शास्त्रीय भौतिक शास्त्रका केही समीकरणहरूमा दुर्यान्तर र प्रदुर्यान्तरको प्रभाव

(यो लेख नयाँ लेख नभई मिति २०१७ जुलाई ७ मा 'भौतिक विज्ञान' नामक ब्लगमा प्रकाशित भइसकेको लेख हो । यस लेखका शब्दकुञ्जीहरू दुर्यान्तर, प्रदुर्यान्तर, सुव्रत समीकरण, सुवात समीकरण, बल, शक्ति, सामथ्र्य आदि रहेका छन् । उक्त लेखको शीर्षक र शब्दकुञ्जी युनिकोडमा परिवर्तन नगरी  प्रकाशित गरिएकाले खोज्न समस्या भएकाले यहाँ सम्पादनबिनै साभार गरिएको हो । उक्त लेखको लिंक यसप्रकार छ : https:// bhautikbigyan.blogspot.com/2017/07/zfqlo-eflts-zfqsf-sxl-dls0fxdf-bofgt.html)

zf:qLo ef}lts zf:qsf s]xL ;dLs/0fx¿df b'of{Gt/ / k|b'of{Gt/sf] k|efj

–भविन्द्र कुँवर

!= kl/ro

ef}lts zf:qdf hl6n / g;'Nem]sf ;d:ofx¿ gjLg ef}lts zf:qLo kl/df0fx¿ b'of{Gt/ / k|b'of{Gt/sf cfwf/df ;dfwfg ug{ ;lsPsf] 5 -#, &, (–!!, !$–!&, !(, @!, @@, @(–#%_ . b'of{Gt/ / k|b'of{Gt/n] rfn;Fu ;DalGwt ljljw ;d:ofx¿sf] ;dfwfg lbPsf 5g\ -!(, @!, @@, @(–#@­_ . ef}lts zf:qsf an, zlQm, ;fdYo{h:tf k|rlnt ljifox¿df klg b'of{Gt/ / k|b'of{Gt/sf] e"ldsf /x]sf] x'G5 -@@, @(–#%_ . o; cg';Gwfgkq d"ntM k|rlnt ef}lts zf:qLo ;dLs/0fx¿df b'of{Gt/ / k|b'of{Gt/n] v]Ng] e"ldsf / k|efj;Fu ;DalGwt 5 .

b'of{Gt/ / k|b'of{Gt/ ;dfg ;dofGt/fndf ;Lwf /]vf nDafOsf] dfqfdf x'g] cGt/;Fu -@@_ dfq geO{ t/u+ z}nLsf] jqm /]vfsf] nDafOdf x'g] cGt/ / s'g} lkG8sf] kl/lwhGo nDafOdf x'g] cGt/;Fu klg ;DalGwt ljifox¿ x'g\ . oL kl/df0fx¿n] e"ldsf lgjf{x ug]{ s'/f yfx gePsf sf/0f ef}lts zf:qdf ;d:of a9\b} uPsf] b]lvG5 . o;af6 ef}lts zf:qdf s'g} dfqfsf] ultsf nflu a/fa/ / s'g} dfqfsf] ultsf nflu a9L kl/df0f lbg] vfnsf tYoOt/ lgod / ;"qx¿sf] k|of]u ul/Fb} cfPsf] b]lvG5 . o; ljifonfO{ klg o; cg';Gwfgkqn] :ki6 kf/]sf] 5 .

o; cg';Gwfgkq ef}lts zf:qdf ePsf tYoOt/ cfwf/x¿sf] vf]hLkl5sf] tYok/s cfwf/ / ;dfwfgx¿sf] k|:t'ltdf s]lGb|t 5 . cg';Gwfgsf qmddf k|o'Qm ;+lIfKt ;+s]tx¿n] ;DalGwt :yfg / ;Gbe{cg';f/sf ljifox¿sf] k|ltlglwTj u/]sf 5g\ . jfSox¿sf sf]i7sdf lbOPsf ;+Vofx¿n] ;"rLdf /x]sf ;Gbe{;fdu|Lx¿nfO{ hgfpFb5g\ . ef}lts zf:qdf k|o'Qm ;+s]tx¿df b]lvPsf] ljljwtf / zfvfut ljljwtfsf sf/0f oxfF k|o'Qm s]xL ;+s]tx¿n] PseGbf km/s ljifonfO{ klg a'emfpFb5g\ . o; cj:yfdf ltgLx¿n] lbg] cy{nfO{ ;DalGwt :yfgdf g} hfgsf/L u/fOPsf] 5 .

ef}lts zf:qsf k|rlnt sltko l;4fGt, lgod, kl/df0f, ;"q jf ;dLs/0fx¿ k/LIf0flagf :yfkgf ul/Psf] k'li6 ePsf] 5 -#–@%, @*–#%_ . :jo+l;4 tYo, ;j{:jLs[t tYo / k/LIf0feGbf aflx/ uPsf ljifox¿nfO{ jf:tljs lj1fgtkm{sf] dfu{ k|bz{g ug{ o; cg';Gwfgn] e"ldsf lgjf{x ub{5 . o; cg';Gwfg s]xL bzsb]lvsf] lg/Gt/ vf]h, ;DalGwt k/LIf0f, Jofks 5nkmn / ax;, tYok/s / lj1f;Ddt cfwf/, :jo+l;4 tYo, ;j{:jLs[t dfGotf / j}1flgs cg;Gwfgk4ltsf cfwf/df ;DkGg ePsf] 5 -@(–#%, %#_ . o;n] cGo ;DalGwt kl/jt{gx¿sf nflu klg cfwf/ lbPsf] 5 .

k6sk6s ul/Psf k/LIf0fkZrft\ tYoOt/ cfwf/df lgdf{0f ePsf] :ki6 k|df0f km]nf k/]kl5 -(–!!, !%–!&_ ef}lts zf:qsf k/Dkl/t lgdf{0fx¿nfO{ jf:tljs ef}lts zf:qsf dfGotfsf] s;Ldf Nofpgsf nflu o; cg';Gwfgkqsf] dxTTj /x]sf] 5 . ljz]iftM zf:qLo ef}lts zf:qdf k|rng / ljZjf;sf ¿kdf /x]sf lj1fgOt/ cg]sg kIfxx¿n] lgDtfpg] ;d:ofx¿sf] ;dfwfgdf o; cg';Gwfg s]lGb|t 5 . k/LIf0fsf cfwf/df k|fKt ePsf kl/0ffdx¿nfO{ Pskl5 csf]{ a'Fbf x'Fb} / tL a'Fbfx¿df cfwfl/t x'Fb} o; cg';Gwfg sfo{ ;DkGg ePsf] xf] . d"ntM b'of{Gt/ / ;dofGt/fndf cfwfl/t k|b'of{Gt/n] zf:qLo ef}lts zf:qdf v]Ng] e"ldsfdf s]lGb|t eO{ k/LIf0fd"ns, :jtGq / j:t'k/s ;dfwfg lbg] sfo{ o; cg';Gwfgkqdf ePsf] 5 .

@= k[i7e"ld

b'of{Gt/ / k|b'of{Gt/sf cfwf/df ;'j|t ;dLs/0fx¿sf] lgdf{0f ePsf] xf] . ;'j|t ;dLs/0fx¿n] k|ydtM ;'jft ;dLs/0fx¿df /x]sf ;d:ofx¿sf] ;dfwfg u/]sf 5g\ -!(, @!, @@, @*_ . oL ;dfwfgx¿af6 sltko ef}lts zf:qLo kl/df0fx¿sf] PsfO, cfoflds kmd'{nf, :j¿k, dfGotf / ;dLs/0fx¿df kl/jt{g ug'{kg]{ clgjfo{tf b]lvPsf] 5 -@, #, $, *, (–!!, !$, !%, !*, @$, @%, @(–#%_ . o;/L ljljw k/LIf0fx¿af6 kl/jt{gsf] of] lgisif{d"ns r/0f;Dd cfOk'Ubf s]xL gjLgtd k|flKtx¿ klg b]vf k/]sf 5g\ . tL k|flKtx¿ o; cg';Gwfgkqdf ;dfj]z 5g\ .

b'of{Gt/ / k|b'of{Gt/sf] k[i7e"ld s]nfpFbf c/:t'n] p7fPsf s]xL ;jfnx¿;Dd k'Ug'kg]{ x'G5 . c/:t'n] xn'sf / u|xf}F j:t'dWo] u|xf}F j:t' l56f] v:5 eGg] s'/f cl3 ;f/]sf] pNn]v 5 -#, *_ . o;;Fu} j:t'sf] ultsf] cWoog x'g yfn]sf] xf] -#, *_ . o;kl5 ljleGg ;dodf pQm ljifosf] vf]h x'Fb} uof] . Uoflnlnof], l:6e]g, af]nL, b]sft{, Go'6g cflbn] o;;DaGwL sfo{x¿ ub} uPsf] b]lvG5 -#, &_ . o; qmddf Uoflnlnof]n] ;g\ !%*# df Pp6f pNn]vgLo k/LIf0f u/]sf] b]lvG5 . o;;Fu} k|j]u / lj:yfkg;Fu ;DalGwt sfo{x¿sf] cWoog sfo{ cl3 a9]sf] b]lvG5 -!(, @!_ .

Go'6gn] Uoflnlnof] / b]sft{sf of]ubfgx¿sf] k[i7e"lddf rfnsf lgodx¿ lbPsf] b]lvG5 -P]=_ . o;nfO{ Opn/, XofldN6g, n]u|]Gh cflbn] klg pNn]vgLo of]ubfg lbPsf] b]lvG5 -#, &_ . o;/L k|j]u / k|j]u ;DalGwt ef}lts zf:qLo kl/df0fx¿sf] lgdf{0f ug]{ sfo{n] ef}lts zf:qdf lg/Gt/tf kfPsf] b]lvG5 . log} kl/df0fx¿sf cfwf/df zf:qLo oflGqsLdfq geO{ ;"IdzlQm oflGqsLsf] ;d]t ljsf; ePsf] kfOG5 .

k|j]usf] :yfkgfdf ePsf] sdhf]/Lsf sf/0f ef}lts zf:qdf 7"nf] ;d:of l;lh{Psf] s'/f …ef}lts zf:qdf ePsf s]xL q'l6x¿sf] vG8g / ;dfwfgÚ zLif{ssf] k':tsdf pNn]v ul/Psf] lyof] -#_ . To;kl5 ePsf k|sfzg tyf /rgfx¿df klg o;af/] pNn]v ul/Psf] 5 -&, !(, @!, @@, @(–#%_ . kl5Nnf] ;dodf x'Fb} cfPsf ljleGg sfo{qmdx¿df ;DalGwt k/LIf0f tyf k|jrgx¿ ;d]t lbOPsf] lyof] -*, (, !), !$, @(–#%_ . o; qmddf oL ljifox¿sf cltl/Qm sfo{sf] ;"qdf b'/Lsf] ;+nUgtf x'g'df klg ;d:of /x]sf] ljifonfO{ phfu/ ul/Psf] lyof] -#, !*_ . k':ts tyf /rgfx¿df klg o;af/] pNn]v ul/Psf] 5 -#, &, !*_ .

lj=;+= @)%% b]lv s'g} klg lkG8n] lglZrt ;+oGqleq /x]/ cj/f]wo'Qm If]qdf klg lg/Gt/ ultjfg x'g] Ifdtf /fVb5 eGg] dfGotfdf cfwfl/t eO{ Pp6f cg';Gwfgsfo{sf] yfngL ul/Psf] lyof] -@, #!, #@_ . lj=;+= @)^& ;Dd cfOk'Ubf ;f] cg';Gwfgn] s]xL km/s df]8 lng k'Uof] . h;cg';f/ …;fdYo{sf] l;4fGtÚ, …ef}lts zf:qdf ePsf s]xL q'l6x¿sf vG8g / ;dfwfgÚ cflb k':tsx¿ k|sfzg x'g'sf ;fy} s]xL gofF vf]hx¿df ;xdlt hgfO{ ljleGg lzIf0f ;+:yfx¿n] lbPsf k|of]uzfnLo l/kf]6{x¿ k|fKt x'g k'u] . -#, %, &, @$, @%_ .

k':tsx¿sf] k|sfzgkl5 Pskl5 csf]{ x'Fb} ljleGg cg';Gwfgkq, sfo{kq tyf n]vx¿ k|sfzg Pjd\ k|:t't ug]{ sfo{ eof] -@(–#%_ . ;Fu;Fu} ;DalGwt k/LIf0f, k|jrg / cleofgd"ns sfo{qmdx¿ klg ;DkGg eP -$, *, (, !), !@, !#, !$, !*_ . o; k[i7e"lddf s]xL cg';Gwfgkqx¿ k|sflzt eO{ ef}lts zf:qdf ;'wf/ ug{'kg]{ kIf / ljifox¿sf cfwf/x¿sf ¿kdf b'of{Gt/ tyf k|b'of{Gt/ nufPtsf ef}lts zf:qLo kl/df0fx¿ tof/ x'g k'u]sf 5g\ -!!, !%–!&, !(, @!, @@, @(, #!, ##_ . t;y{ b'of{Gt/ / k|b'of{Gt/df cfwfl/t eO{ ;'j|t ;dLs/0fx¿ lgdf{0f ul/Psf] xf] . ef}lts zf:qdf ;d:ofsf ¿kdf /x]sf ;'jft ;dLs/0fx¿ o;k|sf/ /x]sf 5g\M

v = u + at                                           -!_

s = (u+v)t/2                                       -@_

s = ut + (at²)/2                                   -#_

v² = u² + 2as                                     -$_

s = vt - (at²)/2                                    -%_

hxfF, s, u, v, a / t n] qmdzM lj:yfkg, ;'¿sf] ult, clGtd ult, k|j]u / ;donfO{ a'emfpFb5g\ -#^, #&, #*, #(_ . oL ;dLs/0fdf ePsf ;DaGwsf cfwf/df ef}lts zf:qdf ;DalGwt kl/df0fx¿ dfkg ug]{, j0f{g ug{] tyf JofVofljZn]if0f ug]{ sfo{ x'Fb} cfPsf] 5 -#, &_ .

