1. xvkv †evw©-2016 · 2017. 3. 24. · 1. xvkv †evw©-2016 1bs cÖ‡kœi dËi k kwvb i...

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

c`v^Æweævb

1. XvKv †evW©-2016

1bs 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 Af¨šÍ‡i †h †KvY Drcbœ K‡i Zv‡K

KwVb I Zi‡ji ¯úk© †KvY e‡j|

L cvLvi cÖwZwU KYv N~Y©b A‡ÿi mv‡c‡ÿ mgvb mg‡q mgvb †KvY Drcbœ K‡i A_©vr mgvb mg‡q mgvb †KŠwYK ~̀iZ¡ AwZµg K‡i|

ZvB cÖwZwU KYvi †KŠwYK †eM GKB _v‡K|

M GLv‡b,

A = A = 5

B = B = 6

I Gi gaëZ©x †KvY, α = 90 + 60 = 150

GLb, Ax = A cos60 = 5 cos60 = 2.5

Ges Ay = A sin60 = 5 sin60 = 2.5 3

A = 2.5î + 2.5 3^j

Avevi, Bx = B cos90 = 0

Ges By = B cos0 = – 6

B = – 6^j

A –

B = 2.5î + 2.5 3^j – (–6^j)

= 2.5î + 10.33^j

myZivs |A –

B| = 2.52 + 10.332 = 10.63 (Ans.)

N (

A

B ) †f±iwU (

A +

B ) Gi Dci j¤̂fv‡e Aew ’̄Z n‡j G‡ì †¯‹jvi ¸Ydj k~b¨ n‡e| A_©vr

(

A

B ) (

A +

B ) = 0

GLb, (

A

B ) (

A +

B ) = (

A

B )

A + (

A

B )

B

†h‡nZz

A

B ,

A I

B Df‡qi Dci j¤̂, †m‡nZz (

A

B )

A = 0 Ges (

A

B )

B = 0

myZivs (

A

B ) (

A +

B ) = 0

AZGe, (

A

B ) †f±iwU (

A +

B ) Gi Dci j¤^fv‡e Aew ’̄Z|

2bs cÖ‡kœi DËi

K †Kv‡bv w¯úªs‡qi ˆ`N©̈ GKK cwigvY e„w× Ki‡Z †h cwigvY ej cÖ‡qvM Ki‡Z nq Zv‡K w¯úªs‡qi ej aªæeK e‡j|

L aiv hvK, GKwU e ‘̄‡K v0 †e‡M Lvov Dc‡ii w`‡K wb‡ÿc Kiv n‡jv| e ‘̄wU me©vwaK D”PZvq †cuŠ‡Q cybivq wb‡ÿ‡ci Ae ’̄v‡b wd‡i

Avm‡Z cÖ‡qvRbxq mgq T = 2v0 g

myZivs T mgq ci e ‘̄i †eM, v = v0 g 2v0 g = v0

myZivs wb‡ÿ‡ci mgq e ‘̄i MwZkw³ 12 mv0

2 Ges m‡e©v”P D”PZvq

†cuŠ‡Q cybivq wb‡ÿ‡ci Ae ’̄v‡b wd‡i G‡j MwZkw³ 12 m(– v0)

2 =

12 mv0

2 | KvR-kw³ Dccv`¨ Abymv‡i AwfKl© ej Øviv K…Z KvR

W = 12 mv0

2 12 mv0

2 = 0

†h‡nZz c~Y© Pµ m¤úbœ K‡i cÖv_wgK Ae ’̄v‡b wd‡i Avmvq AwfKl©

ej Øviv K…Z KvR k~b¨ ZvB AwfKl© ej msiÿYkxj ej|

M 1g †gvUi MvwowU mgZ¡i‡Y P‡j Ges mgZ¡iY,

a = 12 4

10 = 0.8 ms2

†ei Ki‡Z n‡e, t = 5s G AwZµvšÍ ~̀iZ¡, s = ?

GLv‡b, Avw`‡eM, v0 = 4ms1

Avgiv Rvwb, AwZµvšÍ `~iZ¡,

s = x – x0 = v0t + 12 at

2

= (4 ms-1)(5 s) + 12 (0.8 ms

-2)(5 s)2

= 20 m + 10 m

= 30 m (Ans.)

N 1g Mvwoi fi, m1 = 500 kg

1g Mvwoi Z¡iY, a1 = (12 – 4) ms-1

10 s = 0.8 ms–2

myZivs 1g Mvwo KZ©„K wbU ej, F1 = m1a1= (500 kg)(0.8 ms–2)

= 400 N

1g Mvwoi Nl©Y RwbZ euvav, f1 = 120 N

myZivs 1g Mvwo KZ©„K cÖhy³ ej, F1a = F1+ f1 = 400 N + 120 N

= 520 N

2q Mvwoi fi, m2 = 320 kg

2q Mvwoi Z¡iY, a2 = (12 – 2) ms-1

8 s = 1.25 ms-2

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c`v^Æweævb

myZivs 2q Mvwoi Dci wbU ej, F2 = m2a2= (320 kg)(1.25 ms-2)

= 400 N

2q Mvwoi Nl©Y RwbZ euvav, f2 = 120 N

myZivs 2q Mvwo KZ©„K cÖhy³ ej, F2a = F2+ f2 = 400 N + 120 N

= 520 N

myZivs Dfq Mvwo KZ…©K cÖhy³ e‡ji gvb mgvb|

3 bs cÖ‡kœi DËi

K cÖm½ KvVv‡gvi g~jwe›`yi mv‡c‡ÿ Ab¨ †Kv‡bv we›`yi Ae ’̄vb †h †f±i Øviv cÖKvk Kiv nq, Zv‡K H we›`yi Ae ’̄vb †f±i ev e¨vmva©

†f±i e‡j|

L e„ËvKvi c‡_ †Kv‡bv e ‘̄ Nyi‡Z †K›`ªgyLx e‡ji cÖ‡qvRb| euvKv iv Í̄vq Mvwoi MwZI e„ËvKvi| ZvB euvKv iv Í̄vq Mvwo †Nviv‡bvi mgq

†K›`ªgyLx e‡ji cÖ‡qvRb nq| G †K›`ªgyLx ej m„wó Kivi Rb¨ euvKv

iv Í̄vi wfZ‡ii w`K A‡cÿv evB‡ii w`K wKQzUv DPz K‡i ˆZwi Kiv

nq| G‡K iv¯Ívi e¨vswKs e‡j| euvKv iv Í̄vq e¨vswKs _v‡K e‡j Mvwo

†gvo †Nvivi mgq †K›`ªi w`‡K wKQzUv †n‡j c‡o hv‡Z cÖ‡qvRbxq

†K›`ªgyLx ej m„wó Ki‡Z cv‡i|

M GLv‡b, R = 6.4 106 m

f‚c„‡ô AwfKl©R Z¡iY, g = 9.8 ms-2

cvnv‡oi D”PZv, hA = 5 km = 5 103 m

cvnv‡ii Pzovq AwfKl©R Z¡iY, gA = ?

Avgiv Rvwb, f‚c„ô †_‡K h D”PZvq †Kv‡bv ’̄v‡b AwfKl©R Z¡iY,

gA =

R

R + hA

2 g

= 2

36

6

m 105m 104.6

m 104.6

9.8 ms-2Z

= 9.785 ms-2 (Ans.)

N f‚c„ô n‡Z hB = 5 km = 5 103 m Mfx‡i B we›`y‡Z AwfKl©R Z¡iY,

gB =

1

hBR g

= )m 104.6

m 1051(

6

3

9.8 ms-2

= 9.79 ms2

ÔMÕ Ask †_‡K cvB, A ¯’v‡b AwfKl©R Z¡iY, gA = 9.785 ms2

A I B ¯’v‡b GKwU mij †`vj‡Ki †`vjb Kvj h_vµ‡g TA I TB

n‡j, mij †`vj‡Ki Z…Zxq m~Î †_‡K Avgiv cvB,

TATB

= gBgA

= 9.79 ms-2

9.785 ms-2 = 1.000255

myZivs TA > TB

†hLv‡b, mij †`vj‡Ki †`vjbKvj †ewk †mLv‡b mij †`vjK ax‡i

P‡j| myZivs B Ae ’̄v‡bi Zzjbvq A Ae ’̄v‡b mij †`vjK ax‡i

Pj‡e|

4bs cÖ‡kœi DËi

K wbw`©ó ZvcgvÎvq ev®ú m‡e©v”P †h Pvc w`‡Z cv‡i ev wbw`©ó ZvcgvÎvq †Kv‡bv Ave× ’̄v‡b m‡e©v”P †h cwigvY ev®ú aviY Ki‡Z

cv‡i †mB cwigvY ev®ú †h Pvc †`q Zv‡K m¤ú„³ ev®ú Pvc e‡j|

L XvKvq evZv‡mi Av‡cwÿK Av ª̀©Zv 60% Gi Øviv eySv hvq †h,

(i) evZv‡mi ZvcgvÎvq GKwU wbw`©ó AvqZ‡bi XvKvi evZvm‡K m¤ú„³ Ki‡Z †h cwigvY Rjxq ev‡®úi cÖ‡qvRb Zvi kZKiv

60 fvM Rjxq ev®ú XvKvi evZv‡m Av‡Q|

(ii) evZv‡mi ZvcgvÎvq XvKvi evZv‡m Dcw¯’Z Rjxq ev‡®úi Pvc GKB ZvcgvÎvq m¤ú„³ Rjxq ev‡®úi Pv‡ci 100 fv‡Mi 60 fvM A_©vr 3/5 Ask|

(iii) XvKvi evZv‡mi wkwkiv‡¼i m¤ú„³ Rjxq ev‡®úi Pvc evZv‡mi ZvcgvÎvq m¤ú„³ Rjxq ev‡®úi Pv‡ci 100 fv‡Mi 60 fvM|

M

O

mgsin W = mg

mgcos

GLv‡b, e ‘̄i IRb, m = 2 kg

AwfKl©R Z¡iY, g = 10 ms-2

†KvY, θ = 60

myZvi Uvb, T = ?

A Ae¯’v‡b e ‘̄i IRb, W = mg Lvov wb‡Pi w`‡K wµqv Ki‡e|

myZvi Uvb = myZv eivei IR‡bi Dcvs‡ki gvb

myZivs

T = mgcosθ

= (2 kg)(10 ms-2)cos60

= (2 kg)(10 ms-2)(0.5)

= 10 N (Ans.)

N wPÎ †_‡K, OD = OA cos60

= 10 m 0.5


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c`v^Æweævb

= 5 m

myZivs CD = OC – OD

= 10 m – 5 m

= 5 m

Avevi, DE = CD – CE

= 5 m – 3.75 m

= 1.25 m

GLb B we› ỳ‡Z e¯‘wUi MwZkw³,

KB = 12mv

2B

GLv‡b, vB = B we› ỳ‡Z e¯‘wUi †eM

v2B = v

2o + 2gs

ev, v2B = 0 + 2 10 DE

ev, v2B = 2 10 1.25 = 25 m

2s–2

KB = 12 mv

2B

= 12 2 25

= 25J

Avevi, C we›`y‡Z e ‘̄wUi MwZkw³, KC = 12mv

2C

v2C = v

2o + 2gs

ev, v2C = 0 + 2 10 CD

ev, v2C = 2 10 5 = 100 m

2s–2

KC = 12mv

2C =

12 2 100 = 100 J

A_©vr KC > KB

C we›`y‡Z e ‘̄wUi MwZkw³ B we›`y A‡cÿv †ewk n‡e|

5bs cÖ‡kœi DËi

K mg‡qi eëavb k~‡bï KvQvKvwQ n‡j e¯‘i mi‡Yi nvi‡K ZvrÿwbK †eM e‡j|

L Zi‡ji c„‡ô wKQz wefe kw³ Rgv _v‡K| G wefe kw³ Zi‡ji c„‡ôi †ÿÎd‡ji Dci wbf©i K‡i| Zij c„‡ôi †ÿÎdj Kg n‡j

mwÂZ wefe kw³I Kg nq| Zij Pvq Gi wefe kw³‡K me©wb¤œ

ivL‡Z| myZivs me©wb¤œ wefe kw³‡Z _vK‡Z n‡j c„‡ôi †ÿÎdj

me©wb¤œ Ki‡Z n‡e| GKwU wbw ©̀ó cvwbi †duvUv †MvjvK…wZ n‡jB Gi

c„‡ôi †ÿÎdj me©wb¤œ nq| G Kvi‡YB cvwbi †duvUv †MvjvK…wZ aviY

K‡i|

M cÖ_g Zv‡ii e¨vmva©, r1 = 1 mm

2 = 0.5 mm = 0.5 10 – 3 m

cÖ_g Zv‡ii cÖ¯’‡”Q‡ì †ÿÎdj, A1 = π r12 = 3.14 (0.5 10 – 3 m)2 = 0.785 10 – 6 m2

cÖhy³ ej, F = 5 103 N

weK…wZ, l1L1

= 5% = 0.05

Avgiv Rvwb,

GKK AvqZ‡b mwÂZ wefe kw³ = 12 cxob weK…wZ

= 12

FA1

l1L1

= 12

5 103 N

0.785 10 – 6 m2 0.05

= 15.9 107 Jm –3 (Ans.)

N cÖ_g Zv‡ii cxob, F

A1 =

5 103 N

0.785 10 – 6 m2 = 6.37 109 N m–

2

cÖ_g Zv‡ii weK…wZ, l1L1

= 5% = 0.05

cÖ_g Zv‡ii Bqs‡qi ¸Yv¼,

Y1 = F/A1l1/L1

= 6.37 109 N m–2

0.05 = 12.74 1010 N m–2

wØZxq Zv‡ii e¨vmva©, r2 = 2 mm

2 = 1 mm = 10 – 3 m

wØZxq Zv‡ii cÖ¯’‡”Q‡ì †ÿÎdj, A2 = π r22 = 3.14 (10 – 3 m)2

= 3.14 10 – 6 m2

wØZxq Zv‡ii cxob, F

A2 =

5 103 N

3.14 10 – 6 m2 = 1.59 109 N m–2

wØZxq Zv‡ii weK…wZ, l2L2

= 2% = 0.02

wØZxq Zv‡ii Bqs‡qi ¸Yv¼,

Y2 = F/A2l2/L2

= 1.59 109 N m–2

0.02 = 7.96 1010 N m–2

Z…Zxq Zv‡ii e¨vmva©, r3 = 3 mm

2 = 1.5 mm = 1.5 10 – 3 m

Z…Zxq Zv‡ii cÖ¯’‡”Q‡ì †ÿÎdj, A3 = π r32

= 3.14 (1.5 10 – 3 m)2

= 7.065 10 – 6 m2

Z…Zxq Zv‡ii cxob, F

A3 =

5 103 N

7.065 10 – 6 m2 = 0.708 109 N m–2


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c`v^Æweævb

Z…Zxq Zv‡ii weK…wZ, l3L3

= 1% = 0.01

Z…Zxq Zv‡ii Bqs‡qi ¸Yv¼,

Y3 = F/A3l3/L3

= 0.708 109 N m–2

0.01 = 7.08 1010 N m–2

†`Lv hvq, Y1 > Y2 > Y3

myZivs cÖ_g ZviwUi w¯’wZ¯’vcKZvi mxgv me‡P‡q †ewk|


K Zi‡½i Dci Aew¯’Z †Kv‡bv K¤úbkxj KYvi GKwU c~Y© K¤úb m¤úbœ Ki‡Z †h mgq jv‡M Zv‡K ch©vqKvj e‡j|

L ZxeªZv n‡”Q kã mÂvj‡bi c‡_ j¤^fv‡e Aew ’̄Z GKK †ÿÎd‡ji ga¨ w`‡q cÖwZ †m‡K‡Û cÖevwnZ kã kw³| myZivs †Kv‡bv

¯’v‡bi k‡ãi ZxeªZv 10–8 watt m–2 ej‡Z †evSvq H ’̄v‡b kã mÂvj‡bi c‡_ j¤̂fv‡e Aew ’̄Z 1m2 †ÿÎd‡ji ga¨ w`‡q cÖwZ †m‡K‡Û cÖevwnZ kã kw³i cwigvY 10–8 J|

M AMÖMvgx Zi‡½i mgxKiY, y = 0.1 sin

200 t –

2017 x

ev, y = 0.1 sin

200t –

21710

x

G‡K AMÖMvgx Zi‡½i mvaviY mgxKiY, y = a sin

2t –

2

x

Gi mv‡_ Zzjbv K‡i cvB,

we¯Ívi, a = 0.1 m

Zi½‰`N©¨, = 1710 m = 1.7 m

2 = 200

= 100 Hz

GLb, c_ cv_©K¨, x = 1.0 – 0.25 = 0.75 m

`kv cv_©K¨, = ?

