Title Newtonian Mechanics - Solve.pdf ✅

wbDUwbqvb ejwe` ̈v  Final Revision Batch 1 PZz_© Aa ̈vq wbDUwbqvb ejwe` ̈v Newtonian Mechanics Topicwise CQ Trend Analysis UwcK 2016 2017 2018 2019 2021 2022 2023 †gvU wbDU‡bi e‡ji m~‡Îi g‡a ̈ m¤úK© Ñ 1 1 Ñ Ñ 2 2 6 wbDU‡bi e‡ji Z...Zxq m~‡Îi e ̈envi I fi‡e‡Mi wbZ ̈Zv Ñ Ñ Ñ 1 Ñ Ñ Ñ 1 msNl©- (w ̄’wZ ̄’vcK msNl© I Aw ̄’wZ ̄’vcK msNl©) Ñ Ñ Ñ 1 Ñ 1 1 3 RoZvi åvgK I †K.wYK fi‡eM Ñ 1 1 3 7 3 2 17 †K.wYK ivwkgvjv (miY, †eM, Z¡iY, fi‡eM, MwZkw3) 2 5 Ñ 3 4 2 Ñ 16 UK© 1 Ñ Ñ 1 Ñ Ñ Ñ 2 †K›`agyLx I †K›`awegyLx ej Ñ 1 Ñ Ñ 1 3 2 7 iv ̄Ívi e ̈vswKs 2 2 1 2 6 3 4 20 * we.`a.: 2020 mv‡j GBPGmwm cixÿv AbywôZ nqwb| weMZ mv‡j †ev‡W© Avmv m„Rbkxj cÖkœ 1| 5 m cÖk ̄Í GKwU iv ̄Ívi GKwU wbw`©ó ̄’v‡bi euv‡Ki eμZvi e ̈vmva© 80 m| iv ̄Ívi Dfq cv‡ki D”PZvi cv_©K ̈ 0.4 m| euvK AwZμ‡gi c~‡e© GKwU Mvwo 54 kmh–1 †e‡M PjwQj| [Xv. †ev. 23] (K) cÖZ ̈qbx ej Kv‡K e‡j? (L) GKRb bZ©Kx bvPvi mgq nvZ msKzwPZ Ki‡j N~Y©‡b Kx myweav cvq? e ̈vL ̈v K‡iv| (M) euv‡Ki ̄’v‡b iv ̄Ívi e ̈vswKs †KvY wbY©q K‡iv| mgvavb: 0.4 m 5 m  Avgiv Rvwb, e ̈vswKs †KvY,  = sin–1     0.4 5   = 4.588 (Ans.) (N) DÏxcK Abyhvqx MvwowU D3 †e‡M wbivc‡` euvK wb‡Z cvi‡e wK? MvwYwZKfv‡e we‡kølY K‡iv| mgvavb: ÔMÕ n‡Z cvB, e ̈vswKs †KvY,  = 4.588 Avgiv Rvwb, tan = v 2 m rg Mvwoi eZ©gvb †eM, v0 = 54 3.6 ms–1 = 15 ms–1  vm = tan(4.588) × 80 × 9.8  vm = 7.931 ms–1 < 15 ms–1  v0 > vm nIqvq MvwowU wbivc‡` evK wb‡Z cvi‡e bv| (Ans.) 2| 400 kg f‡ii GKwU Mvwo 60 kmh–1 mg‡e‡M 10 †Kv‡Y bZ Zj eivei Dc‡i D‡V| [Nl©Y ̧Yv1⁄4  Gi gvb 0.3 Ges g Gi gvb 9.8 ms–2 ] [Xv. †ev. 23] (K) WvBfvi‡RÝ Kv‡K e‡j? (L) GKwU w ̄úas-†K LwÐZ Ki‡j Gi aaæeK cwieZ©b n‡e wK? e ̈vL ̈v K‡iv| (M) Mvwoi Dci wμqvkxj weiæ× e‡ji gvb wbY©q K‡iv| mgvavb: F f mgsin v Mvwoi Dci wμqvkxj †gvU weiæ× ej = mgsin + f = mgsin + mgcos [R = mgcos] = 400 × 9.8 × sin(10) + 0.3 × 400 × 9.8 × cos(10) = 1838.83 N (Ans.) (N) Mvwoi Bwćbi ÿgZv KZ n‡j MvwowU mg‡e‡M bv P‡j eis Z¡iY cÖvß n‡e? MvwYwZK we‡kølY `vI| mgvavb: F f mgsin v [mg‡e‡M Pjvq, F = mgsin + mgcos] eZ©gv‡b Mvwoi BwÄb KZ...