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দাথবফজ্ঞান প্রশ্ন ৃজনীর ফা তত্ত্বীয় যমভনই যাক না যকন, দাথবফজ্ঞাননয যভৌবরক বফলয়ভূ ফভয় একই থানক । এই যভৌবরক বফলয়ভূ ম্পনক আনায বযস্কায ধাযণা থাকনর ,আবন অফযই যম যকান উদ্দীনকয জ্ঞান, অনু ধাফন, প্রনয়াগ উচ্চতয দক্ষতায প্রশ্নভূ নয উত্তয খুফ ু ন্দয কনয াবজনয় বরনখ আনত াযনফন।এই ই-ফু নক দাথবফজ্ঞাননয যভৌবরক বফলয়ভূ ম্পনক খুফ নজ ধাযনা ফু ঝায জনয দাথবফজ্ঞাননয গুযত্বূ ন বফলয়ভূ নয প্রনয়াজনীয় উদাযণ বিত্র ফযাখযাভূ রক ভাধান কনয যদয়া নয়নে ।
ন োটঃ ‚ভূ র ফই তথা াঠ্য ফইনয়য বফকল্প বকেু নাই ।‛ দাথবফজ্ঞান বানরা বানফ ফু ঝায জনয আনানক অফযই বফববন্ন যরখনকয ফই ড়নত নফ অথাৎ একই টবক্স বফববন্ন ফইনত ড়নত নফ। যম টাকা বদনয় যকাবিিং কযনফন যই টাকা বদনয় বফববন্ন যরখনকয ফই বকনু ন এফিং তা একফায কনয নর ড়ুন , আা কবয যকাবিিং এ ড়ায যিনয় বানরা পরাপর ানফন । বক্ষাথী কানে একান্ত অনু নযাধ , দাথবফজ্ঞান না ফু নঝ ভুখস্ত কযায বফার বুরবট বুনর কযনফন না।নটা বননজয ানয় বননজই কুড়ার ভাযায ভাথক।কাযন এনত ভয় যভধা দু নটাই অিয় য় , এফিং এনত বফলয়টায উয একটা অভূ রক অস্ববস্ত বীবত িনর আন।
অনটানভবটক স্ক্রনরয ভাধযনভ ই-ফুক ড়া / বযনড়য জনযঃ আনায ই−ফুক ফা pdf বযডানযয Menu Bar এয View অনবট যত বিক কনয Auto /Automatically Scroll অনবট বনরক্ট করুন (অথফা যাবয যমনত Ctrl + Shift + H )। এবার ↑ up Arrow ফা ↓ down Arrow যত বিক কনয আনায ড়ায ু বফধা অনু ানয স্ক্রর স্পীড বঠ্ক কনয বনন। যাবয যমনত অধযানয়য নানভয উয বিক করুনঃ 1. যবক্টয (Vector) 2. রযবখক গবত (Linear-Motion) 3. বিভাবত্রক গবত(Motion-In-Two-Dimensions) 4. গবতূ ত্র (Laws-Of-Motion) 5. যকৌবণক গবতূ ত্র (Laws-Of-Angular-Motion) 6. কাজ, বি ক্ষভতা (Work-Energy-And-Power) 7. ভাকল (Gravitation) 8. যর েবন্দত স্পন্দন (Simple-Harmonic-Oscillation) 9. বিবতিাকতা (Elasticity) 10. প্রফাী দাথ (Fluid) 11. তা গযা (Heat-And-Gas) 12. তাভাত্রা (Temperature) 13. তা গবতবফদযায প্রথভ ূ ত্র (First Law Of Thermodynamics) 14. তা বফবকযণ (Heat Radiation) 15. অফিায বযফতন (Change Of State) 16. তাগবতবফদযায 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 যাবঃ ভান বদক উবয়ই আনে এফিং যম যকান একবট ফা উবয়বটয বযফতননয পনর যবক্টয যাব বযফবতত
নত ানয। উদাযণঃ যণ, যফগ, ত্বযণ, ভন্দন, ফর, বযনফগ, অববকলজ ত্বযণ, যিৌম্বক প্রাফরয, রফদু যবতক প্রাফরয, বূ িুম্বনকয অনু বূবভক প্রাফরয, জন, ৃষ্ঠাটান, াদ্রতাতা গুণািংক, ফনরয রামাভক, যিৌম্বক রামাভক, প্রফাভাত্রা ইতযাবদ। অবদক/নস্করাযঃ ভান আনে বকন্তু বদক যনই। উদাযণঃ দ্রুবত, কাজ, ক্ষভতা, বি, তা, িা, যিৌম্বক রফদু যবতক বফবফ,
িাজ, বিবতিাক গুণািংক ইতযাবদ। যবক্টয যাবয যমাজনঃ যনৌকায গবত, িরন্ত গাবড়নত ড়ন্ত ফৃ বি, াবখয উড্ডয়ন। যবক্টয যাবয 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। অিংকঃ যম যবক্টযভূ যমাগ কনয রবধ ায়া মায়, তানদযনক রবধয অিংক ফা উািং 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|
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©
†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, CosGi gvb 2PQ 1 †_‡K +1 Gi g‡a¨ mxgve×| A_©vr, 1 Cos 1 R 2 P 2 Q2 1 1 [CosGi 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
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
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
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,
OP PQ OQ ... ... ... ... (1)
Ges OR RQ OQ ... ... ... ... (2)
(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|
‡f±‡ii †hv‡Mi ms‡hvM m~Î (Associative Law) : (A B) C A (B C)
cÖgvb t aiv hvK OP A , PQ B Ges QR C | GLb OQ, PR Ges OR a‡i
wÎf~R m~Îvbymv‡i cvB, OP PQ OQ (A B)
Ges PQ QR PR (B C)
GLb OQ QR OR A_©vr (A B) C D
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|
e›Ub m~Î: A .(B C) A . B A .C Gi cÖgvY:
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
cvB, A .(B C) A . (OQ QR )
A .(B C) A . OR
A .(B C) A OP Gi Dci OR Gi j¤ ^ Awf‡ ¶ c
A .(B C) A ON
A .(B C) A (OM MN ) Figure 1
A .(B C) A OM A MN
A .(B C) A OP Gi Dci OQ Gi j¤ ^ Awf‡ ¶ c A OP Gi Dci QR Gi j¤ ^ Awf‡ ¶ c
A .(B C) A . OQ A . QR
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
wÎfyR m~Î Abymv‡i, OP ON NP
OP OB BN NP
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|
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 SinGesY = R Sin †h‡nZz +90º 90º Sin Sin () = Cos d‡j, X = R Cos GesY = 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