Avgiv Rvwb, `kv cv_©K¨, = 2

c_ cv_©K¨

= 2

x

= 21.7 0.75

= 2.77 rad (Ans.)

N (M) Ask n‡Z, we Í̄vi, a = 0.1 m

K¤úv¼, = 100 Hz

we¯Ívi I K¤úv¼ wØ¸Y n‡j, a = 2 0.1 = 0.2 m

Ges f = 100 2 = 200 Hz

Avevi, = f

ev, 100 1.7 = 200 [(M) Ask n‡Z = 1.7 m]

= 0.85 m

Zi‡½i mgxKiY,

y1 = 0.2 sin

2 200t –

2x0.85

Ges wecixZgyLx Zi‡½i mgxKiY,

y2 = 0.2 sin

2 200t +

2x0.85

Zi½Øq DcwicvwZZ n‡j,

y = y1 + y2

ev, y = 0.2

sin ( )2 200t – 2x0.85 + sin( )2 200t + 2x0.85

ev, y = 0.2

2sin (2 200t) cos

2x0.85

ev, y = 0.2 2 sin 400t cos 2x0.85

ev, y = 0.4cos 2x0.85 sin 400 t

ev, y = A sin 400t, hv GKwU w ’̄i Zi‡½i mgxKiY

†hLv‡b, A = jwä Zi‡½i we Í̄vi = 0.4 cos 2x0.85

2. ivRkvnx †evW©-2016

1bs cÖ‡kœi DËi K GKB c`v‡_©i wewfbœ AYyi g‡a¨ cvi¯úwiK AvKl©Y ej‡K mskw³ ej ejv nq|


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c`v^Æweævb

L c`v‡_©i wewfbœ Í̄‡ii g‡a¨ Av‡cwÿK MwZ we`¨gvb _vK‡j mv› ª̀Zvi D™¢e nq| A_©vr mv› ª̀Zv m„wói KviY n‡”Q c`v‡_©i wewfbœ

¯Í‡ii gaëZ©x Av‡cwÿK MwZ| KwVb c`v‡_©i wewfbœ Í̄‡ii g‡a¨

†Kv‡bv Av‡cwÿK MwZ _v‡K bv| G Kvi‡Y KwVb c`v‡_©i mv› ª̀Zv

m„wó nq bv| GKB Kvi‡Y w ’̄i cÖevnx (Zij I evqexq) c`v‡_©

mv› ª̀Zv ej wµqv K‡i bv| cÖevnx MwZkxj n‡j wewfbœ Í̄‡ii g‡a¨

Av‡cwÿK MwZ m„wó nq| d‡j mv›`ªZv ej wµqvkxj nq| c`v‡_©i

g‡a¨ ïay cÖevnxB MwZkxj n‡Z cv‡i| G Kvi‡Y cÖevnx c`v_© mv›`ªZv

m„wó nq|

M GLv‡b, OA = 1m.

PB = 0.6m

OB = 1m

OB2 = OP2 + PB2

OP = (1)2 – (0.6)2 = 0.8

AP = h = OA OP = 1 0.8 = 0.2 m

B we›`y‡Z eewUi †eM, vB = 0ms1

A we›`y‡Z eewUi †eM, vA = ?

GLb, vA2 = vB2+ = 2gh

= (0)2 + 2 × 9.8 × 0.2

= 3.92 m2s2

VA = 1.97 ms1 (Ans.)

N DÏxc‡Ki wPÎ †_‡K †bqv Z_¨ n‡Z,

A we›`y‡Z wefekw³, Ep = mgh = mg.0 = 0 J

MwZkw³, Ek = 12 mv

2A

= 12 2 (1.97)

2 = 3.92 J

[M Ask n‡Z vA = 1.97ms1]

A we›`y‡Z †gvU kw³, E = Ep + Ek = 3.92 J

B we›`y‡Z wefekw³, Ep = mgh = 2 9.8 0.2 = 3.92 J

MwZkw³, Ek = 12mv

2B

= 12 m(0)

2 = 0 J

B we›`y‡Z †gvU kw³, E = Ep + Ek = 0 + 3.92 = 3.92 J

C we›`y‡Z wefekw³, Ep = mgh = mg.QA

MwZkw³, Ek = 12 mv

2c =

12 m

2g (PA – QA)

= mg(PA – QA)

C we›`y‡Z †gvU kw³, E = Ep + Ek = mg.PA = 2 9.8 0.2 = 3.92 J

AZGe, †`Lv hv‡”Q †h, A, B, C we›`y‡Z †gvU kw³i cwigvY GKB _v‡K| AZGe, kw³i msiÿYkxjZv bxwZ cÖgvwYZ nq|


K GK cvDÛ f‡ii †Kv‡bv e ‘̄i Ici GK dzU/†m‡KÛ2 Z¡iY m„wó Ki‡Z †h ej cÖhy³ nq Zv‡K GK cvDÛvj ej ejv nq|

L †h‡Kv‡bv ỳBwU e ‘̄i gaëZ©x AvKl©Y ej‡K gnvKl© ej ejv nq| Avi c„w_ex I †h‡Kv‡bv e ‘̄i gaëZ©x AvKl©Y ej‡K AwfKl©

ej e‡j| c„w_ex‡K GKwU e ‘̄i mv‡_ Zzjbv Kiv n‡j c„w_ex I Ab¨

e¯‘i gaëZ©x AvKl©Y n‡”Q gnvKl© ej| A_©vr ejv hvq, AwfKl© ej

GK ai‡bi gnvKl©|

M †Ìqv Av‡Q,

msN‡l©i mgq, t = 4s

cÖwZwµqv ej F1, m2 Gi Ici wµqv K‡i|

myZivs, m2 Gi fi‡e‡Mi cwieZ©‡bi nviB n‡e F1

g‡b Kwi, m2 Gi Avw`‡e‡Mi w`K abvZ¥K|

F1 = m2v2f m2v2i

t

= 0.1 × (90.17) 0.1 × 100

4

= 4.75425 N (Ans.)

GLv‡b, () wPý wb‡`©k K‡i †h, cÖwZwµqv ej wµqv e‡ji wecixZ w`‡K wµqv K‡i|

N g‡b Kwi, m2 Gi Avw`‡e‡Mi w`K abvZ¥K|

fi‡e‡Mi msiÿY m~Îvbymv‡i,

m1v1i + m2v2i = m1v1f + m2v2f

ev, 2 × 0 + 0.1 × 100 = 2 × v1f + 0.1 × (90.17)

ev, 0 + 10 = 2 × v1f 9.017

v1f = 10 + 9.017

2 = 9.5085 ms1

e¯‘Ø‡qi msN‡l©i Av‡Mi MwZkw³i mgwó,

Ek1 = 12 m1v1i

2 + 12 m2v2i

2

= 12 × 2 × (0)

2 + 12 × 0.1 × (100)

2

= 500 J

e¯‘Ø‡qi msN‡l©i c‡ii MwZkw³i mgwó,

Ek2 = 12 m1v1f

2 + 12 m2v2f

2

= 12 × 2 × (9.5085)

2 + 12 × 0.1 × (90.17)

2

= 496.94 J


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c`v^Æweævb

jÿ¨ Kwi, Ek1 Ek2 A_©vr msN‡l©i Av‡Mi Ges c‡ii MwZkw³ mgvb

bq|

myZivs DÏxc‡Ki msNl©wU w¯’wZ¯’vcK bq| msNl©wU Aw¯’wZ¯’vcK|

3bs cÖ‡kœi DËi

K †Kv‡bv ZvcgvÎvq wbw ©̀ó AvqZ‡bi evqy‡Z Dcw¯’Z Rjxqev‡®úi fi Ges H GKB ZvcgvÎvq H AvqZ‡bi evqy‡K m¤ú„³ Ki‡Z

cÖ‡qvRbxq Rjxqev‡®úi f‡ii AbycvZB H ¯’v‡bi Av‡cwÿK

Av`ª©Zv|

L †Kv‡bv mgq †Kv‡bv ’̄v‡bi GKK AvqZ‡bi evqy‡Z †h cwigvY Rjxq ev®ú _v‡K Zv‡K H evqyi cig Av ©̀ªZv e‡j|

cig Av ©̀ªZv evovi mv‡_ mv‡_ †Kv‡bv ̄ ’v‡bi GKK AvqZ‡bi evqy‡Z

M¨vmxq AYyi msL¨v e„w× cvq e‡j M¨vmxq AYyi Mo eM©‡eM I e„w×

cvq|

M GLv‡b,

M¨vm AYy¸‡jvi g~j Mo eM© †eM, c = 500 ms1

M¨v‡mi Pvc, P = 101325 Nm2

M¨v‡mi NbZ¡, = ?

Avgiv Rvwb, c = 3P

ev, = 3Pc2

ev, = 3 101325

(500)2

= 1.2159 kgm3 (Ans.)

N †Ìqv Av‡Q,

M¨vm AYyi e¨vm = 3 10–10 m = 3 × 108 cm Ges cÖwZ Nb †m.wg. G AYyi msL¨v n = 6 1020| ¯^vfvweK ZvcgvÎv I Pv‡c AYy¸‡jvi g~j Mo eM©‡eM 500ms–1|

N msL¨K av°vi †fZi AYy †gvU l `~iZ¡ AwZµg K‡i Z‡e Mo gy³

c_, = lN

GLv‡b, l = vt = 500ms–1 1s = 500m. [t = 1s]

= 500 102 cm

K¬wmqv‡mi c×wZ‡Z, C = 1

n2

= 1

6 1020 3.1416 (3 10–8)2

= 5.89 10–7 cm

†evjR&g¨v‡bi c×wZ‡Z, B = 3

42n

= 3

4 3.1416 (3 10–8)2 6 1020

= 1.47 10–7 cm

K¬wmqv‡mi c×wZ‡Z av°v msL¨v, N = l

C = 8.48 1010

†evjRg¨v‡bi c×wZ‡Z av°vi msL¨v, N = l

B = 3.39 1011

myZivs DÏxc‡Ki Z_¨ n‡Z MvwYwZK we‡køl‡Yi gva¨‡g ejv hvq †h,

†evjR&g¨v‡bi c×wZ‡Z av°v msL¨v K¬wmqv‡mi Zzjbvq †ewk|


K †Kv‡bv e ‘̄i Dci Av‡ivwcZ ch©ve„Ë ¯ú›`‡bi K¤úv¼ e ‘̄wUi ¯^vfvweK K¤úv‡¼i mgvb n‡j e ‘̄wU m‡e©v”P we Í̄v‡i Kw¤úZ nq| G

ai‡bi K¤úb‡K Abybv` e‡j|

L m~Î n‡jv mZ¨ NUbvi ch©‡eÿYg~jK Ges cixwÿZ wee„wZ| Ab¨w`‡K AbyKí Ges m~Î BZ¨vwì wfË‡Z ch©‡ewÿZ cÖK…wZi

myk„•Lj µgaviv Kiv‡K ZË¡ e‡j| A_©vr cixÿv-wbixÿvi gva¨‡g

cÖgvwYZ Z‡Ë¡i g~j K_v¸‡jv †h e³‡eï gva¨‡g cÖKvwkZ Kiv nq

ZvB m~Î Ges cÖKí I wbq‡gi mgš^‡q ZË¡ cÖwZwôZ|

m~Î me©`vB cÖK…wZ Øviv wbqwš¿Z me©`vB Zv mZ¨| wKš‘ ZË¡ n‡”Q

gvby‡li ̂ Zwi hv fzjI n‡Z cv‡i| ZË¡ ch©‡eÿY †_‡K cvIqv hvqwb,

eis Zv †_‡K ch©‡eÿ‡Yi e¨vL¨v Kiv hvq| GKwU cy‡iv‡bv Z‡Ë¡i

†P‡qI GKwU bZzb ZË¡ MÖnY‡hvM¨ n‡Z cv‡i|

M †Ìqv Av‡Q,

B myikjvKv KZ…©K M¨v‡m Drcbœ k‡ãi Zi½‰`N©¨, B = 1.01m

B myikjvKvi K¤úv¼, fB = 512Hz

M¨v‡m k‡ãi †eM, v = ?

M¨v‡mi k‡ãi †eM, v = fBB

= 512 1.01 = 517.12 ms–1 (Ans.)

N fi e„w×i c~‡e©,

A myikjvKvi Zi½‰`N©¨, A = 1m

B myikjvKvi Zi½‰`N©¨, B = 1.01m

B Gi K¤úv¼, fB = 512Hz

A Gi K¤úv¼, fA = ?

†h‡nZz M¨v‡m k‡ãi †eM GKB Ges B > A fA > fB n‡e|

fi e„w×i c~‡e©, fA fB = 6

fA = fB + 6

= (512 + 6) Hz

= 518 Hz.


Teac

hing

bd.co

m



c`v^Æweævb

GLv‡b, †`Lv hv‡”Q †h, fi e„w×i c~‡e© fA > fB| A_©vr ‘A’ myikjvKvi K¤úv¼, fi e„w×i c~‡e© ‘B’ Gi †P‡q †ewk|

fi e„w×i c‡i,

‘A’ myikjvKvi evû‡Z †gvg jvMv‡bv n‡j Gi fi †e‡o hvq Avi fi evov‡j K¤úv¼ K‡g hvq|

fB – fA = 6

ev, 512 – fA = 6

fA = 506 Hz

5bs cÖ‡kœi DËi

K `yB ev Z‡ZvwaK †f±i ivwk †hv‡M †h GKwU bZzb †f±i ivwk nq Zv‡K G‡ì jwä †f±i e‡j|

L †K›`ªgyLx Z¡i‡Yi ivwkgvjv, a = v2

r = 2r|

GB mgxKiY‡K †f±iiƒ‡c wjL‡j cvB, a = 2

r =

v2

r2 r

GLv‡b FYvZ¥K wPý †_‡K †`Lv hvq †K›`ªgyLx Z¡i‡Yi w`K e¨vmva©

†f±i Z_v Ae ’̄vb †f±‡ii wecixZ w`‡K A_©vr e¨vmva© eivei

†K‡›`ªi w`‡K|

M †`qv Av‡Q,

Zi‡ji c„ôUvb, T = 72 10–3 Nm–1

b‡ji e¨vmva©, r = 0.2 10–3 m

¯úk©‡KvY, = 4

†h‡nZz cvwbi c„ôUvb 72 10–3 Nm–1, AZGe D³ ZijwU cvwb|

Zi‡ji NbZ¡ (cvwb), = 1000 kgm–3

Zi‡ji D”PZv, h = ?

Avgiv Rvwb,

T = hrg

2cos

ev, h = 2Tcos

rg

= 2 72 10–3 cos4

0.2 10–3 1000 9.8 m

= 0.073m

ˆKwkK b‡j Zi‡ji DÌvb 0.073m (Ans.)

