Hsc intermediate physics 1st paper by tanbircox

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঩দাথ঱বফজ্ঞান প্রশ্ন ঳ৃজন঱ীর ফা তত্ত্বীয় যমভনই য঴াক না যকন, ঩দাথ঱বফজ্ঞাননয যভৌবরক বফলয়঳ভূ ঴ ঳ফ঳ভয় একই থানক । এই যভৌবরক বফলয়঳ভূ ঴ ঳ম্পনক঱ আ঩নায ঩বযস্কায ধাযণা থাকনর ,আ঩বন অফ঱যই যম যকান উদ্দী঩নকয জ্ঞান, অনু ধাফন, প্রনয়াগ ঑ উচ্চতয দক্ষতায প্রশ্ন঳ভূ ন঴য উত্তয খুফ ঳ু ন্দয কনয ঳াবজনয় বরনখ আ঳নত ঩াযনফন।এই ই-ফু নক ঩দাথ঱বফজ্ঞাননয যভৌবরক বফলয়঳ভূ ঴ ঳ম্পনক঱ খুফ ঳঴নজ ধাযনা ঑ ফু ঝায জনয ঩দাথ঱বফজ্ঞাননয গুযত্ব঩ূ ন঱ বফলয়঳ভূ ন঴য প্রনয়াজনীয় উদা঴যণ ঑ বিত্র঳঴ ফযাখযাভূ রক ঳ভাধান কনয যদ঑য়া ঴নয়নে ।

ন োটঃ ‚ভূ র ফই তথা ঩াঠ্য ফইনয়য বফকল্প বকেু নাই ।‛ ঩দাথ঱বফজ্ঞান বানরা বানফ ফু ঝায জনয আ঩নানক অফ঱যই বফববন্ন যরখনকয ফই ঩ড়নত ঴নফ অথ঱াৎ একই টব঩ক্স বফববন্ন ফইনত ঩ড়নত ঴নফ। যম টাকা বদনয় যকাবি​িং কযনফন য঳ই টাকা বদনয় বফববন্ন যরখনকয ফই বকনু ন এফিং তা একফায কনয ঴নর঑ ঩ড়ুন , আ঱া কবয যকাবি​িং এ ঩ড়ায যিনয় বানরা পরাপর ঩ানফন । ব঱ক্ষাথ঱ী কানে একান্ত অনু নযাধ , ঩দাথ঱বফজ্ঞান না ফু নঝ ভুখস্ত কযায বফ঱ার বুরবট বুনর঑ কযনফন না।ন঳টা বননজয ঩ানয় বননজই কুড়ার ভাযায ঳ভাথ঱ক।কাযন এনত ঳ভয় ঑ যভধা দু নটাই অ঩িয় ঴য় , এফিং এনত বফলয়টায উ঩য একটা অভূ রক অস্ববস্ত ঑ বীবত িনর আন঳।

অনটানভবটক স্ক্রনরয ভাধযনভ ই-ফুক ঩ড়া / বযনড়য জনযঃ আ঩নায ই−ফুক ফা pdf বযডানযয Menu Bar এয View অ঩঱নবট যত বিক কনয Auto /Automatically Scroll অ঩঱নবট ব঳নরক্ট করুন (অথফা ঳যা঳বয যমনত  Ctrl + Shift + H )। এবার ↑ up Arrow ফা ↓ down Arrow যত বিক কনয আ঩নায ঩ড়ায ঳ু বফধা অনু ঳ানয স্ক্রর স্পীড বঠ্ক কনয বনন। ঳যা঳বয যমনত অধযানয়য নানভয উ঩য বিক করুনঃ 1. যবক্টয (Vector) 2. রযবখক গবত (Linear-Motion) 3. বিভাবত্রক গবত(Motion-In-Two-Dimensions) 4. গবত঳ূ ত্র (Laws-Of-Motion) 5. যকৌবণক গবত঳ূ ত্র (Laws-Of-Angular-Motion) 6. কাজ, ঱বি ঑ ক্ষভতা (Work-Energy-And-Power) 7. ভ঴াকল঱ (Gravitation) 8. ঳যর েবন্দত স্পন্দন (Simple-Harmonic-Oscillation) 9. বিবতিা঩কতা (Elasticity) 10. প্রফা঴ী ঩দাথ঱ (Fluid) 11. তা঩ ঑ গযা঳ (Heat-And-Gas) 12. তা঩ভাত্রা (Temperature) 13. তা঩ গবতবফদযায প্রথভ ঳ূ ত্র (First Law Of Thermodynamics) 14. তা঩ বফবকযণ (Heat Radiation) 15. অফিায ঩বযফত঱ন (Change Of State) 16. তা঩গবতবফদযায বিতীয় ঳ূ ত্র (Second Law Of Thermodynamics) 17. তযঙ্গ ঑ ঱ব্দ (Waves & Sound) 18. ঱ব্দ (Sound) 19. ঱নব্দয গবতনফগ (Speed Of Sound) facebook /gmail/skype: -tanbir.cox

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বদক/নবক্টয যাব঱ঃ ভান ঑ বদক উবয়ই আনে এফিং যম যকান একবট ফা উবয়বটয ঩বযফত঱ননয পনর যবক্টয যাব঱ ঩বযফবত঱ত

঴নত ঩ানয। উদা঴যণঃ ঳যণ, যফগ, ত্বযণ, ভন্দন, ফর, বযনফগ, অববকল঱জ ত্বযণ, যিৌম্বক প্রাফরয, রফদু যবতক প্রাফরয, বূ িুম্বনকয অনু বূবভক প্রাফরয, ঑জন, ঩ৃষ্ঠাটান, ঳াদ্রতাতা গুণািংক, ফনরয রামাভক, যিৌম্বক রামাভক, প্রফা঴ভাত্রা ইতযাবদ। অবদক/নস্করাযঃ ভান আনে বকন্তু বদক যনই। উদা঴যণঃ দ্রুবত, কাজ, ক্ষভতা, ঱বি, তা঩, িা঩, যিৌম্বক ঑ রফদু যবতক বফবফ,

িাজ঱, বিবতিা঩ক গুণািংক ইতযাবদ। যবক্টয যাব঱য যমাজনঃ যনৌকায গবত, িরন্ত গাবড়নত ঩ড়ন্ত ফৃ বি, ঩াবখয উড্ডয়ন। যবক্টয যাব঱য বফনয়াজনঃ গুনটানা যনৌকা, রননযারায যঠ্রা, ঳যর যদারনকয গবত। রবধঃ দু ই ফা তনতাবধক যবক্টয যমাগ কনয একবট নতুন যবক্টয ঩া঑য়া মায়। এ নতুন যবক্টযনক রবধ যবক্টয ফনর। অিং঱কঃ যম যবক্টয঳ভূ ঴ যমাগ কনয রবধ ঩া঑য়া মায়, তানদযনক রবধয অিং঱ক ফা উ঩ািং঱ ফনর।

ভনন যাখনত ঴নফঃ

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‡f±i ivwk (Vector Quantities): ‡h mKj †fŠZ ivwk‡K m¤ú~Y©iƒ‡c cÖKvk Kivi Rb¨ gvb I w`K Df‡qi cÖ‡qvRb nq Zv‡`i‡K ‡f±i ivwk e‡j| †hgb- miY, IRb, †eM, Z¡iY, ej BZ¨vw`| ‡¯‹jvi ivwk (Scalar Quantities): ‡h mKj †fŠZ ivwk‡K ïay gvb Øviv m¤ú~Y©iƒ‡c cÖKvk Kiv hvq Zv‡`i‡K ‡¯‹jvi ivwk e‡j| †hgb- `ªæwZ, fi, KvR, ‰`N©¨ BZ¨vw`| ‡¯‹jvi ivwk I ‡f±i ivwki g‡a¨ cv_©K¨ (Distinction between Scalar and Vector Quantities): µwgK ‡¯‹jvi ivwk ‡f±i ivwk ‡h mKj †fŠZ ivwk‡K ïay gvb Øviv m¤ú~Y©iƒ‡c ‡h mKj †fŠZ ivwk‡K m¤ú~Y©iƒ‡c cÖKvk Kivi Rb¨ cÖKvk Kiv hvq Zv‡`i‡K ‡¯‹jvi ivwk e‡j| 1| gvb I w`K Df‡qi cÖ‡qvRb nq Zv‡`i‡K ‡f±i ivwk e‡j| `ªæwZ, fi, KvR, ‰`N©¨ BZ¨vw` †¯‹jvi ivwki miY, IRb, †eM, Z¡iY, ej BZ¨vw` ‡f±i ivwki 2| D`vniY| D`vniY| Kgc‡ÿ GKwU ivwki gvb k~b¨ bv n‡j ¸bdj †Kvb ivwki gvb k~b¨ bv n‡jI ¸bdj k~b¨ n‡Z 3| k~b¨ n‡Z cv‡i bv| cv‡i| 4| gvb Av‡Q wKš‘ w`K bvB| gvb I w`K Av‡Q| ‡hvM, we‡qvM, ¸b, I fvM ¯^vfvweK wbq‡g Kiv ‡hvM, we‡qvM, ¸b, I fvM ¯^vfvweK wbq‡g Kiv hvq 5| hvq| bv| `ywU Aw`K ivwki ¸bdj me©`v GKwU Aw`K ivwk `ywU w`K ivwki myweav RbK ¸bdj GKwU w`K ivwk 6| nq| ev GKwU Aw`K ivwk nq| GKK †f±i (Unit Vector): ‡h †f±‡ii gvb GK Zv‡K GKK †f±i e‡j| gvb k~b¨ bq Ggb †f±i‡K Gi gvb Øviv fvM Ki‡j H w`K ivwkwUi w`‡K

    A GKwU GKK †f±i cvIqv hvq| g‡b Kwi A GKwU †f±i ivwk; A  0  A Gi w`‡K GKK †f±i    aˆ (awi) A

mxgve× †f±i (Restricted Vector): ‡h †f±‡ii Avw` we›`y †Kv_vq _v‡K Zv w¯’i _v‡K Zv‡K mxgve× †f±i e‡j| O we›`y‡Z   P ej OB †iLv eivei wµqv K‡i eySv‡j †f±i P mxgve× †f±i hvi cv` we›`y O | mgvb †f±i (Equal Vector): GKB w`‡K wµqviZ `yÕwU mgRvZxq †f±‡ii gvb mgvb n‡j Zv‡`i‡K mg‡f±i   ev mgvb †f±i e‡j| wP‡Î P I Q mgvb †f±i| k~b¨ ev bvj †f±i (Zero or Null Vector): †h †f±‡ii gvb k~b¨ Zv‡K k~b¨ ev bvj †f±i e‡j| bvj †f±‡ii Avw` I †kl we›`y GKB we›`y‡Z Aew¯’Z|

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01| ‡f±i (Vector)

2

m`„k †f±i ev mgvšÍÍivj †f±i (Like Vector): mgRvZxq `yB ev Z‡ZvwaK †f±i hw` GKB w`‡K wµqv K‡i Z‡e Zv‡`i‡K m`„k †f±i ev mgvšÍÍivj   †f±i e‡j| wP‡Î A I B m`„k †f±i ev mgvšÍÍivj †f±i| wem`„k †f±i (Unlike Vector): mgRvZxq `yB ev Z‡ZvwaK †f±i hw` wecixZ w`‡K wµqv K‡i Z‡e Zv‡`i‡K wem`„k †f±i e‡j|   wP‡Î A I B wem`„k †f±i| mg‡iL †f±i (Co-linear Vector): `yB ev Z‡ZvwaK †f±i hw` GKB mij‡iLv eivei ev ci¯úi mgvšÍÍiv‡j wµqv    K‡i Z‡e Zv‡`i‡K mg‡iL †f±i e‡j| wP‡Î A , B , C mg‡iL †f±i| Ae¯’vb †f±i (Position Vector): cÖm½ KvVv‡gvi g~j we›`yi mv‡c‡ÿ Ab¨ †Kvb we›`yi Ae¯’vb wbY©‡qi Rb¨ †h †f±i e¨envi Kiv nq Zv‡K Ae¯’vb †f±i e‡j| e¨vL¨v t wP‡Î O n‡”Q cÖmsM KvVv‡gvi g~j we›`y Ges †h P †Kvb GKwU we›`y| 

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OP †f±iwU O we›`yi mv‡c‡ÿ P we›`yi Ae¯’vb wb‡`©k Ki‡Q| ZvB OP GKwU Ae¯’vb †f±i|    Ae¯’vb †f±i‡K A‡bK mgq e¨vmva© †f±i r Øviv cÖKvk Kiv nq|  OP  r

AvqZGKK †f±i (Rectangular Unit Vector): wÎgvwÎK ¯’vbv¼ e¨e¯’vq ci¯úi j¤^ wZbwU Aÿ _v‡K| h_v t X,Y Ges Z Aÿ| X A‡ÿi w`‡K wewfbœ †f±i‡K cÖKvk Kivi Rb¨ GKwU GKK †f±i ˆi e¨envi Kiv nq| †Zgwb ˆj I kˆ h_vµ‡g Y I Z A‡ÿi w`‡K GKK †f±i (Wvb cv‡k¦©i wPÎ)| ˆi , ˆj Ges kˆ †K AvqZGKK †f±i e‡j| e¨vmva© †f±i (Radius Vector): A‡bK mgq †Kvb we›`yi Ae¯’vb‡K †h †f±‡ii mvnv‡h¨ cÖKvk 

Kiv nq Zv‡K e¨vmva© †f±i r e‡j| myZivs †Kvb we›`y P-Gi ¯’vbvsK (x,y,z) n‡j, e¨vmva© 

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†f±i r  OP  x i  y j  zk Øviv cÖKvk Kiv nq| Ges Gi gvb nq r  r  x 2  y 2  z 2 ‡f±i ivwki mvgvšÍÍwiK m~Î (Law of Parallelogram) eY©bv I e¨vL¨v : eY©bv: †Kvb KYvi Dci GKB mg‡q wµqvkxj `yÕwU †f±i ivwk‡K hw` †Kvb GK we›`y †_‡K AswKZ mvgvšÍÍwi‡Ki `yÕwU mwbœwnZ evû Øviv wb‡`©k Kiv hvq Z‡e H we›`y ‡_‡K AswKZ mvgvšÍÍwi‡Ki KY©B †f±i `yÕwUi jwäi gvb I w`K wb‡`©k K‡i| g‡bKwi GKwU KYvi Dci GKB mg‡q `yÕwU w`Kivwk P I Q,  †Kv‡Y wµqv Ki‡Q| OA Ges OC †iLv `yÕwU h_vµ‡g P I Q gvb Ges Zxi wPý G‡`i w`K wb‡`©k Ki‡Q| GLv‡b AOC   | GB `yÕwU w`K ivwki jwäi gvb I w`K wbY©q Ki‡Z n‡e| AsKb : mvgvšÍÍwiK OABC AsKb K‡i Kb© OB hy³ Kwi| Zvn‡j OB Kb©B w`Kivwk `yÕwUi gvb I w`K wb‡`©k Ki‡e| g‡b Kwi jwäi gvb R Ges †KvY GKwU myÿ‡KvY| GLb B we›`y †_‡K OA Gi ewa©Z As‡ki Dci BN j¤^ Uvwb|

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01| ‡f±i (Vector)

3

jwäi gvb wbY©qt AB I OC mgvšÍÍivj|  AOC   | OBN wÎfy‡Ri ONB  GK mg‡Kvb|  OB2 = ON2+ BN2  OB 2 =(OA+AN)2 +BN2  OB 2 = OA2 +2OA .AN +AN2 +BN2  OB 2 = OA2 +(AN2 +BN2) +2OA .AN AN  OB 2 = OA2 +AB2 +2OA. AB. AB  OB 2 = OA2 +OC2 +2OA.OC Cos  R2 = P2 +Q2 +2PQ Cos

 R  P 2  Q 2  2PQCos ... ... ... (1)

jwäi w`K wbY©q: g‡bKwi jwä R, P Gi mv‡_  †Kvb Drcbœ K‡i| A_©vr AOB    OBN mg‡KvYx wÎfy‡R tan  

BN ON BN OA  AN BN AB AB  tan   AN OA  AB AB QSin   tan   P  QCos

 tan  

   tan 1

QSin  ... ... ... (2) P  QCos 

(1) bs mgxKiY jwäi gvb I (2) bs mgxKiY jwäi w`K wb‡`©k K‡i|

GKB mg‡q GKB we›`y‡Z wµqviZ `ywU w`Kivwki jwäi m‡e©v”P I me©wb¤§ gvb ivwk `yÕwUi †hvMdj I we‡qvM d‡ji mgvb t P I Q, †Kv‡Y wµqv Ki‡j mvgvšÍÍwi‡Ki m~Îvbyhvqx Avgiv cvB,  R  P 2  Q 2  2PQCos ... ... ... (1)  R2 = P2 +Q2 +2PQ Cos  R2  P2 Q2 = 2PQ Cos R 2  P2  Q2  Cos  Avevi Avgiv Rvwb, CosGi gvb 2PQ 1 †_‡K +1 Gi g‡a¨ mxgve×| A_©vr, 1  Cos  1 R 2  P 2  Q2 1  1 [CosGi gvb ewm‡q] 2PQ  2PQ  R 2  P 2  Q 2  2PQ  P 2  Q 2  2PQ  R 2  P 2  Q 2  2PQ [Dfq c‡ÿ P2+Q2 †hvM K‡i]  ( P  Q) 2  R 2  ( P ~ Q) 2  (P  Q)  R  (P ~ Q) Kv‡RB GKB mg‡q GKB we›`y‡Z wµqviZ `yÕwU w`K ivwki jwäi m‡e©v”P I me©wb¤§ gvb

ivwk `ywUi †hvMdj I we‡qvM d‡ji mgvb| Ab¨ fv‡eI ejv hvq, GKB mg‡q GKB we›`y‡Z wµqviZ `yÕwU w`K ivwki jwäi m‡e©v”P gvb ivwk `ywUi †hvMdj n‡Z eo n‡Z cv‡i bv I me©wb¤§ gvb ivwk `ywUi we‡qvM dj †_‡K †QvU n‡Z cv‡i bv|

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01| ‡f±i (Vector)

4

`yÕwU †f±i ivwk P I Q ci¯úi †Kv‡Y  AvY&Z| G‡`i †¯‹jvi ¸Yb I †f±i ¸Yb t †¯‹jvi ¸Yb ev WU ¸Yb t `yÕwU w`K ivwki †¯‹jvi ¸Yb GKwU Aw`K ivwk Ges Gi gvb w`Kivwk `ywUi gv‡bi ¸Ydj Ges G‡`i ga¨eZ©x †Kv‡Yi Cosine Gi ¸Y d‡ji mgvb n‡e|   e¨vL¨vt g‡b Kwi, P I Q `yÕwU w`K ivwk ci¯úi  †Kv‡Y AvbZ& A_©vr P I Q Gi ga¨eZ©x †KvY | G‡`i WU   ¸Yb A_©vr †¯‹jvi ¸Yb P.Q  PQCos 

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wPÎvbyhvqx, P  OA; Q  OB; BM, OA Gi Dci j¤^ I AN, OB Gi Dci j¤^ | myZivs OM =OB Cos  ev, OM =Q Cos      P.Q  P(QCos) = ( P Gi gvb) (OM)    = ( P Gi gvb) ( P Gi Dci Q Gi j¤^ Awf‡ÿ‡ci gvb)      Abyiƒcfv‡e,  Q.P  QPCos  Q ( PCos ) = ( Q Gi gvb) ( Q Gi Dci P Gi j¤^ Awf‡ÿ‡ci gvb) A_©vr `ywU †f±i ivwki †¯‹jvi A_©vr WU ¸Ydj ej‡Z G‡`i †h †Kvb GKwU †f±i ivwk Øviv Dnviw`‡K Aci †f±i ivwkwUi Awf‡ÿ‡ci ¸Ydj †evSvq|   (K) hw` ‡f±i ivwkØq ci¯úi mg‡Kv‡b AvbZ& _v‡K Z‡e P.Q  PQ Cos 90  0 AZGe `yÕwU †f±iivwk mg‡Kv‡Y AvbZ& _vK‡j G‡`i †¯‹jvi ¸bdj k~b¨ n‡e wecixZµ‡g `ywU †f±i ivwki †¯‹jvi ¸bdj k~b¨ n‡j ivwk `ywU ci¯ú‡ii Dci j¤^ n‡e| awi wZbwU AvqZKvi †f±i ivwk ˆi, ˆj I kˆ cÖ‡Z¨‡K ci¯ú‡ii Dci j¤^| myZivs G‡`i †h †Kvb `yÕwUi †¯‹jvi ¸bdj k~b¨ n‡e| A_©vr iˆ. ˆj  ˆj.kˆ  kˆ.iˆ  0 (L) hw` `ywU †f±i ivwk GKB w`‡K wµqv K‡i Z‡e G‡`i ga¨eZ©x †KvY  = 0º;        †m †ÿ‡Î P.Q  PQ Cos 0  PQ n‡e| GKK w`K ivwki †ÿ‡Î cvB, i .i  j . j  k .k  1   (M) hw` P I Q `ywU †f±i ivwk ci¯úi wecixZgyLx nq Z‡e G‡`i ga¨eZ©x †KvY  = 180º   nq Z‡e †m †ÿ‡Î P.Q  PQ Cos 180   PQ n‡e| †f±i ¸Yb ev µm ¸Yb t `yÕwU w`K ivwki †f±i (µm) ¸Yb GKwU †f±i ivwk hvi gvb w`Kivwk `ywUi gv‡bi ¸Ydj Ges G‡`i ga¨eZ©x †Kv‡Yi Sine -Gi ¸Y d‡ji mgvb Ges GB ¸bdj ivwk `yÕwUi Z‡j j¤^fv‡e ¯’vwcZ GKwU Wvb cv‡Ki KK© ¯Œz‡K 1g †f±i ivwk ‡_‡K 2q †f±i ivwki w`‡K ÿz`ªZi †Kv‡Y Nyiv‡j GUv †h w`‡K AMÖmi nq †mB w`‡K wµqv K‡i|   e¨vL¨vt g‡b Kwi, P I Q `yÕwU w`K ivwk ci¯úi  †Kv‡Y AvbZ& A_©vr P I Q Gi ga¨eZ©x †KvY | G‡`i µm       ¸Yb A_©vr ‡f±i ¸Yb P  Q   PQSin GLv‡b  GKwU GKK w`K ivwk hv ( P  Q ) Gi jwäi w`K wb‡`©k K‡i|      hw` P  Q  R aiv nq Z‡e R Gi AwfgyL P I Q Gi mgZ‡ji mv‡_ Awfj¤^ eivei n‡e|

     R  P  Q  PQSin     ( P Gi gvb) ( P Gi j¤^ eivei Q Gi gvb )    Avevi, Q  P  QPSin ()     Q  P  QPSin      Q  P   ( Q Gi gvb) ( Q Gi j¤^ eivei P Gi gvb )      P  Q  Q  P A_©vr †f±i ¸Yb wewbgq m~Î †g‡b P‡j bv|

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01| ‡f±i (Vector)

5

Avgiv Rvwb, Wvb cv‡Ki KK© ¯Œz‡K evgw`‡K Nyiv‡j GwU Dc‡ii w`‡K Ly‡j Av‡m Avi Wvb w`‡K Nyiv‡j wb‡Pi w`‡K AMÖmi      nq, Z`ªæc P I Q Gi †f±i ¸Yb evgveZ©x n‡j G‡`i jwä R Gi AwfgyL DaŸ© w`‡K nq Ges P I Q Gi †f±i ¸Yb  `wÿbveZ©x n‡j G‡`i jwä R Gi AwfgyL wb¤§gyLx n‡e hv c~‡e©i Awfgy‡Li Dëv w`‡K n‡e| Dc‡iv³ eY©bv Abymv‡i,   1) = 0 n‡j P  Q  0 n‡e KviY Sin 0º = 0     2) P|| Q n‡j P  Q  0 n‡e KviY Sin 0º = 0     3) P  Q n‡j P  Q  PQ n‡e KviY Sin 90º = 1 4) ˆi, ˆj I kˆ mgKv‡Y wµqv Ki‡j ˆi  ˆj  kˆ, ˆj  kˆ  ˆi , kˆ  ˆi  ˆj I ˆi, ˆj I kˆ mgvšÍiv‡j wµqv Ki‡j ˆi  ˆi  ˆj  ˆj  kˆ  kˆ  0 ‡¯‹jvi ¸Yb wewbgq m~Î †g‡b P‡j wKš‘ †f±i ¸Yb wewbgq m~Î †g‡b P‡j bv:     g‡b Kwi `ywU †f±i P I Q ,  †Kv‡Y AvbZ&  P.Q  PQCos ... ... ... (1)   Avevi, Q. P  QPCos

   Q.P  PQCos ... ... ... (2)

        (1) I (2) n‡Z cvB, P. Q  Q. P A_©vr †¯‹jvi ¸Yb wewbgq m~Î †g‡b P‡j| Acic‡ÿ, (P  Q) I (Q  P) Gi gvb GKB    n‡jI G‡`i w`K wecixZ A_©vr P  Q   PQ Sin  ... ... ... (3)    Avevi, Q  P  Q P Sin ( )     Q  P   PQ Sin  ... ... ... (4)     (3) I (4) n‡Z cvB, (P  Q)  (Q  P)     P  Q  Q  P A_©vr †f±i ¸Yb wewbgq m~Î †g‡b P‡j bv|

‡f±i †hv‡Mi wÎfyR m~Î (Triangle Law) t `ywU †f±i‡K GKwU wÎfy‡Ri `ywU mwbœwnZ evû Øviv GKB µ‡g w`‡K I gv‡b wb‡`©k Ki‡j wÎfyRwUi Z…Zxq evû wecixZ µ‡g w`‡K I gv‡b Dnv‡`i jwä wb‡`©k K‡i|   e¨vL¨v t awi P I Q GKB RvZxq `ywU †f±i| ABC wÎfy‡Ri AB Ges BC 

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evû h_vµ‡g †f±i P I Q wb‡`©k K‡i| m~Îvbymv‡i, AC  AB BC    R  P  Q

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[ GLv‡ b, AC  R ]

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01| ‡f±i (Vector)     ‡f±‡ii †hv‡Mi wewbgq m~Î (Commutative Law) t A  B  B  A 

cÖgvb t aiv hvK

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OP  A Ges OR  B `ywU †f±i O we›`y‡Z wµqv K‡i OPQR

mvgvšÍwiK c~Y© K‡i wÎfyR m~Îvbymv‡i cvB, 

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OP PQ  OQ ... ... ... ... (1) 

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Ges OR  RQ  OQ ... ... ... ... (2) 

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(1) I (2) †_‡K cvB, OP PQ  OR  RQ     A_©vr, A  B  B  A myZvivs ‡f±i †hvM wewbgq m~Î †g‡b P‡j| 

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‡f±‡ii †hv‡Mi ms‡hvM m~Î (Associative Law) : (A  B)  C  A  (B  C) 

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cÖgvb t aiv hvK OP  A , PQ  B Ges QR  C | GLb OQ, PR Ges OR a‡i 

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wÎf~R m~Îvbymv‡i cvB, OP PQ  OQ  (A  B) 

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Ges PQ QR  PR  (B C) 

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GLb OQ QR  OR A_©vr (A  B)  C  D 

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Avevi, OP PR  OR A_©vr A  (B  C)  D        (A  B)  C  A  (B  C) myZvivs ‡f±i †hvM ms‡hvM m~Î †g‡b P‡j|

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e›Ub m~Î: A .(B C)  A . B A .C Gi cÖgvY:  

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cÖgvY: g‡b Kwi, A, B I C †f±i wZbwU h_vµ‡g OP,OQI QR Øviv m~wPZ Kiv n‡q‡Q| GLb wPÎ †_‡K Avgiv 

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cvB, A .(B C)  A . (OQ  QR ) 

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 A .(B C)  A . OR 

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 A .(B C)  A  OP Gi Dci OR Gi j¤ ^ Awf‡ ¶ c

 A .(B C)  A  ON 

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 A .(B C)  A  (OM  MN ) Figure 1 

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 A .(B C)  A  OM  A  MN 

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 A .(B C)  A  OP Gi Dci OQ Gi j¤ ^ Awf‡ ¶ c  A  OP Gi Dci QR Gi j¤ ^ Awf‡ ¶ c 

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 A .(B C)  A . OQ A . QR 

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 A .(B C)  A . B A . C (cÖgvwYZ)

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01| ‡f±i (Vector)

wÎgvwÎK ¯’vbv¼ e¨e¯’vvq GKwU Ae¯’vb †f±‡ii gvb wbY©q: A_©vr r  x 2  y 2  z 2 Gi cÖgvY : wÎgvwÎK ¯’vbv¼ e¨e¯’vq ci¯úi wZbwU †iLv OX, OY I OZ h_vµ‡g X, Y I       Z, Aÿ wb‡`©k K‡i Ges i , j I k Aÿ wZbwU eivei GKK †f±i| r GKwU Ae¯’vb †f±i| awi, r  OP Ges P we›`yi ¯’vbv¼ (x,y,z)| XZ Z‡ji Dci PN Ges N we›`y n‡Z X I Z A‡ÿi Dci h_vµ‡g NA I NB j¤^ AvuwK| wPÎ n‡Z, BN =OA= x, AN=OB = z Ges NP = y 

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wÎfyR m~Î Abymv‡i, OP  ON  NP 

 OP  OB BN  NP 

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 OP  BN  NP  OB 

 r  xˆi  yˆj  zkˆ Avevi, OPN wÎfyR n‡Z, OP2=ON2+NP2  OP2 = OB2+BN2+NP2  OP2 = BN2 + NP2 +OB2  r2  x 2  y2  z2

 r  x2  y2  z2

(cÖgvwYZ )

  A.B  A x B x  A y B y  A z B z Gi cÖgvY:   awi, A  A x ˆi  A y ˆj  A z kˆ I B  B x ˆi  B y ˆj  B z kˆ   d‡j, A . B  (A x ˆi  A y ˆj  A z kˆ) . ( Bx ˆi  B y ˆj  B z kˆ)

  A . B  A x B x (ˆi .ˆi )  A x B y ( ˆi .ˆj)  A x B z ( ˆi .kˆ )  A y B x (ˆj.ˆi )  A y B y (ˆj.ˆj)  A y B z (ˆj.kˆ)

 A z B x (kˆ .ˆi )  A z B y (kˆ .ˆj)  A z B z (kˆ .kˆ )

  A . B  Ax Bx (1)  Ax By (0)  Ax Bz (0)  Ay Bx (0)  A y By(1)  A y Bz (0)  Az Bx (0)  Az By (0)  Az Bz (1)   A . B  Ax Bx  0  0  0  Ay By  0  0  0  Az Bz   A . B  Ax Bx  A y By  A z Bz ( Pr oved)   A  A x ˆi  A y ˆj  A z kˆ I B  B x ˆi  B y ˆj  B z kˆ n‡j

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01| ‡f±i (Vector)

  A  B wbY©q :

  A  B  (A x ˆi  A y ˆj  A z kˆ)  (B x ˆi  B y ˆj  B z kˆ )    A  B  A x B x (ˆi  iˆ)  A x B y (ˆi  ˆj)  A x B z (iˆ  kˆ)  A y B x (ˆj  ˆi)  A y B y (ˆj  ˆj)  A y B z (ˆj  kˆ)  A z B x (kˆ  ˆi )  A z B y (kˆ  ˆj)  A z B z (kˆ  kˆ)    A  B  0  A x B y (kˆ)  A x B z (  ˆj)  A y B x (  kˆ)  0  A y B z (ˆi)  A z B x (ˆj)  A z B y (  ˆi )  0    A  B  A y B z (ˆi)  A z B y (ˆi)  A z B x (ˆj)  A x B z (ˆj)  A x B y (kˆ)  A y B x (kˆ)    A  B  (A y B z  A z B y )ˆi  (A z B x  A x B z )ˆj  (A x B y  A y B x )kˆ

ˆi   A  B  Ax Bx

ˆj Ay By

kˆ A z (Ans.) Bz

‡f±i wefvRb ev †f±i we‡køølb (Resolution Of Vectors) t GKwU †f±i ivwk‡K `yB ev Z‡ZvwaK †f±i ivwk‡Z wef³ Kivi c×wZ‡K †f±‡ii wefvRb ev †f±‡ii we‡køølb e‡j|   g‡bKwi, wP‡Î OC †iLv R †f±iwUi gvb I w`K wb‡`©k K‡i| GLb R †f±iwU‡K Ggb `yÕwU As‡k wef³ Ki‡Z n‡e †h, G ¸‡jv OC Gi mv‡_ h_vµ‡g I †KvY Drcbœ K‡i| GLb O we›`y †_‡K OC †iLvi mv‡_ Gi `yB cv‡k I  †KvY K‡i OB I OA †iLv Uvbv nj| OACB mvgvšÍwiKwU c~Y© Kiv n‡j mvgvšÍÍwi‡Ki  m~Îvbymv‡i OA Ges OB evû `ywU R †f±‡ii `ywU Dcvsk wb‡`©k Ki‡e| 

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g‡bKwi, wefvwRZ Dcvsk OA  X Ges OB  Y | (K) wP‡Î,  BOC  OCA   Ges OAC  180º -    GLb OAC wÎfyR we‡ePbv K‡i Avgiv cvB, OA AC OC   Sin  Sin  Sin[180  (  )] X Y R    ( AC  OB  Y) Sin  Sin  Sin (  ) RSin  RSin  X  Ges Y  R †f±i‡K hw` mg‡Kv‡Y wefvwRZ Kiv hvq [wPÎ (L)] A_©vr Sin (  ) Sin (  ) Dcvsk `ywU hw` ci¯úi j¤^ nq Z‡e Zvn‡j,+90º Sin+Sin 90º = 1 X = R SinGesY = R Sin †h‡nZz +90º 90º Sin Sin () = Cos d‡j, X = R Cos GesY = R Sin

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution 1| ‡f±i (Vector)

1| ‡f±i

  1| hw` A  3ˆi  ˆj  2kˆ I B  2ˆi  3ˆj  kˆ nq Z‡e   |A  B|  KZ? ˆi ˆj kˆ   A B  3  1  2

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 iˆ(  1  6 )  ˆj(3  4 )  kˆ (9  2 )  5 iˆ  7 ˆj  11kˆ   | A  B | 52  (7) 2  112  25 49  121 13 . 96(Ans.)

  2| hw` A  6ˆi  3ˆj  2kˆ I B  2ˆi  2ˆj  kˆ nq Z‡e   A.B  KZ?   A.B  A x B x  A y B y  A z B z  (6)(2)  (  3)(2)  (2)(1)  12  6  2  8 (Ans.)   3| A  5ˆi  2ˆj-3kˆ I B  15ˆi  aˆj-9kˆ | a Gi gvb KZ   n‡j A I B ci¯úi mgvšÍivj n‡e?     A I B ci¯úi mgvšÍivj n‡e hw` A  B  0 nq|

ˆi ˆj kˆ   AB  5 2 3 15 a -9    A  B  ˆi(  18  3a)  ˆj(-45  45)  kˆ(5a-30)    A  B  ˆi (3a  18)  kˆ(5a  30)    A  B  (3a  18) 2  (5a  30) 2     cÖkœg‡Z, A I B ci¯úi mgvšÍivj n‡j A  B  0  (3a  18) 2  (5a  30) 2  0  (3a  18) 2  (5a  30) 2  0 [Dfq c¶ ‡ K eM K‡ i]  32 (a  6) 2  5 2 (a  6) 2  0  (a  6) 2 (32  5 2 )  0  (a  6) 2  0 Dfq c¶ ‡ K (32  5 2 ) Øviv fvM K‡ i  a  6 (Ans.)

  A  2ˆi  3ˆj-5kˆ I B  mˆi  2ˆj - 10kˆ | m Gi   gvb KZ n‡j A I B ci¯úi j¤^ n‡e|     A I B ci¯úi j¤^ n‡j A . B  0 n‡e|

4|

  A .B  Ax Bx  Ay By  Az Bz

0

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 (2)(m)  (3)(2)  (5)(10)  2m  6  50  0 - 56 m  m  - 28 (Ans.) 2 



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5| A  2ˆi  2ˆj  kˆ I B  6ˆi  3ˆj  2kˆ n‡j A I B Gi ga¨eZ©x †KvY wbY©q Ki|

  A.B  AB cos      1 A . B A I B Gi ga¨eZ©x †KvY, θ  Cos   AB

  A .B  A x B x  A y B y  A z B z  (2)(6)  (2)(-3)  (-1)(2)  12-6-2  4  2 2 2 A  2  2  (1)  9  3   B  6 2  (3) 2  2 2  36  9  4  49  7  θ  Cos

1

  A .B 4 4 -1  Cos 1    Cos (3)(7) 21 AB

   79 .02 (Ans.)



6. ‡f±i B  6ˆi  3ˆj  2 kˆ Gi Dci ‡f±i

 A  2ˆi  2ˆj  kˆ Gi j¤^ Awf‡ÿc wbY©q Ki|   A.B  AB cos    A.B A Gi j¤^ Awf‡ÿc, A cos    B   A .B  A x B x  A y B y  A z B z  ( 2)(6)  ( 2)(-3)  (1)(2)  12-6  2  8

 B  6 2  (  3) 2  2 2  36  9  4  49  7   A.B 8 A Gi j¤^ Awf‡ÿc =   7 B    7| A  3ˆi  2ˆj  kˆ , B  ˆi  2ˆj  3kˆ I C  ˆi  ˆj  2kˆ  





 

n‡j cÖgvY Ki †h, A.(B  C)  (A  B).C

ˆi ˆj kˆ   B C  1 2  3 1 1 2   ev, B  C  ˆi (4  3)  ˆj(2  3)  kˆ(1-2)  7ˆi-5ˆj -kˆ

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1| ‡f±i (Vector)

   L.H.S  A . (B  C)

 (3)(7)  (2)(-5)  (1)(-1)  21  10  1  10

ˆi ˆj   GLb, A  B  3 2

kˆ 1

  ev, A  B = ˆi (  6  2 )  ˆj(  9  1)  kˆ ( 6 - 2 )   8 ˆi  10 ˆj  4 kˆ     R.H.S  ( A  B ) . C  (- 8 )(1)  (10 )(1)

 ( 4 )( 2 )  - 8  10  8  10  L. H. S  R.H.S       A_©vr, A.(B  C)  (A  B).C ( Pr oved.)    8| A  ˆi  3ˆj  2kˆ , B  ˆi  2ˆj  kˆ I C  2ˆi  3ˆj  4kˆ













n‡j cÖgvY Ki †h, (B  C)  A  B  A  C  A

  (B  C)  (1  2) ˆi  (2  3)ˆj  ( 1  4)kˆ  3ˆi  ˆj  3kˆ ˆi

ˆj

kˆ

1

3

2

    L.H.S  (B  C)  A  3  1 3

 ˆi (  2  9)  ˆj(6  3)  kˆ(9  1)  11 ˆi  3ˆj  10kˆ ˆi ˆj kˆ   Avevi, B  A  1 2  1

1 3

2

 ˆi(4  3)  ˆj(2  1)  kˆ(3-2)  7ˆi  3ˆj  kˆ ˆi ˆj kˆ   C A  2  3 4

1

3

2

 iˆ(  6  12)  ˆj(4-4)  kˆ(6  3)  18ˆi  9kˆ

    R.H.S  B  A  C  A  7ˆi  3ˆj  kˆ  18ˆi  9kˆ   11 ˆi  3ˆj  10kˆ        A_©vr, (B  C)  A  B  A  C  A ( Pr oved.) 

wKš‘ A  0, B  0  CosθC 0 ev , Cos θ  Cos 90  θ  90   AZGe A I Β ci¯ú‡ii Dc‡ii j¤^|   10| hw` A  A x ˆi  A y ˆj  A z kˆ I B  Bx iˆ  B y ˆj  Bz kˆ nq   Z‡e †`LvI †h, A.B  A x B x  A y B y  A z B z

1 2 -3



2



9| A  9ˆi  ˆj-6kˆ Ges B  4ˆi  6ˆj  5kˆ †f±i `ywUi Mybdj wbY©q K‡i †`LvI †h, Giv ci¯ú‡ii Dci j¤^|

  A.B  A x B x  A y B y  A z B z

 (9)(4)  (1)(  6)  (  6)(5)  36  6  30  0   A . B  ABCosθ  0

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  A . B  (A x ˆi  A y ˆj  A z kˆ) . ( Bx ˆi  B y ˆj  B z kˆ)

  A . B  A x B x ( ˆi .ˆi )  A x B y ( ˆi .ˆj)  A x B z (ˆi .kˆ)  A y B x (ˆj.ˆi )  A y B y (ˆj.ˆj)  A y B z (ˆj.kˆ)

 A z B x (kˆ. ˆi )  A z B y (kˆ.ˆj)  A z B z (kˆ.kˆ )

  A . B  Ax Bx (1)  A x By (0)  Ax Bz (0)  Ay Bx (0)  A y By(1)  A y Bz (0)

 Az Bx (0)  Az By (0)  Az Bz (1)   A . B  Ax Bx  0  0  0  Ay By  0  0  0  Az Bz   A . B  Ax Bx  A y By  A z Bz ( Pr oved) 



11| hw` A  2ˆi  4ˆj  5kˆ I B  ˆi  2ˆj  3kˆ ‡f±i ؇qi jwä †f±‡ii mgvšÍivj GKK †f±i wbY©q Ki|    R  A  B  2 iˆ  4 ˆj  5 kˆ  iˆ  2 ˆj  3k

  R  3ˆi  6ˆj  2kˆ

 R R Gi mgvšÍivj GKK †f±i aˆ   R  R  32  6 2  (2) 2  9  36  4  49  7  R 3ˆi  6ˆj  2kˆ 3 ˆ 6 ˆ 2 ˆ  aˆ     i  j  k (Ans.) 7 7 7 7 R

12| GKB we›`y‡Z wµqvkxj `ywU mgvb gv‡bi †f±‡ii ga¨eZ©x †KvY KZ n‡j G‡`i jwäi gvb †h †Kvb GKwU †f±‡ii mgvb n‡e?

R2=P2+Q2+2PQCos ev, X2= X2+X2+2X.X.Cos GLv‡b, ev, X2- X2-X2=2X2Cos awi, ‡f±i, P=Q=X 2 2 ev, -X =2X Cos jwä, R= X X2  ev , Cos   AšÍf © ³ ~ †KvY,  2X 2

1 ev, Cos    2  1 ev ,   Cos 1      2    120  (Ans.)

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1| ‡f±i (Vector)



13| Ae¯’vb †f±i r  x ˆi  y ˆj  z kˆ †K e¨eKjb K‡i wKfv‡e †eM I Z¡iY cvIqv hvq?

Avgiv Rvwb,

 dr dt d  (x ˆi  y ˆj  z kˆ) dt dx ˆ dy ˆ dz ˆ (Ans.) j k i  dt dt dt   dv d  dx ˆ dy ˆ dz ˆ    i j  k Avevi, Z¡iY a  dt  dt dt dt  dt 

‡eM, v 

d 2x ˆ d2 y ˆ d2z ˆ  2 i  2 j 2 k dt dt dt 

(Ans.)



`ywU †f±‡ii µm ¸bdj †f±i `ywU Øviv MwVZ mgZ‡ji Dci j¤^ nq| †mB j¤^ †f±‡ii mgvšÍivj GKK †f±iB n‡e mgZ‡ji Dj¤^ w`‡K GKK

  PQ †f±i| awi, †mB †f±i aˆ , aˆ     PQ

ˆi ˆj kˆ   P  Q  2 3  4  ˆi (9  8)  ˆj(6  4)  kˆ (4  3) 1 2 3

   P  Q  ˆi  10ˆj  7kˆ   P I Q ‡h Z‡j Aew¯’Z Zvi Dj¤^ w`‡K †f±i    (ˆi  10ˆj  7kˆ)  ( P  Q)  aˆ     PQ (1) 2  (10) 2  (7) 2

 aˆ  

14| P  t 2 ˆi  t ˆj  ( 2 t  1) kˆ I Q  5tˆi  tˆj  t 3 kˆ

d   d   (P. Q)  ? (P  Q)  ? dt dt   2 P.Q  ( t )(5t )  ( t )(t )  (2t  1)( t 3 )    P.Q  5t 3  t 2  2t 4  t 3   P.Q  2 t 4  4t 3  t 2 d   d  (P.Q)  (2 t 4  4t 3  t 2 ) dt dt d    (P.Q)  8t 3  12t 2  2t dt ˆi ˆj kˆ   P  Q  t 2  t 2t  1

5t

t

 t3



d   (P  Q)  (4 t 3  4 t  1)ˆi  (5t 4  20 t  5)ˆj dt (Ans.)  (3t 2  10 t )kˆ 



15| P  2ˆi  3ˆj  4kˆ, Q  ˆi - 2ˆj  3kˆ ‡f±i Øq †h Z‡j Ae¯’vb K‡i Zvi Dj¤^w`‡K GKwU GKK †f±i wbY©q Ki| Avgiv Rvwb,

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(ˆi  10ˆj  7kˆ) ( Ans.) 150

16| †Kvb GKwU KYvi Ae¯’vb †f±i

 r  [(3.5ms 1 )t  4.2m] ˆi  [5.3ms 1 ]ˆj n‡j

†eM V wbY©q Ki| Avgiv Rvwb, V 

 dr dt

d [(3.5ms 1 )t  4.2m]ˆi  [5.3ms 1 t]ˆj dt  V  3.5ˆi  5.3ˆj (Ans.)



V



          LHS  A  B   A.B  2

17| cÖgvY Ki t A  B  A.B 2

2

 A 2 B2

2

2

   P  Q  ˆi ( t 4  2 t 2  t )  ˆj(10 t 2  5t  t 5 )  kˆ ( t 3  5t 2 ) 4 2 2 5 d ˆi ( t  2t  t )  ˆj(10t  5t  t ) d   ( P  Q)    dt  dt  kˆ (t 3  5t 2  d   d  (P  Q)  ( t 4  2t 2  t )ˆi dt dt d 5 d  ( t  10 t 2  5t )ˆj  ( t 3  5t 2 ) kˆ dt dt

3

2

 ˆAB sin     AB cos   ˆ 2 A 2 B 2 sin 2   A 2 B 2 cos 2   1. A 2 B 2 sin 2   A 2 B 2 cos2   A 2 B 2 sin 2   cos 2   A 2 B 2 .1  A2 B 2  L.H .S  R.H .S (Proved)









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18| P  ˆi  2ˆj  kˆ Ges Q  3ˆi  6ˆj  3kˆ n‡j †`LvI †h, P I

 Q ci¯úi mgvšÍivj|

    P I Q ci¯úi mgvšÍivj n‡e hw` P  Q  0 nq|

ˆi ˆj kˆ   P  Q  1  2 1  ˆi (6  6)  ˆj(3  3)  kˆ (6  6) 3 6 3    P  Q  ˆi (0)  ˆj(0)  kˆ (0)  0  0  0  0

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1| ‡f±i (Vector)

       P  Q  0  P  Q  0  P I Q ci¯úi mgvšÍivj| (cÖgvwYZ)





19| P  2ˆi  3ˆj  4kˆ Ges Q  2ˆi  ˆj  3kˆ ‡f±i Øq †h Z‡j Aew¯’Z Zvi Dj¤^w`‡K GKwU GKK †f±i wbY©q Ki| Avgiv Rvwb, `ywU †f±‡ii µm ¸bdj †f±i `ywU Øviv MwVZ mgZ‡ji Dci j¤^ nq| †mB j¤^ †f±‡ii mgvšÍivj GKK †f±iB n‡e mgZ‡ji Dj¤^

  P Q w`‡K GKK †f±i| awi, †mB †f±i nˆ , nˆ     P Q ˆi   PQ  2

ˆj kˆ 3  4  ˆi (9  4)  ˆj(6  8)  kˆ (2  6) 2 1 3

   P  Q  13ˆi  2ˆj  8kˆ     Avevi, P  Q  132  2 2  8 2  P  Q  237   PQ 13iˆ  2 ˆj  8kˆ  nˆ       237 PQ

8  2 ˆ  13 ˆ  nˆ   i j ( Ans.) 237  237  237 



20| P  4ˆi  4ˆj  kˆ Ges Q  2ˆi  2ˆj  kˆ †f±iØq GKwU mgvšÍwi‡Ki `ywU mwbœwnZ evû wb‡`©k Ki‡j Gi †ÿÎdj wbY©q Ki| Avgiv Rvwb, `ywU †f±i GKwU mgvšÍwi‡Ki `ywU mwbœwnZ evû wb‡`©k Ki‡j H mgvšÍwi‡Ki †ÿÎdj n‡e †f±i `ywUi µm ¸Yd‡ji gv‡bi





mgvb| P  Q  mgvšÍwi‡Ki †ÿÎdj| GLb,

ˆi ˆj   PQ  4  4

kˆ 1  ˆi (4  2)  ˆj(4  2)  kˆ (8  8)

2  2 1

   P  Q  6ˆi  6ˆj    P  Q  6 2  6 2  72  8.49 GKK (Ans.)

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4

  21| P  2iˆ  mˆj-3kˆ I Q  10ˆi  5ˆj-15kˆ | m Gi gvb KZ n‡j   P I Q ci¯úi mgvšÍivj n‡e?     P I Q ci¯úi mgvšÍivj n‡e hw` P  Q  0 nq|

ˆi   PQ  2

ˆj kˆ m 3 10  5 -15





 P  Q  ˆi (  15m  15)  ˆj(-30  30)  kˆ(  10-10m)

   P  Q  ˆi (  15m  15)  kˆ(  10  10m)    P  Q  (  15m  15) 2  (  10  10m)2     cÖkœg‡Z, P I Q ci¯úi mgvšÍivj n‡j, P  Q  0  (  15m  15) 2  (  10  10m)2  0

 (  15m  15) 2  (  10  10m) 2  0 [Dfq c¶ ‡ K eM  K‡ i]  152 (m  1) 2  102 (m  1) 2  0

 (m  1) 2 ( 152  102 )  0  (m  1) 2  0 Dfq c¶ ‡ K ( 152  102 ) Øviv fvM K‡ i  m  1 (Ans.)





22| A  2iˆ  2 ˆj  kˆ Ges B  6iˆ  3 ˆj  2kˆ `yÕwU ‡f±i ivwk| G‡`i j¤^ Awfgy‡L GKwU GKK †f±i wbY©q Ki| Avgiv Rvwb, `ywU †f±‡ii µm ¸bdj †f±i `ywU Øviv MwVZ mgZ‡ji Dci j¤^ nq| †mB j¤^ †f±‡ii mgvšÍivj GKK †f±iB n‡e mgZ‡ji Dj¤^ w`‡K GKK

  P Q †f±i| awi, †mB †f±i nˆ , nˆ     P Q iˆ   PQ  2

ˆj

2 6 3

kˆ  1  iˆ(4  3)  ˆj (4  6)  kˆ(6  12)

2    P  Q  iˆ  10 ˆj  18kˆ 







Avevi, P  Q  12  (10 2 )  ( 182 )  P  Q  425

  PQ iˆ  10 ˆj  18kˆ  nˆ       (Ans.) 425 PQ

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 ফাস্তফ যক্ষনত্র ঩যভবিবত ঑ ঩যভগবত অ঳ম্ভফ। ঩ৃ বথফীয গবত িরন ঘূ ণ঱ন গবত।  ঘবড়য কাটায গবত ঩ম঱াফৃ ত্ত গবত।  ভ঴াবফনেয ঳ফ ফস্ত্ত্তই গবত঱ীর, তাই বনশ্চর প্র঳ঙ্গ কাঠ্ানভা ঩া঑য়া মায় না।  যনৌকায মাত্রী যনৌকায ঳ান঩নক্ষ আন঩বক্ষক বিবতনত থানক বকন্তু তীনযয ঳ান঩নক্ষ আন঩বক্ষক গবতনত থানক।  ভুিবানফ উ঩য যথনক বননি ঩ড়ন্ত ফস্ত্ত্ত ঳ভত্বযনণয একবট ফাস্তফ উদা঴যণ এফিং খাড়া উ঩নযয বদনক বনবক্ষপ্ত ফস্ত্ত্তয ভন্দন ঳ভভন্দননয একবট ফাস্তফ উদা঴যণ। ঩ড়ন্ত ফস্ত্ত্তয ৩বট ঳ূ ত্র আবফষ্কায কনযনেন গযাবরবর঑ বকন্তু প্রভাণ কনযনেন বনউটন।  অ঳ভ দ্রুবতনত ফা অ঳ভ যকৌবনক যফনগ ঘুযনত থাকনর কণাবটয ৩বট ত্বযণ থানক। মথা-যকদ্রতাভুখী ত্বযণ, স্প঱঱ীত্বযণ ঑ যকৌবনক ত্বযণ। ফরবফদযা ভূ রত দু ই প্রকায। মথা: ১। বিবতবফদযা: (ক) ঩যভবিবত (খ) আন঩বক্ষক বিবত। ২। গবতবফদযা: (ক) ঳ৃ বতবফদযা (খ) ির বফদযা। গবতয প্রকায যবদঃ (i) রযবখক গবত ফা একভাবত্রক গবতঃ যকান ফস্ত্ত্তয গবত মবদ একবট ঳যর যযখায উ঩য ঳ীভাফদ্ধ থানক। তা঴নর তায গবতনক রযবখক গবত ফা একভাবত্রক গবত ফনর। যমভন-য঳াজা ঳ড়নক গাবড়য গবত। (ii) ঳ভতরীয় ফা বিভাবত্রক গবতঃ গবত ঳ভতনরয উ঩য ঳ীভাফদ্ধ। যমভন- ব঩঩ঁড়ায গবত, ভানফ঱নরয গবত। (iii) িাবনক গবত ফা বত্রভাবত্রক গবতঃ যকান ফস্ত্ত্তয গবত মবদ যম যকান বদনক গবত঱ীর ঴নত ঩ানয তনফ তায গবতনক িাবনক গবত ফা বত্রভাবত্রক গবত ফনর। বফববন্ন প্রকায গবতয রফব঱িয ঑ উদা঴যণঃ ১. িরন গবত এই গবতনত ফস্ত্ত্তয প্রবতবট কণা একই বদনক ঳ভান দূ যত্ব অবতক্রভ কযয। উদা঴যণঃ যযর঩নথয উ঩য িরন্ত যযরগাবড়য ফবগয গবত। ২. ঘূ ণন ঱ গবত এই গবতনত ফস্ত্ত্ত একবট বনবদি বফন্দু ফা অক্ষনক যকদ্রতা কনয িক্রাকানয ঩বযরামভন কনয। উদা঴যণঃ রফদু যবতক ঩াখায গবত, ঘবড়য কাঁটায গবত, ৩. িরন ঘূ ণন ঱ ফা বভশ্র ফা জবটর গবত এই গবতনত ফস্ত্ত্তয িরন ঑ ঘূ ণ঱ন দু বট গবতই থানক। উদা঴যণঃ িরন্ত ঳াইনকর ফা গরুয গাবড়য িাকায গবত, ঳ূ নম঱য িাযবদনক ঩ৃ বথফীয গবত এফিং রাবটনভয গবত। ৪. ঩ম঱াফৃ ত্ত গবত এই গবতনত ফস্ত্ত্ত একবট বনবদ঱ি ঳ভয় ঩য ঩য একই ঩থ অবতক্রভ কনয একই বদনক িরনত থানক। উদা঴যণঃ ঘবড়য কাঁটা, ইবিননয ব঩স্টন, রফদু যবতক ঩াখা, যদারক ব঩ন্ড ইতযাবদয গবত। ৫. যদারন গবত এই গবতনত ফস্ত্ত্তবট বনবদ঱ি ঳ভয় অন্তয অন্তয এবদক ঑বদক যদার যদয়। উদা঴যণঃ যদারক ঘবড়য গবত, ঳যর যদারনকয facebook /gmail/skype: -tanbir.cox

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Mo‡eM (Average Velocity) : msÁv: †h †Kvb mgq e¨eav‡b †Kvb e¯‘i M‡o cÖwZ GKK mg‡q †h miY nq Zv‡K e¯‘wUi Mo †eM e‡j|    Δr n‡e| e¨L¨v: t mgq e¨eav‡b †Kvb e¯‘i miY r n‡j Mo †eM v  t

‡eM (Velocity): msÁv: mgq e¨eavb k~‡b¨i KvQvKvwQ n‡j mg‡qi mv‡_ e¯‘i mi‡Yi nvi‡K †eM e‡j|

       Δr Δr wKš‘ e¨L¨v: t mgq e¨eav‡b †Kvb e¯‘i miY r n‡j †eM v  lim n‡”Q Mo †eM v | myZivs v  lim v t 0 t t  0 t

A_©vr mgq e¨eavb k~‡b¨i KvQvKvwQ n‡j Mo †e‡Mi mxgvwšÍK gvb‡KB †eM e‡j| mg‡eM ev mylg †eM (Uniform Velocity) : hw` †Kvb e¯‘i MwZKv‡j Zvi †e‡Mi gvb I w`K AcwiewZ©Z _v‡K Zvn‡j †mB e¯‘i †eM‡K mg‡eM e‡j| A_©vr †Kvb e¯‘ hw` wbw`©ó w`‡K mgvb mg‡q mgvb c_ AwZµg K‡i Zvn‡j e¯‘i †eM‡K mg‡eM e‡j| k‡ãi †eM, Av‡jvi †eM, cÖf„wZ mg‡e‡Mi cÖK…ó cÖvK…wZK D`vniY| Amg‡eM (Variable Velocity) t ‡Kvb e¯‘i MwZKv‡j hw` Zvi †e‡Mi gvb ev w`K ev DfqB cwiewZ©Z nq Zvn‡j †mB †eM‡K Amg‡eM e‡j| Avgiv mPvivPi †h MwZkxj e¯‘ †`wL Zv‡`i †eM Amg‡eM| ZvrÿwbK †eM (Instantaneus Velocity) : GKwU e¯‘ mij ev eµ c‡_ Amg‡e‡M Pj‡j cÖwZwbqZ Gi †e‡Mi cwieZ©b nq| Gfv‡e Amg‡e‡M PjšÍ †Kvb e¯‘i †h †Kvb gyû‡Z©i †eM‡K H e¯‘i ZvrÿwbK †eM e‡j| ZvrÿwbK †e‡Mi w`K e¯‘wUi H gyû‡Z©i Ae¯’v‡b AswKZ MwZc‡_i ¯úk©K eivei| Z¡iY (Acceleration) :

  v  mg‡qi mv‡_ †eM e„w×i nvi‡K Z¡iY e‡j| t mgq e¨eav‡b e¯‘i †e‡Mi cwieZ©b v n‡j Z¡iY a  n‡e| Ab¨fv‡e t ejv hvq mgq e¨eavb k~‡b¨i KvQvKvwQ n‡j mg‡qi mv‡_ e¯‘i †eM e„w×i nvi‡K Z¡iY e‡j| t mgq e¨eav‡b e¯‘i †e‡Mi   v  n‡e| cwieZ©b v n‡j Z¡iY a  lim t  0  t

mgZ¡iY ev mylg Z¡iY (Uniform Acceleration) : GKB w`‡K GKB mgq e¨eav‡b †e‡Mi e„w×i nvi mgvb n‡j Zv‡K mgZ¡iY ev mylg Z¡iY e‡j| AwfK‡l©i Uv‡b gy³fv‡e cošÍ e¯‘i †eM e„w×i nvi‡K AwfKl©R Z¡iY e‡j| AwfKl©R Z¡iY, mgZ¡iY wewkó MwZi GKwU cÖKó… D`vniY| mgZ¡i‡Y, Z¡i‡Yi gvb I w`K mg‡qi mv‡_ AcwiewZ©Z _v‡K| mgZ¡i‡Y MwZkxj e¯‘‡Z mgej wµqvK‡i e¯‘i cici †m‡K‡Ûi †e‡Mi

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2 02| ‰iwLK MwZ (Linear Motion) AšÍiB mgZ¡iY| wP‡Î GKwU mij‡iLv eivei cici †m‡K‡Ûi †eM †`wL‡q Gi Z¡i‡bi cÖK…wZ wb‡`©k Kiv n‡q‡Q| GLv‡b mgZ¡i‡Yi gvb 2ms2 |

miY ( Displacement ) t wbw`©ó w`‡K e¯‘i Ae¯’v‡bi cwieZ©b‡K miY e‡j| miY‡K s ev d Øviv cÖKvk Kiv nq| Gi GKK wgUvi| wbw`©ó w`‡K †Kvb e¯‘ t mgq a‡i v †e‡M Pj‡j, miY s = v t n‡e| miY GKwU †f±i ivwk| Av‡cwÿK †eM: `ywU MwZkxj e¯‘i GKwUi Zzjbvq (mv‡c‡ÿ) AciwUi Ae¯’v‡bi cwieZ©‡bi nvi‡K Av‡cwÿK †eM e‡j| ga¨ †eM: †Kvb GKwU MwZkxj e¯‘i cÖ_g Ges †kl †eM Gi AwfgyL GKB n‡j Zv‡`i †hvM d‡ji A‡a©K‡K ga¨ †eM e‡j| ‡Kvb wbw`©ó w`‡K †Kvb e¯‘i Avw`‡eM viI †kl †eM vf n‡j ga¨‡eM =

vi  v f 2

n‡e|

`ªwZ I †e‡Mi cv_©K¨ (Distinction between Speed and Velocity) : µwgK `ªywZ ‡eM ‡h †Kvb w`‡K e¯‘i ¯’vb cwieZ©‡bi nvi‡K e¯‘i wbw`©ó w`‡K e¯‘i ¯’vb cwieZ©‡bi nvi‡K e¯‘i †eM 1| `ªæwZ e‡j| e‡j| `ªæwZi †Kej gvb Av‡Q, w`K bvB| d‡j `ªæwZ ‡e‡Mi gvb I Av‡Q , w`K I Av‡Q | d‡j, †eM 2| me©`vB abvZ¡K| abvZ¡K I n‡Z cv‡i FbvZ¡K I n‡Z cv‡i| 3| `ªywZ GKwU †¯‹jvi ivwk ev Aw`K ivwk| ‡eM GKwU †f±i ivwk ev w`K ivwk| 4| `ªæwZ cwigv‡ci h‡š¿i bvg w¯ú‡WvwgUvi| ‡eM cwigv‡ci h‡š¿i bvg †fjv‡UvwgUvi| ‡Kvb e¯‘ t mg‡q r `yiZ¡ AwZµg Ki‡j wbw`©ó w`‡K ‡Kvb e¯‘ t mg‡q s `yiZ¡ AwZµg Ki‡j 5| dr ds `ªwZ = n‡e| ‡eM = n‡e| dt

dt

†eM I Z¡i‡Yi cv_©K¨ (Distinction between Velocity and Acceleration) : µwgK ‡eM Z¡iY wbw`©ó w`‡K e¯‘i ¯’vb cwieZ©‡bi nvi‡K e¯‘i mg‡qi mv‡_ †eM e„w×i nvi‡K Z¡iY e‡j| 1| †eM e‡j| †eM‡K v Øviv cÖKvk Kiv nq| Z¡iY‡K a Øviv cÖKvk Kiv nq| 2| ‡e‡Mi GKK ms1| Z¡i‡Yi GKK ms2| 3| ‡e‡Mi gvÎv [LT 1]| Z¡i‡Yi gvÎv [LT 2]| 4| (K) vx  vxo  ax t cÖwZcv`b| g‡bKwi, X Aÿ eivei GKwU e¯‘ mylg Z¡i‡Y MwZkxj| Av‡iv awi, GB MwZi cÖviw¤¢K kZ©vw` nj mgq Mbbvi ïiæ‡Z A_©vr hLb t = 0 ZLb Avw` Ae¯’vb x = 0 Ges Avw`‡eM vx = vxo | Av‡iv awi, t mgq ci e¯‘wUi Ae¯’vb x = x Ges Ges †eM vx = vx| ‡h‡nZz †h †Kvb gyn‡~ Z©i mg‡qi mv‡c‡ÿ †Kvb KYvi †eM e„w×i nvi‡K Z¡iY e‡j| dv x dt  dv x  ax dt

myZivs, ax 

hLb t = 0 ZLb vx = vxo Ges x = xo Avevi, hLb t = t ZLb vx = vx Ges x = x GB mxgvi g‡a¨ mgxKiY‡K mgvKjb K‡i cvB,

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02| ‰iwLK MwZ (Linear Motion)

vx

3

t

 dv

x

v xo

 ax  aª “eK

 a x  dt 0

 v x 

vx

 ax t 

v x0

t o

 v x  v xo  a x t  0   v x  v xo  a x t mgxKiYwU cÖwZcv`b Kiv nj| 1 2

cÖwZcv`b:

(L) x  x o  ( v xo  v x )t

g‡bKwi, X Aÿ eivei GKwU e¯‘ mylg Z¡i‡Y MwZkxj| Av‡iv awi, GB MwZi cÖviw¤¢K kZ©vw` nj mgq Mbbvi ïiæ‡Z A_©vr hLb t = 0 ZLb Avw` Ae¯’vb x = x0 Ges Avw`‡eM vx = vxo | Av‡iv awi, t mgq ci e¯‘wUi Ae¯’vb x Ges Ges †kl‡eM vx | Mo‡e‡Mi msÁv n‡Z Avgiv Rvwb, ÿz`ªvwZÿz`ª mgq e¨eav‡b †eM I mgq e¨eav‡bi ¸bd‡ji mgwó wb‡q Zv‡K †gvU mgq w`‡q fvM Ki‡j H mg‡qi Mo‡eM e‡j| myZivs KYvwUi t mgq ci Mo‡eM v x n‡j, 1t v x   v x dt t0 dx  1 t dx   v x   dt  vx    dt  t 0 dt   x  x o  v x t ... ... ... ... (1)

mymg Z¡i‡Y Pjgvb e¯‘wUi †ÿ‡Î †eM v x mg‡qi mv‡_ mymgfv‡e cwiewZ©Z nq e‡j †h †Kvb mgq e¨eav‡b Zvi Mo gvb H 1 2

mgq e¨eav‡bi ïiæ I †k‡li †e‡Mi gvb؇qi mgwói A‡a©K| A_©vr v x  ( v xo  v x ), v x Gi GB gvb (1) bs 1 2

mgxKi‡Y ewm‡q cvB, x  x o  ( v xo  v x ) t 1 2

(M) x  xo  vxo t  ax t 2

1  x  x o  ( v xo  v x ) t mgxKiYwU cÖwZcv`b Kiv nj| 2

cÖwZcv`b:

g‡bKwi, X Aÿ eivei GKwU e¯‘ ax mylg Z¡i‡Y MwZkxj| Av‡iv awi, GB MwZi cÖviw¤¢K kZ©vw` nj mgq Mbbvi ïiæ‡Z A_©vr hLb t = 0 ZLb Avw` Ae¯’vb x = xo Ges Avw`‡eM vx = vxo | Av‡iv awi, t mgq ci e¯‘wUi Ae¯’vb x = x Ges Ges †eM vx = vx| ‡h‡nZz †h †Kvb gyn‡~ Z©i mg‡qi mv‡c‡ÿ †Kvb KYvi †eM e„w×i nvi‡K Z¡iY e‡j| dvx dt  dvx  ax dt hLb t = 0 ZLb vx = vxo Ges x = xo Avevi, hLb t = t ZLb vx = vx Ges x = x GB

myZivs, ax 

mxgvi g‡a¨ mgxKiY‡K mgvKjb K‡i cvB, vx

 dv

v xo

t x

 ax  aª æeK 

 a x  dt 0

 v x vxxo  ax t o v

t

 vx  vxo  ax t  0

 vx  vxo  ax t ... ... ... ... ... (1)

(1) mgxKi‡Y vx 

dx ewm‡q cvB, dt

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†h †Kvb gyû‡Z© e¯‘i miY e„w×i nvi‡K †eM e‡j| D³

dx  v xo  a x t dt

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02| ‰iwLK MwZ (Linear Motion)

4

 dx  vxo dt  a x t dt t t x   dx  v xo  dt  a x  t dt o o xo t

t2   x   v xo t   ax    2 o 1  x  xo  vxo t  0   a x t 2  0  2 1  x  xo  vxot  a xt 2 mgxKiYwU cÖwZcv`b Kiv nj| 2 1 2 1 2 ev,  x  xo  vxot  a xt  S  v xot  a xt mgxKiYwUI cÖwZcv`b Kiv nj| w¯’i 2 2 1 2 Ae¯’vb †_‡K mgZ¡i‡Y Pjgvb e¯‘i †ÿ‡Î, v xo  0 Ges a x  aª æeK ,d‡j S  0   aª æeK  t  S  aª æeK t 2 2 2  S t Kv‡RB, w¯’i Ae¯’vb †_‡K mgZ¡i‡Y Pjgvb e¯‘i AwZµvšÍ `yiZ¡ mg‡qi e‡M©i mgvbycvwZK| x xo

t o

(N) v 2x  v 2xo  2ax ( x  x o ) cÖwZcv`b : awi, GKwU e¯‘ X Aÿ eivei ax mylg Z¡i‡Y MwZkxj| GB MwZi cÖviw¤¢K kZ©vw` nj hLb mgq Mbbvi ïiæ‡Z hLb t = 0 ZLb Avw` Ae¯’vb x = xo Ges Avw`‡eM vx = vxo Avevi, t mgq ci KYvwUi Ae¯’vb x Ges †eM vx | †h‡nZz ‡h †Kvb gyn‡~ Z© mg‡qi mv‡c‡ÿ e¯‘i †eM e„w×i nvi‡K Z¡iY e‡j| dv x dt dv dx  ax  x  dx dt dv   dx   ax  x  v x  vx   dx   dt  v x dv x  a x dx hLb x = xo ZLb vx = vxo Ges hLb x = x ZLb vx = vx GB mxgvi g‡a¨ Dc‡iv³ mgxKiY‡K mgvKjb K‡i cvB,

myZivs Z¡ iY a x 

vx

x

 v x dv x  ax  dx

v xo

xo

vx

 v2    x   a xx x x  2  vxo o v 2x  v 2xo  ax ( x  x o ) 2  v 2x  v 2xo  2a x ( x  x o ) 

mgxKiYwU cÖwZcv`b Kiv nj|

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02| ‰iwLK MwZ (Linear Motion)

1 2

(O) s t  v o  a (2t  1) ev, s t  v o 

5

(2t  1) a cÖwZcv`b : 2

g‡bKwi, vo Avw`‡e‡M Ges a mgZ¡i‡Y AB mij‡iLv eivei Pjgvb KYvwUi Avw` Ae¯’vb A we›`y‡Z| (t-1) †m‡K‡Û Ges t †m‡K‡Û hw` Gi Ae¯’vb h_vµ‡g C I B we›`y‡Z nq Z‡e t Zg †m‡K‡Û e¯‘ KYvwU BC `~iZ¡ AwZµg Ki‡e| MwZi m~Îvbyhvqx t †m‡K‡Û e¯‘ KZ…K AwZµvšÍ `~iZ¡ AB 1 AB  v0 t  at 2 ... ... ... ... ... (1) 2 (t-1) †m‡K‡Û e¯‘ KZ…K AwZµvšÍ `~iZ¡ AC 1 AC  v 0 ( t  1)  a ( t  1) 2 ... ... ... ... ... (2) 2

Kv‡RB t Zg †m‡K‡Û e¯‘ KZ…K AwZµvšÍ Í `yiZ¡ st n‡j st = CB  s t  AB  AC

1 1      s t  v 0 t  at 2   v 0 (t  1)  a ( t  1) 2  2 2     1 1  s t  v 0 t  at 2  v 0 ( t  1)  a (t  1) 2 2 2 1 2 1 2  s t  v 0 t  at  v 0 t  v 0  a (t  2t  1) 2 2 1 1 1  s t  v 0 t  at 2  v 0 t  v 0  at 2  at  a 2 2 2 1  s t  v 0  at  a 2 1  s t  v 0  at  a 2 1 (2 t  1) a mgxKiYwU cÖwZcv`b Kiv nj| s t  v o  a (2 t  1) ev, s t  v o  2 2

Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î me©vwaK D”PZv wbY©q : g‡bKwi, †Kvb GKwU e¯‘‡K Lvov Dc‡ii w`‡K v0 †e‡M †Qvov nj| G †ÿ‡Î e¯‘i Dci wµqviZ Z¡iY = g, KviY g c„w_exi †K›`ªvwfgyLx| Kv‡RB h D”PZvq v MwZ‡eM AR©b Ki‡j MwZi mgxKiY †_‡K Avgiv cvB, v 2  v 2o  2gH Kv‡RB e¯‘wU D‡aŸ© DV‡Z _vK‡j Gi †kl †e‡Mi gvb µgkt Kg‡Z _vK‡e| Kv‡RB †h we›`y‡Z wbwÿß e¯‘i †eM k~b¨ H we›`yB e¯‘i MwZ c‡_i m‡e©v”P we›`y wb‡`©k K‡i| myZivs †h we›`y‡Z e¯‘i †kl ‡eM k~b¨, †mB we›`yi D”PZv H n‡j , 0 2  v o2  2gH

v 2o v o2 -B nj e¯‘ KZ©…K AwZµvšÍ Í m‡e©v”P H  Kv‡RB Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î 2g 2g

D”PZv| Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î me©vwaK D”PZvq †cŠQ‡Z mgq wbY©q : g‡bKwi, †Kvb GKwU e¯‘‡K Lvov Dc‡ii w`‡K vo †e‡M †Qvov nj| G †ÿ‡Î e¯‘i Dci wµqviZ Z¡iY = -g, KviY g c„w_exi †K›`ªvwfgyLx| t mgq c‡i †Kvb wbw`©ó D”PZvq e¯‘wUi †eM v n‡j, MwZi mgxKiY n‡Z cvB, v = vo  gt Kv‡RB †`Lv hv‡”Q †h, Dc‡ii w`‡K wbwÿß e¯‘i †eM ax‡i ax‡i Kg‡Z _v‡K| †Kvb wbw`©ó D”PZvq hLb gt Gi gvb vo Gi mgvb ZLb †kl †eM v = 0 n‡e| A_©vr e¯‘wU Avi Dc‡i DV‡Z cvi‡e bv| GB D”PZvB m‡Ÿ©v”P D”PZv| awi, m‡e©v”P D”PZvq

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02| ‰iwLK MwZ (Linear Motion)

MgbKvj t, myZiv&s 0 = vo gt  t 

6

vo v Kv‡RB Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î o -B nj e¯‘ KZ©…K AwZµvšÍ g g

m‡e©v”P D”PZvq †cŠQ‡bvi mgq| Lvov Dc‡ii w`‡K wbwÿß e¯‘i †ÿ‡Î DÌv‡bi, cZ‡bi, DÌvb cZ‡bi mgq wbY©q : g‡bKwi, †Kvb GKwU e¯‘‡K Lvov Dc‡ii w`‡K vo †e‡M †Qvov nj| G †ÿ‡Î e¯‘i Dci wµqviZ Z¡iY = g, KviY g c„w_exi †K›`ªvwfgyLx| t mgq c‡i h D”PZvq e¯‘wUi †eM v n‡j MwZi mgxKiY n‡Z cvB, v = vo – gt ... ... ... ... (1) Ges v 2  v 2o  2gH ... ... ... (2)

myZivs †`Lv hv‡”Q †h, Dc‡ii w`‡K wbwÿß e¯‘i †eM ax‡i ax‡i Kg‡Z _v‡K| GB †eM GK mgq 0 (k~b¨) n‡e| A_©vr e¯‘wU Avi Dc‡i DV‡e bv| d‡j e¯‘wU †h D”PZvq D‡V ZvB-B e¯‘ KZ©…K AwZµvšÍ me©vwaK D”PZv H| awi GB D”PZvq †h‡Z mgq jv‡M t1 myZivs (1) bs mgxKiY n‡Z cvB, 0 = vo – gt1 v  t1  o ... ... ... ... ... (3) g v me©vwaK D”PZvq DÌvb Kvj t 1  o , Avevi (2) mgxKiY n‡Z cvB, g 2 0  v o  2gH

v o2 ... ... ... ... (4) H  2g

me©vwaK D”PZvq e¯‘i †kl‡eM k~b¨| Kv‡RB m‡e©v”P D”PZv n‡Z f~wg‡Z

covi mgq Gi Avw`‡eM n‡e k~b¨ Ges ïay AwfKl©R Z¡i‡Y e¯‘wU wb‡P co‡Z _vK‡e| m‡e©v”P we›`y †_‡K f~wg‡Z wd‡i Avm‡Z t2 mgq jvM‡j, MwZi mgxKiY †_‡K cvB, 1 H  0  t 2  gt 22 2 v2 1  gt 22  H  0 2g 2

v o2 g2 v  t 2  o ... ... ... ... (5) g DÌvb cZ‡bi †gvU mgq T n‡j T = t1+t2 v v T o  o g g 2v  T  o BnvB DÌvb cZ‡bi †gvU mgq| g  t 22 

Lvov Dc‡ii w`‡K wbwÿß e¯‘ fywg‡Z wd‡i Avmvi mgq wbY©q : g‡bKwi, †Kvb GKwU e¯‘‡K Lvov Dc‡ii w`‡K vo Avw`‡e‡M †Qvov nj| G †ÿ‡Î e¯‘i Dci wµqviZ Z¡iY = g, KviY g c„w_exi †K›`ªvwfgyLx| d‡j e¯‘wUi †eM ax‡i ax‡i Kg‡Z _vK‡e Ges GKwU wbw`©ó D”PZvq †eM k~b¨ n‡e A_©vr e¯‘wU Avi Dc‡i DV‡ebv| Kv‡RB GB D”PZvq e¯‘ K©Z…K AwZµvšÍ Í m‡e©v”P D”PZv| awi m‡e©v”P D”PZv H, d‡j MwZi mgxKiY †_‡K cvB, v 2  v 2o  2gH  0  v o2  2gH

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02| ‰iwLK MwZ (Linear Motion)

H 

7

2 o

v ... ... ... (1) 2g

Avevi m‡e©v”P D”PZv †_‡K fywg‡Z cÖZ¨veZ©bKv‡j e¯‘i Avw`‡eM k~b¨ n‡e Ges ïay gvÎ AwfKl©R Z¡i‡Y e¯‘wU wb‡P co‡Z _vK‡e| m‡e©v”P we›`y †_‡K fywg‡Z wd‡i Avm‡Z e¯‘i t mgq jvM‡j MwZi mgxKiY †_‡K cvB, 1 H  0  t  gt 2 2 2 v 1  o  gt 2 2g 2 2

v   t   o   g  v  t  o ... ... ... ... (2) BnvB m‡e©v”P D”PZv n‡Z cZbKvj| g awi, fywg‡Z wd‡i Avm‡Z †h mgq jv‡M †mB mg‡qi †k‡l e¯‘i †eM = v myZivs v = 0 + gt v  v  g o g  v  v o myZivs f~wg †_‡K e¯‘‡K †h †e‡M Dc‡ii 2

w`‡K wb‡ÿc Kiv nq, e¯‘wU wd‡i G‡m †mB †e‡M f~wg‡Z AvNvZ K‡i| cošÍ e¯‘i m~Î eY©bv (Laws of falling bodies) : evavnxb fv‡e cošÍ e¯‘ wb‡¤§v³ wZbwU m~Î †g‡b P‡j| 1589 wLª÷v‡ã weÁvbx M¨vwjwjI m~Î wZbwU Avwe®‹vi K‡ib t 1g m~Ît e¯‘ mgvb mg‡q mgvb c_ AwZµg K‡i| 2q m~Ît wbw`©ó mg‡q e¯‘ †h †eM jvf K‡i Zv H mg‡qi mgvbycvwZK| t mg‡q v †eM jvf Ki‡j, m~Îvbyhvqx †eM n‡e, vt

3q m~Ît wbw`©ó mg‡q e¯‘ KZ…K AwZµvšÍ `~iZ¡ H mg‡qi e‡M©i mgvbycvwZK| t mg‡q AwZµvšÍ `yiZ¡ h n‡j, m~Îvbyhvqx D”PZv n‡e, ht 2 ¯^b©gy`ªv I cvjK cixÿv: hš¿cvwZt (K) j¤^v GKwU k³, †gvUv I duvcv `yBgyL †Lvjv KvPbj B| (L) GKwU Uzwc C (M) GKwU ÷c KK© S (N) GKwU cvjK | cixÿvi weeiY: KvPb‡ji GKcÖv‡šÍGKwU Uzwc C Ges Aci cÖv‡šÍGKwU ÷c KK© S _v‡K| Uzwc Ly‡j GKwU ¯^b©gy`ªv G Ges GKwU cvjK F b‡ji g‡a¨ XyKv‡bv nq| ócK‡K©i Pvwe Ly‡j cv‡¤úi mvnv‡h¨ bjwU‡K evqyc~b© ev evqyk~b¨ Kiv hvq| bjwU‡K nVvr Dwë‡q gy`ªv I cvjK‡K wb‡Piw`‡K co‡Z †`Iqv nq| cixÿvq †`Lv hvq †h (1) evqyc~b© Ae¯’vq gy`ªvwU cvj‡Ki Av‡M wb‡Pi cÖv‡šÍc‡o| (2) evqyk~b¨ Ae¯’vq gy`ªv I cvjK GKB mv‡_ wb‡Pi cÖv‡šÍ c Í ‡o| djvdjt evqyk~b¨ ¯’v‡b mKj e¯‘ mgvb mg‡q mgvb c_ AwZµg K‡i|

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02| ‰iwLK MwZ (Linear Motion)

8 1 2

mgZ¡iY MwZi †ÿ‡Î †eM ebvg mgq (v  t)†jLwPÎ AsKb Ges †jLwPÎ n‡Z s  v o t  at 2 mgxKiYwU cÖwZcv`b: mgZ¡i‡Y MwZkxj †Kvb e¯‘i †ÿ‡Î X A‡ÿi w`‡K mgq t Ges Y A‡ÿi w`‡K †eM v wb‡q v ebvg t ‡jL wPÎ AsKb Kiv nj| GwU Y Aÿ‡K †Q`Kvix GKwU mij †iLv nq hv, v = vo+at mgxKiY †g‡b P‡j| GB ‡jLwPÎ †_‡K t mg‡q e¯‘i AwZµvšÍ `~iZ¡ s wbb©q Kiv hvq| AB ‡iLvi Dci †h †Kvb we›`y P †bqv nq| P †_‡K X A‡ÿi Dci PQ j¤^ Uvbv nq| Zvn‡j OQ = t mg‡q AwZµvšÍ `~iZ¡ s n‡e AOQP ‡ÿ‡Îi †ÿÎdj| aiv hvK, KYvwUi mgZ¡iY a Ges Avw`‡eM, vo = AO AwZµvšÍ mgq, t = OQ Ges t mg‡q AwZµvšÍ `~iZ¡ , s = AOQP ‡ÿ‡Îi †ÿÎdj| = AOQR ‡ÿ‡Îi †ÿÎdj  ARP ‡ÿ‡Îi †ÿÎdj| = AO×OQ +

1 2

×AR×PR

s = AO×OQ + 12 ×OQ×PR wKš‘ AB ‡iLvi Xvj n‡”Q KYvwUi Z¡iY a, PR a  AR PR = a×AR = a×OQ s = AO×OQ + 12 ×OQ×a×OQ

[∵ AR = OQ ]

 s = AO×OQ + 12 ×a×OQ2 1  s  v o t  at 2 mgxKiYwU cÖwZcv`b Kiv nj|  2

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

2| ‰iwLK MwZ (Linear Motion) 1| GKwU e›`y‡Ki ¸wj †Kvb †`Iqv‡ji g‡a¨ 0.04m cÖ‡ek Kivi ci A‡a©K †eM nvivq | MywjwU †`Iqv‡ji g‡a¨ Avi KZUzKz cÖ‡ek Ki‡e? g‡b Kwi, jÿ¨¯’‡j cÖ‡e‡ki gyn‡~ Z© ¸wji Avw`‡eM = u Ges ¸wjwU AviI x wgUvi `~iZ¡ cÖ‡ek Ki‡e|

 0.04 m cÖ‡ek Kivi ci †eM n‡e =

u 2

Ges †kl †eM n‡e 0 (k~b¨)| Avgiv Rvwb, cÖ_g As‡ki Rb¨ 2

u 2    u  2a(0.04) 2 u2  0.08a  u 2  4 3u 2 3u 2 a  ..... .......... (1) 4  0.08 0.32

wØZxq As‡ki Rb¨, 2

u 0     2ax  2

3| 20ms-1 †e‡M MwZkxj GKwU e¯‘i †eM cÖwZ †m‡K‡Û 3ms-1 nv‡i n«vm cvq| †_‡g hvIqvi Av‡M e¯‘wU KZ `~iZ¡ AwZµg Ki‡e?

Avgiv Rvwb, v2 = u2 – 2as GLv‡b, ev, 0 = 202 – 2(3)s Avw`‡eM, u = 20 ms-1 ev, 6s = 400 g›`b, a = 3 ms-2 400 ‡kl‡eM, v = 0 ev, s  _vgvi Av‡M e¯‘wU KZ…K 6  s  66.7 m (Ans.) AwZµvšÍ `~iZ,¡ s = ?

4| Dc‡ii w`‡K wbwÿß GKwU ej †Uwj‡dvb Zvi‡K 0.70ms-1 `ªæwZ‡Z AvNvr K‡i| †Qvovi ¯’vb †_‡K ZviwUi D”PZv 5.1m n‡j ejwUi Avw` `ªæwZ KZ wQj? Avgiv Rvwb, GLv‡b, v 2  u 2  2gh D”PZv, h =5.1m 2 2 g = 9.8 ms-2  (0.7)  u  2  9.8  5.1 ‡kl †eM, v =0.70ms-1  u 2  (0.7) 2  2  9.8  5.1 Avw` `ªæwZ, u =?  u 2  0.49  99.96  u 2  100.45 u  100.45  10.02 ms-1 (Ans.)

2

3u 2 u ev , 0     2  x 0.32 2 6u 2 x u 2 ev ,  0.32 4  x

0.32  0.0133 m (Ans.) 6 4

2| 50 wgUvi DPuy †_‡K GKwU e¯‘ f~wg‡Z cwZZ nq| (K) fywg‡Z †cŠuQ‡Z Gi KZ mgq jvM‡e?

(L) fywg‡Z †cŠuQevi c~e© gyn‡~ Z© Gi †eM KZ n‡e?

1 2 gt 2 1  50  0  9.8  t 2 2  50  4.9 t 2 50  t2  4. 9 50 t 4.9 (K) h  ut 

GLv‡b, D”PZv, h =50m Avw`‡eM, u = 0 g = 9.8 ms-2 (K) mgq, t = KZ? (L) ‡kl †eM, v = KZ ?

t = 3.19 s (Ans.) Avevi, (L) v = u + gt ⇒ v = 0 + 9.8 × 3.19  v = 31.26 ms-1 (Ans.)

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5| GKwU †Uªb 3ms-2 mgZ¡i‡b Pj‡Q Ges Avw`‡eM 10m/s †UªbwU hLb 60m c_ AwZµg Ki‡e ZLb Gi †eM KZ n‡e| Avgiv Rvwb, GLv‡b, Z¡iY, a = 3ms-2 2 2 v  u  2as Avw`‡eM, u = 10 ms-1  v 2  10 2  2  3  60 miY, s = 60m  v 2  100  360 ‡kl‡eM, v = ?  v 2  460  v  460  21.447  21.45ms 1 ( Ans ) 6| GKwU e¯‘‡K 98 ms-1 †e‡M Lvov Dc‡ii w`‡K wb‡ÿc Kiv n‡j †`LvI †h, 3 Sec I 17 Sec mg‡q e¯‘i †eMØq mgvb wKš‘ w`K wecixZ gyLx| Avgiv Rvwb, GLv‡b, 3 †mt c‡i †eM Avw`‡eM, u = 98 ms-1 v1 = u gt1 mgq, t1 = 3S ev, v1 = 989.8×3 mgq, t2 = 17S ev, v1 = 9829.4 †kl‡eM, v1 =?  v1 = 68.6ms-1 †kl‡eM, v2 =?

Avevi, 17 †mt c‡i †eM v2 = u gt2 ev, v2 = 98 9.8×17 ev, v2 = 98 166.6  v2 = 68.6 ms-1  3 †mt I 17 †mt c‡i †eR Øq mgvb I wecixZ (cÖgvwYZ)

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2| ‰iwLK MwZ (Linear Motion)

1 7| s  t 3  3t m~Îvbymv‡i GKwU e¯‘ mij †iLvq Pj‡Q| 3 2 †m‡KÛ ci Gi †eM KZ n‡e?

Avgiv Rvwb,

v

GLv‡b, mgq, t = 2 Sec ‡eM, v =?

d 1   v   t 3  3t  dt  3 

1  3t 2  3 3  v  t2  3  v  2 2  3 [t Gi gvb ewm‡q]  v  7 GKK („Ans.) v

8| 54 kmh1 †e‡M PjšÍ GKwU †ij Mvwo‡Z †÷mb †_‡K wKQy `y‡i 0.75ms-2 g›`b m„wóKvix †eªK ‡`Iqvq MvwowU †÷m‡b G‡m †_‡g †Mj| †÷mb ‡_‡K KZ `~‡i †eªK †`Iqv n‡qwQj Ges MvwowU _vg‡Z KZ mgq †j‡MwQj? Avgiv Rvwb, v2 = u22as GLv‡b,

 0  15  2  0.75  s 15 15 m s 2  0.75  s  150 m (Ans.) Avevi, v = u  at

 0  15  0.75  t 15 s t 0.75  t  20s (Ans.)

Avw`‡eM, u = 54 kmh-1



54  1000 1 ms 3600

=15ms-1 g›`b, a = 0.75ms-2 †kl‡eM, v = 0 mgq, t = ? miY, s=?

9| GKwU e¯‘ w¯’i Ae¯’vb n‡Z hvÎv ïiæ K‡i cÖ_g †m‡K‡Û 1m `~iZ¡ AwZµg K‡i| cieZ©x 1m `~iZ¡ AwZµg Ki‡Z KZ mgq jvM‡e| GLv‡b, Avgiv Rvwb, Avw`‡eM, u = 0 1 s1  ut 1  at12 mgq, t1 = 1s 2 miY, s1 =1m 1 Z¡iY, a=?  1  0  a (1) 2

2 a 1 2  a  2ms 2

GLb cÖ_g †_‡K s2 = (1m+1) =2m `~iZ¡ AwZµg Ki‡Z mgq jv‡M = t2

s 2  ut 2 

1  2  0   2  t 22 2 2  t2  2  t 2  2  1.414s

ds dt

2

2

1 2 at 2 2

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‡k‡li 1m `~iZ¡ AwZµg Ki‡Z mgq jv‡M, t  t 2  t1  (1.414  1) s  0.414s (Ans.) 10| GKwU ‡Uªb w¯’i Ae¯’vb n‡Z 10ms-2 Z¡i‡Y Pj‡Z Avi¤¢ Kij| GKB mgq GKwU Mvwo 100ms-1 mg‡e‡M †Uª‡bi mgvšÍiv‡j Pjv ïiæ Kij| †Uªb MvwowU‡K KLb wcQ‡b †dj‡e? g‡b Kwi, t mgq ci †Uªb MvwowU‡K wcQ‡b †d‡j P‡j hv‡e, t mgq †Ub KZ…K AwZµvšÍ `~iZ¡, GLv‡b, 1 x  0  at 2 Mvwoi mg‡eM, V = 100ms-1 2 ‡Uª‡bi Z¡iY, a = 10ms-2 1  x   10  t 2 mgq, t = ?

2  x  5t 2 ... ... ... (1)

t mg‡q Mvwo KZ…K AwZµvšÍ `~iZ¡, x   Vt

 x   100 t ... ... ... (2) kZ©g‡Z †Uªb hLb MvwowU‡K AwZµg Ki‡e ZLb x  x  n‡e|

5t 2  100 t 100 t  t  20s (Ans.) 5 11| w¯’ive¯’v †_‡K Pj‡Z Avi¤¢ K‡i 625m `~iZ¡ AwZµg Ki‡j GKwU e¯‘i †eM 125ms-1 nj| Z¡iY wbY©q Ki| Avgiv Rvwb,

v 2  u 2  2as  125 2  0  2  a  625 125 2 a ms 2 2  625  a  12.5ms 2 (Ans.)

GLv‡b, Avw`‡eM, u = 0 AwZµvšÍ `~iZ¡ , s = 625m ‡kl †eM, v = 125 ms-1 Z¡iY, a =?

12| 64m DuPz `vjv‡bi Qv` †_‡K 5kg f‡ii GKwU cv_i †Q‡o w`‡j f~wg‡Z †cuŠQv‡Z Gi KZ mgq jvM‡e?

Avgiv Rvwb, GLv‡b,

1 h  ut  gt 2 2 1  64  0   9.8  t 2 2  64  4.9t 2 64 t  t  3.61 s.

4.9

Avw`‡eM, u = 0 AwZµvšÍ `~iZ¡ , h = 64m fi, m= 5kg mgq, t =?

(Ans.)

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2| ‰iwLK MwZ (Linear Motion)

13| w¯’i Ae¯’vb n‡Z hvÎv Avi¤¢ K‡i GKwU e¯‘ cÖ_g †m‡K‡Û 2m `~iZ¡ AwZµg K‡i| cieZ©x 2m `~iZ¡ AwZµg Ki‡Z e¯‘wUi KZ mgq jvM‡e| Avgiv Rvwb,

1 s1  ut1  at 12 2 1  2  0  a(1) 2 2

a 2  a  4ms 2 2

GLv‡b, Avw`‡eM, u = 0 mgq, t1 = 1s miY, s1 =1m Z¡iY, a =?

GLb cÖ_g †_‡K s2 = (2m+2m) = 4m `~iZ¡ AwZµg Ki‡Z mgq jv‡M = t2

1 2 at 2 2 1  4  0   4  t 22 2 2  t2  2

s 2  ut 2 

 t 2  2  1.414s ‡k‡li 2m `~iZ¡ AwZµg Ki‡Z mgq jv‡M, t  t 2  t 1  (1.414  1) s  0.414s (Ans.)

3

14| GKwU e¯‘ cÖ_g `yB †m‡K‡Û 30m I cieZ©x Pvi †m‡K‡Û 150m ‡Mj| Z¡iY AcwiewZ©Z _vK‡j e¯‘wU Gi ci GK †m‡K‡Û KZUv c_ AwZµg Ki‡e?

