1. xvkv †evw©-2015 · 01/03/2017 · 1. xvkv †evw ©-2015 1 bs cÖ‡kœi dËi ... l cv‡Îi...

c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb

1. XvKv †evW©-2015

1 bs cÖ‡kœi DËi

K †h mg Í msN‡l©i †ÿ‡Î MwZ kw³ msiwÿZ _v‡K Zv‡K

w¯’wZ¯’vcK msNl© e‡j|

L fi n‡”Q e ‘i RoZvi cwigvc| e¯‘ †h a‡g©i Kvi‡Y †Kv‡bv

wbw`©ó A‡ÿi mv‡c‡ÿ Zvi †KŠwYK MwZi cwieZ©‡b evav †`q Zv‡K

Zvi N~Y©b RoZv ev RoZvi åvgK e‡j| A_©vr îwLK MwZi †ÿ‡Î

fi †h f‚wgKv cvjb K‡i †KŠwYK MwZi †ÿ‡Î N~Y©b RoZv ev

RoZvi åvgK †m f‚wgKv cvjb K‡i| †Kv‡bv e ‘i fi mKj †ÿ‡Î

aªæe Aci c‡ÿ wbw`©ó A‡ÿi mv‡c‡ÿ †Kv‡bv e ‘i N~Y©b RoZv wbw`©ó

wKš‘ wfbœ wfbœ A‡ÿi mv‡c‡ÿ wfbœ wfbœ|

M

gvwS AB eivei †bŠKv Pvjbv K‡i AC eivei Icv‡i †cuŠQvj|

†mÖv‡Zi †eM, u = 4kmh–1

†bŠKvi †eM, v = 3kmh–1

Ges †mÖvZ I †bŠKvi †e‡Mi gaëZx© †KvY, = 90

DÏxc‡Ki wP‡Î,

ABC G, ACB = = 30

sin = ABAC

ev, AC = AB

sin =

1.5km

sin =

1.5km12

= 3km

AC eivei †bŠKvi AwZµvšÍ `~iZ¡ = 3km (Ans.)

N g‡b Kwi, †mÖvZ I evZv‡mi mw¤§wjZ †eM w.

tan30 = vsin90

w + vcos 90

ev, 1

3 =

vw

ev, w = v 3

w = 3 3 km/h

†bŠKvwU‡K AD eivei Pvjbv Ki‡j; g‡b Kwi, jwä †eM R Ges Zv

cv‡oi mv‡_ †KvY Drcbœ K‡i,

R cos = w + vcos 150

ev, R cos = 3 3 + 3 – 3

2

ev, Rcos = 3 3 – 3 3

2

Rcos = 3 3

2 .......... (i)

Rsin = wsin0 + vsin 150

ev, Rsin = 0 + 3 12 = 1.5..........(ii)

(i) I (ii) bs n‡Z,

tan = 1.5

3 3

2

= 1

3

= 30

AD eivei Pvjbv Ki‡j †bŠKvwU B we›`y‡Z †cuŠQ‡e bv|


K ej cÖ‡qv‡M †Kv‡bv e ‘i ˆ`N©¨, AvKvi ev AvqZ‡bi cwieZ©b

NUv‡bv n‡j ej AcmviY Kiv gvÎB e¯‘wU c~e©ve ’vq wd‡i Avmvi

ag©‡K w¯’wZ¯’vcKZv e‡j|

L Lvov Dc‡i wbwÿß e ‘i †ÿ‡Î Abyf‚wgK w`‡K wb‡ÿcY †e‡Mi

Dcvsk k~b¨| ZvB wbwÿß e ‘i Abyf‚wgK ~iZ¡I k~b¨ nq|

M †Ìqv Av‡Q,

wb‡ÿcY †eM, v0 = 20 ms1

wb‡ÿcY †KvY, 0 = 45

AwfKl©R Z¡iY, g = 9.8 ms2

†ei Ki‡Z n‡e, me©vwaK D”PZv, H = ?

Avgiv Rvwb, H = v0

2sin20

2g = (20 ms1)2 (sin 45)2

2 9.8 ms2 = 10.2m (Ans.)

N g‡b Kwi,

ejwU 35m Abyf‚wgK ~iZ¡ AwZµg Kivi gyn~‡Z© f‚wg n‡Z

h D”PZvq _vK‡e| G ~iZ¡ AwZµ‡g t mgq jvM‡j,

x = v0cos0t t = x

v0cso0 =

35m

20 ms1 cos45 = 2.475 sec

h = v0sin0t 12

gt2 = 20 sin45 2.475 12

9.8 (2.475)2

= 4.986 m > 3m

30

d=1.5km

A

3km jwä †eM = R

B C 30

†mÖv‡Zi †eM, u= 4kmh–1

‡bŠK

vi †

eM,

v =

3km

h–

1

= 290

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myZivs H wdìvi K¨vP wb‡Z mg_© n‡Zb bv|


K †Kv‡bv K…wÎg DcMÖ‡ni AveZ©bKvj wbR A‡ÿi Pviw`‡K

N~Y©vqgvb c„w_exi AveZ©b Kv‡ji mgvb Ges AveZ©‡bi w`K c„w_exi

AveZ©‡bi w`‡K n‡j, c„w_exi mv‡c‡ÿ GwU w¯’i _vK‡e| G ai‡bi

DcMÖn‡K f‚-w ’i DcMÖn e‡j|

L GKRb †`Šowe` †`Š‡oi ïiæ‡Z mvg‡bi w`‡K Szu‡K _v‡Kb|

G‡Z gvwU‡Z cv w`‡q wZwb Zxh©Kfv‡e ej cÖ‡qvM K‡ib| d‡j

cÖwZwµqv e‡ji w`KI nq Zxh©K A_©vr mvg‡bi w`‡K| cÖwZwµqv

e‡ji GKwU e„nr Dcvsk mvg‡bi w`‡K KvR K‡i| d‡j ªæZMwZ

AR©b Ki‡Z †`Šowe‡ì †ek myweav nq|

M †Ìqv Av‡Q,

mij †`vj‡Ki †`vjbKvj, T = 2 sec

†ei Ki‡Z n‡e, †KŠwYK K¤úv¼, = ?

Avgiv Rvwb, = 2T =

2 3.14162 = 3.1416 rad.s1 (Ans.)

N cÖv_wgK Ae ’vq †`vjKwUi †`vjbKvj, T1 = 2 s

cwieZ©xZ †`vjbKvj n‡e, T2 = 2 s + 2 s 50% = 3 s

cÖv_wgK Kvh©Ki ˆ`N©¨, L1 = 100 cm

cwieZ©xZ Kvh©Ki = L2

mij †`vj‡Ki wØZxq m~Î †_‡K cvB,

T1

T2 =

L1

L2

ev, L1

L2 =

T12

T22 =

22

32 = 49

ev, L2 = 94 L1 =

94 100 cm = 225 cm

myZivs DÏxc‡Ki kZ©vbymv‡i †`vj‡Ki Kvh©Ki ˆ`N©¨ n‡Z n‡e 225 cm, wKš‘ Kvh©Ki `N©¨ Kiv n‡q‡Q 150 cm| myZivs 150 cm Kvh©Ki

ˆ`‡N©ï †`vjKwU DÏxc‡Ki kZ© c~iY K‡iwb|

4bs cÖ‡kœi DËi

K jf¨ Kvh©Ki kw³ Ges †gvU cÖ`Ë kw³i AbycvZ‡K h‡š¿i

Kg©`ÿZv e‡j|

L ywU e¯‘i w ’wZ¯’vcK msN‡l©i †ÿ‡Î msN‡l©i ci cÖ_g e¯‘i

†eM,

iif vmm

mv

mm

mmv 2

21

21

21

211

2

†`qv‡ji mv‡_ e‡ji msN‡l©i †ÿ‡Î, v2i = 0 Ges m2 » m1| myZivs

v1f = – v1i Ges v2f = 0

A_©vr †`qvj w ’i _vK‡e Ges ejwU GKB †eM wecixZ w`‡K wd‡i

Avm‡e|

M †`qv Av‡Q, cÖ_g Zv‡ii-

ˆ`N©¨ weK…wZ, LL = 10 % = 0.1

e¨vm, D = 2 mm

e¨vm n«vm D n‡j cqm‡bi AbycvZ,

= DD /

LL

ev, DD =

LL = 0.5 0.1 = 0.05

D = 0.05 2 mm = 0.1 mm

myZivs e¨vmva© n«vm, r = 0.1 mm

2 = 0.05 mm (Ans.)

N aiv hvK, Dfq Zv‡ii Avw` ˆ`N©¨ = L

cÖ_g Zv‡ii e¨vmva©, r1 = 1 mm = 10 – 3 m

wØZxq Zv‡ii e¨vmva©, r2 = 2.5 mm = 2.5 10 – 3 m

cÖ_g Zv‡ii ˆ`N©¨ e„w× = l1

wØZxq Zv‡ii ˆ`N© e„w× = l2

kZ©vbymv‡i, l1 = 3 l2

cÖ_g Zv‡ii Bqs‡qi ¸Yv¼, Y1 = F

r12 /

l1L

wØZxq Zv‡ii Bqs‡qi ¸Yv¼, Y2 = F

r22 /

l2L

Y1

Y2 =

r22

r12

l2l1

=

2

3

3

m 10

m 105.2

l2

3l2 =

6.253 = 2.083

Y1 = 2.083 Y2

myZivs cÖ_g Zv‡ii w ’wZ¯’vcKZv †ewk|


K e„Ëc‡_ N~Y©‡bi mgq †Kv‡bv e ‘i Ici e„‡Ëi †K‡›`Öi w`‡K †h

ej wµqv K‡i Zv‡K †K›`ÖgyLx ej e‡j|

L evZv‡mi cÖev‡ni w`‡K †`Šov‡j e¨w³i mv‡c‡ÿ evZv‡mi

Av‡cwÿK †eM, evZv‡mi cÖK…Z †eM A‡cÿv Kg nq| ZvB ZLb

evZv‡mi †eM K‡g †M‡Q e‡j g‡b nq|

M †Ìqv Av‡Q, K…wÎg DcMÖ‡ni MwZkw³, EK = 3.6 109J

K…wÎg DcMÖ‡ni fi, m = 120 kg

K…wÎg DcMÖ‡ni †eM v n‡j, EK = 12 mv2


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v = 2EK

m = 2 3.6 109J

120 kg = 7746 ms1

f‚c„ô n‡Z K…wÎg DcMÖ‡ni D”PZv h n‡j, v = GM

R + h

ev, R + h = GMv2 h =

GMv2 R

= 6.6 1011 Nm2kg2 6 1024 kg

(7746 ms1)2 6.4 106m

2 105m = 200 km (Ans.)

N K…wÎg DcMÖnwU‡K ewntwe‡k¦ Z_v gnvk~‡b¨ †cÖi‡Yi Rb¨

cÖ‡qvRbxq me©wb¤œ kw³,

hRhRhR

drrGMmdrr

GMmFdrW 2

2

hRGMm

rGMm

hR

111=

GMm

R + h

= 6.6 10-11 6 1024 120

6.4 106 + 2 105

= 7.2 109 J wKš‘ K…wÎg DcMÖ‡ni MwZkw³ 3.6 109 J, hv cÖ‡qvRbxq me©wb¤œ

kw³ A‡cÿv Kg| myZivs ejv hvq, mÂvwiZ MwZkw³ K…wÎg

DcMÖnwU‡K ewntwe‡k¦ cvVv‡bvi Rb¨ ch©vß bq|


K mgy`ª c„‡ô 45 Aÿvs‡k 0C ZvcgvÎvq Djø¤^fv‡e Aew ’Z 760 mm D”PZv wewkó ï®‹ I weï× cvi` ¯ Í‡¤¢i Pvc‡K cÖgvY Pvc ejv

nq|

L cv‡Îi †`qvj Zvc mycwievnx Ges M¨v‡mi ms‡KvPb ev cÖmviY

ax‡i ax‡i msMwVZ n‡j M¨vm cwi‡ek‡K Zvc †`q A_ev cwi‡ek

n‡Z Zvc MÖnY K‡i| d‡j ZvcgvÎv AcwiewZ©Z _v‡K| G‡Z M¨v‡mi

Pvc I AvqZb cwiewZ©Z nq| ZvcMZxq G cÖwµqv‡K m‡gvò

cÖwµqv e‡j|

M g‡b Kwi, wbg¾‡bi c~‡e© †ejy‡bi AvqZb wQj = V1

n«‡ì Zj‡`‡k †ejy‡bi cÖviw¤¢K AvqZb, V2 = 1 L

n«‡ì MfxiZv, h = 40.81 m

n«‡ì c„ô‡`‡k Pvc, P1 = 105 Nm–2

n«‡ì Zj‡`‡k Pvc, P2 = P1 + hg

e‡q‡ji m~Îvbymv‡i, P1V1 = P2V2 ev, P1V1 = (P1 + hg)V2

V1 = (P1 + hg)V2

P1

= 105 Nm2 + 40.81m 103 kgm3 9.8 ms2

105 Nm2 1 L

= 5 L

AZGe wbg¾‡bi c~‡e© DÏxc‡Ki †ejy‡bi AvqZb 5 L wQj| (Ans.)

N n«‡ì Zj‡`‡k †ejybwUi bZzb AvqZb, V1 = (1L + 1L) = 2L

awi, c„ô †`‡k Avm‡j †ejy‡bi AvqZb = V2

P2V2 = P1V1

V2 = P1V1

P2 =

(P2 + hg)V1

P2

= 105 Nm2 + 40.81m 103 kgm3 9.8 ms2

105 Nm2 2L

= 10 L > 9 L

myZivs cvwbi DcwiZ‡j †ejybwU AÿZ Ae ’vq †cŠQv‡e bv| †ejybwU

†d‡U hv‡e|

2. ivRkvnx †evW©-2015


K GKwU †f±i‡K hw` yB ev Z‡ZvwaK †f±‡i Ggbfv‡e wef³

Kiv nq, hv‡ì jwä n‡e g~j †f±i, Z‡e G wef³KiY cÖwµqv‡K

†f±‡ii we‡kølY e‡j|

L bvj †f±i n‡jv k~b¨ †f±i| Gi gvb k~b¨ e‡j Gi †Kv‡bv

mywbw`©ó w`K wbY©q Kiv m¤¢e bq| ZvB Gi w`K †h‡Kv‡bv w`‡KB

we‡ePbv Kiv †h‡Z cv‡i|

M †bŠKvi Zj I cvwbi Nl©Y ej D‡cÿYxq n‡j G‡ÿ‡Î

†bŠKvwU‡Z Z¡iY m„wó n‡e|

Avw`‡eM, u = 0 ms1

mgqKvj, t = 5 min = 300 sec

miY, s = 3.6 km = 3600 m

s = ut + 12 at2 ev, a =

2st2 [ u = 0] =

2 3600 m(300s)2 = 0.08 ms2

jwä Uvb, T = ma = 500 kg 0.08 ms2 = 40N

Zvn‡j, T = F2 + F2 + 2F.F cos60

ev, T = F 1 + 1 + 2 12 = F 3


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F = T

3 =

40N

3 = 23.094N (Ans.)