N DÏxc‡Ki 0.2mm e¨vmv‡a©i ˆKwkK b‡j, cÖ_g Zi‡ji †ÿ‡Î 0.073m DÌvb N‡U| [(M) Ask n‡Z]|

A_©vr ZijwU ˆKwkK bj †e‡q 0.073m Dc‡i I‡V|

Avevi, wØZxq Zi‡ji †ÿ‡Î,

b‡ji e¨vmva©, r = 0.2 mm = 0.2 10–3 m

¯úk©‡KvY, = 140

wØZxq Zi‡ji c„ôUvb, T = 465 10–3 Nm–1

†h‡nZz ¯úk©‡KvY 140, ZvB ejv hvq GwU cvi`|

cvi‡ì NbZ¡, = 13.6 103 kgm–3

Rvbv Av‡Q,

T = hrg

2cos

ev, 465 10–3 = h 0.2 10–3 13.6 103 9.8

2cos140

h = – 0.026m.

A_©vr GLv‡b, †evSv hv‡”Q †h, ˆKwkK b‡j cvi‡ì cZb N‡U|

0.026m < 0.073m

ZvB ejv hvq †h, DÏxc‡Ki ˆKwkK b‡j Zi‡ji DÌvb †ewk N‡U|

6bs cÖ‡kœi DËi

K †Kv‡bv e„ËvKvi †¯‹j GKevi Nyiv‡j Zv îwLK †¯‹j eivei †h `~iZ¡ AwZµg K‡i Zv‡K H h‡š¿i cxP e‡j|

L †m‡KÛ †`vj‡Ki †`vjbKvj, T = 2s

Gi K¤úv¼, f n‡j, f = 1T =

12s = 0.5 Hz

gvby‡li kÖve¨Zvi b~¨bZg mxgv 20 Hz. A_©vr †Kv‡bv k‡ãi K¤úv¼ 20 Hz Gi †P‡q Kg n‡j Zv gvbyl ïb‡Z cv‡e bv| †m‡KÛ †`vj‡Ki K¤úv¼ 20Hz Gi †P‡q A‡bK Kg nIqvq, †m‡KÛ †`vjK KZ…©K Drcbœ kã gvbyl ïb‡Z cvq bv| G Kvi‡Y g‡b nq, †m‡KÐ †`vjK

†Kv‡bv kã Drcbœ K‡i bv|

M cÖ̀ Ë miY ebvg mgq †jLwP‡Î x Aÿ eivei mgq Ges y Aÿ eivei miY †`Lv‡bv n‡q‡Q| GLv‡b AB †iLvwU GKwU g~jwe›`yMvgx mij‡iLv hvi Xvj H e ‘̄i †eM wb‡`©k K‡i|

†eM, v = dsdt

= 4 3

4 3 = 1cms1

†h‡nZz AB †iLvwU GKwU mij‡iLv Ges G‡ÿ‡Î †eM aªæeK ZvB e¯‘i †Kv‡bv Z¡iY _vK‡e bv| A_©vr AB As‡k e ‘̄i Z¡iY k~b¨| (Ans.)

N

10 9

5 4

1 0 2 3 4 5 6 7 8 9 10

s (c

m)

3 2 1

t (sec)

B C


Teac

hing

bd.co

m



c`v^Æweævb

DÏxc‡K †jLwP‡Î y Aÿ eivei miY Ges x Aÿ eivei mgq| O †_‡K B ch©šÍ cÖwZ 1s G miY 1cm| wKš‘ B †_‡K C we› ỳ‡Z mg‡qi mv‡_ miY cwiewZ©Z nqbv| ZvB BC w¯’ive ’̄v wb‡`©k K‡i|

3. w`bvRcyi †evW©-2016

1bs cÖ‡kœi DËi

K me©v‡cÿv Kg †h †e‡M †Kv‡bv e ‘̄‡K Lvov Dc‡ii w`‡K wb‡ÿc Ki‡j Zv Avi c„w_ex‡Z wd‡i Av‡m bv †mB †eM‡K gyw³‡eM e‡j|

L hLb †Kv‡bv e ‘̄ GKwU e„ËvKvi c‡_ Nyi‡Z _v‡K ZLb H e„‡Ëi †K›`ª Awfgy‡L †h wbU ej wµqv K‡i e ‘̄wU‡K e„ËvKvi c‡_ MwZkxj

iv‡L Zv‡K †K› ª̀gyLx ej e‡j| e„ËvKvi c‡_i e¨vmva© e„w× †c‡j

†K›`ªgyLx e‡ji gvbI e„w× cvq| KviY, †K› ª̀gyLx ej, F = m2r| GLv‡b m n‡jv e ‘̄i fi, n‡jv †KŠwYK †eM Ges r n‡jv e„ËvKvi c‡_i e¨vmva©| GKwU wbw`©ó f‡ii e¯‘ GKwU wbw ©̀ó †KŠwYK †e‡M

e„ËvKvi c‡_ cwiågY Ki‡j, F r A_©vr e„ËvKvi c‡_ N~Y©bkxj e¯‘i †K›`ªgyLx ej e¨vmv‡a©i cwieZ©‡bi mv‡_ cwiewZ©Z nq|

M DÏxcK n‡Z cvB, e ‘̄i fi, m = 2kg

r = ( î 2 ĵ + bk̂)m

= (2 î 4 ĵ + 2k̂ ) ms1

†KŠwYK fi‡e‡Mi gvb, L = ?

r = ( î 2 ĵ + bk̂) m

p = m

v = 2 (2 î 4 ĵ + 2k̂) ms–1 = (4 î 8 ĵ + 4k̂) ms–1

L =

r

p =

î ĵ k̂

1 –2 b4 –8 4

= î (–8 + 8b) – ĵ (4 – 4b) + k̂ (–8 + 8)

= 8 î (b –1) + 4 ĵ (b–1)

hLb; b = 2 ZLb, L = 8 î (2–1) + 4 ĵ (2–1) = 8 î + 4 ĵ

†KŠwYK fi‡e‡Mi gvb = |L | = 82 + 42 = 4 5 kgm2s–1 (Ans.)

N DÏxcK n‡Z cvB,

r = ( î 2 ĵ + bk̂ ) m

= (2 î 4 ĵ + 2k̂) ms1

r I

ci¯úi mgvšÍivj n‡j,

r

= 0

î

12

ĵ

2

4

k̂b2

= 0

ev, ( 4 + 4b) î + (2b 2) ĵ + ( 4 + 4)k̂ = 0

ev, ( 4 + 4b) î + (2b 2) ĵ = 0

GLb, î I ĵ Gi mnM mgxK…Z K‡i cvB,

4 + 4b = 0 ev, b = 1

Ges 2b 2 = 0 ev, b = 1

r I

ci¯úi mgvšÍivj n‡j, b = 1 n‡e|

Avevi, r I

ci¯úi j¤̂ n‡j,

r .

= 0

ev, ( î 2 ĵ + bk̂ ) . (2 î 4 ĵ + 2k̂) = 0

ev, 2 + 8 + 2b = 0

ev, 2b = 10

b = 5

AZGe, r I

ci¯úi j¤^ n‡j b = 5 n‡e| myZivs

r I

Gi j¤̂ Ae¯’vq b Gi gvb mgvšÍivj Ae¯’vq b Gi gv‡bi †P‡q 1 ( 5) = 6 Kg n‡e|

2bs cÖ‡kœi DËi

K wbwÿß e¯‘ ev cÖvm Avw` D”PZvq wd‡i Avm‡Z †h Abyf‚wgK ̀ ~iZ¡ AwZµg K‡i Zv‡K Abyf‚wgK cvjøv e‡j|

L wbwÿß e ‘̄ ev cÖv‡mi wb‡ÿ‡ci ci Avevi f‚-c„‡ô wd‡i Avm‡Z †h mgq jv‡M Zv‡K cÖv‡mi wePiYKvj e‡j| cÖvm ev wbwÿß e ‘̄i

†ÿ‡Î Zvi Ae ’̄vb †f±‡ii Djø¤^ Dcvsk Ges mg‡qi g‡a¨ m¤úK©

n‡”Q,

y = vyot 12 gt

2

e¯‘ f‚-c„‡ô wd‡i Avm‡j, y = 0| GB kZ© DcwiD³ mgxKi‡Y emv‡j t Gi †h gvb cvIqv hvq ZvB n‡e wePiYKvj| wePiYKvj t n‡j GB mgxKiY n‡Z Avgiv cvB,

0 = v0sin0T 12 gT

2

T = 0 ev, T = 2v0sin0

g

†h‡nZz T = 0 f‚-c„ô †_‡K †h gyn~‡Z© e¯‘wU wb‡ÿc Kiv n‡”Q ZvB wb‡`©k K‡i, myZivs


Teac

hing

bd.co

m



c`v^Æweævb

T = 2v0sin0

g e ‘̄wUi wePiYKvj wb‡ ©̀k K‡i|

†h‡nZz †Kv‡bv ’̄v‡b 2g aªæe ivwk|

AZGe, T v0sin

A_©vr, wePiYKvj Avw`‡e‡Mi Djø¤^ Dcvs‡ki mgvbycvwZK|

M DÏxcK n‡Z cvB,

Avw`‡eM, v0 = 20ms1

wb‡ÿcb †KvY, 0 = 30

D”PZv, h = 30m (wb¤œgyLx)

AwfKl©R Z¡iY, g = 9.8ms2

gvwU‡Z †cuŠQ‡Z mgq, t = ?

Avgiv Rvwb,

h = (v0 sin0)t 12 gt

2

ev, 30 = (20 sin 30)t 12 9.8 t

2

ev, 30 = 10t 4.9t2

ev, 4.9t2 10t 30 = 0

t = ( 10) (10)2 4 4.9 (30)

2 4.9

= 10 688

9.8

t = 3.7s ev, 1.7s

FYvZ¥K mgq MÖnY‡hvM¨ bq|

gvwU‡Z †cuŠQ‡Z mgq, t = 3.7s (Ans.)

N DÏxcK n‡Z cvB,

wb‡ÿcY †eM, v0 = 20 ms1

`vjv‡bi D”PZv, h = 30m

wb‡ÿcY †KvY, 0 = 30

x

0

v0

0

v0sin

h

v0

v0Cos0

R

h =

30

m

Abyf‚wgK cvjøv, R = v02 sin2 0

g

= (20)2 × sin (2 × 30)

9.8

= 35.35 m.

e¯‘wU Qv‡`i mgZ‡j wd‡i Avmvi ciI 30m Dj¤̂ ̀ ~iZ¡ AwZµg Ki‡e| GB Dj¤̂ `~iZ¡ AwZµ‡gi Rb¨ Avbylw½K Abyf‚wgK `~iZ¡ = x Ges GB ~̀iZ¡ AwZµg Kivi mgq t (awi)|

h = v0sin0t + 12 gt

2

ev, 30 = 20 × sin30 × t + 12 × 9.8 × t

2

ev, 30 = 10t + 4.9t2

ev, 4.9t2 + 10t 30 = 0

t = 1.656s ev, 3.696s (hv MÖnY‡hvM¨ bq)

x = v0cos0 × t = 20 × cos30 × 1.656 = 28.68 m

e ‘̄wU gvwU‡Z AvNvZ Kivi c~‡e© †gvU AwZµvšÍ Abyf‚wgK ~̀iZ¡ = R + x = 35.35 + 28.68 = 64.03 m > 35.35 m

AZGe, e ‘̄wU gvwU‡Z AvNvZ Kivi c~‡e© †h Avbyf‚wgK ̀ ~iZ¡ AwZµg

K‡i Zv Avbyf‚wgK cvjøvi †P‡q †ewk|

3bs cÖ‡kœi DËi

K gvby‡li cvVv‡bv †hme e ‘̄ ev gnvKvkhvb c„w_ex‡K †K› ª̀ K‡i wbw`©ó Kÿc‡_ †Nv‡i Zv‡`i K…wÎg DcMÖn e‡j|

L †Kv‡bv KYv GKwU c~Y© Pµ m¤úbœ K‡i Zvi Avw` Ae ’̄v‡b wd‡i Avm‡j KYvwUi Dci gnvKl© ej Øviv m¤úvw`Z Kv‡Ri cwigvY k~b¨

nq e‡j gnvKl© ej msiÿYkxj ej| †hgb, gnvKl©xq †ÿ‡Îi

†Kv‡bv we› ỳ †_‡K †Kv‡bv e ‘̄‡K gnvKl©xq †ÿ‡Îi evB‡i wb‡q †h‡Z

gnvKl©xq ej KZ…©K K…ZKvR abvZ¥K nq| Avevi, H e ‘̄‡K

gnvKl©xq †ÿ‡Îi evB‡i †_‡K H we›`y‡Z Avb‡Z gnvKl©xq ej KZ…©K

mgcwigvY abvZ¥K KvR m¤úbœ n‡e| A_©vr c~Y© P‡µ †gvU Kv‡Ri

cwigvY k~b¨ n‡e|

GQvovI gnvKl© ej Øviv m¤úbœ Kv‡Ri cwigvY KYvwUi MwZc‡_i

Dci wbf©i K‡i bv| ZvB gnvKl© ej GKwU msiÿYkxj ej|

M DÏxcK n‡Z cvB,


Teac

hing

bd.co

m



c`v^Æweævb

Kzqvi MfxiZv, h = 12m

Rvbv Av‡Q, AwfKl©R Z¡iY, g = 9.8 ms–2

cvwbk~b¨ Kzqvi kxl© n‡Z Zjvq †cuŠQv‡Z mgq, t = ?

Avgiv Rvwb, h = 12 gt

2

ev, t = 2hg =

2 129.8

= 1.56s (Ans.)

N DÏxcK n‡Z cvB,

Kzqvi MfxiZv, h = 12m

Kzqvi e¨vm, d = 1.8m

Kzqvi e¨vmva©, r = 1.82 m = 0.9 m

mgq, t = 21 min = 21 60s = 1260s

cvwb DVv‡bvi Kvh©Ki ev Mo D”PZv, s = 0 + 12

2 m = 6m

ÿgZv, P = ?

Avgiv Rvwb, P = Wt

= Fst [ W = Fs]

= mgs

t [ F = mg]

= Vgs

t [ m = V]

= r2hgs

t [ V = r2h]

= 3.1416 (0.9)2 12 1000 9.8 6

1260

= 1425.029 W = 1.91 HP

DÏxc‡K cÖvß Z_¨vbyhvqx KzqvwU‡K cvwb k~b¨ Ki‡Z 1.91 HP Gi

cv¤ú ìKvi| wKš‘ Lvwj‡ì wnmve Abyhvqx 2HP ÿgZvi cv¤ú

ìKvi hv cy‡ivcywi mwVK bq|

4bs cÖ‡kœi DËi

K †h a‡g©i ìæb †Kv‡bv cÖevnxi wewfbœ ¯Í‡ii Av‡cwÿK MwZ‡Z evavi m„wó nq Zv‡K H cÖevnxi mv›`ªZv e‡j|

L cvwbi c„ôUv‡bi Kvi‡Y QvZvi Kvc‡oi †QvU †QvU wQ ª̀ w`‡q cvwb wfZ‡i cÖ‡ek Ki‡Z cv‡i bv|

QvZvi Kvco we‡kl cÖwµqvq cȪ ‘Z Kiv nq Ges G‡Z Lye †QvU †QvU

wQ`ª _v‡K| Gme wQ`ª w`‡q evqy PjvPj Ki‡Z cv‡i wKš‘ cvwb cÖ‡ek

Ki‡Z cv‡i bv| c„ôUv‡bi Kvi‡Y cvwb †MvjvKvi †duvUvq cwiYZ nq

Ges cvwbi †duvUv¸‡jvi AvqZb Kvc‡oi wQ‡ ª̀i AvqZ‡bi Zzjbvq

eo nq| ZvB cvwb QvZvi Dci w`‡q Mwo‡q P‡j Ges QvZvi wfZ‡i

cÖ‡ek Ki‡Z cv‡i bv|

M DÏxcK n‡Z cvB, Kvh©Kwi ˆ`N©¨, L = OA = 2m

c„w_ex‡Z AwfKl©R Z¡iY, g = 9.8ms2

awi, Puv‡ì fi = Mm Ges e¨vmva© = Rm

c„w_exi fi, Me = 81Mm

c„w_exi e¨vmva©, Re = 4Rm

c„w_ex‡Z †m‡KÛ †`vj‡Ki †`vjbKvj, Te = 2s.

