hsc intermediate physics 1st paper

1. , , , , - , , , - / pdf Menu Bar View Auto /Automatically Scroll ( Ctrl + Shift + H ) up Arrow down Arrow 1. (Vector) 2. (Linear-Motion) 3. (Motion-In-Two-Dimensions) 4. (Laws-Of-Motion) 5. (Laws-Of-Angular-Motion) 6. , (Work-Energy-And-Power) 7. (Gravitation) 8. (Simple-Harmonic-Oscillation) 9. (Elasticity) 10. (Fluid) 11. (Heat-And-Gas) 12. (Temperature) 13. (First Law Of Thermodynamics) 14. (Heat Radiation) 15. (Change Of State) 16. (Second Law Of Thermodynamics) 17. (Waves & Sound) 18. (Sound) 19. (Speed Of Sound) Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com

2. / , , , , , , , , , - , , , , , , / , , , , , , , , , , , , , Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 3. fi ivwk (Vector Quantities): h mKj fZ ivwkK m~Yic cKvk Kivi Rb gvb I w`K Dfqi cqvRb nq ZvìK fi ivwk ej| hgb- miY, IRb, eM, ZiY, ej BZvw`| jvi ivwk (Scalar Quantities): h mKj fZ ivwkK ay gvb viv m~Yic cKvk Kiv hvq ZvìK jvi ivwk ej| hgb- `wZ, fi, KvR, `N BZvw`| jvi ivwk I fi ivwki ga cv_K (Distinction between Scalar and Vector Quantities): wgK jvi ivwk fi ivwk 1| h mKj fZ ivwkK ay gvb viv m~Yic cKvk Kiv hvq ZvìK jvi ivwk ej| h mKj fZ ivwkK m~Yic cKvk Kivi Rb gvb I w`K Dfqi cqvRb nq ZvìK fi ivwk ej| 2| `wZ, fi, KvR, `N BZvw` jvi ivwki D`vniY| miY, IRb, eM, ZiY, ej BZvw` fi ivwki D`vniY| 3| Kgc GKwU ivwki gvb k~b bv nj bdj k~b nZ cvi bv| Kvb ivwki gvb k~b bv njI bdj k~b nZ cvi| 4| gvb AvQ wK w`K bvB| gvb I w`K AvQ| 5| hvM, weqvM, b, I fvM ^vfvweK wbqg Kiv hvq| hvM, weqvM, b, I fvM ^vfvweK wbqg Kiv hvq bv| 6| `ywU Aw`K ivwki bdj me`v GKwU Aw`K ivwk nq| `ywU w`K ivwki myweav RbK bdj GKwU w`K ivwk ev GKwU Aw`K ivwk nq| GKK fi (Unit Vector): h fii gvb GK ZvK GKK fi ej| gvb k~b bq Ggb fiK Gi gvb viv fvM Kij H w`K ivwkwUi w`K GKwU GKK fi cvIqv hvq| gb Kwi A GKwU fi ivwk; 0A A Gi w`K GKK fi a A A (awi) mxgve fi (Restricted Vector): h fii Avw` we`y Kv_vq _vK Zv wi _vK ZvK mxgve fi ej| O we`yZ P ej OB iLv eivei wqv Ki eySvj fi P mxgve fi hvi cv` we`y O | mgvb fi (Equal Vector): GKB w`K wqviZ `ywU mgRvZxq fii gvb mgvb nj ZvìK mgfi ev mgvb fi ej| wP P I Q mgvb fi| k~b ev bvj fi (Zero or Null Vector): h fii gvb k~b ZvK k~b ev bvj fi ej| bvj fii Avw` I kl we`y GKB we`yZ AewZ| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 4. 01| fi (Vector) 2 m`k fi ev mgvivj fi (Like Vector): mgRvZxq `yB ev ZZvwaK fi hw` GKB w`K wqv Ki Ze ZvìK m`k fi ev mgvivj fi ej| wP A I B m`k fi ev mgvivj fi| wem`k fi (Unlike Vector): mgRvZxq `yB ev ZZvwaK fi hw` wecixZ w`K wqv Ki Ze ZvìK wem`k fi ej| wP A I B wem`k fi| mgiL fi (Co-linear Vector): `yB ev ZZvwaK fi hw` GKB mijiLv eivei ev cii mgvivj wqv Ki Ze ZvìK mgiL fi ej| wP A ,B ,C mgiL fi| Aevb fi (Position Vector): cm KvVvgvi g~j we`yi mvc Ab Kvb we`yi Aevb wbYqi Rb h fi eenvi Kiv nq ZvK Aevb fi ej| evLv t wP O nQ cmsM KvVvgvi g~j we`y Ges h P Kvb GKwU we`y| OP fiwU O we`yi mvc P we`yi Aevb wb`k KiQ| ZvB OP GKwU Aevb fi| Aevb fiK AbK mgq evmva fi r viv cKvk Kiv nq| rOP AvqZGKK fi (Rectangular Unit Vector): wgvwK vbv eevq cii j^ wZbwU A _vK| h_v t X,Y Ges Z A| X Ai w`K wewfb fiK cKvk Kivi Rb GKwU GKK fi i eenvi Kiv nq| Zgwb j I k h_vg Y I Z Ai w`K GKK fi (Wvb cvki wP)| i , j Ges k K AvqZGKK fi ej| evmva fi (Radius Vector): AbK mgq Kvb we`yi AevbK h fii mvnvh cKvk Kiv nq ZvK evmva fi r ej| myZivs Kvb we`y P-Gi vbvsK (x,y,z) nj, evmva fi kzjyixOPr viv cKvk Kiv nq| Ges Gi gvb nq 222 zyxrr fi ivwki mvgvwiK m~ (Law of Parallelogram) eYbv I evLv : eYbv: Kvb KYvi Dci GKB mgq wqvkxj `ywU fi ivwkK hw` Kvb GK we`y _K AswKZ mvgvwiKi `ywU mwbwnZ ev viv wb`k Kiv hvq Ze H we`y _K AswKZ mvgvwiKi KYB fi `ywUi jwi gvb I w`K wb`k Ki| gbKwi GKwU KYvi Dci GKB mgq `ywU w`Kivwk P I Q, KvY wqv KiQ| OA Ges OC iLv `ywU h_vg P I Q gvb Ges Zxi wP Gì w`K wb`k KiQ| GLvb AOC | GB `ywU w`K ivwki jwi gvb I w`K wbYq KiZ ne| AsKb : mvgvwiK OABC AsKb Ki Kb OB hy Kwi| Zvnj OB KbB w`Kivwk `ywUi gvb I w`K wb`k Kie| gb Kwi jwi gvb R Ges KvY GKwU myKvY| GLb B we`y _K OA Gi ewaZ Aski Dci BN j^ Uvwb| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 5. 01| fi (Vector) 3 jwi gvb wbYqt AB I OC mgvivj| AOC | OBN wfyRi ONB GK mgKvb| OB2 = ON2 + BN2 2 OB =(OA+AN)2 +BN2 2 OB = OA2 +2OA .AN +AN2 +BN2 2 OB = OA2 +(AN2 +BN2 ) +2OA .AN 2 OB = OA2 +AB2 +2OA. AB AN .AB 2 OB = OA2 +OC2 +2OA.OC Cos R2 = P2 +Q2 +2PQ Cos (1).........PQCos2QPR 22 jwi w`K wbYq: gbKwi jw R, P Gi mv_ Kvb Drcb Ki| A_vr AOB OBN mgKvYx wfyR ON BN tan ANOA BN tan AB AN ABOA AB BN AB tan QCosP QSin tan (2)......... QCosP QSin tan 1 (1) bs mgxKiY jwi gvb I (2) bs mgxKiY jwi w`K wb`k Ki| GKB mgq GKB we`yZ wqviZ `ywU w`Kivwki jwi mevP I mewb gvb ivwk `ywUi hvMdj I weqvM dji mgvb t P I Q, KvY wqv Kij mvgvwiKi m~vbyhvqx Avgiv cvB, (1).........PQCos2QPR 22 R2 = P2 +Q2 +2PQ Cos R2 P2 Q2 = 2PQ Cos PQ2 QPR Cos 222 Avevi Avgiv Rvwb, CosGi gvb 1 _K +1 Gi ga mxgve| A_vr, 1Cos1 1 PQ2 QPR 1 222 [CosGi gvb ewmq] PQ2QPRPQ2 222 PQ2QPRPQ2QP 22222 [Dfq c P2 +Q2 hvM Ki] 222 )Q~P(R)QP( )Q~P(R)QP( KvRB GKB mgq GKB we`yZ wqviZ `ywU w`K ivwki jwi mevP I mewb gvb ivwk `ywUi hvMdj I weqvM dji mgvb| Ab fveI ejv hvq, GKB mgq GKB we`yZ wqviZ `ywU w`K ivwki jwi mevP gvb ivwk `ywUi hvMdj nZ eo nZ cvi bv I mewb gvb ivwk `ywUi weqvM dj _K QvU nZ cvi bv| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 6. 01| fi (Vector) 4 `ywU fi ivwk P I Q cii KvY AvY&Z| Gì jvi Yb I fi Yb t jvi Yb ev WU Yb t `ywU w`K ivwki jvi Yb GKwU Aw`K ivwk Ges Gi gvb w`Kivwk `ywUi gvbi Ydj Ges Gì gaeZx KvYi Cosine Gi Y dji mgvb ne| evLvt gb Kwi, P I Q `ywU w`K ivwk cii KvY AvbZ& A_vr P I Q Gi gaeZx KvY | Gì WU Yb A_vr jvi Yb PQCosQ.P wPvbyhvqx, ;OAP ;OBQ BM, OA Gi Dci j^ I AN, OB Gi Dci j^ | myZivs OM =OB Cos ev, OM =Q Cos )QCos(PQ.P = (P Gi gvb) (OM) = (P Gi gvb) ( P Gi DciQ Gi j^ Awfci gvb) Abyicfve, )(. PCosQQPCosPQ = (Q Gi gvb) (Q Gi DciP Gi j^ Awfci gvb) A_vr `ywU fi ivwki jvi A_vr WU Ydj ejZ Gì h Kvb GKwU fi ivwk viv Dnviw`K Aci fi ivwkwUi Awfci Ydj evSvq| (K) hw` fi ivwkq cii mgKvb AvbZ& _vK Ze 090CosPQQ.P AZGe `ywU fiivwk mgKvY AvbZ& _vKj Gì jvi bdj k~b ne wecixZg `ywU fi ivwki jvi bdj k~b nj ivwk `ywU ciii Dci j^ ne| awi wZbwU AvqZKvi fi ivwk j,i I k cZK ciii Dci j^| myZivs Gì h Kvb `ywUi jvi bdj k~b ne| A_vr 0... ikkjji (L) hw` `ywU fi ivwk GKB w`K wqv Ki Ze Gì gaeZx KvY = 0; m PQCosPQQP 0. ne| GKK w`K ivwki cvB, 1... kkjjii (M) hw`P IQ `ywU fi ivwk cii wecixZgyLx nq Ze Gì gaeZx KvY = 180 nq Ze m PQCosPQQP 180. ne| fi Yb ev m Yb t `ywU w`K ivwki fi (m) Yb GKwU fi ivwk hvi gvb w`Kivwk `ywUi gvbi Ydj Ges Gì gaeZx KvYi Sine -Gi Y dji mgvb Ges GB bdj ivwk `ywUi Zj j^fve vwcZ GKwU Wvb cvKi KK zK 1g fi ivwk _K 2q fi ivwki w`K z`Zi KvY Nyivj GUv h w`K AMmi nq mB w`K wqv Ki| evLvt gb Kwi, P I Q `ywU w`K ivwk cii KvY AvbZ& A_vr P I Q Gi gaeZx KvY | Gì m Yb A_vr fi Yb PQSinQP GLvb GKwU GKK w`K ivwk hv )( QP Gi jwi w`K wb`k Ki| hw` RQP aiv nq Ze R Gi AwfgyL P I Q Gi mgZji mv_ Awfj^ eivei ne| PQSinQPR ( P Gi gvb) ( P Gi j^ eivei Q Gi gvb ) Avevi, )(QPSinPQ QPSinPQ PQ ( Q Gi gvb) ( Q Gi j^ eivei P Gi gvb ) PQQP A_vr fi Yb wewbgq m~ gb Pj bv| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 7. 01| fi (Vector) 5 Avgiv Rvwb, Wvb cvKi KK zK evgw`K Nyivj GwU Dcii w`K Lyj Avm Avi Wvb w`K Nyivj wbPi w`K AMmi nq, Z`c P I Q Gi fi Yb evgveZx nj Gì jw R Gi AwfgyL Da w`K nq Ges P I Q Gi fi Yb `wbveZx nj Gì jw R Gi AwfgyL wbgyLx ne hv c~ei AwfgyLi Dv w`K ne| Dciv eYbv Abymvi, 1) = 0 nj 0QP ne KviY Sin 0 = 0 2) QP || nj 0QP ne KviY Sin 0 = 0 3) QP nj PQQP ne KviY Sin 90 = 1 4) j,i I k mgKvY wqv Kij jik,ikj,kji I j,i I k mgvivj wqv Kij 0kkjjii jvi Yb wewbgq m~ gb Pj wK fi Yb wewbgq m~ gb Pj bv: gb Kwi `ywU fi P I Q , KvY AvbZ& (1).........PQCosQ.P Avevi, QPCosP.Q (2).........PQCosP.Q (1) I (2) nZ cvB, P.QQ.P A_vr jvi Yb wewbgq m~ gb Pj| Acic, )QP( I )PQ( Gi gvb GKB njI Gì w`K wecixZ A_vr (3).........SinPQQP Avevi, )(SinPQPQ (4).........SinPQPQ (3) I (4) nZ cvB, )PQ()QP( PQQP A_vr fi Yb wewbgq m~ gb Pj bv| fi hvMi wfyR m~ (Triangle Law) t `ywU fiK GKwU wfyRi `ywU mwbwnZ ev viv GKB g w`K I gvb wb`k Kij wfyRwUi ZZxq ev wecixZ g w`K I gvb Dnvì jw wb`k Ki| evLv t awi P I Q GKB RvZxq `ywU fi| ABC wfyRi AB Ges BC ev h_vg fi P I Q wb`k Ki| m~vbymvi, BCABAC QPR [ RACb,GLv ] Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 8. 01| fi (Vector) 6 fii hvMi wewbgq m~ (Commutative Law) t ABBA cgvb t aiv hvK AOP Ges BOR `ywU fi O we`yZ wqv Ki OPQR mvgvwiK c~Y Ki wfyR m~vbymvi cvB, (1)............OQPQOP Ges (2)............OQRQOR (1) I (2) _K cvB, RQORPQOP A_vr, ABBA myZvivs fi hvM wewbgq m~ gb Pj| fii hvMi mshvM m~ (Associative Law) : )CB(AC)BA( cgvb t aiv hvK AOP , BPQ Ges CQR | GLb OQ, PR Ges OR ai wf~R m~vbymvi cvB, )BA(OQPQOP Ges )CB(PRQRPQ GLb ORQROQ A_vr DC)BA( Avevi, ORPROP A_vr D)CB(A )CB(AC)BA( myZvivs fi hvM mshvM m~ gb Pj| eUb m~: C.AB.A)CB.(A Gi cgvY: cgvY: gb Kwi, CB,A I fi wZbwU h_vg OP,OQI QR viv m~wPZ Kiv nqQ| GLb wP _K Avgiv cvB, )QROQ(.A)CB.(A OR.A)CB.(A cAwf^jGiDciGi OROPA)CB.(A ONA)CB.(A )MNOM(A)CB.(A Figure 1 MNAOMA)CB.(A cAwf^jGiDciGicAwf^jGiDciGi QROPAOQOPA)CB.(A QR.AOQ.A)CB.(A C.AB.A)CB.(A (cgvwYZ) Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 9. 01| fi (Vector) 7 wgvwK vbv eevvq GKwU Aevb fii gvb wbYq: A_vr 222 zyxr Gi cgvY : wgvwK vbv eevq cii wZbwU iLv OX, OY I OZ h_vg X, Y I Z, A wb`k Ki Ges i , j I k A wZbwU eivei GKK fi| r GKwU Aevb fi| awi, OPr Ges P we`yi vbv (x,y,z)| XZ Zji Dci PN Ges N we`y nZ X I Z Ai Dci h_vg NA I NB j^ AvuwK| wP nZ, BN =OA= x, AN=OB = z Ges NP = y wfyR m~ Abymvi, NPONOP NPBNOBOP OBNPBNOP kzjyixr Avevi, OPN wfyR nZ, OP2 =ON2 +NP2 OP2 = OB2 +BN2 +NP2 OP2 = BN2 + NP2 +OB2 2222 zyxr )(222 cgvwYZzyxr zzyyxx BABABAB.A Gi cgvY: awi, kAjAiAA zyx I kBjBiBB zyx dj, )kBjBi) . ( BkAjAi(AB.A zyxzyx )k.k(BA)j.k(BA)i.k(BA )k.j(BA)j.j(BA)i.j(BA )k.i(BA)j.i(BA)i.i(BAB.A zzyzxz zyyyxy zxyxxx )1(BA)0(BA)0(BA)0(BA)1(BA)0(BA)0(BA)0(BA)1(BAB.A zzyzxzzyyyxyzxyxxx zzyyxx BA000BA000BAB.A oved)Pr(BABABAB.A zzyyxx kAjAiAA zyx I kBjBiBB zyx nj Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 10. 