©K cÖhy3 ÿgZv, P = Fv = (mgsin + mgcos)v  P = 1838.83 × 60 3.6  P = 30647.246 W  P > 30647.246 W n‡j MvwowU mg‡e‡M bv P‡j eis Z¡iY cÖvß n‡e| 3| GKwU e„ËvKvi PvKwZ Gi Z‡ji mv‡_ ga ̈we›`yMvgx j¤^ Aÿ AB mv‡c‡ÿ wPÎ-1 Abyhvqx Nyi‡Q| Avevi wPÎ-2 Abyhvqx PvKwZwUi mv‡_ j¤^fv‡e ̄’vwcZ XY A‡ÿi mv‡c‡ÿ Nyi‡Q| [iv. †ev. 23] m = 2 kg r = 0.5 m h = 0.25 m B G A Y G X h (K) RoZvi åvgK Kv‡K e‡j? (L) e„ËvKvi c‡_ euvK cvi nIqvi mgq GKRb mvB‡Kj Av‡ivnx †n‡j hvq †Kb?
2  HSC Physics 1st Paper Chapter-4 (M) 1g wP‡Î PμMwZi e ̈vmva© wbY©q Ki| mgvavb: MK2 = 1 2 MR2  K = R 2 = 0.5 2 = 1 2 2 m (Ans.) (N) †Kvb A‡ÿi mv‡c‡ÿ PvKwZwU Nyiv‡bv mnR n‡e? MvwYwZK we‡køl‡Y gva ̈‡g gZvgZ `vI| mgvavb: wPÎ 1 Gi Rb ̈, I1 = 1 2 MR2 = 1 2 (0.5)2  2 = 0.25 kgm2 wPÎ 2 Gi Rb ̈, I2 = 1 2 MR2 + Md2 = 0.25 + 2  (0.25)2 = 0.375 kgm2  I2 > I1 (ZvB wPÎ-1 Abyhvqx Nyiv‡bv †ewk mnR n‡e|) 4| 500 kg f‡ii GKwU Mvwo 3900 J MwZkw3 wb‡q iv ̄Ívq PjwQj| nVvr MvwowU 120 m e ̈vmv‡a©i GKwU euv‡Ki m¤§yLxb n‡jv| iv ̄Ívq †Kv‡bv e ̈vswKs wQj bv| iv ̄Ívi I Mvwoi PvKvi Nl©Y ̧Yv1⁄4 0.2| [Kz. †ev. 23] (K) PμMwZi e ̈vmva© Kv‡K e‡j? (L) GKRb Av‡ivnx wjd‡U Dc‡i DVvi mgq wb‡R‡K fvix g‡b K‡i, e ̈vL ̈v K‡iv| (M) iv ̄Ívi euv‡K Mvwoi Dci wμqvkxj †K›`awegyLx Z¡i‡Yi gvb wbY©q K‡iv| mgvavb: Avgiv Rvwb, MwZkw3, Ek = 1 2 mv2  3900 = 1 2 × 500 × v2  v = 3.95 ms–1  †K›`awegyLx Z¡iY, a = v 2 r = 3.952 120  a = 0.13 ms–2 (Ans.) (N) m‡e©v”P †eM wb‡q euvK AwZμg Ki‡Z n‡j PvjK‡K Zvi Mvwoi †e‡M Kx cwigvY cwieZ©b Ki‡Z n‡eÑ MvwYwZK we‡køl‡Yi gva ̈‡g wbY©q K‡iv| mgvavb: m‡e©v”P †eM wd‡i AwZμg Ki‡Z PvB‡j, flim = Fc  mg = mv 2 m r  vm = rg  vm = 0.2 × 120 × 9.8  vm = 15.33 