –3°C ls 27°C, rdLFkkukarfjr dh tkrh gS] rks jsfÚtjsVj }kjk izfr lsoaQM yh tkus okyh Å"ek Kkrdhft,A ;g ekfu, fd bldh n{krk ,d iw.kZ n{k batu dh n{krk dk 50% gSA

fp=k 12.8

18-04-2018

88


P

B C

DA

V =VA B

V =VC D

V

fp=k 12.9

1(P V ,�T1, 1 1)

2( )P V ,�T2, 2 2

V1V2

V

P

PV =1/2 ��

12.21 ;fn fdlh jsfÚtjsVj dk fu"iknu xq.kkad 5 gS rFkk ;g d{k rki (27º C) ijizpkfyr gksrk gS rks jsfÚtjsVj osQ Hkhrj dk rki Kkr dhft,A

12.22 ;fn xSl dh vkjafHkd voLFkk (Pi, V

i, T

i) gSA blosQ vk;ru esa V

f gksus rd of¼ gksrh

gSA uhps fn;s x;s nks izdj.kksa ij fopkj dhft,µ

(a) vk;ru o`f¼ fu;r rki ij gksrh gSA(b) vk;ru o`f¼ fu;r nkc ij gksrh gSA

izR;sd izdj.k osQ fy, P-V vkjs[k [khafp,A nksuksa izdj.kksa esa ls fdlesa xSl }kjkvfèkd dk;Z fd;k tkrk gS\


12.23 fdlh csyukdkj ik=k esa Hkjh vkn'kZ xSl osQ 1 eksy dk P-V vkjs[kfp=k 12-9 esa n'kkZ;k x;k gSA

(a) xSl dks voLFkk 1 ls voLFkk 2 esa ys tkus esa fd;k x;k dk;Z Kkrdhft,A

(b) ;fn V2 = 2V

1 gS] rks rkiksa dk vuqikr T

1/T

2 fdruk gS\

(c) fn;k x;k gS fd rki T ij xSl osQ ,d eksy dh vkarfjd ÅtkZ (3/2)

RT gS] rks V2 = 2V

1 osQ lkFk xSl dks voLFkk 1 ls voLFkk 2 rd ys

tkus esa dh xbZ Å"ek vkiwfrZ Kkr dhft,A

12.24 fp=k 12-10 esa fdlh batu (fiLVu flfyaMj esa Hkjh ,d eksy vkn'kZ xSlls cuk) }kjk vuqlj.k fd;k x;k pØ n'kkZ;k x;k gS

A ls B : fu;r vk;ru_B to C : #¼ks"e_C to D : fu;r vk;ruD to A : #¼ks"e

2 2C D A BV V V V= = =

(a) pØ osQ fdl Hkkx esa batu dks ckgj ls Å"ek dh vkiwfrZ dh tkrh gS\(b) pØ osQ fdl Hkkx esa batu ifjos'k dks ÅtkZ ns jgk gS\(c) ,d pØ esa batu fdruk dk;Z djrk gS\ viuk mÙkj P

A, P

B, V

A inksa esa

nhft,A(d) batu dh n{krk D;k gS\

[xSl osQ fy, 53

γ = (,d eksy osQ fy,, (3

2vC R= )

fp=k 12.10

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89

12.25 fp=k 12-11 esa fdlh batu (fiLVu lfgr flfyaMj esa Hkjh ,d eksy vkn'kZ xSl lscuk) }kjk vuqlj.k fd;k x;k pØ n'kkZ;k x;k gSA pØ osQ izR;sd vuqHkkx osQ fy,batu }kjk izfros'k ls Å"ek fofue; Kkr dhft,A (C

v = (3/2) R)

AB µ fu;r vk;ru

BC µ fu;r nkc

CD µ #¼ks"e

DA µ fu;r nkc

12.26 eku yhft, fd fdlh vkn'kZ xSl (n eksy) osQ vk;ru esa P = f ( V ) }kjk fn, x,izØe osQ vuqlkj o`f¼ gks jgh gS] bldh ,d fLFkfr (V

0, P

0 ) ls lalwfpr gksrh gSA

;fn P = f (V) dh izo.krk (P0, V

0 ) ls xqtjus okys #íks"e oØ dh izo.krk ls vfèkd

gS rks n'kkZb, dh xSl (P0, V

0 ) ij Å"ek vo'kksf"kr dj jgh gSA

12.27 fiLVu yxs ,dkad vuqizLFk dkV osQ flfyaMj esa Hkjh ,d eksy vkn'kZ xSl ij fopkjdhft, (fp=k 12.12)A dksbZ dekuh (dekuh fLFkjkad k) bl fiLVu rFkk flfyaMjdh ryh ls tqM+h gS (vrkfur yackbZ L)A vkjaHk esa dekuh vrkfur gS rFkk xSl lkE;esa gSA xSl dks Å"ek dh dksbZ fuf'pr ek=kk Q dh vkiwfrZ djus ij xSl osQ vk;ruesa V

o ls V

1 rd dh o`f¼ gksrh gSA

(a) fudk; dk vkjafHkd nkc D;k gS\(b) fudk; dk vafre nkc D;k gS\(c) rki xfrdh osQ izFke fu;e dk mi;ksx djosQ Q, P

a, V, V

o ,oa k osQ chp

fyf[k,A

fp=k 12.11

fp=k 12.12

�� =�Pa

18-04-2018

90



13.1 fdlh ?kukdkj ik=k (ftlosQ ik'oZ {kSfrt + ÅèokZèkj gSa) esa NTP ij vkn'kZ xSl HkjhgSA ;g ik=k fdlh jkWosQV esa gS] tks 500 m s–1 dh pky ls ÅèokZèkj fn'kk esa xfrdj jgk gSA i`Foh ls ns[kus ij ik=k osQ Hkhrj xSl dk nkc

(a) leku jgrk gS D;ksafd 1500m s− pky xSl dh vrms

ls cgqr de gSA

(b) leku jgrk gS D;ksafd leLr ik=k dh xfr nhokjksa rFkk xSl osQ v.kqvksa dh xfrlkisf{kd xfr dks izHkkfor ugha djrhA

(c) ( ) 22 2 /(500) rmsrmsvv + xquk c<+ tk,xk] ;gk¡ v

rms xSl dk ewy oxZ ekè;

ewy ossx gSA(d) ik=k osQ 'kh"kZ dh nhokj rFkk ryh dh nhokj ij fHkUu&fHkUu gksxkA

13.2 fdlh ?kukdkj ik=k ABCDEFGH esa 300 K rki eksy vkn'kZ xSl Hkjh gS(fp=k 13-1)A bl ?ku dk ,d ik'oZ EFGH fdlh ,sls inkFkZ dk cuk gS tks viusÅij vkifrr xSl osQ fdlh v.kq dks iw.kZr% vo'kksf"kr dj ysrk gSA fdlh Hkh fn,x, le; ij]

vè;k; 13

v.kqxfr fl¼kar

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v.kqxfr fl¼kar

91

(a) EFGH ij nkc 'kwU; gksxkA

(b) lHkh ik'oks± ij nkc leku gksxkA

(c) ABCD dh rqyuk esa EFGH ij nkc nksxquk gksxkA

(d) EFGH ij ABCD dh rqyuk esa nkc vkèkk gksxkA

13.3 ckW;y dk fu;e ykxw gksrk gS(a) #¼ks"e izØe ij(b) lerkih; izØe ij(c) lenkch izØe ij(d) le vk;rfud izØe ij

13.4 fdlh flfyaMj esa ÅèokZèkj fLFkfr esa vkn'kZ xSl Hkjh gS rFkk bl ij M nzO;ekudk fiLVu yxk gS tks fcuk fdlh ?k"kZ.k osQ Åij&uhps xfr dj ldrk gS(fp=k 13-2) ;fn rki esa o`f¼ djsa rks

(a) xSl osQ p rFkk V nksuksa ifjofrZr gks tk,axsA

(b) pkYlZ osQ fu;e osQ vuqlkj osQoy p esa o`f¼ gksxhA(c) V ifjofrZr gksxk ijarq p ughaA(d) p ifjofrZr gksxk ijarq V ughaA

13.5 fdlh vkn'kZ xSl osQ fy, fn, x, nzO;eku osQ fy,] nkc osQ nks fHkUu ekuksaa osQ fy,]vk;ru ,oa rki osQ chp xzkiQ fp=k 13-3 esa n'kkZ;k x;k gSA P

1 rFkk P

2 osQ chp lacaèk

osQ ckjs esa D;k fu"d"kZ fudkyk tk ldrk gS\

(a) P1 > P

2

(b) P1 = P

2

(c) P1 < P

2

(d) vkadM+s i;kZIr ugha gSaA

fp=k 13.1

fp=k 13.2

fp=k 13.3

10

100 200300400500

20

30

40

V

T (K)

( )l P2

P1

18-04-2018

92


fp=k 13.4

13.6 1 eksy H2 xSl T = 300 K rki ij vk;ru V = 1.00 m3 osQ ckWDl esa Hkjh gSA

(a) vkjaHk osQ nkc osQ cjkcj(b) vkjaHk osQ nkc dk nksxquk(c) vkjaHk osQ nkc dk 10 xquk(d) vkjaHk osQ nkc dk 20 xquk

13.7 V vk;ru osQ fdlh ik=k esa 1 eksy gkbMªkstu rFkk 1 eksy vkWDlhtu dk feJ.k(nksuksa xSlksa dks vkn'kZ xSl ekudj) Hkjk gSA eku yhft, f

1(v)dv gkbbªkstu v.kqvksa

osQ ml va'k dks fufnZ"V djrk gS ftudh pky v rFkk (v + dv) osQ chp gS rFkk ,slkf2 (v)dv vkWDlhtu osQ fy, gSA rc

(a) 1 2( ) ( ) ( )f v f v f v+ = eSDlosy&forj.k fu;e dk ikyu djrk gSA

(b) f1 (v), f

2 (v) i`Fkd :i ls eSDlosy&forj.k fu;e dk ikyu djsaxsA

(c) u rks f1 (v), vkSj u gh f

2 (v) eSDlosy&forj.k fu;e dk ikyu djsaxsA

(d) f2 (v) rFkk f

1 (v) leku gksaxsA

13.8 fdlh iwQys gq, jcM+ osQ xqCckjs esa Hkjh 1 eksy xSl dk nkc p, vk;ru V rFkkrki T gSA ;fn rki c<+dj 1-1 T rFkk vk;ru c<+dj 1.05 V gks tkrk gS rks vafrenkc gksxkµ

(a) 1.1 p

(b) p

(c) p ls de

(d) p ,oa 1.1.p osQ chp


13.9 ABCDEFGH Å"ekjksèkh inkFkZ dk cuk dksbZ [kks[kyk ?ku gS (fp=k 13-4)A blosQiQyd ABCD ij èkukos'k gSA ?ku osQ Hkhrj vk;uhÑr gkbMªkstu Hkjh gSA nkc osQ fy,lkekU; v.kqxfr fl¼kar dk O;atd

(a) oSèk gksxkA(b) oSèk ugha gksxk] D;ksafd vk;ru nhokjksa ls la?kV~V osQ vfrfjDr vU; cyksa dk

vuqHko djssaxsA(c) oSèk ugha gksxk] D;ksafd nhokjksa ls la?kV~V izR;kLFk ugha gksaxsA(d) oSèk ugha gksxk D;ksafd lenSf'kdrk yqIr gks tk,xhA

13.10 gkbMªkstu tSls f}ijek.kqd v.kqvksa esa LFkkukarjh; rFkk ?kw.kZu nksuksa xfr;ksa osQ dkj.k ÅtkZ

gksrh gSaA v.kqxfr fl¼kar osQ lehdj.k 2

3pV E= esa E O;Dr djrk gSµ

18-04-2018

v.kqxfr fl¼kar

93

(a) izfr ,dkad vk;ru oqQy ÅtkZ(b) ÅtkZ dk osQoy LFkkukarjh; Hkkx D;ksafd ?kw.khZ ÅtkZ LFkkukarjh; ÅtkZ dh rqyuk

esa cgqr de gSA(c) ÅtkZ dk osQoy LFkkukarjh; Hkkx D;ksafd nhokj ls la?kV~V osQ le; nkc dk lacaèk

jSf[kd laosx esa varj ls gksrk gSA(d) ÅtkZ dk LFkkukarjh; Hkkx D;ksafd ?kw.khZ ÅtkZ,¡ nksuksa fpÉksa dh gks ldrh gS rFkk

lHkh v.kqvksa osQ fy, bldk vkSlr 'kwU; gSA

13.11 fdlh f}&ijek.kqd v.kq esa fdlh fn, x, rki ij ?kw.khZ ÅtkZ

(a) eSDlosy forj.k osQ vuqlkj gksrh gSA(b) izR;sd v.kq osQ fy, leku gksrh gSA(c) izR;sd v.kq osQ fy, LFkkukarjh; xfrt ÅtkZ osQ cjkcj gksrh gSA(d) izR;sd v.kq osQ fy, LFkkukarjh; xfrt ÅtkZ dh (2/3) gksrh gSA

13.12 uhps fn;k dkSu&lk vkjs[k (fp=k 13-5) vkn'kZ xSl O;ogkj n'kkZrk gSµ

13.13 tc dksbZ xSl #¼ks"er% laihfMr dh tkrh gS] rks blosQ rki esa o`f¼ gksrh gSA v.kqvksaesa igys dh rqyuk esa vkSlru vfèkd xfrt ÅtkZ gksrh gSA xfrt ÅtkZ esa o`f¼ dkdkj.k gSµ

(a)

(c)

fp=k 13.5

(b)

(d)

V

P = const

T

T = const

V

P

V = const

T

P

T

PV

18-04-2018

94


(a) osQoy nhokj osQ xfrd Hkkx ls la?kV~VA(b) leLr nhokj ls la?kV~VA(c) vk;ru osQ Hkhrj v.kqvksa dh viuh xfr Rofjr gksukA(d) v.kqvksa osQ chp ÅtkZ dk iqu% forj.kA


13.14 39.4 g xksYM esa ijek.kqvksa dh la[;k ifjdfyr dhft,A xksYM dk eksyj nzO;eku1978 g mole–1 gSA

13.15 fdlh xSl osQ fy, fn, x, nzO;eku dk 27°C rki rFkk 1 atm nkc ij vk;ru100 cc gSA 327°C ij bldk D;k vk;ru gksxk\

13.16 27º C rki rFkk 1.00 ok;qeaMyh; nkc ij fdlh xSl osQ fn, x, nzO;eku esa v.kqvksadh oxZ ekè; ewy pky 100 m s–1 gSA 20º C rki rFkk 2.00 ok;qeaMyh; nkcij xSls osQ v.kqvksa dh oxZ ekè; ewy pky D;k gksxh\

13.17 fdlh xSl osQ nks v.kqvksa dh pky Øe'k% 6 19 10 m s−× rFkk 6 11 10 m s−× gSA bu

v.kqvksa dh oxZ ekè; ewy pky D;k gS\

13.18 fdlh xSlksa osQ feJ.k esa rki T ij 2-0 eksy vkWDlhtu rFkk 4-0 eksy fu;kWu osQgSA lHkh oaQiu fo/kvksa dh mis{kk djrs gq, fudk; dh oqQy vkarfjd ÅtkZ ifjdfyrdhft, (vkWDlhtu dh nks ?kw.khZ foèkk,¡ gksrh gSa)A

13.19 ,slh nks xSlksa osQ v.kqvksa osQ ekè; eqDr inksa dk vuqikr ifjdfyr dhft, ftuosQ

vkf.od O;kl 1A

rFkk 2 A

gSA xSlksa dks rki] nkc rFkk vk;ru dh loZlevoLFkkvksa esa ekuk tk ldrk gSA


13.20 fp=k 13-6 esa n'kkZ;s x, ik=k esa nks pSacj gSa ftUgsa foHkktd }kjk iFkd fd;k x;k gS rFkkbuosQ vk;ru V

1 = 2.0 yhVj ,oa V

2= 3.0 yhVjA bu pSacjksa esa nkc p

1 = 1.00 atm

,oa p2 = 2.00 atm ij xSlksa osQ Øe'k% µ µ= =1 24.0 5.0rFkk eksy gSaA foHkktd

dks gVkus ij feJ.k osQ lkE; esa vkus osQ i'pkr~ nkc ifjdfyr dhft,A

13.21 dksbZ xSl rhu izdkj A, B ,oa C osQ v.kqvksa ftuosQ nzO;eku A B Cm m m> > gS] ls

feydj cuh gSA bu rhuksa izdkj osQ v.kqvksa dks budh (a) vkSlr xfrt ÅtkZ] K.E.,

(b) rms pky osQ vojksgh Øe esa O;ofLFkr dhft,A

fp=k 13.6

V1

V2

µ1, p

1µ

2 ,

p2

18-04-2018

v.kqxfr fl¼kar

95

13.22 gekjs ikl 3 cm lkbt osQ ?kukdkj pSacj esa NTP ij 0.5 g gkbMªkstu xSl gSA pSacjdh bl xSl dks rki fu;r j[krs gq, vafre nkc 100 atm gksus rd laihfMr fd;kx;kA bl vafre voLFkk esa vkn'kZ xSl fu;e dks ekuuk U;k;laxr gSA (gkbMªkstu osQ

v.kqvksa dks 1 Ao

f=kT;k dk xksyk eku ldrs gSa)A

13.23 tc lkbfdy osQ Vk;j esa iai ls ok;q Hkjrs gSa rks Vk;j esa ok;q dk vk;ru ,oa nkcnksuksa esa o`f¼ gksrh tkrh gSA bl izdj.k esa ck;y&fu;e dk D;k gksrk gS\

13.24 fdlh xqCckj esa 7º C ij 5.0g ghfy;e xSl Hkjh gSA ifjdfyr dhft,µ

(a) xqCckjs esa ghfy;e osQ ijek.kqvksa dh la[;k](b) fudk; dh oqQy vkarfjd ÅtkZA

13.25 NTP ij gkbMªkstu xSl osQ 1 cc esa gkbMªkstu osQ v.kqvksa dh Lokra=; dksfV dh la[;kifjdfyr dhft,A

13.26 dksbZ jksèkh ik=k ftlesa m eksyj nzO;eku dh ,dijek.kqd xSl Hkjh gSA ov osx ls

xfreku gSA ;fn ik=k dks ;dk;d jksd fn;k tk,] rks vki esa ifjorZu Kkr dhft,A


13.27 Li"V dhft, fd

(a) panzek ij ok;qeaMy D;ksa ugha gSA(b) rqaxrk c<+us ij rki D;ksa ?kVrk gS\

13.28 fdlh vkn'kZ xSl ij fopkj dhft, ftlesa pkyksa dk forj.k fuEufyf[kr gSµ

(i) rmsV ifjdfyr djosQ T Kkr dhft, 26( 3.0 10 kg)m −= ×

(ii) ;fn 100 m/s dh pky dk lHkh v.kq fudk; ls iyk;u dj tk, rks fudk;dk u;k rmsV vkSj bl izdkj T ifjdfyr dhft,A

pky (m/s) v.kqvksa dk izfr'kr

200 10

400 20

600 40

800 20

1000 10

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96


13.29 20 × 20 × 1.5 km3 vk;ru osQ ok;q osQ {ks=k esa iw.kZ vaèkdkj esa 10 NksVs ok;q;ku150 km/h dh pky ls mM+ jgs gSaA vki buesa ls fdlh ,d ,sls ok;q;ku esa gSa tks;kn`fPNd bl {ks=k esa mM+ jgk gS vkSj ftlosQ ikl ;g tkudkjh djus dk dksbZ lkèkuugha gS fd vU; ok;q;ku dgk¡ gSA vkSlru fdrus le; i'pkr~ vkiosQ ok;q;ku osQlkFk fudV la?kV~V gksus dh laHkkouk gSA eksVs rkSj ij ifjdyu osQ fy, ;g ekfu,fd ok;q;ku osQ pkjksa vksj dk lqjf{kr {ks=k 10 m f=kT;k dk xksyk gSA

13.30 1.00m3 dk dksbZ ckWDl 300K rki ,oa 1.50 atm nkc ij ukbVªkstu xSl ls HkjkgSA bl ckWDl esa 0-010 mm2 dk dksbZ fNnz gSA ;fn ckgj dk nkc 1 atm gS] rksckWDl osQ Hkhrj 0.10 atm nkc esa deh gksus esa fdruk le; yxsxk\

13.31 eku yhft, ydM+h dk dksbZ xqVdk rki T vkSj nzO;eku ?kuRo p dh fdlh xSl esav

0 osx ls xfreku gSA eku yhft, fd osx x-v{k osQ vuqfn'k gS rFkk v osQ yacor~

xqVosQ dh vuqizLFk dkV dk {ks=kiQy A gSA ;g n'kkZb, fd xqVosQ ij d"kZ.k cy

ρ 0

kT4 Av

mgSA ;gka m xSl osQ v.kqvksa dk nzO;eku gSA

18-04-2018


14.1 ,d d.k dk foLFkkiu fuEufyf[kr lehdj.k }kjk O;Dr gksrk gSµ

y = 3 cos 24

tπ

ω

−

d.k dh xfr%

(a) ljy vkorZ xfr gS ftldk nksyu dky 2π/ω.

(b) ljy vkorZ xfr gS ftldk nksyu dky π/ω gSA

(c) vkorhZ gS ijarq ;g ljy vkorZ xfr ugha gSA

(d) vkorhZ xfr ugha gSA

14.2 ,d d.k dk foLFkkiu O;Dr djus osQ fy, lehdj.k gS% 3siny tω= bl d.k dh

xfr

(a) vkorhZ xfr ugha gSA

vè;k; 14

nksyu

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98


(b) vkorhZ rks gS ysfdu ljy vkorZ xfr ugha gSA(c) ljy vkorZ xfr gS ftldk nksyu dky 2π/ω gSA(d) ljy vkorZ xfr gS ftldk nksyu dky π/ω gSA

14.3 pkj d.kksa osQ Roj.k ,oa foLFkkiu osQ chp lacaèk uhps fn, x, gSa%(a) a

x = + 2x .

(b) ax = + 2x 2.

(c) ax = – 2x2 .

(d) ax = – 2x .

buesa fdl d.k dh xfr ljy vkorZ xfr gSA14.4 fdlh U-vkÑfr dh ufydk esa nzo&LraHk dh nksyu xfrµ

(a) vkorhZ xfr gS ijarq ljy vkorZ xfr ugha gSA(b) vukoRkhZ xfr gSA(c) ljy xfr gksrh gS ftldk vkorZ&dky nzo osQ ?kuRo ij fuHkZj ugha djrkA(d) ljy vkorZ xfr gksrh gS ftldk vkorZ&dky nzo osQ ?kuRo osQ vuqØekuqikrh

gksrk gSA14.5 ,d d.k ij ,d lkFk nks ijLij yacor~ ljy vkorZ xfr;k¡ x = a cos ωt ,oa

y = a sin ωt vkjksfir gSaA bl d.k dh xfr dk iFk(a) ,d nh?kZo`Ùk gksxkA(b) ,d ijoy; gksxkA(c) ,d o`Rk gksxkA(d) ,d ljy js[kk gksxhA

14.6 ,d d.k dk le; osQ lkFk foLFkkiu&ifjorZu fuEu lacaèk osQ }kjk O;Dr gksrk gSµy = a sin ωt + b cos ωt.

(a) xfr nksyuh gS ijarq ljy vkorZ xfr ugha gSA(b) xfr (a + b) vk;ke dh ljy vkorZ xfr gSA(c) xfr (a2 + b2 ) vk;ke dh ljy vkorZ xfr gSA

(d) xfr 2 2a b+ vk;ke dh ljy vkorZ xfr gSA

fp=k 14.1

14.7 pkj yksyd A, B C ,oa D ,d gh izR;kLFk vkèkkj lsfp=k 14-1 osQ vuqlkj yVdk;s x;s gSaA A ,oa C dh yackbZcjkcj gS] B dh yackbZ A ls de gS tcfd D dh yackbZ Als vfèkd gSA ;fn A dks ,d vuqizLFk foLFkkiu fn;k tk;s rks(a) D vfèkdre vk;ke osQ nksyu djsxkA(b) C vfèkdre vk;ke osQ nksyu djsxkA(c) B vfèkdre vk;ke osQ nksyu djsxkA(d) lHkh (pkjksa) yksyd leku vk;ke osQ nksyu djsaxsA

18-04-2018

nksyu

99

14.8 fp=k 14-2 esa ,d d.k dh oÙkh; xfr n'kkZbZ xbZ gSA oÙk dh f=kT;k] d.k dk ifjØe.kdky] ifjØe.k dh fn'kk ,oa izkjafaHkd fLFkfr vkÑfr ij vafdr gSaA ifjØe.k djrsd.k P osQ f=kT;k lfn'k osQ x-v{k ij iz{ksi.k dh ljy vkorZ xfr dks O;Dr djldrs gSaA

(a) x (t) = B sin 2

30

tπ

(b) x (t) = B cos15

tπ

(c) x (t) = B sin 15 2

tπ π +

(d) x (t) = B cos 15 2

tπ π +

14.9 ,d d.k dh xfr dk lehdj.k x = a cos (α t )2 gSAbldh xfrµ(a) vkoRkhZ gS ijarq nksyuh ugha gSA(b) vkoRkhZ Hkh gS vkSj nksyuh HkhA(c) nksyuh gS ijarq vkoRkhZ gS u gh nksyuhA(d) u rks vkoRkhZ gS u gh nksyuhA

14.10 ljy vkorZ xfr djrs gq, d.k dh vfèkdre pky 30 cm s–1 rFkk vfèkdreRoj.k 60 cm/s2 gSA bldk vkorZ&dky gSµ

(a) π s (b)2

π s

(c) 2π s (d)t

π s

14.11 ,d nzO;eku m dks tc nks fLizaxksa S1 ,oa S

2 ls i`Fkd&i`Fkd tksM+dj nksyu djk;k

tkrk gS rks nksyu vko`fr ν1 ,oa ν

2. ikbZ tkrh gSA ;fn ml nzO;eku dks mu fLizaxksa

osQ lkFk fp=k 14.3 esa fn[kk;s x;s vuqlkj tksM+dj nksyu djk;k tk, rks nksyuvko`fRk gksxhµ(a) ν

1 + ν

2

(b)2 2

1 2ν ν+

(c)

1

1 2

1 1

ν ν

−

+

(d)2 2

1 2ν ν−

fp=k 14.2

y

P t( = 0)

T = 30s

xB

o

fp=k 14.3

18-04-2018

100

iz'u izn£'kdkµHkkSfrdh��

0T/4

2T/4

3T/4 T 5T/4 �� (s)


14.12 i`Foh dh blosQ v{k osQ ifjr% xfr

(a) vkoRkhZ xfr gksrh gSA(b) ljy vkorZ xfr gksrh gSA(c) vkoRkhZ gksrh gS ijarq ljy vkorZ xfr ugha gksrh gSA(d) vukoRkhZ xfr gksrh gSA

14.13 fdlh ?k"kZ.k jfgr ofØr I;kyh osQ vanj tc fdlh ckWyfo;fjax dks blosQ fuEure¯cnq osQ tjk Åij ls NksM+k tkrk gS rks bldh xfrµ

(a) ljy vkorZ xfr gksrh gSA(b) vukoRkhZ xfr gksrh gSA(c) vkoRkhZ xfr gksrh gSA(d) vkorhZ rks gksrh gS ijarq ljy vkorZ xfr ugha gksrhA

14.14 ljy vkorZ xfr djrs gq, ,d d.k dk foLFkkiu le; oØ fp=k 14-4 esa n'kkZ;kx;k gSA lgh dFku dk p;u dhft,A

fp=k 14.4

fp=k 14.5

(a) t = 0 s ,oa t = 2 s ij nksyu leku dyk esa gSA(b) t = 2 s ,oa t = 6 s ij nksyd leku dyk esa gSA(c) t = 1 s ,oa t = 7 s. ij nksyd leku dyk esa gSA(d) t = 1 s ,d t = 5 s ij nksyd leku dyk esa gSA

14.15 ljy vkoRkhZ nksyd osQ fy, fuEufyf[kr esa dkSu lk@dkSu ls dFku lR; gSaµ

(a) cy ekè; fLFkfr ls foLFkkiu osQ vuqØekuqikrh gksrk gS vkSj blosQ foijhr fn'kkesa izHkkoh gksrk gSA

(b) xfr vkoRkhZ gksrh gSA(c) nksyd dk Roj.k vpj jgrk gSA(d) osx vkoRkhZ gksrk gSA

14.16 ljy vkorZ xfr djrs gq, d.k dk foLFkkiu leqanz xzkiQ fp=k 14-5 esa n'kkZ;k x;kgSA buesa ls dkSu&lk@ ls dFku lR; gaS\

(a)3

4

Tt = ij cy 'kwU; gksrk gSA

(b)4

4

Tt = ij Roj.k vfèkdre gksrk gSA

18-04-2018

nksyu

101

(c)4

Tt = ij osx vfèkdre gksrk gSA

(d)2

Tt = ij nksyd fd fLFkfrt ÅtkZ bldh xfrt ÅtkZ osQ cjkcj gksrh gSA

14.17 dksbZ fiaM ljy vkorZ xfr dj jgk gS rks

(a) blosQ izR;sd pØ dh vkSlr laiw.kZ ÅtkZ] vfèkdre xfr ÅtkZ osQ cjkcj gksrh gSA(b) blosQ izR;sd pØ dh vkSlr laiw.kZ ÅtkZ] vfèkdre xfrt ÅtkZ dh vkèkh gksrh gSA