GLv‡b, Avgiv Rvwb, `~iZ¡, s1 = 30m

1 s1  ut1  at 12 2

1  30  u  2  a  2 2 2 u  a  15..........(1)

mgq, t1 = 2s miY, s2 = (30+150) =180m

miY, s7 =?

cÖ_g †_‡K t2= (2+4)= 6 †m‡K‡Û e¯‘wU hvq s2=(30+150)m=180m

1  s2  ut2  at 22 2 1  180  u  6  a  6 2 2 u  3a  30..........(2) u  a  15..........(1) we‡qvM K‡i, 2a= 15  a  7.5ms 2 GLb (1) bs mgxKi‡Y a Gi gvb ewm‡q,

u  7.5  15 u  7.5ms 1 6 ‡m‡K‡Ûi c‡ii †m‡KÛ A_©vr 7g †m‡K‡Û AwZµvšÍ `~iZ¡,

a st  u  (2t  1) 2 7.5  s7  7.5  ( 2  7  1) 2 7.5  s7  7.5  13 2  s7  7.5  48.75

 s7  56.25m ( Ans)

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cÖkœt wb¤œwjwLZ mgxKiY¸wj mivmwi †f±i iƒ‡c cÖwZcv`b Ki|        ( K) v  v  at ( L) r  r  1 (v  v )t 0 0 0 2    ( K) v  v  at cÖwZcv`b|

    ( M) r  r  v t  1 at 2 0 0 2

   ( N) v 2  v 2  2a (r  r ) 0

0

0

awi, GKwU e¯‘ a mgZ¡i‡Y †Kvb Z‡j MwZkxj| GB MwZi cÖviw¤¢K kZ©vw` nj hLb ti  o ZLb Avw` Ae¯’vb †f±i       r  r0 Ges Avw`‡eM v  v0 Avevi, t f  t mg‡q e¯‘wUi Ae¯’vb †f±i r Ges †eM v | ‡h †Kvb gyn‡~ Z© mg‡qi mv‡c‡ÿ e¯‘i †eM e„w×i nvi‡K Z¡iY e‡j|   dv Z¡ iY a  dt   ev, dv  adt     hLb t = 0 ZLb v  v0 Ges hLb t = t ZLb v  v GB mxgvi g‡a¨ Dc‡iv³ mgxKiY‡K mgvKjb K‡i cvB,  v

 t d v   a  dt

 v0

0

    v  vvo  at  0t     v  v0  a (t  o)     v  v0  a t

    v  v0  at

    (L) r  r  1 (v  v )t cÖwZcv`b| 0 0 2

awi, GKwU e¯‘ a mgZ¡i‡Y †Kvb Z‡j MwZkxj| GB MwZi cÖviw¤¢K kZ©vw` nj hLb ti  0 ZLb Avw` Ae¯’vb †f±i       r  r0 Ges Avw`‡eM v  v0 Avevi, t f  t mg‡q e¯‘wUi Ae¯’vb †f±i r Ges †eM v |  myZivs KYvwUi t mgq e¨eav‡b Mo†eM V n‡j  1t  V   v dt t0  1 t dr dt V   t 0 dt

  dr [ v  ] dt



 1r   V   dr t r0  1    V  [r ]rr0 t  1    V  (r  r0 ) t     r  r0  V t      r  r0  12 (v  v0 )t      r  r0  12 (v  v0 )t

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V 

1 2

  ( v  v0 )



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3| wØgvwÎK MwZ (Motion In Two Dimention)

2

   2 (M) r  r  v t  1 a t cÖwZcv`b | 0 0 2

awi, GKwU e¯‘ a mgZ¡i‡Y †Kvb Z‡j MwZkxj| GB MwZi cÖviw¤¢K kZ©vw` nj hLb ti  o ZLb Avw` Ae¯’vb †f±i       r  r0 Ges Avw`‡eM v  v0 Avevi, t f  t mg‡q e¯‘wUi Ae¯’vb †f±i r Ges †eM v | ‡h †Kvb gyn‡~ Z© mg‡qi mv‡c‡ÿ e¯‘i †eM e„w×i nvi‡K Z¡iY e‡j|   dv Z¡ iY a  dt

  ev, dv  adt     hLb ti = 0 ZLb v  v0 Ges hLb tf = t ZLb v  v GB mxgvi g‡a¨ Dc‡iv³ mgxKiY‡K mgvKjb K‡i cvB,  v

 t d v   a  dt

 v0

0

 v  t  v  vo  at  0

    v  v0  a (t  o)     v  v0  a t     v  v0  a t Avevi, †h‡nZz †Kvb gyn‡~ Z© mg‡qi mv‡c‡ÿ e¯‘i Ae¯’vb †f±i e„w×i nvi‡K †eM e‡j, ZvB msÁvbymv‡i,   dr ewm‡q cvB, v dt





 dr    v0  a t dt     dr  v0 dt  atdt





hLb ti = 0 ZLb r  r0 Ges hLb tf = t ZLb r  r GB mxgvi g‡a¨ Dc‡iv³ mgxKiY‡K mgvKjb K‡i cvB,  r

  t t   d r  v 0  dt  a  tdt  r0

0

0

    [r ]rr0  v0 [t ]t0  a

 t2 2

t 0

    r  r0  v0 (t  0)  a (t 2  0) 2

     r  r0  v 0 t  12 a t 2      r  r0  v 0 t  12 a t 2 (N) v

2

    v 2  2 a ( r  r ) cÖwZcv`b | 0

0

awi, GKwU e¯‘ a mgZ¡i‡Y †Kvb Z‡j MwZkxj| GB MwZi cÖviw¤¢K kZ©vw` nj hLb ti  o ZLb Avw` Ae¯’vb †f±i       r  r0 Ges Avw`‡eM v  v0 Avevi, t f  t mg‡q e¯‘wUi Ae¯’vb †f±i r Ges †eM v | ‡h †Kvb gyn‡~ Z© mg‡qi mv‡c‡ÿ e¯‘i †eM e„w×i nvi‡K Z¡iY e‡j|   dv Z¡ iY a  dt   ev, dv  adt     hLb t = 0 ZLb v  v0 Ges hLb t = t ZLb v  v GB mxgvi g‡a¨ Dc‡iv³ mgxKiY‡K mgvKjb K‡i cvB,  v

 t  dv  a  dt

 v0

0

    v  vvo  at  0t

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    v  v0  a (t  o)     v  v0  a t

3| wØgvwÎK MwZ (Motion In Two Dimention)

3

    v  v0  at Dfq cÿ‡K GB mgxKiY w`‡q ¸Y K‡i cvB,      v .v  (v0  at ).(v0  at )          v .v  v0 .v0  v0 .at  v0 .at  a.at 2        v.v  v0 .v0  2v0 .at  a.at 2      1  v.v  v0 .v0  2a.(v0t  at 2 ) 2          12    v t a t r  v .v  v0 .v0  2a.r  r0    (  r0 )  0  2     2 2  v  v0  2a.r  r0 

cÖkœt cÖvm Kv‡K e‡j?

DËit †Kvb e¯‘‡K Abyfywg‡Ki mv‡_ wZh©Kfv‡e †Kvb ¯’v‡b wb‡ÿc Kiv n‡j Zv‡K cÖvm e‡j| wZh©Kfv‡e wbwÿß wXj, ey‡j‡Ui MwZ BZ¨vw` cÖvm MwZi D`vniY| cÖÖkœt Abyfywg‡Ki mv‡_ wZh©Kfv‡e wbwÿß cÖv‡mi MwZc‡_i mgxKiY wbb©q Ki Ges †`LvI †h, GB MwZc_ Awae„ËvKvi| DËit g‡bKwi, evqyga¨w¯’Z O we›`y n‡Z GKwU cÖvm‡K wb‡ÿc Kiv nj| wb‡ÿc †eM ev Avw`‡eM = vo wb‡ÿc ‡KvY =  g wb‡Pi w`‡K wµqvkxj| AZGe ay = -g; ax = 0; wb‡ÿc we›`y I g~j we›`y GKB nIqvq xo = yo = 0  Avw`‡e‡Mi Abyf~wgK Dcvsk = voCoso Ges Avw`‡e‡Mi Dj¤^ Dcvsk = voSino X Aÿ eivei MwZi cwieZ©b D³ Aÿ eivei Z¡i‡Yi Dci wbf©ikxj| Y Aÿ eivei MwZi cwieZ©b D³ Aÿ eivei Z¡i‡Yi Dci wbf©ikxj| G `ywU Aÿ eivei MwZi cwieZ©b Awbf©ikxj| awi t mg‡q cÖvmwU P(x,y) Ae¯’v‡b _v‡K| ZLb Gi †eM = v Abyf~wg‡Ki w`‡K Z¡iY, ax= 0 Abyf~wg‡Ki w`‡K miY = x x = voCoso t +

1 2

axt2

ev, x = voCoso t + 0 [ax= 0] ev, x = voCoso t x t  .......................(1) vo Cos o

Dj¤^ w`‡K Z¡iY ay=g; Dj¤^ w`‡K miY y; Abyiƒcfv‡e y=voSinot 12 gt2

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3| wØgvwÎK MwZ (Motion In Two Dimention)

ev, y  v Sin  o

o



x



v Cos  o

o

1 2



x

g 

 v Cos   o

o

   

4

2

[t Gi gvb ewm‡q]

  2 g  x ev, y  tan  o x   2 2  2vo Cos  o 

  g θ b awi , aª æeK tan Ges   c o   2 2 2voCos θo  

2

 y  bx  cx

Dc‡iv³ mgxKiYwU GKwU Awae„‡Ëi mgxKiY|  cÖv‡mi MwZc_ GKwU Awae„Ë (c¨viv‡evjv)| cÖkœt cÖgvY Ki, evqynxb Ae¯’vq f~wg n‡Z D”PZvq Aew¯’Z †h †Kvb Ae¯’vb n‡Z Abyf~wgK Awfgy‡L wbwÿß e¯‘i MwZc_ GKwU Awae„Ë| g‡bKwi, k~‡b¨ Aew¯’Z O we›`y n‡Z vo †e‡M f~wgi mgvšÍiv‡j GKwU e¯‘KYv wbwÿß nj| e¯‘ KYvwU g Gi cÖfv‡e bx‡P co‡e| awi cÖ‡ÿcb Z‡j Abyf~wgK OX †iLv X Aÿ Ges OY †iLv Y Aÿ| awi t mgq c‡i e¯‘ KYvwU MwZ c‡_i P(x,y) we›`y‡Z gyn‡~ Zi Rb¨ Ae¯’vb Ki‡e| g bx‡Pi w`‡K wµqvkxj| AZGe ay = g; ax= 0 ; Avw`‡e‡Mi Abyf~wgK Dcvsk = vo Ges Avw`‡e‡Mi Dj¤^ Dcvsk = 0 tmg‡q AwfKl©RZ¡iYnxb Abyf~wgK miY x = vot  x 2  v o2 t 2 ...

...

... ...

... ...

(1 )

tmg‡q Dj¤^ miY y = 0.t + 12 gt2 y=

1 2

gt2...

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

(2)

(1) ‡K (2) Øviv fvM K‡i cvB x2 v 2t 2  1o 2 y 2 gt



x 2 2 v 2o  y g

 2v 2   x 2   o  y  g   x 2  4ay

  2vo2 awi  4a  aª æeK  ,  g  

Dc‡iv³ mgxKiYwU GKwU Awae„‡Ëi mgxKiY| ZvB wbwÿß e¯‘i MwZc_ GKwU Awae„Ë (c¨viv‡evjv)| cÖkœt Abyfywg‡Ki mv‡_ wZh©K fv‡e wbwÿß e¯‘i ‡ÿ‡Î (K) m‡e©v”P D”PZvq †cŠQ‡Z mgq (L) m‡e©v”P D”PZv (M) wePiY Kvj (N) cvjøv (O) me©vwaK cvjøv wbb©q Ki| g‡b Kwi, evqyga¨w¯’Z O we›`y n‡Z GKwU cÖvm‡K vo †e‡M o †Kv‡Y wZh©Kfv‡e wb‡ÿc Kiv nj| cÖvmwU t mg‡q m‡e©v”P D”PZv P(x,y) G Ae¯’vb Ki‡e Ges ZLb Gi †eM n‡e v| (K) m‡e©v”P D”PZvq †cŠQ‡Z mgqt vo †e‡Mi Dj¤^ Dcvsk voSino t mgq c‡i P we›`y‡Z †eM, vy = voSino gt.................(1) P we›`yMvgx m‡e©v”P D”PZvq vy= 0..................................... (2)

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3| wØgvwÎK MwZ (Motion In Two Dimention)

5

(1) bs mgxKi‡Y vy= 0 ewm‡q cvB 0 = voSino gt v Sin o  t o ..................................(3) g

(L) m‡e©v”P D”PZvt g‡bKwi, m‡e©v”P D”PZv = H  H = voSinot  12 gt2

v Sin o 1  vo Sin o    H  vo Sin o  o  2 g  g  g 

H

2 2 vo Sin o  vo Sin o    

g

H 

2g

2

 (3)

bs n‡ Z t Gi gvb ewm‡ q  



v o2Sin 2 o ... ... ... ... ... ... ... (4)  2g

(M) DÇqb (wePiY) Kvj (Time of Flight) t g‡b Kwi wePiY Kvj T A_©vr T mg‡q cÖvmwU mgZ‡j wd‡i Av‡m|  t mg‡q Dj¤^ w`‡K miY y = voSinot  12 gt2GB mgxKi‡Y mgq t = T Ges miY y = 0 ewm‡q cvB, 0 = voSinoT  12 gT2

ev, 12 gT2 = voSinoT 2vo Sinθo ... ... ... ... ... ... (5)  g

T 

(N) cvjøv (Range)t g‡b Kwi cvjøv R A_©vr T mg‡q cÖvmwU Abyfywg‡Ki w`‡K †h `~iZ¡ AwZµg K‡i ZvBB cvjøv R  R = ( voCoso ) × T 2v Sin o  R  voCos o  o g R R 

[(5) bs n‡Z T Gi gvb ewm‡q]

vo2 2Sin o Cos o g

vo2 Sin 2 o ...........................(6) g

(O) me©vwaK cvjøv (Maximum Range) t g‡bKwi me©vwaK cvjøv Rmax| wbw`©ó vo Gi Rb¨, Sin20 Gi gvb me©vwaK n‡j cvjøv n‡e me©vwaK| Sin20 Gi me©vwaK gvb = 1 A_©vr Sin20 = 1 ev, Sin20 = Sin900 ev, 20 = 900  0 = 450 myZivs wb‡ÿc †KvY0 = 450 n‡j cvjøv me©vwaK

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3| wØgvwÎK MwZ (Motion In Two Dimention)

 me©vwaK cvjøv Rmax   Rmax

2 o

v Sin 2  45 g

6

o

vo2 Sin 90 o  g

 Rmax   Rmax 

vo2  1 g

vo2 ... ... ... ... ... (7) g

   cªkœt ˆiwLK †eM I †KŠwbK †e‡Mi msÁv `vI Ges G‡`i g‡a¨ m¤úK© ¯’vcb Ki| ev, v  r ev , v    r cÖgvb Ki |    ev, v    r cÖgvY Ki |

‰iwLK †eM (Linear Velocity)t wbw`©ó w`‡K ˆiwLK c‡_ †Kvb e¯‘ GKK mg‡q †h `yiZ¡ AwZµg K‡i Zv‡K H e¯‘i ‰iwLK †eM e‡j| ˆiwLK †eM‡K v Øviv cÖKvk Kiv nq| wbw`©ó w`‡K e¯‘ t mg‡q d `~iZ¡ AwZµg Ki‡j †eM v 

d n‡e| †eM t

GKwU †f±i ivwk| ˆiwLK †e‡Mi GKK ms-1 ‡KŠwYK †eM (Angular Velocity) t mgq e¨eavb k~‡b¨i KvQvKvwQ n‡j †Kvb we›`y ev Aÿ‡K †K›`ª K‡i e„ËvKvi c‡_ Pjgvb †Kvb e¯‘i mg‡qi mv‡_ †KŠwbK mi‡Yi nvi‡K †KŠwbK †eM e‡j| Ab¨ K_vq e„ËvKvi c‡_ †Kvb e¯‘ GKK mg‡q †h †KŠwbK `~iZ¡ AwZµg K‡i Zv‡K H e¯‘i †KŠwbK †eM e‡j| †KŠwbK †eM‡K  Øviv cÖKvk Kiv nq| wbw`©ó w`‡K e¯‘ t  n‡e| †KŠwbK †e‡Mi GKK rad s-1 t L † KvY Pvc    T-1 Gi gvÎv n‡”Q  mgq e¨vmva©  mgq L  T

mg‡q  ‡KvY Drcbœ Ki‡j †KŠwbK †eM  

m¤úK© (Relation) t g‡bKwi GKwU e¯‘KYv OC= OB = r e¨vmva© wewkó GKwU e„‡Ëi cwiwa eivei‡KŠwbK †e‡M Nyi‡Q| hw` T †m‡K‡Û e¯‘ KYvwU e„‡Ëi cwiwa eivei GKevi Ny‡i Av‡m Z‡e †KŠwbK `~iZ¡  =  †iwWqvb n‡e|  ‡KŠwbK †eM, ω 

ev, T 

2π T

2π ... ... ... ... ...(1) ω

GLb e¯‘ KYvwU hw` e„ËvKvi c‡_ bv Ny‡i H GKB mg‡q mij †iLv eivei PjZ Z‡e T mg‡q e¯‘KYvwU e„ËwUi cwiwai mgvb c_ r `~iZ¡ AwZµg KiZ|  ˆiwLK †eM v  T 

2πr T

2πr ... ... ... ... ...( 2) v

(1) bs I (2) mgxKiYØq n‡Z cvB 2 2 r   v

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3| wØgvwÎK MwZ (Motion In Two Dimention)

7

1 r   v  v = r A_©vr ‰iwLK †eM = †KŠwbK †eM × e„‡Ëi e¨vmva©| 

v = r mgxKi‡Yi ‡f±i iƒc:    g‡b Kwi, u    r ... ... ... (3)     u ‡f±‡ii gvb u  r sin 90  r [  r ]     µm ¸Y‡bi wbqg Abymv‡i,   r ev , u †f±‡ii AwfgyL Ges v †f±‡ii AwfgyL Awfbœ| Avevi v = r| †`Lv hv‡”Q †h,   gvb I w`K we‡ePbvq u I v ‡f±i Awfbœ|    u  v ... ... ... (4)    (3) I (4) n‡Z v    r (cÖgvwYZ)

cÖkœ: †KŠwbK †eM I ˆiwLK †e‡Mi g‡a¨ cv_©K¨ eY©bv Ki| †KŠwbK †eM I ˆiwLK †e‡Mi g‡a¨ cv_©K¨ (Distinction between Angular Velocity and Linear Velocity) µwgK †KŠwbK †eM ˆiwLK †eM †KŠwbK c‡_ e¯‘i †KŠwbK mi‡bi nvi‡K †KŠwbK †eM wbw`©ó w`‡K ‰iwLK c‡_ †Kvb GKwU e¯‘i ¯’vb 1| e‡j| cwieZ©‡bi nvi †K ‰iwLK †eM e‡j| 2| G‡K Øviv v cÖKvk Kiv nq| G‡K Øviv cÖKvk Kiv nq| 3|

Gi mgxKiY  

Gi mgxKiY v 

4| 5|

 t Gi gvÎv mgxKiY [ T-1 ]

s t Gi mgxKiY [ LT-1 ]

Gi GKK †iwWqvb/ †m‡KÛ

Gi GKK wgUvi/ †m‡KÛ

‡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‡_ v2 v mg`ªæwZ‡Z Nyi‡Z _vK‡j Zvi †K›`ªgyLx ej  m | r

†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  m

v2 | r

m fi wewkó GKwU e¯‘ r e¨mv‡a©i e„ËvKvi c‡_ v mg`ªæwZ‡Z Nyi‡Q| (1) ‡`LvI †h, j¤^ Z¡iY a 

j¤^ Z¡i‡Yi ivwkgvjv wbb©q Ki| (3) cÖgvY Ki †h, †K›`ªgyLx ej F  m

v2   2 r ev (2) r

v2  m  2 r ev, (4) e„ËvKvi c‡_ mg`ªæwZ‡Z r

N~b©vqgvb †Kvb e¯‘i Dci wµqvkxj †K›`ªgyLx e‡ji gvb I w`K wbY©©q | aiv hvK, m f‡ii †Kvb e¯‘ r e¨vmv‡a©i e„ËvKvi c‡_ v mg`ªæwZ‡Z Ges  †KŠwbK †e‡M AveZ©biZ Av‡Q| awi AwZ ÿz`ª mgq t e¨eav‡b e¯‘wU A n‡Z B we›`y‡Z A‡m| A we›`y‡Z e¯‘wUi †eM H we›`y‡Z ¯úk©K AC eivei| B we›`y‡Z e¯‘wUi †eM H we›`y‡Z ¯úk©K BD eivei| BD †K †cQ‡b ewa©Z Ki‡j AC I BD Gi wgjb we›`y nq E| facebook /gmail/skype: -tanbir.cox

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3| wØgvwÎK MwZ (Motion In Two Dimention)

8

GLb, OAEB PZzf©~‡R,  AEB+  AOB = `yB mg‡KvY| Avevi,  AEB+  BEC = `yB mg‡KvY|   AOB =  BEC =  awi, A we›`y‡Z e¯‘i †e‡Mi Dj¤^ Dcvsk, vy = 0 Ges AbyfywgK Dcvsk, vx = v B we›`y‡Z e¯‘i †e‡Mi AC eivei †e‡Mi Dj¤^ Dcvsk, v y  vsin 

Ges AbyfywgK Dcvsk, v x  vcos  t AwZ ÿz`ª mgq myZivs AwZ ÿz`ª| sin    Ges cos  1  B we›`y‡Z e¯‘i †e‡Mi †e‡Mi Dj¤^ Dcvsk, v y  v Ges AbyfywgK Dcvsk, v x  v G‡Z †`Lv hv‡”Q, AbyfywgK eivei †e‡Mi Dcvs‡ki †Kvb cwieZ©b nq bv| v  0 t v  t

‡e‡Mi Dj¤^ Dcvs‡ki cwieZ©‡bi Kvi‡Y Z¡iY, a n‡j, a 

      t 

 v 

v     r  v2  2r 2 a     2r r r v2  †K›`ªgyLx ej, F  ma  m  m 2 r (cÖgvwYZ) r v

v r

cÖkœ: Awfj¤^ Z¡iY ev e¨vmva©gyLx Z¡iY ev †K›`ªgyLx Z¡iY: Awfj¤^ Z¡iY ev e¨vmva©gyLx Z¡iY ev †K›`ªgyLx Z¡iY t ‡Kvb e¯‘ hLb e„ËvKvic‡_ Nyi‡Z _v‡K ZLb e„‡Ëi e¨vmva© eivei e„‡Ëi †K‡›`ªi w`‡K wµqvkxj Awf‡K›`ª e‡ji Rb¨ †h Z¡i‡Yi m„wó nq Zv‡K e¨vmva©gyLx Z¡iY ev Awfj¤^ Z¡iY ev †K›`ªgyLx Z¡iY e‡j| Gi GKK wgUvi/†m‡KÛ2| cÖkœ: ‡KŠwbK Z¡iY Kv‡K e‡j? ‡KŠwbK Z¡iYt hLb †Kvb e¯‘KYv Amg †KŠwbK †e‡M Ny‡i, ZLb e¯‘wUi †KŠwbK †eM cwieZ©‡bi nvi‡K †KŠwbK Z¡iY e‡j A_ev, mg‡qi mv‡_ Amg †KŠwbK †eM cwieZ©‡bi nvi‡K †KŠwbK Z¡iY e‡j| G‡K  Øviv cÖKvk Kiv nq| Gi GKK †iwWqvb/†m‡KÛ2| g‡bKwi, eËvKvi c‡_ Nyb©vqgvb e¯‘KYvi Avw`‡KŠwbK †eM i Ges t mgq ci Gi †kl †KŠwbK †eM f Kv‡RB †KŠwbK Z¡iY  

 f  i t

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3| wØgvwÎK MwZ (Motion In Two Dimention)

9

cÖkœ: mg`ªæwZ‡Z Pjgvb e¯‘i Z¡ib _v‡K bv, wKš‘ e„ËvKvi c‡_ mg`ªwZ‡Z Pjgvb e¯‘i Z¡iY _v‡K †Kb? e¨vL¨v Ki| mg`ªæwZ‡Z Pjgvb e¯‘i Z¡ib _v‡K bv, wKš‘ e„ËvKvi c‡_ mg`ªwZ‡Z Pjgvb e¯‘i Z¡iY _v‡K t ‡e‡Mi gvb n‡”Q `ªæwZ Ges †e‡Mi cwieZ©‡bi nvi n‡”Q Z¡iY| †Kvb e¯‘ hLb mij c‡_ mg `ªæwZ‡Z P‡j ZLb †e‡Mi gv‡bi †Kvb cwieZ©b nq bv Avi mij c‡_ Pjvi Rb¨ w`‡Ki I †Kvb cwieZ©b nq bv| d‡j e¯‘i †Kvb Z¡iY _v‡K bv| wKš‘ e„ËvKvi c‡_ Nyievi mgq e¯‘i wbqZ w`‡Ki cwieZ©b nq, KviY †e‡Mi AwfgyL me©`vB e„‡Ëi ¯úk©K eivii nq| Gfv‡e AbeiZ w`K cwiewZ©Z n‡Z _v‡K e‡j e¯‘ mg`ªæwZ‡Z Pj‡jI †eM mgvb _v‡Kbv| †e‡Mi GB cwieZ©‡bi d‡j Z¡i‡Yi m„wó nq| GB Z¡i‡Yi AwfgyL e„ËvKvi c‡_i †K›`ª eivei n‡q _v‡K| G Rb¨ e„ËvKvi c‡_ mg`ªæwZ‡Z Pjgvb e¯‘i Z¡iY _v‡K|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

3| wØgvwÎK MwZ (Motion In Two Dimention) 1| GKwU cÖv‡mi AbyfywgK cvjøv 96m Ges Avw`‡eM 66 ms-1| wb‡ÿc †KvY KZ? Avgiv Rvwb,

R

v o2Sin 2 o g

R g v 20 96  9.8 ev, Sin2 o  66 2 ev , 2θ o  Sin 1 (0.2159) ev, 2θ o  12.47

ev , Sin2θ o 

GLv‡b, AbyfywgK cvjøv, R = 96 m Avw`‡eM, vo = 66ms-1 AwfKl©R Z¡iY, g = 9.8ms-2 wb‡ÿc †KvY,  = ?

2| GKwU e¯‘‡K 40ms-1 †e‡M Ab~fywg‡Ki mv‡_ 60° ‡Kv‡Y wb‡ÿc Kiv nj| me©vwaK D”PZv Ges Abyf~wgK cvjøv wbY©q Ki| Avgiv Rvwb, (v o Sin θ o ) 2 2g

40 Sin

H

2  9.8

40 Sin

 H   H 

 H 

 H 

2 60   2 60  

2  9.8

40  0.86602 2 2  9.8 34 .6408 2

GLv‡b, Avw`‡eM, v0 = 40ms-1 wb‡ÿc †KvY 60º AwfKl©R Z¡iY, g = 9.8ms-2 me©vwaK D”PZv, H = ?

AbyfywgK cvjøv, R = ?

2  9.8 1199 .9850

2  9 .8

 H  61.22 m (Ans .)

Avevi, 2

R

v 0 Sin 2θ o g

40 2 Sin ( 2  60) 9.8 2 40 Sin ( 2  60 )  R 9.8 R

 R 

 R

1385 .632

9.8  R  141.39 m (Ans.) 3| nvB‡Wªv‡Rb cigvbyi g‡W‡ji GKwU B‡jKUªb GKwU †cÖvU‡bi Pviw`‡K 5.2 ×10 -11 m e¨vmv‡a©i GKwU e„ËvKvi c‡_ 2.18 ×106 ms-1 †e‡M cÖ`wÿb K‡i| B‡jKUª‡bi fi 9.1 ×10-31 kg n‡j †K›`ªgyLx ej KZ? Avgiv Rvwb,

F 

mv 2 r

9.1 10 -31  (2.18106 )2 5.2 10-11  F  8. 316  108 N (Ans.) 4| 0.250kg f‡ii GKwU cv_i LÛ‡K 0.75m j¤^v GKwU myZvi GK cÖv‡šÍ †eu‡a e„ËvKvi c‡_ cÖwZ wgwb‡U 90 evi Nyiv‡j myZvi Dci KZ Uvb co‡e| GLv‡b, Avgiv Rvwb, fi, m = 0.250 kg F  m2 r e¨vmva©, r = 0.75 m 2  2πn  mgq t = 1 min.  F m  r = 60s.  t  2 cvKmsL¨v, n = 90 cvK|  23.141690  F  0.25   0.75 Uvb, F = ?

F

 θ o  6.24 (Ans.)

H 

 R

1600 Sin 120 ) 9.8 1600  0.86602

9.8

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

60



 F  16.65 N (Ans.) 5| 9.2 ms-1 †e‡M GKwU ÿz`ª e¯‘‡K Lvov Dc‡ii w`‡K wb‡ÿc Kiv nj| GwU KZ mgq c‡i f~-c„‡ô wd‡i Avm‡e?

Avgiv Rvwb,

2 v oSin  o g 2  9.2  Sin 90 T 9.8 2  9. 2  1 T 9.8 18.4 T 9. 8  T  1.877s (Ans.) T

GLv‡b, Avw`‡eM, vo = 9.2 ms-1 wb‡¶c †KvY, 0º AwfKl©R Z¡iY, g = 9.8ms-2 DÌvb cZ‡bi †gvU mgq T =?

6| Abyfywg‡Ki mv‡_ 30°†KvY f~-c„ô †_‡K 50ms-1 †e‡M GKwU ey‡jU †Qvov nj| ey‡jUwU 50m `~‡i Aew¯’Z GKwU †`Iqvj‡K KZ D”PZvq AvNvZ Ki‡e| GLv‡b, Avgiv Rvwb, Avw`‡eM, v = 50 ms-1

g y  (tan 0 ) x  x2 2 2( v 0 cos  0 )

 y  (tan30 )x 

g (50)2 2(v0 cos30 )2

o

wb‡¶c †KvY, º AwfKl©R Z¡iY, g = 9.8ms-2 AbyfywgK `~iZ¡, x=50m Dj¤^ `~iZ¡, y=?

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3| wØgvwÎK MwZ (Motion In Two Dimention)

 y  (tan30)50

g (50)2 2 2(v0 cos30 )

9.8  (50) 2 2(50 0.866025403 )2  y  28.86751346  6.533333345  y  22.33 m (Ans.)  y  0.577350269  50 

7| GKwU cÖv‡mi AbyfywgK cvjøv 79.53 m Ges wePiYKvj 5.3 s n‡j wb‡ÿc †KvY I wb‡ÿc †eM KZ? Avgiv Rvwb, GLv‡b, v o2Sin 2 o Aby fywgK cvjøv, R = 79.53 m R g wePiYKvj, T=5.3s wb‡¶c ‡eM, vo = ? v 2oSin 2 o  79.53  wb‡¶c †KvY,  = ?

9.8 2 o

 v Sin 2 o  779.394.. ... (1)

2 v oSin  o g 2v Sin  o  5.3  o 9. 8  2v oSin  o  51.94 ... ... ... (2)

Aevi, T 

8| GKwU ej‡K f~wgi mv‡_ 30°†KvY K‡i Dc‡ii w`‡K wb‡ÿc Kiv n‡j GwU 20m `~‡i GKwU `vjv‡bi Qv‡` wM‡q coj| wb‡ÿc we›`y †_‡K Qv‡`i D”PZv 5m n‡j ejwU KZ †e‡M †Qvov n‡qwQj| Avgiv Rvwb,

y  (tan 0 ) x 

g x2 2 2( v 0 cos  0 )

 5  tan30 20

9.8 (20)2 2v cos2 30 2 0

 5  0.577350269  20   5  11.54 

v o2Sin 2 o 779.394  2v oSin  o 51.94 v o2 2Sin  o Coso 779.394  2 v oSin  o 51.94  v 0 Cos o  15 ... ... .. (3) 

(2) bs mgxKiY‡K (3) bs mgxKiY Øviv fvM K‡i cvB,

2v oSin  o 51.94  v 0 Cos o 15 3.463  tan  o  2  tan  o  1.732  o  tan 1 1.732

  o  60(Ans.) (3) bs mgxKi‡Y 0 Gi gvb ewm‡q cvB,

v 0 Cos60  15 1  v 0   15 2  v 0  30ms 1 (Ans.)

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GLv‡b, wb‡¶c †KvY, º AwfKl©R Z¡iY, g = 9.8ms-2 AbyfywgK `~iZ¡, x=20m Dj¤^ `~iZ¡, y=5m Avw`‡eM, vo = ?

9.8  400 2  v02  0.75

3920 v02 1.5

3920  6.54 v 02  1.5 3920  v 02   v 0  20ms 1 (Ans.) 6.54  1.5



9| GKRb †jvK 48 ms-1 †e‡M GKwU ej Lvov Dc‡ii w`‡K wb‡ÿc K‡i| ejwU KZ mgq k~‡Y¨ _vK‡e Ges m‡e©v”P KZ Dc‡i DV‡e?

Avgiv Rvwb,

(1) bs mgxKiY‡K (2) bs mgxKiY Øviv fvM K‡i cvB,

2

2 v oSinθ o T  g 2  48  Sin 90 T 9.8 2  48  1 T 9. 8  T  9.795 s. ( Ans.) 

GLv‡b, †eM, v0 = 48 ms-1 wb‡¶c †KvY, =º DÌvb cZ‡bi †gvU mgq,T =?

D”PZv, H =?

Avevi,

H

v o2Sin 2  o 2g

H

482 (Sin 90) 2  2  9.8

 H  117.5m (Ans.)  10| GKwU MÖv‡gv‡dvb †iKW© cÖwZ wgwb‡U 45 evi Ny‡i| Gi †K›`ª †_‡K 9cm `~‡i †Kvb we›`yi `ªæwZ KZ? Avgiv Rvwb,

v  r 2n r t 2  3.14  45  0.09 v 60 1  v  0.42ms (Ans.) v

GLv‡b, mgq, t = 1m =60s cvKmsL¨v, n=45 e¨vmva©,r =9cm=0.09m `ªæwZ, v =?

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 এযবযস্টটর ভনন কযনতন ফনরয ঳ম্পক঱ গবতয ঳ানথ বকন্তু বনউটননয ঳ূ ত্র অনু মায়ী ফনরয ঳ম্পক঱ গবতয ঩বযফত঱ননয ঳ানথ। 1 বনউটন = 7.2324 ঩াউন্ডার ঑ 1 ঩াউন্ডার= 13825.728 ডাইন।  উৎ঩বত্ত অনু ঳ানয ফর দু ই প্রকায। মথা-১। যভৌবরক ফর ২) মাবিক ফর।  বফনে যভৌবরক ফর ৪বট। মথাঃ (১) ভ঴াকল঱ ফর (২) দু ফ঱র বনউবিয় ফর (৩) তাবড়ৎ যিৌম্বক ফর (৪) ঳ফর বনউবিয় ফর। 30

40

42

 যভৌবরক ফরগুনরায আন঩বক্ষক তীব্রতায অনু ঩াত।মথাক্রনভ 1:10 :10 :10 [নমখানন ভ঴াকল঱ ফনরয তীব্রতা 1]  ঳ারাভ ঑য়ানয়ন ফাগ঱ প্রভান কনযনেন দু ফ঱র ঑ তাবড়ৎ যিৌম্বক ফর একই ফনরয দু বট ববনড়ফরূ঩। ভ঴াকল঱ ফরঃ ঳ফনিনয় দু ফ঱র। Gravitation নাভক এক প্রকায কণায ঩াযষ্পবযক বফবনভনয়য ভাধযনভ ভ঴াকল঱ ফর কাম঱কয।  দু ফ঱র বনউবিয়ায ফরঃ Intermediate vector bosons নাভক কণায ঩াযস্পবযক বফবনভনয়য ভাধযনভ এ ফনরয ঳ৃ বি।  Photon নাভক এক প্রকায কণায ঩াযস্পবযক বফবনভনয়য ভাধযনভ তাবড়ৎ যিৌম্বক ফর কাম঱কয ঴য়।  ঳ফর বনউবিয়ায ফরঃ বনউবিয়ায ফর আকল঱ণ ধভ঱ী। স্বল্প ঩াল্লা বফব঱ি এফিং আধান বনযন঩ক্ষ । যভ঳ন কণায ঩াযস্পবযক বফবনভনয়য ভাধযনভ ঳ফর বনউবিয়ায ফর ঳ৃ বি ঴য়।  ২য় ঳ূ ত্র যথনক ১ভ ঳ূ ত্র ঩া঑য়া মায় অথ঱াৎ ১ভ ঳ূ ত্র ২য় ঳ূ নত্রয একবট রূ঩।  ১ভ ঳ূ ত্র যথনক ঩া঑য়া মায়ঃ ফর ঑ জড়তা  ২য় ঳ূ ত্র যথনক ঩া঑য়া মায়ঃ ফনরয অববভুখ, ফনরয ঩বযভা঩, ফনরয গুণগত রফব঱িয, ত্বযনণয ঳ানথ ফনর ঳ম্পক঱, ফনরয নীযন঩ক্ষ নীবত।  বিবত জড়তায দৃ িান্তঃ ১। কান঩য উ঩য য঩াি কাড঱ ঑ তায উ঩য ভুদ্রা যযনখ যটাকা বদনর ভুদ্রা কান঩ ঩নড় মানফ। ২। ঴ঠ্াৎ গাবড় িরনত শুরু কযনর আনযা঴ী ব঩েনন য঴নর ঩নড়। ৩। কযাযানভয যফানড঱ একবট গুবটয উ঩য আয একবট গুবট যযনখ স্ট্রাইক িাযা আঘাত কযনর নীনিয গুবট িনর মায়। ৪। ধু বরমু ি য঩ালাক দবড় বদনয় আঘাত কযনর ভয়রা দূ য ঴নয় মায়। ৫। কানিয জানারায় ফু নরট যোড়নর একবট বেদ্র বকন্তু বির যোড়নর একাবধক বেদ্র ঴য়। ৬। যঘাড়া ঴ঠ্াৎ যদৌড়ানত শুরু কযনর আনযা঴ী ব঩েনন য঴নর ঩নড়।  গবত জড়তায দৃ িান্তঃ ১। িরন্ত ফা঳ গঠ্াৎ যথনভ যগনর মাত্রী ঳াভনন য঴নর ঩নড়। ২। যদৌড় বদনয় রাপ বদনর। ৩। ধাফভান যঘাড়া যথনক উ঩নযয বদনক রাপ যদয়া। ৪। িরন্ত গাবড়য কাভযায় যকান আনযা঴ী য঳াজা উ঩নযয বদনক বকেু যোড়নর। ৫। ঳যর যদারনক একফায দু বরনয় বদনর অননক্ষণ দু রনত থানক।  ঘাত ফনরয দৃ িান্তঃ ১। ফযাট িাযা ফর আঘাত কযা। ২। ঴াতুবড় বদনয় ব঩ন য঩াতা। ৩। স্ট্রাইকায িাযা গুবটনক আঘাত কযা। ৪। যেনন যেনন ঳িংঘল঱। ৫। ঩া বদনয় পুটফর বকক কযা। ৬। বফনফাযণ। ৭। কাভান ঴নত যগারা যোড়া। facebook /gmail/skype: -tanbir.cox

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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| MwZm~Î (Laws Of Motion)

‡gŠwjK ej mg~n 1) gnvKl© ej 2) ZvwoZ †PŠ¤^K ej 3) mej wbDwK¬q ej 4) `~e©j wbDwK¬q ej

Av‡cwÿK wZeªZv

1 1039 1041 1030

2

cvjøv Amxg Amxg 10-15 m 10-16 m

fi‡eM (Momentum) t fi I †e‡Mi ¸bdj‡K fi‡eM e‡j| 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  F  ma cÖwZcv`b| 



 F  ma

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

  dP n‡e| MwZi 2q m~Îvbymv‡i, fi‡eM cwieZ©‡bi nvi cÖhy³ e‡ji cwieZ©b dP n‡j fi‡eM cwieZ©‡bi nvi dt  dP  F mgvbycvwZK| A_©vr, dt   e¯‘KYvi fi m Ges †eM v n‡j fi‡eM P  mv ewm‡q cvB,  d ( m v)  F dt  dv  m F dt    dv    ma  F   Z¡ iY a  dt    ma  kF ... ... ... .... ... (1) 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|   A_©vr F  1 , m=1 Ges a  1 nq Z‡e mgxKiY (1) n‡Z cvB|   1×1=k×1  k=1 mgxKi‡Y ewm‡q cvB, ma  F

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3 04| MwZm~Î (Laws Of Motion)      A_©vr F  ma †Kvb e¯‘i Dci ¯^Zš¿ ej¸wj hw` nq h_vµ‡g F1 , F2 , F3 ,..Fn BZ¨vw` wµqviZ _vK‡j Avgiv cvB,    F1  F2  F3  ....  Fn  ma          F  ma GLv‡b,  F n‡”Q jwä ej ev bxU ej †hLv‡b,  F  F1  F2  F3  ....  Fn (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 t wbDU‡bi MwZi 1g I 2q m~Î Zzjbv K‡i †`Lv hvq †h, 1g m~Î n‡”Q 2q m~‡Îi GKwU iƒc|





 F  ma

†_‡K †`Lv hvq

   †h, e¯‘i Dci wbU ej k~b¨ n‡j A_©vr hLb  F  0 ZLb ma  0 ,wKš‘ m  0,  a  0 nq| myZivs hLb evB‡i †_‡K

ej cÖ‡qvM bv nq A_©vr wbU ej k~b¨ nq ZLb,  a0  dv  0 dt   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) t 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) t  ‡Kvb ej I e‡ji wµqv Kv‡ji ¸b dj‡K e‡ji NvZ e‡j| †Kvb ej F hw` †Kvb e¯‘i Dci t i mgq †_‡K t f mgq ch©šÍ  tf  wµqv K‡i, Zvn‡j e‡ji NvZ n‡e, J   Fdt ti

 tf  J  F  dt

 [ F  aª æe ]

ti

  tf   J  Ft   F [t f - t i ] ti    J  Ft

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



tf



cÖhy³ ej F n‡j e‡ji NvZ J   Fdt ti

  dP J dt dt ti tf

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   dP   F   dt  

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04| MwZm~Î (Laws Of Motion)

    J   dP  P  pi



4

 Pf  Pi

    J  Pf  Pi    J  P 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,  P  aª æeK 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 P1  Ges wØZxq e¯‘i fi‡eM P2 | Avgiv cÖwZwU e¯‘i †ÿ‡Î wbDU‡bi 2q m~Î cÖ‡qvM K‡i cvB,

   dP1  dP2 F1  Ges F2  dt dt   wbDU‡bi MwZi Z…Zxq m~Î †_‡K cvB, F1   F2   dP1 dP2   dt  dt dP dP  1  2 0 dt dt   d P1  P2  0  dt     dP  0 [ P1  P2 n‡”Q e¨e¯’vi †gvU fi‡eM ev P] dt      ‡h‡nZz mg‡qi mv‡c‡ÿ †gvU fi‡eM P  P1  P2 , P Gi cwieZ©‡bi nvi k~b¨, ZvB †gvU fi‡eM P aªæe _v‡K, A_©vr    P  P1  P2 = aªæe    Pi  Pf      P1i  P2i  P1f  P2f 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| MwZm~Î (Laws Of Motion)

5

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 = Ft (m)v = Ft  m  m  ev , F   n‡”Q R¡vjvwb e¨env‡ii nvi|  v GLv‡b t  t  ‡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,  m  Ma   v  t  1  m   Z¡ iY, a   v M  t 

e‡ji fvimvg¨ (Equilibrium of Force) t  e¯‘i Dci e‡ji jwä k~b¨ nIqv‡K A_©vr,  F  0 nIqv‡K e‡ji fvimvg¨ e‡j| fvimvg¨ Ae¯’vq e¯‘i Dci Z¡iY k~b¨ nq| fvimvg¨ Ae¯’vq e¯‘i †eM aªæe _v‡K| A_©vr †e‡Mi gv‡bi ev w`‡Ki †Kvb cwieZ©b nq bv| fvimvg¨ `yB cÖKvi| h_v(1) w¯’i fvimvg¨ (2) MwZq fvimvg¨ jvwgi Dccv`¨ (Lami's Theorem) t †Kvb e¯‘i Dci wZbwU AmgvšÍivj ej wµqv K‡i fvimvg¨ Ae¯’vi m„wó Ki‡j cÖ‡Z¨KwU ej Aci `ywU e‡ji AšÍÍ©fy³ †Kv‡bi mvB‡bi mgvbycvwZK|      g‡bKwi, P, Q I R wZbwU ej O we›`y‡Z wµqv K‡i fvimvg¨ m„wó K‡i| Q I R Gi     AšÍ©Ífy³ †KvY , R I P Gi AšÍ©Ífy³ †KvY , P I Q Gi AšÍ©Ífy³ †KvY |  myZivs jvwgi Dccv`¨ Abymv‡i,

P Q R   n‡e| Sin  Sin  Sin 

Nl©Y (Friction) t `ywU e¯‘ ci¯ú‡ii ms¯ú‡k© †_‡K hw` G‡Ki Dci w`‡q AciwU Pj‡Z †Póv Ki‡j e¯‘ ؇qi ¯úk© Z‡j GB MwZi weiæ‡× †h evavi DrcwË nq Zv‡K Nl©Y e‡j| Nl©Y mvaviYZ Pvi cÖKv‡ii n‡q _v‡Kt (1) w¯’wZ Nl©Y (Static Friction), (2) MZxq Nl©Y (Kinetic Friction), (3) AveZ© Nl©Y (Rolling Friction), (4) cÖevwn Nl©Y (Fluid Friction),

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04| MwZm~Î (Laws Of Motion)

6

Nl©Y ej t `ywU e¯‘ ci¯ú‡ii ms¯ú‡k© †_‡K hw` G‡Ki Dci w`‡q AciwU Pj‡Z †Póv Ki‡j e¯‘ ؇qi ¯úk© Z‡j GB MwZi weiæ‡× †h e‡ji m„wó nq Zv‡K Nl©Y ej e‡j| w¯’wZNl©Y ev mxgvwšÍK Nl©Y (Static Friction) t cÖhy³ e‡ji m‡e©v”P †h gv‡bi Rb¨ GKwU w¯’wZkxj e¯‘ MwZi Dcµg nq, Zv‡K w¯’wZ Nl©Y ev mxgvwšÍK Nl©Y e‡j&| w¯’wZ Nl©‡bi gvb k~b¨ †_‡K mxgvwšÍK gvb ch©šÍ n‡Z cv‡i| w¯’wZNl©Y ej t ‡h Nl©b ej Øviv †Kvb e¯‘ cÖh³ y ej‡K cÖwZnZ K‡i w¯’i _vK‡Z cÖqvm cvq Zv‡K w¯’wZNl©b ej e‡j|

Nl©Y †KvY I w¯’wZ †Kv‡Yi g‡a¨ m¤úK© t w¯’wZ Nl©Y ¸Yv¼ t w¯’wZ Nl©‡Yi mxgvwšÍK gvb Ges Awfj¤^ cÖwZwµqvi AbycvZ‡K w¯’wZ Nl©Y ¸Yv¼ e‡j| G‡K s Øviv cÖKvk Kiv nq| w¯’wZ Nl©‡Yi mxgvwšÍK gvb fs Ges Awfj¤^ cÖwZwµqv R n‡j w¯’wZ Nl©Y ¸Yv¼ n‡e s 

fs ... ... ... ... ... ... (2) fs I R Gi GKK GKB| Kv‡RB s Gi †Kvb GKK †bB| R

Nl©Y †KvY (Angle of Friction) t w¯’wZ Nl©‡Yi mxgvwšÍK gvb fs Ges Awfj¤^ cÖwZwµqv R †K mgš^q K‡i †h jwä cvIqv hvq Zv‡K jä cÖwZwµqv S e‡j| GB jä cÖwZwµqv Awfj¤^ cÖwZwµqvi mv‡_ †h †KvY K‡i Zv‡K Nl©Y †KvY ev w¯’wZ Nl©Y †KvY e‡j| Nl©Y †KvY‡K  Øviv cÖKvk Kiv nq| Wvbcv‡k¦©i wP‡Î  Gi Ae¯’vb †`Lvb n‡q‡Q| wPÎvbymv‡i cvB, R  S cos  ... ... ... ... ... ... (3) f s  S sin  ... ... ... ... ... ... (4)

(4) bs mgxKiY‡K (3) bs mgxKiY Øviv fvM K‡i cvB,

Avevi,

(2) bs mgxKiY  s 

fs  tan λ ... ... ... ... (5) R

fs †K (5) bs Gi mv‡_ Zzjbv K‡i cvB,  s  tan  ... ... ... (6) A_©vr Nl©Y †Kv‡Yi R

U¨vb‡R›U -Gi gvb w¯’wZ Nl©b ¸bv‡¼i mgvb| BnvB Nl©Y †KvY I w¯’wZ Nl©Y ¸Yv‡¼i g‡a¨ m¤úK©|

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7 04| MwZm~Î (Laws Of Motion) w¯’wZ †KvY ev wbðj †Kvb (Angle of Repose) t ‡Kvb AvYZ Zjw¯’Z e¯‘ Z‡ji †h AvbwZi Rb¨ MwZi Dcµg nq Zv‡K w¯’wZ †KvY ev wbðj †KvY e‡j| G‡K Øviv cÖKvk Kiv nq| wPÎvbymv‡i, OY Z‡ji Dcwiw¯’Z e¯‘ Z‡ji  AvbwZi Rb¨ MwZi Dcµg nq| myZivs n‡e w¯’wZ †KvY| GB Ae¯’vq W Lvov wb‡Pi w`‡K wµqv Ki‡e Ges Z‡ji Awfj¤^ cÖwZwµqv R-Gi wµqv †iLvi mv‡_  †Kv‡Y bZ _vK‡e| wP‡Î cÖ`wk©Z †f±i e¨e¯’v Abymv‡i, R  W Cos f s  W Sin  fs ... ... ... ... (7) R f wKš‘ (2) bs mgxKiY n‡Z cvB,  s  s ...(2) Avevi (7) I (2) bs mgxKiY †_‡K cvB, R  s  tan  ... ... ... (8)  tan  

Avevi,  s  tan  ... ... ... ... (6) d‡j (6) I (8) †_‡K cvB, tan   tan      AZGe, Nl©Y ‡KvY Ges wbðj †Kv‡Yi gvb mgvb| Z‡e Nl©Y †KvY me©‡ÿ‡Î

cÖ‡hvR¨; wKš‘ wbðj †KvY ïaygvÎ AvbZ Z‡ji Rb¨ cÖ‡hvR¨| MwZq Nl©Y (Kinetic Friction) t ‡Kvb Z‡ji Dci w`‡q MwZkxj e¯‘ MwZi weiæ‡× †h evav Abyfe K‡i Zv‡K MwZq Nl©Y ¸bv¼ e‡j Ges †h ej MwZ‡Z evavi m„wó K‡i, Zv‡K MwZq Nl©Y ej e‡j| MwZq Nl©Y ¸bv¼ t MwZq Nl©Y ej Ges Awfj¤^ cÖwZwµqvi AbycvZ‡K MwZq Nl©Y¸bv¼ e‡j| G‡K Øviv cÖKvk Kiv nq|  k 

fk R

GLv‡b, fk = MwZq Nl©Y ej I R Awfj¤^ cÖwZwµqv|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

4| MwZm~Î (Laws Of Motion) 1| 10kg f‡ii GKwU e›`y‡Ki wUªMv‡i Pvc †`Iqv‡Z e›`y‡Ki bj n‡Z 20g f‡ii GKwU ey‡jU 400ms-1 †e‡M †ei n‡q †Mj| e›`yKwUi cðvr †eM KZ? Avgiv Rvwb, GLv‡b, m1v1 = m2v2 e›`y ‡Ki fi, m1 = 10 kg ev, 10v1  20 10 3 400 ey‡j‡Ui fi, 20  10 3  400 m2 = 20×10-3kg ev, v1  ey‡j‡Ui †eM, v2 = 400ms-1 10  v  0.8ms1 (Ans.) e›`y‡Ki cðvr †eM, v1 =?

1

2| 40kg I 60kg f‡ii `ywU e¯‘ h_vµ‡g 10ms-1 I 5ms-1 †e‡M ci¯úi wecixZ w`K †_‡K Avmvi mgq G‡K Aci‡K av°v w`j| av°vi ci e¯‘ Øq GK‡Î hy³ n‡q KZ †e‡M †Kvb w`‡K Pj‡e?

Avgiv Rvwb, GLv‡b, m1u1+m2u2 = m1v1+m2v2 1g e¯‘i fi, m1 = 40kg ev, 40×10+ 60×(-5) 2q e¯‘i fi, m 2 = 60kg = 40v + 60 v 1g e¯‘i Avw`†eM, u1 = 10ms-1 ev, 400300 =100v 2q e¯‘i Avw`†eM, u2 =-5ms-1 ev, 100v =100 hy³ e¯‘؇qi †kl †eM, 100 ev, v   v1 = v2 = v = ?

100  v  1 ms 1

v abvZ¡K †nZz av°vi ci e¯‘ Øq GK‡Î hy³ n‡q 1ms †e‡M 1g e¯‘i Awfgy‡L Pj‡e| -1

3| GKwU e¯‘ w¯’i Ave¯’vq wQj| 15 wbDU‡bi GKwU ej Gi Dci 4 †m‡KÛ wµqv Kivi ci e‡ji wµqv eÜ n‡q hvq| e¯‘wU Gi ci 4 †m‡K‡Û 48m `~iZ¡ †Mj| e¯‘wUi fi KZ? ej _vKv Kvjxb 4 †m‡K‡Û e¯‘wU †h †kl †eM jvf K‡i †mB †kl †eM‡K Mo †eM wb‡q cieZ©x 4 †m‡K‡Û e¯‘wU 48m `~iZ¡ hvq| Avgiv Rvwb, s = v t2

s t2 48 ev, v  4  v  12 ms -1 ev, v 

Avevi, v = u + at1 ev, 12 = 0 + a (4)

ev, a 

12 4

GLv‡b, Avw`‡eM, u = 0 ej, F = 15 N mgq, t1 = 4s mgq, t2 = 4s ‡kl‡eM = Mo‡eM = v miY, s = 48m fi, m = ?

a = 3 ms-2 Avevi, F = ma

ev, m 

 m  5 kg (Ans.) 4| 10N Gi GKwU ej 2kg f‡ii GKwU w¯’i e¯‘i Dci wµqv K‡i| hw` 4s ci e‡ji wµqv e›` n‡q hvq Z‡e cÖ_g †_‡K 8s G e¯‘wU KZ `~i hv‡e? ej _vKv Kvjxb 4 †m‡K‡Û e¯‘wU †h †kl †eM jvf K‡i cieZ©x (8 4) = 4 †m‡KÛ †mB †kl †eM‡K Mo †eM wb‡q e¯‘wU Pj‡e| Avgiv Rvwb, F=ma GLv‡b, F ev, a  Avw`‡eM, u = 0 m ej, F = 10 N 10 ev, a  fi, m = 2 kg 2 mgq, t1 = 4s  a  5 ms -2 mgq, t2 = (84) s = 4s ‡gvU `yiZ,¡ s = s1+s2 = ?

Avevi, s1  ut 1 

1 2 at 1 2

1  5  42 2  s1  40 m Avevi, v  u  at1

 s1  0 

 v  0  5 4  v  20 ms -1 Avevi, s 2  v t 2

 s 2  20  4  s 2  80 m  s = s1+s2 = (40+80)m =120m (Ans.) 5| 100kg Ges 200kg f‡ii `wU e¯‘ ci¯úi wecixZ w`‡K h_vµ‡g 20ms-1 Ges 10ms-1 †e‡M hvIqvi c‡_ G‡K Aci‡K av°v w`j| av°vi ci e¯‘ `ywU GK‡Î hy³ †_‡K KZ †e‡M †Kvb GLv‡b, w`‡K Pj‡e?

1g e¯‘i fi, m1 = 100kg 2q e¯‘i fi, m2 = 200kg Avgiv Rvwb, 1g e¯‘i ‡eM, u1 = 20 ms-1 m1u1+m2u2 = m1v1+m2v2 -1  100×20+200×(-10)=100v+200v 2q e¯‘i ‡eM, u2 = -10 ms GK‡Î hy³ †eM v1=v2=v=?

 100v+200v =2000 2000  300v =0  v = 0

‡h‡nZz †kl †eM k~b¨ d‡j e¯‘ `ywU GK‡Î hy³ n‡q †_‡g hv‡e|

F 15 ev, m  a 3

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4| MwZm~Î (Laws Of Motion)

6| 4 kg f‡ii GKwU e¯‘‡K 10ms-2 Z¡i‡Y MwZkxj Ki‡Z KZ ej cÖ‡qvM Ki‡Z n‡e? c‡_i Nl©b ej 2.5N kg-1 | GLv‡b, Avgiv Rvwb, fi, m = 4 kg F1 = ma Z¡ i Y, a = 10ms-2  F1 = 4×10 N Nl©Y ej, F © =2.5 N kg-1  F1 = 40 N Avevi 4kg †Z Nl©Y ej F2= F ©× 4 N ej, F = ? F2= 2.5 × 4 N = 10 N Avevi, F = F1+F2  F = (40+10)N=50N (Ans.) 7| 70kg f‡ii ev·‡K 500N AbyfywgK e‡j ‡g‡Si Dci w`‡q Uvbv n‡”Q| ev·wU hLb P‡j ZLb ev· I †g‡Si ga¨eZ©x Nl©Y mnM 0.5| ev‡·i Z¡iY wbY©q Ki| Avgiv Rvwb,

F MZxq Nl©Y mnM,  k  k R  Fk   k  R  Fk  0.5  686  Fk  343N

Avevi, F = P-Fk  F =(500 – 343)N  F =157 N Avevi, F = ma  157 = 70 a

GLv‡b, MZxq Nl©Y mnM  k  0.5 ev‡·i fi m = 70kg Awfj¤^ cÖwZwµqv.

R=mg=70×9.8N=686N AbyfywgK ej P=500N ev‡·i Z¡iY a =?

157 1 ms 70 a  2.24 ms 2 (Ans.) 8| 1N Gi GKwU ej 0.1kg f‡ii GKwU w¯’i e¯‘i Dci wµqv K‡i| hw` 1s ci e‡ji wµqv e›` n‡q hvq Z‡e cÖ_g †_‡K 2s G e¯‘wU KZ `~i hv‡e? ej _vKv Kvjxb 1 †m‡K‡Û e¯‘wU †h †kl †eM jvf K‡i cieZ©x (2 1) = 1 †m‡KÛ †mB †kl †eM‡K Mo †eM wb‡q e¯‘wU Pj‡e| GLv‡b, Avgiv Rvwb, Avw`‡eM, u = 0 F=ma ej, F = 1 N F ev, a  fi, m = 0.1 kg m

ev, a 

1 0.1

 a  10 ms -2 Avevi, s1  ut 1   s1  0 

1 2 at 1 2

1  10  12 2

 s1  5 m Avevi, v  u  at 1  v  0  10  1  v  10 ms -1 Avevi, s 2  v t 2

 s 2  10  1  s 2  10 m

 s = s1+s2 = (5+10) m =15m (Ans.) 9| 36 kg f‡ii GKwU e¯‘i Dci KZ ej cÖ‡qvM Ki‡j 1 wgwb‡U

Gi †eM 15kmh-1 e„w× cv‡e?

Avgiv Rvwb, ej F=ma

a

mgq, t1 = 1s mgq, t2 = (21) s = 1s ‡gvU `yiZ,¡ s = s1+s2 = ?

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2

(v  u ) t 15  1000  F  36 N 3600  60  F  2 .5 N ( Ans ).

 F m

GLv‡b, fi, m =36 kg mgq, t = 1m=60s ‡eM e„w×, (v-u)=15kmh-1 15  1000 ms-1  3600 ej,F=?

10| 36 kg f‡ii GKwU e¯‘i Dci KZ ej cÖ‡qvM Ki‡j wgwb‡U Gi †eM N›Uvq 12km e„w× cv‡e?

Avgiv Rvwb, ej F=ma (v  u ) t 12  1000 N  F  36 3600  60  F  2 N ( Ans ).

 F m

GLv‡b, fi, m =36 kg mgq, t = 1m=60s ‡eM e„w×, (v-u)=12kmh-1 12  1000 ms-1  3600 ej,F=?

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1| GKwU PvKvi fi 5kg Ges PµMwZi e¨vmva© 25cm | Gi RoZvi åvgK KZ? PvKvwU‡K 4 rad s-2 †KŠwbK Z¡iY m„wó Ki‡Z KZ gv‡bi UK© cÖ‡qvM Ki‡Z n‡e?

Avgiv Rvwb, GLv‡b, RoZvi åvgK fi, M = 5kg 2 I = MK PµMwZi e¨vmva©, K= 25cm = 5×(0.25)2 2 =0.25m = 0. 3125 kg m RoZvi åvgK, I = ? ( Ans.) ‡KŠwbK Z¡iY,  = 4 rad s-2 Avevi, UK©, ? UK©  0. 3125× 4 Nm =1.25Nm (Ans.) 2| GKwU KYv 1.5m e¨vmv‡a©i e„ËvKvi c‡_ cÖwZ wgwb‡U 120 evi AveZ©b K‡i| Gi mij ˆiwLK †eM wbY©q Ki| GLv‡b, Avgiv Rvwb, e¨vmva©, r = 1.5m v= r mgq, t = 1 min = 60 sec.

2 n ev, v  r cvK msL¨v, n = 120 cvK t ˆiwLK †eM, v =?

2  3.14  120  1.5 60  v  18.84 ms 1 ( Ans.) ev, v 

3| e„ËvKvi c‡_ 3.14ms-1 mg`yªwZ‡Z AveZ©biZ GKwU KYv cÖwZ †m‡K‡Û 10 wU c~b© AveZ©b m¤úbœ K‡i| e„ËvKvi c‡_i e¨vmva© wbY©q Ki| Avgiv Rvwb, v= r GLv‡b, 2πn ˆiwLK `yªwZ, v = 3.14ms-1 ev, v   r t mgq, t = 1 sec. vt cvK msL¨v, n = 10 cvK ev, r  e¨vmva©, r =?

2 n

3.14  1 ev, r  2  3.14  10  r  0.05m (Ans.) -1

4| 13ms †e‡M GKwU Mvwo‡K wbivc‡` 30m e¨vmv‡a©¨i GKwU euvK AwZµg Ki‡Z n‡j euvKwU‡K KZ †Kv‡Y Xvjy Ki‡Z n‡e?

Avgiv Rvwb,

θ  tan  1

v2 rg

132 30  9.8 1  θ  tan 0.57  θ  29.89 (Ans)  θ  tan 1

GLv‡b, †eM, v =13 ms-1 e¨vmva©, r = 30m g = 9.8 ms-2 ‡KvY,  =?

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5| GKwU Zvgvi †Mvj‡Ki fi 0.05 Kg| GwU‡K 2m `xN© GKwU myZvi GK cÖv‡šÍ †e‡a cÖwZ †m‡K‡Û 5 evi Nyivb n‡”Q| †MvjKwUi †KŠwbK fi†eM KZ? Avgiv Rvwb, RoZvi åvgK I = MK2 = 0.05×(2)2 = 0.2 kg-m2

 n t 2  314  5   rad s-1 1

Avevi,  

GLv‡b, fi, M = 0.05 kg PµMwZi e¨vmva©, K= 2m RoZvi åvgK, I = ? cvKmsL¨v, n = 5 mgq, t = 1 Sec ‡KŠwbK fi†eM, L =?

= 31.4 rad s-1 Avevi, L = I  L =0.2×31.4 = 6.28 Kg-m2 s-1 (Ans.)

6| e„ËvKvi c‡_ 72Kmh-1 mg`ªywZ‡Z Pjgvb †Kvb †gvUi Mvoxi †K›`ªgyLx Z¡iY 1ms-2 n‡j e„ËvKvi c‡_i e¨vmva© KZ? Avgiv Rvwb, v2 a r

GLv‡b, v = 72 Kmh-1

20 2 r  r  400m (Ans.)

a = 1ms-2 e¨vmva,© r = ?

v

72  1000

 20ms

-1

3600

1

7| 75m e¨vmv‡a©i e„ËvKvi c‡_ †Kvb gUi mvB‡Kj Av‡ivnx KZ †e‡M Nyi‡j Dj¤^ Z‡ji mv‡_ 30°†Kv‡Y AvbZ _vK‡e?

Avgiv Rvwb,

v2 rg 2  v  tan  rg tan θ 

 v2  v2  v2  v2

 tan 30  75  9.8  tan 30  75  9.8  424.35  424.35

GLv‡b, e¨vmva©, r = 75m g = 9.8 ms-2 ‡KvY,  =30º †eM, v =?

 v  424.35  20.6 ms -1 ( Ans)

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5| ‡KŠwbK MwZm~Î (Laws Of Angular Motion)

2

8| 0.1kg f‡ii GKwU cv_i‡K 0.5m j¤^v myZvi mvnv‡h¨ e„ËvKvi c‡_ Nyivb n‡”Q| cv_iwU cÖwZwgwb‡U 30 evi c~Y© NyY©b m¤úbœ K‡i| myZvi Uvb KZ? Avgiv Rvwb,

10| 200m e¨vmva© wewkó GKwU euvKv c‡_ 50.4 km.h-1 ‡e‡M Mvwo Pvjv‡Z c_wU‡K KZ †Kv‡Y KvZ K‡i ivL‡Z n‡e? iv¯ÍvwUi cÖ¯’ 1m n‡j, evB‡ii cvk¦© wfZ‡ii cvk¦© A‡cÿv KZ DuPz n‡Z n‡e? (g = 9.8 ms-2 )

F  m2 r

Avgiv Rvwb,

GLv‡b, fi, m = 0.1kg  2n  e¨vmva©, r = 0.5m  F  m  r  t  cvKmsL¨v, n = 30 2 mgq, t = 1 min  2  3.14  30  = 60sec 0 .

5 N  F  0.1  60   myZvi Uvb ej, F = ?

2

 F = 0.49298 N (Ans.) 9| c„w_exi Pvwiw`‡K P‡›`ªi Kÿc‡_i e¨mva© cÖvq 3.85×105 km| GKevi cÖ`wÿb Ki‡Z mgq jv‡M 27.3 w`b| Pv‡`i †KŠwbK `ªæwZ wbY©q Ki| GLv‡b, Avgiv Rvwb, e¨mva© r=3.85×105 km 2n =3.85×108 m  t mgq t= 27.3 w`b cvK msL¨v=1cvK 2  3.1416  1 †KŠwbK `ªæwZ =?

27.3  24  3600 6 1    2.66  10 rads (Ans.)



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θ  tan  1

v2 rg

GLv‡b, †eM, v = 50.4 kmh -1

14 2 200  9.8 1  θ  tan 0.1  θ  5.71 (Ans)  θ  tan 1

sin 5 . 71   0 . 0995 

50.4  1000 -1 ms 3600  v = 14 ms -1 v=

e¨vmva©, r = 200m g = 9.8 ms-2 ‡KvY,  =? x=1m, h=?

h 1

h  h  0 . 0994 m  0 . 1m ( Ans .)

1

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 প্রমু ি ফর ঑ ফনরয অববভুনখ প্রনয়াগ বফন্দুয ঳যনণয উ঩ািংন঱য গুণপর িাযা কানজয ঩বযভা঩ কযা ঴য়। কাজ , W = F. S ফা F×S cosθ ঱বি: যকান ফস্ত্ত্ত ফা ফযফিায কাজ কযায ঳াভথ঱যনক ঱বি ফনর।  ক্ষভতা: যকান একবট উৎন঳য কাজ কযায ঴াযনক ক্ষভতা ফনর। ক্ষভতা, p =  ঱বিয বনতযতায ঳ূ ত্র: ভ঴াবফনেয যভাট ঱বিয ঩বযভাণ বনবদ঱ি ঑ অ঩বযফত঱নীয়।  ঳িংযক্ষণ঱ীর ফর: যম ফর িাযা ফস্ত্ত্তনক একবট বনবদ঱ি ঩থ ঘুবযনয় ঐ ঩নথ আননর কৃত কানজয ঩বযভাণ ঱ূ নয ঴য় ঐ ফরনক ঳িংযক্ষণ঱ীর ফর ফরা ঴য়। যমভন: অববকল঱ ফর, রফদু যবতক ফর ঑ আদ঱঱ বরিং এয বফকৃবত প্রবতনযাধ কাযী ফর ইতযাবদ।  অ঳িংযক্ষণ঱ীর ফর: যম ফর িাযা কৃত কাজ ঱ূ নয না ঴য় এফিং ঳ম্পাবদত কাজ শুধু ভাত্র দু বট বফন্দুয অফিাননয উ঩য বনব঱য কনয তানক অ঳িংযক্ষণ঱ীর ফর ফনর। যমভন ঘল঱ণ ফর, ঳াদ্রতা ফর ইতযাবদ। কাজ- ঱বি উ঩঩াদয: যকান ফস্ত্ত্তয উ঩য প্রমু ি ফর িাযা কৃত কাজ ফস্ত্ত্তবটয গবত঱বিয ঩বযফত঱ননয ঳ভান।W = ∆k ১. কাজ, ঱বি ঑ ক্ষভতা এই বতনবট অবদক ফা যস্করায যাব঱। ২. কাজ ফা ঱বিয ঩যভ একক:  F. P. S ঩দ্ধবতনত  Ft-poundal  C. G. S঩দ্ধবতনত  erg  M. K. S ফা S.I ঩দ্ধবতনত  Jole

৩. এক অববকল঱ীয় একক কাজ = এক অববকল঱ীয় একক ফর × একক ঳যণ। ৪. কানজয যক্ষনত্র অববকল঱ীয় এককনক ঩যভ একনক রূ঩ান্তনযয ঳ভয় িাযা গুণ কযনত ঴য় এফিং ঩যভ এককনক অববকল঱ীয় একনক রূ঩ান্তনযয ঳ভয় িাযা বাগ কযনত ঴য়। ৫. ক্ষভতায ঩যভ একক: আন্তজ঱াবতক ঩দ্ধবতনত ক্ষভতায একক Jol/sec ফা watt. ৬. ভাত্রা ঳ভীকযণ: কাজ: [ML2T-2] ঱বি: [ML2T-2] ক্ষভতা: [ML2T-3] ৭. ঳ূ ম঱ প্রবত বভবননট 25 যকাবট টন বয ঴াযায় এফিং ঳ূ নম঱য আয়ু 27 বফবরয়ন ফেয। ৮. একজন ভানু নলয গড় ক্ষভতা 0.143 H.P। ৯. ফর ঑ ঳যনণয ভধযফত঱ী যকাণ 900 ঴নর এফিং তানদয ভান আরদা আরাদা বানফ ঱ূ নয ঴নর কানজয ভান ঱ূ নয ঴য়। ১০. ফৃ ত্ত঩নথ আফত঱নযত ফস্ত্ত্ত ঱ূ নয কানজয উদা঴যণ। 0 <θ < 900 ঴নর ফনরয িাযা কৃতকাজ ধনাত্মক। 900 <θ< 1800 ঴নর ফনরয বফরুনদ্ধ কাজ ফুঝায় এফিং এই কাজ ঋণাত্মক। ১১. মাবিক দক্ষতা = মি িাযা কৃত কানজয ঴ায মনি ঳যফযা঴কৃত ঱বিয ঴ায 

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KvR (Work): ‡Kvb e¯‘i Dci ej cÖ‡qvM Ki‡j hw` e¯‘i miY N‡U Z‡e ej I e‡ji w`‡K mi‡Yi Dcvs‡ki ¸bdj‡K KvR e‡j|  KvR‡K W Øviv cÖKvk Kiv nq| g‡b Kwi, †Kvb KYvi Dci GKwU aªæe ej F Gi wµqvq KYvwUi  e‡ji Awfgy‡L mij‡iLv eivei miY S n‡j KYvwUi Dci ejØviv K…Z KvR W  FS n‡e| KvR GKwU Aw`K ivwk| Kv‡Ri GKK Ryj|  aªæe ej hw` KYvwUi miY S Gi  †KvY Drcbœ K‡i, Zvn‡j GB miY Kv‡j K…Z KvR W  (F cos )S ev, W  F(S cos ) A_©vr, KvR = ej × e‡ji w`‡K mi‡Yi Dcvsk  ‡f±i iƒ‡c wjL‡j, W  F.S

Ryj: GK wbDUb ej cÖ‡qv‡M e¯‘ hw` e‡ji w`‡K GK wgUvi m‡i hvq Z‡e †h KvR nq Zv‡K GK Ryj KvR e‡j| e‡ji Øviv KvR (Work done by the force): hw` ej cÖ‡qv‡Mi d‡j e¯‘ e‡ji w`‡K m‡i hvq Z‡e ev e‡ji w`‡K mi‡Yi Dcvsk _v‡K Z‡e †mB ej Ges e‡ji w`‡K mi‡Yi Dcvs‡ki ¸b dj‡K abvZ¡K KvR ev e‡ji Øviv KvR e‡j| 





e¨vL¨v: W  F.S  FS cos  mgxKiY †_‡K †`Lv hvq †h, cos  abvZ¡K n‡j W abvZ¡K nq| ej F Ges miY S Gi AšÍ©fy³ †KvY  Gi gvb 90º Gi Kg n‡j A_©vr 0    90 n‡j cos  abvZ¥K nq, ZLb e‡ji w`‡K mi‡Yi Dcvsk _v‡K; d‡j e‡ji Øviv KvR ev abvZ¥K KvR nq| D`vniY: GKwU e¯‘ Dci †_‡K gvwU‡Z †d‡j w`‡j e¯‘wU AwfKl© e‡ji cÖfv‡e gvwU‡Z co‡e| G‡ÿ‡Î cÖhy³ ej Z_v e¯‘i 



IRb mg Ges S GKB w`‡K A_©vr wb‡Pi w`‡K nq, d‡j e¯‘i Dci AwfKl© ej Øviv KvR n‡q‡Q ev abvZ¥K KvR n‡q‡Q eySvq| e‡ji weiæ‡× KvR (Work done against the force): hw` ej cÖ‡qv‡Mi d‡j e¯‘ e‡ji wecixZ w`‡K m‡i hvq Z‡e ev e‡ji wecixZ w`‡K mi‡Yi Dcvsk _v‡K Z‡e †mB ej Ges e‡ji wecixZ w`‡K mi‡Yi Dcvs‡ki ¸b dj‡K FbvZ¡K KvR ev e‡ji weiæ‡× KvR e‡j| 





e¨vL¨v: W  F.S  FS cos  mgxKiY †_‡K †`Lv hvq †h, cos  FbvZ¡K n‡j W FbvZ¡K nq| ej F Ges miY S Gi AšÍ©fy³ †KvY  Gi gvb 90º Gi ‡ekx n‡j A_©vr 90 0    180 0 n‡j cos  FbvZ¥K nq, ZLb e‡ji wecixZ w`‡K mi‡Yi Dcvsk _v‡K; d‡j e‡ji weiæ‡× KvR ev FbvZ¥K KvR nq| 

D`vniY: GKwU e¯‘ gvwU †_‡K Dc‡i DVv‡bv nj| Zvn‡j e¯‘i Dci AwfKl© ej Z_v e¯‘i IRb mg Lvov wb‡Pi w`‡K 

Ges S Lvov Dci w`‡K wµqv K‡i| G‡ÿ‡Î AwfKl© ej I miY wecixZ gyLx nIqvq AwfKl© e‡ji weiæ‡× KvR n‡q‡Q ev FbvZ¥K KvR n‡q‡Q eySvq| cwieZ©x ej Øviv K…Z Kv‡Ri ivwkgvjv (GKgvwÎK †ÿ‡Î): aivhvK, †Kvb e¯‘i Dci X Aÿ eivei F ej wµqv Ki‡Q| Zvn‡j e¯‘i miY n‡e x Aÿ eivei| aivhvK, GB ej aªæe bq Ges Gi gvb mi‡Yi Dci wbf©ikxj A_©vr F, x Gi Av‡cÿK ev F(x)| GLb ej ebvg †jLwPÎ wbgœiƒc n‡e, †hLv‡b X A‡ÿi w`‡K miY x I Y A‡ÿi w`‡K ej F †`Lvb n‡q‡Q| facebook /gmail/skype: -tanbir.cox

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06| KvR, kw³ I ¶gZv (Work, Energy And Power)

2

aivhvK, †Kvb e¯‘i Dci F(x) ej cÖ‡qv‡Mi d‡j nq x1 †_‡K x2| GLb GB miY‡K x ˆ`‡N©¨i ÿz`ª ÿy`ª As‡k fvM Kwi| g‡b Kwi fvM msL¨v N (wPÎ-1)| e¯‘ x1 Ae¯’vb †_‡K x1+x Ae¯’v‡b A_©vr x ÿz`ª mi‡Yi Rb¨ ej cÖvq aªæeK we‡ePbv Kiv hvq| g‡bKwi GB aªæe ej F1| Zvn‡j GB As‡k K…Z KvR, W1=F1x Avevi x1+x †_‡K x1+2x Ae¯’v‡b miv‡Z A_©vr x mi‡Yi Rb¨ aªæe ej F2 n‡j G‡ÿ‡Î K…Z KvR, W2=F2x Gfv‡e, cieZ©x x mi‡Yi Rb¨ aªæe ej F3 n‡j G‡ÿ‡Î K…Z Kv‡Ri cwigvb, W3=F3x GLb e¯‘‡K x1 Ae¯’vb †_‡K x2 Ae¯’v‡b miv‡Z †gvU K…Z KvR n‡e G ai‡bi N msL¨K ÿz`ª ÿz`ª Kv‡Ri mgwói mgvb| †gvU KvR W n‡j, W = W1+W2+W3+.........................+WN = F1x + F2x + F3x +......................... +FNx N   F x i i 1 GLb Kv‡Ri mwVK gvb †c‡Z n‡j x1 †_‡K x2 ch©šÍ miY‡K AmsL¨ ÿy`ª ÿz`ª As‡k fvM Ki‡Z n‡e| (wPÎ-2) A_©vr hLb x  0 Ges fvMmsL¨v N   n‡e ZLb K…Z Kv‡Ri cwigvb n‡e, N W  lim F x  x  0 i i 1 x2 F( x )dx K¨vjKzjv‡mi fvlvq  lim x  0  x1 Ges GB K…Z KvR n‡e ej ebvg miY †jLwPÎ Øviv X A‡ÿi mv‡_ Ave× †ÿ‡Îi †ÿÎd‡ji mgvb hv wPÎ-3 G Qvqv hy³ AÂj Øviv †`Lv‡bv n‡q‡Q| †h‡nZz GLv‡b ej I mi‡Yi AšÍ©fy³ ‡KvY 0º ZvB, †jLv hvq, x2 W   Fdx cos 0 x1 x2     F . dx [µm ¸Y‡Yi wbqgvbymv‡i] x1

GwUB GKgvwÎK †ÿ‡Î cwieZ©x ej Øviv K…Z Kv‡Ri ivwkgvjv| w¯cÖs ej Øviv m¤úvw`Z KvR (Work done by the spring force)t w¯cÖs ej Øviv m¤úvw`Z KvR GK gvwÎK Kv‡Ri GKwU cÖK…ó D`vniY| GKwU w¯cÖs‡K †U‡b cÖmvwiZ Ki‡j ev †P‡c msKzwPZ Ki‡j w¯cÖs Gi Af¨šÍ‡i cÖwZµqv e‡ji m„wó nq| G cÖwZwµqv ej‡K w¯cÖs ej e‡j| w¯cÖs ej Fs, w¯cÖs Gi cÖmviY ev ms‡KvPb x Gi mgvbycvwZK I wecixZgyLx A_©vr F  - x s

 F  - k x GLv‡b k GKwU aªæeK| G‡K w¯cÖs aªæeK e‡j| s

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3 06| KvR, kw³ I ¶gZv (Work, Energy And Power) GLb w¯cÖs F  -k x ej Øviv K…Z KvR wn‡me Kiv hvK| aiv hvK, w¯cÖs e‡ji wµqvq GKwU e¯‘‡K x0 Ae¯’vb †_‡K x

s

Ae¯’v‡b Avbv nj| w¯cÖs e‡ji wµqvq e¯‘‡K dx cwigvb miv‡Z K…Z KvR, dW  F dx s

 dW   kx dx AZGe e¯‘‡K x0 Ae¯’vb †_‡K x Ae¯’v‡b wb‡Z †gvU KvR x W    kx dx x 0

x x2   W  k    2  x 0

1 1  W  k  x 2  x 2  GUvB wb‡Y©q Kv‡Ri ivwkgvjv| hw`, x0= 0 nq Z‡e, W   kx 2 | w¯cÖs msKzwPZ ev cÖmvwiZ hvB  2 2  0 1 †nvK A_©vr x abvZ¡K ev FbvZ¡K hvB †nvK bv †Kb Gi eM© me©`v abvZ¡K| d‡j mgxKiY W   kx 2 †_‡K ejv hvq, 2

w¯cÖs ej Øviv KvR me©`v FbvZ¡K n‡e| GKwU w¯cÖs –Gi ms‡KvPb ev cÖmvi‡Yi Rb¨ mwÂZ wefe kw³i ivwkgvjvt g‡b Kwi, GK cÖv‡šÍ `„pfv‡e Ave× GKwU w¯cÖs Gi gy³ cÖv‡šÍ m f‡ii GKwU e¯‘ AvUKvb Av‡Q| e¯‘wU GKwU Nl©Y wenxb Z‡ji Dci PjvPj Ki‡Z cv‡i| Avgiv Rvwb, w¯cÖs‡K Uvb Uvb Ki‡Z w¯cÖs e‡ji weiæ‡× KvR Ki‡Z n‡e| w¯cÖs e‡ji weiæ‡× K…Z GB KvRB w¯cÖs G wefe kw³ wnmv‡e weivR K‡i| w¯cÖswU‡K hLb Zvi wkw_j Ae¯’v x=0 †_‡K x=x Ae¯’v‡b Uvb Uvb Kiv nq, ZLb e¯‘wUi Dci cÖhy³ w¯cÖs Gi ej Fs=  kx| GLb e¯‘wU‡K x `~iZ¡ miv‡Z Zvi Dci Gi mgvb I wecixZgyLx F=ks ej cÖ‡qvM K‡i KvR Ki‡Z n‡e| GB ej Øviv K…Z KvRB n‡e e¯‘wUi mwÂZ wefe kw³| x

wefe kw³ U   Fdx 0 x

ev, U   kx dx 0 x

ev, U  k  x dx 0

x

 x2  ev, U  k    2 0 1  U  kx 2 BnvB wefe kw³i ivwkgvjv| 2

gnvKl© ej Øviv KvR (Work done by the gravitational force) t g‡b Kwi, O we›`y‡Z GKwU e¯‘ Av‡Q| e¯‘i fi M| O we›`y n‡Z r `~‡i A we›`y‡Z Aew¯’Z m f‡ii e¯‘i Dci wµqvkxj ej n‡e, F 

GMm GLv‡b, r2

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4 06| KvR, kw³ I ¶gZv (Work, Energy And Power) G -†K gnvKl© aªæeK e‡j| GB ej AO eivei wµqv Ki‡e| GLb AO Gi ms‡hvM ‡iLvi Dci A †_‡K dr ÿz`ª `~i‡Z¡ B we›`yi Ae¯’vb we‡ePbv Kiv hvK| Wvb cv‡k¦©i wP‡Î AB = dr LyeB ÿz`ª e‡j AB -Gi Gi me©Î gnvKl© e‡ji gvb mgvb we‡ePbv Kiv hvq| myZivs A we›`y †_‡K B we›`y‡Z m f‡ii e¯‘‡K Avb‡Z m¤úvw`Z Kv‡Ri cwigvb, dW  Fdr GMm  dW  dr GLb e¯‘wU‡K ra Ae¯’vb †_‡K rb Ae¯’v‡b miv‡Z m¤úvw`Z r2 r b †gvU KvR, W   dW r a

r b

W  r

GMm

dr

r2

a

r b

 W  GMm  r  2 dr r a

rb

 r 21   W  GMm     1  ra

1 1  W  GMm     rb ra  1 1  W  GMm    BnvB wb‡Y©q Kv‡Ri ivwkgvjv|  ra rb 

kw³ (Energy) t ‡Kvb e¯‘ ev e¨e¯’vi KvR Kivi mvg_©‡K kw³ e‡j| KvR I kw³i cwigvc, GKK I gvÎv GKB| kw³i GKK Ryj, kw³i gvÎv [ML2T-2]| Ae¯’v †f‡` gnvwe‡k¦ kw³ wewfbœ iƒ‡c Ae¯’vb K‡i, †hgbt hvwš¿K kw³, Zvc kw³, kã kw³, we`¨yr kw³, Pz¤^K kw³, Av‡jvK kw³, cvigvYyweK kw³ BZ¨vw`| hvwš¿K kw³ (Mechanical Energy) t ‡Kvb e¯‘ ev e¨e¯’vi ‡fŠZ Ae¯’v‡bi (w¯’wZ ev MwZ) Rb¨ GUv KvR Kivi †h mvg_© AR©b K‡i, Zv‡K hvwš¿K kw³ e‡j| hvwš¿K kw³ 2 cÖKvit 1| wefe kw³ (Potential Energy) 2| MwZ kw³ (Kinetic Energy) wefe kw³ (Potential Energy) t cvwicvwk©¦‡Ki mv‡c‡ÿ w¯’i _vKvi d‡j †Kvb e¯‘ ev e¨e¯’v KvR Kivi †h mvg_© AR©b K‡i ev kw³ mÂq K‡i, Zv‡K wefe kw³ ev w¯’wZkw³ e‡j| wefe kw³i cwigvc (Measurement of Potential Energy) t †Kvb e¯‘‡K GK w¯’wZkxj Ae¯’v †_‡K Ab¨ w¯’wZkxj Ae¯’vq wb‡Z †h cwigvb KvR m¤úv`b Ki‡Z nq Zv‡K H e¯‘i wefe kw³ e‡j| g‡b Kwi, m f‡ii e¯‘ f~ -c„ô n‡Z h D”PZvq †Kvb we›`y B †Z Ae¯’vb Ki‡Q| Gi wefe U kw³ wbY©q Ki‡Z n‡e|

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5 06| KvR, kw³ I ¶gZv (Work, Energy And Power) e¯‘wU‡K f~wg n‡Z h D”PZvq Zzj‡Z Gi Dci wµqviZ AwfKl© ej F= e¯‘i IRb = mg Gi weiæ‡× †h cwigvb KvR Ki‡Z nq, ZvBB B we›`yi wefe kw³i cwigvb wb‡`©k Ki‡e| AZGe, U = ej × miY = mgh | GLv‡b fi m aªæe ivwk Ges †Kvb wbw`©ó ¯’v‡b g aªæe  wbw`©ó ¯’v‡b, U  aª æeK  h  Uh A_©vr wefe kw³ D”PZvi mgvbycvwZK|

MwZkw³ (Kinetic Energy) t cvwicvwk©¦‡Ki mv‡c‡ÿ MwZkxj _vKvi d‡j †Kvb e¯‘ ev e¨e¯’v KvR Kivi †h mvg_© AR©b K‡i, Zv‡K MwZ kw³ e‡j| MwZkw³i cwigvc (Measurement of Kinetic Energy) t g‡b Kwi m f‡ii †Kvb w¯’i e¯‘i Dci GKwU aªæe ej F wµqv Kivq e¯‘wU v †eM cÖvß nj| Gi MwZ kw³ K wbY©q Ki‡Z n‡e| e¯‘wU‡K v †eM w`‡Z F e‡ji Øviv m¤úvw`Z Kv‡Ri cwigvbB Gi MwZkw³i cwigvb wb‡`©k Ki‡e| GLb, F e‡ji wµqvq e¯‘wU‡K dx ÿz`ªvwZÿz`ª `~iZ¡ miv‡Z K…ZKvR, dW  Fdx  dW  ma dx dv  dW  m dx dt dx  dW  m dv dt

 dW  mvdv

 F  ma    dv  dt  a 

  dx  dt  v 

AZGe e¯‘wU‡K †eM v = 0 n‡Z v = v †eM w`‡Z †gvU K…Z KvR A_©vr MwZkw³ W

K   dW 0 v

 K   mvdv 0

v

 v2   K  m   2  0 1  K  mv 2 e¯‘i fi m GKUv aªæeK Kv‡RB 2 K  aª æeK  v 2  K v 2 A_©vr, wbw`©ó f‡ii †Kvb e¯‘i MwZkw³ Gi †e‡Mi e‡M©i mgvbycvwZK|

MwZ kw³i mv‡_ fi‡e‡Mi m¤úK© (Relation with Kinetic Energy and Momentum) t 1 K  mv 2 2 1 m  K  mv 2  m 2

MwZkw³,

K K 

[ni I je‡K m Øviv ¸b]

1 m 2v2 2 m

p2 2m

[mv = fi‡eM p] BnvB MwZkw³ I fi‡e‡Mi g‡a¨ m¤úK©h³ y mgxKiY|

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6 06| KvR, kw³ I ¶gZv (Work, Energy And Power) msiÿbkxj ej (Conservative force) t †Kvb e‡ji Øviv GKwU e¯‘‡K c~Y© P‡µ cwiågb Kiv‡Z K…Z Kv‡Ri cwigvb k~b¨ n‡j, H ej‡K msiÿbkxj ej e‡j| AwfKl© ej GKwU msiÿbkxj ej| GKwU e¯‘‡K Lvov Dci w`‡K wb‡ÿc Ki‡j GUv wKQy mgq ci f~-c„‡ô wd‡i Av‡m| e¯‘wU Dc‡ii MwZi Rb¨ AwfKl© e‡ji weiæ‡× †h cwigvb KvR m¤úvw`Z nq, Zv cZb Kv‡j AwfKl© e‡ji Øviv K…ZKv‡Ri mgvb nq| e¯‘i GB PµvKvi åg‡bi Rb¨ †gvU K…Z KvR k~b¨ nq|

AwfKl© ej GKwU msiÿbkxj ej t Avgiv hw` GKwU e¯‘‡K AwfK‡l©i weiæ‡× Lvov Dc‡ii w`‡K wb‡ÿc Kwi, Z‡e GwU cybivq Avgv‡`i nv‡Z wd‡i Avm‡e| G ‡ÿ‡Î e¯‘wU nvZ †_‡K wbwÿß n‡q cybivq nv‡Z wd‡i Avmv GB c~Y© P‡µ KYvwUi Dci AwfKl© e‡ji m¤úvw`Z Kv‡Ri cwigvb k~b¨| m f‡ii GKwU e¯‘‡K f‚c„‡ôi A we›`y †_‡K h D”PZvq B we›`y‡Z DVv‡j AwfKl© e‡ji Rb¨ K…Z KvR FbvZ¥K nq| e¯‘wU‡K †h c‡_B (Wvb cv‡k¦©i wPÎ) DVv‡bv †nvK bv †Kb mKj †ÿ‡ÎB GB Kv‡Ri cwigvb nq mgh| AZGe AwfKl© ej Øviv m¤úbœ Kv‡Ri cwigvb †Kej we›`y `ywUi Ae¯’v‡bi Dci wbf©ikxj-KYvwUi MwZc‡_i Dci bq| ZvB AwfKl© ej GKwU msiÿbkxj ej| Z`ªæc Zwor ej, †PŠ¤^K ej, GKwU Av`k© w¯úªs- Gi ej cÖf…wZ msiÿbkxj ej| Amsiÿbkxj ej (Non-conservative force) t †Kvb e‡ji Øviv GKwU e¯‘‡K c~Y© P‡µ cwiågb Kiv‡Z K…Z Kv‡Ri cwigvb k~b¨ bv n‡j, H ej‡K Amsiÿbkxj ej e‡j| Nl©Y ej GKwU Amsiÿbkxj ej| mvB‡K¬vUªb h‡š¿ †h Z¡viK ej KvR K‡i, Zv Amsiÿbkxj ej| Nl©Y ej GKwU Amsiÿbkxj ej t g‡bKwi, GKwU e¯‘‡K †Kvb Agm„b Z‡ji Dci w`‡q ACB c‡_ A Ae¯’vb †_‡K B Ae¯’v‡b Ges c‡i BDA c‡_ B Ae¯’vb †_‡K cybivq A Ae¯’v‡b Avbv nj| MwZ c‡_ GKwU ÿz`ª miY dx Ges G miY Mo Nl©Y ej F Gi wecixZ w`‡K NU‡Q|  ACB c‡_ Pjvi Rb¨ Nl©Y e‡ji weiæ‡× KvR   Fdx  ACB c‡_ A n‡Z B we›`y‡Z †cŠQ‡Z e‡ji weiæ‡× KvR W1    Fdx c

Gi ci hw` e¯‘wU‡K cyYivq BDA c‡_ A we›`y‡Z Avbv nq, Z‡e G †ÿ‡ÎI Nl©Y ej e¯‘i wecixZ w`‡K wµqv Ki‡e| Kv‡RB K…Z Kv‡Ri cwigvb, W2    Fdx d

 e¯‘wU A we›`y n‡Z B we›`y‡Z Ges B we›`y n‡Z A we›`y‡Z hvZvqv‡Zi Rb¨ Nl©Y e‡ji weiæ‡× †gvU KvR, W  W1  W2    Fdx   Fdx  0 Kv‡RB Nl©Y ej GKwU Amsiÿbkxj ej| c

d

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7 06| KvR, kw³ I ¶gZv (Work, Energy And Power) msiÿbkxj ej I Amsiÿbkxj e‡ji g‡a¨ cv_©K¨: µwgK msiÿbkxj ej Amsiÿbkxj ej †Kvb e‡ji Øviv GKwU e¯‘‡K c~Y© P‡µ cwiågb †Kvb e‡ji Øviv GKwU e¯‘‡K c~Y© P‡µ cwiågb 1 Kiv‡Z K…Z Kv‡Ri cwigvb k~b¨ n‡j, H ej‡K Kiv‡Z K…Z Kv‡Ri cwigvb k~b¨ bv n‡j, H ej‡K msiÿbkxj ej e‡j| msiÿbkxj ej e‡j| msiÿbkxj ej wµqviZ _vK‡j hvwš¿K kw³i Amsiÿbkxj ej wµqviZ _vK‡j hvwš¿K kw³i 2 msiÿb m~Î Lv‡U| msiÿb m~Î Lv‡U bv| GB e‡ji Øviv K…Z KvR c‡_i cÖK…wZi Dci wbf©i GB e‡ji Øviv K…Z KvR c‡_i cÖK…wZi Dci wbf©i 3 K‡i bv| K‡i| 4 GB e‡ji Øviv K…Z KvR c~biæ×vi Kiv m¤¢e| GB e‡ji Øviv K…Z KvR c~biæ×vi Kiv m¤¢e bq| 5 GB ej Kvh©iZ _vK‡j kw³i AcPq N‡Ubv| GB ej Kvh©iZ _vK‡j kw³i AcPq N‡U|

KvRkw³ Dccv`¨ (Work-Energy theorem) t MwZkxj e¯‘i Dci cÖhy³ ej Øviv K…Z KvR e¯‘i MwZkw³i cwieZ©‡bi mgvb| aªæe e‡ji Rb¨ cÖwZcv`b: awi v0 †e‡M MwZkxj m f‡ii †Kvb e¯‘i Dci F aªæe ej wµqv K‡i| Gi d‡j †eM e¯‘wUi nq v Ges H mg‡q e¯‘wUi e‡ji w`‡K x `~iZ¡ AwZµg K‡i| myZivs ej Øviv K…ZKvR, W=Fx GB ej cÖ‡qv‡Mi d‡j e¯‘i aªæe Z¡iY a n‡j, wbDU‡bi MwZi wØZxq m~Îvbyhvqx, F=ma  W=max

wKš‘ MwZi mgxKiY †_‡K Avgiv Rvwb, v 2  v 2  2ax 0

v

2



v2 0



2ax

v2  v2 0

 ax 

2 v v    0 myZivs, W  m   2    1 1 1 1  W  mv 2  mv02 wKš‘ mv02 n‡”Q e¯‘i Avw` MwZkw³ K0 Ges mv 2 n‡”Q e¯‘i †kl MwZkw³ K | 2 2 2 2  W  K  K 0  K 2

2

A_©vrej Øviv K…Z KvR = e¯‘i MwZkw³ cwieZ©b| cwieZ©bkxj e‡ji Rb¨ cÖwZcv`b: aiv hvK, †KvY KYvi Dci cwieZ©bkxj ej wµqv Ki‡Q| e‡ji gvb cwieZ©bkxj n‡jI Gi w`K AcwieZ©bkxj| †m †ÿ‡Î KYvwUi mi‡Yi AwfgyL e‡ji w`‡KB n‡e| aiv hvK, KYvwU x- Aÿ eivei x

MwZkxj| GLb KYvwU‡K x0 Ae¯’vb †_‡K x Ae¯’v‡b miv‡Z cÖhy³ ej Øviv K…Z Kv‡Ri cwigvY, W   Fdx x0

wKš‘ wbDU‡bi MwZi wØZxq †_‡K Avgiv Rvwb, F=ma Avevi, a 

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dv dv dx dv dv  .  .v  v dt dx dt dx dx

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06| KvR, kw³ I ¶gZv (Work, Energy And Power)

x

x

 m¤úvw`Z KvR, W   madx   mv x0

wKš‘

x0

v

v

2

8

v

v  1 1 dv dx   mvdv m  vdv m    mv 2  mv02 2 dx  2  v0 2 v0 v0

1 1 2 mv0 n‡”Q e¯‘i Avw` MwZkw³ K0 Ges mv 2 n‡”Q e¯‘i †kl MwZkw³ K | 2 2

 W  K  K 0  K

A_©vrcwieZ©bkxj ej Øviv K…Z KvR = e¯‘i MwZkw³i cwieZ©b| MwZkw³ I wefe kw³i g‡a¨ cv_©K¨t µwgK MwZkw³ wefe kw³ cvwicvwk©¦‡Ki mv‡c‡ÿ MwZkxj _vKvi d‡j †Kvb cvwicvwk©¦‡Ki mv‡c‡ÿ w¯’i _vKvi d‡j †Kvb e¯‘ ev 1 e¯‘ ev e¨e¯’v KvR Kivi †h mg_© AR©b K‡i, Zv‡K e¨e¯’v KvR Kivi †h mg_© AR©b K‡i ev kw³ mÂq K‡i, Zv‡K wefe kw³ e‡j| G‡K w¯’wZ kw³I e‡j| MwZ kw³ e‡j| 2

m f‡ii e¯‘i †eMv n‡j Gi MwZ kw³ 

3

e¯‘i †eM bv _vK‡j MwZkw³ _v‡K bv|

1 mv 2 2

m f‡ii e¯‘ h D”PZvq _vK‡j Gi wefe kw³ = mgh

e¯‘i Ae¯’v ev Ae¯’v‡bi cwieZ©b bv n‡j wefe kw³ _v‡K bv| wefe kw³ †K P Øviv cÖKvk Kiv nq|

4 MwZ kw³ †K K Øviv cÖKvk Kiv nq| kw³i wbZ¨Zv m~Î (Conservation law of energy) t kw³i m„wó ev webvk †bB, kw³ †Kej GK iƒc †_‡K Aci GK ev ev GKvwaK iƒ‡c cwiewZ©Z n‡Z cv‡i| gnvwe‡k¦i †gvU kw³i cwigvb wbw`©ó I AcwieZ©bxq| e¨vL¨vt kw³i aŸskI ‡bB m„wóI ‡bB, Av‡Q ïay cwieZ©b| †hgb wnUv‡i we`¨yr kw³ Zvc kw³‡Z, ev‡j¦ we`¨yr kw³ Av‡jvK I Zvc kw³‡Z d¨v‡b we`¨yr kw³ hvwš¿K kw³‡Z iƒcvšÍwiZ nq| hvwš¿K kw³i wbZ¨Zv m~Î (Mechanical Conservation law of Energy) t ‡Kvb e¨e¯’vq †Kej msiÿbkxj ej wµqv Ki‡j e¨e¯’vi MwZ kw³ I wefe kw³i †hvMdj me©`v aªæe _v‡K| A_©vr, MwZkw³ + wefe kw³ = aªæeK cošÍ e¯‘i †ÿ‡Î hvwš¿K kw³i wbZ¨Zv m~Î (Conservation law of energy in case of falling body)t g‡b Kwi, m f‡ii GKwU e¯‘KYv f~-c„‡ôi A Ae¯’vb n‡Z B Ae¯’v‡b Avbv nj| B we›`y‡Z e¯‘wU w¯’i Ae¯’vq _v‡K e‡j Gi †Kvb MwZkw³ _v‡K bv| B we›`y‡Z MwZkw³ = 0 B we›`yi wefe kw³ = mgh B we›`y‡Z †gvU kw³ EB = MwZkw³ + wefe kw³  EB = 0 + mgh EB = mgh GLb aiv hvK, e¯‘wU B we›`y n‡Z Aev‡a cwZZ n‡jv Ges wKQy mgq ci x `yiZ¡ AwZµg K‡i C we›`y‡Z †cŠwQj

Ges v †eM cÖvß n‡jv| Zvn‡j cošÍ e¯‘i MwZi mgxKiY Abymv‡i cvB, v2= o+2gx v2= 2gx 1 2 1  K c  m  2gx 2  K c  mgx

C we›`y‡Z e¯‘i MwZkw³ K c  mv 2

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06| KvR, kw³ I ¶gZv (Work, Energy And Power)

9

f~-c„ô n‡Z C we›`yi D”PZv (h  x) ; Kv‡RB C we›`y‡Z e¯‘i wefe kw³ U c  mg (h  x )  U c  mgh  mgx C we›`y‡Z †gvU kw³ EC = MwZkw³ + wefe kw³ = KC+UC  Ec=mgx + mgh - mgx = mgh A_©vr, B we›`y‡Z †gvU kw³ EB = C we›`y‡Z †gvU kw³ EC

e¯‘wU f~-c„‡ô cwZZ n‡j GB kw³ kã, Zvc I Ab¨vb¨ kw³‡Z iƒcvšÍwiZ nq| Abyiƒc fv‡e †`Lv‡bv hvq †h, f‚wg ¯úk© Kivi gyn~‡Z© A we›`y‡Z †gvU kw³ EA= mgh myZivs AwfK‡l©i cÖfv‡e gy³fv‡e cošÍ e¯‘i †ÿ‡Î me mgq wefe kw³ I MwZ kw³i mgwó mgvb _v‡K|A_©vr cošÍ e¯‘i †ÿ‡Î kw³i wbZ¨Zv m~Î cÖgvwYZ nj| mij ‡`vj‡Ki mvnv‡h¨ hvwš¿K kw³i wbZ¨Zvi m~‡Îi cÖgvY (Conservation law of energy in case of simple pendulum)t aiv hvK, OA GKwU mij †`vj‡Ki mvg¨ve¯’vb| e‡ei fi m| GwU `yj‡Z `yj‡Z †Kvb GK mgq m‡e©v”P we›`y C-‡Z †cuŠwQ‡j ÿwb‡Ki Rb¨ w¯’i n‡e| C we›`y, e‡ei mvg¨ve¯’vb A n‡Z AN D”PZvq Aew¯’Z| C we›`y‡Z mg¯Í kw³ wefe kw³| C we›`y‡Z wefe kw³ EP = mg ×AN Ges C we›`y‡Z MwZ kw³ EK = 0 C we›`y‡Z ‡gvU kw³ EP +EK = mg ×AN + 0 C we›`y‡Z ‡gvU kw³ = mg ×AN ‡`vjKwU †Kvb GK mgq B we›`y‡Z †cŠQ‡e| B we›`y‡Z e‡ei wefe kw³ I MwZkw³ DfqB _vK‡e| B we›`y‡Z wefe kw³ EP = mg ×AM 1 m(u 2  2gh ) 2 1  E K  m(0 2  2gh ) 2 1  E K  m  2gh 2  E K  mgh

B we›`y‡Z MwZ kw³ E K 

 E K  mg  NM

[GLv‡ b, h  NM ]

 EK  mg(AN AM) B we›`y‡Z ‡gvU kw³ = EP +EK = mg ×AM+ mg (AN  AM) = mg ×AM+ mg ×AN  mg × AM = mg ×AN = C we›`y‡Z ‡gvU kw³, Gfv‡e †`Lvb hvq †h, e‡ei MwZ c‡_i me©Î †gvU kw³

aªæeK| AZGe, mij †`vj‡Ki †ÿ‡Î kw³i wbZ¨Zv m~Î cÖgvwYZ nj| ÿgZv (Power) t †Kvb e¨vw³ ev h‡š¿i KvR Kivi nvi‡K ÿgZv e‡j| ÿgZv‡K P Øviv cÖKvk Kiv nq| t mg‡q W KvR Ki‡j ÿgZv P

W t

n‡e| ÿgZvi GKK IqvU| ÿZvi gvÎv [ML2 T 3 ]

IqvU (Watt) t 1 †m‡K‡Û 1 Ryj KvR Kivi ÿgZv‡K GK IqvU ÿgZv e‡j| IqvU †K W Øviv cÖKvk Kiv nq| IqvU = Ryj/ †m‡KÛ|

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06| KvR, kw³ I ¶gZv (Work, Energy And Power)

10

Ak¦ÿgZv (H.P) t 1 †m‡K‡Û 550 dzU cvDÛ KvR Kivi ÿgZ‡K 1 Ak¦ÿgZv e‡j| 1 Ak¦ÿgZv = 550 dzU-cvDÛ/†m‡KÛ| IqvU Gi mv‡_ Gi m¤úK© nj, 1 Ak¦ÿgZv = 746 IqvU| 1 H.P  550ft lb / s  1 H.P  550  30.48  10 -2 m  0.4536Kg / s

 1 H.P  550  30.48  10 -2  0.4536 kg m / s  1 H.P  550  30.48  10 -2  0.4536  g Ryj / † m‡ KÛ  1 H.P  550  30.48  10 -2  0.4536  9.81 Watt  1 Ak¦ÿgZv = 746 IqvU|

Kg©`ÿZv (Efficiency) t ‡Kvb hš¿ †_‡K cÖvß †gvU Kvh©Ki kw³ I h‡š¿ cÖ`Ë †gvU kw³i AbycvZ‡K H h‡›`ªi Kg©`ÿZv e‡j| Kg©`ÿZv‡K  Øviv cÖKvk Kiv nq| Kg©`ÿZv,  

† gvU Kvh Ki kw³ † gvU cÖ`Ë kw³

kZKivq cÖKvk Ki‡j,  

† gvU Kvh Ki kw³  100 % † gvU cÖ`Ë kw³

‡Kvb h‡š¿i Kvh©Ki ÿgZv 50% ej‡Z eywS †h, hw` H h‡š¿ 100 Ryj kw³ †`Iqv nq Z‡e hš¿wU †_‡K 50 Ryj kw³ Kvh©Ki n‡e|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

6| KvR ÿgZv I kw³ (Work, Energy And Power) 1| 100m Mfxi GKwU Kzqv †_‡K Bwćbi mvnv‡h¨ cÖwZ wgwb‡U 1000kg cvwb DVv‡bv nq| hw` Bwćbi ÿgZv 42% bó nq Z‡e Gi Ak¦ ÿgZv wbY©q Ki|

1  Ε k   40  302 J 2

Avgiv Rvwb,

mgh t 1000 9.8 100 W ev, P1  60 ev, P1  16333.33 W 16333 .33 H.P ev, P1  746  P1  21.89 H.P Avevi, P  58%  P1 58 P  21.89 100 21.89  100 H.P P  58  P  37.74 H.P (Ans.) P1 

GLv‡b, MfxiZv, h = 100 m mgq, t = 1min = 60 sec. fi, m = 1000 kg Kvh©Kix ¶gZv, P1 = (10042)% = 58% cÖK…Z ¶gZv, P = ?