01| fi (Vector) 8 BA wbYq : )kBjBi(B)kAjAi(ABA zyxzyx )kk(BA)jk(BA)ik(BA )kj(BA)jj(BA)ij(BA )ki(BA)ji(BA)ii(BABA zzyzxz zyyyxy zxyxxx 0)i(BA)j(BA)i(BA0)k(BA)j(BA)k(BA0BA yzxzzyxyzxyx )k(BA)k(BA)j(BA)j(BA)i(BA)i(BABA xyyxzxxzyzzy k)BAB(Aj)BAB(Ai)BAB(ABA xyyxzxxzyzzy zyx zyx BBB AAA kji BA (Ans.) fi wefvRb ev fi weklb (Resolution Of Vectors) t GKwU fi ivwkK `yB ev ZZvwaK fi ivwkZ wef Kivi cwZK fii wefvRb ev fii weklb ej| gbKwi, wP OC iLv R fiwUi gvb I w`K wb`k Ki| GLb R fiwUK Ggb `ywU Ask wef KiZ ne h, G jv OC Gi mv_ h_vg I KvY Drcb Ki| GLb O we`y _K OC iLvi mv_ Gi `yB cvk I KvY Ki OB I OA iLv Uvbv nj| OACB mvgvwiKwU c~Y Kiv nj mvgvwiKi m~vbymvi OA Ges OB ev `ywU R fii `ywU Dcvsk wb`k Kie| gbKwi, wefvwRZ Dcvsk XOA Ges YOB | (K) wP, OCABOC Ges -180OAC GLb OAC wfyR weePbv Ki Avgiv cvB, )](180[Sin OC Sin AC Sin OA Y)OBAC( )(Sin R Sin Y Sin X )(Sin RSin X Ges )(Sin RSin Y R fiK hw` mgKvY wefvwRZ Kiv hvq [wP (L)] A_vr Dcvsk `ywU hw` cii j^ nq Ze Zvnj,+90 Sin+Sin 90 = 1 X = R SinGesY = R Sin hnZz +90 90 Sin Sin () = Cos dj, X = R Cos GesY = R Sin Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 11. c_g ci AsKi mgvavb First Paper Math Solution 1| fi (Vector) 1| fi 1| hw` k2ji3A I kj3i2B nq Ze |BA| KZ? 132 213 kji BA k11j7i5)29(k)43(j)61(i Ans.)96.13121492511)7(5|BA| 222 ( 2| hw` k2j3i6A I kj2i2B nq Ze KZ?B.A zzyyxx BABABAB.A )(Ans.82612)1)(2()2)(3()2)(6( 3| k3-j2i5A I |k9-jai15B a Gi gvb KZ nj A I B cii mgvivj ne? A I B cii mgvivj ne hw` 0BA nq| 9-a15 325 kji BA )30a-5(k)4545(-ja)318(iBA )30a5(k)18a3(iBA 2)30a5(2)18a3(BA ckgZ, A I B cii mgvivj nj 0BA 0)30a5()18a3( 22 i]KMeKc[Dfq 0)30a5()18a3( 22 0)6(a5)6(a3 2222 0)53()6(a 222 iKfvMvivKcDfq )53(0)6(a 222 .)Ans(6a 4| k5-j3i2A I |k10-j2imB m Gi gvb KZ nj A I B cii j^ ne| A I B cii j^ nj 0B.A ne| 0 zBzAyByAxBxAB.A 0506m2)10)(5()2)(3()m)(2( (Ans.)28-m 2 56- m 5| kj2i2A I k2j3i6B nj BA I Gi gaeZx KvY wbYq Ki| cosABB.A BA I Gi gaeZx KvY, BA B.A Cos 1 42-6-12)2)(1(- )3)(-2()6)(2(BABABABA zzyyxx. 39)1(22A 222 74949362)3(6B 222 (Ans.)02.79 21 4 Cos )7)(3( 4 Cos BA B.A Cos 11-1 6. fi k2j3i6B Gi Dci fi kj2i2A Gi j^ Awfc wbYq Ki| cosABB.A A Gi j^ Awfc, B B.A cosA 826-12)2)(1()3)(-2()6)(2( BABABAB.A zzyyxx 74949362)3(6B 222 A Gi j^ Awfc = 7 8 B B.A 7| k3j2iB,kj2i3A I k2jiC nj cgvY Ki h, C).BA()CB.(A 211 321 kji CB k-j5-i7)2-1(k)32(j)34(iCB ev, Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 12. 1| fi (Vector) 2 1011021)1)(-1()5)(-2()7)(3( )CB. (AL.H.S 3-21 123 kji BA GLb, k4j10i8 )2-6(k)19(j)26(i=BA ev, 108108-)2)(4( )1)(10()1)(8(-C) .BA(R.H.S R.H.SL. H. S A_vr, oved.)Pr(C).BA()CB.(A 8| ,k2j3iA kj2iB I k4j3i2C nj cgvY Ki h, ACABA)CB( k3ji3k)41(j)32(i)21()CB( 231 313 kji A)CB(L.H.S k10j3i11)19(k)36(j)92(i 231 121 kji AB Avevi, kj3i7)2-3(k)12(j)34(i 231 432 kji AC k9i18)36(k)4-4(j)126(i k10j3i11 kj3i7ACABR.H.S k9i18 A_vr, oved.)Pr(ACABA)CB( 9| k6-ji9A Ges k5j6i4B fi `ywUi Mybdj wbYq Ki `LvI h, Giv ciii Dci j^| zzyyxx BABABAB.A 030636)5)(6()6)(1()4)(9( 0ABCosB.A 0CosC0, B0A wK 9090CosCos , ev AZGe A I ciii Dcii j^| 10| hw` kAjAiAA zyx I kBjBiBB zyx nq Ze `LvI h, zzyyxx BABABAB.A )kBjBi) . ( BkAjAi(AB.A zyxzyx )k.k(BA)j.k(BA)i.k(BA )k.j(BA)j.j(BA)i.j(BA )k.i(BA)j.i(BA)i.i(BAB.A zzyzxz zyyyxy zxyxxx )1(BA)0(BA)0(BA )0(BA)1(BA)0(BA )0(BA)0(BA)1(BAB.A zzyzxz zyyyxy zxyxxx zzyyxx BA000BA000BAB.A oved)Pr(BABABAB.A zzyyxx 11| hw` k5j4i2A I k3j2iB fi qi jw fii mgvivj GKK fi wbYq Ki| k3j2ik5j4i2BAR k2j6i3R R Gi mgvivj GKK fi R R a 7494369)2(63R 222 (Ans.)k 7 2 j 7 6 i 7 3 7 k2j6i3 R R a 12| GKB we`yZ wqvkxj `ywU mgvb gvbi fii gaeZx KvY KZ nj Gì jwi gvb h Kvb GKwU fii mgvb ne? R2 =P2 +Q2 +2PQCos ev, X2 = X2 +X2 +2X.X.Cos ev, X2 - X2 -X2 =2X2 Cos ev, -X2 =2X2 Cos 2 2 X2 X Cos, ev 2 1 Cos, ev 2 1 Cos, 1 ev (Ans.)120 GLvb, awi, fi, P=Q=X jw, R= X Af~ KvY, Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 13. 1| fi (Vector) 3 13| Aevb fi kzjyixr K eeKjb Ki wKfve eM I ZiY cvIqv hvq? Avgiv Rvwb, eM, dt rd v )kzjyi(x dt d (Ans.)k dt dz j dt dy i dt dx Avevi, ZiY k dt dz j dt dy i dt dx dt d dt vd a (Ans.)k dt zd j dt yd i dt xd 2 2 2 2 2 2 14| k)1t2(jtitP 2 I ktjtit5Q 3 ?)Q.P( dt d ?)QP( dt d )t)(1t2()t)(t()t5)(t(Q.P 32 3423 tt2tt5Q.P 234 tt4t2Q.P )tt4t2( dt d )Q.P( dt d 234 t2t12t8)Q.P( dt d 23 3 2 ttt5 1t2tt kji QP )t5t(k )tt5t10(j)tt2t(iQP 23 5224 23 5224 t5t(k )tt5t10(j)tt2t(i dt d )QP( dt d k)t5t( dt d j)t5t10t( dt d i)tt2t( dt d )QP( dt d 2325 24 (Ans.)k)t10t3( j)5t20t5(i)1t4t4()QP( dt d 2 43 15| k3j2-iQ,k4j3i2P fi q h Zj Aevb Ki Zvi Dj^w`K GKwU GKK fi wbYq Ki| Avgiv Rvwb, `ywU fii m bdj fi `ywU viv MwVZ mgZji Dci j^ nq| mB j^ fii mgvivj GKK fiB ne mgZji Dj^ w`K GKK fi| awi, mB fi a , QP QP a )34(k)46(j)89(i 321 432 kji QP k7j10iQP P I Q h Zj AewZ Zvi Dj^ w`K fi 222 )7()10()1( )k7j10i( QP )QP( a .)( 150 )k7j10i( a Ans 16| Kvb GKwU KYvi Aevb fi j][5.3msi4.2m])t[(3.5msr 11 nj eM V wbYq Ki| Avgiv Rvwb, dt rd V jt][5.3msi4.2m])t[(3.5ms dt d V 11 j5.3i3.5V (Ans.) 17| cgvY Ki t 22 22 BAB.ABA 22 .BABALHS 22 cossin ABAB 2222222 cosBAsinBA 222222 cosBAsinBA.1 2222 cossinBA 2222 1. BABA (Proved).... SHRSHL 18| kj2iP Ges k3j6i3Q nj `LvI h, P I Q cii mgvivj| P I Q cii mgvivj ne hw` 0QP nq| )66(k)33(j)66(i 363 121 kji QP 0000)0(k)0(j)0(iQP Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 14. 1| fi (Vector) 4 0 QP 0 QP P I Q cii mgvivj| (cgvwYZ) 19| k4j3i2P Ges k3ji2Q fi q h Zj AewZ Zvi Dj^w`K GKwU GKK fi wbYq Ki| Avgiv Rvwb, `ywU fii m bdj fi `ywU viv MwVZ mgZji Dci j^ nq| mB j^ fii mgvivj GKK fiB ne mgZji Dj^ w`K GKK fi| awi, mB fi n , QP QP n )62(k)86(j)49(i 312 432 kji QP k8j2i13QP Avevi, 222 8213QP 237QP 237 8213 kji QP QP n .)( 237 8 237 2 237 13 Ansjin 20| kj4i4P Ges kj2i2Q fiq GKwU mgvwiKi `ywU mwbwnZ ev wb`k Kij Gi dj wbYq Ki| Avgiv Rvwb, `ywU fi GKwU mgvwiKi `ywU mwbwnZ ev wb`k Kij H mgvwiKi dj ne fi `ywUi m Ydji gvbi mgvb| QP mgvwiKi dj| GLb, )88(k)24(j)24(i 122 144 kji QP j6i6QP 49.87266 22 QP GKK (Ans.) 21| k3-jmi2P I |k15-j5i10Q m Gi gvb KZ nj P IQ cii mgvivj ne? P I Q cii mgvivj ne hw` 0QP nq| 15-510 3m2 kji QP m)10-10(k)3030(-j)15m15(iQP m)1010(k)15m15(iQP 2m)1010(2)15m15(QP ckgZ, P IQ cii mgvivj nj, 0QP 02m)1010(2)15m15( i]KMeKc[Dfq 0m)1010()15m15( 22 0)1(m10)1(m215 222 0)1015()1(m 222 iKfvMvivKcDfq )1015(0)1(m 222 .)Ans(1m 22| kjiA 22 Ges kjiB 236 `ywU fi ivwk| Gì j^ AwfgyL GKwU GKK fi wbYq Ki| Avgiv Rvwb, `ywU fii m bdj fi `ywU viv MwVZ mgZji Dci j^ nq| mB j^ fii mgvivj GKK fiB ne mgZji Dj^ w`K GKK fi| awi, mB fi n , QP QP n )126()64()34( 236 122 kji kji QP kjiQP 1810 Avevi, )18()10(1 222 QP 425 QP 425 1810 kji QP QP n (Ans.) Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 15. , - - , : : () () : () () (i) - (ii) - , (iii) . . , , . , . , , , . , Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 16. MoeM (Average Velocity) : msv: h Kvb mgq eeavb Kvb ei Mo cwZ GKK mgq h miY nq ZvK ewUi Mo eM ej| eLv: t mgq eeavb Kvb ei miY r nj Mo eM t r v ne| eM (Velocity): msv: mgq eeavb k~bi KvQvKvwQ nj mgqi mv_ ei miYi nviK eM ej| eLv: t mgq eeavb Kvb ei miY r nj eM t r limv 0t wK t r nQ Mo eM v | myZivs vlimv 0t A_vr mgq eeavb k~bi KvQvKvwQ nj Mo eMi mxgvwK gvbKB eM ej| mgeM ev mylg eM (Uniform Velocity) : hw` Kvb ei MwZKvj Zvi eMi gvb I w`K AcwiewZZ _vK Zvnj mB ei eMK mgeM ej| A_vr Kvb e hw` wbw` w`K mgvb mgq mgvb c_ AwZg Ki Zvnj ei eMK mgeM ej| ki eM, Avjvi eM, cfwZ mgeMi cK cvKwZK D`vniY| AmgeM (Variable Velocity) t Kvb ei MwZKvj hw` Zvi eMi gvb ev w`K ev DfqB cwiewZZ nq Zvnj mB eMK AmgeM ej| Avgiv mPvivPi h MwZkxj e `wL Zvì eM AmgeM| ZvrwbK eM (Instantaneus Velocity) : GKwU e mij ev e c_ AmgeM Pjj cwZwbqZ Gi eMi cwieZb nq| Gfve AmgeM Pj Kvb ei h Kvb gyZi eMK H ei ZvrwbK eM ej| ZvrwbK eMi w`K ewUi H gyZi Aevb AswKZ MwZc_i kK eivei| ZiY (Acceleration) : mgqi mv_ eM ewi nviK ZiY ej| t mgq eeavb ei eMi cwieZb v nj ZiY t v a ne| Abfve ejv hvq mgq eeavb k~bi KvQvKvwQ nj mgqi mv_ ei eM ewi nviK ZiY ej| t mgq eeavb ei eMi cwieZb v nj ZiY t v lim 0t a ne| mgZiY ev mylg ZiY (Uniform Acceleration) : GKB w`K GKB mgq eeavb eMi ewi nvi mgvb nj ZvK mgZiY ev mylg ZiY ej| AwfKli Uvb gyfve co ei eM ewi nviK AwfKlR ZiY ej| AwfKlR ZiY, mgZiY wewk MwZi GKwU cK D`vniY| mgZiY, ZiYi gvb I w`K mgqi mv_ AcwiewZZ _vK| mgZiY MwZkxj eZ mgej wqvKi ei cici mKi eMi Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 17. 02| iwLK MwZ (Linear Motion) 2 AiB mgZiY| wP GKwU mijiLv eivei cici mKi eM `wLq Gi Zibi cKwZ wb`k Kiv nqQ| GLvb mgZiYi gvb 2ms2 | miY ( Displacement ) t wbw` w`K ei Aevbi cwieZbK miY ej| miYK s ev d viv cKvk Kiv nq| Gi GKK wgUvi| wbw` w`K Kvb e t mgq ai v eM Pjj, miY s = v t ne| miY GKwU fi ivwk| AvcwK eM: `ywU MwZkxj ei GKwUi Zzjbvq (mvc) AciwUi Aevbi cwieZbi nviK AvcwK eM ej| ga eM: Kvb GKwU MwZkxj ei c_g Ges kl eM Gi AwfgyL GKB nj Zvì hvM dji AaKK ga eM ej| Kvb wbw` w`K Kvb ei AvwèM viI kl eM vf nj gaeM = 2 fi vv ne| `wZ I eMi cv_K (Distinction between Speed and Velocity) : wgK `ywZ eM 1| h Kvb w`K ei vb cwieZbi nviK ei `wZ ej| wbw` w`K ei vb cwieZbi nviK ei eM ej| 2| `wZi Kej gvb AvQ, w`K bvB| dj `wZ me`vB abvZK| eMi gvb I AvQ , w`K I AvQ | dj, eM abvZK I nZ cvi FbvZK I nZ cvi| 3| `ywZ GKwU jvi ivwk ev Aw`K ivwk| eM GKwU fi ivwk ev w`K ivwk| 4| `wZ cwigvci hi bvg wWvwgUvi| eM cwigvci hi bvg fjvUvwgUvi| 5| Kvb e t mgq r `yiZ AwZg Kij `wZ = dt dr ne| wbw` w`K Kvb e t mgq s `yiZ AwZg Kij eM = dt ds ne| eM I ZiYi cv_K (Distinction between Velocity and Acceleration) : wgK eM ZiY 1| wbw` w`K ei vb cwieZbi nviK ei eM ej| mgqi mv_ eM ewi nviK ZiY ej| 2| eMK v viv cKvk Kiv nq| ZiYK a viv cKvk Kiv nq| 3| eMi GKK ms1 | ZiYi GKK ms2 | 4| eMi gvv [LT 1 ]| ZiYi gvv [LT 2 ]| (K) tavv xxox cwZcv`b| gbKwi, X A eivei GKwU e mylg ZiY MwZkxj| Aviv awi, GB MwZi cviwK kZvw` nj mgq Mbbvi iZ A_vr hLb t = 0 ZLb Avw` Aevb x = 0 Ges AvwèM vx = vxo | Aviv awi, t mgq ci ewUi Aevb x = x Ges Ges eM vx = vx| hnZz h Kvb gyn~Zi mgqi mvc Kvb KYvi eM ewi nviK ZiY ej| myZivs, dt dvx x a dtdv xx a hLb t = 0 ZLb vx = vxo Ges x = xo Avevi, hLb t = t ZLb vx = vx Ges x = x GB mxgvi ga mgxKiYK mgvKjb Ki cvB, Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 18. 