(c) blosQ izR;sd pØ esa ekè; osx] vfèkdre osx dk 2

π xquk gksrk gSA

(d) bldk oxZ ekè; ewy osx blosQ vfèkdre osx dk 1

2 xquk gksrk gSA

14.18 ,d d.k nks ¯cnqvksa A ,oa B osQ chp ljy vkorZ xfr dj jgk gS tks ,d nwljs ls10 cm dh nwjh ij gSaA (fp=k 14-6) A ls B dh vksj èkukRed fn'kk ysa rks lghdFku@dFkuksa dk p;u dhft,µ

(a) tc d.k A ls 3 cm dh nwjh ij gS vkSj B dh vksj tk jgk gS rks blosQ osx]Roj.k vkSj bl ij yxus okys cy osQ fpÉ èkukRed gSaA

(b) C ij O dh vksj tkrs gq, d.k osQ osx dk fpÉ ½.kkRed gSA(c) B ls A dh vksj tkrs gq, vkSj B ls 4 cm nwjh ij d.k osQ osx] Roj.k

,oa cy osQ fpÉ ½.kkRed gSaA(d) tc d.k ¯cnq B ij gS rks blosQ Roj.k vkSj bl ij yxus okys cy

osQ fpÉ ½.kkRed gSaA


14.19 ljy vkorZ xfr djrs gq, ,d d.k dk foLFkkiu le; xzkiQfp=k 14-7 esa n'kkZ;k x;k gSA vkÑfr ij vafdr og ¯cnq igpkfu,ftu ij (i) nksyd dk osx 'kwU; gSA (ii) nksyd dh pky vfèkdregSA

14.20 nks loZle fLiazx ftuesa ls izR;sd dk fLizax fu;rkad K gSAfp=k 14-8 esa n'kkZ, vuqlkj ,d nzO;eku m vkSj nks n`<+ fLFkr vkèkkjksaosQ lkFk tksM+ fn, x, gSaA tc nzO;eku dks bldh ekè; fLFkfr lsnkfguh vksj x-nwjh foLFkkfir fd;k tkrk gS rks bl ij yxus okysizR;ku;u cy dk eku Kkr dhft,A

fp=k 14.6

fp=k 14.8

fp=k 14.7

18-04-2018

102


14.21 ljy vkorZ xfr osQ nks vkèkkjHkwr vfHky{k.k D;k gSa\

14.22 ,d ljy yksyd dh xfr ljy vkorZ dc gksrh gS\

14.23 ljy vkorZ nksyd osQ vfèkdre Roj.k vkSj vfèkdre osx dk vuqikr D;k gS\

14.24 fdlh nksyd }kjk ,d nksyd dky esa pfyr nwjh vkSj blosQ vk;ke dk vuqikrfdruk gksrk gS\

14.25 fp=k 14.9 esa ¯cnq P′ osQ osx dk tks fd R f=kT;k osQ o`Ùk esa nf{k.kkorZ fn'kk esaxfreku lanHkZ d.k P osQ osx dk x-v{k ij iz{ksi.k gS] fpÉ D;k gksxk\

14.26 n'kkZb, fd ljy vkorZ xfr djrs gq, d.k osQ osx esa π/2 dk dyk varj gksrk gSA

14.27 ,d ljy vkorZ nksyd dh fLFkfrt ÅtkZ] xfrt ÅtkZ ,oa oqQy ÅtkZ esa foLFkkiuosQ lkFk gksus okys ifjorZu n'kkZus osQ fy, xzkiQ [khafp,A

14.28 ,d lsoaQM yksyd dh yackbZ i`Foh ij 1 m gSA panzek ij lsoaQM&yksyd dh yackbZfdruh gksxh\


14.29 fp=k 14-10 esa n'kk, x, ra=k osQ fy,] nzO;eku M dks ekè; fLFkfr ls foLFkkfir

fp=k 14.10

M

��

��

djosQ NksM+us ij blosQ nksyu&dky dk O;atd izkIr dhft,A

14.30 n'kkZb, fd fdlh d.k dh y = sinω t – cos ω t }kjk fu:fir xfr] 2π/ω vkorZdky dh ljy vkorZ xfr gksrh gSA

14.31 ,d ljy vkorZ nksyd osQ foLFkkiu dk og eku Kkr dhft, ftl ij bldhfLFkfrt ÅtkZ nksyd dh vfèkdre ÅtkZ dh vkèkh gksrh gSA

y

A�

P

xAP

1o

��t+

fp=k 14.9

18-04-2018

nksyu

103

14.32 m nzO;eku dk ,d fiaM U(x) = U0 (1-cos αx ) foHko {ks=k esa fLFkr gS] tgk¡ U

0

,oa α fLFkjkad gSaA blosQ vYi vk;keh nksyuksa dk nksyudky Kkr dhft,A

14.33 2 kg nzO;eku dk ,d CykWd 50 Nm–1 fLiazx fu;rkad osQ fLizax ls tqM+k gSA CykWddks bldh lkE; fLFkfr (x = 0) ls ,d {kSfrt ?k"kZ.k jfgr lrg ij fojke ls(t = 0) [khapk tkrk gS] blosQ rR{kf.kd foLFkkiu osQ fy, O;atd fyf[k,A

14.34 nks loZle yksydksa ij fopkj dhft, tks ,d nwljs ls Lora=k] leku vk;ke osQ nksyubl izdkj dj jgs gSa fd tc ,d yksyd ÅèokZèkj ls nkfguh vksj 2º dk dks.k cukrsgq, viuh vfèkdre foLFkkiu dh fLFkfr esa gS rks nwljk yksyd ÅèokZèkj fLFkfr lsckb± vksj 1º dk dks.k cukrk gSA bu yksydksa dk dyk&varj D;k gS\


14.35 50 kg Hkkj dk ,d O;fDr ,d ,sls nzO;eku jfgr IysViQkeZ ij [kM+k gS tksÅij&uhps 2.0 s–1 vko`fr rFkk 5.0 cm vk;ke osQ vkorhZ nksyu dj jgk gSAIysViQkeZ ij j[kh ,d Hkkjekid e'khu ml O;fDr dk le; osQ lkFk Hkkjcukrh gSA

(a) D;k nksyu osQ nkSjku O;fDr osQ Hkkj esa dksbZ ifjorZu gksxk\(b) ;fn Hkkx (a) dk mÙkj gk¡ gS rks e'khu esa mlosQ Hkkj osQ vfèkdre vkSj U;wure

Hkkj D;k gksaxs vkSj ;s eku fdu fLFkfr;ksa ij gksaxs\

14.36 m nzO;eku dk ,d fiaM ,d nzO;eku jfgr fLizax osQ ,d fljs ls tqM+k gS tks Lo;a,d fu;r cnq ls ÅèokZèkjr% yVdk gqvk gSA nzO;eku dks gkFk esa idM+k gqvk gS rkfdfLiaazx esa u rks fojyu gks u gh laihMuA vpkud gkFk dk vkèkkj gVk fy;k tkrk gSAnzO;eku ml fLFkfr ls ftl ij og gkFk }kjk jksdk x;k Fkk vfèkdre 4cm uhpsrd tkrk gSA

(a) nksyu dk vk;ke fdruk gS\(b) nksyuksa dh vko`fr Kkr dhft,A

14.37 ,d ydM+h dk yêòk ftldh Å¡pkbZ h rFkk vuqizLFk dkV dk {ks=kiQy A gS ÅèokZjr%ikuh esa rSj jgk gSA bldks nck dj NksM+ fn;k x;kA yêòk ljy vkorZ xfr djsxk]ftldk nksyudky gksxkA

2m

TA g

πρ

=

tgk¡ m yêòs dk nzO;eku rFkk ρ ikuh dk ?kuRo gSA

18-04-2018

104


14.38 ,d V-vkÑfr dh ufydk esa oqQN ikjk Hkjk gSA blosQ ,d fljs dks ,d pwld iails tksM+ fn;k x;k gS vkSj nwljk fljk ok;qeaMy osQ laioZQ esa gSA bldh nksuksa Hkqtkvksaesa izR;sd {kSfrt ls 45º osQ dks.k ij >qdh gSaA nksuksa Hkqtkvksa esa ekewyh nkc&varjmRiUu gksrk gS tc pwld iai dks gVk fy;k tkrk gSA D;k V-ufydk esa ijk ljyvkorZ xfr djsxk\ ;fn gk¡ rks nksyuksa dk nksyu dky Kkr dhft,A dksf'kdkRo ,oa';ku&cy dh vis{kk dj ldrs gSaA

14.39 i`Foh osQ osaQnz esa ls gksrs gq, ,d lqjax cukbZ xbZ gSA n'kkZb, fd bl lqjax osQ ,d fljsij ;fn m nzO;eku dk fiaM fojkekoLFkk ls fxjk;k tk, rks ;g ljy vkorZ xfrdjsxkA

14.40 l yackbZ vkSj 1s nksyu dky dk ,d ljy yksyd ,d lqn`<+ vkèkkj O ls bl izdkjyVdk;k x;k gS fd bldk xksyd Hkw&i`"B osQ cnq A ls ÅèoZèkjr% H Å¡pkbZ ij jgs(fp=k 14-11) nksyuksa dk vk;ke oθ gSA ;fn yksyd dh Mksjh θ = θ

0 /2 ij VwV tk,

rks xksyd dks Hkw&i`"B ls Vdjkus esa yxk le; Kkr dhft,A A ls ml cnq dh nwjhHkh Kkr dhft, tgk¡ xksyd Hkw&i`"B ls Vdjkrk gSA θ

0 dks vR;ar NksVk eku yhft,

rkfd θ θ θ 0 0 0sin cos 1rFkk fy;k tk losQA

fp=k 14-11

Fig. 14.11

��

P

A

H

OP=l

18-04-2018


15.1 ty esa pyrh eksVj cksV }kjk mRiUu ty rjaxsaµ

(a) u rks vuqnS?;Z gksrh gSa u gh vuqizLFkA(b) vuqnS?;Z vkSj vuqizLFk nksuksa gksrh gaSA(c) osQoy vuqnS?;Z gksrh gSaA(d) osQoy vuqizLFk gksrh gSaA

15.2 fdlh ekè;e esa v m/s dh pky ls pyrh gqbZ] λ rjaxnS?;Z dh èofu rjaxsa fdlhvU; ekè;e esa izos'k djrh gSa ftlesa budh pky 2v m/s gksrh gSA nwljs ekè;e esaèofu rjaxksa dk rjaxnS?;Z gksxkA

(a) λ

(b)2

λ

(c) 2λ

(d) 4λ .

vè;k; 15

rjaxsa

18-04-2018

106


15.3 ok;q esa èofu dh rjaxksa dh pky(a) rki ij fuHkZj ugha djrhA(b) nkc osQ lkFk c<+rh gSA(c) vknzZrk c<+us ls c<+rh gSA(d) vknzZrk c<+us ls ?kVrh gSA

15.4 ekè;e osQ rki ifjorZu ls(a) èofu rjaxksa dh vko`fr ifjofrZr gks tkrh gSA(b) èofu rjaxksa dk vk;ke cny tkrk gSA(c) èofu rjaxksa dk rjaxnS?;Z cny tkrk gSA(d) èofu rjaxksa dh izcyrk cny tkrh gSA

15.5 fdlh ekè;e esa vuqnS?;Z rjaxksa osQ izxeu ls tks jkf'k lapfjr gksrh gS og gSµ(a) nzO;(b) ÅtkZ(c) ÅtkZ ,oa nzO;(d) ÅtkZ] nzO; ,oa laosx

15.6 rjax xfr osQ lacaèk esa fuEufyf[k esa ls dkSu&lk dFku lR; gS\(a) ;kaf=kd vuqizLFk rjaxsa lHkh ekè;eksa esa xeu dj ldrh gSaA(b) vuqnS?;Z rjaxsa osQoy Bkslksa esa xeu dj ldrh gSaA(c) ;kaf=kd vuqizLFk rjaxsa osQoy Bkslksa esa xeu dj ldrh gSaA(d) vuqnS?;Z rjaxsa fuokZr esa xeu dj ldrh gSaA

15.7 ,d èofu rjax fdlh ok;qLraHk esa laihMuksa vkSj fojyuksa osQ :i esa xqtj jgh gSaAØfed laihMuksa vkSj fojyuksa esa(a) ?kuRo vpj jgrk gSA(b) ckW;y osQ fu;e dk ikyu gksrk gSA(c) ok;q dk vk;ru izR;kLFkrk xq.kkad nksyu djrk gSA(d) Å"ek dk LFkkukarj.k ugha gksrkA

15.8 ,d lery izxkeh rjax dk lehdj.k 0.6sin22

xy tπ

= − gSA ,d l?ku

ekè;e ls ijkorZu gksus ij bldk vk;ke vkifrr rjax osQ vk;ke dk 2/3 gks tkrkgSA ijkofrZr rjax dk lehdj.k gSµ

(a) 0.6sin22

xy tπ

= +

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(b) 0.4sin22

xy tπ

= − +

(c) 0.4sin22

xy tπ

= +

(d) 0.4sin22

xy tπ

= − − .

15.9 2.5 kg nzO;eku dh ,d Mksjh esa 200N dk ruko gSA rkfur Mksjh dh yackbZ20.0m gSA ;fn Mksjh osQ ,d fljs ij ,d vuqizLFk Lian mRiUu fd;k tk, rksfo{kksHk blosQ nwljs fljs ij igq¡psxkµ(a) ,d lsoaQM esa(b) 0.5 lsoaQM esa(c) 2 lsoaQM esa(d) fn, x, vkadM+s vi;kZIr gSaA

15.10 fu;r vko`fÙk dh lhVh ctkrh gqbZ ,d jsyxkM+h vpj osx v ls LVs'ku dh vksj tkjgh gSA jsyxkM+h LVs'ku ij ,d fLFkj izs{kd osQ ikl ls xqtjrh gSA izs{kd }kjk lquhxbZ èofu dh vko`fr n′ dk le; t osQ iQyu osQ :i esa xzkiQ cuk;k x;k(fp=k 15-1)A visf{kr oØ dks igpkfu,A

(a)

(c) (d)

fp=k 15.1

(b)

n

t

n

t

n

t

n

t

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15.11 fdlh Mksjh ij xfreku vkorhZ rjax dksy (x,t) = 3.0 sin (36t + 0.018x + π/4) }kjk O;Dr fd;k x;k gSAtgk¡ x ,oa y cm esa gSa rFkk t lsoaQM esa gSA x dh èkukRed fn'kk ck,¡ ls nkfguh vksj gSA(a) rjax nkfguh ls ckb± vksj py jgh gSA(b) rjax dh pky 20m/s gSA(c) rjax dh vko`fr 5.7 gVZ~t gSA(d) rjax osQ nks Øekxr Ükax osQ chp dh U;wure nwjh 2.5 cm gSA

15.12 fdlh Mksjh ij rjax dk foLFkkiu gSµy (x,t) = 0.06 sin (2πx/3) cos (120πt)

tgk¡ x ,oa y ehVj esa gS rFkk t lsoaQM esa gSA Mksjh dh yackbZ 1.5m gS vkSj bldk

nzO;eku 23.0 10 kg−× gSA

(a) ;g 60Hz gVZ~t vko`fr dh izxkeh rjax dks fu:fir djrk gSA(b) ;g 60Hz gV~Zt vko`fr dh vizxkeh rjax dks fu:fir djrk gSA(c) ;g 3 m rjaxnS?;Z] 60Hz gV~Zt vko`fr dh nks rjaxksa osQ vè;kjksi.k dk ifj.kke

gS ftuesa ls izR;sd 180 ms–1 dh pky ls foijhr fn'kk esa xfreku FkhA(d) bl rjax dk vk;ke vpj gSA

15.13 rjyksa esa èofu dh pky](a) ekè;e osQ ?kuRo osQ vuqØekuqikrh gksrh gSA(b) ekè;e osQ vk;ru izR;kLFkrk xq.kkad osQ oxZ ij fuHkZj djrk gSA(c) ?kuRo osQ oxZewy osQ O;qRØekuqikrh gksrh gSA(d) ekè;e osQ vk;ru izR;kLFkrk xq.kkad osQ oxZ ewy osQ vuqØekuqikrh gksrh gSA

15.14 lery izxkeh ;kaf=kd rjax osQ xeu osQ nkSjku(a) lHkh d.k ,d gh dyk esa oaQiu djrs gSaA(b) lHkh d.kksa osQ vk;ke cjkcj gksrs gSaA(c) ekè;e osQ d.k ljy vkorZ xfr djrs gSaA(d) rjax osx ekè;e dh izÑfr ij fuHkZj djrk gSA

15.15 nksuksa fljksa ij n`<+rk ls caèkh Mksjh dk vuqizLFk&foLFkkiu fuEufyf[kr lehdj.k }kjkfu:fir gksrk gSAy(x, t) = 0.06 sin (2πx/3) cos (120πt).

nks Øekxr fuLianksa osQ chp Mksjh osQ lHkh ¯cnqvksa osQ oaQiu dh(a) vko`fr leku gksrh gSA(b) dyk leku gksrh gSA(c) ÅtkZ leku gksrh gSA(d) vk;ke leku gksrs gSaA

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15.16 fdlh LVs'ku osQ ;kMZ esa [kM+h ,d jsyxkM+h lhVh ctkrh gS ftldh 'kkar gok esavko`fr 400 gV~Zt gSA ;kMZ ls LVs'ku dh vksj 10 ms–1 dh pky ls iou izokfgrgksus yxrh gSA ;fn ;g fn;k gks fd 'kkar ok;q esa iou dk osx 340 ms–1 gS] rks

(a) IysViQkeZ ij [kM+s fdlh izs{kd }kjk lquh xbZ èofu dh vko`fr 400 gVZ~t gksxhA

(b) IysViQkeZ ij [kM+s issz{kd osQ fy, èofu dk osx 350 m/s gSA

(c) IysViQkeZ ij [kM+s izs{kd osQ fy, èofu dh vko`fr c<+ tk;sxhA

(d) IysViQkeZ ij [kM+s izs{kd osQ fy, èofu dh vko`fr ?kV tk;sxhA

15.17 fdlh vizxkeh rjax osQ fy, fuEufyf[kr esa dkSu ls dFku lR; gaS\

(a) izR;sd d.k dk ,d fu;r vk;ke gksrk gS tks blosQ fudVre iM+kSlh d.k lsfHkUu gksrk gSA

(b) lHkh d.k viuh ekè; fLFkfr dks ,d lkFk ij djrs gSaA

(c) lHkh d.k leku vk;ke ls nksyu djrs gSaA

(d) fdlh Hkh ry osQ vkj ij ÅtkZ dk usV LFkkukarj.k ugha gksrkA

(e) oqQN d.k lnSo fLFkj jgrs gSaA


15.18 fdlh lksuksehVj dk rkj ,d Lofj=k f}Hkqt osQ lkFk vuquknh oaQiu dj jgk gSAlksuksehVj osQ rkj ij yxk, x, ruko dks vifjofrZr j[krs gq, rkj dh yackbZ dksnksxquk dj fn;k tkrk gSA fdu n'kkvksa esa Lofj=k f}Hkqt vHkh Hkh rkj osQ lkFk vuqukndjsxk\

15.19 nksuksa fljksa ij [kqyk L yackbZ dk ,d vkWxZu ikbi 480 gV~Zt vko`fr osQ Lofj=k f}Hkqtls Lofur djus ij izFke gkeksZfud esa oaQiu djrk gqvk ik;k tkrk gSA ,d fljs ijcan vkWxZu ikbi dh yackbZ fdruh gksuh pkfg, rkfd ;g Hkh ml gh Lofj=k f}HkqtosQ lkFk izFke gkeksZfud esa oaQiu djsA

15.20 ,d Lofj=k f}Hkqt A ij 512 Hz vafdr gSA bls tc ,d vU; Lofj=k f}Hkqt B,

ftl ij vko`fr vafdr ugha gS] osQ lkFk èofur djk;k tkrk gS rks izfr lsoaQM 5foLian mRiÂ gksrs gSaA ;fn B dks tjk&lk ekse yxkdj Hkkjh dj fn;k tk, rc Hkh5 foLian izfr lsoaQM gh mRiÂ gksrs gSaA tc Lofj=k f}Hkqt B Hkkfjr ugha gksrk rks bldhvko`fr fdruh gksrh gSA

15.21 ,d izR;kLFk rjax dk foLFkkiu fuEufyf[kr iQyu }kjk O;Dr gksrk gSµy = 3 sin ωt + 4 cos ωt.

ifj.kkeh vk;ke dh x.kuk dhft,A

15.22 flrkj osQ ,d rkj osQ LFkku ij leku yackbZ vkSj inkFkZ dk ,d nwljk rkj yxk;ktkrk gS ftldh f=kT;k igys rkj dh f=kT;k dh rhu xquh gSA ;fn rkj esa ruko Hkhigys rkj ftruk gh gks rks u, rkj dh vko`fr fdrus xquh gks tk,xh\

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15.23 fdl rki (oC esa) ij ok;q esa èofu dh pky OoC ij bldh pky dh rhu xquh gkstk,xh\

15.24 yxHkx cjkcj vko`fr;ksa n 1 ,oa n

2 dh nks rjaxsa fdlh ¯cnq ij ,d lkFk igq¡prh gSa

Øfed mfPp"Bksa osQ chp fdruk le; varjky gSA


15.25 LVhy osQ ,d rkj dh yackbZ 12 m rFkk nzO;eku 2.10 kg gSA ;fn bl rkj esa2.06 × 104N dk ruko yxk;k tk, rks bl rkj ij dksbZ vuqizLFk rjax fdl pkyls pysxh\

15.26 20 lseh- yackbZ dk ,d ikbi ,d fljs ij can gSA 1237.5 gV~Zt vko`fr osQ Ïksrdk blosQ fdl gkeksZfud ls vuqukn gksxk\ (ok;q esa èofu dk osx = 330 m s–1)

15.27 fdlh jsyos LVs'ku osQ ckgjh flXuy ij [kM+h gqbZ ,d jsyxkM+h fLFkj ok;q esa 400

gV~Zt vko`fr dh lhVh ctkrh gSA jsyxkM+h IysViQkeZ dh vksj 10 m s–1 dh pky lspyuk 'kq: djrh gSA IysViQkeZ ij [kM+s fdlh izs{kd dks fdruh vko`fr dh èofulqukbZ nsxhA (èofu dk osx = 330 m s–1)

15.28 fp=k 15-2 esa fdlh rkfur Mksjh ij rjaxksa dk iSVuZ n'kkZ;k x;k gSA le>kb, fd ;gfdl izdkj dh rjax gS vkSj bldk rjaxnS?;Z Kkr dhft,A

foLFkkiu

fp=k 15.2

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15.29 fp=k 15-3 esa nks fHkUu {k.kksa ij ,d rkfur Mksjhij vizxkeh rjaxksa osQ iSVuZ n'kkZ, x, gSaA vè;kjksiudj vixzkeh rjax cukus okyh nks rjaxksa osQ osx360 ms–1 vko`fr;k¡ 256 gV~Zt gSA

(a) og {k.k Kkr dhft, ftl ij nwljk iSVuZizkIr gksrk gSA

(b) oØ ij fuLian ,oa izLian vafdr dhft,A(c) A′ ,oa C′ osQ chp dh nwjh Kkr dhft,A

15.30 512Hz vko`fr osQ oaQiu djrk gqvk ,d Lofj=k f}Hkqt ikuh ls Hkjhuyh osQ [kqys fljs osQ ikl yk;k tkrk gS (fp=k 15-4)A uyh esa ikuh dkry èkhjs&èkhjs de fd;k tkrk gSA tc ikuh dk ry [kqys fljs ls 17cm

uhps gksrk gS rks vfèkdre rhozrk dh èofu lqukbZ nsrh gSA ;fn dejs dkrki 20°C gS rks x.kuk dhft,µ

(a) dejs osQ rki ij èofu dh pky dhA(b) 0°C ij ok;q esa èofu dh pky dhA(c) ;fn uyh esa ty osQ LFkku ij ikjk ys fy;k tk, rks D;k izs{k.kksa esa

dksbZ varj tk,xk\

15.31 n'kkZb, fd tc nksuksa fljksa ij fLFkj dh xbZ dksbZ Mksjh] ,d ywi] nks ywi] rhu ywi]pkj ywi cukrs gq, oaQiu djrh gS rks bldh vkofr;k¡ 1:2:3:4 osQ vuqikr esa gksrh gSaA


15.32 i`Foh dh f=kT;k 6400 km gSA 1000 km f=kT;k vkarfjd ØksM Bksl gSA blosQckgj 100 km rd fi?kyh gqbZ voLFkk esa gS vkSj blosQ Hkh ckgj 3500 km ls64000 km rd i`Foh Bksl gSA nzo esa osQoy vuqnS?;Z (P) rjaxsa gh xeu dj ldrhgSaA eku yhft, fd P rjaxksa dh pky iFoh dh lrg osQ fudV fdlh LFkku rd vkrhgSA ml le; dh x.kuk dhft, ftlosQ ckn i`Foh osQ O;kl osQ vuqfn'k pyrh gqbZrjax i`Foh osQ O;klr% foijhr fcanq ij j[ks lhLeksxzkiQ }kjk fjdkMZ dh tk,xhA

15.33 ;fn xSl osQ v.kqvksa dh oxZ ekè; ewy pky c gS vkSj bl xSl esa èofu rjaxksa dh pkyv gS rks n'kkZb, fd c/v ,d fu;rkad gS vkSj bldk lHkh f}ijek.kqd xSlksa osQ fy,eku rki ij fuHkZj ugha djrkA

15.34 uhps ,d izR;kLFk rjax osQ foLFkkiu dks O;Dr djus osQ fy, x vkSj t osQ oqQN iQyufn, x, gSaA

(a) y = 5 cos (4x ) sin (20t)

fp=k 15.4

��

fp=k 15.3

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(b) y = 4 sin (5x-t/2) + 3 cos (5x-t/2)

(c) y = 10 cos [(252–250) πt ] cos [(252+250)πt ]

(d) y = 100 cos (100πt + 0.5x )

crkb, fd buesa ls fdlosQ }kjk fu:fir gSµ(a) -x fn'kk esa xeu djrh gqbZ izxkeh rjaxA(b) ,d vixzkeh rjaxA(c) foLian(d) +x fn'kk essa xeu djrh gqbZ izxkeh rjaxAvius mÙkj osQ leFkZu esa roZQ Hkh nhft,A

15.35 nh xbZ izxkeh rjax

y = 5 sin (100pt-0.4px)

tgk¡ y ,oa x ehVj esa gSa rFkk t lsoaQM esa gS] Kkr dhft,

(a) vk;ke(b) rjaxnS?;Z(c) vkofr(d) rjax osx(e) d.k osQ osx dk vk;ke

15.36 izxkeh vko`fr rjax y = 2 cos 2π (10t–0.0080x + 3.5) osQ fy,] tgk¡ x ,oay cm esa rFkk t lsoaQM esa gS] nksyu djus okys ,sls nks ¯cnqvksa osQ chp dyk&varjKkr dhft, ftuosQ chp dh nwjh

(a) 4 m gSA(b) 0.5m gSA

(c)2

λgSA

(d)3

4

λ(fdlh {k.k fo'ks"k ij gS)

(e) x = 100cm ij fLFkr d.k osQ nksyuksa esa t = T s ,oa t = 5 ij fdrukdyk&varj gS\

18-04-2018

mÙkj

vè;k; 2

2.1 (b)

2.2 (b)

2.3 (c)

2.4 (d)

2.5 (a)

2.6 (c)

2.7 (a)

2.8 (d)

2.9 (a)

2.10 (a)

2.11 (c)

2.12 (d)

2.13 (b), (c)

2.14 (a), (e)

2.15 (b), (d)

2.16 (a), (b), (d)

2.17 (a), (b)

2.18 (b), (d)

2.19 D;ksafd] fiaMksa osQ ,d gh jkf'k ls lacafèkr vkeki osQ ifj.kkeksa dh dksfV esa egRoiw.kZvarj gksrk gSA mnkgj.kkFkZ] varjkijek.kqd nwfj;k¡ ,axLVªkWe dh dksfV dh gksrh gSaAvarj&uxjh; nwfj;k¡ km dh dksfV dh gksrh gSa rFkk varj&rkjd nwfj;k¡ izdk'ko"kZ dhdksfV dh gksrh gSaA

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2.20 1015

2.21 nzO;eku LisDVªksxzkiQ

2.22 1 u = 1.67 × 10–27 kg

2.23 D;ksafd ( )f θ dks.k θ dh fofHkUu ?kkrksa dk ;ksx gSA ;g vfuok;Zr% foekfoghu gksxkA

2.24 D;ksafd ;kaf=kdh dh vU; lHkh jkf'k;k¡ yackbZ] nzO;eku ,oa le; osQ inksa esa buosQlkFk ljy lacaèkksa osQ :i esa O;Dr dh tk ldrh gSaA

2.25 θ = = �1

( ) 160 60

g o

E

Ra

Rjfs M;u

∴ panzek ls ns[kus ij i`Foh dk O;kl yxHkx 2º gSA

(b) i`Foh&panzek osQ chp dh nwjh ij panzek dk O;kl (1/2)º vkSj i`Foh dk O;kl2º fn[kkbZ iM+rk gSA vr% i`Foh dk O;kl panzek osQ O;kl dk 4 xquk gSA

= 4D

D

i`Foh

panez k

(c) = 400r

r

lw; Z

panez k

(;gk¡ r nwjh rFkk D O;kl fu#fir djrk gS)

i`Foh ls ns[kus ij panzek vkSj lw;Z nksuksa osQ dks.kh; O;kl cjkcj fn[kkbZ iM+rs gSaA

∴ =D D

r r

lw; Z

l;w Z

panez k

pna ez k

∴ = 400D

D

l;w Z

pna ez k

ijarq = ∴ =4 100DD

D D

l;w ZiFoh

pna ez k i`Foh

2.26 ijek.kq ?kM+h lokZfèkd ifj'kq¼ le; ekid ;qfDr gS D;ksafd ijek.kqvksa osQ nksyu1013 s esa 1 s dh ifj'kq¼rk ls nksgjk, tkrs gSaA

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115

2.27 3 × 1016 s

2.28 0.01 mm

2.29 ( ) ( )2 2 2 2e/R /=

s s m emR R Rθ π π

s es

m em

R R

R R⇒ =

2.30 105 kg

2.31 (a) dks.k vFkok ?ku dks.k

(b) vkisf{kd ?kuRo vkfn

(c) Iyk¡d fu;rkad] lkoZf=kd xq#Roh; fu;rkad vkfn

(d) jsukWYM la[;k

2.323.14

31 cm 16.3cm6

= ⇒ = ⇒ = × =l

l r lr

θ θ

2.33 4 × 10– 2

LVsjsfM;u

2.34 ω dk foeh; lw=k = T – 1

k dk foeh; lw=k = L–1

2.35 (a) ifj'kq¼rk midj.k osQ vYirekad }kjk fuèkkZfjr gksrh gSA

20 nksyuksa osQ fy, ifj'kq¼rk = 0.1 s

1 nksyu osQ fy, ifj'kq¼rk = 0.005 s.