N mgvb Uv‡b (F = 23.1N) D³ ~iZ¡ (3.6km) 5 wgwb‡Ui Kg mg‡q

†cuŠQv m¤¢e| †m‡ÿ‡Î iwkØ‡qi ˆ`N©¨ evov‡Z n‡e hv‡Z cÖhy³

UvbØ‡qi ga¨Kvi †KvY 60 A‡cÿv Kg nq|

awi, Gevi = 55

Zvn‡j, F = 23.1N Gi Rb¨ jwä ej, R = F2 + F2 + 2F.F cos

= 2F2 (1 + cos ) = F 2 (1 + cos)

= 23.1N 2 (1 + cos 55) = 40.98N

†bŠKvq Z¡iY, a = Rm

40.98N500kg = 0.082ms–2

G‡ÿ‡Î, S = 3600m `~iZ¡ AwZµ‡g t cwigv‡Y mgq jvM‡j, S = vot

+ 12 at2 = 0.t +

12 at2 =

12 at2

ev, t2 = 2Sa

t = 2Sa =

2 3600 m0.082ms–2

= 296.3 sec = 4 min 56.3 sec < 5 min

myZivs, mgvb Uv‡b D³ ~iZ¡ 5 wgwb‡Ui Kg mg‡q †cuŠ‡Q m¤¢e|


K N~Y©vqgvb e¯‘i N~Y©b A‡ÿi mv‡c‡ÿ N~Y©b RoZv I †KŠwYK

†e‡Mi ¸Ydj‡K H A‡ÿi mv‡c‡ÿ N~Y©vqgvb e¯‘i †KŠwYK fi‡eM

e‡j|

L Avgiv Rvwb, R e¨vmva© I M f‡ii †Kv‡bv e ‘ †_‡K r (r R)

`~i‡Z¡ †Kv‡bv we› y‡Z gnvKl© wefe,

V = – GM

r

myZivs ~iZ¡ e„w×i mv‡_ GM

r Gi gvb `~i‡Z¡i e¨ Ívbycv‡Z Kg‡Z

_vK‡e wKš‘ wefe FYvZ¥K nIqvq V Gi gvb evo‡Z _v‡K Ges

Amxg ~i‡Z¡ gnvKl© wefe k~b¨| ~i‡Z¡i mv‡_ gnvKl© wef‡ei

cwieZ©b wb‡Pi †jLwP‡Îi mvnv‡h¨ †`Lv‡bv hvq-

M †`Iqv Av‡Q,



wKK Kivi t = 0.5 sec ci Abyf‚wgK †eM, vx = vxo = v0cos0

= 25 ms1 cos30 = 21.65 ms1

Ges Djø¤^ †eM, vy = v0sin0 gt = 25 ms1 sin30 9.8 ms2

0.5 sec = 7.6 ms1

wKK Kivi 0.5 sec c‡i e‡ji †e‡Mi gvb, v = vx2 + vy

2

= 21.652 + 7.62 ms1 = 22.94 ms1 Ans.

g‡b Kwi, e‡ji †e‡Mi w`K Abyf‚wg‡Ki mv‡_ †KvY Drcbœ K‡i|

tan = vy

vx

= tan–1 vy

vx

= tan–1

7.6

21.65

= 19.34

0.5s ci ejwUi †e‡Mi gvb 22.94ms–1 Ges GB †e‡Mi w`K n‡e

Abyf‚wg‡Ki mv‡_ 19.34 †KvY K‡i Dc‡ii w`‡K| (Ans.)

N ejwUi Abyf‚wgK cvjøv, R = v0

2 sin20

g =

(25 ms1)2 sin (2 30)

9.8 ms2

= 55.23 m

Ges wePiYKvj, T = 2v0sin0

g = 2 25 ms1 sin30

9.8 ms2 = 2.55 sec

GB mg‡q †MvjwKcvi KZ…©K AwZµvš Í ~iZ¡ = 2.55 sec 10 ms1

= 25.5m

†Mvj‡cvó †_‡K ejwUi cZb we› yi ~iZ¡ = 80 m – 55.23m

= 24.77 m

r

V

R

vy

vx

v


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†h‡nZz †MvjwKcvi ejwU f‚wg‡Z cwZZ nIqvi Av‡MB G‡mwQj, ZvB

†MvjwKcvi ejwU ai‡Z cvi‡e|


K hw` c„w_exi AveZ©‡bi mv‡_ wgwj‡q GKB †KŠwYK MwZ‡Z GKwU

K…wÎg DcMÖn MwZkxj nq, Z‡e Zv c„w_ex‡K 24 N›Uvq GKevi

cÖ wÿY Ki‡e wKš‘ c„w_ex †_‡K GKRb ch©‡eÿi Kv‡Q w¯’i g‡b

n‡e| Giƒc K…wÎg DcMÖn‡K f‚w¯’i DcMÖn e‡j|

L ywU mgvb f‡ii w ’wZ¯’vcK msN‡l©i †ÿ‡Î Avgiv cvB,

mu1 + mu2 = mv1 + mv2

Ges 12 mu1

2 + 12 mu2

2 = 12 mv1

2 + 12 mv2

2

ev, u1 – v1 = v2 – u2 … … … … (1)

Ges u12 – v1

2 = v22 – u2

2 … … … (2)

mgxKiY (2) †K (1) Øviv fvM K‡i

u1 + v1 = v2 + u2 … … … (3)

mgxKiY (1) I (3) †hvM K‡i

2u1 = 2v2

ev, v2 = u1

mgxKiY (1) I (3) we‡qvM K‡i

2u2 = 2v1

ev, v1 = u2

myZivs mgvb f‡ii ywU e ‘i w ’wZ¯’vcK msN‡l© e ‘Øq ci¯úi †eM

wewbgq K‡i|

M awi, f‚wg‡Z AvNv‡Zi c~e© gyn~‡Z© ejwUi †eM, v

DÏxcK †_‡K cvB-

ejwUi AwZµvšÍ `~iæZ¡, h = (20+ 3) cm

= 23cm

= 0.23m

AwfKl©R Z¡iY, g = 9.8 m/s2

m‡e©v”P D”PZvq †eM, v0 = 0ms–1

Avgiv Rvwb, v2 = v02

+ 2gh

= 02 + 2 9.8 0.23

= 0.21 ms–1

AZGe, f‚wg‡Z AvNv‡Zi c~e© gyn~‡Z© ejwUi †e‡Mi gvb 0.21 ms–1 (Ans.)

N w¯úªsej Øviv K…Z KvR k~b¨ KviY ejwU w¯úªswU‡K ¯úk© Kivi

mgq Gi †h †eM _vK‡e, w¯úªs †_‡K gy³ nIqvi mgq †m †eM cÖvß

n‡e| w¯úªs ms‡Kv‡P‡bi mgq w¯úªs ej Øviv FYvZ¥K KvR n‡e Ges

cÖmvi‡Yi mgq mg cwigvY abvZ¥K KvR n‡e d‡j †gvU K…Z KvR

k~b¨ n‡e|

w¯úÖs ej Øviv K…Z KvR = AvNvZ Kivi gyn~‡Z© ejwUi MwZkw³

= 12 mv2

= 12 0.2 kg 2401 m2s2

= 240.1J

myZivs DÏxcK †_‡K w¯úÖs ej Øviv K…ZKvR wbY©q Kiv m¤¢e|


K KwVb I Zi‡ji ¯úk©we›`y n‡Z eµ Zij Z‡j Aw¼Z ¯úk©K

KwVb e ‘i mv‡_ Zi‡j g‡a¨ †h †KvY Drcbœ K‡i, Zv‡K D³ KwVb

I Zi‡ji ga¨Kvi ¯úk© †KvY e‡j|

L Zi‡ji mv› ÖZv Drcbœ nq AvšÍAvYweK e‡ji Kvi‡Y| wKš‘

M¨v‡mi mv›`ÖZv Drcbœ nq AYy¸‡jvi ga¨Kvi msN‡l©i Kvi‡Y|

ZvcgvÎv evo‡j Zi‡ji Avš ÍtAvYweK ej n«vm cvq Ges M¨vm

AYymg~‡ni ga¨Kvi msNl© e„w× cvq| ZvB ZvcgvÎv evov‡j M¨v‡mi

mv› ÖZv ev‡o wKš‘ Zi‡ji mv›`ÖZv K‡g|

M OQ = OAcos30 = 1 m 0.866 = 0.866 m

OP = OBcos15 = 1 m 0.966 = 0.966 m

QP = OP – OQ = 0.966 m – 0.866 m = 0.1 m

B we›`y‡Z e‡ei †eM v n‡j

v2 = 29.8 ms-2 0.1 m = 1.96 m2s-2

myZivs B we› y‡Z e‡ei MwZ kw³,

KB = 12 mv2 =

12 0.02 kg 1.96 m2s-2 = 0.0196 J

N OQ = OAcos30 = 1 m 0.866 = 0.866 m

OP = OBcos15 = 1 m 0.966 = 0.966 m

QP = OP – OQ = 0.966 m – 0.866 m = 0.1 m

A

B C

P

Q

O

vm

30

15

v


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m




QC = OC – OQ = 1 m – 0.866 m = 0.134 m

PC = OC – OP = 1 m – 0.966 m = 0.034 m

myZivs C Gi mv‡c‡ÿ A we›`y‡Z e‡ei wefe kw³,

UA = mgQC = 0.02 kg 9.8 ms-2 0.134 m

= 0.02626 J

Ges MwZkw³, KA = 0 [ÔMÕ Ask †_‡K]

myZivs A we› y‡Z †gvU kw³,

EA = UA + KA = 0.02626 J + 0 = 0.02626 J

Avevi C Gi mv‡c‡ÿ B we› y‡Z e‡ei wefe kw³,

UB = mgPC = 0.02 kg 9.8 ms-2 0.034 m

= 0.00666 J

Ges MwZkw³, KB = 0.0196 J [ÔMÕ Ask n‡Z]]

myZivs B we› y‡Z †gvU kw³,

EB = UB + KB = 0.00666 J + 0.0196 J = 0.02626 J

C Gi mv‡c‡ÿ C we› y‡Z e‡ei wefe kw³, UC = 0

C we›`y‡Z e‡ei †eM vm n‡j

vm2 = 29.8 ms-2 0.134 m = 2.6264 m2s-2

myZivs C we› y‡Z e‡ei MwZ kw³,

KC = 12 mvm

2 = 12 0.02 kg 2.6264 m2s-2 = 0.02626 J

myZivs C we› y‡Z †gvU kw³,

EC = UC + KC = 0 + 0.02626 J = 0.02626 J

GLv‡b, EA = EB = EC

myZivs †`vjKwU hvwš¿K kw³i wbZ¨Zv †g‡b P‡j|


K mgvb ev cÖvq mgvb we Ívi Ges K¤úv‡¼i mvgvb¨ cv_©K¨ wewkó

`ywU kã Zi½ GK mv‡_ GKB mij †iLvq GKB w`‡K mÂvwjZ n‡j

G‡`i DcwicvZ‡bi d‡j kã Zi‡½i ZxeªZvi †h ch©vqµwgK n«vm

e„w× N‡U Zv‡K exU ev ¯iK¤ú e‡j|

L x Gi †h gv‡bi Rb¨ cos2

x = 1 n‡e †mme we› y‡Z we¯ Ívi

m‡e©v”P 2a n‡e A_©vr †mme we› y‡Z my¯c›` we›`y cvIqv hv‡e|

myZivs my¯c›` we›`yi Rb¨,

cos2

x = 1

2

x = n GLv‡b, n = 0, 1, 2, 3, ......

ev, x = n 2 = 0,

2 , ,

32 ..........

A_©vr †h mKj we›`y‡Z x Gi gvb 2 Gi ALÛ ¸wYZK †m mKj

we›`y‡Z my¯ú›` we›`y MwVZ n‡e&

M ¯úxKvi A n‡Z †kÖvZvi ~iZ¡, r = 32 + 22 = 13

¯úxKvi A Gi Rb¨ O we› y‡Z k‡ãi ZxeªZv, I = PA =

P

4r2

= 1 103W

4 3.1416 ( 13)2m2 = 6.12 106 Wm2

N ïay ¯úxKvi A n‡Z cÖvß k‡ãi ZxeªZv †j‡fj,

B = 10 logII0

= 10 log6.12 106 Wm2

1012 Wm2 = 67.88 dB

¯úxKvi A I ¯úxKvi B Df‡qi myBP Ab Ki‡j O we›`y‡Z k‡ãi

ZxeªZv, I = 2 6.12 106 Wm2

G‡ÿ‡Î k‡ãi ZxeªZv †j‡fj, B = 10 logII0

= 10 log2 6.12 106 Wm2

1012 Wm2 = 70.88 dB

†h‡nZz 70.88 dB 2 67.88 dB A_©vr B 2B

myZivs ¯úxKvi A I ¯úxKvi B Df‡qi myBP Ab Ki‡j O we› y‡Z

k‡ãi ZxeªZv †j‡fj c~e©v‡cÿv wØ¸Y n‡e bv|


K †h ZvcgvÎvq GKwU wbw`©ó AvqZ‡bi evqy Zvi †fZ‡ii Rjxq

ev®ú Øviv m¤ú„³ nq Zv‡K H evqyi wkwkivsK e‡j|

L GKB AvqZ‡bi ywU evqyc~Y© †ejyb‡K wfbœ ZvcgvÎvq ivL‡j V2

T2

= V1

T1 m~Îvbymv‡i †ewk ZvcgvÎvi †ejy‡bi AvqZb †ewk n‡e, KviY

Dfq‡ÿ‡Î Pvc evqygÛjxq Pv‡ci mgvb n‡e|

M †`Iqv Av‡Q.