;'jft ;dLs/0fx¿df /x]sf] ;d:of ;dfwfg ug{sf nflu b'of{Gt/ / k|b'of{Gt/sf] e"ldsfsf cfwf/df s]xL ;'j|t ;dLs/0fx¿sf] lgdf{0f ul/Psf] 5 -@(, #)_ . tL kl/df0fx¿sf cfwf/df cGo ef}lts zf:qLo ;dLs/0fx¿ klg lgdf{0f ug{ ;lsg] s'/f ;DalGwt cg';Gwfgkqx¿af6 :ki6 x'G5 . b'of{Gt/ / k|b'of{Gt/sf cfwf/df lgld{t ;'j|t ;dLs/0fx¿ lgDgjdf]lhd 5g\M

v = u+(n-1)r                                      -^_

s = ut+(nrt)/2                                    -&_

s = T[(n-2)u+nv]/2                            -*_

s = vt-(n-2)rt/2                                  -(_

r = (d-d₀)/t                                         -!)_

hxfF, s, u, v, r, t, d₀, d, n / T n] qmdzM lj:yfkg, ;'¿sf] ult, clGtd ult, b'/L kl/jt{gsf] b/, ;do, k|f/lDes b'/L, cflv/L b'/L, b'/Lsf hDdf kb;+Vof / ;dofGt/fnnfO{ a'emfpFb5g\ -!(, @!_ .

s = V_avt = (u+v_av)t                                 -!!_

s = (v₁+v)t/2, V_av =       (v₁+v)/2              -!@_

S = x₀ + d₀(n-1) + Dn(n-1)/2                    -!#_

S = x₀+ut+(rt²)/2T+rt/2                          -!$_

V_av  = (u+v_av)                                     -!%_

d_av = d₀+Dn/2                                     -!^_

d_av = d₀+rnT/2                                    -!&_

V_av = u+(rt/2T + r/2)                            -!*_

V_av = u+nr/2                                      -!(_

D = rT, r = D/T                                   -@)_

oxfF, x₀, u, d₀, v₁, d₁, D₀, D, r₀, r, v, V_av, d_av, v_av, T, t, n / s h:tf ;+s]tx¿n] qmdzM klxnf] ultk"j{sf] b'/Lsf] dfqf, ;'¿sf] ult, nufgL an dfkg ug{ yfNg'k"j{ ;'¿sf] ultdf /xg] elgPsf] ;'¿sf] b'/L jf ;'¿sf] lj:yfkg jf lglxt lj:yfkg -of] b'/L jf lj:yfkgsf] pkl:ylt x'g jf gx'g ;S5 / pkl:ylt ePsf a]nf klxnf] ultk"j{sf] b'/Lsf] dfqfdf ;dfj]z x'G5_, nufgL an dfkg ePkl5sf] klxnf] ult -cyjf klxnf] ult_, nufgL ankl5sf] klxnf] ;dofGt/fndf ePsf] lj:yfkg jf b'/L -cyjf klxnf] lj:yfkg_, klxnf] lj:yfkg / ;'¿sf] b'/Lsf] cGt/ -cyjf ;'¿sf] b'of{Gt/_, bf];|f] ;dofGt/fndf ePsf] lj:yfkg / klxnf] ;dofGt/fndf ePsf] lj:yfkgsf] cGt/ -cyjf klxnf] b'of{Gt/_, ;'¿sf] ;dofGt/fndf ylkg uPsf] lj:yfkgsf] dfqf -cyjf ;'¿sf] k|b'of{Gt/_, bf];|f] ;dofGt/fndf ylkg uPsf] lj:yfkgsf] dfqf -cyjf klxnf] k|b'of{Gt/_, kl5Nnf] ;dofGt/fndf kf/ ePsf] lj:yfkgsf] dfqf -cyjf clGtd ult_, s'g} ;dodf ePsf] lj:yfkgnfO{ -klxnf] ultk"j{sf] lj:yfkg;d]tsf] lj:yfkgnfO{ xf]Og_ pQm ;don] efu ubf{ cfpg] ;/b/ ult -cyjf ;/b/ ult_, ;/b/ ultleqsf] lj:yfkg -cyjf ;/b/ lj:yfkg_, ;/b/ ult / ;'¿sf] ultsf] cGt/ -cyjf lgld{t ;/b/ ult_, ;dofGt/fn, an nufgL ePsf] hDdf ;do -;dofGt/fn / Ps sd kb;+Vofsf] u'0fgkmn_, kb;+Vof / lj:yfkgnfO{ a'emfpFb5g\ -@@_ .

x/]s ;dofGt/fndf x'g] lj:yfkg jf ;/b/ lj:yfkgnfO{ d_av n] ;+s]t ul/Psf] 5 . x_p nfO{ an nufgL ug'{k"j{ klxNo} kf/ eO;s]sf] b'/L jf ;'¿k"j{sf] lj:yfkg dflgPsf vG8df ljleGg ;dofGt/fnx¿sf nflu x₀ Ö d₀ ± x_p x'g hfG5 . To;}n] ;du| lj:yfkgnfO{ ;'¿k"j{sf] lj:yfkg, ;'¿sf] lj:yfkg / lj:yfkgsf] of]ukmnsf ¿kdf lng ;lsG5 . cyf{t\ ;du| lj:yfkg, S = x_p + d₀ + s = x₀ + s x'G5, hxfF S, x_p, d₀ / s n] qmdzM ;du| lj:yfkg, an nufgLk"j{sf] lj:yfkg, ;'¿sf] lj:yfkg / lj:yfkg hgfpF5g\ . lj:yfkgnfO{ klg lglxt lj:yfkg / lgld{t lj:yfkg egL juL{s/0f ug{ ;lsG5 . lglxt lj:yfkg eGgfn] crn dfgsf ¿kdf x/]s cGt/fndf :jtM lglxt eP/ cfpg] lj:yfkg xf] . ;'¿sf] lj:yfkgsf] dfqf x/]s ;dofGt/fndf sfod /lx/x]sf] x'G5 . lgld{t lj:yfkg eGgfn] nufgL ansf sf/0f b]lvg hfg] jf lgdf{0f x'g] b'of{Gt/x¿sf] of]ukmn xf] . o; ljifonfO{ tnsf pbfx/0fx¿n] yk :ki6 ub{5g\ -@@_ .

pko'{Qm ;dLs/0fx¿n] ef}lts zf:qdf ;'¿sf] ult, clGtd ult, lj:yfkg, ;do cflb kl/df0fx¿ klxNofpg / tL;Fu ;DalGwt ljleGg ljifosf] cWoog ug{ 7"nf] e"ldsf lgjf{x ub{5g\ -#), #@, ##–#%_ . pko'{Qm ;dLs/0fx¿ b'of{Gt/ / k|b'of{Gt/sf cfwf/df lgld{t 5g\ . dflysf ;'j|t ;dLs/0fx¿df b'of{Gt/ / k|b'of{Gt/n] j:t'nsf] /]vLo rfnsf] cWoog u/]sf] b]lvG5 . 3'dfp/f] kydf nufgL x'g] annfO{ klg b'of{Gt/ / k|b'of{Gt/sf ;xf/faf6 s]s;/L cWoog ug{ ;lsG5 < eGg] ljifodf o; cg';Gwfgkq s]lGb|t 5 . o;k|sf/ ju{;dosf] tYoOt/ wf/0ff, k|j]usf] cfwf//lxt :yfkgf, df]d]G6 ckm Ogl;{ofsf] c;xh cEof;, cfoflds dfGotfdf lnOPsf] ePsf] tYoOt/ cfwf/, t/+u, zlQm, sfo{, ;fdYo{ cflb;Fu ;DalGwt cgfjZos r/x¿sf] cfof]hg h:tf ljljw ljifo / tL;Fu ;DalGwt ljifox¿df ef}lts lj1fgn] ;xL glthf glbPsf] tYo j:t'k/s, lj1fg;Ddt / ts{;Ddt -@&_ ¿kdf :ki6 x'g k'u]sf] 5 -#, %, ^, &, ^, !!, !(–@@, @#–@%, @(–#%_ .

#= b'of{Gt/ / k|b'of{Gt/

b'of{Gt/ lj:yfkgx¿sf] cGt/af6 lgsflng] ;lbz ef}lts zf:qLo kl/df0f xf] -@@, #), #!_ . o; kl/df0f cj/f]wxLg If]qdf /x]sf] s'g} lkG8df Psgf; an nufgL ePsf a]nf b]lvg k'U5 . Psgf; an nufgL eO/x]sf a]nf ;dfg ;dofGt/fnx¿df kf/ x'g] lj:yfkgx¿sf aLrdf dfq geO{ 3'dfp/f] kydf ;d]t o; kl/df0f b]vf kb{5 -!!, !%–!&_ . cj/f]wxLg If]qdf s'g} kl/df0fsf] Psgf; an nufgL ubf{ Pskl5 csf]{ ;dofGt/fndf kf/ x'g] lj:yfkgsf] dfqf a9\b} hfg] ePsfn] tL lgs6 lj:yfkgx¿sf aLrdf / kl/wLo b'/Lx¿sf] aLrdf cGt/sf ¿kdf pQm kl/df0f b]vf kb{5 .

b'of{Gt/ nDafOdf gflkG5 . ;do crn ePdf nufgL ePsf] ansf] dfqf;Fu b'of{Gt/ ;dfg'kflts x'g] ePsfn] b'of{Gt/sf] dfqfsf] cfwf/df an s]slt dfqfdf nufgL ePsf] 5 eGg] s'/fsf] hfgsf/L xfl;n ug{ ;lsG5 . of] kl/df0f an jf zlQm jf ;fdYo{h:tf kl/df0fx¿df cGtlg{lxt eO{ ljz]if e"ldsf lgjf{x ub{5 . o;afx]s lsg]Dofl6S;\df klg o;n] pNn]vgLo e"ldsf lgjf{x ub{5 .

b'of{Gt/ Ps cy{df ;dofGt/fn pNn]v gul/Psf] k|b'of{Gt/ xf] . b'of{Gt/ s'g} g s'g} kl/df0fsf] ;dofGt/fnsf cfwf/df dfkg ul/Psf] x'G5 t/ ;dofGt/fnsf] dfqf s]slt xf] eGg]af/]df pNn]v ul/Psf] x'Fb}g . k|b'of{Gt/df eg] tTIf0fdf cfwf/sf ¿kdf lnOPsf] ;dofGt/fnsf] dfqf :ki6 ul/Psf] x'G5 -@@_ . t;y{ k|b'of{Gt/ eg]sf] b'of{Gt/sf nflu h'g ;dofGt/fn cfwf/ lnOPsf] xf] ToxL ;dofGt/fnn] efu ubf{ cfpg] kl/df0f xf] . b'of{Gt/ / k|b'of{Gt/ k|fKt x'g] cj:yf Pp6} x'G5 . d"ntM b'of{Gt/ ;dfg ;dofGt/fndf ePsf lgs6 lj:yfkgx¿sf] cGt/af6 k|fKt x'G5 eg] k|b'of{Gt/ lgs6 ultx¿sf] cGt/af6 k|fKt x'G5 .

b'of{Gt/ / k|b'of{Gt/sf af/]df k':ts tyf ljleGg ;fdu|Lx¿df pNn]v ul/Psf] 5 . kl5Nnf] ;dodf k|sflzt …rfnsf /]vLo ;dLs/0fx¿df b'of{Gt/ / k|b'of{Gt/sf] e"ldsfÚ zLif{ssf] cg';Gwfgkqdf b'of{Gt/ / k|b'of{Gt/sf] /]vLo e"ldsfsf af/]df yk :ki6 ul/Psf] 5 . oL kl/df0fx¿n] ef}lts zf:qdf k|j]u / To;;Fu ;DalGwt kl/df0fx¿af6 b]lvg] ;d:ofx¿sf] ;dfwfgdf e"ldsf lgjf{x ub{5g\ . ef}lts zf:qdf zlbof}Fb]lv ;dfwfg x'g g;s]/ /x]sf ;d:ofx¿sf] ;dfwfg u/L gofF, ;/n / j}1flgs pkfo k|:t't ug{ b'of{Gt/ / k|b'of{Gt/sf] e"ldsf /x]sf] 5 .

b'of{/Gt/nfO{ ;Lwf nDafO{sf ;Gbe{df dfq geO{ 3'dfp/f] jf jqm nDafO{sf cfwf/df klg JofVof ug{ ;lsg] b]lvFb} cfPsf] 5 -!!, !%–!&_ . k|b'of{Gt/nfO{ PsfO ;dofGt/fndf b]vf kg]{ ;f]xL b'of{Gt/sf] dfqfsf ¿kdf lng ;lsG5 . o;af6 lkG8n] ;/n /]vfx¿em}F ;Lwfdfq geO{ 3'dfp/f] lbzfsf] ck]Iff klg /fv]sf] b]lvG5 -!!, !%–!&_ . o;f] ubf{ ef}lts zf:qdf ;a} ;Gbe{df ;xL / k|of]u;Ddt kl/0ffd k|fKt ug{ klg ;lsG5 .

$= b'of{Gt/ / k|b'of{Gt/sf] s]xL e"ldsf

b'of{Gt/ / k|b'of{Gt/sf cfwf/df ;'j|t ;dLs/0fx¿ lgdf{0f ul/Psf] 5 . tL ;dLs/0fx¿n] ef}lts zf:qdf k|rlnt ;'jft ;dLs/0fx¿n] lbg g;s]sf ;dfwfgx¿ lbPsf 5g\ . o;af/] dfly hfgsf/L u/fO;lsPsf] 5 . ef}lts zf:qdf k|j]u / k|j]uhGo cGo ef}lts zf:qLo kl/df0fx¿n] ;xL e"ldsf lgjf{x ug{ g;s]sfn] b'of{Gt/ / k|b'of{Gt/sf cfwf/df tL ;dLs/0fx¿sf] klg hfFr u/L gofF ;dfwfg k|:t't ug'{kg]{ b]lvG5 . oxfF ;f]xLjdf]lhd hfFr u/L ;xL ;dfwfg klxNofpg] sfo{ ul/Psf] 5M

rfnsf /]vLo ;dLs/0fx¿df k|j]u gfds ef}lts zf:qLo kl/df0fsf] e"ldsf geO{ b'of{Gt/ / k|b'of{Gt/sf] e"ldsf ePsf] b]lvG5 -#, !(, @!, @@_ . ;'jft ;dLs/0fx¿ -;dLs/0f !–%_ df ;d:of ePsf sf/0f ;'j|t ;dLs/0fx¿ -;dLs/0f ^–@)_ sf] :yfkgf ul/Psf] 5 . ;'j|t ;dLs/0fx¿ k"0f{tM lj1fg;Ddt / tYok"0f{ ePsf] klg l;4 eO;s]sf] 5 -#, !(, @!, @@_ . o;n] xfn;Dd k|rngdf /x]sf k|j]uldl>t s'g} klg ef}lts zf:qLo ;dLs/0f jf kl/df0fx¿ / tL;Fu ldNbfh'Nbf cfoflds ;"q ePsf ;dLs/0fx¿ ;Rrfpg'kg]{ b]vfpF5 -#, !(, @!, @@, @(–#%_ . t;y{ b'of{Gt/ / k|b'of{Gt/sf] ;+nUgtf /xg] ef}lts zf:qLo kl/df0fx¿sf] lgdf{0f ug'{kg]{ clgjfo{ cfjZostf b]lvG5 .

b'of{Gt/ / k|b'of{Gt/ ;;{tL x]bf{ /]vLo rfnsf ;Gbe{df b]lvg] ePsfn] kl/wLo rfnsf ;Gbe{df sf]0ffGt/ / k|sf]0ffGt/sf] k|of]u ug{' plrt x'g]h:tf] k|tLt x'G5 t/ To;f] ug'{ q'l6k"0f{ x'g hfG5 . o;sf nflu sfod ePsf] kl/lwnfO{ g} cfwf/ dfg]/ lkG8sf nflu e"ldsf lgjf{x ug]{ lglxt cw{Jof; / lglxt kl/lwsf] vf]hL ug'{ cfjZos b]lvG5 . ;a} lkG8sf] ;femf kl/lwsf ¿kdf /xg] gofF kl/lw rSsfx¿df b]lvg] kl/lwsf t'ngfdf ;fgf] x'g] ePsfn] ljBdfg jf b]lvg] kl/lwdf ;du| lkG8sf] s]xL c+zsf] j]u x'g'kg]{eGbf s]xL a9L ePsf] kfpg ;lsG5 . pQm j]u rSsfsf] ;Dk"0f{ lkG8sf] j]u eg] xf]Og . To:t} k|rlnt ef}lts zf:qdf k|j]uh;/L g} hs{sf] kl/efiff, a'emfO / ul0ftLo :j¿kdf klg ;'wf/ ug'{kg]{ b]lvG5 .

