04| wbDUwbqb ejwe`¨v

(Newtonian Mechanics)

‡gv: kvn& Rvgvj mnKvix Aa¨vcK (c`v_©weÁvb wefvM ) we G Gd kvnxb K‡jR XvKv ‡dvb: 01670 856105, 9125630, 9115369

E-mail: sjamal59@gmail.com

RoZv (Inertia): c`v_© †h Ae¯’vq Av‡Q †mB Ae¯’vq _vK‡Z PvIqvi †h cÖeYZv ev †mB Ae ’v ivL‡Z PvIqvi †h ag© Zv‡K RoZv e‡j|

RoZv yB cÖKvi: h_v (1) w¯’wZ RoZv (2) MwZ RoZv

(1) w¯’wZ RoZv (Inertia of rest) t w¯’i e ‘ w¯’i _vK‡Z Pvq e ‘i GB ¸b‡K e¯‘i w¯’wZ RoZv e‡j|

D`vniYt w¯’i Mvwo nVvr †Q‡o w`‡j hvwÎiv wcQb w`‡K †n‡j c‡o| KviY Mvwo Pvjy nIqvi ms‡M ms‡M hvw·`i

kix‡ii wb‡Pi Ask Mvwoi mv‡_ mshy³ _vKvq mvg‡bi w`‡K GwM‡q hvq| wKš‘ kix‡ii Dc‡ii Ask w¯’wZ RoZvi Rb¨

wcwQ‡q c‡o|

(2) MwZ RoZv (Inertia of motion) t MwZkxj e¯‘ MwZkxj _vK‡Z Pvq e¯‘i GB ¸b‡K e¯‘i MwZ RoZv e‡j|

D`vniYt PjšÍ Mvwo nVvr †eªK Ki‡j hvwÎiv mvg‡bi w`‡K Sz‡K c‡o| KviY Mvwo ‡eªK Kivi Kvi‡Y Mvwo †_‡g

hvIqvi mv‡_ mv‡_ hvw·`i kix‡ii wb‡Pi Ask Mvwoi mv‡_ mshy³ _vKvq ‡_‡g hvq wKš‘ kix‡ii Dc‡ii Ask MwZ

RoZvi Rb¨ mvg‡bi w`‡K GwM‡q hvq|

ej (Force): hv w¯’i e¯‘i Dci wµqv K‡i e¯‘‡K MwZkxj K‡i ev Ki‡Z Pvq A_ev MwZkxj e¯‘i Dci wµqv K‡i MwZi cwieZ©b K‡i ev

Ki‡Z Pvq Zv‡K ej e‡j| ej‡K F Øviv cÖKvk Kiv nq| m f‡ii e¯‘i Z¡iY a n‡j ej F=ma n‡e| ej GKwU †f±i

ivwk|

‡gŠwjK ej (Fundamental Forces) t ‡h mKj ej ¯vaxb A_©vr ‡h mKj ej Ab¨ †Kvb ej †_‡K Drcbœ nq bv Zv‡K eis Ab¨vb¨ ej GB mKj e‡ji †Kvb bv

†Kvb iƒ‡ci cÖKvk Zv‡`i‡K †gŠwjK ej e‡j| †gŠwjK ej Pvi cÖKvi, h_v: (1) gnvKl© ej (2) ZvwoZ †PŠ¤K ej (3)

mej wbDwK¬q ej I (4) ~e©j wbDwK¬q ej

(1) gnvKl© ej t gnvwe‡k¦i †h †Kvb ywU e¯‘i ga¨Kvi cvi¯úwiK AvKl©b ej‡K gnvKl© ej e‡j|

(2) ZvwoZ †PŠ¤K ej t ywU PvwR©Z KYv Zv‡`i Pv‡R©i Kvi‡Y G‡K Ac‡ii Dci †h AvKl©Y ev weKl©b ej cÖ‡qvM K‡i

Zv‡K ZvwoZ †PŠ¤K ej e‡j|

(3) mej wbDwK¬q ej t cigvbyi wbDwK¬qv‡m wbDwK¬q Dcv`vb Z_v wbDwK¬qb¸‡jv‡K GK‡Î Ave× iv‡L †h kw³kvjx ej

Zv‡K mej wbDwK¬q ej e‡j|

(4) ~e©j wbDwK¬q ej t †h ¯í cvjøvi I ¯^í gv‡bi ej wbDwK¬qv‡mi g‡a¨ †gŠwjK KYv¸wji g‡a¨ wµqv K‡i A‡bK

wbDwK¬qv‡m Aw¯’wZkxjZvi D™¢e NUvq Zv‡K ~e©j wbDwK¬q ej e‡j|

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 2‡gŠwjK ej mg~n Av‡cwÿK wZeªZv cvjøv

1) gnvKl© ej 1 Amxg

2) ZvwoZ †PŠ¤K ej 1039 Amxg

3) mej wbDwK¬q ej 1041 10-15 m 4) ~e©j wbDwK¬q ej 1030 10-16 m

fi‡eM (Momentum) t e ‘i fi I †e‡Mi mgš‡q e¯‘‡Z †h a‡g©i D™¢e nq Zv‡K fi‡eM e‡j| fi I †e‡Mi ¸bdj

Øviv fi‡eM cwigvc Kiv nq| fi‡eM‡K P Øviv cÖKvk Kiv nq| m f‡ii †Kvb e¯‘i †eM v n‡j fi‡eM P = mv n‡e| fi‡e‡Mi GKK kg-ms-1

| Gi gvÎv [MLT-1]

wbDU‡bi MwZi 1g m~Î (Newton's 1st Law of Motion) t

eY©bvt evwn¨K ej cÖ‡qvM bv Ki‡j w¯’i e¯‘ wPiKvj w ’i Ges MwZkxj e¯‘ mg‡e‡M mij c‡_ Pj‡Z _vK‡e|

e¨L¨vt evB‡i †_‡K ej cÖhy³ bv n‡j (1) w¯’i e¯‘ wPiKvj w¯’i _vK‡e Ges

(2) MwZkxj e¯‘ mg‡e‡M mij c‡_ Pj‡Z _vK‡e|

myZivs †`Lv hvq †h, MwZi cÖ_g m~‡Îi ywU Ask| cÖ_g Ask †_‡K w¯’i e¯‘i w¯’wZkxj _vKvi cÖeYZv m¤ú‡K© ¯úó

aviYv cvIqv hvq| GB cÖeYZv‡K w¯’wZ RoZv e‡j| Avevi wØZxq As‡k MwZkxj e¯‘i MwZkxj _vKvi cÖeYZv jÿbxq |

