Aa¨vq-9: Zi½

cÖkœ1 ywU myi kjvKv GKB mgq evRv‡bv n‡q‡Q| hv‡`i Øviv

evqy‡Z Drcbœ kã Zi‡½i mgxKiY h_vµ‡g y1 = 0.5 sin (200t x

3.24 ) Ges y2 = 0.5 sin (210.03t x

3.09 )

[cUzqvLvwj miKvwi gwnjv K‡jR, cÖkœ-5]

K. GK †ej Kv‡K e‡j? 1

L. ÔmKj nvi‡gvwbK Dcmyi wKš‘ mKj Dcmyi nvi‡gvwbK

bqÕe¨vL¨v Ki| 2

M. cÖ_g Zi‡½i mgxKiY †_‡K Zi½‡eM wbY©q Ki| 3

N. DwÏc‡Ki Zi½Øq exU m„wó Ki‡e wKbv? MvwYwZKfv‡e

we‡kølY Ki| 4

1 bs cÖ‡kœi DËi

K cÖgvY ZxeªZv †_‡K 10 ¸Y ZxeªZv m¤úbœ †Kv‡bv k‡ãi ZxeªZv

†j‡fj‡K 1 Bel e‡j|

L †Kvb ¯‡i wewfbœ K¤úvs‡Ki myi _v‡K| G‡`i g‡a¨ †h my‡ii

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

g~j my‡ii †P‡q †ewk Zv‡`i‡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 bq|

M DwÏcK †_‡K cÖ_g Zi‡½i mgxKiYwU cvB,

y1 = 0.5 sin (200t x

3.24 )

= 0.5 sin2 (100t x

6.48 )

= 0.5 sin 2

6.48 (648 t x)

D³ mgxKiYwU‡K y = a sin 2 (vt x) Gi mv‡_ Zzjbv K‡i cvB,

v = 648 ms-1

cÖ_g Zi‡½i mgxKiY †_‡K cvB Zi½‡eM = 648 ms-1 (Ans.)

N ÒMÓ Ask †_‡K cvB,

1g Zi½wU y1 = 0.5 sin 2

6.48 (648 t x)

D³ mgxKiYwU‡K y = a sin 2

(vt x) Gi mv‡_ Zzjbv K‡i cvB,

v1 = 648 ms-1

1 = 6.48 m

Avgiv Rvwb, v1 = f1 1

f1 = v1

1

= 6486.48 s-1

= 100 Hz

Avevi,

2q Zi½wU, y2 = 0.5 sin (210.03t x

3.09 )

= 0.5 sin 2 (105.015 t x

6.18 )

= 0.5 sin 2

6.18 (648.99 t x)

= 0.5 sin 2

6.18 (649 t x)

D³ mgxKiYwU‡K y = a sin 2

(vt x) Gi mv‡_ Zzjbv K‡i cvB,

v2 = 649 ms-1

2 = 6.48 m

f2 = v2

2 =

6496.48 s-1 = 100.15 Hz

myZivs Avgiv †`L‡Z cvw”Q 1g I 2q Zi½Ø‡qi K¤úv¼ h_vµ‡g

100 Hz I 100.15 †hLv‡b my¯úó cv_©K¨ †bB| A_©vr DwÏc‡Ki

Zi½Ø‡qi exU m„wó Ki‡e bv|

cÖkœ2 `ywU ev`¨hš¿ †_‡K wbM©Z kã Zi‡½i mgxKiY h_vµ‡g

y1 = 1.1 sin

(100 t

x3.65 I y2 = 1.1 sin

(110.03 t

x3.15

; †hLv‡b mgq †m‡K‡Ð I mKj `~iZ¡ wgUv‡i cÖKvwkZ| Zi½Øq

gva¨‡gi ga¨ w`‡q AMÖmi nIqvi mgq DcwicvZb N‡U|

[miKvwi gwnjv K‡jR, cvebv]

K. mv› ªZv ¸Yv¼ Kv‡K e‡j? 1

L. ÒmKj nvi‡gvwbK Dcmyi wKš‘ mKj Dcmyi nvi‡gvwbK bqÓ

e¨vL¨v Ki| 2

M. DÏxc‡Ki 1g Zi½wU 10sec-G KZUzKz ~iZ¡ AwZµg

Ki‡e Zv †ei Ki| 3

N. DÏxc‡Ki Zi½ ywUi DcwicvZ‡bi d‡j Drcbœ jwä Zi½

kã weÁv‡bi †Kvb NUbvwU m„wó K‡i Zv MvwYwZK we‡k ølY

c~e©K gZvgZ `vI| 4

2 bs cÖ‡kœi DËi

K wbw`©ó ZvcgvÎvq cÖevnxi ywU ͇ii g‡a¨ †e‡Mi bwZ GKK

ivL‡Z (A_©vr GK ~i‡Z¡ Aew¯’Z ywU ¯ ͇ii g‡a¨ GKK Av‡cwÿK

†eM eRvq ivL‡Z) cÖevnx ¯ ͇ii cÖwZ GKK †ÿÎd‡j †h ¯úk©Kxq

e‡ji cÖ‡qvRb nq Zv‡K H ZvcgvÎvq H cÖevnxi mv› ªZv ¸Yv¼

e‡j|

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Aa¨vq-9: Zi½

L cÖ‡Z¨K ¯^iB yB ev Z‡ZvwaK my‡ii mgwó| †Kv‡bv ¯‡ii g‡a¨

we`¨gvb myi¸‡jvi g‡a¨ hvi K¤úv¼ me‡P‡q Kg Zv‡K g~j myi ev

†gŠwjK myi e‡j|

Ab¨ mKj myi hvi K¤úv¼ g~j my‡ii †P‡q †ewk Zv‡`i 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 mg‡gj ev ni‡gvwbK e‡j| wKš‘

D‡jøL¨ †h, mKj Dcmy‡ii K¤úv¼ g~j my‡ii mij ¸wYZK nq bv|

myZivs, †`Lv hv‡”Q, mKj nvi‡gvwbK Dcmyi, wKš‘ mKj Dcmyi

nvi‡gvwbK bq|

M DÏxcK n‡Z, cÖ_g Zi½wUi mgxKiY,

y1 = 1.1 sin

100 t

x3.65

ev, y1 = 1.1 sin

100 t

x3.65 ..... (i)

Avgiv Rvwb, AMÖMvgx Zi‡½i mgxKiY,

y = A sin

t

2x

..... (ii)

(i) bs I (ii) bs mgxKiY Zzjbv K‡i cvB,

= 2f = 100

ev, f = 50 Hz.

Ges 2 x

=

x3.65

ev, = 7.3 m.

v = f = (50 7.3) ms-1 = 365 ms-1.

myZivs, 10sec G AwZµvšÍ `~iZ¡, s = vt

= (365 10)m

3650 m (Ans.)