(b) vkSlr le;] 39.6 39.9 39.5s 39.6

3

+ += =t s

vkorZ = =39.6

1.9820

s

vfèkdre izsf{kr =kqfV = (1.995 –1.980) = 0.015s

2.36 D;ksafd ÅtkZ dk foeh; lw=k ML2 T

–2 gS] 1J u, ek=kd esa 2 2/γ αβ J gks tk,xkA

vr% 5 J u;s ek=kd esa 2 25 /γ αβ gks tk,xkA

2.37 fn, x, O;atd dk foek;qDr va'k 4ρ

η

r

l gSA vr% nkfgus i{k dh foek,¡

( ) ( )

( ) ( )

3–1 –2 4

–1 –1

LML T L

TLML T= izkIr gksrh gSa] tks vk;ru dks le; ls Hkkx nsus ij izkIr

jkf'k dks fu#fir djrh gSaA vr% lw=k foeh; n`f"V ls lgh gSA

lw;Z

pUnzek

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116

2.38 X esa fHkUukRed =kqfV gSµ

d 2d 3d 2.5d 2d=

X a b c (d)+ + +

X a b c d 0.235 0.24= �

D;ksafd =kqfV izFke n'keyo LFkku ij gSA ifj.kke dks iw.kk±fdr djosQ 2-8 fy[kktkuk pkfg,A

2.39 D;ksafd E;s l vkSj G osQ foeh; lw=k Øe'k% uhps fn, vuqlkj gSaµ

?→ 2 2E [ML T ]

2 –1ML T→l

3 –1 –2G L M T→

vr% P = E l 2 m–5 G–2 dh foek,¡ gksaxh%

[ ][ ][ ][ ]

[ ][ ]

2 –2 2 4 –2 2 4

5 6

ML T M L T M TP

M L=

= M0 L0 T 0

vr%] P foekghu jkf'k gSA

2.40 u, ek=kdksa esa M, L, T, Øe'k% uhps fn, vuqlkj gksaxs]

→ → →3 5

ch hG hGM , L , T

G c c

2.41 fn;k gS% 2 3 3/2T r T r⇒α α A T xq#Ro osQ dkj.k Roj.k g rFkk R dk

iQyu Hkh gS ⇒ T α gx Ry

[ ] [ ][ ] [ ]1 3/2 1 2 1x y

o o o o o o oL M T L M T L M T L M T−∴ =

L osQ fy, 3

02

x y= + +

T osQ fy, 1

1 0 22

x x= − ⇒ = −

blfy, 3 1

0 12 2

y y= − + ⇒ = −

vr% 3

3/2 1/2 1 k rT k r g R

R g

− −= =

2.42 (a) D;ksafd vksyhd vEy vYdksgy esa ?kqy tkrk gS ijarq ty esa ugha ?kqyrkA

(b) tc ykbdksiksfM;e ikmMj dks ty osQ Åij fNM+dk tkrk gS rks ;g mlosQ

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laiw.kZ i`"B ij iSQy tkrk gSA tc rS;kj foy;u dh ,d cw¡n ty osQ i`"B ijMkyh tkrh gS rks vksyhd vEy ty esa ugha ?kqyrk] ;g ty osQ i`"B ijykbdksiksfM;e ikmMj dks ijs gVkrk gqvk tgk¡ cw¡nsa fxjrh gSa mlosQ pkjksa vksj ,do`Ùk {ks=k esa iSQy tkrk gSA blls ge ljyrk ls ml {ks=k dk {ks=kiQy ekildrs gSa ftlesa vksyhd vEy iSQyrk gSA

(c)1 1 1

mL mL20 20 400

× =

(d) C;wVsu vkSj ekid tkj dk mi;ksx djosQ rFkk cw¡nksa dh ifjdfyr la[;k dkvk;ru ekidjA

(e) ;fn foy;u dh n cw¡nksa dk vk;ru 1 mL gks rks bldh ,d cw¡n esa vksyhdvey dk vk;ru (1/400n) mL gksxkA

2.43 (a) ikjlsoQ dh ifjHkk"kk osQ vuqlkj

∴ 1 ikjlsd

=

1A.U.

1pki los Qa M

1 fMxzh = 3600 pki lsoaQM

∴ 1 pki lsoaQM 3600 180

π=

× jsfM;u

∴ 1 ikjlsd 3600 180

π

×= A.U. = 206265 A.U. 52 10≈ × A.U.

(b) 1 A.U. nwjh ij lw;Z dk O;kl 1/2° gSA

blfy, 1 ikjlsd ij rkjs dk O;kl gS 5

1/2

2 10×fMxzh = 15 × 10-5 vkoèkZu 100

gksus ij ;g 5 × 10-3 pki feuV fn[kkbZ nsuh pkfg,A ijarq] ok;qeaMyh; mPpkopuosQ dkj.k ;g 1 pki&feuV gh fn[kkbZ nsxhA VsfyLdksi dk mi;ksx djus ls bldksvkofèkZr ugha fd;k tk ldrk gSA

(c) = =

1 1

2 400

DD

D D

iFoheaxy

iF` oh l;w Z[mÙkj 2.25 (c)] ls

∴ =1

800

D

D

eaxy

l;w Z

A.U. ij lw;Z dk dks.kh; O;kl 1/2 fMxzh fn[kkbZ iM+rk gS vkSj eaxy dk 1/1600 fMxzhA

1/2 A.U. ij eaxy dk dks.kh; O;kl 1/800 fMxzh fn[kkbZ nsxkA 100 xquk vkoèkZu ij

eaxy dk O;kl 1/8 fMxzh60

7.58

= = pki lsoaQM fn[kkbZ nsrkA

pki lsoaQM

ikjlsd

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;g ok;qeaMyh; mPpkopu osQ dkj.k Hksnuh; lhek ls vf/d gS vr% eaxy VsfyLdksils ns[kus ij vkofèkZr utj vkrk gSA

2.44 (a) pw¡fd 1 u = 1.67 × 10–27 kg;s blosQ lerqY; ÅtkZ gSA 1.67×10–27 c2 J eV

vkSj fiQj MeV esa ifjofrZr djus ij 1 u ≡ 931.5 MeV.

(b) 1 u × c2 = 931.5 MeV.

vè;k; 3

3.1 (b)

3.2 (a)

3.3 (b)

3.4 (c)

3.5 (b)

3.6 (c)

3.7 (a), (c), (d)

3.8 (a), (c), (e)

3.9 (a), (d)

3.10 (a), (c)

3.11 (b), (c), (d)

3.12 (a) (iii), (b) (ii), (c) iv, (d) (i)

3.13

3.14 (i) x (t) = t - sin t

(ii) x (t) = sin t

a

t

-g

0

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119

9m/s

9m

10m

3.15 x(t) = A + tBe γ− , A > B, 0γ > mi;qDr :i ls pqus x, èkukRed fu;rkad gSaA

3.16 v = g/b

3.17 xsan dks Å¡pkbZ ls NksM+us ij ;g xq#Ro osQ izHkko esa uhps vkrh gSA ml vYidky osQvfrfjDr tc ;g Hkwry ls la?kV~V djrh gS bldk Roj.k –g gksrk gSA la?kV~V osQ le;

bl ij vkosxdkjh cy dk;Z djrk gS vkSj cgqr vfèkd Roj.k mRiUu gksrk gSA

3.18 (a) x = 0, γ= ov x

3.19 dkjksa dh lkisf{kd pky = 45km/h, muosQ feyus osQ fy, vko';d le;

36 km0.80h

45 km/h= =

vr% i{kh }kjk r; dh xbZ oqQy nwjh = 36 km/h × 0.8h = 28.8 km

3.20 ekuk fd 9 m fxjus esa t le; yxrk gSA

vr% 2

2o oy

gty y v− = −

pw¡fd 0oyv =

2

2( ) 2 9 m1.8 1.34

10 m/s

− ×= → = ≈oy y

tg

s

bl le;kofèk esa {kSfrtr% pfyr nwjhµ

x-xo = v

oxt = 9 ms–1 × 1.34s = 12.06 m

th gk¡] og èkjrh ij fxjsxkA

3.21 nksuksa gh Lora=krkiwoZd fxjrs gSa] vr% ,d osQ lkis{k nwljs dk Roj.k 'kwU; gksrk gSA

blfy, lkisf{kd pky vpj (=40 ms–1 ) jgrh gSA

3.22 v = (-vo/x

o) x + v

o, a = (v

o/x

o)2 x - v

o2/x

o

x osQ lkFk a esa gksus okys ifjorZu dks vko`Qfr esa n'kkZ;k x;k gSA ;g ,d ½.kkRed

var%[kaM rFkk èkukRed izo.krk dh ½tqjs[kk gSA

3.23 (a) = = × × = =–1 –12 2 10 1000 141ms 510km/hv gh

(b)π π

ρ − −= = × = ×3 3 3 3 54 4(2 10 ) (10 ) 3.4 10 kg

3 3m r

− −= ≈ × ≈ ×3 –1 3 –14.7 10 kg ms 5 10 kg msP mv

x

a

0

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120

(c) O;kl 4mm≈

/ 28 s 30 st d v µ µ∆ ≈ = ≈

(d)

−

−

∆ ×= = ≈ ≈ ×

∆ ×

32

6

4.7 10168N 1.7 10 N

28 10

PF

t

(e) vuqizLFk dkV dk {ks=kiQy 2 2/4 0.8mdπ= ≈

vkSlr i`Fkdu 5 cm, gks rks mu cw¡nksa dh la[;k tks yxHkx ,d lkFk fxjrs gSaµ2

2 2

0.8m320

(5 10 )−≈

×

oqQy cy ≈ 54000 N (cw¡nksa dk osx O;ogkjr% ok;q osQ Lianu osQ dkj.k de gksrktkrk gS)A

3.24 tc dkj Vªd osQ ihNs gSµ

Vªd dk osx ßkl –2204ms

5= =

dkj dk osx ßkl = –220s

3m

ekuk fd ftl {k.k czsd yxk, tkrs gSa Vªd dkj ls x nwjh ij gSA

t > 0.5 s ij Vªd dh A ls nwjh x + 20t – 2t2 gSA

A ls dkj dh nwjh gS 10 + 20(t – 0.5) –210

( – 0.5)3

t .

;fn nksuksa okgu fey tkrs gSa rks

x + 20t – 2t2 = 10 + 20t – 10 –210 10 10

– 0.253 3 3

+ ×t t .

x =24 10 5

– –3 3 6

t t+ .

Kkr djus osQ fy, xmin

,

8 10– 0

3 3= + =

dxt

dt

tmin

= 10 5

s8 4

= .

blfy,] xmin

=2

4 10 5 5 55– –

3 3 4 6 44

+ × =

.

vr%] x > 1.25m

nwljh fof/ µ bl fof/ esa O;dyu osQ mi;ksx dh vko';drk ugha gksrhA

;fn dkj Vªd osQ ihNs gSA

vkSj D;ksafd dkj osQoy 0.5 s osQ i'pkr~ Rofjr gksdj izkjaHk djrh gSA t > 0.5 s osQ

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121

fy, Vdkj

= 20 – (20/3)(t – 0.5)

VVªd

= 20 – 4t

nksuksa osxksa dks ckjckj fy[kdj vFkok osx≤ xzkiQ }kjk t dk eku Kkr dhft,] rksizkIr gksxk t = 5/4 s

SVªd

= 20(5/4) – (1/2)(4)(5/4)2 = 21.875m

vr% Scar

– Struck

= 1.25m

;fn izkjaHk esa dkj ;g nwjh cuk, j[krh gS rks 1.25 s osQ i'pkr~ bldh pky Vªd lsges'kk de gksxh vkSj buosQ chp dHkh Hkh la?kV~V ugha gksxkA

3.25 (a) (3/2)s (b) (9/4)s (c) 0,3 s (d) 6 pØ

3.26 v1=20 ms–1, v

2 = 10ms–1, dky&varj = 1s

vè;k; 4

4.1 (b)

4.2 (d)

4.3 (b)

4.4 (b)

4.5 (c)

4.6 (b)

4.7 (d)

4.8 (c)

4.9 (c)

4.10 (b)

4.11 (a), (b)

4.12 (c)

4.13 (a), (c)

4.14 (a), (b), (c)

4.15 (b), (d)

4.16 RO fn'kk esa 2v

R

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122

4.17 fo|kFkhZ vius f'k{kdksa osQ lkFk ppkZ djsa vkSj mÙkj izkIr djsaA

4.18 (a) Hkwry ls Vdjkus ls Bhd igysA

(b) viuh xfr osQ mPpre ¯cnq ijA(c) a = g = fu;rkadA

4.19 Roj.k – g

osx – 'kwU;

4.20 pw¡fd B × C ml ry osQ yacor~ gS ftlesa B ,oa C fLFkr gSaA B × C osQ lkFk fdlh

lfn'k dk otzh; xq.kuiQy B ,oa C osQ ry esa gksxkA

4.21

Hkwfe ij fLFkr isz{kd osQ fy, xsan v0 pky vkSj {kSfrt ls θ dks.k ij izf{kIr iz{ksI;

gSA tSlk Åij vko`Qfr esa n'kkZ;k x;k gSA

4.22

D;ksafd dkj dh pky iz{ksI; dh {kSfrt pky osQ cjkcj gSA dkj esa cSBk gqvk yM+dkosQoy xfr osQ ÅèoZèkj vo;o dks ns[ksxkA tSlk fd vko`Qfr (b) esa n'kkZ;k x;k gSA

4.23 ok;q izfrjksèk osQ dkj.k d.k dh ÅtkZ rFkk osx dk {kSfrt vo;o de gksrs tkrs gSaftlls Åij tkus osQ le; osQ xzkiQ dh rqyuk esauhps vkus osQ le; osQ xzkiQ dh izo.krk vfèkd gkstkrh gSA tSlk fd vko`Qfr esa n'kkZ;k x;k gSA

(a) (b)

y

x0

(mNkyh x;h xsan

dh pky)

(dkj dh pky)

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123

4.241 12 1

, tan tan 23 12 '2

o

o

H gHHR v

g vR

− − = = = = °

φ

4.25 Roj.k 2

2

42v R

R T

π=

4.26 (i) dk esy feyrk gS (d) ls

(ii) dk esy feyrk gS (c) ls

(iii) dk esy feyrk gS (a) ls

(iv) dk esy feyrk gS (b) ls

4.27 (i) dk esy feyrk gS (b) ls

(ii) dk esy feyrk gS (a) ls

(iii) dk esy feyrk gS (d) ls

(iv) dk esy feyrk gS (c) ls

4.28 (i) dk esy feyrk gS (d) ls

(ii) dk esy feyrk gS (c) ls

(iii) dk esy feyrk gS (a) ls

(iv) dk esy feyrk gS (b) ls

4.29 igkM+h dks ikj djus osQ fy, vko';d U;wure ÅèoZèkj osx

2v⊥ ≥ 2gh = 10,000

v1 > 100 m/s

D;ksafd rksi ls 125 m s–1 dh pky ls iSosQV izf{kIr fd, tk ldrs gSa blfy,{kSfrt osx dk vfèkdre eku] v

1 gksxk%

2 1

1125 100 75

−= − = msv

igkM+h osQ f'k[kj rd] v⊥ osx ls igq¡pus esa yxk le; Kkr djus osQ fy, gefy[k ldrs gSa%

110 s

2

2gT =h T =⇒

10s esa pfyr {kSfrt nwjh = 750 m.

blfy, rksi dks Hkwfe ij 50 m LFkkukarfjr djuk iM+sxkAvr% iSosQV dks igkM+h dks ikj djosQ i`Foh rd igq¡pus esa yxk oqQy (vYire)

le; 50

s 10s 10s2

= + + = 45 s

4.31 (i)

2

2

2 sin cos( )

cosov

Lg

β α β

α

+=

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124

(ii)2 sin

cosov

Tg

β

α=

(iii)4 2

= −π α

β

4.3220 sin

Av

gθ

4.33 ˆ ˆ5 5= −rV i j

4.34 (i) mÙkj fn'kk ls 37º ij 5ms–1

(ii) (a) mÙkj fn'kk ls ( )−1 –1tan (b) 7 m/s3/ 7 dk.s k ij

(iii) iz'u (i) dh fLFkfr esa og lcls de vofèk esa foijhr rV ij ig¡qpsxkA

4.35 (i) 1 sintan

coso

o

v

v u

θ

θ− +

(ii)2 sinov

g

θ

(iii)o o2 sin ( cos + u)

g=

v vR

θ θ

(iv) 2 2

1max

8

4

o

o

u u vcos

vθ −

− + +=

(v) = ou v osQ fy, =max 60oθ

= 0u osQ fy, =max 45oθ

ou v<

∴ 1max

1

42 o

ucos

vθ −

−≈

π= �( )4 ou v;fn

( )1max 2

− > ≈ =

�o

oo

vv uu v cos

uπθ

(vi) max 45 .> °�

θ

4.36θ θ

ω ωθ ω θ θ ω

= + = +− +

2 22 2

2 2ˆˆ ˆ 2

d d

dt dtV r a rθθθθ rFkk

4.37 jsr ls gksdj xqtjus okys ljy js[kh; iFk APQC ij fopkj dhft;sµ

bl iFk ij A ls C rd tkus esa yxk le;µ

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125

= Tjsr =

AP + QC PQ+

1 v

= 25 2 +25 2 50 2

+1 v

= 50 2 1

+1v

jsr ls gksdj y?kqre iFk ARC gksxkA bl iFk ls gksdjA ls C rd tkus esa yxk le;

= Tcká

= AR +RC

1s

= 2 22 75 +25

= 2×25 10 s

Tjsr < T

cká osQ fy;s 1

50 2 +1 <2×25 10v

= 1

+1< 5v

= 1

< 5 -1v

;k 1> 0.81

5 -1v ≈ ms–1

vè;k; 5

5.1 (c)

5.2 (b)

5.3 (c)

5.4 (c)

5.5 (d)

5.6 (c)

5.7 (a)

5.8 (b)

5.9 (b)

5.10 (a), (b) ,oa (d)

5.11 (a), (b), (d) ,oa (c)

5.12 (b) ,oa (d)

A

C

P

Q

R

50m

100m

D

B

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126

f

F

5.13 (b), (c)

5.14 (c), (d)

5.15 (a), (c)

5.16 th] gk¡] laosx laj{k.k fu;e osQ dkj.kAizkjafHkd laosx = 50.5 × 5 kg m s–1

vafre laosx = (50 v + 0.5 × 15) kg m s–1

v = 4.9 m s-1, pky esa ifjorZu = 0.1 m s–1

5.17 ekuk fd iSekus ij ikB~;kad R U;wVu gS]

izHkkoh v/ksfn'k Roj.k 50

50

g Rg

−= =

R = 5g = 50N. (rqyk 5 kg Hkkj n'kkZ,xh)

5.18 'kwU;_ 3

2− kg m s-1

5.19 ;fn mlus lhV csYV ugha ck¡èkh gqbZ gSa rks mlosQ Åij yxus okyk ,d ek=k voeandcy lhV }kjk yxus okyk ?k"kZ.k cy gSA tc okgu vpkud jksd fn;k tkrk gS rks ;gcy mlosQ vkxs dh vksj xfr dks jksdus osQ fy, i;kZIr ugha gksrkA

5.20 ˆ ˆ ˆ ˆ8 8 , (4 8 )N= + = +p i j F i j

5.21 tc rd xqVdk fLFkj jgrk gS f = F

bl ¯cnq ls vkxs f dk eku c<+kus ij tc xqVdk xfrdjus yxrk gS rks F dk eku vpj gks tkrk gSA

5.22 eky ys tkrs le; Vªd tSls okgu dks vpkud jksdus dh vko';drk gks ldrh gSAtc dksbZ Hkaxqj nzO;] tSls fd ikslsZysu ls cuh gqbZ dksbZ pyrh gqbZ oLrq vpkud BgjkbZtkrh gS rks bl ij fo'kky cy yxkuk iM+rk gS ftlosQ dkj.k ;g VwV ldrh gSA ;fnbldks Hkwls vkfn ls yisVk gks rks Hkwls osQ dksey gksus osQ dkj.k oLrq #dus ls igysoqQN nwjh py ldrh gSA blosQ fy, de cy dh vko';drk gksrh gS vkSj bl izdkjoLrq osQ VwVus dh laHkkouk de gks tkrh gSA

5.23 tc cPpk lhesaV osQ iQ'kZ ij fxjrk gS rks mldk 'kjhj ,dne fojke dh voLFkk esayk;k tkrk gSA feêðh FkksM+h nc tkrh gS vkSj blfy, fojke esa vkus ls igys mldk 'kjhjoqQN nwjh py ikrk gS ftlesa oqQN le; yxrk gSA bldk vFkZ gS fd feêðh osQ iQ'kZ ijfxjus dh ?kVuk esa D;ksafd laosx ifjorZu dk dky vfèkd gks tkrk gS blfy, cPps dksfojke esa ykus osQ fy, ml ij yxus okys cy dk ifjek.k de gks tkrk gSA

5.24 (a) 12.5 N s (b) 18.75 kg m s–1

5.25 ?k"kZ.k cy gS% f = µ R = µ mg cosθ , tgk¡ <kyw ry }kjk {kSfrt ls cuk dks.k gSA

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127

;fn θ dk eku de gS rks ?k"kZ.k cy vfèkd gksrk gS vkSj fiaM osQ fiQlyus dh laHkkoukde gksrh gSA lhèkh [kM+h lM+d dk <+ky vfèkd gksxkA

5.26 AB, D;ksafd Åijh /kxs ij yxk cy fiaM osQ Hkkj rFkk yxk, x, cy osQ ;ksx osQ cjkcjgksxkA

5.27 ;fn cgqr vfèkd cy >VosQ ls yxk;k tk,xk rks èkkxk CD VwVsxk] D;ksafd] CD dks>Vdk fn;s tkus ij cy rqjar AB dks lapfjr ugha gksxk (cy lapj.k fiaM osQ izR;kLFkrklacaèkh xq.kksa ij fuHkZj djrk gS)A blfy, nzO;eku osQ xfr esa vkus ls igys gh CD VwVtkrk gSA

5.28 T1 = 94.4 N, T

2 = 35.4 N

5.29 W = 50 N

5.30 ;fn F iqLrd ij Å¡xyh }kjk yxk cy gS rks F = N, nhokj dk iqLrd ij yxus okykvfHkyacor~ izfrfØ;k cy gSA Åij dh vksj ftl U;wure ?k"kZ.k cy osQ yxus ls ;glqfuf'pr gks tkrk gS fd iqLrd uhps ugha fxjsxh og gS Mg A ?k"kZ.k cy = µN A vr%

F dk U;wure eku gS MgF

µ= .

5.31 0.4 m s–1

5.32 2,x t y t= =10, 2 m sx ya a −= =

F = 0.5×2 = 1N. y-v{k osQ vuqfn'k

5.332 2 × 20 40 10

= = = = = 3.33s.+ 10 + 2 12 3

Vt

g a

5.34 (a) D;ksafd xfreku fiaM esa dksbZ Roj.k ugha gS vr% cyksa dk lfn'k ;ksx 'kwU; gSµ= 01 2 3F + F + F A ekuk , ,1 2 3F F F ,d gh ¯cnq ij yxus okys rhu cy gSaA ekuk

1F ,oa 2F lery A esa gS (vki nks izfrPNsnh js[kkvksa esa ls xqtjus okyk ,d leryrks cuk gh ldrs gSa fd ;s nksuksa js[kk,¡ ftlesa vofLFkr gksa)A rc +1 2F F lery Aesa gh gksxkA pw¡fd ( )–=3 1 2F F + F , 3F Hkh lery A esa gh gSA

(b) P osQ pkjksa vksj cyksa osQ vk?kw.kZ ij fopkj dhft,A D;ksafd lHkh cy P ls xqtjrsgSa] oqQy cy&vk?kw.kZ 'kwU; gSA vc fdlh vU; fcUnq 0 osQ ifjr% cy&vk?kw.kZ ij fopkjdhft,A 0 osQ ifjr% cy vk?kw.kZA

T ( )= × 1 2 3F + F + FOP

pw¡fd 0=1 2 3F + F + F , T = 0

P

F3

O

F1

F2

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128

5.35 loZlkekU; izdj.k ij fopkj djsa rks21

2 /2

s at t s a= ⇒ =

ty ry fpduk gS

rks Roj.k sin2

ga g θ= =

∴ =1 2 2 /t s g

tc ry [kqjnjk gS

rks Roj.k sin cosa g gθ µ θ= −

(1 ) / 2gµ= −

2 1

2 2 2 2

(1 )

s st pt p

g gµ∴ = = =

−

2

2

1 11

1p

pµ

µ⇒ = ⇒ = −

−

5.36 = < ≤2 0 1xv t t0 1syv t t= < <

= − < <2 (2 ) 1 2t t 11= < t

= 0 2 < t

2; 0 1xF t= < < 1 0 1s= < <yF t

2; 1s 2st= − < < 0 1s t= <

= 0 ; 2s < t

ˆ ˆ2= +F i j 0 1t< < s

ˆ2= − i 1s < t < 2s

= 0 2s < t

5.37 DEF osQ fy,

µ=2v

m m gR

µ −= = = 1max 100 10 m sv g R

ABC osQ fy,

µ −= = =2

1, 200 14.14 m s2

vg v

R

DEF osQ fy, le; = 100

5 s2 10

ππ× =

ABC osQ fy, le; = 3 200 300

s2 14.14 14.14

π π=

FA ,oa DC osQ fy, le; = 100

2 4s50

× =

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129

oqQy le; = 300

5 4 = 86.3s14.14

ππ + +

5.38ˆ ˆsin cos

dA t B t

dtω ω ω ω= = − +

rv i j

2 2;d

mdt

ω ω= = − = −v

a r F r

2 2

2 2cos , sin 1

x yx A t y B t

A Bω ω= = ⇒ + =

5.39 (a) 21

2zv gH= osQ fy;s 2zv gH=

Hkwfe osQ fudV pky = 2 2 2 2s z sv v v gH+ = +

(b) osQ fy, Hkh 21

2smv mgH

+

ml le; xsan dh oqQy ÅtkZ gS tc ;g Hkwfe

ls Vdjkrh gSA

vr% (a) ,oa (b) nksuksa osQ fy, pky cjkcj gksxhA

5.40 3 42

2 1 3

2 2 2

F FF

+ += = = N

3 41

2 2

F FF + =

4 31

1

2 2

F FF

−= = N

5.41 (a) 1tanθ µ−=

(b) mg sin cosmgα µ α−

(c) mg ( )sin cosα µ α+

(d) mg ( )sin cosθ µ θ+ + ma.