GB cÖeYZv‡K MwZ RoZv e‡j| G Kvi‡Y wbDU‡bi MwZi 1g m~·K RoZvi m~Î I ejv nq| G m~Î †_‡K e‡ji msÁv I

cvIqv hvq|

wbDU‡bi MwZi 2q m~Î (NewtonÕs 2nd Law of Motion)t eY©bvt e¯‘i fi †eM cwieZ©‡bi nvi e¯‘i Dci cÖhy³ e‡ji mgvbycvwZK Ges ej †h w`‡K wµqv K‡i e¯‘i fi †e‡Mi

cwieZ©b I †mB w`‡K N‡U|

e¨L¨vt G m~‡Îi e¨vL¨v nj amF cÖwZcv`b|

amF cÖwZcv`b t

g‡b Kwi, †Kvb MwZkxj e¯‘KYvi fi‡eM P

| F

e‡ji wµqvq ÿz ªvwZÿz ª mgq e¨eavb dt AeKv‡k KYvi fi‡e‡Mi

cwieZ©b Pd

n‡j fi‡eM cwieZ©‡bi nvi

dtPd

n‡e| MwZi 2q m~Îvbymv‡i, fi‡eM cwieZ©‡bi nvi cÖhy³ e‡ji

mgvbycvwZK| A_©vr, FdtPd

e¯‘KYvi fi m Ges †eM v n‡j fi‡eM vmP

ewm‡q cvB,

Fdt

)vm(d

Fdtvdm

dtvda Fma

iYZ¡

(1) ... .... ... ... ... Fkma

GLv‡b k GKwU mgvbycvwZK aªæeK; hvi gvb GKK e‡ji

msÁv Øviv wba©viY Kiv hvq| ‡h ej GKK f‡ii Dci wµqv K‡i GKK Z¡iY m„wó K‡i Zv‡K GKK ej e‡j|

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 3A_©vr 1F

, m=1 Ges 1a

nq Z‡e mgxKiY (1) n‡Z cvB|

1×1=k×1 k=1 mgxKi‡Y ewm‡q cvB, Fam

A_©vr amF

†Kvb e¯‘i Dci ¯^Zš¿ ej¸wj hw` nq h_vµ‡g n321 F,..F,F,F

BZ¨vw` wµqviZ _vK‡j Avgiv cvB,

maF....FFF n321

amF GLv‡b, F

n‡”Q jwä ej ev bxU ej †hLv‡b, n321 F....FFFF

(cÖgvwbZ)

e‡ji GKK wbDUb t †h ej 1kg f‡ii Dci wµqv K‡i 1ms -1 Z¡iY m„wó K‡i Zv‡K 1 wbDUb ej e‡j|

A_©vr 1 wbDUb = 1kg ×1ms -2 |

wØZxq m~Î †_‡K cÖ_g m~Î cÖwZcv`b :

wbDU‡bi MwZi 1g I 2q m~Î Zzjbv K‡i †`Lv hvq †h, 1g m~Î n‡”Q 2q m~‡Îi GKwU iƒc| amF †_‡K †`Lv hvq

†h, e¯‘i Dci wbU ej k~b¨ n‡j A_©vr hLb 0F

ZLb 0am

,wKš‘ ,0m 0a

nq| myZivs hLb evB‡i †_‡K

ej cÖ‡qvM bv nq A_©vr wbU ej k~b¨ nq ZLb,

0a

0dtvd

v Gi wWdv‡iw݇qmvb k~b¨ v = aªæeK|

myZivs evB‡i †_‡K e¯‘i Dci ej wbU ej k~b¨ n‡j e ‘i †eM AcwiewZ©Z _v‡K A_©vr e ‘ †h Ae¯’vq wQj †mB Ae¯’vqB

_v‡K| A_©vr evwn¨K ej cÖ‡qv‡M e¯‘ Ae¯’vb cwieZ©b Ki‡Z eva¨ bv Ki‡j w¯’i e¯‘ wPiKvj w ’i Ges MwZkxj e¯‘ wPiKvj

mg‡e‡M mij c‡_ Pj‡Z _vK‡e| GwUB wbDU‡bi MwZi cÖ_g m~Î|

NvZej (Impulsive Force): Lye Aí mgq †h ej cÖhy³ nq Zv‡K NvZ ej e‡j| nvZzwo Øviv Kv‡V †c‡iK †cvZv, e¨vU Øviv wµ‡KU e‡j AvNvr Kiv,

BZ¨vw` NvZ e‡ji D`vniY|

e‡ji NvZ (Impulse of Force): ‡Kvb ej I e‡ji wµqv Kv‡ji ¸b dj‡K e‡ji NvZ e‡j| †Kvb ej F

hw` †Kvb e¯‘i Dci it mgq †_‡K ft mgq ch©šÍ

wµqv K‡i, Zvn‡j e‡ji NvZ n‡e, f

i

t

t

dtFJ

][ æeaª FdtFJf

i

t

t

] t-[t F tFJ if

t

t

f

i

tFJ

e‡ji Nv‡Zi mv‡_ fi‡e‡Mi m¤úK© (Relation with Impulse of Force and Momentum ) t

cÖhy³ ej F

n‡j e‡ji NvZ f

i

t

t

dtFJ

dt

PdF dtdtPdJ

f

i

t

t

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 4

f

i

f

i

P

P

P

p

PPdJ

if PPJ

PJ

A_©vr, e‡ji NvZ fi‡e‡Mi cwieZ©‡bi mgvb|

wbDU‡bi MwZi Z…Zxq m~Î (Newton's 3rd Law of Motion) t eY©bv: cÖ‡Z¨K wµqviB GKwU mgvb I wecixZ cÖwZwµqv Av‡Q|

wbDU‡bi MwZi Z…Zxq m~Îvbymv‡i hw` GKwU e¯‘ A, Aci e¯‘ B Gi Dci ej cÖ‡qvM K‡i, Z‡e B e¯‘I A e¯‘i Dci mgvb

I wecixZ ej cÖ‡qvM Ki‡e| awi A e¯‘ B e¯‘i Dci F1 ej cÖ‡qvM K‡i Ges B e¯‘ A e¯‘i Dci F2 ej cÖ‡qvM K‡i|

m~Îvbymv‡i, F1 = F2 , A_©vr ej ؇qi gvb mgvb wKš‘ wecixZgywL| G ywU e‡ji †h †Kvb GKwU‡K wµqv Ges

Ab¨wU‡K cÖwZwµqv e‡j| Dc‡ii „óv‡šÍÍ wµqv I cÖwZwµqv mgvb I wecixZ|

fi‡e‡Mi msiÿb m~Î (Law of Conservation of Momentum) t GKvwaK e¯‘i g‡a¨ wµqv cÖwZwµqv ej Qvov Ab¨ †Kvb ej wµqv bv Ki‡j, H e¯‘ mg~‡ni †gvU fi‡e‡Mi mgwó aªæe _v‡K|