A cv‡Îi AvqZb, V = 2 cm3 = 2 106m3

Ges Pvc, P = 3 105 Nm2

A cv‡Î M¨v‡mi MwZkw³ EA = 32 PV

= 32 3 105 Nm2 106m3

= 0.9J

x = 0

N N N N A A A A


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ingbd.co

m




N cvÎ B Gi †ÿ‡Î, Pvc, P = 3.66 105 Nm2

Ges AvqZb, V = 2 106m3

cvÎ B Gi M¨v‡mi MwZkw³ = 32 PV = 1.5 3.66 105 Nm2 2

106m3

= 1.098J

GB MwZkw³ Zvciƒ‡c †`Lv w`‡e, hv ¯¯^ cvÎ‡K DËß Ki‡e|

†h‡nZz 1.098J > 0.9J

A_©vr cvÎ B Gi M¨v‡mi MwZkw³ > cvÎ A M¨v‡mi MwZkw³

myZivs cvÎ B †ewk DËß n‡e|

3. w`bvRcyi †evW©-2015


K †Kv‡bv MwZkxj e ‘i †Kv‡bv we‡kl gyn~‡Z© ÿz`ªvwZÿz`ª mgq

eëav‡b †e‡Mi cwieZ©‡bi nvi‡K H we‡kl gyn~‡Z©i ZvrÿwYK Z¡iY

e‡j|

L Dc‡ii w`‡K wbwÿß e ‘i Ici wµqvkxj AwfKl© e‡ji w`K

wb‡Pi w`‡K| ZvB AwfKl©R Z¡i‡Yi w`KI Lvov wb‡Pi w`‡K| G

Z¡i‡Yi Kvi‡Y Dc‡ii w`‡K wbwÿß e ‘i MwZ‡eM n«vm cvq|

M GLv‡b, X we› yi ’vbv¼ (3, 2, 1)

Y we› yi ’vbv¼ (–2, 1, 4)

Zvn‡j,

OX = 3 i + 2 j + k Ges

OY = 2 i + j + 4k

†ei Ki‡Z n‡e, G‡ì gaëZx© †KvY, = ?

Avgiv Rvwb, OX .

OY = |

OX | |

OY | cos

ev, 3 (2) + 2 1 + 1 4 = 32 + 22 + 12 (2)2 + 12 + 42 cos

cos = 0

32 + 22 + 12 (2)2 + 12 + 42 = 0

ev, = cos10 = 90

AZGe, †f±iØ‡qi gaëZx© †KvY 90| (Ans.)

N OX ,

OY Z‡ji Ici j¤ GKK †f±i =

OX

OY

|OX

OY |

GLv‡b, OX

OY =

i

3

2

j21

k14

= i | |21

14 j | |3

2

14 + k

| |3

2

21

= 7 i 14 j + 7k

Avevi, |OX

OY | = 72 + (14)2 + 72 = 7 6

OX

OY

|OX

OY |

= 7 i 14 j + 7k

7 6 =

i 2 j + k

6

wKš‘ OY ,

OX Gi Z‡ji Dci j¤^ GKK †f±i =

OY

OX

|OY

OX |

my¯úóZ: OY

OX = (

OX

OY ) = 7 i + 14 j 7k

Ges |OY

OX | = |

OX

OY | = 7 6

OY

OX

|OY

OX |

= 7 i + 14 j 7k

7 6 =

i 2 j + k

6

A_©vr j¤ GKK †f±iØq gv‡b mgvb n‡jI w`‡K ci¯úi wecixZ|

OX ,

OY Gi Z‡ji Dci j¤ GKK †f±i KvMRc„ô n‡Z

j¤^fv‡e Lvov Ici w`‡K wµqv K‡i Ges OY ,

OX Gi Z‡ji

Ici j¤ GKK †f±i KvMRc„ô n‡Z j¤^fv‡e Lvov wb‡Pi w`‡K wµqv

K‡i|


K †Kv‡bv MwZkxj e ‘i †Kv‡bv we‡kl gyn~‡Z© ÿz`ªvwZÿz`ª mgq

eëav‡b †e‡Mi cwieZ©‡bi nvi‡K H we‡kl gyn~‡Z©i ZvrÿwYK

Z¡iY e‡j|

L †f±‡ii gvb KL‡bvB FYvZ¥K nq bv| KviY †f±‡ii gvb

ej‡Z Avgiv cig gvb‡K eywS Avi cig gvb KL‡bvB FYvZ¥K

bq|

M †Ìqv Av‡Q, c„w_exi e¨vmva©, R = 6.4 103km = 6.4 106m

f‚c„ô n‡Z D we›`yi D”PZv, h = 200 km = 2 105m

Rvbv Av‡Q, f‚c„‡ô AwfKl©R Z¡iY, g = 9.8 ms2

†ei Ki‡Z n‡e, D Ae ’v‡bi AwfKlx©q Z¡i‡Yi gvb, g = ?

Avgiv Rvwb, g = g

R

R + h

2

= 9.8 ms2

6.4 106m

6.4 106m + 2 105m

2


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ingbd.co

m




= 9.215 ms2 (Ans.)

N f‚c„‡ô, m = 10kg f‡ii e ‘i IRb, W = mg = 10 kg 9.8 ms2 = 98N

C Ae ’v‡b AwfKl©R Z¡i‡Yi gvb, g = g

1

dR

= 9.8 ms2

1

100 103m

6.4 106m

= 9.647 ms2

C Ae ’v‡b e ‘wUi IRb, W = mg = 10kg 9.647 ms2 = 96.47N

†h‡nZz 96.47N 98N

A_©vr W W

myZivs DÏxc‡Ki wP‡Î C Ae ’v‡b hw` m = 10kg f‡ii e ‘ wb‡q

hvIqv nq, Z‡e Gi Dci c„w_exi AvKl©Y e‡ji cwieZ©b (n«vm)

NU‡e|


K †Kv‡bv e ‘i Ici ej cÖ‡qv‡Mi d‡j e‡ji wecixZ w`‡K e ‘i

miY NU‡j ev e‡ji wecixZ w`‡K mi‡Yi Dcvsk _vK‡j Zvn‡j ej

I mi‡Yi Dcvs‡ki ¸Ydj‡K FYvZ¥K KvR e‡j|

L †Kv‡bv ^‡i †hme wewfbœ myi _v‡K, Zv‡ì g‡a¨ †h my‡ii K¤úv¼

me‡P‡q Kg, Zv‡K g~j myi e‡j| Ab¨vb¨ myi, hv‡ì K¤úv¼ g~j

my‡ii K¤úv‡¼i †P‡q †ewk, Zv‡ì‡K Dcmyi ejv nq| Avevi,

Dcmyi¸‡jvi K¤úv¼ hw` g~j my‡ii K¤úv‡¼i mij ¸wYZK nq,

Zvn‡j †mB mKj Dcmyi‡K nvi‡gvwbK e‡j| G Kvi‡YB mKj

nvi‡gvwbK Dcmyi n‡jI mKj Dcmyi nvi‡gvwbK bq|

M †Ìqv Av‡Q,

X wmwjÛv‡i M¨v‡mi ZvcgvÎv, Tx = 600 K

Y wmwjÛv‡i M¨v‡mi ZvcgvÎv, Ty = 650 K

†ei Ki‡Z n‡e, G‡ì Mo eM©g~j †e‡Mi Zzjbv ev AbycvZ,

Cx2 t

Cx2 = ?

M¨vmØ‡qi NbZ¡ mgvb weavq Giv g~jZ GKB M¨vm A_©vr MÖvg

AvYweK fi M Gi gvb Dfq‡ÿ‡Î mgvb|

Avgiv Rvwb,

Cx2 =

3RTx

M Ges

Cy2 =

3RTy

M

Cx2

Cy2

= Tx

Ty =

600K650K = 0.9608

ev,

Cx2 = 0.9608

Cy2

Cy2 >

Cx2

AZGe, Y wmwjÛv‡ii M¨v‡mi Mo eM©g~j †e‡Mi gvb †ewk|

N `ywU cv‡Îi g‡a¨ M¨v‡mi Av`vb cÖ`vb wbf©i K‡i M¨v‡mi Pv‡ci

Dci| †h‡nZz Y cv‡Î M¨v‡mi Pvc †ewk ZvB Y cvÎ †_‡K M¨vm X cv‡Î Mgb Ki‡e hZÿY bv Dfq cv‡Îi Pvc mgvb nq| Pvc mgvb

nIqvi ci MwZZË¡ Abymv‡i Dfq cv‡Îi AYy¸‡jv BZ¯ ÍZ

wewÿßfv‡e †QvUvQzwU Ki‡Z _vK‡e d‡j Dfq cv‡Îi g‡a¨ AYyi

MgbvMgb NU‡Z _vK‡e|


K †h ivwk Øviv Zi½ mÂviYKvix KYvi †h‡Kv‡bv gyn~‡Z©i MwZi

mg¨K Ae ’v eySvq Zv‡K `kv e‡j|

L cÖwZ †m‡K‡Û exU 6 ej‡Z eySvq, g~j kã Zi½Ø‡qi

DcwicvZ‡bi d‡j jwä Zi‡½i k‡ãi ZxeªZv cÖwZ †m‡K‡Û 6 evi

n«vm-e„w× N‡U| A_©vr cÖwZ †m‡K‡Û 6wU Zxeª kã †kvbv hvq Ges 6wU

wbtkã †kvbv hvq|

M cÖ_g Zi½wUi mgxKiY: y1 = 0.1 sin

200t

2017 x

= 0.1 sin 2017 (170t x)

G‡K Zi‡½i cÖwgZ mgxKiY, y = asin 2

(vt x) Gi mv‡_ Zzjbv

K‡i cvB,

v = 170 ms1

myZivs cÖ_g Zi½wUi Zi½‡eM, v = 170 ms1

N DÏxc‡Ki Zi½Ø‡qi DcwicvZ‡bi d‡j m„ó jwä Zi‡½i

mgxKiY, y = y1 + y2 = 0.1 sin

200t

2017 x + 0.1 sin

200t +

2017 x

= 0.1 2 sin(200t) cos

20

17 x

= 0.2cos

20

17 x sin (200t)

= A sin (200t)

0.2 cos

20

17 x = cwieZÆbkxj weÕ¦vi A

Dc‡iv³ mgxKi‡Y AMÖMvgx Zi‡½i mgxKi‡Yi b¨vq `kv †Kv‡Yi

†fZi (vt x) RvZxq †Kv‡bv ivwk AšÍfz©³ bvB ZvB GwU AMÖMvgx

Zi‡½i mgxKiY bq| GwU w¯’i Zi‡½i mgxKiY|

myZivs DÏxc‡K Zi½Ø‡qi g‡a¨ DcwicvZ‡bi d‡j w ’i Zi½ m„wó

n‡e|


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ingbd.co

m





K mgvb ev cÖvq mgvb we Ív‡ii wKš‘ K¤úv‡¼i mvgvb¨ cv_©K¨ wewkó

`ywU kã Zi½ GKB mgq GKB mij †iLvq GKB w`‡K mÂvwjZ

n‡j G‡ì DcwicvZ‡bi d‡j k‡ãi ZxeªZvi †h ch©vqµwgK n«vm

e„w× N‡U Zv‡K exU ev ¯iK¤ú e‡j|

L aªæeej I w¯úÖs ¯ú›`b Gi mv‡_ †`vjbKv‡ji m¤úK© wb¤œiƒc :

F = – kx ev ma = kx ev, mk =

xa

†`vjbKvj, T = 2miY

Z¡iY = 2

xa = 2

mk

GLv‡b Z¡iY I miY ci¯úi wecixZ w`‡K nIqvq xa ivwkwU

abvZ¥K|

M †Ìqv Av‡Q, r = 4 i 6 j + 12k

F = 2 i + 3 j 5k

†ei Ki‡Z n‡e, UK©, = ?

Avgiv Rvwb, = r

F = (4 i 6 j + 12k) (2 i + 3 j 5k)

=

i

42

j

63

k

12

5

= i

6

3

12

5 j

4

2 12

5 + k

4

2 63

= i (30 – 36) – j (–20 – 24) + k (12 + 12)

= 6 i + 44 j + 24k

U‡K©i gvb = (–6)2 + 442 + 242 = 50.48 GKK|

mvB‡Kj Pvj‡Ki U‡K©i gvb = 50.48 GKK| (Ans.)

N DE c‡_ mvB‡Kj Pvjv‡Z †M‡j mvB‡Kj Ges wb‡Ri AwfKl©‡K

AwZµg K‡i Ic‡i DV‡Z n‡e| d‡j Gmgq PvjK cÖPÛ Kó Abyfe

Ki‡eb, KviY Zv‡K weivU gv‡bi ÿgZv cÖ‡qvM Ki‡Z n‡e| EG c_

cÖvq Abyf‚wgK, Z‡e DuPz wbPz e‡j Zv h‡_ó gv‡bi Nl©Y cÖ vb Ki‡e|

d‡j G iv Ívq mvB‡Kj Pvjv‡Z †M‡j Pvj‡Ki Abyf‚wZ cy‡ivcywi

myLKi n‡e bv|

Z‡e GH c‡_ bvgvi mgq PvjK‡K c¨v‡Wj Pvc‡Z n‡e bv,

AwfK‡l©i ìæY mvB‡Kj ¯qswµqfv‡e wb‡P bvg‡Z _vK‡e| ïay

Zv‡K mvB‡K‡ji wbqš¿‡Y g‡bv‡hvM w`‡Z n‡e| Gmgq mvB‡Kj

PvjK †ek myLKi Abyf‚wZ cv‡eb|

6bs cÖ‡kœi DËi

K `ywU k‡ãi g‡a¨ hw` GKwUi ZxeªZv Ab¨wUi 100.1 ¸Y ev 1.259

nq, ZLb G‡ì ZxeªZv †j‡fj Gi cv_©K¨ nq 0.1 †ej ev 1

†Wwm‡ej|

L mij †`vjK MwZi †ÿ‡Î,

x = Asin(t + ) ev, dxdt = A cos(t + )

ev, d2xdt2 = A2sin(t + ) = 2x

d2xdt2 + 2x = 0

GwUB mij †`vjb MwZi AšÍiK mgxKiY|

M cÖ_g Zv‡ii †ÿ‡Î,

cqm‡bi AbycvZ, 1 = – DD

L

L =

0.01 mm 1m

5 mm 0.01m = – 0.2

wØZxq Zv‡ii cqm‡bi AbycvZ,

2 = DD

L

L =

0.03 mm 3m

15 mm 0.03m = – 0.2

ZviØ‡qi cqm‡bi AbycvZ mgvb|

N ZviØ‡qi cqm‡bi AbycvZ mgvb nIqvq Avgiv a‡i wb‡Z cvwi

Dfq Zvi GKB Dcv`v‡bi| myZivs G‡ì Amn cxob mgvb| Avgiv

Rvwb, Amn fvi = Amn cxob cÖ ’‡”Q‡ì †ÿÎdj

GLb cÖ_g I wØZxq Zv‡ii Amn fvi h_vµ‡g M1 I M2 n‡j,

M1

M2 =

d12

d22 =

2

mm 15

mm 5

=

19

M2 = 9 M1

A_©vr wØZxq Zv‡ii Amn fvi cÖ_g Zv‡ii Amn fv‡ii 9 ¸Y|

4. Kzwgjøv †evW©-2015


K

V (x, y, z) AšÍwiKiY‡hvM¨ †f±i †ÿÎ n‡j

V †K

VGi

Kvj© e‡j|

L Avgiv Rvwb, †eM GKwU †f±i ivwk| gvb A_ev w`K A_ev

Df‡qi cwieZ©‡b †f±‡ii cwieZ©b nq| †Kv‡bv e ‘ e„ËvKvi c‡_

N~Y©b Kv‡j †e‡Mi gvb cwiewZ©Z bv n‡jI cÖwZ gyn~‡Z© w`‡Ki

cwieZ©b nq Ges †e‡Mi w`K nq †h‡Kv‡bv we› y‡Z e„ËvKvi c‡_i

¯úk©K eivei| myZivs ejv hvq, †Kv‡bv e ‘i e„ËvKvi c‡_ mg‡e‡M

Pjv m¤¢e bq|

M b`xi cÖ ’ eivei †e‡Mi Dcvsk = Vbsin37

= 10 ms1 sin37 = 6.02 ms1


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ingbd.co

m




b`x cvi n‡Z mgq jvM‡e, t = d

6.02 ms1 = 36m

6.02 ms1

= 5.982 sec

b`xi cvo eivei †e‡Mi Dcvs‡ki †hvMdj = Vb cos37 + Vr

= 10 ms1 cos37 + 3 ms1 = 10.986 ms1

`~iZ¡, BC = 10.986 ms1 5.982 sec = 65.72m (Ans.)