02| iwLK MwZ (Linear Motion) 3 t 0 xx v v x dtdv x xo eKaaa t o x v v x tv x x0 a 0tvv xxox a tvv xxox a mgxKiYwU cwZcv`b Kiv nj| (L) t)vv( 2 1 xx xxoo cwZcv`b: gbKwi, X A eivei GKwU e mylg ZiY MwZkxj| Aviv awi, GB MwZi cviwK kZvw` nj mgq Mbbvi iZ A_vr hLb t = 0 ZLb Avw` Aevb x = x0 Ges AvwèM vx = vxo | Aviv awi, t mgq ci ewUi Aevb x Ges Ges kleM vx | MoeMi msv nZ Avgiv Rvwb, z`vwZz` mgq eeavb eM I mgq eeavbi bdji mgw wbq ZvK gvU mgq w`q fvM Kij H mgqi MoeM ej| myZivs KYvwUi t mgq ci MoeM xv nj, dtv t v t 0 xx 1 dt dx xvdt dt dx t 1 v t 0 x (1)............tvxx xo mymg ZiY Pjgvb ewUi eM xv mgqi mv_ mymgfve cwiewZZ nq ej h Kvb mgq eeavb Zvi Mo gvb H mgq eeavbi i I kli eMi gvbqi mgwi AaK| A_vr xxxox v),vv( 2 1 v Gi GB gvb (1) bs mgxKiY ewmq cvB, t)vv( 2 1 xx xxoo t)vv( 2 1 xx xxoo mgxKiYwU cwZcv`b Kiv nj| (M) 2 xxoo 2 1 tatvxx cwZcv`b: gbKwi, X A eivei GKwU e ax mylg ZiY MwZkxj| Aviv awi, GB MwZi cviwK kZvw` nj mgq Mbbvi iZ A_vr hLb t = 0 ZLb Avw` Aevb x = xo Ges AvwèM vx = vxo | Aviv awi, t mgq ci ewUi Aevb x = x Ges Ges eM vx = vx| hnZz h Kvb gyn~Zi mgqi mvc Kvb KYvi eM ewi nviK ZiY ej| myZivs, dt dv a x x dtadv xx hLb t = 0 ZLb vx = vxo Ges x = xo Avevi, hLb t = t ZLb vx = vx Ges x = x GB mxgvi ga mgxKiYK mgvKjb Ki cvB, t 0 xx x xo eKaadtadv v v x t o v v tav x xo xx 0xxox tavv (1)...............xxox tavv h Kvb gyZ ei miY ewi nviK eM ej| D (1) mgxKiY dt dx v x ewmq cvB, tav dt dx xxo Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 19. 02| iwLK MwZ (Linear Motion) 4 xxo dttadtvdx t o t dtxa t o dtxov x xo dx t atvx t o x t oxo x xo 2 2 0 2 1 0 2 tatvxx xxoo 2 2 1 tatvxx xxoo mgxKiYwU cwZcv`b Kiv nj| ev, 2 2 1 tatvxx xxoo 2 2 1 tatvS xxo mgxKiYwUI cwZcv`b Kiv nj| wi Aevb _K mgZiY Pjgvb ei , 0xov Ges eKaxa ,dj 2 2 1 0 tS eKa 2 tS eKa 2 tS KvRB, wi Aevb _K mgZiY Pjgvb ei AwZv `yiZ mgqi eMi mgvbycvwZK| (N) )xx(2vv ox 2 xo 2 x a cwZcv`b : awi, GKwU e X A eivei ax mylg ZiY MwZkxj| GB MwZi cviwK kZvw` nj hLb mgq Mbbvi iZ hLb t = 0 ZLb Avw` Aevb x = xo Ges AvwèM vx = vxo Avevi, t mgq ci KYvwUi Aevb x Ges eM vx | hnZz h Kvb gyn~Z mgqi mvc ei eM ewi nviK ZiY ej| myZivs dt dvx x aiYZ dt dx xd dvx x a xx x x v dt dx v xd dv a dxdvv xxx a hLb x = xo ZLb vx = vxo Ges hLb x = x ZLb vx = vx GB mxgvi ga Dciv mgxKiYK mgvKjb Ki cvB, x x x v v xx o x ox dxdvv a x xx o v v 2 x x 2 v x xo a )xx( 2 vv ox 2 xo 2 x a )xx(2vv ox 2 xo 2 x a mgxKiYwU cwZcv`b Kiv nj| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 20. 02| iwLK MwZ (Linear Motion) 5 (O) )1t2(a 2 1 vs ot ev, a 2 )1t2( vs ot cwZcv`b : gbKwi, vo AvwèM Ges a mgZiY AB mijiLv eivei Pjgvb KYvwUi Avw` Aevb A we`yZ| (t-1) mK Ges t mK hw` Gi Aevb h_vg C I B we`yZ nq Ze t Zg mK e KYvwU BC `~iZ AwZg Kie| MwZi m~vbyhvqx t mK e KZK AwZv `~iZ AB (1)...............at 2 1 tvAB 2 0 (t-1) mK e KZK AwZv `~iZ AC (2)...............)1t(a 2 1 )1t(vAC 2 0 KvRB t Zg mK e KZK AwZv `yiZ st nj st = CB ACABst )1t(a 2 1 )1t(vat 2 1 tvs 2 0 2 0t )1t(a 2 1 )1t(vat 2 1 tvs 2 0 2 0t )1t2t(a 2 1 vtvat 2 1 tvs 2 00 2 0t a 2 1 atat 2 1 vtvat 2 1 tvs 2 00 2 0t a 2 1 atvs 0t a 2 1 atvs 0t )1t2(a 2 1 vs ot ev, a 2 )1t2( vs ot mgxKiYwU cwZcv`b Kiv nj| Lvov Dcii w`K wbw ei mevwaK DPZv wbYq : gbKwi, Kvb GKwU eK Lvov Dcii w`K v0 eM Qvov nj| G ei Dci wqviZ ZiY = g, KviY g cw_exi K`vwfgyLx| KvRB h DPZvq v MwZeM ARb Kij MwZi mgxKiY _K Avgiv cvB, gH2vv 2 o 2 KvRB ewU Da DVZ _vKj Gi kl eMi gvb gkt KgZ _vKe| KvRB h we`yZ wbw ei eM k~b H we`yB ei MwZ c_i mevP we`y wb`k Ki| myZivs h we`yZ ei kl eM k~b, mB we`yi DPZv H nj , gH2v0 2 o 2 g2 v H 2 o KvRB Lvov Dcii w`K wbw ei g2 v2 o -B nj e KZK AwZv mevP DPZv| Lvov Dcii w`K wbw ei mevwaK DPZvq cQZ mgq wbYq : gbKwi, Kvb GKwU eK Lvov Dcii w`K vo eM Qvov nj| G ei Dci wqviZ ZiY = -g, KviY g cw_exi K`vwfgyLx| t mgq ci Kvb wbw` DPZvq ewUi eM v nj, MwZi mgxKiY nZ cvB, v = vo gt KvRB `Lv hvQ h, Dcii w`K wbw ei eM axi axi KgZ _vK| Kvb wbw` DPZvq hLb gt Gi gvb vo Gi mgvb ZLb kl eM v = 0 ne| A_vr ewU Avi Dci DVZ cvie bv| GB DPZvB mvP DPZv| awi, mevP DPZvq Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 21. 02| iwLK MwZ (Linear Motion) 6 MgbKvj t, myZiv&s 0 = vo gt g v t o KvRB Lvov Dcii w`K wbw ei g vo -B nj e KZK AwZv mevP DPZvq cQbvi mgq| Lvov Dcii w`K wbw ei Dvbi, cZbi, Dvb cZbi mgq wbYq : gbKwi, Kvb GKwU eK Lvov Dcii w`K vo eM Qvov nj| G ei Dci wqviZ ZiY = g, KviY g cw_exi K`vwfgyLx| t mgq ci h DPZvq ewUi eM v nj MwZi mgxKiY nZ cvB, v = vo gt ... ... ... ... (1) Ges )2(.........gH2vv 2 o 2 myZivs `Lv hvQ h, Dcii w`K wbw ei eM axi axi KgZ _vK| GB eM GK mgq 0 (k~b) ne| A_vr ewU Avi Dci DVe bv| dj ewU h DPZvq DV ZvB-B e KZK AwZv mevwaK DPZv H| awi GB DPZvq hZ mgq jvM t1 myZivs (1) bs mgxKiY nZ cvB, 0 = vo gt1 )3(............... g v t o 1 mevwaK DPZvq Dvb Kvj g v t o 1 , Avevi (2) mgxKiY nZ cvB, gH2v0 2 o )4(............ g2 v H 2 o mevwaK DPZvq ei kleM k~b| KvRB mevP DPZv nZ f~wgZ covi mgq Gi AvwèM ne k~b Ges ay AwfKlR ZiY ewU wbP coZ _vKe| mevP we`y _K f~wgZ wdi AvmZ t2 mgq jvMj, MwZi mgxKiY _K cvB, 2 22 gt 2 1 t0H g2 v Hgt 2 1 2 02 2 2 2 o2 2 g v t )5(............ g v t o 2 Dvb cZbi gvU mgq T nj T = t1+t2 g v g v T oo g v2 T o BnvB Dvb cZbi gvU mgq| Lvov Dcii w`K wbw e fywgZ wdi Avmvi mgq wbYq : gbKwi, Kvb GKwU eK Lvov Dcii w`K vo AvwèM Qvov nj| G ei Dci wqviZ ZiY = g, KviY g cw_exi K`vwfgyLx| dj ewUi eM axi axi KgZ _vKe Ges GKwU wbw` DPZvq eM k~b ne A_vr ewU Avi Dci DVebv| KvRB GB DPZvq e KZK AwZv mevP DPZv| awi mevP DPZv H, dj MwZi mgxKiY _K cvB, gH2vv 2 o 2 gH2v0 2 o Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 22. 02| iwLK MwZ (Linear Motion) 7 (1)......... g2 v H 2 o Avevi mevP DPZv _K fywgZ cZveZbKvj ei AvwèM k~b ne Ges ay gv AwfKlR ZiY ewU wbP coZ _vKe| mevP we`y _K fywgZ wdi AvmZ ei t mgq jvMj MwZi mgxKiY _K cvB, 2 gt 2 1 t0H 2 2 o gt 2 1 2g v 2 o2 g v t (2)............ g v t o BnvB mevP DPZv nZ cZbKvj| awi, fywgZ wdi AvmZ h mgq jvM mB mgqi kl ei eM = v myZivs v = 0 + gt g v gv o ovv myZivs f~wg _K eK h eM Dcii w`K wbc Kiv nq, ewU wdi Gm mB eM f~wgZ AvNvZ Ki| co ei m~ eYbv (Laws of falling bodies) : evavnxb fve co e wbv wZbwU m~ gb Pj| 1589 wLv wevbx MvwjwjI m~ wZbwU Avwevi Kib t 1g m~t e mgvb mgq mgvb c_ AwZg Ki| 2q m~t wbw` mgq e h eM jvf Ki Zv H mgqi mgvbycvwZK| t mgq v eM jvf Kij, m~vbyhvqx eM ne, tv 3q m~t wbw` mgq e KZK AwZv `~iZ H mgqi eMi mgvbycvwZK| t mgq AwZv `yiZ h nj, m~vbyhvqx DPZv ne, 2 th ^bgy`v I cvjK cixv: hcvwZt (K) j^v GKwU k, gvUv I duvcv `yBgyL Lvjv KvPbj B| (L) GKwU Uzwc C (M) GKwU c KK S (N) GKwU cvjK | cixvi weeiY: KvPbji GKcvGKwU Uzwc C Ges Aci cvGKwU c KK S _vK| Uzwc Lyj GKwU ^bgy`v G Ges GKwU cvjK F bji ga XyKvbv nq| cKKi Pvwe Lyj cvi mvnvh bjwUK evqyc~b ev evqyk~b Kiv hvq| bjwUK nVvr Dwq gy`v I cvjKK wbPiw`K coZ Ìqv nq| cixvq `Lv hvq h (1) evqyc~b Aevq gy`vwU cvjKi AvM wbPi cvco| (2) evqyk~b Aevq gy`v I cvjK GKB mv_ wbPi cv co| djvdjt evqyk~b vb mKj e mgvb mgq mgvb c_ AwZg Ki| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 23. 02| iwLK MwZ (Linear Motion) 8 mgZiY MwZi eM ebvg mgq (v t)jLwP AsKb Ges jLwP nZ 2 o at 2 1 tvs mgxKiYwU cwZcv`b: mgZiY MwZkxj Kvb ei X Ai w`K mgq t Ges Y Ai w`K eM v wbq v ebvg t jL wP AsKb Kiv nj| GwU Y AK Q`Kvix GKwU mij iLv nq hv, v = vo+at mgxKiY gb Pj| GB jLwP _K t mgq ei AwZv `~iZ s wbbq Kiv hvq| AB iLvi Dci h Kvb we`y P bqv nq| P _K X Ai Dci PQ j^ Uvbv nq| Zvnj OQ = t mgq AwZv `~iZ s ne AOQP i dj| aiv hvK, KYvwUi mgZiY a Ges AvwèM, vo = AO AwZv mgq, t = OQ Ges t mgq AwZv `~iZ , s = AOQP i dj| = AOQR i dj ARP i dj| = AOOQ + 2 1 ARPR s = AOOQ + 2 1 OQPR [ AR = OQ ] wK AB iLvi Xvj nQ KYvwUi ZiY a, AR PR a PR = aAR = aOQ s = AOOQ + 2 1 OQaOQ s = AOOQ + 2 1 aOQ2 2 o at 2 1 tvs mgxKiYwU cwZcv`b Kiv nj| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 24. c_g ci AsKi mgvavb First Paper Math Solution 2| iwLK MwZ (Linear Motion) 1| GKwU e`yKi wj Kvb Ìqvji ga 0.04m cek Kivi ci AaK eM nvivq | MywjwU Ìqvji ga Avi KZUzKz cek Kie? gb Kwi, jj ceki gyn~Z wji AvwèM = u Ges wjwU AviI x wgUvi `~iZ cek Kie| 0.04 m cek Kivi ci eM ne = 2 u Ges kl eM ne 0 (k~b)| Avgiv Rvwb, c_g Aski Rb 2a(0.04)u 2 u 2 2 4 u u0.08a 2 2 )1(............... 0.32 3u 0.084 3u a 22 wZxq Aski Rb, ax2 2 u 0 2 x 32.0 u3 2 2 u 0, 22 ev 4 u 0.32 x6u , 22 ev (Ans.)m0.0133 46 0.32 x 2| 50 wgUvi DPuy _K GKwU e f~wgZ cwZZ nq| (K) fywgZ cuQZ Gi KZ mgq jvMe? (L) fywgZ cuQevi c~e gyn~Z Gi eM KZ ne? (K) 2 gt 2 1 uth 2 t9.8 2 1 005 2 t9.405 9.4 50 t2 9.4 50 t t = 3.19 s (Ans.) Avevi, (L) v = u + gt v = 0 + 9.8 3.19 v = 31.26 ms-1 (Ans.) 3| 20ms-1 eM MwZkxj GKwU ei eM cwZ mK 3ms-1 nvi nvm cvq| _g hvIqvi AvM ewU KZ `~iZ AwZg Kie? Avgiv Rvwb, v2 = u2 2as ev, 0 = 202 2(3)s ev, 6s = 400 6 400 s ev, (Ans.)m66.7s 4| Dcii w`K wbw GKwU ej Uwjdvb ZviK 0.70ms-1 `wZZ AvNvr Ki| Qvovi vb _K ZviwUi DPZv 5.1m nj ejwUi Avw` `wZ KZ wQj? Avgiv Rvwb, 2ghuv 22 5.19.82u(0.7) 22 5.19.82(0.7)u 22 96.9949.0u2 45.100u2 (Ans.)ms10.0245.100u -1 5| GKwU Ub 3ms-2 mgZib PjQ Ges AvwèM 10m/s UbwU hLb 60m c_ AwZg Kie ZLb Gi eM KZ ne| Avgiv Rvwb, 2as2u2v 603210v 22 360100v2 460v2 )Ans(1ms45.21447.21460v 6| GKwU eK 98 ms-1 eM Lvov Dcii w`K wbc Kiv nj `LvI h, 3 Sec I 17 Sec mgq ei eMq mgvb wK w`K wecixZ gyLx| Avgiv Rvwb, 3 mt ci eM v1 = u gt1 ev, v1 = 989.83 ev, v1 = 9829.4 v1 = 68.6ms-1 Avevi, 17 mt ci eM v2 = u gt2 ev, v2 = 98 9.817 ev, v2 = 98 166.6 v2 = 68.6 ms-1 3 mt I 17 mt ci eR q mgvb I wecixZ (cgvwYZ) GLvb, AvwèM, u = 20 ms-1 g`b, a = 3 ms-2 kleM, v = 0 _vgvi AvM ewU KZK AwZv `~iZ, s = ? GLvb, DPZv, h =50m AvwèM, u = 0 g = 9.8 ms-2 (K) mgq, t = KZ? (L) kl eM, v = KZ ? GLvb, DPZv, h =5.1m g = 9.8 ms-2 kl eM, v =0.70ms-1 Avw` `wZ, u =? GLvb, ZiY, a = 3ms-2 AvwèM, u = 10 ms-1 miY, s = 60m kleM, v = ? GLvb, AvwèM, u = 98 ms-1 mgq, t1 = 3S mgq, t2 = 17S kleM, v1 =? kleM, v2 =? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 25. 2| iwLK MwZ (Linear Motion) 2 7| t3t 3 1 s 3 m~vbymvi GKwU e mij iLvq PjQ| 2 mK ci Gi eM KZ ne? Avgiv Rvwb, dt ds v t3t 3 1 dt d v 3 3t3 3 1 v 2 3tv 2 32v 2 [t Gi gvb ewmq] 7v GKK (Ans.) 8| 54 kmh1 eM Pj GKwU ij MvwoZ mb _K wKQy `yi 0.75ms-2 g`b mwKvix eK Ìqvq MvwowU mb Gm _g Mj| mb _K KZ `~i eK Ìqv nqwQj Ges MvwowU _vgZ KZ mgq jMwQj? Avgiv Rvwb, v2 = u2 2as s75.02150 2 m 75.02 1515 s (Ans.)m150s Avevi, v = u at t75.0150 s 75.0 15 t (Ans.)s20t 9| GKwU e wi Aevb nZ hvv i Ki c_g mK 1m `~iZ AwZg Ki| cieZx 1m `~iZ AwZg KiZ KZ mgq jvMe| Avgiv Rvwb, 2 111 at 2 1 uts 2 )1(a 2 1 01 2 a 1 2 ms2a GLb c_g _K s2 = (1m+1) =2m `~iZ AwZg KiZ mgq jvM = t2 2 222 at 2 1 uts 2 2t2 2 1 02 2t2 2 s414.12t2 kli 1m `~iZ AwZg KiZ mgq jvM, .)Ans(s414.0s)1414.1(ttt 12 10| GKwU Ub wi Aevb nZ 10ms-2 ZiY PjZ Avi Kij| GKB mgq GKwU Mvwo 100ms-1 mgeM Ubi mgvivj Pjv i Kij| Ub MvwowUK KLb wcQb dje? gb Kwi, t mgq ci Ub MvwowUK wcQb dj Pj hve, t mgq Ub KZK AwZv `~iZ, 2 at 2 1 0x 2 t10 2 1 x (1).........t5x 2 t mgq Mvwo KZK AwZv `~iZ, Vtx (2).........t100x kZgZ Ub hLb MvwowUK AwZg Kie ZLb xx ne| t100t5 2 5 100 t (Ans.)s20t 11| wivev _K PjZ Avi Ki 625m `~iZ AwZg Kij GKwU ei eM 125ms-1 nj| ZiY wbYq Ki| Avgiv Rvwb, as2uv 22 625a201252 2 2 ms 6252 125 a .)Ans(ms5.12a 2 12| 64m DuPz `vjvbi Qv` _K 5kg fii GKwU cv_i Qo w`j f~wgZ cuQvZ Gi KZ mgq jvMe? Avgiv Rvwb, 2 gt 2 1 uth 2 t8.9 2 1 064 2 t9.464 .)Ans(.s61.3t 9.4 64 t GLvb, mgq, t = 2 Sec eM, v =? GLvb, AvwèM, u = 54 kmh-1 1 ms 3600 100054 =15ms-1 g`b, a = 0.75ms-2 kleM, v = 0 mgq, t = ? miY, s=? GLvb, AvwèM, u = 0 mgq, t1 = 1s miY, s1 =1m ZiY, a=? GLvb, Mvwoi mgeM, V = 100ms-1 Ubi ZiY, a = 10ms-2 mgq, t = ? GLvb, AvwèM, u = 0 AwZv `~iZ, s = 625m kl eM, v = 125 ms-1 ZiY, a =? GLvb, AvwèM, u = 0 AwZv `~iZ, h = 64m fi, m= 5kg mgq, t =? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 26. 2| iwLK MwZ (Linear Motion) 3 13| wi Aevb nZ hvv Avi Ki GKwU e c_g mK 2m `~iZ AwZg Ki| cieZx 2m `~iZ AwZg KiZ ewUi KZ mgq jvMe| Avgiv Rvwb, 2 111 at 2 1 uts 2 )1( 2 1 02 a 2 2 a 2 ms4a GLb c_g _K s2 = (2m+2m) = 4m `~iZ AwZg KiZ mgq jvM = t2 2 222 at 2 1 uts 2 24 2 1 04 t 2t2 2 s414.12t2 kli 2m `~iZ AwZg KiZ mgq jvM, .)Ans(s414.0s)1414.1(ttt 12 14| GKwU e c_g `yB mK 30m I cieZx Pvi mK 150m Mj| ZiY AcwiewZZ _vKj ewU Gi ci GK mK KZUv c_ AwZg Kie? Avgiv Rvwb, 2 111 at 2 1 uts 2 2 2 1 230 au )1(..........15 au c_g _K t2= (2+4)= 6 mK ewU hvq s2=(30+150)m=180m 2 222 2 1 atuts 2 6 2 1 6180 au )2(..........303 au )1(..........15 au weqvM Ki, 2a= 15 2 5.7 msa GLb (1) bs mgxKiY a Gi gvb ewmq, 155.7 u 1 5.7 msu 6 mKi cii mK A_vr 7g mK AwZv `~iZ, )12( 2 t a ust )172( 2 5.7 5.77 s 13 2 5.7 5.77 s 75.485.77 s )(25.567 Ansms GLvb, AvwèM, u = 0 mgq, t1 = 1s miY, s1 =1m ZiY, a =? GLvb, `~iZ, s1 = 30m mgq, t1 = 2s miY, s2 = (30+150) =180m miY, s7 =? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 27. ckt wbwjwLZ mgxKiYwj mivmwi fi ic cwZcv`b Ki| )rr(a2vvta 2 1tvrr)tvv( 2 1rrtavv 0 2 000000 22 N)M)L)K) (((( |cwZcv`bK)( tavv 0 awi, GKwU e a mgZiY Kvb Zj MwZkxj| GB MwZi cviwK kZvw` nj hLb oit ZLb Avw` Aevb fi 0rr Ges AvwèM 0vv Avevi, ttf mgq ewUi Aevb fi r Ges eM v | h Kvb gyn~Z mgqi mvc ei eM ewi nviK ZiY ej| dt dv a iYZ dtavd ev, hLb t = 0 ZLb 0vv Ges hLb t = t ZLb vv GB mxgvi ga Dciv mgxKiYK mgvKjb Ki cvB, t 0 dtd 0 av v v tv v tav o 0 )(0 otavv tavv 0 tavv 0 )( 2 1rr 00 |cwZcv`b(L) tvv awi, GKwU e a mgZiY Kvb Zj MwZkxj| GB MwZi cviwK kZvw` nj hLb 0it ZLb Avw` Aevb fi 0rr Ges AvwèM 0vv Avevi, ttf mgq ewUi Aevb fi r Ges eM v | myZivs KYvwUi t mgq eeavb MoeM V nj t dtv t V 0 1 t dt rd vdt dt rd t V 0 ][ 1 r r rd t V 0 1 r rr t V 0 ][ 1 )( 1 0rr t V tVrr 0 tvvrr )( 02 1 0 )( 02 1 vvV tvvrr )( 02 1 0 Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 28. 3| wgvwK MwZ (Motion In Two Dimention) 2 |cwZcv`b(M) 2 00 t 2 1rr atv awi, GKwU e a mgZiY Kvb Zj MwZkxj| GB MwZi cviwK kZvw` nj hLb oit ZLb Avw` Aevb fi 0rr Ges AvwèM 0vv Avevi, ttf mgq ewUi Aevb fi r Ges eM v | h Kvb gyn~Z mgqi mvc ei eM ewi nviK ZiY ej| dt dv a iYZ dtavd ev, hLb ti = 0 ZLb 0vv Ges hLb tf = t ZLb vv GB mxgvi ga Dciv mgxKiYK mgvKjb Ki cvB, t 0 dtd 0 av v v tv v tav o 0 )(0 otavv tavv 0 tavv 0 Avevi, hnZz Kvb gyn~Z mgqi mvc ei Aevb fi ewi nviK eM ej, ZvB msvbymvi, dt rd v ewmq cvB, tav dt rd 0 tdtadtvrd 0 hLb ti = 0 ZLb 0rr Ges hLb tf = t ZLb rr GB mxgvi ga Dciv mgxKiYK mgvKjb Ki cvB, t 0 t 0 0 r r tdtadtvrd 0 tttr r atvr 0200 ][][ 2 0 )0( 2 )0( 2 00 tatvrr 2 2 1 00 tatvrr 2 2 1 00 tatvrr 2 0 2 0 |cwZcv`b(N) )rr(avv2 awi, GKwU e a mgZiY Kvb Zj MwZkxj| GB MwZi cviwK kZvw` nj hLb oit ZLb Avw` Aevb fi 0rr Ges AvwèM 0vv Avevi, ttf mgq ewUi Aevb fi r Ges eM v | h Kvb gyn~Z mgqi mvc ei eM ewi nviK ZiY ej| dt dv a iYZ dtavd ev, hLb t = 0 ZLb 0vv Ges hLb t = t ZLb vv GB mxgvi ga Dciv mgxKiYK mgvKjb Ki cvB, t 0 dtd 0 av v v tv v tav o 0 Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 29. 3| wgvwK MwZ (Motion In Two Dimention) 3 )(0 otavv tavv 0 tavv 0 Dfq cK GB mgxKiY w`q Y Ki cvB, )).((. 00 tavtavvv 2 0000 ..... taatavtavvvvv 2 000 ..2.. taatavvvvv ) 2 1 .(2.. 2 000 tatvavvvv )( 2 1 .2.. 0 2 0000 rrtatvrravvvv 0 2 0 2 .2 rravv ckt cvm KvK ej? Dit Kvb eK AbyfywgKi mv_ wZhKfve Kvb vb wbc Kiv nj ZvK cvm ej| wZhKfve wbw wXj, eyjUi MwZ BZvw` cvm MwZi D`vniY| ckt AbyfywgKi mv_ wZhKfve wbw cvmi MwZc_i mgxKiY wbbq Ki Ges `LvI h, GB MwZc_ AwaevKvi| Dit gbKwi, evqygawZ O we`y nZ GKwU cvmK wbc Kiv nj| wbc eM ev AvwèM = vo wbc KvY = g wbPi w`K wqvkxj| AZGe ay = -g; ax = 0; wbc we`y I g~j we`y GKB nIqvq xo = yo = 0 AvwèMi Abyf~wgK Dcvsk = voCoso Ges AvwèMi Dj^ Dcvsk = voSino X A eivei MwZi cwieZb D A eivei ZiYi Dci wbfikxj| Y A eivei MwZi cwieZb D A eivei ZiYi Dci wbfikxj| G `ywU A eivei MwZi cwieZb Awbfikxj| awi t mgq cvmwU P(x,y) Aevb _vK| ZLb Gi eM = v Abyf~wgKi w`K ZiY, ax= 0 Abyf~wgKi w`K miY = x x = voCoso t + 2 1 axt2 ev, x = voCoso t + 0 [ax= 0] ev, x = voCoso t )1...(.................... ooCosv x t Dj^ w`K ZiY ay=g; Dj^ w`K miY y; Abyicfve y=voSinot 2 1 gt2 Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 30. 3| wgvwK MwZ (Motion In Two Dimention) 4 2 o 2 1 oooo o Cosv x g Cosv x Sinvy ev, [t Gi gvb ewmq] 2 22 2 tan x Cosv g xy oo o ev, c oCosov g bcxbxy o 22 2 2 tan, GeseKaawi Dciv mgxKiYwU GKwU Awaei mgxKiY| cvmi MwZc_ GKwU Awae (cvivevjv)| ckt cgvY Ki, evqynxb Aevq f~wg nZ DPZvq AewZ h Kvb Aevb nZ Abyf~wgK AwfgyL wbw ei MwZc_ GKwU Awae| gbKwi, k~b AewZ O we`y nZ vo eM f~wgi mgvivj GKwU eKYv wbw nj| e KYvwU g Gi cfve bxP coe| awi ccb Zj Abyf~wgK OX iLv X A Ges OY iLv Y A| awi t mgq ci e KYvwU MwZ c_i P(x,y) we`yZ gyn~Zi Rb Aevb Kie| g bxPi w`K wqvkxj| AZGe ay = g; ax= 0 ; AvwèMi Abyf~wgK Dcvsk = vo Ges AvwèMi Dj^ Dcvsk = 0 tmgq AwfKlRZiYnxb Abyf~wgK miY x = vot )(tvx ... ...... ...... ...o 1 222 tmgq Dj^ miY y = 0.t + 2 1 gt2 y = 2 1 gt2 ... .... .... .... .... .... .... (2) (1) K (2) viv fvM Ki cvB 2 2 1 222 gt tv y x o g v2 y x 2 o 2 y g v x o 2 2 2 eKaawi a v ayx 4 g 2 ,4 2 o2 Dciv mgxKiYwU GKwU Awaei mgxKiY| ZvB wbw ei MwZc_ GKwU Awae (cvivevjv)| ckt AbyfywgKi mv_ wZhK fve wbw ei (K) mevP DPZvq cQZ mgq (L) mevP DPZv (M) wePiY Kvj (N) cvjv (O) mevwaK cvjv wbbq Ki| gb Kwi, evqygawZ O we`y nZ GKwU cvmK ov eM o KvY wZhKfve wbc Kiv nj| cvmwU t mgq mevP DPZv P(x,y) G Aevb Kie Ges ZLb Gi eM ne v| (K) mevP DPZvq cQZ mgqt vo eMi Dj^ Dcvsk voSino t mgq ci P we`yZ eM, vy = voSino gt.................(1) P we`yMvgx mevP DPZvq vy= 0..................................... (2) Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 31. 3| wgvwK MwZ (Motion In Two Dimention) 5 (1) bs mgxKiY vy= 0 ewmq cvB 0 = voSino gt )3....(.............................. g Sinv t oo (L) mevP DPZvt gbKwi, mevP DPZv = H H = voSinot 2 1 gt2 qewmgvbGiZnbs t(3) 2 2 1 g Sinv g g Sinv SinvH oooo oo 2 22 g Sinv g Sinv H oooo (4)..................... 2g Sinv H o 22 o (M) Dqb (wePiY) Kvj (Time of Flight) t gb Kwi wePiY Kvj T A_vr T mgq cvmwU mgZj wdi Avm| t mgq Dj^ w`K miY y = voSinot 2 1 gt2 GB mgxKiY mgq t = T Ges miY y = 0 ewmq cvB, 0 = voSinoT 2 1 gT2 ev, 2 1 gT2 = voSinoT (5).................. 2 T g Sinv oo (N) cvjv (Range)t gb Kwi cvjv R A_vr T mgq cvmwU AbyfywgKi w`K h `~iZ AwZg Ki ZvBB cvjv R R = ( voCoso ) T g Sinv CosvR oo oo 2 [(5) bs nZ T Gi gvb ewmq] g CosSinv R ooo 22 .......(6).................... 22 g Sinv R oo (O) mevwaK cvjv (Maximum Range) t gbKwi mevwaK cvjv Rmax| wbw` vo Gi Rb, Sin20 Gi gvb mevwaK nj cvjv ne mevwaK| Sin20 Gi mevwaK gvb = 1 A_vr Sin20 = 1 ev, Sin20 = Sin900 ev, 20 = 900 0 = 450 myZivs wbc KvY0 = 450 nj cvjv mevwaK Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 32. 3| wgvwK MwZ (Motion In Two Dimention) 6 mevwaK cvjv 4522 max g Sinv R o o 092 max g Sinv R o o 12 max g v R o (7)............... 2 max g v R o ckt iwLK eM I KwbK eMi msv `vI Ges Gì ga mK vcb Ki| |Kicgvbev, , rvrv ev |KicgvYev, rv iwLK eM (Linear Velocity)t wbw` w`K iwLK c_ Kvb e GKK mgq h `yiZ AwZg Ki ZvK H ei iwLK eM ej| iwLK eMK v viv cKvk Kiv nq| wbw` w`K e t mgq d `~iZ AwZg Kij eM t d v ne| eM GKwU fi ivwk| iwLK eMi GKK ms-1 KwYK eM (Angular Velocity) t mgq eeavb k~bi KvQvKvwQ nj Kvb we`y ev AK K` Ki evKvi c_ Pjgvb Kvb ei mgqi mv_ KwbK miYi nviK KwbK eM ej| Ab K_vq evKvi c_ Kvb e GKK mgq h KwbK `~iZ AwZg Ki ZvK H ei KwbK eM ej| KwbK eMK viv cKvk Kiv nq| wbw` w`K e t mgq KvY Drcb Kij KwbK eM t ne| KwbK eMi GKK rad s-1 Gi gvv nQ 1-T TL L mgqevmva Pvc mgq KvY mK (Relation) t gbKwi GKwU eKYv OC= OB = r evmva wewk GKwU ei cwiwa eiveiKwbK eM NyiQ| hw` T mK e KYvwU ei cwiwa eivei GKevi Nyi Avm Ze KwbK `~iZ = iwWqvb ne| KwbK eM, T 2 ev, )1...(............ 2 T GLb e KYvwU hw` evKvi c_ bv Nyi H GKB mgq mij iLv eivei PjZ Ze T mgq eKYvwU ewUi cwiwai mgvb c_ r `~iZ AwZg KiZ| iwLK eM 2 T r v )2...(............ 2 v r T (1) bs I (2) mgxKiYq nZ cvB v r22 Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 33. 3| wgvwK MwZ (Motion In Two Dimention) 7 v r 1 v = r A_vr iwLK eM = KwbK eM ei evmva| v = r mgxKiYi fi ic: gb Kwi, (3).........ru u fii gvb ][r90sin rru m Ybi wbqg Abymvi, u, evr fii AwfgyL Ges v fii AwfgyL Awfb| Avevi v = r| `Lv hvQ h, gvb I w`K weePbvq vu I fi Awfb| (4).........vu (3) I (4) nZ rv (cgvwYZ) ck: KwbK eM I iwLK eMi ga cv_K eYbv Ki| KwbK eM I iwLK eMi ga cv_K (Distinction between Angular Velocity and Linear Velocity) wgK KwbK eM iwLK eM 1| KwbK c_ ei KwbK mibi nviK KwbK eM ej| wbw` w`K iwLK c_ Kvb GKwU ei vb cwieZbi nvi K iwLK eM ej| 2| GK viv cKvk Kiv nq| GK viv v cKvk Kiv nq| 3| Gi mgxKiY t Gi mgxKiY t s v 4| Gi gvv mgxKiY [ T-1 ] Gi mgxKiY [ LT-1 ] 5| Gi GKK iwWqvb/ mK Gi GKK wgUvi/ mK K`gyLx ej (Centripetal Force): hLb Kvb e evKvi c_ NyiZ _vK ZLb h ej ei Dci H ei K` AwfgyL wqv Ki ewUK evKvi c_ MwZkxj ivL ZvK K`gyLx ej ej| m fii e r evmva wewk evKvic_ v mg`wZZ NyiZ _vKj Zvi K`gyLx ej r v m 2 | K`wegyLx ej (Centrifugal Force): hLb Kvb e evKvi c_ NyiZ _vK ZLb h ej H ei Kì wecixZ w`K cqvM Ki ZvK K`wegyLx ej ej| m fii e r evmva wewk evKvic_ v mg`wZZ NyiZ _vKj Zvi K`wegyLx ej r v m 2 | m fi wewk GKwU e r emvai evKvi c_ v mg`wZZ NyiQ| (1) `LvI h, j^ ZiY r r v a 2 2 ev (2) j^ ZiYi ivwkgvjv wbbq Ki| (3) cgvY Ki h, K`gyLx ej rm r v mF 2 2 ev, (4) evKvi c_ mg`wZZ N~bvqgvb Kvb ei Dci wqvkxj K`gyLx eji gvb I w`K wbYq | aiv hvK, m fii Kvb e r evmvai evKvi c_ v mg`wZZ Ges KwbK eM AveZbiZ AvQ| awi AwZ z` mgq t eeavb ewU A nZ B we`yZ Am| A we`yZ ewUi eM H we`yZ kK AC eivei| B we`yZ ewUi eM H we`yZ kK BD eivei| BD K cQb ewaZ Kij AC I BD Gi wgjb we`y nq E| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 34. 