Puv‡` †`vjbKvj, Tm = ?

Avgiv Rvwb, g = GMeRe2

Ges

gm = GMmRm2

g

gm =

GMeRe2

Rm2

GMm

= 81Mm Rm2

16 Rm2 Mm

= 8116

Avevi, Te = 2Lg .......... (i)

Tm = 2 L

gm .......... (ii)

(ii) (i) bs mgxKiY n‡Z cvB,

TmTe

= g

gm

ev, Tm = Te g

gm

= 2 8116

= 2 94

= 4.5 s (Ans.)

N DÏxcK n‡Z cvB,

e‡ei fi, m = 5gm = 5 103 kg

†`vjbKvj, T = 2 †m‡KÛ


Teac

hing

bd.co

m



c`v^Æweævb

we¯Ívi, a = 0.5m

Avgiv Rvwb, mij †`vj‡Ki w ’̄wZkw³, EP = 12 m

2x2 Ges

mij‡`vj‡Ki MwZkw³, EK = 12 m

2 (a2 x2)

Avevi, †KŠwYK †eM, = 2T

= 2 3.1416

2

= 3.1416 rads1

A we›`y‡Z, x = 0

A we›`y‡Z w¯’wZkw³, EP = 12 m

2 02 = 0

A we›`y‡Z MwZkw³, EK = 12 m

2(a2 02)

= 12 m

2a2

= 12 5 10

3 (3.1416)2 (0.5)2

= 6.168 103 J

A we›`y‡Z †gvU kw³, EA = EP + Ek

= 0 + 6.168 103

= 6.168 103 J

Avevi, B we›`y‡Z, x = a

B we› ỳ‡Z w¯’wZkw³, EP = 12 m

2a2

= 12 5 10

3 (3.1416)2 (0.5)2

= 6.168 103 J

B we›`y‡Z MwZkw³, Ek = 12 m

2(a2 a2)

= 12 m

2 0

= 0

B we› ỳ‡Z †gvU kw³, EB = EP + Ek = 6.168 103 + 0

= 6.168 10–3 J

AZGe, A we›`y‡Z †gvU kw³ = B we›`y‡Z †gvU kw³|

myZivs A we›`y‡Z †gvU kw³ I B we›`y‡Z †gvU kw³i †Kv‡bv cwieZ©b n‡e bv|


K hLb †Kv‡bv e ‘̄ GKwU e„ËvKvi c‡_ Nyi‡Z _v‡K ZLb H e„‡Ëi †K›`ª Awfgy‡L †h wbU ej wµqv K‡i e ‘̄wU‡K e„ËvKvi c‡_ MwZkxj

iv‡L Zv‡K †K› ª̀gyLx ej e‡j|

L Nl©Y ej I mv›`ª ej DfqB MwZi wecixZ w`‡K KvR Ki‡jI Zv‡`i g‡a¨ wKQy †gŠwjK cv_©K¨ _vKvq Nl©Y ej I mv› ª̀ ej GK

bq| GKwU e ‘̄ hLb Ab¨ GKwU e¯‘i Dci w`‡q MwZkxj nq ev

MwZkxj n‡Z †Póv K‡i ZLb e ‘̄ ỳwUi wgjb Z‡j e¯‘i MwZi

wecix‡Z GKwU evav`vbKvix ej wµqv K‡i| GB e‡ji bvg Nl©Y

ej| †Zgwb †Kv‡bv GKwU cÖevnx Zvi wewfbœ Í̄‡ii Av‡cwÿK MwZi

we‡ivwaZv K‡i †h ej cÖ‡qvM K‡i Zv‡K H cÖevnxi mv› ª̀Zv e‡j|

Nl©Y e‡ji gvb ¯úk©Z‡ji †ÿÎd‡ji Dci wbf©i K‡i bv, mv›`ªZv

e‡ji gvb cÖevnxi Í̄iØ‡qi †ÿÎd‡ji Dci wbf©i K‡i| GQvovI,

mv› ª̀Zv ej cÖevnxi Í̄iØ‡qi †eM I w¯’i Zj †_‡K Gi ~̀i‡Z¡i Dci

wbf©i K‡i| G Kvi‡Y Nl©Y ej I mv›`ª ej GK bq|

M DÏxcK n‡Z cvB,

gnvKl©xq aªæeK, G = 6.7 1011Nm2kg2

c„w_exi e¨vmva©, R = 6.4 106m

DcMÖ‡ni D”PZv, h = 3.6 104 km

= 3.6 107m

Rvbv Av‡Q, c„w_exi fi, M = 6 1024 kg

DcMÖnwUi †eM, = ?

Avgiv Rvwb, = GM

R + h

= 6.7 1011 6 1024

6.4 106 + 3.6 107

= 3.07 103 ms1

= 3.07 kms1 (Ans.)

N DÏxcK n‡Z cvB,

c„w_exi e¨vmva©, R = 6.4 106 m

DcMÖ‡ni D”PZv, h = 3.6 104 km

= 3.6 107 m

ÔMÕ Ask n‡Z cvB,

DcMÖnwUi †eM, = 3.07 103 ms1

DcMÖnwUi AveZ©bKvj, T = ?

Avgiv Rvwb, T = 2(R + h)

= 2 3.14(6.4 106 + 3.6 107)

3.07 103

= 86,733 s


Teac

hing

bd.co

m



c`v^Æweævb

= 24.09 h

= 24h (cÖvq)

Avgiv Rvwb, †hme K…wÎg DcMÖ‡ni AveZ©bKvj c„w_exi AvwýK

MwZi AveZ©bKv‡ji mgvb A_©vr 24 NÈv, Zv‡ì f‚-w¯’i DcMÖn e‡j| DÏxc‡Ki e½eÜy-1 DcMÖnwUi AveZ©bKvj 24 NÈv nIqvq Zv f‚-w¯’i DcMÖn|

6 cÖ‡kœi DËi

K w¯’wZ¯’vcK mxgvi g‡a¨ †Kv‡bv e ‘̄i cxob I weK…wZi AbycvZ GKwU aªæemsL¨v| GB aªæe msL¨v‡K H e ‘̄i Dcv`v‡bi w¯’wZ¯’vcK

¸bvsK e‡j|

L mij †`vj‡Ki MwZ mij Qw›`Z ¯ú›`b MwZ| A_©vr Gi Z¡iY, mi‡Yi mgvb I wecixZgyLx nq| †Kv‡bv †`vjK‡K mvg¨ve ’̄v †_‡K

4 Gi g‡a¨ ỳj‡Z w`‡j GwU mij †`vjK wn‡m‡e KvR K‡i| wKš‘ 4 Gi †P‡q †ewk †Kv‡Y ỳj‡j Gi Z¡iY, mi‡Yi mgvbycvwZK nq bv| A_©vr G‡ÿ‡Î †`vjKwUi MwZ mij Qw›`Z MwZ n‡e bv Ges GwU

mij †`vjK wn‡m‡e KvR Ki‡e bv| GKvi‡Y me †`vjK mij

†`vjK bq|

M DÏxcK n‡Z cvB,

ï®‹ _v‡g©vwgUv‡ii cvV, 1 = 35C

Av`©ª _v‡g©vwgUv‡ii cvV, 2 = 30C

†MøBmv‡ii Drcv`K, G = 1.60

wkwkivsK, = ?

Avgiv Rvwb, = 1 – G(1 2)

= 35C 1.60(35C 30C)

= 27C (Ans.)

N DÏxcK n‡Z cvB,

ivRkvnxi Av‡cwÿK Av ª̀©Zv, R1 = 50%

K·evRvi evqyi ZvcgvÎvq m¤ú„³ Rjxqev‡®úi Pvc,

F = 42.16 mm cvi`

ÔMÕ Ask n‡Z cvB, K·evRv‡i wkwkivsK, = 27C

wkwkivs‡K m¤ú„³ Rjxqev‡®úi Pvc,

f = 25.21 + 28.35 25.21

2

= 26.78 mm cvi`

K·evRv‡i Av‡cwÿK Av ª̀©Zv, R2 = fF 100%

= 26.7842.10 100% = 63.52%

¯úóZB R2 > R1| ZvB ivRkvnx I K·evRv‡ii ZvcgvÎv GK _vK‡jI K·evRv‡ii Av‡cwÿK Av`ª©Zv †ewk| d‡j ivRkvnxi †P‡q

K·evRv‡ii evqy‡Z Rjxqev‡®úi cwigvY †ewk _vK‡e| G Kvi‡Y

K·evRv‡i †Kv‡bv e¨w³i kixi †_‡K wbM©Z Nvg Kg ïKv‡e Ges

Nvg ev®úvq‡bi Rb¨ K·evRv‡i Kg myß Zv‡ci cÖ‡qvRb n‡e| ZvB

ivRkvnxi Zzjbvq K·evRv‡ii e¨w³i kixi Kg Zvc nviv‡e| d‡j

K·evRv‡i e¨w³ AwaK A¯̂w¯Í Abyfe Ki‡e|

4. Kzwgjøv †evW©-2016

1bs cÖ‡kœi DËi

K cÖm½ KvVv‡gvi g~j we› ỳi mv‡c‡ÿ †Kv‡bv we›`yi Ae ’̄vb †h †f±i w`‡q wb‡ ©̀k Kiv nq Zv‡K H we›`yi Ae ’̄vb †f±i e‡j|

L Uªwj e¨v‡Mi nvZj Øviv Uªwj e¨vM‡K mvg‡bi w`‡K †U‡b wb‡q hvIqvi mgq nvZ‡j cÖhy³ ej `yBwU Dcvs‡k wef³ nq| GKwU

Fsin Ges AciwU Fcos| Fsin DcvskwU Dc‡ii w`‡K Kvh©iZ nq, Ges Fcos DcvskwU e¨vM‡K mvg‡bi w`‡K GwM‡q wb‡q hvq| nvZj j¤̂v n‡j Gi gvb Kg nq| G Ae¯ ’vq cos Gi gvb †ewk nq Ges Uªwji †eM aªæe †i‡L Uvb‡Z Kg ej jv‡M| G Kvi‡Y Uªwj

e¨v‡Mi nvZj j¤̂v ivLv nq|

M DÏxcK n‡Z cvB,

C =

î 2

^j + 3

^k

X =

î

C I x A‡ÿi AšÍf‚©³ †KvY, = ?

Avgiv Rvwb,

C .

X = | |C | |X cos

ev, cos =

C .

X

| |C | |X

= 1

1 + 4 + 9 1 =

1

14 = 0.267

= cos1 (0.267) = 74.5 (Ans.)

N DÏxcK n‡Z cvB,

B = î +

^j 2

^k

C = î 2

^j + 3

^k

A = 2î

^j +

^k

B

C =

î

^j

^k

1 1 2

1 2 3

= (3 4) î + (23)

^j + ( 2 1)

^k

= î 5

^j 3

^k


Teac

hing

bd.co

m



c`v^Æweævb

GLb,

A . ( )B C = (2î ^j + ^k ). (î 5^j 3^k )

= 2 + 5 3

= 0

†h‡nZz

A . ( )B C = 0, †m‡nZz B Ges C ‡f±iØ‡qi j¤^w`‡Ki †f±iwU

A Gi mv‡_ GKB mgZ‡j Ae¯’vb K‡i bv eis j¤̂ Z‡j Ae¯’vb K‡i|


K me©v‡cÿv Kg †h †e‡M †Kv‡bv e ‘̄‡K Lvov Dc‡ii w`‡K wb‡ÿc Ki‡j Zv Avi c„w_ex‡Z wd‡i Av‡m bv Zv‡K gyw³‡eM e‡j|

L w¯úÖshy³ †Ljbv Mvwo‡K hLb †cQb w`‡K Uvbv nq ZLb w¯úÖs Gi wecix‡Z ej cÖ‡qvM K‡i KvR Kiv nq| GB KvR w ’̄wZkw³iƒ‡c

w¯úÖs G mwÂZ _v‡K| MvwowU‡K hLb †Q‡o †Ìqv nq, ZLb GB

w¯’wZkw³ MwZkw³‡Z iƒcvšÍwiZ n‡q MvwowU‡K mvg‡bi w`‡K GwM‡q

wb‡q hvq|

M DÏxcK n‡Z cvB,

wb‡ÿcY †KvY, 0 = 35

wb‡ÿcY †eM, v0 = 10ms1

mgq, t = 0.2 sec


†eM, v = ?

Avgiv Rvwb,

vx = v0cos0

= 10 cos 35

= 8.19 ms1

Avevi, vy = v0sin0 gt

= 10 sin35 9.8 0.2

= 3.77 ms1

0.2 †m‡KÛ c‡i †e‡Mi gvb,

v = vx2 + vy2

= (8.19)2 + (3.77)2

= 9.02ms1

g‡b Kwi, v, Abyf‚wg‡Ki mv‡_ †KvY ˆZwi K‡i|

tan = vyvx

= 3.778.19

= 24.72

0.2s c‡i †e‡Mi gvb n‡e 9.02ms1 Ges w`K n‡e Abyf‚wg‡Ki

mv‡_ 24.72 †KvY K‡i Ic‡ii w`‡K| (Ans.)

N GLv‡b, D we› ỳ‡Z Ae¯’vbiZ eÜyi wX‡ji †eM, v0 = 10ms1

Djø¤^ miY, y = 4.9m

Abyf‚wgK miY x n‡j, Avgiv Rvwb,

y = 12 g

x2

v02

ev, 4.9 = 12 9.8

x2

102

x = 10m > 6.3 m

Avevi, y = 4.9 1.5 = 3.4 m Djø¤̂ mi‡Yi Rb¨ Abyf‚wgK miY x

n‡j, y = 12 g

x2

v02

ev, 3.4 = 12 9.8

x2

102

x = 8.33m 6.3 m

A_©vr D we› ỳ‡Z Ae¯’vbiZ eÜyi wXjwU A we›`y‡Z AvNvZ Ki‡e bv|

Avevi, B we› ỳi Rb¨, 6.3m Abyf‚wgK mi‡Yi Rb¨ Djø¤̂ miY y n‡j,

y = x tan g

2(v0cos0)2 x2

ev, y = 6.3 tan 35 9.8 6.32

2 (10 cos 35)2

y = 1.50m

A_©vr B we›`y‡Z Ae ’̄vbiZ eÜyi wXjwU A we›`y‡K ¯úk© Ki‡e| wKš‘

D we›`y‡Z Ae¯’vbiZ eÜyi wXjwU A we›`y‡Z ¯úk©B K‡i bv|


K m~‡h©i Pviw`‡K cÖwZwU MÖ‡ni AveZ©bKv‡ji eM© m~h© †_‡K H MÖ‡ni Mo ~̀i‡Z¡i Nbd‡ji mgvbycvwZK|

L Av‡gi Ici ïay †K›`ªgyLx ej KvR K‡i, †K› ª̀wegyLx ej k~b¨| G Kvi‡Y Avg f‚-c„‡ô AvQ‡o c‡o| wKš‘ K…wÎg DcMÖ‡ni Ici cÖhy³

vy

v

vx


Teac

hing

bd.co

m



c`v^Æweævb

†K›`ªgyLx ej I †K› ª̀wegyLx ej ci¯úi mgvb nIqvq Zv AvQ‡o

c‡o bv|

M DÏxcK n‡Z cvB,


AveZ©bKvj, T = 24 NÈv

= 24 3600 sec

= 86400 sec


Aÿvsk, = 30

P we›`y‡Z Ae¯’vbiZ e ‘̄i fi, m = 1kg

P we›`y‡Z AwfKl©R Z¡iY = g

P we›`y‡Z e ‘̄i Dci Kvh©Ki AwfKl©R ej = F

Avgiv Rvwb, g = g 2Rcos2

= g

2

T2 R cos2

= 9.8

2

864002 6.4 106 cos2 30

= 9.77 ms2

Avevi, F = mg = 1 9.77 = 9.77 N (Ans.)