2
3
1
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 AB 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
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
PQ †f±i| awi, †mB †f±i aˆ , aˆ PQ
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 PQ (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)
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 PQ 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 PQ 13iˆ 2 ˆj 8kˆ nˆ 237 PQ
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 PQ 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 PQ 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 PQ 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
PQ iˆ 10 ˆj 18kˆ nˆ (Ans.) 425 PQ
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ফাস্তফ যক্ষনত্র যভবিবত যভগবত অম্ভফ। ৃ বথফীয গবত িরন ঘূ ণন গবত। ঘবড়য কাটায গবত মাফৃ ত্ত গবত। ভাবফনেয ফ ফস্ত্ত্তই গবতীর, তাই বনশ্চর প্রঙ্গ কাঠ্ানভা ায়া মায় না। যনৌকায মাত্রী যনৌকায াননক্ষ আনবক্ষক বিবতনত থানক বকন্তু তীনযয াননক্ষ আনবক্ষক গবতনত থানক। ভুিবানফ উয যথনক বননি ড়ন্ত ফস্ত্ত্ত ভত্বযনণয একবট ফাস্তফ উদাযণ এফিং খাড়া উনযয বদনক বনবক্ষপ্ত ফস্ত্ত্তয ভন্দন ভভন্দননয একবট ফাস্তফ উদাযণ। ড়ন্ত ফস্ত্ত্তয ৩বট ূ ত্র আবফষ্কায কনযনেন গযাবরবর বকন্তু প্রভাণ কনযনেন বনউটন। অভ দ্রুবতনত ফা অভ যকৌবনক যফনগ ঘুযনত থাকনর কণাবটয ৩বট ত্বযণ থানক। মথা-যকদ্রতাভুখী ত্বযণ, স্পীত্বযণ যকৌবনক ত্বযণ। ফরবফদযা ভূ রত দু ই প্রকায। মথা: ১। বিবতবফদযা: (ক) যভবিবত (খ) আনবক্ষক বিবত। ২। গবতবফদযা: (ক) ৃ বতবফদযা (খ) ির বফদযা। গবতয প্রকায যবদঃ (i) রযবখক গবত ফা একভাবত্রক গবতঃ যকান ফস্ত্ত্তয গবত মবদ একবট যর যযখায উয ীভাফদ্ধ থানক। তানর তায গবতনক রযবখক গবত ফা একভাবত্রক গবত ফনর। যমভন-যাজা ড়নক গাবড়য গবত। (ii) ভতরীয় ফা বিভাবত্রক গবতঃ গবত ভতনরয উয ীভাফদ্ধ। যমভন- বঁড়ায গবত, ভানফনরয গবত। (iii) িাবনক গবত ফা বত্রভাবত্রক গবতঃ যকান ফস্ত্ত্তয গবত মবদ যম যকান বদনক গবতীর নত ানয তনফ তায গবতনক িাবনক গবত ফা বত্রভাবত্রক গবত ফনর। বফববন্ন প্রকায গবতয রফবিয উদাযণঃ ১. িরন গবত এই গবতনত ফস্ত্ত্তয প্রবতবট কণা একই বদনক ভান দূ যত্ব অবতক্রভ কযয। উদাযণঃ যযরনথয উয িরন্ত যযরগাবড়য ফবগয গবত। ২. ঘূ ণন গবত এই গবতনত ফস্ত্ত্ত একবট বনবদি বফন্দু ফা অক্ষনক যকদ্রতা কনয িক্রাকানয বযরামভন কনয। উদাযণঃ রফদু যবতক াখায গবত, ঘবড়য কাঁটায গবত, ৩. িরন ঘূ ণন ফা বভশ্র ফা জবটর গবত এই গবতনত ফস্ত্ত্তয িরন ঘূ ণন দু বট গবতই থানক। উদাযণঃ িরন্ত াইনকর ফা গরুয গাবড়য িাকায গবত, ূ নময িাযবদনক ৃ বথফীয গবত এফিং রাবটনভয গবত। ৪. মাফৃ ত্ত গবত এই গবতনত ফস্ত্ত্ত একবট বনবদি ভয় য য একই থ অবতক্রভ কনয একই বদনক িরনত থানক। উদাযণঃ ঘবড়য কাঁটা, ইবিননয বস্টন, রফদু যবতক াখা, যদারক বন্ড ইতযাবদয গবত। ৫. যদারন গবত এই গবতনত ফস্ত্ত্তবট বনবদি ভয় অন্তয অন্তয এবদক বদক যদার যদয়। উদাযণঃ 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 2ms2 |
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 ms1| Z¡i‡Yi GKK ms2| 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 xx 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, vt
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, ht 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 = 989.8×3 mgq, t2 = 17S ev, v1 = 9829.4 †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 kmh1 †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 = u22as 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 vvo at 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 r0 1 V [r ]rr0 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 vo at 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 ]rr0 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 vvo at 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 12 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 = voCoso Ges Avw`‡e‡Mi Dj¤^ Dcvsk = voSino 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 = voCoso t +
1 2
axt2
ev, x = voCoso t + 0 [ax= 0] ev, x = voCoso t x t .......................(1) vo Cos o
Dj¤^ w`‡K Z¡iY ay=g; Dj¤^ w`‡K miY y; Abyiƒcfv‡e y=voSinot 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 tmg‡q AwfKl©RZ¡iYnxb Abyf~wgK miY x = vot x 2 v o2 t 2 ...
...
... ...
... ...