N †bŠKvwU‡K A †_‡K mivmwi B we›`y‡Z †cuŠQv‡Z n‡j †bŠKv I

†¯ªv‡Zi †e‡Mi jwä Ges † ªv‡Zi †e‡Mi gaëZ©x †KvY = 90 n‡Z

n‡e| †bŠKv I † ªv‡Zi †e‡Mi gaëZ©x †KvY n‡j Avgiv cvB

tan90 = vbsin

vr + vbcos

ev, = vbsin

vr + vbcos

ev, vr + vbcos = 0

ev, cos = – vr

vb = –

3 ms-1

10 ms-1 = – 0.3

= 107.45

myZivs A †_‡K mivmwi B we› y‡Z †cuŠQv‡Z n‡j †bŠKvwU‡K †m&ªv‡Zi

w`‡Ki mv‡_ 107.45 †Kv‡Y Pvjbv Ki‡Z n‡e|


K †Kv‡bv w¯úÖs‡K Gi mvg¨ve ’v n‡Z 1m cÖmvwiZ ev msKzwPZ

Ki‡Z †h cwigvY ej cÖ‡qvM Ki‡Z nq, Zv‡K w¯úÖs aªæeK e‡j|

L GKwU eo e„wói †duvUv †f‡½ A‡bK¸‡jv †QvU †duvUvq cwiYZ

Ki‡j me©‡gvU †ÿÎdj e„w× cvq| c„ôkw³i ìæY G‡ÿ‡Î A‡bK

kw³i ìKvi nq| e„nr cvwbi †duvUv n‡Z G kw³ †kvlY Kiv nq

weavq G‡ÿ‡Î ZvcgvÎvi n«vm NU‡e|

M Avgiv Rvwb,

)y2g(yvv 02y

2y

0

GLv‡b, Kvgv‡bi D”PZv, y0 = 60 m

Ges vy0 = v0 sin0 = 25sin53 = 250.7986= 19.96 ms-1

me©vwaK D”PZvq †e‡Mi Dj¤ Dcvsk k~b¨ A_©vr hLb y = ymax ZLb

vy = 0|

myZivs

0 = (19.96)2 – 29.8(ymax – 60)

ev, 19.6(ymax – 60) = 398.6

ev, (ymax – 60) = 20.34

ymax = 80.34

myZivs me©vwaK D”PZv 80.34 m (Ans.)

N aiv hvK, e› yK I Kvgv‡bi ¸wj f‚wg‡Z co‡Z h_vµ‡g t1 I t2 mgq jvM‡e|

GLv‡b,

Kvgv‡bi I e› y‡Ki Avw` D”PZv, yo = 60 m

[†h‡nZz ¸wjØq f‚wg‡Z c‡o]

y = 0

e›`y‡Ki ¸wji †ÿ‡Î,

vy0 = v0 sin0 = 25sin53 = 250.7986= 19.96 ms-1

myZivs

y = y0 + vy0 t1 – 12 g t1

2

ev, 0 = 60 + 19.96 t1 – 12 9.8 t1

2

ev, 4.9 t12 – 19.96 t1 – 60 = 0

t1 = 9.42

)60(9.4496.1996.19 2

= 9.42

1176398.496.19

=

8.9

68.9396.19

= 6.08 s or – 2.01 s

wKš‘ FYvZ¥K gvb MÖnY‡hvM¨ bq| myZivs t1 = 6.08 s

Kvgv‡bi ¸wji †ÿ‡Î,

vy0 = v0 sin0 = 25sin0 = 0

myZivs,

y = y0 + vy0 t2 – 12 g t2

2

ev, 0 = 60 – 12 9.8 t2

2

ev, 4.9 t22 = 60

ev, t22 = 12.245

t2 = 3.5 s

†h‡nZz, t1 > t2

myZivs Kvgv‡bi ¸wj Av‡M gvwU‡Z AvNvZ Ki‡e|


Teach

ingbd.co

m





K evwn¨K e‡ji cÖfv‡e †Kvb e ‘i gvÎv ev ¯vfvweK Ae ’vi

cwieZ©b NUv‡bvi †Póv Ki‡j ev NUv‡j e ‘i Af¨š Í‡i cÖwZ GKK

†ÿÎd‡j †h evav`vbKvix e‡ji D™¢e nq, Zv‡K cÖZ¨qbx ej e‡j|

L †Kvb wbw ©ó cwigvY M¨v‡mi g‡a¨ M¨vm KwYKv¸‡jvi m¤¢ve¨ †eM

wbY©q Kiv nq M¨v‡mi ZvcgvÎvi Dci wfwË K‡i| M¨v‡mi ZvcgvÎv

Gi AYy¸‡jvi MwZkw³i mv‡_ m¤úwK©Z Ges MwZkw³ AYymg~‡ni

†e‡Mi e‡M©i mv‡_ RwoZ| ZvQvov AYymg~n wewfbœ w`‡K MwZkxj

_vKvq G‡ì Mo‡eM k~bÏ n‡Z cv‡i| wKš‘ †e‡Mi eM©gvb me©`v

abvZ¥K Ges GwU mivmwi M¨v‡mi ZvcgvÎv I cv‡Îi Dci Pv‡ci

mv‡_ m¤úwK©Z| ZvB Mo †eM A‡cÿv g~j Mo eM©‡eM A‡bK †ewk

m½wZc~Y©|

M †Ìqv Av‡Q, e ‘i fi, m = 300g = 0.3 kg

f‚wg n‡Z †njv‡bv Z‡ji D”PZv h n‡j, mgh = Avw` MwZkw³ =

5.88J

h = 5.88Jmg =

5.88J

0.2 kg 9.8 ms2 = 2m

sin30 = h

AE ev, AE = h

sin30 =

2m12

= 4m (Ans.)

N M Abymv‡i h = 2 m Ges AE = 4 m| Avevi †h‡nZz AB = BC = CD = DE †m‡nZz AC = EC = 2 m, AD = 3 m Ges ED = 1 m| Avgiv

cvB,

sinA = h

AE = hC

AC = hD

AD

hC = h

AE AC = 24 2 = 1 m

hD = h

AE AD = 24 3 = 1.5 m

myZivs C I D we› yi wefe kw³ h_vµ‡g UC = (0.39.81) J = 2.94 J I UD = (0.39.81.5) J = 4.41 J|

†njv‡bv Z‡j e ‘i Z¡iY = EA eivei AwfKl©R Z¡i‡Yi Dcvsk

a = gsin30 = 9.8 0.5 = 4.9 ms-2

E †_‡K †njv‡bv Z‡j MwZkxj e¯‘i †eM C I D we› y‡Z h_vµ‡g vC I vD n‡j Avgiv cvB

vC2 = 2aEC = 24.9 ms-22 m = 18.6 m2s-2

Ges vD2

= 2aED = 24.9 ms-21 m = 9.8 m2s-2

C we›`y‡Z MwZkw³

KC = 12 mvC

2 = 12 0.3 kg 18.6 m2s-2 = 2.94 J

D we›`y‡Z MwZkw³

KD = 12 mvD

2 = 12 0.3 kg 9.8 m2s-2 = 1.47 J

C we›`y‡Z †gvU kw³

EC = UC + KC = 2.94 J + 2.94 J = 5.88 J

D we›`y‡Z †gvU kw³

ED = UD + KD = 4.41 J + 1.47 J = 5.88 J

myZivs C we› y‡Z †gvU kw³ = D we›`y‡Z †gvU kw³

myZivs †njv‡bv Zj eivei e ‘wU bvgvi mgq hvwš¿K kw³i wbZ¨Zv

m~Î †g‡b P‡j|


K KwVb Zij ¯úk© we›`y‡Z Zij c„‡ôi ¯úk©K Zi‡ji wfZ‡i

KwV‡bi c„‡ôi mv‡_ †h †KvY Drcbœ K‡i Zv‡K ¯úk©‡KvY e‡j|

L e‡ji NvZ = F t Ges fi‡e‡Mi cwieZ©b = mv

e‡ji Nv‡Zi gvÎv = F Gi gvÎv t Gi gvÎv = MLT2 = MLT1

fi‡e‡Mi cwieZ©‡bi gvÎv = m Gi gvÎv v Gi gvÎv = MLT1

myZivs e‡ji NvZ fi‡e‡Mi cwieZ©‡bi mgvb|

M †Ìqv Av‡Q, †`vj‡bi we Ívi, A = 10 cm = 0.1m

†`vjbKvj, T = 2sec

†ei Ki‡Z n‡e, m‡e©v”P †eM, vmax = ?

Avgiv Rvwb, vmax = A = 2T A

= 2 3.1416

2 sec 0.1m = 0.31416 ms1 (Ans.)

N G‡ÿ‡Î eewU Abyf‚wgKfv‡e wbwÿß cÖv‡mi b¨vq AvPiY Ki‡e|

wb‡ÿcY †eM, v0 = 0.31416 ms1


f‚wg n‡Z Avw` D”PZv, h = 45 cm = 0.45m

GB D”PZv †b‡g Avm‡Z t mgq jvM‡j,

h = v0sin0t + 12 gt2

ev, 0.45m = 0.31416 ms1 sin0 t + 12 9.8 ms2 t2

ev, 4.9t2 = 0.45 [GKKmg~n Dn¨ †i‡L]

t = 0.454.9 = 0.303 sec

GB mgqKv‡j AwZµvšÍ Abyf‚wgK `~iZ¡ = v0cos0 t

= 0.31416 ms1 cos0 0.303 sec

= 0.0952 m


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m




= 9.52 cm

myZivs eewU mvg¨ve ’vb n‡Z 9.52 cm Abyf‚wgK ~i‡Z¡ f‚wg‡Z

cwZZ n‡e|


K †Kvb k‡ãi ZxeªZv Ges cÖgvY ZxeªZvi Abycv‡Zi jMvwi`g‡K

H k‡ãi ZxeªZv †j‡fj e‡j| ZxeªZv †j‡fj, = log10 II0

|

L N~Y©b Kv‡j †KŠwYK fi‡e‡Mi msiÿYkxjZvi Rb¨ m~h© †_‡K

wewfbœ `~i‡Z¡ MÖ‡ni †eM wewfbœ nq| MÖ‡ni †KŠwYK fi‡eM Gi fi

†eM I m~h© †_‡K MÖ‡ni ~i‡Z¡i ¸Ydj| ZvB †KŠwYK fi‡eM

msiwÿZ ivLvi Rb¨ MÖn †_‡K m~‡h©i ~iZ¡ hZ K‡g, Gi †eM GKB

nv‡i ev‡o| GB Kvi‡YB MÖn Zvi N~Y©b Z‡j me©`v mgvb mg‡q mgvb

†ÿÎdj AwZµg K‡i|

M †Ìqv Av‡Q,

c„w_exi AveZ©bKvj, T1 = 365 day

c„w_ex I m~‡h©i ~iZ¡ R1 n‡j, cwiewZ©Z `~iZ¡, R2 = R1

2

†ei Ki‡Z n‡e, cwiewZ©Z AveZ©bKvj, T2 = ?

Avgiv Rvwb, †Kcjv‡ii 3q m~Îvbymv‡i,

T2

2

T12 =

R23

R13

T2 = T1 R2

R1

32

= 365 day

R1

2

R1

1.5

= 129.05 day (Ans.)

N Avgiv Rvwb, c„w_exi †KŠwYK †eM Ges †Kv‡bv ¯’v‡bi

Aÿvsk n‡j, c„w_exi N~Y©b‡e‡Mi ìæb AwfKl©R Z¡iY n«vm cvq

2Rcos2 cwigvY|

wbiÿ‡iLvq, = 0, 2Rcos2 =

2

T2 Rcos2

=

2 3.1416

86400 sec2 6.4 106m (cos0)2 = 0.0338

c„w_ex N~Y©biZ Ae¯’vq wbiÿ‡iLvq AwfKl©R Z¡iY, g = 9.78 ms2

c„w_exi AveZ©b eÜ n‡j wbiÿxq †iLvq Aew ’Z †Kv‡bv e ‘i

IR‡bi cwieZ©b = m 0.0338 ms2

mg 100%

= 0.0338 ms2

9.78 ms2 100%

= 0.346% (e„w× cv‡e)


K GK †gvj Av`k© M¨v‡mi Rb¨ PV/T Abycv‡Zi gvb‡K mve©Rbxb

M¨vm aªæeK e‡j| Gi gvb 8.314 J mol–1 K–1|

L e‡ji NvZ n‡jv e‡ji gvb I wµqvKv‡ji ¸Ydj| Gi GKK

Ns ev kgms1| GwU e¯‘i fi‡e‡Mi cwieZ©‡bi mgvb| Gi gvÎv

MLT1| G¸‡jvB e‡ji Nv‡Zi êwkó¨|

M †Ìqv Av‡Q,

Mo eM©‡e‡Mi eM©g~‡ji Avw` gvb,

C12 = 11.2 kms1

Avw` Pvc P1 n‡j cwiewZ©Z Pvc, P2 = P1/2

†ei Ki‡Z n‡e, Mo eM©‡e‡Mi eM©g~‡ji cwiewZ©Z gvb,

C22 = ?