GLv‡b, awi,fi m I †eM v1

1 m v12 2

d‡j, 16 wU Z³v †f` Kivi kw³, E 2 

1 m v 22 2

cÖkœg‡Z, E2 = 16E1

1 1 m v 22  16  m v12 2 2 2 2  v 2  16 v1 

3| w¯’ive¯’v †_‡K 40kg fi wewkó †Kvb e¯‘ wbw`©ó e‡ji wµqvi d‡j 2s ci 15ms-1 †eM AR©b K‡i| Gi Dci wK cwigvb ej wµqv Ki‡Q Ges 4s ci Gi MwZ kw³ KZ n‡e?

15  7.5ms  2 2

4| GKwU ivB‡d‡ji ¸wj wbw`©ó cyiæ‡Z¡i GKwU Z³v †f` Ki‡Z cv‡i| H iƒc 16 wU Z³v †f` Ki‡Z n‡j Gi †eM KZ ¸b Ki‡Z n‡e?

16 wU Z³v †f` Kivi †eM v2

 Ε133 . 6 1036 J (Ans.)

 a

 Εk 18000 J (Ans.)

d‡j, 1 wU Z³v †f` Kivi kw³ E1 

2| GKwU wbDUª‡bi fi 1.67×10-27 kg Ges GwU 4×10-4 ms-1 †e‡M MwZkxj| Gi MwZ kw³ wbY©q Ki| GLv‡b, Avgiv Rvwb, fi, 1 m = 1.67×10-27 kg MwZkw³, Ε mv 2 ‡eM, v = 4×10-4 ms-1 2 MwZkw³, E = ?

1  Ε 1.6710-27 (410-4 )2 2

Avgiv Rvwb, v1 = u + a t1 ev, 15 = 0 + a×2

1 Avevi, MwZkw³ , Ε k  m  v 22 2

GLv‡b, Avw`‡eM, u = 0 fi, m = 40kg ‡eM, v1 = 15ms-1 mgq, t1 = 2s, ej, F = ? mgq, t2 = 4s, n‡j MwZkw³, Ek=?

Avevi, F = ma  F =40×7.5 N = 300 N (Ans.) Avevi, v2 = u + at2 ev, v2 = 0 +7.5×4  v2 = 30ms-1

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 v 22  ( 4v1 ) 2  v 2  4v1 A_©vr †eM Pvi ¸b Ki‡Z n‡e| 5| GKwU gUi wgwb‡U 5.5×105 kg cvwb 100m Dc‡i DVv‡Z cv‡i| gUiwUi ÿgZv 70 % Kvh©Ki n‡j Gi ÿgZv KZ H.P wbY©q Ki| Avgiv Rvwb,

mgh t 5.5105  9.8100 ev, P1  W 60 ev, P1  8983333.333W 8983333.333 H.P ev, P1  746  P1  12042.00 H.P Avevi, P  70%  P1 70 P  12042.00 100 12042.00  100 H.P P  70  P  17202 . 86 H.P (Ans.) P1 

GLv‡b, MfxiZv, h = 100 m mgq, t = 1min = 60 sec. fi, m = 5.5×105 kg Kvh©Kix ¶gZv, P1 = 70% cÖK…Z ¶gZv, P = ?

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6| KvR ÿgZv I kw³ (Work, Energy And Power)

6| GKwU gUi 4.9 wgwb‡U 10000 wjUvi cvwb 6m Dc‡i DVv‡Z cv‡i| gUiwUi ÿgZv 80 % Kvh©Ki n‡j Gi ÿgZv KZ wbY©q Ki| Avgiv Rvwb, GLv‡b,

mgh t 10000 9.8 6 W ev,P1  60 4.9  P1  2000 W Avevi, P  80%  P1 80 P  2000 100 2000  100 W  P  80  P  2500 W (Ans.) P1 

MfxiZv, h = 6 m mgq, t = 4.9min = 60×4.9s 1wjUvi cvwbi fi, =1 †KwR 10000 wjUvi cvwbi fi, m =10000 †KwR Kvh©Kix ¶gZv, P1 = 80% cÖK…Z ¶gZv, P = ?

7| GKwU gUi N›Uvq 2.5×107 kg cvwb 50m Dc‡i DVv‡Z cv‡i| gUiwUi ÿgZv 45 % Kvh©Ki n‡j Gi ÿgZv wbY©q Ki| Avgiv Rvwb, GLv‡b, mgh P1  MfxiZv, h = 50 m t mgq, t = 3600sec.

7 2.5  10  9 . 8  5 0 W ev, P1  fi, m =2.5×107 kg 3600 Kvh©Kix ¶gZv, ev, P1  3402777.77 W P1 = 45% Avevi, P  45%  P1 cÖK…Z ¶gZv, P = ?

45  3402777.77 100 3402777.77 100 W P 45  P  7561728 . 38 W (Ans.) P

8| 30m D”PZv †_‡K GKwU e¯‘‡K webv evavq co‡Z w`‡j †Kv_vq Dnvi MwZ kw³ wefe kw³i wظb n‡e? g‡b Kwi e¯‘wUi fi m, Ges f‚c„ô †_‡K h D”PZvq e¯‘wUi MwZkw³ wefe kw³i wظb nq| kZ©vbymv‡i, 1 2 mv  2mgh 2  v 2  4 gh  0 2  2 g (30  h)  4 gh

 30  h  2h  3h  30 30 h 10  h  10 m

9| GKRb †jvK I GKRb evjK GK‡Î †`Šov‡”Qb| evjKwUi fi †jv‡Ki f‡ii A‡a©K Ges †jvKwUi MwZkw³ evjKwUi MwZkw³i A‡a©K| †jvKwU hw` Zvi †eM 1ms-1 e„w× K‡ib Z‡e Zvi MwZkw³ evjKwUi MwZkw³i mgvb nq| G‡`i Avw`‡eM wbY©q Ki| g‡b Kwi, evjKwUi Avw`†eM= v1 ‡jv‡Ki Avw`†eM = v2 ‡jv‡Ki fi = m m  evj‡ Ki fi  2

1 cÖkœvbymv‡ i, † jvKwUi MwZkw³   evjKwUi MwZkw³ 2

1 1 1 m mv 22     v12 2 2 2 2 2 v ev , v 22  1 4 v ev, v 2  1 ... ... ... ... (1) 2

ev,

Avevi cÖkœvbymv‡ i,

1 1 m m(v 2  1) 2    v12 2 2 2

ev, (v 2  1) 2 

v12 2

v1 2 v v ev, 1  1  1 2 2 ev, 0.5v1  1  0.707 v1 ev, 0.207v1  1 1  v1   4.83 ms 1 (Ans.) 0.207 ev, v 2  1 

Ges, v 2 

v1 4.83   2.42 ms -1 (Ans.) 2 2



10| GKwU KYvi Dci F  6ˆi  3ˆj  2 kˆ N ej cÖ‡qv‡M KYvwUi

 r  2ˆi  2ˆj  kˆ m miY nq| ej KZ…K m¤úvw`Z Kv‡Ri cwigvb

wbY©q Ki| Avgiv Rvwb,

 W  F.r  W  (6ˆi  3ˆj  2kˆ ). ( 2iˆ  2ˆj  kˆ )  W  (6)(2)  (3)(2)  ( 2)(1)  W  12  6  2  W  4 J (Ans.)

DËi: 10m D”PZvq MwZ kw³ wefe kw³i wظb n‡e|

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6| KvR ÿgZv I kw³ (Work, Energy And Power)

3

11| GKwU cvwbc~Y© Kzqvi MfxiZv 12m Ges e¨vm 1.8m | GKwU cv¤ú 24 wgwb‡U KzqvwU‡K cvwbk~Y¨ Ki‡Z cv‡i| cv¤úwUi Ak¦ÿgZv KZ? Avgiv Rvwb,

13| 200 gm f‡ii GKwU e¯‘ 10m Dci †_‡K bx‡P c‡o hvq| f~c„ô ¯úk© Kivi c~e© gyn‡~ Z© Gi MwZ kw³ wbY©q Ki|

fi m  r 2 l

1 fi, E k  mv 2  mgh m = 0.2 kg 2  0.2  9.8 10  19.6J (Ans.) h=10m

 m  3.14  (0.9) 2  12  10 3 GLv‡b, 1. 8  m  30520.8 Kg e¨vmva© r = = 0.9m 2 Avevi, MfxiZv l = 12m mgh P cvwb DVv‡bvi Mo D”PZv t 12 h= m = 6m 30520.8  908  6 P Watt 2 24  60 mgq t = 24 wgt = 24× 60 †mt 30520.8  908  6 H.P 24  60  746  P  1.67 H.P (Ans.) P

cv¤úwUi Ak¦¶gZv P = ?

12| 6kg fi wewkó GKwU e¯‘ w¯’i Ae¯’vq wQj| 30N ej cÖ‡qvM Kivq 10s ci e¯‘wUi MwZkw³ KZ n‡e?

Avgiv Rvwb, F = ma GLv‡b,  30=6×a mgq, t = 10s.  a = 5ms-2 fi, m = 6kg Avevi, ej F =30N

v  u  at  v  0  5  10  v  50ms 1

MwZkw³, Ek =?

Avgiv Rvwb,

GLv‡b,

MwZkw³, Ek = ?

14| h wgUvi DuPy ¯’vb ‡_‡K GKwU e¯‘ co‡Q| †Kv_vq Zvi MwZ-kw³ w¯’wZ-kw³i A‡a©K n‡e? g‡b Kwi e¯‘wUi fi m, Ges f‚c„ô †_‡K x D”PZvq e¯‘wUi MwZ-kw³ w¯’wZ-kw³i A‡a©K n‡e? kZ©vbymv‡i, 1 2  mv 2  mgx 2  v 2  gx  0 2  2 g (h  x)  gx  2( h  x )  x  2h  2 x  x  3x  2h 2h x  3 DËi: 2h/3 D”PZvq MwZ kw³ wefe kw³i A‡a©K n‡e|

Avevi, Ek 

1 2 mv 2

1  6  502 2  K  7500J(Ans.) K

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 ভ঴াকল঱ঃ ভ঴াবফনেয যম যকান দু বট ফস্ত্ত্তকণয ভধযকায আকল঱ন ফরনক ভ঴াকল঱ ফর ফনর।  বনউটননয ভ঴াকল঱ ঳ূ ত্রঃ ভ঴াবফনেয প্রনতযকবট ফস্ত্ত্তকণা এনক অ঩যনক তানদয ঳িংনমাগ ঳যর যযখা ফযাযফ একবট ফনর আকল঱ণ কনয। এ আকল঱ণ ফনরয ভান, ফস্ত্ত্ত ফা ফস্ত্ত্ত কণািনয়য বনযয গুণপনরয ঳ভানু ঩াবতক এফিং এনদয ভধযকায দূ যনত্বয ফনগ঱য ফযস্তানু ঩াবতক এ গাবণবতকবানফ

ভ঴াকল঱ীয় প্রাফরযঃ ভ঴াকল঱ীয় যক্ষনত্র যকান বফন্দুনত একক বয ঳ম্পনড়ফ একবট ফস্ত্ত্ত িা঩ন কযনর, ফস্ত্ত্তবট যম আকল঱ণ ফর অনু বফ কনয তানক ঐ যক্ষনত্রয দরুন ঐ বফন্দুয তীব্রতা ফা প্রাফরয ফনর। -1

 এবট একবট বদক যাব঱। এয একক (S.I.) Nkg

গাবণবতক বানফ,

 ভ঴াকল঱ীয় বফবফঃ ভ঴াকল঱ীয় যক্ষনত্রয যকান বফন্দুয বফবফ ফরনত একক বয ঳ম্পনড়ফ একবট ফস্ত্ত্তনক অ঳ীভ যথনক অথ঱াৎ যক্ষনত্রয ফাব঴য যথনক ঐ বফন্দুনত আননত ভ঴াকনল঱য বফরুনদ্ধ যম ঩বযভাণ কাজ কযনত ঴য় তানক ফু ঝায়। ভ঴াকল঱ীয় যক্ষত্রঃ ফৃ ঴ৎ বযবফব঱ি যকান ফস্ত্ত্তয ঑জন ফস্ত্ত্তয িাযবদনক যম অঞ্চনরয ভনধয এয আকল঱ণ ফর অনু বূত ঴য়, য঳ অঞ্চরনক ঐ ফস্ত্ত্তয ভ঴াকল঱ীয় যক্ষত্র ফনর।  ভুবিনফগঃ ঳ফ঱বনভড়ফ যম যফনগ যকান ফস্ত্ত্ত বনবক্ষপ্ত ঴নর তা আয ঩ৃ বথফীনত বপনয আ঳নফ না য঳ যফগনক ভুবি যফগ ফনর। এয ঳ফ঱বনভড়ফ -1

-1

ভান ঴নে 11.18kms ফা 6.98mils

যক঩রানযয ঳ূ ত্রঃ প্রভ ঳ূ ত্র (কনক্ষয ঳ূ ত্র): প্রনতযকবট গ্রন঴ই ঳ূ ম঱নক প্রবতবনয়ত নাববনত (Focas) যযনখ উ঩ফৃ ত্তাকায ঩নথ প্রবতবনয়ত ঳ূ ম঱নক প্রদবক্ষন কযনে।

বিতীয় ঳ূ ত্র (নক্ষত্রপনরয ঳ূ ত্র): যম যকান গ্র঴ এফিং ঳ূ নম঱য ঳ানথ ঳িংনমাগকাযী যযখা ঳ভান ঳ভনয় ঳ভান যক্ষত্রপর অবতক্রভ কনয। তৃতীয় ঳ূ ত্র (঩ম঱ায়কানরয ঳ূ ত্র): প্রবতবট গ্রন঴য ঩ম঱ায়কানরয ফগ঱ ঳ূ ম঱ ঴নত ঐ গ্রন঴য গড় দূ যনত্বয ঘনপনরয ঳ভানু ঩াবতক।  ১৯৫৭ ঳ানরয ৪ঠ্া অনক্টাফয য঳ৌববনয়ত ইউবনয়াননয বফজ্ঞানীযা ঳ফ঱ প্র ভ কৃবত্রভ উ঩গ্র঴ স্পু টবনক-১ যপ্রযনণয ভাধযনভ ভ঴া঱ূ নয অববমাননয ঩থ প্রদ঱঱ন কনয। একবট কৃবত্রভ উ঩গ্র঴নক ঩ৃ বথফীনক প্রায় 9.30 বক: বভ: ফা 7 ভাইর যফনগ বননক্ষ঩ কযনত ঩াযনরই তনফ তা ঩ৃবথফীনক ঩বযক্রভন কযনত থাকনফ।

 G এয ভান ঳ফ ঳ভয়ই ধ্রুফ থানক ফনর এনক বফেজনীন ভ঴াকল঱ীয় ধ্রুফক ফনর।

 অববকল঱ ফনরয প্রবানফ ভুিবানফ ঩ড়ন্ত ফস্ত্ত্তয যফগ ফৃ বদ্ধয ঴াযনক অববকল঱জ ত্বযণ ফনর। অববকল঱জ ত্বযণ ‘g’ যকান ধ্রুফ ঳িংখযা নয়। িাননবনদ এয ঩বযফত঱ন ঴য়। g এয ভান যভরুনত যফ঱ী (983.217cm/s2) এফিং বফলূ ফীয় অঞ্চনর কভ (978.039 cm/s2) বূ -঩ৃষ্ঠ যথনক উ঩নযয বদনক ফা বননিয বদনক g এয ভান কভ এফিংনকনদ্রতা ঱ূ নয (0)।  ঩ৃ বথফীয গড় ফযা঳াধ঱, R = 6.37× 106m

 ঩ৃ বথফীয গড় ঘনত্ব, ρ = 5.5 × 103 kgm-3

 ঩ৃ বথফীয বয, M = 5.975 × 1027gm

 ঳ূ নম঱য অববকল঱জ ত্বযনণয ভান ঩ৃবথফীয 27 গুণ এফিং িনদ্রতা g এয ভান ঩ৃবথফীয 1/6 বাগ।

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gnvKl© (Gravitation)t gnvwe‡k¦i cÖvZwU e¯‘ KYvB G‡K Aci‡K wb‡Ri w`‡K AvKl©b K‡i| GB AvKl©b e‡ji gvb ïay e¯‘؇qi fi I G‡`i ga¨Kvi `~i‡Z¡i Dci wbf©i K‡i G‡`i AvK…wZ, cÖK…wZ, AwfgyL I gva¨‡gi cÖK…wZi Dci wbf©i K‡i bv| gnvwe‡k¦i †h †Kvb `ywU e¯‘i ga¨Kvi GB AvKl©Y ej‡K gnvKl© e‡j| wbDU‡bi gnvKl© m~Î (Newton's Law of Gravitation)t gnvwe‡k¦i cÖwZwU e¯‘KYv G‡K Aci‡K wb‡Ri w`‡K AvKl©Y K‡i; GB AvKl©Y e‡ji gvb e¯‘ KYv؇qi f‡ii ¸Yd‡ji mgvbycvwZK Ges G‡`i ga¨eZ©x `~i‡Z¡i e‡M©i e¨v¯ÍvbycvwZK Ges GB ej e¯‘ KYv؇qi ms‡hvRK mij‡iLv eivei wµqv K‡i| e¨vL¨vt aivhvK, m1 I m2 f‡ii `ywU e¯‘ ci¯úi n‡Z d `~i‡Z¡ Aew¯’Z| G‡`i ga¨Kvi AvKl©b ej F n‡j gnvKl© m~Îvbymv‡i, F  m1m 2 hLb d aªæe _v‡K Ges F

1 d2

hLb m1 I m2 aªæe _v‡K|

 Abycv‡Zi m~Îvbymv‡i, F 

ev, F  G

mm 1

2

2

d mm 1

d

2

2

hLb m1, m2 I d cwiewZ©Z nq| ... ... ... (1) GLv‡b G GKwU mgvbycvwZK aªæeK| Gi gvb gnvwe‡k¦i me©Î

GKB †mB Rb¨ G‡K wek¦Rbxb ev me©Rbxb gnvKl©xq aªæeK e‡j| gnvKl©xq aªæeK (Gravitational Constant)t mm FG

1

d2

2

GB mgxKi‡Y m1= m2=1 GKK I d= 1 GKK n‡j F  G

1.1 12

ev, G = F nq|

msÁvt GKK f‡ii `ywU e¯‘ KYv GKK `~i‡Z¡ †_‡K ci¯úi‡K †h e‡j AvKl©b K‡i Zvi gvb‡K gnvKl©xq aªæeK e‡j| G Gi GKK, F  G

mm 1

d2

2

Fd 2 G GB mgxKi‡Y Wvb cv‡k¦©i ivwk¸‡jvi GKK emv‡j G-Gi GKK cvIqv hvq| mm 1

2

Gm AvB c×wZ‡Z G Gi GKK n‡”Q Nm2 kg -2, Gm AvB c×wZ‡Z G Gi gvb me©m¤§Z gvb 6.673× 10-11 Nm2 kg -2, G Gi gvb 6.673× 10-11 Nm2 kg -2 ej‡Z GB eywS ‡h, 1 kg e‡ii `ywU e¯‘ 1m `~‡i †_‡K ci¯úi‡K 6.673× 10-11 N e‡j AvKl©Y K‡i| 2 2  2  L2 G Gi gvÎv: G  Fd  ej  `~ iZ¡  MLT = [M-1L3 T-2]

mm 1

2

fi 2

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07| gnvKl© (Gravitation) 2 K¨v‡fwÛ‡mi c×wZ‡Z G-Gi gvb wbY©q (Determination of Gravitational Constant by Cavendish Method)t K¨v‡fwÛ‡mi h‡š¿ `ywU mgvb f‡ii †QvU †MvjK A I B GKwU nvjKv `Û R ‡K GKwU miæ Zvi F -Gi gva¨‡g GKwU `„p

Aej¤^b n‡Z Ggb fv‡e Szwj‡q †`Iqv nq †hb `ÛwU AbyfywgK _v‡K| j¨v¤ú I †¯‹j e¨e¯’vq R `‡Ûi N~Y©b †KvY cwigv‡ci Rb¨ Szjb Zv‡ii mv‡_ GKwU mgZj `c©b W hy³ _v‡K| cixÿvKv‡j`ywU mgvb f‡ii eo †MvjK C I D †K †QvU †MvjK ؇qi ci¯úi wecixZw`‡K GKB `~i‡Z¡ Ggbfv‡e ¯’vcb Kiv nq, †hb †MvjK PviwUi †K›`ª GKB Abyf~wgK Z‡j _v‡K| eo †MvjKØq ¯’vcb Kivi mv‡_ mv‡_ †QvU †MvjKØq G‡`i cÖwZ AvKwl©Z nq| d‡j R `‡Ûi Dci GKwU Ø›Ø wµqvkxj nq| GB ؇›Øi wµqvq R `Û Ny‡i hvh| d‡j Szjb Zv‡i cvK c‡o| Zv‡ii w¯’wZ¯’vcKZv GB cvK Lyj‡Z cÖqvm cvq; A_©vr Szjb Zv‡i cÖZ¨qbx ؇›Øi m„wó nq| R `‡Ûi mvg¨ve¯’vq GB Dfq ؇›Øi åvg‡Ki gvb mgvb nq| cixÿvq j¨v¤ú I †¯‹j e¨e¯’vq R `‡Ûi N~Y©b †KvY  †g‡c †bIqv nq| wnmve I MYbv t g‡b Kwi, cÖwZwU eo †Mvj‡Ki fi = M Ges cÖwZ †QvU †Mvj‡Ki fi = m| `‡Ûi mvg¨ve¯’vq eo I †QvU †Mvj‡Ki †K›`ªØ‡qi ga¨eZ©x `~iZ¡ = d n‡j, gnvKl© m~Îvbymv‡i cvB, F

GMm d2

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

R `‡Ûi `yB cÖv‡šÍ GB cwigvb ej wµqv Kivq GKwU ؇›Øi m„wó nq| GB ؇›Øi bvg we‡ÿcx Ø›Ø| R `‡Ûi ˆ`N©¨ l n‡j we‡ÿcx ؇›Øi åvgK = F  l GMml ... ... ... (2)  d2 GB ؇›Øi wµqvq R `Û Ny‡i hvq| d‡j Szjb Zv‡i cvK c‡o| Zv‡ii w¯’wZ¯’vcKZv GB cvK Lyj‡Z cÖqvm cvq| myZivs, Szjb GKwU cÖZ¨qbx ؇›Øi m„wó nq| R `‡Ûi we‡ÿc‡KvY n‡j, cÖZ¨qbx ؇›Øi åvgK = ... ... ... ... (3) GLv‡b  n‡jv Szjb Zv‡ii cvK aªªæeK| R `‡Ûi mvg¨ve¯’vq,

we‡ÿcx ؇›Øi åvgK = cÖZ¨qbx ؇›Øi åvgK GMml   d2

G 

 d 2 ... ... ... (4) Mml

GLv‡b, , d, M, m I l -Gi gvb Rvbv Av‡Q|  Gi gvb wbY©q Ki‡Z n‡e| GLb eo †MvjKØq‡K mwi‡q, R `Û‡K e¨eZ© †`vj‡b †`vj †`qv nq Ges †`vjb Kvj T wbY©q Kiv nq| R `‡Ûi RoZvi åvgK I n‡j, T  2

I 

4 2 I  2 4 I

ev, T 2   

GB  Gi gvb (4) bs mgxKi‡Y ewm‡q cvB,

T 2

G

4 I d 2 MmlT

2

2

... ... ... (5) GB mgxKi‡Yi Wvb cv‡k¦i ivwk ¸wji gvb ewm‡q G Gi gvb cvIqv hvq|

K¨v‡fwÛm Zvi cixÿvq †h gvb cÖvß nb Zv nj G = 6.673× 10-11 Nm2 kg -2 |

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07| gnvKl© (Gravitation)

3

AwfKl©R Z¡iY (Acceleration Due to Gravity)t AwfKl© e‡ji cÖfv‡e gy³ fv‡e cošÍ e¯‘i †eM e„w×i nvi‡K AwfKl©R Z¡iY e‡j| G Z¡iY me©`v Lvov wb‡Pi w`‡K wµqv K‡i| G‡K g Øviv cÖKvk Kiv nq| Gi GKK ms -2 | Gi gvÎv [LT -2] cÖkœt AwfKl©R Z¡iY‡K wKfv‡e c„w_exi fi, e¨vmva© Ges gnvKl© aªæe‡Ki mvn‡h¨ cÖKvk Kiv hvq †`LvI| DËit f~-c„ô n‡Z h `~‡i m f‡ii †Kvb e¯‘i Dci AwfKl© e‡ji wµqv we‡ePbv Kiv hvK, c„w_exi fi M Ges e¨vmva© R n‡j, e¯‘wUi Dci AwfKl© ej, F 

GMm Avevi, AwfKl©R Z¡iY g n‡j (R  h ) 2

wbDU‡bi MwZ m~Îvbyhvqx Avgiv cvB, F = mg d‡j, mg  GMm 2 (R  h )

g 

GM ... ... ... (1) (R  h ) 2

GUvB AwfKl©R Z¡i‡Yi Av`k© mgxKiY| G †_‡K †`Lv hvq †h, AwfKl©R Z¡iY mswkøó e¯‘i f‡ii Dci wbf©i K‡i bv| GM ... ... ... (2) mgxKiY (2) n‡Z ejv hvq, g -Gi gvb c„w_exi e¨vmv‡a©i Dci R2 wbf©i K‡i| f~-c„‡ôi †Kvb ¯’v‡b me©Rb M„nxZ gvb 9.8 ms-2| f~-c„‡ôi †Kvb ¯’v‡b g Gi gvb 9.8 ms-2 ej‡Z GB eywS †h, H ¯’v‡b f~-c„‡ô gy³ fv‡e cošÍ e¯‘i †eM cÖwZ †m‡K‡Û 9.8ms-1 nv‡i e„w× cvq| mgxKiY (1) I (2) AwfKl©R Z¡i‡Yi

Avevi f~-c„‡ô h=0 AvZGe, g 

mv‡_ c„w_exi fi, e¨vmva© Ges gnvKl© aªæe‡Ki mv‡_ m¤úK©h³ y mgxKiY| f~-c„ô †_‡K D”PZ¡i I wb¤§Z¡i †Kvb ¯’v‡b AwfKl©xq Z¡iY g Gi gv‡bi wKiƒc cwieZ©b nq MvwbwZK we‡køl‡bi mvnv‡h¨ †`LvI| f~-c„ô n‡Z h D”PZvq AwfKl©R Z¡iY : g‡b Kwi, c„w_exi fi M Ges e¨vmva© R n‡j f~-c„‡ô AwfKl©R Z¡iY, g 

GM ... ... ... (1) Avevi f~-c„ô n‡Z h R2

D”PZvq A_©vr c„w_exi †K›`ª n‡Z (R+h) Dc‡i AwfKl©R Z¡iY g  n‡j (9) bs mgxKiY Abymv‡i, GM ... ... ... (2) (R  h ) 2 (2) bs mgxKiY‡K (1) bs mgxKiY Øviv fvM K‡i cvB, g 

g GM R2   g (R  h ) 2 GM g R2  g (R  h ) 2 g 1   g  R  h 2  R     g 1   2 g  h 1  R   



g  h  2  1   GB mgxKiY‡K evB‡bvwgqvj ZË¡¡vbymv‡i we¯Í…wZ K‡i Ges h ÿz`ª e‡j Gi D”PNvZ AMÖvn¨ K‡i cvB, R g  R g 2h 1 g R



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07| gnvKl© (Gravitation)

4

 2h   g   1   g ... ... ... (3) GUvB wb‡Y©q AwfKl©R Z¡iY|mgxKiY (3) n‡Z †`Lv hvq ‡h, g  g A_©vr f~-c„ô n‡Z R 

hZB Dc‡i DVv hvq AwfKl©R Z¡i‡Yi gvb ZZB Kg‡Z _v‡K| f~-c„ô n‡Z h MfxiZvq AwfKl©R Z¡iYt f‚-‡K‡›`ª g Gi gvb k~b¨| 4 3

g‡b Kwi, c„w_exi e¨vmva© R Ges Gi Mo NbZ¡ | Zvn‡j c„w_exi fi M  R 3 n‡e| 4  G   R 3  GM 3 AZGe, f~-c„‡ô AwfKl©R Z¡iY g n‡j g  2   2  R R 4  g   GR ... ... ... (1) 3 h MfxiZvq c„w_exi e¨vmva© (R-h) Ges Gi Mo NbZ¡ | Zvn‡j (R-h) e¨vmva© ch©šÍ c„w_exi 4 fi M  (R  h )3  n‡e|  c„w_exi †K›`ª n‡Z (R-h) `~‡i AwfKl©R 3 4 G ( R  h ) 3  GM Z¡iY g   3 (R  h ) 2 (R  h ) 2 4  g   G(R  h )... ... ... (2) 3 (2) bs mgxKiY‡K (1) bs mgxKiY Øviv fvM K‡i cvB, 4 G(R  h ) g 3  4 g GR 3 g R  h   g R h   g  1  g ... ... ... (3)  R GUvB wb‡Y©q AwfKl©R Z¡iY| mgxKiY (3) n‡Z †`Lv hvq, g  g A_©vr f~-c„ô n‡Z hZB c„w_exi †K‡›`ªi w`‡K hvIqv hvq, AwfKl©R Z¡i‡bi gvb ZZB Kg‡Z _v‡K| c„w_exi †K‡›`ª h = R;  R  g   1   g  0 ... ... ... (4) AZGe, g  0 c„w_exi †K‡›`ª (f‚-‡K‡›`ª) AwfKl©R Z¡i‡Yi gvb k~b¨|  R

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07| gnvKl© (Gravitation) 5 c„w_exi AvwýK MwZi Rb¨ AwfKl©R Z¡i‡bi cwieZ©b (Variation of acceleration due to gravity due to rotational motion of earth)t

c„w_exi AvwýK MwZi Rb¨ f~-c„‡ôi Dci¯’ †Kvb e¯‘KYv c„w_exi mgvb †KŠwbK †e‡M c„w_exi Aÿ YY' -‡K †K›`ª K‡i e„ËvKvi c‡_ Nyi‡e| d‡j e¯‘ KYvwU GKwU Ac‡K›`ª ej Fc Abyfe Ki‡e| wPÎ Abymv‡i, aiv hvK, m f‡ii †Kvb KYv P, c„w_exi mylg †KŠwYK ‡eM  mnKv‡i t e¨vmv‡a©i e„ËvKvi c‡_ Nyi‡Q| Zvn‡j Ac‡K›`ª ej Fc = mr n‡e| Fc -Gi w`K wP‡Î †`Lvb n‡q‡Q| P KYvwU Aÿvs‡k Ae¯’vb Ki‡j, c„w_exi †K›`ª O n‡Z OP eivei Fc -Gi Dcvsk Fc cos n‡e| e¯‘KYvi IRb mg -Gi wKQz Ask Fc cos -‡K cÖkwgZ Ki‡e| myZivs, c„w_ex KZ…K e¯‘KYvi Dci wµqviZ wbU ej, F= mg  Fc cos Aÿvs‡k AwfKl©R Z¡iY gy n‡j, Dc‡ii mgxKi‡Y, F= mg ewm‡q cvB, mg = mg mr cos ev, g = g r cos c„w_exi e¨vmva© R n‡j, r =Rcos n‡e| AZGe, g = g  R cos2(1) 

GUvB, c„w_exi AvwýK MwZi Rb¨ AwfKl©R Z¡i‡bi cwieZ©b m~PK mgxKiY| welye A‡j  =0 Ges cosd‡j, welye A‡j AwfKl©R Z¡i‡Yi gvb, g = g R (2) Avevi,†giæ A‡j  =90º Ges cosd‡j, †giæ A‡j AwfKl©R Z¡i‡Yi gvb, g = g (3) AZGe, †giæ A‡j AwfKl©R Z¡i‡Yi gvb me©vwaK Ges welye †iLv A‡j AwfKl©R Z¡i‡Yi gvb me©v‡cÿv Kg| AwfKl© †K›`ª (Gravitational Center) t GKwU e¯‘‡K †h fv‡eB ivLv †nvK bv †Kb e¯‘i wfZ‡i Aew¯’Z †h we›`yi ga¨w`‡q †gvU IRb wµqv K‡i †mB we›`y‡K e¯‘i AwfKl© †K›`ª e‡j| gnvKl©xq †ÿÎ (Gravitational Field) t †Kvb e¯‘i Av‡k cv‡k †h AÂje¨vcx Gi gnvKl© cÖfve eRvq _v‡K A_©vr Ab¨‡Kvb e¯‘ ivLv n‡j †mwU AvKl©b ej jvf K‡i, Zv‡K H e¯‘i gnvKl©xq †ÿÎ e‡j| gnvKl©xq †ÿÎ cÖvej¨ (Gravitation Field Strength or Gravitation Intensity)t gnvKl©xq †ÿ‡Îi †Kvb we›`y‡Z GKK f‡ii GKwU e¯‘ ¯’vcb Ki‡j †mwU †h ej jvf K‡i Zv‡K H we›`yi gnvKl©xq †ÿÎ cÖvej¨ e‡j| K…wÎg DcMÖn (Artificial Satellites) t gbyl¨ †cÖwiZ †h mKj gnvk~b¨hvb wbw`©ó Kÿc‡_ †_‡K c„w_ex‡K cwiågb K‡i, Zv‡K K…wÎg DcMÖn e‡j| i‡K‡Ui mvnv‡h¨ K…wÎg DcMÖn‡K wbw`©ó D”PZvq Zz‡j f~-c„‡ôi mgvšÍiv‡j wbw`©ó †e‡M Qy‡o †`qv nq| d‡j DcMÖnwU c„w_ex‡K cÖ`wÿY Ki‡Z _v‡K| f~-w¯’i DcMÖn (Geo-stationary Satellites)t ‡h mKj DcMÖ‡ni AveZ©b Kvj wbR A‡ÿi Pviw`‡K c„w_exi AveZ©b Kv‡ji mgvb, Zv‡`i f~-w¯’i DcMÖn e‡j| Giƒc DcMÖ‡ni MwZ‡eM Ggb †`qv nq, †hb GUv 24 N›Uvq c„w_ex‡K GKevi cÖ`wÿb K‡i| †mB Rb¨ f~-c„ô n‡Z GK Rb ch©‡eÿK DcMÖnwU‡K me mgq GKB RvqMvq †`‡L|

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07| gnvKl© (Gravitation)

6

K…wÎg DcMÖ‡ni †eM, AveZ©b Kvj I f~-c„ô †_‡K `~iZ¡t f~-c„ô n‡Z h D”PZvq m f‡ii GKwU K…wÎg DcMÖn v †e‡M e„ËvKvi c‡_ c„w_ex‡K cÖ`wÿYiZ Av‡Q| c„w_exi e¨vmva© R n‡j, K…wÎg DcMÖ‡ni ‡K›`ª gyLx ej, Fc 

mv 2 ... ... ... ... ... (1) (R  h )

c„w_exi fi M n‡j K…wÎg DcMÖ‡ni Dci wµqviZ AwfKl© ej, GMm ... ... ... ... (2) (R  h ) 2 MwZi mvg¨ve¯’vq, Fc  F n‡e| AZGe (1) I (2) bs mgxKiY e¨envi K‡i cvB, F

mv 2 GMm  (R  h ) (R  h ) 2 GM ... ... ... ... ... (3) BnvB K…wÎg DcMÖ‡ni †eM wbY©‡qi mgxKiY| Rh

v 

AveZ©b Kvj ev ch©vqKvjt ‡h mg‡q †Kvb DcMÖn c„w_ex‡K GKevi cÖ`wÿY K‡i, Zv‡K Gi ch©vqKvj e‡j| K…wÎg DcMÖ‡ni ch©vq Kvj T n‡j, Gi ˆiwLK †eM, v   (R  h ) , v 

2(R  h ) ... ... ... (4) GLb (3) I (4) bs mgxKiY †_‡K Avgiv cvB, T

2(R  h ) GM  T Rh T Rh   2 ( R  h ) GM (R  h )3 ... ... ... ... (5) BnvB ch©vq Kvj wbY©‡qi mgxKiY|  T  2 GM K…wÎg DcMÖ‡ni D”PZv wbY©‡qi Rb¨ (5) bs mgxKiY‡K eM© K‡i cvB, (R  h ) 3 T 2  4 2 GM GMT 2  (R  h )3  4 2

 GMT  R  h   2  4

2

  

1 3

1

 GMT 2  3   R ... ... ... ... ... (6) BnvB D”PZvi mgxKiY|  h   2   4 

K…wÎg DcMÖ‡ni e¨envit 1| AvenvIqv m¤úwK©Z M‡elbv, wbixÿY I c~e©vfvm cÖ`vb| 2| †eZvi I †Uwjwfk‡bi gva¨‡g Z_¨ cÖ`vb| 3| †eZvi I †Uwjwfkb †hvMv‡hv‡Mi †ÿ‡Î ix‡j †ókb wn‡m‡e KvR Kiv| 4| AvšÍgnv‡`kxq †hvMv‡hvM iÿv| 5| cÖwZiÿv, cvnviv I mvgwiK Kv‡R e¨envi BZ¨vw` Kv‡R e¨envi Kiv nq|

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07| gnvKl© (Gravitation)

7

K…wÎg DcMÖ‡n cÖ`wÿbiZ gnvk~b¨Pvix wb‡R‡K IRbnxb g‡b K‡ib †Kb? DËit e¯‘i IRb ej‡Z Gi Dci c„w_exi AvKl©b‡K eySvq| †Kvb ¯’v‡bi AwfKl©R Z¡iY g Ges Gi fi m n‡j, H ¯’v‡b e¯‘i Dci c„w_exi AvKl©b ej n‡e mg Ges GUvB e¯‘i IRb wb‡`©k K‡i| d‡j e¯‘i IRb g Gi wbf©i K‡i| myZivs e¯‘ IRbnxb n‡j g Gi gvb k~b¨ nq| c„w_ex‡K cÖ`wÿbiZ gnvk~b¨hv‡bi hvÎxiv IRbnxbZv Abyfe K‡ib| Gi KviY wbw`©ó D”PZvq Ae¯’vbiZ gnvk~b¨hvb c„w_exi †K‡›`ªi w`‡K wµqvkxj AwfKl©R Z¡iY g -Gi mggv‡bi wecixZgyLx Z¡iY cÖvß nq| GB Ae¯’vq gnvk~b¨hv‡bi cvUvZ‡bi mv‡c‡ÿ gnvk~b¨Pvixi Z¡iY g g = 0 nq, d‡j wZwb gnvk~b¨hv‡bi cvUvZ‡b ‡Kvb ej cÖ‡qvM K‡ib bv Ges †Kvb cÖwZwµqv ej A_©vr IRb Abyfe K‡ib bv| gnvKl©xq wefe (Gravitational Potential) t Amxg `~iZ¡ n‡Z GKK f‡ii †Kvb e¯‘‡K gnvKl©xq †ÿ‡Îi †Kvb we›`y‡Z Avb‡Z gnvKl© ej Øviv m¤úbœ Kv‡Ri cwigvb‡K H we›`yi gnvKl©xq wefe e‡j| G‡K V Øviv cÖKvk Kiv nq| m f‡ii †Kvb e¯‘‡K Amxg `‚iZ¡ n‡Z gnvKl©xq †ÿ‡Îi †Kvb we›`y‡Z Avb‡Z W KvR m¤úbœ n‡j H we›`yi gnvKl©xq wefe V 

W n‡e| KvR I fi DfqB †¯‹jvi ivwk myZivs gnvKl© m

wefe †¯‹jvi ivwk| we›`y f‡ii Rb¨ gnvl©xq †ÿ‡Î ‡Kvb we›`y‡Z gnvKl©xq wef‡ei ivwkgvjv wbY©q (Gravitational Potential due to Point Mass)t g‡b Kwi, A we›`y‡Z GKwU we›`y fi M Aew¯’Z| M fi †_‡K gnvKl©xq ‡ÿ‡Îi g‡a¨ r `yi‡Z¡ B we›`y‡Z wefe wbY©q Ki‡Z n‡e| AB †hvM Kwi| B we›`y‡Z GKK fi ¯’vcb Ki‡j Zvi Dci wµqvkxj gnvKl©xq ej, M  1 GM  2 , BA eivei| aiv hvK, B we›`yi wefe V| GLb r2 r hw` GB GKK f‡ii e¯‘wU M f‡ii e¯‘wUi mv‡_ AvKl©Y ej F Gi d‡j e‡ji Awfgy‡L ÿz`ªvwZÿz`ª `~iZ¡ dr m‡i C we›`y‡Z Av‡m, Zvn‡j KvR n‡e Fdr cos 0º Ges wef‡ei cwieZ©b dV=Fdr cos 0º dV=Fdr×1=Fdr GLb GB mgxKiY‡K r  ‡_‡K r = r GB mxgvi g‡a¨ mgvKjb K‡i B we›`y‡Z wefe V n‡j, FG

r

V   Fdr 

r

GM dr r2 

V

r

dr r2 

 V  GM  r

 V  GM  r  2dr 

r

 1  V  GM    r   1 1  V  GM     r  GM V   GUvB wb‡Y©q wefe| GL‡b FbvZ¡K wPý wb‡`©k K‡i †h, gnvKl© wef‡ei gvb Amx‡g †ekx| Amxg n‡Z r

†ÿ‡Îi w`‡K G¸‡Z _vK‡j wef‡ei gvb Kg‡Z _v‡K|

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07| gnvKl© (Gravitation)

8

gyw³ †eM Kv‡K e‡j? gyw³ †e‡Mi ivwkgvjv cÖwZcv`b Ki t gyw³ †eM (Escape Velocity) t me©v‡cÿv Kg †h †e‡M k~‡b¨ wbwÿß e¯‘ AwfK‡l©i cÖfve KvwU‡q gnvk~‡b¨ P‡j hvq, Zv‡K gyw³ †eM ev wb¯ŒgY †eM e‡j| G‡K ve Øviv cÖKvk Kiv nq| gyw³ †e‡Mi ivwkgvjv (Magnitude of Escape Velocity) t c„w_exi fi M n‡j Gi †K›`ª n‡Z r `~‡i m f‡ii †Kvb e¯‘i Dci wµqviZ AwfKl© ej, F  GMm GB e‡ji wµqvq m f‡ii e¯‘wU‡K dr ÿz`ª `~i‡Z¡ miv‡Z K…Z KvR, 2 r

dW  Fdr GMm  dW  2 dr c„w_exi e¨vmva© R n‡j f~-c„ô n‡Z m f‡ii e¯‘‡K gnvk~‡b¨ A_©vr, Amx‡g †cÖi‡Yi Rb¨ K…Z r 

†gvU KvR, W   dW R



GMm dr r2 R

W



 W  GMm  r 2 dr R



 r 21   W  GMm    1  R 

 1  W  GMm    r R  1 1  W  GMm      R GMm W  ... ... ... ... (1) R

e¯‘wU‡K gnvk~‡b¨ Mg‡bi Rb¨ GB cwigvb Kv‡Ri mgZzj¨ MwZkw³ AR©b Ki‡Z n‡e| AZGe, gyw³‡eM ve n‡j, Avgiv 1 GMm mv e2  2 R 2GM  ve  R

cvB,

cybivq,

ve 

2GMR R2

 ve  2gR ... ... ... ... (2)

GM   g  R 2  BnvB gyw³ †e‡Mi ivwkgvjv|

AwfKl© †K›`ª ev fvi‡K›`ª (Centre of Gravity) t GKwU e¯‘‡K †h fv‡eB ivLv †nvK bv †Kb e¯‘i wfZ‡i Aew¯’Z †h we›`yi ga¨ w`‡q †gvU IRb wµqv K‡i †mB we›`y‡K e¯‘i AwfKl© †K›`ª ev fvi‡K›`ª e‡j|

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07| gnvKl© (Gravitation)

9

MÖ‡ni MwZ msµvšÍ †Kcjv‡ii m~‡Îi eY©bv (Kepler's Laws of Planetary Motion) t cÖ_g m~Ît K‡ÿi m~Ît cÖwZ MÖnB m~h©‡K GKwU †dvKv‡m ( focus) †i‡L Dce„ËvKvi c‡_ Ny‡i| wØZxq m~Ît †ÿÎd‡ji m~Ît MÖn Ges m~‡h©i ms‡hvRK mij‡iLv mgvb mg‡q mgvb †ÿÎdj AwZµg K‡i| Z…Zxq m~Ît AveZ©b Kv‡ji m~Ît m~‡h©i Pviw`‡K cÖwZwU MÖ‡ni AveZ©bKv‡ji eM© m~h©‡_‡K H MÖ‡ni Mo `~i‡Z¡i Nbd‡ji mgvbycvwZK| †Kcjv‡ii m~‡Îi e¨vL¨vt cÖ_g m~‡Îi e¨vL¨vt wP‡Î ABCD GKwU Dce„ËvKvi Kÿc_| F I F GB Dce„‡Ëi `ywU †dvKvm| ‡Kcjv‡ii cÖ_g m~Îvbymv‡i m~h© GB †dvKvm `ywUi †h †Kvb GKwU‡Z _vK‡e Ges Dce„ËvKvi c‡_ Nyi‡e| wØZxq m~‡Îi e¨vL¨vt g‡b Kwi wP‡Î F †dvKv‡m m~h© Aew¯’Z| †Kvb MÖn hw` GB Kÿc‡_i A Ae¯’vb †_‡K B Ae¯’v‡b Avm‡Z t mgq †bq Ges C Ae¯’vb †_‡K D Ae¯’v‡b Avm‡ZI ‡mB GKB mgq t †bq Zvn‡j wØZxq m~Î Abymv‡i AFB †ÿÎdj CFD ‡ÿÎdj mgvb n‡e| Z…Zxq m~‡Îi e¨vL¨vt MÖn¸‡jv Dce„ËvKvi c‡_ m~h©‡K cÖ`wÿb K‡i| myZivs wewfbœ mgq m~h©‡_‡K †h †Kvb MÖ‡ni `~iZ¡ wewfbœ nq| aiv hvK, †h †Kvb MÖ‡ni m~h© †_‡K Mo `~iZ¡ R Ges H MÖ‡ni m~h©‡K GKevi cÖ`wÿb Ki‡Z T mgq jv‡M| †Kcjv‡ii Z…Zxq m~Îvbymv‡i, T 2  R 3 n‡e| A_©vr m~h© †_‡K R1, R2, R2, ... Mo `~i‡Z¡ Aew¯’Z MÖ‡ni AveZ©b Kvj h_vµ‡g T1,T2,T3, ... n‡j,

T2 1

R

3 1

T2

T2 

2

R

3 2

3



R

... ... ... ... 

aªæe n‡e|

3 3

 

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution 7| gnvKl© (Gravitation)

1| c„w_exi e¨vmva© R = 6.4l×103 km I gnvKl©xq aªeK G =6.67×10-11 Nm2kg-2 a‡i Gi Mo NbZ¡ wbY©q Ki| Avgiv Rvwb,

GLv‡b, e¨vmva©, R = 6.41×103km

GM R2

g 

= 6.41×106m

4 G π R3 ρ 3 g  R2

gnvKl©xq aªeK, G =6.67×10-11 Nm2kg-2 Mo NbZ¡,  = ?

AwfKl©R Z¡iY, g = 9.8sms-2

4  g  G π R ρ 3

3  9.8 4  3.14  6.4l106  6.67 1011



   5474 . 87kgm 3

(Ans.)

2| c„w_exi e¨vmva© 6.4l×106m I AwfKl©R Z¡iY 9.8ms-2 n‡j c„w_exc„ô n‡Z †Kvb e¯‘i gyw³ †eM wbY©q Ki| GLv‡b, Avgiv Rvwb, e¨vmva©, R = 6.4l×106m v e  2gR AwfKl©R Z¡i¡ Y,  ve  2  9.8 6.41106 ms-1 g = 9.8ms-2 gyw³†eM, ve = ?

-1

 ve  11208. 75 ms (Ans.) 6

3| c„w_exi e¨vmva© 6.4×10 m I Gi c„‡ô AwfKl©R Z¡iY 9.8 ms-2 | c„w_exc„ô n‡Z 6.4 ×105 m D”PZvq AwfKl©R Z¡i‡Yi gvb wbY©q Ki| Avgiv Rvwb,

R 2g g  (R  h) 2 ev, g 

(6.4106 )2 9.8 (6.4106  6.4105 )2

4.01408  1014 ev , g  4.95616  1013  g  8.099 ms -2 (Ans.)

gR 2 G

 M=6.0×1024 kg (Ans.) 5| c„w_ex‡K 6400Km e¨vmv‡a©i GKwU †MvjK ai‡j f~-c„ô n‡Z KZ D”PZvq AwfKl©R Z¡i‡Yi gvb f~-c„‡ôi AwfKl©R Z¡i‡Yi gv‡bi 641 Ask n‡e| Avgiv Rvwb,

R 2g GLv‡b, (R  h) 2 e¨vmva©, R = 6400Km g R2g = 6400000 m   64 (R  h) 2 D”PZv, h = ? g 1 R2 AwfKl R Z¡ iY, g   64 64 (R  h) 2 1 R   8 Rh  R  h  8R  h  7R  h  7  6400000  h  44800000 m  44800Km (Ans.) 6| e„n¯úwZi fi I e¨vmva© h_vµ‡g 1.9× 1027kg Ges 7× 107m n‡j, Gi gyw³ †eM wbY©q Ki| Avgiv Rvwb,

v e  2gR

GLv‡b, e¨vmva©, R = 6.4×106m AwfKl©R Z¡iY,

2GMR  ve  R2

g = 9.8 ms-2 D”PZv, h = 6.4 ×105 m

 ve 

AwfKl R Z¡ iY, g  

4| c„w_exc„‡ó g Gi gvb 9.8ms-2| c„w_exi e¨vmva© R = 6.4l×103km I gnvKl©xq aªeK G = 6.7×10-11 Nm2kg-2 n‡j c„w_exi fi wbY©q Ki| GLv‡b, Avgiv Rvwb, AwfKl©R Z¡iY, g = 9.8 ms-2 GM e¨vmva©, R = 6.41×103km g 2 =6.41×106 m R

ev, M 

9.8  (6.41  10 6 ) 2 6.7  10 11

g 

3g 4 RG

ρ 

ev, M 

gnvKl©xq aªeK, G=6.7×10-11 Nm2kg-2 c„w_exi fi, M = ?

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2GM R

GLv‡b, fi, M = 1.9× 1027kg e¨vmva©, R = 7× 107m gyw³‡eM, ve = ?

2  6.673  10 -11  1.9  10 27 7  10 7  v e  60187. 08 ms 1 (Ans.)  ve 

7| c„w_ex‡K 6.4×106 m e¨vmv‡a©i Ges 5.5gm/cc Nb‡Z¡i †MvjK g‡b K‡i Gi c„‡ô AwfKl©R Z¡iY wbY©q Ki| G = 6.673×10-11 Nm2kg-2 Avgiv Rvwb, GLv‡b, GM e¨vmva©, R = 6.4×106 m g 2 R gnvKl©xq aªeK, 4 G = 6.673×10-11 Nm2kg-2 G    R 3  c„w_exi NbZ¡ ,  = 5.5gm/cc 3 g 2 = 5.5×103kg/m3 R AwfKl©R Z¡iY, g =?

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7| gnvKl© (Gravitation)

G  4R g 3 6.673 10 11 4  3.14  6.4 10 6 5.5 10 3 g 3 2  g  9.83 ms (Ans.) 8| c„w_ex ‡_‡K 1600 km D”PZvq GKwU K…wÎg DcMÖn c„w_ex‡K †K›`ª K‡i e„ËvKvi c‡_ cÖ`wÿb Ki‡Q| Gi †eM †ei Ki| ‡`qv Av‡Q c„w_exi e¨vmva© 6.4×103 km, c„w_exi fi 6×1024 kg Ges G = 6.67×10-11Nm2kg-2| Avgiv Rvwb,

v

GM Rh -11

24

6.67  10  6.0  10 6.4  10 6  1600000  v  7072.84ms 1 (Ans.) v

GLv‡b, e¨vmva©, R = 6.4×106 m gnvKl©xq aªeK, G = 6.67×10-11 Nm2kg-2 c„w_exi fi, M=6.0×1024 kg D”PZv h = 1600000m DcMÖ‡ni †eM v =?

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9| f‚-c„‡ô †Kvb †jv‡Ki IRb 684N n‡j wZwb Puv‡` wM‡q KZUzKz IRb nviv‡eb? c„w_exi fi I e¨vmva© h_vµ‡g Puv‡`i fi I e¨vmv‡a©i 81 Ges 4 ¸b| Avgiv Rvwb,

GM M ........(1) R 2M GM E ........(2) gE  R 2E gM 



g M M M R 2E  2  gE RM ME

gM M (4R) 2  2 gE R 81M g 16  M  ...........(6) g E 81 ev,

c„w_ex c„‡ô IRb WE = 684N Ges Puv‡` IRb WM n‡j,

WM g M  WE gE

WM 16 16   WM  WE 81 WE 81 16  WM  684  N  135.11N 81  Puv‡` IRb nviv‡e, = (684-135.11) N = 548.89N (Ans.) 

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 ঳যর যদারগবতয রফব঱িযঃ i) ফস্ত্ত্তয গবত ঩ম঱ায় গবত ঴ইনফ;

ii) ফস্ত্ত্তয গবত ঳যর রযবখক গবত ঴ইনফ;

iii) ফস্ত্ত্তয উ঩য ত্বযণ ঳ফ঱দা একবট বনবদ঱ি বফন্দু অববভুনখ বক্রয়া কযনফ, বনবদ঱ি বফন্দুনক উ঴ায ঳াভযফিান ফা ভধযাফিান ফরা ঴য়; iv) ফস্ত্ত্তয উ঩য বক্রয়া঱ীর ত্বযণ ভধযািান ঴ইনত উ঴ানদয ঳যননয ঳ভানু ঩াবতক; v) ফস্ত্ত্তয ত্বযণ ঳যননয বফ঩যীত ভুখী ঴ইনফ; vi) ই঴া একবট যদারন গবত ঴ইনফ।  যদারনকয ফযফ঴াযঃ i) অববকল঱জ ত্বযণ g এয ভান বনণ঱য় ii) ঩া঴ানড়য উচ্চতা বনণ঱য়

iii) ঳ভয় বনণ঱য় iv) ঳যর যদারনকয ঳া঴ানময g এয ভান বনণ঱য়

 য঳নকন্ড যদারকঃ যম যদারনকয যদারনকার 2 য঳নকন্ড উ঴ানক য঳নকন্ড যদারক ফনর।উ঴ায কম্পািংক

cy/s।

একবট বনবদ঱ি িানন একবট য঳নকন্ড যদারনকয T঑ L বনবদ঱ি। ঳কর য঳নকন্ড যদারক অফ঱যই বফন঱ল ঳যর যদারক।  ভনন যাখনত ঴নফ 1. যকৌবণক বফস্তায 40 এয যফ঱ী ঴নর ঳যর যদারক ঳ূ ত্র যভনন িনর না। 3. একবট যদারকনক ঩া঴ানড়য উ঩নয খবনয ভনধয অথফা িদ্রতা ঩ৃ নষ্ঠ বননয় যগনর এয ভান কভ ঴঑য়ায কাযনণ যদারনকার ফাড়নফ এফিং যদারকবট ধীনয িরনফ। 4. একবট যদারকনক বূ -যকনদ্রতা বননয় যগনর এয g ভান ঱ূ নয ঴঑য়ায় যদারন কার অ঳ীভ ঴ইনফ। 5. ঳ভত্বযনণ উ঩নযয বদনক গবত঱ীর বরপনট যদারনকয যদারনকার হ্রা঳ ঩াইনফ এফিং ঳ভত্বযনণ নীনিয বদনক গবত঱ীর বরপনট যদারনকয যদারনকার ফৃ বদ্ধ ঩াইনফ। অথ঱াৎ প্রভ যক্ষনত্র যদারকবট ধীনয িরনফ এফিং বিতীয় যক্ষনত্র উ঴া দ্রুত িরনফ। 6. যভরু অঞ্চনর g এয ভান যফ঱ী, তাই যদারক ঘবড় দ্রুত িনর এফিং বফলু ফীয় অঞ্চনর g কভ, তাই উ঴া ধীনয িনর। আফায, গ্রীষ্মকানর যদারক ঘবড় ধীনয িরনফ এফিং ঱ীতকানর দ্রুত িরনফ। 7. যগারাকায ফফ ব঩ন্ডবট পাঁ঩া ঴নর, বননযট ঴নর ফা পাঁ঩া অিং঱ ঳ম্পূ ণ঱ তযর িাযা ঩ূ ণ঱ কযনর বায যকনদ্রতায যকান ঩বযফত঱ন ঴য় না ফনর এয কাম঱কয রদঘ঱য অ঩বযফবত঱ত থানক মায পনর যদারনকানরয঑ যকান ঩বযফত঱ন ঴য় না। 8. ফফবটনক তযর িাযা অধ঱঩ূণ঱ কযনর বাযনকদ্রতা বকেু টা বননি যননভ মায় পনর L যফনড় মায় এফিং T যফনড় মায় এফিং যদারক ধীনয িরনফ। 9. যদারনকয খবনয ববতয বননর ফা ঩া঴ানড় বননর, ঑খানন g কভ ফনর এয যদারনকার T ফাড়নফ পনর যদারক ধীনয িরনফ। 10. তা঩ভাত্রা ফৃ বদ্ধ য঩নর L ফৃ বদ্ধ ঩ানফ পনর T ফৃ বদ্ধ ঩ানফ এফিং যদারক আনস্ত িরনফ। তা঩ভাত্রা হ্রান঳ বফ঩যীত ঘটনা ঘটনফ। 11. িদ্রতা ঩ৃ নষ্ঠ বননর ফৃ বদ্ধ ঩ানফ এফিং যদারক ধীনয িরনফ। 12. ঳ভত্বযনণ উর্ধ্঱গাভী বরপনট g ফৃ বদ্ধ ঩ানফ ফনর T হ্রা঳ ঩ানফ এফিং ঳ভত্বযনণ বনম্নগাভী বরপনট g হ্রা঳ ঩ানফ ফনর T ফৃ বদ্ধ ঩ানফ পনর যদারক মথাক্রনভ দ্রুত ঑ আনস্ত িরনফ। facebook /gmail/skype: -tanbir.cox

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ch©ve„Ë MwZ (Periodic Motion): ‡Kvb MwZkxj e¯‘KYvi MwZ hw` Ggb nq ‡h, GwU Gi MwZ c‡_ †Kvb wbw`©ó we›`y‡K wbw`©ó mgq cici GKB w`K †_‡K AwZµg K‡i, Zvn‡j †mB MwZ‡K ch©ve„Ë MwZ e‡j| GB MwZc_ e„ËvKvi, Dce„ËvKvi, mij ‰iwLK I RwUj n‡Z cv‡i| Nwoi KuvUvi MwZ, m~‡h©i Pviw`‡K c„w_exi MwZ BZ¨vw` ch©ve„Ë MwZi D`vniY| ch©vqKvj (Time Period) : ch©ve„Ë MwZm¤úbœ †Kvb KYv †h wbw`©ó mgq cici †Kvb wbw`©ó we›`y‡K wbw`©ó w`K †_‡K AwZµg K‡i †mB mgq‡K ch©vqKvj e‡j| ¯ú›`b MwZ (Oscillation Motion): ch©ve„Ë MwZm¤úbœ †Kvb e¯‘ hw` ch©vqKv‡ji A‡a©K mgq †Kvb wbw`©ó w`‡K Ges evwK A‡a©K mgq GKB c‡_ Zvi wecixZ w`‡K P‡j Zv‡K Z‡e Gi MwZ‡K ¯ú›`b MwZ e‡j| mij †`vj‡Ki MwZ, K¤úbgvb myikjvKvi MwZ, MxUv‡ii Zv‡ii MwZ BZ¨vw` ¯ú›`b MwZi D`vniY| mij Qw›`Z ¯ú›`b (Simple Harmonic Oscillation): hw` †Kvb e¯‘i MwZ hw` Ggb nq †h, Gi Dci wµqviZ Z¡iY me©`v mvg¨ve¯’vb n‡Z KYvi mi‡Yi mgvbycvwZK I wecixZgyLx nq, Z‡e e¯‘i H MwZ‡K mij Qw›`Z ¯ú›`b e‡j| Aíwe¯Ív‡i mij †`vj‡Ki MwZ, K¤úbgvb myikjvKvi MwZ, MxUv‡ii Zv‡ii MwZ BZ¨vw` mij Qw›`Z ¯ú›`b MwZi D`vniY| mij Qw›`Z ¯ú›`‡bi ˆewkó (Charecteristics of Simple Harmonic Oscillation): 1| GwU GKwU ch©ve„Ë MwZ 2| GwU GKwU ¯ú›`b MwZ 3| GwU GKwU mij ˆiwLK MwZ 4| †h †Kvb mgq Z¡i‡Yi gvb mvg¨ve¯’vb †_‡K mi‡Yi gv‡bi mgvbycvwZK I wecixZgywL| 5| Z¡iY me©`v GKwU wbw`©ó we›`y AwfgyLx| K¤úv¼ (Frequency): GKK mg‡q hZ¸‡jv c~Y© ¯ú›`b m¤úbœ nq Kv‡K K¤úv¼ e‡j| G‡K f ev n Øviv cÖKvk Kiv nq| Gi GKK Hz| we¯Ívi (Amplitude): mij Qw›`Z ¯ú›`bkxj †Kvb KYv Gi mvg¨ve¯’vb †_‡K †h †Kvb GK w`‡K †h m‡e©v”P `~iZ¡ AwZµg K‡i Zv‡K Gi we¯Ívi e‡j| we¯Ívi‡K a Øviv cÖKvk Kiv nq| Gi GKK wgUvi| `kv (Phase): mij Qw›`Z ¯ú›`bkxj †Kvb KYvi `kv ej‡Z H KYvi †h †Kvb gyn‡~ Z© MwZi mvg¨K Ae¯’v A_©vr KYvwUi miY, †eM, Z¡iY, ej BZ¨vw` eySvq|

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2 08| mij Qw›`Z ¯ú›`b (Simple Harmonic Oscillation) mij Qw›`Z ¯ú›`‡bi e¨eKjbxq mgxKiY (Differential Equation of Simple Harmonic Motion ): Avgiv Rvwb mij Qw›`Z ¯ú›`‡bi Z¡iY mi‡Yi mgvbycvwZK Ges wecixZgyLx| Kv‡RB mij Qw›`Z ¯ú›`‡bi †ÿ‡Î ejI mi‡Yi mgvbycvwZK Ges wecixZgyLx n‡e| †KvY KYvi Dci wµqvkxj ej F Ges miY x n‡j mij Qw›`Z ¯ú›`‡bi †ÿ‡Î F  x  F   kx GLv‡b k GKwU aªæeK| GB k †K ejv nq ej aªæeK| Avevi wbDU‡bi MwZi wØZxq m~Î ‡_‡K Avgiv Rvwb, e¯‘i fi m I Z¡iY a n‡j ej F=ma n‡e| A_©vr, ma = F  ma   kx  F  kx  dv dv   a    m   kx  dt dt   d  dx  dx   v    m     kx  dt  dt  dt   2 d x  m 2  kx dt 2 k d x  2  x m dt 2 d x k  2  x  0 ... ... ... (1) dt m k k Avevi Avgiv Rvwb,   ev,   2 GB gvb mgxKiY (1) G ewm‡q cvB, m m 2 d x   2 x  0 BnvB mij Qw›`Z ¯ú›`‡bi e¨eKjbxq mgxKiY| GLv‡b †K MwZi †KŠwbK K¤úv¼ e‡j| 2 dt

mij Qw›`Z ¯ú›`‡bi e¨eKjbxq mgxKi‡Yi mgvavb (Solution of Differential Equation of SHM) : mij Qw›`Z ¯ú›`‡bi e¨eKjbxq mgxKiY d2x  2 x  0 2 dt a‡i ‡bB, Gi GKwU mgvavb nj, x  A sin (t  ) GLb GwU †h, e¨eKjbxq mgxKi‡Yi GKwU mgvavb Zv Avgiv wb‡b¥v³ Dcv‡q hvPvB Ki‡Z cvwi| mgxKiY x  A sin (t  ) †K mg‡qi mv‡c‡ÿ cici `yB evi e¨eKjb K‡i cvB, dx d  A sin( t  ) dt dt dx    A cos (t  ) dt d2x  2   2 A sin (t  ) dt d2x  A sin (t  )  x   2   2 x dt d2x d2x 2    x GB gvb   2 x  0 mgxKi‡Y ewm‡q cvB, 2 2 dt dt 2 2 evgcÿ =   x   x  0

ev, evgcÿ = Wvbcÿ

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3 08| mij Qw›`Z ¯ú›`b (Simple Harmonic Oscillation) myZivs ‡`Lv hvq e¨eKjbxq mgxKi‡Y x  A sin (t  ) emv‡j mgxKiYwU wm× nq| Kv‡RB GwU mij Qw›`Z ¯ú›`‡bi e¨eKjbxq mgxKi‡Yi GKwU mgvavb|

mij †`vjK (Simple Pendulum) : GKwU fvix AvqZbnxb e¯‘KYv‡K IRbnxb, bgbxq I AcÖmviYkxj myZv w`‡q Szwj‡q w`‡j GwU hw` webv evavq Gw`K Iw`K `yj‡Z cv‡i Z‡e Zv‡K mij †`vjK e‡j| wKš‘ ev¯Í‡e G iKg †`vjK cvIqv m¤¢e bq| ZvB GKwU nvjKv myZvi mvnv‡h¨ †Kvb `„p Aej¤^b †_‡K GKwU fvix e¯‘ Szwj‡q w`‡j e¯‘wU hw` webv evavq Gw`K Iw`K `yj‡Z cv‡i Z‡e myZv mn GB e¨ve¯’v‡K mij †`vjK e‡j| Wvb cv‡k©¦i wPÎwU mij †`vj‡Ki wPÎ| ee (Bob) : ‡h fvix e¯‘‡K myZvi mvnv‡h¨ Szwj‡q mij †`vjK ˆZix Kiv nq Zv‡K ee e‡j| wP‡Î ee C Syjb we›`y (Point of suspension): †h we›`y †_‡K myZvi mvnv‡h¨ ee‡K Szjvb nq ‡mB we›`y‡K Szjb we›`y e‡j| wP‡Î O n‡”Q Szjb we›`y| Kvh©Kix ˆ`N©¨ : Szjbwe›`y †_‡K e‡ei fvi‡K›`ª ch©šÍ `~iZ¡‡K mij †`vj‡Ki Kvh©Kix ˆ`N©¨ e‡j| wP‡Î Kvh©Kix ˆ`N©¨ OC=L | †`vj‡Ki myZvi ˆ`N©¨ l Ges e‡ei e¨vmva© r n‡j Kvh©Kix ˆ`N©¨ L= l+r n‡e| ¯^í we¯Ív‡i ¯úw›`Z mij †`vj‡Ki MwZ mij Qw›`Z ¯ú›`b MwZ : g‡bKwi, AB GKwU mij †`vjK| B Gi fvi‡K›`ª| m Gi fi| †`vjKwU‡K †`vj w`‡j  †KŠwbK miY m„wó K‡i AC Ae¯’v‡b Av‡m| GLb C we›`y‡Z Gi IRb mg Lvov wb‡Pi w`‡K wµqv K‡i| GB IRb‡K `ywU ci¯úi j¤^ Dcvs‡k fvM Kiv hvq| GKwU myZvi ˆ`N©¨ eivei CD -Gi w`‡K mgcos Ges AciwU mgsin Gi mv‡_ j¤^fv‡e ¯úk©K eivei CE Gi w`‡K wµqv K‡i| mgcos DcvskwU myZvi Uvb T Øviv wbw®Œq nq, myZivs GKgvÎ Kvh©Kix ej F n‡”Q mgsin Ges Gi w`K mg¨ve¯’v‡bi w`‡K|  F =  mgsin‡h‡nZz Kvh©Ki ej mi‡Yi wecixZ w`‡K ZvB FbvZ¥K wPý e¨envi Kiv n‡q‡Q| GKvh©Kix e‡ji Rb¨ Z¡iY a n‡j, F = ma  ma =  mgsin  a =  gsin  Gi gvb Lye Kg n‡j, A_©vr 4º Gi †ekx bv n‡j sin†iwWqvb‡jLv hvq|

 a =  g Pvc Pvc      e¨vmva e¨vmva    Pvc BC  a  g  e¨vmva  AC ‡h‡nZz BC n‡”Q miY x Ges AC n‡”Q Kvh©Kix ˆ`N©¨ L| x  a  g  L g g a   x wKš‘ wbw`©ó ¯’v‡b wbw`©ó †`vj‡Ki Rb¨ GKwU aªæeK| G‡K  Øviv cÖKvk Ki‡j L L  a  g 

GwU mij Qw›`Z ¯ú›`‡bi kZ©| myZivs ¯^í we¯Ív‡i ¯úw›`Z mij †`vj‡Ki MwZ mij Qw›`Z ¯ú›`b MwZ, †h †ÿ‡Î

a  2 x ... ... ... (1)

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08| mij Qw›`Z ¯ú›`b (Simple Harmonic Oscillation)

2 

g L

ev,  

4

g L

myZivs mij †`vj‡Ki †`vjbKvj ev ch©vqKvj T 

2 L  2 ... ... ... ... (2) Bnv mij Qw›`Z ¯ú›`b MwZi mgxKiY| d‡j mgxKiY (1) I (2) †_‡K  g

ejv hvq, ¯^í we¯Ív‡i ¯úw›`Z mij †`vj‡Ki MwZ mij Qw›`Z ¯ú›`b MwZ (cÖgvwbZ)| mij‡`vj‡Ki m~Î eY©bv (Laws of Simple Pendulum): 1g m~Î, mgKvj m~Î : †KŠwbK we¯Ívi 4º Gi †ekx bv n‡j Ges †`vj‡Ki Kvh©Kix ˆ`N©¨ AcwiewZ©Z _vK‡j †Kvb wbw`©ó ¯’v‡b GKwU mij †`vj‡Ki cÖwZwU †`vj‡bi Rb¨ mgvb mgq jvM‡e| 2q m~Î, ˆ`‡N©¨i m~Î: †KŠwbK we¯Ívi 4º Gi †ekx bv n‡j †Kvb wbw`©ó ¯’v‡b GKwU mij †`vj‡Ki ‡`vjb Kvj Gi Kvh©Kix ˆ`‡N©¨i eM©g~‡ji mgvbycvwZK| †`vjbKvj T Ges Kvh©Kix ˆ`N©¨ L n‡j m~Îvbyhvqx, T  L n‡e, hLb AwfKl©R Z¡iY g = aªæeK| 3q m~Î, Z¡i‡Yi m~Î: †KŠwbK we¯Ívi 4º Gi †ekx bv n‡j Ges †`vj‡Ki Kvh©Kix ˆ`N©¨ AcwiewZ©Z _vK‡j †Kvb ¯’v‡b GKwU mij †`vj‡Ki ‡`vjb Kvj H ¯’v‡bi AwfKl©R Z¡i‡Yi eM©g‡~ ji e¨v¯ÍvbycvwZK| †`vjbKvj T Ges AwfKl©R Z¡iY g n‡j m~Îvbyhvqx, T 

1 n‡e, hLb Kvh©Kix ˆ`N©¨ L = aªæeK| g

4_© m~Î, f‡ii m~Î : †KŠwbK we¯Ívi 4º Gi †ekx bv n‡j Ges †`vj‡Ki Kvh©Kix ˆ`N©¨ AcwiewZ©Z _vK‡j †Kvb wbw`©ó ¯’v‡b GKwU mij †`vj‡Ki ‡`vjb Kvj e‡ei fi, AvqZb, Dcv`vb BZ¨vw`i Dci wbf©i K‡i bv| wewfbœ fi, AvqZb ev Dcv`v‡bi e‡ei Rb¨ †`vjbKvj GKB n‡e| LT2 ‡jL wP‡Îi cÖK…wZ †Kgb n‡e?

mij †`vj‡Ki 2q m~Î †_‡K Avgiv cvB, T L ev, T 2  L  T 2  aªæeK  L

GKwU QK KvM‡R X A‡ÿi w`‡K L Ges Y A‡ÿi w`‡K T2 Gi Abylw½K gvb ¯’vcb K‡i GKwU †jLwPÎ AsKb Ki‡j †jLwPÎwU g~jwe›`y Mvgx GKwU mij †iLv n‡e| BnvB LT2 ‡jL wP‡Îi cÖK…wZ|

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5 08| mij Qw›`Z ¯ú›`b (Simple Harmonic Oscillation) mij †`vj‡Ki mvnv‡h¨ AwfKl©R Z¡iY ''g"Gi gvb wbY©‡qi c×wZ (Determination of acceleration due to gravity "g" by simple Pendulum):

AwfKl©R Z¡iY (Acceleration due to gravity): AwfKl© e‡ji cÖfv‡e gy³fv‡e cošÍ e¯‘i †eM e„w×i nvi‡K AwfKl©R Z¡iY e‡j| AwfKl©R Z¡iY‡K g Øviv cÖKvk Kiv nq| mij ‡`vj‡Ki †`vjbKv‡ji mgxKiY †_‡K Avgiv Rvwb, L g L  T 2  4 2 g L  g  4 2 2 ... ... ... ... (3) T GB mgxKiY †_‡K †Kvb ¯’v‡b L Kvh©Kix ˆ`‡N©¨i mij †`vj‡Ki †`vjbKvj T wbY©q K‡i H ¯’v‡bi AwfKl©R Z¡iY g wbY©q Kiv hvq| T  2

mij †`vjK ˆZix : ÷v‡Ûi mvnv‡h¨ GKwU ûK †_‡K †Kvb k³ myZv Øviv GKwU ÿz`ª fvix †MvjK Szwj‡q mij †`vjK ˆZix Kiv nq| GB †MvjK‡K ee e‡j| Kvh©Kix ˆ`N©¨ L wbY©q: ‡`vj‡Ki Szjb we›`y †_‡K e‡ei fvi‡K›`ª ch©šÍ `~iZ¡‡K mij †`vj‡Ki Kvh©Kix ˆ`N©¨ L e‡j| cÖ_‡g GKwU wgUvi †¯‹‡ji mvnv‡h¨ myZvi Szjb we›`y A_©vr ûK †_‡K e‡ei Dcwic„ô ch©šÍ `~iZ¡ l †g‡c †bIqv nq| Gici GKwU aªyeK K¨vwjcv‡m©i mvnv‡h¨ e‡ei e¨vm wbY©q K‡i e¨vmva© r wbY©q Kiv nq| Zvn‡j †`vj‡Ki Kvh©Kix ˆ`N©¨ L=l+r | †`vjbKvj T wbY©q: mij †`vj‡Ki GKwU c~Y© †`vj‡bi Rb¨ †h mgq jv‡M Zv‡K †`vjb Kvj e‡j| †`vjKwU‡K mvg¨ve¯’v †_‡K GKcv‡k Ggbfv‡e GKUz †U‡b †Q‡o †`qv nq hv‡Z GwU `yj‡Z _v‡K Ges †KŠwbK we¯Ívi 4º -Gi †ekx bv nq| GKwU _vgv Nwoi mvnv‡h¨ 20 ev 25 wU †`vj‡bi mgq wbY©q K‡i H mgq‡K †`vjb msL¨v w`‡q fvM K‡i GKwU †`vj‡bi mgq A_©vr †`vjbKvj T †ei Kiv nq| Mo

L wbY©q: myZvi ˆ`N©¨ cwieZ©b K‡i †`vj‡Ki Kvh©Kix ˆ`N©¨ L cwieZ©b Kiv nq Ges wewfbœ Kvh©Kix ˆ`‡N©¨i Rb¨ T2

†`vjbKvj L wbY©q K‡i cÖwZ †ÿ‡Î

L L L †ei K‡i Mo 2 wbY©q Kiv nq| GB 2 Gi Mo gvb (3) bs mgxKi‡Y ewm‡q g 2 T T T

Gi gvb wbY©q Kiv nq| ‡jLwPÎ †_‡K L I T2 wbY©q: GKwU QK KvM‡Ri X A‡ÿi w`‡K L -Gi wewfbœ gvb Ges Y A‡ÿi w`‡K Avbylw½K T2 -Gi gvb ¯’vcb K‡i †jL A¼b Kiv nq| †jLwU g~jwe›`y Mvgx GKwU mij †iLv nq| GB mij †iLvi Dci †h †Kvb GKwU we›`y P wb‡q P n‡Z X A‡ÿi Dci PM Ges Y A‡ÿi PN j¤^ Uvbv nq| Zvn‡j †h †Kvb ˆ`N©¨ L=OM -Gi Rb¨ †`vjbKv‡ji eM© T2 = ON cvIqv hvq| djvdj: †jL †_‡K cÖvß GB L I T2 -Gi gvb (3) bs mgxKi‡Y ewm‡q g -Gi gvb wn‡me Kiv nq| L T2 OM  g  4 2 ON g  4 2

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08| mij Qw›`Z ¯ú›`b (Simple Harmonic Oscillation) mZK©Zv : 1| †`vj‡Ki †KŠwbK we¯Ívi hv‡Z 4º Gi †ekx bv nq †m w`‡K jÿ¨ ivLv nq| 2| L Gi gvb h_v m¤¢e †ekx †bIqv nq| 3| †`vjvi mgq †`vjK hv‡Z cvK bv Lvq †m w`‡K jÿ ivLv nq| 4| _vgv Nwo mwVK fv‡e †Lvjv I eÜ Kiv nq|

6

‡m‡KÛ †`vjK: †h †`vj‡Ki †`vjbKvj 2 †m‡KÛ A_©vr †h †`vj‡Ki GK cÖvšÍ †_‡K Aci cÖv‡šÍ †h‡Z GK †m‡KÛ mgq jv‡M Zv‡K †m‡KÛ †`vjK e‡j| †m‡KÛ †`vjK 1 †m‡K‡Û GKwU Aa©‡`vjb m¤úbœ K‡i| w¯cÖs RwbZ ¯ú›`b: GKwU w¯cÖs Gi †ÿ‡Î ch©vq Kv‡ji mv‡_ ej aªæeK I f‡ii g‡a¨ m¤úK© wbb©q: ‡Kvb w¯cÖs-Gi GK cÖvšÍ GKwU `„p Aej¤^‡bi mv‡_ mv‡_ hy³ K‡i G‡Z GKwU fi Szjv‡j w¯cÖswU Uvb Uvb Ae¯’vq _vK‡e| AZtci fiwU‡K GKUz †U‡b †Q‡o w`‡j GUv mij Qw›`Z ¯úw›`Z n‡e| A_©vr, w¯cÖswU ch©vqµ‡g msKzwPZ I cÖmvwiZ n‡e Ges fiwU Dc‡i wb‡P ¯úw›`Z n‡e| A_©vr, w¯cÖswU ch©vqµ‡g msKzwPZ I cÖmvwiZ n‡e Ges fiwU Dc‡i I wb‡P ¯úw›`Z n‡e| wPÎ(K) bs wP‡Î w¯cÖswU ¯^vfvweK Ae¯’vq i‡q‡Q| hw` w¯cÖswUi mv‡_ m fi Szjv‡bv Ae¯’vq w¯cÖswU e cwigv‡Y cÖmvwiZ n‡q mvg¨ve¯’vq _v‡K| [wPÎ (L)], Z‡e GB Ae¯’vq w¯cÖs -Gi Uvb n‡e, To = mg ... ... ... (1)| w¯cÖswU hw` w¯’wZ¯’vcKZvi mxgv AwZµg bv K‡i, Z‡e û‡Ki m~Îvbymv‡i, To  e  To= k e ... ... ... (2) GLv‡b, k n‡jv w¯cÖs Gi ej aªæeK ev w¯cÖs aªæeK| AZGe fi Szjv‡bv Ae¯’vq (1) I (2) n‡Z Avgiv cvB, mg = k e ... ... ... ... ... (3) GLb m fi‡K wb‡Pi w`‡K LvwbKUv †U‡b †Q‡o w`‡j GwU Dc‡i-wb‡P `yj‡Z _v‡K| aiv hvK, †Kvb GK mgq mvg¨ve¯’vb n‡Z fiwUi miY x [wPÎ (M)] Ges GB Ae¯’vq w¯cÖs-Gi Uvb, T  k(x  e) ... ... ... (4) ; fiwUi Z¡iY a n‡j,  F  ma ... ... ... ...(5) mgxKiY e¨envi K‡i cvB, mg  T  ma ... ... ... ...(6) (6) bs mgxKi‡Y mg I T Gi gvb ewm‡q cvB,  k e  k(x  e)  ma  ma   kx k a x m

 a   ω 2 x ... ... ... ... (7)

 k 2  m  ω 

(7) bs mgxKiY mij Qw›`Z MwZi mgxKiY| AZGe m fiwU mij Qw›`Z ¯ú›`‡b ¯úw›`Z nq| G†ÿ‡Î m f‡ii

k Ges ch©vq Kvj T  2π  2π m ... ... ... ... (8) m k ω m e e e¨envi K‡i cvB, T  2π mg = k e ... ... ... (3)   k g g

†KŠwbK K¤úv¼, ω 

(8) bs mgxKiY ch©vq Kv‡ji mv‡_ ej aªæeK I f‡ii g‡a¨ m¤úK© |

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7 08| mij Qw›`Z ¯ú›`b (Simple Harmonic Oscillation) GKwU mij Qw›`Z ¯ú›`‡bi KYvi †ÿ‡Î hvwš¿K kw³i iwkgvjv cÖwZcv`b (Energy of Particle Executing Simple Harmonic Motion): aiv hvK, m f‡ii GKwU e¯‘KYv mij Qw›`Z MwZ‡Z ¯ú›`bkxj| MwZi we¯Ívi A Ges †KŠwbK K¤úv¼ | KYvi miY hLb x ZLb Gi †eM, v   A 2  x 2 ... ... ... ... (1) 1 1 1 2 2 2 1 ev, K  k (A 2 x 2 ) ... ... ... ... (2) 2

MwZ kw³ : x Ae¯’v‡b KYvi MwZkw³, K  mv 2  m 2 (A 2  x 2 )  m 

k (A 2  x 2 ) m

w¯’wZkw³ ev wefe kw³ : ga¨ ¯’vb (x =0) †_‡K x mi‡Yi Rb¨ cÖZ¨qbx e‡ji weiæ‡× †h cwigvY KvR m¤úbœ n‡e Zv e¯‘‡Z w¯’wZkw³ iƒ‡c mwÂZ _vK‡e| cÖZ¨qbx ej, F  kx; awi miY m„wóKvix ej F; hv F Gi mgvb I wecixZgyLx|  F  kx ; AwZ Aí dx mi‡Yi Rb¨ cÖZ¨qbx e‡ji weiæ‡× K…Z KvR  F dx; x

x

x

x2  x †gvU KvR = w¯’wZkw³  U   Fdx  kx dx  k  x dx  k   2o o o o  x2  1 1  k   0  kx 2 ev, U  kx 2 ... ... ... (3) 2 2  2

1 2

1 2

1 2

†gvU kw³, E  K  U  k (A 2  x 2 )  kx 2  kA 2 1  E  kA 2 ... ... ... ... ... (4) 2

AZGe, wbw`©ó we¯Ív‡ii Rb¨ †gvU kw³ aªæe _v‡K| Dc‡iv³ mgxKiY †_‡K Av‡iv Rvbv hvq †h, EA 2 MwZkw³, w¯’wZkw³ I †gvUkw³i †iL wPÎ : 1 2

mgxKiY (2) Abyhvqx, K  k (A 2  x 2 ) KYvwU x  0 Ae¯’vb AwZµg Kv‡j m‡e©v”P MwZkw³ jvf K‡i| 1 kA 2 ... ... ... ... (5) 2 Avevi, MwZkw³i me©wbgœ gvb x  A †Z cvIqv hvq| K min  0 ... ... ... ... (6) 1 mgxKiY (3) Abyhvqx, U  kx 2 , x  A n‡j, A_©vr KYvi miY we¯Ív‡ii mgvb n‡j w¯’wZkw³ m‡e©v”P nq| 2 1 U max  kA 2 ... ... ... ... (7) 2 x  0, Ae¯’v‡b w¯’wZ kw³ me©wbgœ nq| U min  0 ... ... ... ... (8) mgxKiY (2) I (3) e¨envi K‡i, x Gi †h †Kvb gv‡bi Rb¨ K I U wbY©q Kiv hvq| K max 

Dc‡iv³ wPÎ n‡”Q mi‡Yi Av‡cÿK wn‡m‡e w¯’wZkw³, MwZkw³ Ges †gvU kw³|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

8| mij Qw›`Z ¯ú›`b (Simple Harmonic Oscillation) 1| mij Qw›`Z MwZ‡Z Pjgvb GKwU e¯‘i we¯Ívi 0.01m I K¤úv¼ 12Hz | e¯‘wUi 0.005m mi‡Y ‡eM KZ? Avgiv Rvwb,

v  ω A2  x2

ev, v  2 f A2  x 2

GLv‡b, ev , v  2  3.14  12  (0.01)  (0.005) we¯Ívi, A = 0.01 m K¤úv¼, f = 12 Hz ev, v  75.36 0.0001 0.000025 miY, x = 0.005 m ev , v  75.36  0.000075 ‡eM, v = ?