ms–1  Mvwoi †eM e„w× Ki‡Z n‡e = (15.33 – 3.95) ms–1 = 11.38 ms–1 (Ans.) 5| GKwU B‡jKUab GKwU †cÖvU‡bi Pvicv‡k¦© 5.2 × 10–10 m e ̈vmv‡a©i e„ËvKvi c‡_ 6.977 × 105 ms–1 †e‡M AveZ©b Ki‡Q| B‡jKUa‡bi fi 9.1 × 10–31 kg| [h. †ev. 23] (K) UK© Kx? (L) ÒGKK mg‡KŠwYK †e‡M N~Y©vqgvb †Kv‡bv `„p e ̄‘i RoZvi åvgK msL ̈vMZfv‡e Gi †KŠwYK fi‡e‡Mi mgvb|ÓÑ e ̈vL ̈v Ki| (M) DÏxc‡Ki Av‡jv‡K B‡jKUabwUi †KŠwYK fi‡eM KZ? mgvavb: †K.wYK fi‡eM, L = mvr  L = 9.1 × 10–31 × 6.977 × 105 × 5.2 × 10–10  L = 3.3 × 10–34 kgm2 s –1 (Ans.) (N) ÒDÏxc‡Ki B‡jKUabwU Kÿc‡_ wbivc‡` Nyi‡Q|ÓÑ Dw3wU mwVK wKbv Zv hvPvB Ki| mgvavb: B‡jKUa‡bi Dci †K›`agyLx ej, Fc = mv2 r  Fc = 9.1 × 10–31 × (6.977 × 105 ) 2 5.2 × 10–10  Fc = 8.52 × 10–10 N Avevi, †cÖvUb I B‡jKUa‡bi ga ̈Kvi w ̄’i Zworej, F = 1 40 × e 2 r 2  F = 9 × 109 × (1.6 × 10–19) 2 (5.2 × 10–10) 2  F = 8.52 × 10–10 N myZivs, DÏxc‡Ki B‡jKUabwU Kÿc‡_ wbivc‡` Nyi‡Q| (Ans.) 6| 5000 kg f‡ii GKwU evjyfwZ© UavK NÈvq 72 km †e‡M Pj‡Q| UavK n‡Z cÖwZ †m‡K‡Û 200 gm evjy wQ`a c‡_ c‡o hv‡”Q| †eaK †P‡c 20 min c‡i UavKwU‡K 20 m `~i‡Z¡ _vgv‡bv n‡jv| [h. †ev. 23] (K) UK© Kv‡K e‡j? (L) evjyi ga ̈w`‡q Mvwo Pvjv‡Z mgm ̈v nq †Kb? e ̈vL ̈v K‡iv| (M) hvÎv ïiæi 15 min c‡i Uav‡Ki †e‡Mi gvb †ei K‡iv| (N) UavKwU‡K _vgv‡bvi Rb ̈ cÖ‡qvRbxq e‡ji gvb wnmve Kiv m¤¢eÑ we‡kølY K‡i †`LvI| 7| F  r  m = 20 gm N~Y©biZ e ̄‘i e ̈vmva© †f±i r  = (2i )  + 2j  – k  m Ges ej, F  = (i )  + 4j  – 3k  N. e ̄‘wU ïiæ‡Z 200 rpm G NyiwQj| AZci D3 ej 3s e ̈vcx cÖ‡qvM Kiv nj| [h. †ev. 23] (K) msNl© Kv‡K e‡j? (L) MwZkxj wjd‡U e ̄‘i Kvh©Ki IR‡bi ZviZg ̈ e ̈vL ̈v K‡iv|