N DÏxcK n‡Z, y1 = 1.1 sin

100 t

x3.65

ev, y1 = 1.1 sin

100 t

x3.65 .... (i)

Ges y2 = 1.1 sin

110.03 t

x3.15

ev, y2 = 1.1 sin

110.03 t

x3.15 .... (ii)

Avevi, Avgiv Rvwb, AMÖMvgx Zi‡½i mgxKiY,

y = A sin

t

2x

.... (iii)

(iii) bs mgxKi‡Yi mv‡_ (i) bs I (ii) bs Gi Zzjbv K‡i cvB,

cÖ_g Zi‡½i we¯ Ívi, A1 = 1.1 m

wØZxq Zi‡½i we Ívi, A2 = 1.1 m

cÖ_g Zi‡½i K¤úv¼, f1 =

2 =

100

2 = 50 Hz

wØZxq Zi‡½i K¤úv¼, f2 =

2 =

110.03

2 = 55.015 Hz

†h‡nZz, Zi½ ywUi we Ívi mgvb Ges K¤úv‡¼i cv_©K¨ LyeB mvgvb¨|

myZivs, GLv‡b Zi½Ø‡qi DcwicvZ‡bi d‡j exU Drcbœ n‡e|

cÖkœ3 50 cm ˆ`N©¨wewkó GKwU m‡bvwgUv‡ii Zvi 200 Hz

K¤úv‡¼i GKwU myikjvKvi mv‡_ HK¨Zv‡b Av‡Q|

[miKvwi mvi`v my›`ix gwnjv K‡jR, dwi`cyi]

K. w¯úÖs aªæeK Kv‡K e‡j? 1

L. GKwU e ‘i w ’wZkw³ Kxfv‡e k~b¨ nq- e¨vL¨v Ki| 2

M. m‡bvwgUv‡ii Zv‡ii Uvb Pvi¸Y Ki‡j HK¨Zv‡b Avb‡Z

KZ ˆ`‡N©¨i cÖ‡qvRb n‡e? 3

N. Zv‡ii Uvb wVK †i‡L m‡bvwgUv‡ii Zv‡ii ˆ`N©¨ 2% e„w×

Ki‡j cÖwZ †m‡K‡Ð KqwU exU †kvbv hv‡e? 4

3 bs cÖ‡kœi DËi

K †Kv‡bv w¯úÖs Gi ˆ`N©¨ mvg¨ve ’v †_‡K GKK cwigvY cwieZ©‡bi

Rb¨ Gi ˆ`N©¨ eivei †h cwigvY ej cÖ‡qv‡Mi cÖ‡qvRb nq Zv‡K H

w¯úÖs Gi w¯úÖs aªæeK e‡j|

L Avgiv Rvwb, m f‡ii †Kv‡bv e ‘‡K f‚wg †_‡K h D”PZvq DVv‡j

f‚wgi mv‡c‡ÿ H e ‘i w ’wZkw³ = mgh. GLv‡b, g = AwfKl©R

Z¡iY| GLb, m, g I h Gi g‡a¨ †h‡Kv‡bv GKwUi gvb k~b¨ n‡j

w¯’wZkw³ k~b¨ n‡e| e ‘i fi m k~b¨ n‡Z cv‡i bv| myZivs, f‚wg

†_‡K e ‘i D”PZv h k~b¨ n‡j w ’wZkw³ k~b¨ n‡e|

myZivs ejv hvq †h, †h c„‡ôi mv‡c‡ÿ w¯’wZkw³ wnmve Kiv n‡e,

Zvi mv‡c‡ÿ e ‘i D”PZv k~b¨ n‡j e¯‘i w ’wZkw³ k~b¨ n‡e|

M †`Iqv Av‡Q,

m‡bvwgUv‡ii Zv‡ii cÖv_wgK ˆ`N©¨, l1 = 50 cm.

g‡b Kwi, cÖv_wgK Uvb, T1 = T

Pvi¸Y Kivi c‡i Uvb, T2 = 4T

m‡bvwgUv‡ii Zv‡i K¤úv¼, f = myikjvKvi K¤úv¼ = 200 Hz

Uvb Pvi¸Y Kivi c‡i HKZv‡b Avb‡Z cÖ‡qvRbxq Zv‡ii ˆ`N©¨, l2

= ?

Avgiv Rvwb, K¤úv¼ f Ges GKK ˆ`‡N©¨i fi m aªæe _vK‡j,

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Aa¨vq-9: Zi½

T1

T2 =

l12

l22

ev, T

4T = l12

l22

ev, l12

l22 = T

4T

ev, l2l1

= 2

l2 = 2 l1 = 2 50 cm = 100 cm.

N g‡b Kwi, m‡bvwgUv‡ii Zv‡ii Uvb T Ges GKK ˆ`‡N©¨i fi m.

m‡bvwgUv‡ii Zv‡ii Avw` ˆ`N©¨, l1 = 50 cm.

2% e„w× Kivi ci Zv‡ii ˆ`N©¨, l2 = l1 + 2

100 l1

= 50 cm + 2

100 50 cm.

= 51 cm.

ˆ`N©¨ e„w×i Av‡M K¤úv¼, f1 = 200 Hz

ˆ`N©¨ e„w×i c‡ii K¤úv¼, f2 = ?

Avgiv Rvwb, T I m aªæe _vK‡j,

f1l1 = f2l2

ev, f2 = f1 l1l2

f2 = 200 Hz 50 cm

51 cm = 196 Hz.

cÖwZ †m‡K‡Ð Drcbœ weU msL¨v = 200 196 = 4.

myZivs Uvb wVK †i‡L m‡bvwgUv‡ii Zv‡ii ˆ`N©¨ 2% e„w× Ki‡j cÖwZ

†m‡KÐ 4wU weU Drcbœ n‡e|

cÖkœ4 GKwU myikjvKv n‡Z wbM©Z Zi‡½i mgxKiY Y = 3 sin

344 t| GKRb wkÿv_©x myikjvKvwU jw¤^K Ae ’vq †i‡L w ’i Zij

m„wó K‡i DcvË msMÖn Kij Ges wb‡¤œi †Uwe‡j wjwce× Kij|

wkÿK ej‡jb Òjw¤^K Ae ’vq myZvi K¤úvsK n myi kjvKvi

K¤úvsK N Gi A‡a©K nq|Ó

[miKvwi wmwU K‡jR, PÆMÖvg]

K. mij †`vjMwZ Kv‡K e‡j? 1

L. mu~P cvwb‡Z fv‡m †Kb? 2

M. Zi‡½i mgxKiY n‡Z myikjvKvi K¤úvsK wbY©q Ki| 3

N. cixÿvjä DcvË n‡Z wkÿ‡Ki K_vi mZ¨Zv hvPvB Ki|4

4 bs cÖ‡kœi DËi

K ¯ú›`biZ †Kv‡bv e ‘KYvi MwZ hw` Ggb nq †h, 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 cvwb‡Z hLb m~uP ivLv nq ZLb c„‡ôi H ’vbUv Avbyf‚wgK _v‡K

bv eis c„ôUv‡bi Rb¨ GB ej AebwgZ cvwb c„‡ôi mv‡_ wZh©Kfv‡e

wµqvkxj e‡ji Djø¤^ Dcvsk myB Gi IRb‡K cÖkwgZ K‡i, d‡j

myBwU bv Wz‡e mvg¨ve ’vq †f‡m _v‡K|

M †`Iqv Av‡Q,

Zi‡½i mgxKiY,

y = 3 sin 344 t

Zi‡½i mvaviY mgxKiY,

y = a sin 2 ft ‡hLv‡b f myikjvKvi K¤úvsK

2ft = 344 t

ev, f = 344

2

f = 172 Hz

myikjvKvi K¤úvsK 172 Hz (Ans).