MvwbwZK fv‡e, æeKaªP

cÖwZcv`b: 1 I 2 ywU e¯‘ we‡ePbv Kwi| awi †Kvb GK mgq e¯‘Øq

msN‡l© wjß nj| aiv hvK, GB msN‡l© F1 n‡”Q cÖ_g e¯‘i Dci wØZxq

e¯‘ KZ…K cÖhy³ ej Ges F2 n‡”Q wØZxq e¯‘i Dci cÖ_g e¯‘ KZ…K cÖhy³ ej| †Kvb GK mgq t †Z cÖ_g e¯‘i fi‡eM 1P

Ges wØZxq e¯‘i fi‡eM 2P

| Avgiv cÖwZwU e¯‘i †ÿ‡Î wbDU‡bi 2q m~Î cÖ‡qvM K‡i cvB,

dtPdF 1

1

Ges

dtPdF 2

2

wbDU‡bi MwZi Z…Zxq m~Î †_‡K cvB, 21 FF

dtPd

dtPd 21

0dtPd

dtPd 21

0PPdtd

21

0dtPd

21 PP[

n‡”Q e¨e¯’vi †gvU fi‡eM ev ]P

‡h‡nZz mg‡qi mv‡c‡ÿ †gvU fi‡eM 21 PPP

, P

Gi cwieZ©‡bi nvi k~b¨, ZvB †gvU fi‡eM P

aªæe _v‡K, A_©vr

21 PPP

= aªæe fi PP

f2f1i2i1 PPPP

myZivs †`Lv hv‡”Q †h, msN‡l©i Av‡M †Kvb e¨e¯’vi fi‡e‡Mi

†f±i mgwó Avi msN‡l©i c‡i fi‡e‡Mi †f±i mgwó me© v mgvb _v‡K| GwUB fi‡e‡Mi msiÿb m~Î|

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 5wbDU‡bi MwZ m~‡Îi e¨envi:

(1) †Nvovi Mvox:

‡Nvov G¸evi Rb¨ cv Øviv Zxh©Kfv‡e gvwU‡Z AvNvZ K‡i| d‡j gvwUI GKwU cÖwZwµqv ej †Nvovi cv‡qi Dci cÖhy³

e‡ji wecix‡Z cÖ‡qvM K‡i| wbDU‡bi Z…Zxq m~Î Abyhvqx GB wµqv I cÖwZwµqv ej mgvb|

GB cÖwZwµqv e‡ji Dj¤ Dcvsk Rsin †Nvovi IRb‡K cÖkwgZ K‡i Avi Abyf‚wgK Dcvsk | Rcos ‡Nvov‡K Mvoxi PvKv

I f‚wgi Nl©b ej‡K AwZµg K‡i Mvwo‡K mvg‡bi w`‡K GwM‡q hvq|

(2) bUvbv †bŠKv:

b`x ev Lv‡j †mªv‡Zi wecix‡Z †bŠKv Pvjbv Kivi Rb¨ ¸b ev iwk †bŠKvi mv‡_ †eu‡a iwki Ab¨ cÖvšÍ gvby‡li Kuv‡ai mv‡_

AvU‡K a‡i mvg‡b †U‡b wb‡q hvIqv nq| wP‡Î OA eivei ¸‡bi Uvb

F †`Lv‡bv n‡q‡Q| F Gi ci¯úi j¤ Dcvsk ywU n‡jv OB eivei

Fcos Ges BA eivei Fsin | Fcos †bŠKv‡K †mªv‡Zi wecix‡Z

†U‡b mvg‡b wb‡q †h‡Z m‡Pó nq| Gi gvb hZ Kg nq Fcos ZZ

†ekx nq| Ab¨ w`‡K Fsin †bŠKvwU‡K b`x ev Lv‡ji cv‡oi w`‡K

†U‡b wb‡Z Pvq| G‡ÿ‡Î †bŠKvi gvwS †bŠKvi nvj Ggb fv‡e mwµqiv‡L

hv‡Z K‡i †bŠKvi AMÖfv‡M Fsin Gi wecix‡Z GKwU ej jvf K‡i|

d‡j Fsin cÖkwgZ n‡q hvq| Gfv‡e †bŠKvwU ¸‡bi Uv‡b mvg‡bi w`‡K AMÖmi nq|

(3) i‡K‡Ui MwZ (Motion of Rocket) t awi i‡KUwU AwfK‡l©i AvIZvgy³ gnvk~‡b¨ MwZkxj| hLb i‡K‡Ui BwÄb KZ…K M¨vmwbM©Z nq

ZLb †mB M¨v‡mi GKwU fi‡eM _v‡K| ZLb fi‡eM msiwÿZ _v‡K e‡j i‡KU wecixZ w`‡K

MwZkxj nq| i‡KU ‡_‡K M¨vm hLb GKwU wbw ©ó nv‡i wbM©Z n‡Z _v‡K, ZLb M¨v‡mi MwZi

wecixZ w`‡K i‡KUwU GKwU w¯’i ej jvf K‡i| GB ej‡K av°v e‡j|

wnmve: g‡bKwi i‡KU ‡_‡K R¡vjvwb Z_v M¨vm v †e‡M wbM©Z n‡”Q| t mg‡q wbM©Z M¨v‡mi

fi m n‡j, wbM©Z M¨v‡mi fi‡eM n‡e,

P= (m)v

wbM©Z M¨v‡mi GB fi‡eM i‡KU¯’ R¡vjvwbi fi‡e‡Mi cwieZ©‡bi mgvb| fi‡e‡Mi msiÿb

m~Îvbymv‡i R¡vjvwbi fi‡e‡Mi GB cwieZ©b i‡KUwUi fi‡e‡Mi cwieZ©‡bi mgvb| Avgiv Rvwb,

‡Kvb e¯‘i fi‡e‡Mi cwieZ©‡bi nvi Zvi Dci cÖhy³ e‡ji Nv‡Zi mgvb| myZivs i‡K‡Ui Dci

cÖhy³ av°v F n‡j P = Ft (m)v = Ft

vtmF ,

ev GLv‡b

tm n‡”Q R¡vjvwb e¨env‡ii nvi|

‡Kvb gyn~‡Z© i‡K‡Ui fi M n‡j Ges H gyn~‡Z© Zvi Z¡iY a n‡j, F Gi gvb F= Ma ewm‡q cvB,

vtmMa

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 6

vtm

Ma

1 iY,Z¡

(4) e› y‡Ki ¸wj †Qvov:

¸wj †Qvovi mv‡_ mv‡_ e› yKI MwZi wecixZ w`‡K GKwU MwZ jvf K‡i| fi †e‡Mi wbZ¨Zv m~Î cÖ‡qvM K‡i GB cðvr