(1 )
tmg‡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 voSino t mgq c‡i P we›`y‡Z †eM, vy = voSino 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 = voSino 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 = voSinot 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 = voSinot 12 gt2GB mgxKi‡Y mgq t = T Ges miY y = 0 ewm‡q cvB, 0 = voSinoT 12 gT2
ev, 12 gT2 = voSinoT 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 = ( voCoso ) × 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¨, Sin20 Gi gvb me©vwaK n‡j cvjøv n‡e me©vwaK| Sin20 Gi me©vwaK gvb = 1 A_©vr Sin20 = 1 ev, Sin20 = Sin900 ev, 20 = 900 0 = 450 myZivs wb‡ÿc †KvY0 = 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 (2.18106 )2 5.2 10-11 F 8. 316 108 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 m2 r e¨vmva©, r = 0.75 m 2 2πn mgq t = 1 min. F m r = 60s. t 2 cvKmsL¨v, n = 90 cvK| 23.141690 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 Coso 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 2n 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 বযফতননয ানথ। 1 বনউটন = 7.2324 াউন্ডার 1 াউন্ডার= 13825.728 ডাইন। উৎবত্ত অনু ানয ফর দু ই প্রকায। মথা-১। যভৌবরক ফর ২) মাবিক ফর। বফনে যভৌবরক ফর ৪বট। মথাঃ (১) ভাকল ফর (২) দু ফর বনউবিয় ফর (৩) তাবড়ৎ যিৌম্বক ফর (৪) ফর বনউবিয় ফর। 30
40
42
যভৌবরক ফরগুনরায আনবক্ষক তীব্রতায অনু াত।মথাক্রনভ 1:10 :10 :10 [নমখানন ভাকল ফনরয তীব্রতা 1] ারাভ য়ানয়ন ফাগ প্রভান কনযনেন দু ফর তাবড়ৎ যিৌম্বক ফর একই ফনরয দু বট ববনড়ফরূ। ভাকল ফরঃ ফনিনয় দু ফর। Gravitation নাভক এক প্রকায কণায াযষ্পবযক বফবনভনয়য ভাধযনভ ভাকল ফর কামকয। দু ফর বনউবিয়ায ফরঃ Intermediate vector bosons নাভক কণায াযস্পবযক বফবনভনয়য ভাধযনভ এ ফনরয ৃ বি। Photon নাভক এক প্রকায কণায াযস্পবযক বফবনভনয়য ভাধযনভ তাবড়ৎ যিৌম্বক ফর কামকয য়। ফর বনউবিয়ায ফরঃ বনউবিয়ায ফর আকলণ ধভী। স্বল্প াল্লা বফবি এফিং আধান বনযনক্ষ । যভন কণায াযস্পবযক বফবনভনয়য ভাধযনভ ফর বনউবিয়ায ফর ৃ বি য়। ২য় ূ ত্র যথনক ১ভ ূ ত্র ায়া মায় অথাৎ ১ভ ূ ত্র ২য় ূ নত্রয একবট রূ। ১ভ ূ ত্র যথনক ায়া মায়ঃ ফর জড়তা ২য় ূ ত্র যথনক ায়া মায়ঃ ফনরয অববভুখ, ফনরয বযভা, ফনরয গুণগত রফবিয, ত্বযনণয ানথ ফনর ম্পক, ফনরয নীযনক্ষ নীবত। বিবত জড়তায দৃ িান্তঃ ১। কানয উয যাি কাড তায উয ভুদ্রা যযনখ যটাকা বদনর ভুদ্রা কান নড় মানফ। ২। ঠ্াৎ গাবড় িরনত শুরু কযনর আনযাী বেনন যনর নড়। ৩। কযাযানভয যফানড একবট গুবটয উয আয একবট গুবট যযনখ স্ট্রাইক িাযা আঘাত কযনর নীনিয গুবট িনর মায়। ৪। ধু বরমু ি যালাক 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 আনযাী বেনন যনর নড়। গবত জড়তায দৃ িান্তঃ ১। িরন্ত ফা গঠ্াৎ যথনভ যগনর মাত্রী াভনন যনর নড়। ২। যদৌড় বদনয় রাপ বদনর। ৩। ধাফভান যঘাড়া যথনক উনযয বদনক রাপ যদয়া। ৪। িরন্ত গাবড়য কাভযায় যকান আনযাী যাজা উনযয 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 বন যাতা। 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, a0 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 Ft F [t f - t i ] ti J Ft
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 = Ft (m)v = Ft 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 0.8ms1 (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, 400300 =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 = (84) 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 = (21) 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 0.05m (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 2 314 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 m2 r
Avgiv Rvwb,
GLv‡b, fi, m = 0.1kg 2n 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 2n =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 উািংনয গুণপর িাযা কানজয বযভা কযা য়। কাজ , W = F. S ফা F×S cosθ বি: যকান ফস্ত্ত্ত ফা ফযফিায কাজ কযায াভথযনক বি ফনর। ক্ষভতা: যকান একবট উৎনয কাজ কযায াযনক ক্ষভতা ফনর। ক্ষভতা, p = বিয বনতযতায ূ ত্র: ভাবফনেয যভাট বিয বযভাণ বনবদি অবযফতনীয়। িংযক্ষণীর ফর: যম ফর িাযা ফস্ত্ত্তনক একবট বনবদি থ ঘুবযনয় ঐ নথ আননর কৃত কানজয বযভাণ ূ নয য় ঐ ফরনক িংযক্ষণীর ফর ফরা য়। যমভন: অববকল ফর, রফদু যবতক ফর আদ বরিং এয বফকৃবত প্রবতনযাধ কাযী ফর ইতযাবদ। অিংযক্ষণীর ফর: যম ফর িাযা কৃত কাজ ূ নয না য় এফিং ম্পাবদত কাজ শুধু ভাত্র দু বট বফন্দুয অফিাননয উয বনবয কনয তানক অিংযক্ষণীর ফর ফনর। যমভন ঘলণ ফর, াদ্রতা ফর ইতযাবদ। কাজ- বি উাদয: যকান ফস্ত্ত্তয উয প্রমু ি ফর িাযা কৃত কাজ ফস্ত্ত্তবটয গবতবিয বযফতননয 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. Sদ্ধবতনত erg M. K. S ফা S.I দ্ধবতনত Jole