5.42 (a) F - (500 ×10) = (500 × 15) vFkok F = 12.5 × 103 N, tgk¡ F iQ'kZ dk Åijdh vksj vfHkfØ;k cy gS rFkk ;g U;wVu osQ xfr osQ r`rh; fu;e osQ vuqlkj iQ'kZ ijuhps dh vksj yxus okys cy osQ cjkcj gSA(b) R - (2500 × 10) = (2500 × 15) vFkok R = 6.25 × 104 N, ra=k ij Åij dhvksj yxus okyh ok;q dh fØ;k gSA jksVj dh pkjksa vksj dh ok;q ij fØ;k gS6.25 × 104 N Åij dh vksjA

(c) ok;q osQ dkj.k gsfydkWIVj ij cy = 6.25 × 104 N Åij dh vksjA

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130

vè;k; 6

6.1 (b)

6.2 (c)

6.3 (d)

6.4 (c)

6.5 (c)

6.6 (c)

6.7 (c)

6.8 (b)

6.9 (b)

6.10 (b)

6.11 (b) D;ksafd foLFkkiu 3/2tα

6.12 (d)

6.13 (d)

6.14 (a)

6.15 (b)

6.16 (d)

6.17 (b)

6.18 (c)

6.19 (b), (d)

6.20 (b), (d), (f)

6.21 (c)

6.22 th gk¡] th ughaA

6.23 fyÝV dks xq#Ro osQ rgr Lora=krkiwoZd fxjus ls jksdus osQ fy,A

6.24 (a) èkukRed (b) ½.kkRed

6.25 {kSfrt lM+d ij pyus esa xq#Ro osQ vèkhu fd;k x;k dk;Z 'kwU; gksrk gSA

6.26 th ugha] D;ksafd ok;q dk izfrjks/d cy Hkh ¯iM ij dk;Z djrk gS tks v&laj{kh cygSA blfy, xfrt ÅtkZ esa yfCèk fLFkfrt ÅtkZ esa gqbZ gkfu dh vis{kk de gksxhA

6.27 th ugha] can oØh; iFk ij xfr djus esa fd;k x;k dk;Z vfuok;Zr% 'kwU; rHkh gksxktc ra=k ij vkjksfir lHkh cy laj{kh gksaA

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6.28 (b) oqQy js[kh; laosx

tc xsansa laioZQ esa gksrh gSa rks muesa foÑfr gks ldrh gS ftldk vfHkizk; gksrk gS izR;kLFkfLFkfrt ÅtkZ tks xfrt ÅtkZ osQ ,d va'k ls gh izkIr gksrh gSA laosx lnSo lajf{krgksrk gSA

6.29 'kfDr × ×

= = =100 9.8 10

W 490 W20

mgh

T

6.30 0.5 ×72

= =60

EP

t

∆

∆= 0. 6 okV

6.31 ,d leku pqacdh; {ks=k esa xfreku vkosf'kr d.kA

6.32 fd;k x;k dk;Z = xfrt ÅtkZ esa ifjorZunksuksa fiaMksa dh xfrt ÅtkZ leku gS blfy;s cjkcj ifjek.k esa dk;Z fd, tkus dhvko';drk gksrh gS D;ksafd yxk;k x;k cy cjkcj gS os leku nwjh r; djus osQ ckn#osaQxsA

6.33 (a) ljy js[kk% ÅèokZèkj] uhps dh vksj(b) C ijoy;kdkj iFk ftldk 'kh"kZ C ij gksrk gSA(c) ijoy;kdkj iFk ftldk 'kh"kZ C osQ Åij gksrk gSA

6.34

6.35 (a) lEeq[k la?kV~V osQ fy,µ

laosx laj{k.k ⇒ 2mv0 = mv

1 + mv

2

vFkok 2v0 = v

1 + v

2

rFkk 2 1

0

-=

2

v ve

v⇒ v

2 = v

1+ 2v

0e

∴ 2v1 = 2v

0 – 2ev

0

∴

v1 = v

0(1 – e)

pw¡fd e <1 ⇒ v1 dk fpÉ ogh gS tks v

0, dk gS] blfy, xsan la?kV~V osQ ckn Hkh

pyrh jgrh gSA

(b) laosx laj{k.k ⇒ p = p1+ p

2

ijarq xfrt ÅtkZ {kf;r gksrh gS ⇒2 2

1 2> +2 2 2

2 p pp

m m m

C D

F

X

B

Eo

Eo

KE osx

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∴ p2 > p12 + p

22

vr% p, p1 ,oa p

2 vko`Qfr esa n'kkZ, vuqlkj lacafèkr gksrs gSaA

θ U;wu (90° ls de) dks.k gSA 2 22

1 290= + = °( )p p p θgk s rks

6.36 Hkkx A : ugha] D;ksafd ;gk¡ xfrt ÅtkZ ½.kkRed gks tk,xhA

Hkkx B : gk¡ KE 'kwU; u gksus ij oqQy ÅtkZ PE ls vfèkd gks ldrh gSA

Hkkx C : gk¡ PE osQ ½.kkRed gksus ij KE ls oqQy ÅtkZ ls vfèkd gks ldrh gSA

Hkkx D : gk¡] D;ksafd PE dk eku KE ls vfèkd gks ldrk gSA

6.37 (a) xsan A viuk laiw.kZ laosx est ij j[kh xsan dks LFkkukarfjr dj nsrh gS vkSj Lo;a

fcYoqQy Hkh Åij ugha mBrhA

(b) –1= 2 = 4.42m sv gh

6.38 (a) PE esa gkfu = mgh =–3 –3

1×10 ×10 ×10 =10J

(b) KE esa o`f¼ = –31 12 = ×10 × 2500 = 1.25J

2 2mv

(c) th ugha] D;ksafd PE dk ,d va'k ok;q osQ LFkkuh; d"kZ osQ fo#¼ dk;Z djus esa

mi;ksx esa vk tkrk gSA

6.39 (b)

6.40 m = 3.0 × 10 – 5 kg ρ = 10 – 3 kg/m2 v = 9 m/s

A = 1m2 h = 100 cm ⇒ n = 1m3

M = ρ v = 10–3 kg, E = 3 2 41 1

= ×10 ×(9) = 4.05 ×10 J2 2

2Mv .

p

p1

p2

qigys ckn esa

E1

E0

T/4 3 /4T 5 /4Tt

7T/4

E2

E0

T/4 3 /4T 5 /4T 7T/4t

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6.41 2 4 21 1

5 102 2

10= ≅ ×× ×KE mv

= 2.5 × 105 J.

bldk 10% fLiax esa laxzfgr gks tkrk gS1

2.5 ×102

2 4kx =

x = 1 m

k = 5 × 104 N/m–1

6.42 6 km esa 6000 lksiku gSa

∴ E = 6000 (mg)h

= 6000 × 600 × 0.25

= 9 × 105J.

;g xzfgr ÅtkZ dk 10 % gS

∴ xzfgr ÅtkZ = 10 E = 9 ×106J.

6.43 0.5 n{krk ls 1 yhVj ls 1.5 × 107J, ÅtkZ mRiUu gksrh gS tks 15 km rd xkM+h pykus

esa mi;ksx esa ykbZ tkrh gSA

∴ F d = 1.5 ×107J d = 15000 m ysus ij

?k"kZ.k cy F = 1000 N :

6.44 (a) Wg = mg sinθ d = 1×10 × 0.5 ×10 = 50 J.

(b) Wf = µ mg cosθ d = 0.1 ×10 × 0.866 × 10 = 8.66 J.

(c) U = = 1 ×10 × 5 = 50 Jmgh∆

(d) ( ){ } [ ]= - sin + cos = 10 -5.87θ θF mg mga µ

= 4.13 m/s2

v = u + at vFkok v2 = u2 + 2ad

2 21 1= – = = 41.3 J

2 2∆K mv mu mad

(e) W = F d = 100 J

6.45 (a) xsanksa 1 ,oa 3 osQ fy, ÅtkZ lajf{kr gksrh gSA

(b) xsan 1 ?kw.khZ ÅtkZ izkIr dj ysrh gS] xsan 2 esa ?k"kZ.k osQ }kjk ÅtkZ ßkl gksrk gS] xsan

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2 okil A ij ugha igq¡p ldrhA xsan 1 tc B ij igq¡prh gS rks blesa xyr vFkZ

esa ?kw.khZ xfr gksrh gSA xfrt ÅtkZ osQ dkj.k ;g yq<+d dj A ij ugha igq¡p ldrhA

6.46 ( ) 2 21 1( ) ( ) ( )

2 2= + ∆ + ∆ −− ∆+∆KE v v m v uM mt t

jkWosQV xSl21 12

2 2= + ∆ − ∆ + ∆Mv Mv v mvu mu

=1 2( )2

KE Mvt

2 21( ) ( ) ( )

2

1

2− = ∆ − ∆ + ∆+∆ = ∆ =KE KE M v mu v m utt t m u W

(dk;Z&ÅtkZ izes; osQ vuqlkj)

pw¡fd ( ) ( ) 0= ⇒

∆ − ∆ =

Mdv dmu

dt dtM v m u

6.47 gqd osQ fu;e vuqlkj : = YF L

A L

∆

tgk¡ ?ku osQ ik'oZ dk {ks=kiQy rFkk L bldh ,d Hkqtk dh yackbZ gSA ;fn k fLiazxfu;rkad ;k laihMu fu;rkad gS rks F = k ∆ L

∴ A

k = Y = YLL

izkjafHkd KE = 2 –41

2 × =5×10 J2

mv

vafre PE = 21

2 ( )2

k L× ∆

∴ = =Y

∆KE KE

Lk L

=

–4

11

5 ×10

2 ×10 × 0.1=1.58 × 10–7m

6.48 ekuk fd m , V, Heρ Øe'k% ghfy;e osQ xqCckjs osQ nzO;eku] vk;ru rFkk ?kuRo gS

vkSj ρok;q

ok;q dk ?kuRo gSA

xqCckjs dk V vk;ru ok;q dk V vk;ru foLFkkfir djrk gSA

blfy, V ( )ρ − ρ =g m aHeok; q (1)

le; t osQ lkis{k lekdyu djus ij]

( )ρ − ρ =V gt m vHeok; q

( )ρ −ρ⇒ =2 2

2 2 2

2

1 1

2 2 He

Vmv m g t

m ok;q ( )ρ ρ= −

22 2 2

1

2He

V g t

mok;q (2)

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135

;fn xqCckjk h Å¡pkbZ rd Åij tkrk gS rks = + 21

2s ut at ls

gesa izkIr gksrk gS = 21

2h a

( )ρ − ρ= 21

2

V hegt

m

ok;q (3)

lehdj.k (3) ,oa (2) ls

( ) ( )ρ − ρ ρ − ρ=

2 21 1

2 2He Hemv V g V gt

mok;q ok;q

( )= – HeV ghρ ρok;q

inksa dks iqu% leaftr djus ij

⇒ + ρ =1 2

2Hemv V gh V hgok;q

⇒ + =KE PE PExCq ckjk xCq ckjk eas ifjorZuok;q dh

blfy,] tSls&tSls xqCckjk Åij mBrk gS ok;q dk leku vk;ru uhps vkrk gSA xqCckjs

dh PE ,oa KE esa o`f¼ ok;q dh PE osQ ewY; ij gksrh gS (tks fd uhps vkrh gS)A

vè;k; 7

7.1 (d)

7.2 (c)

7.3 izkjafHkd osx gS ˆv yi = ev A nhokj ls ijkorZu osQ i'pkr~ vafre osx gS ˆv yf= − ev A

iz{ksi iFk dk fu:i.k ˆ ˆy ay z= +r e e }kjk fd;k tk ldrk gSA vr% dks.kh; laosx

ifjorZu gS ˆ( ) 2fm mva xi× − =r v v e A vr% mÙkj (b) gksxkA7.4 (d)7.5 (b)7.6 (c)

7.7 tc b →0, ?kuRo ,d leku gks tkrk gS vkSj blfy, nzO;eku osaQnz x = 0.5. ij gksxkAtc b → 0 rks osQoy fodYi (a) eku 0.5 dh vksj izo`Ùk gksrk gSA

7.8 (b) ω7.9 (a), (c)7.10 (a), (d)

7.11 lHkh lR; gS

7.12 (a) vlR;] ;g ˆ.k osQ vuqfn'k gksxkA(b) lR;

(c) lR;

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136

(d) vlR;] nks vyx&vyx v{kksa osQ ikfjr% cy&vk?kw.kks± dks tksM+us dk dksbZ vFkZ

ugha gksrkA

7.13 (a) vlR;] yacor~ v{k izes; osQoy iQydksa osQ fy, gh ykxw gksrk gSA

(b) lR;

(c) vlR;] z ,oa z” lekarj v{k ugha gSaA

(d) lR;

7.14 tc fiaM dh ÅèokZèkj Å¡pkbZ i`Foh dh f=kT;k dh rqyuk esa cgqr de gksrh gS rks ge

mls y?kq fiaM dgrs gSa vU;Fkk ;g foLrkfjr fiaM dgykrk gSA

(a) Hkou vkSj rkykc y?kq fiaM gSaA

(b) ,d xgjh >hy vkSj ioZr foLrkfjr fiaMksa osQ mnkgj.k gSaA

7.15 = ∑2

I m ri iA csyu dk laiw.kZ nzO;eku lefefr v{k ls R nwjh ij gksrk gS ijarq Bksl

xksys dk vfèkdka'k nzO;eku R ls de nwjh ij gksrk gSA

7.16 èkukRed <ky okekoÙkZ ?kw.kZu n'kkZrk gS tks ijaijk osQ vuqlkj èku fy;k tkrk gSA

7.17 (a) ii, (b) iii, (c) i, (d) iv

7.18 (a) iii, (b) iv (c) ii (d) i.

7.19 th ugha] i =F 0i∑ fn;k gS

∴ fdlh ¯cnq '0' osQ ifjr% cy&vk?kw.kks± dk ;ksx

0i ii× =∑r F

fdlh vU; ¯cnq O′, osQ ikfjr% cy&vk?kw.kks± dk ;ksx

( )– –× = × ×∑ ∑ ∑ Fi i i i ii i ir a F r F a

nwljk in 'kwU; gks ;g vko';d ugha gSA

7.20 fdlh ifg;s essa vfHkosaQnzh cy vkarfjd izR;kLFk cyksa osQ dkj.k mRiÂ gksrs gSa tks ,dlefur ra=k dk vax gksus osQ dkj.k ;qXeksa esa ,d nwljs dks fujLr dj nsrs gSaA

vkèks ifg, esa] nzO;eku osaQnz (?kw.kZu v{k) osQ pkjksa vksj nzO;eku&forj.k] lefer ugha

gSA blfy, dks.kh; laosx dh fn'kk dks.kh; osx dh fn'kk osQ laikrh ugha gksrh vkSj

blfy, ?kw.kZu tkjh j[kus osQ fy;s ,d cká cy&vk?kw.kZ dh vko';drk gksrh gSA

7.21 th ugha] dksbZ cy osQoy vius yacor~ fn'kk esa gh cy vk?kw.kZ mRiUu dj ldrk gS

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137

= ×r f x-y-

z y-

7.22 CM ‘b’ ( 1) 10

1

n mb mab a

mn n

7.23 (a)2

2M

a

cos0 0 0 0

a axdm

x r rdrd rdrddm r r

02 sin

0 0

0 0

ar dr

ardr d

sin0 0 0 0

a aydmy r rdrd rdrd

dm r r

30

2

2 sin4 4cos0 0

3 3 3

0 0

./2

ar dr d

a a aardr d a

(b) 2

2

4M

a

7.24 (a)

(b)

1 1 2 2I I I

1 1 2 2

1 2

I I

I I

(c)

2 21 1 2 2 1 1 2 2

1 2 21 21 2

( ) ( )1 1( )

2 2( )f

I I I IK I I

I II I

2 21 1 2 2

1( )

2iK I I

21 21 2

1 2

( )2( )f i

I IK K K

I I

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(d) xfrt ÅtkZ esa ßkl nksuksa pdfr;ksa osQ chp ?k"kZ.k cy osQ fo#¼ fd, x, dk;Z osQ

dkj.k gksrk gSA

7.25 (a) 'kwU; (b) de gksrk gS (c) c<+rk gS (d) ?k"kZ.k (e) .=cmv Rω

(f) ?k"kZ.k osQ dkj.k nzO;eku osaQnz esa mRiUu Roj.k

= =mgR

k

I I

µτα

v u a t v gtcm cm cm cm kµ∴ = + ⇒ =

rFkk ko o

mgRt t

I

µω ω α ω ω= + ⇒ = −

fcuk fiQlys yq<+dus osQ fy,

cm Ko

v mgRt

R I

µω= −

K KO

gt mgRt

R I

µ µω= −

2

1

o

k

Rt

mRg

I

ω

µ

=

+

7.26 (a)

F' F

F''

F

laioZQ fcanq ij

osx

F' F

F''

F

ck¡;sa Mªe ij cy (Åij dh vksj)

nkfgus Mªe ij cy (uhps dh vksj)

F

F

(b) ′ ′′= =F F F tgk¡ F ,oa ′′F vkèkkj ls gksdj xqtjus okys cká cy gSA

FusV

= 0

cká cy vk?kw.kZ = F × 3R, okekoÙkZ

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139

(c) ekuk fd ω1 ,oa ω

2 vafre dks.kh; osx gSa (Øe'k% okekorZ ,oa nf{k.kkorZ)

var esa ?k"kZ.k ugha jgsxk

⇒ R ω1 = 2 R ω

2

1

2

2ω

ω⇒ =

7.27 (i) oxZ dk {ks=kiQy = vk;r dk {ks=kiQy ⇒ c2 = ab

22 2

2 2 21

yRxR

xS yS

II b a ab

I I c c c

× = × = =

(i) ,oa (ii) 1yR yRxR

yS xS yS

I II

I I I> ⇒ >

rFkk 1xR

xS

I

I<

(iii) ( )2 2 2– 2∝ + −zr ZSI I a b c

2 2 2 0= + − >a b ab

∴ − >

∴ >

( ) 0

1

zR zS

zR

zS

I I

I

I

7.28 ekuk fd pdrh osQ nzO;eku osaQnz dk Roj.k ‘a’ gS] rc

Ma = F-f (1)

pdrh dk dks.kh; Roj.k α = a/R gS (;fn pdrh fiQlyrh ugha gS)] rc21

2RfMR α

=

(2)

⇒ Ma = 2f

bl izdkj f = F/3 D;ksafd pdrh fiQlyrh ugha gS rks

⇒ f ≤ µmg

3 .⇒ ≤F Mgµ

vè;k; 8

8.1 (d)

8.2 (c)

8.3 (a)

8.4 (c)

8.5 (b)

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140

8.6 (d)

8.7 (d)

8.8 (c)

8.9 (a), (c)

8.10 (a), (c)

8.11 (a), (c), (d)

8.12 (c), (d)

8.13 (c), (d)

8.14 (a), (c), (d)

8.15 (a), (c)

8.16 (d)

8.17 tSls isM+ ls fxjrk gqvk lso ÅèokZèkj fn'kk esa uhps dh vksj xq#Ro cy dk vuqHko djrkgS oSls gh v.kq Hkh ÅèoZèkjr% uhps dh vksj xq#Ro cy dk vuqHko djrs gSaA rkih; xfrosQ dkj.k] tks fd ;kn`fPNd gksrh gS] buosQ osx ÅèoZèkj fn'kk esa ugha gksrsA xq#Ro osQuhps dh vksj yxus okys cy osQ dkj.k ok;q dk ?kuRo Hkwry osQ fudV vfèkd gksrkgS vkSj tSls&tSls ge i`Foh dh lrg ls Åij tkrs gSa bldk eku de gksrk tkrk gSA

8.18 osaQnzh; cy µ ,d ¯cnq nzO;eku dk xq#Rokd"kZ.k cy] ¯cnq vkos'k osQ dkj.k fLFkj

fo|qrh; cyA

vosaQnzh; cy% fLiu&fuHkZj ukfHkdh; cy] nks èkkjkokgh ywiksa osQ chp pqacdh; cyA

8.19 {ks=kh;

osx

8.20 ;g ml ry osQ vfHkyacor~ gksrk gS ftlesa i`Foh vkSj lw;Z fo|eku gksrs gSa] D;ksafd]

{ks=kh; osx

1.

2

∆= × ∆

∆

Ar v t

t8.21 bldk eku mruk gh cuk jgrk gS D;ksafd xq#Rokd"kZ.k cy nzO;ekuksa osQ chp osQ ekè;e

ij fuHkZj ugha djrkA

8.22 th gk¡] fiaM esa nzO;eku rks ges'kk cuk jgrk gS] ijarq bl ij yxus okyk xq#Rokd"kZ.k

cy 'kwU; gks ldrk gS tSlk fd rc gksrk gS tc bl fiaM dks i`Foh osQ osaQnz ij j[kk

tkrk gSA

8.23 th ughaA

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141

8.24 th gk¡] ;fn varfj{k;ku dk lkbt bruk vfèkd gks fd g esa gksus okys ifjorZu dk irk

py ldrk gksA

8.25

F

0 R r

8.26 milkSj fLFkfr esa] D;ksafd rc i`Foh osQ {ks=kh; osx vpj cuk, j[kus osQ fy, vfèkd

js[kh; nwjh r; djuh gksrh gSA8.27 (a) 90

o (b) 0

o

8.28 izfrfnu i`Foh viuh d{kk esa 1o vkxs c<+ tkrh gSA rc lw;Z dks eè; esa ykus osQ fy,

bldks 361° ?kw.kZu djuk gksxk (ftls ge 1 fnu ifjHkkf"kr djrs gSa)A D;ksafd 361° osQ

laxr 24 ?kaVs gksrs gSa_ vfrfjDr 1° yxHkx 4 feuV (3 feuV 59 lsoaQM) osQ laxr

gksxhA

8.29 ekuk fd eè; esa j[ks nzO;eku m dks ,d vYi nwjh h nkfguh vksj gVk;k

x;kA rks bl ij yxs cy gSa ( )2

G m

−

M

R h nkfguh vksj rFkk ( )2

GMm

R h+ ck;ha

vksjA igys cy dk ifjek.k nwljs cy ls vfèkd gSA vr% oqQy cy nkfguh

vksj gksxkA vr% larqyu vLFkk;h gSA

8.30

8.31 i`Foh osQ xq#Rokd"kZ.k cy osQ vèkhu fdlh d.k (i`Foh osQ ckgj xfr osQ fy;s) dk

xeu iFk ,d 'kkado gksxk ftldk iQksdl i`Foh dk osaQnz gksxkA osQoy (c) gh bl 'krZ

dks iwjk djrk gSA

8.32 mgR/2.

8.33 osQoy {kSfrt ?kVd (vFkkZr~ m rFkk 0 dks feykus okyh js[kk osQ vuqfn'k ?kVd) gh

cpsxkA oy; osQ fdlh ¯cnq ij cy dk {kSfrt ?kVd ftl xq.kd }kjk ifjofrZr gksrk

gS og gSµ

KE

R

M MR

h

10R

m

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142

A

B

CD

E

F

H

G

J

( ) ( )3/2 3/22 2 2 2

2

4

r

r r r r

µ

+ +

8.34 tSls&tSls r dk eku c<+rk gSµ

GMmU

r

= −

dk eku c<+rk gSA

c

GMv

r

=

dk eku ?kVrk gSA

32

1cv

r rω

= ×

dk eku ?kVrk gSA

K dk eku ?kVrk gS] D;ksafd v dk eku c<+rk gSA

E dk eku c<+rk gS] D;ksafd U = 2K rFkk U < O

l dk eku c<+rk gS] D;ksafd mvr ∝ .r

8.35 AB = C

(AC) = 2 AG =3

2. . 32

l l=

AD = AH + HJ + JD

2 2

l ll= + +

= 2l.

3AE AC l= = , AF = l

F ,oa B ij j[ks nzO;eku m osQ dkj.k AD osQ vuqfn'k cy2

2 2

22 2

1 11 1

2 2

GmGm Gm

ll l

= + =

E ,oa C ij j[ks nzO;ekuksa osQ dkj.k AD osQ vuqfn'k cy

( )= + °�

2 2

2 2cos cos(30 )30

3 3

Gm Gm

l l2 2

2 23 .

3 3= =

Gm Gm

l l

D ij j[ks nzO;eku M osQ dkj.k cy

=2

24

Gm

l

∴ oqQy cy =

+ +

2

2

1 11

43

Gm

l

4 2

5 5=

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8.36 (a)

12 3

2

GMTr =

4

π

π

∴

12 3

2

GMTh = – R

4

= 4.23x107 – 6.4x106

= 3.59×107m.

(b) θ = –1 R

cosR + h

= –1

1cos h1+

r

= 1 1

1+5.61

–cos

= 81º18′

∴ 2θ = 162º36′

o3602.21;

2θ≈

vr% U;wure la[;k = 3

8.37 tc i`Foh viuh d{kk esa lw;Z dh ifjØek djrh gS rks bldk dks.kh; laosx rFkk {ks=kh;

osx vpj jgrs gSaA

miHkw fLFkfr esa 2p pr ω viHkw fLFkfr esa 2

a ar ω

;fn ‘a’ i`Foh dh d{kk dh v¼Z eq[;&v{k gks rks ( )1pr = a - e rFkk ( ) .1ar = a + e

a

1

1

2p + e

=- e

ω

ω ∴ tgk¡ e = 0.0167

p

a

=1.0691ω

∴ω

ekuk fd ω dks.kh; pky gS tks fd ωp , ,oa ω

a dk

T;kferh; ekè; gS rFkk ekè; lkSj fnol osQ laxr jkf'k gSA

ωω ωω

ω ω

ω ω

p

a

p

a

=1.0691

= =1.034

∴

∴

q

E

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144

;fn ω 1° izfrfnu (ekè; dks.kh; pky) osQ laxr gks rks ωp = 1.034º izfrfnu rFkk

ωa = 0.967º izfrfnuA pw¡fd 361° = 24 ?kaVs % ekè; lkSj fnol blfy, 361.034º

= 24 ?kaVs 8.14″ (8.1″ vfèkd) gksxk rFkk 360.967º = 23 ?kaVs 59 feuV 52 lsoaQM

(7.9″ de) gksxkA

;g iwjs o"kZ esa fnu dh vofèk esa vkus okys okLrfod varj dh O;k[;k ugha djrkA

8.38 (1 ) 6ar a e R= + =

(1 ) 2pr a e R= − = 1

2e⇒ =

dks.kh; laosx laj{k.k

miHkw fLFkfr esa dks.kh; laosx = viHkw fLFkfr esa dks.kh; laosx

∴ =p p a am v r m v r

∴ =1

3a

p

v

v

ÅtkZ laj{k.k

miHkw fLFkfr esa ÅtkZ = viHkw fLFkfr esa ÅtkZ

2 21 1

2 2p a

p a

GMm GMmmv mv

r r− = −

21 11

219

pa p

v GMr r

−∴ = −−

1 1

2a p

GMr r

−=

1/2 1/2

2

1 1 2 1 122 6

111

9

p a

p

a

p

GMGMr r R

vv

v

− − = = − −

= = =

1/2

–12/3 36.85km s

8/9 4

GM GM

R R

= –16.85km spv , = –12.28km sav

= = = –16 , 3.23km s6

c

GMr R v

RoQs fy,

vr% viHkw fLFkfr esa o`Ùkkdkj d{kk esa LFkkukarj.k djus osQ fy, gesa osx esa

∆ = (3.23 – 2.28) = 0.95 km s–1 dh o`f¼ djuh gksxhA ,slk mixzg ls mi;qDr

jkWosQV nkx dj fd;k tk ldrk gSA

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vè;k; 9

9.1 (b)

9.2 (d)

9.3 (d)

9.4 (c)

9.5 (b)

9.6 (a)

9.7 (c)

9.8 (d)

9.9 (c), (d)

9.10 (a), (d)

9.11 (b), (d)

9.12 (a), (d)

9.13 (a), (d)

