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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: [email protected]
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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