lkG8 ;/n /]vfsf] lbzfdf ultjfg x'Fbf To;sf] ultOgl;{of sd aflwt x'G5 eg] kl/wLdf j]ujfg x'Fbf a9L aflwt eO/x]sf] x'G5 . kl/wLo ansf sf/0f To;f] x'g hfG5 . pxL kl/df0fsf] lkG8 ;fgf] kl/lwdf x'Fbf / 7"nf] kl/lwdf x'Fbf ltgLx¿sf aLrsf] j]udf cGt/ b]vf kb{5 . 7"nf] kl/lwdf lkG8 3'Dbf ;fgf] kl/lwsf t'ngfdf rfnsf] /]vLo :jefj s]xL a9L dfqfdf b]vf kb{5 . o; qmddf k|rlnt k/Dk/fsf] df]d]G6 ckm Ogl;{ofnfO{ lkG8sf ¿kdf lng' k|of]u;Ddt b]lvFb}g .

an, zlQm jf ;fdYo{sf ;"qx¿ km/skm/s g} cfjZos x'g\ jf Pp6} ;"qn] sfd ubf{ub}{ e|djz leGgtf b]lvg uPsf] 5 eGg] ;Gbe{df klg cWoog ul/Psf] 5 -#, &, !*, #!, #@_ . ;Lwf lbzfdf ePsf jf 3'dfp/f] kydf ePsf an / sfo{, ultjfg sfo{, l:y/ sfo{, ;fdYo{ cflbsf nflu ;xL ;dfwfg lbg'kg]{ b]lvPsf] 5 . o;n] ef}lts zf:qdf oL / oL;Fu hf]l8g cfpg] ljljw kl/df0f jf kIfx¿sf] k'glg{df{0f cfjZos ePsf] b]vfpF5 . t;y{ ;+lIfKt ¿kdf lgDgjdf]lhdsf a'Fbfx¿df cfjZos ;dfwfgx¿ k|:t't ul/G5M

$=!= cj/f]wxLg If]qdf Psgf; ansf sf/0f ePsf] lglZrt 36gfsf nflu k|fKt b'of{Gt/;Fu lkG8 / b'of{Gt/;+Vof Jo'Tqmdfg'kflts x'G5g\ eg] ;dofGt/fn cg'qmdfg'kflts x'G5 -#, &, !(, @!, @@, #!, #@_ .

cyjf, F = kmD(n-1)/T                             -@!_

cyjf, F = kmr(n-1)                                  -@@_

hxfF, F, k, m, D, n, n-1, r / T n] qmdzM v'b an, s'g} crn u'0ff+s, lkG8, kb;+Vof, b'of{Gt/;+Vof, k|b'of{Gt/ / ;dofGt/fnnfO{ ;+s]t ub{5g\ .

o;k|sf/ s'g} klg kl/df0fsf] an nufgL ubf{ slt ;do nufgL ul/of] eGg] s'/f ;dfj]z geO{ an, zlQm, sfo{ jf ;fdYo{sf] dfqfaf/] hfGg ;lsFb}g . an nufgLsf a]nf ;+3if{ / ;+3if{;do b]vf kb{5g\ -#, &_ eg] ltgsf] kl/0ffdsf ¿kdf b'of{Gt/, b'of{Gt/ ;+Vof tyf ;dofGt/fnx¿ / ltgsf] ;fk]If e"ldsf b]vf kb{5 -!(, @!, @@_ . lg/Gt/ Psgf; ansf] nufgL ubf{ >f]tn] g} Psgf; ultdf an nufgL u/]sf] 5 eg] cj/f]wxLg If]qdf ;d]t Psgf; an nufgL ePsf] nfu] klg an nfUg gkfO{ b'of{Gt/ jf k|b'of{Gt/ gb]lvg ;S5g\ .

                          

sf]0fLo cfwf/df lj:yfkg ePsf] dfGbf Ps} kl/df0fsf] lkG8 ;fgf] kl/lwdf /x]sf] 5 eg] sd l56f] / 7"nf] kl/lwdf /xFbf a9L l56f] 3'ld/x]sf] x'G5 . o;y{ p;n] kl/lwdf kf/ x'g] b'/Lsf] dfqf klg sd jf a];L eO/x]sf] x'G5 . o;k|sf/n] pxL lkG8df klg ult a9fpg yfn]sf vG8df a9L kl/df0fsf] an nfUg k'U5 . o; cj:yfdf ultsf] ptf/r9fjsf sf/0f lkG8 ;dfg eP klg nufgL ePsf] ansf] dfqfdf cGt/ x'g k'U5 . o;y{ df]d]G6 ckm Ogl;{ofsf] wf/0ff e|djz hlGdPsf] b]lvG5 . o; cj:yfdf klg ;dLs/0f @! jf @@ g} ldNg hfG5 . hxfF, D = C₁-C₀, r = C₁-C₀/T x'G5 / C₁ / C₀ n] qmdzM ;'¿sf] cj:yfsf] b'/L -kl/wLo_ / klxnf] ;dofGt/fndf kf/ ePsf] kl/wLo b'/LnfO{ a'emfpF5g\ .

o; cfwf/df t/u+sf] :jefj, j]u / ;DalGwt zlQmsf] cWoog ug{ klg Ps} ;"qn] e"ldsf lgjf{x ub{5 eGg] s'/f :ki6 x'G5 . k|b'of{Gt/ / o;;Fu hf]l8P/ cfpg] r/x¿sf] ;xf/fn] g} ;a} an / zlQmx¿sf] cWoog ug{ ;lsG5 . ;/n /]vf jf jqm /]vf jf rqmdf x'g] k|b'of{Gt/x¿sf] lbzf jf ky km/s k/] klg anhGo e"ldsf ;dfg k|s[ltsf] b]lvG5 .

$=@= k|j]u tThGo ef}lts zf:qLo kl/df0f xf] . o;nfO{ :yfkgf ug]{ ul0ftLo cfwf/ eg] 5}g / x'Fb}g . tYoOt/ slyt ts{o'Qm jfSosf ¿kdf dfq of] k|rlnt eO/x]sf] b]lvG5 . ef}lts zf:qdf lj1fg;Ddt cfwf/ lnFbf k|j]un] geO{ k|b'of{Gt/n] e"ldsf lgjf{x ub{5 -#, &, !(, @!, @@_ . t;y{ ult k|lt;dofGt/fnsf ¿kdf cfpg] k|j]u lj1fg;Ddt kl/df0f geO{ b'of{Gt/ k|lt;dofGt/fnsf ¿kdf /xg] k|b'ofGt/ lj1fg;Ddt ef}lts zf:qLo kl/df0f xf] . s'g} klg kl/df0f jf:tljs ¿kd} lj1fg;Ddt / tYok"0f{ kl/df0f x'gsf nflu :jo+l;4 tYo / ul0ftLo :jLs[ltx¿sf cfwf/df kl/eflift x'g'sf ;fy} / k|of]uzfnLo k/LIf0faf6 ;d]t k|dfl0ft x'g'kg]{ afWofTdstf x'G5 .

cyjf, a = (v-u)/t geO{ r = (d-d₀)/t ;xL x'g hfG5 . hxfF a, v, u, t, r, d / d₀ n] qmdzM k|j]u, clGtd ult, ;'¿sf] ult, ;do, k|b'of{Gt/, clGtd b'/L / ;'¿sf] b'/LnfO{ hgfpFb5g\ . o;n] k|j]usf] k|To]If jf k/f]If e"ldsf ePsf ef}lts zf:qLo kl/df0f, lgod, l;4fGt tyf ;dLs/0fx¿ tYoOt/ x'g] k|dfl0ft ub{5 -#, &, (–!!, !$–!&, !(, @!, @@, @(–#%_ . tnsf a'Fbfx¿df tLdWo]sf s]xL ;d:of / ;dfwfgx¿ k|:t't ul/Psf] 5 .

$=#= tf}n;Fu lkG8 / u'¿TTjk|b'of{Gt/sf] u'0fgkmn a/fa/ x'g] dfGotf klg $=!= / $=@= df lbOPem}F unt x'g] b]lvG5 . cyf{t\ tf}n klg anh:t} :jefjsf] ef}lts zf:qLo kl/df0f ePsfn] tf}n;Fu lkG8, b'of{Gt/ / b'of{Gt/sf] ;+Vofsf] u'0fgkmnnfO{ ;dofGt/fnn] efu ubf{ cfpg] kl/0ffd a/fa/ x'G5 . t/ s'g} kbfy{ hf]v]/ dfkg ug]{ ;Gbe{df lkG8afx]ssf ;do, b'of{Gt/ cflb ljifonfO{ a]jf:tf ul/G5 . o;sf] eg] 5'6\6} Jojxfl/s a'emfO /x]sf] 5 h'g ef}lts zf:qLo kf6f]eGbf leGg 5 .

cyjf, W = mg geO{ W = F = mD(n-1)/T = mr(n-1) ;xL x'g hfG5 . hxfF W, m, g, F, D, (n-1), T / r n] qmdzM tf}n, lkG8, u'¿TTjk|j]u, an, b'of{Gt/, b'of{Gt/ ;+Vof, ;dofGt/fn / k|b'of{Gt/nfO{ a'emfpF5g\ .

$=$= crn k|j]usf] cj:yfdf /x]sf] j:t'sf nflu cIfx¿df HofldtLo k|ltlglwTj u/fpFbf ;/b/ ult klxNofpg'k5{ . o; qmddf ;'¿sf] ultdf k|j]u / ;dosf] u'0fgkmnsf] cfwf dfqf hf]8\g] ul/G5 . o; tl/sfaf6 ;/b/ ult lgsfNbf s'g} ult @=% ld6/ k|lt;]s]G8 eO{ $ ;]s]G8df !) ld6/ b'/L kf/ ePsf] x'g'kg]{df pQm ult @ ld6/ k|lt;]s]G8 eO{ * ld6/ b'/L kf/ ePsf] x'g] cyjf s'g} ult % ld6/ k|lt;]s]G8 eO{ $ ;s]G8df @) ld6/ b'/L kf/ ePsf] x'g'kg]{df $ ld6/ k|lt;]s]G8 eO{ !^ ld6/ b'/L kf/ ePsf] x'g] h:tf ;d:ofx¿ b]lvg k'U5g\ .

cyjf, V_av = u + at/2 geO{ V_av = u + nr/2 x'g hfG5 . hxfF V_av, u, a, t, n / r n] qmdzM ;/b/ ult, ;'¿sf] ult, k|j]u, ;do, kb;+Vof / k|b'of{Gt/nfO{ hgfpF5g\ .

$=%= 3'dfp/f] an jf 6s{nfO{ df]d]G6 ckm Ogl;{of / sf]0fLo k|j]usf] u'0fgkmnaf6 lgsflnG5 . lkG8sf] dfqf pxL x'g] t/ ansf] dfqf km/s x'g] sf/0f lkG8df ePsf] cGt/ xf]Og . of] t lkG8sf] c+zsf] ult jf j]udf x'g] cGt/sf sf/0f b]vf kg]{ x'G5 . t;y{ km/skm/s cfsf/sf lkG8x¿sf] c+zx¿sf] cjl:yltcg'¿k kl/lwljGb' jf ;DalGwt cw{Jof;sf] dfqfdf km]/abn cfpF5 . To:t} $=! n] lbPem}F sf]0fLo k|j]u klg s'g} lj1fg;Ddt kl/df0f xf]Og . o;/L b'O{ tYoOt/ kl/df0fx¿sf] u'0fgkmnaf6 k|fKt x'g hfg] kl/0ffd klg ;xL x'g ;Sb}g . Ps} kl/df0fsf] an s'g} lkG8df nufpFbf b]lvg] ulteGbf ToxL an Tof]eGbf ;fgf] lkG8df nufpFbf b]lvg] ult a9L x'g] x'G5 . To;/L g} pxL lkG8df x'g hfg] ultsf] ptf/r9fjsf qmddf nufgL ansf] dfqf klg a9L jf sd x'g hfG5 . pxL lkG8nfO{ ;dfg kl/df0fsf] jf:tljs an nufgL u/L 3'dfp/f] kydf j]ujfg agfpFbf kl/lw a9]sf vG8df sd ult / kl/lw 36]sf vG8df a9L ult b]vf kb{5 . o; s'/fnfO{ k|rlnt dfGotfdf eg] sf]0fLo ult jf sf]0fLo k|j]usf cfwf/df kl/eflift ug{ vf]lhPsf] b]lvG5 .

cyjf, 𝝉 = Iα geO{ 𝝉 = F = mD(n-1)/T = mr(n-1) x'g hfG5 . oxfF k|o'Qm 𝝉, I, α, F, m, D, (n-1), r / T n] qmdzM 3'dfp/f] an jf 6s{, df]d]G6 ckm Ogl;{of, sf]0fLo k|j]u, an, lkG8, -3'dfp/f]_ b'of{Gt/, k|b'of{Gt/ ;+Vof, k|b'of{Gt/ / ;dofGt/fnnfO{ hgfpF5g\ .

$=^= sfo{nfO{ ef}lts lj1fgdf k|rlnt an / b'/Lsf] u'0fgkmnaf6 k|fKt ul/G5 . k|rlnt ansf] ;"q ;xL gePsf] dfly pNn]v eO;s]sf] 5 . o;afx]s s'g} cjlwdf nufgL ePsf] anaf6 :jtM sfo{ x'G5 -#, &, !*_ . t;y{ sfo{df k'gM b'/Lsf] ;+nUgtf u/fpg' ;xL x'g ;Sb}g .

cyjf, W = Fd geO{ W = F = mD(n-1)/T = mr(n-1) x'g hfG5 . oxfF k|o'Qm W, F, d, m, D, (n-1), r / T n] qmdzM sfo{, an, b'/L, lkG8, b'of{Gt/, b'of{Gt/ ;+Vof, k|b'of{Gt/ / ;dofGt/fnnfO{ hgfpF5g\ .

o;n] j:t' rNbf jf grNbf b'j} cj:yfdf x'g] sfo{nfO{ j0f{g ub{5 . j:t' grNbf klg sfo{ x'G5 t/ Tolta]nf cj/f]wsf ¿kdf cfpg] k|ltsfo{n] /f]S5 / sfo{ gePsf] k|tLt x'g hfG5 . h:tf] s'g} ef/ s]xL a]/;Dd yfld/fVg' klg sfo{ xf] t/ ;"q gag]sfn] o;nfO{ sfo{ gePsf] dflgPsf] dfq xf] . ultzLn uf8LnfO{ a|]sn] /f]S5 eg] Tof] klg sfo{ xf] . s'g} pks/0fsf leQf jf ljlwsf] ;xf/faf6 slt ;do;Dd slt dfqfdf ca/f]w pTkGg ul/G5 eGg] s'/fsf] olsg ul/of] eg] oL b'j} ljifo Jojxfl/s / k|fof]lus ¿kdf a/fa/ g} x'G5g\ . h:tf] pQm j:t' ca/f]wxLg / u'¿TjxLg If]qdf ePsf] eP yfld/fVg' kb}{gYof] . u'¿TTjIf]qdf s'g} kl/df0fsf] l:ylt zlQmnfO{ a/fa/L u/]/ /f]Sg] an cfkm}df sfo{ xf] . Pp6} ljifonfO{ km/skm/s gfd lbg] / ;Gbe{cg';f/ ;"qx¿ km/skm/s lbg] sfo{ ePsfn] e|d k}bf x'g k'u]sf] dfq xf] .

xfdLn] agfPsf / k|rngdf /x]sf ;"qx¿n] tL k|fs[lts tYox¿nfO{ k|ltlglwTj ug{ g;Sg' ;"qlgdf{0fdf ePsf] sdhf]/L xf] . k|rlnt k/Dk/fsf] an / tf}nsf] lgod Pjd\ cfoflds ;"q tYoOt/ x'G5 . xfd|f] sdhf]/Lsf cfwf/df k|fs[lts / :jfefljs tYox¿nfO{ km/s ¿kn] j0f{g ug'{ lj1fg;Ddt / tYok/s x'Fb}g .