৩. এক অববকলীয় একক কাজ = এক অববকলীয় একক ফর × একক যণ। ৪. কানজয যক্ষনত্র অববকলীয় এককনক যভ একনক রূান্তনযয ভয় িাযা গুণ কযনত য় এফিং যভ এককনক অববকলীয় একনক রূান্তনযয ভয় িাযা বাগ কযনত য়। ৫. ক্ষভতায যভ একক: আন্তজাবতক 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 ূ নময আয়ু 27 বফবরয়ন ফেয। ৮. একজন ভানু নলয গড় ক্ষভতা 0.143 H.P। ৯. ফর যনণয ভধযফতী যকাণ 900 নর এফিং তানদয ভান আরদা আরাদা বানফ ূ নয নর কানজয ভান ূ নয য়। ১০. ফৃ ত্তনথ আফতনযত ফস্ত্ত্ত ূ নয কানজয উদাযণ। 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 যফযাকৃত 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=F1x Avevi x1+x †_‡K x1+2x Ae¯’v‡b miv‡Z A_©vr x mi‡Yi Rb¨ aªæe ej F2 n‡j G‡ÿ‡Î K…Z KvR, W2=F2x Gfv‡e, cieZ©x x mi‡Yi Rb¨ aªæe ej F3 n‡j G‡ÿ‡Î K…Z Kv‡Ri cwigvb, W3=F3x 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 = F1x + F2x + F3x +......................... +FNx 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 21 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 Uh 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_©vrej 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_©vrcwieZ©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 = (10042)% = 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
Ε133 . 6 1036 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 1.6710-27 (410-4 )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 5.5105 9.8100 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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ভাকলঃ ভাবফনেয যম যকান দু বট ফস্ত্ত্তকণয ভধযকায আকলন ফরনক ভাকল ফর ফনর। বনউটননয ভাকল ূ ত্রঃ ভাবফনেয প্রনতযকবট ফস্ত্ত্তকণা এনক অযনক তানদয িংনমাগ যর যযখা ফযাযফ একবট ফনর আকলণ কনয। এ আকলণ ফনরয ভান, ফস্ত্ত্ত ফা ফস্ত্ত্ত কণািনয়য বনযয গুণপনরয ভানু াবতক এফিং এনদয ভধযকায দূ যনত্বয ফনগয ফযস্তানু াবতক এ গাবণবতকবানফ
ভাকলীয় প্রাফরযঃ ভাকলীয় যক্ষনত্র যকান বফন্দুনত একক বয ম্পনড়ফ একবট ফস্ত্ত্ত িান কযনর, ফস্ত্ত্তবট যম আকলণ 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,
ভাকলীয় বফবফঃ ভাকলীয় যক্ষনত্রয যকান বফন্দুয বফবফ ফরনত একক বয ম্পনড়ফ একবট ফস্ত্ত্তনক অীভ যথনক অথাৎ যক্ষনত্রয ফাবয যথনক ঐ বফন্দুনত আননত ভাকনলয বফরুনদ্ধ যম বযভাণ কাজ কযনত য় তানক ফু ঝায়। ভাকলীয় যক্ষত্রঃ ফৃ ৎ বযবফবি যকান ফস্ত্ত্তয জন ফস্ত্ত্তয িাযবদনক যম অঞ্চনরয ভনধয এয আকলণ ফর অনু বূত য়, য অঞ্চরনক ঐ ফস্ত্ত্তয ভাকলীয় যক্ষত্র ফনর। ভুবিনফগঃ ফবনভড়ফ যম যফনগ যকান ফস্ত্ত্ত বনবক্ষপ্ত নর তা আয ৃ বথফীনত বপনয আনফ না য যফগনক ভুবি যফগ ফনর। এয ফবনভড়ফ -1
-1
ভান নে 11.18kms ফা 6.98mils
যকরানযয ূ ত্রঃ প্রভ ূ ত্র (কনক্ষয ূ ত্র): প্রনতযকবট গ্রনই ূ মনক প্রবতবনয়ত নাববনত (Focas) যযনখ উফৃ ত্তাকায নথ প্রবতবনয়ত ূ মনক প্রদবক্ষন কযনে।
বিতীয় ূ ত্র (নক্ষত্রপনরয ূ ত্র): যম যকান গ্র এফিং ূ নময ানথ িংনমাগকাযী যযখা ভান ভনয় ভান যক্ষত্রপর অবতক্রভ কনয। তৃতীয় ূ ত্র (মায়কানরয ূ ত্র): প্রবতবট গ্রনয মায়কানরয ফগ ূ ম নত ঐ গ্রনয গড় দূ যনত্বয ঘনপনরয ভানু াবতক। ১৯৫৭ ানরয ৪ঠ্া অনক্টাফয যৌববনয়ত ইউবনয়াননয বফজ্ঞানীযা ফ প্র ভ কৃবত্রভ উগ্র স্পু টবনক-১ যপ্রযনণয ভাধযনভ ভাূ নয অববমাননয থ প্রদন কনয। একবট কৃবত্রভ উগ্রনক ৃ 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 ভাকলীয় ধ্রুফক ফনর।
অববকল ফনরয প্রবানফ ভুিবানফ ড়ন্ত ফস্ত্ত্তয যফগ ফৃ বদ্ধয াযনক অববকলজ ত্বযণ ফনর। অববকলজ ত্বযণ ‘g’ যকান ধ্রুফ িংখযা নয়। িাননবনদ এয বযফতন য়। g এয ভান যভরুনত যফী (983.217cm/s2) এফিং বফলূ ফীয় অঞ্চনর কভ (978.039 cm/s2) বূ -ৃষ্ঠ যথনক উনযয বদনক ফা বননিয বদনক g এয ভান কভ এফিংনকনদ্রতা ূ নয (0)। ৃ বথফীয গড় ফযাাধ, R = 6.37× 106m
ৃ বথফীয গড় ঘনত্ব, ρ = 5.5 × 103 kgm-3
ৃ বথফীয বয, M = 5.975 × 1027gm
ূ নময অববকলজ 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 FG
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 GR ... ... ... (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 GR 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 = mr 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 mr 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 cosd‡j, welye A‡j AwfKl©R Z¡i‡Yi gvb, g = g R (2) Avevi,†giæ A‡j =90º Ges cosd‡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| Rh
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 Rh T Rh 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, FG
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 21 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 6.4l106 6.67 1011
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 6.41106 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
(6.4106 )2 9.8 (6.4106 6.4105 )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 Rh 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 4R 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 Rh -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 রফবিযঃ i) ফস্ত্ত্তয গবত মায় গবত ইনফ;
ii) ফস্ত্ত্তয গবত যর রযবখক গবত ইনফ;