N DÏxcK n‡Z cvB,


f‚-c„ô n‡Z K…wÎg DcMÖ‡ni D”PZv, h = 3.2 106m

c„w_exi GKwU c~Y© N~Y©‡bi AveZ©bKvj, T = 24 NÈv|

Rvbv Av‡Q, gnvKl©xq aªæeK, G = 6.7 1011Nm2kg2

c„w_exi fi, M = 6 1024kg

awi, K…wÎg DcMÖ‡ni AveZ©bKvj = T

T = 2 (R+h)3

GM

= 2 (6.4 106 + 3.2 106)3

6.7 1011 6 1024

= 9321.24 sec

= 2.58 hr

†h‡nZz K…wÎg DcMÖ‡ni AveZ©bKvj (T), c„w_exi AvwýK MwZi

AveZ©bKv‡ji (T) mgvb bq| ZvB f‚-c„ô n‡Z K…wÎg DcMÖnwU‡K

w¯’i e‡j g‡b n‡e bv|

4bs cÖ‡kœi DËi

K cÖevnxi ga¨ w`‡q cošÍ e ‘̄i Ici cÖhy³ ejmg~‡ni jwä k~b¨ n‡j, e ‘̄wU †h aªæe †e‡M cÖevnxi ga¨ w`‡q co‡Z _v‡K ZvB

AšÍt‡eM|

L cvwbi AYy I KPz cvZvi AYyi ga¨Kvi AvmÄb ej A‡cÿv cvwbi Abymg~‡ni ga¨Kvi msmw³ ej e„nËi gv‡bi| ZvB KPz cvZvi Mv‡q

cvwb †j‡M _v‡K bv| cÿvšÍ‡i cvwbi AYy I Kv‡Pi AYyi ga¨Kvi

AvmÄb ej A‡cÿv cvwbi AYymg~‡ni ga¨Kvi msmw³ ej ÿz`ªZi

gv‡bi| ZvB Kv‡Pi Mv‡q cvwb †j‡M _v‡K|

M DÏxcK n‡Z cvB,

e¯‘i fi, m = 0.1 kg

Zv‡ii Avw` ˆ`N©¨, L = 0.50m

Zv‡ii cÖ¯’‡”Q‡`i †ÿÎdj, A = 106m2

Zv‡ii Dcv`v‡bi Bqs-Gi ¸YvsK, Y = 2 1011Nm2

Rvbv Av‡Q, AwfKl©R Z¡iY, g = 9.8ms2

Zv‡ii ˆ`N©¨ e„w×, l = ?

Avgiv Rvwb, Y = FLAl

ev, l = mgLYA

= 0.1 9.8 0.50

2 1011 106

= 2.45 106m (Ans.)

N DÏxcK n‡Z cvB,

e¯‘i fi, m = 0.1kg

Zv‡ii ˆ`N©¨ Z_v e„ËvKvi c‡_i e¨vmva©, r = 0.50m

N~Y©b msL¨v, N = 600

mgq, t = 1 min = 60 sec.

myZvi Uvb, F = ?

†KŠwYK †eM n‡j,

F = m2r

= m

2N

t2r

= 0.1

2 3.1416 600

60

2

0.50

= 197.39N

Avevi, Zv‡ii Amn cxob = Amn ej

†¶Îdj


Teac

hing

bd.co

m



c`v^Æweævb

ev, Amn ej = Zv‡ii Amn cxob †ÿÎdj

= 4.8 107 106 = 48N

jÿ Kwi, F>48

AZGe, iZ‡bi N~Y©b msL¨vi aviYv mwVK bq| KviY, N~Y©bmsL¨v

600 r.p.m n‡j ZviwU wQu‡o hv‡e|

5bs cÖ‡kœi DËi

K ỳwU my‡ii K¤úvs‡Ki AbycvZ‡K myi weivg e‡j|

L Avgiv A_©en †hme kã ïwb Zvi †ewkifvMB A‡bK¸‡jv K¤úv‡¼i mgš̂‡q m„wó| †Kv‡bv e¯‘i wbR¯̂ K¤úv¼ Avi Zvi Dci

Av‡ivwcZ ch©ve„Ë ¯ú›`‡bi K¤úv¼ mgvb n‡j e ‘̄wU m‡e©v”P we Í̄vi

mnKv‡i Kw¤úZ nq Ges †Rviv‡jv kã m„wó nq| Zejvq AvNvZ

Ki‡j Zejvi wbR¯^ K¤úv¼ Ges Zvi Dci Av‡ivwcZ ch©ve„Ë

¯ú›`‡bi K¤úv¼ mgvb nq e‡j Zejvq AvNvZ Ki‡j †Rviv‡jv kã

m„wó nq| Aci w`‡K †`qv‡j AvNvZ Ki‡j †`qv‡ji wbR¯^ K¤úv¼

I Zvi Dci Av‡ivwcZ ch©ve„Ë K¤ú‡bi K¤úv¼ mgvb nq bv e‡j

†`qv‡j AvNvZ Ki‡j ZZUv †Rviv‡jv kã m„wó nq bv|

M Avgiv Rvwb,

y = a sint

= a sin

2

T t

= 0.16 sin

2 180

5

DÏxcK n‡Z cvB,

we¯Ívi, a = 0.16m

ch©vqKvj, T = 5 2

= 10 sec

mgq, t = 2 sec

miY, y = ?

= 0.16 sin72

= 0.152 m (Ans.)

N DÏxcK Abymv‡i,

mvg¨ve ’̄vb †_‡K Q we›`yi miY, y = 0.08m

awi, Q we›`y‡Z Aew¯’Z †Kv‡bv KYvi fi = m

†KŠwYK K¤úv¼ =

we¯Ívi, a = 0.16m

Q we›`y‡Z, w¯’wZkw³, Ep = 12 m

2y2

Q we›`y‡Z MwZkw³, Ek = 12 m

2 (a2 y2)

EpEk

= y2

a2y2

= (0.08)2

(0.16)2 (0.08)2 =

6.4 103

0.0192

EpEk

= 13

AZGe, Q we›`y‡Z w¯’wZkw³, MwZkw³i GK-Z…Zxqvsk n‡e|

6bs cÖ‡kœi DËi

K †hme M¨vm mKj ZvcgvÎv I Pv‡c e‡q‡ji m~Î I Pvj©‡mi m~Î c~Y©iƒ‡c †g‡b P‡j Zv‡`i‡K Av`k© M¨vm e‡j|

L w¯’i Pv‡c M¨v‡mi NbZ¡ Gi cig ZvcgvÎvi e¨ Í̄vbycvwZK|

M¨v‡mi NbZ¡ Ges cig ZvcgvÎv T Gi g‡a¨ m¤úK© n‡jv, 1T

| GB mgxKiY n‡Z †`Lv hvq ZvcgvÎv e„w× †c‡j NbZ¡ K‡g|

†jLwPÎwU n‡e wb¤œiƒc

T(K) O

aË‚eK P

(g

/L)

M awi, cvÎ B †Z iwÿZ M¨v‡mi MwZkw³ E

DÏxcK n‡Z, †gvjvi M¨vm aªæeK, R = 8.314 J mol1K1

STP ‡Z ZvcgvÎv, T = 273K

†gvj msL¨v, n =2

Avgiv Rvwb, E = 32 n RT

= 32 × 2 × 8.314 mol

1K1 × 273 K

= 6809.166 J

STP †Z cvÎ B †Z iwÿZ M¨v‡mi MwZkw³ 6809.166 J (Ans.)

N DÏxcK n‡Z cvB, A cv‡Î M¨v‡mi

Pvc, P1 = 42MPa = 42 × 106 Pa

AvqZb, V1 = 103m3

†gvj msL¨v, n = 2 mole

g~j Mo eM©‡eM, Crms1= 1500ms1

Avgiv Rvwb, P1V1 = nRT1

RT1 = P1V1

n = 42 106 × 103

2 = 21 × 103 Jmol1

Crms1= 3 RT1M1

ev, (1500)2 = 3 21 103

M1

M1 = 0.028 kg = 28 gm

A cv‡Î iwÿZ M¨v‡mi MÖvg AvbweK fi = 28 gm


Teac

hing

bd.co

m



c`v^Æweævb

Abyiƒcfv‡e cvIqv hvq, B cv‡Î iwÿZ M¨v‡mi MÖvg AvbweK fi = 30.47 gm

Avgiv Rvwb, bvB‡Uªv‡Rb M¨v‡mi MÖvg AvbweK fi 28 gm Ges A

cv‡Î iwÿZ M¨v‡mi MÖvg AvbweK fiI 28 gm.

AZGe A cv‡Î Rvbv M¨vmwU Av‡Q|

5. PÆMÖvg †evW©-2016

1bs cÖ‡kœi DËi

K †gŠwjK GKK n‡Z †h GKK cvIqv hvq Zv‡K jwä ev †hŠwMK GKK e‡j|

L ỳBwU Amgvb mgRvZxq †f±‡ii jwä k~b¨ n‡Z cv‡i bv|

`yBwU †f±‡ii hw` gvb mgvb nq Ges Zv‡ì w`K hw` wecixZ nq

Z‡e Zv‡ì jwä k~b¨ nq| †hgb: A

+ ( )– A = 0| wKš‘ ̀ yBwU Amgvb wKš‘ mgRvZxq †f±i †hvM Ki‡j Zv‡ì mgwó A_©vr jwä k~b¨ n‡Z

cv‡i bv|

M †`qv Av‡Q,

A

= 2i^ + 2j

^ – k

^

B

= 6i^ – 3j

^ + 2k

^

A = |A

| = (2)2 + (2)2 + (–1)2 = 3

B = |B

| = (6)2 + (–3)2 + (2)2 = 7

A

.B

= (2i^ + 2j

^ – k

^). (6i

^ – 3j

^ + 2k

^)

= 12 – 6 – 2

= 4

Avgiv Rvwb,

A

.B

= ABcos

ev, 4 = 3 7 cos

ev, cos = 4

21

ev, = cos–1

4

21

= 79.02 (cÖvq)

A I B

Gi AšÍM©Z †KvY, = 79.02 (cÖvq)| (Ans.)

N g‡b Kwi,

Gi cwie‡Z© †Kv‡Yi gvb Ki‡j A

Gi Ici B Gi Awf‡ÿc

GK PZz_©vsk n‡e|

†h‡nZz, = 79.02 [(M) Ask n‡Z cÖvß]

A Gi Ici B

Gi Awf‡ÿc,

Bcos = A

.B

|A

|

= 43; [A

.B Ges |A

| Gi gvb (M) n‡Z]

A Gi Ici B

Gi Awf‡ÿ‡ci GK PZz_©vsk =

14

43 =

13

GLb,

Bcos = 13

ev, |B

| cos = 13

ev, 7cos = 13

ev, = cos–1

1

21

= 87.27

†Kv‡Yi gvb 87.27 n‡j A

Gi Ici B Gi Awf‡ÿc c~‡e©i GK

PZz_©vsk n‡e|

Gi gv‡bi cwieZ©b = 87.27 – 79.02

= 8.25

myZivs Gi gvb 8.25 evov‡j A

Gi Dci B Gi Awf‡ÿc c~‡e©i

GK PZz_©vsk n‡e|


K †h mKj ej g~j ev AK…wÎg A_©vr Ab¨ †Kv‡bv ej †_‡K Drcbœ nq bv eis Ab¨vb¨ ej †Kv‡bv bv †Kv‡bv fv‡e G mKj e‡ji cÖKvk

Zv‡ì‡K †gŠwjK ej e‡j|

L GKwU `„p e ‘̄ †Kv‡bv GKwU w¯’i A‡ÿi Pviw`‡K AvewZ©Z n‡Z _vK‡j H A‡ÿi mv‡c‡ÿ e ‘̄wUi RoZvi åvgK ej‡Z Aÿ n‡Z

cÖwZwU KYvi ~̀i‡Z¡i eM© I KYvwUi f‡ii ¸Yd‡ji mgwó †evSvq|

I = mr2

wKš‘ hw` e ‘̄wUi mgMÖ fi GKwU we›`y‡Z †K› ª̀xf‚Z e‡j aiv nq Ges

N~Y© Aÿ mv‡c‡ÿ H we›`y‡Z RoZvi åvgK mgMÖ e ‘̄wUi RoZvi


Teac

hing

bd.co

m



c`v^Æweævb

åvg‡Ki mgvb nq, Z‡e Aÿ n‡Z H we›`yi ̀ ~iZ¡‡K PµMwZi e¨vmva©

e‡j|

I = mr2 = MK2; K = PµMwZi e¨vmva©

K = I

M

RoZvi åvg‡Ki mv‡_ PµMwZi e¨vmv‡a©i m¤úK© n‡jv PµMwZi

e¨vmva© RoZvi åvg‡Ki eM©g~‡ji mgvbycvwZK|

M ỳB cÖv‡šÍi ga¨eZ©x ~̀iZ¡, d = 8m

D”PZv, h = 0.4m

sin = hd

ev, = sin–1h

d

= 2.86

D‡jøwLZ iv Í̄vi e¨vswKs †KvY, = 2.86 (Ans.)

N GLv‡b, D³ iv Í̄vi e¨vswKs †KvY, = 2.86

e¨vmva©, r = 100m

†eM, v = ?

Rvbv Av‡Q, tan = v2

rg

v = rg. tan

v = 7.004 ms–1

D³ iv Í̄vi e¨vswKs †KvY Abyhvqx m‡e©v”P 7.004ms–1 †e‡M H iv Í̄vq Mvox Pvjv‡bv hv‡e| wKš‘ PvjK 30kmh–1 = 8.33ms–1 (>7.004ms–1) †e‡M Mvwo Pvjv‡bvq evmwU Lv‡` c‡o hvq|


K me©wb¤œ †h †e‡M †Kv‡bv e ‘̄‡K Lvov Ic‡ii w`‡K wb‡ÿc Ki‡j Zv Avi c„w_ex‡Z wd‡i Av‡m bv †mB †eM‡K gyw³ †eM e‡j|

L †Kv‡bv KYv GKwU c~Y© Pµ m¤úbœ K‡i Avw` Ae ’̄v‡b wd‡i Avm‡j KYvwUi Ici †h ej Øviv m¤úvw`Z Kv‡Ri cwigvY k~b¨ nq

bv, †mB ej‡K AmsiÿYkxj ej e‡j|

Nl©Y ej me©`v MwZi weiæ‡× wµqv K‡i| ZvB GKwU c~Y© P‡µi

cÖwZwU As‡k Nl©Y e‡ji Øviv K…Z KvR FYvZ¥K, d‡j GKwU c~Y©

P‡µ Nl©Y ej Øviv m¤úvw`Z Kv‡Ri cwigvY KL‡bv k~b¨ n‡Z cv‡i

bv| ZvB Nl©Y ej GKwU AmsiÿYkxj ej|

M GLv‡b, c„w_exi fi, M = 6 1024 kg

e¨vmva©, R = 6.4 106 m

f‚w¯’i DcMÖ‡ni ch©vqKvj, T = 24 hr

= 86400 s

D”PZv, h = ?

Avgiv Rvwb,

h =

GMT

2

42

13

– R

=

6.7 10–11 6 1024 (86400)2

4 (3.1416)2

13

– (6.4 106)

= 3.596 107 m

myZivs, f‚-w¯’i DcMÖnwU 3.596 107m D”PZvq Dr‡ÿcY Ki‡Z n‡e| (Ans.)