N †`Iqv Av‡Q,

mg = Zv‡ii Uvb, T = 0.55 9.8 = 5.39 N

Zv‡ii ˆ`N©¨, l = 0.573 m

myZvi GKK ˆ`‡N©¨i fi, m = 0.005 kg

myZvi K¤úvsK = n

jyc msL¨v, a = 3

ÒMÓ bs n‡Z myikjvKvi K¤úvsK, N = f = 172 Hz

Avgiv Rvwb,

Uvbv Zv‡i AbycÖ ’ Zi‡½i †eM v = Tm

1 ch©‡e¶K msL¨v

0.55 †gvU fi m.kg

0.573 `yB w`‡bi ga¨eZx© ~iZ¡

L

3 `yB w`‡bi ga¨eZx© jyc

msL¨v a

0.005 myZvi GKK ˆ`‡N©¨i fi

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Aa¨vq-9: Zi½

ev, = 5.39.005

ev, v = 32.83 ms–1

Avevi,

n = av2L [jyc msL¨v a n‡j]

ev, n = 86 Hz

ev, n = 12 172

n = 12 N

myZivs jw¤^K Ae ’vq myZvi K¤úv¼ n myikjvKvi K¤úvsK N Gi

A‡a©K nq|

cÖkœ5 exU MV‡bi j‡ÿ¨ Zv‡iK ywU myikjvKv‡K GK‡Î evqy‡Z

kãvwqZ Ki‡jv| Zv‡i‡Ki 1g myikjvKvi kã Zi½‡K y = 0.1 sin

2512t (me¸‡jv ivwk Gm. AvB GK‡K cÖ`Ë Ges = 3.14) Ges

2q myikjvKvi Zi½‡K wb‡¤œv³ wP‡Îi mvnv‡h¨ cÖKvk Kiv n‡q‡Q|

evqy‡Z k‡ãi †eM 350ms-1.

0.43 m

1.72 m

P Q

Y

X

[K¨v›Ub‡g›U cvewjK ¯‹zj I K‡jR, exi DËg kwn` gvneye †mbvwbevm,

cveZ©xcyi, w`bvRcyi]

K. exU Kx? 1

L. Drm؇qi K¤úv‡¼i cv_©K¨ bv n‡j ev A‡bK †ewk n‡j exU

†kvbv hvq bv †Kb? 2

M. P I Q we› yi ga¨Kvi `kv cv_©K¨ wbY©q Ki| 3

N. Zv‡iK Kx cÖK…Zc‡ÿ exU ïb‡Z cvi‡e?- MvwYwZK

we‡køl‡Y gZvgZ `vI| 4

5 bs cÖ‡kœi DËi

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‡`i 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 Drm؇qi K¤úv‡¼i cv_©K¨ Lye †ewk n‡j cÖwZ †m‡K‡Ð Drcbœ

weU msL¨vI Lye †ewk nq, d‡j ZxeªZvi n«vm-e„w× GZ `ªæZ nq, Zv

Dcjwä Kiv hvq bv| Ab¨w`‡K K¤úv‡¼i cv_©K¨ bv _vK‡jI

ZxeªZvi n«vme„w× Dcjwä Kiv hvq bv| Dfq‡ÿ‡ÎB Kv‡b GKUvbv

kã †kvbv hvq|

M †`Iqv Av‡Q,

y = 0.1sin2512 t

mgxKiYwU‡K y = a sin 2f t Gi mv‡_ Zzjbv K‡i cvB,

a = 0.1 m

2f1 = 2512

f1 = 400 Hz

Avevi, wPÎ n‡Z, P I Q Gi gv‡S c_ cv_©K¨ = 0.43m.

Ges 22 = 1.72 m

2 = 0.86 m.

Avgiv Rvwb, kv cv_©K¨ = 2

2 c_ cv_©K¨

= 2

2 0.43 m.

= 2

0.86 0.43

`kv cv_©K¨ = rad (Ans.)

N †`Iqv Av‡Q,

k‡ãi †eM, v = 350 ms-1

Avevi, 2q Zi‡½i †ÿ‡Î,

v = f2 2

ev, f2 =

2 =

350 0.86 Hz

f2 = 407 Hz

(M) Ask †_‡K cvB,

f1 = 400Hz

Ges f2 = 407 Hz

†h‡nZz Zi½Ø‡qi g‡a¨ K¤úv‡¼i cv_©K¨ i‡q‡Q †m‡nZz Zv‡iK weU

ïb‡Z cvi‡e| we‡Ui msL¨v n‡e Zi½Ø‡qi K¤úv‡¼i cv_©‡K¨i (f2

f1) mgvb|

f2 f1 = 407 Hz 400 Hz = 7 Hz| A_©vr Zv‡iK cÖwZ †m‡K‡Ð 7 wU

weU ïb‡Z cv‡e|

cÖkœ6 2000 gyiwMi aviYÿgZv m¤úbœ GKwU †cvwëª dv‡g© 500wU

gyiwM i‡q‡Q| †cvwëª dv‡gi eZ©gvb k‡ãi ZxeªZv 3.2 10-4 Wm-

2| †cvwëª dv‡g©i gvwjK gyiwMi msL¨v evwo‡q 2000 Kivi wm×všÍ

wb‡jb| [wek¦bv_ wWMÖx K‡jR, wm‡jU]

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Aa¨vq-9: Zi½

K. †f±i wefvRb Kx? 1

L. mKj nvi‡gvwbKB Dcmyi wKš‘ mKj Dcmyi nvi‡gvwbK

bq e¨vL¨v Ki| 2

M. †cvwëª dvg©wUi k‡ãi eZ©gvb ZxeªZv †j‡fj wbY©q Ki| 3

N. gyiwMi msL¨v evov‡j dvg©wU‡Z Kx ai‡bi mgm¨v n‡Z

cv‡i MvwYwZKfv‡e we‡kølY Ki| 4

6 bs cÖ‡kœi DËi

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

cÖwµqvB n‡jv †f±i wefvRb|

L †Kvb ¯‡i wewfbœ K¤úvs‡Ki myi _v‡K| G‡`i g‡a¨ †h my‡ii

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

g~j my‡ii †P‡q †ewk Zv‡`i‡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 †`Iqv Av‡Q,

k‡ãi ZxeªZv, I = 3.2 10-4 Wm-2

cÖgvY ZxeªZv, Io = 1 10-12 Wm-2

ZxeªZv †j‡fj, = ?

Avgiv Rvwb,

= 10 log I

Io dB

= 10 log

3.2 10-4

1 10-12

= 85.05 dB. (Ans.)

N †`Iqv Av‡Q,

eZ©gvb k‡ãi ZxeªZv, I = 3.2 10-4Wm-2

gyiMxi msL¨v e„wׇZ k‡ãi ZxeªZv, I = 2000500 I

I = 4 3.2 10-4 Wm-2

cwiewZ©Z ZxeªZv †j‡fj, = ?