iii) ফস্ত্ত্তয উয ত্বযণ ফদা একবট বনবদি বফন্দু অববভুনখ বক্রয়া কযনফ, বনবদি বফন্দুনক উায াভযফিান ফা ভধযাফিান ফরা য়; iv) ফস্ত্ত্তয উয বক্রয়াীর ত্বযণ ভধযািান ইনত উানদয যননয ভানু াবতক; v) ফস্ত্ত্তয ত্বযণ যননয বফযীত ভুখী ইনফ; vi) ইা একবট যদারন গবত ইনফ। যদারনকয ফযফাযঃ i) অববকলজ ত্বযণ g এয ভান বনণয় ii) াানড়য উচ্চতা বনণয়
iii) ভয় বনণয় iv) যর যদারনকয াানময g এয ভান বনণয়
যনকন্ড যদারকঃ যম যদারনকয যদারনকার 2 যনকন্ড উানক যনকন্ড যদারক ফনর।উায কম্পািংক
cy/s।
একবট বনবদি িানন একবট যনকন্ড যদারনকয T L বনবদি। কর যনকন্ড যদারক অফযই বফনল যর যদারক। ভনন যাখনত নফ 1. যকৌবণক বফস্তায 40 এয যফী নর যর যদারক ূ ত্র যভনন িনর না। 3. একবট যদারকনক াানড়য উনয 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 অীভ ইনফ। 5. ভত্বযনণ উনযয বদনক গবতীর বরপনট যদারনকয যদারনকার হ্রা াইনফ এফিং ভত্বযনণ নীনিয বদনক গবতীর বরপনট যদারনকয যদারনকার ফৃ বদ্ধ াইনফ। অথাৎ প্রভ যক্ষনত্র যদারকবট ধীনয িরনফ এফিং বিতীয় যক্ষনত্র উা দ্রুত িরনফ। 6. যভরু অঞ্চনর g এয ভান যফী, তাই যদারক ঘবড় দ্রুত িনর এফিং বফলু ফীয় অঞ্চনর g কভ, তাই উা ধীনয িনর। আফায, গ্রীষ্মকানর যদারক ঘবড় ধীনয িরনফ এফিং ীতকানর দ্রুত িরনফ। 7. যগারাকায ফফ বন্ডবট পাঁা নর, বননযট নর ফা পাঁা অিং ম্পূ ণ তযর িাযা ূ ণ কযনর বায যকনদ্রতায যকান বযফতন য় না ফনর এয কামকয রদঘয অবযফবতত থানক মায পনর যদারনকানরয যকান বযফতন য় না। 8. ফফবটনক তযর িাযা অধূণ 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. তাভাত্রা ফৃ বদ্ধ যনর L ফৃ বদ্ধ ানফ পনর T ফৃ বদ্ধ ানফ এফিং যদারক আনস্ত িরনফ। তাভাত্রা হ্রান বফযীত ঘটনা ঘটনফ। 11. িদ্রতা ৃ নষ্ঠ বননর ফৃ বদ্ধ ানফ এফিং যদারক ধীনয িরনফ। 12. ভত্বযনণ উর্ধ্গাভী 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| LT2 ‡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 LT2 ‡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 Fdx kx dx k x dx k 2o 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, EA 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 29 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 0.52
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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 ABGi 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 1106 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 11010 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 22 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 BC 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 2r A_©vr b‡ji wfZ‡ii ¯úk© †iLvi ˆ`N©¨ n‡e 2r| d‡j c„ôUv‡bi Rb¨ b‡ji †`Iqvj ¯úk©K eivei wfZ‡ii w`‡K 2rT ej Abyfe Ki‡e| †`IqvjI Zi‡ji Dci Gi wecix‡Z evB‡ii w`‡K mgvb ej 2rT cÖ‡qvM Ki‡e| GB ej 2rT †K `ywU j¤^ Dcvs‡k wefvwRZ Ki‡j Lvov Dc‡ii w`‡K 2rTcos Ges Gi mv‡_ j¤^fv‡e AbyfywgK eivei 2rTsin evB‡ii w`‡K Dc‡ii wPÎ Abyhvqx wµqv Ki‡e| b‡ji e¨v‡mi wecixZ w`‡K wµqv Kivq 2rTsin e‡ji Dcvsk ¸wj ci¯úi‡K bvKP K‡i †`q| AZGe Zi‡ji Dci †gvU DaŸ©gyLx ej n‡e 2rTcos|
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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, 2rTcos(r2h + v)g
(r 2 h v) g ... ... ... (1) 2r 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 2r cos r r 2 (h ) g 3 T 2r 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 rhg 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 rhg ... ... ... (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 rg 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 rg 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
rhg ... ... ... (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, FA
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 ML1T 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, 1z= 2 z =1 (3) bs mgxKi‡Y yI z Gi gvb ewm‡q cvB, x1 +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 6r 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 = 6rv
‡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 6rv r 3 f g r 3 s g 3 3 4 4 6rv r 3 s g r 3 f g 3 3 4 3 6rv 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‡if 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 7.30×103 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 = 6rv †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
hgr 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
6.75310 13.6 10 9.8 0.8 10 2cos 3 6.75310 13.6 103 9.8 0.8 103 cos 2 4.7 101 cos 0.765991353 cos 1 (0.765991353) 140 (Ans.) 4.7 101