N (M) Ask n‡Z cvB,

f‚-w¯’i DcMÖ‡ni D”PZv, h = 3.596 107 m

ch©vqKvj, T = 24h = 86400 s

f‚-w¯’i DcMÖ‡ni †eM, v = ?

Avgiv Rvwb, v = 2T (R + h)

= 2 3.1416

86400 (6.4 106 + 3.596 107)

= 3080.57 ms–1

hw` h Gi gvb wØ¸Y n‡j,

h = 2 3.596 107 m = 7.192 107 m

D”PZv wØ¸Y n‡j cÖ‡qvRbxq †eM, v = ?

v = 2T (R + h)

= 2 3.1416

86400 (6.4 106 + 7.192 107)

= 5695.59 ms–1

†eM e„w×, v = v – v

= 2615.02 ms–1

h Gi gvb wØ¸Y n‡j DcMÖnwUi †eM 2615.02 ms–1 cwigvY

evov‡Z n‡e|

4bs cÖ‡kœi DËi

K †Kv‡bv Zij c„‡ôi Dci hw` GKwU †iLv Kíbv Kiv nq Z‡e H †iLvi cÖwZ GKK ̂ `‡N©¨ †iLvi mv‡_ j¤^fv‡e Ges c„‡ôi ̄ úk©Kiƒ‡c

†iLvi Dfq cv‡k †h ej wµqv K‡i Zv‡K H Zi‡ji c„ôUvb e‡j|

L ¯úk©‡KvY wbf©i K‡i KwVb I Zi‡ji cÖK…wZi Dci| msmw³ ej Zi‡ji Zj‡K Abyf‚wgK ivLvi †Póv K‡i| cÿvšÍ‡i, AvmÄb

d = 8m h

= 0

.4m


Teac

hing

bd.co

m



c`v^Æweævb

ej Zij Zj‡K Dc‡i DVv‡Z †Póv K‡i| Kv‡P ˆZjv³ c`v_©

jvMv‡j Zi‡ji msmw³ ej AvmÄb ej A‡cÿv e„nËi nq| d‡j

¯úk©‡KvY e„w× cvq|

M †`Iqv Av‡Q,

D”PZv, h = 0.73m

ˆKwkK b‡ji e¨vm, d = 0.04mm

e¨vmva©, r = 0.02mm

= 0.02 10–3 m

cvwbi NbZ¡, = 1000 kgm–3

cvwbi ZjUvb,

T = hrg

2cos

= 0.73 0.02 10–3 1000 9.8

2 1 ; [cos = 1]

= 0.07154 N (Ans.)

N g‡b Kwi,

ˆKwkK b‡ji cwiewZ©Z e¨vmva© = rm

cvwbi D”PZv, h = 0.80m

cvwbi ZjUvb, T = 0.07154N [(M) D: n‡Z]

cvwbi NbZ¡, = 1000 kgm–3

T = hrg

2

r = 2T

hg =

2 0.07154

0.80 1000 9.8

= 1.825 10–5m

e¨vmv‡a©i cwieZ©b, r = r – r

= (0.02 10–3 – 1.825 10–5)m

= 1.75 10–6 m

e¨vmv‡a©i cwigvY 1.75 106m Kgv‡bv n‡j cvwbi D”PZv 0.80m

n‡e|

5bs cÖ‡kœi DËi

K †Kv‡bv †`vjbiZ KYvi Z¡iY mvg¨ve ’̄vb †_‡K mi‡Yi mgvbycvwZK I me mgq mvg¨ve ’̄v‡bi AwfgyLx n‡j H KYvi MwZ‡K

mij Qw›`Z MwZ e‡j|

L Zi‡½i ZxeªZv, I = 22n2a2v

Zi‡½i we Í̄v‡ii mv‡_ ZxeªZvi m¤úK© n‡jv, Zi‡½i ZxeªZv Zi‡½i

we¯Ív‡ii e‡M©i mgvbycvwZK| A_©vr Zi‡½i we Í̄vi hZUzKz cwieZ©b

nq, Zvi e‡M©i mgvbycv‡Z ZxeªZv cwiewZ©Z nq|

M Zi½wUi mgxKiY, y1 = 15 sin 26 (100t x)

Zi½wU cÖwZdj‡bi ci cÖwZdwjZ Zi‡½i mgxKiY

y2 = 15 sin 26 (100t + x)

D³ Zi½Øq DcwicvwZZ n‡q w ’̄i Zi½ Drcbœ Ki‡e| w¯’i Zi‡½i

Dci ’̄ †Kv‡bv KYvi jwä miY y n‡j,

y = y1 + y2

= 15 sin 26 (100t x) + 15 sin

26 (100t + x)

= 15 2sin 26 100t cos

26 x

= 30 cos 26 x sin

26 100t

= A sin 26 100t

A_©vr cÖwZdj‡bi ci jwä Zi‡½i mgxKiY

y = A sin 26 100t

†hLv‡b, A = jwä Zi‡½ we Í̄vi = 30cos 26 x

N ÔMÕ Ask n‡Z cvB, jwä w¯’i Zi‡½i mgxKiY, y = A sin 26

100t

†hLv‡b, A = 30 cos 26 x

GB mgxKiY‡K, y = A sin 2

(vt x) mgxKi‡Yi mv‡_ Zzjbv K‡i

cvB, v = 100 ms1 Ges = 6m

f = v

=

1006 Hz

T = 1f =

6100 sec

GLb, x =

2 = 62 = 3m n‡j,

A = 30 cos

2

6 3 = 30

y = 30 sin 26 100t

t Gi wewfbœ gv‡bi Rb¨ y Gi gvb wb‡Pi Q‡K †`qv n‡jv :

t (s) 0 T4 = 0.015

T2 = 0.03

3T4 = 0.045

T = 0.06


Teac

hing

bd.co

m



c`v^Æweævb

y(m) 0 30 0 30 0

cÖvß Z_¨mg~n wb‡P †jLwP‡Îi gva¨‡g Dc¯’vcb Kiv n‡jv :

T

4 3T

4

Y

y(m

)

–30

T t(s)

T

2

O

30

6bs cÖ‡kœi DËi

K mgy`ªc„‡ô 45 Aÿvs‡k 273.15K ZvcgvÎvq Djø¤^fv‡e Aew ’̄Z 0.76m D”PZvwewkó weï× cvi` Í̄¤¢ †h Pvc †`q Zv n‡jv cÖgvY 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 µvwšÍ

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

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

= 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 cvi` Pvc

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

Avgiv Rvwb,

Av‡cwÿK Av`©ªZv, R = fF 100% =

7.5672 103 cvi` Pvc

17.4 103 cvi` Pvc

= 43.49%

Av‡cwÿK Av ©̀ªZv 43.49%| ZvB ejv hvq H w`b H ’̄v‡bi AvenvIqv ï®‹ I †iŠ‡ ª̀v¾¡j _vK‡e|

6. wm‡jU †evW©-2016

1bs cÖ‡kœi DËi

K hv †Kv‡bv AN~Y©bkxj e ‘̄‡Z NyY©b m„wó K‡i ev NyY©vqgvb e ‘̄i †KŠwYK †e‡Mi cwieZ©b K‡i Zv‡K UK© e‡j|

L î . î = 0 bq 1|

î Ges î Gi ga¨eZ©x †KvY 0

î . î = 1 1 cos0 = 1 1 1 = 1|

M †`Iqv Av‡Q, Abyf‚wg‡Ki mv‡_ †KvY, = 45

cÖhy³ ej, F = 20N

Abyf‚wgK Dcvsk = F cos

= 20cos 45

= 20

2 N

= 10 2 N (Ans.)

N wPÎ †_‡K ¯úó †h, 1 > 2

cos1 < cos2

Fcos1 < Fcos2

L e¨w³ mn‡RB †bŠKvwU Pvjv‡Z cvi‡e|

2bs cÖ‡kœi DËi

2

F F 1

L e¨w³

K e¨w³


Teac

hing

bd.co

m



c`v^Æweævb

K hLb †Kv‡bv Zi½ we Í̄…Z gva¨‡gi ga¨ w`‡q µgvMZ AMÖmi nq ZLb Zv‡K AMÖMvgx Zi½ e‡j|

L †Kv‡bv ¯̂‡i wewfbœ K¤úvs‡Ki myi _v‡K| G‡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 †P‡q †ewk Zv‡ì‡K Dcmyi e‡j| Avevi Dcmyi¸‡jvi

K¤úv¼ hw` g~j my‡ii K¤úv‡¼i mij ¸wYZK nq Zvn‡j †mB mKj

Dcmyi‡K e‡j mg‡gj ev nvi‡gvwbK| myZivs ejv hvq, mKj

nvi‡gvwbKB Dcmyi wKš‘ mKj Dcmyi nvi‡gvwbK bv|

M †Ìqv Av‡Q,

†m‡KÛ †`vj‡Ki †`vjbKvj, T = 2s

K AÂ‡ji AwfKl©R Z¡iY, gK = 9.78ms2

Avgiv Rvwb, T = 2 L

g K

ev, T2 = 42 L

g K

ev, L = g K T2

42

L = 0.9909 m (Ans.)

N †Ìqv Av‡Q, L AÂ‡ji AwfKl©R Z¡iY, gL = 9.83ms2

M n‡Z cvB, LK = 0. 9909m, TK = 2s

Avgiv Rvwb, T = 2 Lg

ev,

2 Lg

2

= T2

ev, g = 42LT2

K AÂ‡ji Rb¨, gK = 42L

T2K ................. (i)

L AÂ‡ji Rb¨, gL = 42L

T2L .................... (ii)

(ii) †K (i) w`‡q fvM K‡i,

gLgK

= T2K T2L

ev, gL = T2K T2L

gK

ev, T2L = gKgL

T2K

= 9.789.83 2

= 3.979 s

TL = 1.995 s

TL < TK

†`vjbKvj K‡g hv‡e|

3bs cÖ‡kœi DËi

K GKwU †f±i‡K hw` ỳB ev Z‡ZvwaK †f±‡i Ggbfv‡e wef³ Kiv nq hv‡ì jwä g~j †f±‡ii mgvb nq, Z‡e GB wef³KiY

cÖwµqv‡K †f±i wefvRb e‡j|

L `ywU e¯‘i g‡a¨ me©`v AvKl©Y ej we`¨gvb _vKvq GKK f‡ii e¯‘‡K e„nr fim¤úbœ e ‘̄i w`‡K wb‡Z ewntkw³ ev evB‡ii †Kv‡bv

G‡R›U‡K cÖK…Zc‡ÿ †Kv‡bv KvR Ki‡Z nq bv| ewnt ’̄ G‡R›U

KZ…©K K…Z KvR abvZ¥K| †h‡nZz G‡ÿ‡Î ewnt ’̄ G‡R›U‡K †Kv‡bv

KvR Ki‡Z nq bv| myZivs G‡ÿ‡Î m¤úbœ KvR n‡e FYvZ¥K|

Kv‡RB †Kv‡bv we› ỳ‡Z GKwU e ‘̄ ev e ‘̄ mgwó KZ…©K m„ó gnvKl©xq

wef‡ei gvb me©`v FYvZ¥K nq|

M ỳcy‡i Av‡cwÿK Av ª̀Zv, R = 75%

wkwkivs‡K Rjxq ev®úPvc, = 9.22 103m Hg

evqyi ZvcgvÎvq m¤ú„³ evqyi Pvc, F = ?

Avgiv Rvwb, Av‡cwÿK Av ª̀Zv,

R = fF 100%

ev, 75 = fF 100

ev, F = f

75 100 = 9.22 103m Hg

75 100

= 12.29 103m Hg (Ans.)

N †Ìqv Av‡Q,

20C ZvcgvÎvq m¤ú„³ ev®úPvc = 17.54 103m Hg

10C ZvcgvÎvq m¤ú„³ ev®úPvc = 9.22 103m Hg

mÜ¨vq Av‡cwÿK Av ª̀©Zv = 9.22 103

17.54 103 100%

= 52.565%

52.565 < 75

Av‡cwÿK Av ª̀©Zv K‡g‡Q|

GRb¨ ZvovZvwo Nvg ïKvw”Qj|

4bs cÖ‡kœi DËi

K †Kv‡bv e ‘̄i Dci cÖhy³ ej Øviv K…ZKvR e ‘̄i MwZ kw³i cwieZ©‡bi mgvb|


Teac

hing

bd.co

m



c`v^Æweævb

L GKwU fvwi AvqZbnxb e ‘̄KYv‡K IRbnxb bgbxq I AcÖmviYkxj myZv w`‡q Szwj‡q w`‡j hw` GwU Nl©Y Gwo‡q ỳj‡Z

cv‡i Z‡e Zv‡K mij †`vjK e‡j| †Kv‡bv mij †`vj‡Ki

†`vjbKvj wbw`©ó bq|

wKš‘ †m‡KÛ †`vj‡Ki †`vjbKvj wbw`©ó Ges Zv ỳB †m‡KÛ A_©vr

mKj †m‡KÛ †`vjK mij †`vjK| wKš‘ mKj mij †`vjK †m‡KÛ

†`vjK bq|

M †Ìqv Av‡Q, d = 4mm = 4 103 m

r = 2 103 m

= 4 102 ms1

= 7800 kgm3

= 800 kgm3

Avgiv Rvwb,

= 2r2( )g

9

= 2 (2 103)2 (7800 800) 9.8

9 4 102 Nms2

= 1.5244Nms2

Avevi, F = 6rv

= (6 3.1416 1.5244 2 103 4 102)N

= 2.29987 103N (Ans.)

N †Ìqv Av‡Q, †jvnvi NbZ¡, = 7800 kgm3

wMømvwi‡bi NbZ¡, = 1250 kgm3

wMømvwi‡bi mv›`ªZvsK, = 1.6 Nms2

e¨vmva©, r = 2 103 m

cÖvšÍ‡eM, v = ?

v = 2r2( )g

9

= 2 (2 103)2 (7800 1250) 9.8

9 1.6

= 3.56 102ms1

= 3.65 102 < 4 102

wiwgi aviYv mwVK bq|


K †Kv‡bv hš¿ KZ…©K K…ZKvR Ges H mgq mieivnK…Z kw³i AbycvZ‡K H h‡š¿i Kg©`ÿZv e‡j|

L †h‡Kv‡bv Zi‡ji †MvjvKvi Ae ’̄vq c„‡ôi †ÿÎdj me©wb¤œ nq| Avi c„‡ôi †ÿÎdj me©wb¤œ nIqvi A_© n‡jv c„ôkw³ me©wb¤œ| c„ô

kw³ me©wb¤œ n‡j †mUv †ewk w ’̄wZkxj _vK‡e| GRb¨ e„wói †duvUv

†MvjvKvi AvKvi aviY K‡i|

M †Ìqv Av‡Q, e¨vmva©, r = 200m

†eM, v = 60 kmh1

= 60 1000

3600 ms1 =

503 ms

1

e¨vwKs †KvY, = ?

Avgiv Rvwb, tan = 2

rg

= (50/3)2

200 9.8

= 0.1417

= 8.06 (Ans.)

N †Ìqv Av‡Q, DÏxc‡Ki e¨w³wUi †eM 50 kmh1 ev 13.88ms1 GB †e‡M Mvwo Pvjv‡j 200m e¨vmv‡a© e„ËvKvi †gvo †bIqvi Rb¨

e¨vswKs †KvY cÖ‡qvRb tan1(13.88)2

200 9.8 ev 5.6

M n‡Z cvB D³ iv Í̄vi e¨vswKs †KvY 8.06

5.6 < 8.06

A_©vr 50 km/h †e‡M †gvo wb‡jI †Kv‡bv ̀ yN©Ubv NUvi m¤¢vebv †bB|

DÏxc‡Ki †e‡M Mvwo Pvjv‡jI PvjK wbivc‡` †gvo wb‡Z

cvi‡e|


K †h mKj GKK †gŠwjK GKK mgš^‡q MwVZ nq Zv‡ì‡K jä GKK ev †hŠwMK GKK e‡j|

L †Ìqv Av‡Q, A I

B Gi gaëZ©x †KvY 45

A .