$=&= k|rlnt k/Dk/fsf] ;fdYo{ s'g} ;dofGt/fndf ePsf] sfo{ xf] . t/ dfly $=^ / $=!= df elgPem}F sfo{ cfkm}df s'g} ;dofGt/fndf eO/x]sf] x'G5 . o;df km]l/ s'g} ;dosf] e"ldsf yk u/]df emg\ ;d:of lglDtG5 . a9L an nufgL x'Fbf pxL ;dofGt/fndf a9L kl/df0fsf] sfd x'g] / sd an nufgL x'Fbf sd kl/df0fsf] sfd x'g] x'G5 . To:tf] a]nf b'of{Gt/ jf k|b'of{Gt/sf] dfqf a9\g] jf 36\g]dfq ePsf] x'G5 . jfik OlGhgdf pxL rfk sfod /fvL gf]hn 36fPdf d]lzgn] sd ult lnG5 / a9fPsf vG8df OlGhgn] a9L ult lnG5 . o;sf] cy{ s] xf] eg] b'j} cj:yfdf jfik s0fx¿sf] ult pxL eP klg ltgsf] dfqf a9L x'Fbf;fy cj/f]wnfO{ l56f] l5rf]Ng ;lsG5 t/ o;sf] ;Ldf s0fx¿sf] clwstd ultsf cfwf/df lglZrt x'G5 .

cyjf, P = W/t geO{ P = F = W = mD(n-1)/T = mr(n-1) x'g hfG5 . oxfF k|o'Qm P, W, F, t, m, D, (n-1), r / T n] qmdzM ;fdYo{, sfo{, an, ;do, lkG8, b'of{Gt/, b'of{Gt/ ;+Vof, k|b'of{Gt/ / ;dofGt/fnnfO{ hgfpF5g\ .

$=*= Olkml;oG;LnfO{ sfo{ k|flKt / zlQm k|j]zsf] cg'kftsf ¿kdf j0f{g ul/G5 . sfo{ jf zlQm eGg' jf an eGg' Pp6} kl/df0f ePsfn] o;nfO{ ca/f]w cflbsf afjh'b k|fKt x'g] v'b an / nufgL ansf] cg'kftsf ¿kdf j0f{g ug'{ cfjZos b]lvG5 .

cyjf, Efficiency = Work out/Energy in geO{ Efficiency = Force out/Force in x'g hfG5 . an, sfo{ jf zlQm Pp6} eP klg gofF ;"qx¿sf t'ngfdf k|rlnt k/Dk/fsf ;"qx¿ km/s 5g\ . $=& df klg ansf] ;"q lbOPsf] 5 .

$=(= oflGqs lskmfotnfO{ lgisfl;t an / nufgL ansf] cg'kftsf ¿kdf j0f{g ul/G5 . oflGqs lskmfotdf klg cj/f]w gePsf vG8df nufgL an / k|ltkmn an a/fa/dfq x'g] b]lvG5 . t/ k|rlnt k/Dk/fdf nufgL an / k|fKt sfo{ km/s x'g] dflgG5 . lkG8 / ult jf ;Defljt ultsf] u'0fgkmn g} an x'g] ePsfn] pQm ultdf dfq km]/abn x'g k'u]sf] x'G5, ansf] dfqfdf ptf/r9fj cfpg] x'Fb}g . oflGqs lskmfotdf klg an;Fu an a/fa/ g} eO/xsf] x'G5 .

cyjf, Mechanical Advantage = Force out/Force in -hxfF nufgL an / k|fKt an c;dfg klg x'G5g\_ geO{ Mechanical Advantage = Force out/Force in = 1 cyjf Force out = Force in x'g hfG5 . h:tf], pQf]nsdf km/skm/s glthf x'G5 t/ oflGqs lskmfot geO{ ult Go"gLs/0f x'g hfG5 . k/:k/df ljk/Lt x'g] u/L Ps};fy anx¿ nufgL ePsf vG8df o; vfnsf] cj:yf l;h{gf ePsf] x'G5 . o; ;Gbe{df pxL kl/df0fsf] ef/ lsg sd jf a];L cg'ej x'G5 eGg] s'/fn] e"ldsf lgjf{x u/]sf] b]lvG5 . s'g} ;do;Dd nufgL u/]sf] ann] cj/f]wsf sf/0f s'g} lkG8 rNg jf grNg ;S5 t/ ;DefJo ultn] o;df e"ldsf lgjf{x u/]sf] x'G5 . pQf]nssf ;jfndf tf}n / tf}nsf] b'/Lsf] u'0fgkmn;Fu an / ansf] b'/Lsf] u'0fgkmn a/fa/ x'G5 eGg] ;"q ag] klg lkG8 / ult;Fu csf{kl6\6sf] lkG8 / ult -ansf] :j¿ksf]_ a/fa/ x'G5 .

$=!)= ;'¿sf] ultdf k|j]u / ;dosf] u'0fgkmn yKbf cfpg] kl/df0f clGtd ult;Fu a/fa/ x'G5 eGg] k|rlnt dfGotf 5 . Psflt/ k|j]u cfkm}df s'g} lj1fg;Ddt kl/df0f gePsfn] Tof];Fu u'0fg ubf{ cfpg] glthf ;xL x'g kfpFb}g eg] csf{lt/ PsfOafx]s cGo ;dofGt/fnsf nflu k|j]un] ;xL dfg glbg] x'Fbf o; ;"qaf6 clGtd ult g} km/s b]lvg cfpF5 -#, !(, @!, @@, ##–#%_ .

cyjf, v = u+at geO{ v = u+r(n-1) x'g hfG5 . hxfF v, u, a, t, r / (n-1) n] qmdzM clGtd ult, ;'¿sf] ult, k|j]u, ;do, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ a'emfpFb5g\ .

$=!!= lj:yfkg klxNofpg] ;"qdf klg ;d:of /x]sf] b]lvPsf] 5 -#, &, !(, @!, @@, @(, #)_ . o;sf nflu ;'j|t ;dLs/0fx¿af6 ;dfwfg lbOPsf] 5 -!(, @!, @@, @(, #)_ .

cyjf, s = ut+(1/2)at² geO{ s = ut + nrt/2 x'g hfG5 . hxfF s, u, a, t, r / n n] qmdzM lj:yfkg, ;'¿sf] ult, k|j]u, ;do, k|b'of{Gt/ / kb;+VofnfO{ a'emfpFb5g\ .

$=!@= v² = u² + 2as cfkm}df ;xL b]lvFb}g -@!, @(, #)_ . hxfF v, u, a / s n] qmdzM clGtd ult, ;'¿sf] ult, k|j]u / lj:yfkgnfO{ hgfpF5g\ .

                                       

cyjf, v² = {u+r(n-1)}² x'g hfG5 . ;DalGwt ;+s]tx¿n] $=!) dfem}F ;+s]t ub{5g\ .

$=!#= ;dLs/0f s = (u + v)t/2, hxfF v, u / t n] qmdzM clGtd ult, ;'¿sf] ult, / ;donfO{ hgfpF5g\, ;xL b]lvFb}g -!(, @!_ .

cyjf, s = (u₁ + v)t/2 x'g hfG5 . hxfF v, u₁ / t n] qmdzM clGtd ult, klxnf] ult, / ;donfO{ hgfpF5g\ .

$=!$= ;dLs/0f s = vt - (1/2)at², hxfF s, v, a / t n] qmdzM lj:yfkg, clGtd ult, k|j]u / ;donfO{ hgfpF5g\, tYoOt/ 5 -!(, @!_ .

cyjf, s = vt - nrt/2 x'g hfG5 . hxfF s, v, t, n / r n] qmdzM lj:yfkg, clGtd ult, ;do, kb;+Vof / ;donfO{ hgfpF5g\ .

$=!%= sf]0fLo j]usf] dfGotfn] sf]0fsf cfwf/df 3'dfp/f] rfnsf] cWoog ub{5 . o;/L cWoog ubf{ lkG8sf nflu df]d]G6 ckm Ogl;{ofsf] agfj6L cfwf/ lnPsf] b]lvG5 . t/ lkG8nfO{ h]sf]t] dfg]/ g} ;xL ;dfwfg lgsfn]df dfq ;xL ;dfwfg k|fKt ug{ ;lsG5 . o;sf nflu kl/wLo j]usf] dfGotf cl3 ;fg'{kg]{ x'G5 .

cyjf, v = rω geO{ v = d/t x'g hfG5 . hxfF v, r, ω, d / t n] qmdzM clGtd ult, cw{Jof;, sf]0fLo j]u, 3'dfp/f] b'/L / ;donfO{ hgfpF5g\ .

$=!^= l:ylt oflGqs zlQmnfO{ lkG8, u'¿TTjk|j]u / prfOsf] u'0fgkmn;Fu a/fa/ x'g] dflgG5 . t/ of] klg dfly JofVof ul/Pem}F an g} ePsfn] k|rlnt k/Dk/fsf] km/s ;"qn] ;xL ;dfwfg lbg ;Sb}g .

cyjf, P = mgh geO{ P = F = mr(n-1) x'g hfG5 . hxfF P, m, g, h, F, r / (n-1) n] qmdzM l:ylt zlQm, lkG8, u'¿TTjk|j]u, prfO, an, cw{Jof; / k|b'of{Gt/;+VofnfO{ hgfpF5g\ .

$=!&= ultsf] ju{ / lkG8sf] u'0fgkmnsf] cfwf dfqfnfO{ ultzlQm dflgG5 . of] ;"q ;'jft ;dLs/0fdf ePsf] uNtL / b'/LnfO{ ;+nUg u/fpg] sfo{;DaGwL tYoOt/ dfGotfsf cfwf/df lgdf{0f ul/Psf] ;"q xf] -#, !(, @!, @@_ . t;y{ o; ;"q ;xL ePsf] kfpg ;lsFb}g .

cyjf, K = (1/2)mv² geO{ K = F = mr(n-1) x'g hfG5 . hxfF K, m, F, r / (n-1) n] qmdzM ultzlQm, lkG8, an, cw{Jof; / k|b'of{Gt/;+VofnfO{ hgfpF5g\ .

$=!*= ODkN;nfO{ an / ;dosf] u'0fgkmnsf ¿kdf j0f{g ul/G5 . cfkm}df ;xL gePsf] k|j]udf cfwfl/t an / ;dosf] u'0fgkmnaf6 df]d]G6ddf ePsf] kl/jt{g yfx nfUg] s'/f pNn]v ePsf] kfOG5 . t/ Tof] ljleGg ;dofGt/fnsf nflu ul/g] k/LIf0fx¿df ;xL b]lvFb}g -#, &_ . o;n] ubf{ PsfOafx]ssf ;dofGt/fnsf ;Gbe{df clGtd ult g} ;xL glgl:sg] ;d:of b]vf kb{5 eg] k|j]u klg ;xL x'Fb}g -@@_ . o;y{ ODkN;sf] dfGotf klg ;xL 5}g .

cyjf, Impulse = Ft geO{ Impulse = F = mr(n-1) x'g hfG5 . hxfF m, F, r / (n-1) n] qmdzM lkG8, an, cw{Jof; / k|b'of{Gt/;+VofnfO{ hgfpF5g\ .

$=!(= ω = ω₀ + αt sf]0fLo j]u lgsfNg] ;"q xf] . pxL sf]0fdf klg Pp6} kl/df0fsf] lkG8sf nflu kl/lw a9L jf sd x'g ;Sg] ePsfn] sf]0fLo a]usf] dfGotf d]n vfg] b]lvFb}g . To;sf nflu df]d]G6 ckm Ogl;{ofsf] jgfj6L / tYoOt/ Joj:yf ug]{ ul/G5 . To;dfly s'g} klg k|j]u jf ju{;doo'Qm kl/df0fsf ¿kdf /x]sf] sf]0fLo k|j]u cfkm}df ;xL gePsf] s'/f dfly g} :ki6 eO;s]sf] 5 . t;y{ sf]0fLo j]usf nflu kl/wLo j]unfO{ g} cfwf/ dfGg'kg]{ x'G5 .

                              

cyjf, v = u + r(n-1) n] kl/wLo clGtd j]unfO{ j0f{g ub{5 . hxfF v, u, r / (n-1) n] qmdzM clGtd ult, ;'¿sf] ult, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=@)= 𝛉-𝛉₀ = ω₀t + (1/2)αt² sf]0fLo lj:yfkg lgsfNg] ;"q xf] . sf]0fLo k|j]u / of] ;"q ;xL gePsfn] o;sf] ;dfwfgsf nflu ;'j|t ;dLs/0fsf] k|of]u ug'{kg]{ x'G5 .  

cyjf, s = ut + nrt/2 n] 3'dfp/f] lj:yfkgnfO{ j0f{g ub{5 . hxfF s, u, n, r / t n] qmdzM lj:yfkg, ;'¿sf] 3'dfp/f] j]u, kb;+Vof, k|b'of{Gt/ / ;donfO{ hgfpFb5g\ .

$=@!= ω² = ω₀² + 2α(𝛉-𝛉₀), hxfF ω, ω₀, α, 𝛉 / 𝛉₀  n] clGtd sf]0fLo j]u, ;'¿sf] sf]0fLo j]u, sf]0fLo k|j]u, /]l8og-%&=#+_ / ;'¿sf] sf]0fnfO{ a'emfpF5g\, h'g sf]0fLo j]usf] ju{kmn lgsfNgsf nflu k|of]u ul/g] ;"q xf] . sf]0fLo k|j]u / of] ;"q ;xL gePsfn] o;sf] ;dfwfgsf nflu ;'j|t ;dLs/0fsf] k|of]u ug'{kg]{ x'G5 . 

cyjf, v² = {u + r(n-1)}² n] sf]0fLo ultsf] ju{kmn lgsfNb5 . hxfF v, u, r / (n-1) n] qmdzM kl/wLo j]u, ;'¿sf] j]u, kl/wLo k|b'of{Gt/ / k|b'of{Gt/;+VofnfO{ hgfpFb5g\ .

$=@@= 𝛉-𝛉₀ = ωt - (1/2)αt² sf]0fLo lj:yfkg lgsfNg] ;"q xf] . dfly lbOPem}F sf]0fLo k|j]u / of] ;"q ;xL gePsfn] o;sf] ;dfwfgsf nflu ;'j|t ;dLs/0fsf] k|of]u ug'{kg]{ x'G5 . 

cyjf, s = vt - nrt/2 n] 3'dfp/f] lj:yfkgnfO{ j0f{g ub{5 . hxfF s, v, n, r / t n] qmdzM lj:yfkg, 3'dfp/f] clGtd ult, kb;+Vof, k|b'of{Gt/ / ;donfO{ hgfpFb5g\ .