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¡ , s7.8×103 kgm-3 Avgiv Rvwb, ‡Z‡ji NbZ¡ , f0.87×103 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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তাভাত্রায রু যস্কর : ুরু ূ নয তাভাত্রানক ূ নয ধনয রতবয তাভাত্রায যস্করনক তাভাত্রায যভ যস্কর ফনর। যভ যস্কনরয াঠ্ = যবিনগ্রনড াঠ্ + ২৭৩ নভাষ্ণ বযফতন : বিয তাভাত্রায় িানয ানথ একবট বনবদি বনযয যকান ফস্ত্ত্তয আয়তননয বযফতননক ভতাীয় ফা নভাষ্ণ বযফতন ফনর। এনক্ষনত্র গযা ফনয়নরয ূ ত্র যভনন িনর অথাৎ PV= ধ্রুফক। রুদ্ধতাীয় বযফতন : ফাবয নত যকান তা গ্রন ফা ফজন না কনয যকান ফস্ত্ত্তয যবৌত অফিায বযফতননক রুদ্ধতাীয় বযফতন ফনর। অথাৎ Pvγ= ধ্রুফক, যমখানন, γ =
বিয িান গযানয আনবক্ষক তা বিয আয়তনন গযানয আনবক্ষক তা
আনবক্ষক আদ্রতা : যকান বনবদি তাভাত্রায় একবট বনবদি আয়তননয ফায়ু নত যম বযভাণ জরীয় ফাষ্প থানক ঐ তাভাত্রায় ঐ আয়তননয ফায়ু নক ম্পৃ ি কযনত যম বযভাণ জরীয় ফানষ্পয প্রনয়াজন য় তানদয অনু াতনক আনবক্ষক আদ্রতা ফনর। আনবক্ষক আদ্রতা=
ববযানে জরীয় ফানষ্পয িা ফায়ু য তাভাত্রায় ম্পৃি জরীয় ফানষ্পয িা
× 100%
ববযািংকঃ যম তাভাত্রায় ফায়ু ভন্ডনরয যকান বনবদি আয়তননয ফায়ু এয ভনধয অফবিত জরীয় ফাষ্প িাযা ম্পৃ ি য়, য তাভাত্রানক ববযািংক ফনর। ক্রাবন্ত তাভাত্রাঃ নফাচ্চ যম তাভাত্রায় যকান গযানক শুধু িা প্রনয়াগ কনয তযনর বযণত কযা মায় তানক ক্রান্তি তাভাত্র ফনর। ক্রাবন্ত আয়তনঃ ক্রান্তি তাভাত্রা ক্রান্তি িান এক গ্রাভ বনযয যকান গযানয আয়তননক ঐ গযানয ক্রান্তি আয়তন ফনর। আদ গযাঃ যমকর গযা কর তাভাত্রায় ফনয়র িার এয ূ নত্র যভনন িনর তানদযনক আদ গযা ফনর। যকান গযাই আদ গযা নয়। রত্রধ বফন্দুঃ যম তাভাত্রায় বফশুদ্ধ ফযপ, াবন ম্পৃ ি জরীয় ফাষ্প তাগত অফিানন থানক তানক াবনয রত্রধ বফন্দু ফনর। াবনয রত্রধবফন্দু 0.160C ফা 273.16k (4.58mm াযদ)। স্বাবাবফক িান কক্ষ তাভাত্রায় গযানয অনু গুনরায ভনধয িংঘনলয িংখযা প্রবত যনকনন্ড প্রায় 109 তা প্রনয়ানগ গযানয প্রাযণ তযনরয যিনয় অননক যফী এফিং একই তাভাত্রা ফৃ বদ্ধনত কর গযানয প্রাযণ একই য়। -23
PV= KT ভীকযণ K যক যফাল্টজভযান ধ্রুফক ফনর। এই ধ্রুফনকয ভান 1.38×10 ক্রাবন্ত তাভাত্রা
-1
JK |
ক্রাবন্ত িা
ক্রাবন্ত আয়তন
Co2 এয ক্রাবন্ত তাভাত্রা 31.1 C
Co2 এয ক্রাবন্ত িা 73 atm
Co2 এয ক্রাবন্ত আয়তন 2.17 ঘন য.বভ.
O2 এয ক্রাবন্ত তাভাত্রা 1190C
O2 এয ক্রাবন্ত িা 50 atm
O2 এয ক্রাবন্ত আয়তন 2.320 ঘন য.বভ.
H2 এয ক্রাবন্ত তাভাত্রা -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 VT 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` 273C nq Z‡e H ZvcgvÎvq M¨v‡mi AvqZb nq
V-273 Vo 1
273 0 273
2
myZivs †`Lv hvq,
273C ZvcgvÎvq †Kvb M¨v‡mi AvqZb k~b¨| ZvcgvÎv Av‡iv Kwg‡q hw` 274C 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 274C _vK‡Z cv‡i bv| myZivs me©wb¤œ Kíbv‡hvM¨ †h ZvcgvÎv Zv nj 273C Gi wb‡P †Kvb ZvcgvÎv _Kv m¤¢e bq| GRb¨ GB 273C 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 PT 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‡eT = () |
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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 VT
[ 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)
4
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, mum(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 ET 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 nl | A_©vr l `~iZ¡ AwZµg Kivi mgq C AYywU nl 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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11| Zvc I M¨vm (Heat and Gas)
7
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 5.8910 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, =3×1010cm =3×10 12 m
1 3.14 6 10 (310 -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, 13.64103 67.3% 100 % F = ? F Av‡cw¶K Av`ªZv, R = 67.30%
R
3
13.6410 100 67.3 F 20.27 103 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 = (8.27.5)10-3 m cvi`| 0.4°CÓ 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°CÓ 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 = (28.4525.25)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 + hg)V = P2×5V n«‡`i Zj‡`‡k ey`ey‡`i AvqZb, V1 = V ev, P2 + hg = 5P2 n«‡`i c„‡ô ey`ey‡`i AvqZb, V2 = 5 V ev, hg = 5P2 P2 cvwbi NbZ¡, 103 kg m-3 AwfKl©R Z¡iY, g = 9.8 ms-2 ev, hg = 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 + hg
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 + hg)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 + hg = 27P2 n«‡`i c„‡ô ey`ey‡`i AvqZb, V2 43 π(3x) 3 ev, hg = 27P2 P2 V2 27 43 πx 3 27V ev, hg = 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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P1 = P2 + h MfxiZvq cvwbi Pvc P1 = P2 + hg
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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 = (87.5)10-3 m cvi`| 0.6°CÓ 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 1.9410 cm λ 1.9510 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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ZvcgvÎv, T = 0ºC =273 K NbZ¡, = 1.25Kg-m-3 P =1.013×105Nm-2 C=?