Avgiv Rvwb,

C2 = 3P

Av‡jvP¨ †ÿ‡Î,

C12 =

3P1

Ges

C22 =

3P2

[ NbZ¡ AcwiewZ©Z]

C22

C1

2

= 3P2

3P1 =

P2

P1 =

P1

2

P1 = 0.707

C22 = 0.707

C1

2 = 0.707 11.2 kms1 = 7.92 kms1 (Ans.)

N T1 = 27C = 300 K ZvcgvÎvq Aw·‡Rb AYyi Mo eM©‡e‡Mi

eM©g~j, C1

2 = 3RT1

M1 =

3R 300K

32 103 kg

[Aw·‡R‡bi †gvjvi AvYweK fi, M1 = 32 10–3 kg]

g‡b Kwi, T2 ZvcgvÎvq bvB‡Uªv‡Rb AYyi Mo eM©‡e‡Mi eM©g~j =

C22

C22 =

3RT2

28 103 kg

cÖkœg‡Z,

C12 =

C22 ev,

3RT

28 103 kg =

3R 300 K

32 103 kg

T2 = 300 K 2832 = 262.5 K

myZivs 262.5 K ZvcgvÎvq bvB‡Uªv‡Rb AYyi Mo eM©‡e‡Mi eM©g~j

gvb 27C ZvcgvÎvq Aw·‡Rb AYyi Mo eM©‡e‡Mi eM©g~j gv‡bi

mgvb n‡e|


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m




5. PÆMÖvg †evW©-2015


K †h cÖPÛ gv‡bi ej AwZ Aí mgq a‡i wµqv K‡i Zv‡K NvZej

e‡j|

L mylg e„ËvKvi MwZi êwkó¨ n‡jv :

1. G‡Z mg`ªæwZ we`¨gvb|

2. G‡Z mg‡KŠwYK †eM we`¨gvb|

3. Gi †KŠwYK Z¡iY k~b¨|

4. †K›`ªgyLx Z¡iY _v‡K |

M †Ìqv Av‡Q, iwki ˆ`N© Z_v e„ËvKvi c‡_i e¨vmva©, r = 2m

†KŠwYK †eM, = 2N

t = 2 3.1416 20 rad

60 sec = 2.0944 rads1

†Mvj‡Ki fi, m = 5 kg

†ei Ki‡Z n‡e, †K‡› Öi w`‡K Abyf‚Z ej = †K› ÖgyLx ej, Fc = ?

Avgiv Rvwb, Fc = m2r = 5kg (2.0944 rads1)2 2m

= 43.865 N (Ans.)

N N~Y©vqgvb Ae ’vq iwkwU wQ‡o †M‡j †MvjKwU Abyf‚wgK fv‡e

wbwÿß e¯‘i b¨vq AvPiY Ki‡e| Gi Dj ø¤^ †eM, vy0 = 0 Ges

Abyf‚wgK †eM vx0 = r = 2.0944 rads-1 2 m = 4.1888 ms-1

†MvjKwU f‚wg‡Z co‡Z t mgq jvM‡j,

y = y0 + vy0 t – 12 g t2

ev, 0 = 1.5 – 12 9.8 t2

ev, 4.9 t2 = 1.5

ev, t2 = 0.3061

t = 0.5533 s

G mgq AwZµvš Í Abyf‚wgK `~iZ¡,

x = vx0 t = 4.1888 ms-1 0.5533 s = 2.3177 m

myZivs cvidg©vi n‡Z `k©‡Ki ~iZ¡ 2.3177 m A‡cÿv †ewk n‡j

†MvjKwU `k©K‡K AvNvZ Ki‡e bv|


K gnvwe‡k¦i †h‡Kv‡bv ywU KYv G‡ì ms‡hvRK mij †iLv eivei

ci¯úi‡K GKwU ej Øviv AvKl©Y K‡i, G e‡ji gvb KYvØ‡qi

f‡ii ¸Yd‡ji mgvbycvwZK Ges G‡ì gaëZ©x ~i‡Z¡i e‡M©i

e¨¯ÍvbycvwZK|

L Avgiv Rvwb, N~Y©b MwZi †ÿ‡Î,

†KŠwYK fi‡eM = RoZvi åvgK †KŠwYK †eM ev, L = I

†KŠwYK †eM GKK gv‡bi A_©vr = 1 n‡j, L = I 1 = I

myZivs GKK mg‡KŠwYK †e‡M N~Y©biZ †Kv‡bv e ‘i RoZvi åvgK

Gi †KŠwYK fi‡e‡Mi mgvb|

M AD n‡Z BC Z‡ji D”PZv h n‡j, h

AB = sin60

h = AB sin60 = 4m 3

2 = 3.464m

A n‡Z C we› y‡Z †cŠQv‡Z AwfKl© e‡ji weiæ‡× K…ZKvR,

EP = mgh = 30kg 9.8 ms2 3.464m = 1018.4J (Ans.)

N CD c‡_ †Kv‡bv Nl©Y bv _vK‡j CD Zj eivei wb‡Pi w`‡K

evjKwUi Z¡iY n‡Zv, g = g sin

n‡jv f‚wgi mv‡_ CD Z‡ji AvbwZ

sin = h

CD = 3.464m

5m = 0.6928

= sin1 (0.6928) = 43.85

g = 9.8 ms2 sin 43.85 = 6.79 ms2 < 9.8 ms2

myZivs †Kv‡bv Nl©Y bv _vK‡j CD eivei wb‡Pi w`‡K Z¡iY n‡Zv

6.79 ms2, Avi Nl©Y _vK‡j Z¡iY Av‡iv Kg n‡e|

AZGe, CD c‡_ bvgvi mgq evjKwUi Z¡iY AwfKl©R Z¡i‡Yi †P‡q

Kg n‡e|


K †h mKj M¨vm mKj ZvcgvÎv I Pv‡c e‡qj I Pvj©‡mi m~Î †g‡b

P‡j, Zv‡ì‡K Av`k© M¨vm e‡j|

L Avgiv Rvwb, PV = nRT

P = 1 Ges n = 1 n‡j, V = RT ev, VT = R

myZivs GKK Pv‡c GK †gvj †Kv‡bv M¨v‡mi AvqZb ebvg cig

ZvcgvÎv †jLwP‡Îi Xvj Av`k© M¨vm aªæeK wb‡ ©k K‡i|

M †Ìqv Av‡Q,

XvKvq ï®‹ ev‡j¦i ZvcgvÎv, 1 = 28.6C

Ges Av ª© ev‡j¦i ZvcgvÎv, 2 = 20C

evqyi ZvcgvÎvq †Møwmqv‡mi Drcv`K, G = 1.664

wkwkivsK n‡j, = 1 G(1 2)

= 28.6C 1.664 (28.6C 20C)

= 14.29C (Ans.)

GLv‡b,

y0 = 1.5 m


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m




N XvKvq wkwkivs‡K (14.29C) m¤ú„³ ev®úPvc,

f = 11.99 + (13.63 11.99) 0.29

2 mm HgP

= 12.228 mmHgP

evqyi ZvcgvÎvq (28.6C) m¤ú„³ ev®ú Pvc,

F = 28.35 + (31.83 28.35) 0.6

2 mm HgP

= 29.394 mm HgP

XvKvq Av‡cwÿK Av ªZv, R = fF =

12.22829.394 100% = 41.6%

ivRkvnx‡Z wkwkivsK, = 1 G(1 2)

= 32.5C 1.625 (32.5C 22C)

= 15.437C

ivRkvnx‡Z evqyi ZvcgvÎvq (32.5C) m¤ú„³ ev®ú Pvc,

F = 35.66 + (39.90 35.66) 0.5

2 mm HgP

= 36.72 mmHgP

wkwkivs‡K (15.437C) m¤ú„³ ev®ú Pvc,

f = 11.99 + (13.63 11.99) 1.4375

2 mm HgP

= 13.169 mmHgP

ivRkvnx‡Z Av‡cwÿK Av`ª©Zv, R = f

F 100% =

13.16936.72 100%

= 35.86%

†h‡nZz 35.86% < 41.6% A_©vr ivRkvnx‡Z Av‡cwÿK Av ª©Zv Kg|

ZvB H e¨w³ ivRkvnx‡Z AwaKZi ¯^w¯ Í †eva Ki‡eb|


K wbw`©ó K¤úv‡¼i Rb¨ †h Av`k© ZxeªZvi mv‡_ Zzjbv K‡i wewfbœ

ZxeªZvi k‡ãi k‡ãv”PZvi gvÎv wbY©q Kiv nq, Zv‡K H K¤úvs‡Ki

k‡ãi Rb¨ cÖgvY ZxeªZv ejv nq|

L †Ìqv Av‡Q, A I

B Gi gaëZx© †KvY, = 45

evgcÿ = A .

B = AB cos = AB cos45 =

AB

2

Ges Wvbcÿ = |A

B | = ABsin = ABsin45 =

AB

2

A .

B = |

A

B |

M †Ìqv Av‡Q, f‚c„ô n‡Z K…wÎg DcMÖ‡ni D”PZv, h = 2 106m

Rvbv Av‡Q, f‚c„‡ô AwfKl©R Z¡iY, g = 9.8 ms2

Ges c„w_exi e¨vmva©, R = 6.4 106m

K…wÎg DcMÖ‡ni Ae ’v‡b AwfKl©R Z¡iY, gh = g

R

R + h2

= 9.8 ms2

6.4 106m

6.4 106m + 2 106m

2 = 5.69 ms2 Ans.

N f‚c„ô n‡Z d = 3 106m MfxiZvq AwfKl©R Z¡iY,

gd = g

1

dR = 9.8 ms2

1

3 106m

6.4 106m = 5.206 ms2

K…wÎg DcMÖ‡ni Ae ’v‡b Ges Lwbi Af¨šÍ‡i mij †`vj‡Ki

†`vjbKvj h_vµ‡g Th Ges Td n‡j, mij †`vj‡Ki Kvh©Kix

Z¡i‡Yi m~Îvbymv‡i,

Th

Td =

gd

gh =

5.206 ms2

5.69 ms2 = 0.9565

Th < Td

myZivs Lwbi Af¨šÍ‡i †`vjKwU ax‡i Pj‡e|


K †h MvwYwZK wP‡ýi Øviv GKwU †f±i ivwk‡K Ab¨ GKwU †¯‹jvi

ev †f±i ivwk‡Z iƒcvšÍi Kiv hvq ev †Kv‡bv cwieZ©bkxj †f±i

ivwki e¨vL¨v †`qv hvq Zv‡K †f±i Acv‡iUi e‡j|

L ej = fi Z¡iY = fi miY

mgq2

wbDUb (e‡ji GKK) = †KwR wgUvi

†m‡KÛ2 = †KwR

wgUvi/†m.2

ev, N = kgm s–2

M †Ìqv Av‡Q,

ek©vwUi wb‡ÿcY †eM, v0 = 30 ms1

ek©vwUi wb‡ÿcY †KvY, 0 = 30

AwfKl©R Z¡iY, g = 10 ms2

†ei Ki‡Z n‡e, me©vwaK D”PZv, ymax = ?

Avgiv Rvwb, ymax = v0

2sin20

2g = (30 ms1) (sin30)2

2 10 ms2 = 11.25m

(Ans.)

N ek©vi Abyf‚wgK cvjøv, R = v0

2sin20

g = (30 ms-1)2sin(230)

10 ms-2

= 77.94 m


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m




Ges ek©vi DÇqb Kvj, T = 2v0sin0

g = 230 ms-1sin30

10 ms-2

= 3 s

3 s ci wkKvix †_‡K nwi‡Yi ~iZ¡,

x = x0 + vx0 t + 12 a t = 32.94 m +

12 10 ms-2 (3 s)2

= 32.94 m + 45 m = 77.94 m

myZivs ek©vwU nwiY‡K AvNvZ Ki‡e|


K AwfK‡l©i cÖfv‡e †Kv‡bv cÖevnxi ga¨ w`‡q MwZkxj †Kv‡bv e ‘

m‡e©v”P †h †e‡M DcbxZ n‡j wbU ej k~b¨ nq Ges e ‘wU mg‡e‡M

Pj‡Z _v‡K, †m †eM‡K ejv nq cÖvwš ÍK‡eM ev AšÍ‡eM|

L wb‡¤œ ciek I Abybv` K¤úv‡¼i cv_©K¨ e¨vL¨v Kiv n‡jv-

ciek K¤úb Abybv` K¤úb

1. GKwU K¤úbiZ e ‘‡K

Ab¨ GKwU K¤úb ÿq

e¯‘i wbKU Avb‡j 2q

e¯‘wU‡Z K¤úb ïiæ nq

G‡K ciek K¤úb e‡j|

1. ciek K¤úbkxj e ‘

¯^vfvweK K¤úv‡¼i mgvb

n‡j e ‘ AwaK we¯ Ív‡i

Kw¤úZ nq| G ai‡bi

K¤úb‡K Abybv` e‡j|

2. ciek K¤ú‡b ch©ve„Ë

e‡ji ewkó¨ _vK‡Z nq|

2. wbR¯^ K¤úv¼ I ciek

m„wóKvix ch©vqe„Ë e‡ji

K¤úv‡¼i mgvb êwkó¨

AR©b K‡i|

M Y = 6 sin

8t

x25

= 6 sin250 (200t x)

G‡K cÖwgZ mgxKiY Y = Asin2

(vt x) Gi mv‡_ Zzjbv K‡i cvB,

Zi½‡eM, v = 200 cms1 Ges Zi½‰`N©¨, = 50 cm

K¤úv¼, f = v

=

200 cms1

50 cm = 4 Hz

N Zi‡½i we Ívi, a = 6 cm = 0.06 m

†Ìqv Av‡Q, gva¨‡gi NbZ¡, = 0.09 kgm3

Zi‡½i ZxeªZv, I = 22f2a2v

= 2 9.87 (4 Hz)2 (0.06m)2 0.09 kgm3 2 ms1

= 0.2047 Wm2

ZxeªZv †j‡fj, = 10 logII0

dB

= 10 log0.2047 Wm2

1012 Wm2 dB

= 113.1 dB < 120 dB

†h‡nZz gvbe KY© kÖæZ m‡e©v”P ZxeªZv 120 dB Gi †ewk, myZivs

DÏxc‡Ki Zi½wU kÖve¨|

6. wm‡jU †evW©-2015


K GKwU †f±i ivwk‡K yB ev Z‡ZvwaK Dcvs‡k wef³ Kivi

cÖwµqvB n‡jv †f±i wefvRb|

L Avgiv Rvwb, †f±‡ii gvb A_ev w`K A_ev Df‡qi cwieZ©‡b

†f±i cwieZ©xZ nq| †eM n‡”Q †f±i ivwk| myZivs gvb cwieZ©b

bv n‡jI w`‡Ki cwieZ©‡b †eM cwieZ©xZ n‡e| mg ªæwZ‡Z eµc‡_

Pjvi mgq †e‡Mi gvb cwieZ©xZ bv n‡jI w`‡Ki cwieZ©b nq| Avi

†e‡Mi cwieZ©‡bi nvi‡K Z¡iY e‡j| myZivs Avgiv ej‡Z cvwi,

mij c‡_ mg ªæwZ‡Z Pjgvb †Kv‡bv e ‘i Z¡iY bv _vK‡jI eµ c‡_

mg`ªæwZ‡Z Pjgvb e ‘i Z¡iY _v‡K|

M K…wÎg DcMÖnwUi †K› ÖgyLx Z¡iY, Kÿc‡_ Gi Ici cÖhy³

AwfKl©R Z¡i‡Yi mgvb|

†Ìqv Av‡Q,

MÖ‡ni fi, M = 6 1024 kg

MÖ‡ni e¨vmva©, R = 6.4 106m

MÖ‡ni c„ô n‡Z Kÿc‡_i D”PZv, h = 700 km = 700 103m

Kÿc‡_i Ae ’v‡b AwfKl©R Z¡iY, g = GM

(R + h)2

= 6.673 1011 Nm2kg2 6 1024 kg

(6.4 106 m + 700 103 m)2

= 7.942 ms2 (Ans.)