†e‡Mi gvb wbY©q Kiv hvq|

g‡bKwi, e› y‡Ki fi M, ¸wji fi m| ¸wj †Qvovi Av‡M

e› y‡Ki †eM Vi = 0, ¸wji †eM vi = 0|

Zvn‡j e› yK I ¸wji †gvU Avw` fi‡e‡Mi mgwó Pi = MVi + mvi =0

¸wj †Qvovi mv‡_ mv‡_ e› y‡Ki †eM Vf =V I ¸wji †eM vf = v, e› yK I ¸wji †gvU ‡kl fi‡e‡Mi mgwó

Pf =MVf + mvf = MV + mv fi †e‡Mi wbZ¨Zv m~Î Abymv‡i, Pi =Pf

ev, 0 = MV + mv ev, MV = - mv

)1....(.......... vMmV

(1) bs mgxKiY n‡Z †`Lv hvq †h, e› yK I ¸wji †eM ci¯úi wecixZ gyLx|

RoZvi åvgK (Moment of Inertia)t ‡Kvb N~Y©vqgvb e¯‘i cÖ‡Z¨KwU KYvi fi Ges N~Y©b Aÿ‡_‡K G‡`i cÖ‡Z¨KwU ~i‡Z¡i e‡M©i ¸bd‡ji mgwó‡K H A‡ÿi

mv‡c‡ÿ H e ‘wUi RoZvi åvgK e‡j| G‡K I Øviv cÖKvk Kiv nq|

e¨vL¨vt g‡b Kwi, †Kvb e¯‘i KYv¸‡jvi fi h_vµ‡g m1, m2, m3 .......................|

GB KYv¸‡jvi N~Y©b Aÿ ‡_‡K h_vµ‡g r1, r2, r3, .................... BZ¨vw` j¤ ~i‡Z¡ Aew¯’Z| myZivs H A‡ÿi mv‡c‡ÿ

e¯‘wUi RoZvi åvgK ... ... ... ... 2233

222

211 nnrmrmrmrmI

2 ii rmI Gi gvÎv ML2 Ges Gi GKK Kgm2 MwZkw³ cÖKv‡ki ivwkgvjv t

g‡b Kwi, M fi wewkó †Kvb e¯‘‡Z N msL¨K KYv Av‡Q| e¯‘wU BD A‡ÿi PZ©y„w`‡K mg‡KŠwbK †e‡M N~Y©vqgvb| GB

N~Y©‡bi Kvi‡Y e¯‘wU †h MwZ kw³ jvf K‡i Zv‡K N~Y©b MwZkw³ e‡j| g‡b Kwi M f‡ii e¯‘wU m1, m2, m3............AmsL¨

e¯‘ KYvi mgwó Ges N~Y©b Aÿ †_‡K h_vµ‡g r1, r2, r3, .................. rn ~i‡Z¡ Aew¯’Z| †Kvb Aÿ †_‡K N~Y©bkxj †Kvb e¯‘i

~iZ¡ ej‡Z Avgiv j¤ ~iZ¡ eywS| KYv ¸wji mg‡KŠwbK †eM GKB A_©vr wKš‘ N~Y©b Aÿ †_‡K KYv¸wji ~iZ¡ GK bq

e‡j G‡`i ˆiwLK †eM GK n‡e bv| m1 f‡ii KYvi ˆiwLK †eM v1 = r1 2

12

1212

1121

11 E , m rmvm MwZkw³ KYvi Avevi m2 f‡ii KYvi ˆiwLK †eM v2 = r2

22

222

12222

122 E , m rmvm MwZkw³ KYvi

Gfv‡e Avgiv cÖ‡Z¨K KYvi MwZ kw³ wbY©q Ki‡Z cvwi Ges GB KYv¸‡jvi MwZ

kw³ †hvM K‡i Avgiv mgMÖ e¯‘i MwZ kw³ wbY©q Ki‡Z cvwi| myZivs mgMÖ e¯‘i MwZkw³,

E = E1+E2+E3+ ..................... +En

22212

32

3212

22

2212

12

121 .................. nn rmrmrmrmE

2233

222

211

221 .................. nnrmrmrmrmE

åvgK RoZvib,GLv‡ I ............... I 22233

222

211

221

iinn rmrmrmrmrmIE

e¯‘wU GKK mg‡KŠwbK †e‡M Nyi‡j =1 mgxKi‡Y ewm‡q cvB, IE I1E 2

121 A_©vr GKK mg‡KŠwbK †e‡M Nyb©vqgvb †Kvb e¯‘i MwZkw³ Gi RoZvi åvg‡Ki A‡a©K|

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 7

2E I A_©vr GKK mg‡KŠwbK †e‡M Nyb©vqgvb †Kvb e¯‘i RoZvi åvgK e¯‘i MwZkw³i wظb|

‡KŠwbK fi‡eM (Angular Momentum) t N~Y©i MwZi Rb¨ GKwU e¯‘KYv †h fi‡eM cÖvß nq Zv‡K H e ‘KYvi ‡KŠwbK fi‡eM e‡j|

Ab¨ K_vq, †Kvb e¯‘KYvi ‰iwLK fi‡eM Ges †Kvb we› y †_‡K fi‡e‡Mi wµqv †iLvi j¤

~i‡Z¡i ¸bdj‡K H we› yi mv‡c‡ÿ e¯‘KYvwUi †KŠwbK fi‡eM e‡j|

e¨vL¨v t N~Y©b †K‡› ªi mv‡c‡ÿ †Kvb KYvi e¨vmva© †f±i ev Ae ’vb †f±i Ges

H KYvi ˆiwLK fi‡e‡Mi †f±i ¸bdj‡K †KŠwbK fi‡eM e‡j| Ae¯’vb †f±i r Ges

H KYvi ˆiwLK fi‡eM p n‡j, H we› yi mv‡c‡ÿ KYvwUi †KŠwbK fi‡eM prL

rpSinL

‡KŠwbK fi †e‡Mi ivwkgvjvt

g‡b Kwi, GKwU e¯‘ †Kvb GKwU A‡ÿi mv‡c‡ÿ mg‡KŠwbK ªæwZ‡Z N~Y©vqgvb| D³ e¯‘i †h †Kvb GKwU KYvi

fi 1m , N~Y©b Aÿ †_‡K LYvwUi j¤ ~iZ¡ 1r Ges KYwUi †eM 1v n‡j

N~Y©b Aÿ mv‡c‡ÿ KYvwUi †KŠwbK fi‡eM 111 rvm

]r v[ r m 112

11

Abyiƒ‡c, N~Y©b Aÿ mv‡c‡ÿ 2m KYvwUi †KŠwbK fi‡eM

222 r m Gfv‡e cÖwZwU

e¯‘ KYvi Rb¨ †KŠwbK fi‡eM †ei K‡i Zv‡`i mgwó wb‡j m¤ú~Y© e¯‘wUi I KŠwbK

fi‡eM L cvIqv hv‡e|

................r mr mr mL 233

222

211

......)..........rmrmrm( L 233

222

211

..)]..........([ 233

222

211 rmrmrmIIL

‡KŠwbK miY:

g‡b Kwi GKwU „oe¯‘ O we› y mv‡c‡ÿ xy Z‡j N~Y©vqgvb| GB N~Y©vqgvb e¯‘wU AmsL¨ KYv Øviv