2

2

ev , v  75.36  0.00866 ev , v  75.36  0.00866  v  0.6526 m s -1 (Ans.) 2| ‡Kvb w¯cÖs Gi cÖv‡šÍ GKwU e¯‘ Szjv‡j GwU 20cm cÖmvwiZ nq| e¯‘wU‡K GKUz †U‡b †Q‡o w`‡j K¤úv¼ KZ n‡e?

Avgiv Rvwb, T  2π

e g

gM M (4R) 2 ev,   g E R 2 81M g 16  M  ...........(6) g E 81 GLb (3) bs mgxKi‡Y gvb ewm‡q cvB,

2  TM 2   TM

16 81 4 9 29  TM  4  T M  4 . 5 s (Ans.)



4| GKwU mij †`vj‡Ki †`vjbKvj 25% evov‡Z Gi Kvh©Kix ˆ`‡N©¨i wKiƒc cwiKZ©b Ki‡Z n‡e?

0.2 9.8 ev, T  6.28  0.142857s  T  0.897142 s

ev , T  2  3 . 14 

Avevi, f 

g M M M R 2E   2  gE RM ME

GLv‡b, cÖmviY, e =20cm= 0.2m K¤úv¼, f = ?

1 T

1 Hz 0.897142  f  1 . 11 Hz (Ans.) f 

3| GKwU †m‡KÛ †`vjK f~-c„‡ô mwVK mgq †`q| P›`ª c„‡ô wb‡q †M‡j Gi †`vjbKvj KZ n‡e? c„w_exi e¨vmva© P‡›`ªi e¨vmv‡a©i 4 ¸b Ges c„w_exi fi P‡›`ªi f‡ii 81 ¸b| Avgiv Rvwb, TE  2 π

Ges, TM  2π

T  E  TM

L .....( 1 ) gE

L ....(2) gM

gM ......( 3 ) gE

GLv‡b, f~-c„‡ô †`vjbKvj,TE = 2s P›`ªc„‡ô †`vjbKvj, TM = ? awi, P‡›`ªi e¨vmva©, RM = R c„w_exi e¨vmva©,RE = 4R P‡›`ªi fi, MM = M c„w_exi fi, ME = 81M

GM M ........(4) R 2M GM E Ges, g E  ........(5) R 2E Avevi, g M 

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T1 L1  T2 L2 

GLv‡b, ‰`N©¨, L1 = L ‰`N©¨, L2 = ? ‡`vjbKvj, T1 =T ‡`vjbKvj, T2 =T+T/4=5T/4

L T  5T L2 4



4T  5T



16 L  25 L 2

 L2 

L L2

25 L 16

 L 2  1.5625L DËit ˆ`N©¨ 1.5625 ¸b Ki‡Z n‡e?

5| 1m ˆ`N©¨ wewkó GKwU mij †`vjK cÖwZ †m‡K‡Û 2 wU †`vjb m¤úbœ K‡i| AwfKl©R Z¡i‡Yi gvb wbY©q Ki| Avgiv Rvwb,

GLv‡b,

L T  2 g

‰`N©¨, L =1wgUvi ‡`vjbKvj,

L g L  g  4 2 2 T

 T 2  4 2

 g  4  9.87 

T

1 s  0.5s 2

AwfKl©R Z¡iY, g= ?

1 0.52

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8| mij Qw›`Z ¯ú›`b (Simple Harmonic Oscillation)

 g = 157.92 ms-2 (Ans.)

TA LA  TB LB

6| GKwU †m‡KÛ †`vj‡Ki ˆ`N©¨ wbY©q Ki| Avgiv Rvwb,

T  2

L g

 T 2  4 2

†`vjb Kvj, T =2 Sec AwfKl©R Z¡iY, g = 9.8ms-2 †`vj‡Ki ˆ`N©¨, L=?

L g

gT 2 L 2 4 9.8  2 2 L m 4  9.87  L  0.99 m (Ans.) 7| ‡Kvb mij Qw›`Z ¯ú›`b KYvi we¯Ívi 3cm Ges m‡ev©”P †eM 6.24cms-1 n‡j, KYvwUi ch©vqKvj KZ? Avgiv Rvwb, Vmax = A GLv‡b,  

Vmax 6.24  10 2 rad s 1  A 3  10  2

= 2 .08 rad s-1

Avivi, ch©vqKvj, T 

2 

we¯Ívi, A = 3 cm = 3×10-2 m m‡eŸ©v”P ‡eM, Vmax= 6.24cms-1 = 6.24×10-2 ms-1 ch©vqKvj,T = ?

2  3.14 s 2.08  T  3 s (Ans.) T

8| mij Qw›`Z MwZ m¤úbœ GKwU KYvi MwZi mgxKiY Y=10sin (t + ), ch©vqKvj 30s Ges Avw` miY 0.05m n‡j KYvwUi (i) †KŠwbK K¤úv¼; (ii) Avw``kv wbY©q Ki| Avgiv Rvwb,

2 GLv‡b, T ch©vqKvj , T =30s 2  3.14 rad s 1  miY, Y = 0.05m 30 (i) ‡KŠwbK K¤úv¼, =?    0 . 21 rad s 1 (ii) Avw` `kv  =? (ii) Y = 10sin (t + )  0.05 =10× sin (×0 + ) 0.05  Sin   0.005 10    0.286 (Ans.) 9| GKwU mij †`vjK A Gi ˆ`N©¨ Aci GKwU mij †`vjK B Gi ˆ`‡N©¨i 4 ¸Y| †`vjK B Gi †`vjb Kvj 2s n‡j A Gi †`vjb Kvj KZ? Avgiv Rvwb, (i)  

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2

GLv‡b, ‡`vj‡Ki ˆ`N©¨, LB=L awi ‡`vj‡Ki ˆ`N©¨, LA=4L ‡`vjb Kvj, TB = 2 s ‡`vjb Kvj, TA =?

TA 4L  2 L T  A 2 2  TA  4s (Ans.) 

10| mij Qw›`Z MwZ‡Z Pjgvb GKwU KYvi m‡e©Ÿv”P †eM 0.02ms-1Ges we¯Ívi 0.004m n‡j KYvwUi ch©vq Kvj KZ? Vmax = A V GLv‡b, 0.02 rad s 1    max  we¯Ívi, A = 0.004 m A 0.004 = 5 rad s-1 m‡eŸ©v”P ‡eM, Vmax = 0.02ms-1 2 ch©vqKvj T =?

Avivi, ch©vqKvj, T 



2  3.14 T s 5  T  1.256 s (Ans.) 11| 40cm `xN© GKwU mij †`vjK cÖwZ wgwb‡U 40 evi †`vj Lvq| hw` Gi ˆ`N©¨ 160cm nq Z‡e 60 evi `yj‡Z KZ mgq †b‡e?

Avgiv Rvwb,

T1  T2

L1 L2

GLv‡b, ‡`vj‡Ki ˆ`N©¨, L1= 40cm= 0.4m ‡`vjb Kvj, T1 

60 3 s s 40 2

3 0. 4  ‡`vj‡Ki ˆ`N©¨, L2= 160cm= 1.6m 2T2 1.6 ‡`vjb Kvj, T2 =?

3   0.5 60 evi `yj‡Z mgq 60T2 =?

2T2  T2  3s  60T2  3  60s  180s  3 Minute (Ans.) 

12| †Kvb ¯’v‡b GKwU †m‡KÛ †`vj‡Ki ˆ`N©¨ 1m| †h †`vjK H ¯’v‡b cÖwZ wgwb‡U 25 evi †`vj †`q Zvi ˆ`N©¨ †ei Ki| Avgiv Rvwb, GLv‡b, †`vjb Kvj, T1 =2s T1 L1  ˆ`N©¨, L1=1m T2 L2 

2  25 1  60 L2

†`vjb Kvj, T2 =

60 s 25

ˆ`N©¨, L2=?

25 1  36 L 2 36  L2   1.44 m (Ans.) 25



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cøvRgv Ae¯’v (Plasma State): AZ¨šÍÍ D”P ZvcgvÎvq c`v_©‡K DËß Ki‡j c`v‡_©i cigvbyi wbDwK¬qvm‡K wN‡i _vKv B‡jKUª‡bi GK ev GKvwaK B‡jKUªb gy³ n‡q hvq| B‡jKUªb gy³ nIqvi d‡j c`v_©wU FbvZ¥K PvR© hy³ gy³ B‡jKUªb I abvZ¥K PvR©hy³ Avq‡bi mgwó nq| †Kvb c`v‡_©i G ai‡bi mgvb msL¨K abvZ¥K Avqb I FbvZ¥K Avqbhy³ AwZ D”P AvqwbZ M¨vm‡K ejv nq cøvRgv Ae¯’v| AvšÍtAvbweK ej: (Intermolecular force): mKj c`v_©B ÿz`ª ÿz`ª KYv w`‡q MwVZ hv c`v‡_©i me ¸Y eRvq iv‡L| G me ÿy`ª KYv‡K AYy e‡j| c`v_© MV‡bi mgq AYy¸‡jv ci¯ú‡ii cvkvcvwk _v‡K Ges Zv‡`i g‡a¨ AwZ ÿz`ª cwigv‡bi duvKv ¯’vb _v‡K| GB duvKv ¯’vb‡K AvšÍtAvbweK ej ¯’vb e‡j| AvšÍtAvbweK `~i‡Z¡i cwigvb 10-9 †_‡K 10-10m| AYy¸‡jv GB cwigvb `~i‡Z¡ †_‡K ci¯úi‡K GKwU e‡j AvKl©Y K‡i| GB AvKl©Y ej‡K AvšÍtAvbweK ej e‡j| w¯’wZ¯’vcKZv (Elasticity): †h a‡g©i d‡j e‡ji wµqvq weK…Z e¯‘ cÖhy³ ej Acmvi‡b cye©ve¯’vq wd‡i Av‡m ev Avm‡Z Pvq, Zv‡K w¯’wZ¯’vcKZv e‡j Ges H e¯‘‡K w¯’wZ¯’vcK e¯‘ e‡j| c~Y© w¯’wZ¯’vcK e¯‘ (Absolute Elastic Body): evwn¨K ej AcmvwiZ n‡j hw` weK…Z e¯‘ wVK Av‡Mi AvKvi I AvqZb wd‡i cvq Zvn‡j H e¯‘‡K c~Y© w¯’wZ¯’vcK e¯‘ e‡j| w¯’wZ¯’vcK mxgv (Elastic Limit): †h gv‡bi ej ch©šÍ †Kvb e¯‘ c~Y© w¯’wZ¯’vcK e¯‘i b¨vq AvPiY K‡i A_©vr me©v‡cÿv †ekx †h ej cÖ‡qvM K‡i ej AcmviY Ki‡j e¯‘wU c~e©ve¯’vq wd‡i hvq Zv‡K e‡ji †mB m‡e©v”P mxgv‡K w¯’wZ¯’vcK mxgv e‡j| `„p e¯‘ (Rigid Body): evwn¨K e‡ji wµqvq hw` †Kvb e¯‘i weKvi bv nq Z‡e Zv‡K `„p e¯‘ e‡j| cøvw÷K e¯‘ (Plastic Body): evwn¨K e‡ji wµqvq †Kvb e¯‘i weK…wZ NU‡j Ges cÖhy³ e‡ji AcmvwiZ n‡j hw` e¯‘i weK…Z Ae¯’v eRvq _v‡K Z‡e Zv‡K cøvw÷K e¯‘ e‡j| weK…wZ (Strain): evwn¨K e‡ji wµqvq †Kvb e¯‘i KvwqK cwieZ©b n‡j e¯‘wU weK…Z n‡q‡Q ejv nq| e¯‘i GKK gvÎvq cwieZ©‡bi cwigvb‡K weK…wZ e‡j| †Kvb e¯‘i Avw` gvÎv x Ges weK…wZi ci gvÎv y n‡j Gi weK…wZ 

y~x n‡e, x

weK…wZi †Kvb GKK ev gvÎv †bB KviY Giv `yBwU GKB cÖKvi ivwki AbycvZ| cxob (Stress): evwn¨K e‡ji wµqvq †Kvb e¯‘i weK…wZ NU‡j Gi Af¨šÍ‡i GKUv cÖwZwµqvg~jK e‡ji D™¢e N‡U, hv e¯‘Uv‡K c~e©ve¯Ívq wdwi‡q Avb‡Z cÖqvm cvq| e¯‘i GKK †ÿ‡Îi Dci wµqviZ GB cÖwZwµqv g~jK ej‡K cxob e‡j| A †ÿÎd‡j F ej cÖhy³ n‡j cxob =

F n‡e| †ÿÎd‡ji w`K I e‡ji w`K GKB| ZvB ej I †ÿÎd‡ji fvMdj w`K A

wbi‡cÿ nq| ZvB cxob †¯‹jvi ivwk| e‡ ji gvÎv   MLT 2  1  2    ML T  2  † ¶ Îd‡ ji gvÎv L    

cxo‡ bi gvÎv  

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09| w¯’wZ¯’vcKZv

cxo‡ bi GKK 

2

(Elasticity)

e‡ ji GKK N  2  Nm 2 ev, Pa † ¶ Îd‡ ji GKK m

cxo‡bi Zvrch© (Significance of Stress): †Kvb e¯‘i cxob 5×105 Nm-2 ej‡Z GB eywS †h, e¯‘i cÖwZ 1m2 †ÿÎd‡ji Dci D™¢yZ cÖwZ‡ivaKvix e‡ji gvb 5×105 N | w¯’wZ¯’vcK K¬vwšÍ (Elastic Fatigue): †Kvb Zv‡ii Dci µgvMZ cÖhy³ cxo‡bi n«vm e„w× Ki‡j e¯‘i w¯’wZ¯’vcKZv ag© n«vm cvq| Gi d‡j ej Acmvi‡Yi mv‡_ mv‡_ e¯‘ Av‡Mi Ae¯’v wd‡i cvq bv wKQyUv †`ix nq| e¯‘i GB Ae¯’v‡K w¯’wZ¯’vcK K¬vwšÍ e‡j| ZLb Amn fv‡ii †P‡q Kg fv‡i GgbwK w¯’wZ¯’vcK mxgvi g‡a¨B e¯‘wU wQ‡o †h‡Z cv‡i| Amnfvi: me©v‡cÿv Kg †h fv‡ii wµqvq †Kvb e¯‘ †f‡½ ev wQ‡o hvq, Zv‡K Amnfvi e‡j| Amncxob: Amnfv‡ii wµqvq e¯‘‡Z †h cxob nq, Zv‡K Amncxob e‡j| A_©vr, Amncxob  Amnfvi

† ¶ Îdj

û‡Ki m~Î (Hooke's Law): w¯’wZ¯’vcK mxgvi g‡a¨ e¯‘i cxob Gi weK…wZi mgvbycvwZK| A_©vr, cxob  weK…wZ| ev, cxob = aªyeK × weK…wZ| 

cxob  aª æeK weK … wZ

GB aªæe‡Ki gvb e¯‘i Dcv`vb Ges GK‡Ki c×wZi Dci wbf©i K‡i| GB aªæe msL¨v‡K e¯‘i Dcv`v‡bi w¯’wZ¯’vcK ¸Yv¼ ev w¯’wZ¯’vcK aªæeK e‡j| w¯’wZ¯’vcK ¸bv¼ (Elastic Constant) : w¯’wZ¯’vcK mxgvi g‡a¨ e¯‘i cxob I weK…wZi AbycvZ GKwU aªæe msL¨v| GB aªæe msL¨v‡K e¯‘i Dcv`v‡bi w¯’wZ¯’vcK ¸Yv¼e‡j| Bqs Gi ¸bv¼ (Young's Modulus) : w¯’wZ¯’vcK mxgvi g‡a¨ e¯‘i ‰`N©¨ cxob I ‰`N©¨ weK…wZi AbycvZ GKwU aªæe msL¨v| GB aªæe msL¨v‡K e¯‘i Dcv`v‡bi BqsGi ¸Yv¼e‡j| BqsGi ¸bv¼‡K Y Øviv cÖKvk Kiv nq| ‰`N©¨ cxob ‰`N©¨ weK … wZ Bqs Gi ¸Yv¼ GKwU †¯‹jvi ivwk| A cÖ¯’‡”Q‡`i †ÿÎdj I L ‰`N©¨ wewkó GKwU Zvi †Kvb `„p Aej¤^b n‡Z Szwj‡q hw` ZviwUi wb‡Pi cÖv‡šÍ F ej cÖ‡qvM Kiv nq Zvn‡j Zv‡ii ˆ`N©¨ l e„w× †c‡j, F l F FL hw` ZviwUi wb‡P M fi Szjvb nq ‰`N©¨ weK…wZ  Ges ‰`N©¨ cxob  myZivs Y  A  l L A Al L Zvn‡j. F=Mg, GLv‡b g=AwfKl©R Z¡iY| Avevi ZviwUi e¨vmva© r n‡j †ÿÎdj A = r2| †m‡ÿ‡Î MgL Y hw` A =1GKK Ges l =L nq Z‡e mgxKiY Abymv‡i, F = Y nq| r 2 l

BqsGi ¸bv¼ Y = 

myZivs GKK cÖ¯’‡”Q‡`i †ÿÎdjwewkó †Kvb Zv‡ii ˆ`N©¨ eivei †h ej cÖ‡qvM Ki‡j ˆ`N©¨ weK…wZ GKK nq A_©vr ZviwUi ˆ`N©¨ e„w× Avw` ‰`‡N©¨i mgvb nq Zv‡K Bqs Gi ¸Yv¼ e‡j| Gi GKK Nm-2 ev Pa| Gi gvÎv [ML-1T -2]

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3 09| w¯’wZ¯’vcKZv (Elasticity) 2 11 BqsGi ¸Yv¼ 2×10 Nm ej‡Z GB eywS †h, 1m cÖ¯’‡”Q‡`i †ÿÎdjwewkó B¯úv‡Zi `‡Ûi ˆ`N©¨ eivei 2×10 N ej cÖ‡qvM Ki‡j Gi ˆ`N©¨ e„w× Avw` ˆ`‡N©¨i mgvb nq| 11

-2

fvwb©qvi c×wZ‡Z B¯úv‡Zi Zv‡ii BqsGi ¸Yv¼ wbY©q (Determination of Young's Modulus of Steel by Verneer Method )t ZË¡t w¯’wZ¯’vcK mxgvi g‡a¨ e¯‘i ‰`N©¨ cxob I ‰`N©¨ weK…wZi AbycvZ GKwU aªæe msL¨v| GB aªæe msL¨v‡K e¯‘i Dcv`v‡bi BqsGi ¸Yv¼e‡j| BqsGi ¸bv¼‡K Y Øviv cÖKvk Kiv nq| BqsGi ¸bv¼ Y = Bqs Gi ¸Yv¼ GKwU †¯‹jvi ivwk| A cÖ¯’‡”Q‡`i †ÿÎdj I L ‰`N©¨ wewkó GKwU Zvi †Kvb `„p Aej¤^b n‡Z Szwj‡q hw` ZviwUi wb‡Pi cÖv‡šÍ F ej cÖ‡qvM Kiv nq Zvn‡j Zv‡ii ˆ`N©¨ l e„w× †c‡j, F l F FL ‰`N©¨ weK…wZ  Ges ‰`N©¨ cxob  myZivs Y  A  hw` ZviwUi wb‡P M fi Szjvb nq l L A Al L Zvn‡j. F=Mg, GLv‡b g = AwfKl©R Z¡iY| Avevi ZviwUi e¨vmva© r n‡j †ÿÎdj A = r2| †m‡ÿ‡Î Y

MgL r 2 l

DcwiD³ mgxKi‡Y Wvbw`‡Ki ivwk¸‡jv cixÿvi mvnv‡h¨ †ei K‡i Avgiv Bqs Gi ¸Yv¼ wbY©q Ki‡Z cvwi| h‡š¿i eY©bv: †h c`v‡_©i BqsGi ¸Yv¼ wbY©q Ki‡Z n‡e †mB c`v‡_©i GKB e¨v‡mi `ywU Zvi AB I CD †K GKwU `„p Aej¤^b ‡_‡K Szjvb nq| AB cixÿvaxb Zvi I CD mnvqK Zvi| CD Zv‡ii mv‡_ wgwjwgUv‡i `vMvw¼Z GKwU cÖavb †¯‹j Ges AB Zv‡ii mv‡_ GKwU fvwb©qvi †¯‹j Ggbfv‡e AvUKvb _v‡K hv‡Z fvwb©qvi †¯‹jwU cÖavb †¯‹‡ji Mv †e‡q euvavnxb fv‡e DVvbvgv Ki‡Z cv‡i| CD Zv‡ii †¯‹‡ji wb‡P GKwU ûK jvMv‡bv _v‡K| GB û‡Ki mv‡_ GKwU IRb Szwj‡q CD ZviwU‡K UvbUvb K‡i ivLv nq| ‰`N©¨ wbY©q : GKwU wgUvi †¯‹‡ji mvnv‡h¨ cixÿvaxb Zvi A_©vr ABGi Szjb we›`y A †_‡K fvwb©qvi ‡¯‹‡ji k~b¨ `vM O ch©šÍ ˆ`N©¨ AO †g‡c †bqv nq| GwU nj Zv‡ii Avw` ˆ`N©¨, L| e¨vmva© wbY©q : ¯Œz M‡Ri mvnv‡h¨ AB Zv‡ii wewfbœ RvqMvq e¨vm †g‡c wb‡q Mo e¨vm d wbY©q Kiv nq| GLb Mo e¨vm‡K `yB w`‡q fvM K‡i e¨vmva© r cvIqv hvq| Amnfvi wbY©q: e¨vmva© wbY©‡qi ci cÖ¯’‡”Q‡`i †ÿÎdj A=r2 †ei Kiv nq| GLb cÖ¯’‡”Q‡`i ‡ÿÎdj‡K Zv‡ii Amn cxob w`‡q ¸b Ki‡j Amn fvi cvIqv hvq| AB Zv‡i Amn fv‡ii A‡a©K ev ZviI Kg fi Pvcv‡j, w¯’wZ¯’vcK mxgv AwZµg Ki‡e bv| fvi I ˆ`N©¨ m¤cÖmviY wbY©q: GLb cÖavb †¯‹j I fvwY©qvi †¯‹‡ji cvV †`‡L †bqv nq| GwU n‡”Q Avw` cvV| Gici AB Zv‡ii û‡K Avav kg IRb Szjvb nq| d‡j AB Zv‡ii ˆ`N©¨ e„w× nIqvq fvwb©qvi †¯‹jwU wb‡P †b‡g hvq| GB Ae¯’vq cÖavb †¯‹j I fvwb©qvi †¯‹‡ji cvV †bIqv nq| GB cvV I Avw` cv‡Vi cv_©K¨B nj Avav kg fv‡ii Rb¨ Zv‡ii ˆ`N©¨ cÖmviY| Giƒc fv‡e µgš^‡q Avav kg K‡i fvi e„w× K‡i cÖwZ †ÿ‡Î cÖavb †¯‹j I fvwb©qvi †¯‹‡ji cvV †bIqv nq| cÖwZeviB cÖvß cvV †_‡K Avw` cvV we‡qvM K‡i †gvU fv‡ii Rb¨ m¤cÖmviY wbY©q Kiv nq| Gici GKwU GKwU K‡i Avav kg fvi bvwg‡q cÖ‡Z¨K evi cvV †bqv nq| G‡Z ˆ`N©¨ n«vm cvIqv hvq| Gi d‡j cÖ‡Z¨K fv‡ii Rb¨ `ywU K‡i cvV cvIqv hv‡e| GKwU fvi e„w×i mgq AciwU fvi n«v‡mi mgq| GB `yB cv‡Vi Mo ‡_‡K mswkøó fv‡ii Rb¨ ˆ`N©¨ m¤cÖmviY cvIqv hvq| Gfv‡e wewfbœ fvi Ges Zvi Rb¨ ˆ`N©¨ e„w× wbY©q Kiv nq|

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09| w¯’wZ¯’vcKZv

4

(Elasticity)

‡jL AsKb: X Aÿ eivei fvi M Ges Y Aÿ eivei ˆ`N©¨ eivei l wb‡q †jLwPÎ A¼b Ki‡j g~jwe›`y Mvgx GKwU mij †iLv cvIqv hv‡e| GB †j‡Li Dci †h †Kvb GKwU we›`y P †bIqv nq| P †_‡K OX Gi Dci PQ j¤^ Uvb‡j OQ=M fv‡ii Rb¨ PQ=l ˆ`N©¨ e„w× cvIqv hv‡e| djvdjt GLb Y 

MgL mgxKi‡Y , M, g, L, r I l Gi ewm‡q wnmve Ki‡j Y Gi gvb cvIqv hv‡e| r 2l

mZK©Zv: 1| Zvi `ywU GKB c`v‡_©i Ges GKB ˆ`‡N©¨i nIqv DwPZ| 2| cÖ_‡g wKQy IRb Pvwc‡q UviwU‡K UvbUvb K‡i wb‡Z nq| 3| wcQb ÎæwU cwinvi K‡i ¯ŒzMR‡K GKB w`‡K Nyivb nq| 4| Amn fv‡ii A‡a©K ev Zvi Kg fvi Pvcvb nq| AvqZb ¸bv¼ (Bulk Modulus) t w¯’wZ¯’vcK mxgvi g‡a¨ e¯‘i AvqZb cxob I AvqZb weK…wZi AbycvZ GKwU aªæe msL¨v| GB aªæe msL¨v‡K e¯‘i Dcv`v‡bi AvqZb ¸Yv¼e‡j| AvqZb ¸bv¼‡K B Øviv cÖKvk Kiv nq| AvqZb ¸bv¼,

F FV pV AvqZb cxob  A  B v Av v AvqZb weK … wZ V

  F  A  p 

msbg¨Zv (Compressibility) t w¯’wZ¯’vcK mxgvi g‡a¨ e¯‘i AvqZb weK…wZ I AvqZb cxo‡bi AbycvZ‡K msbg¨Zv e‡j| msbg¨Zv, AvqZb weK … wZ  AvqZb cxob

1 1  AvqZb cxob B AvqZb weK … wZ ¸bv¼ (Modulus of rigidity)

`„pZvi ¸bv¼ ev KvwV‡b¨i t w¯’wZ¯’vcK mxgvi g‡a¨ e¯‘i e¨eZ©b cxob I e¨eZ©b weK…wZi AbycvZ GKwU aªæe msL¨v| GB aªæe msL¨v‡K e¯‘i Dcv`v‡bi `„pZvi ¸bv¼ ev KvwV‡b¨i ¸bv¼ e‡j| `„pZvi ¸bv¼ ev KvwV‡b¨i ¸bv¼ ‡K n Øviv cÖKvk Kiv nq| `„pZvi ¸bv¼,

F e¨eZ© b cxob F n  A e¨eZ© b weK … wZ  A

cqm‡bi AbycvZ (Poisson's Ratio): w¯’wZ¯’vcK mxgvi g‡a¨ e¯‘i cvk©¦ weK…wZ I ‰`N©¨ weK…wZi AvbycvZ GKwU aªæe msL¨v| GB aªæe msL¨v‡K e¯‘i Dcv`v‡bi cqm‡bi AbycvZ e‡j| A_©vr cqm‡bi AbycvZ †K Øviv cÖKvk Kiv nq| A_©vr, cqm‡bi AbycvZ,  = e¨vL¨v: e„ËvKvi cÖ¯’‡”Q` wewkó ‡Kvb Zv‡ii ˆ`N©¨ L I e¨vm D n‡j Ges evwn¨K e‡ji cÖfv‡e Gi ˆ`N©¨ e„w× l I e¨vm d cwigvb K‡g †M‡j, ˆ`N©¨ weK…wZ 

l d Ges cvk¦© weK…wZ  L D

d dL e¨v‡mi cwie‡Z© e¨vmva© w`‡qI cqm‡bi AbycvZ‡K cÖKvk Kiv hvq|  cqm‡ bi AbycvZ,   D  l Dl L aivhvK, Zv‡ii Avw` ˆ`N©¨ L0 Ges e¨vmva© r| evwn¨K e‡ji cÖfv‡e Gi ˆ`N©¨ e„w× L Ges e¨vmva© n«vm r n‡j,

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09| w¯’wZ¯’vcKZv

(Elasticity)

5

Δr Δr L 0  cqm‡ bi AbycvZ , σ   r   GLv‡b FbvZ¥K wPý Øviv ΔL r ΔL L0 eySvb n‡”Q †h, L abvZ¥K n‡j r FbvZ¥K n‡e Ges L FbvZ¥K n‡j r abvZ¥K n‡e| A_©vr ej cÖ‡qv‡M ˆ`N©¨ e„w×

L r ˆ`N©¨ weK…wZ  Ges cvk¦© weK…wZ  L0 r

†c‡j e¨vmva© n«vm cv‡e Avi ˆ`N©¨ n«vm †c‡j e¨vmva© e„w× cv‡e| weK…wZ GKB cÖKvi `ywU ivwki AbycvZ e‡j Gi †Kvb GKK I gvÎv †bB| Avevi cqm‡bi AbycvZ I `yBwU GKB cÖKvi ivwki AbycvZ e‡j GiI †Kvb GKK I gvÎv †bB| Zv‡ii m¤cÖmvi‡b K…ZKvR (Work done in stretching wire) ev mwÂZ wefe kw³ (Elastic Potential Energy) wbY©q ev, †Kvb e¯‘i GKK AvqZ‡b mwÂZ w¯’wZ¯’vcK wefe kw³ cxob I weK…wZi ¸bd‡ji A‡a©K Gi cÖgvYt g‡b Kwi, L ˆ`N©¨ I A cÖ¯’‡”Q‡`i †ÿÎdj wewkó GKwU Zvi‡K †Kvb `„p Aej¤^b †_‡K Szwj‡q G‡Z F ej cÖ‡qvM Kivi d‡j Gi ˆ`N©¨ dl cwigvb e„w× †cj| myZivs Zv‡i mwÂZ wefe kw³i cwieZ©b ev K…ZKvR n‡e = ej × ˆ`N©¨ e„w×  K…Z KvR dW  Fdl GB mgxKiY‡K l=0 †_‡K l=l GB mxgvi g‡a¨ mgvKjb K‡i mwÂZ †gvU wefe kw³ ev K…ZKvR cvB, l

W   Fdl ... ... ... ... (1) 0

Bqs Gi ¸bv¼ †_‡K Avgiv Rvwb †h, Y 

FL Al

†hLv‡b L Zv‡ii Avw` ˆ`N©¨ Ges A Zv‡ii cÖ¯’‡”Q‡`i ‡ÿÎdj Ges l

YAl (1) bs mgxKi‡Y F Gi gvb emv‡j, L l YAl W  dl L 0

ˆ`N©¨ e„w×, F ej  F 

l

W 

YA l dl L 0 l

YA  l 2  W  L  2  0

YA l 2 W   L 2 1 YAl 2 W  2 L

GKK AvqZ‡b mwÂZ wefe kw³ ev GKK AvqZ‡b K…Z KvR     

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W V 1 YAl 2 1  2 L V 2 1 YAl 1  2 L AL 2 1 Yl 2 L2 1 Yl l   2 L L

[ V  AL]

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09| w¯’wZ¯’vcKZv

(Elasticity)

6

1 F Yl l  cxob × weK…wZ KviY   cxob Ges = weK…wZ 2 A L L 1 GKK AvqZ‡b mwÂZ wefe kw³ ev GKK AvqZ‡b K…Z KvR   cxob × weK…wZ 2 

A_©vr GKK AvqZ‡b mwÂZ wefe kw³ cxob I weK…wZi ¸bd‡ji A‡a©K (cÖgvwYZ) B¯úvZ ivev‡ii †P‡q †ekx w¯’wZ¯’vcK (Steel is more elastic than Rubber): GK UzKiv ivev‡ii wd‡Z Uvb‡j mn‡RB j¤^v nq| wKš‘ GKwU B¯úv‡Zi Zvi Uvb‡j Zv mn‡R j¤^v nq bv| GKB cÖ¯’‡”Q‡`i †ÿÎdj I ˆ`N©¨ wewkó `ywU wfbœ e¯‘i g‡a¨ †h e¯‘‡Z hZ †ekx cÖwZ‡iva e‡ji m„wó nq †mB e¯‘i w¯’wZ¯’vcKZv ZZ †ekx| cÖwZ‡iva ej cÖhy³ e‡ji mgvb e‡j wbw`©ó weK…wZ m„wó Ki‡Z †h e¯‘‡Z hZ †ekx ej cÖ‡qvM Ki‡Z nq Zv‡K ZZ †ekx w¯’wZ¯’vcK ejv nq| GB wn‡m‡e †`Lv hvq †h, GKB ˆ`N©¨ I cÖ¯’‡”Q` wewkó ivevi I B¯úv‡Zi Zv‡i mgvb ˆ`N©¨ e„w× Ki‡Z ivev‡ii Zzjbvq B¯úv‡Zi Zv‡i ej cÖ‡qvM Ki‡Z nq A‡bK †ekx| G Rb¨ ivev‡ii Zzjbvq B¯úv‡Zi w¯’wZ¯’vcKZv A‡bK †ekx| awi, GKB ˆ`N©¨ L Ges GKB cÖ¯’‡”Q‡`i †ÿÎdj A wewkó GKwU B¯úvZ I GKwU ivev‡ii Zv‡i mgcwigvb ej F cÖ‡qvM Kivq Zv‡`i ˆ`N©¨ e„w× h_vµ‡g ls I lr nj| FL ... ... ... ... (1) Al s FL ... ... ... ... (2) Ges ivev‡ii Bqs-Gi ¸Yv¼, Yr  Alr (1) bs mgxKiY‡K (2) bs mgxKiY Øviv fvM K‡i cvB, Ys FL Al r   Yr Al s FL Y l  s  r wKš‘ l r  l s  Ys  Yr myZivs B¯úvZ ivevi A‡cÿv †ekx w¯’wZ¯’vcK| Yr ls

myZivs B¯úv‡Zi Bqs-Gi ¸Yv¼, Ys 

w¯’wZ¯’vcK wefe kw³: †Kvb e¯‘i weK…wZ NUv‡bvi Rb¨ e¯‘wUi Dci ewn¯’ ej cÖ‡qvM Ki‡Z nq| ej cÖ‡qv‡M e¯‘ weK…Z n‡j H ej Øviv KvR m¤úvw`Z nq| m¤úvw`Z G KvR e¯‘‡Z wefe kw³ iƒ‡c mwÂZ _v‡K| G kw³‡K w¯’wZ¯’vcK wefe kw³ e‡j|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

9| w¯’wZ ¯’vcKZv (Elasticity) 4| GKwU Zv‡ii Bqs Gi ¸bv¼ 2.35×1011 Nm-2 Ges ZviwUi e¨vm 2mm| ZviwUi ˆ`N©¨ 0.25% e„w× Ki‡Z n‡j KZ ej cÖ‡qvM Ki‡Z n‡e?

GLv‡b, Avgiv Rvwb,

1| 1 eM© wgwjwgUvi cÖ¯’‡”Q‡`i †ÿÎdj wewkó GKwU B¯úv‡Zi Zv‡ii ˆ`N©¨ kZKiv 2 fvM e„w× Ki‡Z KZ ej cÖ‡qvM Ki‡Z n‡e?

B¯úv‡Zi Bqs Gi ¸bv¼ 2×1011 Nm-2 | Avgiv Rvwb, GLv‡b, Y

FL Al

YA l L 2 1011 1106  2x F x 100 F

2

cÖ¯’‡”Q‡`i †¶Îdj, A = 1 mm =1×10-6 m2 Avw` ˆ`N©¨, L = x (awi) ‰`N©¨ e„w×, l  x  2 100 Bqs Gi ¸bv¼, Y = 2×1011 Nm-2 ej F = ?

 F  4 0 00 N (Ans.) 2| 1×10-10 m2 cÖ¯’‡”Q‡`i †ÿÎdjwewkó GKwU B¯úv‡Zi Zv‡i KZ ej cÖ‡qvM Ki‡j Gi ˆ`N©¨ wظb n‡e [ Y=2×1011 Nm-2 ] Avgiv Rvwb, GLv‡b,

FL Al YA l F L 2 1011 11010  x F x Y

 F  2 0 N (Ans.)

cÖ¯’‡”Q‡`i †¶Îdj, A = 1×10-10 m2 Avw` ˆ`N©¨, L = x (awi) ‰`N©¨ e„w×,

l  (2 x  x )  x Bqs Gi ¸bv¼, Y = 2×1011 Nm-2 ej, F = ?

3| 6 m `xN© Ges 1mm2 cÖ¯’‡”Q‡`i †ÿÎdj wewkó GKwU Zv‡ii cÖv‡šÍ 20kg Gi GKwU fi Szwj‡q †`Iqv nj| Zv‡ii Dcv`v‡bi Bqs Gi ¸bv¼ 2.35×1011 Nm-2 n‡j ZviwU KZUzKz e„w× cv‡e?

FL Y Al FL l  YA 20  9.8  6 2.35  1011  1 10 6 l  5  10 3 m (Ans.) l 

GLv‡b, Avw` ˆ`N©¨, L = 6 m cÖ¯’‡”Q‡`i †¶Îdj, A = 1mm2 = 1×10-6 m2 Bqs Gi ¸bv¼, Y = 2.35×1011 Nm-2 ej, F = 20kg-Wt = 20 × 9.8 N ‰`N©¨ e„w×, l  ?

e¨vm, d = 2mm e¨vmva©, r=1mm=0.001m cÖ¯’‡”Q‡`i †¶Îdj A=r × (0.001)2 m2 =3.14×10-6 m2 11 6 2.35  10  3.14  10  0.25x Avw` ˆ`N©¨, L=x (awi) F x  0.25 x  100 ‰`N©¨ e„w×, l  m  F  1844 . 75 N (Ans.) 100 Y  2.35  1011 Nm 2 ej, F  ?

FL Al YA l F L Y

5| GKwU Zv‡ii Bqs Gi gvbv¼ 2×1011 Nm-2 ZviwUi ˆ`N©¨ 15% e„w× Ki‡Z cÖh³ y cxob wbY©q Ki| FL Al F Yl   A L F 2  1011  x  15   A x  100 F   3  1010 Nm  2 (Ans) A Y

GLv‡b, awi Avw` ˆ`N©¨, L = x ˆ`N©¨e„w×, l  x  15 100

Y  2 1011 Nm 2 cxob,

F ?

A

6| 200cm j¤^v Ges 1mm2 cÖ¯’‡”Q‡`i †ÿÎdj wewkó GKwU B¯úv‡Zi Zv‡ii ˆ`N©¨ 1×10-3 m e„w× Ki‡Z cÖ‡qvRbxq Kv‡Ri cwigvb 0.05J | Zv‡ii Dcv`v‡bi BqsGi ¸bv¼ wbY©q Ki| Avgiv Rvwb,

1 YAl 2 W 2 L 1 Y  10 6  (10 3 ) 2  0.05   2 2

0.05  2  2 Nm 2 1  10 -12  Y  2  1011 Nm -2 (Ans.) Y

GLv‡b, Avw` ˆ`N©¨, L = 200cm=2m cÖ¯’‡”Q‡`i †¶Îdj, A = 1mm2 = 1×10-6 m2 KvR, W = 0.05 J ‰`N©¨ e„w×, l = 1×10-3m Bqs Gi ¸bv¼, Y = ?

7| 6m `xN© Ges 2mm2 cÖ¯’‡”Q‡`i †ÿÎdj wewkó GKwU Zv‡i 10kg Gi GKwU fi Szjvb Av‡Q| hLb fiwU mwi‡q †bIqv nq ZLb Zv‡ii ˆ`N©¨ nq 5.9975 m| Zv‡ii Dcv`v‡bi BqsGi ¸bv¼ wbY©q Ki| GLv‡b, Avgiv Rvwb, Avw` ˆ`N©¨, L = 5.9975 m FL Y ‰`N©¨ e„w×, Al l = (6 5.9975)m MgL = 0.0025m Y Al cÖ¯’‡”Q‡`i †¶Îdj, 10  9.8  5.9975 A = 2mm2 = 2×10-6 m2 Y fi M = 10 kg 2  10 6  0.0025 Bqs Gi ¸bv¼, Y = ?

11 -2

 Y  1.18  10 Nm facebook /gmail/skype: -tanbir.cox

(Ans.)

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9| w¯’wZ ¯’vcKZv (Elasticity)

8| 1m `xN© †Kvb Zv‡ii e¨vm 5×10-3m Zv‡ii ˆ`N©¨ eivei GKwU ej cÖ‡qv‡M Gi ˆ`N©¨ 1×10-2m e„w× cvq| cqm‡bi AbycvZ 0.2 n‡j Zv‡ii e¨vm -Gi n«vm wbY©q Ki| GLv‡b, Avgiv Rvwb,



dL Dl

d 1 5  10  1  10 2  d  0.2  5  10 3  1  10 2  d  10 5 m(Ans.)  0.2 

3

Avw` ˆ`N©¨, L = 1m e¨vm, D = 5×10-3m ‰`N©¨ e„w×, l = 1×10-2m cqm‡bi AbycvZ,  = 0.2 e¨vm n«vm d=?

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2

9| 200cm j¤^v Ges 1mm2 cÖ¯’‡”Q‡`i †ÿÎdj wewkó GKwU B¯úv‡Zi Zv‡ii ˆ`N©¨ 1×10-3 m e„w× Ki‡Z cÖ‡qvRbxq Kv‡Ri cwigvb wbY©q Ki| Zv‡ii Dcv`v‡bi BqsGi ¸bv¼ Y  2 1011 Nm-2  Avgiv Rvwb, GLv‡b,

1 YAl 2 2 L 2 1011 10-6  (10-3 )2 W 22 W  0.05 J(Ans.) W

Avw` ˆ`N©¨, L = 200cm=2m cÖ¯’‡”Q‡`i †¶Îdj, A = 1mm2 = 1×10-6 m2 ‰`N©¨ e„w×, l = 1×10-3m  Y  2  1011 Nm -2

KvR, W = ?

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c„ôUvb (Surface Tension): ‡Kvb Zij c„‡ôi Dci hw` GKwU †iLv Kíbv Kiv nq Z‡e H †iLvi cÖwZ GKK ˆ`‡N©¨ †iLvi mv‡_ j¤^fv‡e Ges c„‡ôi ¯úk©Kiƒ‡c †iLvi Dfq cv‡k †h ej wµqv K‡i Zv‡K H Zi‡ji c„ôUvb e‡j| ‡Kvb Zi‡ji c„‡ôi Dci l ˆ`‡N©¨i †iLvi mv‡_ j¤^fv‡e Ges c„‡ôi ¯úk©Kiƒ‡c †iLvi Dfq cv‡k F ej wµqv K‡i Z‡e H Zi‡ji c„ôUvb T 

F n‡e| Gi GKK Nm-1 | cvwbi c„ôUvb 72×10-3 Nm -1 ej‡Z l

GB eywS †h, cvwb c„‡ô 1m j¤^v GKwU †iLv Kíbv Ki‡j H †iLvi mv‡_ j¤^fv‡e Ges c„‡ôi ¯úk©K iƒ‡c †iLvi Dfq cv‡k 72×10 -3N ej wµqv K‡i| c„ôUv‡bi gvÎv n‡”Q

MLT 2 ej  MT 2 Gi gvÎv| A_©vr, L ˆ`N ¨





c„ôkw³ (Surface Energy): m‡gvò Ae¯’vq †Kvb Zi‡ji gy³Z‡ji GKK †ÿÎdj e„w×i Rb¨ m¤úbœ Kv‡Ri cwigvb Z_v GKK †ÿÎd‡j mwÂZ wefekw³‡KB c„ôkw³ e‡j| †Kvb Zi‡ji gy³ Z‡ji †ÿÎdj A cwigvb e„w× Ki‡Z hw` W cwigvb KvR m¤úbœ nq W n‡e| Gi GKK Jm-2 ev Nm-1| myZivs †`Lv hv‡”Q c„ôkw³ Avi c„ôUv‡bi GKK GKB| c„ôkw³i A ML2T 2 KvR gvÎv n‡”Q Gi gvÎv| A_©vr,  MT  2 2 † ¶ Îdj L c„ôkw³ I c„ôUv‡bi g‡a¨ m¤úK© (Relation between Surface Energy and Surface Tension):

Zn‡j c„ôkw³ E 





g‡b Kwi, ABCD GKwU Zv‡ii †d«g| Gi BC evûwU AB I DC evû eivei Aev‡a PjvPj Ki‡Z cv‡i| ZviwU‡K mvevb cvwb‡Z Wywe‡q Zz‡j Avb‡j Gi gvSLv‡b GKwU cvZjv c`©v AvU‡K _vK‡e| GB c`©v c„ôUv‡bi Rb¨ †d«‡gi cÖ‡Z¨K evû‡K wfZ‡ii w`‡K Uvb‡Z _v‡K wKš‘ BC evû Qvov Aci evû¸‡jv AvUK‡bv _vKvq †m¸‡jv w¯’i _vK‡e| Gi d‡j c„ôUv‡bi Rb¨ BC evûwU AD evûi w`‡K AMÖmi n‡e| myZivs BC evû‡K Gi wbR ¯’v‡b ivLvi Rb¨ wecwiZ w`‡K ej cÖ‡qvM Ki‡Z n‡e| BC evûi ˆ`N©¨ l Ges Zi‡ji c„ôUvb T n‡j, BC Zv‡ii Dci AD Gi w`‡K †gvU ej n‡e, F=l×T+l×T=2lT (KviY Dc‡i I wb‡P `ywU c„ô Av‡Q)| myZivs BC evû‡K Gi Ae¯’v‡b w¯’i ivL‡Z n‡j Gi Dci c„ôUv‡bi wecixZgyLx F=2lT ej cÖ‡qvM Ki‡Z n‡e| GLb BC Zvi‡K ÿz`ª `~iZ¡ b mwi‡q BC  Ae¯’v‡b Avb‡Z m¤úvw`Z KvR n‡e, W=Fb ev, W=2lTb Gi d‡j c`©vi Dci Ges wbP Dfq c„‡ôi cÖwZwUi †ÿÎdj lb cwigvb K‡i e„w× cv‡e| myZivs ABCD c`©vi †gvU †ÿÎdj e„w× cv‡e A=2lb| W 2lTb c„ôUv‡bi weiæ‡× cÖwZ GKK †ÿÎdj e„wׇZ K…ZKvR ev, c„ôkw³, E    T GB kw³ c„‡ô mwÂZ n‡e| A 2lb

myZivs, †Kvb Z‡ji c„ôkw³ Zvi c„ôUv‡bi mgvb| c„ôUv‡bi AvbweK gZev` (Molecular Theory on Surface Tension): c„ôUvb GKwU AvbweK NUbv e‡j G‡K AvbweK gZev` Øviv e¨vL¨v Kiv hvq| weÁvbx j¨vcøvm me©cÖ_g me©v‡cÿv wbf©i‡hvM¨ ZË¡ cÖ`vb K‡ib e‡j G‡K j¨vcøv‡mi AvbweK ZË¡ e‡j| j¨vcjv‡mi aviYv g‡Z, Zij c`v‡_©i AYy mg~‡ni AvKl©Y e‡ji R‡b¨B c„ôUv‡bi D™¢e nq| me©vwaK †h `~iZ¡ ch©šÍ `ywU AYyi ga¨Kvi cvi¯úwiK AvKl©Y ej A_©vr mshyw³ ej Abyf~Z nq Zv‡K AvYweK cvjøv e‡j| †Kvb GKwU AYy‡K †K›`ª K‡i AvYweK cvjøvi mgvb e¨vmva© wb‡q †Kvb †MvjK Kíbv Ki‡j, H †MvjK‡K H AYyi cÖfve †MvjK e‡j| ‡KvY AYyi cÖfve †Mvj‡Ki g‡a¨ Ab¨ †h AYy _v‡K H AYy Zv‡`i AvKl©Y K‡i| Ges

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2 10| cÖevnx c`v_© (Fluid Substance) wb‡RI Zv‡`i Øviv AvK„ó nq| cÖfve †Mvj‡Ki evB‡ii †Kvb AYy cÖfve †Mvj‡Ki cÖfve ‡Mvj‡Ki ga¨w¯’Z AbywU Øviv AvK…ó nq bv| g‡b Kwi, A, B I C GKwU Zi‡ji cÖfve †MvjK mn wZbwU AYy| A AbywU cÖfve †MvjKmn Zi‡ji wfZ‡i, B AYywU Zi‡ji gy³ Z‡ji GKUz wb‡P Ges C AYywU wVK Zij Z‡j Aew¯’Z| A AYyi cÖfve †MvjK Zi‡j wbgw¾Z _vKvi d‡j †Mvj‡Ki ga¨Kvi Ab¨vb¨ AYy Øviv Pvwiw`‡K mgfv‡e AvK…ó nIqvq H AYyi Dci mskw³ e‡ji jwä k~b¨ nq| B AYyi cÖfve †MvjK Zi‡j AvswkK wbgw¾Z _vKvi d‡j cÖfve †Mvj‡Ki Dc‡ii Aa©vs‡k wb‡Pi Aa©vsk A‡cÿv Kg AYy _v‡K| Gi d‡j B AYyi Dci wµqvkxj wb¤œgyLx msmw³ ej DaŸ©gyLx msmw³ ej A‡cÿv †ekx n‡e| Kv‡RB B AYyi jwä mskw³ ej wb¤œgyLx| C AYywU Zij c`v‡_©i gy³ Z‡j Aew¯’Z| Kv‡RB C AYyi Dc‡i Ab¨ †Kvb AYy bv _vKvq H AYywU wb‡Pi w`‡K AvK…ó n‡e| A_©vr Gi Dci jwä ej wb¤œgyLx|

Zij c„ó PQ n‡Z cÖfve †Mvj‡Ki e¨vmv‡a©i mgvb `~i‡Z¡ RS Zj Kíbv Ki‡j G `yÕZ‡ji gv‡Si mKj AYyB mskw³ e‡ji Rb¨ wb‡Pi w`‡K AvKl©Y Abyfe Ki‡e| Kv‡RB PQ Z‡ji Dci Aew¯’Z AYymgy‡ni Dci wb¤œgyLx ej me©vwaK nIqvq GwU wb¤œgyLx ej Abyfe Ki‡e| d‡j ZjwU Zvi wbR¯^ †ÿÎdj Kgv‡Z Pvq Ges msKzwPZ n‡Z cÖqvm cvq| G cÖqvm †_‡KB Zij c`v‡_©i ¯úk©K eivei GKwU Uvb ej m„wó nq| GB Uvb ejB Zi‡ji c„ôUvb ev Zj Uvb| ¯úk© ‡KvY (Angle of Cotact) : KwVb I Zi‡ji ¯úk© we›`y †_‡K eµ Zij Z‡j Aw¼Z ¯úk©K KwVb c`v‡_©i mv‡_ Zi‡ji wfZ‡i †h †KvY Drcbœ K‡i Zv‡K D³ KwVb I Zi‡ji ¯úk© †KvY e‡j| wP‡Î  n‡”Q ¯úk© †Kvb| †h me Zij c`v_© KwVb c`v_©‡K wfRvq Zv‡`i †ejvq ‡ejvq ¯úk© †Kvb m~ÿ‡KvY nq| KvP I cvwbi †ejvq ¯úk© †Kv‡Yi gvb cÖvq 8º nq| †h me Zij c`v_© KwVb c`v_ †K wfRvq bv Zv‡`i †ejvq ¯úk© ‡KvY ¯’yj †Kvb nq| KvP I weï× cvi‡`i †ejvq ¯úk© †Kv‡Yi gvb cÖvq 139º nq| ˆKwkK bj I ‰KwkKZv (Capillary tube and Capillarity) : AwZ m~ÿè I mylg wQ`ªwewkó bj‡K ˆKwkK bj e‡j| †Kvb ˆKwkK KvP b‡ji GK cÖvšÍ Zi‡ji g‡a¨ Lvov K‡i XyKv‡j b‡ji wfZi wKQy Zij Zi‡ji gy³ Zj †_‡K Dc‡i D‡V hvq A_©vr Zi‡ji DaŸ‡ivnY A_©vr Awa‡ÿc nq ev wb‡P †b‡g Av‡m A_©vr Zi‡ji Aebgb A_©vr Ae‡ÿc nq | ˆKwkK b‡j Zi‡ji G iKg Awa‡ÿc ev Ae‡ÿc‡K Zi‡ji ˆKwkKZv e‡j| Zi‡ji c„ôUv‡bi Rb¨ G iKg n‡q _v‡K| Zij c`v‡_©i c„ôUvb wbY©‡qi ZË¡ (Theory of meserment of Surface tension): GKwU ˆKwkK bj‡K cvwb ev H RvZxq †Kvb Zi‡ji wfZi Wyev‡j †`Lv hvq †h b‡ji g‡a¨ Zij LwbKUv Dc‡i D‡V Ges Gi Zj AeZj AvKvi aviY K‡i| g‡b Kwi, Zij I KwV‡bi ¯úk©‡KvY = | Zij Zj †hLv‡b b‡ji g‡a¨ bj‡K ¯úk© K‡i‡Q †mLv‡b b‡ji e¨vmva© =r| Zij Zj †_‡K b‡ji wfZ‡ii Zi‡ji wb¤œ cÖvšÍ ch©šÍ D”PZv = h| Zi‡ji NbZ¡ =Ges Zi‡ji c„ô Uvb =T| b‡ji wfZ‡ii †`Iqvj Ges Zi‡ji ¯úk© we›`y n‡Z eµ Zij Z‡j ¯úk©K Uvb‡j H ¯úk©K eivei c„ôUvb T wfZ‡ii w`‡K wµqv Ki‡e| ‰KwkK b‡ji wfZ‡ii e¨mva© r n‡j cwiwa n‡e 2r A_©vr b‡ji wfZ‡ii ¯úk© †iLvi ˆ`N©¨ n‡e 2r| d‡j c„ôUv‡bi Rb¨ b‡ji †`Iqvj ¯úk©K eivei wfZ‡ii w`‡K 2rT ej Abyfe Ki‡e| †`IqvjI Zi‡ji Dci Gi wecix‡Z evB‡ii w`‡K mgvb ej 2rT cÖ‡qvM Ki‡e| GB ej 2rT †K `ywU j¤^ Dcvs‡k wefvwRZ Ki‡j Lvov Dc‡ii w`‡K 2rTcos Ges Gi mv‡_ j¤^fv‡e AbyfywgK eivei 2rTsin evB‡ii w`‡K Dc‡ii wPÎ Abyhvqx wµqv Ki‡e| b‡ji e¨v‡mi wecixZ w`‡K wµqv Kivq 2rTsin e‡ji Dcvsk ¸wj ci¯úi‡K bvKP K‡i †`q| AZGe Zi‡ji Dci †gvU DaŸ©gyLx ej n‡e 2rTcos|

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3 10| cÖevnx c`v_© (Fluid Substance) ‡h‡nZz GB DaŸ©gyLx e‡ji cÖfv‡e Zij ¯Í¤¢ ˆKwkK b‡ji g‡a¨ Dc‡i DV‡Z _v‡K myZivs hLb Zij ¯Í‡¤¢i IRb GB DaŸ©gyLx e‡ji mgvb nq ZLb mvg¨ve¯’v m„wó nq A_©vr b‡ji g‡a¨ Zij ¯Í¤¢ w¯’i n‡q hvq| b‡ji eµ As‡ki AvqZb v n‡j Zij ¯Í‡¤¢i †gvU AvqZb = r2h + v Ges GB Zi‡ji IRb = (r2h + v)g AZGe mvg¨ve¯’vq, 2rTcos(r2h + v)g

(r 2 h  v) g ... ... ... (1) 2r cos  GLb, v= ABCD wmwjÛv‡ii AvqZb  AEB Aa©‡Mvj‡Ki AvqZb 1 4  v  r 2 . r  . r 3 2 3 2  v  r 3  r 3 3 1 3  v  r 3 (1) bs mgxKi‡Y v Gi gvb ewm‡q cvB, 1 (r 2 h  r 3 ) g 3 T 2r cos  r r 2 (h  ) g 3 T  2r cos r r (h  ) g r 3 T  GLb ˆKwkK bj hw` miæ nq A_©vr r Gi gvb LyeB Kg nq, Zvn‡j h Gi Zzjbvq †K 2 cos 3 rhg D‡cÿv Kiv hvq| †m‡ÿ‡Î, T  , weï× cvwb I cwi®‹vi Kv‡Pi ga¨Kvi ¯úk©‡KvY cÖvq 0º nIqvq cvwbi Rb¨ 2 cos  rhg ... ... ... (2) cos   1 aiv nq| †m †ÿ‡Î c„ôUvb T  2 myZivs r, h, I g Gi gvb Dc‡iv³ mgxKi‡Y ewm‡q c„ôUvb T wbY©q Kiv hvq| hw` Zi‡ji c„ôUvb Rvbv _v‡K Zvn‡j 2T ... ... ... (3) ‰KwkK b‡j †h D”PZv ch©šÍ cvwb DV‡Z cv‡i Zvi mgxKiY, h  rg T 

‰KwkK bj c×wZ‡Z Zij c`v‡_©i c„ôUvb wbY©q (Determination of Surface Tension of liquid by capillary Tube Method) : ‰KwkK bj c×wZ‡Z Zij c`v‡_©i c„ôUvb wbY©‡qi mvaviY m~Î nj  r  rg  h   3  T 2 cos 

GLv‡b, T = c„ôUvb r = ˆKwkK b‡ji e¨vmva©  =Zij c`v‡_i NbZ¡  = ¯úk© †KvY h = ˆKwkK b‡ji wfZi Zij c`v‡_©i D”PZv, g = AwfKl©R Z¡iY ‡h me Zij KvP wfRvq Ges †h me Zij c`v‡_©i †ejvq ¯úk© †Kvb cÖvq k~b¨ Zv‡`i c„ôUvb wbY©‡q ˆKwkK bj

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10| cÖevnx c`v_© (Fluid Substance) e¨envi Kiv nq| cvwb I Kv‡Pi †ÿ‡Î ¯úk© †KvY cÖvq k~b¨| myZivs cos= cos0º = 1| mvaviYZ h>>r myZivs h Gi Zzjbvq Avgiv r/3 †K D‡cÿv Ki‡Z cvwi| T 

4

rhg ... ... ... (1) cixÿvq GB mgxKiY n‡Z c„ôUvb wbY©q Kiv nq| 2

Kvh©c×wZ: GKwU weKv‡i cvwb wb‡q ÷¨vÛ I avi‡Ki mvnv‡h¨ `yÕwZbwU Kv‡Pi ˆKwkK bj we‡kl e¨e¯’vq hy³ Kiv nq, †hb ˆKwkK bj ¸‡jvi wKQy Ask weKv‡ii cvwb‡Z wbgw¾Z nq Ges bj¸‡jv wPÎvbyhvqx Lvovfv‡e Ae¯’vb K‡i| GKwU muyP‡K avi‡Ki mv‡_ Ggb fv‡e hy³ Kiv nq †hb my‡Pui AMÖfvM cvwbi Zj ¯úk© K‡i| Gici GKwU åvg¨gvb `~iweÿb h‡›`ªi mvnv‡h¨ ˆKwkK b‡j cvwb¯Í‡¤¢i wb¤œ Z‡j †dvKvm Kiv nq Ges `~iweÿb h‡›`ªi Dj¤^ †¯‹‡ji cvV †bIqv nq| Gfv‡e cÖwZwU ˆKwkK b‡j DwÌZ cvwb¯Í‡¤¢i wb¤œ Z‡ji cvV †bIqv nq| Gici weKviwU mwi‡q my‡u Pi AMÖfv‡Mi cvV †bIqv nq| Gici cvwb¯Í‡¤¢i D”P Z‡ji cvV n‡Z m~u‡Pi AMÖfv‡Mi cvV we‡qvM K‡i cvwb ¯Í‡¤¢i D”PZv h wbY©q Kiv nq| Gi ci cvwb¯Í‡¤¢i D”PZvi ¯’v‡b avivj dvBj w`‡q bj †K‡U †djv nq| AZtci `~iweÿb h‡›`ªi mvnv‡h¨ b‡ji e¨vm †g‡cwb‡q Zv n‡Z e¨vmva© r cwigvc Kiv nq| cixÿvMv‡ii cvwbi ZvcgvÎvq -Gi gvb †R‡b wb‡q (1) bs mgxKi‡Y ewm‡q cvwbi c„ôUvb T wbY©q Kiv nq| Gfv‡e cÖwZwU ˆKwkK b‡ji Rb¨ c„ôUvb wbY©q K‡i Mo c„ôUvb wbY©q Kiv nq| mZK©Zv (Caution) : 1| mylg wQ`ªhy³ ˆKwkK bj ‡bIqv DwPZ| 2| cixÿvaxb Zi‡j †hb †Kvb cÖKvi †Zj ev Pwe© RvZxq c`v_© bv _v‡K †mw`‡K jÿ¨ ivLv DwPZ| 3| ˆKwkK bj¸‡jv‡K Zi‡j Lvov fv‡e Wyevb DwPZ| c„ôUv‡bi Dci ZvcgvÎvi cÖfve (Effect of Temperature on Surface Tension): ZvcgvÎv e„wׇZ Zi‡ji NbZ¡ n«vm cvq A_©vr AYymgy‡ni ga¨Kvi `~iZ¡ e„w× cvq| d‡j Zi‡ji mskw³ ej n«vm cvq| myZivs ZvcgvÎv e„wׇZ Zi‡ji c„ôUvb n«vm cvq| hw` 0ºC ZvcgvÎv I ºC ZvcgvÎvq Zi‡ji c„ôUvb h_vµ‡g T0 I T nq Z‡e, T  T0 (1   ) GLv‡b  GKUv aªæeK, G‡K c„ôUv‡bi DòZv ¸Yv¼ e‡j| c„óUv‡bi msKU ZvcgvÎv (Critical Temperature): †h ZvcgvÎvq †Kvb GKwU Zi‡ji c„ôUvb k~b¨ nq Zv‡K H Zi‡ji c„óUv‡bi msKU ZvcgvÎv e‡j| mv›`ªZv (Viscosity): ‡h a‡g©i `iæb cÖevwni wewfbœ ¯Í‡ii Av‡cwÿK MwZ‡Z evavi m„wó nq Zv‡K H cÖevwni mv›`ªZv e‡j| Nl©Y †hgb `ywU KwVb c`v‡_©i Av‡cwÿK MwZ‡K evav †`q, mv›`ªZv †Zgwb cÖevwni `ywU ¯Í‡ii Av‡cwÿK MwZ‡Z evav †`q Ges MwZ e¨vnZ Ki‡Z †Póv K‡i| mv›`ªZv‡K ZvB AšÍt¯’ Nl©YI ejv nq| mv›`ªZv ¸bv¼ (Coefficient of Viscosity) : g‡b Kwi, AB GKwU w¯’i KwVb Zj| Gi Dci w`‡q GKwU cÖevnx MwZkxj| cÖevnxi AwZ wbKUeZ©x `ywU ¯Íi CD I EF Kíbv Kwi| AB n‡Z CD Gi `~iZ¡ = x AB n‡Z EF Gi `~iZ¡ = (x+dx) CD ¯Í‡ii †eM= v EF ¯Í‡ii †eM= v+dv cÖwZ GKK `~i‡Z¡ †e‡Mi cwieZ©‡bi nvi ev †eM Aeµg =

dv dx

awi, mv›`ª ej F| mv›`ª ej msµvšÍ wbDU‡bi m~Î wb¤§iƒc t (1) mv›`ª ej Zij ¯Í‡ii †ÿd‡ji mgvbycvwZK| A_©vr, F  A hLb,

dv aªæe| dx

(2) mv›`ª ej †eM Aeµ‡gi mgvbycvwZK|

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10| cÖevnx c`v_©

A_©vr, F 

(Fluid Substance)

5

dv hLb, A aªæe| dx

m~ÎØq GKwÎZ K‡i cvB, dv dx dv  F  A ... ... ... (1) GLv‡b,  GKwU aªæe msL¨v| GB aªæe msL¨v‡K mv›`ªZv¼ ev mv›`ªZv ¸bv¼ ev mv›`ªZv mnM dx (Coefficient Of Viscocity) e‡j| F dv dv (1) bs mgxKiY n‡Z cvB,   ... ... ... (2) hw` A=1 GKK Ges †eM Aeµg  1 GKK nq Z‡e (2) bs A dx dx mgxKiY n‡Z cvB, F    mv›`ªZv ¸bv‡¼i msÁv wb¤œiƒc, FA

GKK †eM Aeµg eRvq _vKv Ae¯’vq AwZ wbKUeZ©x `ywU Zij ev M¨vm ¯Í‡ii g‡a¨ cÖwZ GKK †ÿÎd‡j †h cwigvb mv›`ªej ¯úk©K eivei wµqv K‡i, Zv‡K H cwievnxi mv›`ªZv ¸bv¼ e‡j| F dx  ej  `~ iZ¡   MLT 2  L  1 1    2   ML T  A dv  † ¶ Îdj  † eM   L  L / T  F dx N m ej  `~ iZ¡  2  Gi GKK t      Nm 2 s ev, Pa s A dv † ¶ Îdj  † eM m  ms 1 Zvrch© (Significance): cvwbi mv›`ªZv mnM 10 -3 Nm -2 s ej‡Z GB eywS †h, 1 m2 †ÿÎdj wewkó cvwbi `ywU ¯Íi ci¯úi 1m `~i‡Z¡ Aew¯’Z n‡j G‡`i wfZi 1ms -1 Av‡cwÿK †eM eRvq ivL‡Z 10 -3 N e‡ji cÖ‡qvRb nq|



 Gi gvÎv t  



mv›`ªZvi Dci ZvcgvÎvi cÖfve (Effect of Temperature on Viscosity): ZvcgvÎv evo‡j Zi‡ji mv›`ªZv K‡g| †`Lv‡M‡Q †h, 10º C ZvcgvÎvq cvwbi mv›`ªZv mn‡Mi †h gvb cvIqv hvq, 80º C ZvcgvÎvq †m gvb nq GK Z…Zxqvsk| ZvcgvÎvi Dci mv›`ªZvi wbf©ikxjZv wb¤œiƒct log   A 

B T

GLv‡b, nj Zi‡ji mv›`ªZv mnM, T Zi‡ji †Kjwfb ZvcgvÎv Ges A I B nj aªæeK|

wKš‘ ZvcgvÎv evo‡j M¨v‡mi mv›`ªZv ev‡o| M¨v‡mi mv›`ªZv mnM Zvi †Kjwfb ZvcgvÎvi eM©g~‡ji mgvbycvwZK| A_©vr,   T mv›`ªZvi Dci Pv‡ci cÖfve (Effect of Pressure on Viscosity): Pvc e„w× †c‡j Zij c`v‡_©i mv›`ªZv e„w× cvq| M¨v‡mi mv›`ªZv Pv‡ci Dci wbf©jkxj bq| ‡÷vK&m Gi m~Î (Stoke's Law) : AwaK mgmZ¡ I Ams‡KvPbxq †Kvb cÖevnxi ga¨w`‡q cošÍ GKwU ÿz`ª †Mvj‡Ki Dci g›`b ej, cÖevnxi cÖvšÍ‡eM v Gi mgvbycvwZK, mv›`ªZv ¸bv¼ -Gi mgvbycvwZK Ges †Mvj‡Ki e¨vmva© r Gi mgvbycvwZK| e¨vL¨v t †Kvb e¯‘ Zij ev M¨v‡mi ga¨ w`‡q AwfKl© e‡ji cÖfv‡e wb‡P co‡Z _vK‡j e¯‘wU Zvi ms¯ú‡k© _vKv Zij ev M¨v‡mi ¯Íi‡K wb‡Ri mv‡_ †U‡b wb‡q P‡j| d‡j M¨v‡mi wewfbœ ¯Í‡ii g‡a¨ Av‡cwÿK †eM m„wó nq| wKš‘ gva¨‡gi mv›`ªZv H Av‡cwÿK MwZ‡K g›`xf~Z Kivi †Póv K‡i d‡j e¯‘i Dci g›`b wµqv K‡i| cošÍ e¯‘i †eM hZB e„w× cvq mv›`ªZv RwbZ wecixZgyLx ej Z_v g›`b ej I ZZ e„w× cvq| ¯^í mg‡qi g‡a¨ evav `vbKvix ej MwZ m„wóKvix e‡ji mgvb nq d‡j e¯‘ mg‡e‡M co‡Z _v‡K| GB mg‡eM †K cÖvwšÍK †eM e‡j| r e¨vmv‡a©i †Kvb ÿz`ªKvi †MvjK  mv›`ªZv ¸bv‡¼i †Kvb Zij ev M¨vmxq c`v‡_©i ga¨w`‡q v cÖvwšÍK †e‡M co‡Z _vK‡j †÷vK&m Gi m~Îvbymv‡i, mv›`ªZvi Rb¨ evav`vb Kvix ej ev g›`b ej, F  r v  F  k r v GLv‡b k mgvbycvwZK aªæeK| †÷vKm& MvwbwZK we‡køl‡bi mvnv‡h¨ cÖgvb K‡ib †h, k = 6  F  6 r v

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10| cÖevnx c`v_©

(Fluid Substance)

6

‡÷vK&m Gi m~Î cÖwZcv`b t awi, r e¨vmv‡a©i †Kvb ÿz`ª †MvjK mv›`ªZv ¸bv‡¼i †Kvb cÖevnx (Zij ev M¨vm) Gi ga¨ w`‡q v cÖvwšÍK †e‡M co‡Z _vK‡j mv›`ªZvi Rb¨ GKwU evav`vbKvix ej MwZi wecix‡Z wµqv Ki‡e| hyw³ hy³ fv‡e ejv hvq †h, g›`b ej F wb¤œwjwLZ ivwk¸‡jvi Dci wbf©ikxjt F  e¯‘i e¨vmva©, r F  gva¨‡gi mv›`ªZv¼ , F  cÖvšÍ‡eM, v g‡bKwi, F  kr x y v z ... ... ... (1) GLv‡b k GKwU gvÎv wenxb aªæeK Ges x, y I z AÁvZ m~PK| GLb GB mgxKi‡Yi Dfq cv‡k¦©i gvÎv mgvb n‡e| AZGe

F   r x   y  v z  y z x  MLT  2   L  ML1T 1  LT 1   MLT 2  Lx M y L yT  y Lz T  z  MLT 2  Lx  y  z M y T  y  z M, L I T -Gi Rb¨ Dfq cv‡k¦©i m~PK mgvb a‡i cvB, y =1 ... ... ... ... (2) x y+ z = 1 ... ... (3) y z= 2 ... ... ... (4) (4) bs mgxKi‡Y y Gi gvb ewm‡q cvB, 1z= 2  z =1 (3) bs mgxKi‡Y yI z Gi gvb ewm‡q cvB, x1 +1=1  x =1 (1) bs mgxKi‡Y x, y I z Gi gvb ewm‡q cvB, F  kr 1 1v1  F  kr v  F  6r v GB mgxKiYwU †÷vKm& me©cÖ_g cÖwZcv`b K‡ib ZvB Gi bvg nq †÷vKm& Gi mgxKiY|

cÖvšÍ-‡eM ev AšÍ-†e‡Mi ivwk gvjv: †Kvb e¯‘ Zij ev M¨v‡mi ga¨ w`‡q AwfKl© e‡ji cÖfv‡e wb‡P co‡Z _vK‡j e¯‘wU Zvi ms¯ú‡k© _vKv Zij ev M¨v‡mi ¯Íi‡K wb‡Ri mv‡_ †U‡b wb‡q P‡j| d‡j M¨v‡mi wewfbœ ¯Í‡ii g‡a¨ Av‡cwÿK †eM m„wó nq| wKš‘ gva¨‡gi mv›`ªZv H Av‡cwÿK MwZ‡K g›`xf~Z Kivi †Póv K‡i d‡j e¯‘i Dci g›`b wµqv K‡i| cošÍ e¯‘i †eM hZB e„w× cvq mv›`ªZv RwbZ wecixZgyLx ej Z_v g›`b ej I ZZ e„w× cvq| ¯^í mg‡qi g‡a¨ evav `vbKvix ej MwZ m„wóKvix e‡ji mgvb nq d‡j e¯‘ mg‡e‡M co‡Z _v‡K| GB mg‡eM †K AšÍ-‡eM ev cÖvwšÍK †eM e‡j| awi, AšÍ-‡eM v | v Gi Rb¨ ivwkgvjv cÖwZcv`b Ki‡Z PvB| g‡bKwi, wb¤œgyLx ej Z_v †Mvj‡Ki IRb W DaŸ©gyLx ej Z_v cøeZv U Ges DaŸ©gyLx evav`vbKvix ej Z_v mv›`ª-cðvrUvb F| Avw`‡Z wb¤œgyLx ej W DaŸ©gyLx ej (U+F) Gi †P‡q eo| d‡j †MvjKwUi wb¤œgyLx Z¡iY _v‡K| ‡MvjKwUi †eM e„w×i mv‡_ mv›`ª-cðvrUvb I e„w× cvq| d‡j (U+F) GK mgq W Gi mgvb nq| ZLb †MvjKwUi bxU Z¡iY k~b¨ nq| †MvjKwU ZLb aªæe ev mg‡e‡M wb‡P cwZZ nq| G aªæe ‡eM‡K ejv nq AšÍ‡eM v|

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10| cÖevnx c`v_© GLb hw` cÖevnxi NbZ¡ f I †Mvj‡Ki NbZ¡ s nq Zvn‡j,

(Fluid Substance)

4 4 3 3 cÖevnxi mv›`ªZv mnM  n‡j, F = 6rv

‡Mvj‡Ki IRb, W  r 3  s g Ges U  r 3  f g ‡÷vK&‡mi m~Îvbymv‡i †MvjKwU AšÍ‡eM cÖvß n‡j, F+U=W 4 4  6rv  r 3  f g  r 3  s g 3 3 4 4  6rv  r 3  s g  r 3  f g 3 3 4 3  6rv  r g (  s   f ) 3 2 2r (  s   f ) g v  BnvB AšÍ †e‡Mi ivwkgvjv| 9

‡÷vK&‡mi m~‡Îi mvnv‡h¨ Zi‡ji mv›`ªZv mnM wbY©q (Determinition of Co-efficient of Viscosity by Stoke's law): ‡h Zi‡ji mv›`ªZv mnM  wbY©q Ki‡Z n‡e Zv GKwU eo AvKv‡ii gvc †Pv‡O †bqv nq| GLb r e¨vmv‡a©i †QvU †MvjK (†hgb ejweqvwis) GB Zi‡j Lyeax‡i †Q‡o †`qv nq| †MvjKwU GK mgq AšÍ-‡eM cÖvß n‡q wb‡P bvg‡Z _v‡K| A `vM †_‡K B `vM ch©šÍ c_ †h‡Z †MvjKwUi †h mgq jv‡M Zv _vgv Nwoi mvnv‡h¨ wbY©q Kiv nq| awi GB mgq t| g‡b ivL‡Z n‡e A †hb Zi‡ji †ek wfZ‡i nq hv‡Z †MvjKwU A Ae¯’v‡b ‡cŠQvi c~‡e©B AšÍ-†eM cÖvß nq| AB , GLb hw` cÖevnxi NbZ¡ f I †Mvj‡Ki NbZ¡ s nq Zvn‡j Avgiv cvB, t 2 AB 2r (  s   f ) g  GLv‡b, g AwfKl©R Z¡iY| t 9 GLb ¯Œ M‡Ri mvnv‡h¨ †Mvj‡Ki e¨vmva© r Gi gvb †ei K‡if Ges s Gi gvb †R‡b 2r 2 (  s   f ) gt G ewm‡q  Gi gvb mn‡RB wbY©q Kiv hvq| wb‡q  Gi gvb wb‡¤§v³ mgxKiY   9 AB mskw³ ej (Cohesive Force) :

G‡ÿ‡Î, v 

GKB c`v‡_©i wewfbœ AYyi g‡a¨ cvi¯úwiK AvKl©Y ej‡K mskw³ ej ejv nq| AvmÄb-ej (Adhesive Force): wewfbœ c`v‡_©i AYyi wfZi cvi¯úwiK AvKl©Y ej‡K AvmÄb ej e‡j|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution 10| cÖevwn c`v_© (Fluid)

1| GKwU Zv‡ii IRb bMb¨ a‡i G‡K 25°C ZvcgvÎvi cvwbi DcwiZj †_‡K 0.05m j¤^v GKwU AbyfywgK Zvi‡K me©vwaK 7.30×103 N e‡j †U‡b DVvb hvq| cvwbi c„óUvb wbY©q Ki| Avgiv Rvwb, GLv‡b, F T ej, F = 7.30×10 3N 2L Zv‡ii ˆ`N©¨, L=0.05m 7.30  10 -3 2  0.05  T  0.073 Nm -1 (Ans.) T

cvwbi c„ôUvb, T=?

2T rρg

ev, h 

2  72  10 -3 0.1  10 -3  10 3  9.8

 h  0.1469 m (Ans.)

r = 0.1 mm = 0.1×10 -3 m cvwbi c„ôUvb, T=72×10 -3 Nm -1 cvwbi NbZ¡ kgm -3 cvwbi D”PZv, h = ?

3| 200mm e¨vmv‡a©i GKwU avZe †MvjK GKwU Zi‡ji ga¨w`‡q 2.1×10-2ms-1 cÖvšÍ †e‡M co‡Q| Zi‡ji mv›`ªZvsK 0.003 kg m-1s-1| Zi‡ji mv›`ª ej wbY©q Ki| GLv‡b, Avgiv Rvwb, e¨vmva©, r =200 mm=0.2m F = 6rv †eM, v=2×10-2ms-1  F = 6×3.14×0.2×0.003× 2.1×10-2 N mv›`ªZvsK. =0.003 kg m-1s-1 F = 2.37×10-4 N (Ans.) mv›`ªej, F  ?

4| 2mm e¨v‡mi GKwU cvwbi †MvjK‡K †f‡O `k jÿ mgAvqZb ÿz`ª †duvUv ‰Zix Ki‡j wK cwigvb KvR m¤úbœ n‡e| [cvwbi c„ôUvb =72× 10 -3 Nm-1] Avgiv Rvwb.

KvR W= ‡ÿÎd‡ji cwieZ©b × c„ôUvb

 KvRW  A  T  W  4( Nr 2 - R 2 )T GLb,

4 4 10  r 3  R 3 3 3 6





3

 10 2 r  R 3 R r 100 10 3 r m  10 5 m 100

5| 0.8× 10-3 m e¨vmv‡a©i GKwU ˆKwkK KvPbj cvi‡` Wyev‡j b‡ji g‡a¨ cvi‡`i Aebgb 6.753× 10-3 m nq| Kv‡Pi mv‡_ cvi‡`i ¯úk© †KvY KZ? cvi‡`i c„ôUvb 4.7× 10-1 Nm-1 Ges NbZ¡ 13.6× 103 kg m-3 | GLv‡b, e¨vmva© r = 0.8× 10-3 m Avgiv Rvwb, cvi‡`i Aebgb, h =  6.753× 10-3 m

T

2| 0.2mm e¨v‡mi GKwU b‡j cvwbi Av‡ivnb wbY©q Ki| cvwbi c„ôUvb =72 ×10-3 Nm -1I cvwbi NbZ¡ 103 kgm -3| Avgiv Rvwb, GLv‡b, rhρg T b‡ji e¨vm, d = 0.2 mm 2 b‡ji e¨vmva©, ev, h 

 W  8.95  10 5 J (Ans.)

GLv‡b, eo †dvUvi e¨vm D = 2mm eo †dvUvi e¨vmva©, R= 1mm = 1×10 -3m

†QvU †dvUvi e¨vmva©, r =? cvwbi c„ôUvb, T=72×10 -3 Nm -1 m¤úvw`Z KvR, W=? ‡dvUvi msL¨v, N=106

hgr 2 cos 

cvi‡`i c„ôUvb, T = 4.7× 10-1 Nm-1 cvi‡`i NbZ,¡ 13.6× 103 kgm-3 cvi‡`i 3 ¯úk© †KvY,3 = KZ? 3

 6.75310 13.6 10  9.8 0.8 10 2cos 3  6.75310 13.6 103  9.8  0.8 103  cos  2  4.7 101  cos   0.765991353    cos 1 (0.765991353)    140 (Ans.)  4.7 101 

6| 2× 10-4 m e¨vmv‡a©i GKwU †jvnvi ej Zvwc©b †Z‡ji wfZi w`‡q 4× 10-2 ms-1 cÖvšÍ ‡e‡M co‡Q| hw` †jvnv I Zvwc©b ‡Z‡ji NbZ¡ h_vµ‡g 7.8×103 kgm-3 Ges GLv‡b, e¨vmva© r = 2× 10-4 m 3 -3 0.87×10 kgm nq, Z‡e Zvwc©b cÖvšÍ †eM, v = 4× 10-2 ms-1 †Z‡ji mv›`ªZvsK wbY©q Ki| ‡jvnvi NbZ¡ , s7.8×103 kgm-3 Avgiv Rvwb, ‡Z‡ji NbZ¡ , f0.87×103 kgm-3 mv›`ªZvsK=?  2 r 2 ( s   f )g  9v 2(2  10 4 ) 2 (7.8  103  0.87  10 3 )  9.8  9  4  10 2    0.0151 kg m -1s 1 (Ans.)

7| GKwU b‡ji e¨vmva© 0.1mm | G‡K 60 ×10-3 Nm -1 c„ôUvb Ges 800 kgm -3 Nb‡Z¡i GKwU †Z‡j Wyev‡j ‰KwkK b‡j KZ D”PZvq †Zj DV‡e| ¯úk© †KvY 20°| GLv‡b, rhρg b‡ji e¨vmva©, r = 0.1mm T = 0.1×10 -3 m 2cos -3 c„ ô Uvb, T=60×10 Nm -1 T2cos  h NbZ¡ kgm -3 rρg ¯úk© †KvY,  =20º| 60  10 -3  2  cos 20 0 cvwbi D”PZv, h = ? h 0.1 10 -3  800  9.8 60  10 -3  2  0.93969262 h 0.1 10 -3  800  9.8  h  0.1438m (Ans.)

 W  4  3.14{10 6  (10 5 ) 2 - (10 -3 ) 2 }  72  10 -3 J

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 তা঩ভাত্রায ঩রু যস্কর : ঩ুরু ঱ূ নয তা঩ভাত্রানক ঱ূ নয ধনয রতবয তা঩ভাত্রায যস্করনক তা঩ভাত্রায ঩যভ যস্কর ফনর। ঩যভ যস্কনরয ঩াঠ্ = য঳বিনগ্রনড ঩াঠ্ + ২৭৩  ঳নভাষ্ণ ঩বযফত঱ন : বিয তা঩ভাত্রায় িান঩য ঳ানথ একবট বনবদ঱ি বনযয যকান ফস্ত্ত্তয আয়তননয ঩বযফত঱ননক ঳ভতা঩ীয় ফা ঳নভাষ্ণ ঩বযফত঱ন ফনর। এনক্ষনত্র গযা঳ ফনয়নরয ঳ূ ত্র যভনন িনর অথ঱াৎ PV= ধ্রুফক।  রুদ্ধতা঩ীয় ঩বযফত঱ন : ফাব঴য ঴নত যকান তা঩ গ্র঴ন ফা ফজ঱ন না কনয যকান ফস্ত্ত্তয যবৌত অফিায ঩বযফত঱ননক রুদ্ধতা঩ীয় ঩বযফত঱ন ফনর। অথ঱াৎ Pvγ= ধ্রুফক, যমখানন, γ =

বিয িান঩ গযান঳য আন঩বক্ষক তা঩ বিয আয়তনন গযান঳য আন঩বক্ষক তা঩

 আন঩বক্ষক আদ্র঱তা : যকান বনবদ঱ি তা঩ভাত্রায় একবট বনবদ঱ি আয়তননয ফায়ু নত যম ঩বযভাণ জরীয় ফাষ্প থানক ঐ তা঩ভাত্রায় ঐ আয়তননয ফায়ু নক ঳ম্পৃ ি কযনত যম ঩বযভাণ জরীয় ফানষ্পয প্রনয়াজন ঴য় তানদয অনু ঩াতনক আন঩বক্ষক আদ্র঱তা ফনর। আন঩বক্ষক আদ্র঱তা=

ব঱ব঱যানে জরীয় ফানষ্পয িা঩ ফায়ু য তা঩ভাত্রায় ঳ম্পৃি জরীয় ফানষ্পয িা঩

× 100%

 ব঱ব঱যািংকঃ যম তা঩ভাত্রায় ফায়ু ভন্ডনরয যকান বনবদ঱ি আয়তননয ফায়ু এয ভনধয অফবিত জরীয় ফাষ্প িাযা ঳ম্পৃ ি ঴য়, য঳ তা঩ভাত্রানক ব঱ব঱যািংক ফনর।  ক্রাবন্ত তা঩ভাত্রাঃ ঳নফ঱াচ্চ যম তা঩ভাত্রায় যকান গযা঳নক শুধু িা঩ প্রনয়াগ কনয তযনর ঩বযণত কযা মায় তানক ক্রান্তি তা঩ভাত্র ফনর। ক্রাবন্ত আয়তনঃ ক্রান্তি তা঩ভাত্রা ঑ ক্রান্তি িান঩ এক গ্রাভ বনযয যকান গযান঳য আয়তননক ঐ গযান঳য ক্রান্তি আয়তন ফনর।  আদ঱঱ গযা঳ঃ যম঳কর গযা঳ ঳কর তা঩ভাত্রায় ফনয়র ঑ িার঳঱ এয ঳ূ নত্র যভনন িনর তানদযনক আদ঱঱ গযা঳ ফনর। যকান গযা঳ই আদ঱঱ গযা঳ নয়।  রত্রধ বফন্দুঃ যম তা঩ভাত্রায় বফশুদ্ধ ফযপ, ঩াবন ঑ ঳ম্পৃ ি জরীয় ফাষ্প তা঩গত ঳঴অফিানন থানক তানক ঩াবনয রত্রধ বফন্দু ফনর। ঩াবনয রত্রধবফন্দু 0.160C ফা 273.16k (4.58mm ঩াযদ)।  স্বাবাবফক িান঩ ঑ কক্ষ তা঩ভাত্রায় গযান঳য অনু গুনরায ভনধয ঳িংঘনল঱য ঳িংখযা প্রবত য঳নকনন্ড প্রায় 109  তা঩ প্রনয়ানগ গযান঳য প্র঳াযণ তযনরয যিনয় অননক যফ঱ী এফিং একই তা঩ভাত্রা ফৃ বদ্ধনত ঳কর গযান঳য প্র঳াযণ একই ঴য়। -23

 PV= KT ঳ভীকযণ K যক যফাল্টজভযান ধ্রুফক ফনর। এই ধ্রুফনকয ভান 1.38×10 ক্রাবন্ত তা঩ভাত্রা

-1

JK |

ক্রাবন্ত িা঩

ক্রাবন্ত আয়তন

Co2 এয ক্রাবন্ত তা঩ভাত্রা 31.1 C

Co2 এয ক্রাবন্ত িা঩ 73 atm

Co2 এয ক্রাবন্ত আয়তন 2.17 ঘন য঳.বভ.

O2 এয ক্রাবন্ত তা঩ভাত্রা 1190C

O2 এয ক্রাবন্ত িা঩ 50 atm

O2 এয ক্রাবন্ত আয়তন 2.320 ঘন য঳.বভ.

H2 এয ক্রাবন্ত তা঩ভাত্রা -2400C

H2 এয ক্রাবন্ত িা঩ 13 atm

H2 এয ক্রাবন্ত আয়তন 32.2 ঘন য঳.বভ.