N Kÿc‡_ cwiågYKv‡j K…wÎg DcMÖnwUi

MwZ‡eM, v = GM

R + h

= 6.673 1011 Nm2kg2 6 1024 kg

6.4 106m + 700 103 m

= 7509 ms1

K…wÎg DcMÖnwU‡K gnvk~‡b¨ wgwj‡q hvIqvi Rb¨ cÖ‡qvRbxq me©wb¤œ

kw³

hRhRhR

drrGMmdrr

GMmFdrW 2

2


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m




hR

GMm

hRGMm

rGMm

hR

111

= GMm

R + h

G cwigvY KvR Ki‡Z cÖ‡qvRbxq me©wb¤œ †eM ve n‡j

12 mve

2 = GMmR + h

ve = hR

GM

2= 2

hR

GM

= 1.415709 ms–1

= 8049 ms–1

myZivs ejv hvq, DcMÖnwU k~‡b¨ wgwj‡q hvIqvi m¤¢vebv †bB|


K AwfK‡l©i cÖfv‡e k~b¨ ’v‡b f‚wgi mv‡_ Zxh©Kfv‡e Dc‡ii w`‡K

wbwÿß e¯‘‡K cÖwÿß e¯‘ ev cÖvm e‡j|

L †h †f±‡ii cv`we› y mywbw`©ó bq, Zv‡K ¯vaxb †f±i e‡j|

†h‡nZz ¯vaxb †f±‡ii cv`we› y mywbw`©ó bq, ZvB GB cv`we› y

g~jwe›`y‡Z Aew¯’Z nIqvi cÖ‡qvRb †bB| G Kvi‡YB ^vaxb †f±‡ii

cv`we›`y g~jwe›`y‡Z bq|

M †Ìqv Av‡Q,

cÖhy³ ej, F = 500N

j¤^ `~iZ¡, r = 1m

myZivs wb‡Y©q UK©, = Fr = 500 m 1m = 500 Nm (Ans.)

N g‡b Kwi, mgMÖ `‡Ði fi M Ges ˆ`N©¨

l

Zvn‡j GKK ˆ`‡N©ï fi = Ml

Ges dx ÿz`ªvwZÿz`ª As‡ki fi = Ml dx

N~Y©b Aÿ `‡Ði cÖv‡šÍ Aew¯’Z n‡j Ges ˆ`‡N©ï mv‡_ j¤ n‡j Gi

mv‡c‡ÿ `‡Ði RoZvi åvgK, I = lo(x2)

Ml dx

= Ml

l o

x2dx = Ml x3

3

l o

= Ml

l3

3 = Ml2

3

wKš‘ N~Y©b Aÿ `‡Ði ga¨we›`yMvgx n‡j Ges

`‡Ði ˆ`‡N©ï mv‡_ j¤ n‡j RoZvi åvgK,

I =

l/2

l/2

x2 Ml dx

= Ml x3

3

l/2

l/2

= Ml

l3

24 + l3

24

= Ml2

12

†h‡nZz Ml2

3 > Ml2

12

A_©vr I > I

myZivs N~Y©b Aÿ ‡Ði cÖvš Íwe›`y‡Z Aew¯’Z n‡j RoZvi åvgK †ewk

n‡e|


K †Kv‡bv hš¿ mieivnK…Z kw³i hZ Ask Kv‡R iƒcvš ÍwiZ Ki‡Z

cv‡i Zv‡K H h‡š¿i Kg©`ÿZv e‡j| A_ev †Kv‡bv hš¿ KZ…©K

K…ZKvR I mieivnK…Z kw³i AbycvZ‡K H h‡š¿i Kg©`ÿZv e‡j|

L †Kv‡bv e¯‘i Ici ej cÖ‡qv‡Mi d‡j e‡ji w`‡K mi‡Yi Dcvsk

_vK‡j e‡ji Øviv KvR ev abvZ¥K KvR nq| e‡ji Øviv Kv‡Ri

d‡j e ‘i MwZkw³ e„w× cvq| †Kv‡bv e ‘ AwfK‡l©i cÖfv‡e wb‡P

†b‡g Avmv-e‡ji Øviv Kv‡Ri D`vniY|

M h D”PZvq wefe kw³ mgh Ges MwZkw³ 0| myZivs †gvUkw³

mgh| aiv hvK, y D”PZvq wefe kw³ MwZkw³i wØ¸Y n‡e| y D”PZvq wefe kw³ mgy Ges MwZkw³ Ek n‡j kw³i wbZ¨Zvi bxwZ

†_‡K cvB,

mgy + Ek = mgh

Ek = mgh mgy

kZv©bymv‡i, 2(mgh mgy) = mgy

ev, 3mgy = 2mgh

y = 2h 3 =

2375 m3 = 250 m

myZivs f‚wg n‡Z 250 m D”PZvq e¯‘i wefe kw³ MwZ kw³i wØ¸Y

n‡e|

N †h‡nZz Kv‡mg Ges gwb‡ii fi wfbœ, ZvB Kv‡m‡gi mgvb ÿgZv

cÖ‡qvM Ki‡j gwbi GKB mg‡q KvRwU Ki‡Z cvi‡e bv| Z‡e gwbi

wKQyUv Kg ÿgZv cÖ‡qvM Ki‡j GKB mg‡q KvRwU Ki‡Z cvi‡e|

GLv‡b, e ‘mn gwb‡ii fi,m = 55 kg + 10kg = 65 kg


AwZµvš Í D”PZv, h = 375m

mgqKvj, t = 40 min = 40 60 sec = 2400 sec

gwb‡ii CwáZ ÿgZv P n‡j,

P = mgh

t

= 65 kg 9.8 ms2 375m

2400 sec

l

dx

x

dx

x


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m




= 99.53 watt

myZivs gwbi 99.53 watt ÿgZv cÖ‡qvM Ki‡j GKB mg‡q KvRwU

Ki‡Z cvi‡e|

4bs cÖ‡kœi DËi

K KwVb I Zi‡ji ¯úk©we›`y †_‡K eµ Zij Z‡j Aw¼Z ¯úk©K

KwVb c`v‡_©i mv‡_ Zi‡ji wfZi †h †KvY Drcbœ K‡i Zv‡K D³

KwVb I Zi‡ji ¯úk© †KvY e‡j|

L cvwbi AYy I KPzcvZvi AYyi ga¨Kvi AvmÄb ej A‡cÿv cvwbi

AYymg~‡ni ga¨Kvi mskw³ ej e„nËi gv‡bi| ZvB e„wói †duvUv

KPzcvZv‡K wfRvq bv| cÿvšÍ‡i cvwbi AYy I Avg cvZvi AYyi

ga¨Kvi AvmÄb ej A‡cÿv cvwbi AYymg~‡ni ga¨Kvi mskw³ ej

ÿz`ªZi gv‡bi| ZvB e„wói †duvUv AvgcvZv‡K wfRvq|

M †`Iqv Av‡Q,

B ˆKwkK b‡ji e¨vmva©, r = 0.4 mm

2 = 0.2 103m

cvwbi ¯úk©‡KvY, = 2

c„ô Uvb, T = 72 103 Nm1


cvwbi NbZ¡, = 1000 kgm3

†ei Ki‡Z n‡e, B b‡j cvwbi D”PZv, h = ?

Avgiv Rvwb, T = rgh

2 cos

h = 2T cos

rg

= 2 72 103 Nm1 cos2

0.2 103 m 1000 kgm3 9.8 ms2

= 0.0734m

= 7.34 cm (Ans)

N ˆKwkKZvi ZË¡ n‡Z Avgiv Rvwb, Zi‡ji c„ôUvb,

T = r g h

2cos

ev, h = 2Tcos

r g

A I B b‡ji e¨vmva© h_vµ‡g rA I rB Ges cvwbi D”PZv hA I hB n‡j,

hA = 2Tcos

rA g

hB = 2Tcos

rB g

hA

hB =

rB rA

†h‡nZz rB < rA †m‡nZz hA > hB

myZivs Avgiv ej‡Z cvwi b‡ji e¨vmv‡a©i wfbœZvB b‡ji wfZi

Zi‡ji D”PvZvi wfbœZvi KviY| †h b‡ji e¨vmva© hZ Kg †m b‡j

Zi‡ji D”PZv ZZ †ewk|


K Zi½ mÂvj‡bi Awfgy‡Li mv‡_ GKK †ÿÎd‡ji ga¨ w`‡q cÖwZ

†m‡K‡Û †h cwigvY kw³ cÖevwnZ nq Zv‡K Zi‡½i ZxeªZv e‡j|

L ˆmb¨iv eªx‡Ri Dci w`‡q gvP© K‡i †M‡j eªx‡Ri Ici cÖhy³ ej

AZ¨waK gv‡bi nq| G e‡ji K¤úv¼ eªx‡Ri ¯vfvweK K¤úv‡¼i

mgvb ev KvQvKvwQ n‡j eªxRwU‡Z Abybv` m„wó n‡e Ges GwU m‡e©v”P

we¯Ívi mnKv‡i Kw¤úZ n‡e| ZLb eªxRwU †f‡½ hvevi m¤¢vebv

_v‡K| G Kvi‡Y GK mv‡_ A‡bK¸‡jv ˆmb¨ eªx‡Ri Dci w`‡q gvP©

K‡i hvIqv mwVK bq|

M evqy gva¨‡g k‡ãi †eM, va = 352 ms1

k‡ãi K¤úv¼, f = 300 Hz

evqy‡Z Zi½‰`N©¨, a = va

f = 352 ms1

300 Hz = 1.173m

cvwb‡Z Zi½‰`N©¨ †ewk n‡e

cvwb‡Z Zi½‰`N©¨, w = a + 4.16m

= 1.173m + 4.16m

= 5.33 m

cvwb‡Z k‡ãi †eM, vw = fw

= 300 Hz 5.33m

= 1600 ms1 (Ans.)

N ÔMÕ As‡ki we‡kølY g‡Z,

cvwb‡Z k‡ãi †eM > evqy‡Z k‡ãi †eM

GLb cÖgvY Ki‡ev †h,

cvwb‡Z k‡ãi ZxeªZv > evqy‡Z k‡ãi ZxeªZv

evZv‡mi kãZi‡½i ZxeªZv, Ia = 22n2a2ava

= 2 9.87 (300 Hz)2 (0.25 102m)2 1.293 kgm3 352 ms1

= 5054 Wm2

cvwb‡Z kãZi‡½i ZxeªZv,

Iw = 22n2a2 wvw

= 2 9.87 (300 Hz)2 (0.25 102m)2 1000 kgm3 1600 ms1

= 1.7766 107 Wm2 >> 5054 Wm2 (= Ia)

MvwYwZK we‡køl‡Y †`Lv hvq,


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m




Iw >> Ia

A_©vr cvwb‡Z k‡ãi ZxeªZv > evqy‡Z k‡ãi ZxeªZv

myZivs mvjv‡gi e³e¨ mwVK|


K †h mKj M¨vm mKj ZvcgvÎv I Pv‡c e‡qj I Pvj©‡mi m~Î †g‡b

P‡j, Zv‡ì‡K Av`k© M¨vm e‡j|

L †Kv‡bv ¯’v‡b evZv‡mi Av‡cwÿK Av ª©Zv 70% ej‡Z eySvq, H

ZvcgvÎvq H ’v‡bi evZvm‡K m¤ú„³ Ki‡Z †h cwigvY Rjxq ev®ú

ìKvi Zvi kZKiv 70 fvM Rjxq ev®ú H gyn~‡Z© H ¯’v‡bi evqy‡Z

i‡q‡Q|

M †Ìqv Av‡Q,

Avw` AvqZb, V1 = 5.7 104m3

Avw` Pvc, P1 = 0.64 m HgP

Avw` ZvcgvÎv, T1 = 39C = (39 + 273)K = 312 K

P‚ovš Í Pvc, P2 = 0.76 m HgP

P‚ovš Í ZvcgvÎv, T2 = 273 K

†ei Ki‡Z n‡e, P‚ovšÍ AvqZb, V2 = ?

Avgiv Rvwb, P1V1

T1 =

P2V2

T2

V2 = P1V1 T2

P2T1

= 0.64m 5.7 104m3 273K

0.76m 312K

= 4.2 104m3 (Ans.)