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তা এফিং তাভাত্রায ভনধয তুরনাঃ ১) তা এক প্রকায বি, বকন্তু তাভাত্রা ফস্ত্ত্তয একবট তাীয় অফিা। ২) তা ফস্ত্ত্তবিত ফ অণুয যভাট গবতবিয ভানু াবতক এফিং তাভাত্রা ফস্ত্ত্তবিত একবট অণুয গড় গবতবিয ভাণুাবতক। ৩) দু ইবট ফস্ত্ত্ত একই তাভাত্রায় থাবকনর উানদয তানয বযভাণ বফববন্ন ইনত ানয। ৪) একবট ফস্ত্ত্ত ইনত অনয ফস্ত্ত্তনত তানয প্রফা উানদয তাভাত্রায উয বনবয কনয, তানয বযভানণয উয বনবয কনয না। 0
0
0
0
৫) তানয একক কযারবয, B.T.U বকনরা-কযারবয বকস্ত্ত্ত তাভাত্রায একক- C, F, K R। তাভাত্রায বফববন্ন যস্করঃ 0
0
১) যবিনগ্রড যস্করঃ ইায বনম্নবিযািংক 0 C এফিং উর্ধ্বিযািংক 100 C 0
0
২) পানযনাইট যস্করঃ ইায বনম্নবিযািংক 32 F এফিং উর্ধ্বিযািংক 212 F। 0
0
৩) যযাভায যস্করঃ ইায বনম্ন উর্ধ্ বিযািংক মথাক্রযভ 0 R 80 R। ৪) যকরববন যস্করঃ তাভাত্রায যম যস্কর ফস্ত্ত্তয যবৌত গুণাফরীয উয বনবযীর নয় তাানক তাভাত্রায যকরববন ফা যভ যস্কর 0
0
ফনর। ইায বনম্ন বিযািংক 237 K, উর্ধ্ বিযািংক 373 K 0
0
ভানফনদনয স্বাবাবফক তাভাত্রা 98.4 F ফা 36.89 C 0
0
0
0
0
0
াবনয ফু টনািংক 100 C ফা 373 K এফিং বভািংক 0 C ফা 273 K, যভ ূ নয তাভাত্রা -273 C ফা 0 K. 2 -2
তা এক প্রকায বি। তাই উায ভাত্রা ভীকযণ [ML T ]| বকন্তু তাভাত্রায যকান ভাত্রা ভীকযণ যনই। তনফ উাযা উবয়ই অবদক যাব। াযদ থানভাবভটানযয ু বফধাঃ াযদ তা ু বযফাী, উজ্জ্বর-অস্বে, তাধাযণ ক্ষভতা খুফ কভ এফিং কভ উিায়ী। 0
0
াযনদয ফু টনািংক 357 C এফিং বভািংক -39 C, তাই উচ্চ তাভাত্রা বযভান াযদ থানভাবভটায ফযফহৃত য়। অযারনকার থানভাবভটানযয ু বফধাঃ 0
0
অযারনকানরয বভািংক -130 C এফিং ফু টনািংক 78 C, তাই ইা বনম্ন তাভাত্রা বযভান ফযফহৃত য়। থাবভিযঃ তাভাত্রা বযভানয জনয অধবযফাক িাযা রতযী ু নফদী যযাধনক থাবভিয ফনর।
যজনন যাখা বার তযর থানভাবভটানয তাভাত্রা বযফতননয ানথ তযনরয আয়তননয বযফতন ঘনট। াযদ থানভাবভটায, অযারনকার থানভাবভটায ইতযাবদ তযর থানভাবভটায। গযা থানভাবভটানয বিয আয়তনন বনবদি বরুাণ গযানয িা তাভাত্রায ানথ বযফবতত য়, এই ধভনক ফযফায কনয গযা থানভাবভটায ফযফায কযা য়। যযাধ থানভাবভটানয ‘‘তাভাত্রা ফৃ বদ্ধয ানথ অবধকািং ধাতফ দানথয যযাধ ফৃ বদ্ধ ায়’’ এই ধভনক কানজ রাগাননা য়। প্লাবটনাভ থানভাবভটায এই থানভাবভটানযয উদাযণ।
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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 EGi 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 C0 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 6.510 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 4.5I 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 32C (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 C0 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 122F (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 122F
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 17.5F 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 62.5F
DËit -27.5°C I -17.5°F Ges -52.5°C I -62.5°F
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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 nT
‡gvjvi Zvc aviY ÿgZv C Q
nT
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 mT
M¨v‡mi Av‡cwÿK Zvc Q
mT
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 dUdW Cv dT =dU dWewn¯’t Kv‡Ri cwigvb I (1) bs mgxKiY n‡Z dQ Gi gvbewm‡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‡ZdU 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 CpI Cv g‡a¨ m¤úK©| CpI Cv I AYycvZ‡K Øviv cÖKvk Kiv nq| A_©vr C P †h‡nZz Cp Cv = R Avevi R abvZ¡K msL¨v CV
CpCv , Avevi CpCv 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, PVaªæeK ... ... ... ... ... (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©): CpI 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 CpCv , Avevi CpCv 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, PV 1dV V dP 0
PV 1dV V PV 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 = PV 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 V2γ1 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 - 45.64C (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 = PV Pvc, P = 1.013×105 Nm-2 Avw` AvqZb, V1 = 100 m3 W = PV2V1) 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 65.70C ( 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 = eT4 | 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 eT4| e¯‘wU †_‡K t mg‡q weKxY© Zv‡ci cwigvb n‡e, E = A eT4t | 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 = T4T04) | K…òKvqv bv n‡j E Gi gvb n‡e, E = eT4T04)
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 AT 4 t E 4 109 5.7 108 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 5.7 108 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 = AT4 †¶Îdj, A= 4r2 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 AeT 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 AT 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 15.66108 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 832110 10 3473 b 2.9 10 3 mK (Ans.)