MwVZ| Giƒc GKwU KYv P, O n‡Z r ~i‡Z¡ †_‡K e„ËvKvi c‡_ Pjgvb| e¯‘wUi cÖwZwU KYv GKB

mg‡q mgvb †KŠwYK ~iZ¡ AwZµg Ki‡Q|

aiv hvK, t1 mg‡q N~Y©vqgvb „p e¯‘i P KYvi †KŠwYK Ae¯’vb x A‡ÿi mv‡c‡ÿ Ges t2 mg‡q

| d‡j 12 ttt mg‡q P KYvi †KŠwYK miY nj 12 | †KŠwbK mi‡Yi GKK

†iwWqvb ev wWwMÖ|

‡KŠwbK ‡eM:

‡KŠwbK mi‡Yi nvi‡K †KŠwYK †eM e‡j|

t mgq e¨eav‡b P we› yi Mo †KŠwYK †eM

ttt

12

12

Ges t k~‡b¨i KvQvKvwQ n‡j ZvrÿwYK †KŠwbK †eM

dtd

tLt

t

0

†KŠwbK mi‡Yi GKK rads-1|

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 8‡KŠwbK Z¡iY:

‡KŠwbK ‡e‡Mi cwieZ©‡bi nvi‡K †KŠwYK Z¡iY e‡j|

aiv hvK, t1 mg‡q P KYvi †KŠwbK †eM 1 Ges t2 mg‡q †KŠwYK †eM 2 |

myZivs t mgq e¨eav‡b †KŠwYK †e‡Mi cwieZ©b 12 Zvn‡j Mo †KŠwYK Z¡iY

)1.(....................12

12

ttt

Ges t k~‡b¨i KvQvKvwQ n‡j ZvrÿwbK †KŠwYK Z¡iY

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

2

0 dtd

dtd

dtd

dtd

tLt

t

‡KŠwYK Z¡i‡Yi GKK rads-2| wbw ©ó mgq e¨eav‡b „p e¯‘i cÖwZwU KYvi †KŠwYK miY GKB| d‡j cÖwZwU KYvi †KŠwYK

†eMI GKB n‡e|

N~Y©b MwZi mgxKiY:

(1) mgxKiY t 0 cÖwZcv`b

g‡b Kwi, GKwU e¯‘ AwZ ÿz ª e¯‘ †Kvb we› y‡K ev Aÿ‡K †K› ª K‡i N~Y©vqgvb| t = 0 mg‡q e¯‘wUi Avw` †KŠwYK †eM 0 ,

t mg‡q †KŠwbK †eM I mg‡KŠwYK Z¡iY n‡j †KŠwYK Z¡i‡Yi msÁvbymv‡i,

dtd

„p e¯‘wU mgZ¡i‡Y N~Y©vqgvb e‡j †KŠwYK Z¡iY GKwU aªæe msL¨v| G mgxKiY n‡Z wjLv hvq

)3.....(.......... dtd

aiv hvK, t = 0 mg‡q e¯‘KYv P Gi †KŠwYK ¯’vbv¼ 0 Ges t mgq c‡i e¯‘ KYv P Gi ’vbv¼

mgxKiY (3) †K t = 0 n‡Z t mgq ch©šÍ mgvKjb K‡i cvB,

t

dtd0

0

tt 00

)0(0 t

)4.(.......... 0 t cÖwZcv`b Kiv nj|

(2) mgxKiY

200 2

1 tt cÖwZcv`b

g‡b Kwi, GKwU e¯‘ AwZ ÿz ª e¯‘ †Kvb we› y‡K ev Aÿ‡K †K› ª K‡i N~Y©vqgvb| t = 0 mg‡q e¯‘wUi Avw` †KŠwYK †eM 0 ,

I †KŠwYK miY 0 Ges t mg‡q †KŠwbK †eM I mg‡KŠwYK miY n‡j †KŠwYK Z¡iY n‡j †KŠwYK †e‡Mi

msÁvbymv‡i,

.dtd

)........(5.......... dtd

mgxKiY (5) †K t = 0 n‡Z t mgq ch©šÍ mgvKjb K‡i cvB,

00

t

dtd

mgxKiY (4) †_‡K 0 t ewm‡q cvB,

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 9

t

dttd0

0 )(0

t t

tdtdtd0 0

0

0

t

t tt0

2

00 20

200 2

1 tt cÖwZcv`b Kiv nj|

(3) mgxKiY )(2 020

2 cÖwZcv`b

Avgiv Rvwb,

dtd

dd

dtd

dtd

dd

.......(6).......... dd

mgxKiY (6) †K t = 0 n‡Z t mgq ch©šÍ mgvKjb K‡i cvB,

00

dd

0

0 2

2

)(21)( 2

02

0

20

20 )(2

)(2 020

2 cÖwZcv`b Kiv nj|

(Torque) t ‡Kvb ej †Kvb e¯‘i Dci cÖhy³ n‡q †Kvb we› y ev A‡ÿi PZzw ©‡K †h cwigvb Nyivevi

ÿgZv iv‡L Zv‡K H we› y ev A‡ÿi mv‡c‡ÿ H e‡ji UK© e‡j| GK K_vq, N~Y©vqgvb

‡Kvb KYvi e¨vmva© †f±i I KYvi Dci cÖhy³ e‡ji ‡f±i ¸Ydj‡K UK© e‡j| UK©‡K Øviv cÖKvk Kiv nq| N~Y©b †K‡› ªi mv‡c‡ÿ †Kvb KYvi e¨vmva© †f±i r

Ges H KYvi Dci

cÖhy³ ej F

n‡j H †K‡› ªi mv‡c‡ÿ KYvwUi Dci cÖhy³ UK© Fr

rFSin GLv‡b n‡”Q r I F

Gi ga¨eZ©x †KvY| UK© GKwU ‡f±i ivwk|

U‡K©i GKK Nm I Gi gvÎv = e‡ji gvÎv × ~i‡Z¡i gvÎv = MLT-2×L= ML2T-2

UK© I ‡KŠwbK Z¡i‡Yi g‡a¨ m¤úK© A_©vr I Gi cÖgvb (Relation with Torque and angular acceleration) t ‡Kvb w¯’i A‡ÿi Pvwiw`‡K N~b©vqgvb e¯‘i UK© Zvi RoZvi åvgK I †KŠwbK Z¡i‡Yi ¸Yd‡ji mgvb|

cÖgvY: g‡b Kwi, GKwU „p e¯‘i Dci UK© cÖ‡qvM Kivq Dnv XY A‡ÿi Pvwiw`‡K mg‡KŠwbK Z¡i‡Y Nyi‡Q|

e¯‘wU‡Z n msL¨K KYv Av‡Q, hv‡`i fi h_vµ‡g m1, m2, m3, ................ mn Ges XY Aÿ n‡Z KYv¸‡jvi ~iZ¡ h_vµ‡g

r1, r2, r3, ............... rn awi KYv¸‡jvi Dci wµqviZ UK© h_vµ‡g h_vµ‡g 123n GLb

m1 f‡ii KYvi †KŠwbK Z¡iY =

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 10m1 f‡ii KYvi ‰iwLK Z¡iY = r1 m1 f‡ii KYvi Dci Kvh©Ki ej = m1r1 m1 f‡ii KYvi Dci Kvh©Ki UK© 2