$=@#= pmWj{rfknfO{ ult, 3gTTj / u'¿TTjk|j]usf] u'0fgkmnaf6 lgsflnG5 . s'g} klg k|j]u cfkm}df ;xL gx'g] ePsfn] u'¿TTjk|j]udf klg ;d:of ePsf] b]lvG5 -@@, ##–#%_ .

cyjf, F_B = vρg geO{ F_B = F = mr(n-1) x'g hfG5 . hxfF F_B, v, ρ, g, m, r / (n-1) n] qmdzM pmWj{rfk, cfotg, 3gTTj, u'¿TTjk|j]u, lkG8, k|b'of{Gt/ / k|b'of{Gt/;+VofnfO{ hgfpFb5g\ .

cfsif{0f ann] s'g} kl/df0fsf] lkG8 tnlt/ tfGg] / ;Fu} t/n kbfy{ klg tfGg] ubf{ cfkm\gf] an;Fu pxL kl/df0fsf] cfkm\gf] an cj/f]wsf ¿kdf pleg k'U5 . j:t'sf] cwf]an pWj{aneGbf a9L ePdf j:t' tnlt/ hfG5 eg] sd ePdf j:t' 8'Ab}g .

$=@$= u'¿TTjk|j]usf] dfg (=* ld6/ k|lt;]s]G8k|lt;]s]G8 dflgG5 . t/ of] cfkm}df k|b'of{Gt/ ePsfn] b'of{Gt/ (=* ld6/ x'g] cj:yfdfdfq k|b'of{Gt/ (=* ld6/ k|lt;]s]G8 n]Vg'k5{ . gq o;n] PsfO ;do tyf cGo u/L ;a} ;dofGt/fndf ;xL dfg lbg ;Sb}g .

cyjf b'of{Gt/ (=* ld6/ ePsf] cj:yfdf, g = 9.8 m/s² geO{ g = r = 9.8 m/s x'g hfG5 . hxfF g, m, s / r n] qmdzM u'¿TTjk|j]u, ld6/, ;]s]G8 / k|b'of{Gt/nfO{ hgfpFb5g\ .

$=@%= s'g} j:t' t/n kbfy{df 8'Abf pQm j:t'n] cg'ej ug]{ pmWj{rfk To; j:t'n] lj:yfkg ug]{ t/n kbfy{sf] tf}n;Fu 36L, a9L jf hlt;'s} x'g ;S5 t/ ef}lts lj1fgdf a/fa/ g} x'g] eGg] dfGotf /x]sf] b]lvG5 -%, ^, @)_ . o;n] ;xL ;dfwfg glbg] x'Fbf ;xL ;dfwfgsf nflu gofF l;4fGt k|ltkfbg ePsf] 5 .

cyjf, U = W geO{ U ⋚ W x'g hfG5 . hxfF U / W n] qmdzM pmWj{rfk / lj:yflkt t/n kbfy{nfO{ hgfpFb5g\ -%, ^, @), @$, @%_ . ;fy} $=@# jdf]lhd U = mr(n-1) x'G5 .

$=@^= ef}lts lj1fgdf $=@% jdf]lhdsf] cj:yfdf pmWj{rfk;Fu lj:yflkt t/n kbfy{sf] tf}n a/fa/ x'G5 eGg] wf/0ff /x]sf] kfOG5 . t/ k/LIf0fdf To;f] ePsf] kfpg ;lsFb}g . a¿ pmWj{rfk eGg' / c6fpg] t/n kbfy{sf] tf}n eGg' ul0ftLo cfwf/df Pp6} ljifo ePsf] b]lvG5 . h'g prfO{, If]qkmn, 3gTTj / u'¿TTjk|j]usf] u'0fgkmnaf6 pmWj{rfk lgsflnG5 / ltg} prfO{, If]qkmn, 3gTTj / u'¿TTjk|j]usf] u'0fgkmnaf6 g} c6fpg] t/n kbfy{sf] tf}n lgsflnG5 . t;y{ k|rlnt dfGotfsf cfwf/df klg pmWj{rfk / c6fpg] t/n kbfy{nfO{ Pp6} kl/df0fsf ¿kdf j0f{g ug{ ;lsG5 . t/ o;nfO{ ans} ¿kdf lnO{ j0f{g ug'{kg]{ x'G5 .

cyjf, U = W_displaced ;xL geO{ U = mr(n-1) = W_occupied ;xL x'g hfG5 -%, ^, @), @$_ .

$=@&= $=@# / $=@^ sf] cj:yfdf s'g} efF8f] t/n kbfy{n] k"0f{tM el/Psf] 5 eg] j:t' 8'a]kl5 U = mr(n-1) = W_overflowed x'g hfG5 -%, ^, @), @$_ .

$=@*= $=@# jdf]lhdsf] ;Gbe{df t/n kbfy{ aflx/ gkf]lvg] cj:yf 5 eg] pmWj{rfksf] k"j{;txeGbf tNnf] efusf] dfqf / lj:yflkt t/n kbfy{sf] k"j{;txeGbf dflyNnf] efusf] tf}n;Fu a/fa/ x'G5 . oL b'j} kl/df0fx¿ klg an ePsfn] ;f]xL cfwf/df af]w ug'{kg]{ x'G5 .

cyjf, U = mr(n-1) = W_displaced geO{ U_{underthepreviouslevel} = mr(n-1) = W_{abovethepreviouslevel} x'g hfG5 . oxfF kf]lvg] t/n kbfy{sf t'ngfdf ;txeGbf dflysf] t/n kbfy{sf] dfqf sd x'g k'U5 .

$=@(= x'ssf] lgod s'g} j:t'sf] tGsfO;Fu ;DalGwt 5 . s'g} C0ffTds crn / tGsfO jf v'DrfOsf] dfqfsf] u'0fgkmnsf ¿kdf annfO{ JofVof ug]{ lgodsf ¿kdf o;nfO{ lnOG5 . pQm crnnfO{ PsfO an k|lt ld6/sf ¿kdf JofVof ul/G5 . s'g} tlGsg] j:t'nfO{ tGsfpFb} hfg] xf] eg] ;dfg kl/df0fsf] ann] sfd ub}{g, ansf] dfqf a9fpFb} n}hfg'k5{ . o; cj:yfdf tGsfO jf v'DrfOsf] dfqf k|lt ;dofGt/fnsf cfwf/df j0f{g ug'{kg]{ x'G5 . o;f] ubf{ ;dfg ;dodf pxL j:t' ;dfg nDafOdf tGsfPdf dfq tL an a/fa/ x'g hfG5g\ . j:t' tlGsFb} hfg] ;Gbe{df x/]s ;dofGt/fndf nufgL x'g] ansf] dfqf km/s x'g'df k|b'of{Gt/sf] dfqfdf km/s x'g k'U5 . a9\bf] k|b'of{Gt/sf nflu ;/b/ k|b'of{Gt/ lgsfNg ;lsG5 / o;sf nflu kb;+Vof / k|b'of{Gt/aLrsf] ;femf k|b'of{Gt/sf] u'0fgkmnsf] cfwf dfqfaf6 ;/b/ k|b'of{Gt/ lgsfNg ;lsG5 .

cyjf, F = ma = -kx geO{ F_total = mr(n-1) x'g hfG5 . hxfF F, m, a, k, x, F_total, r / (n-1) n] qmdzM an, lkG8, k|j]u, crn, tGsfOsf] dfqf, hDdf an, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpF5g\ . ansf] dfqf a9\b} hfg] ePsfn] oxfF lbOPsf] k|b'of{Gt/ ljleGg k|b'of{Gt/x¿sf] of]ukmnsf ¿kdf /xG5, cyjf lkG8sf] dfqfdf ptf/r9fj cfPsf] dfGg klg ;lsG5 .

$=#)= s'g} /a/ jf sdfgL df]6f] ePdf sd tflgg' / dl;g' eP a9L tflgg'df PsfO If]qkmndf kg]{ ansf] dfqfn] e"ldsf lgjf{x u/]sf] x'G5 . pxL kl/df0fsf] an df]6f] sdfgLdf nfUbf PsfO ;"Id s0fx¿df nfUg hfg] ansf] dfqf ljtl/t x'G5 eg] dl;g' sdfgLdf nfUbf t'ngfTds ¿kdf a9L Pslqt x'G5 . oxL sf/0f sdfgL jf /a/ cflb tlGsg k'U5g\ . tlGsO;s]sf] cj:yfkl5 v'lDrFbf klg df]6f] sdfgLn] a9L / dl;gf] sdfgLn] sd an ;Dk|]if0f ub{5g\ . kbfy{leq tlGsg] kgsf] dfqfsf nflu tfkqmd, nDafO, 3gTTj, kbfy{sf] ;+/rgf cflb ljifon] e"ldsf lgjf{x ub{5g\ . oL ljljw kIfx¿ ;+nUg ePsf] lglZrt ePkl5 pQm j:t'df s]slt tGsfO x'g ;S5 < eGg] s'/fsf] olsg ug{sf nflu ansf] dfqf / If]qkmnn] e"ldsf lgjf{x ub{5g\ . slt If]qkmndf slt kl/df0fsf] an nufgL x'Fbf slt dfqf tlGsG5 < eGg] ljifosf] cWoogdf klg x'ssf] lgodnfO{ s]lGb|t ug'{kg]{ b]lvG5 . tlGsg' / dl;gf]kg a9\g' b'O{ leGg r/sf ¿kdf b]lvFb}gg\ . of] :j¿k lau|g';Fu klg ;DalGwt x'g cfpF5 . t;y{ of] slt If]qkmndf slt kl/df0fsf] an nufgL ubf{ slt ulx/f] vfN8f] aG5 < eGg] ljifoh:t} cWoog x'g hfG5 . rfk / tGsfOsf dfqfx¿ s'g} anhGo ljifo geO{ cj:yfcg';f/ x'g hfg] ansf] k|efjsf dfqfx¿ x'g\ . s'g kbfy{df slt If]qdf s'g kl/df0fsf] an nufpFbf s]slt k|efj kb{5 eGg] s'/fsf] cWoog ug{ rfkn] ;xof]u k'/fPemF} tGsfOn] ;f]xLadf]lhd tGsfOsf] dfqfsf] hfgsf/L lbG5 .

$=#!= tlGsPsf] sdfgLsf] l:ylt zlQmsf] dfGotf klg dfly elgPem}F aneGbf km/s ljifo xf]Og . k|rngcg';f/ cGo ;"qx¿df zlQmsf cfwf/df cWoog u/]sf] ljifonfO{ o; ;DaGwdf klg lgs6 u/fpg vf]h]sf] b]lvG5 . o;df klg ;'wf/ ug'{kg]{ b]lvG5 . s'g} sdfgL tfg]/ 5f]8\bf clGtd cj:yf;Dd nufgL ansf] dfqf sd x'g k'U5 . t/ ;/b/ Psgf; ansf] sNkgf ubf{ clGtd ult klxNofpg ;lsG5 .

cyjf, W = (1/2)kx² = Work done on spring geO{ W = F = mr(n-1) x'g hfG5 . hxfF W, k, x, F, m, r / (n-1) n] qmdzM sfo{, crn, tGsfOsf] dfqf, an, lkG8, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=#@= s]Gb|d'vL ansf nflu klg k|rlnt dfGotf ;xL ePsf] b]lvFb}g . o; ;jfndf klg k|j]udf cfwfl/t eO{ ;f]xLjdf]lhdsf] cfofdcg'¿ksf] lgod agfpg] sfo{ ePsf] 5 . o;nfO{ ;RrfO{ gofF ;dfwfg lbg'kg]{ b]lvG5 .

cyjf, F = mv²/radius geO{ F = cmr(n-1) x'g hfG5 . hxfF F, m, v, c, r / (n-1) n] qmdzM s]Gb|d'vL an, lkG8, ult, s'g} crn, -3'dfp/f]_ k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=##= $=#@ df em}F kl/lwd'vL annfO{ klg j0f{g ug{ ;lsG5 . h;cg';f/ 3'dfp/f] kydf b]lvg hfg] ansf] dfqf / To;sf] kl/lwd'vL c;/nfO{ cWoog ug{ ;lsG5 .

cyjf, F = mv²/radius geO{ F = cmr(n-1) x'g hfG5 . hxfF F, m, v, c, r / (n-1) n] qmdzM kl/lwd'vL an, lkG8, ult, crn, 3'dfp/f] k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=#$= nufgLstf{ / k|fksn] k|fKt ug]{ Ogl;{ofTds wSsfsf] dfqf cfkm}df an xf] . 3'dfp/f] ;Gbe{df klg o;nfO{ k|fKt ug{ ;lsG5 . 3'dfp/f] ;Gbe{sf nflu lkG8 lgsfNbf df]d]G6 ckm Ogl;{ofsf] k|of]u x'G5 . o; dfGotfcg';f/ klg pxL lkG8sf nflu cfsf/sf cfwf/df km/skm/s kl/df0fsf] df]d]G6 ckm Ogl;{of x'g] dfgL sf]0fLo an klxNofOG5 . vf;df lkG8df s'g} kl/jt{g geO{ To;sf] lglxt jf:tljs cw{Jof;sf cfwf/df x'g] kl/wLo j]udf kl/jt{g ePsf] x'G5 .

cyjf, I = mr₁² geO{ m = m x'g hfG5 / nuftf/ gylkg] lg/Gt/ Psgf; ansf nflu Ogl;{of cfkm}df an g} xf] jf F = kmr(n-1) g} x'G5 . hxfF I, m, r₁, F, k, r / (n-1) n] qmdzM df]d]G6 ckm Ogl;{of, lkG8, cw{Jof;, an, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

pbfx/0fsf ¿kdf s'g} 3'Dg] /8sf nflu F = mr(n-1): r₁ = kr₂: k = n/2(n-1) x'G5, hxfF F, r, r₁, r₂, k, m, v / n n] qmdzM an, kl/wLo k|b'of{Gt/, cw{Jof;, lkG8sf] lglxt jf:tljs cw{Jof;M h;sf cfwf/df kl/wLo k|b'of{Gt/ lgsflnG5, s'g} crnM h;n] cw{Jof; / lglxt jf:tljs cw{Jof;aLr ;DaGw agfpF5, lkG8, s'g} ;dofGt/fndf k"/f ePsf] kl/wLo b'/L / ljGb'x¿sf] ;+Vof -h:tf] % ;]=ld= cw{Jof; ePdf ^ j6f ljGb'x¿ x'G5g\_ nfO{ hgfpFb5g\ .

$=#%= k|j]unfO{ 3'dfp/f] ;Gbe{df ju{ultnfO{ cw{Jof;n] efu ubf{ cfpg] efukmnsf ¿kdf j0f{g ul/G5 . t/ PsfO ;dofGt/fndf b]lvg] 3'dfp/f] b'of{Gt/ jf k|b'of{Gt/ g} j:t'ut / tYok"0f{ x'G5 . o; ljifodf dflysf a'Fbfx¿n] :ki6 ub{5g\ .

cyjf, a = v²/r₁ geO{ r = kD/T x'g hfG5 . hxfF a, v, r₁, r, k, D, t n] qmdzM k|j]u, ult, cw{Jof;, k|b'of{Gt/, crn, b'of{Gt/ / ;dofGt/fnnfO{ hgfpF5g\ .