9.14 bLikr

9.15 th] ugha

9.16 rk¡ck

9.17 vuar

9.18 vuar

9.19 ekuk fd inkFkZ dk ;ax xq.kkad y gS] rc2/

/=

f rY

l L

π

ekuk fd nwljs rkj dh yackbZ esa o`f¼ l ′ gS] rc

2

2

4/2

f

r Yl L

π =′

;k, 2

1 22

4

fl L

Y rπ′ =

π

π= × =

2

2

22

4

l r fL l

L f r

18-04-2018


146

9.20 rki o`f¼ osQ dkj.k NM+ dh izfr bdkbZ yackbZ esa o`f¼ gksrhµ

5 2 –3

0

10 2 10 2 10l

Tl

α − −∆= ∆ = × × = ×

ekuk fd NM+ esa laihMd ruko T gS vkSj blosQ vuqizLFk dkV dk {ks=kiQy a gS] rc

0

/

/=

∆

T aY

l l

11 3 –42 10 2 10 10−∆∴ = × = × × × ×

lT Y a

lo44 10 N= ×

9.21 ekuk fd xgjkbZ h gS] rks nkc gS

P = ρ gh = 103 × 9.8 × h

vc /

=∆

PB

V V

8 –29.8 10 0.1 10V

P BV

∆∴ = = × × ×

8 –22

3

9.8 10 0.1 1010 m

9.8 10

× × ×∴ = =

×h

9.22 ekuk fd yackbZ esa o`f¼ l∆ gS] rc

( )11

–6

8002 10

( 25 10 )/ /9.1= ×

× × ∆lπ

–6 11

9.1 800m

25 10 2 10

×∴ ∆ =

× × × ×l

π

–30.5 10 m×�

9.23 pw¡fd xhyh feêðh dh xsan dh rqyuk esa gkFkh nk¡r dh xsan vfèkd izR;kLFk gSA la?kV~V

osQ i'pkr~ ;g rqjar viuk iwoZ vkdkj izkIr djus dh vksj izo`r gksxhA blfy, xhyh

feV~Vh dh xsan dh rqyuk esa blesa ÅtkZ ,oa laosx dk LFkkukarj.k vfèkd gksxkA vr%

la?kV~V osQ i'pkr~ gkFkh nk¡r dh xsan vfèkd Åij mBsxhA

9.24 ekuk fd NM+ dh vuqizLFk dkV {ks=kiQy A gSA lery aa ′ osQ larqyu ij fopkj

dhft,A bl lery ij vfHkyac ON ls dks.k (2

πθ− ) cukrk gqvk dksbZ cy F dk;Z

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147

djuk pkfg,A F dks lery osQ vuqfn'k vkSj blosQ vfHkyacor~ fo;ksftr djus ij%

cos=PF F θ

sin=NF F θ

ekuk fd lery aa ′ dk {ks=kiQy ′A gS] rc

sin=′

A

Aθ

sin′∴ =

AA

θ

ruu&izfrcy 2sin

sin= =′

F FT

A A

θθ rFkk vi:id izfrcy

cos=

′

FZ

A

θ cos sin=

F

Aθ θ θ

= 2

2

F sin

AA

vr% vfèkdre ruu izfrcy osQ fy, πθ =2 rFkk vfèkdre vi:id izfrcy osQ

fy, 2 /2=θ π vFkkZr~ /4=θ π

9.25 (a) Hkkj ls x nwjh ij (x = 0) ,d lw{eka'k dx yhft,A ;fn T (x) ,oa T (x + dx) Øe'k%

dx nwjh }kjk i`FkDÑr nks ifjPNsnksa ij rukoksa osQ eku gksa] rc

T (x +dx) – T(x) = gdxµ (tgk¡ µ bdkbZ yackbZ dk nzO;eku)

dTdx = gdx

dxµ

( )⇒ = +T x gx Cµ

= = ⇒ =0, (0)x T Mg C Mgij

( )T x gx Mgµ∴ = +

ekuk dh x fLFkfr ij yackbZ dx esa dr o`f¼ gksrh gS] rc

=d

d

d 1 =d

T(x)A Y

rx

rT(x)

x YAvFkok

a

a¢

FN

FP

N

FO

q

pq

/2 –

x = 0

Mg

dx

x

18-04-2018


148

L

0

L2

0

1= ( + )d

1= +

2

1= +

2

∫

r gx Mg xYA

gxMgx

YA

mglMgL

YA

µ

µ

⇒

(m rkj dk nzO;eku gS)

–3 2 2× (10 ) mA = π , 9 –2200 ×10 NmY =

–3 2×(10 ) ×10 × 7860kgm =π

π11 –6

1r =

2 ×10 × ×10∴

–7× 786 ×10 ×10 ×10+ 25 ×10 ×10

2

π

[ ]–6 –3196.5×10 +3.98×10= -34×10 m∼

(b) vfèkdre ruko x = L ij gksxk

T= µ gL+Mg = (m+M)g

uE;d cy

–3 2(10 ) 250 Nπ π6= 250 ×10 × × = ×

uE;u dh fLFkfr esa

(m + M)g = 250 × π

–3 2= × (10 ) ×10 ×7860 <<m Mπ 250Mg π∴ ×∼

vr% M = 250

25 75kg.10

×= × ∼

ππ

9.26 r ij dksbZ lw{eka'k dr yhft,A ekuk T (r) ,oa T (r+dr) blosQ nksuksa fljksa ij ruko osQ

eku gSaA

– T (r+dr) + T (r) = 2µω rdr tgk¡ µ izfr bdkbZ yackbZ nzO;eku gSA2dT

– dr = rdrdr

µω

2 2

22 2

At r = l T =0

lC =

2

T(r) = (l - r )2

µω

µω

⇒

∴

18-04-2018

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149

ekuk fd lw{eka'k dr dh yackbZ esa o`f¼ d ( δ ) gSµ

( ) ( )2 2 2 /A/2 -Y =

d( )

d

µω

δl r

r2

2 2

22 2

22 2

0

2 32 3 2 23

d( ) 1= ( - )

d 2

1d( ) = ( - )

2

1= ( - )

2

1 1 1= = =-

2 3 33

δ µω

µωδ

µωδ

µωµω µω

∴

∴

∴ ∫

l

l rr YA

l r drYA

l r drYA

ll ll

YA YA YA

yackbZ eas oqQy o`f¼ 2 222

3l

YAδ µω=

9.27 ekuk l1 = AB, l

2 = AC, l

3 = BC

2 2 23 1 2

3 1

cos2

l l l

l lθ

+ −=

;k 2l3l1 cos θ = l

32 + l

12 - l

22

Differenciating

( )3 1 1 3 1 32 cos 2 sinl dl l dl l l dθ θ θ+ −

3 3 1 3 1 1 2 22 2 2 2l dl l dl l lα α= + + −

vc]1 1 1

2 2 1

3 3 2

dl l t

dl l t

dl l t

α

α

α

= ∆

= ∆

= ∆

rFkk l1 = l

2 = l

3 = l

( ) 2 2 2 22 21 1 21 1

cos sinl d l t l t l tl t l t θ θ θ α α αα α + = ∆ + ∆ − ∆∆ + ∆

vFkok 1 2sin (1 cos )d t tθ θ α θ α= 2 ∆ − − ∆

θ = 60o j[kus ij

( )1 2

32 1/2

2= ∆ × − ∆d t tθ α α

vFkok 1 22( )

3

td

α αθ

− ∆=

r

drl

B C

A

l

(A )l

(Cu)

(Cu)

60°

l1

l

l1

l2

l

( )1 2 tα α= − ∆

18-04-2018


150

h

W

r

h/2

d A

R

d BC

9.28 tc o`{k >qdus gh okyk gksrk gS ml le;4

4=

Y rWd

R

π

;fn R � h, xq#Ro osaQnz dh Hkwfe ls Å¡pkbZ gksxh �1

2l h

∆ ABC ls

− +

�

212 2( )2

R R d h

;fn �d R

− +�12 2 224

R R Rd h

∴ =2

8

hd

R

;fn 0w Hkkj/vk;ru dks O;Dr djs rksπ

π=4 2

20 ( )

4 8

Y r hw r h

R R

1/3

2/32r

o

Yh

w

⇒

�

9.29 (a) L yackbZ rd iRFkj xq#Ro osQ vèkhu Lora=krkiwoZd fxjrk gSA blosQ i'pkr~ Mksjh dh

izR;kLFkrk bldks SHM djus osQ fy, ckè; djsxhA ekuk fd iRFkj y ij rkR{kf.kd :i

ls fojke esa vkrk gSA iRFkj dh P.E. esa vkus okyh deh rkfur Mksjh esa P.E. osQ :i

esa laxzfgr gks tk,xhA21

( )2

mgy k y L= −

vFkok 2 21 1

2 2mgy ky kyL kL= − +

vFkok − + + =2 21 1( ) 0

2 2ky kL mg y kL

2 2 2( ) ( )kL mg kL mg k Ly

k

+ ± + −=

2 2( ) 2kL mg mgkL m g

k

+ ± +=

èkukRed fpÉ cuk, j[ksa rks

2 2( ) 2kL mg mgkL m gy

k

+ + +∴ =

18-04-2018

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151

(b) iRFkj vfèkdre osx ml le; izkIr djrk gS tc ;g ^larqyu voLFkk** ls xqtjrk gS

vFkkZr~ tc rkR{kf.kd Roj.k 'kwU; gksrk gSA vFkkZr~

mg - kx = 0, tgk¡ x yackbZ L esa o`f¼ gSA

⇒ mg = kx

ekuk fd osx v gS] rc

2 21 1

( )2 2

mv kx mg L x+ = +

2 21 1( )

2 2mv mg L x kx= + −

vc mg = kx, =2m g

xk

2 22

2

1 1

2 2

m gmgmv mg kL

kk

∴ = −+

2 2 2 21

2

m g m gmgL

k k= + −

2 221 1

2 2

m gmv mgL

k= +

2 22 /v gL mg k∴ = +

2 1/2(2 / )v gL mg k= +

(c) ekuk fd fdlh {k.k fo'ks"k ij d.k y fLFkfr ij gS] rc

2

2( )= − −

md ymg k y L

dt

2

2( ) 0

d y ky L g

mdt⇒ + − − =

( )k

z y L gm

= − − j[kdj pj jkf'k dks :ikarfjr djus ijµ

2

20

d z kz

mdt+ =

cos( )z A tω φ∴ = + tgk¡ k

mω =

cos( )m

y A tL gk

ω φ ′⇒ = + ++

vr% iRFkj ¯cnq 0

my L g

k= + osQ ifjr% dks.kh; vko`fÙk ω ls SHM djsxkA

18-04-2018


152

vè;k; 10

10.1 (c)

10.2 (d)

10.3 (b)

10.4 (a)

10.5 (c)

10.6 (a), (d)

10.7 (c), (d)

10.8 (a), (b)

10.9 (c), (d)

10.10 (b), (c)

10.11 th] ughaA

10.12 th] ughaA

10.13 ekuk fd I;koh fge'kSy dk vk;ru V gSA bl fge'kSy dk Hkkj iρ Vg gksxkA ;fn bldk

fuefTtr va'k x gks] rks foLFkkfir ty dk vk;ru xV gksxkA rc bl ij vkjksfir

mRIykod cy ρwxVg gksxk] ρw ty dk ?kuRo gSA

mRIykou fu;e vuqlkjµ

iρ Vg = ρw

xVg

w

0.917ix∴ = =ρ

ρ

10.14 ekuk fd x fLizax esa laihMu gSA D;ksafd xqVdk larqyu esa gSA

Mg – (kx + ρwVg) = 0

tgk¡ ρw ty dk ?kuRo rFkk V xqVosQ dk vk;ru gSA rqyk dk ikB ty }kjk iyM+s ij

yxk, cy dk eki gS] vFkkZr~

mik=k

+ mty

+ ρwVg crkrk gSA

D;ksafd xqVosQ dks ty esa ykus ls igys rqyk dks 'kwU; ikB osQ fy, leaftr fd;k x;k

Fkk] u;k ikB gksxk ρwVg

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153

10.15 ekuk fd ty dk ?kuRo ρw gSA

rc ρaL3 + ρL3g = ρw

xL3 (g + a)

w

x∴ =ρ

ρ

vr% xqVosQ dk fuefTtr va'k Roj.k ij fuHkZj ugha djrk gS] pkgs ;g xq#Ro osQ vèkhu

gks ;k fyÝV esaA

10.16 jl ftl Å¡pkbZ rd Åij mBsxk og gSµ2

3 5

2 cos0 2(7.2 10 )

10 9.8 2.5 10ρ

−

−

° ×= =

× × ×

Th

gr 0.6m�

;g og vfèkdre Å¡pkbZ gS ftl rd bl i`"B ruko osQ vèkhu Åij mB ldrk gS]

D;ksafd vusd o`{kksa dh Å¡pkbZ blls vfèkd gksrh gS osQoy dksf'kdRo osQ izHkko ls lHkh

o`{kksa esa ty Åij rd p<+us dh O;k[;k ugha dh tk ldrhA

10.17 ;fn VSadj èku x fn'kk esa Rofjr gksrk gS rks ty VSad esa ihNs dh vksj ,df=kr gks tk,xkA

eqDr i`"B ,slk gksxk fd blosQ fdlh Hkh va'k ij [khaph xbZ Li'kZ js[kk osQ vuqfn'k yxus

okyk cy 'kwU; gksxkA

i`"B ij bdkbZ vk;ru osQ ,d va'k ij fopkj dhft,A rjy ij yxus okys cy gSaµ

ˆ ˆg a− −y xρ ρ,oa

i`"B osQ vuqfn'k Hkkj dk ?kVd gS% ρ g sin θ

i`"B osQ vuqfn'k Roj.k cy dk ?kVd gSµ

g a cos θ

sin cosg aρ θ ρ θ∴ =

vr% tan θ =a/g

18-04-2018


154

10.18 ekuk fd v1 ,oa v

2 ¯cnqdkvksa osQ vk;ru gSa vkSj v ifj.kkeh cw¡n dkA

rc v = v1+v

2

( )3 3 3 3 31 2 cm 0.009cm0.001 0.008r r r⇒ = + = =+

0.21cmr∴ �

( )( )2 2 21 2

4U T r r rπ∴ ∆ = − +

( )3 424 435.5 10 10 J0.21 0.05π − −= × × ×−

32−� × 10–7 J

10.19 3 3R Nr=

1/3

Rr

N⇒ =

2 24 ( )∆ = −U T R Nrπ

ekuk fd laiw.kZ ÅtkZ osQ foeqDru ls rki esa deh vkrh gSA ;fn s fof'k"V Å"ek gks

rks rkikarj gksxkµ

( )2 2

3

4

4

3

TU R Nr

msR s

∆ −∆ = =

πθ

π ρ , tgk¡ ρ ?kuRo gSA

2

3

3 1T rN

s R Rθ

ρ

∴ ∆ = −

2 3

3 3

3 3 1 11T Tr R

s s R rR R rρ ρ

= = −−

10.20 cw¡n rHkh ok"i esa cnysxh tc ty dk nkc ok"i nkc ls vfèkd gks tk,xkA f>fYydk

nkc (ty osQ dkj.k)µ32

2.33 10 PaT

pr

= = ×

2

3

2 2(7.28 10 )

2.33 10

Tr

p

−×∴ = =

× = 6.25 × 10-5 m

10.21 (a) ifjPNsn&{ks=k A rFkk Å¡pkbZ ∆h osQ ok;q osQ ,d {kSfr”k y?kq va'k ij fopkj

dhft,A ekuk fd blosQ Åijh rFkk fupys i`"B ij nkc Øe'k% p ,oa P + dp

gSA ;fn ;g va'k larqyu voLFkk esa gS rks bl ij Åij dh vksj yxk cy blosQ

Hkkj }kjk larqfyr gksuk pkfg,A

p

p + dp

18-04-2018

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155

vFkkZr (p+dp)A–pA = – PgA dh

⇒ dp = – ρ gdh.

(b) ekuk fd Hkw&ry osQ fudV ok;q dk ?kuRo ρo gS] rc

o o

p

p

ρ

ρ=

o

o

pp

ρρ⇒ =

o

o

gdp pdh

p

ρ∴ = −

o

o

gdpdh

p p

ρ⇒ = −

o

p ho

op o

gdpdh

p p

ρ⇒ = −∫ ∫

ln o

o o

gph

p p

ρ⇒ = −

exp oo

o

ghp p

p

ρ −⇒ =

(c)1

ln10

oo

o

gh

p

ρ= −

1ln

10o

o

o

ph

gρ∴ = −

2.303o

o

p

gρ= ×

55 31.013 10

2.303 0.16 10 m 16 10 m1.29 9.8

×= × = × = ×

×

(d) ;g ekU;rk fd p ρ∝ osQoy lerkih; izdj.kksa osQ fy, gh ;qfDr;qDr gS tks

osQoy vYi nwfj;ksa osQ fy, gh oSls jg ikrs gSaA

10.22 (a) 1 kg ty dks Lv fdyks oSQyksjh dh vko';drk gksrh gS

∴MA kg ty dks M

AL

v fdyks oSQyksjh dh vko';drk gksxhA

pw¡fd MA kg ty esa N

A v.kq gSa] v.kq osQ ok"ihdj.k osQ fy, vko';d ÅtkZ gksxhµ

JA v

A

M Lu

N=

×=

90

18 540 × × 34.2 10

6 × 26J

10

18-04-2018


156

= 90 × 18 × 4.2 × 10–23 J

� 6.8 × 10–20 J

(b) ty osQ v.kqvksa dks ¯cnqvksa dh Hkk¡fr eku yhft, tks ,d nwljs ls d nwjh ij gSa]

NA v.kq

A

w

M

ρ yhVj vk;ru ?ksjrs gSa

vr% ,d v.kq osQ pkjksa vksj dk vk;ru A

A w

M

N ρ yhVj gksxkA

∴ ,d v.kq osQ pkjksa vksj dk vk;ru d3 = (MA/N

A ρ

w)

1/ 3 1/3

26 3

18

6 10 10

A

A w

Md

N ρ

∴ = =

× ×

( )1/3 1030 m 3.1 10 m30 10

−−= ×× �

(c) 1 kg ok"i 1601 × 10-3 m3 vk;ru ?ksjrh gS

∴18 kg ok"i 18 × 1601 × 10–3 m3 vk;ru ?ksjsxh

⇒ 6 × 1026 v.kq 18 × 1601 × 10–3 m3 vk;ru ?ksjrs gSa

∴ 1 v.kq -3

3

26

18 × 1601 × 10 m

6×10 vk;ru ?ksjxk]

;fn d ′ varj vkf.od nwjh gks] rks

′ 3( )d = ( 3 × 1601 × 10-29)m3

∴ d ′ = (30 × 1601)1/3 × 10-10 m

= 36.3 × 10-10 m

(d) F (d ′ -d) = u

2010

10

6.8 100.2048 10 N

' (36.3 3.1) 10

uF

d d

−−

−

×⇒ = = = ×

− − ×

(e)

10–1 2 –1

10

0.2048 10/ 0.066N m 6.6 10 N m

3.1 10F d

−−

−

×= = = ×

×

10.23 ekuk fd xqCckjs osQ vanj nkc Pi vkSj ckgj P

o gS] rks

Pi – P

o

2

r

γ=

ok;q dks ,d vkn'kZ xSl eku ysa rks

18-04-2018

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157

Pi V= n

i R T

i tgk¡ V xqCckjs osQ vanj dk vk;ru gS] n

i blosQ vanj eksyksa dh la[;k

gS vkSj Ti vanj dk rki gS rFkk P

o V= n

o R T

o tgk¡ V foLFkkfir ok;q dk vk;ru gS n

o

foLFkkfir ok;q esa eksyksa dh la[;k gS rFkk To ckgj dk rki gSA

ni

i i

i A

P V M

R T M= = tgk¡ M

i vanj dh ok;q dh nzO;eku gS rFkk M

A ok;q dk eksyj nzO;eku

gSA no o o

o A

P V M

R T M= = tgk¡ M

o foLFkkfir ckgjh gok dk nzO;eku gSA ;g blosQ }kjk

W nzO;eku dks Åij mBk ldrk gS] rks

W + Mi g = M

o g

⇒ W= Mo g – M

i g

ok;q esa 21% O2 rFkk 79% N

2 gksrh gSA

∴ ok;q dk eksyj nzO;eku MA = 0.21×32 + 0.79×28 = 28.84 g.

o iA

o i

P PM VW g

R T T

⇒ = −

=

340.02884 8 9.8

38.314

π× × ×

5 51.013 10 1.013 10 2 5

293 333 8 313

× × ×− −

× N

3

5

40.02884 8

1 13 1.013 10 9.8N8.314 293 333

× ×

× × − ×

�

π

= 3044.2 N

vè;k; 11

11.1 (d)

11.2 (b)

11.3 (b)

11.4 (a)

11.5 (a)

11.6 (a)

11.7 (d)

ewy vk;ru 4 3

o3

V R= π

jSf[kd o`f¼ xq.kkad = α

∴ vk;ru o`f¼ xq.kkad 3α�

18-04-2018


158

α=1

3dV

V dT

3dV V dTα⇒ = 34 R Tπ α ∆�

11.8 (c)

11.9 (b), (d)

11.10 (b)

11.11 (a), (d)

11.12 (b), (c), (d)

11.13 Å"ek&ik;Z

11.14 2 ,oa 3 xyr gSa] 4 lgh gSA

11.15 pkydrk esa varj osQ dkj.k] dk"B dh rqyuk esa /krqvksa dh pkydrk vfèkd gksrh gSA

Å¡xyh ls Li'kZ djus ij vkl&ikl dh Å"ek èkkrq esa gksdj rs”kh ls Å¡xyh esa igq¡prh

gS blfy, èkkrq xeZ vuqHko gksrh gSA blh izdkj tc O;fDr BaMh èkkrq dks Nwrk gS rks

Å¡xyh ls Å"ek èkkrq esa gksdj rsth ls okrkoj.k esa tkrh gS blfy, mls èkkrq BaMh

yxrh gSA

11.16 40 C 40 F− = −� �

11.17 D;ksafd rk¡cs dh pkydrk LVhy dh rqyuk esa

vfèkd gksrh gS] rk¡cs vkSj LVhy dh lafèk

rs”kh ls xeZ gks tkrh gS] ijarq mruh gh rs”kh

ls LVhy Å"ek dk pkyu ugha dj ikrk

ifj.kkeLo:i vanj j[kk Hkkstu ,d leku

:i ls Å"ek izkIr djrk gSA

11.1821

12=I Ml

2 2 21 1 1 1' ( ) 2 ( )

12 12 12 12I M l l Ml Ml l M l= + ∆ = + ∆ + ∆ α

212

12I Ml T≈ + ∆α

2I I Tα= + ∆

2I I Tα∴ ∆ = ∆

11.19 ty osQ P-T vkjs[k ij è;ku nhft,

vkSj blesa nksuksa vksj uksad okys rhj osQ

fpÉ dk lanHkZ yhft,A 0°C ij vkSj

1 atm. {ks=k dh voLFkk ls nkc

lafèk

LVhy

rkack

ykS

Bksl

nzo

xSl

18-04-2018

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159

c<+kus ij ciZQ nzo voLFkk esa cny tkrh gS vkSj blh voLFkk ls nkc ?kVkus ij ty

ciZQ esa cny tkrh gSA

tc oqQpyh gqbZ ciZQ dks nck;k tkrk gS rks bldk oqQN Hkkx fi?ky dj ty cu tkrk

gS tks fged.kksa osQ chp esa lek tkrk gSA nkc gVkus ij ;g ty fgehHkwr gksdj lHkh

fged.kksa dks vkil esa la;ksftr dj nsrk gS vkSj xsan dks vfèkd LFkk;h cuk nsrk gSA

11.20 ifj.kkeh feJ.k dk rki 0oC gks tkrk gSA 12.5 g ciZQ esa cny tkrk gS rFkk 'ks"k ty

osQ :i esa jgrk gSA

11.21 igys fodYi osQ vuqlj.k osQ ty vfèkd xeZ jgsxk] D;ksafd U;wVu osQ 'khryu fu;e

osQ vuqlkj] Å"ek gkfu dh nj fiaM osQ rki rFkk okrkoj.k osQ rki osQ varj osQ

lekuqikrh gksrh gS vkSj igys fodYi esa rkikarj de gS blfy, gkfu dh nj de jgrh gSA

11.22 lHkh rkiØeksa ij lyksgk –l

ihry = 10 cm

∴ lºyksgk (1 + a

yksgk ∆t) –lº

ihry(1 + a

ihry ∆t) = 10cm

lºyksgk a

yksgk = lº

ihry a

ihry

∴ = =�

�

1.8 3

1.2 2

l

l

yksgk

ihry

110cm 20 cm

2l l∴ = ⇒ =� �

i hry i hry

rFkk = 30 cm°l yksgk

11.23 yksgs osQ ik=k ftlesa vanj ihry dh NM+ yxh gks6

3.55

V

V=vk;ju

ihry

100ccV V Vo− = =vk;ju ihry

= =144.9cc 244.9ccV VNM+ vanjihry vk;ju

11.24 izfrcy = K × foÑfr

V

=KV

∆

=K(3 ) tα ∆

ihryvk;ju

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160

9 5140 10 3 1.7 10 20−= × × × × ×

8 21.428 10 N/m= ×

;g ok;qeaMyh; nkc ls yxHkx 103 xquk gSA

11.25

2 2

2 2 2

L L Lx

∆ = −+

1

22

L L≈ ∆

α∆ = ∆L L t

22

Lx tα∴ ≈ ∆

0.11m 11cm≈ →

11.26 fofèk - I

,d fljs (ftl ij rki θ 1 gS) ls x nwjh ij rki θ 1

gSµ

( )1 2 1

o

x

Lθ θ θ θ= + − : jSf[kd foHko izo.krk

dx0 yackbZ osQ y?kq va'k dh ubZ yackbZ

( )1odx dx αθ= +

( )1 2 1o oo

xdx dx

Lθ θ θα

+ −= +

vc = =∫ ∫o odx L dx LrFkk : ubZ yackbZ

lekdyu djus ijµ( )2 1

1o o o

o

L L L x dxL

θ θα θ α

−∴ = + + ∫

( )2 1

11

2oL α θ θ

= + +

D;kasfd

0L

20

0

1

2=∫ xdx L

fofèk - II

;fn rki yackbZ osQ lkFk jSf[kdr% c<+rk gS rks ge eku ldrs gSa fd vkSlr rki

( )1 2

1

2θ θ+ gS vkSj blfy, ubZ yackbZ ( )2 1

11

2oL L α θ θ

= + +

gSA

11.27 (i) 1.8 × 1017 J/S (ii) 7 × 109 kg (iii) 47.7 N/m2

dx

xq q1 2

18-04-2018

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161

vè;k; 12

12.1 (c) fLFkjks"e

A lenkfcd izØe gS] D le vk;rfud gSA B ,oa C esa ls] B dh izo.krk (dk

ifj.kke) de gSA vr% ;g lerkih; izØe 'ks"k cpk izØe fLFkjks"e gSA

12.2 (a) 0.251g

12.3 (c)

12.4 (b) –2 PV

12.5 (a)

12.6 (b)

12.7 (a), (b) ,oa (d).

12.8 (a), (d)

12.9 (b), (c)

12.10 (a), (c)

12.11 (a), (c)

12.12 ;fn ra=k izfros'k osQ fo#¼ bl izdkj dk;Z djas fd iznÙk Å"ek ls ÅtkZ iwfrZ gksrh jgs

rks rki vpj cuk jg ldrk gSA

12.13 p QU U− = iFk&1 esa ra=k }kjk fd;k x;k dk;Z + 1000 J

= iFk–2 esa ra=k }kjk fd;k x;k dk;Z + Q

(–100 1000)J 900 J= + =Q

12.14 ;gk¡ yh xbZ Å"ek nh xbZ Å"ek ls de gS] vr% jsfÚtjsVj (tks dejs ls foyfxr ugha

gS) lfgr dejs dk rki c<+ tk,xkA

12.15 th gk¡] tc xSl esa fLFkjks"e laihMu gksrk gS rks bldk rki c<+ tkrk gSA

dQ = dU + dW

D;ksafd (fLFkjks"e izØe esa) dQ = 0

∴ dU = -dW

18-04-2018


162

laihMu esa dk;Z ra=k osQ Åij fd;k tkrk gS blfy, dW ½.kkRed gksrk gSA

⇒ dU èkukRed gS

vr% xSl dh vkarfjd ÅtkZ esa o`f¼ gks tkrh gS vFkkZr bldk rki c<+ tkrk gSA

12.16 okgu pykrs le; xSl dk vk;ru rks fLFkj jgrk gS ijarq bldk rki c<+ tkrk gSA vr%

pkYlZ fu;e osQ vuqlkj fLFkj V ij P α T A

blfy, xSl dk nkc c<+ tkrk gSA

12.1732

1 2

1 1

3, 10 J

5

TQQ Q

Q T= = − =

3 21 1

5310 J 10 J1

25Q Q

= ⇒ = ×−

= 2500 J, 2 1500JQ =

12.18 35 7000 10 4.2J 60 15 10 N× × × = × × ×

63 321 7 10 147

10 16.3 10900 9

× ×= = × = ×N times.

12.19 ( ) ( )P V v P p Vγ γ+ ∆ = + ∆

1 1v p

P PV P

γ∆ ∆

=+ +

;v p dv V

V P dp pγ

γ

∆ ∆= =

W.D. ( )2 2

1 1

2 1

γ γ

−= = =∫ ∫

P P

P P

P PVp dv p dp V

p

12.20270 1

1300 10

η = − =

jsfÚtjsVj dh n{krk 1

0.520

η= =

;fn Q Å"ek dh og ek=kk gS tks izfr lsoaQM mPprj rki dh vksj LFkkukarfjr dh tkrh

gS] rks 1

20

W

Q= vFkok Q = 20W = 20kJ, rFkk fuEurj rki osQ dks"B ls gVkbZ xbZ

Å"ek =19 kJ.