111 rm Abyiƒcfv‡e,

m2 f‡ii KYvi Dci Kvh©Ki UK© 2222 rm

....... ....... ....... ....... ...... ...... ....... ........ ....... .......

mn f‡ii KYvi Dci Kvh©Ki UK© 2nnn rm

myZivs †gvU UK©, 1n

2233

222

211 ............. nnrmrmrmrm

rm.............rmrmrm 2nn

233

222

211

2233

222

211 ............. ( nnrmrmrmrmII cÖgvwYZ)

Avevi ]d[ ( dtd

dtI cÖgvwYZ)

GKwU miæ I mylg `‡Ûi ga¨we› y w`‡q Ges ˆ`‡N© i j¤fv‡e MgbKvix A‡ÿi mv‡c‡ÿ `‡›`i RoZvi åvgK Ges PµMwZi

e¨vmva© wbY©q t

g‡b Kwi, AB GKwU miæ `Û| Gi ˆ`‡N¨©i ga¨we› y O w`‡q Ges ˆ`‡N© i j¤fv‡e MgbKvix PQ A‡ÿi mv‡c‡ÿ H `‡Ûi

RoZvi åvgK wbY©q Ki‡Z n‡e| aiv hvK, `‡Ûi ˆ`N© l Ges fi M |

`‡Ûi cÖwZ GKK ˆ`‡N©¨i fi

lM

myZivs Aÿ †_‡K x ~i‡Z¡ dx ˆ`‡N© i GKwU ÿz ªvwZÿz ª As‡ki fi, dxMdm

l

GLb PQ A‡ÿi mv‡c‡ÿ dx ˆ`‡N© i As‡ki RoZvi åvgK dxxMdxMxdmxdI 222

ll

GLb GB mgxKi‡Yi Wvb cvk‡K

2lx †_‡K

2lx mxgvi g‡a¨ mgvKjb Ki‡j mgMÖ `‡Ûi RoZvi åvgK I cvIqv

hvq, 2

2-

2dxxMI

l

l l

2

2

2dxxMI

l

l-l

2

2

12

3xMI

l

ll

33

223MI ll

l

883MI

33 lll

12MI

2l

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 11

awi PµMwZi e¨vmva© K

12K

12MMK

22 ll

GKwU miæ I mylg `‡Ûi GK cÖvšÍ w`‡q Ges ˆ`‡N© i j¤fv‡e MgbKvix A‡ÿi mv‡c‡ÿ `‡›`i RoZvi åvgK Ges PµMwZi

e¨vmva© wbY©q t

g‡b Kwi, AB GKwU miæ `Û| Gi GK cÖvš A w`‡q Ges ˆ`‡N© i j¤fv‡e MgbKvix PQ A‡ÿi mv‡c‡ÿ H `‡Ûi RoZvi

åvgK wbY©q Ki‡Z n‡e| aiv hvK, `‡Ûi ˆ`N© l Ges fi M |

`‡Ûi cÖwZ GKK ˆ`‡N©¨i fi =

lM

myZivs Aÿ †_‡K x ~i‡Z¡ dx ˆ`‡N© i GKwU ÿz ªvwZÿz ª As‡ki fi, dxMdm

l

GLb PQ As‡ki mv‡c‡ÿ dx ˆ`‡N© i As‡ki RoZvi åvgK dxxMdxMxdmxdI 222

ll

GLb GB mgxKi‡Yi Wvb cvk‡K x = 0 †_‡K x = l mxgvi g‡a¨ mgvKjb Ki‡j mgMÖ `‡Ûi RoZvi åvgK I cvIqv hvq,

dxxMI 2

0l

l

l

l 0

2dxxMI

l

l 0

12

3xMI

33 03MI l

l

3

l3MI l

2M31I l

awi PµMwZi e¨vmva© K

3KM

31MK 22 ll

cvZjv e„ËvKvi PvKwZi c„‡ôi Awfj¤fv‡e MgbKvix A‡ÿi mv‡c‡ÿ PvKwZi RoZvi åvgK Ges PµMwZi e¨vmva© wbY©q t

aiv hvK BCD GKwU e„ËvKvi PvKwZ| Gi fi‡K› ª O Ges c„‡ôi mv‡_ j¤fv‡e

MgbKvix PQ A‡ÿi mv‡c‡ÿ H PvKwZi RoZvi åvgK wbY©q Ki‡Z n‡e|

aiv hvK PvKwZi fi M Ges e¨vmva© r|

myZivs PvKwZi †ÿÎdj A=r2 |

PvKwZi cÖwZ GKK †ÿÎd‡ji fi, 2r M

AM

|

GLb Aÿ †_‡K x ~i‡Z¡ dx cÖ‡¯’i GKwU miæ ejq Kíbv Kiv hvK|

GB ej‡qi †ÿÎdj = ej‡qi cwiwa × ej‡qi cÖ ’

= 2xdx

myZivs GB ÿz ªvwZÿz ª Ask 2xdx ‡ÿÎd‡ji fi, xdx2r

Mdm 2

xdxrM2dm 2

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 12GLb PQ A‡ÿi mv‡c‡ÿ GB dx cÖ‡¯’i miæ ej‡qi RoZvi åvgK dI

dx xrM2xdmxdI 2

22

ev, dx xrM2dI 32

GLb GB mgxKi‡Yi Wvb cvk‡K x = 0 ‡_‡K x = r mxgvi g‡a¨ mgvKjb Ki‡j mgMÖ cv‡Zi RoZvi åvgK I

cvIqv hvq|

r

0

32 dx x

rM2I

r

0

32 dx x

rM2

r

0

13

2 13x

rM2

r042 x

r4M2

442 0r

r4M2

2Mr21I

awi PµMwZi e¨vmva© K

2rKrM

21MK 22

e¨envwiKt GKwU d¬vB ûB‡ji RoZvi åvgK wbY©qt

ZË¡t g‡b Kwi, GKwU fvix PvKvi Aÿ`‡Û GKwU iwk Rov‡bv Av‡Q| iwki GK cÖv‡šÍ m f‡ii GKwU e¯‘ mshy³