GLv‡b, ZvcgvÎv,T3200ºC =(3200+273)K =3473K Zi½ ˆ`N¨© m8321×10-10 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 AeT 4 t E 0.1 0.7 5.7 108 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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বভােঃ বিয িান যকান তযর দাথনক ক্রভাগত ীতর কযনত থাকনর যম তাভাত্রায় দাথবট কবঠ্ন অফিায় রূান্তবযত নত থানক এফিং কবঠ্নীবফন যল না য়া মন্ত ঐ তাভাত্রায যকান বযফতন য় না যই তাভাত্রানক বভাে ফনর। অনকরাবত দানথয বনবদি যকান গরনাে ফা বভাে যনই। যমভন- িবফ, ভাখন, কাঁি, বি, যভাভ। এই কর দাথ জভফায ফা গরফায ূ নফ আঠ্ানরা অফিা ধাযণ কনয। 0
াবনয বভাে 0 C। বকন্তু ফায়ু ভুি বফশুদ্ধ াবননক াফধানতায ানথ ধীনয ধীনয ীতর কযনত থাকনর (-100C) মন্ত াবন তযর যথনক মায়।
দ্রফনণয বভািংক বফশুদ্ধ দ্রাফনকয বভািংক অনক্ষা কভ। গরনােঃ স্বাবাবফক িান কবঠ্ন দাথ যম তাভাত্রায় গরনত শুরু কনয তানক গরনািংক ফনর। যমভন- ফযপ। 0
0
0
কনয়কবট দানথয গরােঃ াযদঃ 39 C, যভাভঃ 52 C – 56 C গরনানেয উয িানয প্রবাফঃ দানথয গরনািংক আয়তন গরনকানর হ্রা ায়, যফ দানথয গরনািংক িা প্রনয়াগ কযনর কনভ মায়। যমভনঃ ফযপ, িারাই যরাা, এবিভবন, বফভাথ। যমফ দানথয আয়তন গরননয পনর ফৃ বদ্ধ ায়, উানদয গরনাে িা প্রনয়ানগ ফৃ বদ্ধ ায়। যমভন- যভাভ, যাযাবপন। ফু টনািংকঃ যম বনবদি তাভাত্রায় যকান তযর ফানষ্প বযণত নত শুরু কনয এফিং উায ম্পৃ ি ফাষ্প িা ফায়ু ভন্ডনরয িানয ভান য় তানক ফু টনািংক ফনর। * াধাযণতঃ িা ফাড়নর ফু টনািংক ফানড় এফিং িা কভনর ফু টনাে কনভ। ু প্ততাঃ যম তা ফস্ত্ত্তয তাভাত্রায বযফতন না ঘবটনয় অফিায বযফতন ঘটায় তানক ু প্ত তা ফনর। এনক L িাযা প্রকা কযা য়। কনমকবট দানথয ু প্ত তাঃ
বনবদি গরনািংক থাকা যকরাবনত দানথয একবট ধভ। রত্রধ বফন্দুনত াবন ফযপ জরীয় ফাষ্প একই তাীয় া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)
তাভাত্রায অনক্ষক।
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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‡ZQ 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‡ZQ 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 × (1000) 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 = (263273)°C = 10°C 373K = (373273)°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 + 211890-21 2118901680-52.5 42157.5 θ
157.5 C 3.75C (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×(10030) 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 + 8820991200 24780 882024780991200 705600 33600285600
285600 8.5C(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 (0.020.01875) 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, Q1T1 Ges Q2T2 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 584.14C (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 21000lnT
=(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 4200lnT
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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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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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
2x 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 2x 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 wa = 4.18 m f f v 350 w 4.18 300 300 vw 4.18 1.1666666300 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 = f11= f22
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 1t2 =?
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¨, QPm
PQ 50Q= ?
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 = f100 × 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 = f1f22 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(200t1.57x) ; 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(200t1.57x) .............(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 1002=?
2
m 350 332.5 0.05 100 m 1002 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 1.078125320
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 wa = 3.9m vw va
2 320 480 160
V 1920 ms1 (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 100A= ?
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x t 0.5 50 2t 2x y 0.5 sin ‡K 0.5 50 2x Av`k© mgxKiY, y a sin 2ft 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 4.910-3 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 T2 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 pV 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 ms1 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 pV 1 V 0 dV dV dp pV 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©¨ IK¤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 31 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=nn‡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 V0 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 ) V0 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, TºC=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¨, f1f2 =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 যযহতু এই যখা পড়হেি , তাই আনম ধহর নিনি যয আপনি কনিউটার ও ইন্টারহিট বযবাহর অনভজ্ঞ ,কাহেই কনিউটাহরর প্রহয়ােিীয় নবয় গুহা িহকে ভাহা খারাপ নবহবচিা করারা ক্ষমতা অবশ্যই আহে … তাই আপিাহের কাহে একান্ত অনুহরাধ “ আপিারা ামান্য একটু ময় বযয় কহর ,শুধু এক বার নিহচর নিংহক নিক কহর এই DVD গুহার মহধয অবনিত বই ও ফটওয়যার এর িাম মূহর উপর যচাখ বুনহয় নিি।”তাহই বুহে যহবি যকি এই DVD গুহা আপিার কাহকলহি রাখা েরকার!আপিার আেহকর এই বযয়কৃত ামান্য ময় ভনবষ্যহত আপিার অহিক কষ্ট াঘব করহব ও আপিার অহিহক ময় বাাঁনচহয় নেহব। নবশ্বা করুি আর িাই করুিঃ- “নবনভন্ন কযাটাগনরর এই DVD গুহার মহধয যেওয়া বািংা ও ইিংনল বই , ফটওয়যার ও নটউহটানরয়া এর কাহকলি যেহখ আপনি তবাক হয় যাহবি !” আপনি যনে বতেমাহি কনিউটার বযবার কহরি ও ভনবষ্যহতও কনিউটার াহে যুক্ত োকহবি তাহ এই নিনভনি গুহা আপিার অবশ্যই আপিার কাহকলহি রাখা েরকার........ কারিঃ এই নিনভনি গুহা যকাি যোকাহি পাহবি িা আর ইন্টারহিহটও এহতা ইিরটযান্ট কাহকলি একাহে পাহবি বহ মহি য় িা।তাোড়া এত বড় াইহের ফাই যিট যেহক িামাহিা খুবই কষ্টাধয ও ময়াহপক্ষ বযাপার।এোড়া আপনি যযই ফাইটা িামাহবি তা ফু ভােি িাও হত পাহর .. এই নিনভনি গুহা আপিার কাহকলহি োকহ আপিাহক আর যকাি কনিউটার নবহলজ্ঞহের কাহে নগহয় টাকার নবনিমহয় বা বন্ধুহের খানতহর “ভাই একটু যল্প করুি” বহ অন্যহক নবরক্ত করা াগহব িা ... ও নিহেহকও য়রানি হত হব িা । এই নিনভনি গুহার মহধয অবনিত আমার করা ৩০০ টা বািংা ই-বুক (pdf ) ও যোট াইহের প্রহয়ােিীয় ফটওয়যার আপিাহের েন্য নবিামূহয আমার াইহট যলয়ার কহর নেহয়নে । নকন্তু প্রহয়ােিীয় বড় াইহের বই, নটহটানরয়া ও ফু ভােি ফটওয়যার গুহা যলয়ার াইট গুহার ীমাবদ্ধতা ও ইন্টারহিহটর যলা আপহাি গনতর েন্য যলয়ার করহত পারাম িা । তাোড়া এই বড় ফাই গুহা িাউিহাি 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 নবয় নচন্তা কহর আপিাহের েন্য এই নিনভনি পযাহকে চাু 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 গুহা ... এই ক ক http://tanbircox.blogspot.com/2013/07/My-DVD-Collection-4-U.html
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