B = AB cos 45

= AB

2

| A

B | = |AB (sin 45)|

= AB

2

A .

B = |

A

B | [†`Lv‡bv n‡jv]

M †Ìqv Av‡Q, wUwfi k‡ãi ZxeªZv, I1 = 1 106 Wm2

cÖgvY ZxeªZv, Io = 1 1012 Wm2


Teac

hing

bd.co

m



c`v^Æweævb

ZxeªZv †j‡f‡ji cwieZ©b, = ?

Avgiv Rvwb, = 10 log I1Io

= 10 log 1 106

1 1012

= 60 dB

bvwdm ZxeªZv †j‡fj e„w× K‡iwQj, = (78 60) = 18 dB (Ans.)

N †Ìqv Av‡Q,

†eøÛv‡ii ZxeªZv †j‡fj, 1 = 85 dB

wUwfi ZxeªZv †j‡fj, 2 = 78 dB

aiv hvK, †eøÛv‡ii ZxeªZv = I1

Ges wUwfi ZxeªZv = I2

Avgiv Rvwb, 1 = 10 log I1Io

ev, 85 = 10 log I1

1012

ev, I1

1012 = 108.5

ev, I1 = 108.5 1012

I1 = 103.5

Avevi, 2 = 10 log I2Io

ev, 78 = 10 log I2

1012

ev, I2

1012 = 107.8

ev, I2 = 107.8 1012

I2 = 104.2

†gvU ZxeªZv, I = I1 + I2 = 103.5 + 104.2

= 3.79 104

ZxeªZv †j‡fj, = 10 log IIo

= 10 log 3.79 104

1012

= 10 log 3.79 108 = 85.79 dB

ZxeªZvi †j‡fj 120 dB n‡j Avgv‡ì Kv‡b kã kÖæwZ hš¿Yvi m„wó K‡i| cÖ̀ Ë †ÿ‡Î MvwYwZKfv‡e Avgiv †`L‡Z cvB †gvU ZxeªZv = 85.79 dB < 120 dB|

†eøÛvi Pvjy Ae ’̄vq mw¤§wjZ ZxeªZv †j‡fj A¯^w¯ÍKi n‡e bv|

7. h‡kvi †evW©-2016

1bs cÖ‡kœi DËi

K wÎgvwÎK ’̄vbv¼ eë ’̄vq wZbwU abvZ¥K Aÿ eivei †h wZbwU GKK †f±i we‡ePbv Kiv nq, Zv‡ì‡K AvqZ GKK †f±i e‡j|

L cÖv‡mi †eM mgZ¡i‡Y wØ-gvwÎK MwZi GKwU DrK…ó D`vniY|

g‡b Kwi, f‚wgi Dci¯’ O we›`y †_‡K vo †e‡M Abyf‚wg‡Ki mv‡_ o †Kv‡Y GKwU cÖvm‡K wb‡ÿc Kiv n‡jv| x I y Aÿ eivei Avw`‡e‡Mi Dcvsk¸‡jv n‡jv h_vµ‡g

vxo = vocoso

vyo = vosino

e¯‘wU t †m‡K‡Û P Ae ’̄v‡b †cuŠQv‡j Zvi †eM v Gi Abyf‚wgK I

Djø¤^ Dcvsk h_vµ‡g,

vx = vxo = vocoso Ges

vy = vyo – gt = vosino – gt

vo

vyj^

vxi^ P

y

x o

O


Teac

hing

bd.co

m



c`v^Æweævb

myZivs t mg‡q ev P Ae ’̄v‡b, cÖv‡mi †eM v Gi gvb n‡jv | v

| = v

= vx2 + vy2 Ges †eM v †h‡nZz x Aÿ Z_v Abyf‚wg‡Ki mv‡_

†KvY Drcbœ K‡i, myZivs

tan = vyvx

M DÏxcK n‡Z cvB,

e„wói †eM, v = 6 kmh–1

QvZv I Djø‡¤î gaëZ©x †KvY, = 33.8

cvq †nu‡U Pjv e¨w³i †eM, u = ?

wPÎ n‡Z cvB,

tan = uv

ev, u = vtan

= 6 tan 33.8

= 4kmh–1 (Ans.)

N mvB‡K‡j Pjv e¨w³i QvZv I Djø‡¤î mv‡_-

Drcbœ †KvY, = 53.06

e„wói †eM, v = 6kmh–1

mvB‡K‡j Pjv e¨w³i †eM, u = ?

wPÎ n‡Z cvB,

tan = uv

ev, u = vtan

= 6 tan53.06

= 7.98 kmh–1

ÔMÕ Ask n‡Z cvB, cv‡q nuvUv e¨w³i †eM, u = 4kmh–1

†h‡nZz e¨w³Ø‡qi †eM GK bq, †mKvi‡Y e„wó †_‡K iÿv cvIqvi

Rb¨ e¨w³Ø‡qi wfbœ †Kv‡Y QvZv ai‡Z n‡qwQj|

2bs cÖ‡kœi DËi

K †h †Kv‡bv mgq eëav‡b †Kv‡bv e¯‘i †gvU miY‡K H mgq eëavb w`‡q fvM Ki‡j †h ivwk cvIqv hvq Zv‡KB e ‘̄wUi Mo †eM

e‡j|

L NvZ e‡ji ZviZ‡gï Kvi‡Y Kuv‡P ¸wj Ki‡j wQ ª̀ nq wKš‘ wXj Qyo‡j KuvP P‚Y© wePzY© nq| Lye Kg mg‡qi Rb¨ NvZ ej cÖhy³ nq|

Kuv‡P ¸wj Ki‡j ¸wj KZ©„K cÖhy³ ej F, Kuv‡Pi fi‡eM cwieZ©b K‡i| †h mgq a‡i KuvP ¸wji ms¯ú‡k© _v‡K †m mg‡q ¸wj KZ©„K

cÖhy³ ej Ab¨vb¨ e‡ji Zzjbvq A‡bK eo nq Ges ¸wjwU KuvP wQ ª̀

K‡i †ei n‡q hvq| wKš‘ wXj Gi fi‡eM Ges wµqvKvj †ewk nIqvq

Kuv‡P cÖhy³ ej Pviw`‡K Qwo‡q wM‡q KuvP‡K P‚Y© weP‚Y© K‡i|

M DÏxcK n‡Z cvB,

wb‡ÿcY †eM, vo = 30ms–1

wb‡ÿcY †KvY, o = 60

Rvbv Av‡Q, AwfKl©R Z¡iY, g = 9.8ms–2

cvjøv, R = ?

Avgiv Rvwb,

R = vo2 sin2o

g

= (30)2 sin(2 60)

9.8

= 79.53m (Ans.)

N DÏxcK n‡Z cvB,

wb‡ÿcY †eM, vo = 30ms–1

wb‡ÿcY †KvY, o = 60

Rvbv Av‡Q, AwfKl©R Z¡iY, g = 9.8 ms–2

†`qv‡ji D”PZv, h = 25m

Abyf‚wgK ~̀iZ¡, x = 20m

awi, Djø¤^ ~̀iZ¡ = y

Avgiv Rvwb,

y = x tano – gx2

2(vocoso)2

= 20 tan 60 – 9.8 (20)2

2(30cos60)2

= 34.64 – 8.71

= 25.93m

†h‡nZz y > h, †m‡nZz cÖvmwU †`qvj AwZµg Ki‡Z cvi‡e|


K hLb †Kv‡bv eë ’̄vi Ici cÖhy³ wbU evwn¨K ej k~b¨ nq, ZLb eë ’̄vwUi †gvU fi‡eM msiwÿZ _v‡K|

L †Kv‡bv †gvUi ev †ijMvwo hLb euvK †bq ZLb G euvKv c‡_ Nyivi Rb¨ GKwU †K›`ªgyLx e‡ji cÖ‡qvRb nq| G †K› ª̀gyLx ej

cvIqv bv †M‡j Mvwo RoZvi Kvi‡Y euvKv c‡_i ¯úk©K eivei P‡j

hv‡e| A‡bK mgq Mvwo D‡ë c‡o hvq| mgZj c‡_ euvK †bIqvi

mgq Mvwoi PvKv I iv Í̄vi gaëZ©x Nl©Y ej G †K›`ªgyLx e‡ji

†hvMvb †`q| wKš‘ Nl©Y e‡ji gvb Z_v †K› ª̀gyLx e‡ji gvb Lye Kg

nIqvq Mvwo †ewk †Rv‡i euvK wb‡Z cv‡i bv| †ewk †Rv‡i euvK wb‡Z

†M‡j †K›`ªgyLx ej Z_v Nl©Y e‡ji gvb evov‡Z n‡e| Avi †m Rb¨

euv‡Ki gy‡L iv Í̄vi Zj‡K Abyf‚wgK Z‡ji mv‡_ †nwj‡q ivL‡Z nq|

ZvB iv Í̄vi ev‡Ki wfZ‡ii cÖvšÍ †_‡K evB‡ii cÖvšÍ DuPz _v‡K|

u –u

v = 6kmh–1

u –u

v = 6kmh–1

Vy V

Vx


Teac

hing

bd.co

m



c`v^Æweævb

M DÏxcK n‡Z cvB, e ‘̄i fi, m = 8kg

N~Y©b Aÿ n‡Z e ‘̄wUi j¤^ `~iZ¡, r = 0.2m

†KŠwYK †eM, = 2rads–1

awi, RoZvi åvgK = I Ges †KŠwYK fi‡eM = L

Avgiv Rvwb,

L = I

= mr2

= 8 (0.2)2 2

= 0.64 kgm2s–1 (Ans.)

N DÏxcK Abymv‡i,

e¯‘i cÖv_wgK fi, m1 = 8kg

e ‘̄i cwiewZ©Z fi, m2 = 82 = 4kg

N~Y©b Aÿ †_‡K e ‘̄i ~̀iZ¡, r = 0.2m

awi, †KŠwYK Z¡iY = rad s–2

cÖv_wgK UK©, 1 = I1 = m1r2 = 8 (0.2)2 = 0.32 Nm

cwiewZ©Z UK©, 2 = I2 = m2r2 = 4 (0.2)2 = 0.16 Nm

ev, 2

1 =

12

2 = 12 1

AZGe, e ‘̄wUi fi A‡a©K Kiv n‡j UK© A‡a©K n‡q hv‡e|

4bs cÖ‡kœi DËi

K GKK f‡ii ỳwU e¯‘KYv GKK ~̀i‡Z¡ †_‡K †h e‡j ci¯úi‡K AvKl©Y K‡i Zv‡K gnvKl©xq aªæeK e‡j|

L g½j MÖ‡n †Kv‡bv e ‘̄i gyw³‡eM 4.77 kms–1 ej‡Z eySvq g½j MÖ‡ni c„ô n‡Z 4.77 kms–1 †e‡M †Kv‡bv e ‘̄‡K Lvov Dc‡ii w`‡K wb‡ÿc Ki‡j Zv Avi g½jMÖ‡n wd‡i Av‡m bv| A_©vr e ‘̄wU g½j

MÖ‡ni AvKl©Y KvwU‡q gnvk~‡b¨ P‡j hv‡e|

M

DÏxcK n‡Z cvB, Mvwoi fi, m = 250 kg

Djø‡¤^i mv‡_ Z‡ji AvbwZ, = 66.42

Avw`‡eM, v0 = 12.393 ms–1

miY, s = 30m

Zj eivei Mvwoi IR‡bi Dcvsk,

Fg = mg cos

= 250 9.8 cos66.42

= 980.07N

MvwowU _vgv‡Z evav`vbKvix e‡ji gvb F n‡j,

g›`b, a = F – Fg

m

Avgiv Rvwb,

v2 = v20 – 2as

ev, 0 = v20 – 2as

ev, 2as = v20

ev, 2.F – Fg

m .s = v2

0

ev, F – Fg = mv20

2s

ev, F = Fg + mv20

2s

ev, F = 980.07 + 250 (12.393)2

2 30

F = 1620 N (Ans.)

N DÏxcK Abymv‡i,

f‚wg n‡Z Z‡ji kxl©we›`yi D”PZv, h = s cos

= 30cos66.42

= 12m

AvbZ Z‡ji kxl© we›`y‡Z Mvwoi †eM, v0 = 12.393 ms–1

AvbZ Z‡ji kxl© we›`y‡Z MwZkw³,

Ek = 12 mv0

2

= 12 250 (12.393)

2

= 19198.306 J

h

30 m

mgcos mg


Teac

hing

bd.co

m



c`v^Æweævb

AvbZ Z‡ji kxl© we›`y‡Z wefekw³,

Ep = mgh

= 250 9.8 12

= 29,400 J

AvbZ Z‡ji kxl© we›`y‡Z †gvU kw³, E = Ek + Ep

E1 = (19198.306 + 29,400)

= 48598.306

= 48600 J

AvbZ Zj eivei 30m ~̀iZ¡ AwZµg Kivi ci †eM, v = 0

AZGe, AvbZ Z‡ji wb¤œ we›`y‡Z MwZkw³,

Ek = 12 mv

2

= 12 m(0)

2

= 0

Avevi, AvbZ Z‡ji wb¤œ we›`y‡Z h = 0

wefe kw³, Ep = mg 0 = 0

Mvwoi Dci K…Z KvR, Z_v e¨wqZ kw³,

W = Fs

= 1620 30

= 48600 J

AvbZ Z‡ji wb¤œ we›`y‡Z †gvU kw³,

E2 = 0 + 0 + 48600

= 48600 J

†h‡nZz E1 = E2

†m Kvi‡Y DÏxc‡K msiÿYkxjZvi bxwZ iwÿZ n‡e|

5 bs cÖ‡kœi DËiW

K hw` †Kv‡bv e ‘̄i Z¡iY GKwU wbw`©ó we›`y †_‡K Gi mi‡Yi mgvbycvwZK Ges me©`v H we›`y AwfgyLx nq, Zvn‡j e ‘̄i GB MwZ‡K

mij Qw›`Z MwZ e‡j|

L mij †`vj‡Ki †KŠwYK we Í̄vi 4 Gi †ewk bv n‡j mij †`vj‡Ki MwZc_ mij‰iwLK nq| †m‡ÿ‡Î, mij †`vj‡Ki Z¡i‡Yi

mgxKiY nq a = – 2x ev a – x|

A_©vr Z¡iY mi‡Yi mgvbycvwZK I wecixZgyLx, hv mij Qw›`Z MwZi

ˆewkó¨ cÖKvk K‡i| G Kvi‡Y mij †`vj‡Ki MwZ mij Qw›`Z MwZ|

M DÏxc‡K cÖ`Ë AMÖMvgx Zi‡½i mgxKiY,

y = 0.5sin (200t – 0.602x) †K AMÖMvgx Zi‡½i cÖwgZ mgxKiY,

y = asin

t –

2x

Gi mv‡_ Zzjbv K‡i cvB,

2

= 0.602

ev, = 2

0.602 = 3.322m (Ans.)

Avevi, = 200

ev, 2f = 200

f = 100 Hz

Avevi,

Avgiv Rvwb,

v = f = 100 3.322 = 332.22 ms–1 (Ans.)