Av‡Q| e¯‘wU‡K ¯vaxbfv‡e h D”PZv †_‡K co‡Z w`‡j e¯‘wUi w¯’wZkw³ PvKvi N~Y©bRwbZ MwZ kw³‡Z, cošÍ e¯‘i ˆiwLK

MwZkw³‡Z Ges Nl©‡bi cÖwZK‚‡j m¤úbœ Kv‡R iƒcvšÍwiZ nq|

PvKvwUi ‡KŠwYK †eM Ges Aÿ`‡Ûi e¨vmva© r n‡j m f‡ii e¯‘wUi ˆiwLK †eM, v = r| hw` N~Y©b PvKvi

RoZvi åvgK I nq Z‡e Zvi N~Y©b MwZkw³ =

2

21 I | cošÍ e¯‘wUi ˆiwLK MwZkw³ =

2

21 mv | PvKvwUi cÖwZ N~Y©‡bi Rb¨

Nl©‡bi weiæ‡× m¤úbœ Kv‡Ri cwigvb w n‡j Ges m f‡ii e¯‘wU f‚wg‡Z covi c~‡e© PvKvi N~Y©bmsL¨v n1 n‡j †gvU Kv‡Ri

cwigvb W= n1w n‡e| m f‡ii e¯‘wU h cwigvb †b‡g Avm‡j ZLb Zvi w¯’wZkw³i n«vm = mgh

kw³i wbZ¨Zv kZ© n‡Z wjL‡Z cvwi, wnImvmgh 122

21

21

)1...(.................... 21

21

1222 wnImrmgh

k³ iwki mv‡_ mshy³ e¯‘wU iwkmn Aÿ`Û n‡Z wew”Qbœ nevi ci PvKvwU n2 evi Ny‡i †_‡g †M‡j Zvi N~Y©b

MwZkw³ Nl©‡bi weiæ‡× KvR m¤úv`b Ki‡Z e¨vwqZ nq| ZvB,

22 2

1 Iwn |

Zvn‡j, )2......(....................1.21

2

2

nIw

(1) I (2) bs mgxKiY n‡Z cvB,

2

21

222 1.21

21

21

nInImrmgh

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 13

2

1222 121

21

nnIrmmgh

2

1222 12nnIrmmgh

)3.......(..........1

2

2

12

22

nn

rmmghI

GLb N~Y©b PvKvwU n2evi Nyi‡Z hw` t †m‡K›U mgq jv‡M Z‡e Zvi Mo †KŠwYK †eM

tn22

| wKš‘ PvKvwU

†KŠwbK †eM n‡Z mgg›`‡b k~b¨ †eM AR©b K‡i e‡j Mo †KŠwbK †eM

220

1

22 2

tn

ev, )4.......(..........4 2

tn

(4) bs mgxKiY n‡Z Gi gvb (3) bs mgxKi‡Y ewm‡q cvB,

)5.....(..........8

22

22

2

21

rn

nght

nnmI

hš¿cvwZ:

1. N~Y©vqgvb PvKv

2. k³ iwk

3. wgUvi †¯‹j

4. ¯øvBW K¨vwjcvm©

5. _vgv Nwo BZ¨vw`|

h‡› ªi eY©bv: Wvb cv‡k¦i wP‡Îi N~Y©vqgvb d¬vB ûBjwU GKwU †gvUv

k³ PvKwZ| GUv ga¨w¯’Z GKwU A‡ÿi Dci Ggb fv‡e emvb _v‡K

†h, mvgvb¨ GKUz ej cÖ‡qv‡M PvKwZwU Aev‡a Nyi‡Z cv‡i| GKwU k³

`woi GK cÖvšÍ A‡ÿi Dci PvKwZi GKcvk¦© euvav _v‡K Ges `woi

A‡ÿi Dci K‡qK cvK w`‡q Ab¨ cÖv‡šÍ GKwU evULviv Szjvb _v‡K|

AÿwU Abyf‚wgKfv‡e KvVv‡gvi Dci emv‡bv _v‡K| `woi ˆ`N© Ggb

wb‡Z nq †h, GUv Dci ‡_‡K †Q‡o †`qvi ci gvwU ¯úk© Kivi mv‡_ mv‡_

Aÿ n‡Z Qz‡U hvq|

cixÿv c×wZ:

1. N~Y©vqgvb PvKvi Mv‡q PK Øviv `vM †`Iqv nq|

2. GKwU k³ iwki GK cÖv‡šÍ wbw ©ó f‡ii evULviv hy³ K‡i Aÿ`‡Û c¨vPvb nq| iwkwUi ˆ`N© Ggbfv‡e †bIqv nq

†hb evULviv gvwU‡Z co‡j iwk Aÿ cwiZ¨vM K‡i|

3. iwk‡K Aÿ`‡Û Ggbfv‡e c¨vPvb nq †hb evULvivwU A Ae¯’v‡b _v‡K| A Ae¯’vb n‡Z evULviv wU‡K †g‡Si B

Ae¯’v‡b bvg‡Z †`Iqv nq| A n‡Z B Ae¯’v‡b evULvivwU bvg‡Z †h mgq jv‡M ‡mB mg‡q PvKvwUi N~Y©bmsL¨v n1

wbY©q Kiv nq|

4. k³ iwkwU‡K cybivq Aÿ `‡Û c¨vPvb nq Ges evULvivwU†K A Ae¯’v‡b ivLv nq| evULvivwU†K wb‡Pi w`‡K bvg‡Z

†`Iqv nq| †h gyn~‡Z© evULvivwU †g‡S‡Z c‡o Ges iwkwU Aÿ `Û n‡Z wew”Qbœ nq †mB gyn~‡Z© _vgv Nwo Pvjy Kiv

nq|

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 145. N~Y©vqgvb PvKvwU w¯’i Ae¯’vq Avmvi mv‡_ mv‡_ _vgv Nwo eÜ Kiv nq| †gvU mgq t Ges GB mg‡q PvKvwUi N~Y©b

msL¨v n2 wbY©q Kiv nq|

6. wgUvi †¯‹‡ji mvnv‡h¨ AB = h cwigvc Kiv nq|

7. mgxKiY (5) mvnv‡h¨ RoZvi åvgK wbY©q Kiv nq|

fvwY©qvi aªæeK wbY©q:

cÖavb †¯‹‡ji 1 Ni = S wgwg

fvwY©qvi †¯‹‡ji fvM msL¨v n =

fvwYqvi aªæe VC = ...nS

wgwg=...‡mwg

hvwš¿K ÎæwU (IE) =... †mwg

cÖK…Z cvV = AvcvZcvV - ( hvwš¿K ÎæwU)

cixÿvjä DcvË :

Aÿ `‡Ûi e¨vmva© (r) wbY©‡qi QK:

cvV msL¨v cÖavb

†¯‹‡ji

cvV

M †mwg.

fvwY©qvi

†¯‹j cvV

V

fvwY©qvi

aªæeK

VC ‡mwg.

fvwb©qvi

†¯‹j cvV

C=V×VC †mwg.