0

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e‡qj -Gi m~Î ev, w¯’i DòZvq Pvc I AvqZ‡bi g‡a¨ m¤úK© (Boyle's Law or, Relation between Pressure and Volume at constant temperature):

w¯’i DòZvq GKwU wbw`©ó f‡ii M¨v‡mi AvqZb Gi Pv‡ci e¨v¯ÍvbycvwZK| e¨vL¨v: aivhvK, T †Kjwfb ZvcgvÎvq P Pv‡c GKwU wbw`©ó f‡ii M¨v‡mi AvqZb V; Zvn‡j e‡qj Gi eY©bv Abymv‡i,

V

1 P

1 P  PV  K ... ... ... ... (1) GLv‡b, K GKwU mgvbycvwZK aªæeK| hw` w¯’i ZvcgvÎvq P1, P2, P3, ........, Pn Pv‡c GKwU wbw`©ó f‡ii M¨v‡mi AvqZb h_vµ‡g V1, V2, V3, ..............,Vn nq, Z‡e mgxKiY (1) Abymv‡i Avgiv wjL‡Z cvwi, P1V1=P2V2=P3V3=.........= PnVn = K ... ... ... ... (2) BnvB e‡qj- Gi m~‡Îi MvwYwZK e¨vL¨v| V  K

Pvj©m -Gi m~Î (Charle's Law): m~‡Îi eY©bv: w¯’i Pv‡c cÖwZ wWwMÖ DòZv cwieZ©‡b GKwU wbw`©ó f‡ii M¨v‡mi AvqZb, Gi k~b¨ wWwMÖ AvqZ‡bi

1 ev, 273

0.00366 Ask cwiewZ©Z nq| e¨vL¨v: aiv hvK, w¯’i Pv‡c 0ºC DòZvq GKwU wbw`©ó f‡ii M¨v‡mi AvqZb Vo; Zvn‡j cÖwZ 1ºC DòZv e„wׇZ M¨v‡mi 1 1 Ask e„w× cvq| hw` DòZv 1ºC nv‡i K‡g hvq, Z‡e M¨v‡mi AvqZb Ask K‡g hv‡e| AvqZb 273 273 V 1ºC DòZvq M¨v‡mi AvqZb  Vo  o

2ºC DòZvq M¨v‡mi ºC DòZvq M¨v‡mi

273 2Vo Abyiƒcfv‡e, AvqZb  Vo  273 θV AvqZb  Vo  o 273 θV  Vθ  Vo  o 273

θ    Vθ  Vo 1    273   273  θ   Vθ  Vo   GLb ZvcgvÎvi AvšÍ©RvwZK GK‡K 273    ºC = T  273 

Ges 273ºC =To K wjL‡j Dc‡iv³ mgxKiY `vovq,

T Vθ  Vo  To

ev,

Vθ T  Vo To

ev mvaviY fv‡e VT A_©vr w¯’i Pv‡c

GKwU wbw`©ó f‡ii M¨v‡mi AvqZb -Gi cig ZvcgvÎvi mgvbycvwZK| cig k~b¨ ZvcgvÎv (Absolute Zero temperature): †h ZvcgÎvq ZvwË¡K fv‡e M¨v‡mi AvqZb k~b¨ nq, hvi wb‡P †Kvb ZvcgvÎv _vKv m¤¢e bq, KviY Zvn‡j M¨v‡mi AvqZb FbvZ¥K n‡Z nq, hv Am¤¢e, †mB Kíbv †hvM¨ ZvcgvÎv‡K cig k~b¨ ZvcgvÎv e‡j| aiv hvK, w¯’i Pv‡c 0ºC DòZvq GKwU wbw`©ó f‡ii M¨v‡mi AvqZb Vo; Zvn‡j cÖwZ 1ºC DòZv n«v‡m M¨v‡mi AvqZb 1 Ask K‡g hv‡e| d‡j  C ZvcgvÎvq H M¨v‡mi AvqZb V n‡j Pvj©‡mi m~Î †_‡K cvIqv hvq, 273

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θ Vo Vθ  Vo  273

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11| Zvc I M¨vm (Heat and Gas)

wKš‘  hw`  273C nq Z‡e H ZvcgvÎvq M¨v‡mi AvqZb nq

 

V-273  Vo 1 

273  0 273 

2

myZivs †`Lv hvq,

 273C ZvcgvÎvq †Kvb M¨v‡mi AvqZb k~b¨| ZvcgvÎv Av‡iv Kwg‡q hw`  274C Kiv nq Z‡e Dc‡iv³ m~Îvbymv‡i

M¨v‡mi AvqZb n‡e

 

V- 274  Vo 1 

V 274  o A_©vr  273 273 

AvqZb FbvZ¥K nq| wKš‘ FbvZ¥K AvqZb A_©nxb, GwU Am¤¢e I

Aev¯Íe, Zv n‡Z cv‡i bv| Kv‡RB †Kvb ZvcgvÎvB  274C _vK‡Z cv‡i bv| myZivs me©wb¤œ Kíbv‡hvM¨ †h ZvcgvÎv Zv nj  273C Gi wb‡P †Kvb ZvcgvÎv _Kv m¤¢e bq| GRb¨ GB  273C ZvcgvÎv‡K me©wb¤œ ZvcgvÎv ev cigk~b¨ ZvcgvÎv e‡j| ‡i‡bvi m~Î ev, Pvc Gi m~Î (Law of Pressure): m~‡Îi eY©bv: w¯’i AvqZ‡b cÖwZ wWwMÖ DòZv cwieZ©‡b GKwU wbw`©ó f‡ii M¨v‡mi Pvc, Gi k~b¨ wWwMÖi Pv‡ci

1 ev, 273

0.00366 Ask cwiewZ©Z nq|

e¨vL¨v: aiv hvK, w¯’i AvqZ‡b 0ºC DòZvq GKwU wbw`©ó f‡ii M¨v‡mi Pvc Po; Gi Zvn‡j cÖwZ 1ºC DòZv e„wׇZ M¨v‡mi Pvc

1 1 Ask e„w× cv‡e| hw` DòZv 1ºC nv‡i K‡g hvq, Z‡e M¨v‡mi Pvc Ask K‡g hv‡e| 273 273 P 1ºC DòZvq M¨v‡mi Pvc  Po  o

2ºC DòZvq M¨v‡mi Pvc

273 2Po  Po  273

ºC DòZvq M¨v‡mi Pvc  Po 

Abyiƒcfv‡e,

θ Po 273

θ Po 273 θ    Pθ  Po 1    273   273  θ   Pθ  Po   GLb ZvcgvÎvi AvšÍ©RvwZK GK‡K 273  θ  ºC = T  273  P T T ev mvaviY fv‡e PT A_©vr w¯’i Ges 273ºC =To K wjL‡j Dc‡iv³ mgxKiY `vovq, Pθ  Po  ev, θ  To Po To  Pθ  Po 

AvqZ‡b GKwU wbw`©ó f‡ii M¨v‡mi Pvc -Gi cig ZvcgvÎvi mgvbycvwZK| Av`k© M¨vm (Ideal Gas): †h me M¨vm MwZ Z‡Ë¡i ‡gŠwjK ¯^xKvh©mg~n, e‡qj, Pvj©m I Pv‡ci m~Î cy‡ivcywi †g‡b P‡j Ges †gqv‡ii cÖKí Abyhvqx evqyk~b¨ ¯’v‡b cÖmvi‡Yi `iæb ZvcgvÎvi †Kvb n«vm N‡Ubv; Zv‡`i‡K Av`k© M¨vm e‡j| ZvcgvÎvi cig †¯‹j (Absolute scale of temperature): cig k~b¨ ZvcgvÎv‡K k~b¨ a‡i ZvcgvÎvi †h †¯‹j cÖYqb Kiv n‡q‡Q, Zv‡K ZvcgvÎvi cig †¯‹j e‡j| weÁvbx jW© †Kjwfb me©cÖ_g GB †¯‹j cÖYqb K‡ib e‡j †K ZvcgvÎvi †Kjwfb †¯‹j I e‡j| ZvcgvÎvi GB †¯‹j AvšÍR © vwZK fv‡e ¯^xK…Z| GB †¯‹j Abymv‡i ZvcgvÎvi GK‡K wWwMÖ †Kjwfb ejv (K) nq| ‡mjwmqvm †¯‹‡j †Kvb ZvcgvÎv ºC n‡j †Kjwfb †¯‹‡j n‡eT = ()  |

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11| Zvc I M¨vm (Heat and Gas)

3

cÖgvY ZvcgvÎv ev ¯^vfvweK ZvcgvÎv (Standard temperature): ‡h ZvcgvÎvq ¯^vfvweK Pv‡c cvwb R‡g eid nq, A_ev eid M‡j cvwb‡Z cwibZ nq Zv‡K cÖgvY ZvcgvÎv ev ¯^vfvweK ZvcgvÎv e‡j| cÖgvY ZvcgvÎv nj 0ºC ev, 273.16 K, Z‡e e¨envwiK †ÿ‡Î 273 K we‡ePbv Kiv nq| cÖgvY Pvc ev ¯^vfvweK Pvc (Standard Pressure) : mgy`ª mgZ‡ji 45º Aÿvs‡k 273 K DòZvq 1 eM©wgUvi †ÿ‡Îi Dci `Ûvqgvb 0.76 wgUvi weï× cvi`¯Í‡¤¢i IRb‡K cÖgvY Pvc ev ¯^vfvweK Pvc e‡j| ¯^ vfvweK Pvc, P  0.76 m Hg Pvc|  P = 0.76 ×13.6×103×9.8 Nm-2 P = 1.013×105 Nm-2 = 1.013×105 Pa| PV=RT ev, PV= nRT Gi cÖgvb ev, Av`k© M¨v‡mi mvaviY mgxKiY cwZcv`b:

mvaviY fv‡e †h mKj M¨vm e‡qj I Pvj©m-Gi m~Î †g‡b P‡j Zv‡`i‡K Av`k© M¨vm e‡j| g‡b Kwi P Pv‡c T ZvcgvÎvq GKwU wbw`©ó f‡ii M¨v‡mi AvqZb V, Zvn‡j Avgiv cvB, 1 P VT

[ e‡q‡ji m~Î ]

V

[Pvj©m-Gi m~Î]

T P PV  KT ... ... ... ... ... (3) GLv‡b K GKwU mgvbycvwZK aªæeK| 1 MÖvg AYy ev, 1 †gvj mKj M¨v‡mi Rb¨ K Gi gvb AcwiewZ©Z _v‡K| ZvB weÁvbxiv G‡K mve©Rbxb M¨vm aªæeK R Øviv cÖKvk K‡ib| myZivs 1 †gvj M¨v‡mi Rb¨ PV  RT ... ... ... ... (4) GLb 1 †gvj M¨v‡mi cwie‡Z© m f‡ii M¨vm we‡ePbv Kwi, hvi AvqZb V Ges

m~ÎØq GKÎ K‡i cvB, V

AvbweK fi M ; Zvn‡j (4) bs mgxKi‡Y V Gi cwie‡Z© P

V m

V m

M

e¨envi K‡i cvB,

M  RT

 PV 

m M

RT

 PV  nRT ... ... ... ... (5)

  m   M  n  M¨v‡ mi † gvj msL¨v 

(5) bs mgxKiY Av`k© M¨v‡mi mvaviY mgxKiY|

ZvcgvÎv I Pv‡ci mv‡_ M¨v‡mi Nb‡Z¡i m¤úK©: w¯’i ZvcgvÎvq †Kvb M¨v‡mi NbZ¡ Gi Pv‡ci mgvbycvwZK I w¯’i Pv‡c †Kvb M¨v‡mi NbZ¡ Gi cigZvcgvÎvi e¨v¯ÍvbycvwZK: m fiwewkó †Kvb M¨v‡mi p1 Pv‡c Ges T1K ZvcgvÎvq hw` AvqZb V1 Ges NbZ¡  nq, Ges H M¨v‡mi p2 Pv‡c Ges T2K ZvcgvÎvq AvqZb V2 Ges NbZ¡  nq Z‡e, 1  m / V1 ev, V1  m / 1 Ges  2  m /V2 ev, V2  m /  2 Avgiv Rvwb,

p1V1 p2V2  GB mgxKi‡Y V1 Ges V2 Gi ewm‡q cvB, T1 T2 p1m p2 m  T1 1 T2  2

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11| Zvc I M¨vm (Heat and Gas)

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p1 p  2 1T1  2T2 T  T  1 1  2 2  aª æeK ... ... ... (1) p1 p2 T  aª æeK GB m¤úK© Pvc I ZvcgvÎvi mv‡_ Nb‡Z¡i cwieZ©b wb‡`©k K‡i| A_©vr, p hw` ZvcgvÎv w¯’i _v‡K A_©vr T1  T2 nq Z‡e (1) bs mgxKiY †_‡K cvIqv hvq, 1  2   aª æeK p1 p2 

  aª æeK    p  aª æeK    p myZivs w¯’i ZvcgvÎvq †Kvb M¨v‡mi NbZ¡ Gi Pv‡ci mgvbycvwZK| p Avevi hw` Pvc w¯’i _v‡K, A_©vr p1  p2 nq Z‡e, (1) bs mgxKiY †_‡K cvB, 

1T1   2T2  aª æeK 1   myZivs w¯’i Pv‡c †Kvb M¨v‡mi NbZ¡ Gi cigZvcgvÎvi e¨v¯ÍvbycvwZK| T M¨v‡mi MwZZ‡Ë¡¡i †gŠwjK ¯^xKvh© mg~n (Fundamental Assumption of kinetic theory of Gas) :

(1) cÖwZwU M¨vm mgvb f‡ii AmsL¨ KYvi mgš^‡q MwVZ| KYv¸‡jv‡K M¨v‡mi AYy e‡j| M¨vwmq Aby¸‡jv me©`v MwZkxj _v‡K Ges Giv wbDU‡bi MwZm~Î †g‡b P‡j| (2) GKB M¨v‡mi AYy¸‡jv m`„k; wKš‘ wewfbœ M¨v‡mi Abymgyn ci¯úi †_‡K wfbœ| (3) AYy¸‡jvi ci¯ú‡ii cÖwZ Ges cv‡Îi †`Iqv‡ji cÖwZ †Kvb AvKl©b ev weKl©b †bB| G‡`i kw³ m¤úyY©UvB MwZkw³| (4) cv‡Îi AvqZ‡bi Zzjbvq M¨vwmq AYy¸‡jvi AvqZb AwZ bMY¨| (5) M¨vwmq AYy¸‡jvi cvi¯úwiK av°v m¤úyY© ZvrÿwbK| A_©vr, msN‡l©i d‡j mg‡qi AcPq nq bv| (6) M¨vwmq Aby¸‡jv ci¯ú‡ii mv‡_ av°v Lvq| cici `ywU av°vi ga¨eZ©x `~iZ¡ AYy¸‡jv mg‡e‡M mij c‡_ P‡j| 1 3

MwZZË¡ †_‡K Pv‡ci iwkgvjv A_©vr PV  mnc 2 Ges P 

2E Gi cÖgvY (Expression of pressure form kinetic 3V

theory of gas) :

g‡b Kwi, mgvb ‰`N¨© cÖ¯’ I D”PZv l wewkó cy‡ivcywi w¯’wZ¯’vcK †Kvb Nb‡K n msL¨K Av`k© M¨v‡mi AYy i‡q‡Q| cÖwZwU AYyi fi m Ges G‡`i Mo eM©‡e‡Mi eM©g~j c| GKwU AYyi K_v we‡ePbv Kiv hvK| AYywUi MwZ‡eM c1| hw` X, Y I Z Aÿ eivei c1 Gi Dcvsk h_vµ‡g u, v I w nq, Z‡e Avgiv cvB, c12  u 2  v 2  w 2 ... ... (6) AZGe, AYywU X Aÿ eivei u †e‡M Pj‡e Ges †Kvb GK mgq Nb‡Ki A †`Iqv‡j [wPÎ bs -1] av°v w`‡e| w¯’wZ ¯’vcKZvi Kvi‡Y AYywU u †e‡MB wecixZ w`‡K wd‡i Avm‡e Ges Zvn‡j

l u

l †m‡KÛ ci Nb‡Ki B ‡`Iqv‡j av°v w`‡e| u

‡m‡K‡Û fi‡e‡Mi cwieZ©b `vovq, mum(u) = 2mu| myZivs AYywUi fi‡e‡Mi cwieZ©‡bi nvi

2mu 2mu 2  , GKB fv‡e †`Lv‡bv hvq †h, Y I Z Aÿ eivei AYywUi fi‡eM cwieZ©‡bi nvi h_vµ‡g l/u l 2mv 2 2mw 2 Ges | AYywUi †gvU fi‡e‡Mi cwieZ©‡bi l l 2mu 2 2mv 2 2mw 2 nvi    l l l



2m 2 u  v2  w 2 l





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11| Zvc I M¨vm (Heat and Gas)



2m 2 c1 l

5

[mgxKiY (6) Abymv‡i]

hw` 2q, 3q, 4_©,............., n Zg AYyi MwZ‡eM h_vµ‡g, c2, c3, c4, ............cn nq, Z‡e GKB fv‡e †`Lvb hvq †h, 2m 2 2m 2 2m 2 2m 2 c2 , c3 , c 4 .............. , cn l l l l 2m 2 2m 2 2 m 2 2 m 2 2m 2 cn | c4  ..............  c3  c2  c   AYy¸‡jvi †gvU fi‡e‡Mi cwieZ©‡bi nvi, F  l l l l l 1

AYy¸‡jvi fi‡e‡Mi cwieZ©‡bi nvi h_vµ‡g



2m 2 c1  c22  c32  c42  ..............  cn2 l



2mn  c12  c22  c32  c42  ..............  cn2    l  n 





 c 2  c 22  c32  c42  ..............  c n2 GLv‡ b, c  1 n  c-‡K AYyy¸‡jvi Mo eM©‡e‡Mi eM©g~j e‡j| GLb M¨vwmq AYy¸‡jvi Dci wµqviZ †gvU ej F, 

2mn 2 c l

M¨v‡mi Pvc P n‡j msÁvbymv‡i, P 

  

† gvU ej † gvU † ¶ Îdj

F 6l 2 2mnc 2 P 6l 3

P

wKš‘ l3 = Nb‡Ki AvqZb= M¨v‡mi AvqZb = V awi,

P 

mnc 2 3V

1 (cÖgvwYZ|) 3 2 1 1M 2 1 2 ev, PV   Mc 2  fi M  mn ev, P  c  c 3 2 3 3V 2E 1    E  Mc 2  (cÖgvwYZ)  P  3V 2  

ev, PV  mnc 2

2 E A_©vr M¨v‡mi Pvc Gi GKK AvqZ‡bi MwZkw³i `yB -Z„Zxqvsk| 3 wKš‘ GKwU M¨vm AYyi Rb¨, PV  KT Avevi, GK †gvj M¨v‡mi Rb¨, PV  RT 2E 2E   KT [GLv‡b, K †evëRg¨vb aªæeK]   RT 3 3 3 3  E  KT  E  RT 2 2  ET  M¨v‡mi Mo MwZkw³ M¨v‡mi cig ZvcgvÎvi mgvbycvwZK|

V GKK n‡j P 

Mo eM© ‡eM I g~j Mo eM© ‡eMt †Kvb M¨v‡mi mKj AYyi †e‡Mi e‡M©i Mo‡K Mo eM©‡eM e‡j| ‡Kvb M¨v‡mi n msL¨K c12  c22  c32  c42  ..............  c n2 n Avevi †Kvb M¨v‡mi mnj AYyi †e‡Mi e‡M©i Mogv‡bi eM©g~j‡K g~j Mo eM© ‡eM e‡j|‡Kvb M¨v‡mi n msL¨K

AYyi cÖwZwUi †eM h_vµ‡g, c1, c2, c3, .........cn n‡j AYy¸‡jvi Mo eM© †eM c 2 

AYyi cÖwZwUi †eM h_vµ‡g, c1, c2, c3, .........cn n‡j, c 2 

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c12  c22  c32  c42  ..............  c n2 n

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11| Zvc I M¨vm (Heat and Gas) Mogy³ c_ ev Mo wbe©va `~iZ¡ (Mean free path): cici `ywU msN‡l©i ga¨eZ©x `~iZ¡ †K Mogy³ c_ ev Mo wbe©va `~iZ¡ e‡j| wP‡Î A we›`y †_‡K GKwU AYy B †Z wM‡q Ab¨ AYyi mv‡_ av°v Lv‡”Q Ges BC c‡_ hv‡”Q| C we›`y‡Z Avevi Avi GKwU AYyi mv‡_ av°v †L‡q CD c‡_ hv‡”Q| AB, BC, CD, DE BZ¨vw` cÖ‡Z¨KwU `~iZ¡B gy³ c_| †ewki fvM †ÿ‡ÎB †h †Kvb `ywU gy³ c_ mgvb nq bv| ZvB, †Kvb AYyi K‡qKwU msN‡l©i ga¨eZ©x `~i‡Z¡i Mo wb‡j †h `yiZ¡ cvIqv hvq Zv‡K H AYyi Mogy³ c_ ev Mo wbe©va `~iZ¡ e‡j|

6

l | N Mogy³ c‡_i (Mo wbe©va `~iZ¡) ivwkgvjv (Equation of mean free path) : Mo gy³ c_ M¨v‡mi Nb‡Z¡i e¨v¯ÍvbycvwZK: g‡b Kwi, †Kvb M¨v‡mi cÖwZ GKK AvqZ‡b AYyi msL¨v n Ges cÖwZwU AYyi e¨vm | Avgiv †h AYywUi Mo gy³ c_ wbY©q

hw` N av°vq AYy †gvU l `yiZ¡ AwZµg K‡i Z‡e Mo gy³ c_, λ 

Ki‡Z PvB wnmv‡ei myweavi Rb¨ †KejgvÎ †mB AYywU‡K MwZkxj a‡i evwK AYy¸wj‡K w¯’i we‡ePbv KiwQ| wPÎvbyhvqx Avgiv C AYywUi Mo gy³ c_ wbY©q Ki‡Z PvB| C AYywU l `~iZ¡ AwZµg Kivi ci Ab¨ †h mKj AYyi †K›`ª C AYywUi †K›`ª †_‡K  `~‡i [wP‡Î A I B AYyØq] A_ev,  A‡cÿv Kg `~i‡Z¡ _vK‡e Zv‡`i mv‡_ av°v Lv‡e| A_©vr  e¨vmva© I l ˆ`‡N¨©i GKwU wmwjÛv‡ii g‡a¨ ‡h mKj AYyi †K›`ª _vK‡e Zv‡`i mv‡_ av°v Lv‡e| GB wmwjÛv‡ii AvqZb l| GLb GKK AvqZ‡b AYyi msL¨v n I l AvqZ‡b Abyi msL¨v n‡e nl | A_©vr l `~iZ¡ AwZµg Kivi mgq C AYywU nl msL¨K evi av°v Lv‡e| †h‡nZz †h †Kvb AYyi cici `ywU msN‡l©i ga¨eZ©x `~iZ¡¸‡jvi Mo wb‡j †h AwZµvšÍ `~ iZ¡ 1 l     2 ... ... ... (7) 2 πσ n av°v msL¨v nπσ l BnvB K¬wmqv‡mi g‡Z Mo gy³ c‡_i ivwkgvjv| GLb aiv hvK, GKwU AYyi fi m| †h‡nZz GKK AvqZ‡b AYyi msL¨v n| Kv‡RB GKK AvqZ‡b AYyi fi = mn = M¨v‡mi = , (7) bs mgxKi‡Yi ni I je‡K m w`‡q ¸Y K‡i cvB,

`~iZ¡ cvIqv hvq Zv‡K Mo gy³ c_ e‡j| AZGe Mo gy³ c_,  

λ

1 m m m 1  2  2  λ KviY m, I  2 nπσ  m πσ mn πσ ρ ρ

aªæe| myZivs Mo gy³ c_ M¨v‡mi Nb‡Z¡i e¨v¯ÍvbycvwZK| wKš‘

M¨v‡mi NbZ¡, M¨v‡mi Pv‡ci mgvbycvwZK Ges cig ZvcgvÎvi e¨v¯ÍvbycvwZK| Kv‡RB Mo gy³ c_ M¨v‡mi Pv‡ci e¨v¯ÍvbycvwZK Ges cig ZvcgvÎvi mgvbycvwZK| G Kvi‡Y Mo gy³ c_ M¨v‡mi NbZ¡, Pvc I cig ZvcgvÎvi Dci wbf©i K‡i| Mo gy³ c‡_i Ab¨vb¨ ivwkgvjv: K¬wmqv‡mi c×wZ‡Z Mo gy³ c_ MYbv wbf©yj bq| KviY, †h AYyi Mo gy³ c_ wbY©q Kiv n‡q‡Q †mwU Qwow Ab¨ AYy¸wj‡K w¯’i aiv n‡q‡Q| wKš‘ cÖK…Zc‡ÿ mKj AYyB MwZkxj| ‡evj&Rgvb mKj Mo †eM mgvb a‡i Mo gy³ c‡_i †h 3 ... ... ... (8) c‡i g¨v·I‡qj Zuvi †eM e›U‡bi m~‡Îi mvnv‡h¨ Mo gy³ c‡_i 4πσ 2 n 1 ... ... ... (9) †h ivwkgvjv wbY©q K‡ib Zv nj,   2 πσ 2 n

ivwkgvjv wbY©q K‡ib Zv nj,  

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m¤ú„³ ev®ú t ‡Kvb Ae× ¯’v‡b Zij msjMœ ev®ú‡K H ZvcgvÎvq m¤ú„³ ev®ú e‡j| Am¤ú„³ ev®ú t hw` ‡Kvb Ae× ¯’v‡b wKQy ev®ú _v‡K wKš‘ †Kvb Zij bv _v‡K Z‡e H ev®ú‡K Am¤ú„³ ev®ú e‡j| µwgK 1| 2| 3| 4|

m¤ú„³ ev®ú I Am¤ú„³ ev‡®úi g‡a¨ cv_©K¨ : m¤ú„³ ev®ú Am¤ú„³ ev®ú ‡Kvb Ae× ¯’v‡b Zij msjMœ ev®ú‡K H hw` ‡Kvb Ae× ¯’v‡b wKQy ev®ú _v‡K wKš‘ †Kvb Zij bv ZvcgvÎvq m¤ú„³ ev®ú e‡j| _v‡K Z‡e H ev®ú‡K Am¤ú„³ ev®ú e‡j| m¤ú„³ ev®ú e‡q‡ji m~Î †g‡b P‡j bv| Am¤ú„³ ev®ú e‡q‡ji m~Î †g‡b P‡j| m¤ú„³ ev®ú Pvj©m-Gi m~Î †g‡b P‡j bv| Am¤ú„³ ev®ú Pvj©m-Gi m~Î †g‡b P‡j| ZvcgvÎv e„w× K‡i m¤ú„³ ev®ú‡K Am¤ú„³ ZvcgvÎv Kwg‡q Am¤ú„³ ev®ú‡K m¤ú„³ ev‡®ú ev‡®ú iƒcvšÍwiZ Kiv hvq| iƒcvšÍwiZ Kiv hvq|

wkwkivsK (Dew point) : ‡h ZvcgvÎvq ‡Kvb wbw`©ó AvqZ‡bi evqy Gi g‡a¨ Aew¯’Z Rjxq ev®ú Øviv m¤ú„³ nq, A_©vr †h ZvcgvÎvq wkwki m„wó nq ev A`„k¨ nq Zv‡K †mB ZvcgvÎv‡K wkwkivsK e‡j| †Kvb ¯’v‡b evqyi wkwkivsK 16ºC ej‡Z GB eywS ‡h, D³ ¯’v‡b 16ºC DòZvq Gi g‡a¨ Aew¯’Z Rjxq ev®ú Øviv evqy m¤ú„³ nq, A_©vr 16ºC DòZvq wkwki Rg‡Z ev A`„k¨ n‡Z ïiæ K‡i| cig Av`ªªZ © v (Absolute Humidity): ‡Kvb ¯’v‡b GKK AvqZ‡bi evqy‡Z †h cwigvb Rjxq ev®ú _v‡K Zv‡K H ¯’v‡bi cig Av`ªªZ © v e‡j| †Kvb ¯’v‡bi cig -3 -3 3 -3 Av`ªªZ © v 5×10 Kgm ej‡Z GB eywS †h, H ¯’v‡bi 1m evqy‡Z 5×10 Kg Rjxq ev®ú Av‡Q| Av‡cwÿK Av`ªªZ © v (Relative Humidity): wbw`©ó ZvcgvÎvq wbw`©ó AvqZ‡bi evqy‡Z †h cwigvb Rjxq ev®ú Av‡Q Ges H ZvcgvÎvq D³ AvqZ‡bi evqy‡K Rjxqev®ú Øviv m¤ú„³ Ki‡Z AviI †h cwigvb Rjxq ev‡®úi cÖ‡qvRb Zv‡`i AbycvZ‡K D³ ¯’v‡bi Av‡cwÿK Av`ª©Zv e‡j| G‡K R Øviv cÖKvk Kiv nq Ges kZKivq cÖKvk Kiv nq| msÁvbymv‡i, Av‡cwÿK Av`©ªZv

wKš‘ wbw`©ó ZvcgvÎvq †Kvb ¯’v‡bi Rjxq ev‡®úi Pvc H ¯’v‡bi Rjxq ev‡®úi f‡ii mgvbycvwZK

  wKš‘ †Kvb ZvcgvÎvq †Kvb ¯’v‡b Rjxq ev‡®úi Pvc H ¯’v‡b wkwkivs‡K m¤ú„³ Rjxq ev‡®úi Pv‡ci mgvb|   wkwkivs‡K m¤ú„³ Rjxq ev‡®úi Pvc‡K f, evqyi ZvcgvÎvq m¤ú„³ Rjxq ev‡®úi Pvc‡K F w`‡q cÖKvk Ki‡j, f f n‡e| Av‡cwÿK Av`ª©Zv‡K mvaviYZ kZKiv wn‡m‡e cÖKvk Kiv nq, R  100 % F F Zvrch© : evqyi Av‡cwÿK Av`ª©Zv 60% ej‡Z GB eywS †h, evqyi ZvcgvÎvq GKwU wbw`©ó AvqZ‡bi evqy‡K m¤ú„³ Ki‡Z †h cwigvb Rjxq ev®ú cÖ‡qvRb, Zvi kZKiv 60 fvM Rjxq ev®ú evZv‡m Av‡Q| R

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11| Zvc I M¨vm (Heat and Gas)

8

nvB‡MÖvwgUvi (Hygrometer) : ‡h h‡š¿i mvnv‡h¨ †Kv ¯’v‡bi Av‡cwÿK Av`ª©Zv wbY©q Kiv nq Zv‡K nvB‡MÖvwgUvi ev Av‡cwÿK Av`ª©Zv gvcK hš¿ e‡j| Av`ª I ﮋ evj¦ nvB‡MÖvwgUv‡ii mvnv‡h¨ Av‡cwÿK Av`ªZv wbY©q (Determination of Relative Humidity by Wet and Dry Bulb Hygrometer) : h‡š¿i eY©bv : GB h‡š¿i cÖavb Ask n‡jv GKB iKg `yBwU cvi` _v‡g©vwgUvi T1 I T2| _v‡g©vwgUviØq‡K GKwU Kv‡Vi ‡d«‡g Dj¤^ fv‡e cvkvcvwk ¯’vcb Kiv nq| T1 _v‡g©vwgUv‡ii evj¦‡K ﮋ ivLv nq| GwU evqyi DòZv wb‡`©k K‡i| T2 _v‡g©vwgUv‡ii ev‡j¦i mv‡_ gmwjb ev wj‡j‡bi cj‡Z Rov‡bv _v‡K Ges GB cj‡Z GKwU cv‡Î ivLv cvwbi g‡a¨ Wyevb _v‡K| cvwb gmwjb ev wj‡jb †e‡q Dc‡i D‡V T2 Ges _v‡g©vwgUv‡ii evj¦‡K me mgq wfRv iv‡L| Kvh©cÖYvjx t ‡h ¯’v‡bi Av‡cwÿK Av`ª©Zv wbY©q Ki‡Z n‡e †mB ¯’v‡b Av‡cwÿK Av`ªZv gvcK hš¿‡K †bIqv nq Ges _v‡g©vwgUv‡ii cvV †iKW© Kiv nq| hw` ﮋ I Av`ª evj¦ _v‡g©vwgUv‡ii cvV h_vµ‡g 1ºC I 2ºC mswkøó ¯’v‡bi wkwkivsK ºC nq, Z‡e †Mømv‡ii m~Îvbymv‡i cvB, 1  =G(1  2)  = 1 G(1  2) ... ... ... (8) GL‡b G †K †MøBmvi Drcv`K e‡j| GLb 1ºC A_©vr evqyi ZvcgvÎvq †MøBmv‡ii ZvwjKv n‡Z G Gi gvb †R‡b wb‡q (8) bs mgxKi‡Yi evqyi wkwkivsK ºC wbY©q Kiv nq| Gici †i‡bvi m¤ú„³ Rjxqev®ú Pvc ZvwjKv n‡Z wkwkivs‡K A_©vr ºC G m¤ú„³ Rjxqev®ú Pvc f Ges f evqyi ZvcgvÎvq A_©vr 1ºC G m¤ú„³ Rjxqev®ú Pvc F †R‡b wb‡q, R  100% F mgxKi‡Y f I F Gi gvb ewm‡q Av‡cwÿK Av`ªZv R wbY©q Kiv nq| mZvK©Zv (Caution) : GB cixÿvq wb¤§iƒc mZK©Zv Aej¤^b Kiv nq:

1| my‡e`x _v‡g©vwgUvi e¨envi Kiv nq| 2| _v‡g©vwgUv‡ii cvi` w¯’i n‡j cvV wb‡Z nq| 3| cj‡Zi wb¤§ cÖvšÍ hv‡Z me©`v cvwb‡Z wbgw¾Z _v‡K †mw`‡K jÿ ivL‡Z nq|

GKB ZvcgvÎvq XvKv A‡cÿv PÆMÖv‡g †ekx A¯^w¯ÍKi †eva nq t PÆMÖvg mgy‡`ªi wbK‡U Aew¯’Z e‡j †mLv‡b Av‡cwÿK Av`©ªZv †ewk| XvKv mg~`ª‡_‡K A‡bK`~‡i nIqvq ¯^vfvweK fv‡e †mLvbKvi Av‡cwÿK Av`ª©Zv Kg| Avgiv Rvwb, evqygÛ‡ji Av‡cwÿK Av`©ªZv †e‡o †M‡j ev®úvq‡bi nvi K‡g hvq| d‡j XvKvq kixi †_‡K wbM©Z Nvg `ªæZ ïKv‡e I kixi †_‡K †ekx myßZvc MÖnb Ki‡e; d‡j †`n kxZj nq I ¯^w¯Í jv‡M| PÆMÖv‡g kixi †_‡K wbM©Z Nvg Kg ïKv‡e I ev®úvq‡bi Rb¨ Kg myßZv‡ci cÖ‡qvRb n‡e| d‡j XvKv A‡cÿv PÆMÖv‡g †ekx A¯^w¯ÍKi †eva n‡e| A_©vr †hLv‡b Av‡cwÿK Av`ªZv †ekx †mLv‡b †ekx A¯^w¯Í †eva n‡e| el©vKvj A‡cÿv kxZ Kv‡j wfRv Kvco ZvovZvwo ïKvq : kxZKv‡ji ZvcgvÎv el©vKvj A‡cÿv Kg nIqv m‡Z¡I ev®úvqb `ªæZ nq e‡j ZvovZvwo Kvco ïKvq| ev®úvqb wbf©i K‡i Av‡cwÿK Av`©ªZvi Dci| el©vKv‡j Av‡cwÿK Av`ª©Zv †ekx _v‡K| kxZKv‡j evZv‡m Av‡cwÿK Av`©ªZv Kg _v‡K e‡j wfRv Kvco †_‡K ev®úvqb `ªæZ nq, d‡j Kvco ZvovZvwo ïKvq|

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11| Zvc I M¨vm (Heat and Gas)

9

`yc‡y ii Av‡MB wkwki wZ‡ivwnZ nq : m~‡h©v`‡qi ci mgq evovi mv‡_ mv‡_ evqy DËß n‡Z _v‡K| d‡j evqy Am¤ú„³ n‡q c‡o| Am¤ú„³ evqy wkwki‡K ï‡l †bq| G Kvi‡Y `ycy‡ii Av‡MB wkwki wZ‡ivwnZ nq| kxZKv‡j †Vv‡U wMømvwib jvMv‡bv nq : kZKv‡j evqygÛ‡j Rjxq ev‡®úi cwigvb AZ¨šÍ Kg _v‡K| d‡j evqy †h †Kvb ¯’vb †_‡K Rjxq ev®ú msMÖn K‡i| †`‡ni Avbve„Z As‡ki A‡cÿvK…Z †Kvgj ¯’vb ¸‡jv †_‡K evqygÛj Rjxq ev®ú †U‡b †bq| Gi d‡j Avgv‡`i †VuvU ‡d‡U †h‡Z Pvq| †Vv‡U wMømvwib jvMv‡j cvwbi ev®úvq‡b evav m„wó K‡i d‡j †VuvU dv‡U bv|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

11| Zvc I M¨vm (Heat and Gas) 1| ‡Kvb M¨vm AYyi e¨vm 3×10-10 m Ges cÖwZ NbwgUv‡i AYyi msL¨v 6×1020 n‡j AYyi Mo gy³ c_ wbY©q Ki| Avgiv Rvwb, GLv‡b,

1 π n σ2

λ

λ

e¨vm, σ  3  10

1 3.14  6  1020  (3  10-10 ) 2

-10

m

cÖwZ NbwgUv‡i Abyi msL¨v, n = 6×10 20 /m3 Mo gy³c_, ?

 λ  5.8910 3 m (Ans.) 2| ‡Kvb M¨vm AYyi e¨vm 3×10-10 cm Ges cÖwZ NbwgUv‡i AYyi msL¨v 6×1020 n‡j AYyi Mo gy³ c_ wbY©q Ki| Avgiv Rvwb, GLv‡b,

λ

1 π n σ2

λ

e¨vm,  =3×1010cm =3×10 12 m

1 3.14 6 10  (310 -12) 2 20

 λ  58 .976 m (Ans.)

cÖwZ NbwgUv‡i Abyi msL¨v, n = 6×1020 /m3 Mo gy³c_ ?

3| †Kvb GKw`‡bi wkwkiv¼ 10°C I Av‡cwÿK Av`©ªZv 67.30% | H w`‡bi evqyi m¤ú„³ ev®ú Pvc KZ? [10°C ZvcgvÎvq m¤ú„³ Rjxq ev®ú Pvc 13.64×10-3 mHg] Avgiv Rvwb, GLv‡b,

f  100% F

wkwkiv‡¼ ev®úPvc, f = 13.64×10-3 mHg evqyi ZvcgvÎvq m¤ú„³ ev®úPvc, 13.64103  67.3%  100 % F = ?

F Av‡cw¶K Av`ªZv, R = 67.30%

R

3

13.6410 100 67.3  F  20.27 103 mHg (Ans.)

F

4| 0.64m cvi` ¯Í¤¢ Pv‡c Ges 39°C ZvcgvÎvq †Kvb M¨v‡mi AvqZb 5.7 ×10-4 m3 | cÖgvY Pvc I ZvcgvÎvq M¨v‡mi AvqZb KZ? Avgiv Rvwb, GLv‡b,

P1V1 P2 V2  T1 T2 PVT  V2  1 1 2 T1P2

Pvc, P1 = 0.64 m Hg ZvcgvÎv, T1 =39ºC=(39+273)= 312K AvqZb, V1 = 5.7 ×10-4 m3 Pvc, P2 =0.76 m Hg 0.64 5.7 10 4 273 ZvcgvÎv, T2 = 273K  V2  AvqZb, V2 =?

312 0.76

V2  4.2 10 4 m3 (Ans.)

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5| ‡Kvb GKw`‡bi wkwkivsK 7.4°C Ges Kÿ ZvcgvÎv 18.6°C| Av‡cwÿK Av`©ªZv wbY©q Ki| [7°C, 8°C, 18°C I 19°C ZvcgvÎvq m¤ú„³ Rjxq ev®ú Pvc h_vµ‡g 7.5 ×10-3 m, 8.2 ×10-3 m, 15.6 ×10-3 m Ges 16.5 ×10-3 m cvi`|] 7°C ‡_‡K 8°C A_©vr 1°C ZvcgvÎv evo‡j ev®ú Pvc ev‡o = (8.27.5)10-3 m cvi`|  0.4°CÓ Ó Ó Ó Ó = 0.7×10-3 ×0.4 m cvi`| 0.28×10-3 m cvi`|   7.4°C ZvcgvÎvq A_©vr wkwkivs‡K m¤ú„³ Rjxq ev®ú Pvc f = (7.5+0.28)10-3 m cvi`| = 7.78×10-3m cvi`| Avevi, 18°C ‡_‡K 19°C A_©vr 1°C ZvcgvÎv evo‡j ev®ú Pvc ev‡o = (16.5 -15.6)10-3 m cvi`|  0.6°CÓ Ó Ó Ó Ó = 0.9×10-3×0.6 m cvi`| 0.54×10-3 m cvi`|   18.6°C ZvcgvÎvq A_©vr evqyi ZvcgvÎvq m¤ú„³ Rjxq ev®ú Pvc F = (15.6+0.54)10-3m cvi` = 16.14 ×10-3 m cvi`|

f  100%  F 7.78  10 -3 R  100%   16.14  10 -3  R  48.2 % (Ans.)

 Av‡cwÿK Av`©ªZv R 

6| 0°C ZvcgvÎvq ‡Kvb M¨v‡mi Pvc 3×105 Pa n‡j 60°C ZvcgvÎvq Gi Pvc KZ? Avgiv Rvwb,

P1V1 P2 V2  T1 T2 PVT  P2  1 1 2 V2 T1

3  105  V  333 V  273  P2  3.66  105 Pa (Ans.)  P2 

GLv‡b, Pvc, P1 = 3×105 Pa ZvcgvÎv, T1 =0ºC=(0+273)= 273K AvqZb, V1 = V2 = V= ?

ZvcgvÎv, T2 =60ºC=(60+273)= 333K Pvc, P2=?

7| ‡Kvb GKw`b wm³ I ﮋ evj&e Av`©ªZvgvcK h‡š¿i ﮋ evj&e Gi cvV 30°C Ges wm³ evj&e Gi cvV 28°C| Av‡cwÿK Av`ª©Zv wbY©q Ki| 30°C G †MøBmv‡ii Drcv`K 1.65 Ges 26°C, 28°C Ges 30°C ZvcgvÎvq m¤ú„³ Rjxqev®ú Pvc h_vµ‡g 25.25×10-3m, 28.45×103 m Ges 31.85×10-3m cvi` Pvc | ﮋ evj&e Gi cvV 1= 30°C A_©vr evqyi ZvcgvÎv = 30°C wm³ evj&e Gi cvV 2 = 28°C †MøBmv‡ii Drcv`K G = 1.65

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11| Zvc I M¨vm (Heat and Gas)

awi wkwkivsK =  Avgiv Rvwb, 1  G (1-2) ev, 1 G (1-2) ev,   1.65 () ev,      °C evqyi ZvcgvÎv 30°C G m¤ú„³ Rjxqev®ú Pvc F = 31.85×103 m cvi` 26°C ‡_‡K 28°C A_©vr 2°C ZvcgvÎv evo‡j ev®ú Pvc ev‡o = (28.4525.25)10-3 m cvi`| ev,1°C

Ó

ev, 0.7°C Ó

Ó

Ó Ó

Ó

Ó

Ó Ó

ÕÕ

-3

3.2  10 m cvi`| 2 -3 = 3.2  10  0.7 m cvi`| 2

=

×10-3 m cvi`|   °C ZvcgvÎvq A_©vr wkwkivs‡K m¤ú„³ Rjxq ev®ú Pvc f = (25.25 + )10-3 m cvi`| = 26.37× 10-3 m cvi`|

f  Av‡cwÿK Av`©ªZv, R   100%  F 26.37  10 -3  100% ev, R  31.85  10 -3  R  82.79 % (Ans.) 8| ‡Kvb n«‡`i Zj‡`k †_‡K cvwbi DcwiZ‡j Avmvq GKwU evqy ey`&ey&`& AvqZ‡b cuvP¸b nq| evqygÛ‡ji Pvc 105 Nm-2 n‡j n«‡`i MfxiZv KZ? Avgiv Rvwb, P1V1 = P2V2 GLv‡b awi, ev, ( P2 + hg)V = P2×5V n«‡`i Zj‡`‡k ey`ey‡`i AvqZb, V1 = V ev, P2 + hg = 5P2 n«‡`i c„‡ô ey`ey‡`i AvqZb, V2 = 5 V ev, hg = 5P2  P2 cvwbi NbZ¡, 103 kg m-3 AwfKl©R Z¡iY, g = 9.8 ms-2 ev, hg = 4P2

ev, h 

4P2 ρg

4  10 5 ev, h  3 10  9.8  h  40.81 m (Ans.)

n«‡`i Zj‡`‡k Pvc, = P1 n«‡`i c„‡ô evqygÛ‡ji Pvc, P2 = 105 Pa n«‡`i MfxiZv, h = ?

P1 = P2 + h MfxiZvq cvwbi Pvc  P1 = P2 + hg

9| hw` R = 8.31 JK-1mol-1 nq Z‡e 72cm cvi` Pv‡c Ges 27°C ZvcgvÎvq 20g Aw·‡R‡bi AvqZb wbY©q Ki| GLv‡b, M¨vm ayªeK, R=8.31 JK-1mol-1 Pvc, P = 72cmHg = 0.72 mHg = 0.72×13.6×103×9.8 Pa = 95961.6 Pa ZvcgvÎv, T = 27 ºC = (27+273)K=300K

Avgiv Rvwb, PV  nRT ev, V  n ev, V 

RT P

20 8.31  300  32 95961.6

‡ gvjmsL¨v, n

 V  16.236963 10 3 m 3 (Ans.)

20  32

AvqZb, V = ?

10| ¯^vfvweK ZvcgvÎv I Pv‡c bvB‡UÖv‡R‡bi NbZ¡ 1.25Kg- m-3 n‡j 100°C ZvcgvÎvq bvB‡Uªv‡Rb AYyi Mo eM©‡e‡Mi eM©g~j wbY©q Ki| Avgiv Rvwb,   2 

1T1 T2

1.25  273 373 3  2 = 0.914879356 kg m Avevi,  2 

C

GLv‡b, ZvcgvÎv, T1 = 0ºC =273 K NbZ¡,  = 1.25Kg-m-3 ZvcgvÎv T2 = 100ºC=373 K P =1.013×105Nm-2 C= ?

3P 3  1.013  10 5   576.34 ms 1 (Ans.) ρ2 0.91487935 6

11| 27°C ZvcgvÎvq cÖwZ †gvj wnwjqvg M¨v‡mi MwZkw³ wbY©q Ki| (R= 8.31 JK-1 mol-1) GLv‡b, Avgiv Rvwb, ZvcgvÎv T1 = 27ºC 3 E  RT =(273+27) K=300 K 2 R= 8.31 JK-1 mol-1 3 cÖwZ †gv‡j MwZkw³ E= ?  E   8.31  300J

2  E = 3739.5 J

12| ‡Kvb n«‡`i Zj‡`k †_‡K cvwbi DcwiZ‡j Avmvq GKwU evqy ey`&e‡y& `i& e¨vm wZb ¸b nq| e¨v‡ivwgUv‡i cvi` ¯Í‡¤¢i D”PZv 75cm n‡j n«‡`i GLv‡b awi, MfxiZv KZ? n«‡`i Zj‡`‡k ey`ey‡`i e¨vm = 2x Avgiv Rvwb,  n«‡`i Zj‡`‡k ey`ey‡`i e¨vmva©= x P1V1 = P2V2 n«‡`i Zj‡`‡k ey`ey‡`i AvqZb, V1  4 x 3  V 3 ev, ( P2 + hg)V = P2×27V n«‡`i c„‡ô ey`ey‡`i e¨vm = 6x n«‡`i c„‡ô ey`ey‡`i e¨vmva©=3x ev, P2 + hg = 27P2 n«‡`i c„‡ô ey`ey‡`i AvqZb, V2  43 π(3x) 3 ev, hg = 27P2  P2 V2  27  43 πx 3  27V ev, hg = 26P2 cvwbi NbZ¡, 103 kg m3 26P2 AwfKl©R Z¡iY, g = 9.8 ms-2 ev, h  n«‡`i Zj‡`‡k Pvc, = P1 ρg n«‡`i c„‡ô evqygÛ‡ji Pvc, 26  99930.6 P2 =0 .75×13596×9.8 Pa ev, h  = 99930.6Pa 3 10  9.8 n«‡`i MfxiZv, h = ?

h  265.122 m (Ans.)

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2

P1 = P2 + h MfxiZvq cvwbi Pvc  P1 = P2 + hg

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11| Zvc I M¨vm (Heat and Gas)

13| ¯^vfvweK ZvcgvÎv I Pv‡c †Kvb Ave× M¨v‡mi NbZ¡ 0.0892Kg m-3 n‡j M¨v‡mi AYyi Mo eM©‡e‡Mi eM©g~j wbY©q Ki| Avgiv Rvwb, 3P GLv‡b, C ρ ZvcgvÎv, T = 0ºC =273 K 3  1.013  105 C 0.0892 3 1.013 105 C 0.0892  C  1845.79 ms  1 (Ans.)

NbZ¡,  = 0.0892Kg m-3 P =1.013×105Nm-2 Mo eM©‡e‡Mi eM©g~j C =?

14| ‡Kvb M¨vm AYy¸‡jvi Mo gy³ c_ 6×10-8 m I AYyi e¨vm 2.5×10-10 m, cÖwZ NbwgUv‡i AYyi msL¨ wbY©q Ki| Avgiv Rvwb, GLv‡b, 1 λ 2 e¨vm, σ  2.5 10 -10 m πnσ Mo gy³c_, 6×10-8 m 1 n cÖwZ NbwgUv‡i Abyi msL¨v, n= ? π λσ 2 n

1 3.14  6 10  (2.5  10-10 ) 2 -8

 n  8 . 49  10

25

/m 3 (Ans.)

15| ‡Kvb GKw`‡bi wkwkivsK 7.6°C Ges Kÿ ZvcgvÎv 16°C| Av‡cwÿK Av`©ªZv wbY©q Ki| [7°C, 8°C I 16°C ZvcgvÎvq m¤ú„³ Rjxq ev®ú Pvc h_vµ‡g 7.5 ×10-3 m, 8 ×10-3 m Ges 13.5 ×10-3 m cvi`|] 7°C ‡_‡K 8°C A_©vr 1°C ZvcgvÎv evo‡j ev®ú Pvc ev‡o = (87.5)10-3 m cvi`|  0.6°CÓ Ó Ó Ó Ó = 0.5×10-3 ×0.6 m cvi`| 0.3×10-3 m cvi`|   7.6°C ZvcgvÎvq A_©vr wkwkivs‡K m¤ú„³ Rjxq ev®ú Pvc f = (7.5+0.3)10-3 m cvi`| = 7.8×10-3m cvi`| 16°C ZvcgvÎvq A_©vr evqyi ZvcgvÎvq m¤ú„³ Rjxq ev®ú Pvc F = 13.5 ×10-3 m cvi`|

f  100%  F 7.8 10 -3   R  100% 13.5 10 -3

 Av‡cwÿK Av`©ªZv R 

 R  57.78 % (Ans.)

16| GKwU ﮋ I Av`ª evj&e nvB‡MÖvwgUv‡i h‡š¿i ﮋ I Av`ª ev‡j¦i ZvcgvÎv h_vµ‡g 20°C I 12°C n‡j wkwkivsK I Av‡cwÿK Av`ª©Zv wbY©q Ki| (20°C G †MøBmv‡ii Drcv`K 1.79 Ges 20°C Ges 5.68°C ZvcgvÎvq Rjxq ev‡®úi m‡e©v”P Pvc h_vµ‡g 17.6 mmHg Ges 6.856 mmHg )

ﮋ evj&e Gi cvV 1= 20°C A_©vr evqyi ZvcgvÎv = 20°C wm³ evj&e Gi cvV 2 = 12°C †MøBmv‡ii Drcv`K G = 1.79 awi wkwkivsK =  Avgiv Rvwb, 1  G (1-2) ev, 1 G (1-2) ev,  20 -1.79 (20-12) ev, 20 14. 32   5.68°C evqyi ZvcgvÎv 20°C G m¤ú„³ Rjxqev®ú Pvc, F = 17.6 mmHg wkwkivsK 5.68 °C G m¤ú„³ Rjxq ev®ú Pvc, f = 6.856 mmHg

f  100%  F 6.856 ev, R   100% 17.6  R  38.95 % (Ans.)

 Av‡cwÿK Av`©ªZv, R 

&DËi : wkwkivsK =5.68°C I Av‡cwÿK Av`©ªZv = 38.95 % 17| w¯’i Pv‡c KZ ZvcgvÎvq †Kvb M¨vm AYyi Mo eM©‡e‡Mi eM©g~j ¯^vfvweK Pvc I ZvcgvÎvi Mo eM©‡e‡Mi eM©g‡~ ji wظY n‡e?

3RT1 .........(1) M 3RT2 I C2  ....(2) M

Avgiv Rvwb, C1 



C2 T2 2C1    C1 T1 C1

T2 T1

GLv‡b, cÖv_wgK ZvcgvÎv , T1 = 273K †kl ZvcgvÎv, T2 =?

C2 = 2C1

T2 T 4 2 T1 273 T2  4  273K  1092 K ( Ans.) 4

18| ‡Kvb M¨vm AYyi e¨vmva© 3.9×10-10 m Ges cÖwZ Nb †mw›UwgUv‡i AYyi msL¨v 2.69×1019 n‡j AYyi Mo gy³ c_ wbY©q Ki| Avgiv Rvwb, GLv‡b,

λ

1 π n σ2

λ

1 3.14  2.69  1019  (7.8  10-8 ) 2 6

 λ  1.9410 cm  λ  1.9510 8 m (Ans.)

e¨vm, σ  2  3.9  10 -10 m  σ  2  3.9  10-10  100 cm  σ  7.8  10-8 cm cÖwZ Nb †mw›UwgUv‡i Abyi msL¨v, n = 2.69×1019/cm3 Mo gy³c_, ?

19| ¯^vfvweK ZvcgvÎv I Pv‡c bvB‡UÖv‡R‡bi NbZ¡ 1.25Kg- m-3 n‡j Gi Mo eM©‡e‡Mi eM©g~j wbY©q Ki| GLv‡b, Avgiv Rvwb, 3P 3  1.013  10 5 C  ρ 1.25 C  493 . 07 ms 1 (Ans.)

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3

ZvcgvÎv, T = 0ºC =273 K NbZ¡,  = 1.25Kg-m-3 P =1.013×105Nm-2 C=?

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 তা঩ এফিং তা঩ভাত্রায ভনধয তুরনাঃ ১) তা঩ এক প্রকায ঱বি, বকন্তু তা঩ভাত্রা ফস্ত্ত্তয একবট তা঩ীয় অফিা। ২) তা঩ ফস্ত্ত্তবিত ঳ফ অণুয যভাট গবত঱বিয ঳ভানু ঩াবতক এফিং তা঩ভাত্রা ফস্ত্ত্তবিত একবট অণুয গড় গবত঱বিয ঳ভাণু঩াবতক। ৩) দু ইবট ফস্ত্ত্ত একই তা঩ভাত্রায় থাবকনর঑ উ঴ানদয তান঩য ঩বযভাণ বফববন্ন ঴ইনত ঩ানয। ৪) একবট ফস্ত্ত্ত ঴ইনত অনয ফস্ত্ত্তনত তান঩য প্রফা঴ উ঴ানদয তা঩ভাত্রায উ঩য বনব঱য কনয, তান঩য ঩বযভানণয উ঩য বনব঱য কনয না। 0

0

0

0

৫) তান঩য একক কযারবয, B.T.U ঑ বকনরা-কযারবয বকস্ত্ত্ত তা঩ভাত্রায একক- C, F, K ঑ R।  তা঩ভাত্রায বফববন্ন যস্করঃ 0

0

১) য঳বিনগ্রড যস্করঃ ই঴ায বনম্নবিযািংক 0 C এফিং উর্ধ্঱বিযািংক 100 C 0

0

২) পানযন঴াইট যস্করঃ ই঴ায বনম্নবিযািংক 32 F এফিং উর্ধ্঱বিযািংক 212 F। 0

0

৩) যযাভায যস্করঃ ই঴ায বনম্ন ঑ উর্ধ্঱ বিযািংক মথাক্রযভ 0 R ঑ 80 R। ৪) যকরববন যস্করঃ তা঩ভাত্রায যম যস্কর ফস্ত্ত্তয যবৌত গুণাফরীয উ঩য বনব঱য঱ীর নয় তা঴ানক তা঩ভাত্রায যকরববন ফা ঩যভ যস্কর 0

0

ফনর। ই঴ায বনম্ন বিযািংক 237 K, উর্ধ্঱ বিযািংক 373 K 0

0

 ভানফনদন঴য স্বাবাবফক তা঩ভাত্রা 98.4 F ফা 36.89 C 0

0

0

0

0

0

 ঩াবনয ফু টনািংক 100 C ফা 373 K এফিং ব঴ভািংক 0 C ফা 273 K, ঩যভ ঱ূ নয তা঩ভাত্রা -273 C ফা 0 K. 2 -2

 তা঩ এক প্রকায ঱বি। তাই উ঴ায ভাত্রা ঳ভীকযণ [ML T ]| বকন্তু তা঩ভাত্রায যকান ভাত্রা ঳ভীকযণ যনই। তনফ উ঴াযা উবয়ই অবদক যাব঱।  ঩াযদ থানভ঱াবভটানযয ঳ু বফধাঃ ঩াযদ তা঩ ঳ু ঩বযফা঴ী, উজ্জ্বর-অস্বে, তা঩ধাযণ ক্ষভতা খুফ কভ এফিং কভ উিায়ী। 0

0

঩াযনদয ফু টনািংক 357 C এফিং ব঴ভািংক -39 C, তাই উচ্চ তা঩ভাত্রা ঩বযভান঩ ঩াযদ থানভ঱াবভটায ফযফহৃত ঴য়।  অযারনকা঴র থানভ঱াবভটানযয ঳ু বফধাঃ 0

0

অযারনকা঴নরয ব঴ভািংক -130 C এফিং ফু টনািংক 78 C, তাই ই঴া বনম্ন তা঩ভাত্রা ঩বযভান঩ ফযফহৃত ঴য়।  থাবভ঱িযঃ তা঩ভাত্রা ঩বযভান঩য জনয অধ঱঩বযফা঴ক িাযা রতযী ঳ু নফদী যযাধনক থাবভ঱িয ফনর।

যজনন যাখা বার  তযর থানভ঱াবভটানয তা঩ভাত্রা ঩বযফত঱ননয ঳ানথ তযনরয আয়তননয ঩বযফত঱ন ঘনট। ঩াযদ থানভ঱াবভটায, অযারনকা঴র থানভ঱াবভটায ইতযাবদ তযর থানভ঱াবভটায।  গযা঳ থানভ঱াবভটানয বিয আয়তনন বনবদ঱ি ঩বরুাণ গযান঳য িা঩ তা঩ভাত্রায ঳ানথ ঩বযফবত঱ত ঴য়, এই ধভ঱নক ফযফ঴ায কনয গযা঳ থানভ঱াবভটায ফযফ঴ায কযা ঴য়।  যযাধ থানভ঱াবভটানয ‘‘তা঩ভাত্রা ফৃ বদ্ধয ঳ানথ অবধকািং঱ ধাতফ ঩দানথ঱য যযাধ ফৃ বদ্ধ ঩ায়’’ এই ধভ঱নক কানজ রাগাননা ঴য়। প্লাবটনাভ থানভ঱াবভটায এই থানভ঱াবভটানযয উদা঴যণ।

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ZvcgvÎv (Temperature): e¯‘i Zvcxq Ae¯’v‡K e¯‘i ZvcgvÎv e‡j| ZvcgvÎv ej‡Z e¯‘ KZUzKz Mig ev VvÛv Zv eySvq| ZvcgvÎv‡K  ev T Øviv cÖKvk Kiv nq| ZvcgvÎvi GKK ºC, K, ºF BZ¨vw`| Zvc (Heat): Zvc GK cÖKvi kw³| hv cÖ‡qvM Ki‡j e¯‘ Mig nq Ges wbM©Z Ki‡j e¯‘ VvÛv nq Zv‡K Zvc e‡j| Zvc‡K Q ev H Øviv Kiv nq| Zv‡ci GKK Ryj ev K¨vjwi| 1 K¨vjwi = 4.2 Ryj| c`v‡_©i DòZvwgwZ ag© I DòZvwgwZK c`v_© (Thermometric Property and Thermometric Substance ): c`v‡_©i KZK¸‡jv ag© ZvcgvÎv cwieZ©‡bi mv‡_ mv‡_ cwiewZ©Z nq| †hgbt ZvcgvÎv e„wׇZ Zi‡ji AvqZb ev‡o, M¨v‡mi Pvc e„w× cvq, cwievnxi †iva e„w× cvq BZ¨vw`| c`v‡_©i †h me ag© ZvcgvÎvi mv‡_ wbqwgZ cwiewZ©Z nq †mB me ag©‡K DòZvwgwZ ag© e‡j| †h mKj c`v‡_©i g‡a¨ DòZvwgwZ ag© we`¨gvb Zv‡`i‡K DòZvwgwZK c`v_© e‡j| c`v‡_©i DòZvwgwZ ag© Kv‡R jvwM‡q _v‡g©vwgUvi wbg©vb Kiv nq| ˆÎa we›`y (Triple Point): GKwU wbw`©ó Pv‡c †h ZvcgvÎvq †Kvb c`v_© KwVb, Zij I evqexq GB wZb iƒ‡c mvg¨ve¯’vq _v‡K Zv‡K H c`v‡_©i ˆÎa we›`y e‡j| cvwbi ˆÎa we›`y (Triple Point of water): ‡h wbw`©ó ZvcgvÎvq I Pv‡c cvwb KwVb (eid), Zij (cvwb) I evqexq (Rjxqev®ú) GB wZb Ae¯’vq _vK‡Z cv‡i Zv‡K cvwbi ˆÎawe›`y e‡j| cvwbi ˆÎa we›`yi ZvcgvÎv 0ºC ev, 273.16K Ges GB we›`y‡Z wbw`©ó Pvc 4.58 mmHg| GLv‡b K nj †Kjwfb, Avi †Kjwfb nj Gm AvB c×wZ‡Z ZvcgvÎvi GKK| 1 †K 1K ev GK †Kjwfb e‡j| †Kjwfb ZvcgvÎvi Dci wfwË K‡i 273.16 cigk~b¨ ZvcgvÎv n‡”Q 0 K, eid we›`y 273.16K Ges ÷xg we›`y 373.16K|

‡Kjwfb (Kelvin) : cvwbi ‰Îa we›`y ZvcgvÎvi

ZvcgvÎvi cig †¯‹j ev, ZvcgvÎvi AvšÍ©RvwZK †¯‹j (Absolute scale of Temperature or, International Scale of Temperature): cigk~b¨ ZvcgvÎv A_©vr  273ºC ZvcgvÎv‡K 0 (k~b¨) a‡i Ges †gŠwjK e¨eavb‡K 100 fvM K‡i ZvcgvÎvi †h †¯‹j Kíbv Kiv nq Zv‡K ZvcgvÎvi cig †¯‹j ev ZvcgvÎvi AvšÍ©RvwZK †¯‹j e‡j| ZvcgvÎvi cig †¯‹j Abyhvqx ˆÎa we›`yi ZvcgvÎv 273K I cvwbi ùzUbv¼ 373K| wb¤œ w¯’i we›`y (Lower fixed point): †h ZvcgvÎvq cÖgvb Pv‡c weï× eid cvwbi mv‡_ mvg¨ve¯’vq _vK‡Z cv‡i A_©vr †h ZvcgvÎvq weï× eid Mj‡Z ïiæ K‡i Zv‡K wb¤œ w¯’i we›`y ev eid we›`y (Ice Point) e‡j| DaŸ© ©w¯’i we›`y (Upper fixed point): †h ZvcgvÎvq cÖgvb Pv‡c weï× cvwb Rjxq ev‡®úi mv‡_ mvg¨ve¯’vq _vK‡Z cv‡i A_©vr †h ZvcgvÎvq weï× cvwb Rjxq ev‡®ú cwiYZ n‡Z ïiæ K‡i Zv‡K DaŸ© w¯’i we›`y ev ÷xg (Steam Point) we›`y e‡j| ‡gŠwjK e¨eavb (Fundamental Interval): DaŸ© w¯’i we›`y I wb¤œ w¯’i we›`yi ga¨eZ©x ZvcgvÎvi e¨veavb‡K †gŠwjK e¨eavb e‡j| GB e¨eavb‡K myweav RbK KZK¸wj mgvbfv‡M wef³ K‡i GK GKwU fvM‡K DòZv ÁvcK msL¨v Øviv wPwýZ K‡i ZvcgvÎvi †¯‹j wba©viY Kiv nq| _v‡gv©wgUvi (Thermometer) : †h h‡š¿i mvnv‡h¨ †Kvb e¯‘i ZvcgvÎv ev DòZv cwigvc Kiv nq Zv‡K _v‡g©vwgUvi e‡j| _v‡g©vwgUvi A‡bK cÖKv‡ii n‡q _v‡K, Gi g‡a¨ cÖavbZt †mjwmqvm, †Kjwfb, dv‡ibnvBU D‡jøL‡hvM¨|

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2 12| ZvcgvÎv (Temperature) cvi` _v‡g©vwgUv‡ii MVb (Construction of Mercury Thermometer ): †h _v‡g©vwgUv‡i cvi‡`i AvqZb cÖmviY Kv‡R jvwM‡q ZvcgvÎv gvcv nq Zv‡K cvi` _v‡g©vwgUvi e‡j| ÓZvcgvÎv e„w× ‡c‡j cvi‡`i AvqZb e„w× cvq Ges ZvcgvÎv n«vm ‡c‡j AvqZb n«vm cvqÓ G bxwZi Dci cvi` _v‡g©vwgUvi cÖwZwôZ|

MVb : GwU miæ wQ`ªwewkó Ges mgvb e¨v‡mi GKwU k³ KvP bj we‡kl| b‡ji GK cÖv‡šÍ GKwU †Pv½vK…wZ KzÛ _v‡K Ges Aci cÖvšÍ eÜ _v‡K| KzÛ I b‡ji wKQy Ask weï× cvi‡` c~Y© _v‡K| b‡ji Mv‡q †¯‹j `vMvw¼Z _v‡K| †h e¯‘i ZvcgvÎv gvc‡Z n‡e Zv‡K Kz‡Ûi ms¯ú‡k© Avb‡j cvi` AvqZ‡b †e‡o †h `vM ch©šÍ †cŠQvq Zv-B e¯‘i ZvcgvÎv| cÖ¯‘Z cÖYvjx : cÖ_‡g m~ÿè I mymg wQ`ªwewkó GKwU KvP bj †bIqv nq| b‡ji GKcÖv‡šÍ GKwU evj¦ B Ges Aci cÖv‡šÍ A †Lvjv _v‡K| bjwU‡K GKwU ÷¨v‡Ûi mvnv‡h¨ Lvov fv‡e ¯’vcb K‡i †Lvjv gy‡L GKwU dv‡bj F emv‡bv nq| dv‡b‡j wKQy ﮋ I weï× ci` Xvjv nq| b‡ji wQ`ª Lye m~ÿè nIqvq cvi` b‡ji wfZi cÖ‡ek Ki‡Z cv‡i bv| GKwU w¯úwiU j¨v¤ú ev eyb‡mb evwZ w`‡q evj¦wU‡K Aí Zvc cÖ‡qvM Ki‡j evj¦ I b‡ji wfZ‡ii evqyi AvqZb e„w× cvq Ges wKQy evqy cvi‡`i ga¨w`‡q ey`ey` AvKv‡i †ei n‡q hvq| Ÿvj¦wU‡K VvÛv Ki‡j evj¦ I b‡ji Aewkó evqyi AvqZb K‡g hvq Ges evqyi Pv‡c wKQy cvi` b‡ji wfZi cÖ‡ek K‡i| Gfv‡e ch©vqµ‡g Mig I VvÛv K‡i bj I evj¦ m¤ú~Y© fwZ© Kiv nq| Gi ci dv‡bj‡K mwi‡q wb‡q cvi` bv †dvUv ch©šÍ evj¦wU‡K Mig Kiv nq| dzUšÍ Ae¯’vq cvi‡`i AvqZb e„w× cvIqvq DwÌZ cvi` ev®ú b‡ji wfZ‡ii evqy‡K †ei K‡i †`q| Zxeª I miæ AwMœwkLvi mvnv‡h¨ bj Mwj‡q eÜ K‡i †`Iqv nq| evj¦ VvÛv n‡q KÿZvcgvÎvq G‡j cvi` m¤ú~Y© evj¦ I b‡ji wKQy Ask c~Y© K‡i iv‡L Ges b‡ji evKx Ask evqyk~b¨ Ae¯’vq _v‡K| `vMv¼b : ZvcgvÎv cwigv‡ci Rb¨ _v‡g©vwgUv‡ii b‡j GKwU `vMKvUv †¯‹‡ji cÖ‡qvRb nq| †¯‹j ˆZwii Rb¨ `ywU wbw`©ó ZvcgvÎv‡K w¯’i aiv nq| G‡`i‡K _v‡g©vwgUv‡ii w¯’iv¼ e‡j| cvi` _v‡g©vwgUv‡i ei‡di Mjbv¼‡K wb¤œw¯’iv¼ Ges cvwbi ùzUbv¼‡K Ea©w¯’iv¼ aiv nq| wb¤œ w¯’iv¼ wbY©q : GKwU dv‡b‡j wKQy weï× MjšÍ eid UzKiv wb‡q Zvi g‡a¨ _v‡g©vwgUv‡ii evj¦ cÖ‡ek Kiv‡bv nq| G‡Z ev‡j¦i cvi` m¼zwPZ nq, d‡j ˆKwkK b‡j cvi‡`i D”PZv Kg‡Z _v‡K| b‡j cvi‡`i D”PZv GKwU wbw`©ó ¯’v‡b G‡m w¯’i n‡j Kv‡Pi Mv‡q `vM †K‡U H ¯’vb‡K wPwýZ Kiv nq| cvi` kx‡l©i GB Ae¯’vbB wb¤œw¯’iv¼ wb‡`©k K‡i| DaŸ©©w¯’iv¼ wbY©q : cvi` _v‡g©wgUv‡ii DaŸ©w¯’iv¼ wbY©‡qi Rb¨ Rb¨ wnc‡mvwgUvi bvgK hš¿ e¨envi Kiv nq| Gici _v‡g©vwgUvi‡K wnc‡mvwgUv‡ii Zvgvi cv‡Î Ggbfv‡e ivLv nq hv‡Z Gi evj¦ cv‡Î iwÿZ cvwb ¯úk© bv K‡i| cvwb fwZ© cv·K eyb‡mb evwZ ev w¯úwiU j¨v‡¤úi mvnv‡h¨ Zvc †`Iqv nq d‡j wKQyÿ‡Yi g‡a¨ cvwb dzU‡Z _v‡K Ges Drcbœ Rjxq ev®ú _v‡g©vwgUv‡ii evj¦‡K DËß K‡i| b‡ji Af¨šÍi¯’ cvi` cÖmvwiZ nq Ges GK mgq ev‡®úi ZvcgvÎvi mgvb nq, ZLb cvi` b‡ji g‡a¨ GK ¯’v‡b w¯’i nq| G Ae¯’vq cvi`kx‡l©i D³ Ae¯’v‡b b‡ji Mv‡q GKwU `vM KvUv nq| GUvB _v‡g©vwgUv‡ii DaŸ© w¯’iv¼| DaŸ© I wb¤œ DaŸ©w¯’iv‡¼i ga¨eZ©x ¯’vb‡K 100 I 180 fv‡M fvM K‡i h_vµ‡g †mjwmqvm I dv‡ibnvBU _v‡g©vwgUvi wnmv‡e e¨envi Kiv hvq| _v‡g©vwgUv‡i cvi` e¨env‡ii myweav (Advantage to use Mercury in a Thermometer) t 1| GwU GKwU D¾¡j PKP‡K c`v_©, ZvB mn‡R Kv‡Pi b‡ji wfZi †`Lv hvq| 2| cvi` KvP b‡ji Mv‡q †j‡M _v‡K bv| 3| cvi` -39ºC ZvcgvÎvq R‡g Ges 357ºC ZvcgvÎvq dz‡U| myZivs ZvcgvÎvi `xN© cwim‡i GwU Zij Ae¯’vq _v‡K| d‡j †gvUvgywU - 30ºC †_‡K 350ºC ch©šÍ ZvcgvÎv gvcv hvq| 4| cvi` weï× Ae¯’vq cvIqv hvq| 5| cvi` Zvc mycwievnx c`v_© e‡j AwZ`ªæZ †Kvb e¯‘ †_‡K Zvc †kvlb K‡i cÖmvwiZ nq| 6| cvi‡`i cÖmviY mylg ZvB ZvcgvÎv e„w×i Rb¨ Gi AvqZb cÖmviY me©Î mgvb nq| facebook /gmail/skype: -tanbir.cox

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12| ZvcgvÎv (Temperature)

3

_vwg©÷i (Thermistor): _vwg©÷i GKwU Aa©cwievnx c`v‡_©i ˆZix e¨e¯’v hv w`‡q ZvcgvÎv cwigvc Kiv hvq| GB ZvcgvwÎK ¸b nj ˆe`¨ywZK †iva| DòZv e„w×i mv‡_ mv‡_ _vwg©÷‡ii ˆe`¨ywZK †iva myPKxq nv‡i n«vm cvq| GB _v‡g©vwgUvi Ab¨vb¨ †iva _v‡g©vwgUv‡ii Zzjbvq A‡bK †ekx my‡e`x nq| GB _v‡g©vwgUv‡ii mvnv‡h¨70ºC †_‡K 300º C ch©šÍ ZvcgvÎv cwigvc Kiv hvq| ZvcgvÎv cwigvc Kivi Rb¨ _vwg©ói‡K µgv¼b K‡i wb‡Z nq| Zvc-hyMj (_v‡g©vKvcj) I mx‡eK wµqv (Thermocouple & Seebeck-effect): `ywU wfbœ avZzi Zv‡ii `yB cÖvšÍ †Rvov jvwM‡q Gi g‡a¨ GKwU M¨vjfv‡bvwgUvi AšÍf©y³ K‡i hw` eZ©bx ˆZix Kiv hvq Ges Zvi `ywUi ms‡hvM ¯’j `ywU‡Z (Junctions) ZvcgvÎvi e¨eavb m„wó Kiv hvq, Zvn‡j H eZ©bx‡Z Zwor cÖevwnZ n‡e, GB Zwor cÖevn‡K Zvc Zwor cÖevn e‡j| 1821 wLª÷v‡ã mx‡eK me© cÖ_g NUbvwU cÖ_g cÖZ¨ÿ K‡ib| ZvB G NUbv‡K mx‡eK wµqv (Seebeck-effect) e‡j| `ywU wfbœ Zvi Øviv m„ó G e¨e¯’v‡K ejv nq Zvc-hyMj (Thermocouple) Ges eZ©bx‡Z †h Zwo”PvjK kw³i D™¢e nq Zv‡K Zvcxq Zwo”PvjK kw³ (Thermo electromotive force) ejv nq| wbi‡cÿ ZvcgvÎv (Neutral Temperature): Zvc-hyM‡ji kxZj ms‡hvM‡K 0ºC ZvcgvÎvq †i‡L Dò ms‡hvM‡K †h, ZvcgvÎvq ivL‡j eZ©bx‡Z Zvcxq Zwo”PvjK kw³i gvb me©vwaK nq †mB ZvcgvÎv‡K wbi‡cÿ ZvcgvÎv e‡j| Zvgv I †jvnv hyM‡ji Rb¨ G ZvcgvÎv 275º C| Drµg ZvcgvÎv (Inversion Temperature): Zvc-hyM‡ji kxZj ms‡hvM‡K 0ºC ZvcgvÎvq †i‡L Dò ms‡hvM‡K †h ZvcgvÎvq ivL‡j Zvc Zwo”PvjK kw³i gvb k~b¨ nq Zv‡K H hyM‡ji Drµg ZvcgvÎv e‡j| Zvgv I †jvnv hyM‡ji Rb¨ G ZvcgvÎv 550º C| cÖgvY ZvcgvÎv ev ¯^vfvweK ZvcgvÎv (Standard temperature): ‡h ZvcgvÎvq ¯^vfvweK Pv‡c cvwb R‡g eid nq, A_ev eid M‡j cvwb‡Z cwibZ nq Zv‡K ¯^vfvweK ZvcgvÎv e‡j| ¯^vfvweK ZvcgvÎv nj 0ºC ev, 273.16 K, Z‡e e¨envwiK †ÿ‡Î 273 K we‡ePbv Kiv nq| Zvc Zwor _v‡gv©wgUvi ev Zvc hyMj ev _v‡g©vKvc‡ji mvnv‡h¨ ZvcgvÎv wbY©q (Determination of Temperature By the help of Thermocouple): g~jbxwZ: _v‡g©vKvcj‡K _v‡g©vwgUvi wn‡m‡e e¨envi Kiv hvq| Zvcxq Zwo”PvjK kw³ _v‡g©vKvc‡ji ms‡hvM؇qi ZvcgvÎvi cv_©‡K¨i Dci wbf©i K‡i| hw` _v‡g©vKvc‡ji GKwU ms‡hvM‡K ei‡di ms¯ú‡k© 0º C ZvcgvÎvq ivLv nq Ges Aci ms‡hvM‡K †Kvb D”PZi ZvcgvÎv ºC G ivLv nq Zvn‡j eZ©bx‡Z †h Zwo”PvjK kw³i D™¢e nq Zv wb‡¤œv³ mgxKiY †_‡K cvIqv hvq| Zvcxq Zwo”PvjK kw³ E n‡j, E=+ ... ... ... (1) GLv‡b, I `ywU aªæeK hv‡`i gvb _v‡g©vKvc‡ji cÖK…wZi Dci wbf©i K‡i| E,I Gi gvb †R‡b Gi gvb wbY©q Kiv nq| eZ©bx ms‡hvM : wPÎvbyhvqx c‡UbwkIwgUv‡ii A I B we›`yi mv‡_ GKwU A¨vwgUvi A, e¨vUix Ba, Pvwe K Ges cwieZ©bkxj †iva Rh †kªYx mgev‡q hy³ Kiv nq| e¨vUix Ba Gi abvZ¥K cÖvšÍ A we›`yi mv‡_ hy³ _v‡K| GLb Zvgv I †jvnvi Zvi w`‡q ‰Zix _v‡g©vKvc‡ji Dò ms‡hvM ¯’‡ji mv‡_ †h Zvgvi Zvi _v‡K, †mB

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4 12| ZvcgvÎv (Temperature) Zv‡ii gy³ cÖvšÍ‡K A -‡Z Ges kxZj ms‡hvM ¯’‡ji mv‡_ hy³ Zvgvi Zv‡ii gy³ cÖvšÍ‡K M¨vjfv‡bvwgUv‡ii G -Gi ga¨w`‡q RwK‡Z hy³ Kiv nq|

cixÿv: Pvwe K eÜ K‡i c‡UbwkIwgUv‡i AB Zv‡i Zwor cÖevn Pvjbv Kiv nq| Gi ci cwieZ©bkxj †iv‡ai gvb Ggb fv‡e mgwš^Z Kiv nq hv‡Z RwKwU‡K A we›`y‡Z ¯úk© Kiv‡j M¨vjfv‡bvwgUv‡ii KuvUv †h we‡ÿc ‡`q B we›`y‡Z ¯úk© Kiv‡j Zvi wecixZ w`‡K we‡ÿc †`q| aiv hvK, RwKwU‡K C we›`y‡Z ¯úk© Kiv‡j M¨vjfv‡bvwgUv‡ii †Kvb we‡ÿc nq bv A_©vr C we›`yB fvimvg¨ we›`y| GLb A I C Gi ga¨Kvi Zv‡ii ˆ`N©¨ l cwigvc Kiv nq Ges A¨vwgUvi †_‡K we`¨yr cÖev‡ni gvb I wbY©q Kiv nq| wnmve (Calculation) : Zvcxq Zwo”PvjK kw³ E n‡j E= l ˆ`‡N©¨i As‡ki Zv‡ii wefe cv_©K¨ =I× l ˆ`‡N©¨i As‡ki Zv‡ii ‡iva c‡UbwkIwgUv‡ii cÖwZ GKK ˆ`‡N©¨i †iva  n‡j l ˆ`‡N©¨i Zv‡ii †iva n‡e  l | E=I l Avevi c‡UbwkIwgUv‡ii m¤ú~Y© Zv‡ii A_©vr L ˆ`‡N©¨i Zv‡ii †iva R n‡j  

R  L

IlR GB mgxKi‡Yi Wvb w`‡Ki me ivwk Rvbv _vKvq m„ó Zvcxq Zwo”PvjK kw³i gvb wbY©q Kiv hv‡e| Gevi (1) L bs mgxKi‡Y E,I Gi gvb ewm‡q Gi gvb wbY©q Kiv nq| 

E 

cvB‡ivwgUvi (Pyrometer) : †h h‡š¿i mvnv‡h¨ D”P ZvcgvÎv cwigvc Kiv nq Zv‡K cvB‡ivwgUvi e‡j| cvB‡ivwgUvi mvaviYZt `yB ai‡bi n‡q _v‡Kt h_v  (1) c~Yw© ewKiY cvB‡ivwgUvi (Total radiation pyrometer ) (2) Av‡jvK cvB‡ivwgUvi (Optical pyrometer) c~Y© wewKiY cvB‡ivwgUvi (Total radiation pyrometer) t c~Y© wewKiY cvB‡ivwgUv‡ii mvnv‡h¨ †Kvb e¯‘ n‡Z wewKwiZ Zvckw³ cwigvc K‡i w÷dv‡bi m~Î cÖ‡qvM K‡i ZvcgvÎv wbY©q Kiv nq| †dix (Fery) cÖ_g cÖ_g GB ai‡bi cvB‡ivwgUi ˆZix K‡i ZvcgvÎv cwigvc K‡ib e‡j G‡K †dixi cvB‡ivwgUvi I ejv nq| h‡š¿i MVbt GB h‡š¿ GKwU AeZj `c©Y M i‡q‡Q hv Zvgvi cvZ w`‡q ˆZix| cv‡Zi Dci Zj wb‡Kj avZzi cÖ‡jc †`Iqv| `c©‡bi gvSLv‡b GKwU wQ`ª Av‡Q hvi wcQ‡b Awf‡bÎ E hy³ _v‡K| M-Gi m¤§y‡L GKwU †QvU wQ`ª D i‡q‡Q hvi wcQ‡bB GKwU avZe djK S _v‡K| `c©Y AwfgyLx dj‡Ki c„‡ô Kv‡jv cÖ‡jc †`qv _v‡K| wQ`ª D `ywU Aa©e„ËvKvi `c©Y Øviv MwVZ| S-Gi wcQb c„†ô _v‡g©vKvcj T hy³ _v‡K| _v‡g©vKvc‡j Drcbœ Zwo”PvjK ej cwigv‡ci Rb¨ GwU wgwj‡fvë wgUv‡ii mv‡_ hy³ _v‡K| djKwUi Dci e¯‘i wewKwiZ iwk¥ hv‡Z mivmwi AvcwZZ bv n‡Z cv‡i †mRb¨ djKwU GKwU ev‡· Ave× ivLv nq| GKwU ¯Œzi mvnv‡h¨ m¤úyY© e¨e¯’vwU mvg‡b wcQ‡b miv‡bv hvq| Kvh©bxwZ t †h e¯‘i ZvcgvÎv wbY©q Kiv nq †mwU n‡Z AvMZ iwk¥ AeZj `c©‡bi mvnv‡h¨ cÖwZwe¤^ wQ`ª D-Gi ga¨w`‡q S-Gi Dci AvcwZZ nq| Awf‡bÎ E-G †PvL †i‡L ZvKv‡j mwVK †dvKvwms n‡j wQ`ª D e„ËvKvi †`Lv‡e| mwVK †dvKvm bv n‡j D `yB Aa©vs‡k m‡i hvq| `c©b mvg‡b wcQ‡b mwi‡q †dvKvwms Kiv nq| ZvcgvÎv wbY©q t wgwj‡fvë wgUv‡ii cvV V, Dr‡mi ZvcgvÎv T Ges djK S -Gi ZvcgvÎv T0 n‡j, w÷dv‡bi m~Î Abymv‡i, V   (T 4  T 4 ), GLv‡b,  w÷dvb aªæeK| 0

myZivs, V I T0 cwigvc K‡i GB cvB‡ivwgUv‡ii mvnv‡h¨ ARvbv ZvcgvÎv wbY©q Kiv hvq|

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5 12| ZvcgvÎv (Temperature) Av‡jvK cvB‡ivwgUvi (Optical pyrometer) : G cvB‡ivwgUv‡ii mvnv‡h¨ †h e¯‘i ZvcgvÎv wbY©q Ki‡Z n‡e †mB e¯‘i cÖwZwe‡¤^i D¾¡jZvi mv‡_ ˆe`¨ywZK evwZi wdjv‡g›U F-Gi mv‡_ D¾¡jZvi Zzjbv Kiv nq| O G h‡š¿i Awfjÿ¨ †jÝ I E Awf‡bÎ| wdjv‡g›U F-Gi ga¨w`‡q wewfbœ Zwor cÖevn cvwV‡q wdjv‡g›U‡K wewfbœ ZvcgvÎvq DËß Kiv nq| Zwor cÖevn cwieZ©b Kivi Rb¨ GKwU wiI÷¨vU (Rh) Ges cÖevn gvÎv cwigv‡ci Rb¨ GKwU A¨vwgUvi A _v‡K| ¯Œ S Gi mvnv‡h¨ Awfjÿ¨ O Gi Ae¯’vb Ggb fv‡e wbqwš¿Z Ki‡Z n‡e hv‡Z e¯‘ ‡_‡K weKxY© iwk¥ F wdjv‡g‡›Ui Dci †K›`ªxf~Z nq Ges †mLv‡b Dr†mi GKwU cÖwZwe¤^ m„wó nq| wdjv‡g‡›Ui g‡a¨ Zwor cÖevn Pvwj‡q R jvj Kv‡Pi gva¨‡g Awf‡bÎ EGi ga¨w`‡q wdjv‡g›U‡K jÿ¨ Kiv nq| Zwor cÖevngvÎv †ekx n‡j wdjv‡g›U cÖwZwe‡¤^i †P‡q D¾¡j n‡e| Avi hw` Zwor cÖev‡ni cwigvb Kg nq Zvn‡j cÖwZwe‡¤^i Zzjbvq wdjv‡g›U Kv‡jv †`Lv‡e| wiI÷¨v‡Ui mvnv‡h¨ cÖevngvÎv wbqš¿b K‡i wdjv‡g›U A`„k¨ Ki‡Z nq| G Ae¯’vq wdjv‡g‡›Ui ZvcgvÎv Dr‡mi ZvcgvÎvi mgvb n‡e| G Ae¯’vq A¨vwgUvi †_‡K cÖevngvÎvi cvV †bIqv nq| wdjv‡g‡›Ui ga¨w`‡q cÖevwnZ ZworcÖevn gvÎvi gvb hw` I nq, Zvn‡j wdjv‡g‡›Ui ZvcgvÎv Z_v Dr‡mi ZvcgvÎv TK wb‡¤§i mgxKiY n‡Z Rvbv hvq| I = a+bT+cT2 GLv‡b a, b I c wZbwU aªæeK| hš¿wU‡K wZbwU Rvbv ZvcgvÎvq µgvw¼Z K‡i a, b I c Gi gvb wbY©q Ki‡Z nq| A¨vwgUvi‡K µgvw¼Z K‡iI mivmwi Dr‡mi ZvcgvÎv wbY©q Kiv hvq| G h‡š¿i mvnv‡h¨ 600º C †_‡Z 1500º C ch©šÍ ZvcgvÎv gvcv hvq| N~b©vqgvb e„ËKjv (Rotating sector) e¨envi K‡i G cvB‡ivwgUv‡ii mvnv‡h¨ AviI D”P ZvcgvÎv cwigvc Kiv hvq|

ZvcgvÎvi wewfbœ †¯‹‡ji g‡a¨ m¤úK© (Relation among different scales of temperature) t ‡mjwmqvm, dv‡ibnvBU I †Kjwfb †¯‹‡ji cvi¯úwiK m¤úK© wbY©‡qi Rb¨ GKwU cvi` _v‡g©vwgUvi AB ‡bB hvi eid we›`y I ÷xg we›`y h_vµ‡g A I B `v‡Mi mv‡_ wg‡j hvq| aiv hvK, †Kvb ZvcgvÎvq AB _v‡g©vwgUv‡ii cvi`kxl© hLb M Ae¯’v‡b Av‡m ZLb †mjwmqvm, dv‡ibnvBU I †Kjwfb †¯‹‡j ZvcgvÎv h_vµ‡g C, F I K|  Avgiv wjL‡Z cvwi, MA C0 F  32 K  273    BA 100  0 212  32 373  273 C F  32 K  273    100 180 100



C F  32 K  273   BnvB ‡mjwmqvm, dv‡ibnvBU I †Kjwfb †¯‹‡ji cv¯úwiK m¤ú‡K©i ivwkgvjv| 5 9 5

GK w¯’i we›`yi cwi‡cÖwÿ‡Z _v‡g©vwgwZi g~j mgxKiY cÖwZcv`b t aivhvK, †h †Kvb GKwU _v‡g©vwgUv‡i ZvcwgwZK ag© X1 ZvcgvÎv T1 Gi mgvbycvwZK| MvwbwZKfv‡e, T1  X1  T1  aX1 GLv‡b a GKwU aªæeK| AZGe, GKB _v‡g©vwgUv‡ii `ywU ZvcgvÎvi ZvcwgwZK ag© Zzjbv K‡i cvB, T1 X1  ... ... ... ... (1) GLv‡b, T1 ZvcgvÎvq ZvcwgwZK ag© X1 I T2 ZvcgvÎvq ZvcwgwZK ag© X2 T2 X 2 cvwbi ‰Îawe›`y‡Z A_©vr 273.16K ZvcgvÎvq ZvcwgwZK ag© Xtr Ges ‡h †Kvb ZvcgvÎv T †Z ZvcwgwZK ag© X n‡j (1) T X X bs mgxKiY Abymv‡i cvB,  T   273.16 K 273.16 X tr X tr l (K) cvi` _v‡g©vwgUv‡ii †ÿ‡Î X Gi ¯’‡j cvi` ¯Í‡¤¢i ˆ`N©¨ l n‡j, T   273.16 K l tr

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12| ZvcgvÎv (Temperature)

6

V  273.16 K Vtr P (M) w¯’i AvqZb M¨vm _v‡g©vwgUv‡ii †ÿ‡Î X Gi ¯’‡j ZvcwgwZK ag© PvcP n‡j, T   273.16 K Ptr R (N) ‡iva _v‡g©vwgUv‡ii †ÿ‡Î X Gi ¯’‡j ZvcwgwZK ag© ‡iva R n‡j, T   273.16 K R tr E (O) Zvc Zwor _v‡g©vwgUv‡ii †ÿ‡Î X Gi ¯’‡j ZvcwgwZK ag© Zwo”PvjK ej E n‡j, T   273.16 K E tr

(L) w¯’i Pvc M¨vm _v‡g©vwgUv‡ii †ÿ‡Î X Gi ¯’‡j ZvcwgwZK ag© AvqZb V n‡j, T 

`yB w¯’i we›`yi cwi‡cÖwÿ‡Z _v‡g©vwgwZi g~j mgxKiY cÖwZcv`b : †h ZvcgvÎvq cÖgvb Pv‡c weï× eid cvwbi mv‡_ mvg¨ve¯’vq _vK‡Z cv‡i A_©vr †h ZvcgvÎvq weï× eid Mj‡Z ïiæ K‡i Zv‡K wb¤œ w¯’i we›`y (Lower fixed point) ev eid we›`y (Ice Point) ev wngv¼ (Freezing Point) e‡j| †h ZvcgvÎvq cÖgvb Pv‡c weï× cvwb Rjxq ev‡®úi mv‡_ mvg¨ve¯’vq _vK‡Z cv‡i A_©vr †h ZvcgvÎvq weï× cvwb Rjxq ev‡®ú cwiYZ n‡Z ïiæ K‡i Zv‡K DaŸ© w¯’i we›`y (Upper fixed point) ev ÷xg (Steam Point) we›`y ev ùzUbv¼ (Boiling Point) e‡j| DaŸ© w¯’i we›`y I wb¤œ wb¤œ w¯’i we›`yi ga¨eZ©x ZvcgvÎvi e¨veavb‡K †gŠwjK e¨eavb (Fundamental Interval) e‡j| G e¨eavb‡K KZK¸‡jv mgvb fv‡M fvM K‡i GK GKwU †¯‹j MVb Kiv nq| cÖwZwU fvM 1º ZvcgvÎv wb‡`©k K‡i| aiv hvK, eid we›`y I ÷xg we›`y‡Z †Kvb GKwU ZvcwgwZK c`v‡_©i a‡g©i gvb h_vµ‡g Xice I Xsteam| Ab¨ †h †Kvb () ZvcgvÎvq H a‡g©i gvb X| †gŠwjK e¨eav‡bi fvM msL¨v =n ; X Gi cwieZ©b ZvcgvÎvi cwieZ©‡bi mgvbycvwZK| n fvM Z_v nº ZvcgvÎv e„wׇZ X Gi e„w× X steam  X ice  n  X steam  X ice  kn ... ... ... (1) Avevi,  fvM Z_v º ZvcgvÎv e„wׇZ X Gi e„w× X   X ice    X   X ice  k ... ... ... (2) (1) I (2) n‡Z

X   X ice X   X ice    n `yB w¯’i we›`yi cwi‡cwÿ‡Z GwUB _v‡g©vwgwZi g~j mgxKiY|   n X steam  X ice X steam  X ice

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution 12| ZvcgvÎv (Tmperature)

1| GKwU aªæe AvqZb _v‡g©vwgUv‡i T †Kjwfb ZvcgvÎvq Pvc cvIqv †Mj 6.5×104 Pa| cvwbi ˆÎa we›`y‡Z Pvc 5×103Pa n‡j T Gi gvb wbY©q Ki| Avgiv Rvwb, GLv‡b, P T  T  273.16 K cvwbi ‰Îa we›`y‡Z Pvc, Ptr Ptr = 5×103Pa 4 6.510 ev, T   273.16K wb‡b©q DòZvq Pvc, 4 3 PT = 6.5×10 Pa 5 10 wb‡b©q ZvcgvÎv, T = ? T  3551.08 K (Ans.) 2| GKwU †iva _v‡g©vwgUvi eid I wóg we›`y‡Z h_vµ‡g 4.5I 9.5 †iva cÖ`k©b K‡i| GwU GKwU Zi‡j ¯’vcb Ki‡j 6.1 †iva cÖ`k©b K‡i| ZijwUi ZvcgvÎv wbY©q Ki| Avgiv Rvwb, GLv‡b, Rθ  R0 θ  100 C  eid we›`y‡Z †iva, R0 = 4.5 R 100  R 0 wóg we›`y‡Z †iva, R100 =9.5 6.1  4.5 Zi‡j †iva R = 6.1  100 C θ ZijwUi ZvcgvÎv ?

9.5  4.5

θ

1.6  100 C  θ  32C (Ans.) 5

3| GKwU ÎæwUc~b© _v‡g©vwgUvi mvaviY evq~Pv‡c MwjZ ei‡d 4°C Ges ﮋ ev‡®ú 98°C cvV †`q| _v‡g©vwgUviwU 42°C cvV w`‡j cÖK…Z ZvcgvÎv KZ? g‡bKwi, cÖK…Z ZvcgvÎv = C cÖvß ZvcgvÎv  wb¤ œ C0 cÖkœg‡Z,  DaŸ©  wb¤ œ 100  0

100  38 C 42  4 C  ev, C  94 100 98  4 C= 40.42 °C ( Ans.)

4| †Kvb ZvcgvÎvq †mjwmqvm I dv‡ibnvBU †¯‹‡j GKB cvIqv hvq?