wbDUwbqvb ejwe` ̈v  Final Revision Batch 3 (M) e ̄‘i Dci wμqvkxj U‡K©i gvb †ei Ki| mgvavb: Avgiv Rvwb, UK©,   = r  × F     = (2i )  + 2j  – k  × (i )  + 4j  – 3k    =        i   2 1 j  2 4 k  –1 –3    = i  (–6 + 4) + j  (–1 + 6) + k  (8 – 2)    = –2i  + 5j  + 6k   |  | = 2 2 + 52 + 62 = 8.06 Nm (N) ej cÖ‡qv‡Mi c~‡e©i I c‡ii ch©vqKvj Zzjbv K‡iv| mgvavb: ej cÖ‡qv‡Mi c~‡e©, i = 2N t = 2 × 200 60 = 20.944 rads–1  Ti = 2 i = 2 20.944 = 0.3 s ej cÖ‡qv‡Mi c‡i,  = I   =  I =  mr2 = 8.06 20 × 10–3 × (2 2 + 22 + 12 )   = 44.778 rad s–2  f = i +   f = 20.944 + 44.778 × 3 = 155.278 rad s–1  ch©vqKvj, Tf = 2 f = 0.04 s myZivs ej cÖ‡qv‡Mi c‡i ch©vqKvj n«vm cv‡e| 8| 3 ms–1 †e‡M 2 kg f‡ii GKwU e ̄‘ 0.5 kg f‡ii Ab ̈ GKwU w ̄’i e ̄‘i m‡1⁄2 †mvRvmywR w ̄’wZ ̄’vcK msN‡l© wjß nq| [e. †ev. 23] (K) wbðj †KvY Kx? (L) •iwLK fi †e‡Mi wbZ ̈Zvi bxwZ‡Z w`‡Ki ̧iæZ¡ Av‡Q wK bv? Av‡jvPbv Ki| (M) msN‡l©i ci w ̄’i e ̄‘i †kl †eM KZ? wbY©q Ki| mgvavb: Avgiv Rvwb, v2 =    m2  – m1 m2 + m1 u2 +    m1  m2 + m1 u1  v2 = 0.5 – 2 0.5 + 2 × 0 + 2 × 2 0.5 + 2 × 3  v2 = 4.8 ms–1 (Ans.) (N) DÏxc‡Ki MwZkxj e ̄‘i fi w ̄’i e ̄‘i f‡ii Zzjbvq A‡bK †ewk n‡j msN‡l©i ci e ̄‘؇qi cwiYwZ Kx n‡e? MvwYwZKfv‡e hvPvB Ki| mgvavb: When m1 > m2 m1 – m2  m1 m1 + m2  m1 hw` m1 >> m2 nq Z‡e, v1 =    m1  – m2 m1 + m2 u1 + 2m2 m1 + m2 u2  v1 = m1 m1 × u1 + 0 [∵ u2 = 0]  v1 = u1 = 3 ms–1 v2 =    m2  – m1 m2 + m1 u2 + 2m1 m2 + m1 u1  v2 = 0 + 2m1 m1 × u1  v2 = 2u1 = 2 × 3 = 6 ms–1 (Ans.) 9| 2 m cÖk ̄Í Ges 200 m e ̈vmva© wewkó GKwU e ̈vswKs hy3 euvKv c‡_ GKwU Mvwo 50.4 kmh–1 †e‡M P‡j wbivc‡`i euvK wb‡Z cv‡i| [g = 9.8 ms–2 Ges iv ̄Ívi Nl©Y ̧Yv1⁄4  = 0.5] [e. †ev. 23] (K) j ̈vcøvwmqvb Acv‡iUi Kv‡K e‡j? (L) `yB‡qi AwaK †f±‡ii jwä mvgvšÍwi‡Ki m~‡Îi mvnv‡h ̈ wbY©q Kiv hvq wK? e ̈vL ̈v Ki| (M) iv ̄Ívi e ̈vswKs D”PZv wbY©q Ki| mgvavb: h 2 m  tan = v 2 rg   = tan–1     142 200  9.8   = 5.71  sin = h d  h = dsin = 2  sin (5.71)  h = 0.2 m (Ans.) (N) DÏxc‡Ki iv ̄ÍvwU e ̈vswKsnxb n‡j ZLb MvwowU wbivc‡` euvK wb‡Z cvi‡e wK bv? MvwYwZKfv‡e hvPvB K‡i †Zvgvi gZvgZ `vI| mgvavb: DÏxc‡Ki iv ̄ÍvwU e ̈vswKsnxb n‡j,  = v 2 m rg  