$=#^= lg/Gt/ crn an nufgL ug{ ;s]df u'¿TTjfTds If]qdf klg k|b'of{Gt/ ;Dej x'G5 . l8h]n OlGhgn] lbg] an, lah'nL df]6/n] lbg] an, kfgL jf jfik OlGhgn] 6jf{Ogdf lbg] an cflb anx¿ u'¿TTj anh:tf lg/Gt/ Psgf; an xf]Ogg\ . ;'¿sf] cj:yfaf6 cfkm\gf] clwstd Ifdtfsf] cj:yf;Dddf NofpFbf;Dd ult a9fpFb} uP klg oL anx¿ s]xL a]/kl5 ;Lldt x'g k'Ub5g\ . clwstd ult k|fKt u/]kl5 km]l/ ultdf j[l4 x'g] ;Defjgf o; cj:yfsf anx¿df x'Fb}g . lglZrt rfkaf6 lgl:sPsf s0fx¿sf] ult cj/f]wxLg If]qdf klg crn /xG5, qmdzM a9\b}g . s'g} kl/df0fsf] lg/Gt/ an nufgL u/]sf] s]xL ;dokl5 k|fKt x'g] clGtd ult / lkG8sf] u'0fgkmnsf ¿kdf o;nfO{ lng ;lsG5 . oxfF k|fKt x'g] clGtd ult k|b'of{Gt/ / k|b'of{Gt/;+Vofsf] u'0fgkmnaf6 k|fKt x'G5 .

$=#&= $=#$ jdf]lhd ;f]ln8 l;lnG8/sf] df]d]G6 ckm Ogl;{of klg ;xL cfwf/af6 klxNofOPsf] b]lvFb}g . o; dfGotfdf klg ;'wf/ ug'{kg]{ x'G5 . cGo cj:yfdf em}F o; cj:yfdf klg ansf] dfqf lgsfNg ;lsG5 .

cyjf, I = (1/2)mr₁² geO{ F = cmr(n-1) x'g hfG5 . hxfF I, c, m, r₁, F, r / (n-1) n] qmdzM df]d]G6 ckm Ogl;{of, s'g} crn, lkG8, cw{Jof;, an, crn, lglxt cw{Jof;df cfwfl/t kl/wLo k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ . o; cj:yfsf nflu b]lvg] cw{Jof;;Fu crn / lglxt cw{Jof;sf] u'0fgkmn a/fa/ x'g] ePsfn] crnsf] dfgsf cfwf/df pQm lglxt jf:tljs cw{Jof; klxNofpg ;lsG5 .                                    

$=#*= $=#$ jdf]lhd aLrdf 5]l8Psf] /8sf] df]d]G6 ckm Ogl;{ofn] lj1fg;Ddt e"ldsf lgjf{x ug{ ;Sg] cfwf/ b]lvFb}g . o; dfGotfdf klg ;'wf/ ug'{kg]{ x'G5 . cGo cj:yfdf em}F o; cj:yfdf klg ansf] dfqf lgsfNg ;lsG5 .

cyjf, I = (1/12)mL² geO{ F = cmr(n-1) x'g hfG5 . hxfF I, m, L, F, c, r / (n-1) n] qmdzM df]d]G6 ckm Ogl;{of, lkG8, nDafO, an, crn, kl/wLo k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ . an ;Dk|]if0f ubf{ lkG8sf] b'/L lgsfNg ;d:of x'g k'Ubf unt k|rng /xg uPsf] b]lvG5 .

$=#(= To:t}, I = (1/3)mL² geO{ F = cmr(n-1) x'g hfG5 . hxfF I, m, L, F, c, r / (n-1) n] qmdzM df]d]G6 ckm Ogl;{of, lkG8, nDafO, an, crn, kl/wLo k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ . an ;Dk|]if0f ubf{ lkG8sf] b'/L lgsfNg ;d:of x'g k'Ubf unt k|rng /xg uPsf] b]lvG5 .

$=$)= lyg ;s'{n/ x'kdf, I_z = mr₁² geO{ F = cmr(n-1) x'g hfG5 . hxfF I_z, m, r₁, F, c, r / (n-1) n] qmdzM df]d]G6 ckm Ogl;{of, lkG8, cw{Jof;, an, crn, kl/wLo k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ . an ;Dk|]if0f ubf{ lkG8sf] b'/L lgsfNg ;d:of x'g k'Ubf unt k|rng /xg uPsf] b]lvG5 .

$=$!= lyg ;s'{n/ x'kdf, I_x = I_y = (1/2)mr₁² geO{ F = cmr(n-1) x'g hfG5 . hxfF I_x (=I_y), m, r₁, F, c, r / (n-1) n] qmdzM df]d]G6 ckm Ogl;{of, lkG8, cw{Jof;, an, crn, kl/wLo k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ . an ;Dk|]if0f ubf{ lkG8sf] b'/L lgsfNg ;d:of x'g k'Ubf unt k|rng /xg uPsf] b]lvG5 .

$=$@= lyg ;f]ln8 l8:sdf, I_z = (1/2)mr₁² geO{ F = cmr(n-1) x'g hfG5 . hxfF I_z, m, r₁, F, c, r / (n-1) n] qmdzM df]d]G6 ckm Ogl;{of, lkG8, cw{Jof;, an, crn, kl/wLo k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpF5g\ . an ;Dk|]if0f ubf{ lkG8sf] b'/L lgsfNg ;d:of x'g k'Ubf unt k|rng /xg uPsf] b]lvG5 .

$=$#= lyg ;f]ln8 l8:sdf, I_z = I_y = (1/4)mr₁² geO{ F = cmr(n-1) x'g hfG5 . hxfF I_x, m, r₁, F, c, r / (n-1) n] qmdzM df]d]G6 ckm Ogl;{of, lkG8, cw{Jof;, an, crn, kl/wLo k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ . an ;Dk|]if0f ubf{ lkG8sf] b'/L lgsfNg ;d:of x'g k'Ubf unt k|rng /xg uPsf] b]lvG5 .

$=$$= dflysf a'Fbfx¿df em}F cGo ;Gbe{df klg df]d]G6 ckm Ogl;{of km/skm/s cfsf/sf lkG8sf nflu km/skm/s kl/df0fsf] x'g] ub{5 . hals lkG8sf] dfqf Pp6} xF'bf klg cfsf/cg';f/ df]d]G6 ckm Ogl;{ofsf] dfqf km/s x'g] ub{5 . oxL km/s dfqfnfO{ sf]0fLo lkG8 klg eGg] ul/G5 . lkG8 pxL x'Fbf cfsf/ / :j¿kcg';f/ pQm lkG8sf] kl/wLo ult jf s'g} ;dofGt/fndf x'g] kmGsfGt/ km/s x'g k'Ub5 t/ lkG8 g} km/s x'g] u/L JofVof ul/Psf] b]lvG5 . o; qmddf df]d]G6 ckm Ogl;{of;Fu ;DalGwt cGo, emG8} yk @ bh{g -#^–#(_ ;"qx¿df ;d:of /x]sf] b]lvG5 . tL ;a}df dfly pNn]]v ul/Pem}F F = cmr(n-1) x'g hfG5 .

$=$%= 3'dfp/f] j]u;Fu ;DalGwt zlQm lgsfNgsf nflu df]d]G6 ckm Ogl;{of / sf]0fLo ultsf] ju{kmnsf] u'0fgkmnnfO{ cfwf ug]{ k|rng /x]sf] 5 . df]d]G6 ckm Ogl;{of / sf]0fLo ultsf] wf/0ff ;xL gePsf] af/]df dfly g} rrf{ ul/;lsPsf] 5 . o;afx]s an Psgf; sfod /fv]/ oL b'O{r/x¿nfO{ :jtGq ePsf] kfpg klg ;lsFb}g . t;y{ o; lgodnfO{ klg ;Rrfpg'kg]{ b]lvG5 .

cyjf, KE_rotational = (1/2)Iω² geO{ F = cmr(n-1) x'g hfG5 . hxfF, I, ω, F, c, m, r / (n-1) n] qmdzM df]d]G6 ckm Ogl;{of, sf]0fLo ult, an, crn, lkG8, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=$^= 3'dfp/f] / lrlKng] ult zlQmsf] of]ukmn lgsfNbf /]vLo ult zlQm / sf]0fLo ult zlQmsf] of]ukmn lgsflnG5 . o; ;Gbe{df klg ;xL cEof; ePsf] b]lvFb}g . t;y{ o; ;"qnfO{ ;Rrfpg'kg]{ b]lvG5 .

cyjf, KE_{rolling&slipping} = (1/2)mv² + (1/2)Iω² geO{ F_{rolling&slipping} = mr₁(n-1) + mr₂(n-1) x'g hfG5 . hxfF m, v, I, ω, r₁, r₂ / (n-1) n] qmdzM lkG8, ult, df]d]G6 ckm Ogl;{of, sf]0fLo ult, k|b'of{Gt/, kl/wLo k|b'of{Gt/ / -b'j}sf nflu_ k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=$&= sf]0fLo df]d]G6d lgsfNbf df]d]G6 ckm Ogl;{of / sf]0fLo ultsf] u'0fgkmnaf6 klxNofOG5 . vf;df ultsf] j[l4;Fu} a9\g] jf 36\g] ansf] dfqf lkG8sf cfwf/df dfq hfGg ;lsG5 .

                                                                   

cyjf, L = Iω geO{ F = mr(n-1) = mv x'g hfG5 . hxfF m, v, I, ω, r / (n-1) n] qmdzM lkG8, ult, df]d]G6 ckm Ogl;{of, kl/wLo ult, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=$*= sf]0fLo ODkN;nfO{ klg ;'wf/ ug'{kg]{ b]lvG5 . o;n] klg df]d]G6ds} e"ldsf lgjf{x ub{5 . df]d]G6ddf ePsf] kl/jt{g eGgfn] yk x'g uPsf] jf a9L cjlwsf] df]d]G6d g} xf] . o;nfO{ ansf] ;"qn] g} j0f{g ub{5 .

cyjf, 𝝙L = 𝝙Iω geO{ $=$& jdf]lhd F = mr(n-1) = mv x'g hfG5 . hxfF L, I, ω, F, m, r / (n-1) n] qmdzM sf]0fLo df]d]G6d, df]d]G6 ckm Ogl;{of, sf]0fLo j]u, an, lkG8, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=$(= s'g} klg ;"Id s0fdf s'g} kl/df0fsf] Psgf; an lg/Gt/ nfu]sf] 5 / cfGtl/s jf afx\o jf l;lh{t s'g} klg cj/f]w 5}g eg] nufgL ePsf] hDdf zlQm pQm s0fsf] clGtd ultdf b]vf kb{5 . kbfy{hGo k|ltlqmofsf ;Gbe{x¿df h]h:tf] x'g] eP klg ulthGo :jefjsf] j0f{g ubf{ o;af/]df gofF sf]0faf6 cWoog ug'{ cfjZos 5 . o; qmddf t/u+sf ;Gbe{df cfofdsf] dfqfsf] ptf/r9fjnfO{ ;d]t ;+nUg u/fpg'kg]{ x'G5 . cfj[lQnfO{ x/]s ;dofGt/fndf k"/f x'g] rqmsf ¿kdf lnOG5 / o;nfO{ t/u+nDafOn] u'0fg u/L t/u+j]u lgsflnG5 . t/ ;+uLtdf k|of]u x'g] ;'/x¿sf] cfj[lQ crn klg dflgG5 . olb Wjlgsf] j]u crn x'g] xf] eg] cfofddf lsg ptf/r9fj x'G5 < cfofdsf] ptf/ r9fjn] cfj[lQsf] dfqfdf km]/abn gug]{ xf] / j]udf klg cGt/ gx'g] xf] eg] cfofdsf] ;+nUgtf 5'6]sf] x'g'k5{ .

h:tf], Wjlgsf ;Gbe{df ;f sf nflu klxnf] ;f df x'g] xh{ hlt 5, Tof]eGbf bf]Aa/ bf];|f] ;f df x'g] x'G5 . t/+unDafO klg klxnf]sf] t'ngfdf bf];|f]df cfwf x'g] dflgG5 -$)_ . t;y{ km/skm/s ;f x¿sf] xh{df ptf/r9fj eP klg t/u+j]u ;dfg /xG5 . k|rlnt dfGotfn] o;} cg';f/ Wjlgt/u+sf] cWoog u/]sf] b]lvG5 . t/ o; ;dodf nufgL ansf cfwf/df cfofd eg] a9\g jf 36\g ;Sg] pNn]v 5 . o;n] j]usf] cWoogdf t/+u cfofdsf] ;+nUgtf 5'6]sf] k|dfl0ft ub{5 .

cyjf, v = Lf geO{ v = 2kr₁/T x'g hfG5 . hxfF v, L, f, k, r₁ / T n] qmdzM ult, t/+unDafO, cfj[lQ, cfj[lQ;+Vof, s'g} crn ;+Vof -jf kfO_, ;/b/ cfofd -jf ;/b/ cw{Jof; jf ;/b/ PlDKnRo'8_ / ;dofGt/fnnfO{ hgfpFb5g\ .

$=%)= $=$( df lbOPem}F E = hf = mc² geO{ F = mr(n-1) = m(2kr₁/T) = mv x'g hfG5 . hxfF E, h, f, m, c, v, F, r / (n-1) n] qmdzM zlQm, crn, af/Daf/tf, lkG8, k|sfzsf] ult, cflv/L jf sfod /xg] pRrtd ult, an jf zlQm jf ;fdYo{, k|b'of{Gt/ / k|b'of{Gt/;+VofnfO{ hgfpFb5g\ . k|rlnt k/Dk/fsf] df]d]G6dnfO{ cfwf/ dfg]/ x]bf{ klg an jf zlQmsf] gjk|ltkflbt ;"q k"0f{tM ;xL x'g hfG5 . k|rlnt k/Dk/fdf k|sfzsf] ultsf] ju{kmn k|j]u / ;'jftsf] tYoOt/ dfGotf / To;df cfwfl/t zlQm;DaGwL e|dsf sf/0f lgdf{0f x'g k'u]sf] xf] .