Avgiv Rvwb, GLv‡b, C F  32 C= F = x awi, 

5

9 x x  32   5 9  9 x  5x  160  9x  5x  160  4x  160

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 160 4  x  40 C  40 F (Ans.) x

5| GKwU ÎæwUc~b© _v‡g©vwgUv‡i wb¤§w¯’i we›`y 4°C Ges DaŸ© w¯’i we›`y 98°C| _v‡g©vwgUviwU 51°C cvV w`‡j dv‡ibnvBU †¯‹‡j ZvcgvÎv KZ n‡e? g‡bKwi, cÖK…Z ZvcgvÎv = F

F  32 cÖvß ZvcgvÎv  wb¤ §  212  32 DaŸ©  wb¤ § F  32 51  4   212  32 98  4 F  32 47   180 94 47  180  F  32  94  F  90  32  F  122F (Ans.) cÖkœg‡Z,

6| GKwU ÎæwUc~b© _v‡g©vwgUv‡i mvaviY evqyPv‡c MwjZ ei‡d 2°CGes ﮋ ev‡®ú 96°C cvV †`q| _v‡g©vwgUviwU 49°C cvV w`‡j dv‡ibnvBU I †Kjwfb †¯‹‡j KZ cvV cvIqv hv‡e? g‡bKwi, cÖK…Z ZvcgvÎv h_vµ‡g = F I K

cÖvß ZvcgvÎv  wb¤ œ K  273 F  32   Da   wb¤ œ 212  32 373  273 K  273 F  32 49  2    212  32 96  2 373  273 F  32 47 K  273    180 94 100 F  32 1 K  273    180 2 100 180  F  32  2  F  32  90  F  90  32  122F

cÖkœg‡Z,

Avevi,

1 K  273  2 100  K  273  50  K  50  273  K  323K

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cÖ_g c‡Îi As‡Ki mgvavb

7| †Kvb& ZvcgvÎvq †mw›U‡MÖW I dv‡ibnvBU †¯‹‡j cv‡Vi cv_©K¨ 10°nq? awi, †mjwmqvm †¯‹‡j cvV, C= x  dv‡ibnvBU †¯‹‡j cvV = x  10  x  10 ev, x  10 Avgiv Rvwb,

C F  32  1g †ÿ‡Î, †mjwmqvm †¯‹‡j cvV, C= x 5 9 Ges dv‡ibnvBU †¯‹‡j cvV, F = x  10 x x  10  32   5 9  9 x  5x  110  9 x  5x  110  4x  110  110 x 4  x  27.5 C

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 dv‡ibnvBU †¯‹‡j, x  10  27.5  10  17.5F wØZxq †ÿ‡Î, †mjwmqvm †¯‹‡j cvV, C= x Ges dv‡ibnvBU †¯‹‡j cvV = x  10

x x  10  32  5 9  9 x  5x  210  9 x  5x  110  4x  210  210 x 4  x  52.5 C  dv‡ibnvBU †¯‹‡j, x  10  52.5  10  62.5F 

DËit -27.5°C I -17.5°F Ges -52.5°C I -62.5°F

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2

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Zvc MwZwe`¨vi 1g m~Î (First Law of Thermodynamics): Ry‡ji g‡Z ÒhLbB †Kvb Kv‡Ri d‡j Zvc Drcbœ nq ev Zv‡ci d‡j KvR mvwaZ nq ZLb KvR I Zvc ci¯ú‡ii mgvbycvwZKÓ| e¨vL¨v: g‡b Kwi, W cwigvb Kv‡Ri d‡j Q cwigvb Zvc Drcbœ nj| G‡ÿ‡Î m~Î g‡Z, W∞Q W = J Q GLv‡b J GKwU aªæe msL¨v| GB aªæe msL¨v‡K Zv‡ci hvwš¿K mgZv ev Ryj aªyeK e‡j|

weÁvbx K¬wmqv‡mi g‡Z  hLbB †Kvb e¯‘ ev ms¯’vq Zvc mieivn Kiv nq ZLbB GB Zv‡ci wKQy Ask e¯‘ ev ms¯’vi AšÍw© bwnZ kw³ e„wׇZ e¨vq nq evKx Ask evwn¨K KvR m¤úv`‡b LiP nq| MwY‡Zi fvlvq dQ = dU + dW GLv‡b dQ = †gvU cÖ`Ë Zvc dU = AšÍw© bwnZ kw³i cwieZ©b I dW = evwn¨K m¤úvw`Z KvR| weÁvbx †nj¥R Gi g‡Z  Ò kw³ Awebk¦i Ó | kw³ m„wó ev webvk Kiv hvq bv| †Kej GK kw³ Ab¨ GK ev GKvwaK kw³‡Z iƒcvšÍwiZ Kiv hvq| hLbB GK kw³ A`„k¨ n‡e ZLbB Aci kw³i Avwef©ve NU‡e| ZvB GB we‡k¦i †gvU kw³i cwigvb AcwieZ©bxq| Dc‡iv³ msMv n‡Z †h, kw³i e¨q Qvov KvR Kiv m¤¢e bq| Ggb †Kvb hš¿ Avwe®‹vi Kiv m¤¢e bq hv kw³ Qvov Pj‡Z cv‡i| Zv‡ci hvwš¿K mgZv (Mechanical equivalent of heat) : Ry‡ji Zvc MwZwe`¨vi 1g m~Î n‡Z Avgiv †h mgxKiY cvB Zv nj W = JQ D³ mgxKi‡Y Q = 1 K¨vjwi n‡j W = J nq | A_©vr 1 K¨vjwi Zvc m¤úyb©iƒ‡c Kv‡R iƒcvšÍwiZ n‡j hZUzKz KvR nq Zv‡K Zv‡ci hvwš¿K mgZv e‡j| hvwš¿K mgZv‡K J Øviv cÖKvk Kiv nq| J Gi gvb 4.2 Ryj/ K¨vjwi| J = 4.2 Ryj/ K¨jwi ej‡Z GB eywS †h, 1 K¨vjwi Zvc Kv‡R iƒcvšÍwiZ n‡j 4.2 Ryj KvR m¤úvw`Z nq| ZvcMZxq cÖwµqv (Thermodynamic Process ): Pvc P, ZvcgvÎv T I AvqZb V ivwk¸‡jv‡K Zvc MZxq ¯’vbvsK ev Zvc MZxq Pj ivwk e‡j| Zvc MZxq ¯’vbvs‡Ki mvnv‡h¨ †KvY e¨e¯’vi Ae¯’v cÖKvk Ki‡j, †m Ae¯’v‡K Zvc MZxq Ae¯’v e‡j| †h cwieZ©‡bi Kvi‡Y Zvc MZxq ¯’vbvs‡Ki gvb cwiewZ©Z nq †m cwieZ©b‡K ZvcMZxq cÖwµqv e‡j| ZvcMZxq ¯’vbvsK (Thermodynamic co-ordinates): ZvcMZxq Av‡jvPbvi Rb¨ †Kvb e¨e¯’vi Ae¯’v Pvc P , AvqZb V I cig ZvcgvÎv T Gi mvnv‡h¨ cÖKvk Kiv nq| GB ivwk ¸wj‡K ZvcMZxq ¯’vbvsK e‡j| ZvcMZxq Ae¯’v (Thermodynamic State): ZvcMZxq Av‡jvPbvi Rb¨ †Kvb e¨e¯’vi Ae¯’v Pvc P , AvqZb V I cig ZvcgvÎv T Gi mvnv‡h¨ cÖKvk Kiv nq| GB ivwk ¸wj‡K ZvcMZxq ¯’vbvsK ev Zvc MZxq Pj ivwk e‡j| ZvcMZxq ¯’vbvs‡Ki mvnv‡h¨ †KvY e¨e¯’vi Ae¯’v cÖKvk Ki‡j †m Ae¯’v‡K ZvcMZxq Ae¯’v e‡j| ZvcMZxq mvg¨ve¯’v (Thermodynamic Equilibrium): ‡Kvb wew”Qbœ e¨e¯’vi P‚ovšÍ AwePj Ae¯’v‡K ZvcMZxq mvg¨ve¯’v e‡j| mvg¨ve¯’vq e¨e¯’vi mKj we›`y‡Z ZvcMZxq ¯’vbv¼ A_©vr Pvc P , AvqZb V I cig ZvcgvÎv T Gi gvb mgvb| ZvcMZxq wm‡÷g ev e¨e¯’v (Thermodynamic system): ZvcMZxq wm‡÷g ej‡Z Zj ev †eóbx Øviv mxgve× †Kvb wbw`©ó cwigvb e¯‘‡K eySvq| †hgb GKwU wc÷b hy³ wmwjÛv‡i A_ev GKwU †ejy‡b Ave× M¨vm|

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2 13| Zvc MwZwe`¨vi 1g m~Î (First Law Of Thermodynamics) AšÍ¯’ kw³ (Internal Energy): †Kvb e¨e¯’vi Af¨šÍixb kw³B AšÍ¯’ kw³| e¨e¯‘vi GB kw³ Ab¨vb¨ kw³‡Z iæcvšÍwiZ n‡Z cv‡i| GB kw³ c`v‡_©i Af¨šÍixb MV‡bi Dci wbf©i K‡i| AšÍ¯’ kw³ AYy¸‡jvi wefe kw³ I MwZkw³i ‡hvMdj| Zvc cÖ‡qvM ev Zvc Acmvib Øviv AšÍ¯’ kw³i cwieZ©b NUv‡bv hvq| Avevi e¯‘i Dci KvR K‡i ev e¯‘i Øviv KvR m¤úvw`Z n‡j AšÍ¯’ kw³i cwieZ©b nq| AšÍ¯’ kw³i cwieZ©b ïaygvÎ e¨e¯’vi cÖv_wgK I PzovšÍ Ae¯’vi Dci wbf©i K‡i| M¨vm‡K msbwgZ Ki‡j AšÍ¯’ kw³i e„w× cvq Avevi M¨vm‡K cªmvwiZ Ki‡j AšÍ¯’ kw³ n«vm cvq|

iæ× Zvc cÖwµqvq M¨vm‡K msbwgZ Ki‡j Gi ZvcgvÎv e„w× cvq - Gi KviY: iæ× Zvc cwieZ©‡b M¨v‡mi ZvcgvÎv KLbB w¯’i _v‡K bv| A_v©r iæ×Zvc cÖwµqvq M¨vm †Kvb Zvc MÖnb ev eR©b K‡i bv| wKš‘ M¨vm Zvc MÖnb eR©b bv Ki‡jI M¨v‡mi AšÍt¯’ kw³ w¯’i _v‡K bv| hLb M¨vm‡K msbwgZ Kiv nq ZLb M¨v‡mi KvR m¤úv`b Kiv nq| G‡Z M¨v‡mi kw³ e„w× cvq A_©vr M¨v‡mi AšÍt¯’ kw³i e„w× N‡U| KviY G †ÿ‡Î M¨vm Zvc eR©b Ki‡Z cv‡i bv| G Rb¨ Gi ZvcgvÎv e„w× cvq| M¨v‡mi Av‡cwÿK Zvc I †gvjvi ZvcaviY ÿgZvi (†gvjvi Av‡cwÿK Zvc) g‡a¨ m¤úK© (Relation between specific heat of gas and molar thermal capacity of gas): 1 †gvj M¨v‡mi ZvcgvÎv 1K evov‡Z ‡h cwigvb Zv‡ci cÖ‡qvRb Zv‡K †gvjvi Zvc aviY ÿgZv ev †gvjvi Av‡cwÿK Zvc e‡j| †Kvb M¨v‡mi n msL¨K †gv‡ji ZvcgvÎv T cwigvb evov‡Z hw` Q Zv‡ci cÖ‡qvRb nq Z‡e 1 †gvj M¨v‡mi ZvcgvÎv 1K evov‡Z Zvc jvM‡e

Q nT

‡gvjvi Zvc aviY ÿgZv C   Q

nT

1 ‡KwR M¨v‡mi ZvcgvÎv 1K evov‡Z ‡h cwigvb Zv‡ci cÖ‡qvRb Zv‡K M¨v‡mi Av‡cwÿK Zvc e‡j| †Kvb M¨v‡mi m ‡KwR M¨v‡mi ZvcgvÎv T cwigvb evov‡Z hw` Q Zv‡ci cÖ‡qvRb nq Z‡e Q 1 kg M¨v‡mi ZvcgvÎv 1K evov‡Z Zvc jvM‡e mT

 M¨v‡mi Av‡cwÿK Zvc   Q

mT

GLv‡ b † gvj msL¨v n 

ΔQ m e¯‘ i fi  C  m M AvbweK fi ΔT M ΔQ ev, C  M mΔT ev, C  M  M¨v‡ mi Av‡ cw¶ K Zvc

A_©vr †gvjvi Zvc aviY ÿgZv = AvbweK fi × M¨v‡mi Av‡cwÿK Zvc M¨v‡mi Av‡cwÿK Zvc: 1 ‡KwR M¨v‡mi ZvcgvÎv 1K evov‡Z ‡h cwigvb Zv‡ci cÖ‡qvRb Zv‡K M¨v‡mi Av‡cwÿK Zvc e‡j| w¯’i Pv‡c M¨v‡mi Av‡cwÿK Zvc (Specific heat of Gas at constant Pressure): Pvc w¯’i †i‡L 1 ‡KwR f‡ii M¨v‡mi ZvcgvÎv 1K e„w× Ki‡Z hZUzKz Zvc cÖ‡qvM Ki‡Z nq Zv‡K w¯’i Pv‡c M¨v‡mi Av‡cwÿK Zvc e‡j|

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3 13| Zvc MwZwe`¨vi 1g m~Î (First Law Of Thermodynamics) w¯’i AvqZ‡b M¨v‡mi Av‡cwÿc Zvc (Specific heat of Gas at constant Volume): AvqZb w¯’i †i‡L 1 ‡KwR f‡ii M¨v‡mi ZvcgvÎv 1K e„w× Ki‡Z ‡h cwigvb Zv‡ci cÖ‡qvRb nq Zv‡K w¯’i AvqZ‡bi M¨v‡mi Av‡cwÿK Zvc e‡j|

w¯’i Pv‡c M¨v‡mi †gvjvi Av‡cwÿK Zvc ev †gvjvi Zvc aviY ÿgZv (Molar specific heat of Gas at constant Pressure): Pvc w¯’i †i‡L 1 †gvj M¨v‡mi ZvcgvÎv 1K e„w× Ki‡Z hZUzKz Zvc cÖ‡qvM Ki‡Z nq Zv‡K w¯’i Pv‡c M¨v‡mi †gvjvi Av‡cwÿK Zvc ev †gvjvi Zvc aviY ÿgZv e‡j| G‡K Cp Øviv cÖKvk Kiv nq|

w¯’i AvqZ‡b M¨v‡mi †gvjvi Av‡cwÿc Zvc ev †gvjvi Zvc aviY ÿgZv (Molar specific heat of Gas at constant Volume): AvqZb w¯’i †i‡L 1 ‡gvj M¨v‡mi ZvcgvÎv 1K e„w× Ki‡Z ‡h cwigvb Zv‡ci cÖ‡qvRb nq Zv‡K w¯’i AvqZ‡bi M¨v‡mi ‡gvjvi Av‡cwÿK Zvc ev †gvjvi Zvc aviY ÿgZv e‡j| G‡K Cv Øviv cÖKvk Kiv nq| GKwU Av`k© M¨v‡mi Rb¨ Cp I Cv Gi g‡a¨ m¤úK© (Relation between Cp and Cv ) ev , Cp  Cv = R Gi cÖgvY: g‡b Kwi wc÷b hy³ GKwU wmwjÛv‡ii g‡a¨ P Pv‡c, T ZvcgvÎvq, V AvqZ‡b 1 †gvj Av`k© M¨vm Av‡Q| (wPÎ wb‡¤§)

Gevi M¨v‡m wKQy Zvc cÖ`vb Kiv nj| G Zvc M¨v‡mi ZvcgvÎv I Pvc e„w× Ki‡e Ges AvqZb I e„w× †c‡Z PvB‡e| d‡j wc÷bwU Dc‡ii w`‡K DV‡Z †Póv Ki‡e| wc÷b‡K c~‡e©i Ae¯’v‡b ivL‡Z n‡j A_©vr AvqZb w¯’i ivLvi Rb¨ wc÷‡bi Dci wKQy IRb Pvcv‡Z n‡e| g‡b Kwi dQ Zvc cÖ‡qvM Kiv nj Ges wc÷‡b dP IRb Pvwc‡q M¨v‡mi AvqZb V AcwiewZ©Z ivLv nj| dQ Zvc MÖnb K‡i M¨v‡mi ZvcgvÎv hw` dT cwigvb e„w× cvq| Zvn‡j, dQ = 1. Cv dT [ M„nxZ Zvc = fi × Avt Zvc × DòZv e„w× ] ev, dQ = Cv dT... ... ... ... ... (1) Avevi Zvc MwZwe`¨vi 1g m~Î dQ dUdW Cv dT =dU dWewn¯’t Kv‡Ri cwigvb I (1) bs mgxKiY n‡Z dQ Gi gvbewm‡q]

 dU =Cv dT ... ... ... ... ...  Gevi wc÷‡bi Dci †_‡K cÖv_wgK IRb †i‡L AwZwi³ dP IRb Zz‡j wb‡j wc÷b Dc‡ii w`‡K DV‡Z _vK‡e A_©vr AvqZb e„w× cv‡e| [wPÎ - M]| AvqZb e„w×i Rb¨ M¨v‡mi wKQy KvR Ki‡Z n‡”Q| M¨v‡mi AšÍt¯’ kw³ e¨v‡q G KvR Ki‡Z nq e‡j AšÍt¯’ kw³ n«vm cvq d‡j ZvcgvÎv I n«vm cvq| w¯’i Pv‡c M¨v‡mi ZvcgvÎv dT e„w×i Rb¨ M¨vm †h cwigvb KvR Ki‡Q Zvi mgZzj¨ Zvc cÖ‡qvM Ki‡Z n‡e| A_©vr G‡ÿ‡Î †h AwZwi³ Zvc mieivn Ki‡Z n‡e Zvi cwigvb PdV| GLv‡b M¨v‡mi AvqZb e„w×i cwigvb dV Ges cÖ`Ë w¯’i Pvc P| G‡ÿ‡Î mieivnK…Z Zvc dQ

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13| Zvc MwZwe`¨vi 1g m~Î

(First Law Of Thermodynamics)

4

dQ = 1. Cp dT  dQ = Cp dT... ... ... ... ... (3)

Avevi Zvc MwZwe`¨vi 1g m~ÎwU nj dQ = dU + dW [ dW = PdV ]  dQ = dU + PdV [3 bs n‡Z dQ Ges 2 bs n‡ZdU Gi gvb ewm‡q|]  Cpd Cv dT + PdV ev, Cpd Cv dT = PdV ev, Cp Cv d = PdV ... ... ... ... ... ... (4) Avevi, †gvjvi M¨vm aªæeK R n‡j, 1 †gvj M¨v‡mi Rb¨ Avgiv Rvwb, PV = RT ... ... ... ... ... ... ... (5) (5) bs mgxKiY‡K T Gi mv‡c‡ÿ e¨eKjb K‡i cvB, d PV   d RT  dT dT dV dT ev, P R [P I R aªæe] dT dT dV R ev, P dT PdV = RdT ... ... ... ... ... ... ... (6) (4) bs mgxKi‡Y (6) bs mgxKi‡Yi gvb ewm‡q cvB, Cp Cv d = RdT Cp Cv = R ... ... ... ... ... ... ... (7) BnvB CpI Cv g‡a¨ m¤úK©| CpI Cv I AYycvZ‡K  Øviv cÖKvk Kiv nq| A_©vr   C P †h‡nZz Cp Cv = R Avevi R abvZ¡K msL¨v CV

 CpCv , Avevi CpCv e‡j  Gi gvb me mgq 1 Gi †ekx|

iæ× Zvcxq cwieZ©‡b M¨v‡mi Pvc I AvqZ‡bi g‡a¨ m¤úK© (Relation between Pressure and volume in an adiabatic Process) A_©vr PV = aªæeK Gi cÖgvb I ZvcgvÎv I AvqZ‡bi g‡a¨ m¤úK© TV γ 1  aª æeK | g‡b Kwi, 1 †gvj M¨vm‡K dQ cwigvb Zvc †`Iqvi d‡j Gi AvqZb I DòZvi cwieZ©b h_vµ‡g dV I dT nq| Zvc MwZwe`¨vi 1g m~Î †_‡K cvB, dQ = dU +dW dQ = dU + PdV ... ... ... ... ... ... (1) [ dW = PdV ] wKš‘ Avgiv Rvwb, 1 †gvj M¨v‡mi AšÍt¯’ kw³i e„w× dU nj w¯’i AvqZ‡b †gvjvi Av‡cwÿK Zvc Cv Ges ZvcgvÎv e„w× dT Gi ¸Y d‡ji mgvb, A_©vr dU = Cv dT ... ... ... ... ... ... ... (2) (1) bs mgxKi‡Y dU = Cv dT ewm‡q cvB,  dQ = Cv dT + PdV ... ... ... ... ... ... (3) wKš‘ iæ× Zvcxq cÖwµqvq dQ = 0, (3) bs mgxKi‡Y dQ = 0 ewm‡q cvB, Cv dT + PdV = 0 ... ... ... ... ... ... ... ... (4)

1 ‡gvj Av`k© M¨v‡mi Rb¨ Avgiv Rvwb, PV = RT ... ... ... ... ... ... ... (5)

GB mgxKiY‡K AšÍiKjb K‡i cvB, PdV +VdP = RdT PdV  VdP ev, dT  Gi GB gvb (4) bs mgxKi‡Y ewm‡q cvB, R

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13| Zvc MwZwe`¨vi 1g m~Î

ev, ev, ev, ev,

5

 PdV  VdP  Cv    PdV  0 R   CvPdV + CvVdP + RPdV = 0 CvPdV + CvVdP + (Cp Cv)PdV = 0 CvPdV + CvVdP + CpPdV CvPdV = 0 CvVdP + CpPdV = 0

ev, VdP 

Cp Cv

[ Dfq cÿ‡K Cv Øviv fvM K‡i|]

PdV  0



ev, VdP + PdV = 0 ev ,

(First Law Of Thermodynamics)

dV dP  γ  0 V P

CP CV





[ Dfq cÿ‡K PV Øviv fvM K‡i| ]

Gevi Dfq cÿ‡K mgvKjb K‡i cvB, LogeP + LogeV = aªæeK ev, LogeP +LogeV= LogeK ev, LogePV = LogeK ev, PV   K  PV   aª æeK (cÖgvwYZ|) ... ... ... ... ... ... (6)

Avgiv Rvwb, PV = RT P

RT ... ... ... ... ... ... (7) V

Avevi Avgiv Rvwb, PVaªæeK ... ... ... ... ... (6) 

RT  V  aª æeK V

[(7) bs n‡Z gvb ewm‡q|]

 RTV  1  aª æeK TV γ 1  aª æeK ( cÖgvwYZ )

[R me©Rbxb M¨vm aªæeK]

BnvB M¨v‡mi iæ× Zvc cwieZ©‡b AvqZb I ZvcgvÎvi g‡a¨ m¤úK©|

Cp I Cv Gi gvb wbY©q (Calculation of Cp and Cv): T cwigvb DòZv e„w×i Rb¨ GKwU Av`k© M¨v‡mi AšÍt¯’ kw³i e„w× U n‡j, U = Cv T GLv‡b Cv w¯’i AvqZ‡b M¨v‡mi †gvjvi Av‡cwÿK Zvc| GKwU GK civgvYyK M¨v‡mi AšÍt¯’ kw³ ïaygvÎ Gi AcmviY MwZ kw³i Kvi‡Y nq| M¨v‡mi MwZZË¡ n‡Z Avgiv Rvwb, 3 2

AšÍt¯’ kw³ U  RT dT DòZv e„w×i Rb¨ AšÍt¯’ kw³ dU cwigvb e„w× Ki‡j, 3 RdT 2 3  Cv dT  R dT 2 dU 

 Cv   Cv 

 dU  Cv dT 

3 R 2

3  8.31 J mol 1 K 1 2

[ R = 8.31 J mol-1 K-1]

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13| Zvc MwZwe`¨vi 1g m~Î

1

(First Law Of Thermodynamics)

6

1

 C v  12.5 J mol K Avevi, Cp  Cv = R  Cp = R +Cv  Cp = (8.31 +12.5) J mol-1 K-1  Cp =20.81 J mol-1 K-1  Gi ¸iæZ¡ (Zvrch©): CpI Cv I AYycvZ‡K  Øviv cÖKvk Kiv nq| A_©vr   C P , †ÿÎ we‡k‡l  ¸iæZ¡c~Y© f~wgKv cvjb K‡i| M¨v‡m k‡ãi CV

†eM wbY©‡q  Gi ¸iæZ¡ i‡q‡Q| iæ× Zvc cwieZ©‡b -Gi cÖ‡qvRbxZv Av‡Q| Zv Qvov M¨v‡mi †hvR¨Zv m¤úwK©Z Z_¨  †_‡K cvIqv hvq| GK cvigvYyweK M¨v‡mi Rb¨ =1.33 Ges wØcvigvYyweK M¨v‡mi Rb¨ =1.41I wZb cvigvYyweK M¨v‡mi Rb¨ =1.66|  -Gi M¨v‡mi cigvbyi msL¨vi Dci wbf©ikxj e‡j †Kvb M¨v‡mi AYy‡Z KqwU cigvYy Av‡Q Zv Gi gvb wbY©q K‡i Rvbv hvq| †h‡nZz Cp Cv = R Avevi R abvZ¡K msL¨v  CpCv , Avevi CpCv e‡j  Gi gvb me mgq 1 Gi †ekx|

iƒ×Zvc †jL m‡gvò †jL A‡cÿv Lvov (An adiabatic Carve is Steeper than Isothermal Curve) : iæ×Zvc cwieZ©‡bi †ÿ‡Î, Avgiv Rvwb, PV = aªæeK, GB mgxKiY †K e¨eKjb K‡i cvB, PV  1dV  V  dP  0

PV  1dV V PV  1  dP     V  dV  Q P  dP   dP  

... ... ... (1)    V  dV  Q

cybivq, m‡gvò cwieZ©‡b, PV = aªæeK, GB mgxKiY †K e¨eKjb K‡i cvB, PdV + VdP = 0 P  dP      ... ... ... (2) V  dV T

mgxKiY (1) ‡K (2) Øviv fAM K‡i cvB,

AZGe,

 dP  P     dV  Q  V P  dP     V  dV T

 dP     dV  Q    dP     dV T

Zvcxq † iLvi Xvj   ev, iæ× Zvcxq‡iLvi Xvj = m‡gvò †iLvi Xvj ×  m‡ gvò † iLvi Xvj

wKš‘ Gi gvb 1 A‡cÿv †ekx| myZivs, mgxKiY (3) †_‡K ejv hvq †h, iæ×Zvc †j‡Li Xvj ev bwZ (Slope) m‡gvò †j‡Li Xvj ev bwZi | †h †jL wP‡Îi Xvj hZ †ekx †m †jLwPÎ ZZ †ekx Lvov| myZivs iƒ×Zvc †jL m‡gvò †jL A‡cÿv Lvov|

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7 13| Zvc MwZwe`¨vi 1g m~Î (First Law Of Thermodynamics) m‡gvò cÖwµqv (Isothermal Process): Pvc cwieZ©b Kivi d‡j hLb †Kvb M¨v‡mi AvqZ‡bi cwieZ©b nq, wKš‘ ZvcgvÎvi †Kvb cwieZ©b nq bv, †m cwieZ©b‡K m‡gvò cwieZ©b e‡j| †h cÖwµqvq G cwieZ©b msNwUZ nq Zv‡K m‡gvò cÖwµqv e‡j|

iæ×Zvcxq cÖwµqv (Adiabatic Process): ‡h cwieZ©‡b M¨v‡mi Pvc I AvqZ‡bi cwieZ©b nq wKš‘ cÖwµqvaxb M¨vm Zvc MÖnY ev eR©b Ki‡Z cv‡i bv, †m cwieZ©b‡K iæ× Zvcxq cwieZ©b e‡j Ges †h cÖwµqvq G cwieZ©b msNwUZ nq Zv‡K iæ×Zvcxq cÖwµqv e‡j| m‡gvò I iæ×Zvc cÖwµqvi g‡a¨ cv_©K¨t µwgK m‡gvò cÖwµqv iæ×Zvc cÖwµqv w¯’i ZvcgvÎvq †h cÖwµqv m¤úvw`Z nq Zv‡K ‡h cÖwµqv m¤úv`b Kv‡j cwi‡e‡ki mv‡_ Zv‡ci 1| m‡gvò cÖwµqv e‡j| Av`vb cÖ`vb nqbv Zv‡K iæ×Zvc cÖwµqv e‡j| 2| m‡gvò cÖwµqvq ZvcgvÎv w¯’i _v‡K| iæ×Zvc cÖwµqvq ZvcgvÎv cwiewZ©Z nq| 3| GUv axi cÖwµqv GUv `ªæZ cÖwµqv| 4| m‡gvò cÖwµqvq PV = aªæeK nq| iæ×Zvc cÖwµqvq PV = aªæeK nq| 5| m‡gvò †jL Kg Lvov| iæ×Zvc †jL AwaK Lvov| m‡gvò cÖwµqvq †Kvb wm‡÷g KZ…K K…Z KvR wm‡÷‡g mieivnK…Z Zvc kw³i mgvbt aiv hvK, Avgv‡`i wm‡÷g n‡”Q GKwU M¨vmfwZ© wmwjÛvi hvi mv‡_ GKwU bobÿg wc÷b jvMv‡bv Av‡Q| wmwjÛvi †`Iqv‡ji ga¨ w`‡q kw³ wm‡÷‡g cÖ‡ek Ki‡Z wK¤^v wm‡÷g †_‡K †ewi‡q †h‡Z cv‡i| hw` wm‡÷‡g Lye ax‡i ax‡i kw³ mieivn Kiv hvq Zvn‡j M¨v‡m Pvc I AvqZ‡bi cwieZ©b n‡e hw`I Gi ZvcgvÎvi †Kvb cwieZ©b n‡e bv| Giƒc cwieZ©b‡K m‡gvò cwieZ©b e‡j| Gi d‡j M¨v‡mi †h cÖmviY nq m‡gvò cÖmviY e‡j| m‡gvò cÖwµqvq pV ˆiwLK wP‡Î †`Lvb n‡q‡Q| GB ˆiwLK wP·K m‡gv‡iL e‡j| m‡gvò cÖwµqvq DòZv w¯’i _v‡K e‡j wm‡÷‡gi AšÍt¯’ kw³i †Kvb cwieZ©b nq bv| A_©vr dU = 0 | myZivs, Zvc MwZwe`¨vi cÖ_g m~Î †_‡K cvB, dQ = 0 + dW  dQ  dW A_©vr m‡gvò cÖwµqvq †Kvb wm‡÷g KZ…K K…Z KvR wm‡÷‡g mieivnK…Z Zvc kw³i mgvb| m‡gvò cÖwµqvq p1V1  p2V2 n‡e|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

13| ZvcMwZwe`¨vi 1g m~Î (13. First Law Of Thermodynamics)

1| wcóbhy³ GKwU wmwjÛv‡i wKQy M¨vm Ave× Av‡Q| M¨‡mi Pvc 400Pa-G w¯’i †i‡L wm‡ó‡g ax‡i ax‡i 800J Zvckw³ mieivn Kivq 1200J KvR m¤úvw`Z nq| M¨v‡mi AvqZb cwieZ©b Ges AšÍt¯’ kw³i cwieZ©b wbY©q Ki| GLv‡b, Avgiv Rvwb, Pvc, P = 400Pa Q = U +W Zvc, Q = 800J KvR,W = 1200 J ev, U =Q W U = ? ev, U =800J 1200 J AvqZb cwieZ©b, V =?  U =400 J  AšÍ¯’t kw³i cwieZ©b FbvZ¥K nIqvi A_© wm‡÷‡gi AšÍ¯’t kw³ n«vm cv‡e| Avevi, W = PV W P 1200 3  V  m 400  V  3m 3 (Ans.)  V 

2| 27°C ZvcgvÎvi †Kvb wbw`©ó cwigvb M¨vm nVvr cÖmvwiZ n‡q wظb AvKvi jvf K‡i| P~ovšÍ ZvcgvÎv KZ? [ =1.4] Avgiv Rvwb, GLv‡b, T1V1γ 1  T2 V2γ1 T2 V1γ 1   T1 V2γ 1 T V   2   1  T1  V2 

γ 1

T  V   2   300  2V 

ZvcgvÎv, T1 = 27 ºC = (27+273)K = 300K AvqZb, V1 = V (awi) AvqZb, V2 = 2V ZvcgvÎv, T2 = ?  =1.4

1.4 1

 T2  300  0.5 0.4 K  T2  227.36 K  (227.36 - 273)C  - 45.64C (Ans.)

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3| 1 evqygÛjxq Pv‡c †Kvb M¨vm‡K iæ×Zvcxq cÖwµqvq AvqZb wظb Ki‡j H M¨v‡mi P~ovšÍ Pvc KZ n‡e?[   ] Avgiv Rvwb, P1V1γ = P2 V2γ V  ev, P2  P1  1   V2 

GLv‡b, evqygÛ‡ji Pvc, P1 =1.013×105 Pa AvqZb, V1 = V (awi) AvqZb, V2 = 2V Pvc, P2 =?  



V  ev, P2  1.013  10    2V 

1.41

5

ev, P2  1.013  105  0.51.41  P2 = 38120.37 Pa (Ans.)

4| ¯^vfvweK ZvcgvÎv I Pv‡c †Kvb M¨vm‡K m‡gvò cÖwµqvq wظb AvqZ‡b cÖmvwiZ Ki‡j H M¨v‡mi P~ovšÍ Pvc KZ n‡e?

Avgiv Rvwb, P1V1 P2 V2  T1 T2 PVT  P2  1 1 2 T1V2 0.76  V  273  P2  273 2V  P2  0.38mHg (Ans.)

GLv‡b, Pvc, P1 = 0.76 m Hg ZvcgvÎv, T1 =0ºC= (0+273) = 273K = T2 AvqZb, V1 = V (awi) AvqZb, V2 = 2V Pvc, P2 =?

5| ¯^vfvweK Pv‡c 100 m3 AvqZ‡bi GKwU M¨v‡m 5×103 J Zvc w`‡j M¨v‡mi AvqZb 100.2 m3 nq| H M¨v‡mi K…Z Kv‡Ri gvb wbY©q Ki| Avgiv Rvwb, GLv‡b, K…ZKvR W = PV Pvc, P = 1.013×105 Nm-2 Avw` AvqZb, V1 = 100 m3  W = PV2V1) 3  W =1.013×105(100.2 100)J

W =1.013×105 × 0.2 J 5  W =1.013×10 × 0.2 J  W = 20260 J (Ans.) 

‡kl AvqZb, V2 = 100.2 m cÖhy³ Zvc, Q = 5×103 J K…ZKvR W =?

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13| ZvcMwZwe`¨vi 1g m~Î (13. First Law Of Thermodynamics)

6| GKwU wmwjÛv‡ii g‡a¨ ivLv wKQy cwigvb M¨vm cwi‡e‡ki Dci 200 J KvR m¤úv`‡bi mgq cwi‡ek †_‡K 500 J Zvckw³ †kvlY K‡i| M¨v‡mi AšÍ¯’t kw³i cwieZ©b KZ n‡e?

Avgiv Rvwb, GLv‡b, Q = U +W ‡kvwlZ Zvc, Q =500 J  U = Q W K…ZKvR, W =200 J  U =   AšÍ¯’t kw³i cwieZ©b, U= ?  U = J (Ans.) AšÍ¯’t kw³i cwieZ©b abvZ¥K nIqvi A_© wm‡÷‡gi AšÍ¯’t kw³ e„w× cv‡e|

2

7| 27°C ZvcgvÎvi †Kvb wØcvigvbweK M¨v‡mi Pvc nVvr wظb Kiv nj, Pvc cwieZ©‡bi ci ZvcgvÎvi cwieZ©b KZ n‡e?

Avgiv Rvwb, 1-γ γ

T1P1

1-γ γ 2 2

T P

T P  2   1 T1  P2

  

1- γ γ

1-1.4

T  P  1.4  2   1  T1  2P2 

GLv‡b, ZvcgvÎv, T1 = 27 ºC = (27+273)K = 300K Pvc, P1 = P (awi)  Pvc, P2 = 2P ZvcgvÎv gvÎvi cwieZ©b ,

T  T2  T1  ? wØcvigvbweK M¨v‡mi  

0.2857

T 1  2   300  2   T2  300  0.5 0.2857 K

 T2  365.70 K

T  T2  T1  (365.70  300) K  65.70 K  65.70C ( Ans.)

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Zvc wewKiY (Heat Radiation): ZvcgvÎvi Kvi‡Y †Kvb e¯‘ ‡_‡K wbtm„Z wewKiY‡K ejv nq Zvc wewKiY (Thermal Radiation)| e¯‘ Zvc ïay wewKiYB K‡i bv, †kvlY I K‡i| †Kvb e¯‘i ZvcgvÎv AwePjfv‡e e„w× Kiv n‡j wb¤œwjwLZ NUbv ¸wj N‡U  (1) ZvcgvÎv hZ e„w× cv‡e wbtm„Z Zvc wewKi‡Yi cwigvb I ZZ evo‡Z _vK‡e| (2) ZvcgvÎv hZ †ekx e„w× cv‡e eY©vjxi D¾j Ask †_‡K wb:m„Z wewKi‡Yi Zi½‰`N©¨ I ZZ †ekx n«vm cv‡e| (3) DËß e¯‘ †_‡K wb:m„Z wewKiY e¯‘wU †Kvb c`v_© w`‡q ˆZix Ges Gi AvK…wZ I c„‡ôi cÖK…wZi Dci I wbf©ikxj| D`vniY ¯^iƒc ejv hvq, 2,000K ZvcgvÎvq cwjk Kiv †PÞv UvO‡÷‡bi c„ô †_‡K wbM©Z wewKi‡Yi nvi 23.5×104Wm-2 | gwje‡Wbv‡gi Rb¨ GB nvi 19.2×104Wm-2| cÖwZ†ÿ‡Î c„ôwU Agm„Y n‡j wb:mi‡Yi nvi e„w× cvq| weKxY© Zvc (Radiant Heat ): wewKiY c×wZ‡Z †h Zvc GK e¯‘ †_‡K Ab¨ e¯‘‡Z ev GK¯’vb †_‡K Ab¨ ¯’v‡b mÂvwjZ nq, Zv‡K weKxY© Zvc (Radiant Heat ) ejv nq| weKxY© Zv‡ci ˆewkó¨ (Characteristics of Radiant Heat) : weKxY© Zvc kw³i g‡a¨ wb¤œwjwLZ ˆewkó cwijwÿZ nq| (1) weKxY© Zvc k~Y¨ ¯’v‡bi ga¨w`‡q Pjv Pj Ki‡Z cv‡i| (2) Av‡jvK kw³i gZ weKxY© Zvc I mij‡iLvq P‡j| (3) Av‡jK iwk¥i gZ weKxY© Zvc cÖwZdjb I cÖwZmi‡Yi m~Î †g‡b P‡j| (4) weKxY© Zvc Av‡ji †e‡M P‡j| (5) Av‡jvK iwk¥i gZ weKxY© ZvcI wecixZeM©xq m~Î (Inverse square law) †g‡b P‡j| A_©vr †Kvb Zvc Drm †_‡K weKxY© Zv‡ci cÖvej¨ Drm †_‡K e¯‘i `~i‡Z¡i e‡M©i e¨v¯ÍvbycvwZK| R1 `~i‡Z¡ cÖvej¨ I1 I R2 `~i‡Z¡ cÖvej¨ I2 n‡j I 1 R22  I 2 R12

(6) Av‡jvK iwk¥i gZ weKxY© Zvc kw³i I e¨wZPvi (Interference) AceZ©b (Diffraction) I †cvjvivqb (Polarization) nq| Av`k© K…ò e¯‘ (Perfect Black Body): hLb †Kvb weKxY© Zvc †Kvb e¯‘i Dci AvcwZZ nq ZLb Gi wKQy Ask e¯‘i c„‡ô cÖwZdwjZ nq, wKQy Ask †kvwlZ nq, Ges evKx Ask e¯‘i wfZi w`‡q evB‡i P‡j hvq | hw` weKxY© Zv‡ci r fMœvsk cÖwZdwjZ nq, a fMœvsk †kvwlZ nq Ges t fMœvsk evB‡i P‡j hvq Z‡e, r+a+t=1. hLb r = 0, t = 0, ZLb a = 1, A_©vr mg¯Í wewKiYB †kvwlZ nq| Gi †_‡K c~Y© wewKiK ev Av`k© K…òe¯‘i msÁv cvIqv hvq| Av`k© K„ò e¯‘i msÁv (Definition of Perfect Black Body): hw` †Kvb e¯‘i AvcwZZ weKxY© Zv‡ci †Kvb Ask cÖwZdwjZ ev mÂvwjZ bv K‡i meUzKzB †kvlY K‡i, Zvn‡j †m e¯‘‡K Av`k© K…òe¯‘ ejv nq| Zi½ ˆ`‡N©i Av‡jv‡K ejv hvq, †h e¯‘ mKj Zi½ ˆ`‡N©¨i weKxY© Zvc †kvlb K‡i, Zv‡K Av`k© K…òe¯‘ e‡j| K…òe¯‘ DËg †kvlK I DËg wewKiK|

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2 14| Zvc wewKiY (Heat Radiation) Av`k© K…ò e¯‘i D`vniY (Example of Perfect Black Body): Av`k© K…òe¯‘ wn‡m‡e †h me e¯‘‡K we‡ePbv Kiv nq, Zv‡`i †KvbwUBAv`k© K…òe¯‘ bq| G‡`i †kvlYÿgZv 100% Gi KvQvKvwQ e‡j G‡`i K…òe¯‘ e‡j we‡ePbv Kiv hvq| †hgb Kv‡jv cøvwUbvg 98% AvcwZZ wewKiY †kvlY Ki‡Z cv‡i| weÁvbx †dix GKwU K…òe¯‘i cwiKíbv K‡ib| wb‡P Zv eY©bv Kiv nj| GwU GKwU `yÕqvj wewkó duvcv †MvjK| †fZ‡ii †`qv‡j fylv Kvwj gvLv‡bv _v‡K Ges evB‡ii w`KUv wb‡Kj cvwjk Kiv _v‡K| `yÕ†`qv‡ji ga¨eZ©x ¯’vb evqyk~b¨ Kiv nq hv‡Z cwienb I cwiPjb c×wZ‡Z Zvc bó bv nq| GB †Mvj‡Ki GKwU m~ÿè wQ`ª nj A | wQ‡`ªi wecixZ w`‡K GKwU wXwe C _v‡K †hb AvMZ weKxY© Zvc mivmwi cÖwZdwjZ n‡q evB‡i †h‡Z bv cv‡i| A wQ`ª w`‡q wewKiY †Mvj‡Ki g‡a¨ cÖ‡ek K‡i| H wewKiY cv‡Îi †`Iqvj n‡Z evi evi cÖwZdwjZ n‡q †kvwlZ nq| hLb GB †MvjK‡K GKwU wbw`©ó DòZvq DËß Kiv nq ZLb GwU GKwU Av`k© K…òe¯‘i gZ AvPiY K‡i Ges A wQ`ªc‡_ †h wewKiY †ewi‡q Av‡m Zv‡K K…ò wewKiY ev c~Y© wewKiY ejv nq|

wewKiY ÿgZv: wbw`©ó ZvcgvÎvi †Kvb e¯‘i GKK †ÿÎdj n‡Z cÖwZ †m‡K‡Û weKxY© Zv‡ci cwigvb I GKB ZvcgvÎvq K…òe¯‘i GKK †ÿÎdj n‡Z cÖwZ ‡m‡K‡Û weKxY© Zv‡ci cwigvb, G `yÕGi AbycvZ‡K wewKiY ÿgZv e‡j| Av‡cwÿK wewKiY ÿgZv: †Kvb e¯‘i wewKiY ÿgZv Ges GKwU K…ò e¯‘i wewKiY ÿgZvi AbycvZ‡K H e¯‘i Av‡cwÿK wewKiY ÿgZv e‡j| ‡kvlY ÿgZvt †Kvb wbw`©ó mg‡q ‡Kvb e¯‘ weKxY© Zv‡ci †h cwigvb †kvlY K‡i Ges H mg‡q e¯‘i Dci †h cwigvb weKxY© Zvc AvcwZZ nq, Zv‡`i AbycvZ‡K †kvlb ÿgZv e‡j| w÷dv‡bi m~Î (Steafan's Law): weÁvbx w÷dvb K…òKvqv †_‡K weKxY© Zv‡ci †ÿ‡Î GKwU m~Î †`b| Zuvi m~ÎwU njÑ ‡Kvb K…òKvqv e¯‘i GKK †ÿÎdj †_‡K cÖwZ †m‡K‡Û weKxY© Zv‡ci cwigvb Gi cig ZvcgvÎvi PZz_© Nv‡Zi mgvbycvwZK| e¨vL¨v: ‡Kvb K…òKvqv e¯‘i cÖwZ GKK †ÿÎdj †_‡K cÖwZ †m‡K‡Û weKxY© Zv‡ci cwigvb E Ges e¯‘i cig ZvcgvÎv ev †Kjwf‡b cÖKvwkZ ZvcgvÎv T n‡j w÷dv‡bi m~Îvbyhvqx, E ∞ T4 ev, E = T4, GLv‡b  w÷dvb aªæeK| Gi gvb, =5.7×10-8 Wm-2K-4| e¯‘ Av`k© K…òKvqv bv n‡j E Gi gvb n‡e, E = eT4 | GLv‡b e nj e¯‘wUi Av‡cwÿK wewKiY ÿgZv| e gvb 0 ‡_‡K 1 Gi g‡a¨| Av`k© K…òKvqvi †ÿ‡Î, e=1 | e¯‘wUi A †ÿÎdj †_‡K cÖwZ †m‡K‡Û weKxY© Zv‡ci cwigvb n‡e, E = A eT4| e¯‘wU †_‡K t mg‡q weKxY© Zv‡ci cwigvb n‡e, E = A eT4t | T ZvcgvÎvi K…òKvqvwU hw` T0 ZvcgvÎvi e¯‘ Øviv cwi‡ewóZ _v‡K Z‡e cÖwZ GKK †ÿÎdj †_‡K cÖwZ †m‡K‡Û weKxY© Zvckw³i cwigvb n‡e E = T4T04) | K…òKvqv bv n‡j E Gi gvb n‡e, E = eT4T04)

wbDU‡bi kxZjxKiY m~Î (Newton's Law of cooling): ‡Kvb e¯‘ wewKi‡Yi Rb¨ †h nv‡i Zvc nvivq Zv e¯‘i ZvcgvÎv I cvwicvwk¦©K ZvcgvÎvi cv_©‡K¨i mgvbycvwZK| ZvcgvÎvi cv_©K¨ hLb Kg n‡e ZLb G m~Î cÖ‡hvR¨ n‡e| e¨vL¨v: ‡Kvb e¯‘i ZvcgvÎv , Gi cvwicvwk¦©‡Ki ZvcgvÎv n‡j Zvc eR©‡bi nvi, 

dQ  (θ  θ0 ) dt

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14| Zvc wewKiY (Heat

ev, 

Radiation)

3

dQ  K(θ  θ0 ) ... ... ... ... ... (1) dt

GLv‡b K nj GKwU mgvbycvwZK aªæeK| Gi gvb e¯‘i †ÿÎdj I cÖK…wZi Dci wbf©i K‡i| hw` e¯‘i fi m, Av‡cwÿK Zvc s Ges dt mg‡q e¯‘i ZvcgvÎv hw` d n«vm cvq, Z‡e e¯‘ KZ…K ewR©Z Zvc, dQ=msd e¯‘ KZ…K ewR©Z Zv‡ci nvi ,

dQ dθ  ms dt dt

(1) bs mgxKi‡Y D³ gvb ewm‡q cvB,

dθ  K(θ  θ0 ) dt dθ K ev,   (θ  θ0 ) dt ms dθ K ev,   (θ  θ0 ) dt ms dθ ev,   K (θ  θ0 ) dt

 ms



[K  

K  aª æeK ] ms

dθ (θ  θ0 ) dt

A_©vr, †Kvb e¯‘i ZvcgvÎv n«v‡mi nvi, Z_v kxZjxKi‡Yi nvi Gi wbR¯^ ZvcgvÎv Ges cwicv‡k©¦i ZvcgvÎvi cv_©‡K¨i mgvbycvwZK| G Rb¨ GB m~·K wbDU‡bi kxZjxKiY m~Î e‡j| w÷dv‡bi m~Î †_‡K wbDU‡bi kxZjxKiY m~Î (Newton's law of cooling from Steafan's law) cÖwZcv`b: w÷dv‡bi m~Îvbyhvqx, T ZvcgvÎvi †Kvb e¯‘‡K hw` To ZvcgvÎvi †Kvb e¯‘ Øviv cwi‡ewóZ _v‡K, Zvn‡j K…òKvqvwU †h nv‡i Zvc nvivq Zv nj, E  Ae (T 4  T04 ) GLb ( T 4  T04 ) †K Drcv`‡K we‡kølY Ki‡j cvIqv hvq, E  Ae (T 2  T02 )(T 2  T02 )  E  Ae (T  To )(T  To )(T 2  T02 )  E  Ae (T  To )(T 3  TT02  T 2T0  T03 ) ... ... ... ... (2)

wKš‘ w÷dv‡bi m~Î mKj ZvcgvÎvq cÖ‡hvR¨ n‡jI wbDU‡bi kxZjxKiY m~Î ZLb cÖ‡hvR¨ n‡e n‡e hLb (T-To) Gi gvb mvgvb¨ nq| A_©vr, (T  To )  0 , A_©vr T I To Gi gvb cÖvq mgvb nq| †m †ÿ‡Î To = T ai‡j, (2) bs mgxKiY `vovq,  E  Ae (T  To )(T 3  TT 2  T 2T  T 3 )  E  Ae (T  To )(4T 3 )  E  Ae 4T 3 (T  To )

 E  K (T  To ), †hLv‡b Ae 4T 3  K (GKwU aªæeK)|  E  (T  To ) A_©vr, ZvcgvÎvi cv_©K¨ Kg n‡j e¯‘i ewR©Z Zv‡ci nvi ev kxZjxKi‡Yi nvi ev

kxZjxKi‡Yi nvi e¯‘i ZvcgvÎv I cvwicvwk©¦‡Ki ZvcgvÎvi cv_©‡K¨i mgvbycvwZK| GwUB wbDU‡bi kxZjxKiY m~Î| AZGe w÷dv‡bi m~Î †_‡K wbDU‡bi kxZjxKiY m~Î cÖwZcvw`Z|

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14| Zvc wewKiY (Heat

Radiation)

4

wf‡q‡bi miY m~Î: Rvg©vb weÁvbx wf‡qb 1896 wLªóv‡ã K…òe¯‘ wewKi‡Yi †ÿ‡Î ZvcMwZwe`¨vi m~Î cÖ‡qvM K‡i Zvcxq wevKi‡Yi wewfbœ Zi½ ˆ`‡N©¨i g‡a¨ kw³i e›Ub m¤^‡Ü m~Î cÖYqY K‡ib| m~ÎwU wb¤œiƒc: K…òe¯‘ n‡Z wewKwiZ me©vwaK Zvckw³i Rb¨ Zi½‰`N©¨ K…òe¯‘i cig ZvcgvÎvi e¨v¯ÍvbycvwZK| hw` K…òe¯‘i me©vwaK kw³i Rb¨ Zvi Zi½‰`N©¨ mGes cig ZvcgvÎv T K nq Z‡e wf‡q‡bi m~Îvbymv‡i, m 

1 T

1 [GLv‡b, b = fxb aªæeK] T  m  T  b

 m  b 

GLv‡b, aªæeK b =2.898×10-3 mK  m  T  2.898  10 3 G mgxKiY †K wf‡q‡bi miY m~Î ejv nq| wkZjxKiY c×wZ‡Z Zij c`v‡_©i Av‡cwÿK Zvc wbY©‡qi c×wZ eY©bv (Determination of Specific heat of liquid by cooling method): Kvh©cÖYvjxt GKwU ﮋ I cwi®‹vi K¨vjwiwgUvi wb‡q fi wbb©q Kiv nq| K¨vjwiwgUv‡ii cÖvq A‡a©K Kÿ ZvcgvÎvi †P‡q 25ºC †_‡K 30ºC †ekx ZvcgvÎvi Zij c`v_© Øviv c~b© Kiv nq| Gici K¨vjwiwgUvi‡K Acwievnx c`v‡_©i Dci ¯’vcb K‡i Zij c`v_©wU‡K ºC †_‡K ºC ZvcgvÎvq bv bvgv ch©šÍ A‡cÿv Kiv nq| Zij c`v_©wUi ZvcgvÎv ºC †_‡K ºC ch©šÍ †b‡g Avm‡Z †h mgq jv‡M Zv ÷c Iqv‡Pi mvnv‡h©¨ wbb©q Kiv nq| Gi ci Zijmn K¨vjwiwgUv‡ii fi wbY©q Kiv nq| 2q I 1g f‡ii cv_©K¨ †_‡K Zij c`v‡_©i fi cvIqv hvq| Gici mgAvqZb Mig cvwb wb‡q Gi ZvcgvÎv I ºC †_‡K ºC G †b‡g Avm‡Z †h mgq jv‡M Zv cwigvc Kiv nq| Gi fi wbb©q K‡i Lvwj K¨vjwiwgUv‡ii fi we‡qvM K‡i cvwbi fi †ei Kiv nq| wnmve I Mbbv: g‡bKwi, K¨vjwiwgUv‡ii fi = m1 Kg Zij c`v‡_©i fi = m Kg cvwbi fi = m2Kg K¨jwiwgUv‡ii Dcv`v‡bi Av‡cwÿK Zvc = S1 JKg-1K-1 Zij c`v‡_©i wb‡b©q Av‡cwÿK Zvc = S JKg-1K-1 cvwbi Av‡cwÿK Zvc = S2 JKg-1K-1 ºC †_‡K ºC ZvcgvÎvq †b‡g kxZj n‡Z Zij c`v‡_©i mgq jv‡M = t1 Sec ºC †_‡K ºC ZvcgvÎvq †b‡g kxZj n‡Z cvwbi mgq jv‡M = t2 Sec ms (1   2 )  m1s1 (1   2 ) 1 JS t1 (ms  m1s1 )(1   2 ) 1  JS t1 m s (   )  m1s1 (1   2 ) 1 JS Ges cvwb I K¨vjwiwgUv‡ii Zvc eR©‡bi nvi  2 2 1 2 t2 (m s  m1s1 )(1   2 ) 1  2 2 JS t2 Zij c`v_© I K¨vjwiwgUv‡ii Zvc eR©‡bi nvi 

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14| Zvc wewKiY (Heat Radiation) kxZjxKiY Ae¯’v GKBiƒc _vKvq Zvc eR©‡bi GB nvi mgvb, A_©vr

5

(ms  m1s1 )(1   2 ) (m2 s2  m1s1 )(1   2 )  t1 t2 (ms  m1s1 ) (m2 s 2  m1s1 ) ev,  t1 t2 ( m s  m s )t ev, ms  m1s1  2 2 1 1 1 t2 (m s  m s )t ev, ms  2 2 1 1 1  m1s1 t2 s 

 1  (m2 s 2  m1s1 )t1  m1s1  Jkg 1 K 1  m t2 

mZK©Zv (Caution): (1) †h AvqZ‡bi Zij wVK †mB GKB AvqZ‡bi cvwb Aek¨B wb‡Z n‡e| (2) mwVK ZvcgvÎv cÖvwßi mv‡_ mv‡_ ÷c IqvP Pvjy I eÜ Kiv DwPr| (3) my‡e`x wbw³ Øviv mwVK fv‡e IRb Kiv DwPr| (4) my‡e`x _v‡g©vwgUvi Øviv ZvcgvÎv †bIqv DwPr| MÖxb nvDR I Gi cÖ‡qvRbxqZv (Green House and its necessity): kxZ cÖavb †`‡k MvQcvjv msiÿ‡bi Rb¨ KvP w`‡q wbwg©Z Ni‡K MÖxb nvDm e‡j| GwU G cÖKvi Zvc duv` (Heat trap)| MÖxb nvD‡m GKwU wbw`©ó ZvcgvÎv eRvq †i‡L kvK mwâ Drcv`b Kiv nq| G Qvov MÖxb nvDm e¨eüZ nq `y®úªvc¨ Dw™¢`, jZv¸j¥ BZ¨vw` msiÿ‡bi Kv‡R| kxZ cÖavb †`‡k Db¥y³ Dw™¢` I jZv¸j¥ m~h© †_‡K weKxb© Zvc MÖnb K‡i DËß n‡jI evqy cÖevn ev cwiPjb evqy‡mªvZ Øviv VvÛv nq| MÖxb nvD‡m evqyi ZvcgvÎv me©Î cÖvq GKB mgvb _v‡K e‡j Gi g‡a¨ evqy cÖevn _v‡K bv| KviY, Kv‡Pi N‡ii evB‡i †_‡K evqy cÖ‡ek †hgb Ki‡Z cv‡i bv, †Zgwb †ei n‡ZI cv‡i bv| myZivs Gi †fZi ivLv Dw™¢` I jZv¸j¥ Ges wfZiKvi evqygÛj m~‡h©i Zv‡c Mig n‡jI Zv Avi VvÛv nq bv| KviY m~h© †_‡K Avmv ÿz`ª Zi½‰`‡N©¨i Zvc wewKiY Kv‡Pi N‡i cÖ‡ek Ki‡Z cv‡i, wKš‘ Kv‡Pi N‡i Gi cÖK…wZ cwiewZ©Z n‡q `xN© Zi½‰`N©¨ wewkó nq e‡j Kv‡Pi †`Iqvj †f` K‡i †h‡Z cv‡i bv| d‡j MÖxb nvD‡mi wfZiKvi mewKQy Mig _v‡K| fx‡bi miY m~Î ØvivI Gi e¨vL¨v Kiv hvq| fx‡bi m~Î †_‡K Avgiv Rvwb †h, K„ò e¯‘ †_‡K weKxY© Zvc kw³i Rb¨ Zi½‰`N¨© K„òe¯‘i cig ZvcgvÎvi e¨v¯ÍvbycvwZK| weKxY© me©vwaK kw³i Rb¨ Zv‡ci Zi½‰`N¨© m Ges cig ZvcgvÎv T n‡j fx‡bi m~Îvbymv‡i m  1 | Kv‡Pi N‡ii †ejvq, m~h© †_‡K Avmv wewKi‡Y ZvcgvÎv T †ekx _v‡K d‡j m T

ÿz`ª _v‡K| GB Zi½ ¸‡jvi †f`b ÿgZv (Penetrating power) †ekx nIqvq Giv Abvqv‡m KvP †f` K‡i MÖxb nvD‡mi wfZi †h‡Z cv‡i| H Zv‡c DËß n‡q †fZ‡ii MvQcvjv BZ¨vw` †h Zvc wewKiY K‡i Zvi T K‡g hvq d‡j m ‡e‡o hvq| GB `xN© Zi½‰`‡N¨©i Zv‡ci †f`b ÿgZv Kg nIqvq Giv KvP †f` K‡i evB‡i †h‡Z cv‡i bv| d‡j Ni Mig _v‡K Ges †fZ‡ii Dw™¢` I evqygÛj‡K GKwU wbw`©ó ZvcgvÎvq _v‡K| MÖxb nvDm wµqv (Green House effect) mn‡RB fx‡bi miY m~Î Øviv e¨vL¨v Kiv hvq| MÖxb nvIm wµqv ej‡Z AwZwi³ Kve©b WvBA·vBW wbtmi‡Yi d‡j c„w_exi `xN©‡gqv`x DòKiY ev, DËß nIqv‡K eySvq| Kve©b WvBA·vB‡Wi ag© nj ¯^í ev ÿz`ª Zi½‰`‡N©¨i wewKiY Gi wfZi w`‡q †h‡Z cv‡i bv| m~h© AZ¨šÍ DËß Ae¯’vq Zvc wewKiY K‡i| d‡j T †ekx nIqvq GB wewKi‡Y m ÿz`ª nq| GB ÿz`ª Zi½‰`‡N©¨i wewKiY mn‡R evqygÛ‡ji Kve©b WvBA·vB‡Wi wfZiw`‡q c„w_ex‡Z Av‡m| c„w_ex GB wewKiY †kvlY K‡i DËß nq| c„w_ex hLb c~bivq Zvc wewKiY K‡i ZLb Zvi ZvcgvÎv, T m~‡h©i ZvcgvÎv †_‡K A‡bK Kg _vKvq GB wewKi‡Y m `xN© nq| GB `xN© Zi½‰`‡N©¨i wewKiY c„w_exi Kve©b WvBA·vB‡Wi duv`‡K †cwi‡q †h‡Z cv‡i bv, G‡Z c„w_ex‡Z Zvc AvUKv c‡o, d‡j MÖxb nvDm wµqv msNwUZ n‡q `xN©‡gqv`x DòKiY N‡U|

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6 14| Zvc wewKiY (Heat Radiation) Av`k© K…ò e¯‘ DËg †kvlK I DËg wewKiK Gi e¨vL¨v t ‡Kvb e¯‘i Dci AvcwZZ wewKiY hw` e¯‘ KZ…K m¤úyY© †kvwlZ nq, Z‡e H e¯‘‡K K…ò e¯‘ ev K…òKvqv e‡j| K…òKvqv wewKiY wbtm„Z K‡i, Zv‡K K…òKvqv wewKiY e‡j| K…òKvqvi GKwU we‡kl ˆewkó¨ n‡jv Gi Dci AvcwZZ wewKiY K…òKvqv KZ…K †hgb m¤ú~Y© †kvwlZ nq, †Zgwb kxZj cwi‡e‡k K…òKvqv Zvi †kvwlZ wewKiY m¤ú~Yi© ƒ‡c wewKwiZ K‡i| ZvB Av`k© K…ò e¯‘ DËg †kvlK I DËg wewKiK|

ivbœvi cvÎ wn‡m‡e bZzb PKP‡K cv‡Îi †P‡q cyiv‡bv KvwjgvLv cvÎ †ekx Dc‡hvMx : Kvwj gvLv cvÎ †ekxi fvM Zvc †kvlb K‡i DËß nq| wKš‘ bZzb PKP‡K cvÎ †ekx Zvc †kvlb Ki‡Z cv‡i bv; †ekxi fvM Zvc PKP‡K Zj †_‡K cÖwZdwjZ n‡q hvq| KvwjgvLv cv‡Îi Zvc †kvlb ÿgZv †ekx e‡j ivbœvi Kv‡R †ekx Dc‡hvMx| Mi‡gi w`‡b mv`v †cvkvK I kx‡Zi w`‡b Kv‡jv †cvkvK AvivgcÖ`: mv`v is Zvcxq wewKi‡Yi DËg cÖwZdjK| mv`v †cvkvK m~‡h©i Zvc‡K cÖwZdwjZ K‡i wdwi‡q †`q, ZvB mv`v †cvkvK mn‡R DËß nq bv| Avi Kv‡jv is DËg †kvlK| d‡j Kv‡jv †cvkvK m~‡h©i Zvc †kvlb K‡i DËß n‡Z cv‡i| ZvB Mi‡gi w`‡b mv`v †cvkvK Ges kx‡Zi w`‡b Kv‡jv †cvkvK AvivgcÖ`| giæ A‡j w`‡b cÖPÛ Mig I iv‡Zi †ejv Zxeª kxZ Abyf~Z nq : giæ A‡ji evqy ﮋ _v‡K| ﮋ evqy Zvc ¯^”Q| w`‡bi †ejv m~‡h©i Zvc mn‡RB f~-c„‡ô †cuŠQvq I f~-c„ô AZ¨šÍ DËß nq| d‡j giæ A‡j w`‡b cÖPÛ Mig †eva nq| Avevi iv‡Zi †ejv f~-c„ô Zvc wewKiY K‡i| ﮋ evqyi ga¨w`‡q mn‡RB Zvc evqygÛj †f` K‡i P‡j hvq| d‡j f~-c„ô kxZj n‡q hvq| ZvB iv‡Zi †ejv giæ A‡j Zxeª kxZ Abyf~Z nq| ‡gNgy³ ivZ A‡cÿv †gNv”Qbœ iv‡Z †ekx Mig Abyf~Z nq: w`‡bi †ejv f~-c„ô m~h©Zvc †kvlb K‡i DËß nq Ges iv‡Zi †ejv Zvc wewKiY K‡i kxZj nq| †gNgy³ iv‡Z f~-c„ô mn‡RB Zvc wewKiY K‡i kxZj n‡q hvq| wKš‘ Rjxq ev®ú Zvc Am”Q e‡j f~-c„‡ôi Zvc †g‡Ni ga¨ w`‡q mÂvwjZ n‡Z cv‡i bv| d‡j †gNv”Qbœ iv‡Z †ekx Mig †eva nq|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

14| Zvc wewKiY (Heat Radiation) 1| 250gm f‡ii GKwU Zvgvi K¨vjwiwgUv‡i ivLv 5gm cvwb 60°C n‡Z 40°C ZvcgvÎvq kxZj n‡Z 80 †m‡KÛ mgq jv‡M| GKB K¨vjwiwgUv‡i mgvb AvqZ‡bi 6gm f‡ii †Kvb Zij c`v_© 60°C n‡Z 40°C ZvcgvÎvq kxZj n‡Z mgq jv‡M 70 †m‡KÛ| Zij c`v_©wUi Av‡cwÿK Zvc wbY©q Ki| [Zvgvi Av‡cwÿK Zvc 380Jkg-1K-1 cvwbi Av.Zvc 4200Jkg-1K-1] GLv‡b, K¨vjwiwgUv‡i fi, m1 = 250gm = 0.25 Kg cvwbi fi, m2 = 5gm = 0.005 kg Zvgvi Av‡cwÿK Zvc, s1 = 380 Jkg-1K-1 cvwbi Avt Zvc, s2 = 4200 Jkg-1K-1 cvwbi mgq, t2 = 80 Sec K¨jwiwgUvi I cvwbi ZvcgvÎv cv_©K¨, = °C = K¨jwiwgUvi I Zi‡ji ZvcgvÎv cv_©K¨,°C = 20°C Zi‡ji fi, m = 6gm = 0.006kg Zi‡ji mgq, t1 = 70 Sec Zi‡ji Avt Zvc, s = ?

Avgiv Rvwb, K¨jwiwgUvi I Zij KZ…K Zvc n«v‡mi nvi = K¨jwiwgUvi I cvwb KZ…K Zvc n«v‡mi nvi m 1 s1 (θ1  θ 2 )  ms(θ1  θ 2 ) m 1 s1 (θ1  θ 2 )  m 2s 2 (θ1  θ 2 )  t1 t2 m 1 s1  ms

m 1s1  m 2 s 2 t1 t2 0.25  380  0.006  s 0.25  380  0.005  4200   70 80 95  0.006  s 95  21   70 80



K¨jwiwgUvi I Zi‡ji ZvcgvÎv cv_©K¨, °C = 2°C Zi‡ji fi, m= 0.2kg Zi‡ji mgq, t1 = 300 Sec Zi‡ji Avt Zvc, s =?

Avgiv Rvwb, K¨jwiwgUvi I Zij KZ…K Zvc n«v‡mi nvi = K¨jwiwgUvi I cvwb KZ…K Zvc n«v‡mi nvi m 1 s1 (θ1  θ 2 )  ms(θ1  θ 2 ) t1



m 1 s 1  ms t1





m 1 s1 (θ1  θ 2 )  m 2 s 2 (θ1  θ 2 ) t2

m 1s1  m 2s 2 t2

42  0.2  s 42  0.3  4200  600 300 42  0.2  s 42  1260   1 2 1302  42  0.2s  2 

 0.2s  651  42 609 s 0.2  s  3045 Jkg 1K 1 (Ans.)



116  70 80  0.006s  101.5  95 6. 5 s 0.006  s  1083.33Jkg 1K 1 (Ans.)  95  0.006s 

2| mgAvqZ‡bi cvwb I GKwU Zij c`v‡_©i fi h_vµ‡g 0.3kg Ges 0.2kg| Zv‡`i GKB K¨vjwiwgUv‡i ci ci †i‡L 50°C †_‡K 30°C-G kxZj Ki‡Z h_vµ‡g 600s Ges 300s mgq jv‡M| K¨vjwiwgU‡ii ZvcaviKZ¡ 42 Jkg-1 n‡j Zi‡ji Av‡cwÿK Zvc wbY©q Ki| [ cvwbi Av‡cwÿK Zvc 4200Jkg-1K-1 ] GLv‡b, K¨vjwiwgUv‡ii ZvcaviKZ¡, m1s1 = 42 JK-1 cvwbi fi, m2 = 0.3 kg cvwbi Avt Zvc, s2 = 4200 Jkg-1K-1 cvwbi mgq, t2 = 600 Sec K¨jwiwgUvi I cvwbi ZvcgvÎv cv_©K¨ =°C =

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3| GKwU K…ò Kvqvi †ÿÎdj 4×10-9 m2| GwU 1500K ZvcgvÎvq wK nv‡i kw³ wewKiY Ki‡e? [  5.7×10-8 Wm-2K-4 ] GLv‡b Avgiv Rvwb,

E  AT 4 t E  4 109  5.7 108 15004 t E   1.15425  10 -3 W (Ans.) t

†¶Îdj, A= 4×10-9 m2 ZvcgvÎv, T = 1500K   5.7×10-8 Wm-2K- 4 wewKwiZ kw³i nvi, E/t = ?

4| 400K ZvcgvÎvq GKwU e¯‘ 300K ZvcgvÎvq GKwU K…ò e¯‘ Øviv cwi‡ewóZ| e¯‘؇qi ga¨eZ©x ¯’vb evqy k~b¨| cÖ_g e¯‘wUi cÖwZ GKK †ÿÎdj †_‡K Zvc wewKi‡Yi nvi wbY©q Ki| GLv‡b Avgiv Rvwb, †¶Îdj, A= 1 m2 E  A(T 4  To4 ) ZvcgvÎv, T = 400K t ZvcgvÎv, To= 300K E   1 5.7 108 4004 - 3004    5.7×10-8 Wm-2K- 4 t wewKwiZ kw³i nvi, E/t = ?

E -8 10   5.7  10  1.75  10 W t



E  997.5W (Ans.) t Web: http://tanbircox.blogspot.com

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14| Zvc wewKiY (Heat Radiation)

5| 0.3m e¨vmv‡a©i GKwU avZe †MvjK 25W ÿgZv wewkó Zvc wewKiY K‡i| Gi ZvcgvÎv wbY©q Ki| [×10-8 Wm-2K-4 ] Avgiv Rvwb, GLv‡b E/t = AT4 †¶Îdj, A= 4r2  25 = 1.1304× 5.67×10-8×T4 = 4×3.14×0.32m2 25 = 1.1304 m2  T4  8 wewKwiZ kw³i nvi, 6.409  10 4 E/t = 25W  T  390054058.4   5.67×10-8 Wm-2K- 4  T  140.53K (Ans.) ZvcgvÎv T = ?

6| 2000K ZvcgvÎvi GKwU fv¯^i evwZi wdjv‡g›U kw³ wewKiY Ki‡Q| j¨v‡¤úi c„‡ôi †ÿÎdj 5×10-5m2 Ges Gi Av‡cwÿK wewKiY ÿgZv 0.85 n‡j wewKwiZ kw³i nvi wbY©q Ki| Avgiv Rvwb,

E  AeT 4 t E  5  10  5  0. 85  5. 67  10 8  2000 4 t



E  38.556W (Ans.) t

GLv‡b †¶Îdj, A= 5×10-5m2 ZvcgvÎv, T = 2000K   5.67×10-8 Wm-2K- 4 wewKwiZ kw³i nvi,E/t = ?

Av‡cw¶K wewKiY ¶gZv, e = 0.85

7| 5×10-5m2 †ÿÎd‡ji GKwU K…òKvqv 2000 K ZvcgvÎvq cÖwZ †m‡K‡Û KZUv kw³ wewKiY Ki‡e?

GLv‡b [  5.7×10-8 Wm-2K- 4] †¶Îdj, A= 5×10-5m2 Avgiv Rvwb, 4

E  AT t E  5  10  5  5 .7  10 8

 E  45.6 J (Ans.)

ZvcgvÎv, T = 2000K   5.7×10-8 Wm-2K- 4 4  2000  1 mgq t = 1 s wewKwiZ kw³E = ?

8| 127°C ZvcgvÎvq GKwU e¯‘ 27°C ZvcgvÎvq GKwU K…ò e¯‘ Øviv cwi‡ewóZ| e¯‘؇qi ga¨eZ©x ¯’vb evqy k~b¨| cÖ_g e¯‘wUi cÖwZ GKK †ÿÎdj †_‡K Zvc wewKi‡Yi nvi wbY©q Ki| [  5.66×10-8 Wm-2K- 4 ] GLv‡b Avgiv Rvwb,

E  A(T 4  To4 ) t E  15.66108 4004 - 3004  t E   990.5W (Ans.) t

2

†¶Îdj, A= 1 m ZvcgvÎv, T = 127ºC =400K ZvcgvÎv, To= 27ºC =300K   5.66×10-8 Wm-2K- 4 wewKwiZ kw³i nvi, E/t = ?

2

9| †Kvb MÖxb nvI‡mi g‡a¨ 3200°C ZvcgvÎvq 8321×10-10 m Zi½ ˆ`‡N©¨i m‡eŸv”P cwigvY kw³ weKxY© n‡j fx‡bi aªæeK KZ n‡e?

Avgiv Rvwb,

m T  b  b  m T  b  832110 10  3473 b  2.9  10 3 mK (Ans.)

GLv‡b, ZvcgvÎv,T3200ºC =(3200+273)K =3473K Zi½ ˆ`N¨© m8321×10-10 m fx‡bi aªæeK, b=?

10| ‡Kvb e¯‘i c„‡ói †ÿÎdj 0.1m2| e¯‘wU‡K 1000K ZvcgvÎvq DËß Ki‡j GwU N›Uvq KZ kw³ wewKiY Ki‡e? (e = 0.7) Avgiv Rvwb, GLv‡b

E  AeT 4 t  E  0.1  0.7  5.7  108  10004  3600

 E  14364000 J (Ans.)

†¶Îdj, A= 0.1m2 e=0.7 ZvcgvÎv, T = 1000K   5.7×10-8 Wm-2K- 4 mgq t = 3600 s wewKwiZ kw³ E = ?

11| `ywU K…ò e¯‘i wbM©Z Zvckw³i AbycvZ 16 t 1| wØZxq e¯‘i ZvcgvÎv 3000K n‡j cÖ_g e¯‘i ZvcgvÎv KZ?

E1 : E 2  16 : 1 , T2  3000K , T1  ?

Avgiv Rvwb, E1  AT14 I E 2  AT24  

E1 AT14  E 2 AT24

16 T4  1 4 1 3000 4

4

 2  T     1   1   3000   T1  6000K (Ans.) 12| ‡Kvb e¯‘ †_‡K m‡e©v”P wewKi‡bi m‡Ÿ©v”P wewKi‡Yi Zi½ ˆ`N©¨ 20×10-6m | e¯‘wUi ZvcgvÎv wbY©q Ki| fx‡bi aªæeK 2.898×10-3mK

Avgiv Rvwb,

GLv‡b fx‡bi aªæeK, b=2.898×10-3mK Zi½ ˆ`N©¨,  m = 20×10-6m ZvcgvÎv T=?

mT  b b T b T m

 m 

2.898 × 10 -3 T 20 × 10 -6  T  144.9K (Ans.)

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 ব঴ভােঃ বিয িান঩ যকান তযর ঩দ঱াথনক ক্রভাগত ঱ীতর কযনত থাকনর যম তা঩ভাত্রায় ঩দাথ঱বট কবঠ্ন অফিায় রূ঩ান্তবযত ঴নত থানক এফিং কবঠ্নীবফন য঱ল না ঴঑য়া ঩ম঱ন্ত ঐ তা঩ভাত্রায যকান ঩বযফত঱ন ঴য় না য঳ই তা঩ভাত্রানক ব঴ভাে ফনর।  অনকরাব঳ত ঩দানথ঱য বনবদ঱ি যকান গরনাে ফা ব঴ভাে যনই। যমভন- িবফ঱, ভাখন, কাঁি, ব঩ি, যভাভ। এই ঳কর ঩দাথ঱ জভফায ফা গরফায ঩ূ নফ঱ আঠ্ানরা অফিা ধাযণ কনয। 0

 ঩াবনয ব঴ভাে 0 C। বকন্তু ফায়ু ভুি ঑ বফশুদ্ধ ঩াবননক ঳াফধানতায ঳ানথ ধীনয ধীনয ঱ীতর কযনত থাকনর (-100C) ঩ম঱ন্ত ঩াবন তযর যথনক মায়।

 দ্রফনণয ব঴ভািংক বফশুদ্ধ দ্রাফনকয ব঴ভািংক অন঩ক্ষা কভ।  গরনােঃ স্বাবাবফক িান঩ কবঠ্ন ঩দাথ঱ যম তা঩ভাত্রায় গরনত শুরু কনয তানক গরনািংক ফনর। যমভন- ফযপ। 0

0

0

 কনয়কবট ঩দানথ঱য গরােঃ ঩াযদঃ 39 C, যভাভঃ 52 C – 56 C গরনানেয উ঩য িান঩য প্রবাফঃ ঩দানথ঱য গরনািংক আয়তন গরনকানর হ্রা঳ ঩ায়, য঳঳ফ ঩দানথ঱য গরনািংক িা঩ প্রনয়াগ কযনর কনভ মায়। যমভনঃ ফযপ, িারাই যরা঴া, এবিভবন, বফ঳ভাথ।  যম঳ফ ঩দানথ঱য আয়তন গরননয পনর ফৃ বদ্ধ ঩ায়, উ঴ানদয গরনাে িা঩ প্রনয়ানগ ফৃ বদ্ধ ঩ায়। যমভন- যভাভ, ঩যাযাবপন।  ফু টনািংকঃ যম বনবদ঱ি তা঩ভাত্রায় যকান তযর ফানষ্প ঩বযণত ঴নত শুরু কনয এফিং উ঴ায ঳ম্পৃ ি ফাষ্প িা঩ ফায়ু ভন্ডনরয িান঩য ঳ভান ঴য় তানক ফু টনািংক ফনর। * ঳াধাযণতঃ িা঩ ফাড়নর ফু টনািংক ফানড় এফিং িা঩ কভনর ফু টনাে কনভ।  ঳ু প্ততা঩ঃ যম তা঩ ফস্ত্ত্তয তা঩ভাত্রায ঩বযফত঱ন না ঘবটনয় অফিায ঩বযফত঱ন ঘটায় তানক ঳ু প্ত তা঩ ফনর। এনক L িাযা প্রকা঱ কযা ঴য়।  কনমকবট ঩দানথ঱য ঳ু প্ত তা঩ঃ

 বনবদ঱ি গরনািংক থাকা যকরাব঳নত ঩দানথ঱য একবট ধভ঱।  রত্রধ বফন্দুনত ঩াবন ফযপ ঑ জরীয় ফাষ্প একই তা঩ীয় ঳াভযফিায় থাকনত ঩ানয, 273.16 k যক রত্রধ বফন্দু ফনর। -2  ঩াবনয রত্রধ বফন্দুনত িা঩ 4.6 mm (Hg) ফা 610.616 Nm iv) উধক্ষ঱ ঩াতন িা঩ (Sub Imitation Pressure)

তা঩ভাত্রায অন঩ক্ষক।

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Mjbv¼ (Melting Point): ‡Kvb KwVb c`v‡_© Zvc cÖ‡qvM Ki‡Z _vK‡j cÖgvY Pv‡c †h wbw`©ó ZvcgvÎvq KwVb c`v_©wU Mj‡Z ïiæ K‡i Ges m¤úyY© Mjb †kl bv nIqv ch©šÍ H ZvcgvÎv w¯’i Zv‡K H KwVb c`v‡_©i Mjbv¼ e‡j| Mjbv‡¼i Zvrch© (Significance of Melting Point) : ei‡di Mjbv¼ 0ºC ej‡Z GB eywS †h, cÖgvY Pv‡c 0ºC ZvcgvÎvq eid M‡j cvwb‡Z cwibZ nIqv ïiæ K‡i Ges m¤ú~Y© eid bv Mjv ch©šÍ Gi ZvcgvÎv ev‡o bv| wngv¼ (Freezing Point) : cÖgvY Pv‡c †h wbw`©ó ZvcgvÎvq Zij c`v_© Rg‡Z ïiæ K‡i Ges m¤úyY© Zij bv Rgv ch©šÍ H ZvcgvÎv w¯’i Zv‡K cÖgvY Pv‡c H Zij c`v‡_©i wngv¼ e‡j| wngv‡¼i Zvrch© (Significance of Freezing Point) : cvwbi wngv¼ 0ºC ej‡Z GB eywS †h, cÖgvY Pv‡c 0ºC ZvcgvÎvq cvwb R‡g ei‡d cwibZ nIqv ïiæ K‡i Ges m¤ú~Y© cvwb RgvU †e‡au ei‡d cwibZ bv nIqv ch©šÍ Gi ZvcgvÎv K‡g bv| ùzUbv¼ (Boiling Point) : Zij c`v‡_© Zvc cÖ‡qvM Ki‡Z _vK‡j cÖgvY Pv‡c †h wbw`©ó ZvcgvÎvq Zij c`v_© ev‡®ú cwiYZ n‡Z ïiæ K‡i Ges m¤úyY© Zij ev‡®ú cwiYZ bv nIqv ch©šÍ H ZvcgvÎv w¯’i Zv‡K cÖgvY Pv‡c H Zij c`v‡_©i ùzUbv¼ e‡j| ùzbv‡¼i Zvrch© (Significance of Boiling Point) : cvwbi ùzUbv¼ 100ºC ej‡Z GB eywS †h, cÖgvY Pv‡c 100ºC ZvcgvÎvq cvwb Rjxq ev‡®ú cwibZ nIqv ïiæ K‡i Ges m¤ú~Y© cvwb ch©šÍ ev®úxf~Z bv nIqv ch©šÍ Gi ZvcgvÎv ev‡o bv| myßZvc ev Av‡cwÿK myßZvc (Latent Heat or Specific Latent Heat ) t ZvcgvÎvi cwieZ©b bv NwU‡q GKK f‡ii †Kvb e¯‘i GK Ae¯’v †_‡K Ab¨ Ae¯’vq iƒcvšÍwiZ n‡Z †h Zvc MÖnb ev eR©b K‡i Zv‡K H c`v‡_©i Ae¯’v cwieZ©‡bi myßZvc ev Av‡cwÿK myßZvc e‡j| m f‡ii †Kvb e¯‘i Ae¯’v cwieZ©‡bi Rb¨ M„nxZ ev ewR©Z Zvc Q n‡j, Gi Dcv`v‡bi Av‡cwÿK myßZvc Q , Gi GKK Jkg-1 | Av‡cwÿK myßZvc PvicÖKv‡ii n‡q _v‡K| h_v t m 1. Mj‡bi Av‡cwÿK myßZvc (Specific Latent Heat of Fusion): 2. KwVbxfe‡bi Av‡cwÿK myßZvc (Specific Latent Heat of Solidification): 3. ev®úxfe‡bi Av‡cwÿK myßZvc (Specific Latent Heat of Vaporization): 4. Nbxfe‡bi Av‡cwÿK myßZvc (Specific Latent Heat of condensation): l

Mj‡bi Av‡cwÿK myßZvc (Specific Latent Heat of Fusion): Mjbv‡¼ ZvcgvÎv w¯’i †i‡L GKK f‡ii †Kvb KwVb c`v_©‡K KwVb Ae¯’v †_‡K Zij Ae¯’vq iƒcvšÍwiZ Ki‡Z †h Zv‡ci cÖ‡qvRb nq Zv‡K H c`v‡_©i Mj‡bi Av‡cwÿK myßZvc e‡j| G‡K lf Øviv cÖKvk Kiv nq| ZvcgvÎvi cwieZ©b bv NwU‡q m f‡ii †Kvb KwVb c`v_©‡K Zi‡j iƒcvšÍwiZ Q n‡e| m Mj‡bi Av‡cwÿK myßZv‡ci Zvrch© (Significance of Specific Latent Heat of Fusion): eid Mj‡bi Av‡cwÿK myßZvc 3,36,000 J kg-1| eid Mj‡bi Av‡cwÿK myßZvc 3,36,000 J kg-1 ej‡Z GB eywS †h, 0ºC ZvcgvÎvi 1kg eid‡K 0ºC ZvcgvÎvi 1kg cvwb‡Z cwiYZ Ki‡Z 3,36,000 J Zvc cÖ‡qvM Ki‡Z nq|

Ki‡ZQ Zv‡ci cÖ‡qvRb n‡j Mj‡bi Av‡cwÿK myßZvc, l f 

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2 15| Ae¯’vi cwieZ©b (Change Of State) ev®úxfe‡bi Av‡cwÿK myßZvc: ùzUbv‡¼ ZvcgvÎv w¯’i †i‡L GKK f‡ii †Kvb Zij c`v_©‡K Zij Ae¯’v †_‡K evqexq Ae¯’vq iƒcvšÍwiZ Ki‡Z †h Zv‡ci cÖ‡qvRb nq Zv‡K H c`v‡_©i ev®úxfe‡bi Av‡cwÿK myßZvc e‡j| G‡K lv Øviv cÖKvk Kiv nq| ZvcgvÎvi cwieZ©b bv NwU‡q m f‡ii †Kvb Zij c`v_©‡K M¨vmxq Ae¯’vq iƒcvšÍwiZ Q n‡e| m ev®úxfe‡bi Av‡cwÿK myßZv‡ci Zvrch© (Significance of Specific Latent Heat o vaporization): cvwbi ev®úxfe‡bi Av‡cwÿK myßZvc 22,68,000 J kg-1| cvwbi ev®úxfe‡bi Av‡cwÿK myßZvc 22,68,000J kg-1 ej‡Z GB eywS †h, 100ºC ZvcgvÎvi 1kg cvwb‡K 100ºC ZvcgvÎvi 1kg Rjxqev‡®ú cwiYZ Ki‡Z 22,68,000 J Zvc cÖ‡qvM Ki‡Z nq|

Ki‡ZQ Zv‡ci cÖ‡qvRb n‡j Mj‡bi Av‡cwÿK myßZvc, lv 

eid Mj‡bi Av‡cwÿK myßZvc wbY©q (Determination of Specific Latent Heat of Fusion of Ice): eb©Yv t GKwU ﮋ I cwi®‹vi K¨vjwiwgUvi †bIqv nq| Gici Gi fi cwigvc Kiv nq| K¨vjwiwgUv‡ii `yB-Z„Zxqvsk mvgvb¨ Mig cvwb Øviv c~Y© K‡i Avevi fi gvcv nq| GB `yB f‡ii cv_©K¨ ‡_‡K Mig cvwbi fi cvIqv hvq| GKwU my‡e`x _v‡g©vwgUv‡ii mvnv‡h¨Mig cvwbi ZvcgvÎv wbY©q Kiv nq| Gici K‡qK UzKiv eid‡K cwi®‹vi cvwb w`‡q ay‡q †Pvl KvMR w`‡q ïwK‡q wb‡q cvwb‡Z ZvovZvwo †Q‡o †`qv nq| Gici bvobx Øviv Av‡¯Í Av‡¯Í bvovb nq| GB Ae¯’vq eid Mj‡Z _v‡K Ges cvwbi ZvcgvÎv µgk Kg‡Z _v‡K| m¤úyY© eid M‡j hIqvi ci wgkª‡Yi me©wb¤œ ZvcgvÎv my‡e`x _v‡g©vwgUv‡ii mvnv‡h¨ gvcv nq| Gici cvwbi ZvcgvÎv Kÿ ZvcgvÎvq †cuŠQ‡j cybivq K¨vjwiwgUvi¯’ cvwbmn K¨vjwiwgUv‡ii fi gvcv nq| Z…Zxq fi †_‡K wØZxq fi we‡qvM K‡i ei‡di fi wbY©q Kiv nq| wnmve I MYbvt g‡bKwi, K¨vjwiwgUv‡ii fi = m1 kg Mig cvwbi fi = m2 kg ei‡di fi = m kg K¨vjwiwgUvi I Migcvwbi ZvcgvÎv = ºC wgkª‡Yi me©wb¤œ ZvcgvÎv =  ºC K¨vjwiwgUv‡ii Dcv`v‡bi Av‡cwÿK Zvc = S1 Jkg-1K-1 cvwbi Av‡cwÿK Zvc = S2 Jkg-1K-1 eid Mj‡bi Av‡cwÿK myßZvc = lf Jkg-1 GLv‡b K¨vjwiwgUvi I Migcvwb Zvc eR©b K‡i Avi eid Zvc MÖnb K‡i| K¨vjwiwgUv‡ii ZvcgvÎv ºC †_‡K ºC G †b‡g Avm‡Z ewR©Z Zvc, H1=m1S1() J Ges Mig cvwbi ZvcgvÎv ºC †_‡K ºC G †b‡g Avm‡Z ewR©Z Zvc, H2=m2S2() J eid KZ…K Zvc MÖnY `yB ch©v‡q m¤úbœ nq| cÖ_gZ, ºC ZvcgvÎvi eid ºC ZvcgvÎvi cvwb‡Z cwiYZ n‡Z I ºC ZvcgvÎvi eid Mjv cvwb ºC ZvcgvÎvi cvwb‡Z cwiYZ n‡Z| ºC ZvcgvÎvi m kg eid ºC ZvcgvÎvi cvwb‡Z cwiYZ n‡Z eid KZ…K M„nxZ Zvc, H3=mlf J ºC ZvcgvÎvi m kg eid Mjv cvwb ºC ZvcgvÎvi cvwb‡Z cwiYZ n‡Z eid Mjv cvwb KZ…K M„nxZ Zvc, H4=mS2() J = mS2 J

Ab¨‡Kvb Dcv‡q Zv‡ci Av`vb cÖ`vb bv n‡j, ‡gvU M„nxZ Zvc= †gvU ewR©Z Zvc H3 + H4 =H1 + H2 mlf + mS2 m1S1() + m2S2()   m(lf + S2)= (m1S1+ m2S2) () (m S  m2 S 2 ) (1 -  )  (m S  m2 S 2 ) (1 -  )   (l f  S 2 )  1 1 l f   1 1  S 2  Jkg 1 m m  

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15| Ae¯’vi cwieZ©b

(Change Of State)

3

mZK©Zv (Caution) : 1| eid cvwb‡Z †djvi mgq †hb nvZ w`‡q bv aiv nq, Zv bv n‡j nvZ †_‡K Zvc wb‡q wKQy eid M‡j †h‡Z cv‡i| 2| eid Mj‡bi mgq K¨jwiwgUv‡i Rvwj †`Iqv bvobx e¨envi Kiv DwPr KviY Gi mvnv‡h¨ eid cvwbi bx‡P ivL‡Z nq| 3| K¨jwiwgUvi‡K AcwievnK c`v_© Øviv †X‡K ivL‡Z n‡e hv‡Z cwienb RwbZ Kvi‡Y †Kvb Zvc ÿq bv nq| 4| K¨vjwiwgUv‡i my‡e`x _v‡g©vwgUvi e¨venvi Ki‡Z n‡e hv‡Z ZvcgvÎvi cwieZ©b mv‡_ mv‡_ aiv c‡o| cvwbi ev®úxfe‡bii Av‡cwÿK myßZvc wbY©q (Determination of Specific Latent Heat of Vaporization of Water) t eb©Yv t GKwU eqjvi B -†Z cvwb Mig Kiv nq| eqjv‡ii gyL KK© w`‡q AvUKvb _v‡K Ges euvKv bj A K‡K©i g‡a¨ cÖ‡ek Kiv‡bv _v‡K| A Gi Aci cÖvšÍ GKwU ev®ú duv‡`i g‡a¨ cÖ‡ek Kiv‡bv _v‡K| ev®ú duv`wU GKwU Kv‡Pi bj we‡kl| Gi Dc‡i GKwU Ges bx‡P `yBwU wQ`ª Iqvjv KK© jvMvb _v‡K| Dc‡ii KK© w`‡q A bjwU ev®ú duv‡` cÖ‡ek Kiv‡bv nq| wb‡Pi KK© w`‡q bj C I D cÖ‡ek Kiv‡bv nq| C bj w`‡q ev®ú duv‡` Rgv nIqv cvwb †ei Kiv nq| D bjwUi Aci cÖvšÍ K¨vjwiwgUvi K †Z cÖ‡ek Kiv‡bv _v‡K| K¨vjwiwgUvi I eqjv‡ii gvSLv‡b GKwU c`©v P ¯’vcb Kiv nq †hb K¨vjwiwgUviwU eqjvi DËß Kivi mgq Zvc MÖnb Ki‡Z bv cv‡i| bvowb S mn K¨jwiwgUvi K wb‡q Gi fi †ei Kiv nq| Gi `yB-Z„Zxqvsk cvwbc~Y© K‡i Gi fi †bIqv nq| GB `yB f‡ii cv_©K¨ †_‡K cvwbi fi †ei Kiv nq| GKwU _v‡g©vwgUvi T Gi mvnv‡h¨ Gi ZvcgvÎv wbY©q Kiv nq| Gi ci ev®ú A b‡ji ga¨w`‡q ev®ú dv‡` cÖ‡ek K‡i Ges D b‡ji g‡a¨ w`‡q AvMZ ev®ú K¨jwiwgUv‡ii cvwbi ms¯ú‡k© G‡m Nbxf~Z nq Ges Zvc eR©b K‡i| GB Zvc MÖnY K‡i K¨jwiwgUvi I cvwbi ZvcgvÎv e„w× cvq| Gi wKQyÿb ci D bj mwi‡q cvwbi m‡eŸ©v”P ZvcgvÎv †`‡L †bqv nq| Gici cvwb mn K¨vjwiwgUv‡ii fi †bIqv nq| Z…Zxq fi †_‡K wØZxq fi ev` w`‡j Nbxf~Z ev‡®úi fi cvIqv hvq| wnmve I MYbv : g‡b Kwi, K¨jwiwgUv‡ii fi = m1 Kg K¨jwiwgUv‡i iwÿZ cvwbi fi = m2 Kg Nbxf~Z ev‡®úi fi = m Kg K¨jwiwgUvi I cvwbi cÖv_wgK ZvcgvÎv = ºC wgkª‡bi m‡eŸ©v”P ZvcgvÎv = ºC K¨jwiwgUv‡ii Dcv`v‡bi Av‡cwÿK Zvc = S1 J kg-1K-1 cvwbi Av‡cwÿK Zvc = S2 J kg-1K-1 cvwbi ev®úxfe‡bi Av‡cwÿK myßZvc = lv J kg-1 m f‡ii ev®ú `yB ch©v‡q Zvc eR©b K‡i| cÖ_gZ, 100ºC ZvcgvÎvi ev®ú 100ºC ZvcgvÎvi cvwb‡Z cwiYZ n‡Z Zvc eR©b K‡i| wØZxqZ, 100ºC ZvcgvÎvi cvwb ºC ZvcgvÎvi cvwb‡Z cwiYZ n‡Z Zvc eR©b K‡i| 100ºC ZvcgvÎvi ev®ú 100ºC ZvcgvÎvi cvwb‡Z cwiYZ n‡Z ewR©Z Zvc, H1 = mlv J 100ºC ZvcgvÎvi cvwb ºC ZvcgvÎvi cvwb‡Z cwiYZ n‡Z ewR©Z Zvc, H2 = mS2( ) J K¨jwiwgUvi KZ…K M„nxZ Zvc, H3 = m1S1(1) J cvwb KZ…K M„nxZ Zvc, H4 = m2S2(1) J Ab¨‡Kvb Dcv‡q Zv‡ci Av`vb cÖ`vb bv n‡j,

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15| Ae¯’vi cwieZ©b

(Change Of State)

4

M„nxZ Zvc = ewR©Z Zvc H1+H2=H3+H4  mlv+ mS2( )= m1S1(1)+ m2S2(1)  m{lv+ S2( )}= (m1S1+ m2S2)(1) (m S + m2 S 2 )( - 1 )  lv + S 2 (100 - θ ) = 1 1 m (m S + m2 S 2 )(θ - θ1 )   lv =  1 1 - S 2 (100 - θ) Jkg 1 m  

mZK©Zv (Caution) t 1| ev®ú Lye ax‡i ax‡i ˆZix Ki‡Z n‡e Zv bv n‡j ev®ú duv‡`i AvMg bj w`‡q ev‡®úi mv‡_ wKQy cvwb ev®ú duv‡` cÖ‡ek Ki‡Z cv‡i | 2| ev®ú duv‡`i AvMvg bjwU‡K fvj fv‡e Zzjv ev Ab¨‡Kvb AcwievnK c`v_© Øviv †X‡K w`‡Z n‡e hv‡Z c‡_ †Kvb ev®ú Nbxf~Z bv n‡Z cv‡i | 3| K¨jwiwgUvi‡K AcwievnK c`v_© Øviv †X‡K ivL‡Z n‡e hv‡Z cwienb RwbZ Kvi‡Y †Kvb Zvc ÿq bv nq| 4| K¨vjwiwgUv‡i my‡e`x _v‡g©vwgUvi e¨venvi Ki‡Z n‡e hv‡Z ZvcgvÎvi cwieZ©b mv‡_ mv‡_ aiv c‡o| cvwbi ˆÎawe›`y (Triple tip of Water) t ‡h wbw`©ó ZvcgvÎvq I Pv‡c cvwb KwVb (eid), Zij (cvwb) I evqexq (Rjxhev®ú) GB wZb Ae¯’vq _vK‡Z cv‡i Zv‡K cvwbi ˆÎawe›`y e‡j| GB we›`y‡Z wbw`©ó ZvcgvÎv 0ºC ev, 273.16K Ges GB we›`y‡Z wbw`©ó Pvc 4.58 mmHg| msKU ZvcgvÎv (Critical temperature) t m‡eŸ©v”P ‡h ZvcgvÎvq _vK‡j †Kvb M¨vm‡K ïaygvÎ Pvc cÖ‡qvM K‡i Zi‡j cwiYZ Kiv hvq †mB ZvcgvÎv‡K msKU ZvcgvÎv e‡j| cvwbi msKU ZvcgvÎv 647K| `kv (Phase) : †Kvb wm‡÷‡gi GKwU wbw`©ó Ask, hvi Dcv`vbMZ MVb meÎ Awfbœ Ges †fŠZ wePv‡i Ab¨vb¨ Ask †_‡K c„_K‡hvM¨ A_©vr †Kvb hvwš¿K cÖwµqvq Ab¨vb¨ Ask n‡Z mn‡R c„_K Kiv hvq, Zv‡K `kv e‡j| `kv wPÎ (Phase Diagram) t ‡Kvb c`v‡_©i m¤ú„³ ev®úPvc ebvg ZvcgvÎv wb‡q †jLwPÎ A¼b Ki‡j †h †jLwPÎ cvIqv hvq Zv‡Z `kv wPÎ e‡j| cvwbi `kv wPÎ (Phase Diagram of water) t GKwU QK KvM‡R X -A‡ÿ ZvcgvÎv Ges Y-A‡ÿ m¤ú„³ ev®úPvc ¯’vcb K‡i `kvwPÎ A¼b Kiv nj| ‡h ZvcgvÎvq cvwb Ges Rjxq ev®ú mn-Ae¯’v‡b _v‡K, †m ZvcgvÎv ebvg m¤ú„³ Rjxq ev‡®úi Pvc wb‡q ‡jLwPÎ A¼b Ki‡j ‡h †jLwPÎ cvIqv hvq Zv‡K OP †iLv Øviv m~wPZ Kwi| G †iLvi bvg ev®ú †iLv| †h ZvcgvÎvq eid Ges Rjxq ev®ú mn-Ae¯’v‡b _v‡K, †m ZvcgvÎv ebvg m¤ú„³ Rjxq ev‡®úi Pvc wb‡q ‡jLwPÎ A¼b Ki‡j ‡h †jLwPÎ cvIqv hv‡e, Zv‡K OQ †iLv Øviv m~wPZ Kwi| G †iLvi bvg †nvi‡d«v÷ †iLv| Avevi †h ZvcgvÎvq eid Ges Zij cvwb mn-Ae¯’v‡b _v‡K †m ZvcgvÎv ebvg m¤ú„³ Rjxq ev‡®úi Pvc wb‡q ‡jLwPÎ A¼b Ki‡j ‡h †jLwPÎ cvIqv hv‡e Zv‡K OR †iLv Øviv m~wPZ Kwi| GB †iLvi bvg eid †iLv|

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5 15| Ae¯’vi cwieZ©b (Change Of State) Dc‡iv³ wZbwU †iLv GKwU mvaviY we›`y O- †Z †Q` K‡i‡Q| GB mvaviY we›`y‡K ˆÎa we›`y e‡j| mvaviY fv‡e ejv hvq †h, †h ZvcgvÎvq †Kvb c`v‡_©i KwVb, Zij Ges ev®ú GKwU wbw`©ó Pv‡c ZvcMZ mnAe¯’v‡b _v‡K Zv‡K D³ c`v‡_©i ˆÎa we›`y e‡j|

msKU ZvcgvÎvt m‡e©v”P †h ZvcgvÎvq _vK‡j GKwU M¨vm‡K ïaygvÎ Pvc cÖ‡qv‡M Zi‡j cwiYZ Kiv hvq Zv‡K H M¨v‡mi msKU ZvcgvÎv e‡j| ev®ú I M¨vm t ev®út evqexq Ae¯’vq †Kvb c`v‡_©i ZvcgvÎv hw` msKU ZvcgvÎvi †P‡q Kg nq Z‡e Zv‡K ev®ú e‡j| †hgbt Rjxq ev®ú, †eªvwgb ev®ú, B_vbj ev®ú BZ¨vw`| M¨vmt evqexq Ae¯’vq †Kvb c`v‡_©i ZvcgvÎv hw` msKU ZvcgvÎvi †P‡q ‡ekx nq Z‡e Zv‡K M¨vm e‡j| †hgbt Aw·‡Rb M¨vm, nvB‡Wªv‡Rb M¨vm BZ¨vw`| ev®úvqb I ùzUb I G‡`i g‡a¨ cv_©K¨ t ev®úvqb t †h cÖwµqvq †Kvb wbw`©ó Pv‡c †Kvb Zij c`v_© †h †Kvb ZvcgvÎvq Gi Dcwi gy³ Zj n‡Z ax‡i ax‡i ev‡®ú cwiYZ nq Zv‡K ev®úvqb e‡j| ùzUb : †h cÖwµqvq †Kvb wbw`©ó Pv‡c †Kvb Zij c`v_© Gi ùzUzbv‡¼ ZvcgvÎv w¯’i ‡i‡L Zi‡ji mgMÖ Ask n‡Z `ªæZ ev‡®ú cwiYZ nq Zv‡K ùzUb e‡j| cv_©K¨ ev®úvqb ùzUb 1| †Kej gvÎ gy³ Zj n‡Z ax‡i ax‡i ev‡®ú cwiYZ nq| 1| Zi‡ji mgMÖ Ask n‡Z `ªæZ ev‡®ú cwiYZ nq| 2| †h †Kvb ZvcgvÎvq N‡U| 2| wbw`©ó Pvc I wbw`©ó ZvcgvÎvq N‡U| 3| ev®ú Pvc, evwn¨K Pv‡ci †P‡q Kg| 3| ev®ú Pvc, evwn¨K Pv‡ci mgvb 4| ey`ey` m„wó nq bv| 4| ey`ey` m„wó nq| 5| ev®úvq‡bi nvi Zi‡ji gy³ Z‡ji †ÿÎd‡ji Dci 5| ùzU‡bi nvi, Zvc mieiv‡ni nv‡ii Dci wbf©i K‡i| wbf©i K‡i| 6| ev®úvq‡bi mgq Zi‡ji ZvcgvÎv n«vm cvq| 6| ùzU‡bi mgq Zi‡ji ZvcgvÎv w¯’i _v‡K|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

15| Ae¯’vi cwieZ©b (Change Of State) 1| 0°C ZvcgvÎvi 0.02kg eid‡K 100°C ZvcgvÎvi ev‡®ú cwibZ Ki‡Z KZ Zvc jvM‡e? [eid Mj‡bi myßZvc =336000Jkg-1 , cvwbi Av‡cwÿK Zvc =4200Jkg-1K-1 Ges ev‡®úi myßZvc = 2260000 Jkg-1] 0°C ZvcgvÎvi eid‡K 0°C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc jv‡M Q1 = fi × Mj‡bi myßZvc = 0.02×336000 J = 6720 J 0°C ZvcgvÎvi cvwb‡K 100°C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc jv‡M Q2 = fi × Avt Zvc × ZvcgvÎvi cv_©K¨ = 0.02 × 4200 × (1000) J = 8400 J 100°C ZvcgvÎvi cvwb‡K 100°C ZvcgvÎvi ev‡®ú cwibZ Ki‡Z Zvc jv‡M Q3 = fi × ev®úxfe‡bi myßZvc = 0.02 × 2260000 J = 45200 J ‡gvU cª‡qvRbxq Zvc Q = Q1 + Q2 + Q3 = (6720+8400+45200) J = 60320 J (Ans.) 2| 263K ZvcgvÎvi 0.02kg eid‡K 373K ZvcgvÎvi ev‡®ú cwibZ Ki‡Z KZ Zvc jvM‡e? [ei‡di Av‡cwÿK Zvc =2100Jkg-1K-1 eid Mj‡bi myßZvc = 336000Jkg-1 , cvwbi Av‡cwÿK Zvc =4200Jkg-1K-1 Ges ev®úxfe‡bi Av‡cwÿK myßZvc = 2260000 Jkg-1] 263K = (263273)°C = 10°C 373K = (373273)°C = °C 10°C ZvcgvÎvi eid‡K 0°C ZvcgvÎvi ei‡d cwibZ Ki‡Z Zvc jv‡M Q1 = fi × Avt Zvc × ZvcgvÎvi cv_©K¨ = 0.02 × 2100 × {0 (10)} J = 420J 0°C ZvcgvÎvi eid‡K 0°C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc jv‡M Q2 = fi × Mj‡bi myßZvc = 0.02×336000 J = 6720 J 0°C ZvcgvÎvi cvwb‡K 100°C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc jv‡M Q3 = fi × Avt Zvc × ZvcgvÎvi cv_©K¨ = 0.02×4200×(100  0) J = 8400 J 100°C ZvcgvÎvi cvwb‡K 100°C ZvcgvÎvi ev‡®ú cwibZ Ki‡Z Zvc jv‡M, Q4 = fi × ev®úxfe‡bi myßZvc = 0.02 × 2260000 J = 45200 J ‡gvU cª‡qvRbxq Zvc, Q = Q1 + Q2 + Q3 + Q4 = (420+6720+8400+45200) J = 60740 J (Ans.)