vm = rg = 0.5  200  9.8  vm = 31.305 ms–1  v < vm nIqvq MvwowU wbivc‡` euvK wb‡Z cvi‡e| (Ans.) 10| 2 kg Ges 3 kg f‡ii `ywU e ̄‘ h_vμ‡g 8.8 ms–1 Ges 1.2 ms–1 †e‡M wecixZ w`K n‡Z G‡m msN‡l©i ci e ̄‘ `ywU GK‡Î wgwjZ n‡q wbw`©ó w`‡K Pj‡Z jvMj| [wm. †ev. 23] (K) PμMwZi e ̈vmva© Kv‡K e‡j? (L) †Kv‡bv Bwćbi `ÿZv 50% ej‡Z Kx eySvq? e ̈vL ̈v K‡iv| (M) mshy3 e ̄‘ `ywUi P‚ovšÍ †eM KZ? mgvavb: •iwLK fi‡e‡Mi msiÿYkxjZv bxwZ Abymv‡i, m1u1 – m2u2 = (m1 + m2)v  2 × 8.8 – 3 × 1.2 = (2 + 3) × v  v = 2.8 ms–1 (Ans.) (N) D3 msNl© w ̄’wZ ̄’vcK bv Aw ̄’wZ ̄’vcKÑ MvwYwZK we‡køl‡Yi gva ̈‡g †Zvgvi gZvgZ `vI| mgvavb: msN‡l©i c~‡e© MwZkw3, Ei = 1 2 m1u 2 1 + 1 2 m2u 2 2  Ei = 1 2 × 2 × 8.82 + 1 2 × 3 × 1.22  Ei = 79.6 J msN‡l©i c‡i MwZkw3, Ef = 1 2 × m1v 2 + 1 2 m2v 2  Ef = 1 2 × 2.82 × (2 + 3)  Ef = 19.6 J †h‡nZz Ei  Ef myZivs msNl©wU Aw ̄’wZ ̄’vcK| (Ans.)
4  HSC Physics 1st Paper Chapter-4 11| [w`. †ev. 23] A P Q 1m B P Q 0.5 kg f‡ii AB `ÐwU PQ Aÿ mv‡c‡ÿ cÖwZ wgwb‡U 60 evi †Nviv‡bv n‡jv| cieZ©x‡Z `ÐwU‡K Gi •`‡N© ̈i mgvb e ̈v‡mi GKwU cvZjv PvKwZ‡Z cwiYZ K‡i PQ Aÿ mv‡c‡ÿ wgwb‡U 70 evi †Nviv‡bv n‡jv| (K) GK wbDUb ej Kx? (L) e ̈vswKs †Kv‡Yi gvb evov‡j iv ̄Ívi euv‡K Mvwo Pvjv‡bvi MwZmxgv ev‡oÑe ̈vL ̈v K‡iv| (M) PvKwZwUi †KŠwYK fi‡eM wbY©q K‡iv| mgvavb: †K.wYK fi‡eM, L = I  L = 1 2 mr2 × 2N t  L = 1 2 × 0.5 × 0.52 × 2 × 70 60  L = 0.458 kgm2 s –1 (Ans.) (N) †Kvb †ÿ‡Î N~Y©‡bi Rb ̈ Kg kw3i cÖ‡qvRb n‡eÑ MvwYwZKfv‡e e ̈vL ̈v K‡iv| mgvavb: `‡Ði †ÿ‡Î, MwZkw3, E1 = 1 2 I1 2 1  E1 = 1 2 ×     1 3 mL2 ×     2N t 2  E1 = 1 2 ×     1 3 × 0.5 × 12 ×     2 × 60 60 2  E1 = 3.289 J PvKwZi †ÿ‡Î, MwZkw3, E2 = 1 2 I2 ×  2 2  E2 = 1 2 ×     1 2 mr2 ×     2 × 70 60 2  E2 = 1.68 J  PvKwZwU Nyiv‡Z Kg kw3 cÖ‡qvM Ki‡Z n‡e| (Ans.) 12| 8 m cÖ‡ ̄’i iv ̄Ív w`‡q GKwU Mvwo h_vμ‡g 100 m I 80 m e ̈vmv‡a©i `yBwU euvK AwZμg Ki‡jv| iv ̄Ívi