3| wgvwK MwZ (Motion In Two Dimention) 8 GLb, OAEB PZzf~R, AEB+ AOB = `yB mgKvY| Avevi, AEB+ BEC = `yB mgKvY| AOB = BEC = awi, A we`yZ ei eMi Dj^ Dcvsk, vy = 0 Ges AbyfywgK Dcvsk, vx = v B we`yZ ei eMi AC eivei eMi Dj^ Dcvsk, vsinvy Ges AbyfywgK Dcvsk, vcosvx t AwZ z` mgq myZivs AwZ z`| sin Ges 1c os B we`yZ ei eMi eMi Dj^ Dcvsk, vvy Ges AbyfywgK Dcvsk, vvx GZ `Lv hvQ, AbyfywgK eivei eMi Dcvski Kvb cwieZb nq bv| eMi Dj^ Dcvski cwieZbi KviY ZiY, a nj, t v a 0 t v v t r v v r v r r r r v a 2 222 K`gyLx ej, rm r v mmaF 2 2 (cgvwYZ) ck: Awfj^ ZiY ev evmvagyLx ZiY ev K`gyLx ZiY: Awfj^ ZiY ev evmvagyLx ZiY ev K`gyLx ZiY t Kvb e hLb evKvic_ NyiZ _vK ZLb ei evmva eivei ei Kì w`K wqvkxj AwfK` eji Rb h ZiYi mw nq ZvK evmvagyLx ZiY ev Awfj^ ZiY ev K`gyLx ZiY ej| Gi GKK wgUvi/mK2 | ck: KwbK ZiY KvK ej? KwbK ZiYt hLb Kvb eKYv Amg KwbK eM Nyi, ZLb ewUi KwbK eM cwieZbi nviK KwbK ZiY ej A_ev, mgqi mv_ Amg KwbK eM cwieZbi nviK KwbK ZiY ej| GK viv cKvk Kiv nq| Gi GKK iwWqvb/mK2 | gbKwi, evKvi c_ Nybvqgvb eKYvi Avw`KwbK eM i Ges t mgq ci Gi kl KwbK eM f KvRB KwbK ZiY t if Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 35. 3| wgvwK MwZ (Motion In Two Dimention) 9 ck: mg`wZZ Pjgvb ei Zib _vK bv, wK evKvi c_ mg`wZZ Pjgvb ei ZiY _vK Kb? evLv Ki| mg`wZZ Pjgvb ei Zib _vK bv, wK evKvi c_ mg`wZZ Pjgvb ei ZiY _vK t eMi gvb nQ `wZ Ges eMi cwieZbi nvi nQ ZiY| Kvb e hLb mij c_ mg `wZZ Pj ZLb eMi gvbi Kvb cwieZb nq bv Avi mij c_ Pjvi Rb w`Ki I Kvb cwieZb nq bv| dj ei Kvb ZiY _vK bv| wK evKvi c_ Nyievi mgq ei wbqZ w`Ki cwieZb nq, KviY eMi AwfgyL me`vB ei kK eivii nq| Gfve AbeiZ w`K cwiewZZ nZ _vK ej e mg`wZZ PjjI eM mgvb _vKbv| eMi GB cwieZbi dj ZiYi mw nq| GB ZiYi AwfgyL evKvi c_i K` eivei nq _vK| G Rb evKvi c_ mg`wZZ Pjgvb ei ZiY _vK| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 36. c_g ci AsKi mgvavb First Paper Math Solution 3| wgvwK MwZ (Motion In Two Dimention) 1| GKwU cvmi AbyfywgK cvjv 96m Ges AvwèM 66 ms-1 | wbc KvY KZ? Avgiv Rvwb, g 2Sinv R o 2 o 2 0 o v gR Sin2, ev 2o 66 9.896 Sin2 ev, (0.2159)Sin2, 1 o ev )Ans.(24.6 47.122 o o ev, 2| GKwU eK 40ms-1 eM Ab~fywgKi mv_ 60 KvY wbc Kiv nj| mevwaK DPZv Ges Abyf~wgK cvjv wbYq Ki| Avgiv Rvwb, g2 )Sin (v H 2 oo 8.92 2 60Sin40 H 8.92 2 60Sin40 H 8.92 2 86602.040 H 8.92 2 6408.34 H 8.92 9850.1199 H .)m (Ans22.61H g o2Sinv R 2 0 Avevi, 8.9 )602Sin (40 R 2 8.9 )602Sin (40 R 2 8.9 )120Sin1600 R 8.9 86602.01600 R 8.9 632.1385 R m (Ans.)39.141R 3| nvBWvRb cigvbyi gWji GKwU BjKUb GKwU cvUbi Pviw`K 5.2 10 -11 m evmvai GKwU evKvi c_ 2.18 106 ms-1 eM c`wb Ki| BjKUbi fi 9.1 10-31 kg nj K`gyLx ej KZ? Avgiv Rvwb, r mv F 2 11- 2631- 102.5 )1018.2(101.9 F N (Ans.)10316.8F 8 4| 0.250kg fii GKwU cv_i LK 0.75m j^v GKwU myZvi GK cv eua evKvi c_ cwZ wgwbU 90 evi Nyivj myZvi Dci KZ Uvb coe| Avgiv Rvwb, rmF 2 r t n2 mF 2 75.0 60 901416.32 25.0F 2 (Ans.)N65.16F 5| 9.2 ms-1 eM GKwU z` eK Lvov Dcii w`K wbc Kiv nj| GwU KZ mgq ci f~-c wdi Avme? Avgiv Rvwb, g Sinv2 T oo 8.9 90Sin2.92 T 8.9 12.92 T 8.9 4.18 T .)Ans(s877.1T 6| AbyfywgKi mv_ 30KvY f~-c _K 50ms-1 eM GKwU eyjU Qvov nj| eyjUwU 50m `~i AewZ GKwU ÌqvjK KZ DPZvq AvNvZ Kie| Avgiv Rvwb, 2 2 00 0 x )cosv(2 g x)(tany 2 2 0 )50( )30cosv(2 g x)30(tany GLvb, AbyfywgK cvjv, R = 96 m AvwèM, vo = 66ms-1 AwfKlR ZiY, g = 9.8ms-2 wbc KvY, = ? GLvb, AvwèM, v0 = 40ms-1 wbc KvY 60 AwfKlR ZiY, g = 9.8ms-2 mevwaK DPZv, H = ? AbyfywgK cvjv, R = ? GLvb, fi, m = 0.250 kg evmva, r = 0.75 m mgq t = 1 min. = 60s. cvKmsLv, n = 90 cvK| Uvb, F = ? GLvb, AvwèM, vo = 9.2 ms-1 wbc KvY, 0 AwfKlR ZiY, g = 9.8ms-2 Dvb cZbi gvU mgq T =? GLvb, AvwèM, vo = 50 ms-1 wbc KvY, AwfKlR ZiY, g = 9.8ms-2 AbyfywgK `~iZ, x=50m Dj^ `~iZ, y=? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 37. 3| wgvwK MwZ (Motion In Two Dimention) 2 2 2 0 )50( )30cosv(2 g 50)30(tany 2 2 )50( )866025403.050(2 8.9 50577350269.0y 533333345.686751346.28y (Ans.)m33.22y 7| GKwU cvmi AbyfywgK cvjv 79.53 m Ges wePiYKvj 5.3 s nj wbc KvY I wbc eM KZ? Avgiv Rvwb, g 2Sinv R o 2 o 8.9 2Sinv 53.79 o 2 o (1).....394.7792Sinv o 2 o Aevi, g Sinv2 T oo 8.9 Sinv2 3.5 oo (2).........94.51Sinv2 oo (1) bs mgxKiYK (2) bs mgxKiY viv fvM Ki cvB, 94.51 394.779 Sinv2 2Sinv oo o 2 o 94.51 394.779 Sinv2 CosSin2v oo oo 2 o (3)........15Cosv o0 (2) bs mgxKiYK (3) bs mgxKiY viv fvM Ki cvB, 15 94.51 Cosv Sinv2 o0 oo 2 463.3 tan o 732.1tan o 732.1tan 1 o .)Ans(60o (3) bs mgxKiY 0 Gi gvb ewmq cvB, 1560Cosv0 15 2 1 v0 (Ans.)ms30v 1 0 8| GKwU ejK f~wgi mv_ 30KvY Ki Dcii w`K wbc Kiv nj GwU 20m `~i GKwU `vjvbi Qv` wMq coj| wbc we`y _K Qvì DPZv 5m nj ejwU KZ eM Qvov nqwQj| Avgiv Rvwb, 2 2 00 0 x )cosv(2 g x)(tany 2 22 0 )20( 30cos2 8.9 2030tan5 v 75.02 4008.9 02577350269.05 2 0 v 5.1 3920 54.115 2 0 v 54.6 5.1v 3920 2 0 5.154.6 3920 v2 0 .)Ans(ms20v 1 0 9| GKRb jvK 48 ms-1 eM GKwU ej Lvov Dcii w`K wbc Ki| ejwU KZ mgq k~Y _vKe Ges mevP KZ Dci DVe? Avgiv Rvwb, g Sinv2 T oo 8.9 90Sin482 T 8.9 1482 T .)(.795.9T Anss Avevi, 2g Sinv H o 22 o 9.82 )90Sin(48 H 22 (Ans.)117.5mH 10| GKwU Mvgvdvb iKW cwZ wgwbU 45 evi Nyi| Gi K` _K 9cm `~i Kvb we`yi `wZ KZ? Avgiv Rvwb, rv r t n2 v 60 09.04514.32 v .)Ans(ms42.0v 1 GLvb, AbyfywgK cvjv, R = 79.53 m wePiYKvj, T=5.3s wbc eM, vo = ? wbc KvY, = ? GLvb, wbc KvY, AwfKlR ZiY, g = 9.8ms-2 AbyfywgK `~iZ, x=20m Dj^ `~iZ, y=5m AvwèM, vo = ? GLvb, eM, v0 = 48 ms-1 wbc KvY, = Dvb cZbi gvU mgq,T =? DPZv, H =? GLvb, mgq, t = 1m =60s cvKmsLv, n=45 evmva,r =9cm=0.09m `wZ, v =? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 38. 1 = 7.2324 1 = 13825.728 - ) () () () () 1:10 30 :10 40 :10 42 [ 1] Gravitation Intermediate vector bosons Photon , , , , Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 39. RoZv (Inertia): c`v_ h Aevq AvQ mB Aevq _vKZ PvIqvi h ceYZv ev mB Aev ivLZ PvIqvi h ag ZvK RoZv ej| RoZv `yB cKvi: h_v (1) wwZ RoZv (2) MwZ RoZv (1) wwZ RoZv (Inertia of rest) t wi e wi _vKZ Pvq ei GB bK ei wwZ RoZv ej| D`vniYt wi Mvwo nVvr Qo w`j hvwiv wcQb w`K nj co| KviY Mvwo Pvjy nIqvi msM msM hvwì kixii wbPi Ask Mvwoi mv_ mshy _vKvq mvgbi w`K GwMq hvq| wK kixii Dcii Ask wwZ RoZvi Rb wcwQq co| (2) MwZ RoZv (Inertia of motion) t MwZkxj e MwZkxj _vKZ Pvq ei GB bK ei MwZ RoZv ej| D`vniYt Pj Mvwo nVvr eK Kij hvwiv mvgbi w`K SzK co| KviY Mvwo eK Kivi KviY Mvwo _g hvIqvi mv_ mv_ hvwì kixii wbPi Ask Mvwoi mv_ mshy _vKvq _g hvq wK kixii Dcii Ask MwZ RoZvi Rb mvgbi w`K GwMq hvq| ej (Force): hv wi ei Dci wqv Ki eK MwZkxj Ki ev KiZ Pvq A_ev MwZkxj ei Dci wqv Ki MwZi cwieZb Ki ev KiZ Pvq ZvK ej ej| ejK F viv cKvk Kiv nq| m fii ei ZiY a nj ej F=ma ne| ej GKwU fi ivwk| gwjK ej (Fundamental Forces) t h mKj ej ^vaxb A_vr h mKj ej Ab Kvb ej _K Drcb nq bv ZvK eis Abvb ej GB mKj eji Kvb bv Kvb ici cKvk ZvìK gwjK ej ej| gwjK ej Pvi cKvi, h_v: (1) gnvKl ej (2) ZvwoZ P^K ej (3) mej wbDwKq ej I (4) `~ej wbDwKq ej (1) gnvKl ej t gnvweki h Kvb `ywU ei gaKvi cviwiK AvKlb ejK gnvKl ej ej| (2) ZvwoZ P^K ej t `ywU PvwRZ KYv Zvì PvRi KviY GK Acii Dci h AvKlY ev weKlb ej cqvM Ki ZvK ZvwoZ P^K ej ej| (3) mej wbDwKq ej t cigvbyi wbDwKqvm wbDwKq Dcv`vb Z_v wbDwKqbjvK GK Ave ivL h kwkvjx ej ZvK mej wbDwKq ej ej| (4) `~ej wbDwKq ej t h ^ cvjvi I ^ gvbi ej wbDwKqvmi ga gwjK KYvwji ga wqv Ki AbK wbDwKqvm AwwZkxjZvi De NUvq ZvK `~ej wbDwKq ej ej| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 40. 04| MwZm~ (Laws Of Motion) 2 gwjK ej mg~n AvcwK wZeZv cvjv 1) gnvKl ej 1 Amxg 2) ZvwoZ P^K ej 1039 Amxg 3) mej wbDwKq ej 1041 10-15 m 4) `~ej wbDwKq ej 1030 10-16 m fieM (Momentum) t fi I eMi bdjK fieM ej| fieMK P viv cKvk Kiv nq| m fii Kvb ei eM v nj fieM P = mv ne| fieMi GKK kg-ms-1 | Gi gvv [MLT-1 ] wbDUbi MwZi 1g m~ (Newton's 1st Law of Motion) t eYbvt evwnK ej cqvM bv Kij wi e wPiKvj wi Ges MwZkxj e mgeM mij c_ PjZ _vKe| eLvt evBi _K ej chy bv nj (1) wi e wPiKvj wi _vKe Ges (2) MwZkxj e mgeM mij c_ PjZ _vKe| myZivs `Lv hvq h, MwZi c_g m~i `ywU Ask| c_g Ask _K wi ei wwZkxj _vKvi ceYZv mK aviYv cvIqv hvq| GB ceYZvK wwZ RoZv ej| Avevi wZxq Ask MwZkxj ei MwZkxj _vKvi ceYZv jbxq | GB ceYZvK MwZ RoZv ej| G KviY wbDUbi MwZi 1g m~K RoZvi m~ I ejv nq| G m~ _K eji msv I cvIqv hvq| wbDUbi MwZi 2q m~ (Newtons 2nd Law of Motion)t eYbvt ei fi eM cwieZbi nvi ei Dci chy eji mgvbycvwZK Ges ej h w`K wqv Ki ei fi eMi cwieZb I mB w`K NU| eLvt G m~i evLv nj amF cwZcv`b| amF cwZcv`b t gb Kwi, Kvb MwZkxj eKYvi fieM P | F eji wqvq z`vwZz` mgq eeavb dt AeKvk KYvi fieMi cwieZb Pd nj fieM cwieZbi nvi dt Pd ne| MwZi 2q m~vbymvi, fieM cwieZbi nvi chy eji mgvbycvwZK| A_vr, F dt Pd eKYvi fi m Ges eM v nj fieM vmP ewmq cvB, F dt )vm(d F dt vd m dt vd aFma iYZ (1)................Fkma GLvb k GKwU mgvbycvwZK aeK; hvi gvb GKK eji msv viv wbaviY Kiv hvq| h ej GKK fii Dci wqv Ki GKK ZiY mw Ki ZvK GKK ej ej| A_vr 1F , m=1 Ges 1a nq Ze mgxKiY (1) nZ cvB| 11=k1 k=1 mgxKiY ewmq cvB, Fam Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 41. 04| MwZm~ (Laws Of Motion) 3 A_vr amF Kvb ei Dci ^Z ejwj hw` nq h_vg n321 F,..F,F,F BZvw` wqviZ _vKj Avgiv cvB, maF....FFF n321 amF GLvb, F nQ jw ej ev bxU ej hLvb, n321 F....FFFF (cgvwbZ) eji GKK wbDUb t h ej 1kg fii Dci wqv Ki 1ms -1 ZiY mw Ki ZvK 1 wbDUb ej ej| A_vr 1 wbDUb = 1kg 1ms -2 | wZxq m~ _K c_g m~ cwZcv`b t wbDUbi MwZi 1g I 2q m~ Zzjbv Ki `Lv hvq h, 1g m~ nQ 2q m~i GKwU ic| amF _K `Lv hvq h, ei Dci wbU ej k~b nj A_vr hLb 0F ZLb 0am ,wK ,0m 0a nq| myZivs hLb evBi _K ej cqvM bv nq A_vr wbU ej k~b nq ZLb, 0a 0 dt vd v Gi wWdviwqmvb k~b v = aeK| myZivs evBi _K ei Dci ej wbU ej k~b nj ei eM AcwiewZZ _vK A_vr e h Aevq wQj mB AevqB _vK| A_vr evwnK ej cqvM e Aevb cwieZb KiZ eva bv Kij wi e wPiKvj wi Ges MwZkxj e wPiKvj mgeM mij c_ PjZ _vKe| GwUB wbDUbi MwZi c_g m~| NvZej (Impulsive Force) t Lye A mgq h ej chy nq ZvK NvZ ej ej| nvZzwo viv KvV ciK cvZv, evU viv wKU ej AvNvr Kiv, BZvw` NvZ eji D`vniY| eji NvZ (Impulse of Force) t Kvb ej I eji wqv Kvji b djK eji NvZ ej| Kvb ej F hw` Kvb ei Dci it mgq _K ft mgq ch wqv Ki, Zvnj eji NvZ ne, f i t t dtFJ ][ ea FdtFJ f i t t ]t-[tFtFJ if t t f i tFJ eji NvZi mv_ fieMi mK (Relation with Impulse of Force and Momentum ) t chy ej F nj eji NvZ f i t t dtFJ dt Pd Fdt dt Pd J f i t t Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 42. 