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3| –5°C ZvcgvÎvi 0.005kg ei‡di mv‡_ 90°C ZvcgvÎvq 0.005 kg cvwb wgkv‡j wgkª‡bi PzovšÍ ZvcgvÎv KZ n‡e? [ei‡di Av‡cwÿK Zvc =2.1 × 103Jkg-1K-1 eid Mj‡bi myßZvc =3.36 × 105 Jkg-1 , cvwbi Av‡cwÿK Zvc = 4.2 × 103 Jkg-1K-1 ] g‡b Kwi, wgkª‡bi PyovšÍ ZvcgvÎv n‡e °C  5°C ZvcgvÎvi eid‡K 0°C ZvcgvÎvi ei‡d cwibZ Ki‡Z Zvc jv‡M Q1 = fi×Avt Zvc×ZvcgvÎvi cv_©K¨ = 0.005×2.1×103×{0(5)} J = 52.5 J 0°C ZvcgvÎvi eid‡K °C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc jv‡M Q2 = fi × Mj‡bi myßZvc = 0.005×3.36 ×105 J = 1680 J 0°C ZvcgvÎvi cvwb‡K °C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc jv‡M Q3 = fi×Avt Zvc×ZvcgvÎvi cv_©K¨ = 0.005×4.2×103×(-0) J = 21 J 90°C ZvcgvÎvi cvwb‡K °C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc eR©b K‡i Q4 = fi×Avt Zvc×ZvcgvÎvi cv_©K¨ = 0.005×4.2×103×(90- J = (1890-21 J cÖkœg‡Z, Q1 + Q2 + Q3 = Q4  52.5 + 1680 + 211890-21  2118901680-52.5  42157.5 θ 

157.5 C  3.75C (Ans.) 42

4| 100°C ZvcgvÎvi 500g Rjxqev®ú Nbxf~Z n‡q 30°C ZvcgvÎvi cvwb‡Z cvwb‡Z cwibZ nIqvi Rb¨ KZ Zvc eR©b Ki‡Z n‡e? [cvwbi Av‡cwÿK Zvc = 4200 Jkg-1K-1 ev®úxfe‡bi Av‡cwÿK myßZvc = 2.26 ×106 Jkg-1] 500g = 0.5Kg 100°C ZvcgvÎvi Rjxqev®ú‡K 100°C cvwb‡Z cwibZ Ki‡Z Zvc eR©b K‡i Q1 = fi × ev®úxfe‡bi myßZvc = 0.5×2.26×106 J = 1130000 J 100°C ZvcgvÎvi cvwb‡K 30°C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc eR©b K‡i Q2 = fi × Avt Zvc × ZvcgvÎvi cv_©K¨ = 0.5 ×4200×(10030) J = 147000 J ‡gvU cÖ‡qvRbxq Zvc = (1130000 + 147000) J = 1277000 J (Ans.)

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cÖ_g c‡Îi As‡Ki mgvavb

5| 0°C ZvcgvÎvi 2.1 Kg eid 40°C ZvcgvÎvi 5.9 Kg cvwbi mv‡_ wgwkªZ Kiv nj| wgkª‡bi P~ovšÍ ZvcgvÎv KZ n‡e? g‡bKwi wgkª‡bi ZvcgvÎv 

0°C ZvcgvÎvi eid‡K 0°C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc jv‡M Q1 = fi × Mj‡bi myßZvc = 2.1×336000 J = 705600 J 0°CZvcgvÎvi cvwb‡K °C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc jv‡M Q2 = fi × Avt Zvc × ZvcgvÎvi cv_©K¨ =2.1× 4200 × (0) J = 8820 J 40°C ZvcgvÎvi cvwb‡K °C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc eR©b K‡i Q3 = fi × Avt Zvc × ZvcgvÎvi cv_©K¨ = 5.9×4200×(40) J =(991200 24780 J cÖkœg‡Z, M„nxZ Zvc = ewR©Z Zvc Q1 + Q2 = Q3 705600 + 8820991200 24780  882024780991200 705600   33600285600

 

285600  8.5C(Ans) 33600

6| mgcwigvb Mig cvwb Ges 0°C ZvcgvÎvi eid GKmv‡_ wgkvb nj| m¤úyY© eid M‡j cvwb nIqvi ci wgkª‡bi DòZv 0°C nj| Mig cvwbi DòZv KZ wQj? (eid Mj‡bi Av‡cwÿK myßZvc =3.36×105 Jkg-1Ges cvwbi Av‡cwÿK Zvc = 4.2 × 103 Jkg1 -1 K ) g‡b Kwi, ei‡di I Mig cvwbi fi wQj = m Kg I Mig cvwbi DòZv KZ wQj°C 0°C ZvcgvÎvi eid‡K °C ZvcgvÎvi cvwb‡Z cwiYZ Ki‡Z Zvc jv‡M Q1 = fi × Mj‡bi myßZvc = m×3.36 ×105 J = 3.36 ×105 m J °C ZvcgvÎvi Mig cvwb‡K °C ZvcgvÎvi cvwb‡Z cwiYZ Ki‡Z Zvc eR©b K‡i Q2 = fi×Avt Zvc×ZvcgvÎvi cv_©K¨ = m ×4.2×103×( 0) J = m ×4.2×103× J cÖkœg‡Z, M„nxZ Zvc = ewR©Z Zvc 3.36 ×105 m J= m ×4.2×103× J

 

3.36  105 C 4.2  103

°C (Ans.)

7| 50°C ZvcgvÎvi 0.03 kg cvwb‡Z 0°C ZvcgvÎvi 0.02kg eid wgkvb n‡j wgkª‡bi djvdj wK n‡e? [cvwbi Av: Zv: 4200 Jkg-1K-1 Ges eid Mj‡bi myßZvc =3.36×105 Jkg-1] 50 °C ZvcgvÎvi cvwb‡K 0 °C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z ewR©Z Zvc, Q1 = fi×Avt Zvc×ZvcgvÎvi cv_©K¨ = 0.03×4200×(50-0) J = 6300J 0°C ZvcgvÎvi eid‡K 0°C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z

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Zvc jv‡M Q2 = fi × Mj‡bi myßZvc = 0.02×3.36×105 J = 6720 J cvwb KZ…K ewR©Z Zvc, eid Mj‡Z cÖ‡qvRbxq Zv‡ci †P‡q Kg nIqvq mg¯Í eid Mj‡e bv| g‡b Kwi m kg eid Mj‡e| cÖkœg‡Z, m×3.36×105 = 6300

m 

6300 kg  0.01875 kg 3.36  105

DËi wgkª‡bi P~ovšÍ ZvcgvÎv n‡e 0 °C Ges 0.01875 kg eid Mj‡e, evKx (0.020.01875) kg = 0.00125kg eid (0.03+0.01875)kg=0.04875kg cvwbi Dci fvm‡e|

8| 50°C ZvcgvÎvi 0.03 kg cvwb‡Z -10°C ZvcgvÎvi 0.02kg eid wgkvb n‡j wgkª‡bi djvdj wK n‡e? [cvwbi Av: Zv: 4200 Jkg-1K-1 Ges eid Mj‡bi myßZvc = 3.36×105 Jkg-1] 50 °C ZvcgvÎvi cvwb‡K 0 °C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z ewR©Z Zvc, Q1 = fi×Avt Zvc×ZvcgvÎvi cv_©K¨ = 0.03×4200×(50 0) J = 6300J -10°C ZvcgvÎvi eid‡K 0°C ZvcgvÎvi ei‡d cwibZ Ki‡Z Zvc jv‡M Q2 = fi×Avt Zvc×ZvcgvÎvi cv_©K¨ = 0.02×2100×{0(10)} J = 420J 0°C ZvcgvÎvi eid‡K 0°C ZvcgvÎvi cvwb‡Z cwibZ Ki‡Z Zvc jv‡M Q3 = fi × Mj‡bi myßZvc = 0.02×3.36×105 J = 6720 J cvwb KZ…K ewR©Z Zvc 6300J, eid Mj‡Z cÖ‡qvRbxq Zvc (420+6720) J=7140J Gi †P‡q Kg nIqvq mg¯Í eid Mj‡e bv| g‡b Kwi m kg eid Mj‡e| cÖkœg‡Z, 420 + m×3.36×105 = 6300

m 

6300 - 420 kg  0.0175 kg 3.36  10 5

DËi: wgkª‡bi P~ovšÍ ZvcgvÎv n‡e 0 °C Ges 0.0175 kg eid Mj‡e, evKx (0.02-0.0175) kg = 0.0025kg eid (0.03+0.0175)kg = 0.0475kg cvwbi Dci fvm‡e| 9| A_ev, mgvb f‡ii 0°C ZvcgvÎvi eid I dzUšÍ cvwb GK‡Î wgwkÖZ Kiv nj| G‡Z m¤ú~Y© eid cvwb‡Z cwiYZ nj Ges wgkÖ‡bi ZvcgvÎv 10°C nj| eid Mj‡bi Av‡cwÿK myßZvc wbY©q Ki| [cvwbi Av‡cwÿK Zvc 4200Jkg-1K-1] g‡b Kwi, eid I cvwbi fi m, I eid Mj‡bi Av‡cwÿK myßZvc=lf eid KZ…K M„nxZ Zvc= mlf+m×4200(10-0) cvwb KZ…K ewR©Z Zvc= m×4200×(100-10) cÖkœg‡Z, mlf+m×4200(10-0) = m×4200×(100-10)  lf+4200×10 = 4200×90  lf= 378000-42000=336000Jkg-1 (Ans.)

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cÖZ¨vMvgx cÖwµqv (Reversible Process): ‡h cÖwµqv wecixZgyLx n‡q cÖZ¨veZ©b K‡i Ges m¤§yLeZ©x I wecixZgyLx cÖwµqvi cÖwZ ¯Í‡i Zvc I Kv‡Ri djvdj mgvb I wecixZ †mB cÖwµqv‡K cÖZ¨vMvgx cÖwµqv e‡j| eid Zvc †kvlY K‡i cvwb‡Z cwiYZ nq| Avevi †mB cvwb †_‡K Zvc mgcwigvb Zvc AcmviY Ki‡j Zv cybivq ei‡d cwiYZ nq| GwU cÖZ¨vMvgx cÖwµqvi GKwU D`vniY| AcÖZ¨vMvgx cÖwµqv (Irreversible Process): ‡h cÖwµqv wecixZgyLx n‡q cÖZ¨veZ©b Ki‡Z cv‡i bv A_©vr m¤§yLeZ©x I wecixZgyLx cÖwZ ¯Í‡i Zvc I Kv‡Ri djvdj mgvb I wecixZ nq bv †mB cÖwµqv‡K AcÖZ¨vMvgx cÖwµqv e‡j| `ywU e¯‘i g‡a¨ Nl©‡Yi d‡j †h Zvc m„wó nq Zv GKwU AcÖZ¨vMvgx cÖwµqv KviY Nl©‡bi weiæ‡× †h KvR nq ZvB Zv‡c cwiYZ nq Ges H Zvc‡K †Kvbfv‡eB Kv‡R iƒcvšÍwiZ Kiv hvq bv| cÖZ¨vMvgx cÖwµqv I AcÖZ¨vMvgx cÖwµqvi g‡a¨ cv_©K¨ (Distinction between reversible and irreversible process): cÖZ¨vMvgx cÖwµqv AcÖZ¨vMvgx cÖwµqv 1| ‡h cÖwµqv wecixZgyLx n‡q cÖZ¨veZ©b K‡i Ges 1| ‡h cÖwµqv wecixZgyLx n‡q cÖZ¨veZ©b Ki‡Z cv‡i bv m¤§yLeZ©x I wecixZgyLx cÖwµqvi cÖwZ ¯Í‡i Zvc I Kv‡Ri A_©vr m¤§yLeZ©x I wecixZgyLx cÖwZ ¯Í‡i Zvc I Kv‡Ri djvdj mgvb I wecixZ †mB cÖwµqv‡K cÖZ¨vMvgx cÖwµqv djvdj mgvb I wecixZ nq bv †mB cÖwµqv‡K e‡j| AcÖZ¨vMvgx cÖwµqv e‡j| 2| Kvh©wbe©vnK e¯‘ cÖv_wgK Ae¯’vq wd‡i Av‡m| 2| Kvh©wbe©vnK e¯‘ cÖv_wgK Ae¯’vq wd‡i Avm‡Z cv‡i bv| 3| GwU GKwU axi cÖwµqv| 3| GwU GKwU `ªæZ cÖwµqv| 4| GB cÖwµqvq wm‡÷‡gi ZvcMZxq mvg¨ve¯’v eRvq 4| GB cÖwµqvq wm‡÷‡gi ZvcMZxq mvg¨ve¯’v eRvq _v‡K| _v‡K& bv | BwÄb (Engine): ‡h h‡š¿ Zvc kw³ hvwš¿K kw³‡Z iƒcvšÍwiZ nq Zv‡K Zvc BwÄb e‡j| BwÄb cÖavYZ `yB cÖKvi h_vt- 1| AšÍ©`nb BwÄb I 2| ewn`©nb BwÄb| ZvcMwZwe`¨vi wØZxq m~Î (Second Law of Thermodynamics): K‡b©vi wee„wZ: †Kvb wbw`©ó cwigvb Zvc kw³‡K m¤úyY©iƒ‡c hvwš¿K kw³‡Z iƒcvšÍ‡i mÿg Ggb hš¿ wbg©vb m¤¢e bq| K¬wmqv‡mi wee„wZ: evB‡ii kw³i mvnvh¨ Qvov †Kvb ¯^qswµq h‡š¿i c‡ÿ wb¤œ DòZvi e¯‘ n‡Z D”PZi DòZvi e¯‘‡Z Zv‡ci ¯’vbvšÍi m¤¢e bq| cøv‡¼i wee„wZ: Ggb †Kvb BwÄb ˆZix m¤¢e bq, †hUv †Kvb e¯‘ †_‡K Zvc MÖnY K‡i Aweivgfv‡e Kv‡R cwiYZ Ki‡e A_P cwi‡e‡ki †Kvb cwieZ©b n‡e bv| ‡Kjwf‡bi wee„wZ: ‡Kvb e¯‘‡K Gi cwicv‡k¦©i kxZjZg Ask n‡Z AwaKZi kxZj K‡i kw³i Aweivg mieivn cvIqv m¤¢e bq|

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2 16| Zvc MwZwe`¨vi 2q m~Î (Second Law Of Thermodynamics) Bwćbi Zvcxq `ÿZv (Efficiency of a Engine): ‡Kvb Zvc BwÄb Øviv Kv‡R iƒcvšÍwiZ Zvc kw³i cwigvb Ges BwÄb Øviv †kvwlZ Zvc kw³i AbycvZ‡K Bwćbi Zvcxq `ÿZv e‡j| G‡K  Øviv cÖKvk Kiv nq| msÁv g‡Z,

Zvcxq `ÿZv, 

BwÄb Øviv iƒcvšÍwiZ Zvc kw³ BwÄb Øviv † kvwlZ Zvc kw³

e¨vL¨v t †Kvb BwÄb hw` T1 ZvcgvÎvq Zvc Drm ‡_‡K Q1 Zvc ‡kvlY K‡i Ges T2 ZvcgvÎvq Q2 Zvc eR©b K‡i, Zvn‡j BwÄb Øviv Kv‡R iƒcvšÍwiZ Zvc kw³i cwigvb = (Q1-Q2)|  

Q1  Q2 | Bwćbi Zvcxq `ÿZv‡K kZKivq cÖKvk Ki‡j Zvcxq `ÿZvi ivwkgvjv n‡e, Q1

Q Q   Q  ev,    1  2   100 %   1  2   100 % Q1   Q1 Q1    T  Zvc ZvcgvÎvi mgvbycvwZK, A_©vr, Q1T1 Ges Q2T2 d‡j,   1  2   100 %  T1 



Q1  Q2  100 % Q1

K‡Y©vi BwÄb (Carnot's Engine): Zvckw³‡K hvwš¿K kw³‡Z iƒcvšÍwiZ Kivi Rb¨ mv`x K‡b©v mKj †`vlΤœwU gy³ †h Av`k© h‡š¿i Kíbv K‡ib Zv‡K K‡b©v BwÄb e‡j| K‡b©v BwÄb GKwU Av`k© Bwćbi aviYvgvÎ, ev¯Í‡e Gi iƒcvšÍi m¤¢e nqwb| K‡b©v Bwćbi PviwU Ask| Ask PviwU (1) wmwjÛvi (2) ZvcDrm (3) Zvc MÖvnK (4) Zvc AšÍiK e¯‘ ; wb‡¤œ wPÎmn wewfbœ Ask eY©bv Kiv nj| wmwjÛvi (Cylinder): GKwU wmwjÛvi C| wmwjÛviwUi †`Iqvj m¤ú~Y© Zvc AšÍiK c`v_© Ges Zj‡`k m¤ú~yY© Zvc cwievnK c`v_© Øviv ˆZix| Gi wfZ‡i m¤ú~Y© Zvc AšÍiK c`v‡_© ˆZix GKwU wc÷b P Nl©Ynxbfv‡e PjvPj Ki‡Z cv‡i| wmwjÛv‡ii wfZi Kvh©wbe©vnK e¯‘ wn‡m‡e Av`k© M¨vm †bqv nq|

ZvcDrm: D”P Zvc aviYÿgZv m¤úbœ GKwU DËß e¯‘ hv T1 ZvcgvÎvq Av‡Q Ges Zv‡ci Drm wn‡m‡e KvR K‡i| Gi ZvcgvÎv me©`v w¯’i _v‡K, Zv‡ci Av`vb cÖ`v‡b ZvcgvÎv KL‡bv cwieZ©b nq bv| Zvc MÖvnK: D”P Zvc aviYÿgZv m¤úbœ T2 ZvcgvÎvq GKwU kxZj e¯‘ hv Zvc MÖvnK wn‡m‡e KvR K‡i| Gi ZvcgvÎvI me©`v w¯’i _v‡K, Zv‡ci Av`vb cÖ`v‡b ZvcgvÎv KL‡bv cwieZ©b nq bv| Zvc AšÍiK e¯‘: m¤ú~Y© Zvc AšÍiK c`v‡_©i ˆZix GKwU Avmb hvi Dci wmwjÛviwU emvb _v‡K|

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3 16| Zvc MwZwe`¨vi 2q m~Î (Second Law Of Thermodynamics) K‡b©vi Pµ (Carnot's Cycle): †h we‡kl cÖwµqvq KvR Ki‡j GKwU Av`k© Zvc BwÄb Z_v K‡b©v BwÄb Aweivg kw³ mieivn Ki‡Z cv‡i Zv‡K K‡b©v Pµ e‡j| K‡b©v P‡µ cÖZ¨vMvgx cÖwµqvi gva¨‡g Kvh©wbe©vnK e¯‘ Drm †_‡K Zvc MÖnb K‡i GKwU wbw`©ó Pvc, AvqZb I ZvcgvÎv n‡Z Avi¤¢ K‡i GKwU m‡gvò cÖmviY I GKwU iæ×Zvcxq cÖmviY Ges GKwU m‡gvò m‡¼vPb I GKwU iæœ×Zvcxq m‡¼vP‡bi gva¨‡g Zv‡ci wKQy Ask Kv‡R iƒcvšÍi K‡i Ges evKx Ask Zvc MÖvn‡K eR©b K‡i Avw` Ae¯’vq wd‡i Av‡m| Av`k© M¨vm‡K PviwU ch©v‡qi g‡a¨w`‡q AwZµg Kiv‡bv nq|

cÖ_g ch©vq (First Operation): cÖ_‡g wmwjÛvi C †K Zvc Dr‡mi Dci emv‡bv nq| Lye Aímg‡qi g‡a¨ wmwjÛv‡i Ave× M¨v‡mi ZvcgvÎv Dr‡mi ZvcgvÎvi mgvb nq|aiv hvK, GB Ae¯’vq M¨v‡mi Pvc I AvqZb h_vµ‡g P1I V1 nq hv wb‡`©kK wP‡Îi A we›`y Øviv wb‡`©k Kiv n‡q‡Q| GLb Av`k© M¨vm‡K m‡gvò cÖwµqvq cÖmvwiZ n‡Z w`‡j cÖwµqv †k‡l Gi Pvc I AvqZb h_vµ‡g P2I V2 nq hv wb‡`©kK wP‡Îi B we›`y Øviv wb‡`©k Kiv n‡q‡Q| AB †iLv Øviv M¨v‡mi m‡gvò cÖmviY †`Lv‡bv n‡q‡Q Ges K…Z KvR, W1=ABGE †ÿ‡Îi †ÿÎd‡ji mgvb| wØZxq ch©vq (Second Operation): Gevi wmwjÛvi C †K Zvc-AšÍiK Avm‡bi Dci emv‡bv nq Ges Ave× M¨vm‡K iæ×Zvcxq fv‡e cÖmvwiZ n‡Z †`Iqv nq| cÖwµqv †k‡l M¨v‡mi Pvc I AvqZb h_vµ‡g P3I V3 nq hv wb‡`©kK wP‡Îi C we›`y Øviv wb‡`©k Kiv n‡q‡Q| iæ× Zvcxq cÖmvi‡Yi d‡j M¨v‡mi ZvcmvÎv n«vm †c‡q Zvc MÖvn‡Ki ZvcgvÎv T2 Gi mgvb nq| wP‡Î BC †iLv Øviv M¨v‡mi iæ× Zvcxq cÖmviY †`Lv‡bv n‡q‡Q Ges GB cÖmvi‡Y K…Z KvR, W2=BCHG †ÿ‡Îi †ÿÎd‡ji mgvb| Z…Zxq ch©vq (Third Operation): Gevi wmwjÛvi C †K ZvcMÖvn‡Ki Dci ewm‡q Avevi M¨v‡mi Pvc e„w× Ki‡j wcób Øviv M¨v‡mi Dci KvR m¤úvw`Z n‡e| d‡j M¨v‡mi AšÍt¯’ kw³ e„w× cv‡e| wKš‘ Ave× M¨vm e„w× cÖvß AšÍt¯’ kw³ Zvciƒ‡c ZvcMÖvn‡K eR©b K‡i ZvcgvÎv ZvcMÖvn‡Ki mgvb A_©vr T2 eRvq ivL‡e| GB Ae¯’vq Ave× M¨v‡mi Pvc I AvqZb h_vµ‡g P4 I V4 nq hv wb‡`©kK wP‡Îi D we›`y Øviv wb‡`©k Kiv n‡q‡Q| wP‡Î CD †iLv m‡gvò ms‡KvPb wb‡`©k K‡i Ges GB m‡¼vP‡bi d‡j K…Z KvR, W3=CDFH †ÿ‡Îi †ÿÎd‡ji mgvb| PZz_© ch©vq (Fourth Operation): ‡kl ch©v‡q wmwjÛvi C †K cybivq Zvc-AšÍiK Avm‡bi Dci emv‡bv nq Ges iæ×Zvcxqfv‡e Ave× M¨v‡mi Dci Pvc evov‡bv nq| d‡j Gi AvqZb n«vm cvq| Ave× M¨v‡mi Dci KvR m¤úvw`Z nIqvq Gi ZvcgvÎv e„wׇc‡q cybivq Dr‡mi ZvcgvÎv T1 Gi mgvb nq| GB Ae¯’vh M¨v‡mi Pvc I AvqZb cybivq h_vµ‡g P1 I V1 nq hv wb‡`©kK wP‡Îi A we›`y wb‡`©k K‡i A_©vr Ave× M¨vm Avw` Ae¯’vq wd‡i Av‡m| wP‡Î DA †iLv Ave× M¨v‡mi iæ× Zvcxq m‡¼vPb †`Lv‡bv n‡q‡Q d‡j K…Z KvR, W4=DAEF †ÿ‡Îi †ÿÎd‡ji mgvb| GLv‡b W1 I W2 Ave× M¨vm Øviv K…Z KvR e‡j abvZ¥K n‡e Ges W3 I W4 Ave× M¨v‡mi Dci K…Z KvR e‡j FbvZ¥K n‡e| d‡j Ave× e¯‘ Øviv †gvU K…Z KvR, W= W1+ W2 - W3 - W4  W= ABGE + BCHG - CDFH - DAEF W= ABCD †ÿ‡Îi †ÿÎd‡ji mgvb| myZivs GKwU K‡b©v P‡µ Kvh©wbe©vnK e¯‘ Øviv m¤úvw`Z KvR wb‡`©kK wP‡Î `ywU

m‡gvò †iLv I `ywU iæ×Zvcxq †iLv Øviv Ave× †ÿ‡Îi †ÿÎd‡ji mgvb| GbUªwc (Entropy): ‡Kvb wm‡÷‡g kw³ _vK‡jB †h Zv‡K cÖ‡qvRbxq Kv‡R jvMv‡bv hv‡e Ggb †Kvb wbðqZv †bB| GUv wbf©i K‡i wm‡÷‡gi ZvrÿwYK Ae¯’vi Dc‡i| Avgiv Rvwb †Kvb M¨vm‡K iæ×Zvcxq cÖwµqvq m¼zwPZ Ki‡j M¨v‡mi Dci †h KvR Kiv nq Zvi d‡j M¨v‡mi Aš’t¯’kw³ I ZvcgvÎv †e‡o hvq| Avevi iæ×Zvcxq cÖwµqvq †Kvb M¨vm‡K cÖmvwiZ

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4 16| Zvc MwZwe`¨vi 2q m~Î (Second Law Of Thermodynamics) n‡Z w`‡j M¨vm‡K wKQy KvR Ki‡Z nq| d‡j M¨v‡mi Aš’t¯’kw³ I ZvcgvÎv K‡g hvq| myZivs †`Lv hvq †h, iæ×Zvcxq cÖwµqvq M¨v‡mi AšÍt¯’kw³ I ZvcgvÎv Df‡qiB cwieZ©b nq| wKš‘ m‡gvò cÖwµqvq †hgb ZvcgvÎv w¯’i _v‡K iæ×Zvcxq cÖwµqvq †Zgwb †Kvb GKwU ivwk w¯’i _v‡K| K¬wmqvm GB ivwkwUi bvg †`b GbUªwc| iæ×Zvcxq cÖwµqvq e¯‘i †h Zvcxq ag© w¯’i _v‡K Zv‡K GbUªwc e‡j| GbUªwc GKwU †fŠZ ivwk, G‡K S Øviv cÖKvk Kiv nq| hw` †Kvb wm‡÷g T ZvcgvÎvq dQ cwigvb Zvc MÖnY ev eR©b Kivi d‡j GbUªwci cwieZ©b ds nq Z‡e ds 

dQ n‡e| T

‡Kvb wm‡÷‡gi kw³ iƒcvšÍ‡ii AÿgZv ev Am¤¢ve¨Zv‡K ev iƒcvšÍ‡ii Rb¨ kw³i AcÖvßZv‡K GbUªwc e‡j| GbUªwci e¨vL¨v nj e¯‘i iæ×Zvcxq cÖwµqvq GbUªwc w¯’i _v‡K: †Kvb e¯‘i GbUªÖwci cig gvb AvRI Rvbv m¤¢e nqwb| Z‡e †Kvb e¯‘ hw` Zvc MÖnb ev eR©b K‡i, Zvn‡j e¯‘i GbUªwci cwieZ©b nq| †Kvb wm‡÷‡gi ZvcgvÎv mv‡c‡ÿ M„nxZ ev ewR©Z Zvc cwieZ©‡bi nvi Øviv GbUªwci cwieZ©b cwigvc Kiv nq| hw` †Kvb wm‡÷g T ZvcgvÎvq dQ cwigvb Zvc MÖnY ev eR©b Kivi d‡j GbUªwci cwieZ©b ds nq Z‡e ds 

dQ n‡e| T

Dc‡iv³ mgxKiY n‡Z †`Lv hvq †h, iæ×Zvcxq cÖwµqvq †h‡nZz Kvh©wbe©vnK e¯‘i mv‡_ evB‡ii Zv‡ci †Kvb Av`vb cÖ`vb nqbv Kv‡RB, dQ=0 | myZivs iæ×Zvcxq cÖwµqvq GbUªwci cwieZ©b, dS 

dQ 0   0 A_©vr iæ×Zvcxq cÖwµqvq T T

GbUªwci cwieZ©b nq bv A_©vr iæ×Zvcxq cÖwµqvq GbUªwci w¯’i _v‡K| myZivs †Kvb e¯‘i GbUªwc ej‡Z Avgiv Ggb GKUv ‡fŠZ ivwk‡K eywS hv e¯‘i iæ×Zvcxq cÖZ¨vMvgx cÖwµqvq me©`v w¯’i _v‡K| GbUªwci cwieZ©‡bi GKK JK-1 | cÖZ¨veZ©x cÖwµqvq GbUªwc w¯’i _v‡K I AcÖZ¨veZ©x cÖwµqvq GbUªwc e„w× cvq t cÖZ¨veZ©x cÖwµqvq GbUªwc w¯’i _v‡K: K‡b©vi Pµ GKwU cÖZ¨veZ©x cÖwµqv| G‡Z `ywU m‡gvò †iLv I `ywU iæ×Zvcxq †iLv Øviv m„ó AB I CD `ywU m‡gvò †iLv Ges BC I DA `ywU iæ× Zvcxq †iLv| ‡h‡nZz iæ×Zvcxq cÖwµqvq GbUªwc w¯’i _v‡K myZivs m‡gvò AB I CD Gi †ÿ‡Î GbUªwci cwieZ©b NU‡e| AB Gi †ÿ‡Î Zvc Q1 I ZvcgvÎv T1 n‡j GbUªwci cwieZ©b, ds1 

Q1 T1

Ges CD Gi †ÿ‡Î Zvc Q2 I ZvcgvÎv T2 n‡j GbUªwci cwieZ©b, ds2    †gvU GbUªwci cwieZ©b ds  ds1  ds2 

wKš‘ cÖZ¨vMvgx cÖwµqvq K‡b©vP‡µ,

Q2 T2

Q1 Q2  T1 T2

Q1 Q2  T1 T2

 G‡ÿ‡Î †gvU GbUªwci cwieZ©b, ds  ds1  ds2 

Q1 Q2   0 A_©vr cÖZ¨veZ©x cÖwµqvq GbUªwci cwieZ©b k~b¨| T1 T2

myZivs cÖZ¨veZ©x cÖwµqvq GbUªwc w¯’i _v‡K|

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5 16| Zvc MwZwe`¨vi 2q m~Î (Second Law Of Thermodynamics) AcÖZ¨veZ©x cÖwµqvq GbUªwc e„w× cvq: aiv hvK, `ywU e¯Íy A I B cwicv‡k¦©i cÖfve †_‡K gy³ Ae¯’vq ci¯ú‡ii ms¯ú‡k© Av‡Q| e¯‘Øq Ges G‡`i ms‡hvM¯’j Zvc mycwievnx| e¯‘؇qi ZvcgvÎv h_vµ‡g T1 I T2 ; T1  T2 ; Zvc DËß e¯‘ †_‡K kxZj e¯‘‡Z mÂvwjZ n‡e| †m Zvc Avi DËß e¯‘‡Z wd‡i Avm‡e bv| AZGe GwU GKwU AcÖZ¨veZ©x cÖwµqv| awi, AwZ Aí mg‡q dQ cwigvb Zvc A †_‡K B †Z mÂvwjZ nq| A e¯‘ dQ cwigvb Zvc nvivq, B

e¯‘ dQ cwigvb Zvc AR©b K‡i| A e¯‘i GbUªwc n«vm  mgMÖ e¨e¯’vq †gvU GbUªwci cwieZ©b, ds 

dQ dQ ; B e¯‘i GbUªwc e„w×  ; T1 T2

dQ dQ  T2 T1

G‡ÿ‡Î, †gvU GbUªwci cwieZ©b GKwU abvZ¥K ivwk; KviY, T1  T2 | AZGe, AcÖZ¨veZ©x cÖwµqvq GbUªwc e„w× cvq| A_ev Ab¨fv‡e ejv hvq, ¯^ZtùzZ© cÖwµqv †mw`‡KB msNwUZ nq †h w`‡K GbUªwc e„w× cvq|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

16| ZvcMwZwe`¨vi 2q m~Î (Second Law Of Thermodynamics) 1| GKwU K‡b©v Bwćbi Dr‡mi ZvcgvÎv 400K, GB ZvcgvÎvq GKwU Drm ‡_‡K GwU 840J Zvc Mªnb K‡i Ges wms‡K 630J Zvc eR©b K‡i| wms‡Ki ZvcgvÎv I Bwćbi Kg©`ÿZv wbY©q Ki| Avgiv Rvwb, Q1  Q 2  100% Q1 840  630   100% 840 



210  100% 840

   25% (Ans)

GLv‡b, ZvcgvÎv, T1 = 400K Zvc, Q1 = 840 J Zvc, Q2 = 630 J Kg©`¶Zv,  wm‡¼i ZvcgvÎv, T2 = ?

Avevi,Avgiv Rvwb, 

T1  T2  100% T1

400  T2  100% 400 400  T2  25  4  100  400  T2  T2  400  100  T2  300K (Ans.)  25% 

1000 100% 3400   29.41% (Ans)

3| GKwU Bwćbi Kg© `ÿZv 65%| Gi wb¤§ Zvcvav‡ii (MÖvn‡Ki) ZvcgvÎv 27°C | Gi D”P Zvcvav‡ii (Dr‡mi) ZvcgvÎv wbY©q Ki| Avgiv Rvwb, GLv‡b,

T1  T2  100% T1 65 T1  300   100 T1  100T1  30000  65T1  100T1  65T1  30000  35T1  30000

4| GKwU BwÄb 3400J Zvc MÖnb K‡i I 2400J Zvc eR©b K‡i| BwÄbwU Øviv m¤úvw`Z Kv‡‡Ri cwigvb I Bwćbi `ÿZv wbY©q Ki| Avgiv Rvwb, GLv‡b, W  Q Q 1 2 Zvc, Q1 = 3400 J  W  (3400  2400 ) J Zvc, Q2 = 2400 J  W  1000 J ( Ans .)

KvR, W =?

Q Q Kg©`¶Zv,  2  100%  1 Q 1 3400  2400  100% 3400



2| GKwU K‡b©v BwÄb 327°C Ges 27°C DòZvi g‡a¨ KvR Ki‡Q| Gi Kg© `ÿZv KZ? GLv‡b, Avgiv Rvwb, D”P Zvcvav‡ii ZvcgvÎv, T1  T2 T1 = 327ºC   100% =(327+273)K T1 = 600 K 600  300 wb¤§ Zvcvav‡ii ZvcgvÎv,   100% 600 T2 = 27ºC = (27+273)K    0.5  100% = 300 K Kg©`¶Zv,     50 % (Ans.)

65% 

30000 35  T1  857.14K  T1  (857.14 - 273)C  584.14C (Ans.)  T1 

wb¤§ Zvcvav‡ii ZvcgvÎv, T2 = 27ºC = (27+273)K = 300 K Kg©`¶Zv,  D”P Zvcvav‡ii ZvcgvÎv, T1 = ?

5| GKwU BwÄb 25°C Ges 225°C ZvcgvÎvi g‡a¨ Kvh©iZ| Bwćbi Zvc Drm †_‡K 4200J Zvc MÖnb K‡i| BwÄb Øviv m¤úvw`Z Kv‡Ri cwigvb wbY©q Ki| GLv‡b, Avgiv Rvwb, D”P Zvcvav‡ii ZvcgvÎv, T1 = 225ºC Q1 T1  = (225+273)K Q 2 T2 = 498 K Q1T2 wb¤§ Zvcvav‡ii ZvcgvÎv,  Q2  T2 = 25ºC T1 = (25+273)K 4200  298 = 298 K  Q2  498 M„nxZ Zvc, Q1 = 4200J  Q 2  2513.25J K…Z KvR, W = ?

Avevi,

W  Q1  Q 2  W  4200  2513.25  W  1686.74 J (Ans.) 6| GKwU K‡Y©v BwÄb 800K I 400K Zvc gvÎvq †h `ÿZvq KvR K‡i, wVK mg`ÿZvq KvR K‡i T K Ges 900K ZvcgvÎv| ZvcgvÎv T wbY©q Ki| GLv‡b, 1g †¶‡Î Avgiv Rvwb, D”P Zvcvav‡ii ZvcgvÎv, T1  T2 T1 = 800K   100% T1 wb¤§ Zvcvav‡ii ZvcgvÎv, T2 = 400K 800  400   100% Kg©`¶Zv, 

800

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16| ZvcMwZwe`¨vi 2q m~Î (Second Law Of Thermodynamics)

   0.5  100%    50% Avevi,

T T   4 3 100% T4 T  900  50%   100% T T  900 1 2 T  2T  1800  T  T  1800 K ( Ans.)

GLv‡b, 2q †¶‡Î wb¤§ Zvcvav‡ii ZvcgvÎv, T3 = 900K Kg©`¶Zv,  D”P Zvcvav‡ii ZvcgvÎv, T4 = T=? 

7| 10°C ZvcgvÎvi 5kg cvwb‡K 100°C ZvcgvÎvq DËxY© Ki‡Z GLv‡b, G›Uªwci cwieZ©b wbY©q Ki| Avgiv Rvwb, fi, m = 5kg T2 dQ ZvcgvÎv,T1 = 10ºC

dS  

T1

=(273+10)K=283K

T

ZvcgvÎv,T2 = 100ºC

msdT T1 T T2 5  4200  dT  dS   T1 T

 dS  

T2

 dS  21000lnT 

=(273+100KC=373K

G›Uªwci cwieZ©b dS=? 

T2 T1

 dS  21000  (ln T2  ln T1 ) T  dS  21000  ln 2 T1 373  dS  21000  ln 283  dS  21000  0.276131522  dS  5798.76 JK 1 (Ans.) 8| 0°C ZvcgvÎvi 3kg eid‡K 0°C ZvcgvÎvi cvwb‡Z cwiYZ Ki‡Z GbUªwci cwieZ©b KZ n‡e? [eid Mj‡bi Av‡cwÿK myßZvc =336000 Jkg-1]

dQ Avgiv Rvwb, dS  T mlf ev, dS  T

2

dQ dS   T1 T T2 msdT  dS   T1 T T2 1  4200  dT  dS   T1 T T2

GLv‡b, ei‡di fi, m = 3kg Mj‡bi Av‡cw¶K myßZvc lf=336000 Jkg-1| ZvcgvÎv T= (0+273)K=273 K GbUªwci cwieZ©b dS=?

ev, dS  3  336000 273  dS  3692.3JK 1 (Ans) 9| 0°C ZvcgvÎvi 1kg cvwb‡K 100°C ZvcgvÎvq DËxY© Ki‡Z G›Uªwci cwieZ©b wbY©q Ki| Avgiv Rvwb,

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 dS  4200lnT 

T2 T1

 dS  4200  (ln T2  ln T1 ) T  dS  4200  ln 2 T1 373  dS  4200  ln 273  dS  4200  0.312106624  dS  1310.84JK 1 (Ans.)

GLv‡b, fi, m = 1kg ZvcgvÎv, T1 = 0ºC =(273+0)ºK=273ºK ZvcgvÎv, T2 = 100ºC =(273+100)ºK=373ºK G›Uªwci cwieZ©b dS=? 

10| GKwU Kv‡b©v Bwćbi Zvc MÖvn‡Ki ZvcgvÎv 7°C Ges `ÿZv 50%| Bwćbi `ÿZv 60% Ki‡Z n‡j Zvc Dr‡mi ZvcgvÎv KZ e„w× Ki‡Z n‡e?

Avgiv Rvwb,

T1  T2  100% T1 T  280 50%  1  100% T1 50 T1  280   100 T1  100T1  28000  50T1  50T1  28000 1 

 T1 

28000 50

GLv‡b, MÖvn‡Ki ZvcgvÎv, T2 = 7ºC = (7+273)K = 280 K Kg©`¶Zv,  Dr‡mi ZvcgvÎv, T1 = ?

Kg©`¶Zv,  T1/  T1  ? 

 T1  560K T1/  T2  100% T1/ T /  280 60%  1 /  100% T1

2 



60 T1/  280  100 T1/

 100T1/  28000  60T1/  40T1/  28000 28000  T1/  40 /  T1  700K ZvcgvÎv e„w× = T1/  T1  (700  560) K  140K ev, 140°C (Ans.)

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kã (Sound): †h evwn¨K KviY Avgv‡`i Kv‡bi kªeb Bw›`ª‡q kÖæwZi Abyf~wZ RvMvq ev RvMv‡Z †Póv K‡i Zv‡K kã e‡j| kã GK cÖKvi kw³ hv †Kvb K¤úbkxj e¯‘ †_‡K Drcbœ n‡q Ro gva¨‡gi mvnv‡h¨ Avgv‡`i Kv‡b †cŠwQ‡q kÖæwZi Abyf~wZ RvMvq ev RvMv‡Z †Póv K‡i| kã ‡hfv‡e Drcbœ nq (Production of Sound): evZv‡m †Kvb e¯‘i K¤ú‡bi d‡j kã Drcbœ nq| Z‡e hw` GB k‡ãi K¤úv¼ 20Hz †_‡K 20,000Hz ch©šÍ nq Z‡e Avgiv Drcbœ kã ïb‡Z cvB| Avi hw` Drcbœ 20Hz Gi Kg ev 20,000Hz Gi †ekx nq Z‡e Drcbœ kã Avgiv ïb‡Z cvB bv| GB Rb¨ 20Hz †_‡K 20,000Hz ch©šÍ K¤úv¼ wewkó kã Zi½‡K AwWI ev kªebxq kã Zi½ e‡j| kªve¨Zvi mxgv, K¤úv¼ ev wdª‡Kv‡qwÝ, Zi½, Avo Zi½, jw¤^K Zi½, Zi½ ‰`N©¨, Zi½ †eM, we¯Ívi, †`vjb Kvj,`kv, Zi½ †eM, k‡ãi †eM Ges ‡KŠwbK K¤úv¼ Gi msÁv I e¨vL¨v: kªve¨Zvi mxgv (Limit of Audibility): ‡h kã Avgiv ïb‡Z cvB Zvi K¤úv¼ 20 Hz ‡_‡K 20,000 Hz Gi g‡a¨ nq| G Rb¨ G‡K kªe¨Zvi mxgv e‡j| A_©vr kªe¨Zvi mxgv nj †h kã Zi‡½i K¤úv¼ 20 Hz ‡_‡K 20,000 Hz Gi g‡a¨| K¤úv¼ (Frequency): ‡Kvb K¤úgvb e¯‘ cÖwZ †m‡K‡Û hZwU c~b© K¤úb m¤úbœ K‡i Zv‡K Zv‡K H K¤úgvb e¯‘i K¤úv¼ e‡j| G‡K n ev f Øviv cÖKvk Kiv nq| K¤úv¼ ev wd«‡Kv‡qÝxi GKK Hz ev, cy/s| Zi½ (Wave): w¯’wZ¯’vcK gva¨‡gi KYv ¸wji mgwóMZ K¤ú‡bi d‡j m„ó Av‡›`vjb‡K Zi½ e‡j| Zi½ cÖavbZ `yB cÖKvi (1) Avo Zi½ ev AbycÖ¯’ Zi½ (2) jw¤^K Zi½ ev Aby‰`N©¨ Zi½ (1) Avo Zi½ ev AbycÖ¯’ Zi½ (Transverse Wave):

‡h mg¯Í Zi‡½i †ÿ‡Î Ro gva¨‡gi KYv ¸wji K¤ú‡bi w`K Zi½ cÖev‡ni w`‡Ki mv‡_ mg‡KvY ev j¤^ fv‡e nq †mB mg¯Í Zi½‡K Avo Zi½ ev AYycÖ¯’ Zi½ e‡j| Av‡jvK Zi½ AYycÖ¯’ Zi‡½i D`vniY|

(2) jw¤^K Zi½ ev Aby‰`N©¨ Zi½ (Longitudinal Wave):

‡h mg¯Í Zi‡½i †ÿ‡Î Ro gva¨‡gi KYv ¸wji K¤ú‡bi Avwfgy‡L ev mgvšÍiv‡j Zi½ cÖevwnZ nq †mB mg¯Í Zi½‡K jw¤^K

Zi½ ev Aby‰`N©¨ Zi½ e‡j| kã Zi½ jw¤^K Zi‡½i D`vniY|

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Zi½ ˆ`N©¨ (Wave Length): K¤úgvb e¯‘i GKwU c~Y© K¤ú‡b m„ó Zi½ †h `~iZ¡ AwZµg K‡i Zv‡K Zi‡½i Zi½ ˆ`N©¨ e‡j| Zi½

ˆ`N©¨‡K  Øviv cÖKvk Kiv nq| Zi½ ˆ`‡N©¨i GKK wgUvi ev wdU| Zi½ cÖevnKv‡j Zi½w¯’Z GKwU KYvi N wU c~Y© K¤ú‡bi mgq AeKv‡k Zi½ S `~iZ¡ AwZµg Ki‡j λ 

S n‡e| N

Zi½ †eM (Wave Velocity): wbw`©ó w`‡K Zi½ GKK mg‡q †h `~iZ¡ AwZµg K‡i Zv‡K Zi½ †eM e‡j| G‡K V Øviv cÖKvk Kiv nq| t mg‡q wbw`©ó w`‡K †Kvb Zi½ d `~iZ¡ AwZµg Ki‡j Zi½ †eM V 

d n‡e| Gi GKK ms-1| t

we¯Ívi (Amplitude): ‡Kvb GKwU K¤úgvb KYvi mvg¨ve¯’vb †_‡K †h †Kvb GKw`‡K me©vwaK †h `~iZ¡ AwZµg K‡i Zv‡K Gi we¯Ívi e‡j|

we¯Ívi‡K a Øviv cÖKvk Kiv nq| we¯Ívi Gi GKK wgUvi| ‡`jbKvj ev ch©vqKvj (Time Period): ‡Kvb GKwU gva¨‡g GKwU c~Y© Zi½ m„wó Ki‡Z K¤úbiZ Zi½ Dr‡mi †h mgq jv‡M Zv‡K H K¤úgvb e¯‘i †`vjbKvj ev ch©vqKvj e‡j| G‡K T Øviv cÖKvk Kiv nq| t †m‡K‡Û Zi½ Drm N wU c~b© K¤úb w`‡j †`vjbKvj T 

t N

n‡e|

`kv (Phase): Zi‡½i g‡a¨ GKwU KYvi †Kvb gyn‡~ Z©i Ae¯’vb Ges Gi MwZi Ae¯’v I w`K hv Øviv cÖKvk Kiv nq Zv‡K `kv e‡j| GRb¨ `kv †Kvb GKwU K¤úgvb e¯‘i Ae¯’v cÖKvk K‡i| Zi½ gyL (Wave Front):

‡Kvb Zi‡½i Dc‡i Aew¯’Z mg`kv m¤úbœ KYv ¸wj †h Z‡j Ae¯’vb K‡i Zv‡K Zi½gyL e‡j| Avo Zi‡½i †ÿ‡Î Zi½ kx‡l© I Zi½ cv‡` Aew¯’Z mKj KYv mg `kvq _v‡K| A_©vr Zv‡`i `kv GKB| k‡ãi †eM (Velocity of Sound): kã GKK mg‡q wbw`©ó w`‡K †h `~iZ¡ AwZµg K‡i Zv‡K k‡ãi †eM e‡j| G‡K V Øviv cÖKvk Kiv nq| t mg‡q d `~iZ¡ AwZµg Ki‡j †eM V 

d n‡e| Gi GKK ms-1| t

‡KŠwbK K¤úv¼ (Angular Frequency): mg‡qi mv‡_ `kvi cwieZ©‡bi nvi‡K †KŠwbK K¤úv¼ e‡j| GKwU c~b© K¤ú‡b T mg‡q `kvi cwieZ©b myZivs †KŠwbK K¤úv¼ ω 

2π  2πf ‡KŠwbK K¤úv‡¼i GKK rad s-1 T

v = f  m¤úK© wU cÖgvb A_ev, k‡ãi †e‡Mi mv‡_ K¤úv¼ I Zi½ ‰`‡N©¨i m¤úK© ¯’vcb (Relation Between v and  ): Zi½ m„wóKvix †Kvb K¤úbkxj KYvi GKwU c~Y© K¤úb w`‡Z †h mgq jv‡M, †mB mgq‡K ch©vqKvj T e‡j| T mg‡q Zi½ †h `~iZ¡ AwZµg K‡i Zv‡K Zi½ ˆ`N©¨  e‡j|

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T mg‡q Zi½ AwZµg K‡i  `~iZ¡   GKK mg‡q Zi½ AwZµg K‡i `~iZ¡ T

wKš‘ Zi½ GKK mg‡q †h `~iZ¡ AwZµg K‡i Zv‡K Zi½ †eM e‡j| Zi½ †eM v n‡j v 

 T

Avevi, K¤úbkxj e¯‘ GKK mg‡q hZ¸‡jv c~Y© K¤úb m¤úbœ K‡i Zv‡K K¤úv¼ e‡j| ch©vqKvj T n‡j, T ‡m‡K‡Û †`q GKwU †`vjb d‡j 1 †m‡K‡Û †`q 1/T wU †`vjb | K¤úv¼‡K f Øviv cÖKvk Ki‡j, f

myZivs v 

1 T

 A_©vr v = f (cÖgvwYZ) T

mij †`vjMwZ (Simple Harmonic Motion): †Kvb wbw`©ó mgq ci ci GKB cÖKvi MwZi cybive„wË NU‡j †mB MwZ‡K mij †`vj MwZ e‡j| hw` †Kvb e¯‘i †`vj MwZ GKwU c‡_ Ggb fv‡e nq †h, †`vjvqgvb e¯‘wUi Z¡iY Gi MwZc‡_i GKwU wbw`©ó we›`ygywL nq Ges GB Z¡iY we›`ywU †_‡K †`vjvqgvb e¯‘wUi mi‡bi mgvbycvwZK nq Z‡e H iƒc MwZ‡K mij †`vj MwZ e‡j| mij †`vjMwZi ‰ewkó¨ mg~n (Characteristics Simple Harmonic Motion): (1) G MwZ mij ˆiwLK MwZ (2) G MwZ †`vj MwZ (3) G MwZ m¤úbœ e¯‘i Z¡iY me©`v mvg¨ve¯’vb †_‡K mi‡Yi mgvbycvwZK (4) G MwZ ch©vq MwZ (5) G MwZ m¤úbœ e¯‘i Z¡i‡Yi w`K me©`v mvg¨ve¯’vb †_‡K mi‡Yi wecixZ gywL| (6) G MwZ m¤úbœ e¯‘i Z¡iY me©`v Gi MwZ c‡_i GKwU wbw`©ó we›`y AwfgywL| Avo Zi½ I jw¤^K (`xNj) Zi‡½i cv_©K¨ (Distinction between Transverse wave and longitudinal wave) t µwgK Avo ev AbycÖ¯’ Zi½ jw¤^K ev `xNj Zi½ w¯’wZ ¯’vcK gva¨‡gi KYv¸wji Av‡›`vj‡bi d‡j m„ó w¯’wZ ¯’vcK gva¨‡gi KYv¸wji Av‡›`vj‡bi d‡j m„ó 1 Zi½ hw` KYv¸wji MwZc‡_i mv‡_ mg‡Kv‡Y Zi½ hw` KYv¸wji MwZc‡_i mv‡_ mgvšÍiv‡j cÖevwnZ cÖevwnZ nq Z‡e H Zi½‡K Avo Zi½ e‡j| nq Z‡e H Zi½‡K jw¤^K Zi½ e‡j| 2 Zi½ cÖev‡ni d‡j gva¨‡g Zi½ P~ov I Zi½ cv` Zi½ cÖev‡ni d‡j gva¨‡g msKzwPZ ¯Íi I cÖmvwiZ ¯Íi Gi m„w÷ nq| m„wó nq| 3 gva¨‡g Zi½ cÖev‡n m„ó ci ci `ywU Zi½ P~ov ev Zi½ cÖev‡n m„ó GKwU msKzwPZ I GKwU cÖmvwiZ ¯Íi Zi½ cv` Gi ga¨eZ©x `~iZ¡‡K Zi½ ˆ`N©¨ e‡j| Gi wgwjZ ˆ`N©¨‡K Zi½ ˆ`N©¨ e‡j| 4 AvK…wZi ag© m¤úbœ gva¨‡g GB Zi‡½i DrcwË nq| AvqZ‡bi w¯’wZ¯’vcK ag© m¤úbœ gva¨‡g GB Zi‡½i DrcwË nq| 5 gva¨‡g Gi mgeZ©b n‡Z cv‡i| gva¨‡g Gi mgeZ©b nq bv| Avo Zi‡½i MVb wPÎ wb¤§iƒc jw¤^K Zi‡½i MVb wPÎ wb¤§iƒc 6

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4

w¯’i Zi½ (Stationary wave): ‡Kvb gva¨‡gi GKwU mxwgZ As‡k ci¯úi wecixZ gyLx Zi‡½i we¯Ívi I †`vjbKvj hw` mgvb nq Z‡e G‡`i wgwjZ wµqvq H As‡k †h b~Zb Zi‡½i DrcwË nq Zv‡K w¯’i Zi½ e‡j| GKwU Uvbv Zv‡ii †Kv_vI AvNvZ Ki‡j G ai‡bi Zi½ Zvi †e‡q `yB cÖv‡šÍi w`‡K AMÖmi nq Ges `yB cÖvšÍ ‡_‡K cÖwZdwjZ n‡q wd‡i Av‡m| D³ Zv‡i m„ó Zi½ GKwU w¯’i Zi‡½i D`vniY|w¯’i Zi‡½i †Kvb †Kvb we›`yi we¯Ívi k~b¨ Ges †Kvb †Kvb we›`yi we¯Ívi me©vwaK| †h we›`y ¸wji we¯Ívi me©vwaK [A wPwýZ we›`y ¸wj] Zv‡`i‡K my¯ú›` we›`y Ges †h we›`y¸wji we¯Ívi k~b¨ [N wPwýZ we›`y ¸wj] Zv‡`i‡K wb¯ú›` we›`y e‡j| w¯’i Zi‡½i cv_©K¨ ‰ewkó¨t (K) GB Zi½ gva¨‡gi †Kvb GKwU mxwgZ As‡k Drcbœ nq| (L) AMÖmi bv n‡q GKB As‡k mxgve× _v‡K| (M) Zi‡½i wewfbœ we›`y‡Z K¤ú‡bi we¯Ívi mgvb bq| (N) Zi‡½i my¯ú›` we›`yi we¯Ívi Zi½ m„wóKvix g~j Zi‡½i we¯Ív‡ii wظb-Gi mgvb| (O) ci ci `ywU jy‡ci miY ci¯úi wecixZ w`‡K nq| (P) w¯’i we›`y¯’ KYv¸‡jv Qvov mKj KYvi MwZ mij Qw›`Z ¯ú›`b MwZ| Pjgvb ev AMÖMvgx Zi½ (Progressive wave or travelling wave) : ‡Kvb Zi½ hw` we¯Í…Z gva¨‡gi GK ¯Íi n‡Z Ab¨¯Í‡i µgvMZ mÂvwjZ n‡q m¤§y‡Li w`‡K AMÖmi nq Z‡e †mB Zi½‡K AMÖMvgx Zi½ e‡j| gy³ evqy‡Z cÖevngvb kã Zi½ Ges mvaviY cvwbi Zi½ AMÖMvgx jw¤^K Zi‡½i D`vniY| Pjgvb ev AMÖMvgx Zi‡½i cv_©K¨ ‰ewkó¨t (K) †Kvb gva¨‡gi GKB cÖKvi K¤ú‡b GB Zi‡½i DrcwË nq| (L) Gi †eM gva¨‡gi NbZ¡ I w¯’wZ¯’vcKZvi Dci wbf©i K‡i| (M) gva¨‡gi KYv ¸‡jv K‡Lv‡bv w¯’i _v‡K bv| (O) Zi½ cÖev‡n gwa¨‡gi wewfbœ As‡ki Pvc I Nb‡Z¡i GKB cÖKvi cwieZ©b N‡U| (P) Zi½ gy‡Li Awfj¤^ eivei kw³ enb K‡i G Zi½ cÖevwnZ nq| (Q) gva¨‡gi cÖwZwU KYvi K¤úv¼ I we¯Ívi GKB nq Ges Zviv GKB ai‡bi K¤ú‡b Kw¤úZ nq| w¯’i I Pjgvb (AMÖMvgx) Zi‡½i cv_©K¨ (Distinction between Stationary and progressive wave : w¯’i Zi½ Pjgvb (AMÖMvgx) Zi½ 1| ‡Kvb gva¨‡gi GKwU mxwgZ As‡k ci¯úi wecixZ 1| Kvb Zi½ hw` we¯Í…Z gva¨‡gi GK ¯Íi n‡Z Ab¨¯Í‡i gyLx Zi‡½i we¯Ívi I †`vjbKvj hw` mgvb nq Z‡e µgvMZ mÂvwjZ n‡q m¤§y‡Li w`‡K AMÖmi nq Z‡e †mB G‡`i wgwjZ wµqvq H As‡k †h bZzb Zi‡½i DrcwË Zi½‡K AMÖMvgx Zi½ e‡j| nq Zv‡K w¯’i Zi½ e‡j| 2| Zi‡½i wb¯ú›` we›`y Qvov Ab¨ we›`yi MwZ mij †`vj 2| Zi½w¯’Z wewfbœ we›`yi MwZ mij †`vj MwZ| MwZ| 3| Zi‡½i AvKvi GK ¯’v‡b w¯’i _v‡K| 3| Zi‡½i AvKvi GK ¯’v‡b w¯’i _v‡K bv| 4| Zi½ cÖev‡n gva¨‡gi KYv ¸wj cÖ‡Z¨K c~Y© K¤ú‡b 4| Zi½ cÖev‡n gva¨‡gi KYv ¸wj KL‡bv w¯’i Ae¯’v cÖvß nq `yBevi w¯’i Ae¯’v cÖvß nq| bv| 5| w¯’i Zi‡½i wPÎt 5| Pjgvb Zi‡½i wPÎt

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Pjgvb (AMÖMvgx) Zi‡½i mgxKiY cÖwZcv`b Ki ev,

5

2π y  A sin (vt  x) mgxKiY cÖwZcv`b : λ

Pjgvb (AMÖMvgx) Zi‡½i mgxKiY (Equation of travelling or progressive wave) : gva¨‡gi KYv ¸‡jv mij Qw›`Z ¯úw›`Z n‡j Pjgvb Zi‡½i D™¢e nq| GKwU KYv †_‡K Av‡›`vjb cieZ©x KYv‡Z †cŠQ‡Z wKQy mgq jv‡M| myZivs Zi‡½i AwfgyL eivei KYv¸wji `kvi cwieZ©b NU‡Z _v‡K| g‡bKwi, GKwU Pjgvb Zi½ O †_‡K C eivei G¸‡”Q| [cvk©¦ wPÎ] †h‡nZz gva¨‡gi KYv¸‡jv mij Qw›`Z ¯ú›`‡b G¸‡”Q, †m‡nZz O KYvi MwZ‡K wb‡Pi mgxKiY Øviv cÖKvk Kiv hvq| y = a sin t GLv‡b, y = t mg‡q OBC †iLv ev mvg¨ve¯’v †_‡K KYvwUi miY| a = KYvi we¯Ívi  = KYvi †KŠwbK K¤úv¼ hw` KYvwUi K¤úv¼ f nq Z‡e f  y = a sin ft .........................................(1) Avevi O KYvwU hLb mvg¨ve¯’v AwZµg K‡i ZLb B KYvwUI GKB w`‡K mvg¨ve¯’v AwZµg K‡i| myZivs Giv mg`kv m¤úbœ| mg`kv m¤úbœ ci¯úi `ywU KYvi ga¨eZ©x `~iZ¡ n‡”Q Zi½ ˆ`N©¨ | GLv‡b  = OB| GLv‡b O we›`y †_‡K B

we›`y‡Z hvIqvi mgq `kvi cwieZ©b nq | AZGe, O we›`y †_‡K x `~i‡Z¡ P we›`y‡Z hvIqvi mgq `kvi cv_©K¨   2π x λ

GLb P we›`y‡Z Aew¯’Z KYvi miY y n‡j,

y = a sin (t – )

2π    y  a sin  ωt  x   λ   2π   x  y  a sin  2πft  λ    2πvt 2π   y  a sin   x λ    2 vt  x   y  a sin  2 Zi½ Wvb w`K †_‡K evg w`‡K †M‡j KYvwUi miY n‡e, y  a sin vt  x  

w¯’i Zi‡½i mgxKiY cÖwZcv`b ev, y  A sin

 ω  2πf   v   f   

2π vt mgxKiY cÖwZcv`b : λ

w¯’i Zi½: ‡Kvb gva¨‡gi GKwU mxwgZ As‡k mgvb we¯Ívi I Zi½ ‰`‡N©¨i `ywU Pjgvb Zi½ GKB gv‡bi †e‡M wecixZ w`K †_‡K AMÖmi n‡q G‡K Ac‡ii Dci AvcwZZ n‡q †h Zi‡½i D™¢e nq Zv‡K w¯’i Zi½ e‡j|

GKwU Zv‡ii GK cÖvšÍ GKwU `„p Aej¤^‡b †e‡a Ab¨ cÖvšÍ a‡i Dci wb‡P AvovAvwofv‡e †`vjv‡j GKwU Zi½ Zvi †e‡q AMÖmi n‡e Ges e×cÖv‡šÍ cÖwZdwjZ n‡q wd‡i Avm‡e| GB cÖwZdwjZ Zi½ hLb bZzb Pjgvb Zi‡½i Dci AvcwZZ n‡e ZLb w¯’i Zi‡½i D™¢e n‡e| GB Zi½ Zvi †e‡q AMÖmi bv n‡q eis Zv‡ii H As‡ki g‡a¨ Drcbœ I wejyß nq| Zi‡½i D™¢‡ei mgq †`Lv hvq †h, Zv‡ii †Kvb †Kvb RvqMvq, †hgb N1, N2, N3 BZ¨vw` we›`y‡Z †Kvb ¯ú›`b †bB, Avevi †Kvb †Kvb hvqMvq, †hgb A1, A2, A3 BZ¨vw` ¯ú›`b me mgq me©vwaK| †h mg¯Í we›`y‡Z †Kvb ¯ú›`b †bB Zv‡`i‡K wb¯ú›` we›`y (Nodes ) e‡j Ges †h mg¯Í we›`y‡Z ¯ú›`b me©vwaK †m mKj we›`y‡K my¯ú›` we›`y (Antinodes ) e‡j|

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6

g‡b Kwi, GKB we¯Ívi a Ges GKB Zi½ ˆ`N©¨  wewkó `ywU Pjgvb Zi½ †eM v wb‡q GKB Aÿ X eivei ci¯úi wecixZ w`‡K AMÖmi n‡”Q| †h Zi½wU X Aÿ eivei Wvbw`‡K MwZkxj Zvi Rb¨ x we›`y‡Z t mg‡q †Kvb KYvi miY y1 Ges evg w`‡K MwZkxj Zi‡½i Rb¨ H KYvi miY y2 n‡j Avgiv Rvwb, 2 (vt  x )  2 I y 2  a sin (vt  x ) myZivs KYvwUi jwä miY y n‡e  y1  a sin

y = y1 + y2 2 2 ( vt  x )  a sin ( vt  x )   2 2    y  a  sin (vt  x )  sin ( vt  x )    

 y  a sin

 2  vt  x  vt  x  2  vt  x  vt  x   y  2a  sin   cos     2 2     

 2   2   y  2a sin  vt  cos x        2   y  A sin  vt  ... ... ... ... (1)   

GLv‡b we¯Ívi, A  2a Cos

2π x λ

Dc‡iv³ mgxKiYwU w¯’i Zi½ cÖKvk K‡i| wewfbœ we›`y‡Z x Gi gv‡bi Dci wbf©i K‡i we¯Ív‡ii gvb I wewfbœ n‡e| my¯ú›` we›`y (Anti Node): †h mKj we›`y‡Z jwä we¯Ívi m‡Ÿ©v”P A_©vr A  2a n‡e †mB mKj my¯ú›` we›`y ˆZix n‡e| A_©vr †h mKj we›`y‡Z cos

2 x  1 n‡e †mB mKj we›`y‡Z my¯ú›` we›`y ˆZix n‡e| myZivs †h mKj we›`y‡Z 

2x  0, , 2 ............., n, n‡e| †hLv‡b, (n = 0, 1, 2, 3 . . ....... . . .) |  n  2 ev, x  0, , , . . .. . . . . . . , †hLv‡b, (n = 0, 1, 2, 3 . .. . .. . .)

2 2 2 2 4 2n , †hLv‡b, (n = 0, 1, 2, 3 . .. . .. . .) †mB mKj we›`y‡Z my¯ú›` we›`y ˆZix ev, x  0, , , . . .. . . . . . .

4 4 4  n‡e| myZivs w¯’i Zi‡½i †h mKj we›`y Gi ‡Rvo ¸wbZK `~‡i Aew¯’Z †mB mKj we›`y‡Z my¯ú›` we›`y m„wó n‡e| 4

wb¯ú›` we›`y (Node): †h mKj we›`y‡Z jwä we¯Ívi k~Y¨ A_©vr A  0 n‡e †mB mKj we›`y‡Z wb¯ú›` we›`y ˆZix n‡e| A_©vr 2 x  0 n‡e †m mKj we›`y‡Z wb¯ú›` we›`y ˆZix n‡e| myZivs †h mKj we›`y‡Z  2x  3 5   , , ............., (2n  1) , n‡e| †hLv‡b, (n = 0, 1, 2, 3 . . . . . . .) |  2 2 2 2  3 5  ev, x  , , . . .. . . . . . . (2n  1) , †hLv‡b, (n = 0, 1, 2, 3 . . . . . . .) †mB mKj we›`y‡Z wb¯ú›` we›`y 4 4 4 4  ˆZix n‡e| myZivs †h mKj we›`y Gi †e‡Rvo ¸wbZK `~‡i Aew¯’Z †mB mKj we›`y‡Z wb¯ú›` we›`y m„wó n‡e| 4

†h mKj we›`y‡Z cos

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k‡ãi e¨wZPvi (Interference of Sound): GKB w`‡K Pjgvb mgvb K¤úv¼ I we¯Ívi wewkó `ywU kã Zi‡½i DcwicvZ‡bi d‡j †Kvb ¯’v‡b bxieZv I †Kvb ¯’v‡b cÖejZ¡i k‡ãi m„wó n‡j H NUbv‡K k‡ãi e¨wZPvi e‡j| e¨wZPvi `yB cÖKvi| (1) MVbg~jK e¨wZPvi I (2) aŸsmvZ¥K e¨wZPvi (1) MVbg~jK e¨wZPvi (Constructive Interference): k‡ãi e¨wZPv‡i cÖejZ¡i kã n‡j Zv‡K MVbg~jK e¨wZPvi e‡j| (2) aŸsmvZ¥K e¨wZPvi (Destructive Interference): k‡ãi e¨wZPv‡i Av‡`Š‡Kvb kã bv n‡j Zv‡K aŸsmvZ¥K e¨wZPvi e‡j| kã Zi‡½i e¨wZPv‡ii Rb¨ kã kw³i webvk nqbv eis kw³i cybe©›Ub nq| e¨wZPv‡ii MvwbwZK we‡kølb (Mathematical deduction of Interference): g‡bKwi GKB we¯Ívi a Ges GKB Zi½ ˆ`N©¨  wewkó `ywU Pjgvb Zi½ GKB †eM v wb‡q Pj‡Z _vK‡j †Kvb GK mgq Zviv GK we›`y‡Z wgwjZ nq| Zi½ `ywUi Rb¨ H we›`y‡Z Aew¯’Z †Kvb KYvi t mgq c‡i miY h_vµ‡g y1 I y2 n‡j, 2 ( vt  x 1 );  2 y 2  a sin ( vt  x 2 ); GLv‡b, cÖ_g Zi½wU H we›`y‡Z †h‡Z x1 c_ Ges wØZxq Zi½ x2 c_ AwZµg K‡i|  GLb `ywU Zi‡½i DcwicvZ‡bi d‡j G‡`i jwä miY y n‡j, y = y1 + y2 2 2 ev, y  a sin ( vt  x 1 )  a sin ( vt  x 2 )   2 2 ev, y  a sin (vt  x1 )  sin ( vt  x 2 )     2  vt  x1  vt  x 2  2  vt  x 1  vt  x 2  ev, y  2a sin   cos   2 2       2 x x  ev, y  2a cos x 2  x 1  sin  vt  1 2  2     2  x x   ev, y  A sin  vt  1 2  GLv‡b, we¯Ívi A  2a cos x 2  x 1     2  y1  a sin

Dc‡iv³ mgxKiY †_‡K †`Lv hvq †h, `ywU Zi‡½i DcwicvZ‡bi d‡j GKwU bZzb Zi½ Drcbœ nq hvi we¯Ívi A  2a cos

 x 2  x 1  Ges gvb Drm؇qi c_ cv_©‡K¨i Dci wbf©i K‡i| 

MVbg~jK e¨wZPvi (Constructive Interference) : †h mKj we›`y‡Z Zi½ `ywU GKB `kvqA_©vr mg `kvq wgwjZ n‡e, †m me we›`y‡Z jwä we¯Ívi m‡e©v”P n‡e A_©vr A  2a n‡e| d‡j k‡ãi ZxeªZv m‡e©v”P n‡e ev MVbg~jK e¨wZPvi Drcbœ n‡e|  

myZivs †h mKj we›`y‡Z, cos ( x 2  x 1 )  1 n‡e|  (x 2  x 1 )  0, , 2, 3. ... ....... n n‡e [ n = 0, 1, 2.3, ........]  ev, (x 2  x1 )  0, , 2, 3, ... ....... n n‡e [ n = 0, 1, 2, 3, ........]

2 4 6   ( x 2  x1 )  0, , , , ... ....... 2n n‡e [ n = 0, 1, 2, 3, ........]

2 2 2 2  A_©vr †h mKj we›`y‡Z Zi½ `ywUi c_ cv_©K¨ Gi †Rvo ¸wbZK n‡e †mB mKj we›`y‡Z MVbg~jK e¨wZPvi Drcbœ n‡e| 2

ev,

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aŸsmvZ¥K e¨wZPvi (Destructive Interference) : †h mKj we›`y‡Z Zi½ `ywU wecixZ `kvq wgwjZ n‡e, †m me we›`y‡Z jwä we¯Ívi k~b¨ n‡e A_©vr A  0 n‡e| d‡j k‡ãi ZxeªZv k~b¨ n‡e ev aŸsmvZ¥K e¨wZPvi Drcbœ n‡e|     3 5  ev, ( x 2  x 1 )  , , , . ... .......(2n  1) n‡e [ n = 0, 1, 2, 3, ........]

2 2 2  2  3 5   ( x 2  x 1 )  , , , ... ....... (2n  1) n‡e [ n = 0, 1, 2, 3, ........]

2 2 2 2  A_©vr †h mKj we›`y‡Z Zi½ `ywUi c_ cv_©K¨ Gi we‡Rvo ¸wbZK n‡e, †mB mKj we›`y‡Z aŸsmvZ¥K e¨wZPvi m„wó n‡e| 2

myZivs †h mKj we›`y‡Z, cos ( x 2  x 1 )  0 n‡e|

k‡ãi e¨wZPvi cÖ`k©‡bi cixÿv (Experiment on Interference) : KzB‡Ki cixÿv Øviv GKwU kã Zi½‡K †Kvb GKwU we›`y ‡_‡K `ywU Lv‡Z cÖevwnZ n‡Z w`‡q Dchy³ `kv cv_©‡K¨ cybivq Aci GKwU we›`y‡Z AvcwZZ K‡i k‡ãi e¨wZPvi m„wó Kiv nq| [wPÎ wb‡¤§] GB cixÿvq U AvK…wZi `yB gyL †Lvjv bj ABP I DEF jIqv nq| ABP b‡ji `yB evû‡Z `ywU cvk¦ bj M I N Av‡Q| DEF b‡ji `yB evûi g‡a¨ ABP b‡ji evû `ywU cÖ‡ek Kiv‡bv hvq| cixÿv: GKwU myikjvKv‡K kãvwqZ K‡i M b‡ji gy‡L aiv nq| G‡Z myi kjvKv †_‡K kã Zi½ AB I DEF cÖevwnZ n‡q N bj w`‡q †ei n‡q hvevi mgq P we›`y‡Z wgwjZ n‡e| GB `yB c‡_ cÖevngvb Zi‡½i K¤úv¼, we¯Ívi I RvwZ Awfbœ nIqvq Giv N b‡j GKB †iLvq miY m„wó Ki‡e| GLb DEF bjwU‡K evB‡ii w`‡K †U‡b A_ev wfZ‡ii w`‡K †V‡j ABP I AEP c‡_i `~i‡Z¡i cv_©K¨ evov‡j ev Kgv‡j N b‡ji gy‡L k‡ãi wZeªZvi wb¤§wjwLZ cwieZ©b jÿ¨ Kiv hvq| (K) hLb ABP I AEP Gi g‡a¨ c‡_i ˆ`‡N©¨i cv_©K¨ Zi½ ˆ`‡N©¨i AhyM¥ ¸wbZK n‡e ZLb (AEP~ABP) = /2, 3/2, 5/2 BZ¨vw` n‡e ZLb ZLb Zi½ `ywU P we›`y‡Z wecixZ `kvq wgwjZ nIqvq N b‡ji gy‡L †Kvb kã †kvbv hv‡ebv| GUvB k‡ãi aŸskKvix e¨wZPvi| (L) hLb ABP I AEP Gi g‡a¨ c‡_i ˆ`‡N©¨i cv_©K¨ k~b¨ ev Zi½ ˆ`‡N©¨i hyM¥ ¸wbZK n‡e ZLb (AEP~ABP) = 0, 2/2, 4/2, BZ¨vw` n‡e ZLb Zi½ `ywU P we›`y‡Z mg`kvq wgwjZ nIqvq N b‡ji gy‡L ‡Rvivj kã †kvbv hv‡e| GUvB k‡ãi MVb g~jK e¨wZPvi| †h †h Kvi‡Y kã (Pjgvb) AMÖMvgx jw¤^K Zi½ ev, kã GKwU AMÖMvgx jw¤^K Zi‡½i cÖgvY (Prof of Sound is a Kind of longitudinal wave): kã GKwU AMÖMvgx jw¤^K Zi½ Gi ¯^c‡ÿ wb¤§wjwLZ NUbv mg~n cwijwÿZ nq| (1) Zi½ m„wói Rb¨ e¯‘i K¤úb cÖ‡qvRb kã m„wói Rb¨ I e¯‘i K¤úb cÖ‡qvRb| (2) Zi½ mÂvj‡bi Rb¨ gva¨g ¯’vbvšÍwiZ nq bv kã mÂvj‡bi Rb¨ I gva¨g ¯’vbvšÍwiZ nq bv| (3) Zi½ mÂvj‡bi Rb¨ mg‡qi cÖ‡qvRb kã mÂvj‡bi Rb¨ I mg‡qi cÖ‡qvRb| (4) cÖ‡Z¨K Zi‡½i †ÿ‡Î cÖwZdjb, cÖwZmiY, e¨wZPvi Ges AceZ©b nq| k‡ãi †ÿ‡Î I G me nq| (5) M¨vwmq gva¨‡g kã Zi½vKv‡i mÂvwjZ nq| Gi mgeZ©b nq bv| mgeZ©b †Kej gvÎ Avo Zi‡½ nq| Dc‡iv³ NUbv Øviv cÖgvwYZ nq †h, kã GKwU AMÖMvgx jw¤^K Zi½|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

17| Zi½ I kã (Waves & Sound) 1| evqy I cvwb‡Z 300 Hz K¤úv‡¼i GKwU kã Zi‡½i Zi½ ˆ`‡N¨©i cv_©K¨ 4.18m| evqy‡Z k‡ãi †eM 350ms-1 n‡j cvwb‡Z k‡ãi †eM KZ? GLv‡b, Avgiv Rvwb,

K¤úv¼, f = 300Hz  w   a  4.18 evqy‡Z k‡ãi v v  w  a  4.18 †eM, va = 350 ms-1 f f cwb‡Z k‡ãi †eM, Vw = ? vw va   4.18  wa = 4.18 m f f v 350  w  4.18  300 300  vw  4.18  1.1666666300 1

 v w  1604 ms (Ans.) 2| `yÕwU myikjvKvi K¤úv¼ h_vµ‡g 128Hz Ges 384Hz| evqy‡Z kjvKv `zÕwU n‡Z m„ó Zi½ ˆ`‡N©¨i AbycvZ wbY©q Ki| Avgiv Rvwb, GLv‡b, V = f11= f22

λ1 f 2  λ 2 f1 λ 384  1  λ 2 128  λ1 : λ 2  3 : 1 (Ans.) 

K¤úv¼, f1= 128Hz K¤úv¼, f2 = 384 Hz 1t2 =?

f 



V 

VP VQ  λP Q

300 350  λ Q  0.1 λ Q

 350λ Q  35  300λ Q

y  ASin(2πft 

GLv‡b, k‡ãi †eM, VP=300ms-1 k‡ãi †eM,VQ=350ms-1 Zi½ ‰`N©¨ cv_©K¨, QPm

 PQ 50Q= ?

 50λ Q  35m (Ans.)

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2π x) .......(2) Av`k© mgxKiY Gi mv‡_ λ

Zzjbv K‡i cvB, we¯Ívi, A = 5m 2f = 200  K¤úv¼, f = 200/2 Hz = 100 Hz

2π 2  3.14 2π  1.57  λ   1.57 1.57 λ

 Zi½ ˆ`N©¨, m  Zi½ †eM, V = f100 × 4 m /s = 400 m/s (Ans.) 1 1 s  0.01s (Ans) ch©vqKvj, T   f 100 5| evZv‡m `yÕwU myi kjvKvi Øviv m„ó k‡ãi Zi½ ˆ`N©¨ h_vµ‡g 50cm Ges 70cm cÖ_g myi kjvKvi K¤úvsK 350Hz n‡j wØZxq myi kjvKvi K¤úv¼ KZ n‡e?

GLv‡b, Avgiv Rvwb, Zi½ ‰`N©¨, 1= 50cm V = f1f22 f1 λ 1 Zi½ ‰`N©¨, 2 = 70cm  f2  λ2 K¤úvsK, f1 = 350Hz  f2 

3| P I Q `yÕwU gva¨‡g k‡ãi †eM h_vµ‡g 300ms-1 Ges 350ms-1 | gva¨g `ywU‡Z k‡ãi Zi½ ˆ`‡N©¨i cv_©K¨ 0.1m n‡j myi kjvKvi 50 K¤ú‡b kã Q gva¨‡g KZ`yi hv‡e?

Avgiv Rvwb, V = f  f 

4| GKwU Pjgvb Zi‡½i mgxKiY t y = 5 Sin(200t1.57x) ; GLv‡b me KqwU ivwk Gm AvB GK‡K cÖ`Ë| Zi½wUi we¯Ívi, K¤úv¼, †eM I ch©vq Kvj wbY©q Ki| y = 5 Sin(200t1.57x) .............(1) cÖ`Ë mgxKiY

350  50 Hz 70

K¤úvsK, f2 = ?

 f 2  250 Hz (Ans.)

6| †Kvb myi kjvKv GKwU gva¨‡g 5cm ˆ`‡N©¨i Ges 350ms-1 †e‡Mi Zi½ Dcbœ K‡i| Aci GKwU gva¨‡g Zi½ †eM hw` 332.5ms-1 nq Z‡e H gva¨‡g myi kjvKvi 100 K¤ú‡b kã KZ`yi hv‡e| Avgiv Rvwb, GLv‡b, k‡ãi †eM, V1=350ms-1 V1 V2  f   Zi½ ‰`N©¨ 1= 5cm λ1 λ 2 =0.05m 350 332.5 -1 k‡ãi †eM, V =332.5ms 2   0.05 λ2 100 K¤ú‡b AwZµvšÍ `~iZ¡ 332.5  0.05 1002=?

 2 

m 350 332.5  0.05  100 m  1002  350 100λ 2  4.75 m (Ans.)

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cÖ_g c‡Îi As‡Ki mgvavb 100  50m (Ans.)

7| †Kvb gva¨‡g 480Hz Ges 320Hz K¤úv‡¼i `yÕwU k‡ãi Zi½ ˆ`‡N©¨i cv_©K¨ 2m n‡j gva¨‡g k‡ãi †eM KZ| Avgiv Rvwb, GLv‡b, 2  1  2 K¤úv¼, f1= 480Hz V V K¤úv¼, f2 = 320Hz   2 f 2 f1 λ  λ  2m 2

V V   2 320 480 480 V  320 V  2  320  480 160V  2 320  480

1

k‡ãi †eM, V=?



 3. 9  f f vw 345   3. 9  320 320  v w  3.9  1.078125320 

 160V  2  320  480 V

10| 320 Hz K¤úv‡¼i myi kjvKv n‡Z cvwb‡Z I evqy‡Z Drcbœ Zi‡½i Zi½ ˆ`‡N¨©i cv_©K¨ 3.9m| evqy‡Z k‡ãi †eM 345ms-1 n‡j cvwb‡Z k‡ãi †eM KZ? Avgiv Rvwb, GLv‡b,  w   a  3.9 K¤úv¼, f = 320Hz v w va evqy‡Z k‡ãi †eM, Va = 345 ms-1    3. 9 cwb‡Z k‡ãi †eM, Vw = ? f f wa = 3.9m vw va

2  320  480 160

V  1920 ms1 (Ans.)

 v w  1593 ms 1 (Ans.)

8| †Kvb myi kjvKvi K¤úv¼ 700Hz, evqyi ZvcgvÎv 30°C n‡j 100 K¤ú‡b kã KZ`yi hv‡e| 0°C ZvcgvÎvq k‡ãi †eM 332ms-1 GLv‡b, v   v 0 (1  ) k‡ãi †eM, vo = 332ms-1 1  ZvcgvÎv,  = 30ºC   30  v   332 1  100 = ?

11| `yÕwU myi kjvKvi K¤úv‡¼i cv_©K¨ 118 Hz| evqy‡Z kjvKv `yÕwU ‡h Zi½ m„wó K‡i, Zv‡`i GKwUi `yÕwU c~Y© Zi½ ˆ`N©¨ AciwUi wZbwU c~Y© Zi½ ˆ`‡N©¨i mgvb| kjvKv ؇qi K¤úv¼ wbY©q Ki|



273



 v   332 (1  0.109890109 )  v   332 1.10989011

 v   349.76ms 1 Avevi,

v   f  349.76  700 349.76  700 349.76  100  100  700 100  49.96m (Ans.)

VA VB  A B V  B  A  A VB 5V  0. 1  A  V   A  0.5m  100  100  0.5m

Avgiv Rvwb,

V  f1 1  f 2  2 1 f 2 2 f  118    1  2 f1 3 f1  3f1  354  2f1  f1  354Hz I f 2  (354  118)Hz  236Hz 12| GKwU Zv‡ii Dc‡ Drcbœ AbycÖ¯’ Zi‡½i

x   t  , GLv‡b xGes y †mw›UwgUv‡i  0.5 50 

mgxKiY y  0.5 sin 2 

Ges t †m‡K‡Û cÖKvk Kiv n‡q‡Q| Zi½wUi we¯Ívi, Zi½ ˆ`N©¨, K¤úvsK I ch©vqKvj wbY©q Ki|

9| A gva¨‡g k‡ãi †eM B gva¨‡g k‡ãi †e‡Mi †P‡q 5 ¸Y †ewk| B gva¨‡g GKwU k‡ãi Zi½ ˆ`N©¨ 10cm n‡j A gva¨‡g Dr†mi 100 evi K¤ú‡b kã KZ`yi hv‡e?

Avgiv Rvwb, f 

f1  f 2  118Hz  f 2  (f1  118)Hz  2 3 1  2 2  1  2 3

GLv‡b, awi, k‡ãi †eM, VB=V  k‡ãi †eM,VA=5V Zi½ ‰`N©¨,B=10cm=0.m 100A= ?