N †h‡nZz 4.2 104m3 < 5.7 104m3

myZivs M¨vmwUi AvqZb n«vm †c‡q‡Q|

T cig ZvcgvÎvq n †gvj M¨v‡mi MwZkw³, E = 32 nRT

n (†gvj msL¨v) AcwiewZ©Z _vK‡j, E T

DÏxc‡Ki NUbvq, M¨v‡mi fi Z_v †gvj msL¨v (n) AcwiewZ©Z|

myZivs cig ZvcgvÎvi n«v‡m (T1 = 312K n‡Z T2 = 273K) MwZkw³I

n«vm cv‡e| A_©vr †bnv‡ji e³e¨ mwVK|

7. h‡kvi †evW©-2015


K yB ev Z‡ZvwaK GKB RvZxq †f±i †hvM Ki‡j †h †f±i cvIqv

hvq Zv‡K †f±i¸‡jvi jwä †f±i e‡j|

L AwfKl© e‡ji †ÿ‡Î GK we›`y n‡Z Aci we› y‡Z e¯‘i Mg‡bi

d‡j K…ZKvR, c‡_i Ici wbf©i K‡i bv eis Avw` I Aš Íwe›`yi Ici

wbf©i K‡i| e ‘wU cybivq Avw` we›`y‡Z wd‡i G‡j K…Z KvR k~b¨ nq

Ges kw³i AcPq N‡U bv| ZvB AwfKl© ej AmsiÿYkxj ej bq

A_©vr msiÿYkxj ej|

M

Uªwji MwZ m„wóKvix ej = cÖhy³ e‡ji Abyf‚wgK Dcvsk|

= Fsin30 = 10 N 0.5 = 5 N (Ans.)

N

wPÎ-1 Abymv‡i hLb Uªwj‡K †Vjv n‡”Q ZLb e‡ji Dj¤ Dcvsk

Fcos30 wb‡Pi w`‡K wµqv Ki‡Q| Uªwji IRb W n‡j, wb¤œgyLx †gvU

ej = W + Fcos30

G‡Z f‚wgi cÖwZwµqv ej e„w× cvq d‡j Nl©Y ej †ewk nq KviY

Nl©Y ej Awfj¤^ cÖwZwµqvi mgvbycvwZK| Aci c‡ÿ UªwjwU‡K wPÎ-

30

Fsin30

Fcos30

Uªwji Dci cÖhy³ ej, F

30

Fsin30

Fcos30

F

wPÎ-1

30

Fsin30

Fcos30

F

wPÎ-2


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m




2 Abymv‡i Uvbv n‡j e‡ji Dj¤^ Dcvsk Fcos30 Dc‡ii w`‡K wµqv

K‡i| d‡j wb¤œgyLx †gvU ej = W – Fcos30

G‡Z f‚wgi cÖwZwµqv ej n«vm cvq d‡j Nl©Y ej Kg nq| G Kvi‡Y

Uªwj †Vjvi †_‡K Uvbv mnR nq| G Kvi‡YB †`vKvb`vi mvwenv‡K

Uªwj Uvb‡Z e‡jwQj|


K †Kv‡bv e ‘i Ici Av‡ivwcZ ch©ve„Ë ¯ú›`‡bi K¤úv¼ e ‘i

¯^vfvweK K¤úv‡¼i †P‡q wfbœZi n‡j cÖ_‡g AwbqwgZ fv‡e Kw¤úZ

nq c‡i Av‡ivwcZ K¤ú‡bi K¤úv‡¼ Kw¤úZ nq GB ai‡bi

K¤úb‡K ciek K¤úb e‡j|

L GKwU ¯‡i †h GKvwaK myi _v‡K Zvi g‡a¨ me‡P‡q wb¤œ

K¤úv‡¼i myi‡K †gŠwjK myi e‡j Ges Ab¨ myi¸‡jv‡K Dcmyi e‡j|

†h mKj Dcmy‡ii K¤úv¼ †gŠwjK my‡ii mij ¸wYZK Zv‡ì‡K

mg‡gj e‡j| myZivs mKj mg‡gjB Dcmyi wKš‘ mKj Dcmyi

mg‡gj bq|

M †Ìqv Av‡Q,

A Zv‡ii Uvb, T1 = 100N

B Zv‡ii Uvb, T2 = 200N

A Zv‡ii K¤úv¼, f1 = 50 Hz

†ei Ki‡Z n‡e, B Zv‡ii K¤úv¼, f2 = ?

Zvi¸‡jvi ˆ`N©¨ Ges fi mgvb e‡j, f T

ev, f2

f1 =

T2

T1

f2 = f1

T2

T1 = 50 Hz

200 N100 N = 70.7 Hz

N A Zv‡ii Rb¨ 1 = 12l

T1

C Zv‡ii Rb¨ 3 = 12l

T3

GLb, 3

1 =

T3

T1 ev, 3 = 1

T3

T1 = 50

250100

3 = 79 Hz

AZGe, C Zv‡ii K¤úv¼, 3 = 79 Hz

DÏxcK n‡Z, A Zv‡ii K¤úv¼, 1 = 50 Hz

A I C Zvi ywU‡K GK‡Î Kw¤úZ Ki‡j Drcbœ weU, N = 3 – 1 = 79Hz – 50 Hz = 29 Hz

cÖwZ †m‡K‡Û Drcbœ weU msL¨v Lye †ewk n‡j, k‡ãi ZxeªZvi n«vm-

e„w× G‡Zv ªæZ nq †h Zv Dcjwä Kiv hvq bv| Kv‡b GKUvbv kã

†kvbv hvq bv| cixÿv K‡i †`Lv †M‡Q †h, we‡Ui msL¨v †m‡K‡Û 15

Gi †ewk n‡j Kv‡b Zv Dcjwä Kiv m¤¢e bq|

†h‡nZz 29Hz > 15 Hz

AZGe, A I C Zvi‡K GK‡Î Kw¤úZ Ki‡j weU †kvbv hvq bv|


K mg‡qi eëavb k~‡bï KvQvKvwQ n‡j e ‘i mi‡Yi nvi‡K Gi

ZvrÿwYK †eM e‡j|

L GKRb A¨v_‡jU js Rv¤ú †`qvi c~‡e© †ek wKQy ~i †`Šo †`b|

Gi D‡Ï‡k¨ n‡jv, MwZRoZv AR©b Kiv hvi ìæb †m Rv¤ú †`qvi

ci †ek LvwbKUv ~iZ¡ AwZµg Ki‡Z mÿg n‡eb|

M †Ìqv Av‡Q,

ey‡j‡Ui Avw`‡eM, 0 = 0 ms1

†kl †eM, v = 750 ms1

miY, s = 0.6m

†ei Ki‡Z n‡e, Mo Z¡iY, a = ?

Avgiv Rvwb, v2 = 02 + 2as

ev, 2as = v2 02

a = v2 0

2

2s = (750 ms1)2 (0 ms1)2

2 0.6m = 468750 ms2 (Ans.)

N Avgiv Rvwb, Abyf‚wgK cvj øv, g

vR 0

2

0 2sin

ey‡jUwU cÖvm we‡ePbv K‡i cvB,

Gi wb‡ÿcY †eM, v0 = 750 ms–1

GLb Abyf‚wg‡Ki mv‡_ †KvY,1 = 30 Ges 2 = 60 we‡ePbv K‡i

AwZµvš Í `~iZ¡ †ei Kwi,

1 = 30 Gi Rb¨, R1 = v0

2 sin 21

g

= (750ms–1)2 sin (2 30)

9.8ms–1

= 49708.09 m

Avevi, 2 = 60 Gi Rb¨, R2 = (750ms–1)2 sin (2 30)

9.8ms–2

= 49708.09m

GLv‡b,R1 = R2

AZGe, wfbœ wfbœ †Kv‡Yi Rb¨ GKB †e‡M wbwÿß e ‘i AwZµvšÍ

`~iæZ¡ mgvb|


K †Kv‡bv e ‘i Ici cÖhy³ ej Øviv K…ZKvR e ‘i MwZkw³i

cwieZ©‡bi mgvb|


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m




L †Kv‡bv e ‘i MwZkw³ = 12 mv2; e¯‘i fi m KL‡bvB FYvZ¥K

nq bv Ges v abvZ¥K ev FYvZ¥K hvB †nvK bv †Kb v2 me©`vB

abvZ¥K nq| ZvB †Kv‡bv e ‘i MwZkw³ FYvZ¥K n‡Z cv‡i bv|

M †Ìqv Av‡Q,

c„w_exi fi, M = 6 1024 kg

c„w_exi e¨vmva©, R = 6.4 106m

f‚c„ô n‡Z K…wÎg DcMÖ‡ni D”PZv, h = 690 km = 6.90 105m

†ei Ki‡Z n‡e, K…wÎg DcMÖnwUi îwLK †eM, v = ?

Avgiv Rvwb, v = GM

R + h =

6.673 1011 Nm2kg2 6 1024 kg

6.4 106m + 6.90 105 m

= 7514.7 ms1 (Ans.)

N DÏxc‡K ewY©Z Ae¯’vq K…wÎg DcMÖnwUi ch©vqKvj T n‡j,

T = 2 (R + h)

v = 2 3.1416 (6.4 106m + 6.9 105m)

7514.7 ms1

= 5928 sec

f‚c„ô n‡Z K…wÎg DcMÖnwUi D”PZv h = 800 km = 8 105m n‡j,

Gi ch©vqKvj n‡e, T = 2(R + h)3

GM

= 2 3.1416 (6.4 106m + 8 105m)3

6.673 1011 Nm2kg2 6 1024 kg

= 6066. 56 sec

†h‡nZz 6066 sec 5928 sec

A_©vr T T

myZivs DcMÖnwU‡K c„w_ex c„ô †_‡K 800 km miv‡j †mwUi

cwiågYKv‡ji cwieZ©b NU‡e (e„w× cv‡e)|


K ¯ú›`biZ †Kv‡bv e ‘KYvi MwZ hw` Ggb nq †h, Gi MwZc_

mij‰iwLK Ges Gi †h‡Kv‡bv gyn~‡Z©i Z¡iY, mvg¨ve ’vb n‡Z mi‡Yi

mgvbycvwZK wKš‘ wecixZgyLx nq, Z‡e H e ‘KYvi MwZ‡K mij

Qw›`Z MwZ e‡j|

L GKwU w¯úÖs Gi w¯úÖs aªæeK 2.5 N/m ej‡Z eySvq, G‡K Gi

mvg¨ve ’vb n‡Z 1m cÖmvwiZ Ki‡Z 2.5 N ej cÖ‡qvRb nq|

M †Ìqv Av‡Q,

cvnv‡oi P‚ovq cÖwZ NÈvq cÖvß Aa©‡`vjb msL¨v = 3600 30

= 3570

†h‡nZz 3570 ¸‡jv Aa©‡`vjb †`q 3600 †m‡K‡Û

2wU Aa©‡`vjb †`q 3600 2

3570 †m‡K‡Û

= 2.0168 sec

BnvB wb‡Y©q †`vjbKvj| (Ans.)

N c„w_exi e¨vmva©, R = 6400 km = 6.4 106m

f‚c„‡ô AwfKl©R Z¡iY, g = 9.8 ms2

awi, cvnv‡oi D”PZv = hm

f‚c„‡ô I cvnv‡oi P‚ovq †`vjbKvj h_vµ‡g T1 I T2 n‡j,

T1

T2 =

g2

g1 =

R2

(R + h)2 = R

R + h

ev, R + h

R = 1 + hR =

T2

T1 =

2.01682 = 1.0084

hR = 1.0084 – 1

h = 0.0084 R = 0.0084 6.4 106m = 53760 m

myZivs DÏxc‡Ki Z‡_ï wfwË‡Z cvnv‡oi D”PZv wbY©q Kiv m¤¢e|

6bs cÖ‡kœi DËi

K mgy`ªc„‡ô 45 Aÿvs‡k 273.15K ZvcgvÎvq Dj¤^fv‡e Aew ’Z

0.76m D”PZvwewkó weï× cvi` Í¤¢ †h Pvc †`q Zv n‡jv cÖgvY ev

¯^vfvweK Pvc|

L M¨vm ej‡Z Ggb c`v_© †evSvq hvi ¯vfvweK Ae ’v ev®úxq|

†hgb: nvB‡Wªv‡Rb, Aw·‡Rb| Avi ¯vfvweKfv‡e ev®ú ej‡Z

†Kv‡bv KwVb ev Zij c`v_©‡K Zvc w`‡j †h Ae¯’v cvIqv hvq Zv‡K

†evSvq|

GKB ZvcgvÎv e„w×‡Z mKj M¨v‡mi cÖmviY GKB nq| ev‡®úi

†ÿ‡Î Ggb †`Lv hvq bv| †Kv‡bv M¨vmxq c`v‡_©i ZvcgvÎv Gi µvwš Í

ZvcgvÎv A‡cÿv Kg n‡j Zv‡K ev®ú e‡j| †Kv‡bv c`v_© Gi µvwš Í

ZvcgvÎv A‡cÿv AwaK ZvcgvÎvq _vK‡j Zv‡K M¨vm e‡j| mvaviY

ZvcgvÎvq M¨vm‡K Pvc cÖ‡qv‡M Zi‡j cwiYZ Kiv hvq bv, ev®ú‡K

hvq|

M †`qv Av‡Q, ï®‹ ev‡j¦i ZvcgvÎv, 1 = 20C

Av`©ª ev‡j¦i ZvcgvÎv, 2 = 12.8C

20C G †MøBmvi Drcv`K, G = 1.79

wkwkiv¼, = ?

Rvbv Av‡Q,

= 1 – G(1 – 2)

= 20 – 1.79 (20 – 12.8)

= 7.112C

myZivs H w`‡bi wkwkiv¼ 7.112C| (Ans.)