Avgiv Rvwb,

= 10 log I

Io dB

= 10 log

4 3.2 10-4

1 10-12 dB

= 10 log (1.28 109) dB

= 10 9.107 dB

= 91.7 dB

ÔMÕ n‡Z hLb dv‡g© 500 gyiwM wQj ZLb ZxeªZv †j‡fj 85.05 dB

hv KviLvbvi †Kvjvnj ZxeªZv †j‡f‡ji †P‡q Kg|

wKš‘ hLb msL¨v †e‡o 2000 nj ZLb ZxeªZv †j‡fj 91.07 dB hv

KviLvbvi †Kvjvnj ZxeªZv †j‡f‡ji †P‡q †ewk Ges †RU †c ø‡bi

k‡ãi KvQvKvwQ| GwU c~‡e©i Zzjbvq †ewk kã ~lY m„wó K‡i|

myZivs gyiwMi msL¨v evo‡j dvg©wU‡Z k‡ãi mn¨ gvÎv AwZµg

Kivi cÖeYZv †e‡o hv‡e|

cÖkœ7 e¸ov kn‡ii mvZgv_vq †gvevBj †KvU© k‡ãi ZxeªZv

cwigvc Ki‡Qb| †ejv 12Uvq mvZgv_vq k‡ãi ZxeªZv wQj 10–5

W/m2| [miKvwi kvn& myjZvb K‡jR, e¸ov]

K. myihy³ kã Kx? 1

L. k‡ãi ZxeªZv ej‡Z Kx †evS? 2

M. mvZgv_vq k‡ãi ZxeªZv †j‡fj KZ? 3

N. nVvr K‡i †gvevBj †Kv‡U©i A ~‡i Mvwoi PvKv we‡ùvwiZ

nIqvq k‡ãi ZxeªZv 3 ¸Y †e‡o †M‡j| H mgq ZxeªZvi

†j‡fj mnbxq gvÎvq wQj wKbv? MvwYwZK e¨vL¨v `vI| 4

7 bs cÖ‡kœi DËi

K †h †hŠwMK k‡ãi g‡a¨ we`¨gvb †gŠwjK Dcv`vbmg~n cÖ‡Z¨K

ch©ve„Ë kãZi½ A_©vr wbw ©ó K¤úv¼ Ges Zi½‰`N©¨ wewkó, Zv‡K

myihy³ kã e‡j|

L †Kv‡bv Zi‡½i mg‡Kv‡Y GKK †ÿÎd‡ji ga¨ w`‡q GK

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

G‡K I Øviv m~wPZ Kiv nq|

Zi‡½i ZxeªZv, I = kw³ NbZ¡ Zi½ †eM

MvwYwZKfv‡e †`Lv‡bv hvq †h, I = 22a2n2v

GLv‡b, = gva¨‡gi NbZ¡

n = Zi‡½i K¤úv¼

a = Zi‡½i we Ívi Ges

v = Zi‡½i †eM|

M †`Iqv Av‡Q,

k‡ãi ZxeªZv, I = 10-5 W/m2

ZxeªZv †j‡fj, = ?

cÖgvY ZxeªZv, I0 = 10–12 W/m2

Avgiv Rvwb,

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Aa¨vq-9: Zi½

= 10 log I

Io dB

= 10 log

10–5

10–12 dB

= 70 dB (Ans.)

N GLv‡b, ZxeªZv 3 ¸Y †e‡o nj,

I = 3I = 3 10–5 W/m2

cÖgvY ZxeªZv, Io = 1 10-12 W/m2

ZxeªZv †j‡fj, = ?

Avgiv Rvwb, = 10 log

I

1o dB

ev, = 10 log

3 10-5

1 10-12 dB

= 10 log (3 107) dB

= 10 7.477 dB

= 74.77 dB

ÔMÕ n‡Z mvZgv_vq k‡ãi ZxeªZv †j‡fj 70 dB Ges ZxeªZv 3 ¸Y

evovq ZxeªZv †j‡fj 74.77 nj hv ¯^vfvweK e¨ ÍZg iv Ívi kã

A‡cÿv mvgvb¨ †ewk|

myZivs, Mvwoi PvKv we‡ùvwiZ nIqvq ZxeªZv †j‡fj mnbxq gvÎvqB

wQj Z‡e k‡ãi ZxeªZv †j‡fj e„w× †c‡q‡Q|

cÖkœ8

wP‡Î 40W ÿgZvi GKwU ¯úxKvi A n‡Z 1.5km I 2km ~‡i

h_vµ‡g B I C ywU Ae¯’vb|

[miKvwi †Zvjvivg K‡jR, bvivqYMÄ; cÖkœ-3]

K. cqm‡bi AbycvZ Kv‡K e‡j? 1

L. cvBc jvB‡bi ga¨ w`‡q Mig cvwb I kxZj M¨vm `ªæZ

cÖevwnZ nq †Kb? 2

M. ¯úxKvi A-Gi Rb¨ C Ae ’v‡b k‡ãi ZxeªZv KZ? 3

N. B Ae ’vb I C Ae¯’v‡b k‡ãi ZxeªZvi †j‡f‡ji ZviZ‡g¨i

MvwYwZK we‡kølY `vI| 4

8 bs cÖ‡kœi DËi

K w¯’wZ¯’vcK mxgvi g‡a¨ †Kv‡bv e¯‘i cvk¦© weK…wZ I Aby‰`N©¨

weK…wZi AbycvZ‡K cqm‡bi AbycvZ e‡j|

L Zi‡ji ZvcgvÎv e„w× †c‡j Zi‡ji AYy¸‡jv Zvc †_‡K kw³

MÖnY K‡i †ewk kw³ cvq Ges G‡`i MwZ †e‡o hvq G‡Z AYy¸‡jvi

Mo gy³ c_ e„w× cvq d‡j G‡`i g‡a¨ Nl©Y Kg nq| Mo gy³ c_

e„w×i d‡j Zi‡ji ͇ii Av‡cwÿK evav K‡g hvq| d‡j Zi‡ji

mv› ªZv n«vm cvq| Avevi ZvcgvÎv n«vm †c‡j M¨v‡mi mv›`ªZv n«vm

cvq| G Kvi‡Y cvBc jvB‡bi ga¨ w`‡q Mig cvwb I kxZj M¨vm

`ªæZ cÖevwnZ nq|

M †`Iqv Av‡Q, w¯úKv‡ii ÿgZv, P = 40W

A n‡Z C we› yi ~iZ¡, r = 2km = 2 103m|

†ei Ki‡Z n‡e C we›`yi k‡ãi ZxeªZv, I = ?

Avgiv Rvwb,

I = PA

ev, I = P

4r2

ev, I = 40

4 3.1416 (2 103)2

I = 7.95 10-7 Wm-2 (Ans.)

N †`Iqv Av‡Q,

B we›`yi ~iZ¡, r = 1.5km

ev, r = 1.5 103 m.

cÖgvY ZxeªZv, Io = 10-12 Wm–2

†ei Ki‡Z n‡e,

A we›`yi ZxeªZv †j‡fj, 1 = ?

B we›`yi ZxeªZvi †j‡fj, 2 = ?