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x   t    0.5 50   2t 2x   y  0.5 sin    ‡K  0.5 50  2x   Av`k© mgxKiY, y  a sin  2ft   Gi mv‡_ Zzjbv    1 K‡i cvB, we¯Ívi a=0.5cm,I f  Hz  2 Hz 0. 5 1 1 1 1     50cm T    0.5Sec( Ans.) I f 2  50

GLv‡b cÖ`Ë mgxKiY: y  0.5 sin 2 

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k‡ãi ZxeªZv (Intensity of Sound): kã mÂvj‡bi c‡_ j¤^fv‡e Aew¯’Z GKK †ÿÎd‡ji ga¨w`‡q cÖwZ †m‡K‡Û cÖevwnZ kã kw³i cwigvb †K k‡ãi ZxeªZv e‡j| A ‡ÿÎd‡j cÖwZ †m‡K‡Û kã kw³i cwigvb P n‡j k‡ãi ZxeªZv I 

P | Gi GKK Wm-2 A

cÖgvY ZxeªZv (Standard Intensity): 1000 Hz K¤úv¼ wewkó 10 -12 Wm-2 ZxeªZv‡K cÖgvY ZxeªZv e‡j| ZxeªZv †j‡ej (Intensity Level): †Kvb k‡ãi ZxeªZv I cÖgvY ZxeªZvi Abycv‡Zi jMvwi`g‡K †ej GK‡K H k‡ãi ZxeªZv †j‡ej e‡j| †Kvb k‡ãi ZxeªZv I Ges cÖgvY ZxeªZv Io n‡j ‡ej GK‡K ZxeªZv †j‡ej n‡e,   log I Bell I0

ZxeªZv †j‡e‡ji GKK †ej ev †Wwm‡ej| †ej ev †Wwm‡ej (Bel or Desibel): †Kvb k‡ãi ZxeªZv I cÖgvY ZxeªZvi Abycv‡Zi jMvwi`g‡K †ej GK‡K H k‡ãi ZxeªZv †j‡ej e‡j| ZxeªZv †j‡e‡ji GKK †ej ev †Wwm‡ej| cÖgvY ZxeªZv †_‡K 10 ¸b ZxeªZv m¤úbœ †Kvb k‡ãi ZxeªZv †j‡ej †K 1 †ej e‡j| GK †e‡ji `k fv‡Mi GK fvM‡K GK †Wwm‡ej e‡j| †Kvb k‡ãi ZxeªZv I Ges cÖgvY ZxeªZv Io n‡j †Wwm‡ej GK‡K ZxeªZv †j‡ej n‡e,   10log I dB I0

myi (Tone) : ‡Kvb Drm †_‡K wbtm„Z k‡ã hw` GKwU gvÎ K¤úv¼ _v‡K Zvn‡j †mB kã‡K myi (Tone) e‡j| †hgb, myikjvKv †_‡K wbtm„Z kã, KviY Gi GKwUB K¤úv¼| ¯^i (Note) : ‡Kvb Drm †_‡K wbtm„Z k‡ãi g‡a¨ hw` GKvwaK K¤úv¼ _v‡K Zvn‡j †mB kã‡K ¯^i (Note) e‡j| A_©vr ¯^i n‡”Q GKvwaK my‡ii mgwó| Avgiv †h Mvb evRbv ïwb Zv ¯^i, KviY Zv A‡bK ¸‡jv my‡ii mgwó| ‡gjwW ev ¯^igvayh© (Melody): hw` K‡qKwU kã G‡Ki ci GK D”PvwiZ n‡q GKwU myihy³ k‡ãi m„wó K‡i Z‡e Zv‡K ‡gjwW ev ¯^igvayh© e‡j| k‡ãi ¸Y ev RvwZ: (Quality or Timbre of sound): ‡h ‰ewk‡ó¨i Øviv GKB ZxeªZv I ZxÿœZvi `ywU kã‡K ci¯úi †_‡K Avjv`v Kiv hvq Zv‡K k‡ãi ¸Y ev RvwZ e‡j? GK mv‡_ K‡qKwU ev`¨hš¿ †hgb †envjv, wMUvi, euvwk BZ¨vw` hw` GKB ZxeªZv I ZxÿèZvq evRv‡bv nq Zvn‡jI Avgiv †Kvb& myiwU †Kvb& h‡š¿i Zv Avgiv mn‡RB eyS‡Z cvwi| k‡ãi ¸Y ev RvwZi Rb¨ GUv n‡q _v‡K| †gŠwjK myi (Fundamental Tone): †Kvb ¯^‡ii g‡a¨ we`¨gvb myi ¸‡jvi g‡a¨ hvi K¤úv¼ me‡P‡q Kg Zv‡K g~j myi ev †gŠwjK myi e‡j| †hgb †Kvb AM©vb †_‡K wbtm„Z wb‡b¥v³ K¤úv¼ ¸‡jv h_vµ‡g 256, 268, 502, 512, 620, 768, 1280 Hz | Gi g‡a¨ 256 Hz g~j myi| Dcmyi I nvi‡gvwbK (Overtone and Harmonic): †Kvb ¯^‡ii g‡a¨ we`¨gvb myi ¸‡jvi g‡a¨ hvi K¤úv¼ me‡P‡q Kg Zv‡K g~j myi ev †gŠwjK myi e‡j| Ab¨ mKj myi hvi K¤úv¼ g~j my‡ii †P‡q †ekx Zv‡`i‡K Dcmyi e‡j| †hgb †Kvb AM©vb †_‡K wbtm„Z wb‡b¥v³ K¤úv¼ ¸‡jv h_vµ‡g 200, 250, 400, 475, 600, 720, 800, 1280 Hz | Gi g‡a¨ 200 Hz g~j myi| 200 Hz K¤úv¼ Qvov Ab¨ mKj K¤úv¼ †hgb 250, 400, 475, 600, 720, 800, 1280 Hz K¤úv¼ Dcmyi| Avevi Dc myi ¸‡jvi K¤úv¼ hw` g~jmy‡ii K¤úv‡¼i mij ¸wbZK nq, Zvn‡j †mB mKj Dcmyi‡K mg‡gj ev nvi‡gvwbK e‡j, †hgb 200 Hz Gi wظb 400 Hz, wZb ¸b 600 Hz, Pvi ¸b 800 Hz, Kv‡RB 400 Hz, 600 Hz I 800 Hz K¤úv¼ ¸wj nvi‡gvwbK| d‡j mKj nvi‡gvwbK Dcmyi wKš‘ mKj Dcmyi nvi‡gvwbK bq| Dcmyi hw` g~j my‡ii K¤úv‡¼i wظb nq Z‡e wØZxq nvi‡gvwbK, wZb¸b n‡j Z…Zxq nvi‡gvwbK, Pvi¸b n‡j PZz_© nvi‡gvwbK BZ¨vw` e‡j| Avevi †Kvb my‡ii K¤úv¼ hw` Ab¨ GKwU my‡ii K¤úv‡¼i wظb nq Zvn‡j wØZxqwU‡K cÖ_gwUi AóK (Octave) e‡j †hgb, 400 Hz, 200 Hz Gi AóK, 800 Hz, 400 Hz Gi AóK|

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2 18| kã (Sound) myi hy³ I myi ewR©Z kã (Musical Sound and Noise): ‡h mg¯Í kã Avgv‡`i ïb‡Z fvj jv‡M Zv‡`i‡K Avgiv mykªve¨ ev myimg„× kã e‡j| Avevi †h ¸‡jv Avgv‡`i Kv‡Q weiw³Ki Ae¯’vi m„wó K‡i Zv‡`i‡K Kjie ev myi ewR©Z kã e‡j| myi weivg (Musical Intervel): `ywU my‡ii K¤úv‡¼i AbycvZ‡K myi weivg e‡j| aiv hvK, A,B,I C wZbwU my‡ii K¤úv¼

h_vµ‡g f1,f2I f3 BZ¨vw`| Zv n‡j, BI AGi g‡a¨ myi weivg  g‡a¨ myi weivg n‡e,

f f2 CI BGi g‡a¨ myi weivg  3 GB Ae¯’vq C I AGi f1 f2

f3 f f  3  2 myZivs †`Lv hv‡”Q †h, `ywU k‡ãi myi weivg G‡`i ga¨eZ©x myi weivg ¸‡jvi ¸b f1 f 2 f1

d‡ji mgvb| mgZvb (Harmony): KZ¸‡jv kã hw` GK m‡½ Drcv`b n‡q HK¨Zv‡bi m„wó K‡i, Z‡e Zv‡K mgZvb e‡j| ¯^iMÖvg (Musical Scale): ¯^iMÖvg ej‡Z Avgiv wbw`©ó K¤úv¼ ev ZxÿèZvi K‡qKwU mvRv‡bv myi‡K eywS| †h †Kvb myi I Zvi AóK ev wظY K¤úv¼wewkó my‡ii g‡a¨ K‡qKwU wbw`©ó myi Avgv‡`i Kv‡b mn‡R mvov †`q| GB myi ¸‡jvi g‡a¨ mgmsMwZ eRvq _v‡K e‡j Giv m½xZ ¸Ym¤úbœ nq| Giƒc mgmsMwZc~Y© myimgwó‡K ¯^iMÖvg e‡j| ¯^iMÖv‡gi me‡P‡q †QvU K¤úv‡¼i m~Pbv myi‡K †UvwbK ev cÖavb myi e‡j| GB ¯^iMÖv‡g AvUwU µgea©gvb K¤úv‡¼i mgmsMwZc~Y© myi _v‡K e‡j G‡K Wvqv‡UvwbK ¯^iMÖvg e‡j| evsjvq ¸‡jv h_vµ‡g mv †i Mv gv cv av wb mv | exU (Beat): cÖvq mgvb ZxeªZv Ges K¤úvsK wewkó `yÕwU Drm †_‡K GKB mgq kã Drcbœ Ki‡j Giv ci¯ú‡ii mv‡_ wg‡j GKwU jwä kã m„wó K‡i| GB jwã k‡ãi ZxeªZv †Kvb g~û‡Z© n«vm cvq Ges †Kvb g~û‡Z© e„w× cvq| k‡ãi ZxeªZvi GB ch©vqµwgK n«vm e„wׇK exU ev ¯^iK¤ú ev AwaK¤ú e‡j| exU†K N Øviv cÖKvk Kiv nq| `yÕwU K¤úbkxj e¯‘i K¤úv¼ h_vµ‡g n1 I n2 n‡j Ges n1 > n2 n‡j exU N = n1 - n2 n‡e| Avi hw` n2> n1nq Z‡e exU N = n2 - n1 n‡e| A_©vr exU N = n1 ~ n2| `yÕwU K¤úbkxj e¯‘i K¤úv‡¼i cv_©K¨B exU| exU MV‡bi †KŠkj (Formation of Beat): cÖvq mgvb ZxeªZv I K¤úv¼ wewkó `ywU myikjvKv †bIqv nq| GLb G‡`i‡K ivevi c¨vWØviv AvNvZ Ki‡j kã

[wPÎ bs - 1]

Drcbœ n‡q gva¨‡gi wfZi w`‡q mÂvwjZ n‡Z _vK‡e| Gi d‡j gva¨‡gi †Kvb GK we›`y‡Z Zi½ `ywU †Kvb GK mgq mg`kvq wgwjZ n‡e [ wPÎ bs - 1]| gva¨‡gi †h we›`y‡Z Zi½ `ywU GKB `kvq wgwjZ nq †mLv‡b Dcwi cvZ‡bi d‡j jwä Zi‡½i we¯Ívi Zi½ ؇qi we¯Ív‡ii †hvM d‡ji mgvb n‡e d‡j k‡ãi cÖvej¨ ‡e‡o hv‡e| wP‡Î Zi½ `ywU‡K miæ †iLv Øviv I jwä Zi½‡K †gvUv †iLv Øviv †`Lv‡bv n‡q‡Q| †h‡nZz mg‡qi mv‡_ mv‡_ Zi½Øq GwM‡q hvq ZvB cªwZwbqZ Zi½Ø‡qi `kvi cwieZ©b nq| ZvB hLb wecixZ `kvq wgwjZ n‡e ZLb jwä Zi‡½i we¯Ívi Zi½ ؇qi we¯Ív‡ii we‡qvM d‡ji mgvb n‡e d‡j k‡ãi cÖvej¨ K‡g hv‡e| Gfv‡e jwä Zi‡½i ch©vq µwgK n«vm e„w× N‡U exU Drcbœ nq|

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18| kã (Sound)

3

MvwbwZK we‡køl‡bi Øviv exU m„wói e¨L¨v (Mathematical Analysis of Beat): g‡bKwi mgvb we¯Ívi I K¤úv‡¼i mvgvb¨ cv_©K¨ wewkó `ywU Zi½ GKB w`‡K AMÖmi n‡”Q| t mgq c‡i †Kvb wbw`ó we›`y‡Z Zi½Ø‡qi miY h_vµ‡g Y1 I Y2 n‡j, Y1  a sin 2 n1 t Y2  a sin 2 n 2 t GLv‡b, a = Zi‡½i we¯Ívi, n1 I n2 n‡jv Zi½ `ywUi K¤úv¼ (n1 > n2) DcwicvZ‡bi bxwZ Abymv‡i, mgq t †Z jwä miY Y = Y1+Y2

 Y  a sin 2 n 1 t  a sin 2 n 2 t  Y  a( sin 2  n 1 t  sin 2  n 2 t)  n  n2   n  n2   Y  2a sin 2 1  t cos 2 1 t  2   2   n  n2   Y  A sin 2 1  t ......................(1)  2  n1  n 2  t  2 

(1) bs mgxKiY mij Qw›`Z ¯ú›`b Zi‡½i mgxKiY hvi we¯Ívi A  2a cos 2π

 n1  n 2   t  1 n‡e|  2 

GLb jwä we¯Ívi A Gi gvb me©vwaK n‡e A_©vr A=  2a n‡e hLb cos 2π

 n  n2   2π 1  t  0, π,2π .............mπ BZ¨vw`  2  1 2 m  t  0, BZ¨vw` , ............. n 1  n 2 n1  n 2 n1  n 2 G mgq kã Zi‡½i we¯Ívi me‡P‡q †ekx n‡e A_©vr  2a n‡e| n n  Avevi jwä Zi‡½i we¯Ívi me‡P‡q Kg n‡e A_©vr A= 0 n‡e, hLb cos 2π 1 2  t  0 n‡e|  2  π 3π 5π π  n  n2  ( m = 0,1,2,3 BZ¨vw` ) 2π 1  t  , , ..................(2m  1) 2 2 2 2  2  1 3 5 t , ...... BZ¨vw` | , 2(n 1  n 2 ) 2(n 1  n 2 ) 2(n 1  n 2 )

G mgq kã Zi‡½i we¯Ívi me †P‡q Kg n‡e A_©vr k~b¨ n‡e Ges kã Kv‡b †kvbv hv‡e bv| 3 1 1   2(n1  n 2 ) 2(n 1  n 2 ) n 1  n 2 1 1 †m‡KÛ| 0  n1  n 2 n1  n 2

AZGe cici `ywU wbtkã AeKvk = Avevi cici `ywU cÖej k‡ãi AeKvk = A_©vr

1 n1  n 2

†m‡KÛ|

†m‡K‡Û nq 1 wU exU

1''

'' (n1  n2) wU exU| A_©vr exU msL¨v K¤úv‡¼i cv_©‡K¨i mgvb|

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18| kã (Sound)

4

exU Mbbv K‡i myikjvKvi ARvbv K¤úv¼ wbb©q (Determination of unknown frequency by counting Beat): ARvbv K¤úv¼ wbb©‡qi Rb¨ `ywU myikjvKv †bIqv nq| G‡`i K¤úv¼ h_vµ‡g n1 I n2| n2 Rvbv Av‡Q| n1 AvRvbv K¤úv¼ | n1 wbb©q Ki‡Z n‡e|

cixÿv c×wZ (Procedure): cÖ_‡g myikjvKv `yÕwU‡K GKB mv‡_ AvNvZ K‡i †Uwe‡ji Dci aiv nq| exU ev ¯^iK¤ú m„wó n‡j cÖwZ †m‡K‡Û exU Mbbv Kiv nq| wnmve I Mbbv (Calculation): g‡bKwi cÖwZ †m‡K‡Û Drcbœ ex‡Ui msL¨v = N  Avgiv cvB, N = n1 ~ n2 GLb ARvbv K¤úv¼ n1 Rvbv K¤úv¼ n2 A‡cÿv †QvU ev eo n‡Z cv‡i| myZivs ARvbv K¤úv¼ n1 = n2 ± N

GLb ARvbv K¤úv¼ n1 Gi gvb n 2  N ev n 2  N †KvbwU n‡e Zv wbb©‡qi Rb¨ fvi evov‡bv c×wZ wb‡¤§ ewb©Z nBj| fvi evwo‡q ev †gvg jvwM‡q: cixÿvaxb myikjvKvi Mv‡q wKQy †gvg jvwM‡q kjvKv `ywU‡K GK‡Î kãvwqZ K‡i cÖwZ †m‡K‡Û Drcbœ ex‡Ui msL¨v Mbbv Kiv nq| †gvg jvMv‡bvi d‡j myi kjvKvi fvi evo‡e d‡j Gi ¯^vfvweK K¤úv¼ Kg‡e| G‡Z exU N Gi †P‡q Kg‡ZI cv‡i evo‡ZI cv‡i| GB cixÿvq ex‡Ui msLv N Gi †P‡q evo‡j eyS‡Z n‡e ARvbv K¤úv¼ n1 Av‡MB Kg wQj GLb †gvg jvMv‡bvi d‡j K¤úv¼ AviI K‡g hvIqvq `yB myikjvKvi K¤úv‡¼i cv_©K¨ e„w× cvq| d‡j n1<n2 , AZGe n1 = n2 N n‡e| Avevi †gvg jvMv‡bvi d‡j ex‡Ui msLv N Gi †P‡q Kg‡j eyS‡Z n‡e ARvbv K¤úv¼ n1 Av‡M †ekx wQj GLb †gvg jvMv‡bvi d‡j K¤úv¼ K‡g hvIqvq `yB myikjvKvi K¤úv‡¼i cv_©K¨ K‡g hvq| d‡j n1>n2 AZGe n1 = n2 N n‡e| wm×všÍ : AÁvZ K¤úv‡¼i myi kjvKvi Mv‡q fi hy³ Ki‡j hw` ex‡Ui msL¨v e„w× cvq Z‡e ARvbv K¤úv¼ Rvbv K¤úv¼ A‡cÿv Kg n‡e Ges ex‡Ui msL¨v Kg n‡j ARvbv K¤úv¼ Rvbv K¤úv‡¼i †P‡q †ekx n‡e| evqyi ga¨w`‡q kã mÂvj‡bi †KŠkj (Mechanism for the propagation of sound in air): GKwU wUDwbs dK© ev myi kjvKv jB| wUDwbs dK© Øviv ivevi c¨v‡W AvNvr Ki‡j Gi evû `yÕwU mvg‡b I wcQ‡b AMÖmi nq| hLb †Kvb GKwU evû OB (wPÎ bs -1 ) mvg‡bi w`‡K AMÖmi nq A_©vr A Ae¯’vb †_‡K C Ae¯’v‡b hvq ZLb Bnv mvg‡bi evqy¯Íi¸wj‡K µ‡g µ‡g AvNvZ K‡i| d‡j evqy¯Íi ¸wjI H fv‡e msKzwPZ nq Ges kw³ †kl bv nIqv ch©šÍ GB ms‡KvPb GK ¯Íi †_‡K Ab¨ ¯Í‡i mÂvwjZ n‡Z _v‡K| ÔKÕ wP‡Î OB evûi mvg‡b KZK¸wj mg`~ieZ©x mgvšÍivj †iLv AsKb K‡i KZK¸wj GKB ai‡bi evqy¯Íi wb‡`©k Kiv n‡q‡Q| ÔLÕ wP‡Î Nb ‡iLv ¸wj Øviv PQ As‡k kã

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18| kã (Sound)

5

cÖev‡n ms‡KvPb N‡U‡Q wb‡`©k Kiv n‡q‡Q| evûwU hLb wcQ‡bi w`‡K AMÖmi nq A_©vr C Ae¯’vb †_‡K A Ae¯’v‡b hvq ZLb AvNvZ ev av°v AcmvwiZ nq| evqy¯Íi ¸wj w¯’wZ¯’vcKZv a‡g©i Rb¨ c~‡e©i Ae¯’v‡b wd‡i Av‡m| ÔMÕ wP‡Î PQ As‡k cÖmviY N‡U‡Q †`Lv‡bv n‡q‡Q Ges c~‡e©i PQ As‡ki we¯Z…Z msKzwPZ ¯Íi AMÖmi n‡q QR As‡k wM‡q‡Q †`Lv‡bv n‡q‡Q| GB ms‡KvPb I cÖmviY evq~i NbZ¡ I w¯’wZ ¯’vcKZvi wbf©i K‡i| wUDwbs d‡K©i Ab¨evûi †ÿ‡ÎI GKB Ae¯’v N‡U| A_©vr kã mÂvj‡bi mgq evqy ¯Íi ¸wj GKevi msKywPZ Avi GKevi cÖmvwiZ nq| AZGe wm×všÍ GB †h, Ò evqy¯Í‡ii ms‡KvPb I cÖmvi‡b kã evqy¯Íi †K Aej¤^b K‡i jw¤^K Zi½vKv‡i AMÖmi nq Ges †kªvZvi Kv‡b †cŠwQ‡q kÖæwZi Abyf~wZ RvMvq ev RvMv‡Z †Póv K‡i| BnvB evqyi ga¨ w`‡q kã mÂvj‡bi †KŠkj| UvbvZv‡i Avo K¤ú‡bi †ÿ‡Î, V 

T 1 T cÖgvY: ev, f  μ 2l μ

T Uv‡b ivLv CC ZviwU‡K ˆ`‡N©¨i mv‡_ mg‡Kv‡Y †U‡b †Q‡o w`‡j Zv‡i Avo K¤ú‡bi D™¢e n‡e| d‡j Zv‡ii weP¨yZ As‡ki kxl© AEB GKwU e„ËPv‡ci Ask aviY Ki‡e [wPÎ cv‡k¦©] awi Avo Zi½ evg †_‡K Wv‡b V ‡e‡M cÖevwnZ n‡”Q| GLb kxl© E Gi wbKU¯’ we›`yi e„ËvKvi MwZi Rb¨ cÖ‡qvRbxq †K›`ªgyLx ej A I B we›`yi Uvb T †_‡K cvIqv hvq| A I B we›`y‡Z cÖhy³ Uvb؇qi PO eivei wµqvkxj cÖ‡Z¨K Dcvs‡ki gvb TSin | A_©vr PO eivei ‡gvU wµqvkxj ej 2TSin| PO-Gi j¤^ eivei wµqvkxj T Gi Dcvsk `ywU ci¯úi mgvb I wecixZgyLx nIqvq ci¯úi‡K bvKP K‡i †`‡e| GLb AE = BE Ges AEB Pv‡ci eµZvi †K›`ª O, OA Ges OB †hvMK‡i A I B we›`y‡Z `ywU ¯úk©K Uvbv nj| ¯úk©KØq‡K wcQ‡b ewa©Z Kivq Zviv OE-Gi ewa©Zvs‡ki Dci P we›`y‡Z wgwjZ nq|

awi ∠AOE = AEB Pv‡ci ˆ`N©¨ = S, Zv‡ii GKK ˆ`‡N©¨i fi =  , AEB Pv‡ci eµZvi e¨vmva© = R myZivs Zv‡ii Dci m„ó Zi½ MwZi mv‡c‡ÿ Zv‡ii e„ËvKvi MwZi Rb¨ cÖ‡qvRbxq †K›`ªgyLx e‡ji gvb n‡e μSV 2  2Tsinθ R μSV 2   2Tθ R μSV 2 S2   2T R R T  V2  μ

[ θ Lye † QvU ]

Pvc AE S 2      θ  e¨vmva ' R R  

T Mg  ‡ h‡ nZz T  Mg   ................................(1) μ μ  hLb ZviwU g~j myi Drcbœ K‡i ZLb l  ev,  = 2 l | wKš‘ Avgiv Rvwb V = f V = 2fl 2 bs mgxKi‡Y V Gi gvb ewm‡q cvB, V

2fl 

T μ

K¤úv¼ f 

1 2l

T μ

... ... ... ... ... ... (3) (cÖgvwYZ|)

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18| kã (Sound)

6

UvbvZv‡ii Avo K¤ú‡bi m~Î ¸wj (Laws of transverse vibration of stretched string) eY©bv I e¨vL¨v: T Uv‡b Uvbv l ˆ`‡N©¨i †Kvb Zv‡i Avo K¤úb m„wó Ki‡j Zv‡ii K¤úv¼, f 

1 2l

T hLb m Zv‡ii GKK ˆ`‡N¨©i fi| m

1g m~Ît ˆ`‡N©¨i m~Î : †Kvb K¤úgvb Zv‡ii Uvb (T) Ges cÖwZ GKK ˆ`‡N©¨i Zv‡ii fi (m) AcwiewZ©Z _vK‡j, Zv‡ii K¤úv¼ (f) Gi ˆ`N©¨ (l) Gi e¨v¯ÍvbycvwZK n‡e| A_©vr ˆ`N©¨ evo‡j K¤úv¼ Kg‡e Ges ˆ`N©¨ Kg‡j K¤úv¼ evo‡e| A_©vr

f

1 l

hLb T I m

aªæe _v‡K| 2q m~Ît Uv‡bi m~Î: †Kvb K¤úgvb Zv‡ii ‰`N©¨ (l) Ges cÖwZ GKK ˆ`‡N©¨i Zv‡ii fi (m) AcwiewZ©Z _vK‡j, Zv‡ii K¤úv¼ (f) Gi Uvb (T) Gi eM©g~‡ji mgvbycvwZK n‡e| A_©vr Uvb Pvi¸b Ki‡j K¤úv¼ wظb n‡e A_©vr f  T hLb l I m aªæe _v‡K| 3q m~Ît f‡ii m~Î: †Kvb K¤úgvb Zv‡ii ‰`N©¨ (l) Ges Uvb (T) AcwiewZ©Z _vK‡j, Zv‡ii K¤úv¼ (f) Gi GKK ˆ`‡N©¨i Zv‡ii fi (m)Gi eM©g‡~ ji e¨v¯ÍvbycvwZK n‡e| A_©vr GKK ˆ`‡N©¨i Zv‡ii fi Pvi¸b Ki‡j K¤úv¼ A‡a©K n‡e A_©vr

f

1 m

hLb l

I T aªæe _v‡K| UvbvZv‡ii Avo K¤ú‡bi K¤úv¼ †h †h wel‡qi Dci wbf©i K‡i : Uvbv Zv‡ii Avo K¤ú‡bi K¤úv¼ wZbwU wel‡qi Dci wbf©i K‡i h_v: 1| Zv‡ii ˆ`N©¨ 2| Zv‡ii Uvb I 3| Zv‡ii GKK ‰`‡N©¨i fi| UvbvZv‡ii Avo K¤ú‡bi †h †Kvb `ywU m~Î cÖgv‡Yi c×wZ eY©bv : m‡bvwgUv‡ii eY©bv (Description of Sonometer) : 1 wgUvi j¤^v GKwU duvcv AvqZKvi Kv‡Vi ev· Øviv m‡bvwgUvi ˆZix| GB ev‡·i Dcwifv‡M GKwU Zvi Uvbvb _v‡K| Zv‡ii GK cÖvšÍ ev‡·i cÖv‡šÍ AvUKv‡bv KwcK‡ji Dci w`‡q P‡j †M‡Q| Zv‡ii GB cÖv‡šÍ GKwU ûK AvUKv‡bv _v‡K| hvi d‡j cÖ‡qvRbxq IRb Pvwc‡q ZviwU‡K Uvb Kiv nq| Zv‡ii bx‡P `ywU w¯’i †mZz B I B1 `ywU Pjbkxj †mZz C I C1 | †mZz `ywU Zv‡ii bxP †N‡l P‡j Ges mvnv‡h¨ K¤úgvb Zv‡ii ˆ`N©¨ cwieZ©b Kiv hvq| ev‡·i Dci Avb¨ Avi GKwU Zvi _v‡K, hv cÖ‡qvR‡b Zzjbvi Kv‡R e¨envi Kiv nq|2q Zvi‡K cÖ_g Zv‡ii mgvšÍivj K‡i Uvbvb nq| †mZzi ga¨eZ©x wPÎt m‡bvwgUvi `~iZ¡ cwieZ©b K‡i Abybv` m„wó Kiv nq| 1g m~Ît ˆ`‡N©¨i m~‡Îi cÖgvY (Varification of 1st law i.e. law of length) : cÖ_‡g ARvbv K¤úv‡¼i K‡qKwU myi kjvKv jIqv nq| hv‡`i K¤úv¼ h_vµ‡g f1, f2 I f3 BZ¨vw`| Gici m‡bvwgUv‡ii cixÿvg~jK Zv‡i wKQy IRb Pvcvb nq| Gici GKwU nvjKv KvM‡Ri UzKiv C I C1 †mZzi gv‡S ewm‡q wUDwbs d‡K© AvNvr K‡i m‡bvwgUv‡ii Dci †P‡c a‡i C I C1 Gi ga¨eZ©x `~iZ¡ Kg‡ekx K‡i Abybv` m©wó Kiv nq| G mgq KvM‡Ri UzKiv wQUwK‡q gvwU‡Z c‡o †M‡j Zv‡ii K¤úv¼ wUDwbs d‡K©i K¤úv‡¼i mgvb nq| Gici C I C1 Gi ga¨eZ©x `~iZ¡ †g‡c †bIqv nq| Abyiƒcfv‡e Ab¨vb¨ myi kjvKvi †ejvq I GKB fv‡e ˆ`N©¨ wbb©q Kiv nq|

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18| kã (Sound)

7

wnmve I MYbv (Calculation): awi f1, f2, I f3 BZ¨vw` K¤úv‡¼i myi kjvKv¸wj h_vµ‡g Zv‡`i l1, l2, I l3, BZ¨vw` ‰`‡N©¨ Abybv` m„wó K‡i| cixÿvq †`Lv hvq †h, f1 l1 = f2 l2= f3 l3 ...........................= aªyeK| ev, f l = aªyeK

ev, f  aª yeK 

1 l

f 

1 l

G‡Z ˆ`‡N©¨i m~Î cÖgvwYZ nq|

2q m~Ît Uv‡bi m~‡Îi cÖgvY (Varification of 2nd law i.e. law of tension): G cixÿvq cÖ_‡g mvnvh¨Kvix Zv‡ii GK cÖvšÍ Dc‡hvMx Uv‡b †U‡b ivLv nq| c‡i cixÿbxq ZviwUi Szjvb cÖv‡šÍ GKwU IRb Pvcvb nq| awi, ZLb cixÿbxq Zv‡i Uvb T1| GLb cixÿbxq ZviwU‡K †U‡b †Q‡o w`‡j GwU Kw¤úZ n‡e Ges mvnvh¨Kvix Zvi‡K e¨va¨K K¤ú‡b Kw¤úZ Ki‡e| GB Ae¯’vq mvnvh¨Kvix Zv‡ii Pjgvb †mZz Gw`K Iw`K mwi‡q Ggb GK ¯’v‡b Avbv nq †hb Zv‡ii ¯úw›`Z As‡ki K¤úv¼ cixÿbxq Zv‡‡ii K¤úv‡¼i mgvb nq| awi G Ae¯’vq mvnvh¨Kvwi Zv‡ii ˆ`N©¨ l1| cybivq cixÿbxq Zv‡ii ˆ`N©¨ wVK ‡i‡L Gi Uvb cwieZ©b K‡i T2 Kiv nq Ges c~‡e©i b¨vq mvnvh¨ Kvix Zv‡ii l2 ˆ`‡N©¨i mv‡_ mgmy‡i Avbv nq| wnmve I MYbv (Calculation): cixÿvq †`Lv hv‡e †h,

T1 l22  ........................(1) T2 l12

wKšÍ K¤úv¼ ˆ`‡N©¨i e¨v¯ÍvbycvwZK A_©vr

f 2 l2 f1 l 2 l2  ev, 12  22 .........................(2) (1) bs mgxKiY †_‡K 22 Gi f 2 l1 l1 f 2 l1

gvb (2) bs mgxKi‡Y ewm‡q cvB

f 12 2

f2



T1 T2



f1 T1 Av_©vr K¤úv¼ Uv‡bi eM©g~‡ji mgvbycvwZK (cÖgvwYZ)|  f2 T2

exU I e¨wZPv‡ii g‡a¨ cv_©K¨ exU 1| cÖvq mgvb ZxeªZv Ges K¤úvsK wewkó `yÕwU Drm †_‡K GKB mgq kã Drcbœ Ki‡j Giv ci¯ú‡ii mv‡_ wg‡j GKwU jwä kã m„wó K‡i| GB jwã k‡ãi ZxeªZv †Kvb g~û‡Z© n«vm cvq Ges †Kvb g~û‡Z© e„w× cvq| k‡ãi ZxeªZvi GB ch©vqµwgK n«vm e„wׇK exU e‡j| 2| Zi½ `ywUi g‡a¨ `kv cv_©K¨ mg‡qi mv‡_ cwiewZ©Z nq| 3| jwä Zi‡½i K¤úv¼ exU m„wóKvix Zi½Ø‡qi K¤úv‡¼i Mo gv‡bi mgvb| 4| k‡ãi cÖvej¨ mg‡qi mv‡_ cwiewZ©Z nq|

e¨wZPvi 1| GKB w`‡K Pjgvb mgvb K¤úv¼ I we¯Ívi wewkó `ywU kã Zi‡½i DcwicvZ‡bi d‡j ïay c_ cv_©‡K¨i Rb¨ †Kvb ¯’v‡b bxieZv I †Kvb ¯’v‡b cÖejZ¡i k‡ãi m„wó n‡j H NUbv‡K k‡ãi e¨wZPvi e‡j| 2| Zi½ `ywUi g‡a¨ `kv cv_©K¨ mg‡qi mv‡_ AcwiewZ©Z _v‡K| 3| jwä Zi‡½i K¤úv¼ e¨wZPvi m„wóKvix Zi½Ø‡qi Df‡qiB K¤úv‡¼i mgvb| 4| k‡ãi cÖvej¨ mg‡qi mv‡_ cwiewZ©Z nq bv|

gy³ K¤úb: †h †Kvb AvKvi, MVb ev AvK…wZi e¯‘‡K Av‡›`vwjZ Ki‡j Zv GKwU wbR¯^ K¤úv¼ iÿv K‡i ¯úw›`Z nq| GB ¯ú›`b‡K wbe©va ev gy³ K¤úb e‡j| Av‡ivwcZ ev ciek K¤úb: †Kvb e¯‘i Dci Av‡ivwcZ ch©ve„Ë ¯ú›`‡bi K¤úv¼ e¯‘i ¯^vfvweK K¤ú‡bi K¤úv‡¼i †P‡q wfbœZi n‡j e¯‘wU cÖ_‡g AwbqwgZfv‡e Kw¤úZ nq c‡i Av‡ivwcZ K¤ú‡bi K¤úv‡¼ Kw¤úZ n‡Z _v‡K| GB ai‡bi K¤úb‡K Av‡ivwcZ ev ciek K¤úb e‡j|

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18| kã (Sound)

8

m‡bvwgUv‡ii mvnv‡h¨ myikjvKvi ARvbv K¤úv¼ wbY©q: ZË¡ t ‡Kvb myikjvKvi K¤úv¼ hw` GKwU Uvbv Zv‡ii wbR¯^ K¤úv‡¼i mv‡_ wg‡j hvq Zvn‡j Abybv` m„wó nq, d‡j ZviwU m‡e©v”P we¯Ív‡i Kuvc‡Z _v‡K| GLb Zv‡ii Uvb T, GKK ˆ`‡N©¨i Zv‡ii fi  n‡j hw` Zv‡ii l ˆ`‡N¨©i mv‡_ myi kjvKvi K¤úv‡¼i Abybv` m„wó nq Zvn‡j Zv‡ii K¤úv¼ Z_v myikjvKvi K¤úv¼, f 

1 2l

T 

cixÿv: cÖ_‡g m‡bvwgUv‡ii Zv‡i M fi Szwj‡q Uvb Kiv nq| Gevi GKwU myikjvKv‡K Kuvwc‡q m‡bvwgUv‡ii ev‡·i Dci aiv nq| GZ K‡i myikjvKvi K¤úb Zv‡i mÂvwjZ nq Ges Zv‡i Avo K¤ú‡bi m„wó nq| GLb mÂviYkxj †mZzwU‡K mwi‡q Ggb Ae¯’v‡b ivLv nq hv‡Z K‡i myikjvKv I Zv‡ii K¤ú‡bi g‡a¨ mgmyi m„wó nq A_©vr Abybv` m„wó nq| mgmyi m„wó n‡q‡P wKbv Zv wbY©‡qi Rb¨ †QvU GK UzKiv KvMR Zv‡ii Dci fvuRK‡i ewm‡q w`‡j hw` myikjvKvi K¤ú‡bi Rb¨ KvM‡Ri UzKiv c‡o hvq Zvn‡j mgmyi m„wó n‡q‡Q aiv nq| GB AŸ¯’vq Zv‡ii ˆ`N©¨ †g‡c †bIqv nq| aiv hvK GB ˆ`N©¨ l| GLb cixÿYxq ZviwU m‡bvwgUvi †_‡K Ly‡j wb‡q wgUvi ‡¯‹‡ji mvnv‡h¨ Zv‡ii ˆ`N©¨ L Ges wbw³i mvnv‡h¨ fi m wbY©q Kiv nq| wnmve I MYbv: Zv‡ii Uvb, T  mg m L T 1 MgL GB mgxKi‡Y l,M,g,L I m Gi gvb ewm‡q f wbY©q Kiv nq|   2l m

GKK ˆ`‡N©¨i Zv‡ii fi,   GLb K¤úv¼, f 

1 2l

mZK©Zv: (1) myikjvKv‡K Lye Avj‡Zvfv‡e m‡bvwgUv‡ii ev‡·i Dci aiv nq| (2) myikjvKv hv‡Z K¤úgvb Zvi‡K ¯úk© bv K‡i ‡m w`‡K jÿ¨ ivLv nq| (3) myikjvKvi mv‡_ Abybv`x Zv‡ii ˆ`N©¨ mwVK fv‡e wbY©q Kiv nq| (4) Zv‡ii ˆ`N©¨ I fi mwVK fv‡e wbY©q Kiv nq| (5) Lye †QvU AvKv‡ii KvM‡Ri UyKiv †bIqv nq|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution 18| kã (Sound)

1| 1m I 1.01m Zi½ ˆ`‡N©i `yÕwU kã Zi½ †Kvb M¨vmxq gva¨‡g 6 †m‡K‡Û 20 wU exU Drcbœ K‡i| D³ M¨vmxq gva¨‡g k‡ãi †eM wbY©q Ki| Avgiv Rvwb, GLv‡b,

f1 

V λ1

 f1 

V Hz 1

Ges

f2 

V V Hz f2  1.01 λ2

Zi½ ‰`N©¨, 1m Zi½ ‰`N©¨, 1.01m 20 10 exU, Ν  6 3 k‡ãi †eM, V= ?

Avevi, N = f1  f2

10 V V   3 1 1.01 10 1.01V  1V   3 1  1.01 10 0.01 V   3 1  1.01 10  1.01 1 ms V 3  0.01  V  336.67ms 1 (Ans.) 

2| 60cm I 60.5cm Zi½ ˆ`‡N©i `yÕwU kã Zi½ †Kvb M¨vmxq gva¨‡g 4 †m‡K‡Û 19 wU exU Drcbœ K‡i| D³M¨vmxq gva¨‡g k‡ãi †eM wbY©q Ki| Avgiv Rvwb,

f1 

V λ1

 f1 

V Hz 60

Ges

f2 

V V f 2  Hz λ2 60.5

Avevi, N = f1  f2

19 V V   4 60 60.5 19 60.5V  60V   4 60  60. 5 19 0.5 V   4 60  60. 5 19  60  60.5 cm s 1 V 4  0.5  V  34485 cm s 1  V  344.85m s 1 (Ans.) 

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3| ‡Kvb †kªYx K‡ÿ k‡ãi ZxeªZv 10-8 Wm-2 n‡j k‡ãi ZxeªZv †j‡ej †Wwm‡e‡j wbY©q Ki| GLv‡b, Avgiv Rvwb, cÖgvY ZxeªZv, Io = 10-12Wm-2 I ‡kªYx K‡¶i ZxeªZv, I=10-8Wm-2 β  10log dB Io ZxeªZv †j‡ej, ?

10 -8 10 -12  β  10 log 10 4  β  40 dB (Ans.)  β  10log

4| 20cm `xN© GKwU Zvi †Kvb GKwU myikjvKvi mv‡_ HK¨Zv‡b Av‡Q| Uvb w`¡¸b Ki‡j HK¨Zv‡b Avb‡Z KZ ˆ`‡N©¨i cÖ‡qvRb n‡e?

Avgiv Rvwb, GLv‡b, 1 T1 Zv‡ii cÖv_wgK ˆ`N©¨, l1=20cm f1  Ges 2l1 μ Zv‡ii cÖv_wgK Uvb, T1=T (awi) Zv‡ii ‡kl Uvb, T2=2T 1 T2 f2  Zv‡ii ‡kl ˆ`N©¨, l2 = ?

2l μ 2

cÖkœvbymv‡i, f1 = f2

T1



1 2l1



T 2T  20 l2



1 2  20 l2

μ



1 2l2

T2 μ

 l2  20 2  20  1.414 l 2  28.28 Cm (Ans) 5| 0.5m j¤^v GKwU Zvi‡K 50N ej Øviv Uvbv nj| hw` Zv‡ii fi 0.005kg nq Z‡e Gi †gŠwjK K¤úv¼ wbY©q Ki| Avgiv Rvwb,

1 f  2l

GLv‡b, Zv‡ii ˆ`N©¨, l = 0.5m Zv‡ii Uvb, T = 50N GKK ˆ`‡N©¨i Zv‡ii fi,

T μ

1 50 Hz 2  0.5 0.01 1 f   5000 Hz 2  0.5  f  70.71Hz (Ans.)  f 

0.005 kg m 1  0.5  μ  0.01 kg m 1   μ 

K¤úv¼, f =?

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cÖ_g c‡Îi As‡Ki mgvavb  f1  96  ( f1  4)  97  96 f1  97 f1  388 GLv‡b,   f1   388 Zv‡ii ˆ`N©¨, l = 25cm  f1  388 Hz

2

6| 25cm ‰`‡N©¨i GKwU Zvi 5kg-wt e‡ji Øviv Uvbv nj| ZviwU †_‡K Drcbœ g~jmy‡ii K¤úv¼ wbY©q Ki| [ZviwUi 1wgUvi ˆ`‡N©¨i fi = 4.9 gm Ges g = 9.8ms-2] Avgiv Rvwb,

1 f  2l

= 0.25m Zv‡ii Uvb, T = 5 kg wt. = 5×9.8N GKK ˆ`‡N©¨i Zv‡ii fi,

T μ

1 5 9.8 Hz f  2  0.25 4.910-3 1 f  10000Hz 2  0.25  f  200 Hz (Ans.)

 μ  4.9  10 K¤úv¼, f =?

3

Ges f2 =(f1 – 4) Hz = (388 – 4) Hz = 384 Hz (Ans.)

1

kg m 

7| `ywU GKB iKg Uvbv Zvi mg K¤úv‡¼i Avo K¤ú‡b Kw¤úZ n‡”Q| GKwU Zv‡ii Uvb 2% e„w× K‡i Kw¤úZ Ki‡j cªwZ †m‡K‡Û 3 wU exU Drcbœ nq| Zvi `ywUi cÖviw¤¢K K¤úv¼ KZ? Avgiv Rvwb,

T2 f2  T1 f1

GLv‡b, Zv‡ii Uvb T1=T awi T2 Zv‡ii Uvb, T2  T  100 T 51T T   1. 02 T 50 50 exU N=3 K¤úv¼, f1=?

f2 1.02T  f1 T f  2  1.02 f1 f  2  1.009950494 f1  f 2  1.009950494f1.......(1) 

Avevi, f2 – f1 = N  1.009950494f1 – f1 = 3  0.009950494f1 = 3

3  f1  301.49 Hz (Ans.) 0.009950494 Avevi, f 2  1.009950494f1  f 2  1.009950494 l  301.49Hz  304.49Hz (Ans.)  f1 

8| `ywU myikjvKv‡K GKB mg‡q kãvwqZ Ki‡j cÖwZ †m‡K‡Û 4 wU exU Drcbœ nq| GKwU wbw`©ó Uvb Kiv Zv‡ii 96cm ˆ`‡N©¨i mv‡_ GKwU myikjvKv Ges 97cm ˆ`‡N©¨i mv‡_ Aci myikjvKv HK¨ZvwbK nq| myikjvKv `yÕwUi K¤úv¼ wbY©q Ki| l1 = 96 cm ˆ`‡N©¨i mv‡_ HK¨ZvwbK myikjvKvi K¤úv¼ awi f1 Hz l2 = 97 cm ˆ`‡N©¨i mv‡_ HK¨ZvwbK myikjvKvi K¤úv¼ f2 = (f1 – 4) Hz KviY ˆ`N©¨ †ekx n‡j ex‡Ui mgvb K¤úv¼ Kg nq| Avgiv Rvwb, f1l1 = f2l2

 f1  96  ( f1  4)  97

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9| `yBwU m`„k Zvi HK¨Zv‡b Av‡‡Q| 0.36m ˆ`N©¨ wewkó GKwU Zvi 100kg IRb Øviv Uvbv †`Iqv Av‡Q| Aci ZviwU 220kg IRb Øviv Uvbv †`Iqv _vK‡j Gi ‰`N©¨ †ei Ki| Avgiv Rvwb, f1  f 2 1  2 l1

T1 1  μ 2l2

T2 μ

GLv‡b, Zv‡ii ˆ`N©¨, l1 = 0.36m Zv‡ii Uvb, T1= 100 kgwt Zv‡ii Uvb, T2= 220 kgwt.

K¤úv¼, f1 = f2 Zv‡ii ˆ`N©¨, l2=?

1 100 1 220  0.36  l2  1 1   10  14.83 0.36 l2 0.36 14.83 l2  m  0.5339m  0.534m (Ans) 10 

10| `yÕwU myi-kjvKv GK‡Î kãvwqZ Ki‡j Giv cÖwZ †m‡K‡Û 5 wU exU m„wó K‡i| hw` G†`i GKwUi K¤úv¼ 275Hz nq Z‡e AciwUi K¤úv¼ KZ? Avgiv Rvwb, GLv‡b,

f 2  f1  N  f 2  275  5  f 2  280 or 270Hz

exU, N = 5 K¤úv¼, f1 =275Hz K¤úv¼, f2 =?

11| `yÕwU myi-kjvKv A I B GK‡Î kãvwqZ nIqvq cÖwZ †m‡K‡Û 5 wU exU m„wó K‡i| A †Z LvwbKUv †gvg jvwM‡q IRb evov‡j exU msL¨v K‡g hvq| B Gi K¤úv¼ 256Hz n†j A Gi K¤úv¼ KZ? Avgiv Rvwb, GLv‡b, fA  fB  N ‡h‡nZz A †Z †gvg jvwM‡q IRb evov‡j exU, N = 5 K¤úv¼, fB =256Hz exU K‡g| d‡j †gvg jvMv‡bvi c~‡e© A K¤úv¼, fA =?

Gi K¤úv¼ †ekx wQj| †gvg jvMv‡bvi Kvi‡Y K¤úv¼ K‡g hvIqvq `yB myi kjvKvi K¤úv‡¼i cv_©K¨ K‡g e‡j exU K‡g| d‡j fA > fB A_©vr fA = fB + N  fA = 256 + 5  fA = 261 Hz (Ans.) 12| ‡Kvb K‡ÿi k‡ãi ZxeªZv 1×10-7 Wm-2| k‡ãi ZxeªZv wظb n‡j bZzb ZxeªZv †j‡ej wbY©q Ki| GLv‡b, Avgiv Rvwb, cÖgvY ZxeªZv, I β 2  10log 2 dB Io = 10-12Wm-2 Io ‡kªYx K‡¶i ZxeªZv, I1 = 1×10-7 Wm-2 2  10 -7  β 2  10log wظb ZxeªZv, 10-12 I2 = 2×10-7 Wm-2 bZzb ZxeªZv †j‡ej, ? Web: http://tanbircox.blogspot.com

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cÖ_g c‡Îi As‡Ki mgvavb

 β 2  10 log200000  β 2  53.01 dB (Ans.)

13| 50cm I 51cm ˆ`‡N©¨ wewkó GKgyL eÜ `ywU b‡j AM©vb b‡j cÖwZ †m‡K‡Û 3 wU exU m„wó K‡i| evqy‡Z k‡ãi †eM wbY©q Ki| Avgiv Rvwb, GLv‡b, 1 gyL eÜ b‡j K¤úv¼ b‡ji ˆ`N©¨, l1 = 0.5m V V b‡ji ˆ`N©¨, l2 = 0.51m f1  I f2  4 l1 4l2 exU, N =3 Avevi Avgiv Rvwb, k‡ãi †eM, V=? f1  f2 = N V V   3 4l1 4l 2 V V   3 4  0.5 4  0.51 V V   3 4  0.5 4  0.51 V  0.51  0.50    3 4  0.5  0.51 

V  0.01 3 4  0.5  0.51 3  4  0.5  0.51 V 0.01  V  306 ms 1 

14| wbw`©ó ˆ`‡N©¨i GKwU m‡bvwgUv‡ii Zvi wbw`©ó ej Øviv Uvbv Av‡Q| hw` Uvbv ej Pvi¸b Ges GKB mv‡_ Zv‡ii ‰`N©¨ w`¡¸b Kiv nq Z‡e K¤úv‡¼i wKiƒc cwieZ©b n‡e| Avgiv Rvwb, GLv‡b, 1 T1 Zv‡ii cÖv_wgK Uvb, T1=T (awi) f1  Zv‡ii ‡kl Uvb, T2=4T 2l1 μ Zv‡ii cÖv_wgK ˆ`N©¨, l1=l (awi) 1 T T Zv‡ii ‡kl ˆ`N©¨, l2 = 2l  f1   2l μ 2l  f1t f2=?

Avevi, f 2 

 f2 

1 2l 2

1 2  2l

T2 μ 4T 2 T  μ 4l 

f1 T 4l   f1 : f 2  1 : 1 A_©vr K¤úv‡¼i †Kvb   f 2 2l  2 T

15| GKwU Uvbv Zv‡ii ˆ`N©¨ cwieZ©b bv K‡i Gi Uvb Pvi¸b Kiv nj| Zv‡ii K¤úv‡¼i KZ cwieZ©b n‡e?

Avgiv Rvwb, GLv‡b, Zv‡ii cÖv_wgK Uvb, T1=T (awi) 1 T1 f1  Zv‡ii ‡kl Uvb, T2=4T 2l1 μ Zv‡ii ˆ`N©¨, l1= l2 = l (awi) 1 T T f2=? f1

 f1 

2l

Avevi, f 2 

f 2 

1 2l

μ

1 2l 2



2l 

T2 μ

4T 2 T  μ 2l 

2l  1 f1 T    f 2 2l  2 T 2  f 2  2f1 A_©vr K¤úv¼ wظb n‡q hv‡e| 16| GKwU myi-kjvKv 512Hz K¤úv‡¼i GKwU myi-kjvKvi mv‡_ 4 wU exU Ges 514Hz K¤úv‡¼i Aci GKwU myi-kjvKvi mv‡_ 6wU exU Drcbœ K‡i| myi-kjvKvwUi K¤úv¼ KZ? g‡b Kwi myi kjvKvi K¤úv¼ = f| cÖ_g k‡Z©, 512  f  4..........(1) ØZxq k‡Z©, 514  f  6..........(2) ‡hvM K‡i, 1026 – 2f = 10 1026  10 1016  f  f   508Hz (Ans.) 2 2 17| `yÕwU myi-kjvKv A I B GK‡Î kãvwqZ nIqvq cÖwZ †m‡K‡Û 5 wU exU m„wó K‡i| A -Gi evû‡Z GKLÛ Zvi Rov‡j Avevi Zviv 5 wU exU m„wó K‡i| B Gi K¤úv¼ 320Hz n†j A Gi K¤úv¼ KZ? Avgiv Rvwb,

fA  fB  N GLv‡b, ‡h‡nZz A †Z Zvi Rwo‡q IRb evov‡j exU c~‡e©i mgvb nq| d‡j Zvi Rov‡bvi exU, N = 5 c~‡e© A Gi K¤úv¼ †ekx wQj| K¤úv¼, fB =320Hz d‡j fA > fB K¤úv¼, fA =?

A_©vr fA = fB + N  fA = 320 + 5  fA = 325 Hz (Ans.)

cwieZ©b n‡e bv|

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k‡ãi `ªæwZ m¤úwK©Z wbDU‡bi m~Î (Newton's Formula for Speed of Sound): m~Ît †Kvb gva¨‡g k‡ãi `ªæwZ gva¨‡gi w¯’wZ¯’vcK ¸bv‡¼i eM©g~‡ji mgvbycvwZK Ges gva¨‡gi Nb‡Z¡i eM©g~‡ji e¨¯ÍvbycvwZK| wbDUb MvwYwZK fv‡e GB m~·K wb‡b¥v³ mgxKi‡Yi mvnv‡h¨ cÖKvk K‡ib t E ... ... ... ... (1) GLv‡b, E = gva¨‡gi w¯’wZ¯’vcK ¸Yv¼,  = gva¨‡gi NbZ¡, v = k‡ãi †eM  M¨vmxq ev evqy gva¨‡g m~‡Îi iƒct M¨vmxq c`v‡_©i †ÿ‡Î w¯’wZ ¯’vcK ¸Yv¼ E, AvqZ‡bi w¯’wZ¯’vcK ¸Yv¼ B wb‡`©k K‡i| v

myZivs M¨vmxq gva¨‡g k‡ãi `ªæwZ m¤úwK©Z wbDU‡bi m~‡Îi iƒc n‡e, v 

B GLv‡b, B = gva¨‡gi AvqZ‡bi w¯’wZ¯’vcK 

¸Yv¼,  = M¨v‡mi NbZ¡, v = k‡ãi †eM B = p cÖwZcv`b t awi, †Kvb wbw`©ó f‡ii M¨v‡mi Pvc p Ges AvqZb V | Zi½ wµqvq GB Pvc I AvqZ‡bi cwieZ©b N‡U| wbDU‡bi avibv wQj GB Pvc I AvqZ‡bi cwieZ©b m‡gvò cÖwµqv| myZivs e‡q‡ji m~Îvbyhvqx, pV =aªæeK GB mgxKiY‡K V Gi mv‡c‡ÿ e¨eKjb Ki‡j, A_©vr d (pV)  0 dV dV dp p V 0 dV dV dp pV 0 dV p

dp AvqZb cxob AvqZ‡bi   dV AvqZb weK … wZ V

w¯’wZ¯’vcK ¸Yv¼ = B

p = B

GLv‡b FbvZ¥K wPý Pvc e„wׇc‡j AvqZb n«vm ev Pvc n«vm †c‡j AvqZb e„w× †evSvq| m~Î g‡Z evqy‡Z k‡ãi †eM v

B p   

p 1.013  105   280 ms1 GLv‡b, p = evqyi Pvc=1.013×105 Nm-2,  = evqyi NbZ¡ =1.293 kgm-3 wKš‘ 1.293  cÖgvY ZvcgvÎv I Pv‡c evqy‡Z k‡ãi †eM 332ms-1| GLvb †_‡K avibv Kiv nq †h, wbDU‡bi m~‡Îi †Kv_vI fyj Av‡Q| j¨vcjvm KZ…K wbDU‡bi m~‡Îi ms‡kvabx (Correction of Newton's law by Laplace): wbDU‡bi m~Î g‡Z evqy‡Z k‡ãi `ªæwZ 280 ms-1| GB gvb cixÿv jä gv‡bi †P‡q A‡bK Kg nIqvq ¯úó eySv hvq †h, Zvi m~Î cÖ‡qv‡Mi mgq wbDUb †h avibv K‡i wQ‡jb Zv‡Z wKQy fyj wQj| 1817 mv‡j weÁvbx j¨vcjvm wbDU‡bi Abygv‡bi v

fyj †ei Ki‡Z mg_© nb Ges M¨vmxq gva¨‡g k‡ãi `ªæwZ m¤úwK©Z m~‡Îi ms‡kvab K‡ib| j¨vcjv‡mi g‡Z M¨vmxq gva¨‡g kã mÂvj‡bi mgq wbDUb gva¨‡gi ZvcgvÎvi †Kvb ZvcgvÎvi †Kvb cwieZ©b nqbv e‡j †h Abygvb K‡i wQ‡jb Zv wVK wQjbv| Avi ZvB M¨v‡mi †h AvqZb ¸Yv¼ B, Pvc p-Gi mgvb †ei K‡i wQ‡jb Zv wVK nqwb| j¨vcjvm e‡jb Pvc e„w×i d‡j evqy¯Í‡ii m‡¼vPb nIqvi d‡j †h Zv‡ci D™¢e nq, †mB Zvc wewKwiZ nIqvi Av‡MB cieZ©x cÖmviY ïiæ nq| d‡j ZvcgvÎv †Kvb g‡ZB w¯’i _vK‡Z cv‡i bv| myZivs GB cwieZ©b m‡gvò cwieZ©b bq e‡j GLv‡b e‡q‡ji m~Î pV = aªæeK cÖ‡hvR¨ n‡Z cv‡i bv| GB cwieZ©b iæ× Zvcxq cixeZ©b | GB cÖwµqvq Pvc I AvqZ‡bi m¤úK© pV=aªæeK| GLv‡b  = w¯’i Pv‡c I w¯’i AvqZ‡b M¨v‡mi Av‡cwÿK Zv‡ci AbycvZ| wØ cigvbyK M¨v‡mi Rb¨ Gi gvb 1.41| pV=aªæeK, GB mgxKiY‡K V Gi mv‡c‡ÿ e¨eKjb K‡i cvB,

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k‡ãi `ªæwZ (Speed Of Sound)

2 d (pV  )  0 dV dV dp  pV  1 V  0 dV dV dp  pV  1  V  dV  V dp  p    1 V dV dp  p   V   1 dV dp dp AvqZb cxob = AvqZ‡bi w¯’wZ¯’vcK ¸Yv¼=B GLv‡b FbvZ¥K wPý Pvc  p   V  p    dV dV AvqZb weK … wZ V B p e„wׇc‡j AvqZb n«vm ev Pvc n«vm †c‡j AvqZb e„w× †evSvq| m~Î g‡Z evqy‡Z k‡ãi †eM v     p 1.41  1.013  10 5 v   332.4 ms 1 GLv‡b, p = evqyi Pvc=1.013×105 Nm-2,  = evqyi NbZ¡ =1.293  1.293 -3 kgm | myZivs j¨vcjv‡mi ms‡kva‡bi d‡j m~‡Îi mvnv‡h¨ ZvwZ¡K fv‡e k‡ãi `ªæwZi †h gvb cvIqv hvq Zv cixÿv jä

gv‡bi mv‡_ wg‡j hvq| GB fv‡e j¨vcjvm wbDU‡bi m~‡Îi ms‡kvab K‡ib| k‡ãi `ªæwZi Dci Pv‡ci cÖfve (Effect of Pressure on speed of Sound): m f‡ii †Kvb M¨v‡mi Dci Pvc hw` p1 †_‡K p2 †Z AvqZb h_vµ‡g V1 †_‡K V2 -‡Z cwiewZ©Z nq| GB cwieZ©‡bi mgq hw` ZvcgvÎv w¯’i _v‡K, Zvn‡j e‡q‡ji m~Îvbyhvqx, p1V1= p2V2 m m  p1  p2 1 2 p p  1  2  aªæeK 1  2

GLb k‡ãi `ªæwZ, v 

 m m  I V2   V1   1 2  

p 1.41p m~‡Î †h‡nZz AbycvZwU me©`v w¯’i _v‡K, myZivs w¯’i ZvcgvÎvq †Kvb M¨v‡mi Pvc  

cwiewZ©Z n‡j Zv‡Z k‡ãi `ªæwZi †Kvb cwieZ©b nq bv| A_©vr w¯’i ZvcgvÎvq k‡ãi `ªæwZi Dci Pv‡ci Kvb cÖfve †bB| k‡ãi `ªæwZi Dci ZvcgvÎvi cÖfve (Effect of Temperature on speed of Sound): M¨v‡mi ZvcgvÎvi cwieZ©b n‡j Gi Nb‡Z¡i I cwieZ©b nq| myZivs k‡ãi `ªæwZi I cwieZ©b nq| awi, p1 Pv‡c, T1 †Kjwfb ZvcgvÎvq †Kvb M¨v‡mi NbZ¡  Ges D³ M¨v‡m k‡ãi `ªywZ v1| GLb p2 Pv‡c, T2 †Kjwfb ZvcgvÎvq hw` H M¨v‡mi NbZ¡  nq Ges ZLb M¨v‡m k‡ãi `ªywZ v2 n‡j, v1 



p1 Ges v  2 1

v1 p1  2    v2 1 p 2

p 2 2 p1  2  ... ... ... (1) p2 1

GLb M¨v‡mi cÖmviY †_‡K Avgiv Rvwb, p1 p  2 1T1  2T2

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k‡ãi `ªæwZ (Speed Of Sound)

3

 pT  2  2 1 1 p1T2

GB gvb (1) bs mgxKi‡Y ewm‡q cvB, v1 p pT  1 2 1 v2 p 2 p1T2 

v1 T1  v2 T2

 v T A_©vr M¨v‡m k‡ãi `ªæwZ †Kjwfb ZvcgvÎvi ev cig ZvcgvÎvi eM©g~‡ji mgvbycvwZK|

Avevi, 0ºC ZvcgvÎvq ev, ToK ZvcgvÎvq evZv‡m k‡ãi `ªæwZ v0 Ges ºC ev, TK ZvcgvÎvq evZv‡m k‡ãi `ªæwZ v n‡j v T wKš‘ T=(Ges T0=  v0 T0 1

v   273   2     1   wØc`x Dccv‡`¨i mvnv‡h¨ we¯Ívi K‡i Ges D”PZi NvZ wewkó c` mg~n D‡cÿv K‡i cvB, v0 273  273    1  v   v 0 1  .  ...............   2 273     v   v 0 1    546   v   v 0 1  0.00183    v   332ms 1 1  0.00183  ( 0 ºC ZvcgvÎvq evZv‡m k‡ãi `ªæwZ 332ms-1|  v   332ms 1  (0.61 ms 1 )  myZivs †`Lv hv‡”Q †h, evqy‡Z cÖwZ wWwMÖ †mjwmqvm ZvcgvÎv e„w×i Rb¨ k‡ãi `ªæwZ 0.61ms-1 ev 61 cm s-1 e„w× cvq| k‡ãi `ªæwZi Dci Av`ªZvi cÖfve (Effect of Humidity on speed of Sound):

Av`ª evqy A_©vr evqy‡Z Rjxqev®ú _vK‡j Gi Nb‡Z¡iI cwieZ©b nq; myZivs k‡ãi `ªwZiI cwieZ©b nq| Av`ª evqy ev Rjxq ev®úc~Y© evqyi NbZ¡ ﮋ evqyi Nb‡Z¡i Zzjbvq Kg A_©vr evqy‡Z Rjxq ev®ú hZ †e‡o hvq Gi NbZ¡ ZZ K‡g hvq| Avevi k‡ãi `ªæwZ Nb‡Z¡i eM©g~‡ji e¨¯ÍvbycvwZK| myZivs evqy‡Z Rjxq ev®ú †ewk _vK‡j k‡ãi `ªæwZ †e‡o hvq| g‡bKwi, wbw`©ó Pvc p I ZvcgvÎv -‡Z ﮋ I Av`ª evhyi NbZ¡ h_vµ‡g d I m Ges ﮋ I Av`ª evqy‡Z k‡ãi `ªæwZ h_vµ‡g vd I vm| myZivs v d 

p Ges v m  d 

p m

vm p  d    vd  m p

ev, v m  v d

d m

d m

d , m Gi †P‡q eo nIqvq vm , vd Gi †P‡q eo|

myZivs Av`ª evqy‡Z k‡ãi `ªæwZ ﮋ evqy‡Z k‡ãi `ªæwZi †P‡q †ekx| GK gyL eÜ b‡ji †Lvjv gy‡L Av‡jvob m„wó Ki‡j Gi ga¨¯’ evqy¯Í‡¤¢i K¤ú‡bi cÖK„wZ (Vibration of Air Column in a Closed Pipe): GK gyL eÜ b‡ji †Lvjv gy‡L duz w`‡j ev Av‡jvob m„wó Ki‡j b‡ji wfZ‡ii evqy¯Í‡¤¢i ga¨w`‡q kã jw¤^K Zi½vKv‡i eÜ gy‡Li w`‡K mÂvwjZ n‡e Ges eÜ gyL †_‡K cÖwZdwjZ n‡q †Lvjv gy‡Li w`‡K AMÖmi n‡e| dzu Gi g~j ¯ú›`b I evqy¯Í‡¤¢i K¤ú‡bi g‡a¨ Abybv` n‡j evqy¯Í¤¢ AwaK we¯Ív‡i Kuvc‡Z _v‡K Ges myi †Rviv‡jv n‡e| GB Ae¯’vq b‡ji †Lvjv gy‡L me©`v GKwU my¯ú›` we›`y (A) Ges eÜ gy‡L GKwU wb¯ú›` we›`y (N) wPÎvbyhvqx m„wó n‡e| evqy¯Í‡¤¢i K¤úb Abymv‡i b‡ji wfZi GKvwaK my¯ú›` I wb¯ú›` we›`yi m„wó n‡Z cv‡i| evqy¯Í‡¤¢i mnRZ¡i K¤ú‡b [wPÎ bs-(K)] ïaygvÎ eÜ gy‡L GKwU wb¯ú›`

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k‡ãi `ªæwZ (Speed Of Sound)

4

we›`yI †Lvjv gy‡L GKwU my¯ú›` we›`y MwVZ n‡e| wKš‘ ci¯úi msjMœ GKwU wb¯ú›` we›`y I GKwU my¯ú›` we›`yi ga¨eZ©x `~iZ¡ Zi½ ‰`‡N©¨i ¼ As‡ki mgvb| myZivs b‡ji ˆ`N©¨ l Ges K¤ú‡b m„ó k‡ãi Zi½ ˆ`N©¨ IK¤úv¼ n‡j l  Ges N 0 

V V  ...........(2)  0 4l



0

4

l ......(1)

[∵ V = f]

GLv‡b V k‡ãi †eM | b‡ji GB myiB g~j myi ev cÖ_g nvi‡gvwbK| GB b‡j AviI †Rv‡i dzu w`‡j b‡ji evqy¯Í‡¤¢ m„ó jw¤^K Zi‡½i ˆ`N©¨ n«vm cv‡e Ges evqy¯Í‡¤¢i K¤úb e„w× cv‡e| evqy¯Í‡¤¢i 2q m¤¢ve¨ K¤ú‡b [wPÎ bs-(L)] †Lvjv gy‡Li my¯ú›` we›`y Ges eÜ gy‡Li wb¯ú›` we›`yi g‡a¨ GKwU my¯ú›` we›`y I GKwU wb¯ú›` we›`y MwVZ n‡e| awi evqy ¯Í‡¤¢i GB K¤ú‡b m„ó my‡ii Zi½ ˆ`N©¨ 1 Ges K¤úv¼ N1, Zvn‡j l  31  1  4l   0 Ges 4

3

3

N1 

V V 3V    3N 0  1 0 0 3

GB

myi‡K cÖ_g Dcmyi ev Z„Zxq nvi‡gvwbK e‡j| b‡ji m¤¢ve¨ K¤ú‡b [wPÎ bs-(M)] †Lvjv gy‡Li my¯ú›` we›`y Ges eÜ gy‡Li wb¯ú›` we›`yi g‡a¨ `ywU my¯ú›` we›`y I `ywU wb¯ú›` we›`y MwVZ n‡e| Kv‡RB GB K¤ú‡b m„ó my‡ii Zi½ ˆ`N©¨ 2 Ges K¤úv¼ N2, n‡j l  Ges

N2 

V V 5V    5N 0  2 0 0 5

5 2 4l   2   0 4 5 5

GB myi‡K wØZxq Dcmyi ev cÂg nvi‡gvwbK e‡j|

Dc‡iv³ mgxKib †_‡K †`Lv hvq †h, GKgyL eÜ b‡j †h mg¯Í myi m„wó nq Zv‡`i Zi½ ˆ`N©¨  0  Nn 

4l Ges K¤úv¼ (2n  1)

V V (2n  1)   (2n  1) N 0 GLv‡b n = 0, 1, 3, ........BZ¨vw` †h ‡Kvb c~b© msL¨v| GK gyL eÜ b‡j ïaygvÎ 0 4l

‡e‡Rvo nvi‡gvwbK ¸wj Drcbœ n‡Z cv‡i A_©vr 2q, 4_©, 6ô BZ¨vw` nvi‡gvwbK ¸wj Abycw¯’Z _v‡K| `yB gyL †Lvjv b‡ji GKwU †Lvjv gy‡L Av‡jvob m„wó Ki‡j Gi ga¨¯’ evqy¯Í‡¤¢i K¤ú‡bi cÖK„wZ (Vibration of Air Column in a Open Pipe): Dfq gyL ‡Lvjv b‡ji GK cÖv‡šÍ duz w`‡j ev Av‡jvob m„wó Ki‡j b‡ji wfZ‡ii evqy¯Í‡¤¢i ga¨w`‡q kã jw¤^K Zi½vKv‡i Ab¨ cÖv‡šÍ mÂvwjZ nq| Ab¨ cÖv‡šÍ Dcw¯’Z n‡j GB Zi½ nVvr cÖmvwiZ nIqvi my‡hvM cvq| GB Kvi‡Y b‡ji †Lvjv gy‡L me©`v my¯ú›` we›`y (A) Ges K¤úb †f‡` b‡ji gvSLv‡b GK ev GKvwaK wb¯ú›` we›`y (N) wPÎvbyhvqx m„wó n‡e| evqy¯Í‡¤¢i mnRZ¡i K¤ú‡b [wPÎ bs-(K)] `yB gy³ cÖv‡šÍ my¯ú›` we›`y Ges `yB my¯ú›` we›`yi gv‡S GKwU wb¯ú›` we›`y _vK‡e| G‡ÿ‡Î Zi½ ˆ`N©¨ o Ges b‡ji ˆ`N©¨ l I n‡j K¤úv¼ n‡j l  Ges N 0 

V V  ...........(2)  0 2l



0

2

l ......(1)

[ V  f]

GLv‡b V k‡ãi †eM | b‡ji GB myiB g~j myi ev cÖ_g nvi‡gvwbK| GB b‡j AviI †Rv‡i dzu w`‡j b‡ji evqy¯Í‡¤¢ m„ó jw¤^K Zi‡½i Zi½ˆ`N©¨ n«vm cv‡e Ges evqy¯Í‡¤¢i K¤úb e„w× cv‡e| evqy¯Í‡¤¢i 2q m¤¢ve¨ K¤ú‡b [wPÎ bs-(L)] ‡gvU wZbwU my¯ú›` we›`y Ges gvS Lv‡b `ywU wb¯ú›` we›`y MwVZ n‡e| awi evqy ¯Í‡¤¢i GB K¤ú‡b m„ó my‡ii Zi½ ˆ`N©¨ 1 Ges K¤úv¼ N1, Zvn‡j l  1  1  2l   0 Ges 2

2

N1 

V V 2V    2N 0  1 0 0 2

GB myi‡K cÖ_g Dcmyi

ev wØZxq nvi‡gvwbK e‡j|

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k‡ãi `ªæwZ (Speed Of Sound)

5

Z…Zxq m¤¢ve¨ K¤ú‡b [wPÎ bs-(M)] †‡gvU PviwU my¯ú›` we›`y Ges gvS Lv‡b wZbwU wb¯ú›` we›`y MwVZ n‡e| Kv‡RB GB K¤ú‡b m„ó my‡ii Zi½ ˆ`N©¨ 2 Ges K¤úv¼ N2, n‡j l 

V V 3V 2l  3 2    3 N 0 GB   2   0 Ges N 2  2 0 0 3 3 2 3

myi‡K wØZxq Dcmyi ev Z…Zxq nvi‡gvwbK e‡j| Dc‡iv³ mgxKib †_‡K †`Lv hvq †h, `yB gyL †Lvjv b‡j †h mg¯Í myi m„wó nq Zv‡`i Zi½ ˆ`N©¨  n  K¤úv¼ N n 

 2l  o Ges n 1 n 1

V V(n  1)   (n  1) N 0 n‡e| GLv‡b n = 0, 1, 3, ........BZ¨vw` †h ‡Kvb c~b© msL¨v| `yB gyL †Lvjv b‡j n 2l

‡e‡Rvo I ‡Rvo mKj cÖKvi nvi‡gvwbK Drcbœ n‡Z cv‡i| Abybv`x evqy¯Í¤¢ wK? Abybv` evqy¯Í¤¢ c×wZ‡Z k‡ãi †eM wbY©q Ki: Abybv` evqy¯Í¤¢ (Resonance Air Column): †Kvb GKwU GKgyL †Lvjv b‡ji g‡a¨ Ave× evqy¯Í‡¤¢i GKwU ¯^fvweK K¤úv¼ _v‡K| GB K¤úv¼ evqy¯Í‡¤¢i ˆ`‡N©¨i Dci wbf©i K‡i| Giƒc GKwU †Lvjv b‡ji gy‡L GKwU K¤úgvb wUDwbs dK© ai‡j d‡K©i K¤úb evqy¯Í‡¤¢ ciek K¤úb m„wó K‡i| d‡K©i K¤úv¼ hw` evqy¯Í‡¤¢i K¤úv‡¼i mgvb nq Z‡e evqy¯Í¤¢ cÖejfv‡e Kuvc‡Z _v‡K| G‡K Abybv`x evqy¯Í¤¢ e‡j| Abybv` evqy¯Í¤¢ c×wZ‡Z k‡ãi †eM wbY©q (Determination of Speed of Sound by Resonance Air Column Method): h‡š¿i eY©bvt Abybv` evqy¯Í¤¢ c×wZ‡Z AvMv‡Mvov mgvb cÖ¯’‡”Q‡`i GKwU KvPbj _v‡K| GB bjwU‡K GKwU cvwbi cv‡Î Wywe‡q `yBZ…Zxqvsk cvwb‡Z fwZ© Kiv nq| Kvh©c×wZ (Procedure): wbw`©ó K¤úv‡¼i GKwU myikjvKv F †bIqv nq| G‡K ivevi c¨v‡W AvNvZ K‡i Abybv`x b‡ji Db¥y³ cÖv‡šÍ aiv nq| G‡Z b‡ji ga¨¯’ evqy‡Z ciek K¤ú‡bi m„wó nq| G K¤úb wb‡Pi w`‡K mÂvwjZ nq Ges cvwbi DcwiZj n‡Z cyYivq cÖwZdwjZ n‡q wd‡i Av‡m| bjwU‡K DVvbvgv K‡i b‡ji evqy¯Í‡¤¢i ˆ`N©¨‡K Ggb fv‡e Dc‡hvRb Kiv nq hv‡Z evqy¯Í‡¤¢i me‡P‡q Kg ˆ`‡N©¨ Abybv` m„wó nq| GB Ae¯’vq cvwbi Dcwi Zj n‡Z b‡ji Db¥y³ cÖvšÍ ch©šÍ Abybv`x ˆ`N©¨ wbY©q Kiv nq| wnmve I MYbv (Calculation): evqy¯Í‡¤¢i GB Abybv` b‡ji †Lvjv gy‡L GKwU my¯ú›` we›`y I cvwbi Dcwi Z‡j GKwU wb®ú›` we›`yi DrcwË n‡e| GB Ae¯’vq evqy¯Í‡¤¢i K¤úv¼ myikjvKvi K¤úv‡¼i mgvb n‡e| g‡bKwi, Abybv`x evqy¯Í‡¤¢i ˆ`N©¨ l1| hw` evqy‡Z k‡ãi †eM v Ges evqy¯Í‡¤¢i K¤úv¼ n nq Z‡e, l1 

    4l1 KviY 4

b‡ji eÜ gy‡L wb¯ú›` we›`y Ges †Lvjv gy‡L my¯ú›` we›`y Drcbœ& nq Ges G‡`i  4

ga¨eZ©x `~iZ¡ l1  | v =n mgxKiY n‡Z cvB, v =n4l1 A_©vr v =4nl1 cixÿb n‡Z n I l Gi gvb †R‡b v †ei Kiv nq| cÖvšÍ ms‡kvab K‡i k‡ãi †eM wbY©q (Determination of Sound of Velocity with end Correction): Abybv` c×wZ‡Z evZv‡m k‡ãi †eM wbY©‡qi mgq a‡i †bIqv nq †h, my¯ú›` we›`y b‡ji Db¥y³ cÖv‡šÍ m„wó nq| wKš‘ weÁvbx i¨v‡j cÖgvb K‡ib †h, my¯ú›` we›`y b‡ji †Lvjv gy‡L bv n‡q wKQyUv Dc‡i nq| ZvB Gi R‡b¨ GKwU ms‡kvab cÖ‡qvRb| Gi bvg cÖvšÍ ms‡kvab| Kv‡RB GKwU Abybv`x evqy ¯Í‡¤¢i evB‡ii gy³ cÖvšÍ n‡Z by¨bZg †h `~i‡Z¡ my¯ú›` we›`y Ae¯’vb K‡i Zv‡K cÖvšÍ ms‡kvab e‡j| awi cÖvšÍ ms‡kvab = x Abybv`x evqy¯Í‡¤¢i ˆ`N©¨ = (l1+x) b‡ji AšÍt e¨vm d n‡j x=0.3d Ges b‡ji AšÍt e¨vmva© r n‡j x=0.6r| Kv‡RB k‡ãi †eM v=nn‡Z cvB,

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k‡ãi `ªæwZ (Speed Of Sound)

6

v = 4n(l1+x) ev, v = 4n(l1+0.6r) ev, v = 4n(l1+0.3d) BnvB cÖvšÍ ms‡kva‡bi ci k‡ãi †eM|

cÖvšÍ ms‡kvab cwinvi K‡i k‡ãi †eM wbY©q: cÖvšÍ ms‡kvab cwinvi K‡iI k‡ãi †eM wbY©q Kiv hvq| cÖ_g Abybv` m„wó Kivi ci Abybv`x b‡j evqy¯Í‡¤¢i ˆ`N©¨ hw` µ‡g AviI e„w× Ki‡Z _vwK Zvn‡j cÖ_g Abybv‡`i cÖvq wZb ¸b ˆ`‡N¨© wØZxq Abybv` m„wó n‡e| aiv hvK, wØZxq Abybv‡`i mgq evqy¯Í‡¤¢i ˆ`N©¨ = l2 3  ... ... (2) (2) †_‡K (1) we‡qvM K‡i cvB, 4 4  l2  l1     2(l2  l1 ) GLb, v  n 2  v  n  2(l2  l1 ) A_©vr  v  2n(l2  l1 ) GLb l1, l2 Ges n Gi gvb ewm‡q k‡ãi †eM wbY©q Kiv hvq| Wcjvi cÖfve (Doppler Effect):

Zvn‡j, l1  x  ... ... (1) Ges l 2  x 

‡kªvZv I Dr‡mi Av‡cwÿK MwZi d‡j kÖæZ k‡ãi K¤úv‡¼i Z_v ZxÿèZvi AvcvZ cwieZ©b‡K Wcjvi cÖfve e‡j| GKwU A¨v¤^y‡jÝ hLb mvB‡ib evwR‡q Avm‡Z _v‡K ZLb mvB‡i‡bi ZxÿèZv e„w× †c‡Z _v‡K Avevi hLb †mwU P‡j †h‡Z _v‡K ZLb ZxÿèZv µgkt Kg‡Z _v‡K| hw` †kªvZv I Drm ci¯ú‡ii w`‡K AMÖmvi nq ZLb kÖæZ k‡ãi K¤úv‡¼i AvcvZ e„w× nq Ges †kªvZv I Drm ci¯úi †_‡K `y‡i m‡i †M‡j kÖæZ k‡ãi K¤úv‡¼i AvcvZ n«vm cvq| w¯’i †kªvZvi w`‡K MwZkxj Drm (Source Moving Towards Stationary Observer): g‡bKwi, GKRb †kªvZv O Ae¯’v‡b w¯’i Av‡Q Ges S kã Dr‡mi Avw` Ae¯’vb| DrmwU f K¤úv‡¼i kã Drcbœ Ki‡Z Ki‡Z us †e‡M w¯’i †kªvZv O Gi w`‡K G¸‡”Q Ges v kã Zi‡½i †eM| GLb Dr‡mi K¤úv¼ f e‡j cÖwZ †m‡K‡Û f msL¨K Zi½ wbM©Z nq Ges Zi½ †eM v e‡j cÖwZ †m‡K‡Û Zi½ v `~iZ¡ AwZµg K‡i| myZivs GB f msL¨K Zi½ vˆ`‡N©¨i g‡a¨ Ae¯’vb K‡i| (wPÎ cv‡k¦©) Kv‡RB Drm hLb w¯’i v | wKš‘ DrmwU hw` †m‡K‡Û f msL¨K Zi½ f wbM©Z Ki‡Z Ki‡Z us †e‡M w¯’i †kªvZv O Gi w`‡K AMÖmi n‡q GK †m‡K‡Û S  Ae¯’v‡b Av‡m Zvn‡j †mB msL¨K Zi½ GLb (v  us)) ˆ`‡N©¨i g‡a¨ Vvmv Vvwm

_v‡K ZLb k‡ãi g~j Zi½ ˆ`N©¨  

K‡i Ae¯’vb Ki‡e| d‡j Zi½ ˆ`N©¨ n«vm cv‡e| GLb Gi Zi½ ‰`N¨© n‡e  

v  us v  us v | w¯’i †kªvZvi Kv‡Q k‡ãi Zi½ ˆ`N©¨   K‡g   f f f

gv‡bi e‡j g‡b n‡e| myZivs †kªvZvi w`‡K Dr‡mi MwZkxjZvi Rb¨ w¯’i †kªvZvi Kv‡Q kÖæZ k‡ãi AvcvZ K¤úv¼ f  n‡j| AvcvZ K¤úv¼ f  

v kã Zi‡½i † eM   Zi½ ˆ`N ¨ cwiew ZZ 

v (v  u s ) / f v f   f GB mgxKiY †_‡K †`Lv hvq †h, †h‡nZz ( v  u s )  v  f   f (v  u s )  f 

myZivs Drm hLb †Kvb w¯’i †kªvZvi w`‡K MwZkxj _v‡K ZLb †kªvZvi Kv‡Q kã Zi‡½i K¤úv¼ e„w× †c‡q‡Q e‡j g‡b n‡e| Ges kÖæZ k‡ãi ZxÿèZviI AvcvZ e„w× n‡e| DrmwU hw` w¯’i †kªvZv n‡Z `~‡i m‡i hvq Zvn‡j us-†K FbvZ¥K ai‡Z nq| A_©vr †m‡ÿ‡Î, f 

v v f f   f v  (u s ) v  us

G‡ÿ‡Î f   f myZivs Drm †kªvZv †_‡K `~‡i m‡i †M‡j kÖæZ

k‡ãi K¤úv¼ Z_v ZxÿèZv AvcvZ n«vm cv‡e|

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k‡ãi `ªæwZ (Speed Of Sound)

7

Drm w¯’i wKš‘ †kªvZv Dr‡mi w`‡K MwZkxj ( Source Stationary But Observer Moving Towards It ): aiv hvK, kã Drm s Ae¯’v‡b w¯’i Av‡Q Ges †kªvZv O Ae¯’vb n‡Z Dr‡mi w`‡K uo `ªæwZ‡Z AMÖmi n‡”Q| Drm KZ…K wbm„Z k‡ãi K¤úv¼ f, Zi½ ‰`N©¨  Ges `ªæwZ v| GK †m‡K‡Û f msL¨K Zi½ Drcbœ Ki‡Q †kªvZv w¯’i _vK‡j cÖwZ †m‡K‡Û Zvi Kv‡b f msL¨K Zi½B †cŠQvZ| A_©vr 1 †m‡K‡Û f Zi½ v `~iZ¡ Ry‡o _vKZ| myZivs †kªvZv O Gi Kv‡Q †h Zi½gvjv †cŠQv‡e v | GLb †h‡nZz †kªvZv Dr‡mi w`‡K uo `ªæwZ‡Z MwZkxj Kv‡RB †kªvZvi f v  uo † kª vZvi mv‡ c‡ ¶ k‡ ãi `ª æwZ  mv‡c‡ÿ k‡ãi `ªæwZ (v+uo) n‡e| myZivs AvcvZ K¤úv¼, f   O we›`y‡ Z † cuŠQvb Zi‡½i Zi½ ˆ`N ¨ v f v  uo  f  f ... ... ... ... (1) v Dc‡iv³ mgxKiY †_‡K †`Lv hvq †h †h‡nZz (v+uo) >v  f   f

Zvi ˆ`N© AcwiewZ©Z _vK‡e Ges Zv n‡e  

myZivs AvcvZ K¤úv¼ cÖK…Z K¤úv‡¼i †P‡q †ekx n‡e| A_©vr hLb †Kvb †kªvZv w¯’i Dr‡mi w`‡K MwZkxj _v‡K ZLb kÖæZ k‡ãi K¤úv¼ Z_v ZxÿèZvi AvcvZ e„w× nq| hw` †kªvZv w¯’i Drm †_‡K `~‡i m‡i hvq Zvn‡j uo †K FbvZ¥K ai‡Z n‡e| A_©vr †m‡ÿ‡Î  v  uo   f ... ... ... ... (2) f     v u  s 

G †ÿ‡Î †h‡nZz f   f myZivs †kªvZv w¯’i Drm †_‡K `~‡i m‡i †M‡j kÖæZ k‡ãi K¤úv‡¼i Z_v ZxÿèZvi AvcvZ n«vm n‡e|

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cÖ_g c‡Îi As‡Ki mgvavb First Paper Math Solution

19| k‡ãi `ªæwZ (Speed of Sound) 1| 261Hz K¤úvs‡Ki GKwU myikjv‡K AvNvZ K‡i Abybv`x b‡ji Db¥y³ cÖv‡šÍi wbKU ai‡j evZv‡mi 0.30m Ges 0.94m ˆ`‡N©¨ Abybv` cvIqv †Mj| k‡ãi `ªæwZ I cÖvšÍ ms‡kvab wbY©q Ki| Avgiv Rvwb, V=2 f (l2 - l1) GLv‡b, K¤úv¼, f = 261 Hz ev, V=2 ×261(0.94-0.30) -1 ev, V=334.08ms m/s(Ans.) cÖ_g Abybv`x ˆ`N©¨, l1 = 0.30 m 2q Abybv`x ˆ`N©¨, l2 = 0.94 m Avevi, k‡ãi †eM, V= ? ev, V= 4 f(l1 + x) ev, 334.08=4×261×(0.3+ x) cÖvšÍ ms‡kvab, x = ?

334 . 08 4  261  x = 0.32-0.30 =0.02 m (Ans.) ev, 0.3  x 

4| GKwU †Uªb euvwk evRv‡Z evRv‡Z GKwU cøvUd‡g©i w`‡K 90kmh-1 `ªæwZ‡Z AMÖmi n‡”Q| euvwki K¤úbv¼ 600Hz Ges k‡ãi `ªæwZ 325ms-1 n‡j cøvUd‡g© `Ûvqgvb †Kvb †kªvZvi Kv‡b H k‡ãi AvcvZ K¤úv¼ KZ g‡b n‡e?

GLv‡b, Avgiv Rvwb,

v  uo f v  us 325  0  600  f  325  25 325  f   600 300  f   650Hz (Ans.) f 

†Uª‡bi `ªæwZ u s  90km/h 90  1000 1  ms  25ms 1 3600 †Uª‡bi euvwki K¤úv¼ f = 600 Hz k‡ãi `ªæwZ v = 325ms-1 ‡mªvZvi `ªæwZ, uo = 0 AvcvZ K¤úv¼ f' =?

f 2| 512Hz K¤úv‡¼i GKwU myi myikjvKv‡K AvNvZ K‡i GKwU Abybv`x b‡ji Db¥y³ cÖv‡šÍ aivq evZv‡mi 0.15m ˆ`‡N©¨ cÖ_g Abybv` cvIqv †Mj| evZv‡m k‡ãi `ªæwZ 350ms-1 n‡j b‡ji e¨vm KZ? Avgiv Rvwb, GLv‡b, V = 4f(l1+0.3d) K¤úv¼, f = 512 Hz V cÖ_g Abybv`x ˆ`N©¨,  l1  0.3d  l1 = 0.15 m 4f k‡ãi `ªæwZ, V=350ms-1 V  0.3d   l1 b‡ji e¨vm, d = ?

4f

350  0.15 4  512  0.3d  0.170898437  0.15  0.3d  0.020898437 0.020898437 m d 0.3  d  0.069 m (Ans.)  0.3d 

3| KZ ZvcgvÎvq evqy‡Z k‡ãi †eM 0°C (cÖgvY) ZvcgvÎvi ‡e‡Mi GLv‡b, wظb n‡e?

V1 T1  V2 T2 

V 273  2V T2



1 273  4 T2

k‡ãi †eM, V1= V k‡ãi †eM, V2= 2V ZvcgvÎv, T1 =0ºC = 273K ZvcgvÎv, T2 =?

5| †`LvI †h, Drm hw` w¯’i †kªvZv †_‡K k‡ãi `ªæwZ‡Z m‡i hvq, Z‡e kÖæZ k‡ãi AvcvZ K¤úvsK A‡a©K n‡e| Avgiv Rvwb,

v  uo f v  us v1  0  f  f v1  ( v1 ) v 0  f  1 f 2v1 f f   (cÖgvwYZ|) 2 f 

awi, k‡ãi `ªæwZ v = v1 Dr‡mi `ªæwZ us = v1 ‡mªvZvi `ªæwZ, uo = 0 K¤úv¼ f n‡j, cÖgvY Ki‡Z n‡e †h, AvcvZ K¤úv¼, f   f 2

6| GKwU BwÄb Pj‡Z Pj‡Z 300 Hz K¤úv‡¼i eskxaŸwb Kij wKš‘ GKRb ch©‡eÿ‡Ki wbKU H k‡ãi K¤úv¼ 305Hz g‡b nj| †Kvb w`‡K Ges KZ `ªæwZ‡Z BwÄbwU MwZkxj wQj| [k‡ãi `ªæwZ 332 ms-1] GLv‡b, Avgiv Rvwb, †Uª‡bi euvwki K¤úv¼, v  uo f = 300 Hz f v  us k‡ãi `ªæwZ, v = 332ms-1 ‡mªvZvi `ªæwZ, uo = 0 332  0  305   300 AvcvZ K¤úv¼, f' = 305 Hz 332  u s †Uª‡bi `ªæwZ, us=?

332  332  u s   300  326.55 305  u s  326.55  332  5.45  us= 5.45ms-1

f 

DËit 5.45ms-1 †e‡M ‡kªvZvi w`‡K AMÖmi n‡e|

 T2 = 4×273 K = 1092K= (1092-273) °C = 819°C (Ans.)

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19| k‡ãi `ªæwZ (Speed of Sound)

7| GKwU BwÄb w¯’i `k©K AwZµg Kv‡j Gi AvcvZ cÖwZqgvb K¤úv¼ 6:5 Abycv‡Z cwieZ©b nq| hw` evZv‡m k‡ãi †eM 352ms-1 nq, Z‡e BwÄbwUi †eM wbYq Ki| Avgiv Rvwb,

V  uo f V  us V0  f  f V  us Vf ... ... ... (1) f   V  us f 

Avevi,

GLv‡b, ‡kªvZvi †eM, uo= 0 k‡ãi †eM, V =352ms-1 awi, cÖK…Z K¤úv¼ = f K¤úv‡¼i AbycvZ, f ©: f © ©= 6:5 BwÄbwUi †eM, us= 0

V  uo f V  (u s ) V0  f   f V  us Vf ... ... ... (2)  f   V  us f  

T 273

165  332 2

T  165      273  332  165  165  273  T  K 332  332  T  67.43K  (67.43  273)C

 T  205.56 C (Ans.) 9| cÖwZ †m‡K‡Û 200 P‡µi Wcjvi cwieZ©b Drcbœ Ki‡Z n‡j 1050Hz K¤úvsKwewkó kã Drm‡K †Kvb w¯’i `k©‡Ki w`‡K †h †e‡M AvMgb Ki‡Z n‡e Zvi wnmve `vI| [evZv‡m k‡ãi †eM=330ms-1] Avgiv Rvwb,

V  uo f V  us 330  0  1250   1050 330  u s  330  u s 

f 6  f  5 V  us 6 Vf    V  us Vf 5 V  us 6   V  us 5 352  u s 6   352  u s 5  5  352  5u s  6  352  6u s

330  1050 1250

 u s  330  277.2



 u s  52.8 ms 1 (Ans.)

10| GKwU myi kjvKv †h mg‡q 200 evi K¤úb †`q †mB mg‡q GwU Øviv m„ó Zi½ evZv‡m 140 m `~iZ¡ AwZµg K‡i| myi kjvKvi K¤úv¼ 500Hz n‡j evqy‡Z k‡ãi †eM KZ? GLv‡b, Avgiv Rvwb, 200=140m V=f

 V  500  0.7 ms 1  V  350 ms 1 ( Ans.)

 5u s  6u s  6  352  5  352  11 u s  352 352  us  11  u s  32ms 1 (Ans.) 8| Av‡jv †`Lvi 10 sec c‡i eRª wb‡N©v‡li kã †kvbv †Mj| †g‡Ni `~iZ¡ hw` 1650m Ges 0°†m. ZvcgvÎvq k‡ãi `ªæwZ 332ms-1 nq, Z‡e H mgqKvi ZvcgvÎv wbY©q Ki| Avgiv Rvwb, GLv‡b, S= V t mgq, t = 10s.  1650= V ×10 `~iZ¡, S = 1650 m  V =165ms -1 k‡ãi †eM, V =?

-1

k‡ãi †eM, V0 = 332 ms ZvcgvÎv, TºC=273K ZvcgvÎv, T

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GLv‡b, k‡ãi †eM, V =330ms-1 cÖK…Z K¤úv¼, f =1050 Hz Drm `k©‡Ki w`‡K AvMgb Ki‡j K¤úv¼ †ekx n‡e, d‡j AvcvZ K¤úvs&K, f © =(1050+200) Hz = 1250Hz BwÄbwUi †eM, us =? ‡kªvZvi †eM, uo= 0

f 

cÖkœg‡Z, f ©: f © ©= 6 : 5

V T Avevi,   V0 T0



2

 

140 m  0. 7 m 200

K¤úv¼ f=500Hz V=?

11| N.T.P. ‡Z k‡ãi †eM 332ms-1 n‡j 50°C ZvcgvÎvq I 70 cm cvi` Pv‡c k‡ãi †eM wbY©q Ki| k‡ãi †e‡Mi Dci Pv‡ci †Kvb cÖfve †bB, Avgiv Rvwb, GLv‡b, v   v 0 (1  ) k‡ãi †eM, vo = 332ms-1 1   ZvcgvÎv,  = 50ºC  50  v   332 1  273   v =?

 v   332 (1  0.183150183)  v   332 1.183150183

 v   361.12 ms -1 (Ans.)

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19| k‡ãi `ªæwZ (Speed of Sound)

12| `yÕwU nb© enb K‡i GKwU †gvUi Mvox 36kmh-1 †e‡M `Ûvqgvb GKRb ch©‡eÿ‡Ki w`‡K avweZ n‡”Q| nb© `yÕwUi k‡ãi K¤úv‡¼i cv_©K¨ 320Hz n‡j, ch©‡eÿK KZ…K kÖæZ k‡ãi K¤úv‡¼i cv_©K¨ KZ n‡e? evZv‡m k‡ãi †eM 350ms-1| Avgiv Rvwb, GLv‡b, 1g n‡b©i †ÿ‡Î,

v  u0 f1 v  us v f1 /  f1...............(1) v  us

f1/ 

2q n‡b©i †ÿ‡Î,

k‡ãi †eM, v = 350ms-1 ‡kªvZvi †eM, u0=0 ‡kªvZvi †eM, us=36 kms-1



36000 -1 ms  10ms -1 3600

nb©`yÕwUi K¤úv‡¼i cv_©K¨, f1f2 =320Hz kÖæZk‡ãi K¤úv‡¼i cv_©K¨,

v  u0 f2 v  us f1/ - f 2/  ? v f 2/  f 2 ...............(2) v  us vf1 vf 2  f1/  f 2/   v  us v  us v  f1/  f 2/  ( f1  f 2 ) v  us 350  f1/  f 2/   320 350  10  f1/  f 2/  329.41Hz ( Ans.) f 2/ 

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3

13| GKwU K¤úgvb my‡ijx KuvUv cÖ_‡g GKgyL eÜ Kuv‡Pi b‡j 33cm. `xN© evqy¯‡Í ¤¢i mv‡_ Abybv` m„wó K‡i; H GKB b‡j evqy¯‡Í ¤¢i ˆ`N©¨ 100.5cm. n‡j my‡ijx KuvUvwU cybivq Abybv` m„wó K‡i| hw` evqy‡Z k‡ãi †eM 350ms-1 nq, Z‡e cÖvšÍ ms‡kvab KZ? Avgiv Rvwb, V=2 f (l2 - l1) = 4f(l1+x) GLv‡b,

 l1 ) 4f (l  l 1 ) - l1 ev, x  2 2 ev, l1  x 

2f(l

2

cÖ_g Abybv`x ˆ`N©¨, l1 = 0.33 cm 2q Abybv`x ˆ`N©¨, l2 = 100.5 cm k‡ãi †eM, V= 350ms-1 cÖvšÍ ms‡kvab, x = ?

(100.5  3 3 ) - 33 2 67 . 5 x  - 3 3  0 .75cm (Ans.) 2 ev, x 

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আপনি যযহ঵তু এই য঱খা পড়হেি , তাই আনম ধহর নিনি যয আপনি কনিউটার ও ইন্টারহিট বযব঵াহর অনভজ্ঞ ,কাহেই কনিউটাহরর প্রহয়ােিীয় নব঳য় গুহ঱া ঴িহকে ভাহ঱া খারাপ নবহবচিা করারা ক্ষমতা অবশ্যই আহে … তাই আপিাহের কাহে একান্ত অনুহরাধ “ আপিারা ঴ামান্য একটু ঴ময় বযয় কহর ,শুধু এক বার নিহচর ন঱িংহক নিক কহর এই DVD গুহ঱ার মহধয অবনিত বই ও ঴ফটওয়যার এর িাম ঴মূহ঵র উপর যচাখ বুন঱হয় নি​ি।”তা঵হ঱ই বুহে যহবি যকি এই DVD গুহ঱া আপিার কাহ঱কলহি রাখা েরকার!আপিার আেহকর এই বযয়কৃত ঴ামান্য ঴ময় ভনবষ্যহত আপিার অহিক কষ্ট ঱াঘব করহব ও আপিার অহিহক ঴ময় বা​াঁনচহয় নেহব। নবশ্বা঴ করুি আর িাই করুিঃ- “নবনভন্ন কযাটাগনরর এই DVD গুহ঱ার মহধয যেওয়া বািং঱া ও ইিংন঱ল বই , ঴ফটওয়যার ও নটউহটানরয়া঱ এর কাহ঱কলি যেহখ আপনি ঵তবাক ঵হয় যাহবি !” আপনি যনে বতেমাহি কনিউটার বযব঵ার কহরি ও ভনবষ্যহতও কনিউটার ঴াহে যুক্ত োকহবি তা঵হ঱ এই নিনভনি গুহ঱া আপিার অবশ্যই আপিার কাহ঱কলহি রাখা েরকার........ কারিঃ  এই নিনভনি গুহ঱া যকাি যোকাহি পাহবি িা আর ইন্টারহিহটও এহতা ইিরটযান্ট কাহ঱কলি এক঴াহে পাহবি বহ঱ মহি ঵য় িা।তাোড়া এত বড় ঴াইহের ফাই঱ যিট যেহক িামাহিা খুবই কষ্ট঴াধয ও ঴ময়঴াহপক্ষ বযাপার।এোড়া আপনি যযই ফাই঱টা িামাহবি তা ফু঱ ভা঴েি িাও ঵হত পাহর ..  এই নিনভনি গুহ঱া আপিার কাহ঱কলহি োকহ঱ আপিাহক আর যকাি কনিউটার নবহল঳জ্ঞহের কাহে নগহয় টাকার নবনিমহয় বা বন্ধুহের খানতহর “ভাই একটু য঵ল্প করুি” বহ঱ অন্যহক নবরক্ত করা ঱াগহব িা ... ও নিহেহকও ঵য়রানি ঵হত ঵হব িা ।  এই নিনভনি গুহ঱ার মহধয অবনিত আমার করা ৩০০ টা বািং঱া ই-বুক (pdf ) ও যোট ঴াইহের প্রহয়ােিীয় ঴ফটওয়যার আপিাহের েন্য নবিামূহ঱য আমার ঴াইহট যলয়ার কহর নেহয়নে । নকন্তু প্রহয়ােিীয় বড় ঴াইহের বই, নটহটানরয়া঱ ও ফু঱ ভা঴েি ঴ফটওয়যার গুহ঱া যলয়ার ঴াইট গুহ঱ার ঴ীমাবদ্ধতা ও ইন্টারহিহটর যলা আপহ঱াি গনতর েন্য যলয়ার করহত পার঱াম িা । তাোড়া এই বড় ফাই঱ গুহ঱া িাউিহ঱াি করহত যগহ঱ আপিার ইন্টারহিট পযাহকহের ক নেনব খরচ করহত ঵হব ... যযখাহি ১ নেনব পযাহকে েন্য ঴বেনিম্ন ৩৫০ টাকা যতা খরচ ঵হব , এর ঴াহে ঴ময় ও ইন্টারহিট গনতরও একটা বযাপার আহে। এই ঴ব নব঳য় নচন্তা কহর আপিাহের েন্য এই নিনভনি পযাহকে চা঱ু কহরনে ... যমাট কো আপিাহের কনিউটাহরর নবনভন্ন ঴মস্যার নচরিায়ী ঴মাধাি ও কনিউটাহরর েন্য প্রহয়ােিীয় ঴ব বই, ঴ফটওয়যার ও নটউহটানরয়া঱ এর ঴ানবেক ঴াহপাটে নেহত আমার খুব কাযেকর একটা উহেযাগ ঵হি এই নিনভনি পযাহকে গুহ঱া ... এই ক ক  http://tanbircox.blogspot.com/2013/07/My-DVD-Collection-4-U.html

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