N (8 – 7)C = 1C Gi Rb¨ m¤ú„³ Rjxq ev®úPv‡ci cv_©K¨

= (8.1 – 7.5) 10–3


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m




= 0.6 10–3 cvi`Pvc|

0.112C Gi Rb¨ ev®úPv‡ci e„w×

= 0.0672 10–3 cvi`Pvc

wkwkiv¼ = 7.112C [(M) DËi: †_‡K]

wkwkiv¼ 7.112C G m¤ú„³ Rjxq ev®ú Pvc,

f = (7.5 + 0.0672) 10–3

= 7.5672 10–3 Hg

evqyi ZvcgvÎv 20C G Rjxq ev®ú Pvc, F = 17.4 10–3 mHg

Avgiv Rvwb,

Av‡cwÿK Av`©ªZv, R = fF 100%

= 43.49%

Av‡cwÿK Av ©ªZv 43.49%| ZvB ejv hvq H w`b H ’v‡bi

AvenvIqv ï®‹ I †iŠ‡ ªv¾¡j _vK‡e|

8. ewikvj †evW©-2015


K hLb †Kv‡bv e ‘ GK Ae ’vb †_‡K Ab¨ †Kv‡bv Ae ’v‡b Mgb

K‡i ZLb Avw` Ae ’vb‡K cv`we›`y Ges †kl Ae ’vb‡K kxl© we› y

we‡ePbv K‡i †h †f±i cvIqv hvq Zv‡K miY †f±i e‡j|

L aiv hvK,

†bŠKvi B we›`y‡Z

¸Y †eu‡a GK e¨w³

BM eivei F e‡j

Uvb‡Q| G ej ywU

Dcvs‡k wef³

n‡e| GKwU Dcvsk

Fsin, hv †bŠKv‡K K‚‡ji w`‡K wb‡q †h‡Z _vK‡e| wKš‘ gvwS b`xi

†mªvZ‡K eënvi K‡i êVvi mvnv‡h¨ wecixZ w`‡K GKwU ej Drcbœ

K‡i hv Fsin AskwU‡K cÖkwgZ Ki‡e| Aci Dcvsk Fcos, hv

†bŠKv‡K mvg‡bi w`‡K wb‡q hv‡e|

M †Ìqv Av‡Q,



AwfKl©R Z¡iY, g = 10 ms2

†ei Ki‡Z n‡e, me©vwaK D”PZv, ymax = ?

Avgiv Rvwb, ymax = v0

2sin2 0

2g

= (30 ms1)2 (sin30)2

2 10 ms1

= 11.25 m (Ans.)

N Abyf‚wgK cvjøv, R = v0

2 sin20

g

= (30 ms1)2 sin (2 30)

10 ms2

= 77.94m

wb‡ÿc‡Yi t mgq c‡i cÖvmwU y = 4m D”PZvq Avm‡j,

y = v0sin0t 12 gt2

ev, 4 = 30sin30t 12 10t2 [GKKmg~n Dn¨ †i‡L]

ev, 5t2 15t + 4 = 0

t = 52

4541515 2

=

10

8022515

= 10

14515=

10

04.1215 = 0.296 s or 2.704 s

e¯‘wU t1 = 0.296 s mg‡q a eivei Ges t2 = 2.704 s mg‡q b eivei

Dc‡i Ae ’vb Ki‡e| myZivs ab ~iZ¡ AwZµg Ki‡Z cÖ‡qvRbxq

mgq t = (2.704 – 0.296) s = 2.408 s

myZivs ab As‡ki ˆ`N©¨ = †e‡Mi Abyf‚wgK Dcvsk mgq

= 30cos30 2.408 = 30 0.8660254 2.408

= 62.56 m

myZivs R t ab = 77.94 t 62.56


K †Kv‡bv e ‘i Ici cÖhy³ ej Øviv K…ZKvR e ‘i MwZ kw³i

cwieZ©‡bi mgvb|

L mij †`vj‡Ki †KŠwYK we¯ Ívi AbwaK 4 n‡j Gi MwZ mij

Qw›`Z nq| KviY mij Qw›`Z MwZi GKwU êwkó¨ n‡jv- GwU

mij‰iwLK MwZ| wKš‘ †KŠwYK we¯ Ívi 4 Gi †ewk n‡j mij

†`vj‡Ki MwZc_ Avi mij‰jwLK _v‡K bv| myZivs GKwU mij

†`vj‡Ki †KŠwYK we Ívi 3 n‡j Gi MwZ mij Qw›`Z n‡e|

M e¯‘wUi fi K‡g hvevi c~‡e©,

myZvi ˆ`N©¨ Z_v e„ËvKvi c‡_i e¨vmva©,

r = 90cm = 0.9m

†KŠwYK †eM, = 2N

t = 2 100

60 rad.s1 = 10.472 rad.s1

F

Fcos

Fsin

M

B


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m



†ei Ki‡Z n‡e, †K› ÖgyLx Z¡iY, ac = ?

Avgiv Rvwb, ac = 2r

= (10.472 rad.s1)2 0.9m

= 98.7 ms2 (Ans.)

N †L‡jvqv‡oi nvZ Øviv cÖhy³ Uvb Z_v †K› ÖgyLx ej AcwiewZ©Z

gv‡bi|

g‡b Kwi, e ‘i Avw` fi = m

Zvn‡j Avw` Ae ’vq,

†K›`ÖgyLx ej Z_v myZvi Uvb, Fc = mac = m 98.7

= 98.7m N

fi GK-Z…Zxqvsk K‡g †M‡j Aewkó fi, m = m m3 =

2m3

G‡ÿ‡Î myZvi bZzb ˆ`N©¨ r n‡j, m2r = m2r

ev, mr = mr

ev, r = mr

m =

mr2m/3 =

32 r

myZivs myZvi ˆ`‡N©¨ cwieZ©b (e„w×) = r r

r =

32 r r

r

= 12 = 50%


K wbw`©ó ¯’v‡bi g‡a¨ mxgve× †Kvb Zi½ Ges H mxgvi g‡a¨ Gi

cÖwZdwjZ Zi‡½i DcwicvZ‡bi d‡j m„ó jwä Zi½‡K w¯’i Zi½

e‡j|

L †kÖwYK‡ÿi k‡ãi ZxeªZv 10-6 Wm-2 ej‡Z eySvq, †kÖwYK‡ÿi

†h †Kv‡bv ’v‡b k‡ãi w`‡Ki mv‡_ j¤^ GK eM©wgUvi †ÿÎd‡ji

ga¨ w`‡q cÖwZ †m‡K‡Û 10-6 J kã kw³ mÂvwjZ nq|

M †Ìqv Av‡Q,

1g myikjvKv n‡Z m„ó k‡ãi ZxeªZv, I = 107 Wm2

†ei Ki‡Z n‡e, ZxeªZv †j‡fj, = ?

Avgiv Rvwb, = 10 logII0

dB

= 10 log107 Wm2

1012Wm2 dB

= 50 dB

k‡ãi ZxeªZv = 50 dB (Ans.)

N cÖ_g myikjvKvi K¤úv¼, f1 = 450 Hz

wØZxq myikjvKvi K¤úv¼, f2 > 450 Hz

[ Gi evû ÿ‡q †M‡Q]

†h‡nZz cÖ_g myikjvKv ywU GKB mv‡_ úw›`Z Ki‡j cÖwZ †m‡K‡Û

N = 6wU exU Drcbœ nq|

myZivs wØZxq myikjvKvi K¤úv¼, f2 = f1 + N

= 450 Hz + 6 Hz

= 456 Hz

G‡ÿ‡Î Z…Zxq myikjvKvi K¤úv¼ 450 Hz n‡Z e„nËi wKš‘ 456 Hz n‡Z ÿz`ªZi|

Gi K¤úv¼ f3 n‡j, cÖ`Ë kZ©g‡Z,

Drcbœ exU msL¨v = f3 f1 = f3 f2

ev, f3 f1 = f2 f3

ev, 2f3 = f1 + f2

f3 = f1 + f2

2 = 450 Hz + 456 Hz

2

= 453 Hz

myZivs DÏxc‡Ki Z_¨mg~n n‡Z 3q myikjvKvwUi K¤úv¼ wbY©q Kiv

m¤¢e|

4bs cÖ‡kœi DËi

K †Kv‡bv Zij c„‡ôi †ÿÎdj cÖwZ eM© GKK (m2) e„w× Ki‡Z

†h cwigvY kw³i cÖ‡qvRb nq Zv‡K H Zi‡ji c„ôkw³ e‡j|

L c„w_ex‡Z w`‡bi msL¨v ej‡Z GLv‡b m~‡h©i PZzw`©‡K c„w_exi

AveZ©bKvj (T) eySv‡bv n‡q‡Q| c„w_ex I m~‡h©i gaëZx© Mo ~iZ¡

R n‡j, MÖn m¤úwK©Z †Kcjv‡ii 3q m~Îvbymv‡i, T2 R3

M †Ìqv Av‡Q,

A Zv‡ii Avw` ˆ`N©¨, L = 0.80m

Avw` e¨vm, D = 2 0.5 mm = 1 mm = 103m

ˆ`‡N©ï e„w×, l = 7 mm = 7 103m

e¨v‡mi n«vm, d = 0.005 mm = 0.005 103m

†ei Ki‡Z n‡e, cqm‡bi AbycvZ, = ?

Avgiv Rvwb, = d/Dl/L

= dLDl


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m



= 0.005 103m 0.80m

103m 7 103m = – 0.57 (Ans)

N A Zv‡ii Dcv`v‡bi Bqs Gi ¸Yv¼,

YA = FALA

rA2lA

= 5N 0.80m

3.1416 (0.5 103m)2 7 103m

= 7.28 108Nm2

B Zv‡ii Dcv`v‡bi Bqs Gi ¸Yv¼,

YB = FBLB

rB2lB

= 6N 0.75m

3.1416 (0.6 103m)2 8 103m

= 4.97 108 Nm2

†h‡nZz 7.28 108 Nm2 > 4.97 108 Nm2

A_©vr YA > YB

A_©vr A I B ZviwUi g‡a¨ A ZviwU †ewk w¯’wZ¯’vcK|


K GKwU ej‡K AmsiÿYkxj ejv n‡e hw` GKwU e ‘ c~Y© Pµ

m¤úbœ K‡i c~‡e©i Ae¯’v‡b wd‡i G‡j H ej Øviv K…ZKvR k~b¨ bv

nq| hw` †Kv‡bv e¯‘‡K GK we› y †_‡K Aci GK we›`y‡Z wb‡Z H

ej Øviv K…ZKvR e ‘‡K †Kvb c‡_ †bqv n‡q‡Q Zvi Dci wbf©i

K‡i Z‡e H ej GKwU AmsiÿYkxj ej|

L g‡b Kwi, ywU e ‘i fi m1 I m2 (m2 > m1) Ges MwZ‡eM v1

I v2| G‡ì fi‡eM mgvb n‡j, m1v1 = m2v2

ev, v2

v1 =

m1

m2

G‡ì MwZkw³i AbycvZ =

EK1

EK2 =

12 m1v1

2

12 m2v2

2

= m1

m2

m2

m1

2 =

m2

m1

m2 > m1

EK1 > EK2

A_©vr nvjKv e ‘i MwZkw³ †ewk|

M †Ìqv Av‡Q,

MÖnwUi fi, M = 12 1024 kg

e¨vmva©, R = 8 106m

gnvKlx©q aªæeK, G = 6.67 1011 Nm2kg2

†ei Ki‡Z n‡e, MÖ‡ni c„‡ô gyw³‡eM, ve = ?

Avgiv Rvwb, ve = 2GM

R

= 2 6.67 1011 Nm2kg2 12 1024 kg

8 106 m

= 14.146 103 ms1

= 14.146 kms1 (Ans.)

N ao = ab = bc kZ©g‡Z,

a, ob Gi ga¨we›`y

ab = bc = R2 =

8 106m2 = 4 106m

MÖ‡ni c„‡ô AwfKl©R Z¡iY g n‡j,

MÖnc„ô n‡Z d = 4 106m MfxiZvq AwfKl©R Z¡iY,

g = g

1

dR = g

1

4 106m

8 106m

= g2

Ges c„ô n‡Z h = 4 106m D”PZvq AwfKl©R Z¡iY,

g = g

R

R + h2 = g

R

R + R/22 = g

R

3R/22 =

49 g

g2 >

49 g

A_©vr g > g

myZivs a we› yi AwfKl© Z¡i‡Yi gvb, c we› yi AwfKl© Z¡iY

A‡cÿv †ewk|


K †Kv‡bv ’v‡bi evZv‡m cÖwZ NbwgUv‡i †h cwigvY Rjxq ev®ú

Av‡Q Zv‡K H ’v‡bi cig Av ª©Zv e‡j|

L M¨v‡mi AYy¸‡jv mew`‡KB G‡jv‡g‡jvfv‡e PjvPj Ki‡Z cv‡i

Ges G‡ì g‡a¨ msNl© N‡U| M¨vm AYy¸‡jvi g‡a¨ ~iZ¡ Zi‡ji

Zzjbvq A‡bK †ewk nIqvq AvšÍtAvYweK ej †bB ej‡jB P‡j|

ZvcgvÎv e„w× †c‡j AYymg~‡ni Mo †eM e„w× cvq, d‡j msNl©I

ev‡o| msNl© evovi Kvi‡Y wewfbœ ¯ Í‡ii cÖev‡n evavi cwigvY e„w×

cvq| A_©vr mv› ÖZv e„w× cvq|

M GLv‡b Mo MwZkw³ ej‡Z cÖwZwU M¨vm AYyi Mo MwZkw³

eySv‡bv n‡q‡Q|

2q cv‡Îi cÖwZ †gvj M¨v‡mi MwZkw³, E = 32 RT

= 1.5 8.314 Jmol1K1 273K

= 3403 J.mol1


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m



myZivs cÖwZwU M¨vm AYyi Mo MwZkw³ = E

NA

= 3403 J.mol1

6.023 1023 mol1 = 5.65 1021J (Ans.)

N cvÎ ywU n‡Z M¨vm e¨vc‡bi gva¨‡g †ei n‡e| e¨vcb nvi mgvb

n‡jB GKB mg‡q cvÎ ywU Lvwj n‡e| Avi e¨vcb nvi wbf©i K‡i

eM©g~j Mo eM©‡e‡Mi Dci|

DÏxc‡Ki Z_¨ n‡Z cvB,

1g cv‡Îi M¨v‡mi AvYweK fi, M1 = 2 g = 0.002 kg

2q cv‡Îi M¨v‡mi AvYweK fi, M2 = 32 g = 0.0032 kg

1g cv‡Îi M¨v‡mi ZvcgvÎv, T1 = 273 K

†gvjvi M¨vm aªæeK, R = 8.31 Joule mol–1 K–1

2q cv‡Îi M¨v‡mi ZvcgvÎv, T2 = ?

1g I 2q cv‡Îi M¨v‡mi eM©g~j Mo eM©‡eM h_vµ‡g c1 I c2 n‡j,

c1 = c2 n‡Z n‡e|

3RT1

M1 =

3RT2

M2

ev, T1

M1 =

T2

M2

ev, T2 = M2

M1 T1 =

0.0320.002 273 = 4368K

2q cv‡Îi M¨v‡mi ZvcgvÎv evov‡Z n‡e (4368 – 273) K = 4095 K

myZivs cvÎ ywU GKB mv‡_ Lvwj n‡Z n‡j wØZxq cv‡Îi ZvcgvÎv

4095K evov‡Z n‡e|


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1. xvkv †evw©-2015 · 01/03/2017 · 1. xvkv †evw ©-2015 1 bs cÖ‡kœi dËi ... l cv‡Îi...

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