12.21 2 5=Q

W, 2 15W, 6W= =Q Q

18-04-2018

mÙkjekyk

163

22

1

5, 250K 23 C

6 300= = = = − �T T

TT

12.22 izR;sd izdj.k osQ fy, P-V vkjs[k vko`Qfr esa n'kkZ;k x;k gSA izdj.k - (i) osQ fy,Pi Vi = Pf Vf = Pg Vg ; blfy, izØe-(i) lerkih; gSA fd;k x;k dk;Z P-V oØosQ uhps dk {ks=kiQy] vr% fd;k x;k dk;Z ml fLFkfr dh rqyuk esa vfèkd gSftlesa xSl fLFkj nkc ij foLrkfjr gksrh gSA

12.23 (a) xSl }kjk fd;k x;k dk;Z (ekuk PV1/2 = A)

( )22 2

1 1 1

2 12

1/2

∆ = = = = −

∫ ∫

VV V

V V V

dV VW pdv A A A V V

V

= 2 1/2 1/2 1/2

1 1 2 1P V V – V

(b) pw¡fd / .A

T pV nR VnR

= =

vr% 2 2

1 1

2T V

T V= =

(c) rc] vkarfjd ÅtkZ esa ifjorZu

( ) ( )2 1 2 1 1

3 32 1

2 2U U W R T T RT∆ = − = − = −

( ) ( )1 12 22 1 2 1W A V RT∆ = =− −

( )1(7/2) 2 1Q RT∆ = −

12.24 (a) A ls B

(b) C ls D

(c) WAB

= 0; 0= =∫B

CD

A

p dV W .

blh izdkj 1

1

C

B

VC C r

BC r

B B V

dV VW pdV k k

RV

− + = = =

− + ∫ ∫

1

( )1

c c B BP V P Vγ

= −−

blh izdkj γ

= −−

1( )

1DA A A D DW P V P V

vc] 2BC B B

C

VP P P

V

γ

γ− = =

izdj.k

izdj.k

18-04-2018


164

blh izdkj] PD = P

A 2-γ

fd;k x;k oqQy dk;Z = WBC

+WDA

( ) ( )1 11

2 1 2 11

B B A AP V P Vγ γ

γ− + − += − − −

−

11

(2 1)( )1

B A AP P Vγ

γ−= − −

−

( )2 /33 1

12 2

B A AP P V = −−

(d) izØe A, B osQ nkSjku nh xbZ Å"ek

dQAB

= dUAB

= − = −3 3

( ) ( )2 2

AB B A B A AQ nR T T P P V

n{krk =

231

12

= −

fd;k x;k oqQy dk;ZiznÙk Å"ek

12.253 3

( ) ( )2 2

AB AB B A A B AQ U R T T V P P= = − = −

Bc BC BCQ U W= +

( ) ( ) ( )3/2 B C B B C BP V V P V V= − + −

( )5/2 ( )B C AP V V= −

QCA

= 0

QDA

=( 5/2) PA (V

A-V

D)

12.26 (Vo, P

o) ij oØ P = f (V) dh izo.krk

= f (Vo)

(Vo, P

o) ij fLFkjks"e dh izo.krk

= k (-γ) Vo–1– γ = -γ P

o/V

o

izØe P = f (V) esa vo'kksf"kr Å"ek]

dQ = dV + dW

= nCvdT + P dV

18-04-2018

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165

pw¡fd T = (1/nR) PV = (1/nR) V f (V)

dT = (1/nR) [f (V) + V f ′ (V)] dV

vr%]

dQ

dV V = Vo

[ ]( ) '( ) ( )o o o o

CVf V V f V f V

R+= +

= 1 '( )

1 ( )1 1

o oo

V f Vf V

γ γ

+ + − −

'( )1 1

oo o

VP f V

γ

γ γ= +

− −

Å"ek rc vo'kksf"kr gksrh gS tc dQ/dV > 0, tc xSl dk vk;ru c<+rk gS vFkkZr tc

γ Po + V

of ′ (V

o) >0

f ′ (Vo) > -γ P

o/V

o

12.27 (a) i aP P=

(b) ( ) ( )= + − = + −f a o a o

kP P V V P k V V

A

(c) nh xbZ laiw.kZ Å"ek ;kaf=kd ÅtkZ esa :ikarfjr gks tkrh gSA (vkn'kZ xSl osQ

fy,) vkarfjd ÅtkZ esa dksbZ ifjorZu ugha gksrk

21( ) ( ) ( )

2∆ = − + − + −a o o V oQ P V V k V V C T T

tgk¡ To = P

a V

o/R,

T = [Pa+(R/A)-(V-V

o)]V/R

vè;k; 13

13.1 (b)

ppkZ osQ fy, fVIif.k;k¡µ ;gk¡ lkis{k xfr dh ladYiuk dk vfoHkkZo gksrk gS vkSj ;g

rF; egRoiw.kZ gks tkrk gS fd tc Hkh la?kV~V gksrk gS rks lkis{k osx esa gh ifjorZu gksrk

gSA

13.2 (d)

ppkZ osQ fy, fVIif.k;k¡µ vkn'kZ fLFkfr esa tks izk;% gekjh ppkZ osQ fo"k; gksrs gSa

izR;sd la?kV~V esa vfHkyacor~ laosx osQ ifj.kke esa nks ckj ifjorZu gksrk gSA iQyd

18-04-2018


166

EFGH ij bldk osQoy vkèkk Hkkx gLrkarfjr gksrk gSA

13.3 (b)

13.4 (c) ;g ,d fLFkj nkc ( )/p Mg A= O;oLFkk gSA

13.5 (a)

13.6 (d)

ppkZ osQ fy, fVIif.k;k¡µ vkn'kZ xSl fu;e dk lkekU; dFku ewyr% v.kqvksa osQ fy,

izLrqr fd;k x;k gSA ;fn dksbZ xSl ijek.kq :i esa gks (ftldh iwjh laHkkouk gS) vFkok

v.kqvksa vkSj ijek.kqvksa ls feydj cuh gks rks mlosQ fy, bl fu;e dks Li"V dFku

osQ :i esa izLrqr ugha fd;k x;k gSA

13.7 (b)

fVIif.k;k¡µ fdlh feJ.k esa vkSlr xfrt ÅtkZ,¡ cjkcj gksrh gSaA vr% muosQ osxksa dk

forj.k fcYoqQy fHkUu gksrk gSA

13.8 (d)

ppkZ osQ fy, fVIif.k;k¡µ bl vè;k; eas fLFkj nkc rFkk fLFkj vk;ru dh fLFkfr;ksa

ij ppkZ dh xbZ gSA ijarq okLrfod thou esa ,slh vusd fLFkfr;k¡ vkrh gSa ftuesa nksuksa

,d lkFk ifjofrZr gksrs gSaA ;fn lrgsa n`<+ gksa rks p c<+dj 1.1 p gks tk,xk rFkkfi]

tSls&tSls nkc c<+rk gS V Hkh c<+rk gS ftlls pv var esa 1.1 RT gks tkrk gS] tgk¡ pvafre

> p rFkk Vvafre

> VA vr% (d)A

13.9 (b),(d)

13.10 (c)

13.11 (a), (d)

fVIif.k;k¡µ vkius lehdj.k <LFkkukarjh; xfrt ÅtkZ> = ( )32

RT , <?kw.khZ ÅtkZ> =

RT i<+h gSA ij vHkh rd bl rF; ij tksj ugha fn;k x;k Fkk fd ;s nksuksa ,d nwljs

ls Lora=k gksrh gSaA ,d nwljs ls Lora=k :i ls os eSDlosy osQ fu;eksa dk vuqlj.k djrh gSaA13.12 (a), (c)13.13 (a)

fVIif.k;k¡µ izk;% fo|kfFkZ;ksa dks ;g ladYiuk Li"V ugha gksrh fd xfreku fiaM osQ

lkFk izR;kLFk la?kV~V djus ls fiaM dh ÅtkZ esa ifjorZu gks tkrk gSA

13.14 ∴ Lo.kZ dk eksyj nzO;eku 197 g mole–1 gS vkSj 1 eksy esa ijek.kqvksa dh la[;k

= 6.0 × 1023 gSA

∴ 39.4g esa ijek.kqvksa dh la[;k 23

236.0 10 39.41.2 10

197

× ×= ×

13.15 P dks vpj j[kus ls gesa izkIr gksrk gSµ

1 22

1

100 600200cc

300

×= = =

V TV

T

18-04-2018

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167

13.16 1 1 2 2

1 2

P V P V

T T=

1 2 1

2 1 2

2 300 3

400 2

V P T

V P T

×= = =

2 21 1 2 2

1 2

1 1;

3 3

− −= =M M

P c P cV V

2 2 2 22 1

1 1

V Pc c

V P∴ = × ×

2 2

(100) 23

= × ×

–12

200m s

3c =

13.17

2 21 2

2rms

v vv

+=

6 2 6 2(9 10 ) (1 10 )

2

× + ×=

−+ ×= = ×

126 1(81 1) 10

41 10 m s2

13.18 O2 ik¡p Lokra=;&dksfV dk v.kq gSA blfy, bldh izfr eksy ÅtkZ

5

2RT=

∴ O2 osQ 2 eksyksa dh ÅtkZ = 5RT

fu;kWu dh Lokra=; dksfV 3 gS ∴ bldh izfr eksy ÅtkZ 3

2RT=

∴ fu;kWu osQ 4 eksyksa dh ÅtkZ 34 6

2RT RT= × =

∴ oqQy ÅtkZ = 11RT.

13.19 2

1l

dα

1 21 2o o

d A Aα= =

1 2: 4 :1l l =

13.20 V1 = 2.0 litre V

2 = 3.0 litre

1µ = 4.0 moles 2µ = 5.0 moles

P1 = 1.00 atm P

2 = 2.00 atm

1 1 1 1 2 2 2 2P V RT P V RTµ µ= =

18-04-2018


168

1 2 1 2V V Vµ µ µ= + =

1 eksy osQ fy, 2

3PV E=

1µ eksy osQ fy, 1 1 1 1

2

3P V Eµ=

2µ eksy osQ fy, 2 2 2 2

2

3P V Eµ=

oqQy ÅtkZµ 1 1 2 2 1 1 2 2

3( ) ( )

2E E P V P Vµ µ+ = +

2 2

3 3total per molePV E Eµ= =

1 2 1 1 2 2

2 3( ) ( )

3 2P V V P V P V+ = × +

1 1 2 2

1 2

P V P VP

V V

+=

+ *

1.00 2.0 2.00 3.0

atm2.0 3.0

× + × =

+

8.0

1.60atm.5.0

= =

fVIif.k;k¡µ vafdr lehdj.k }kjk fu:fir vkn'kZ xSl fu;e dk ;g :i fLFkjks"e

ifjorZuksa osQ fy, cgqr mi;ksxh gSA

13.21 D;ksafd rki vkSj nkc dh n'kk,¡ leku gSa blfy, vkSlr K.E Hkh leku gksaxhA1

rmsvm

α

A B cm m m> >∵

C B Av v v> >

13.22 0.25 × 6 × 1023 v.kq gSa ftuesa ls izR;sd dk vk;ru 10–30m3 gSA

vkf.od vk;ru = 2.5 × 10–7m3

;g eku ysa fd vkn'kZ xSl fu;e ykxw fd;k tk ldrk gS rks

vafre vk;ru 3 6

7 3(3) 102.7 10 m

100 100inV −

−×= = ≈ ×

tks yxHkx mruk gh gS ftruk vkf.od vk;ruA vr% var%vkf.od cyksa dks ux.; ugha

ekuk tk ldrk gSA blfy, ;gk¡ vkn'kZ xSl dh fLFkfr ugha gSA

13.23 tc gok Hkjh tkrh gS rks vfèkdkfèkd v.kq vanj izos'k djrs gSaA ckW;y osQ fu;e dk

dFku ml fLFkfr osQ fy, fd;k x;k gS ftlesa v.kqvksa dh la[;k fLFkj jgrh gSA

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13.24 5.0µ =

T = 280K

ijek.kqvksa dh la[;k 235.0 6.02 10ANµ= = × ×

= 30 × 1023

izfr v.kq vkSlr xfrt ÅtkZ 3

2= kT

∴ oqQy vkarfjd ÅtkZ 3

2= ×kT N

23 233

30 10 1.38 10 2802

−= × × × × ×

= 1.74 × 104 J

13.25 NTP ij xSl osQ 1g eksy dk vk;ru = 22400cc

∴ gkbMªkstu osQ 1 CC esa v.kqvksa dh la[;k

23196.023 10

2.688 1022400

×= = ×

D;ksafd izR;sd f}&ijekf.od v.kq dh 5 Lokra=; dksfV gksrh gaS] gkbMªkstu dh Hkh

D;ksafd ;g f}&ijekf.od gS 5 Lokra=; dksfV gksrh gSa]

∴ oqQy Lokra=; dksfV;ksa dh la[;k = 5 × 2.688 × 1019

= 1.344 × 1020

13.26 xSl dh xfrt ÅtkZ esa ßkl 21

( )2

oE mn v= ∆ =

tgk¡ n = eksyksa dh la[;k

;fn blosQ rki esa T∆ ifjorZu gksrk gS] rks23 1

2 2on R T mn v∆ = . ∴

2

3omv

TR

∆ =

13.27 panzek dk xq#Rokd"kZ.k cy cgqr {kh.k gksrk gS blfy, blosQ i`"B ls iyk;u osx dk

eku cgqr de gSA lw;Z ls ns[kus ij panzek i`Foh osQ cgqr fudV gSA panzek Hkh izfr bdkbZ

{ks=kiQy ij mruh gh Å"ek xzg.k djrk gS ftruh iFohA ok;q osQ v.kqvksa dk pky&ijkl

cgqr vfèkd gksrk gSA ;|fi ok;q osQ v.kqvksa dh rms pky panzek ij iyk;u osx ls

cgqr de gksrh gS fiQj Hkh ,d cM+h la[;k esa ,sls v.kq gksrs gSa ftudh pky iyk;u

osx ls vfèkd gksrh gS vkSj os iyk;u dj tkrs gSaA vc 'ks"k cps v.kqvksa esa larqyu rki

osQ fy, pky forj.k gksrk gSA fiQj ls cM+h la[;k esa v.kqvksa dh pky iyk;u osx ls

vfèkd gks tkrh gS vkSj ;s iyk;u dj tkrs gSaA bl izdkj nh?kZdky r panzek dk lkjk

ok;qeaMy lekIr gks x;kA

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170

300 K ij 23

26

3 3 1.38 10 300=1.7 km/s

7.3 10

−

−

× × ×= =

×rms

kTV

m

panzek ij Viyk;u

= 4.6 km/s

(b) tSls&tSls v.kq Åij mBrs gSa mudh fLFkfrt ÅtkZ esa o`f¼ gksrh gSA blfy, xfrt

ÅtkZ esa deh vkrh gS ftlls rki de gks tkrk gSA vfèkd Å¡pkbZ ij xSl dks iSQyus

osQ fy, vfèkd vk;ru miyCèk gksrk gSA xSl osQ iSQyus ls oqQN rki de gks tkrk gSA

13.28 (;g iz'u ok"iu ls gksus okys 'khryu dh èkkj.kk le>kus osQ fy, vfHkdfYir fd;k

x;k gS)

(i)

210 100 (1 4 2 16 4 36 2 64 1 100)

100

× × × + × + × + × + ×=

2 21000 (4 32 144 128 100) 408 1000m /s= × + + + + = ×

639m/s∴ =rmsv

21 3

2 2rmsmv kT=

2 26 5

23

1 1 3.0 10 4.08 10

3 3 1.38 10rmsmv

Tk

−

−

× × ×∴ = = ×

×

22.96 10 K 296K= × =

(ii)

2 2 2 22 10 (200) 20 (400) 40 (600) 20 (800)

90rmsV

× + × + × + ×=

210 100 (1 4 2 16 4 36 2 64)

90

× × × + × + × + ×=

2 2308

10000 342 1000 m /s9

= × = ×

584m/srmsv =

21248K

3rmsmV

Tk

= =

13.29 le; tv

λ=

=

× + × + × + × + ×=

2

2

2 2 2 2 210 (200) 20 (400) 40 (600) 20 (800) 10 (1000)

100

i i

irms

i

n v

Vn

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171

2

1

2 d nλ

π= , d = tgk¡ d O;kl gS vkSj n la[;k ?kuRo

3100.0167 km

20 20 1.5

−= = =× ×

Nn

V

2

1

2 ( / )t

d N V vπ=

×

2 3

1

1.414 3.14 (20) 0.0167 10 150−=

× × × × ×

= 225 h

13.30 V1x

= cDls osQ vanj x fn'kk osQ vuqfn'k v.kqvksa dh pky

n1 = izfr bdkbZ vk;ru esa v.kqvksa dh la[;k

∆ t, le; esa nhokj dh vksj pyus okys d.k blls Vdjk,¡xs ;fn os 1( )xV t∆ nwjh ij

gSaA ekuk fd a = nhokj dk {ks=kiQy] ∆t le; esa nhokj ls Vdjkus okys d.kksa dh l[;k1

( )2

i ixn V t a= ∆ (xq.kd1/2 nhokj dh vksj xfr osQ dkj.k)

lkekU; :i ls xSl larqyu esa gksxh D;ksafd fNnz dh rqyuk esa nhokj dk vkdkj cgqr

cM+k gSA

2 2 2 21 1 1x y z rmsV V V V∴ + + =

22

13rms

x

VV∴ =

2 21 3 3

2 2rms rms

kTmV kT V

m= ⇒ =

21x

kTV

m∴ =

∴ ∆t le; esa la?kV~V djus okys d.kksa dh la[;k 1

1

2

kTn t a

m= ∆ ;fn d.k fNnz

ij Vdjkrs gSa rks os blls ckgj fudy tkrs gSaA blh izdkj fNnz ij Vdjkus okys ckgjh

d.k vanj vk tk,¡xsA

∴ ∆t le; esa oqQy izokfgr gksus okys d.kksa dh la[;k 1 2

1( )

2

kTn n t a

m= − ∆ ]

D;ksafd vanj vkSj ckgj dk rki leku gSAPV

pV RTRT

µ µ= ⇒ =

A AN PNn

V RT

µ= =

oqQN le; τ i'pkr~ vanj dk nkc cny dj 1′p gks tkrk gS

11

AP Nn

RT

′′∴ =

18-04-2018


172

1 1′−n V n V = ckgj tkus okys d.kksa dh la[;k 1 2

1( )

2

kTn n a

mτ= −

1 11 2

1( )

2τ

′∴ − = −A A AP N P N N kT

V V P P aRT RT RT m

1 1

1 2

2P P V m

P P a kTτ

′− ∴ = −

27

6 23

5 1.00 46.7 101.5 1.42

0.01 10 1.38 10 3001.5 1.0

−

− −

× ×− =

× × ×−

= 1.38 ×105 s

13.31 n = bdkbZ vk;ru esa v.kqvksa dh la[;k

vrms

= xSl v.kqvksa dh rms pky

tc xqVdk vo pky ls py jgk gS rks lEeq[k i`"B osQ lkis{k v.kqvksa dh pky = v + v

o

(tc os lEeq[k la?kV~V osQ fy, vkrs gSa)

∴ izfr la?kV~V gLrkarfjr laosx = 2m (v+vo),

∆t le; esa gksus okys la?kV~Vksa dh la[;k 1

( )2

ov v n tA= + ∆ tgk¡ A = xqVosQ dh

vuqizLFk dkV dk {ks=kiQy gS rFkk xq.kd 1/2 xqVosQ dh vksj xfreku d.kksa osQ

dkj.k gSA

∴∆t le; esa lEeq[k i`"B ls gLrkarfjr laosx 2( )om v v nA t+ ∆

blh izdkj ∆t le; esa i'p i`"B ls gLrkarfjr laosx 2( )om v v nA t= − ∆

∴oqQy cy (d"kZ.k cy) 2( ) ( )o omnA v v v v= + − − lEeq[k i`"B ls

= mnA (4vvo) = (4mnAv)v

o

(4 ) oAv vρ=

geas ;g Hkh Kkr gS fd1 1

=2 2

2mv kT (v -, x-v{k osQ vuqfn'k gS)

∴ =kT

vm

.

vr% d"kZ.k cy ρ 0

kT= 4 A v

m.

18-04-2018

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173

vè;k;&14

14.1 (b)

14.2 (b)

14.3 (d)

14.4 (c)

14.5 (c)

14.6 (d)

14.7 (b)

14.8 (a)

14.9 (c)

14.10 (a)

14.11 (b)

14.12 (a), (c)

14.13 (a), (c)

14.14 (d), (b)

14.15 (a), (b), (d)

14.16 (a), (b), (c)

14.17 (a), (b) (d)

14.18 (a), (c), (d)

14.19 (i) (A),(C),(E),(G) (ii) (B), (D), (F), (H)

14.20 ckb± vksj 2kx.

14.21 (a) Roj.k foLFkkiu osQ vuqØekuqikrh gksrk gSA

(b) Roj.k dh fn'kk foLFkkiu dh fn'kk osQ foijhr gksrh gSA

14.22 tc yksyd osQ xksyd osQ ekè; fLFkfr ls bruk foLFkkfir fd;k tk, fd sinθ ≅ θ

14.23 +ω

14.24 pkj

18-04-2018


174

14.25 ½.kkRed

14.27

14.281 1

m6 6

l lm E= =

14.29 ;fn nzO;eku m uhps dh vksj h nwjh pyrk gS rks fLiazx esa 2h yackbZ dh o`f¼ gksrh gS

(D;ksafd izR;sd Hkkx esa h o`f¼ gksrh gS) Mksjh vkSj fLiazx nksuksa esa leku ruko gSA

larqyu voLFkk esa

mg = 2 (k. 2h)

tgk¡ k fLiazx fu;rkad gSA

nzO;eku dks x nwjh uhps [khapus ij

F = mg - 2k 2h( )+ 2x

= – 4kx

blfy, = 24

mT

kπ

14.30 ( ) π ωω π= =−2 sin ; 2 //4y Tt

14.312

A

14.32 ( )o= 1 cosU U xα−

( )coso o

–dU –dU – U a xF = =

dx dx

o= – sin xU α α

oU xαα−� ( )α α α∼sinx x xosQ y?kqeku oQs fy,

2o= –U xα

F = –kx ls

2ok U α=

22

o

mT

Uπ

α=

18-04-2018

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175

14.33 x = 5 sin 5t.

14.34 ( )1 1sino tθ θ ω δ= +

( )2 2sino tθ θ ω δ= +

izFke izdj.k esa] ( )12 , sin 1tθ ω δ= ° ∴ + =

f}rh; izdj.k ( )21 , sin 1/2tθ ω δ= − ° ∴ + = −

1 290 , 30t tω δ ω δ∴ + = ° + = − °

1 2 120δ δ∴ − = °

14.35 (a) th gk¡A

(b) vfèkdre Hkkj = Mg +MAω2

( )2550 9.8 50 2 2

100π= × + × × ×

= 490 + 400 = 890N

U;wure Hkkj = Mg –MAω2

( )2550 9.8 50 2 2

100= × − × × ×π

= 490–400

= 90 N

vfèkdre Hkkj mPpre fLFkfr esa gksrk gSA

U;wure Hkkj fuEure fLFkfr esa gksrk gSA

14.36 (a) 2cm (b) 2.8 s–1

14.37 ekuk fd yV~Bs dks nck;k tkrk gS vkSj larqyu voLFkk esa vfèkdre foLFkkiu xo gSA

larqyu esa mg = mRIykou cy= oAx gρ

tc bldks vkSj vfèkd foLFkkiu x fn;k tkrk gS rks mRIykou cy gS ( ) .+oA x x gρ

oqQy izR;ku;u cy

= mRIykou cy – Hkkj= ( )oA x x gρ+ – mg

( )A g xρ= vFkkZr~ F α x.

2m

TA g

πρ

∴ =

14.38 dx yackbZ esa Hkjs nzo ij fopkj dhft,A x Å¡pkbZ ij bldk nzO;eku A dxρ gSA

PE = A dxρ gx

18-04-2018


176

45° 45°

h1 h2l

dx

lx

oke&LrEHk dh P.E.

1h

o

A gxdxρ= ∫

1 2 2 221 sin 45

2 22

h

o

h A glxA gA g

ρρρ

°= ==

blh izdkj nf{k.k LraHk dh P.E. 2 2 2

2 sin 45

2 2

h A glA g

ρρ

°= =

h1 = h

2 = l sin 45° tgk¡ l uyh dh ,d Hkqtk esa nzo dh yackbZ gSA

oqQy P.E. 2 2 2sin 45A gh A glρ ρ= = °2

2

A glρ=

;fn ufydk osQ vuqfn'k blosQ ck;ha vksj

osQ Hkkx esa nzo Lrj esa ifjorZu y gks rks

ck;ha vksj nzo dh yackbZ (l–y) gksxh vkSj

nkfguh vksj (l + y) gksxhA

oqQy P.E. 2 2 2 2( – ) sin 45 ( ) sin 45A g l y A g l yρ ρ= ° + + °

P.E. esa ifjorZu = (PE)f – (PE)

i

( ) ( )2 2 2–

2

A gll y l y

ρ = + −+

2

A gl

ρ=

2 2 2y ly+ − 2 2 2l y ly+ + + l−2

2 2A g y lρ= +

K.E. esa ifjorZu 212

2A lyρ= �

oqQy ÅtkZ ifjorZu = 0

( . ) ( . ) 0P E K E∆ + ∆ =

22 2 0A g A lyl yρ ρ+ = + �

nksuksa i{kksa dks le; osQ iQyu osQ :i esa O;dfyr djus ij

2 00 2dy

A g A lyyydt

ρ ρ + =+

� ��

2 2 0A gy A lyρ ρ+ =��

18-04-2018

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177

0ly gy+ =��

0g

y yl

+ =��

2 g

lω =

g

lω =

2l

Tg

π=

14.39 P ij xq#Ro osQ dkj.k Roj.k .g x

R, tgk¡ g i`Foh osQ i`"B ij xq#Ro osQ dkj.k Roj.k

dk eku gSA

cy mgx

R= – .

mgk x k

R= =tgk¡

xfr SHM gksxhA ftldk vkorZ dky gksxkµ π π= =2 2m R

TK g

14.40 eku yhft, fd tc θ = θ0 rks t = 0 gSA

rc] θ = θ0 cos ωt

lsoaQM yksyd osQ fy, ω = 2π

le; t1 ij ekuk θ = θ

0/2

∴ cos 2πt1 = 1/2 1

1

6⇒ =t

0– 2 sinθ θ π•

= 2πt d

dt

θθ•

=

t1 = 1/6 ij

0 0

2– 2 sin – 3

6

πθ θ π πθ•

= =

vr% jSf[kd osx gS]

0u – 3πθ= l tks Mksjh osQ yacor~ gSA

bl osx dk ÅèoZèkj ?kVd gSµ

0– 3 sinπθ θ=y 0u l

rFkk {kSfrt ?kVd gS%

0– 3 cosπθ θ=x 0u l

q /2o

qo

A

H

18-04-2018


178

ftl le; ;g VwVrh gS ÅèokZèkj Å¡pkbZ gSµ

( )( )0 /21 – cos θ′ =H H + l

ekuk fd fxjus esa yxk le; t gS] rc

( )1/2′ = 2yH u t + gt (è;ku nas fd g Hkh ½.kkRed fn'kk esa gS)

vFkok sinπθ θ ′20 0

1gt + 3 l t – H = 0

22 2 2 2

0 0 0 0– 3 sin 3 e sin 2 ′± +∴ =

l gHt

g

πθ θ π θ θ

2 2 40 0– 3 3 2π θ π θ ′± +

�

2l l gH

g

20θ ,oa mPprj ?kkrkadksa osQ inksa dks mis{k.kh; ekusa rks

′�

2Ht

g.

vc ( ) =�H' H + l H1 – 1 ∴ �2H

tg

x fn'kk esa pfyr nwjh uxt gS tks ml fLFkfr osQ ck;ha vksj gS tgk¡ Mksjh VwVh FkhA

cosπθ θ0 0

2HX = 3 l

g

0θ dh izFke dksfV osQ fy;s

πθ θ=0 0

2H 6HX = 3 l l

g g.

Mksjh VwVus osQ le; xksYiQ dh fLFkfr Fkh A ls

0 0sinθ θ�l l nwjh ij]

vr% A ls nwjh gS%

θ θ0 0

6Hl – l

g = θ −(1 6 / )0l H g

vè;k;&15

15.1 (b)

15.2 (c)

15.3 (c)

15.4 (c)

18-04-2018

mÙkjekyk

179

15.5 (b)

15.6 (c)

15.7 (d)

15.8 (b)

15.9 (b)

15.10 (c)

15.11 (a), (b), (c)

15.12 (b), (c)

15.13 (c), (d)

15.14 (b), (c), (d)

15.15 (a), (b), (d)

15.16 (a), (b)

15.17 (a), (b), (d), (e)

15.18 nksxquh yackbZ dk rkj f}rh; gkeksZfud esa oaQiu djrk gSA vr% ;fn f}Hkqt Lofj=k

L yackbZ osQ fy, vuqukn iznf'kZr djrk gS rks ;g 2L yackbZ osQ fy, Hkh vuqukn iznf'kZr djsxkA

15.19 L/2 D;ksafd λ vpj gSA

15.20 517 Hz

15.21 5 cm

15.22 1/3, pw¡fd vko`fr 21

m rm

α π ρ=

15.23 2184oC, pw¡fd C Tα

15.241 2

1

n n−

15.25 343 m s–1. 1

2

Tn

l m

=

15.26 r`rh; gkeksZfud

= = = 412.5 330 m/s

4no

vv

lpf¡w d tgk¡

15.27 412.5 Hz 'c

n nc v

= −

18-04-2018


180

15.28 vizxkeh rjaxsa_ 20cm

15.29 (a) 9.8 × 10-4s. (b) fuLian-A, B, C, D, E. izLian-A1, C1. (c) 1.41m

15.30 (a) 348.16 ms-1

(b) 336 m s–1

(c) ok;q LraHk 17cm yackbZ ij vuqukn izsf{kr fd;k tk,xk] osQoy ikjs dh lrg ls

vfèkd vPNk ijkorZu gksus osQ dkj.k lquh tkus okyh èofu dh rhozrk vfèkd gksxhA

15.31 okafNr ifj.kke lacaèk 2

nv

Lν = ls izkIr gksrk gSA

15.326400 3500 2500 1000

28 5 8

t−

= ×+ +

= 1975 s.

= 32 feuV 55 lsoaQM

15.333 3

,P RT P RT

c vM M

γ γ

ρ ρ= = = =

3c

v=

γrFkk f}&ijek.kqd xSlksa osQ fy,

7

5=γ

15.34 (a) (ii), (b) (iv), (c) (iii), (d) (i).