GLb, B we›`yi †ÿ‡Î,

I = P

4r2

ev, I = 40

4 3.1416 (1.5 103)2

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Aa¨vq-9: Zi½

I = 1.4 10-6 Wm-2

GLb, B we›`yi ZxeªZvi †j‡fj,

1 = 10 log IIo

ev, 1 = 10 log 1.42 10-6

10-12

1 = 61.5 dB.

C we›`yi ZxeªZvi †j‡fj,

2 = 10 log IIo

ev, 2 = 10 log 7.95 10-7

10-12

2 = 59 dB.

Dc‡iv³ MvwYwZK we‡køl‡Yi gva¨‡g †`Lv hv‡”Q C we›`yi k‡ãi

ZxeªZv 59dB Ges B we›`yi k‡ãi ZxeªZv 61.5 dB.

cÖkœ9 ywU myi kjvKv †_‡K evqy gva¨‡g ywU kã Zi‡½i

mgxKiY h_vµ‡g y1 = 0.5 sin

200 t

x1.75 Ges y2 = 0.5 sin

210 t

x1.667 GLv‡b ivwk¸‡jv S.I GK‡K cÖ wk©Z, myi kjvKv

`ywU GKB mg‡q evRv‡bv n‡qwQj| [U½x miKvwi K‡jR, MvRxcyi; cÖkœ-2]

K. †gjwW Kv‡K e‡j? 1

L. Abybv` Av‡ivwcZ K¤úY wKš‘ Av‡ivwcZ K¤úY Abybv` bq

e¨vL¨v Ki| 2

M. Zi½wUi K¤úv¼, †eM, Zi½‰`N©¨ I ch©vqKvj wbY©q Ki|

3

N. DÏxc‡Ki Z_¨ Abymv‡i †Kvb exU Drcbœ n‡qwQj wKbv Zv

MvwYwZKfv‡e †`LvI| 4

9 bs cÖ‡kœi DËi

K hw` K‡qKwU kã G‡Ki ci GKK D”PvwiZ n‡q GKwU kÖæwZgayi

k‡ãi m„wó K‡i Z‡e Zv‡K †gjwW e‡j|

L GKwU ch©ve„Ë ej cÖ‡qvM K‡i †Kvb e¯‘‡K Kw¤úZ Ki‡j e ‘wU

cÖ_‡g Zvi wbR¯^ ¯^vfvweK K¤úv‡¼ Kw¤úZ nIqvi †Póv K‡i| wKš‘

ax‡i ax‡i †`Lv hv‡e †h, e ‘wU ch©ve„Ë e‡ji K¤úv¼ Abyhvqx

¯úw›`Z n‡”Q| e ‘wUi ¯vfvweK K¤úv¼ hvB †nvK bv †Kb, ch©ve„Ë

ejwU hZÿY wµqvkxj _v‡K e ‘wUI ch©ve„Ë e‡ji K¤úv¼ Abymv‡i

Kw¤úZ n‡e| G ai‡bi K¤úb‡K Av‡ivwcZ K¤úb e‡j|

Ab¨w`‡K Abybv` we‡kl ai‡bi Av‡ivwcZ K¤úb| e ‘i wbR¯^

K¤úv¼ Ges Zvi Dci Av‡ivwcZ ch©ve„Ë ¯ú›`‡bi K¤úv¼ mgvb

n‡j e¯‘wU m‡e©v”P we Ívi mnKv‡i Kw¤úZ n‡Z _v‡K| G ai‡bi

K¤úb‡K Abybv` e‡j| myZivs mKj Abybv`B Av‡ivwcZ K¤úb wKš‘

mKj Av‡ivwcZ K¤úb Abybv` bq|

M GLv‡b,

kã Zi‡½i cÖ_g mgxKiY,

y1 = 0.5 sin

200 t

x1.75

= 0.5 sin 2

100 t

x3.5

= 0.5 sin 23.5 (350 t x)

Dc‡iv³ mgxKiY‡K AMÖMvgx Zi‡½i cÖgvY mgxKiY,

y = a sin 2

(vt x) Gi mv‡_ Zzjbv K‡i cvB,

Zi½ ˆ`N©¨, 1 = 3.5 m

Zi½ †eM, v1 = 350 ms–1

K¤úv¼, f1 = v1

1 =

3503.5

= 100 Hz

ch©vqKvj, T1 = 1f1

= 1

100 = 0.01 s

A_©vr, K¤úv¼, †eM, Zi½‰`N©¨ I ch©vqKvj h_vµ‡g 100 Hz, 350

ms-1, 3.5m I 0.01 s.

N GLv‡b,

cÖ_g Zi‡½i †ÿ‡Î, [M †_‡K cvB]

Zi½‰`M©¨, = 3.5 m

evqy‡Z k‡ãi †eM, v1 = 350 ms-1

cÖ_g Zi‡½i K¤úv¼, f1 = 100 Hz

wØZxq Zi‡½i †ÿ‡Î,

y2 = 0.5 sin

210t

x1.667

= 0.5 sin 2

105t

x3.334

= 0.5 sin 2

3.334 (350.07t x)

wØZxq Zi‡½i mgxKiY‡K AMÖMvgx Zi‡½i cÖgvY mgxKiY,

y = a sin 2

(vt x) Gi mv‡_ Zzjbv K‡i cvB,

wØZxq Zi‡½i Zi½ ˆ`N©¨, 2 = 3.334 m

Ges †eM v2 = 350.07 ms–1.

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m

Aa¨vq-9: Zi½

wØZxq Zi‡½i K¤úv¼, f2 = v2

2 =

350.073.334 = 105 Hz > f1.

f1 f2 Ges f1 I f2 Gi g‡a¨ cv_©K¨ Kg|

myZivs exU Drcbœ n‡qwQj|

cÖwZ †m‡K‡Ð Drcbœ exU msL¨v = f2 f1

= (105 100) Hz

= 5 Hz.

cÖkœ10 ai †Zvgvi wbKU `kwU myi kjvKv i‡q‡Q| G‡`i‡K

µgea©gvb K¤úvsK Abymv‡i mvRv‡j| G‡Z †klwUi K¤úv¼

cÖ_gwUi wظY Ges ci ci †h †Kvb ywU kjvKv‡K GK‡Î evRv‡j

cÖwZ †m‡K‡Ð 6wU exU m„wó nq| [exi‡kÖô gyÝx Avãyi iDd cvewjK

K‡jR, cÖkœ-8]