$=%!= hDdf zlQmnfO{ klg k|rlnt k/Dk/fcg';f/ l:ylt / ult zlQmsf] of]ukmnaf6 lgsflnG5 . t/ l:ylt zlQm / ult zlQmsf k|rlnt ;"q agfj6L / 36gf;Fu c;DalGwt ePsfn] ltgsf] k'glg{df{0f ug'{kg]{ cfjZostf ePsf] dfly g} :ki6 ul/Psf] 5 . s'g} j:t' dflylt/ p7fpFbf an jf zlQm nufOG5 / j:t' 5f]8]kl5 ult zlQm k}bf ePsf] dflgG5 . olta]nf hDdf zlQm ult zlQmsf ¿kdf b]vf kb{5 . h:tf], tf]/L k]Nbf ;fdfGotM t]n / lkgf kl/0flt x'G5 . tf]/L, t]n / lkgfsf] of]ukmn g} ;du| k|flKt xf]Og . t/ hf]8\g} k/]sf vG8df dfq To;f] x'G5 . k]G8'nddf klg x/If0f l:ylt zlQm / ult zlQm k/:k/df kl/jt{g x'g] qmd hf/L /xG5 . t/ cj/f]wn] lj:tf/} sd ul/lbG5 .

cyjf, Total energy = KE + PE geO{ F = mr(n-1) = mv x'g hfG5 . hxfF F, m, r / (n-1) n] qmdzM zlQm jf an, lkG8, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=%@= hs{nfO{ s'g} ;dofGt/fndf x'g] k|j]usf] kl/jt{gsf ¿kdf lnOG5 . o;n] cem cGof]ntf / ;d:of l;h{gf ub{5 . o;nfO{ ;dofGt/fn k'gM glnOsg b'O{ k|b'of{Gt/sf] cGt/af6 lgsfNg ;lsG5 . hs{ Psgf;n] a9\b} guPsf vG8df k|b'of{/Gt/x¿aLr x'g] cGt/x¿sf] ;/b/ cGt/ lgsfNg ;lsG5 . Psgf; j[l4sf nflu eg] kb;+Vof / k|b'of{Gt/x¿ aLrsf] cGt/sf] u'0fgkmnnfO{ cfwf u/L ;/b/ k|b'of{Gt/ k|fKt ug{ ;lsG5 .

cyjf, J = da/dt geO{ r' = r₂-r₁ x'g hfG5 . hxfF J, a, t, r', r₁ / r₂ n] qmdzM hs{, k|j]u, ;do, b'O{ k|b'of{Gt/x¿ aLrsf] cGt/, klxnf] k|b'of{Gt/ / bf];|f] k|b'of{Gt/nfO{ hgfpFb5g\ .

$=%#= 3if{0f an / cj/f]w ljleGg sf/0fn] 36a9 x'G5 . of] ;fdfGotM l:yltd"ns / ultd"ns k|s[ltsf] x'G5 . ;tx, wft'sf] :jefj, Ogl;{of, tfkqmd cflb ljljw kIfn] o;df e"ldsf lgjf{x ub{5g\ . 3if{0f an klg an jf zlQmnfO{ k|ltafb ug{] Ps k|sf/sf] an jf zlQm g} xf] . 3if{0f an jf cj/f]w ansf] dfqfnfO{ ;dofGt/fndf ;/b/ kl/df0f lgsfNg ;lsG5 . To;f] ePdf anh;/L g} o;nfO{ klg j0f{g ug{ ;lsG5 .

cyjf, F_f = ma = 𝞵F geO{ F_f = mr(n-1) x'g hfG5 . hxfF F_f, m, a, 𝞵, r / (n-1) n] qmdzM 3if{0f an, lkG8, k|j]u, crn, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=%$= s]Gb|d'vL annfO{ sf]0fLo j]u jf ultsf cfwf/df j0f{g ug]{ ul/G5 . o;af6 lkG8n] lbg] aneGbf km/s j0f{g x'g] x'Fbf o;nfO{ klg k'glg{df{0f ug'{ cfjZos 5 .

cyjf, F = -mω² geO{ F = -mr(n-1) x'g hfG5 . hxfF F, m, ω, r / (n-1) n] qmdzM s]Gb|d'vL an, lkG8, sf]0fLo j]u, k|b'of{Gt/ / k|b'of{Gt/ ;+VofnfO{ hgfpFb5g\ .

$=%%= n]u|]lGhogdf ult zlQm / l:ylt zlQmsf] cGt/sf] j0f{g / cWoog ul/G5 . t/ o;cg';f/ j0f{g ubf{ ult zlQm / l:ylt zlQmsf k|rlnt tYoOt/ ;"qx¿sf] k|of]u ePsf] kfOG5 . o; ljifonfO{ klg ;'wf/ ug'{kg]{ b]lvG5 .

cyjf, L = T-V geO{ 0 =. F_k - F_p x'g hfG5 . hxfF L, T, V, F_k / F_p n] qmdzM n]u|]lGhog, ult zlQm, l:ylt zlQm, ult an / l:ylt annfO{ hgfpF5g\ . oxfF Pp6} / ;DalGwt 36gfsf ;Gbe{df ult an / l:ylt an a/fa/ 5g\ . k|rlnt k/Dk/fdf ;"q ;xL gePsf sf/0f s'g} cGt/ b]lvg] k|tLt x'g uPsf] xf] . u'¿TTjIf]qdf prfO{cg';f/ clGtd ult lgsfNg ;lsg] ePsfn] jf dflysf $=! / cGo a'Fbfh;/L ansf] gofF ;"q nufpg ;lsg] ePsfn] l:ylt zlQm lgsfNg s'g} ;d:of x'Fb}g .

$=%^= XofldN6lgogdf klg $=%% sf] h:t} t/ of]ukmnsf] ;Gbe{ ldNg hfG5 . k|rlnt k/Dk/fsf ;"q jf lgodx¿df tYoOt/ cfwf/sf sf/0f ;d:of pTkGg eO{ b'O{ zlQmsf leGgleGg glthf xft nfUg] u/]sf] lyof] . hf]8g' k/]df eg] hf]8]/ jf bf]Aa/ u/]/ ;d:ofsf] ;dfwfg lgsfNg ;lsG5 .

$=%&= ult zlQm;Fu df]d]G6dsf] ju{sf] cfwf dfqfnfO{ lkG8n] efu ubf{ cfpg] kl/df0f a/fa/ x'g] dflgG5 . t/+uzlQmsf ;Gbe{df of] lgod nfu' ul/G5 . t/ o; ;"q klg tYoOt/ dfGotfdf cfwfl/t k|j]u, ;'jft / ultzlQmsf ;"qx¿sf cfwf/df lgdf{0f ul/Psfn] ;xL b]lvFb}g .

cyjf, KE = p²/2m geO{ KE = F = mr(n-1) x'G5 . hxfF p, F, r, (n-1) / m n] qmdzM df]d]G6d, an, lglxt cw{Jof;df cfwfl/t k|b'of{Gt/, k|b'of{Gt/ ;+Vof / lkG8nfO{ ;+s]t ub{5g\ .   

$=%*= To:t} b'O{ zlQmnfO{ hf]8\g' k/]df, KE+PE (= p²/2m+kx²/2 etc.) geO{ KE+PE = KF+PF = mr(n-1)+mr(n-1) x'G5 . hxfF p, F, r, k, x, n / m n] qmdzM df]d]G6d, an, lglxt cw{Jof;df cfwfl/t k|b'of{Gt/, crn, tGsfO -;lGrt nDafO{_, k|b'of{Gt/ ;+Vof / lkG8nfO{ ;+s]t ub{5g\ . 

$=%(= ;dLs/0f E = mc² geO{ F = mr(n-1) x'G5 . hxfF E, c, F, r, (n-1) / m n] qmdzM zlQm, k|sfzsf] ult, an, k|b'of{Gt/, k|b'of{/Gt/ ;+Vof / lkG8nfO{ hgfpF5g\ . k|rlnt ;"qsf] cfofd klg tYoOt/ 5 .

$=^)= b|jl:ylt lj1fgsf ljljw ;"qx¿df ;d:of /x]sf] b]lvPsf] 5 . h;df Opn/sf] ;dLs/0f, ag{nLsf] ;dLs/0f cflb kb{5g\ . oL b'j} ;dLs/0fdf $=! / To;} jdf]lhdsf] ;DalGwt ansf] lgodn] g} sfd ub{5 . To;} u/L dfly lbOPsf ;a} ;dLs/0f tyf kmd'{nfx¿sf] km]/abn;Fu} ;DalGwt P;= cfO{= PsfO tyf cfoflds kmd'{nfx¿sf] klg cfof]hg ug'{kg]{ x'G5 .

%= ljZn]if0f

d"ntM k|j]u / To;df cfwfl/t ;'jftsf ;dLs/0fx¿sf] k|rngn] sfnfGt/df lgod / cfwf/sf] :j¿k wf/0f ub}{ uPkl5 ef}lts zf:qn] tL ljifox¿df k'glj{rf/ ug]{tkm{ ;f]r]sf] b]lvFb}g . o;/L x/]s o'udf gofF vf]h ug]{ vf]hstf{x¿n] k|rngx¿nfO{ cfwf/ dfgL To;d} cfwfl/t cg';Gwfgx¿ yKb} hfFbf ef}lts zf:qn] emg\eGbf emg\ tYoOt/ dfu{ ;dfTg k'u]sf] b]lvPsf] 5 .

ljut nfdf] ;dob]lv ef}lts zf:qsf ;a} If]qdf ;femf e"ldsf lgjf{x ug{] lgod x'g'kg]{ xf] eGg] wf/0ff ljleGg j}1flgsx¿n] /fVb} cfPsf] / To;af/]df k|of; u/]sf] klg pNn]v ePsf] kfOG5 . t/ s'g} ;dfwfg lbPsf] b]lvFb}g . a¿ pgLx¿n] agfPsf lgodx¿ g} tYoOt/ cfwf/x¿df lgld{t x'g k'u]sf] b]lvG5 . o;af6 Psflt/ tYoOt/ kl/0ffd k|fKt ePsf] b]lvG5 eg] cGo vf]hstf{x¿nfO{ ;d]t To;n] ulx/f] 5fk 5f]8]sf] b]lvG5 . o; vfnsf ;d:ofnfO{ ;dod} lrg{ g;s]df vf]hstf{x¿ emg\eGbf emg\ lbUe|ldt x'g] / unt ¿kn] dfu{lgb]{lzt x'g] vt/f /xG5 .

ef}lts zf:qdf ljleGg wf/0ffx¿ sfnqmd;Fu} ljsl;t ePsf] b]lvG5 . o; qmddf klxn]sf vf]hLstf{x¿n] eljiodf x'g hfg] vf]hLtkm{ klxNo} cg'dfg nufpg] ;Defjgf sd x'g' :jfefljs lyof] . t/ kl5Nnf vf]hstf{x¿n] tL cfwf/x¿nfO{ lagfk/LIf0f cufl8 a9fpg] sfo{ ug'{ nfek|b x'g g;s]sf] b]lvG5 . o;k|sf/sf] k/Dk/fsf sf/0f lglDtg k'u]sf ;d:ofx¿sf] ;femf ;dfwfgsf] vf]hL ug]{ sfo{ o; cg';Gwfgkqdf ePsf] 5 .

vf]hsf qmddf k|j]u, ansf] lgod lgdf{0f, zlQmsf ;"qx¿sf] lgdf{0f, ;fdYo{sf ;"qx¿sf] lgdf{0f, 3'dfp/f] j]u, 3'dfp/f] k|j]u, df]d]G6 ckm Ogl;{of, Ogl;{on lkG8, t/+uj]usf] wf/0ff, k|j]unfO{ cfwf/df lgld{t sngsf lgodx¿, df]d]G6dnfO{ k'gM ;dofGt/fnsf ¿kdf dfkg ug]{ ;d:of, sfo{x¿df b'/Lsf] cgfjZos e"ldsf lgjf{x, k|rlnt cfoflds dfGotf cflb sfo{ / vf]hx¿ tYoOt/ ePsf] b]lvG5 . oxL sf/0f tL ;"qx¿ tYo;Fu c;DalGwt aGg k'u]sf 5g\ .

k/LIf0fdf l;4 ug}{ g;lsg] / s'g} cfwf/lagf :yflkt kl/df0fx¿nfO{ cfwf/ dfg]/ ef}lts zf:q cufl8 al9/x]sf] b]lvG5 . o;n] x/]s o'udf ePsf vf]hstf{x¿sf k|of;x¿ ;fy{s x'g ;Dej x'g g;s]sf] b]lvG5 . b'of{Gt/ / k|b'of{Gt/sf] e"ldsf x'g'kg]{df cGo kl/df0fsf] e"ldsf ePsf] dfGbf ef}lts lj1fg ;d:ofdf k/]sf] b]lvG5 . Pskl5 csf]{ x'Fb} :jo+l;4 tYo / ;j{:jLs[t tYox¿ lj1fgsf jf:tljs cfwf/x¿ x'g\ eGg] s'/fdf Wofg lbg lal;{g] k/Dk/fsf] ljsf; xF'b} hfg' lj1fgnfO{ lg/fwf/ zf:q agfpg] k/Dk/f cl3 a9fpg' xf] . oxL ;d:ofsf] olsg ug]{ / j:t'k/s ;dfwfg lbg] sfo{ o; cg';Gwfgkqdf ePsf] 5 .

o; cg';Gwfgkqdf d"ntM zf:qLo ef}lts zf:qdf b'of{Gt/ / k|b'of{Gt/n] kfg]{ k|efjsf] cWoog ul/Psf] 5 . o;afx]s tL kl/df0fx¿n] ;DaGw /fVg] cGo kIfsf] klg cWoog ul/Psf] 5 . o; qmddf ;fdfGo Jojxfl/s an, 3'dfp/f] an, t/+u an, zlQm, ;fdYo{ cflb / tL;Fu ;DalGwt x'g cfpg] r/ tyf kIfx¿nfO{ cGt/;DalGwt u/fpg] sfo{ klg ePsf] 5 . o;af6 ef}lts zf:qsf] j:t'k/s :j¿ksf] 9fFrf tof/ ePsf] 5 . cg';Gwfgstf{x¿n] casf cg';Gwfgx¿df j:t'k/s tYox¿nfO{ cfwf/sf ¿kdf lng'kg]{ b]lvG5 .

k|rng, k|efj / ljZjf;eGbf dfly tYo / k|df0fx¿ x'G5g\ . ef}lts zf:qsf vf]hx¿ s'g} g s'g} k|sf/sf k/LIf0faf6 dfq l;4 x'G5g\ . o;af/] hfgsf/L /fVg] s'g} vf]hstf{df :jtGq cWoog ug]{ ulx/f] cle¿lr jf rfxgf x'Fb}df p;n] cfkm"nfO{ cfkm\gf] rfxgfcg'¿k cl3 a9fpg ;xh x'G5 jf cg's"n jftfj/0f ldN5 eGg] lglZrt x'Fb}g . o;sf nflu k|rng, k|efj / dfxf]nn] k|an e"ldsf lgjf{x ug{ ;Sg] vt/f x'G5 . o;tkm{ klg ;a} vf]hstf{n] Wofg lbg'kg]{ x'G5 .

^= lgisif{

o; cg';Gwfgkqn] zf:qLo ef}lts zf:qdf b'of{Gt/ / k|b'of{Gt/n] v]Ng] e"ldsfaf/] hfgsf/L u/fPsf] 5 . ef}lts zf:qdf /x]sf] k|j]usf] :yfkgfdf ul0ftLo cfwf/ gePsf] :ki6 b]lvPsf] 5 -!(, @!, @@, ##–#%_ . cfoflds kmd'{nfx¿sf] lgdf{0f klg tYoOt/ cfwf/df ul/Psf] 5 . o;n] w]/}h;f] ;"qx¿n] j:t'ut / k/LIf0fd"ns oyfy{eGbf leGg glthf lbFb} cfPsf] b]lvG5 . o;af6 oflGqs lskmfot, an / zlQm;DaGwL lgodx¿, 3'dfp/f] an jf zlQm, t/+u;DaGwL lgod cflbdf lbOPjdf]lhd k'glg{df{0f ug'{kg]{ clgjfo{tf b]lvg uPsf] 5 . oltdfq geO{ cGo ;"qx¿df klg ;f]xLjdf]lhdsf] k'glg{df{0f k|lqmof cl3 a9fpg'kg]{ b]lvG5 .

o; cg';Gwfgaf6 zlQm, an, ;fdYo{, t/+uo'Qm zlQm cflbsf nflu Pp6} lgodn] sfd ug]{ t/ pQm lgod s] lyof] eGg] xfn;Dd yfx gnfu]sf] :ki6 ePsf] 5 . o; vf]haf6 Pp6} ;"qn] ;a} If]qdf sfd ug]{ b]lvg'sf ;fy} Go'6gL lkG8sf] dfGotf ;xL ePsf] b]lvPsf] 5 . oflGqs lskmfotsf] /x:o klg an jf zlQmsf] ;femf ;"qdf lglxt ePsf] tYo v'Ng k'u]sf] 5 . o;/L ef}lts zf:qdf nfdf] ;dob]lv /x]sf] ;d:ofsf] ;dfwfg ePsf] 5 . 3'dfp/f] jf jqm j]usf cj:yfsf nflu lkG8 s] xf] < zlQm s] xf] < cflb ljifodf klg k/Dkl/t k|rngx¿ ;d:ofu|:t ePsf] b]lvG5 .