wfZ‡ii I evB‡ii cÖv‡šÍi D”PZvi euvK AwZμg Ki‡jv| iv ̄Ívi wfZ‡ii I evB‡ii cÖv‡šÍi D”PZvi cv_©K ̈ 0.4 m| [g. †ev. 23] (K) PμMwZi e ̈vmva© Kv‡K e‡j? (L) U‡K©i w`K Kxfv‡e wbY©q Ki‡e? (M) DÏxc‡K DwjøwLZ iv ̄Ívi e ̈vswKs †KvY wbY©q K‡iv| mgvavb: 0.4 m 8 m  C A B e ̈vswKs †KvY,  = sin–1     BC AC   = sin–1     0.4 8   = 2.87 (Ans.) (N) MvwowU Dfq euvK wK mgvb †e‡M AwZμg Ki‡Z cvi‡eÑ MvwYwZKfv‡e e ̈vL ̈v K‡iv| mgvavb: 100 m e ̈vmv‡a©i iv ̄Ívi †ÿ‡Î, tan = v 2 1 r1g  v1 = tan × r1 × g  v1 = tan(2.87) × 100 × 9.8  v1 = 7 ms–1 Avevi, 80 m e ̈vmv‡a©i ev‡Ki †ÿ‡Î, tan = v 2 2 r2g  v2 = tan × r2 × g = tan(2.87) × 80 × 9.8  v2 = 6.27 ms–1  Mvwoi †eM  6.27 ms–1 n‡j Dfq evK mgvb †e‡M AwZμg Ki‡Z cvi‡e| 13| 1000 kg f‡ii GKwU evm 75000 J MwZkw3 wb‡q Pjvi mgq 100 m e ̈vmva© wewkó GKwU evu‡Ki m¤§yLxb n‡jv| iv ̄Ívi cÖ ̄’ 10 m Ges iv ̄Ívi cÖvšÍ؇qi ga ̈eZ©x D”PZvi e ̈eavb 0.2m. [Xv. †ev. 22] (K) e‡ji NvZ Kv‡K e‡j? (L) big gvwU‡Z jvd w`‡j AvNvZ cvIqvi m¤¢vebv KgÑ e ̈vL ̈v Ki| (M) evmwUi fi‡eM wbY©q Ki| DËi: 12247.45 kgms–1 (N) MwZ‡eM bv Kwg‡q evmwU wbivc‡` evuKwU AwZμg Ki‡Z cvi‡e wK? MvwYwZKfv‡e we‡kølY Ki| DËi: cvi‡e bv 14| 72 kmh–1 †e‡M Pjgvb 1800 kg f‡ii GKwU eo Mvwo mvg‡b `vuwo‡q _vKv 1000 kg f‡ii GKwU †QvU Mvwo‡K wcQb w`K †_‡K m‡Rv‡i av°v w`‡jv| av°vi ci Mvwo `yÕwU GKwÎZ n‡q 60 m wM‡q †_‡g †M‡jv| iv ̄Ívi cv‡k `vuwo‡q _vKv weÁv‡bi QvÎ gvnx, `yN©UbvwU ch©‡eÿY K‡i ejj GwU GKwU Aw ̄’wZ ̄’vcK msNl©| [P. †ev. 22] (K) UK© Kx? (L) iv ̄Ívi evu‡K mvB‡Kj Av‡ivnx‡K †fZ‡ii w`‡K †n‡j evuK wb‡Z nq †Kb? e ̈vL ̈v Ki| (M) DÏxc‡Ki Mvwo `yÕwU _vgv‡Z †h evuav`vbKvix ej wμqvkxj wQj Zvi gvb wbY©q Ki| mgvavb: •iwLK fi‡e‡Mi wbZ ̈Zv Abymv‡i, m1u1 + m2u2 = (m1 + m2)v  1800 × 72 3.6 + 1000 × 0 = (1800 + 1000) × v  v = 12.857 ms–1  F = (m1 + m2) × a  F = (m1 + m2) × v 2 2s  F = (1800 + 1000) × 12.8572 2 × 60  F = 3857.05 N (Ans.)