15.35 (a) 5m (b) 5m (c) 50 gV~Zt (d) 250ms-1 (e) 500π ms-1

15.36 (a) 6.4π jsfM;u (b) 0.8π jsfM;u (c) π jsfM;u (d) 3π /2 jsfM;u (e) 80π jsfM;u

18-04-2018

HkkSfrdhd{kk 11

le; µ 3 ?kaVs vf/dre vad µ 70

iz'u i=k osQ fofHkUu vk;keksa ij vadksa dk forjk fuEuor~ gksxk %

A. fo"k;oLrq@ikB&bdkbZ;ksa osQ vuqlkj vad forj.k

Ø- la- bdkbZ vad

1. HkkSfrd txr ,oa ekiu 03

2. 'kq¼ xfrdh 10

3. xfr osQ fu;e 10

4. dk;Z] ÅtkZ ,oa 'kfDr 06

5. d.kksa dh iz.kkyh ,oa n`<+ fi.Mksa dh xfr 06

6. xq#Rokd"kZ.k 05

7. LFkwy nzO; osQ xq.k 10

8. rki xfrdh 05

9. vkn'kZ xSl dk O;ogkj rFkk xSlksa dk v.kq xfrdh fl¼kar 05

10. nksyu ,oa rjaxsa 10

;ksx 70

B. iz'uksa osQ izdkj osQ vuqlkj vad forj.ks

Ø-la- iz'u dk izdkj izR;sd iz'u osQ iz'uksa dh la[;k oqQy vad

vad

1. nh?kZ mÙkjh; iz'u (LA) 5 3 15

2. y?kq mÙkjh; iz'u I (SAI) 3 09 27

3. y?kq mÙkjh; iz'u II (SAII)/ 2 10 20

cgq fodYih iz'u (MCQ)

4. vfr y?kq mÙkjh; iz'u (VSA)/ 1 08 08

cgq fodYih iz'u (MCQ)

;ksx – 30 70

iz'u i=k dk fM”kkbu

izfrn'kZ iz'u i=k

18-04-2018


182

1 vad osQ iz'u y?kq mÙkjh; (VSA) izdkj osQ gks ldrs gSa ;k fiQj ,sls cgq fodYih iz'u (MCQ) gks ldrsgSa ftuesa osQoy ,d gh fodYi lgh gksA

2 vad osQ iz'u y?kq mÙkjh (SAII) izdkj osQ gks ldrs gSa ;k fiQj ,sls cgq fodYih iz'u (MCQ) gks ldrsgSa ftuesa ,d ls vf/d fodYi lgh gksaA

C. oSdfYid iz'uksa dh ;kstuk

1. iz'u&i=k esa lexz ij dksbZ fodYi (vksoj vky p;u) ugha gSA

2. oqQN iz'uksa esa vR;ar p;ukRed vk/kj ij vkarfjd p;u (;g ;k fiQj ;g izdkj) dh lqfo/k nh xbZ gSA

D. iz'uksa osQ dfBukbZ Lrj osQ vuqlkj vad forj.k

Ø-la- vuqekfur dfBukbZ Lrj izfr'kr

1. vklku 15

2. eè;e 70

3. dfBu 15

18-04-2018

183

izfrn'kZ iz'u i=k

fo"k;

VSA

(1 v

ad)

SA

I (2

vad

)SA

II

(3 v

ad)

LA

(5 v

ad)

;ksx

IHkkSfrd t

xr v

kSj ekiu

1(1

)2 (

1)

——

3 (

2)

II'kq¼ xfrdh

1 (

1)

4 (

2)

—5 (

1)

10 (

4)

III

xfr osQ0;

e1 (

1)

—9 (

3)

—10 (

4)

IVdk;Z] ÅtkZ ,o

a 'kfD

r1 (

1)

2 (

1)

3 (

1)

—6 (

3)

Vd.kksa dh iz.kky

h ,o

a n`<+

1 (

1)

2 (

1)

3 (

1)

—6 (

3)

¯iMksa d

h xfr

VI

xq#Rokd

"kZ.k

—2 (

1)

3 (

1)

—5 (

2)

VII

LFkwy nzO; osQ xq.

k—

2 (

1)

3 (

1)

5 (

1)

10 (

3)

VII

Irki xfrd

h—

2 (

1)

3 (

1)

—5 (

2)

IXvkn'kZ xSl

dk O;og

kj ,o

a

1 (1)

4 (

2)

——

5 (

3)

xSlksa dk v.kqxfrd

fl¼

kar

Xnksyu ,o

a rjaxsa

2

(2)

—3 (

1)

5 (

1)

10 (

4)

;ksx

8 (

8)

20 (1

0)

27 (

9)

15 (

3)

70 (3

0)

izfrn

'kZ i

z'u i

=k &

1

Cyw ¯iV

(:ij

s[kk)

18-04-2018


184

lkekU; funsZ'kµ

(a) lHkh iz'u vfuok;Z gSaA

(b) oqQy 30 iz'u fn, x, gSaA iz'u 1 ls 8 rd izR;sd iz'u 1 vad dk gS] iz'u 9 ls 18 rd izR;sd iz'u 2 vad dkgS] iz'u 19 ls 27 rd izR;sd iz'u 3 vad dk gS rFkk iz'u 28 ls 30 rd izR;sd iz'u 5 vad dk gSA

(c) iw.kZ :i ls p;u dk izko/ku ugha gSA rFkkfi] 2 vadksa osQ ,d iz'u esa] 3 vadksa osQ ,d iz'u esa rFkk 5 vadksaosQ lHkh iz'uksa esa vkarfjd p;u dh lqfo/k nh xbZ gSA bu lHkh iz'uksa esa fn, x, nks fodYiksa esa ls vkidks dksbZ,d djuk gSA

(d) ifjdfy=kksa osQ mi;ksx dh vuqefr ugha gSA

(e) vko';drk iM+us ij vki uhps fn, x, HkkSfrd fu;rkadksa dk mi;ksx dj ldrs gSa %

c = 3 × 108ms-1

h = 6.6 × 10-34Js

µo = 4π × 10–7 TmA–1

cksYRleku fu;rkad] k = 1.38 × 1023 JK-1

,oksxknzks la[;k] NA = 6.023 × 1023/eksy

1. ;fn laosx (P), {ks=kiQy (A) rFkk le; (T) dks ewy jkf'k;k¡ ys ysa rks ÅtkZ dk foeh; lw=k gksxk %

(a) [P1 A-1 T1]

(b) [P2 A1 T1]

(c) [P1 A-1/2 T1]

(d) [P1 A1/2 T-1]

2. fdlh d.k dk vkSlr osx blosQ rkR{kf.kd osx osQ cjkcj gSA bldh xfr fdl izo`Qfr dh gS\

3. ( )= N6 - 3F i j cy 2kg nzO;eku ij vkjksfir gSA Roj.k dk ifjek.k Kkr dhft,A

4. fdlh ¯iM }kjk ?k"kZ.k osQ fo:¼ fd;k x;k dk;Z lnSo &

(a) xfrt ÅtkZ esa ßkl dk dkj.k curk gSA(b) fLFkfrt ÅtkZ esa ßkl dk dkj.k curk gSA(c) xfrt ÅtkZ esa o`f¼ dk dkj.k curk gSA(d) fLFkfrt ÅtkZ esa o`f¼ dk dkj.k curk gSA

izfrn'kZ iz'u i=k IHkkSfrdh – 11


18-04-2018

185

izfrn'kZ iz'u i=k

5. fuEufyf[kr esa dkSu&lk ¯cnq fp=k&1 esa n'kkZbZ xbZ O;oLFkk dk nzO;eku&osaQnzgksus dh laHkkouk gS\

(a) A

(b) B

(c) C

(d) D

6. fdlh xSl osQ nks v.kqvksa dh pkysa Øe'k% 9 × 106 m/s rFkk 1.0 × 106 m/s

gSA budh oxZ ekè; ewy (rms) pky fdruh gS\

7. ljy vkorZ xfr (S.H.M) djrs gq, fdlh d.k osQ foLFkkiu x dks x = 3 cos (5πt+π) }kjk fu#fir fd;k tkldrk gS] tgk¡ x ehVj esa rFkk t lsoaQM esa gSA t = 0 vkSj t = 1/2 s ij ;g d.k dgk¡ gS\

8. ljy vkoÙkZ xfr esa tc fdlh d.k dk foLFkkiu mlosQ vk;ke dk ,d pkSFkkbZ gksrk gS rks bldh xfrt ÅtkZ]bldh oqQy ÅtkZ dk dkSu lk va'k gksrk gS\

9. fdlh izxkeh rjax dk foLFkkiu

y = A sin(ωt – kx), }kjk fu:fir fd;k tkrk gS tgk¡ x nwjh rFkk t le; gSA

(i) ω, ,oa (ii) k osQ foeh; lw=k fyf[k,A

10. 100 g ty dks –10oC rd vfr'khrfyr fd;k tkrk gSA fdlh izdkj osQ fo{kksHk osQ dkj.k bldk oqQN Hkkxvpkud tedj ciZQ esa cny tkrk gSA ifj.kkeh feJ.k dk rki D;k gksxk rFkk fdruk nzO;eku ciZQ esa cnysxk\

ow FusionS =1cal/g/ C and = 80cal/gwL

vFkok

,d fnu izkr% Luku osQ fy, eSaus ,d frgkbZ ckYVh xhtj ls xeZ ikuh ysdj HkjhA 'ks"k nks frgkbZ ckYVh dks(dejs osQ rki osQ) BaMs ikuh ls Hkj dj feJk dk rki lguh; cukuk FkkA Luku ls igys vpkud eq>s oqQNvko';d dk;Z djus dh vko';drk Fkh ftlesa 5-10 feuV yxus FksA vc esjs lkeus nks fodYi Fks % (i) eSackYVh dks BaMs ikuh ls Hkj yw¡ vkSj fiQj dke d:¡] (ii) igys dke [kRe dj yw¡ vkSj Luku ls Bhd igys ckYVhesa ikuh Hk:¡A nksuksa esa fdl fodYi esa Luku osQ le; ckYVh esa ty vf/d xeZ jgsxk\ Li"V dhft,A

11. fuEufyf[kr dks fl¼ dhft,µ{kSfrt ls ‘θ ’ ,oa (90-θ ) dks.kksa ij leku izkjafHkd osx ls izf{kIr nks ¯iMksa osQ

(a) ijkl leku gksaxs

(b) Å¡pkbZ;k¡ tan2 θ:1 osQ vuqikr esa gksaxhA

12. ^iy;ku osx* ls D;k rkRi;Z gS\ i`Foh osQ i`"B ls izf{kIr fdlh ¯iM osQ iyk;u osx osQ fy, O;atd O;qRiUudhft,A

fp=k 1

R/2

R/2

A

B

C

D

��

��

��

18-04-2018


186

13. ,d yM+kowQ foeku 1.5 km dh Å¡pkbZ ij 720 km/h osQ osx ls {kSfrtr% mM+ jgk gSA y{; dks ns[kus osQckn fdl n`'; dks.k ({kSfrt osQ lkis{k) ij ik;yV dks ce fxjkuk pkfg, fdog y{; ls Vdjk,\

14. fdlh {kSfrt lM+d ij R f=kT;k dk ,d xksyk fcuk fiQlys yq<+drk gSA laioZQ¯cnq A ls xqtjus okyh ÅèokZ/j js[kk ij pkj ¯cnq gSa A, B, C ,oa D(fp=k 2)A bu ¯cnqvksa ij fo|eku d.kksa osQ LFkkukarjh; D;k&D;k gS\ nzO;ekuosaQnz dk osx V

cm gSA

15. fdlh m"ekxfrdh; ra=k dks ewy voLFkk D ls fp=k&3 esa n'kkZ, x,js[kh; izØe }kjk ekè;fed voLFkk rd ys tk;k tkrk gSA fiQj bldkvk;ru ,d lenkch izØe }kjk de djosQ ewy vk;ru osQ cjkcjdjus osQ fy, E ls F voLFkk esa yk;k tkrk gSA D ls E vkSj fiQj FvoLFkk rd vkus esa xSl }kjk fd, x, oqQy dk;Z dk ifjdyu dhft,A

16. fdlh ÝykLd esa vkxZu ,oa Dyksjhu xSl Hkjh gS ftuosQ nzO;eku 2:1

vuqikr esa gSaA feJ.k dk rki 27°C gSA nksuksa xSlksa osQ (i) izfr v.kq dhvkSlr xfrt ÅtkZ dk vuqikr (ii) nksuksa xSlksa osQ v.kqvksa dh oxZ ekè;ewy pkyksa v

rms dk vuqikr Kkr dhft,A vkxZu dk ijek.kq&nzO;eku

= 39.9u, Dyksjhu dk v.kq nzO;eku = 70.9uA

17. ok;q esa fo|eku 5 × 10-17 kg nzO;eku osQ /wez d.kksa dk lkekU; rki ,oa nkc (NTP) ij oxZ ekè; ewy (rms)

osx ifjdfyr dhft,A

18. 9 m s–1 dh pky ls pyrh gqbZ dksbZ xsan fojke esa j[kh ,d loZle xsan ls bl izdkj Vdjkrh gS fd la?kV~VosQ i'pkr~ izR;sd xsan izkjafHkd fn'kk ls 30° dk dks.k cukrh gqbZ tkrh gSA nksuksa xsanksa dh la?kV~V i'pkr~ pkyksadk ifjdyu dhft,A D;k bl la?kV~V izfØ;k esa xfrt ÅtkZ lajf{kr gksrh gS\

19. ml vf/dre osx osQ fy, O;atd O;qRiUu dhft, ftlls dksbZ dkj θ dks.k ij cafdr (cSaDM)] r f=kT;kosQ o`Ùkkdkj eksM+ dh lM+d ij lqjf{kr xqtj ldrh gSA dkj osQ Vk;jksa vkSj lM+d osQ chp ?k"kZ.k xq.kkad µ

gSA

20. fuEufyf[kr osQ fy, dkj.k crkb,µ

(a) xsan dks yidrs le; dksbZ fØosQV f[kykM+h vius gkFk ihNs dh vksj [khaprk gSA(b) ykWu yfo=k (Lawn Mower) dks /osQyus dh vis{kk [khapuk vf/d vklku gksrk gSA(c) njh esa ls /wy fudkyus osQ fy, bls NM+h ls ihVk tkrk gSA

21. 1000 kg lagfr dk dksbZ gsyhdkWIV 15 m s–2 osQ ÅèokZ/j Roj.k ls Åij mBrk gSA pkyd ny rFkk ;kf=k;ksadh lagfr 300 kg gSA fuEufyf[kr cyksa dk ifjek.k vkSj fn'kk fyf[k,µ

(a) pkyd ny rFkk ;kf=k;ksa }kjk iQ'kZ ij vkjksfir cy](b) pkjksa vksj dh ok;q ij gsyhdkWIVj osQ jksVj dh fØ;k] rFkk(c) pkjksa vksj dh ok;q osQ dkj.k gsyhdkWIVj ij vkjksfir cyA

fp=k 3

fp=k 2

18-04-2018

187

izfrn'kZ iz'u i=k

22. dksbZ efgyk LVs'ku osQ IysViQkeZ osQ [kqjnjs iQ'kZ ij j[ks ,d Vªad dks /osQyrh gSA og bl ij 10 m dhnwjh rd 100 N dk cy yxkrh gSA blosQ i'pkr~ og yxkrkj Fkdrh tkrh gS vkSj mlosQ }kjk yxk;k cynwjh osQ lkFk ,d leku :i ls de gksrk gqvk var esa 50 N jg tkrk gSA Vªad dks oqQy 20 m dh nwjh rd/osQyk x;k gSA efgyk }kjk yxk, x, cy rFkk ?k"kZ.k cy esa] tks fd 50 N gS] xzkiQ cukb,A nksuksa cyksa }kjk20m dh nwjh pyus esa fd, x, dk;Z dk ifjdyu dhft,A

23. fdlh n`<+ iM }kjk fu;r dks.kh; Roj.k ‘α’ rFkk izkjafHkd osx ωo ls ?kw.kZu djrs iM dh xfr osQ fy, O;atd

O;qRiUu dhft,A

24. fdlh xzg osQ pkjksa vksj fu;r d{kk esa ifjØe.k djrs mixzg dh xfrt ÅtkZ ,oa fLFkfrt ÅtkZ osQ fy,O;atd O;qRiUu dhft,A ;fn dksbZ 200kg nzO;eku dk mixzg 5 × 1030 kg nzO;eku osQ xzg osQ pkjksa vksj6.6 × 106 m f=kT;k dh d{kk esa ifjØe.k dj jgk gks rks mixzg dh ca/u ÅtkZ (B.E.) dk ifjdyudhft,A -11 2 2G = 6.6 ×10 Nm /kg A

25. cuwZyh dk izes; fyf[k, vkSj mldh miifÙk nhft,A

26. ekuk fd fdlh lkbfdy Vk;j esa iai }kjk gok Hkjh tk jgh gSA bl Vk;j dk vk;ru V gS (tks fu;r gS)vkSj iai izR;sd ckj ok;q dk ( )V V∆ vk;ru fLFjks"e voLFkk esa Vk;j esa LFkkukarfjr djrk gSA o`Qr dk;Zdk ifjdyu dhft, tcfd V~;wc esa nkc c<+dj P

1 ls P

2 gks tkrk gSA

vFkok

jsfizQtjsVj esa Å"ek fuEure rki ls ysdj mPp rki ij ifjos'k esa fu{ksfir dh tkrh gSA bl izfØ;k esa;kaf=kd dk;Z djuk gksrk gS tks fo|qr eksVj }kjk iznku fd;k tkrk gSA ;fn eksVj 1kW dh gks vkSjÅ"ek o o-3 C 27 Cls , ij LFkkukarfjr dh tkrh gS rks ;g ekurs gq, fd ml jsfizQtjsVj dh n{krk vkn'kZ batudh n{krk dh 50% gS blls izfr lsoaQM fu"Øfer gksus okyh Å"ek dk ifjdyu dhft,A

27. n'kkZb, fd nksuksa fljksa ij fLFkj dksbZ Mksjh tc 1 ywi] 2 ywi] 3 ywi vkSj 4 ywi esa dEiu djrh gS rks bldhvko`fr;ksa dk vuqikr 1:2:3:4 gksrk gSA

28. (a) ';kurk xq.kkad dh ifjHkk"kk vkSj bldk SI ek=kd fyf[k,A

(b) vafre (V£euy) osx dh ifjHkk"kk dhft,A fdlh ';ku nzo esa fxjrs gq, xksys osQ vafre osx osQ fy,O;atd O;qRiUu dhft,A

vFkok

fdlh /krq osQ rkj osQ fy, izfrcy&foo`Qfr xzkiQ fp=k 4 esa n'kkZ;k x;kgSA /hjs&/hjs rkj ij yxk Hkkj de djus ij rkj oØ EFO osQ vuqfn'kviuh izkjafHkd voLFkk esa ykSVrk gSA ¯cnq B ij rkj VwV tkrk gSA

(i) oØ osQ fdl ¯cnq rd gqd osQ fu;e dk vuqikyu gksrk gS\

(ii) oØ dk dkSu&lk ¯cnq rkj dh izR;kLFkrk lhek ;k uE;rk ¯cnq dksfu:fir djrk gS\

(iii) izfrcy&foo`Qfr xzkiQ osQ izR;kLFkrk ,oa IykfLVd {ks=k dkSu&ls gSa] crkb,A

(iv) rkj osQ xzkiQ ij n'kkZ, ¯cnq A osQ laxr izfrcy yxkus osQ ckn fiQj /hjs&/hjs Hkkj de djus ij D;kgksrk gS] le>kb,A fo'ks"k :i ls ¯cnqfdr oØ dh O;k[;k dhft,A

��

��

B

C

A

E

F

fp=k 4

18-04-2018


188

(v) izfrcy&foo`Qfr xzkiQ osQ C ls B osQ chp osQ Hkkx osQ laca/ esa D;k foy{k.k ckr gS\ rkj ij yxk, x,fdrus izfrcy rd ;g ugha VwVrk gS\

29. ;g ,d lkekU; izs{k.k gS fd o"kkZ es?k Hkwry ls yxHkx ,d fdyksehVj dh Å¡pkbZ ij gks ldrs gSaA

(a) ;fn o"kkZ dh dksbZ cw¡n bruh Å¡pkbZ ls xq#Ro osQ rgr eqDr :i ls fxjs rks mldh pky D;k gksxh\ ;geku kmh–1 esa Hkh ifjdfyr dhft,A (g = 10 m s–2)

(b) fdlh izk:fid o"kkZ dh cw¡n dk O;kl 4mm gSA ;g vkidks Vdjk, rks fdrus laosx ls Vdjk,xh\

(c) cw¡n dks piVk gksus dk vFkkZr~ igys vkSj vafre laioZQ osQ chp dk le;&varjky D;k gksxk\

(d) vkdyu dhft, fd ,slh dksbZ cw¡n vkiosQ Åij fdruk cy vkjksfir djsxh\

(e) fdlh Nkrs osQ Åij yxus okys cy dh ifjek.k dksfV dk vkdyu dhft,A nks o"kkZ cw¡nksa osQ chpizk:fid ik£'od i`Fkd 5 cm gSA

(;g eku yhft, fd Nkrs osQ diM+s ls cw¡n Nu dj ikj ugha tkrhA)

vFkok

fØosQV osQ [ksy esa ,d {ks=k j{kd xsan dks vo pky ls isaQd ldrk gSA ;fn og u pky ls nkSM+rk gqvk xasn

dks {kSfrt ls θ dks.k cukrs gq, isaQosq rks Kkr dhft,

(i) fdlh n'kZd osQ fy, n`f"Vxr {kSfrt ls cuk og izHkkoh dks.k ftl ij ok;q esa ;g xsan izf{kIr gqbZ gSA

(ii) xsan dk mM~M;u dky D;k gksxk\

(iii) iz{ksi.k ¯cnq ls fdruh nwjh ({kSfrt ijkl) ij ;g xsan i`Foh ls Vdjk,xh\

(iv) og dks.k Kkr dhft, ftl ij isaQdus ls (iii) esa izkIr {kSfrt ijkl vf/dre gksxkA

(v) ;fn u >vo, u = v

o, u < v

o rks vf/dre ijkl osQ fy, θ dk eku fdl izdkj ifjo£rr gksxk\

30. (a) n'kkZb, fd ljy vkoÙkZ xfr (S.H.M.) esa fdlh fn, x, {k.k ij Roj.k dk eku foLFkkiu osQvuqØekuqikrh gksrk gSA

(b) h Å¡pkbZ vkSj A vuqizLFk dkV dk dk"B dk ,d csyukdkj yV~Bk ÅèokZ/jr% ikuh esa rSj jgk gSA bldksuhps dh vksj nck dj NksM+ fn;k x;k gSA n'kkZb, fd dk"B dk ;g yV~Bk ljy vkoÙkZ xfr (S.H.M.)

djsxk ftldk vkoÙkZdky 2m

TA g

= πρ

gksxkA ;gk¡ m yV~Bs dk nzO;eku rFkk ρ nzo dk ?kuRo gSA

18-04-2018

189

izfrn'kZ iz'u i=k

vFkok

dksbZ izxkeh rjax y = 5 sin (100πt-0.4πx) }kjk fu:fir dh tkrh gS] tgk¡ y ,oa x ehVj esa gS rFkk t lsoaQM esa gSAfuEufyf[kr dk eku D;k gS\

(a) vk;ke(b) rjaxnS?;Z(c) vkofÙk(d) rjax osx(e) d.k osx osQ ifjek.k

18-04-2018


190

izfrn'kZ iz'u i=k & Igy ,oa vadu ;kstuk

1. (d) (1)

2. ,dleku xfr (1)

3. ( ) 2 2ˆ ˆ m/s ; 3.35m/s3 1.5= =−a ai j (½)+(½) (1)

4. (a) (1)

5. (c) (1)

6. 6.4 × 106 m s–1 (lw=k ½, mÙkj ½) (1)

7. –3m; O m. (½)+(½) (1)

8.. . 15

16

K E

E= (lw=k ½ , vuqikr ½) (2)

9. (i) [ ]–1M L T° ° , (ii) [ ]–1M L T° ° 1 + 1 (2)

10. ifj.kkeh feJ.k dk rki 0oC gks tkrk gSA 12.5 g ciZQ] 'ks"k tyA (1+1)

vFkok

igyk fodYi ls ty vfèkd xeZ jgrk gS D;ksafd U;wVu osQ 'khryu fu;e osQ vuqlkj Å"ek gkfu dh nj fiaM

vkSj ifjos'k osQ rkikUrj osQ vuqØekuqikrh gksrh gSA igys fodYi esa rkikUrj de gSA vr% Å"ek gkfu de gksxhA

(2)

11. mRifÙk R1 = R

2 rFkk

21

2

tan

1

h

h=

θ1 + 1 (2)

12. fdlh fiaM dk og U;wure iz{ksi.k osx ftlls fdlh xzg osQ i`"B ls isaQdus ij ;g Bhd ml xzg osQ xq#Rokd"kZ.k

{ks=k ls ckgj gks tk,A (1)

21 2

2 e e

GMm GMmv or v

R R= = (1)

13. ekuk fd ce }kjk y{; ls tkdj Vdjkus esa fy;k x;k le; t gSA

18-04-2018

191

izfrn'kZ iz'u i=k

211500

2gt=

Or, 300 17.32 st = = (1)

ce }kjk r; dh xbZ {kSfrt nwjh = 17.32 × v

= 17.32 × 200 = 3464m.