N ÔMÕ Ask n‡Z cvB,

1g gva¨‡gi K¤úvsK, f1 = 100 Hz

1g gva¨‡gi Zi½ †eM, v1 = 332.22ms–1

1g gva¨‡gi Zi½ ˆ`N©¨, 1 = 3.322m

gva¨gØ‡q k‡ãi Zi½‰`‡N©¨i cv_©K¨, = 0.2m

2q gva¨‡g K¤úvsK, f2 = f1 = 100Hz

GLb,

= 2 – 1

ev, 0.2 = 2 – 1

ev, 2 = 0.2 + 3.322

= 3.522m

Avevi,

v2 = f22

= 100 3.522

= 352.2 ms–1

†eM e„w×, v = v2 – v1

= (352.2 – 332.22) ms–1

= 19.98 ms–1

AZGes wØZxq gva¨‡g Zi½‡eM cÖ_g gva¨‡gi †P‡q 19.98 ms–1 e„w× cv‡e|

6bs cÖ‡kœi DËi

K w¯’wZ¯’vcK mxgvi g‡a¨ e ‘̄i cxob Gi weK…wZi mgvbycvwZK|


Teac

hing

bd.co

m



c`v^Æweævb

L Bqs Gi ¸YvsK, Y = 2 1011 Nm–2 ej‡Z †evSvq 1m2 cȪ ’‡”Q‡ì †ÿÎdj wewkó †Kv‡bv w¯’wZ¯’vcK c`v‡_©i ˆ`N©¨ eivei

2 1011 N ej cÖ‡qvM Kiv n‡j Gi ˆ`N©¨ e„w× Avw` ˆ`‡N©ï mgvb n‡e|

M DÏxcK n‡Z cvB,

A cv‡Îi M¨v‡mi Pvc, P = 4 105 Nm–2

A cv‡Îi AvqZb, V = 3cm3 = 3 10–6 m3

A cv‡Îi M¨v‡mi MwZkw³, E = ?

Avgiv Rvwb,

E = 32 PV

= 32 4 10

5 3 10–6

= 1.8 J (Ans.)

N DÏxcK Abymv‡i,

A cv‡Îi M¨v‡mi Pvc, P1 = 4 105 Nm–2

B cv‡Îi M¨v‡mi Pvc, P2 = 4.7 105 Nm–2

A cv‡Îi AvqZb = B cv‡Îi AvqZb = V

awi, †gvjvi M¨vm aªæeK = R

cÖwZwU cv‡Îi M¨v‡mi †gvj msL¨v = n

A cv‡Îi M¨v‡mi ZvcgvÎv = T1

B cv‡Îi M¨v‡mi ZvcgvÎv = T2

Avgiv Rvwb,

P1V1 = nRT1 .......... (i)

Ges P2V2 = nRT2 .......... (ii)

(i) (ii) bs mgxKiY n‡Z cvB,

P1V1P2V2

= nRT1nRT2

ev, 4 105 V

4.7 105 V =

T1T2

ev, T2 = 1.175 T1

¯úóZB T2 > T1

AZGe, B cvÎwU †ewk DËß n‡e|

8. ewikvj †evW©-2016

1bs cÖ‡kœi DËi

K Lye Aí mg‡qi Rb¨ Lye eo gv‡bi †h ej †Kv‡bv e ‘̄i Dci cÖhy³ nq Zv‡K NvZ ej e‡j|

L GKwU BwÄ‡bi Kg©`ÿZv 60% ej‡Z eySvq, hw` GB BwÄ‡b 100J kw³ †`qv nq Zvn‡j †mB BwÄb †_‡K cÖvß †gvU Kvh©Ki kw³ n‡e 60 J|

M †Ìqv Av‡Q, A = î ĵ + k̂

Ges B = 2 î 3 ĵ + 6 k̂

Avgiv Rvwb, A .

B = AB cos1

cos1 =

A .

B

AB

GLb, A .

B = Ax Bx + AyBy + AzBz

= 1 2 + ( 1) ( 3) + 1 6

= 2 + 3 + 6

= 11

Ges A = Ax2 + Ay2 + Ax2 = 12 + ( 1)2 + 12 = 3

B = Bx2 + By2 + Bz2 = 22 + ( 3)2 + 62 = 49 = 7

cos1 = 11

7 3 = 0.9073

1 = cos1(0.9073) = 24.87 (Ans.)

N DÏxc‡Ki ÔMÕ bs cÖ‡kœi Av‡jv‡K Avgiv 1 Gi gvb cvB 24.87|

Avevi wPÎ2 †_‡K cvB,

2

A +

B =

Q (a

wi)

P (awi )

Q (a

wi)

A B

GLv‡b, P =

A +

B = ( î ĵ + k̂) + (2 î 3 ĵ + 6k̂)

= 3 î 4 ĵ + 7 k̂

Ges Q =

A

B = ( î ĵ + k̂) – (2 î 3 ĵ + 6k̂)


Teac

hing

bd.co

m


= î ĵ + k̂ 2 î + 3 ĵ 6k̂

= î + 2 ĵ 5k̂

GLb, P .

Q = P Q cos2 cos2 =

P .

Q

PQ

P .

Q = 3 ( 1) + ( 4) (2) + 7 ( 5)

= 3 8 35

= 46

P = 32 + ( 4)2 + 72

= 74

Q = ( 1)2 + 22 + ( 5)2

= 30

cos2 = 46

74 30

= 0.9763

2 = cos1 ( 0.9763) = 167.5

Kv‡RB 2 > 1

MvwYwZK we‡køl‡Yi gva¨‡g †`Lv hvq †h 2 = 2 nIqv m¤¢e bq|

2bs cÖ‡kœi DËi

K KvR m¤úv`bKvix †Kv‡bv e¨w³ ev h‡š¿i KvR Kivi nvi ev kw³ mieiv‡ni nvi‡K ÿgZv e‡j|

L ¯ú›`b mij c‡_ I e„ËvKvi c‡_ n‡Z cv‡i| mijc‡_ ¯ú›`b n‡j Zv‡K mij Qw›`Z ¯ú›`b e‡j|

myZivs ejv hvq, mKj mij Qw›`Z ¯ú›`bB ch©vqe„Ë ¯ú›`b wKš‘

mKj ch©vqe„Ë ¯ú›`b mij Qw›`Z ¯ú›`b bq|

M †Ìqv Av‡Q, Avw`‡eM, o = 20 ms1

wb‡ÿcY †KvY, = 30

ejwU †h mgq k~‡b¨ _vK‡e Zv Zvi wePiY Kvj, T Gi mgvb n‡e|

Avgiv Rvwb,

wePiYKvj, T = 2o sin

g [ g = AwfKl©R Z¡iY = 9.8 ms2]

= 2 20 sin 30

9.8

= 2.04 sec (Ans.)

N †Ìqv Av‡Q, iæ‡ej e¨vUmg¨vb n‡Z 60m `~‡i i‡q‡Q| K¨vP aivi Rb¨ iæ‡ej‡K Aek¨B ejwU f‚wg ¯úk© Kivi c~‡e© ejwUi

cÖ‡ÿcY mxgv ev cvjøvi g‡a¨ †cuŠQv‡Z n‡e|

Avgiv Rvwb, cÖ‡ÿcY mxgv ev cvjøv, R = o2

g sin2

= 202

9.8 sin (2 30)

= 202

9.8 sin 60

= 35.35 m

Avevi, †h‡nZz ejwUi wePiY Kvj 2.04 sec ZvB K¨vP ai‡Z n‡j

iæ‡ej‡K 2.04 sec Gi g‡a¨ (60 35.35) ev, 24.65 ~̀iZ¡ AwZµg

Ki‡Z n‡e|

†Ìqv Av‡Q, iæ‡e‡ji †eM, R = 8 ms1

Zvn‡j, 2.04 sec G Zvi AwZµvšÍ `~iZ¡ d n‡j,

d = 8 2.04 sec

= 16.32 m

†`Lv hv‡”Q †h, ejwU f‚wg ¯úk© Kivi c~‡e© iæ‡ej ejwUi Ae ’̄v‡b

†cuŠQv‡Z cvi‡e bv| ZvB ejv hvq, iæ‡e‡ji c‡ÿ K¨vPwU aiv m¤¢e

bq|


K evB‡i †_‡K ej cÖ‡qv‡Mi d‡j †Kv‡bv e ‘̄i AvKvi ev ˆ`N©¨ ev AvqZ‡bi cwieZ©b NU‡j w¯’wZ¯’vcKZvi Rb¨ e ‘̄i †fZi †_‡K

GKwU cÖwZwµqv e‡ji D™¢e nq| e ‘̄i GKK †ÿÎd‡ji Dci

j¤^fv‡e D™¢‚Z GB weK…wZ cÖwZ‡ivaKvix e‡ji gvb‡K cxob e‡j|

L e„wói †duvUv cZ‡bi mgq †MvjvKvi AvKvi aviY K‡i cvwbi c„ôUv‡bi R‡b¨| c„ôUv‡bi Rb¨ e„wói †duvUvwU Ggb GKwU AvKv‡i

_vK‡Z Pvq †hLv‡b Zvi c„‡ôi †ÿÎdj me©v‡cÿv Kg nq| c„‡ôi

†ÿÎdj me©wb¤œ Kivi Rb¨B e„wói †duvUv †MvjvKvi AvKvi aviY

K‡i|

M †Ìqv Av‡Q, c„w_exi fi, M = 5.98 1024 kg

gnvKl©xq aªæeK, G = 6.67 1011 Nm2 kg2

c„w_exi e¨vmva©, R = 6400 km

= 6400 103 m

K…wÎg DcMÖ‡ni D”PZv, h = 650 km

= 650 103 m

Avgiv Rvwb, K…wÎg DcMÖ‡ni †eM,

= GM

(R + h)


Teac

hing

bd.co

m


= 6.67 1011 5.98 1024

(6400 103 + 650 103)

= 7521.75 ms1

DÏxc‡K K…wÎg DcMÖnwUi †eM n‡e 7521.75 ms1 (Ans.)

N †h‡nZz D”PZv evo‡j DcMÖnwU‡K †ewk c_ cÖ`wÿY Ki‡Z n‡e ZvB mvaviYfv‡e ejv hvq D”PZv e„w× †c‡j DcMÖnwUi ch©vqKvjI

e„w× cv‡e| wb‡P MvwYwZK we‡kølYwU †`qv n‡jv

awi, DÏxc‡Ki K…wÎg DcMÖnwUi AveZ©bKvj T Ges GwU hw` 700

km Dc‡i n‡Zv Zvn‡j Zvi AveZ©bKvj n‡Zv T'|

Avgiv Rvwb, T = 42(h + R)3

GM

T = 4 (3.1416)2 (650 103 + 6400 103)3

6.67 1011 5.98 1024

= 5889.13 sec

GLb, hw` h = 700 km ev 700 103m nq

Zvn‡j, bZzb AveZ©bKvj,

T' = 42 (700 103 + 6400 103)3

6.67 1011 5.98 1024

= 5951.90 sec

†`Lv hv‡”Q †h, DcMÖnwU hw` 700 km Dc‡i n‡Zv Z‡e Zvi bZzb

AveZ©bKvj T' c~‡e©i AveZ©bKvj T n‡Z (5951.90 5889.13) sec

ev, 62.77 sec †ewk n‡Zv|

4bs cÖ‡kœi DËi

K †h mKj ivwk ¯̂vaxb, Ab¨ †Kv‡bv ivwki Dci wbf©i K‡i bv Zv‡K †gŠwjK ivwk e‡j|

L euvK †bqv iv Í̄vq Pjgvb Mvwoi †ÿ‡Î Avgiv Rvwb,

tan = 2

rg

G mgxKiY †_‡K †`Lv hvq †h, Mvwoi †eM hZ †ewk n‡e Ges

euv‡Ki e¨vmva© hZ Kg n‡e Zv‡K ZZ †ewk †nj‡Z n‡e|

ZvB euvK †bqv iv Í̄vq Mvwoi MwZ‡eM 60kmh1 Gi A_© n‡jv GB †eM †_‡K †ewk †e‡M euvK wb‡j Mvwo‡K A‡bK †ewk †Kv‡Y †nj‡Z

n‡e hv AZ¨šÍ wec¾bK Ges hvi d‡j ỳN©Ubv NU‡Z cv‡i|

M DÏxcK n‡Z cvB,

Zvwc©b †Z‡ji mv› ª̀ZvsK, = 1.5 102 Pas

eo †Mvj‡Ki e¨vmva©, r = 6cm

2 = 3cm = 3 102 m

avZe c`v‡_©i NbZ¡, = 8 103 kgm3

Zvwc©b †Z‡ji NbZ¡, = 8.9 102 kgm3

Rvbv Av‡Q, AwfKl©R Z¡iY, g = 9.8 ms2

eo †MvjKwUi cÖvwšÍK †eM, V = 29

r2 ( ) g

= 29

(3 102)2 (8 103 8.9 102) 9.8

1.5 102

= 929.04 ms1

GLb, eo †Mvj‡Ki Dci cÖhy³ mv›`ª ej, F n‡j,

F = 6rv

= 6 3.1416 1.5 102 3 102 929.04

= 7.88 N (Ans.)

m~Î †_‡K Avgiv Rvwb,

cÖhy³ mv›`ª j, F = 6r

GLv‡b, = mv› ª̀ZvsK = 1.5 102 Pa.s

r = eo †Mvj‡Ki e¨vmva© = 6cm

2 = 3 cm

= 3 102 m

= eo †Mvj‡Ki cÖvwšÍK †eM = AwZµvš• `•iZ¡

mgq

= 21 cm3 sec

= 21 102 m

3 sec

= 7 102 ms1

Kv‡RB eo †Mvj‡Ki Dci cÖhy³ mv›`ª ej,

F = 6 3.1416 1.5 102 3 102 7 102

= 5.94 104 N (Ans.)

N ÔMÕ n‡Z cvB, eo †MvjKwUi cÖvwšÍK †eM n‡”Q 929.04ms1| hw` †QvU †MvjKwUi cÖvwšÍK †eM G gvb †_‡K †ewk nq Zvn‡j †QvU

†MvjKwU Av‡M wb‡P co‡e Ab¨_vq eo †MvjKwU Av‡M wb‡P co‡e|

†`Iqv Av‡Q,

avZe c`v_© ev †Mvj‡Ki NbZ¡, = 8 103 kgm3

Zvwc©b †Z‡ji NbZ¡, = 8.9 102 kgm3

†QvU †Mvj‡Ki e¨vmva©, r2 = 2cm

= 2 102m

Zvwc©b †Z‡ji mv› ª̀ZvsK, = 1.5 102 Pa.s


Teac

hing

bd.co

m


Avgiv Rvwb, cÖvwšÍK †eM, = 2r22( )g

9

= 2 (2 102)2 (8 103 8.9 102) 9.8

9 1.5 102

= 412.91 ms1 < 929.04 ms1

†`Lv hv‡”Q †h, eo †MvjKwUi cÖvšÍ †eM †QvU †Mvj‡Ki cÖvwšÍK †eM

†_‡K †ewk ZvB eo †MvjKwU Av‡M wb‡P co‡e|


K †h mKj GKK †gŠwjK GKK mgš^‡q MwVZ nq Zv‡ì‡K jä GKK ev †hŠwMK GKK e‡j|

L †Ìqv Av‡Q, A I

B Gi gaëZ©x †KvY 45

A .

B = AB cos 45

= AB

2

| A

B | = |AB (sin 45)|

= AB

2

A .

B = |

A

B |

[†`Lv‡bv n‡jv]

M †Ìqv Av‡Q, wUwfi k‡ãi ZxeªZv, I1 = 1 106 Wm2

cÖgvY ZxeªZv, Io = 1 1012 Wm2

ZxeªZv †j‡f‡ji cwieZ©b, = ?

Avgiv Rvwb, = 10 log I1Io

= 10 log 1 106

1 1012

= 60 dB

bvwdm ZxeªZv †j‡fj e„w× K‡iwQj, = (78 60) = 18 dB (Ans.)

N †Ìqv Av‡Q,

†eøÛv‡ii ZxeªZv †j‡fj, 1 = 85 dB

wUwfi ZxeªZv †j‡fj, 2 = 78 dB

aiv hvK, †eøÛv‡ii ZxeªZv = I

1. xvkv †evw©-2016 · 2017. 3. 24. · 1. xvkv †evw©-2016 1bs cÖ‡kœi dËi k kwvb i...

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