K. w¯’i Zi½ Kv‡K e‡j? 1

L. AbycÖ¯’ Zi½ Ges Aby‰`N©¨ Zi‡½i cv_©K¨ Kx? 2

M. DÏxd‡Ki wØZxq kjvKvi K¤úvsK KZ? 3

N. cÂg Ges mßg kjvKv ywU GK‡Î evRv‡j exU ïb‡Z cv‡e

wKbv? KviY e¨vL¨v Ki| 4

10 bs cÖ‡kœi DËi

K †Kv‡bv gva¨‡gi GKwU mxwgZ As‡k mgvb we¯Ívi I Zi½‰`‡N©¨i

`ywU AMÖMvgx Zi½ GKBgv‡bi †e‡M wecixZ w`K †_‡K AMÖmi n‡q

G‡K Ac‡ii Dci AvcwZZ n‡j †h Zi‡½i D™¢e nq Zv‡K w ’i

Zi½ e‡j|

L

AYycÖ ’ Zi½ AYy‰`N©¨ Zi½

i. GB Zi½ Ro gva¨‡gi

KYv¸wji K¤ú‡bi w`K

Zi½ cÖev‡ni w`‡Ki

mg‡KvYx nq|

i. GB Zi½ Ro gva¨‡gi

KYv¸wji K¤ú‡bi w`K

Zi½ cÖev‡ni w`‡Ki

mgvšÍivj nq|

ii. Zi½ cÖev‡n gva¨‡g Zi½

kxl© Ges Zi½ cv` m„wó nq|

ii. Zi½ cÖev‡n gva¨‡g

ms‡KvPb I cÖmviY m„wó

nq|

iii. gva¨‡g Gi mgeZ©b ev

†cvjvivqY N‡U|

iii. gva¨‡g Gi mgeZ©b ev

†cvjvivqY N‡U bv|

M DÏxcK †_‡K cvB,

`kwU myi kjv‡K µgea©gvb K¤úv¼ Abymv‡i mvRv‡j †klwUi

K¤úv¼ cÖ_gwUi wظY nq|

g‡b Kwi,

cÖ_g myi kjvKvi K¤úv¼ = x Hz

†kl myi kjvKvi K¤úv¼ = 2x Hz

†h‡nZz, cici †h‡Kv‡bv ywU kjvKv‡K GK‡Î evRv‡j cÖwZ †m‡K‡Ð

6wU exU m„wó nq| †m‡nZz

wØZxq myi kjvKvi K¤úv¼ = x + 6 Hz

Z…Zxq myi kjvKvi K¤úv¼ = x + (6 2) Hz

PZz_© myi kjvKvi K¤úv¼ = x + (6 3) Hz

`kg myi kjvKvi K¤úv¼ = x + (6 9) Hz

DÏxcK g‡Z,

x + (6 9) = 2x

ev, 2x x = 6 9

x = 54

wØZxq myi kjvKvi K¤úv¼ = (x + 6) Hz

= (54 + 6) Hz

Ges mßg myi kjvKvi K¤úv¼ = 60 Hz (Ans.)

N ÒMÓ Ask †_‡K cvB,

cÂg myi kjvKvi K¤úv¼ = x + (6 4) Hz

= (54 + 24) Hz = 78 Hz

Ges mßg myi kjvKvi K¤úv¼ = x + (6 6) Hz

= (54 + 36) Hz

= 90 Hz

cÂg I mßg kjvKv ywU GK‡Î evRv‡j

Drcbœ exU n‡e = (90 78) Hz

= 12 Hz

wKš‘, Avgiv Rvwb exU 10 Hz Gi †ewk n‡j Avgiv Zv ïb‡Z cvB

bv| KviY exU 10 Hz n‡j ch©vqKvj nq 1

10 s| Avi †Kvb kã †kvbvi

ci Zvi †ik Avgv‡`i gw ͇®‹ 1

10 s ’vqx nq, Gi gv‡S Avgiv wØZxq

†Kv‡bv kã ïb‡Z cvB bv|

A_©vr, Avgiv cÂg I mßg kjvKv ywU GK‡Î evRv‡bvi iæY Drcbœ

exU ïb‡Z cv‡ev bv|

cÖkœ11 GKB mg‡q evRv‡bv ywU myikjvKv †_‡K evqy gva¨‡g

wbM©Z `ywU kã Zi‡½i mgxKiY h_vµ‡g,

y1 = 0.5 sin

200t +

x1.75

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m

Aa¨vq-9: Zi½

y2 = 0.5 sin

210t +

x1.667 [Puv`cyi miKvwi gwnjv K‡jR, cÖkœ-6]

†hLv‡b me¸‡jv ivwk SI GK‡K cÖ`Ë

K. mij Qw›`Z ¯ú›`b Kv‡K e‡j? 1

L. m~‡h©i Pviw`‡K AveZ©biZ MÖn¸‡jvi Kÿc_ Dce„ËvKvi

nevi KviY e¨vL¨v Ki| 2

M. cÖ_g Zi½wUi †KŠwYK K¤úv¼, ch©vqKvj I †eM wbY©q Ki|

3

N. DÏxc‡Ki Z_¨vbymv‡i we‡Ui nvi 6 Hz Gi Kg _vK‡e-

MvwYwZK we‡køl‡Yi gva¨‡g e³e¨wUi mZ¨Zv hvPvB Ki| 4

11 bs 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 MÖn m„wói cÖv°v‡j MÖn¸‡jv hLb m~h© †_‡K wew”Qbœ nq ZLbKvi

wb‡ÿcY †e‡Mi gv‡bi Dci wbf©i K‡i MÖn¸‡jvi Kÿc_ wK

AvKv‡ii n‡e|

Avgiv Rvwb, Dr‡ÿcY ev wb‡ÿcY †e‡Mi †ÿ‡Î, ve

2 < v < ve,

†hLv‡b, ve n‡jv hv †_‡K wbwÿß n‡”Q Zvi gyw³‡eM Ges v n‡jv

wb‡ÿcY †eM| ZvB m~h© †_‡K wew”Qbœ neviKv‡j m~‡h©i Pvwiw`‡K

AveZ©biZ MÖn¸‡jvi †eM ve

2 †_‡K eo Ges ve †_‡K †QvU nevi

`iæY Zv‡`i Kÿc_ Dce„ËvKvi nq|

G‡ÿ‡Î ve †_‡K e¨wZµg ïay eya| wb‡ÿcYKv‡j eyaMÖn m~‡h©i

gyw³‡e‡Mi 1

2 ¸Y †e‡M wbwÿß nq d‡j eya MÖ‡ni Kÿc_

e„ËvKvi|

M DÏxcK †_‡K cÖ_g Zi‡½i mgxKiY cvB,

y1 = 0.5 sin

200t +

x1.75

mgxKiY †_‡K †`Lv hv‡”Q GwU x A‡ÿi FYvZ¡K w`‡K AMÖMvgx|

GLb,

y1 = 0.5 sin

200t +

x1.75

ev, y1 = 0.5 sin

200t +

x1.75

D³ mgxKiY‡K y = a sin

2ft +

2x

Gi mv‡_ Zzjbv K‡i cvB,

2f1 = 200

f1 = 200

2

f1 = 100 Hz

ch©vqKvj, T1 = 1f1

= 1

100 = 0.01 s

2x

1 =

x1.75 ; [D³ mgxKi‡Yi mv‡_ Zzjbv K‡i]

ev, 1 = (2 1.75) = 3.5 m

Zi½‡eM, v1 = f1 1 = (100 3.5)ms-1 = 350 ms-1. (Ans.)