1500tan

3464θ∴ =

or θ = tan –1 0.43 (1)

14. A CM= – = 0v v Rω

vCM

= ωR (½)

ωB CM CM= – = /2

2

Rv v v (½)

C CM CM

3= =

2 2

Rv v v

ω+ (½)

D CM CM= + = 2v v R vω (½)

15. fd;k x;k dk;Z = P-V oØ osQ uhps dk {ks=kiQy (1)

= 1

(300)(30) = 450J2

(1)

16. pw¡fd vkxZu ,oa Dyksjhu nksuksa ,d gh rki ij gS] mudh izfr v.kq vkSlr xfrt ÅtkZvksa dk vuqikr 1:1 1

2(½)

½MV 2rms

=K.E. izfr v.kq 3

2= kT. (½)

( )( )

( )∴ = =

( ) 70.9

39.9rms

rms

V M Cl

V M Ar

vkxZuDykjs hu

= 1.77 1.33= (½+½)

18-04-2018


192

F

F

FF

F

( )�� ( )��

F

mgmg

17.21

3rms

mPV mV RT

M= = (1)

( )3 3rms

T kTNkV

Nµ µ= = (½)

23

–17

3 1.38 10 273

5 10

−× × ×=

×(½)

= 15 × 10–3 m s–1

= 1.5 cm s–1 (1)

18. 1 29 cos30 cos30m mv mv× = ° + ° (½)

1 20 sin30 – sin30mv mv= ° ° (½)

1 2 6 3v v+ =

1 2v v=

–11 2 3 3 m sv v= =

( )2 2

f i

1 1– = × 2 – 9 = –13.5 joule3 3

2 2T T m m × m (½)

m izR;sd xsan dk nzO;eku gSA vr% K.E. lajf{kr ugha gksrhA (½)

19. fp=k] lw=k ( )tan +

=1 – tan

rgV

θ µ

µ θ dh O;qRifÙk (1) + (2)

20. (a) og ,slk oSQp idM+us esa yxus okys le; dks c<+kus osQ fy, djrk gSA pw¡fd F = Ma = dv

Mdt

,

vr% oSQp djus esa yxk le; c<+us ls xsan }kjk gkFkksa ij yxus okys cy dk izHkko de gks

tkrk gSA

(b) tSlk fd fp=k ls Li"V gS tc ykWu eksoj dks {kSfrt ls θ dks.k ij F cy yxk dj [khapk tkrk

gS rks F dk {kSfrt ?kVd F cos θ ykWu eksoj dks vkxs c<+kus osQ fy, iz;qDr gksrk gS tcfd ÅèokZèkj

?kVd F sin θ, blosQ Hkkj dks de djrk gSA

;fn ykWu eksoj dks {kSfrt ls θ dks.k ij F

cy yxkdj èkosQyk tkrk gS rc Hkh {kSfrt

?kVd rks F cos θ gh gksrk gS ijUrq ÅèokZèkj

?kVd F sin θ eksoj osQ Hkkj esa tqM+ tkrk gS

ftlls bls èkosQyuk dfBu gks tkrk gSA

(1)

(1)

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izfrn'kZ iz'u i=k

100

50

0

–50 Gf ( )�� H

20mxI

E

CFB

x D10m

(c) U;wVu osQ tM+Ro osQ fu;e osQ vuqlkj tc njh dks MaMs ls ihVk tkrk gS rks ;g vpkud vkxs c<+

tkrk gS tcfd èkwy osQ d.k viuh iwoZfLFkfr esa fojke esa cus jgrs gSa blfy, os xq#Ro osQ rgr

uhps fxj tkrs gSaA (1)

21. (a) 7.5 × 103 N, uhps dh vksj

(b) 3.25 × 104N, uhps dh vksj

(c) 3.25 × 104N, Åij dh vksj (1+1+1)

22. efgyk }kjk fd;k x;k dk;Z = 1750 J

?k"kZ.k cy }kjk fd;k x;k dk;Z = –1000 J (1+1+1)

23.2 2 21

; ; 22

f i i f it t t= + = + = +ω ω α θ ω α ω ω αθ (1+1+1)

24. xfrt ÅtkZ dh O;qRifÙk= ,GMm

2r

= –GMm

rxfrt ÅtkZ

15–5 ×10 J2B

1 1 GMmV = – mv = – =

2 2 r (1+1+1)

25. cuwZyh izes; dk dFku ,oa mRifÙk (1 + 2)

26. ( ) ( )P V v P p Vγ γ+ ∆ = + ∆

1 1v p

P PV P

γ∆ ∆

=+ + (1)

;v p dv V

V P dp pγ

γ

∆ ∆= =

W.D. ( )2 2

1 1

2 1P P

P P

P PVP dv P dp V

pγ γ

−= = = (2)

vFkok

270 11

300 10η = − = (1)

jsfÚtsjsVj dh n{krk 1

0.520

η= = (1)

;fn Q mPp rki ij izfr lsoaQM LFkkukUrfjr Å"ek gS%

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194

rks 1

20

W

Q= or Q = 20W = 20µKJ

rFkk fuEurj rki ls yh xbZ Å"ek = 19 kJ. (1)

27. lacaèk 2

nv

Lν = , ls gesa tks ifj.kke pkfg, izkIr gks tkrk gSA

vko`fÙk;ksa osQ vuqikr dk ifjdyuµ

+ + + +

1 1 1 11

2 2 2 2

28. (a) fdlh rjy osQ ';kurk xq.kkad dks vi#i.k izfrcy vkSj foÑfr nj osQ vuqikr osQ :i esa ifjHkkf"kr dj

ldrs gSaµ

η = =/

/

F A Fl

v l vA (1)

';kurk dk SI ek=kd iks;lqys (Pl) gS (½)

(b) fdlh ';ku rjy esa fxjrs fiaM dk vafre osx og vpj vfèkdre osx gS ftlij ';ku cy uhps dh

vksj yxus okys oqQy cy dks fu"izHkkfor dj nsrk gSA (1)

( )2 –2

9

LTv r=

ρ ρ

ηdh O;qRifÙk (2½)

vFkok

(i) fcUnq P rd (1+1+1+1+1)

(ii) fcUnq E(iii) izR;kLFk {ks=k µ O ls E

IykfLVd {ks=k µ E ls B(iv) P rd foÑfr Hkkj c<+kus ls c<+rh gS vkSj blosQ vuqØekuqikrh gksrh gSA P ls vkxs blosQ eku esa

o`f¼ Hkkj esa o`f¼ ls vfèkd gksrh gSA izR;kLFkrk lhek E ikj gksus ij ;g oØ dks okil iqu%vuqjsf[kr ugha djrk cfYd Hkkj gVkus ij fcUnqfdr js[kk AO′ osQ vuqfn'k okil ykSVrkA fcUnq O′

'kwU; Hkkj osQ laxr gS vkSj rkj esa ,d LFkkbZ foÑfr dk fu#i.k djrk gSA(v) C ls B rd rkj ij ls Hkkj gVkus ij Hkh foÑfr esa o`f¼ gksrh gS vkSj B ij ;g VwV tkrk gSA fcUnq

C osQ laxr izfrcy rd vkjksfir djus ij rkj ugha VwVrkA

29. (a) 2 2 10 1000 141m/s 510km/hv gh= = × × = = (1+1+1+1+1)

(b)3 3 3 3 54 4

(2 10 ) (10 ) 3.4 10 kg3 3

m rπ π

ρ − −= = × = ×

3 34.7 10 kg m/s 5 10 kg m/sP mv − −= ≈ × ≈ ×

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izfrn'kZ iz'u i=k

(c) O;kl 4mm≈

/ 28 s 30 st d v µ µ∆ ≈ = ≈

(d)

32

6

4.7 10168N 1.7 10 N

28 10

PF

t

−

−

∆ ×= = ≈ ≈ ×

∆ ×

(e) ,d izk:fid Nkrs dk O;kl 1m gksrk gSA

∴vuqizLFkdkj dk {ks=kiQy 2 2/4 0.8mdπ= ≈

5cm osQ vkSlr i`Fkdu ij] yxHkx ,d lkFk fxjus okyh cw¡nksa dh la[;k 2

2 2

0.8m320

(5 10 )−≈

×

vFkok

(i)1 sin

tancoso

o

v

v u

θ

θ− +

(1+½+1+1+½+½+½)

(ii)2 sinov

g

θ

(iii)2 ( )o ov v u

Rg

+=

��

(iv)

2 21

max

8

4

o

o

u u vcos

vθ −

− + +=

(v) θ = =max 60 ,oou v oQs fy, , θ = =max 45 0o u osQ fy, A

< ou v osQ fy, :

1max

1

42 o

ucos

vθ −

−≈

= ( )4 oif u vπ

( )θ π−> ≈ = 1max f: /2o

oo

vi v uu v cos

uosQ fy,

30. (a) S.H.M. esa fdlh fn, x, {k.k ij d.k dk foLFkkiu gksrk gSµ

y = sinr tω

osx, cosdy

v r tdt

= = ω ω

Roj.k 2– sindv

a r tdt

= = ω ω 2– yω= (1)

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196

blfy, SHM djrs fdlh fiaM dk Roj.k ml {k.k blosQ ekè; fLFkfr ls foLFkkiu osQ vuqØekuqikrh gksrk gSA

(b) ekuk fd CykWd dks nckus ij larqyu voLFkk esa bldk ÅèokZèkj foLFkkiu xo gSA

larqyu voLFkk esa

mg = mRIykou cy

. . .oA x gρ=

blls vkxs vkSj vfèkd foLFkkiu x, nsus ij mRIykou cy gksxk ( )oA x x gρ+

oqQy izR;ku;u cy = mRIykou cy — Hkkj= ( )oA x x gρ+ — mg

( )A g xρ= . vFkkZr x osQ vuqØekuqikrh

2m

TA g

πρ

∴ =

vFkok

(a) 5m (b) 5m (c) 50Hz (d) 250ms-1 (e) 500π ms-1 A (1+1+1+1+1)

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197

izfrn'kZ iz'u i=k

izzfrn'kZ iz'ui=k IICyw&fizUV (#ijs[kk)

fo"k;fo"k;fo"k;fo"k;fo"k; VSA (1 vad) SAI (2 vad) SA II (3 vad) LA (5 vad) ;ksx

I HkkSfrd txr vkSj ekiu 1(1) 2(1) — — 3(2)

II 'kq¼ xfrdh — 4(2) 6(2) — 10(4)

III xfr osQ fu;e — 2(1) 3(1) 5(1) 10(3)

IV dk;Z] ÅtkZ ,oa 'kfDr 1(1) 2(1) 3 (1) — 6(3)

V d.kksa dh iz.kkyh rFkk n`<+ — — 6 (2) — 6(2)

fiaMksa dh xfr

VI xq#Rokd"kZ.k 1(1) 4(2) — — 5(3)

VII LFkwy nzO; osQ xq.k 2(2) — 3(1) 5(1) 10(4)

VIII rki xfrdh 2(2) — 3(1) — 5(3)

IX vkn'kZ xSl dk O;ogkj ,oa 1(1) 4(2) — — 5(3)

xSl dk v.kq xfrd fl¼karX nksyu ,oa rjaxs — 2(1) 3(5) 5(1) 10(3)

;ksx 8(8) 20(10) 27(9) 15(3) 70(30)

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198

izfrn'kZ iz'u i=k IIHkkSfrdh – 11


lkekU; funsZ'kµ

(a) lHkh iz'u vfuok;Z gSaA

(b) oqQy 30 iz'u fn, x, gSaA iz'u 1 ls 8 rd izR;sd iz'u 1 vad dk gS] iz'u 9 ls 18 rd izR;sd iz'u 2 vad dkgS] iz'u 19 ls 27 rd izR;sd iz'u 3 vad dk gS rFkk iz'u 28 ls 30 rd izR;sd iz'u 5 vad dk gSA

(c) iw.kZ :i ls p;u dk izko/ku ugha gSA

(d) ifjdfy=kksa osQ mi;ksx dh vuqefr ugha gSA

(e) vko';drk iM+us ij vki uhps fn, x, HkkSfrd fu;rkadksa dk mi;ksx dj ldrs gSaµ

c = 3 × 108ms-1

h = 6.6 × 10-34Js

µo = 4π × 10–7 TmA–1

Boltzmann constant k = 1.38 × 1023 JK-1

Avogadro’s number NA = 6.023 × 1023/mole

1. nzo dk vi:i.kkad gksrk gSµ

(a) vuUr (b) 'kwU; (c) ,d (d) dksbZ lkUr] y?kq] 'kwU;rj] fu;reku

2. ;fn nks fiaMksa osQ fy, uhps fn, izR;sd fodYi esa mfYyf[kr izkpy osQ vfrfjDr vU; lHkh izkpy ,d lekugksa rks fdl izdj.k esa mudh xfrt ÅtkZ leku gksxh\

(a) fiaM Adk nzO;eku B osQ nzO;eku dk nksxquk gSA

(b) A dk vk;ru B osQ vk;ru dk vkèkk gSA

(c) fiaM A uhps dh vksj Lora=kkiwoZd fxj jgk gS tcfd B fdlh {k.k mruh gh pky ls Åij dh vksj tk jgk gSA

(d) fiaM A fu;r pky ls {kSfrt fn'kk esa py jgk gS tcfd B eqDr :i ls fxj jgk gSA

3. ;fn lw;Z ,oa xzgksa ij foijhr vkos'kksa dh fo'kky ek=kk,a gksrh rks

(a) osQIyj osQ lHkh rhu fu;e rc Hkh ekU; gksrsA

(b) osQoy rhljk fu;e gh ekU; gksrkA

(c) f}rh; fu;e esa dksbZ ifjorZu u gksrkA

(d) izFke fu;e rc Hkh ekU; gksrkA

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izfrn'kZ iz'u i=k

4. uhps nh xbZ HkkSfrd jkf'k;ksa osQ tksM+ksa esa fdl dk foHkh; lw=k ,d tSlk ugha gS\

(a) dk;Z ,oa cy vk?kw.kZ

(b) dks.kh; laosx ,oa Iykad fu;rkad

(c) ruko ,oa i`"B ruko

(d) vkosx ,oa js[kh; laosx

5. dksbZ vkn'kZ xSl ,d gh izkjafHkd fLFkfr ls izkjaHk djosQ pkj fofHkUu izØeksals xqtjrh gSA ;s pkj izØe gSa%

:¼ks"eh;] lerkih;] lenkch; rFkk levk;rfudA A, B, C ,oa D esa ls:¼ks"eh; dkSu&lk gS\

(a) (B)

(b) (A)

(c) (C)

(d) (D)

6. leku eksVkbZ osQ diM+ksa dh nks ijrksa osQ oL=k iguus ls nksxquh eksVkbZ osQ vosQys diM+s osQ oL=k dh rqyuk esavfèkd xehZ D;ksa izkIr gksrh gS\

7. vkn'kZ xSl osQ fn, x, nzO;eku osQ fy, vk;ru vkSj rki dk xzkiQ esa nkc osQnks fofHkUu fu;r ekuksa osQ fy, fp=k&2 esa n'kkZ;k x;k gSA P

1 ,oa P

2 osQ chp lacaèk

osQ fo"k; esa D;k fu"d"kZ fudkyk tk ldrk gS\?

(a) 1 2P P>

(b) 1 2P P=

(c) 1 2P P<

(d) dksbZ Hkh fu"d"kZ fudkyus osQ fy, i;kZIr vkadM+s miyCèk ugha gSA

8. fdlh èkkjk&js[kk osQ vuqfn'kµ

(a) rjy d.kksa dk osx fu;r jgrk gSA(b) fdlh nh xbZ fLFkfr osQ ikj tkus okys lHkh rj d.kksa dk osx fu;r gksrk gSA(c) fdlh fn, x, {k.k ij rjy osQ lHkh d.kksa dk osx fu;r gksrk gSA(d) fdlh rjy d.k dh pky fu;r gksrh gSA

9. U;wVu osQ xfr osQ r`rh; fu;e dk dFku fyf[k, vkSj bldk mi;ksx djosQ jSf[kd laosx laj{k.k fu;e O;qRiUudhft,A

fp=k 1

P

D

C

B

A

V

10

100 200300400500

20

30

40

V

T (K)

( )l P2

P1

fp=k 2

18-04-2018


200

10. x vkSj t osQ chp ,d xzkiQ fp=k&3 esa n'kkZ;k x;k gSA uhps fn, x,fodYiksa esa ls lgh fodYiksa dk p;u dhft,%

(a) t = 0, ij d.k dks fojke voLFkk ls eqDr fd;k x;k FkkA

(b) B ij] Roj.k a > 0

(c) C ij, osx ,oa Roj.k 'kwU; gks tkrs gSaA

(d) A ,oa D osQ chp xfr osQ fy, vkSlr osx dk eku èkukRed gksrkgSA

(e) D ij pky] E ij pky ls vfèkd gksrh gSA

11. dksbZ okgu nwjh L dh vk/h nwjh rks pky V1 lsA r; djrk gS vkSj 'ks"k vkèkh nwjh pky V

2 ls] bldh

vkSlr pky gS%

(a)1 2

2

V V+

(b)1 2

1 2

2V V

V V

+

+

(c)1 2

1 2

2V V

V V+

(d)1 2

1 2

( )L V V

V V

+

12. fp=k&4 esa n'kkZ, x, vkÑfr;ksa esa dkSu&lk i`Foh dk lw;Z osQ ifjr% viuh nh?kZo`Ùkh; d{kk esa ,d ifjØe.kosQ nkSjku xfrt ÅtkZ esa ifjorZu lokZfèkd fudVre :i esa n'kkZrk gS\

x

AB

C

t

E

DO

fp=k 4

fp=k 3

(a)(b)

(c) (d)

18-04-2018

201

izfrn'kZ iz'u i=k

13. fdlh py&lw{en'khZ osQ ofuZ;j iSekus ij 50 Hkkx gSa tks eq[; iSekus osQ 49 Hkkxksa osQ vuq:i gSaA ;fn eq[;iSekus osQ ,d Hkkx dk eku 0.5 mm gS rks nwjh osQ ekiu esa U;wure viFkkFkZrk dk ifjdyu dhft,A

14. fdlh ik=k esa nks ,d&ijek.kqd xSlsa Hkjh gSa ftudk nzO;ekuqlkj vuqikr 1:1 gSA feJ.k dk rki 27°C gSA ;fnmuosQ ijek.kq nzO;ekuksa dk vuqikr 7:4 gks] rks (i) izfr v.kq vkSlr xfrt ÅtkZ (ii) xSl ijek.kqvksa dh rms pky]D;k gS\

15. 500kg dk dksbZ mixzg i`Foh osQ pkjksa vksj Re f=kT;k dh o`Ùkkdkj d{kk esa ifjØek dj jgk gSA bldks

4 Re f=kT;k dh o`Ùkkdkj d{kk esa LFkkukarfjr djus osQ fy, fdruh ÅtkZ dh vko';drk gS\ bldh xfrt vkSj

fLFkfrt ÅtkZ esa D;k ifjorZu gksrs gSa\ ( )6 –2= 6.37 ×10 m, = 9.8 × m seR g

16. 17 cm yEckbZ dk ,d fljs ij cUn ikbi 1.5 kHz lzksr osQ lkFk vuqukfnr gksrk gqvk ik;k tkrk gSA (a)

ikbi dk dkSu lk gkeksZfud mijksDr lzksr osQ lkFk vuqukn djrk gS\ (b) ;fn ikbi nksuksa fljksa ij [kqyk gks rksD;k rc Hkh ml lzksr osQ lkFk vuqukn izsf{kr fd;k tk ldrk gS\ vius mÙkj osQ leFkZu esa roZQ nhft,A(ok;q esa èofu dh pky = 340 m s-1)

17. n'kkZb, fd fdlh vkn'kZ xSl osQ ,d v.kq dh vkSlr xfrt ÅtkZ xSl osQ ije rki osQ vuqØekuqikrh gksrh gSA

18. i`Foh osQ i`"B osQ uhps h xgjkbZ ij xq#Ro osQ dkj.k Roj.k osQ fy, U;×td O;qRiUu dhft,A

19. fdlh xfreku d.k dh fLFkfr dks 2ˆ ˆ ˆ6 4 10t t= + +r i j k tgk¡ r esa vkSj t lsoaQM esa gSA

(a) le; osQ iQyu osQ :i esa osx ,oa Roj.k Kkr dhft,A

(b) t = 2s ij osx dk ifjek.k ,oa fn'kk Kkr dhft,A

20. ,d unh 3m s–1 dh fu;r pky ls iwoZ dh vksj izokfgr gks jgh gSA dksbZ rSjkd'kkar ty esa 4 m s–1 dh pky ls rSj ldrk gSA (fp=k&5)

(a) ;fn ;g rSjkd mÙkj dh vksj rSjuk 'kq# djs rks bldk ifj.kkeh osx(ifjek.k ,oa fn'kk) D;k gksxk\

(b) ;fn og nf{k.kh rV osQ fcUnq A ls izkjaHk djosQ mÙkjh rV ij fLFkfr AosQ foijhr fcUnq B ij igqapuk pkgs rks

(i) mldks fdl fn'kk esa rSjuk pkfg,\

(ii) mldh ifj.kkeh pky d;k gksxh\

(c) Åij mfYyf[kr nks fofHkUu izdj.kksa (a) ,oa (b) esa ls fdl esa og foijhrrV ij de le; esa igq¡psxk\

21. (a) 1 g nzO;eku dh o"kkZ dh ,d cw¡n 1 km Å¡pkbZ ls fojke voLFkk ls fxjrh gS vkSj Hkwry dks50m s–1 dh pky ls Vdjkrh gSA

(i) cw¡n dh vafre xfrt ÅtkZ (K.E.) D;k gS vkSj izkjafHkd fLFkfrt ÅtkZ (P.E.) fdruh gS ?(ii) bu nks ekuksa esa vUrj dh O;k[;k vki oSQls djsaxs\

(g = 10m s–2 yhft,)A

fp=k 5

N

E

B

A

3m/s

18-04-2018


202

fp=k 8

(b) nks loZle cky cs;fjax ,d nwljs dks Li'kZ djrs gq, ,d ?k"kZ.k jfgr est ij j[ks gSa vkSj buls leku nzO;ekudk ,d vU; cky cs;fjax V osx ls pyrk gqvk vkdj lEeq[k la?kV~V djrk gS] tSlk fd fp=k&6 esa n'kkZ;kx;k gSA

;fn la?kV~V izR;kLFk gks rks uhps (fp=k&7) n'kkZ, x, fp=kksa esa dkSu&lk la?kV~V osQ i'pkr dh laHkkfor fLFkfrfu:fir djrk gS\

22. (a) le>kb, fd D;ksa gFkxksys dks èkosQyus dh vis{kk [khapuk vklku gksrk gSA

(b) fp=k&8 esa f}foHkh; xfr djrs ,d d.k osQ (x, t), (y,t) xzkiQ n'kkZ, x, gSaA ;fn d.k dk nzO;eku500 g gks rks bl ij vkjksfir cy (ifjek.k ,oa fn'kk) Kkr dhft,A

fp=k 6

x

t1s 2s 3s

(c)

(d)

(a)

(b)

1

V = 0 V/2

1 2 3

V /3

1 2 3

V /1 V /2 V /3

fp=k 7

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203

izfrn'kZ iz'u i=k

23. (a) lekUrj v{k ,oa yEcor v{k izes;ksa osQ dFku fyf[k,A

(b) fdlh xksys dks Li'kZ djrh gqbZ v{k osQ ifjr% bl xksys dk tM+Ro&vk?kw.kZ Kkr dhft,A ;g fn;k gSfd blosQ fdlh O;kl osQ ifjr% xksys dk tM+Ro&vk?kw.kZ 2/5(MR2) gS] tgk¡ M xksys dk nzO;eku vkSjR bldh f=kt;k gSA

24. ,d 3m yEch lh<+h ftldk nzO;eku 20 kg gS] ,d ?k"kZ.k jfgr nhokj ij vkur gSA bldk fupyk fljk iQ'kZij nhokj ls 1 m dh nwjh ij fVdk gSA nhokj vkSj iQ'kZ osQ izfrfØ;k cyksa dk ifjdyu dhft,A

25. ,d iwjh rjg ejs gq, cksbax foeku dk nzO;eku 3.3×105 kg gSA blosQ i{kksa dk oqQy {ks=kiQy 500 m2 gSA;g ,d {kSfrt ry esa 960 km h–1 dh leku pky ls mM+ jgk gSA (a) i{kksa osQ uhps vkSj Åij osQ i`"BksaosQ chp nkckUrj dk ifjdyu dhft,] rFkk (b) uhps osQ i`"B dh rqyuk esa Åij osQ i`"B ij c<+s gq,ok;q osQ osx dks fHkUukRed o`f¼ osQ :i esa vkadfyr dhft,A(ok;q dk ?kuRo –31.2kg mρ = )

26. jsfizQtsjsVj dk dk;Z fl¼kar la{ksi esa le>kb, vkSj blosQ ifjpkyu xq.kkad osQ fy, O;×td O;qRiUu dhft,A

27. tc èofulzksr vkSj Jksrk nksuksa ,d gh fn'kk esa xfreku gksa rks Jkzrk }kjk lquh xbZ èofu dh vkHkklh vko`fÙk osQfy, O;×td O;qRiUu dhft,A

28. (a) n'kkZb, fd NksVs vk;keksa osQ fy, ljy yksyd xfr ljy vkorhZ gksrh gS vkSj blosQ vkoÙkZdky osQ fy,O;×td Hkh O;qRiUu dhft,A

(b) nks loZld yksydksa osQ ,d tksM+s ij fopkj dhft, tks ,d nwljs ls Lora=k bl izdkj nksyu djrs gSa fdtc ,d yksyd viuh vUR; fLFkfr esa ÅèokZèkj ls nkfguh vksj 2° dks dks.k cukrk gS rks nwljk yksydviuh vUR; fLFkfr esa ck¡bZa vksj ÅèokZèkj ls 1° dk dks.k cukrk gSA bu nksuksa yksydksa osQ chp fdrukdyk&vUrj gS\

29. (a) osQf'kdh; mUu;u D;k gksrk gS\ r f=kT;k dh dksf'kdk uyh esa dksbZ nzo ftl Å¡pkbZ rd Åij mBrk gSmlosQ fy, O;a×td O;qRiUu dhft,A

(b) fdlh nzo dh NksVh cw¡nsa lnZo xksykdj D;ksa gksrh gS\

30. (a) ml vfèkdre lqjf{kr pky osQ fy, O;×td O;qRiUu dhft, ftlls dksbZ dkj fdlh ,sls cafdr(cSaDM)iFk ij lqjf{kr xqtj ldrh gS tks {kSfrt ls α dks.k ij ur gSA lM+d vkSj Vk;jksa osQ chp ?k"kZ.k

xq.kkad µ gSA

(b) ,d 100 kg dh rksi 500 m Å¡ph pV~Vku osQ f'k[kj ls 1 kg dk xksyk nkxrh gSA ;g xksyk Hkwfr ij pV~Vkudh ryh ls 400m dh nwjh ij fxjrk gSA rksi osQ fjdkW;y (izfr{ksi) osx dh x.kuk dhft,A (xq#Roh;Roj.k = 10 m s-2)

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204

izfrn'kZ iz'u i=k & IIgy ,oa vadu ;kstuk

1. (b) (1)

2. (e) (1)

3. (c) (1)

4. (c) (1)

5. (c)

6. diM+s dh nks ijrksa osQ chp iaQlh gqbZ gok gekjs 'kjhj ls Å"ek dks ckgj tkus ls jksdrh gSA (1)

7. (a) (1)

8. (b) (2)

9. dFku (1)

( )1 21 2 0

d d dor

dt dt dt= − + =

p pp p (½)

+ =1 2p p fu;rkda (½)

10. (a), (c), (e) (2)

11. (c) (2)

12. (d) (2)

13. 0.01 mm (2)

14. (i) 1:1, (ii) 1.32:1 (2)

15. ∆ 9=11.75 ×10 JE (1)

9KE = -11.75 ×10 J∆ (½)

9PE = -23.475 ×10 J∆ (½)

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205

izfrn'kZ iz'u i=k

16. (a) 2× 340 ×10

5004 17

nn=

×, tgk¡ n can ikbi esa gkeksZfud dh la[;k gSA 1.5 kHz lzksr osQ lkFk ikbi dk

rhljk gkeksZfud vuqukn djsxkA (1)

(b) nksuksa fljksa ij [kqys ikbi osQ fy,

23× 340 ×10

102 17

nn=

× tgk¡ n gkeksZfud dh la[;k gSA n osQ fdlh Hkh iw.kkZad eku osQ fy,1.5 kHz

osQ lkFk vuqukn laHko ugha gSA vr% vuUr gS ughaA (1)

17.21

3

MCP

V= (½)

21 2.

3 3PV M C K E= = (½)

PV = nRT (½)

K.E ∝ T (½)

18. AP = h (½)

( )2

e

GMg

R h

′′ =

− (½)

( )34

3eR hM π ρ′ −= (½)

e

1R

hg g

−′ =

(½)

19. (a) = 6 + 8tv i j (1+1+1)

8=a j

(b) 6 16 or 36 256 19.8m/sv= + = + =v i j .

v x-v{k osQ lkFk tan-1 (8/3) dks.k cukrk gSA

20. (i) 5 m/s 37at ° to N. (1)

(ii) (a) ( )1tan of N, (b) 7 m/s3/ 7− (½ + ½)

(iii) fLFkfr (1) esa og foijhr rV ij U;wure le; esa igq¡psxkA (1)

A

ReP

hP

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206

21. (i) (a) 1.25 J , 10J (1)

(b) vUrj ok;q osQ ';ku&cy }kjk fd, x, dk;Z osQ dkj.k gSA (1)

(ii) (b) (1)

22. (a)

([khapuk)(èkosQyuk)

[khapus dh fLFkfr esa F sinθ uhps dh vksj uxus okys oqQy cy dks de dj nsrk gSA (1)

(b) 2,x t y t= =

10, 2 m sx ya a −= =

F = 0.5×2 = 1N. y-v{k osQ vuqfn'k (2)

23. (a) lekarj v{k izes; dk dFku (1)

(b)27

5MR (lekarj v{k izes; dk mi;ksx djosQ) (1)

yacor~ v{k izes; dk dFku (1)

24. ekuk F1 ,oa F

2 Øe'k% nhokj ,oa iQ'kZ osQ izfrfØ;k cy gSaA

N–W = 0 (1)

F–F1 = 0 (½)

( )12 2 – 01/2F W = (½)

W = N = 20 × 9.8 N = 196 N

F = F1 = /4 2w = 34.6N (½)

F2 = 2 2F N+ = 199.0N (½)

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207

izfrn'kZ iz'u i=k

cy F2 {kSfrt ls α dks.k cukrk gSA

–1tan / 4 2, tan 4 2N Fα = = α = (½)

25. (a) cksbax foeku dk Hkkj nkckUrj osQ dkj.k Åij dh vksj yxus okys cy }kjk larqfyr gksrk gSA

5 –23.3 10 kg 9.8m sP A∆ × = × × (½)

∆ = × ×5 –2 2(3.3 10 kg 9.8m s )/500mP (½)

= 6.5×103 N m–2

(b) i{kksa osQ fupys vkSj Åijh i`"Bksa osQ chp nkckUrj gS%

( ) ( )2 22 1/2 –P v v∆ = ρ (½)

tgk¡ v2 Åijh i`"B ij ok;q dh pky rFkk v

1 fupys i`"B osQ uhps ok;q dh pky gSA

( )2 12 1

2–

pv v

v v

∆=

ρ + (½)

( ) –11 2 /2 960km/h 267m savv v v= + = = (½)

( ) 22 1– / / 0.08

avavv v v P v= ∆ ρ ≅

(½)

26. (a) O;qRØe Å"ek batu dk fl¼kar (1)

(b) (1)

��

�� T2

�� T2

W�=�Q -Q1 2

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208

(c) 2

1 2

T

T Tβ =

− (1)

27.+

= + o

oS

v vv v

v v . (3)

28. (a) ljy yksyd dk fp=k ftl ij cy n'kkZ, x, gksaA (1)

lw=k 2l

Tg

π= dh O;qRifÙk (2)

(b) ( )1 1sino tθ θ ω δ= +

( )2 2sino tθ θ ω δ= +

igys osQ fy,] ( )12 , sin 1tθ ω δ= ° ∴ + =

nwljs osQ fy, ( )21 , sin 1/2tθ ω δ= − ° ∴ + = −

1 290 , 30t tω δ ω δ∴ + = ° + = − °

1 2 120δ δ∴ − = ° (2)

29. (a) osQf'kdk O;ogkj dh ifjHkk"kk (1)

osQf'kdk p<+ko osQ fy, fp=kA (½)

O;qRifÙk (1½)

(b) i`"B ruko osQ dkj.k nzo dh c¡n U;wure i`"B {ks=k xzg.k djrh gSa tks

xksys dk gksrk gSA (2)

30. (a) fp=k (1)

lw=k 1/2

tan

1 tansV rg+

= −

µ α

µ α dh O;qRifÙk (2)

(b) 0.4 m s-1 (2)

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