04| MwZm~ (Laws Of Motion) 4 f i f i P P P p PPdJ if PPJ PJ A_vr, eji NvZ fieMi cwieZbi mgvb| wbDUbi MwZi ZZxq m~ (Newton's 3rd Law of Motion) t eYbv: cZK wqviB GKwU mgvb I wecixZ cwZwqv AvQ| wbDUbi MwZi ZZxq m~vbymvi hw` GKwU e A, Aci e B Gi Dci ej cqvM Ki, Ze B eI A ei Dci mgvb I wecixZ ej cqvM Kie| awi A e B ei Dci F1 ej cqvM Ki Ges B e A ei Dci F2 ej cqvM Ki| m~vbymvi, F1 = F2 , A_vr ej qi gvb mgvb wK wecixZgywL| G `ywU eji h Kvb GKwUK wqv Ges AbwUK cwZwqv ej| Dcii `v wqv I cwZwqv mgvb I wecixZ| fieMi msib m~ (Law of Conservation of Momentum) t GKvwaK ei ga wqv cwZwqv ej Qvov Ab Kvb ej wqv bv Kij, H e mg~ni gvU fieMi mgw ae _vK| MvwbwZK fve, eKaP cwZcv`b: 1 I 2 `ywU e weePbv Kwi| awi Kvb GK mgq eq msNl wj nj| aiv hvK, GB msNl F1 nQ c_g ei Dci wZxq e KZK chy ej Ges F2 nQ wZxq ei Dci c_g e KZK chy ej| Kvb GK mgq t Z c_g ei fieM 1P Ges wZxq ei fieM 2P | Avgiv cwZwU ei wbDUbi 2q m~ cqvM Ki cvB, dt Pd F 1 1 Ges dt Pd F 2 2 wbDUbi MwZi ZZxq m~ _K cvB, 21 FF dt Pd dt Pd 21 0 dt Pd dt Pd 21 0PP dt d 21 0 dt Pd 21 PP[ nQ eevi gvU fieM ev ]P hnZz mgqi mvc gvU fieM 21 PPP , P Gi cwieZbi nvi k~b, ZvB gvU fieM P ae _vK, A_vr 21 PPP = ae fi PP f2f1i2i1 PPPP myZivs `Lv hvQ h, msNli AvM Kvb eevi fieMi fi mgw Avi msNli ci fieMi fi mgw me`v mgvb _vK| GwUB fieMi msib m~| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 43. 04| MwZm~ (Laws Of Motion) 5 iKUi MwZ (Motion of Rocket) t awi iKUwU AwfKli AvIZvgy gnvk~b MwZkxj| hLb iKUi Bwb KZK MvmwbMZ nq ZLb mB Mvmi GKwU fieM _vK| ZLb fieM msiwZ _vK ej iKU wecixZ w`K MwZkxj nq| iKU _K Mvm hLb GKwU wbw` nvi wbMZ nZ _vK, ZLb Mvmi MwZi wecixZ w`K iKUwU GKwU wi ej jvf Ki| GB ejK avv ej| wnmve: gbKwi iKU _K Rvjvwb Z_v Mvm v eM wbMZ nQ| t mgq wbMZ Mvmi fi m nj, wbMZ Mvmi fieM ne, P=(m)v wbMZ Mvmi GB fieM iKU Rvjvwbi fieMi cwieZbi mgvb| fieMi msib m~vbymvi Rvjvwbi fieMi GB cwieZb iKUwUi fieMi cwieZbi mgvb| Avgiv Rvwb, Kvb ei fieMi cwieZbi nvi Zvi Dci chy eji NvZi mgvb| myZivs iKUi Dci chy avv F nj P = Ft (m)v = Ft v t m F, ev GLvb t m nQ Rvjvwb eenvii nvi| Kvb gyn~Z iKUi fi M nj Ges H gyn~Z Zvi ZiY a nj, F Gi gvb F= Ma ewmq cvB, v t m Ma v t m M a 1 iY,Z eji fvimvg (Equilibrium of Force) t ei Dci eji jw k~b nIqvK A_vr, 0F nIqvK eji fvimvg ej| fvimvg Aevq ei Dci ZiY k~b nq| fvimvg Aevq ei eM ae _vK| A_vr eMi gvbi ev w`Ki Kvb cwieZb nq bv| fvimvg `yB cKvi| h_v- (1) wi fvimvg (2) MwZq fvimvg jvwgi Dccv` (Lami's Theorem) t Kvb ei Dci wZbwU Amgvivj ej wqv Ki fvimvg Aevi mw Kij cZKwU ej Aci `ywU eji Afy Kvbi mvBbi mgvbycvwZK| gbKwi, Q,P I R wZbwU ej O we`yZ wqv Ki fvimvg mw Ki| Q I R Gi Afy KvY , R I P Gi Afy KvY , P I Q Gi Afy KvY | myZivs jvwgi Dccv` Abymvi, Sin R Sin Q Sin P ne| NlY (Friction) t `ywU e ciii msk _K hw` GKi Dci w`q AciwU PjZ Pv Kij e qi k Zj GB MwZi wei h evavi Drcw nq ZvK NlY ej| NlY mvaviYZ Pvi cKvii nq _vKt (1) wwZ NlY (Static Friction), (2) MZxq NlY (Kinetic Friction), (3) AveZ NlY (Rolling Friction), (4) cevwn NlY (Fluid Friction), Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 44. 04| MwZm~ (Laws Of Motion) 6 NlY ej t `ywU e ciii msk _K hw` GKi Dci w`q AciwU PjZ Pv Kij e qi k Zj GB MwZi wei h eji mw nq ZvK NlY ej ej| wwZNlY ev mxgvwK NlY (Static Friction) t chy eji mevP h gvbi Rb GKwU wwZkxj e MwZi Dcg nq, ZvK wwZ NlY ev mxgvwK NlY ej&| wwZ Nlbi gvb k~b _K mxgvwK gvb ch nZ cvi| wwZNlY ej t h Nlb ej viv Kvb e chy ejK cwZnZ Ki wi _vKZ cqvm cvq ZvK wwZNlb ej ej| NlY KvY I wwZ KvYi ga mK t wwZ NlY Yv t wwZ NlYi mxgvwK gvb Ges Awfj^ cwZwqvi AbycvZK wwZ NlY Yv ej| GK s viv cKvk Kiv nq| wwZ NlYi mxgvwK gvb fs Ges Awfj^ cwZwqv R nj wwZ NlY Yv ne (2).................. R fs s fs I R Gi GKK GKB| KvRB s Gi Kvb GKK bB| NlY KvY (Angle of Friction) t wwZ NlYi mxgvwK gvb fs Ges Awfj^ cwZwqv R K mg^q Ki h jw cvIqv hvq ZvK j cwZwqv S ej| GB j cwZwqv Awfj^ cwZwqvi mv_ h KvY Ki ZvK NlY KvY ev wwZ NlY KvY ej| NlY KvYK viv cKvk Kiv nq| Wvbcvki wP Gi Aevb `Lvb nqQ| wPvbymvi cvB, (3)..................cosSR (4)..................sin Sfs (4) bs mgxKiYK (3) bs mgxKiY viv fvM Ki cvB, (5)............tan R fs Avevi, (2) bs mgxKiY R fs s K (5) bs Gi mv_ Zzjbv Ki cvB, (6).........tans A_vr NlY KvYi UvbRU -Gi gvb wwZ Nlb bvi mgvb| BnvB NlY KvY I wwZ NlY Yvi ga mK| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 45. 04| MwZm~ (Laws Of Motion) 7 wwZ KvY ev wbj Kvb (Angle of Repose) t Kvb AvYZ ZjwZ e Zji h AvbwZi Rb MwZi Dcg nq ZvK wwZ KvY ev wbj KvY ej| GK viv cKvk Kiv nq| wPvbymvi, OY Zji DcwiwZ e Zji AvbwZi Rb MwZi Dcg nq| myZivs ne wwZ KvY| GB Aevq W Lvov wbPi w`K wqv Kie Ges Zji Awfj^ cwZwqv R-Gi wqv iLvi mv_ KvY bZ _vKe| wP c`wkZ fi eev Abymvi, CosWR SinWfs (7)............ R f tan s wK (2) bs mgxKiY nZ cvB, ...(2) R fs s Avevi (7) I (2) bs mgxKiY _K cvB, (8).........tans Avevi, (6)............tans dj (6) I (8) _K cvB, tantan AZGe, NlY KvY Ges wbj KvYi gvb mgvb| Ze NlY KvY me chvR; wK wbj KvY aygv AvbZ Zji Rb chvR| MwZq NlY (Kinetic Friction) t Kvb Zji Dci w`q MwZkxj e MwZi wei h evav Abyfe Ki ZvK MwZq NlY bv ej Ges h ej MwZZ evavi mw Ki, ZvK MwZq NlY ej ej| MwZq NlY bv t MwZq NlY ej Ges Awfj^ cwZwqvi AbycvZK MwZq NlYbv ej| GK viv cKvk Kiv nq| R fk k GLvb, fk = MwZq NlY ej I R Awfj^ cwZwqv| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 46. c_g ci AsKi mgvavb First Paper Math Solution 4| MwZm~ (Laws Of Motion) 1| 10kg fii GKwU e`yKi wUMvi Pvc ÌqvZ e`yKi bj nZ 20g fii GKwU eyjU 400ms-1 eM ei nq Mj| e`yKwUi cvr eM KZ? Avgiv Rvwb, m1v1 = m2v2 40010200v1 3 1 ev, 10 4001020 v 3 1 ev, (Ans.)0.8msv 1 1 2| 40kg I 60kg fii `ywU e h_vg 10ms-1 I 5ms-1 eM cii wecixZ w`K _K Avmvi mgq GK AciK avv w`j| avvi ci e q GK hy nq KZ eM Kvb w`K Pje? Avgiv Rvwb, m1u1+m2u2 = m1v1+m2v2 ev, 4010+ 60(-5) = 40v + 60 v ev, 400300 =100v ev, 100v =100 100 100 vev, ms1v 1 v abvZK nZz avvi ci e q GK hy nq 1ms-1 eM 1g ei AwfgyL Pje| 3| GKwU e wi Avevq wQj| 15 wbDUbi GKwU ej Gi Dci 4 mK wqv Kivi ci eji wqv e nq hvq| ewU Gi ci 4 mK 48m `~iZ Mj| ewUi fi KZ? ej _vKv Kvjxb 4 mK ewU h kl eM jvf Ki mB kl eMK Mo eM wbq cieZx 4 mK ewU 48m `~iZ hvq| Avgiv Rvwb, s = v t2 t s v 2 ev, 4 48 v ev, ms12v -1 Avevi, v = u + at1 ev, 12 = 0 + a (4) 4 12 aev, a = 3 ms-2 Avevi, F = ma a F m ev, 3 15 m ev, (Ans.)kg5m 4| 10N Gi GKwU ej 2kg fii GKwU wi ei Dci wqv Ki| hw` 4s ci eji wqv e` nq hvq Ze c_g _K 8s G ewU KZ `~i hve? ej _vKv Kvjxb 4 mK ewU h kl eM jvf Ki cieZx (8 4) = 4 mK mB kl eMK Mo eM wbq ewU Pje| Avgiv Rvwb, F = m a m F aev, 2 10 aev, -2 ms5 a 2 111 at 2 1 uts Avevi, 2 1 45 2 1 0s m40s1 1- 1 ms20v 450v tuv aAvevi, m80s 420s tvs 2 2 22 Avevi, s = s1+s2 = (40+80)m =120m (Ans.) 5| 100kg Ges 200kg fii `wU e cii wecixZ w`K h_vg 20ms-1 Ges 10ms-1 eM hvIqvi c_ GK AciK avv w`j| avvi ci e `ywU GK hy _K KZ eM Kvb w`K Pje? Avgiv Rvwb, m1u1+m2u2 = m1v1+m2v2 10020+200(-10)=100v+200v 100v+200v =2000 2000 300v =0 v = 0 hnZz kl eM k~b dj e `ywU GK hy nq _g hve| GLvb, e`yKi fi, m1 = 10 kg eyjUi fi, m2 = 2010-3 kg eyjUi eM, v2 = 400ms-1 e`yKi cvr eM, v1 =? GLvb, 1g ei fi, m1 = 40kg 2q ei fi, m 2 = 60kg 1g ei AvwèM, u1 = 10ms-1 2q ei AvwèM, u2 =-5ms-1 hy eqi kl eM, v1 = v2 = v = ? GLvb, AvwèM, u = 0 ej, F = 10 N fi, m = 2 kg mgq, t1 = 4s mgq, t2 = (84) s = 4s gvU `yiZ, s = s1+s2 = ? GLvb, AvwèM, u = 0 ej, F = 15 N mgq, t1 = 4s mgq, t2 = 4s kleM = MoeM = v miY, s = 48m fi, m = ? GLvb, 1g ei fi, m1 = 100kg 2q ei fi, m2 = 200kg 1g ei eM, u1 = 20 ms-1 2q ei eM, u2 = -10 ms-1 GK hy eM v1=v2=v=? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 47. 4| MwZm~ (Laws Of Motion) 2 6| 4 kg fii GKwU eK 10ms-2 ZiY MwZkxj KiZ KZ ej cqvM KiZ ne? c_i Nlb ej 2.5N kg-1 | Avgiv Rvwb, F1 = ma F1 = 410 N F1 = 40 N Avevi 4kg Z NlY ej F2= F 4 N F2= 2.5 4 N = 10 N Avevi, F = F1+F2 F = (40+10)N=50N (Ans.) 7| 70kg fii evK 500N AbyfywgK ej gSi Dci w`q Uvbv nQ| evwU hLb Pj ZLb ev I gSi gaeZx NlY mnM 0.5| evi ZiY wbYq Ki| Avgiv Rvwb, MZxq NlY mnM, R Fk k RF kk 6865.0Fk N343Fk Avevi, F = P-Fk F =(500 343)N F =157 N Avevi, F = ma 157 = 70 a 1 ms 70 157 a (Ans.)ms24.2 2 a 8| 1N Gi GKwU ej 0.1kg fii GKwU wi ei Dci wqv Ki| hw` 1s ci eji wqv e` nq hvq Ze c_g _K 2s G ewU KZ `~i hve? ej _vKv Kvjxb 1 mK ewU h kl eM jvf Ki cieZx (2 1) = 1 mK mB kl eMK Mo eM wbq ewU Pje| Avgiv Rvwb, F = m a m F aev, 0.1 1 aev, -2 ms01 a 2 111 at 2 1 uts Avevi, 2 1 101 2 1 0s m5s1 1- 1 ms10v 1010v tuv aAvevi, m10s 110s tvs 2 2 22 Avevi, s = s1+s2 = (5+10) m =15m (Ans.) 9| 36 kg fii GKwU ei Dci KZ ej cqvM Kij 1 wgwbU Gi eM 15kmh-1 ew cve? Avgiv Rvwb, ej F=ma t uv mF )( NF 603600 100015 36 ).(5.2 AnsNF 10| 36 kg fii GKwU ei Dci KZ ej cqvM Kij wgwbU Gi eM NUvq 12km ew cve? Avgiv Rvwb, ej F=ma t uv mF )( NF 603600 100012 36 ).(2 AnsNF GLvb, fi, m = 4 kg ZiY, a = 10ms-2 NlY ej, F =2.5 N kg-1 ej, F = ? GLvb, MZxq NlY mnM 5.0k evi fi m = 70kg Awfj^ cwZwqv. R=mg=709.8N=686N AbyfywgK ej P=500N evi ZiY a =? GLvb, AvwèM, u = 0 ej, F = 1 N fi, m = 0.1 kg mgq, t1 = 1s mgq, t2 = (21) s = 1s gvU `yiZ, s = s1+s2 = ? GLvb, fi, m =36 kg mgq, t = 1m=60s eM ew, (v-u)=15kmh-1 3600 100015 ms-1 ej,F=? GLvb, fi, m =36 kg mgq, t = 1m=60s eM ew, (v-u)=12kmh-1 3600 100012 ms-1 ej,F=? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 48. 1| GKwU PvKvi fi 5kg Ges PMwZi evmva 25cm | Gi RoZvi vgK KZ? PvKvwUK 4 rad s-2 KwbK ZiY mw KiZ KZ gvbi UK cqvM KiZ ne? Avgiv Rvwb, RoZvi vgK I = MK2 = 5(0.25)2 = 0. 3125 kg m2 ( Ans.) Avevi, UK 0. 3125 4 Nm =1.25Nm (Ans.) 2| GKwU KYv 1.5m evmvai evKvi c_ cwZ wgwbU 120 evi AveZb Ki| Gi mij iwLK eM wbYq Ki| Avgiv Rvwb, v = r r t n2 v ev, 1.5 60 1203.142 v ev, Ans.)(ms18.84v 1 3| evKvi c_ 3.14ms-1 mg`ywZZ AveZbiZ GKwU KYv cwZ mK 10 wU c~b AveZb mb Ki| evKvi c_i evmva wbYq Ki| Avgiv Rvwb, v = r r t n2 v ev, n2 v t r ev, 103.142 13.14 rev, r 0.05m (Ans.) 4| 13ms-1 eM GKwU MvwoK wbivc` 30m evmvai GKwU euvK AwZg KiZ nj euvKwUK KZ KvY Xvjy KiZ ne? Avgiv Rvwb, rg v tan 2 1 9.830 13 tan 2 1 0.57tan 1 (Ans)29.89 5| GKwU Zvgvi MvjKi fi 0.05 Kg| GwUK 2m `xN GKwU myZvi GK cv ea cwZ mK 5 evi Nyivb nQ| MvjKwUi KwbK fieM KZ? Avgiv Rvwb, RoZvi vgK I = MK2 = 0.05(2)2 = 0.2 kg-m2 Avevi, t n 1 51432 rad s-1 = 31.4 rad s-1 Avevi, L = I L =0.231.4 = 6.28 Kg-m2 s-1 (Ans.) 6| evKvi c_ 72Kmh-1 mg`ywZZ Pjgvb Kvb gvUi Mvoxi K`gyLx ZiY 1ms-2 nj evKvi c_i evmva KZ? Avgiv Rvwb, .)Ans(m400r r 220 1 r 2v a 7| 75m evmvai evKvi c_ Kvb gUi mvBKj Avivnx KZ eM Nyij Dj^ Zji mv_ 30KvY AvbZ _vKe? Avgiv Rvwb, rg v tan 2 rgtanv2 8.97503tanv2 8.97503tanv2 35.424v2 35.424v2 )Ans(ms20.635.424v -1 GLvb, fi, M = 5kg PMwZi evmva, K= 25cm =0.25m RoZvi vgK, I = ? KwbK ZiY, = 4 rad s-2 UK, ? GLvb, iwLK `ywZ, v = 3.14ms-1 mgq, t = 1 sec. cvK msLv, n = 10 cvK evmva, r =? GLvb, evmva, r = 1.5m mgq, t = 1 min = 60 sec. cvK msLv, n = 120 cvK iwLK eM, v =? GLvb, eM, v =13 ms-1 evmva, r = 30m g = 9.8 ms-2 KvY, =? GLvb, fi, M = 0.05 kg PMwZi evmva, K= 2m RoZvi vgK, I = ? cvKmsLv, n = 5 mgq, t = 1 Sec KwbK fieM, L =? GLvb, v = 72 Kmh-1 1- 20ms 3600 100072 v a = 1ms-2 evmva, r = ? GLvb, evmva, r = 75m g = 9.8 ms-2 KvY, =30 eM, v =? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 49. 5| KwbK MwZm~ (Laws Of Angular Motion) 2 8| 0.1kg fii GKwU cv_iK 0.5m j^v myZvi mvnvh evKvi c_ Nyivb nQ| cv_iwU cwZwgwbU 30 evi c~Y NyYb mb Ki| myZvi Uvb KZ? Avgiv Rvwb, rmF 2 r t n2 mF 2 N5.0 60 3014.32 1.0F 2 F = 0.49298 N (Ans.) 9| cw_exi Pvwiw`K Pì Kc_i emva cvq 3.85105 km| GKevi c`wb KiZ mgq jvM 27.3 w`b| Pvì KwbK `wZ wbYq Ki| Avgiv Rvwb, t n2 3600243.27 11416.32 .)Ans(rads1066.2 16 10| 200m evmva wewk GKwU euvKv c_ 50.4 km.h-1 eM Mvwo PvjvZ c_wUK KZ KvY KvZ Ki ivLZ ne? ivvwUi c 1m nj, evBii cvk wfZii cvk Acv KZ DuPz nZ ne? (g = 9.8 ms-2 ) Avgiv Rvwb, rg v tan 2 1 9.8200 14 tan 2 1 0.1tan 1 (Ans)71.5 1 h 71.5sin 1 h 0995.0 .)(1.00994.0 Ansmmh GLvb, fi, m = 0.1kg evmva, r = 0.5m cvKmsLv, n = 30 mgq, t = 1 min = 60sec myZvi Uvb ej, F = ? GLvb, emva r=3.85105 km =3.85108 m mgq t= 27.3 w`b cvK msLv=1cvK KwbK `wZ =? GLvb, eM, -1 kmh50.4=v 1- ms 3600 100050.4 =v -1 ms14=v evmva, r = 200m g = 9.8 ms-2 KvY, =? x=1m, h=? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 50. , W = F. S FS cos : : , p = : : : , : , - : W = k . , . : F. P. S Ft-poundal C. G. S erg M. K. S S.I Jole . = . . : Jol/sec watt. . : : [ML 2 T -2 ] : [ML 2 T -2 ] : [ML 2 T -3 ] . 25 27 . 0.143 H.P . 900 . 0 < < 900 900 fB A_vr fA = fB + N fA = 256 + 5 fA = 261 Hz (Ans.) 12| Kvb Ki ki ZxeZv 110-7 Wm-2 | ki ZxeZv wb nj bZzb ZxeZv jej wbYq Ki| Avgiv Rvwb, dB I I 10log o 2 2 12- -7 2 10 102 10log GLvb, Zvii `N, l = 25cm = 0.25m Zvii Uvb, T = 5 kg wt. = 59.8N GKK `Ni Zvii fi, 13 mkg109.4 Kv, f =? GLvb, Zvii `N, l1 = 0.36m Zvii Uvb, T1= 100 kgwt Zvii Uvb, T2= 220 kgwt. Kv, f1 = f2 Zvii `N, l2=? GLvb, exU, N = 5 Kv, f1 =275Hz Kv, f2 =? GLvb, exU, N = 5 Kv, fB =256Hz Kv, fA =? GLvb, Zvii Uvb T1=T awi Zvii Uvb, 100 2T TT2 T02.1 50 T51 50 T T exU N=3 Kv, f1=? GLvb, cgvY ZxeZv, Io = 10-12 Wm-2 kYx Ki ZxeZv, I1 = 110-7 Wm-2 wb ZxeZv, I2 = 210-7 Wm-2 bZzb ZxeZv jej, ? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 176. c_g ci AsKi mgvavb 3 log200000102 (Ans.)dB01.352 13| 50cm I 51cm `N wewk GKgyL e `ywU bj AMvb bj cwZ mK 3 wU exU mw Ki| evqyZ ki eM wbYq Ki| Avgiv Rvwb, 1 gyL e bj Kv 2 2 1 1 44 l V f l V f I Avevi Avgiv Rvwb, f1 f2 = N 3 l4 V l4 V 21 3 51.04 V 5.04 V 3 51.04 V 5.04 V 3 51.05.0 50.051.0 4 V 3 51.05.04 01.0V 01.0 51.05.043 V 1 ms306V 14| wbw` `Ni GKwU mbvwgUvii Zvi wbw` ej viv Uvbv AvQ| hw` Uvbv ej Pvib Ges GKB mv_ Zvii `N w`b Kiv nq Ze Kvi wKic cwieZb ne| Avgiv Rvwb, T 2 1 f 1 1 1 l 2 T T 2 1 f1 ll Avevi, T 2 1 f 2 2 2 l 4 T2 4T 22 1 f2 ll T2 4 2 T f f 2 1 l l 1:1f:f 21 A_vr Kvi Kvb cwieZb ne bv| 15| GKwU Uvbv Zvii `N cwieZb bv Ki Gi Uvb Pvib Kiv nj| Zvii Kvi KZ cwieZb ne? Avgiv Rvwb, T 2 1 f 1 1 1 l 2 T T 2 1 f1 ll Avevi, T 2 1 f 2 2 2l 2 T2 4T 2 1 f2 ll 2 1 T2 2 2 T f f 2 1 l l 12 f2f A_vr Kv wb nq hve| 16| GKwU myi-kjvKv 512Hz Kvi GKwU myi-kjvKvi mv_ 4 wU exU Ges 514Hz Kvi Aci GKwU myi-kjvKvi mv_ 6wU exU Drcb Ki| myi-kjvKvwUi Kv KZ? gb Kwi myi kjvKvi Kv = f| c_g kZ, )1(..........4512 f Zxq kZ, )2(..........6514 f hvM Ki, 1026 2f = 10 2 101026 f .)Ans(Hz508 2 1016 f 17| `ywU myi-kjvKv A I B GK kvwqZ nIqvq cwZ mK 5 wU exU mw Ki| A -Gi evZ GKL Zvi Rovj Avevi Zviv 5 wU exU mw Ki| B Gi Kv 320Hz nj A Gi Kv KZ? Avgiv Rvwb, Nff BA hnZz A Z Zvi Rwoq IRb evovj exU c~ei mgvb nq| dj Zvi Rovbvi c~e A Gi Kv ekx wQj| dj fA > fB A_vr fA = fB + N fA = 320 + 5 fA = 325 Hz (Ans.) GLvb, bji `N, l1 = 0.5m bji `N, l2 = 0.51m exU, N =3 ki eM, V=? GLvb, Zvii cv_wgK Uvb, T1=T (awi) Zvii kl Uvb, T2=4T Zvii cv_wgK `N, l1=l (awi) Zvii kl `N, l2 = 2l f1t f2=? GLvb, Zvii cv_wgK Uvb, T1=T (awi) Zvii kl Uvb, T2=4T Zvii `N, l1= l2 = l (awi) f2=? f1 GLvb, exU, N = 5 Kv, fB =320Hz Kv, fA =? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 177. ki `wZ mwKZ wbDUbi m~ (Newton's Formula for Speed of Sound): m~t Kvb gvag ki `wZ gvagi wwZvcK bvi eMg~ji mgvbycvwZK Ges gvagi NbZi eMg~ji evbycvwZK| wbDUb MvwYwZK fve GB m~K wbbv mgxKiYi mvnvh cKvk Kib t (1)............ E v GLvb, E = gvagi wwZvcK Yv, = gvagi NbZ, v = ki eM Mvmxq ev evqy gvag m~i ict Mvmxq c`v_i wwZ vcK Yv E, AvqZbi wwZvcK Yv B wb`k Ki| myZivs Mvmxq gvag ki `wZ mwKZ wbDUbi m~i ic ne, B v GLvb, B = gvagi AvqZbi wwZvcK Yv, = Mvmi NbZ, v = ki eM B = p cwZcv`b t awi, Kvb wbw` fii Mvmi Pvc p Ges AvqZb V | Zi wqvq GB Pvc I AvqZbi cwieZb NU| wbDUbi avibv wQj GB Pvc I AvqZbi cwieZb mgv cwqv| myZivs eqji m~vbyhvqx, pV =aeK GB mgxKiYK V Gi mvc eeKjb Kij, A_vr 0)pV( dV d 0 dV dp V dV dV p 0 dV dp Vp wZweKAvqZb cxobAvqZb V dV dp p AvqZbi wwZvcK Yv = B p = B GLvb FbvZK wP Pvc ewcj AvqZb nvm ev Pvc nvm cj AvqZb ew evSvq| m~ gZ evqyZ ki eM pB v 1 5 ms280 293.1 10013.1p v GLvb, p = evqyi Pvc=1.013105 Nm-2 , = evqyi NbZ =1.293 kgm-3 wK cgvY Zvcgvv I Pvc evqyZ ki eM 332ms-1 | GLvb _K avibv Kiv nq h, wbDUbi m~i Kv_vI fyj AvQ| jvcjvm KZK wbDUbi m~i mskvabx (Correction of Newton's law by Laplace): wbDUbi m~ gZ evqyZ ki `wZ 280 ms-1 | GB gvb cixv j gvbi Pq AbK Kg nIqvq eySv hvq h, Zvi m~ cqvMi mgq wbDUb h avibv Ki wQjb ZvZ wKQy fyj wQj| 1817 mvj wevbx jvcjvm wbDUbi Abygvbi fyj ei KiZ mg_ nb Ges Mvmxq gvag ki `wZ mwKZ m~i mskvab Kib| jvcjvmi gZ Mvmxq gvag k mvjbi mgq wbDUb gvagi Zvcgvvi Kvb Zvcgvvi Kvb cwieZb nqbv ej h Abygvb Ki wQjb Zv wVK wQjbv| Avi ZvB Mvmi h AvqZb Yv B, Pvc p-Gi mgvb ei Ki wQjb Zv wVK nqwb| jvcjvm ejb Pvc ewi dj evqyii mvPb nIqvi dj h Zvci De nq, mB Zvc wewKwiZ nIqvi AvMB cieZx cmviY i nq| dj Zvcgvv Kvb gZB wi _vKZ cvi bv| myZivs GB cwieZb mgv cwieZb bq ej GLvb eqji m~ pV = aeK chvR nZ cvi bv| GB cwieZb i Zvcxq cixeZb | GB cwqvq Pvc I AvqZbi mK pV =aeK| GLvb = wi Pvc I wi AvqZb Mvmi AvcwK Zvci AbycvZ| w cigvbyK Mvmi Rb Gi gvb 1.41| pV =aeK, GB mgxKiYK V Gi mvc eeKjb Ki cvB, Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 178. ki `wZ (Speed Of Sound) 2 0)pV( dV d 01 dV dp V dV dV Vp dV dp VVp 1 dV dp V V p 1 dV dp Vp 1 dV dp Vp wZweKAvqZb cxobAvqZb V dV dp p = AvqZbi wwZvcK Yv=B GLvb FbvZK wP Pvc ewcj AvqZb nvm ev Pvc nvm cj AvqZb ew evSvq| m~ gZ evqyZ ki eM pB v 1 5 ms4.332 293.1 10013.141.1p v GLvb, p = evqyi Pvc=1.013105 Nm-2 , = evqyi NbZ =1.293 kgm-3 | myZivs jvcjvmi mskvabi dj m~i mvnvh ZvwZK fve ki `wZi h gvb cvIqv hvq Zv cixv j gvbi mv_ wgj hvq| GB fve jvcjvm wbDUbi m~i mskvab Kib| ki `wZi Dci Pvci cfve (Effect of Pressure on speed of Sound): m fii Kvb Mvmi Dci Pvc hw` p1 _K p2 Z AvqZb h_vg V1 _K V2 -Z cwiewZZ nq| GB cwieZbi mgq hw` Zvcgvv wi _vK, Zvnj eqji m~vbyhvqx, p1V1= p2V2 m V m V m p m p 2 2 1 1 2 2 1 1 I pp 2 2 1 1 aeK GLb ki `wZ, p41.1 v m~ hnZz p AbycvZwU me`v wi _vK, myZivs wi Zvcgvvq Kvb Mvmi Pvc cwiewZZ nj ZvZ ki `wZi Kvb cwieZb nq bv| A_vr wi Zvcgvvq ki `wZi Dci Pvci Kvb cfve bB| ki `wZi Dci Zvcgvvi cfve (Effect of Temperature on speed of Sound): Mvmi Zvcgvvi cwieZb nj Gi NbZi I cwieZb nq| myZivs ki `wZi I cwieZb nq| awi, p1 Pvc, T1 Kjwfb Zvcgvvq Kvb Mvmi NbZ Ges D Mvm ki `ywZ v1| GLb p2 Pvc, T2 Kjwfb Zvcgvvq hw` H Mvmi NbZ nq Ges ZLb Mvm ki `ywZ v2 nj, 1 p v 1 1 Ges 2 2 2 p v (1)......... 1 2 2 1 2 2 1 1 2 1 p p p p v v GLb Mvmi cmviY _K Avgiv Rvwb, 22 2 11 1 T p T p Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 179. ki `wZ (Speed Of Sound) 3 21 12 1 2 Tp Tp GB gvb (1) bs mgxKiY ewmq cvB, 21 12 2 1 2 1 Tp Tp p p v v 2 1 2 1 T T v v Tv A_vr Mvm ki `wZ Kjwfb Zvcgvvi ev cig Zvcgvvi eMg~ji mgvbycvwZK| Avevi, 0C Zvcgvvq ev, ToK Zvcgvvq evZvm ki `wZ v0 Ges C ev, TK Zvcgvvq evZvm ki `wZ v nj 00 T T v v wK T=(Ges T0= 2 1 273 1 273 273 v v 0 wc`x Dccvì mvnvh wevi Ki Ges DPZi NvZ wewk c` mg~n Dcv Ki cvB, ............... 273 . 2 1 1vv 0 546 1vv 0 00183.01vv 0 00183.01ms332v 1 ( 0 C Zvcgvvq evZvm ki `wZ 332ms-1 | )ms61.0(ms332v 11 myZivs `Lv hvQ h, evqyZ cwZ wWwM mjwmqvm Zvcgvv ewi Rb ki `wZ 0.61ms-1 ev 61 cm s-1 ew cvq| ki `wZi Dci Av`Zvi cfve (Effect of Humidity on speed of Sound): Av` evqy A_vr evqyZ Rjxqev _vKj Gi NbZiI cwieZb nq; myZivs ki `wZiI cwieZb nq| Av` evqy ev Rjxq evc~Y evqyi NbZ evqyi NbZi Zzjbvq Kg A_vr evqyZ Rjxq ev hZ eo hvq Gi NbZ ZZ Kg hvq| Avevi ki `wZ NbZi eMg~ji evbycvwZK| myZivs evqyZ Rjxq ev ewk _vKj ki `wZ eo hvq| gbKwi, wbw` Pvc p I Zvcgvv -Z I Av` evhyi NbZ h_vg d I m Ges I Av` evqyZ ki `wZ h_vg vd I vm| myZivs d d p v Ges m m p v m dd md m p p v v ev, m d dm vv d , m Gi Pq eo nIqvq vm , vd Gi Pq eo| myZivs Av` evqyZ ki `wZ evqyZ ki `wZi Pq ekx| GK gyL e bji Lvjv gyL Avjvob mw Kij Gi ga evqyi Kbi cKwZ (Vibration of Air Column in a Closed Pipe): GK gyL e bji Lvjv gyL duz w`j ev Avjvob mw Kij bji wfZii evqyi gaw`q k jw^K ZivKvi e gyLi w`K mvwjZ ne Ges e gyL _K cwZdwjZ nq Lvjv gyLi w`K AMmi ne| dzu Gi g~j `b I evqyi Kbi ga Abybv` nj evqy AwaK wevi KuvcZ _vK Ges myi Rvivjv ne| GB Aevq bji Lvjv gyL me`v GKwU my` we`y (A) Ges e gyL GKwU wb` we`y (N) wPvbyhvqx mw ne| evqyi Kb Abymvi bji wfZi GKvwaK my` I wb` we`yi mw nZ cvi| evqyi mnRZi Kb [wP bs-(K)] aygv e gyL GKwU wb` Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 180. ki `wZ (Speed Of Sound) 4 we`yI Lvjv gyL GKwU my` we`y MwVZ ne| wK cii msjM GKwU wb` we`y I GKwU my` we`yi gaeZx `~iZ Zi `Ni Aski mgvb| myZivs bji `N l Ges Kb m ki Zi `N IKv nj 4 0 l l ......(1) Ges )2.(.......... 4 VV N 0 0 l [ V = f] GLvb V ki eM | bji GB myiB g~j myi ev c_g nvigvwbK| GB bj AviI Rvi dzu w`j bji evqy m jw^K Zii `N nvm cve Ges evqyi Kb ew cve| evqyi 2q mve Kb [wP bs-(L)] Lvjv gyLi my` we`y Ges e gyLi wb` we`yi ga GKwU my` we`y I GKwU wb` we`y MwVZ ne| awi evqy i GB Kb m myii Zi `N 1 Ges Kv N1, Zvnj 33 4 4 3 0 1 1 l l Ges 0 001 1 N3 V3 3 VV N GB myiK c_g Dcmyi ev ZZxq nvigvwbK ej| bji mve Kb [wP bs-(M)] Lvjv gyLi my` we`y Ges e gyLi wb` we`yi ga `ywU my` we`y I `ywU wb` we`y MwVZ ne| KvRB GB Kb m myii Zi `N 2 Ges Kv N2, nj 55 4 4 5 0 2 2 l l Ges 0 002 2 N5 V5 5 VV N GB myiK wZxq Dcmyi ev cg nvigvwbK ej| Dciv mgxKib _K `Lv hvq h, GKgyL e bj h mg myi mw nq Zvì Zi `N )1n2( 4 0 l Ges Kv 0 0 n N)1n2( 4 )1n2(VV N l GLvb n = 0, 1, 3, ........BZvw` h Kvb c~b msLv| GK gyL e bj aygv eRvo nvigvwbK wj Drcb nZ cvi A_vr 2q, 4_, 6 BZvw` nvigvwbK wj AbycwZ _vK| `yB gyL Lvjv bji GKwU Lvjv gyL Avjvob mw Kij Gi ga evqyi Kbi cKwZ (Vibration of Air Column in a Open Pipe): Dfq gyL Lvjv bji GK cv duz w`j ev Avjvob mw Kij bji wfZii evqyi gaw`q k jw^K ZivKvi Ab cv mvwjZ nq| Ab cv DcwZ nj GB Zi nVvr cmvwiZ nIqvi myhvM cvq| GB KviY bji Lvjv gyL me`v my` we`y (A) Ges Kb f` bji gvSLvb GK ev GKvwaK wb` we`y (N) wPvbyhvqx mw ne| evqyi mnRZi Kb [wP bs-(K)] `yB gy cv my` we`y Ges `yB my` we`yi gvS GKwU wb` we`y _vKe| G Zi `N o Ges bji `N l I nj Kv nj 2 0 l l ......