‡gvU cvV

(M+C) ‡mwg.

Mo e¨vm

D © †mwg.

ï× Mo

e¨vm D = D ©-( IE)

†mwg.

e¨vmva©

r = D/2 ‡mwg.

1. 2. 3. 4.

n1, n2, Ges t wbY©‡qi QK:

cvV

msL¨v

e¯‘i fi

m MÖvg

P‡µi N~Y©b

msL¨v

n1

Mo

n1

P‡µi N~Y©b

msL¨v

n2

Mo

n2

E¯‘i

D”PZv

h †mwg.

N~Y©‡b

cÖ‡qvRbxq

mgq t ‡m.

Mo

mgq

t ‡m.

1. 2. 3. 4.

djvdj: wb‡Y©q N~Y©b A‡ÿi mv‡c‡ÿ P‡µi RoZvi åvgK= .... MÖvg-‡mwg.

2

wnmve I MYbv: m =... MÖvg n1 =...

n2 =... h =...‡mwg.

t =...‡m‡KÛ r =...‡mwg.

g =...‡mwg ‡m.

-2

.8

22

22

2

21

rn

nght

nnmI

mZK©Zv I Av‡jvPbv:

1. D”PZv h wbY©q Kivi mgq evULvivi Zj c„ô n‡Z †g‡Si Dci ch©šÍ Lvov fv‡e ‰`N©¨ wbY©q Kiv nq|

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04| wbDUwbqb ejwe`¨v (Newtonian Mechanics) 152. `woi ˆ`N©¨ cwigvc Kivi mgq jÿ ivLv nq hv‡Z fiwU †g‡S ¯úk© Kivi mv‡_ mv‡_ GUv Aÿ n‡Z Qz‡U hvq

Ges †m gyn~Z© n‡ZB P‡µi AveZ©b msL¨v MYbv Kiv nq|

3. PµwU †hb Kgc‡ÿ 100 evi Ny‡i eÜ nq †m j‡ÿ †Zj, wMÖR BZ¨vw` e¨envi K‡i PvKvwU‡K Nl©Ywenxb Kiv

nq|

4. `wowUi e¨vm Ggb fv‡e †bIqv nq †hb GUv PvKvi A‡ÿi e¨v‡mi Zzjbvq bMY¨ nq|

5. Aÿ †_‡K `wo Qz‡U hvIqvi mv‡_ mv‡_ PvKvi N~Y©b msL¨v Ges Zvi mgqKvj Le mveavbZvi mv‡_ MYbv Kiv

nq|

‡K› ªgyLx ej (Centripetal Force): hLb †Kvb e¯‘ e„ËvKvi c‡_ Nyi‡Z _v‡K ZLb †h ej e ‘i Dci H e„‡Ëi †K› ª

Awfgy‡L wµqv K‡i e¯‘wU‡K e„ËvKvi c‡_ MwZkxj iv‡L Zv‡K †K› ªgyLx ej e‡j| m f‡ii e¯‘ r e¨vmva© wewkó e„ËvKvic‡_

v mg ªæwZ‡Z Nyi‡Z _vK‡j Zvi †K› ªgyLx ej

rvm

2

|

†K› ªwegyLx ej (Centrifugal Force): hLb †Kvb e¯‘ e„ËvKvi c‡_ Nyi‡Z _v‡K ZLb †h ej H e„‡Ëi †K‡› ªi wecixZ

w`‡K cÖ‡qvM K‡i Zv‡K †K› ªwegyLx ej e‡j| m f‡ii e¯‘ r e¨vmva© wewkó e„ËvKvic‡_ v mg ªæwZ‡Z Nyi‡Z _vK‡j Zvi

†K› ªwegyLx ej

rvm

2

|

Avgiv Rvwb, ‡K›`ªgyLx Z¡iY, rrr

rva 2

222

†K› ªgyLx ej, rmr

vmmaF 22

euvKv c‡_ mvB‡Kj Av‡ivnxi Dj‡¤i mv‡_ †njvb †Kv‡Yi ivwkgvjv t

‡Kvb mvB‡Kj Av‡ivnx hLb †mvRv c‡_ P‡j ZLb mvB‡Kjmn Av‡ivnxi IRb W=mg Lvov wb‡Pi w`‡K Ges iv¯Ívi

cÖwZwµqv R Lvov Dci w`‡K wµqv K‡i| iv¯Ívi euvK wb‡Z n‡j mvB‡Kj Av‡ivnx‡K euv‡Ki wfZ‡ii w`‡K KuvZ n‡q euvK

wb‡Z n‡e| iv¯Ívi euvK wb‡Z n‡j mvB‡Kj Av‡ivnx‡K e„ËvKvi MwZ m¤úbœ n‡Z nq Ges Gi Rb¨ †K› ª gyLx e‡ji cÖ‡qvRb

nq| GB †K› ª gyLx ej IRb W Ges iv¯Ívi cÖwZwµqv R n‡Z msM„nxZ nq| ‡h‡nZz IRb me© v fvi‡K› ª n‡Z Lvov wb‡Pi

w`‡K wµqv K‡i, Kv‡RB cÖwZwµqv R -†K Awfj‡¤i mv‡_ bZ n‡Z nq| G‡Z R Gi ywU Dcvsk cvIqv hvq| Dj¤ Dcvsk

f~wgi mgvšÍiv‡j ‡K› ªgyLx e‡ji †RvMvb †`q Ges Abyf~wgK Dcvsk Lvov Dc‡ii w`‡K wµqv K‡i IRb W -‡K cÖkwgZ

K‡i|

hw` R Awfj‡¤i mv‡_ †KvY Drcbœ K‡i, Z‡e Gi Abyf~wgK Dcvsk

Rcos= W = mg n‡e Ges Dj¤ Dcvsk Rsin †K› ªgyLx e‡ji cwigvc wb‡ ©k K‡i| hw`

iv¯Ívi euvK wb‡Z mvB‡Kj v ªæwZ‡Z r e¨vmv‡a©i e„ËvKvi c_ AbymiY K‡i, Z‡e Avgiv cvB,

‡K› ªgyLx ej,

rmvsinR

2

Ges mgcosR

d‡j,

mgr

mv

cosRsinR

2

mg1

rmvtan

2

rgvtan

21 †`Lv hvq †h, euvK wb‡Z n‡j †h †Kv‡Y †n‡j _vK‡Z nq Zv mvB‡Kj

Av‡ivnxi f‡ii Dci wbf©i K‡i bv| wbf©i K‡i Av‡ivnxi MwZ‡eM I MwZ c‡_i e¨vmv‡a©i Dci|

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