;'jft ;dLs/0fx¿n] ;d:of lgDtfPsf] / To;sf] abnfdf ;'j|t ;dLs/0fx¿ cufl8 ;fl/Psf] ljifosf] hfgsf/L klg o; kqdf lbOPsf] 5 . ;a} ;dofGt/fnsf nflu ;dfwfg lbg g;Sg] k|rlnt ljljw lgod / ;dLs/0fx¿ klg ;'wf/ ug'{kg]{ hfgsf/L o;df lbOPsf] 5 . o;af6 n]u|]Gh, XofldN6g, Opn/ cflbsf k|z+;gLo ;dLs/0fx¿ k|rlnt k|efj / e|dsf sf/0f lgdf{0f ePsf] yfx nfUg'sf ;fy} an, zlQm jf sfo{sf] ;xL / tYok"0f{ lgodaf/] klg :ki6 ePsf] 5 . zf:qLo oflGqsL / SofG6d d]sflgS;sf nflu ;d]t dfu{ k|z:t ePsf] 5 .

gofF lgodx¿ tYok"0f{ ePsfn] o;af6 ;Dk"0f{ vf]hstf{x¿nfO{ ef}lts zf:qtkm{sf] vf]hL cl3 a9fpg cfjZos kg]{ cfwf/x¿ tof/ ePsf 5g\ . ljleGg ;+:yf / cg';Gwfgstf{x¿n] tYoOt/ / cfwf/xLg ljifodf cfwfl/t eO{ vf]h ug'{sf] ;f6f] ;dod} ;xL cfwf/af6 cg';Gwfg ug]{ cEof; cl3 a9fpg'kg]{ ePsf] 5 . o;sf nflu ef}lts zf:q / pko'{Qm cfljisf/ tyf vf]hx¿;Fu ;DalGwt If]qsf kf7\ok':tsx¿ ;RRofpg'kg]{, ljZj rlr{t hg{n tyf tL;Fu ;DalGwt ;Dkfbs Pjd\ ;Grfnsx¿n] cg';Gwfgkqx¿ k|sflzt ubf{ ;xL / tYok"0f{ cfwf/x¿nfO{ ckgfpg'kg]{, ljZjljBfnox¿n] unt eO;s]sf ljifoj:t'nfO{ ;dod} ;'wf/ u/L ;xL dfu{ k|z:t ug'{kg]{, vf]hstf{x¿n] cfkm\gf cg';Gwfgx¿sf] bL3{sfnLgtf / j:t'k/stfdfly Wofg lbg'kg]{, /fHon] tYok/s cg';Gwfgx¿nfO{ cufl8 a9fpg ;dod} kxn ug'{kg]{, cg';Gwfgs]Gb|x¿n] ;dod} ;xL ljifosf] cg';Gwfgdf nufgL ug{ hf]8 lbg'kg]{ / ;Grf/dfWodx¿n] j:t'ut ljifosf] k|j4{gsf nflu oyf]lrt e"ldsf lgjf{x ug'{kg]{ b]lvG5 . tYok"0f{ gofF cfwf/x¿ e]l6Pkl5 klg To;af/] yfx gkfO{ k/Dkl/t ¿kdf tYoOt/ cfwf/df g} cg';Gwfg ug]{ k|rng lg/Gt/ sfod /xg' cg';Gwfgstf{ tyf ;Dk"0f{ dfgjhfltsf nflu xflgsf/s x'g hfG5 eGg] s'/fdf ;DalGwt ;a}n] Wofg lbg' cfjZos 5 .

&= ;Gbe{ ;fdu|L

&=!= ;Gbe{;fdu|LM -k':ts tyf n]vx¿_

!= OG;fOSnf]k]l8of la|6flgsf, OG;fOSnf]k]l8of la|6flgsf nfOa]|/L, ;g\ @))( .

@= e'iffn, afa'/fd=, of] ztfAbLs} 7"nf] pknAwL -cGtjf{tf{_, ;+of]u b\j}dfl;s, kj{tM d'l8s'jf, ;g\ @))% .

#= ________ ef}lts zf:qdf ePsf s]xL q'l6x¿sf vG8g / ;dfwfg, sf7df8f}FM sf7df8f}F l/;r{ PG8 klAns];G; k|f= ln=, cfOP;laPgM (&*((#&@@#^&^, k|ltlnlk clwsf/ ;g\ @)!! .

$= ________ jQmf= cfls{ldl8hsf] l;4fGtdf ePsf] ;d:of / k/LIf0f, nlntk'/M g]kfn lj1fg tyf k|ljlw k|1f k|lti7fg, v'dn6f/, ;g\ @)!@ .

%= ________ ;fdYo{sf] l;4fGt < -bf]=;+=_, sf7df8f}FM sf7df8f}F l/;r{ PG8 klAns];G; k|f= ln=, cfOP;laPgM (&*((#&@@#^&^, k|ltlnlk clwsf/ ;g\ @)!! .

^= ________ cfls{ldl8hsf] l;4fGt s;/L unt 5 <, sf7df8f}FM sf7df8f}F l/;r{ PG8 klAns];G; k|f= ln=, ;g\ @)!@ .

&= ________ Go'6gsf] rfnsf] bf];|f] lgod s;/L unt 5 <, sf7df8f}FM sf7df8f}F l/;r{ PG8 klAns];G; k|f= ln=, cfOP;laPgM (&*((#&*%%^)), k|ltlnlk clwsf/ ;g\ @)!@ .

*= ________ jQmf= Uoflnlnof]k"j{sf] ef}lts zf:qLo Oltxf; / s]xL cfnf]rgfx¿, sf7df8f}FM jflNdsL ljBfkL7, k|bz{gLdfu{, k|jrg, ;g\ @)!@ .

(= ________ jQmf= -s_Go'6gk"j{sf] ljZn]if0ffTds Oltxf; -v_ Go'6gsf] rfnsf] bf];|f] lgodsf] JofVof / v08g, sf7df8f}FM lqe'jg ljZjljBfno, sLlt{k'/, k|jrg, ;g\ @)!@ .

!)= ________ jQmf= -s_Go'6gk"j{sf] Oltxf;sf] ljj]rgf -v_ Go'6gsf] rfnsf] bf];|f] lgodsf] ljZn]if0f / v08g, sf7df8f}FM o'gfO6]8 :sn;{ Ps]8]dL, sn+sL, k|jrg, ;g\ @)!@ .

!!= ________ k/LIf0fstf{= P;Pl6Pd–! sf] lgdf{0f / k/LIf0fsf ljljw r/0fx¿, sf7df8f}FM sf7df8f}F l/;r{ PG8 klAns];G; k|f= ln=, ljljw cfGtl/s k/LIf0fx¿, ;g\ @)!@ .

!@= ________ jQmf= cfls{ldl8hk"j{sf] Oltxf; / cfls{ldl8hsf] l;4fGt, sf7df8f}FM lqe'jg ljZjljBfno, Pd=lkmn= ljefu, lbNnLahf/, k|jrg, ;g\ @)!@ .

!#= ________ jQmf= cfls{ldl8hk"j{sf] Oltxf; / cfls{ldl8hsf] l;4fGt, sf7df8f}FM jfNdLsL ljBfkL7, k|bz{gLdfu{, k|jrg, ;g\ @)!@ .

!$= ________ jQmf= Go'6gk"j{sf] ef}lts zf:qLo Oltxf;sf] ljZn]if0f / rfnsf lgodx¿sf] ljZn]if0f, sf7df8f}FM lqe'jg ljZjljBfno, Pd=lkmn= ljefu, lbNnLahf/, k|jrg, ;g\ @)!@ .

!%= ________ jQmf÷k/LIf0fstf{= P;Pl6Pd–! sf] k|of]uaf6 Go'6gsf] rfnsf] bf];|f] lgodsf] 7f8f] ;txsf ;Gbe{df k/LIf0f / ljZn]if0f, sf7df8f}FM o'gfO6]8 :sn;{ Ps]8]dL, sn+sL, k|jrg tyf k/LIf0f k|bz{g, ;g\ @)!# .

!^= ________ P;Pl6Pd–! / Go'6gsf] rfnsf] bf];|f] lgodsf] ;dtn ;txsf] k/LIf0f, sf7df8f}FM g]kfn k|ltlnlk clwsf/ /lhi6\/f/sf] sfof{no, k|ltlnlk clwsf/ ;g\ @)!# .

!&= ________ jQmf÷k/LIf0fstf{= P;Pl6Pd–! / Go'6gsf] rfnsf] bf];|f] lgodsf] ;dtn ;txsf] k/LIf0f / ljZn]if0f, nlntk'/M c/lgsf] OG6/g]zgn Ps]8]dL, tfNrLv]n, k|jrg÷k/LIf0f k|bz{g ;g\ @)!# .

!*= ________ jQmf÷k/LIf0fstf{= sfo{sf] ;"qsf] k/LIf0f, nlntk'/M c/lgsf] OG6/g]zgn Ps]8]dL, tfNrLv]n, k|jrg÷k/LIf0f k|bz{g @)!# .

!(= ________ = ;'jftb]lv ;'j|t;Dd, AnuM PSo'/]6 lkmlhS; 86 Anu:k6 86 sd, leGg, ;g\ @)!^ .

@)= ________ = kf]lvg] c6fpg] / lj:yflkt x'g] t/n kbfy{, AnuM PSo'/]6 lkmlhS; 86 Anu:k6 86 sd, leGg, ;g\ @)!& .

@!= ________ = ;'j|tsf s]xL k|efjx¿, AnuM PSo'/]6 lkmlhS; 86 Anu:k6 86 sd, leGg, ;g\ @)!& .

@@= ________ = rfnsf /]vLo ;dLs/0fx¿df b'of{Gt/ / k|b'of{Gt/sf] e"ldsf, AnuM PSo'/]6 lkmlhS; 86 Anu:k6 86 sd, leGg, ;g\ @)!& .

@#= uf}td, O{Zj/rGb|, afnljqmd vqL / >Ls[i0f kf}8\ofn= k|of]uzfnLo l/kf]6{ / l;kmfl/; kqx¿, sf7df8f}FM sf7df8f}F df]8]n pRr dfWolds ljBfno, x]/fN8 OG6/g]zgn sn]h / o'lge;{n pRr dfWolds ljBfno, lj= ;+= @)^^ .

@$= uf}td, ejfgLk|;fb, g]kfndf gofF cfls{ldl8h, gofF klqsf -b}lgs_, sf7df8f}FM gofF k|sfzg ldl8of k|f= ln=, ;g\ @))( .

@%= b]jsf]6f s[i0fHjfnf -;Dkf=_= dnhn u/ j}1flgs k|ltefnfO{, gofF klqsf -;DkfbsLo_, sf7df8f}FM gofF k|sfzg ldl8of k|f= ln=, ;g\ @))( .

@^= kfs{/, ;fOljn kL / cGo -;Dkf=_= Pd;L u|fjlxn OG;fOSnf]k]l8of ckm lkmlhS;, bf]= ;+=, jfl;+6gM Pd;L u|fjlxn cfP;L= ;g\ !((# .

@&= kf]v/]n, afns[i0f / cGo= g]kfnL a[xt\ zAbsf]z -kfF= ;+=_, sf7df8f}FM g]kfn /fhsLo k|1f k|lti7fg, lj+= ;+= @)%% .

&=@= cGo ;Gbe{;fdu|LM -ln+sx¿_

@*= https://en.wikipedia.org/wiki/Equations_of_motion

@(= http://gorkhapatraonline.com/news/38086 (Suvrt)

#)= http://www.ratopati.com/news/130831/ (suvrt) 

#!= http://www.onlinekhabar.com/2017/01/533506/ (duryantar)

#@= http://gorkhapatraonline.com/news/35475 (duryantar)

##= http://www.sikaionline.com/2017/06/23/1000/ (praduryantar)

#$= http://www.setoghar.com/news/21439/ (praduryantar)

#%= http://gorkhapatraonline.com/news/41448 (praduryantar)

#^= https://en.wikipedia.org/wiki/List_of_equations_in_classical_mechanics

#&= http://www.buzzle.com/articles/physics-formulas-list.html

#*= http://faculty.trinityvalleyschool.org/hoseltom/handouts/Formula%20Sheet-2003-05-07-8pg.pdf

#(= https://en.wikipedia.org/wiki/List_of_moments_of_inertia

$)= http://www.phy.mtu.edu/~suits/notefreqs.html

&=#= ;xfos ;fdu|Lx¿M -k':ts tyf n]vx¿_

$!= UjLg, /f]af]6{ kL= / cGo= b Go" OG;fOSnf]k]l8of la|6flgsf, lzsfuf]M o'lgel;{6L ckm lzsfuf], OG;fOSnf]k]l8of la|6flgsf, cfO{=Pg=;L=, ;g\ !(() .

$@= hh{ Go"g]; lnld6]8= RofDa;{ OG;fOSnf]k]l8of, nG8gM hg{ Go"g]; lnld6]8, ;g\ !(^$ .

$#= h]D;÷h]D;, Dofyd]l6S; l8S;g]/L, lbNnLM l;laP; klAn;;{ PG8 l8l:6Ao'6;{, k'gd'{b|0f, ;g\ !(** .

$$= 8]lG6y, hf]g / cGo= ;+= cS;kmf]8{ l8S;g]/L ckm lkmlhS;, lbNnLM cS;kmf]8{ o'lgel;{6L k]|;\, ;g\ @))( .

$%= zdf{, df]xg/fh / n'O{F6]n, vu]Gb|= zf]wljlw, nlntk'/M ;femf k|sfzg, lj=;+= @)^@ .

$^= lzjfsf]6L, uf]kfn / rf}nfufO{F, ltns= cg';Gwfgk4ltsf] kl/ro, sf7df8f}FM kf7\o;fdu|L k;n, k'gd'{b|0f, lj=;+= @)%% .

&=$= cGo ;xfos ;fdu|Lx¿ -ln+sx¿_

$&= http://aboutphysicsreformationcampaign.blogspot.com

$*= https://www.facebook.com/physicsreformationcampaign

$(= https://www.facebook.com/krpnepal

%)= https://www.youtube.com/watch¿v=-VXDbFELld8

%!= https://www.youtube.com/watch¿v=eixEdgu5iDA

%@= https://www.youtube.com/watch¿v=fHDB7PMUdZE

%#= http://www.janamanch.com/¿p=8523

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