(1) Ges )2.(.......... 2 VV N 0 0 l ]fV[ GLvb V ki eM | bji GB myiB g~j myi ev c_g nvigvwbK| GB bj AviI Rvi dzu w`j bji evqy m jw^K Zii Zi`N nvm cve Ges evqyi Kb ew cve| evqyi 2q mve Kb [wP bs-(L)] gvU wZbwU my` we`y Ges gvS Lvb `ywU wb` we`y MwVZ ne| awi evqy i GB Kb m myii Zi `N 1 Ges Kv N1, Zvnj 22 2 0 11 l l Ges 0 001 1 N2 V2 2 VV N GB myiK c_g Dcmyi ev wZxq nvigvwbK ej| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 181. ki `wZ (Speed Of Sound) 5 ZZxq mve Kb [wP bs-(M)] gvU PviwU my` we`y Ges gvS Lvb wZbwU wb` we`y MwVZ ne| KvRB GB Kb m myii Zi `N 2 Ges Kv N2, nj 33 2 2 3 0 2 2 l l Ges 0 002 2 N3 V3 3 VV N GB myiK wZxq Dcmyi ev ZZxq nvigvwbK ej| Dciv mgxKib _K `Lv hvq h, `yB gyL Lvjv bj h mg myi mw nq Zvì Zi `N 1n1n 2 o n l Ges Kv 0 n n N)1n( 2 )1n(VV N l ne| GLvb n = 0, 1, 3, ........BZvw` h Kvb c~b msLv| `yB gyL Lvjv bj eRvo I Rvo mKj cKvi nvigvwbK Drcb nZ cvi| Abybv`x evqy wK? Abybv` evqy cwZZ ki eM wbYq Ki: Abybv` evqy (Resonance Air Column): Kvb GKwU GKgyL Lvjv bji ga Ave evqyi GKwU ^fvweK Kv _vK| GB Kv evqyi `Ni Dci wbfi Ki| Gic GKwU Lvjv bji gyL GKwU Kgvb wUDwbs dK aij dKi Kb evqy ciek Kb mw Ki| dKi Kv hw` evqyi Kvi mgvb nq Ze evqy cejfve KuvcZ _vK| GK Abybv`x evqy ej| Abybv` evqy cwZZ ki eM wbYq (Determination of Speed of Sound by Resonance Air Column Method): hi eYbvt Abybv` evqy cwZZ AvMvMvov mgvb cQì GKwU KvPbj _vK| GB bjwUK GKwU cvwbi cv Wyweq `yBZZxqvsk cvwbZ fwZ Kiv nq| KvhcwZ (Procedure): wbw` Kvi GKwU myikjvKv F bIqv nq| GK ivevi cvW AvNvZ Ki Abybv`x bji Dby cv aiv nq| GZ bji ga evqyZ ciek Kbi mw nq| G Kb wbPi w`K mvwjZ nq Ges cvwbi DcwiZj nZ cyYivq cwZdwjZ nq wdi Avm| bjwUK DVvbvgv Ki bji evqyi `NK Ggb fve DchvRb Kiv nq hvZ evqyi mePq Kg `N Abybv` mw nq| GB Aevq cvwbi Dcwi Zj nZ bji Dby cv ch Abybv`x `N wbYq Kiv nq| wnmve I MYbv (Calculation): evqyi GB Abybv` bji Lvjv gyL GKwU my` we`y I cvwbi Dcwi Zj GKwU wb` we`yi Drcw ne| GB Aevq evqyi Kv myikjvKvi Kvi mgvb ne| gbKwi, Abybv`x evqyi `N l1| hw` evqyZ ki eM v Ges evqyi Kv n nq Ze, 4 1 l 14l KviY bji e gyL wb` we`y Ges Lvjv gyL my` we`y Drcb& nq Ges Gì gaeZx `~iZ 4 1 l | v =n mgxKiY nZ cvB, v =n4l1 A_vr v =4nl1 cixb nZ n I l Gi gvb Rb v ei Kiv nq| cv mskvab Ki ki eM wbYq (Determination of Sound of Velocity with end Correction): Abybv` cwZZ evZvm ki eM wbYqi mgq ai bIqv nq h, my` we`y bji Dby cv mw nq| wK wevbx ivj cgvb Kib h, my` we`y bji Lvjv gyL bv nq wKQyUv Dci nq| ZvB Gi Rb GKwU mskvab cqvRb| Gi bvg cv mskvab| KvRB GKwU Abybv`x evqy i evBii gy cv nZ bybZg h `~iZ my` we`y Aevb Ki ZvK cv mskvab ej| awi cv mskvab = x Abybv`x evqyi `N = (l1+x) bji At evm d nj x=0.3d Ges bji At evmva r nj x=0.6r| KvRB ki eM v=nnZ cvB, Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 182. ki `wZ (Speed Of Sound) 6 v = 4n(l1+x) ev, v = 4n(l1+0.6r) ev, v = 4n(l1+0.3d) BnvB cv mskvabi ci ki eM| cv mskvab cwinvi Ki ki eM wbYq: cv mskvab cwinvi KiI ki eM wbYq Kiv hvq| c_g Abybv` mw Kivi ci Abybv`x bj evqyi `N hw` g AviI ew KiZ _vwK Zvnj c_g Abybvì cvq wZb b `N wZxq Abybv` mw ne| aiv hvK, wZxq Abybvì mgq evqyi `N = l2 Zvnj, (1)...... 4 x1 l Ges (2)...... 4 3 x2 l (2) _K (1) weqvM Ki cvB, 2 12 ll )(2 12 ll GLb, nv )(2 12 llnv A_vr )(2 12 llnv GLb l1, l2 Ges n Gi gvb ewmq ki eM wbYq Kiv hvq| Wcjvi cfve (Doppler Effect): kvZv I Drmi AvcwK MwZi dj kZ ki Kvi Z_v ZxZvi AvcvZ cwieZbK Wcjvi cfve ej| GKwU Av^yj hLb mvBib evwRq AvmZ _vK ZLb mvBibi ZxZv ew cZ _vK Avevi hLb mwU Pj hZ _vK ZLb ZxZv gkt KgZ _vK| hw` kvZv I Drm ciii w`K AMmvi nq ZLb kZ ki Kvi AvcvZ ew nq Ges kvZv I Drm cii _K `yi mi Mj kZ ki Kvi AvcvZ nvm cvq| wi kvZvi w`K MwZkxj Drm (Source Moving Towards Stationary Observer): gbKwi, GKRb kvZv O Aevb wi AvQ Ges S k Drmi Avw` Aevb| DrmwU f Kvi k Drcb KiZ KiZ us eM wi kvZv O Gi w`K GQ Ges v k Zii eM| GLb Drmi Kv f ej cwZ mK f msLK Zi wbMZ nq Ges Zi eM v ej cwZ mK Zi v `~iZ AwZg Ki| myZivs GB f msLK Zi v`Ni ga Aevb Ki| (wP cvk) KvRB Drm hLb wi _vK ZLb ki g~j Zi `N f v | wK DrmwU hw` mK f msLK Zi wbMZ KiZ KiZ us eM wi kvZv O Gi w`K AMmi nq GK mK S Aevb Avm Zvnj mB msLK Zi GLb (v us)) `Ni ga Vvmv Vvwm Ki Aevb Kie| dj Zi `N nvm cve| GLb Gi Zi `N ne f uv s | wi kvZvi KvQ ki Zi `N f v Kg f uv s gvbi ej gb ne| myZivs kvZvi w`K Drmi MwZkxjZvi Rb wi kvZvi KvQ kZ ki AvcvZ Kv f nj| AvcvZ Kv v f N`ZiZZwcwie eMZiik f/)uv( v f s f )uv( v f s GB mgxKiY _K `Lv hvq h, hnZz ffv)uv( s myZivs Drm hLb Kvb wi kvZvi w`K MwZkxj _vK ZLb kvZvi KvQ k Zii Kv ew cqQ ej gb ne| Ges kZ ki ZxZviI AvcvZ ew ne| DrmwU hw` wi kvZv nZ `~i mi hvq Zvnj us-K FbvZK aiZ nq| A_vr m, f )u(v v f s f uv v f s G ff myZivs Drm kvZv _K `~i mi Mj kZ ki Kv Z_v ZxZv AvcvZ nvm cve| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 183. ki `wZ (Speed Of Sound) 7 Drm wi wK kvZv Drmi w`K MwZkxj ( Source Stationary But Observer Moving Towards It ): aiv hvK, k Drm s Aevb wi AvQ Ges kvZv O Aevb nZ Drmi w`K uo `wZZ AMmi nQ| Drm KZK wbmZ ki Kv f, Zi `N Ges `wZ v| GK mK f msLK Zi Drcb KiQ kvZv wi _vKj cwZ mK Zvi Kvb f msLK ZiB cQvZ| A_vr 1 mK f Zi v `~iZ Ryo _vKZ| myZivs kvZv O Gi KvQ h Zigvjv cQve Zvi `N AcwiewZZ _vKe Ges Zv ne f v | GLb hnZz kvZv Drmi w`K uo `wZZ MwZkxj KvRB kvZvi mvc ki `wZ (v+uo) ne| myZivs AvcvZ Kv, fv uv f o O N`ZiZiicuQvbZwe`y wZìkcmvvZvik (1)............f v uv f o Dciv mgxKiY _K `Lv hvq h hnZz (v+uo) >v ff myZivs AvcvZ Kv cKZ Kvi Pq ekx ne| A_vr hLb Kvb kvZv wi Drmi w`K MwZkxj _vK ZLb kZ ki Kv Z_v ZxZvi AvcvZ ew nq| hw` kvZv wi Drm _K `~i mi hvq Zvnj uo K FbvZK aiZ ne| A_vr m (2)............f uv uv f s o G hnZz ff myZivs kvZv wi Drm _K `~i mi Mj kZ ki Kvi Z_v ZxZvi AvcvZ nvm ne| Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 184. c_g ci AsKi mgvavb First Paper Math Solution 19| ki `wZ (Speed of Sound) 1| 261Hz KvsKi GKwU myikjvK AvNvZ Ki Abybv`x bji Dby cvi wbKU aij evZvmi 0.30m Ges 0.94m `N Abybv` cvIqv Mj| ki `wZ I cv mskvab wbYq Ki| Avgiv Rvwb, V=2 f (l2 - l1) ev, V=2 261(0.94-0.30) ev, V=334.08ms-1 m/s(Ans.) Avevi, ev, V= 4 f(l1 + x) ev, 334.08=4261(0.3+ x) 2614 08.334 x0.3 ev, x = 0.32-0.30 =0.02 m (Ans.) 2| 512Hz Kvi GKwU myi myikjvKvK AvNvZ Ki GKwU Abybv`x bji Dby cv aivq evZvmi 0.15m `N c_g Abybv` cvIqv Mj| evZvm ki `wZ 350ms-1 nj bji evm KZ? Avgiv Rvwb, V = 4f(l1+0.3d) 4f V 0.3d1 l 1 4f V 0.3d l 15.0 5124 350 0.3d 15.0170898437.00.3d 020898437.00.3d m 3.0 020898437.0 d (Ans.)m069.0d 3| KZ Zvcgvvq evqyZ ki eM 0C (cgvY) Zvcgvvi eMi wb ne? 2 1 2 1 T T V V 2T 273 V2 V 2T 273 4 1 T2 = 4273 K = 1092K= (1092-273) C = 819C (Ans.) 4| GKwU Ub euvwk evRvZ evRvZ GKwU cvUdgi w`K 90kmh-1 `wZZ AMmi nQ| euvwki Kbv 600Hz Ges ki `wZ 325ms-1 nj cvUdg `vqgvb Kvb kvZvi Kvb H ki AvcvZ Kv KZ gb ne? Avgiv Rvwb, f uv uv f s o 006 52253 0253 f 006 003 253 f (Ans.)650Hzf 5| `LvI h, Drm hw` wi kvZv _K ki `wZZ mi hvq, Ze kZ ki AvcvZ KvsK AaK ne| Avgiv Rvwb, f uv uv f s o f )v(v 0v f 11 1 f 2v 0v f 1 1 2 f f (cgvwYZ|) 6| GKwU Bwb PjZ PjZ 300 Hz Kvi eskxawb Kij wK GKRb cheKi wbKU H ki Kv 305Hz gb nj| Kvb w`K Ges KZ `wZZ BwbwU MwZkxj wQj| [ki `wZ 332 ms-1 ] Avgiv Rvwb, f uv uv f s o 300 u233 0233 053 s 55.326300 305 233 u233 s 45.533255.326us us= 5.45ms-1 Dit 5.45ms-1 eM kvZvi w`K AMmi ne| GLvb, Kv, f = 261 Hz c_g Abybv`x `N, l1 = 0.30 m 2q Abybv`x `N, l2 = 0.94 m ki eM, V= ? cv mskvab, x = ? GLvb, ki eM, V1= V ki eM, V2= 2V Zvcgvv, T1 =0C = 273K Zvcgvv, T2 =? awi, ki `wZ v = v1 Drmi `wZ us = v1 mvZvi `wZ, uo = 0 Kv f nj, cgvY KiZ ne h, AvcvZ Kv, 2 f f GLvb, Ubi `wZ 11 s 25msms 3600 100090 90km/hu Ubi euvwki Kv f = 600 Hz ki `wZ v = 325ms-1 mvZvi `wZ, uo = 0 AvcvZ Kv f' =? f GLvb, Kv, f = 512 Hz c_g Abybv`x `N, l1 = 0.15 m ki `wZ, V=350ms-1 bji evm, d = ? GLvb, Ubi euvwki Kv, f = 300 Hz ki `wZ, v = 332ms-1 mvZvi `wZ, uo = 0 AvcvZ Kv, f' = 305 Hz Ubi `wZ, us=? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 185. 19| ki `wZ (Speed of Sound) 2 7| GKwU Bwb wi `kK AwZg Kvj Gi AvcvZ cwZqgvb Kv 6:5 AbycvZ cwieZb nq| hw` evZvm ki eM 352ms-1 nq, Ze BwbwUi eM wbYq Ki| Avgiv Rvwb, f uV uV f s o f uV 0V f s (1)......... uV Vf f s Avevi, f )u(V uV f s o f uV 0V f s (2)......... uV Vf f s ckgZ, f : f = 6 : 5 5 6 f f 5 6 Vf uV uV Vf s s 5 6 uV uV s s 5 6 u352 u352 s s ss u63526u53525 35253526u6u5 ss 352u11 s 11 352 us (Ans.)ms32u 1 s 8| Avjv `Lvi 10 sec ci eR wbNvli k kvbv Mj| gNi `~iZ hw` 1650m Ges 0m. Zvcgvvq ki `wZ 332ms-1 nq, Ze H mgqKvi Zvcgvv wbYq Ki| Avgiv Rvwb, S= V t 1650= V 10 V =165ms -1 Avevi, 00 T T V V 273 T 332 165 273 T 332 165 2 K 332332 273165165 T C)27343.67(K43.67T .)Ans(C56.205T 9| cwZ mK 200 Pi Wcjvi cwieZb Drcb KiZ nj 1050Hz KvsKwewk k DrmK Kvb wi `kKi w`K h eM AvMgb KiZ ne Zvi wnmve `vI| [evZvm ki eM=330ms-1 ] Avgiv Rvwb, f uV uV f s o 1050 u330 0330 1250 s 1050 1250 330 u330 s 277.2330us (Ans.)ms52.8u 1 s 10| GKwU myi kjvKv h mgq 200 evi Kb `q mB mgq GwU viv m Zi evZvm 140 m `~iZ AwZg Ki| myi kjvKvi Kv 500Hz nj evqyZ ki eM KZ? Avgiv Rvwb, V = f 1 ms7.0500V .)Ans(ms350V 1 11| N.T.P. Z ki eM 332ms-1 nj 50C Zvcgvvq I 70 cm cvi` Pvc ki eM wbYq Ki| ki eMi Dci Pvci Kvb cfve bB, Avgiv Rvwb, )1(vv 0 50 273 1 1332v )183150183.01(332v 183150183.1332v (Ans.)ms361.12v -1 GLvb, kvZvi eM, uo= 0 ki eM, V =352ms-1 awi, cKZ Kv = f Kvi AbycvZ, f : f = 6:5 BwbwUi eM, us= 0 GLvb, mgq, t = 10s. `~iZ, S = 1650 m ki eM, V =? ki eM, V0 = 332 ms-1 Zvcgvv, TC=273K Zvcgvv, T GLvb, 200=140m m7.0m 200 140 Kv f=500Hz V=? GLvb, ki eM, V =330ms-1 cKZ Kv, f =1050 Hz Drm `kKi w`K AvMgb Kij Kv ekx ne, dj AvcvZ Kvs&K, f =(1050+200) Hz = 1250Hz BwbwUi eM, us =? kvZvi eM, uo= 0 GLvb, ki eM, vo = 332ms-1 Zvcgvv, = 50C v =? Want more Updates: https://www.facebook.com/tanbir.ebooks facebook /gmail/skype: -tanbir.cox Web: http://tanbircox.blogspot.com 186. 19| ki `wZ (Speed of Sound) 3 12| `ywU nb enb Ki GKwU gvUi Mvox

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