N DÏxc‡Ki 2q Zi½wUi mgxKiY,

y2 = 0.5 sin

210t +

x1.667

= 0.5 sin

210 t +

x1.667

D³ mgxKiY‡K y = asin

2f t +

2x

Gi mv‡_ Zzjbv K‡i cvB,

2f2 = 210

ev, f2 = 210

2 Hz

f2 = 105 Hz

ÒMÓ Ask †_‡K cvB, f1 = 100 Hz

Drcbœ we‡Ui nvi, N = f1 f2

= f2 f1

= (105 100) Hz

= 5 Hz , hv 6Hz Gi Kg

DÏxc‡Ki e³e¨wU mwVK|

cÖkœ12 ywU myikjvKvi A I B GKB mv‡_ kãvwqZ n‡j 0.50 I

0.505m Zi½ ˆ`‡N©¨i kã Drcbœ nq †mB mv‡_ 5wU exU Drcbœ nq|

[Bmjvwgqv wek¦we`¨vjq K‡jR, PÆMÖvg; Aa¨vq-9]

K. †Wwm‡ej Kx? 1

L. w¯’i Zi‡½i Av‡jv‡K Kxfv‡e Zi½ `N©¨‡K msÁvwqZ Ki‡e

Zv wP‡Îi mvnv‡h¨ e¨vL¨v Ki| 2

M. M¨vmwU‡Z k‡ãi †eM wbY©q Ki| 3

N. M¨vmwU‡Z 1wU exU cvIqvi Rb¨ Kx e¨e ’v MÖnY Ki‡Z n‡e

MvwYwZKfv‡e e¨vL¨v Ki| 4

12 bs cÖ‡kœi DËi

K †ej GK‡Ki GK-`kgvsk‡K GK †Wwm‡ej e‡j|

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Aa¨vq-9: Zi½

L w¯’i Zi‡½i Dcwiw¯’Z cici wZbwU my¯ú›` we›`y ev wb¯ú›`

we›`yi ga¨eZ©x ~iZ¡‡K Zi½‰`N©¨ e‡j|

wb‡¤œi wP‡Î w ’i Zi‡½i Dcwiw ’Z me¸‡jv A we› y‡K my¯úm›` we›`y

Ges me¸‡jv N we› y‡K wb¯ú›` we›`y e‡j|

x = 0

A

A A A A N N N

x

wPò

Dc‡ii wPÎ ~iZ¡ n‡jv w ’i Zi‡½i Av‡jvi †ÿ‡Î Zi½‰`N©¨|

M DÏxcK n‡Z cvB,

A myikjvKvi Zi½‰`N©¨, A = 0.50m

B myikjvKvi Zi½‰`N©¨, B = 0.505m

exU msL¨v, N = 5

awi, A myikjvKvi K¤úv¼ = fA

B myikjvKvi K¤úv¼ = fB

†h‡nZz, B > A †m‡nZz, fA > fB n‡e|

fA fB = N

ev, v

A

v

B = 5

ev, v

1

A

1

B = 5

ev,

B A

A B = 5

ev, v = 5A B

B A

= 5 0.50 0.505

0.505 0.50

= 252.5 ms-1 (Ans.)

N DÏxcK n‡Z cvB,

A myikjvKvi K¤úv¼, A = 0.50m

B myikjvKvi K¤úv¼, B = 0.505m

exU msL¨v, N = 1

k‡ãi †eM = v

A myikjvKvi K¤úv¼ = fA

B myikjvKvi K¤úv¼ = fB

†h‡nZ, B < A †m‡nZz fA > fB n‡e|

fA fB = 1

ev, v

A

v

B = 1

ev, v

B A

A B = 1

ev, v = AB

B – A

ev, v = 0.50 0.505

0.505 0.50

v = 50.5 ms-1

Avevi, v = f1 1

ev, f1 = v

1 =

50.5 0.50 = 101 Hz

Ges f2 = v

2 =

50.5 0.505 = 100 Hz.

AZGe, M¨vmwU‡Z 1 exU cvIqvi Rb¨ 100 Hz I 101 Hz K¤úvs‡Ki

m yikjvKv‡K GK‡Î kãvwqZ Ki‡Z n‡e|

cÖkœ13 bxjv 0.4kgm3 Nb‡Z¡i GKwU gva¨‡g GKwU myZv UvbUvb

K‡i GKwU Zi½ m„wó Kij| Zi‡½i mgxKiY y = 0.9 sin

x

15 + 2t0.3 S.I GKK| Aciw`‡K w`bv I webv ywU ¯’v‡bi k‡ãi

ZxeªZv †j‡fj 68 dB Ges 74 dB cwigvc Kij|

[Kzgyw`bx miKvwi K‡jR, Uv½vBj]

K. exU Kv‡K e‡j? 1

L. Abybv` ciek K¤úb wKš‘ mKj ciek Abybv` bq e¨vL¨v

K‡iv| 2

M. DcwiD³ mgxKiY †_‡K k‡ãi †eM wbY©q K‡iv| 3

N. bxjvi k‡ãi ZxeªZv †j‡fj, w`bv I webvi mw¤§wjZ k‡ãi

ZxeªZv †j‡f‡ji †P‡q wظY wKbv hvPvB K‡iv| 4

13 bs cÖ‡kœi DËi

K mgvb ev cÖvq mgvb ZxeªZv Ges cÖvq mgvb K¤úv¼ wewkó GKB

w`‡K AMÖMvgx ywU kã Zi‡½i DcwicvZ‡bi d‡j jwä cÖve‡j¨i

n«vm-e„w×i NUbv‡K exU e‡j|

L GKwU K¤úgvb e¯‘‡K Ab¨ GKwU e¯‘i wbKU ai‡j wØZxq e ‘wU

Kuvc‡Z ïiæ K‡i; G‡K ciek K¤úb e‡j| hw` e ‘i ¯vfvweK

ch©vqKvj I cÖhy³ e‡ji ch©vqKvj wfbœ nq Z‡e e ‘ ÿz`ª we Ív‡i

Kuvc‡e| wKš‘ e ‘i ¯^vfvweK ch©vqKvj I Zvi Dci cÖhy³ e‡ji

ch©vqKvj mgvb n‡j e ‘wU e„nËi we¯ Ív‡i Kuvc‡Z eva¨ nq Ges

k‡ãi cÖvej¨ e„w× cvq| G cÖwµqv‡K Abybv` e‡j| myZivs ejv hvq,

Abybv` ciek K¤úb wKš‘ mKj ciek Abybv` bq|†Kbbv, mKj

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Aa¨vq-9: Zi½

ciek K¤ú‡b cÖhy³ e‡ji ch©vqKvj e ‘i ¯^vfvweK ch©vqKv‡ji

mgvb n‡e bv, ïay mgvb n‡j †mB ciek K¤úb‡KB Avgiv Abybv`

ej‡ev| myZivs, Abybv` ciek K¤ú‡bi GKwU we‡kl Ae ’v|

M cÖ Ë Zi‡½i mgxKiY, y = 0.9 sin

x

15 + 2t0.3 GB mgxKi‡Yi

meKUv ivwk Gm.AvB GK‡K cÖKvwkZ| cÖ`Ë mgxKiYwU‡K,

AMÖMvgx Zi‡½i mvaviY mgxKiY,

y = a sin

2x

+ t Gi mv‡_ Zzjbv K‡i cvB,

we¯Ívi, a = 0.9 m

2

=

115

= 2 15 = 30

Zi½‰`N©¨, = 30 m

Ges †KŠwYK K¤úv¼, = 2

0.3 = 203 rads1

Avgiv Rvwb, K¤úv¼ n‡j,

= 2

=

2

=

203

2

= 203

1

2

= 10

3 Hz

Zi½wUi †eM, v =

= 10

3 30 ms1

= 100 ms1 (Ans.)

N †`Iqv Av‡Q,

bxjv †h gva¨‡g Zi½ Drcbœ K‡i Zvi NbZ¡, = 0.4 kgm3

ÔMÕ Ask †_‡K cvB, bxjv KZ©„K Drcbœ Zi‡½i,

we¯Ívi, a = 0.9 m

K¤úv¼, = 10

3 Hz.

Ges †eM, v = 100 ms1

bxjv KZ©„K Drcbœ Zi‡½i ZxeªZv,

I = 22a22v

= 22 0.4 (0.9)2

10

3

2

100 Wm2

= 22 0.4 (0.9)2 102

92 100 Wm2

= 720 Wm2

Avgiv Rvwb, cÖgvY ZxeªZv, Io = 1012Wm2

bxjvi k‡ãi ZxeªZv †j‡fj, = 10 log IIo

= 10 log

720

1012 dB

= 148.57 dB

GLb, g‡bKwi, w`bvi k‡ãi ZxeªZv = I1 Ges webvi k‡ãi ZxeªZv =

I2

kZ©g‡Z,

68 = 10 log

I1

1012

I1 = 6.31 106Wm2

Ges 74 = 10 log

I2

1012

I2 = 2.51 105Wm2

w`bv I webvi mw¤§wjZ k‡ãi ZxeªZv,

I3 = I1 + I2

= (6.31 106 + 2.51 105) Wm2

= 3.141 105 Wm2

w`bv I webvi mw¤§wjZ k‡ãi ZxeªZv †j‡fj,

1 = l0 log

I3

1o

= 10 log

3.14 105

1012 dB

= 74.97 dB

2

= 21

bxjvi k‡ãi ZxeªZv †j‡fj, w`bv I webvi mw¤§wjZ k‡ãi ZxeªZv

†j‡f‡ji †P‡q wظY n‡e|

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Aa¨vq-9: Zi½

cÖkœ14 y = 10 sin(10t – 2x) Zi½wU euvav †c‡q cÖwZdwjZ

n‡q cybivq GKB c‡_ wecixZ w`‡K wd‡i bZzb Zi½ m„wó K‡i|

[miKvwi Gg.Gg. K‡jR, h‡kvi, cÖkœ-4]

K. ¯^iK¤ú Kv‡K e‡j? 1

L. mKj Dcmyi mg‡gj bq e¨vL¨v Ki| 2

M. DÏxc‡Ki Av‡jv‡K Zi½‡eM wbY©q Ki| 3

N. DÏxcK Abymv‡i Drcbœ bZzb Zi½ †Kvb ai‡Yi Ges †mB

bZzb Zi‡½i my¯ú›` I wb¯ú›` we›`yi Ae¯’vb wK GKB

n‡e? MvwYwZKfv‡e we‡kølY Ki| 4

14 bs cÖ‡kœi DËi

K mgvb ev cÖvq mgvb we Ív‡ii ywU kã Zi½ GKB mgq GKB

mij‡iLv eivei GKB w`‡K mÂvwjZ n‡Z _vK‡j G‡`i

DcwicvZ‡bi d‡j k‡ã ZxeªZvi †h ch©vqµwgK n«vm e„w× N‡U Zv‡K

¯^iK¤ú e‡j|

L †Kv‡bv ¯‡i we`¨gvb myi¸‡jvi g‡a¨ †hwUi K¤úv¼ me‡P‡q Kg

Zv‡K g~j myi ev †gŠwjK myi e‡j| Ab¨vb¨ myi, hv‡`i K¤úv¼ g~j

my‡ii †_‡K †ewk Zv‡`i‡K Dcmyi e‡j| Avevi †Kv‡bv †Kv‡bv

Dcmy‡ii K¤úv¼ g~j my‡ii K¤úv‡¼i mij ¸wYZK| Zv‡`i‡K H

g~j my‡ii mg‡gj e‡j| Kv‡RB mKj mg‡gj Dcmyi wKš‘ mKj

Dcmyi mg‡gj bq|

M †`Iqv Av‡Q, GKwU Zi‡½i mgxKiY

y = 10 sin(10t – 2x)

= 10 sin 2

10

2t – x .......(i)

(i) bs mgxKiYwU‡K y = asin 2

(vt – x) mgxKi‡Yi mv‡_ Zzjbv

K‡i cvB, v = 10

2 ms–1 = 1.6 ms–1 (Ans.)

N aiv hvK, y = 10 sin(10t – 2x) Zi½wU abvZ¥K X-A‡ÿi

Awfgy‡L Pj‡Q|

Zi½wU euvav †c‡q cÖwZdwjZ n‡q –X A‡ÿi Awfgy‡L Pj‡Q|

Zvn‡j, cÖwZdwjZ Zi‡½i mgxKiY, y = 10sin(10t + 2x)|

aiv hvK, Zi½ ywU GKwU AciwUi Dci AvcwZZ n‡jv| GLb GB

`ywU Zi‡½i jwä miYÑ

y = y + y

= 10 sin (10t – 2x) + 10 sin (10t + 2x)

= 10 sin 2

10

2t – x + 10 sin2

10

2t + x

= 2 10 sin2. 10

2t. cos2.x

= 20 cos2x. sin10t

= A sin 10t

GLv‡b, A = 20 cos 2x = jwä Zi‡½i x ~i‡Z¡ Aew¯’Z KYvi

we¯Ívi|

Dc‡iv³ jwä Zi‡½i mgxKi‡Y `kvi †Kv‡bv cv_©K¨ †bB| A_©vr

AMÖMvgx Zi‡½i b¨vq `kv †Kv‡Yi wfZi (vt – x) RvZxq †Kv‡bv

ivwki AšÍf©yw³ †bB| myZivs DÏxcK Abymv‡i Drcbœ Zi½wU GKwU

w¯’i Zi½|

GLb, x Gi †h gv‡bi Rb¨ cos 2x = 0 n‡e †mme we› y‡Z we¯ Ívi

k~b¨ n‡e A_©vr †m me we› y‡Z wb¯ú›` we›`y cvIqv hv‡e| myZivs

wb¯ú›` we›`yi Rb¨,

cos 2x = 0

ev, 2x = (2n – 1)

2 GLv‡b, n = 1, 2, 3, .....

ev, x = (2n – 1) 14 =

14,

34,

54, .....

Avevi, x Gi †h gv‡bi Rb¨ cos 2x = 1 n‡e †mme we›`y‡Z

we¯Ívi m‡e©v”P 20 n‡e A_©vr †mme we›`y‡Z my¯ú›` we›`y cvIqv

hv‡e| myZivs my¯ú›` we›`yi Rb¨,

cos 2x = 1

ev, 2x = n GLv‡b, n = 0, 1, 2, 3, .....

ev, x = n12 = 0,

12, 1,

32, .....

Dc‡ii MvwYwZK we‡k ølY †_‡K †`Lv hvq †h, bZzb Zi‡½i my¯ú›`

I wb¯ú›` we›`yi Ae ’vb GKB bq|

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